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THE ELECTRONIC “OTHER’: A STUDY OF CALCULATOR-BASED SYMBOLIC MANIPULATION UTILITIES WITH SECONDARY SCHOOL MATHEMATICS STUDENTS
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
By
Michael Todd Edwards, B.A., B.S., M.A., M.S., M.Ed.
*****
The Ohio State University May 18,2001
Dissertation Committee: Approved by
Douglas T. Owens, Chair
Bert K. Waits Advisor College of Education Suzanne K. Damarin UMI Number; 3011050
UMI'
UMI Microform 3011050 Copyright 2001 by Bell & Howell Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code.
Bell & Howell Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Copyright by Michael Todd Edwards 2001
All Rights Reserved ABSTRACT
This study investigated two groups of suburban high school students using computer algebra systems (CAS) with graphing calculators. The CAS group (n = 25) used symbolic manipulation utilities systematically throughout the school year. The non-
CAS (n - 22) group were taught using graphing calculators without CAS.
Group performances on a year-end Advanced Algebra Final Examination were analyzed, using scores on two pretests as separate covariates. An analysis of covariance
(ANCOVA) revealed that non-CAS students significantly outperformed CAS students
(p < 0.000). Additional applications of ANCOVA revealed that low-performing non-
CAS students significantly outperformed their CAS counterparts, (p = 0.029). Middle- performing non-CAS students significantly outperformed their CAS counterparts (p =
0.033). Differences in the performances of high-performing students were not statistically significant (p = 0.463).
Additionally, chapter test and quiz means were examined. Of the nineteen tests and quizzes administered, separate analyses of variance (ANOVA) revealed no statistically significant differences. Non-CAS students outperformed CAS students on all tests and quizzes until the fifteenth week - suggesting that CAS posed challenges for students at the beginning of the year. On specific test and quiz items, CAS students employed symbolic methods more frequently than non-CAS counterparts (51 percent to 42 ii percent). CAS students used tables 14.1 percent of the time, compared to 19.4 percent for non-CAS. The CAS group used graphs 32 percent of the time, compared to 31.3 percent for non-CAS.
Several conclusions can be drawn from student essays, short writing items, and a teacher journal. First, CAS students’ perception that CAS would be useless in subsequent coursework interfered with their mathematical performance. Secondly, CAS students indicated that by-hand manipulation helped them understand algebraic concepts more clearly than CAS. Awkwardness of calculator output and the calculator’s tendency to perform "too many steps,” contributed to students’ preference for by-hand methods.
Focusing student attention on the theory of equation solving is the real promise of s}Tnbolic manipulation with CAS. CAS enables students to choose transformations to apply to equations without worrying about arithmetic errors. However, until CAS utilities exist allowing students to choose transformations for themselves, the use of
CAS as a means of teaching equation solving to secondary school students is not recommended.
Ill To my loving wife, Jennifer, and my daughter, Cassady, for their patience, guidance, and love. I would have never survived this process without their support. To my parents, Charles and Francine, for instilling within me a passion for learning.
IV ACKNOWLEDGMENTS
A pivotal moment in my career as a mathematics student and educator occurred
when I was only eight years old. For my birthday, I received two gifts that dramatically
influenced the course of my education and professional life. My parents bought me a
Texas Instruments TI-1600 pocket calculator, and my Aunt Georgia bought me a small
tape recorder. For hours, I lay on the floor of my bedroom, intently pressing buttons on
the calculator. While carefully examining the results of various computations on the
calculator’s red LED screen, 1 recorded my discoveries onto cassette tape. Each
evening I played edited versions of the tape-recorded lectures back to my parents,
sharing my findings eagerly with them. Although my early calculator activities were
purely recreational — a diversion from cartoons and kickball — they developed within me
a strong interest in the interplay of mathematics, technology, and learning that continues to be my passion to this day. The activities also fostered within me a curiosity regarding research and calculators that has nourished me as I continue work with calculator-based research in mathematics education.
As I grew older, I became fascinated with sports — particularly baseball — both as a player and as a hobbyist. I spent countless hours poring over statistics of my favorite players — comparing batting averages, home runs, and earned run averages of the all- time greats. In a thirst for more statistics and more colorful player profiles, I soon began producing my own baseball cards. Using fictitious athletes and teams, I drew action "photos" on the front of heavy cardboard cutouts while composing extensive statistics on the back. With the assistance of my calculator, I compiled statistics not commonly found on baseball cards — on-base percentages, slugging percentages, and homerun-to-hit ratios. After creating teams of fictitious players, I developed statistics- based card games that allowed my athletes to compete against each other. The creation of these card games provided me with invaluable practice for the cross-curricular lesson planning 1 would find myself engaged in two decades later and provided me with a view of mathematics as a creative enterprise with numerous connections to art, writing, and atheletics.
This document is dedicated to my parents — Charles and Francene Edwards — for their tireless support of my academic pursuits; to my wife — Jennifer — for her patience regarding my calculator tirades; and to my daughter Cassady — the best CAS with which
I’ve ever worked. A special thanks also goes to all of my math teachers, both good and bad, for helping me build my own view of what mathematics instruction should be. In particular, thanks to Dr. Dermis Davenport — professor of Mathematics at Miami
University — for turning me onto higher level mathematics and Dr. Bert Waits — professor emeritus at Ohio University — for introducing me to the “power of visualization.” I am forever grateful to Bert and Dr. Ed Laughbaum for their help with
VI this study. Last, but not least, thanks to Dr. Douglas T. Owens — my Ph.D. adviser at
Ohio State University — for his kind words of encouragement and his tireless editing of my dissertation drafts.
VII VITA
1990 ...... B.A. and B.S. in Mathematics, Miami University, Oxford, Ohio
1991-199 2 ...... Computer Programmer, Cincinnati Bell Information Systems, Cincinnati, Ohio
1992-199 5 ...... Graduate Teaching Associate, Department of Mathematics, Ohio University, Athens, Ohio
1995 ...... M.S. Mathematics, Ohio University, Athens, Ohio
1995 ...... M.Ed. Education with Secondary Certification, Ohio University, Athens, Ohio
1995-2000 ...... Mathematics and Computer Science Teacher, Upper Arlington High School, Upper Arlington, Ohio
2000-2001...... Mathematics Teacher, Linworth Alternative School, Worthington, Ohio
FIELDS OF STUDY Major Field: Education
via TABLE OF CONTENTS
ABSTRACT...... ,...... ii
ACKNOWLEDGMENTS...... v
VITA ...... viii
LIST OF TABLES...... xiii
LIST OF FIGURES...... xvi
CHAPTER PAGE
1. STATEMENT OF THE PROBLEM...... 1
Introduction ...... 1 Background of CAS Debate ...... 4 Statement of the Problem ...... 6
2. THE CASE FOR COMPUTER SYMBOLIC ALGEBRA...... 7
Introduction ...... 7 Symbolic Algebra and Hand-Held Graphers: An Historical Perspective ...... 9 The Development of Computer Algebra Systems ...... 9 The Development of Graphing Calculators ...... 14 Examples of Meaningful Algebra with CAS ...... 21 Encouraging Student Conjecturing with Exponents ...... 21 Linear Equations from Multiple Perspectives ...... 24 The Nature and Limitations of Computer Algebra ...... 27 Summary...... 31
IX 3. THEORETICAL FRAMEWORK...... 32
Introduction ...... 32 The Nature of Knowledge ...... 34 Knowledge is Constructed ...... 34 The Importance of Prior Knowledge in MatDiematics Learning ...... 39 Meaningful Learning is facilitated through interactions with Learned Others ...... 48 Emergent Theory of CAS Usage in tDie Secondary Mathematics Classroom ...... 60 Summary...... 65
4. REVIEW OF THE LITERATURE...... 66
Introduction ...... 66 Mathematics in the Information Age ...... 67 The Beginnings of Algebra Reform ...... 69 Research Findings ...... 80 Studies of CAS in Secondary Schools ...... 81 Studies of Remedial College Algebra studemts ...... 96 Studies of University Calculus students ...... 102 Summary...... 105
5. METHODOLOGY...... 106
Introduction ...... 106 Description of Setting ...... 110 The School and Community ...... 110 The Curriculum ...... 113 Data Collection ...... 122 Rationale for Mixed-Methodology Study ...... 143 Limitations of this Study ...... 145 6- QUANTITATIVE DATA ANALYSIS...... 148
Introduction ...... 148 Analysis of Pretest Data ...... 151 Algebraic Skills Pretest ...... 151 Technology Literacy Pretest ...... 152 Student Attitudinal Surveys ...... 155 Analysis of Posttest Data ...... 164 Analysis of Student Tests and Quizzes ...... 174 Summary...... 184
7. QUALITATIVE DATA ANALYSIS...... 187
Introduction ...... 187 Analysis of Student Essays ...... 189 Solving Linear Equations Essay ...... 190 Teacher Response to Solving Linear Equations Essays ...... 203 Systems Essay...... 222 Calculator Algebra Questionnaire ...... 232 End of the Year Short Answer Essays ...... 248 Summary...... 266 CAS Activities...... 266 Student Attitudes ...... 268
8. SUMMARY AND CONCLUSIONS ...... 271
Summary...... 271 Hypotheses tested and summary outcomes ...... 274 Discussion ...... 276 Conclusions ...... 287 Implications ...... 302 Implications for Practice ...... 302 Theoretical Implications ...... 305 Recommendations for Future Research ...... 313
BIBLIOGRAPHY...... 314
XI APPENDICES...... 324
A. Letters of Permission ...... 324 B. Pre- and Post-Tests ...... 328 G. Student attitudinal survey results by item (Fall, non-CAS group) ...... 355 D. Student attitudinal survey results by item (Fall, CAS group) ...... 358 E. Student attitudinal survey results by item (Spring, non-CAS group) 361 F. Student attitudinal survey results by item (Spring, CAS group) ...... 364 G. Item analysis of Advanced Algebra Final Examination by class and topic 367 H. Test and quiz items designed for multiple solution strategies ...... 369 I. Selected items from Solving Linear Equations Essay...... 373 J. Systems warm-up and skills-based warm-up activities ...... 375 K. Calculator Algebra Questionnaire writing prompts ...... 379
XU LIST OF TABLES
TABLE PAGE
4.1 : Summary of CAS dissertation findings ...... 104
5.1: Comparison of statewide 9th grade proficiency test score averages with scores of students at Midvale High ...... I l l
5.2: Research questions and data collection methods ...... 123
5.3 : Reliability coefficient of instrument: Student Attitudinal Survey ...... 129
5.4: Items on Advanced Algebra Final Examination organized by topic ...... 135
6.1: Descriptive statistics of Algebraic Skills Pretest data...... 151
6.2: Descriptive statistics of Technology Literacy Pretest data...... 153
6.3 : Overall student calculator attitudes (Fall) ...... 158
6.4: ANOVA for Calculator Attitude (Fall Student Attitudinal Survey) ...... 159
6.6: Overall student calculator attitudes (Spring) ...... 161
6.7: ANOVA for calculator attitude (Spring Student Attitudinal Survey) ...... 161
6.8: ANOVA for change in calculator attitude (CAS Group) ...... 163
6.9: ANOVA for change in calculator attitude (non-CAS group) ...... 164
6.10: Descriptive statistics of Advanced Algebra Final Examination ...... 164
Xlll 6.11 : ANCOVA comparison of treatment. Technology Literacy Pretest scores, and Algebraic Skills Pretest scores ...... 166
6.12: Lower third and upper third quantités for Combined Pretest scores ...... 168
6.13: Descriptive statistics of Advanced Algebra Final Examination by performance group ...... 170
6.14: ANCOVA comparison of treatment and Combined Pretest Scores for low performing students ...... 171
6.15: ANCOVA comparison of treatment and Combined Pretest Scores for middle performing students ...... 172
6.16: ANCOVA comparison of treatment and Combined Pretest Scores for high performing students ...... 173
6.17: Class test and quiz means with respect to various mathematics topics ...... 176
6.18: Percentage of agreement between raters ...... 179
6.19: Popularity of various solution techniques on various test and quiz items ...... 180
7.1 : Student writing assignments ...... 189
7.2: Student responses to item 3 of Solving Linear Equations Essay ...... 201
7.3 : Summary of student responses to item 4 of Systems Essay assignment .223
7.4: Student response to Item 1 of Calculator Algebra Questionnaire ...... 233
7.5: Student responses regarding possible class changes as a response to item 6 of Calculator Algebra Questionnaire ...... 234
7.6: Student responses regarding comfort level with calculator ...... 238
7.7: CAS Student comfort level with calculator ...... 239
XIV 7.8: Non-CAS Student comfort level with calculator ...... 241
7.9: Student responses regarding item 2(b) of Calculator Algebra Questionnaire ...... 241
7.10: Student responses regarding uses of calculator ...... 244
7.11: Student career interests of non-CAS students as reported in items 3 and 4 of Calculator Algebra Questionnaire ...... 247
7.12: Student Responses regarding previous calculator use ...... 256
7.13: Student response regarding obsolesence of paper-and-pencil mathematics ...... 258
7.14: Student Responses regarding calculator satisfaction as indicated on item 4 of End o f Year Short Answer Essay ...... 262
XV LIST OF FIGURES
FIGURE PAGE
2.1 : Sample output from Mathematica ...... 11
2.2: Sample output from Mathcad ...... 12
2 I 2.3: Solution of x* using MathCad ...... 13
2 2 2.4: Solution of jt'* = using TI-92 Plus...... 13
2 1 2.5: Solution of x^ using Maple ...... 13
2.6: Y I defined as 2 • sin(j:) — 1 on the T l-81 ...... 16
2.7: Zooming tools and graphics window of the Tl-81 ...... 16
2.8: The Tl-81 TRACE utility ...... 16
2.9: The ZOOM IN utility of the Tl-81 ...... 17
2.10: Results of successive approximations of root on a Tl-81 ...... 17
2.11: Root finding on the TI-83 ...... 18
2.12: An approximation of a root ...... 18
2.13: Exact roots calculated on the Tl-92 ...... 19
XVI 2.14: Some exact calculations possible on the TI-89 ...... 20
2.15: Examples of integer exponentation using the TI-92 Plus CAS ...... 22
2.16: Numerous examples of exponent multiplication encourage students to generalize patterns they see ...... 22
2.17: Examples of exponent division encourage students to generalize patterns they see ...... 23
2.18: Examples of negative integer exponents encourage students to generalize patterns illustrated by input/output pairs ...... 24
2.19: Equations represented as procedures and functions on the TI-89 ...... 25
2.20: The expressions 2x+ 7 and 3 viewed as functions of x within the TI- 89 table utility ...... 26
2.21: Solving equations by locating intersections of functions on a graphing calculator ...... 27
2.22: A binary tree structure for 2{x + 5) ...... 28
2.23: An unexpected answer from the TI-89 resulting from limitations of CAS design ...... 29
2.24: Limitations of CAS design are evidenced by the TI-89’s tendency to skip intermediate steps and ignorance of domain restrictions ...... 30
3.1: Entering an equation to solve on the TI-89 ...... 36
3.2: Using the TI-89 to promote a procedural view equations ...... 37
3.3: Using the TI-89 to promote an object-based view of equations ...... 38
3.4: Using the TI-89 to highlight an error common among novice equation solvers ...... 38
xv ii 3.5: Solving a linear equation successfully using the TI-89 CAS ...... 39
3.6: Graphs of non-equivalent functions that appear to be equivalent ...... 40
3.7: Table of values suggest that non-equivalent functions are equivalent ...... 41
3.8: Kutzler’s "house of mathematics” of a typical high school student ...... 42
3.9: The house of mathematics of a student possessing a solid foundation on which to build advanced mathematical concepts ...... 42
3.10: Mathematical understanding of typical high school students contains holes and misconceptions ...... 43
3.11: Success in basic trigonometry without knowledge of arithmetic, algebra, and equations is improbable ...... 43
3.12: CAS enables students to study more advanced concepts while resolving misconceptions with earlier material ...... 44
3.13: The three fundamental base ten blocks described by Bruner and Dienes...... 46
3.14: Several possible squares constructed with base ten blocks ...... 46
3.15: Equivalent algebraic names for squares generated by Bruner’s students ...... 47
3.16: A graphical interpretation of Vygotsky’s ZPD ...... 50
3.17: Student A enters a higher order polynomial equation into the Tl-92 CAS...... 51
3.18: Students add 1 to each side of equation using Tl-92 CAS ...... 52
3.19: Students factor each side of equation using Tl-92 fa c to r command ...... 52
X V lll 3.20: Using the TI-92 CAS, students calculate the square root of both sides of an equation ...... 53
3.21: Student finds solutions for x by subtracting 1 and taking square root of each side of equation ...... 53
3.22: Student checks algebraic solutions using graphing capabilites of the TI-92 CAS ...... 54
3.23: Using the solve function on the TI-89 CAS to determine the solution for a basic linear equation ...... 56
3.24: A flowchart depicting Buchberger’s white box / black box principle ...... 59
3.25: Edwards ’ s model of student/calculator interaction ...... 61
3.26: Interactions in a CAS-equipped classroom ...... 63
4.1: Monomial expansions and exact trigonometric values are routine tasks with the TI-89 ...... 71
4.2: Study design of Wain, et al. (1993) ...... 83
4.3 : Third and Fourth form curriculum in Austrian schools ...... 89
4.4: Posttest from EClinger (1994) ...... 90
4.5: Algebra problem from French CAS study (Rousselet, 1996) ...... 93
4.6: An example of a “long term” problem by Aldon ...... 95
5.1: Physical features of Room 202 of Midvale High school ...... 116
5.2: Attitude-specific items from Student Attitudinal Survey ...... 126
5.3 : Authority-specific tems from Student Attitudinal Survey ...... 126
XIX 5.4: Concept-specific items from Student Attitudinal Survey ...... 127
5.5: Skill specific items from Student Attitudinal Survey ...... 127
5.6: Items from Student Attitudinal Survey dealing specifically with leaming/doing mathematics with a calculator ...... 128
5.7: Distribution of core topics on Advanced Algebra Final Examination ...... 136
5.8: Items 1 and 2 from the Advanced Algebra Final Examination ...... 136
5.9: Item 43 from the Advanced Algebra Final Examination ...... 137
5.10: Schemata of Quasi-Experimental Design ...... 137
6.1: Distributions of all pretest scores by group ...... 155
6.2: Distributions of Student Attitudinal Survey scores ...... 158
6.3 : Plot of mean calculator attitude of CAS and non-CAS students during Fall and Spring ...... 162
6.4: Performance groups by class as determined by Combined Pretest scores ...... 169
6.5: Plot of test and quiz performance of CAS and non-CAS groups with respect to time (in weeks) ...... 177
6.6: Plot of symbolic manipulation usage (as a percentage of overall problem solving techniques) with respect to time by class ...... 181
6.7: Plot of table usage (as a percentage of overall problem solving techniques) with respect to time by class ...... 182
6.8: Overall popularity of solution strategies (non-CAS group) ...... 183
6.9: Overall popularity of solution strategies (CAS group) ...... 183
XX 7.1: Statistical options on the TI-83 ...... 191
7.2: Data entry using TI-83 ...... 191
7.3: Regression options available to TI-83 users...... 192
7.4: Calculations resulting from a linear regression on the TI-83 ...... 192
7.5: Option to invoke data matrix editor on the TI-92 ...... 193
7.6: Creating a new data file on the TI-92 ...... 193
7.7: Creation of data file lin e d a ta using the TI-92 ...... 194
7.8: Entering data into lists using the TI-92 ...... 194
7.9: Menu option to perform linear regression on TI-92 ...... 195
7.10: Dialog box in which regression parameters are entered on the TI-92 ...... 195
7.11: Calculations resulting from a linear regression on the TI-92 ...... 196
7.12: The TI-92 does not allow use of familiar slope formula...... 197
7.13: A modified formula for slope works successfully on the TI-92 ...... 197
7.14: A numerical value for slope obtained by substituting values into the into a general slope formula on TI-92 ...... 198
7.15: Replacing variables with constants using the TI-92's command line editor ...... 198
7.16: First step of solving a linear equation from the general slope formula using Kutzler’s methods on the TI-92 ...... 199
7.17: Application of the distributive property using EXPAND function on the TI-92 ...... 199
xxi 7.18: Using TI-92 to manipulate a linear equation into standard form ...... 200
7.19: Verification of linear equation solution using boolean algebra capabilities of TI-92 ...... 200
7.20: A student’s first step at solving a system of equations using by-hand methods ...... 205
7.21: Mechanical errors involving the distributive property introduces error into the student’s solution ...... 205
7.22: An incorrect solution for variable y found with by-hand methods ...... 206
7.23 : An incorrect solution for variable x found with by-hand methods ...... 206
7.24: Both equations of the system typed into the y= editor of the TI-92 ...... 207
7.25: The TI-92 ZOOMFIT menu option ...... 207
7.26: Misleading results produced by TI-92 ZOOMFIT utility...... 208
7.27: System of equations entered directly onto TI-92 home screen ...... 209
7.28: Isolating y variable using traditional by-hand procedures within TI- 92 environment ...... 209
7.29: Solving for y using traditional by-hand procedures on the TI-92 ...... 210
7.30: Solving for x using traditional by-hand procedures on the TI-92 ...... 210
7.31 : Window settings for graph based on CAS-derived solutions ...... 211
7.32: Graphical depiction of system solution using TI-92 ...... 211
7.33: Simultaneous graphs of both equations using ZOOMFIT reveals little information regarding the solution of the system on the TI-83 ...... 212
x x ii 7.34: Equations for both equations of the system typed into the TI-83 y= editor ...... 213
7.35: Values of various x-y pairs that satisfy the system of equations as shown on the TI-83 table ...... 213
7.36: Window values suggested by non-CAS students ...... 214
7.37: Graphical depiction of system solution using TI-83 ...... 214
7.38: Cost functions typed into y= editor of TI-83 ...... 216
7.39: TI-83 TBLSET (i.e. table set) utility ...... 217
7.40: An initial table generated by students using the TI-83 consisting of prices of both phone plans ...... 217
7.41: A table suggesting that Plan B is a better option for consumers making 93 or more local calls per month ...... 218
7.42: The graphs associated with the costs of calling plans A and B intersect between 92 and 93 calls ...... 218
7.43: Definition of cost function using DEFINE command on TI-92 ...... 219
7.44: Evaluation of student-defined cost function using TI-92 with various number of minutes ...... 220
7.45: A solution to the phone call problem using evaluation of cost function on TI-92 ...... 220
7.46: Student-built equation representing two phone call plans with equal costs ...... 221
7.47: Equation solving on the TI-92 for phone call problem ...... 221
7.48: Calculation highlighting inability of TI-92 to simplify fractional expressions with common factors ...... 230
xxiii 8.1: Steps automatically performed upon entering equation into TI-92 CAS...... 296
8.2: More calculations automatically performed by TI-92 CAS ...... 297
8.3: Algebraic output is read like “sentences in a book” on the TI-92 home screen ...... 297
8.4: The TI-92 does not allow use of familiar slope formula...... 299
8.5: A modified formula for slope works successfully on the TI-92 ...... 300
8.6: An example of student work illustrating Bruner’s Symbolic level o f understanding ...... 308
8.7: An example of student work illustrating Calculator Symbolic level of understanding ...... 308
8.8: CAS enables students to study more advanced concepts while resolving misconceptions with earlier m aterial...... 310
8.9: Optimal conditions for acquiring new mathematical knowledge are not met when one has weak by-hand manipulative skills ...... 311
8.10: Optimal conditions for acquiring new mathematical knowledge are not met when one has weak calculator skills ...... 311
8.11 : Optimal mathematical knowledge-building is possible in technology- rich classrooms only when students have strong by-hand manipulative skills and strong calculator skills...... 312
A. 1 : Question 3 from Quiz 1 covering functions (week 2) ...... 370
A.2: Question 6 from Test 1 (part 1) covering functions (week 4) ...... 370
A.3 : Question 4 from Test 1 (part 2) covering functions (week 4) ...... 370
A.4: Question 4 from Quiz 2 covering direct variation (week 7) ...... 370
XXIV A.5: Question 2 from Test 3 covering linear relations (week 11) ...... 371
A.6: Question 4 from Quiz 6 covering Quadratics (week 23) ...... 371
A.7: Question 2 from Test 6 covering Quadratics (week 26) ...... 371
A.8: Question 1 from Test 7 covering power functions and exponents (week 31)...... 372
A.9: Question 8 from Test 9 covering logarithms (week 38) ...... 372
A. 10: Item 1 from Solving Linear Equations essay...... 374
A. 11 : Item 2 from Solving Linear Equations essay...... 3 74
A. 12: Item 3 from Solving Linear Equations essay...... 3 74
A. 13: Skills-based manipulation problem involving systems of equations (November 10, 1999) ...... 376
A. 14: Phone bill application problem (November 17, 1999) ...... 376
A. 15: Item 4 from Systems Essay assignment ...... 378
A. 16: Item 5 from Systems Essay assignment ...... 378
A. 17: Item 3 from Calculator Algebra Questionnaire ...... 380
A. 18 : Item 4 from Calculator Algebra Questionnaire ...... 380
A. 19: Item 2(b) from Calculator Algebra Questionnaire ...... 380
A.20: Item 1 from Calculator Algebra Questionnaire ...... 380
A.21 : Item 6 from Calculator Algebra Questionnaire ...... 380
A.22: Item 2(a) from Calculator Algebra Questionnaire ...... 380
XXV A.23 : Item 5 from Calculator Algebra Questionnaire ...... 381
A.24: Item 1 from End o f Year Short Answer Essay...... 383
A.25 : Item 2 from End o f Year Short Answer Essay...... 3 83
A.26: Item 3 from End o f Year Short Answer Essay...... 383
A.27 : Item 4 from End o f Year Short Answer Essay...... 383
XXVI CHAPTER 1
STATEMENT OF THE PROBLEM
A problem is a chance for you to do your best.
Duke Ellington
Introduction
Since their introduction into mathematics classes more than a decade ago, graphing calculators and Computer Algebra Systems (CAS) — Maple (Waterloo Maple, 2000),
Derive (Soft Warehouse, 1995), Mathematica (Wolfram Research, 1997) - have garnered considerable attention from the mathematics education community. In the past, the types and levels of problems that could be explored in each environment were distinct, due in large part to the greater power and higher cost of desktop computers. As
Demana and Waits (1992) note:
CAS systems like Derive require either desktop computers and regular access to computer laboratories or expensive pocket computers and software that are not available on a regular basis to most high school mathematics students (p. 180) For this reason, a number of critical research questions regarding technology’s role in
secondary mathematics classrooms have remained unanswered, particularly those that
have required student access to computers.
On the other hand, because hand-held graphing utilities are relatively inexpensive,
many researchers have investigated the impact of calculators on secondary mathematics
teaching and learning. For instance, student conceptualization of function and variable
have been examined in numerous studies (Carter, 1995; Hollar, 1996; Martinez-Cruz,
1993; Slavit, 1994). Research, as well as anecdotal evidence from classroom teachers,
suggests that calculators enable students to interpret graphical information more
accurately (Boers-van Oosterum, 1990), to more fully realize the multiple
representational characteristics of functions (Browning, 1989; Hart, 1991), and to feel
more positive about their ability to do mathematics.
Features not typically found on graphing calculators have received far less attention from secondary educators. For instance, the potential impact of computer symbolic algebra — a powerful feature of Computer Algebra Systems such as Maple (Waterloo
Maple, 2000) or Mathematica (Wolfram Research, 1997) — remains largely unknown to high school teachers, despite the fact that documents such as NCTMs Algebra in a
Technological World (Heid, 1995) advocate the use of such tools at the secondary level.
The technological world in which students and teachers now operate demands a radical transfiguration of algebra in schools. As we enter the twenty-first century, the study of algebra in schools must focus on helping students describe and explain the world around them rather than on developing and refining their execution of by-hand symbolic- manipulation procedures — procedures that are better accomplished through the informed use of computing tools (p. 143).
Up to now, in-depth examinations of CAS and computer symbolic algebra in American classrooms have been limited primarily to university studies (Crocker, 1991; Heid,
1988; Judson, 1989; Mayes, 1995; Palmiter, 1991; Roddick, 1997). Although a number of CAS studies have been conducted with secondary students in Austria (Aspetsberger,
Fuchs, & Watkins, 1996; Fuchs, 1996; Klinger, 1996; Kutzler, 1996), the inevitable debate regarding symbolic manipulation’s importance to high school mathematics has been largely postponed by cost and portability constraints.
Now, however, as miniaturization of calculator-based technologies continue to narrow the gap between “calculator” and “computer,” computer symbolic algebra utilities are beginning to appear in devices no larger than the traditional graphing calculator. For instance, Texas Instruments describes its TI-89 (Texas Instruments,
1997a) as a “vertical format, portable symbolic, numeric and graphing solution for advanced mathematics and engineering coursework” (Texas Instruments, 1997b). In the 1982 article “The disk with a college education,” Wilf (1982) poses the following question: “what happens when $29.95 pocket calculators can . . . solve standard forms of differential equations, do multiple integrals, vector analysis, and what-have-you?”
Little more than a fifteen years after Wilf initially asked the question, math educators must now come to terms with the existence of such technologies in their own classrooms. Background of CAS Debate
The introduction of CAS into secondary mathematics classrooms promises to shift the focus of high school math away from symbolic, pencil-and-paper computation.
Advocates of CAS contend that the technology “promote[s] a deeper conceptual understanding” among algebra students, “reduce[ing] tedious manipulations so students and instructors can focus more on problem formulations and problem solving strategies”
(Boyce & Ecker, 1995). Several post-secondary studies have illustrated that CAS may enhance student motivation, to leam algorithmic processes by allowing teachers to examine applications with students prior to a presentation of the pencil-and-paper algorithms required for solving them. As Judson (1990) states:
Students are more interested in course material when this [applications- first] sequence is followed. The old question ‘Why do we have to leam this stuff?’ is replaced by wonder at the many applications of differentiation (p. 154).
Skeptics fear that students, given the opportunity to solve problems with a symbolic manipulator, “will push buttons blindly” (Hillel, Laborde, Lee, & Linchevski, 1992), not bothering to leam concepts behind their calculations. Other critics suggest that procedural skills and conceptual skills cannot be separated. Such a view posits that students must first possess pencil-and-paper skills before really understanding secondary-level mathematics. Gibb (1977) asserts that:
Efforts for developing understandings alone are not effective unless they are tempered with drill and practice to build proficiency in problem solving and in thinking logically (p. 390). Advocates of traditional teaching methods (such as Gibb) point out that pencil-and- paper exercises have worked well for many students, whereas the effects of calculators on student achievement remain unclear. They warn against premature conclusions regarding the worth of technology in mathematics education. Krantz (1993) purports this view eloquently.
Consider this: in my view, a student is better off spending an hour with a pencil [emphasis added] — graphing functions just as you and I learned — than generating fifty graphs on a computer screen in the same time period. From first hand experience, I am absolutely sure what the first exercise will teach the student. It is not at all clear what the student gains from the second (p. 26).
Unfortunately, a number of international assessments suggest that United States mathematics students may not be "better off’ utilizing traditional paper-and-pencil techniques. Initial findings from the Third International Mathematics and Science
Study (National Center for Education Statistics, 1997) paint a dismal picture of the mathematical performance of average students in the United States.
On the mathematics portion of the general knowledge assessment, U.S. students scored below the international average, and among the lowest of the 21 countries (p. 26).
Advocates of algebra reform feel that, for the majority of high school mathematics students, something is wrong with current mathematics instruction. In their opinion, the current system represents a pedagogical failure for far too many students. A growing number of educators see CAS as a vehicle for improving student performance and conceptual understanding at the secondary level. Statement of the Problem
The aim of this study is to investigate the impact that hand-held computer algebra environments, such as the TI-89 (Texas Instruments, 1997a), may have on student learning at the secondary level — particularly with novice algebra students. Specifically, the study aims to shed light on the following five questions:
1. How does CAS affect overall student algebraic understanding?
2. How does CAS affect student algebraic understanding in specific mathematical topics?
3. How do students use CAS utihties? Does this use differ from non-CAS use?
4. What attitudes do intermediate algebra students have regarding graphing calculators and symbolic manipulation utilities? Do their attitudes change as they gain experience with the utilities? How do attitudes differ among CAS and non-CAS students?
5. Which types of problems are well-suited for use with CAS-equipped utilities? Which are not?
A valuable by-product of the study will be the creation of teaching materials that are compatible with CAS at the secondary level. CHAPTER2
THE CASE FOR COMPUTER SYMBOLIC ALGEBRA
Example is not the main thing in influencing others. It is the only thing.
Albert Schweitzer
Introduction
After considering hand-held CAS utilities and their potential for transforming the secondary school mathematics curriculum, a first encounter with a CAS-equipped calculator may seem surprisingly uneventful. On the surface, a device such as the TI-89
(Texas Instruments, 1997) appears similar to any other graphing calculator. Its size, screen dimensions, and keypad are virtually identical to any number of earlier TI models. Needless to say, appearances can be quite deceiving. Unlike its predecessors, the TI-89 (Texas Instruments, 1997a) is a hand-held computer algebra system. Demana and Waits (1998) identify four necessary components for CAS:
• Graphical software capabilities • Numerical solvers Exact (rational, real, complex) arithmetic • Computer symbolic algebra utilities for solving of algebraic equations and manipulation of algebraic expressions (p. 3) Early graphers were equipped with only the first two of these components, while newer calculators such as the TI-89 (Texas Instruments, 1997) and TI-92 (Texas Instruments,
1995) contain all four.
This chapter presents sample screens from various CAS (as well as several non-
CAS graphing calculators). Such a discussion is important for several reasons. First of all, descriptions of technology help the reader to better understand motivations for the algebra reform efforts of the last two decades (and help to anticipate those which lie ahead). Secondly, illustrations calculator activities encourage an appreciation of both the power and limitations of current technology with regard to teaching and learning school mathematics. The main objectives of this chapter are the following:
To highlight similarities and differences among CAS and non-CAS utilities, with emphasis placed on symbolic manipulation functionality of CAS calculators.
To informally investigate ways in which early graphers heightened the importance of graphing in the secondary mathematics curriculum.
• To envision analogous changes that CAS-calculators may have upon the role of symbolic computation in the secondary mathematics curriculum.
• To foster an understanding of the nature and limitations of computer algebra systems as they currently exist. Symbolic Algebra and Hand-Held Graphers: An Historical Perspective
The Development of Computer Algebra Systems
The idea of using a computer to simplify algebraic expressions is not new. In fact,
as Hillel, et al. (1992) point out, ‘‘the field of computer algebra which deals with
automated symbolic manipulations is nearing its fortieth birthday — routines for analjdic
differentiation having first appeared in 1953” (p. 119). Algebraic manipulation utilities
were originally created as research tools for mathematicians and computer scientists,
not as pedagogical tools for secondary school teachers. Lengthy problems have inspired
mathematicians to write computer programs to solve algebraic expressions
symbolically. As Pavelle, Rothstein, and Fitch (1981) note:
In 1973 one of us (Pavelle) undertook an algebraic calculation pertaining to the general theory of relativity; the calculation required three months of work with pencil and paper. The following year a more formidable problem arose . . . instead of attempting another calculation by hand, Pavelle decided to construct a computer program for manipulating mathematical expressions of the kind that commonly appear in gravitational theories. The program was then written in the computer language of a powerful system of algebraic programs called MACSYMA, the 1973 problem was solved as a test; the computer confirmed the results o f the three-month calculation in two minutes [emphasis added] (p. 136).
Palmiter (1986) states that modem computer algebra systems have been in existence
“since the early 1970s,” with MACSYMA being “one o f the first” (p. 12). Although
CAS have existed in commercial versions since that time, the systems did not gain widespread popularity for over a decade. A partial explanation for this rests with the relative scarcity of personal computers until the early 1980s. One of the first CAS specifically designed for use with personal computers was
muMATH{Soft Warehouse, 1983). Although muM ATH{Soü. Warehouse, 1983) was
significantly less powerful than systems running on large mainframe computers, it was
responsible for introducing CAS to many new users. As Hillel, et al. (1992) note:
At the time of its appearance, miiMATH did not cause a great stir among mathematicians. From today's perspective, it was a rather impoverished CAS, limited in scope, without graphing and without “pretty printing” (i.e. outputs were not converted into the standard mathematical form). However, it was a sign of things to come and by the mid 1980s, powerful, elegant, and more user-friendly systems such as Maple, Derive, and Mathematica became available (p. 120).
Among muMath’s (Soft Warehouse, 1983) most vocal users was Herbert Wilf, a mathematics faculty member at the University of Pennsylvania. W ilfs 1982 article
“The disk wdth the college education” introduced many mathematicians to the notion of symbolic algebra. Hillel, et al. (1992) comment that “the mathematics community at large probably first became aware of CAS through W ilf s 1982 article” (p. 120).
In the paper, Wilf (1982) demonstrates some of the major features of m uM ATH
(Soft Warehouse, 1983), including addition of fractions, simplifying of square roots, differentiation, and rudimentary programming in the software’s built-in computer language, miiSIMP. Wilf writes his article as a “distant early warning signal” to college mathematics teachers, stating that “the sudden mass availability of a program with these
[CAS] capabilities” is heading “in the direction of college mathematics” (p. 4).
Insightfully, he envisions a future in which such tools are available to students in hand held versions.
10 Will we allow students to bring them into exams? Use them to do homework? How will the content of calculus courses be affected? Will we take the advice that we have been dispensing to teachers in the primary grades: that they should teach more of concepts and less of mechanics. What happens when $29.95 pocket computers can do all of the above and solve standard forms of differential equations, do multiple integrals, vector analysis, and what-have-you? (p. 8)
Since W ilf s article was first published, a variety of other CAS utilities have become available commercially, both for desktop computers in packages such as M aple
(Waterloo Maple, 2000), Mathematica (Wolfram Research, 1997), and M athcad
(MathSoft, 1996) and more recently as a feature of hand-held calculators such as the TI-
92 (Texas Instruments, 1995) and TI-89 (Texas Instruments, 1997). Although each system has its own set of strengths and weaknesses - from ease of use to computational power and speed — all share the ability to produce exact solutions for various symbolic manipulation problems. Below are sample screens from several CAS.
□ = Mathematica Fartnrin«| g] g f/}{ //:= Factor[ ( (2y+6)/ (y-2-2y-24) )»( (y-6)/ (y+3) ) ]
2 4 + y
1100% V 4 I ► I
Figure 2.1: Sample output from Mathematica
11 [ MathCad Factoring I m m (2 7 + 6) j ^ —Zy — ZA L(7 + 3)J
+ (y + 4)
Figure 2.2: Sample output from Mathcad
One may be somewhat surprised to discover that the computational abilities of various CAS differ from product to product. For instance, when attempting to solve the
following equation:
the TI-92 CAS, Maple, and MathCad Plus 6 CAS produce different solutions. As the screens in Figures 2.3 through 2.5 indicate, MathCad Plus 6 finds no solutions to the equation. The TI-92 Plus finds an infinite number of solutions over a restricted domain.
Maple, on the other hand, indicates that the equation is true for any value of x. Three
CAS provide the user with three distinctly different solutions. The user is left wondering which solution (if any) is correct.
12 0
m No answer found.
Figure 2.3: Solution of jr* = j"* using MathCad
(fT^int F Z ^ Y F3V Y FMV Y FË Y F S ^ ▼ Algebra Calc Other PrgmIO Clean Up
■ solue[x^"^‘^ =[x^) , x] X > Q soloe
Figure 2.4: Solution of using TI-92 Plus
□ ....— Maple Example - Student Edition @ B rSCUDENX > soive(x (2/4)=(x^2) (1/4)r X); L ^ [ SXUnENI > 1 <
Figure 2.5: Solution of using Maple
The previous example suggests that teachers and their students will need guidance — and critical eyes — when using CAS.
13 The Development of Graphing Calculators
Although symbolic algebra capabilities of computer algebra systems promise to influence the teaching and learning of school mathematics, they have had virtually no impact on school programs to date (at least in the United States). This is due in part to significant portability and cost constraints which have characterized the technology until quite recently. Non-CAS graphing calculators, on the other hand, have played a significant role in mathematics reform efforts — providing teachers and students with portable utilities to graph functions, solve equations, and approximate roots more efficiently than possible with by-hand techniques.
Before the advent of graphing calculators, much of the content of secondary school algebra dealt with by-hand graphing procedures and pencil-and-paper symbolic manipulation. In the late-1980s, the emergence of hand-held graphing utilities transformed the role of graphing in secondary school classrooms. As Demana and
Waits (1998) note:
We estimate that at least one-fourth of the material that was typically taught in a high school “trig/functions” course or college “precalculus/college or algebra/trig” course before 1972 is simply no longer taught today in similar courses. Many sections and even some chapters in textbooks dealing with paper-and-pencil computation methods became obsolete and disappeared from the curriculum — period! Why? Because hand-held scientific calculators provided better ways to “compute” than pencil and paper methods (p. 3)
14 It is important to note that the calculators have not diminished the importance of
graphing in mathematics classrooms. However, the technology has changed the nature
o f graphing in school classrooms in the following ways:
Students can now produce graphs more quickly and more accurately.
• The study of families of graphs is now possible. Thus, the transformational qualities of graphs have become more central to instruction.
• Emphasis has shifted away from the production of graphs, with increased attention placed on interpretation of graphical data.
To illustrate the curriculum changes that graphing calculators precipitated, several
features of one of the first graphing calculators — the TI-81 — are discussed below.
Although the TI-81 is not equipped with CAS, knowledge of the TI-81 is useful from a historical perspective. A careful examination of early graphing calculators and their
impact on algebra reform provides the reader with insight regarding the possible influence of hand-held CAS on subsequent reform efforts. The examples below also establish a context from which to examine various algebra reform efforts that appear in subsequent chapters of this document.
The capabilities of the TI-81 are compared with more recent calculators by means of the following problem: Find a root o f the equation y = 'l- sm(x) — 1.
Root-finding of non-CAS calculators
An investigation of the problem begins after one types the function y = 2- sin(.r) — 1 into the TI-81 equation editor.
15 YiHZsin X-1
: Vh =
Figure 2.6: Y i defined as 2 - sin(jr) — 1 on the TI-81
To produce a graph of Yi, the user selects the TRIG option from the calculator’s ZOOM menu. The ZOOM>TRIG utility automatically adjusts window settings for optimal viewing of trigonometric functions.
:Box 2:Zoom In 3:Zoom Out 4:Set Factors 5: S-=iuare 6!Standard MITrig
Figure 2.7: Zooming tools and graphics window of the TI-81
After a graph is produced, a student may use the TRACE utility to find various (x,y) coordinate pairs that lie on the graph of y = 2- sin(.r) — 1.
/ /! \ / K=.H6ag?1S5: V=-.in67B2H K=.ggg£Hgi3A/ V=.121H3D11
Figure 2.8: The TI-81 TRACE utility
The TRACE feature provides the student with a useful method for finding roots. On the
TI-81, the X and y coordinates of each point are displayed on the bottom of the
16 calculator’s viewing screen. A user may focus attention to a region near a root by using the ZOOM utility to obtain a more detailed view of the graph close to an x-intercept.
: Box _Zoom In : Z o o m Out 4: Set Factors 5:Square 6:Standard 74rTri9
Figure 2.9: The ZOOM IN utility of the TI-81
Subsequent iterations of zooming and tracing allow the user to approximate roots with a greater degree of precision. The graph displayed in Figure 2.10 was obtained utilizing such a technique.
/ X=. K353B7B Y=2.1HZE-g
Figure 2.10: Results of successive approximations of root on a TI-81
Despite their revolutionary status in the late 80s, the root-finding capabilities of the TI-
81 are crude relative to those found on newer graphing calculators. For sake of comparison, the TI-83 ZERO utility is described below.
Unlike the TI-81, the TI-83 features root-finding routines. Using the ZERO utility, the user selects left and right bounds of a specific root, then makes a guess regarding the
17 value of the root. The calculator uses numerical methods to approximate the nearest zero of the function.
a assiH ais V1=£ïiri Figure 2.11: Root finding on the TI-83 Y=0 Figure 2.12: An approximation of a root TI-83 users may approximate roots to a function f(x) with any of the following techniques. 1. Tracing along a graph. 2. Using a table of values for f(x). 3. Using the calculator’s SOLVER features. As Kutzler (1996) suggests, technology shifts the emphasis of classroom dialog from finding the answer to a comparison of various problem-solving strategies. Although the first hand-held graphers were quite powerful, they were unable to manipulate expressions symbolically. The first graphing calculators (e.g. TI-81 and TI- 18 83) performed all computations using floating-point approximations. Because of their inability to produce exact solutions, first generation graphing calculators have failed to impact significant portions of the traditional mathematics curriculum (notably the “algebraic” portions of the curriculum). Root Finding using CAS calculators It was not until the mid-1990s that hand-held graphing utilities would enjoy the precise algebraic solving features that computer algebra systems such as Derive or Mathematica long possessed. However, with the introduction of the TI-92, calculators equipped with computer symbolic algebra utilities were now able to produce exact solutions in the same way that larger, more expensive computer-based always had. In the example below, the TI-92 computes all of the zeros of y = 2- sin(x) — 1. As Figure 2.13 illustrates, the TI-92 produces a solution of the form 2n7t + tc/6 or 2n7t 4- 57t/6, where n is an arbitrary integer. (T IT ^T Y F3V Y Flv Y P 5 Y F6^ ' ▼ Algebra Calc Other Prgw 1 0 C le a n Up s o l v e ( 2 s i n ( x ) - 1 = 0 , x) X = 2 - g n l -n + o r x = 2 (Enl-jt+-^ solve<2sin Figure 2.13: Exact roots calculated on the TI-92 19 The influence that hand-held CAS utilities "nvill exert on the teaching of symbolic manipulation may very well parallel the influence that graphing calculators exhibited on graphing a decade earlier. 1. Students may produce simplified algebraicc expressions and solutions more quickly and more accurately. 2. The study of families of solutions/simpliffications will be possible. Thus, pattern recognition may become a focus of algebra instruction. 3. Emphasis may shift away from the by-hatnd production of algebraic forms, with increased attention placed on interpretation of algebraic data. When equipped with symbolic manipulation utilities, time-intensive and error-prone algebraic tasks (e.g. factoring, rationalizing excpressions) become automatic processes for students. This is illustrated in Figure 2.14. r Fi- ■ F2- F3- FI- F5 FG-r ' FI* F2* F3* F4* F5 FG* Tools A13«bro Cole CthOf FrSmlD CToan Up TTooTs ftl34bro Calc Othor Pr3min Clton Up -(.J3 - 2) 2+j3 factor(x^ + 5 • X + ô) '■ expand((x + 3) (x + 2) (x - !► ,3 j. . ^,2 ______(x + 2) (x + 3) X + 4 - X + X - 6 eexpard((x+3)+(x+2)*(x-l)) MrtIN FtftP ftUTO FUNC RAD ftUTQ FUNC 1 / 3 0 Figure 2.14: Some exact calculartions possible on the TI-89 Although the above discussion suggests that a mumber of skills currently taught in high school algebra classrooms may now be relegatesd to the calculator, none of the examples have recommended types of activities that m ight replace traditional ones. In the next section, examples of reform-minded algebra letssons that use CAS and calculator-based 20 symbolic manipulation are illustrated. The first activity, adapted from Edwards and Reinhardt (1999), uses a graphical interpretation of matrices to explore algebraic transformation, trigonometry, and pattern recognition. The second activity illustrates ways in which technology may be used to transform the nature of traditional algebra by examining of various solution strategies of simple linear equations. Examples of Meaningful Algebra with CAS Encouraging Student Conjecturing with Exponents The ability of hand-held CAS devices to generate numerous solutions quickly and accurately enables students to build conjectures regarding algebraic patterns that they see on the calculator’s screen. As Drijvers (1995) notes: This exploration phase can lead to interesting discoveries and forms the basis of the explanation phases. In the later phase the results of the explorations will be sorted and proved, or they will lead to the development of new concepts (p. 5). In the paragraphs that follow, CAS is used as a tool to help students uncover symbol patterns. Exponent rules are examined with CAS from an exploratory point of view. Rules for Exponents Exploring exponent rules with students using the TI-92 CAS highlights the pedagogical usefulness of using CAS as an “answer generator.” By examining numerous instances of exponent simplification, calculator output encourages students to build conjectures regarding general exponent rules. As Figure 2.15 suggests, an 21 examination of numerous input/output pairs encourages students to consider integer exponents and their relationship to repeated multiplication. FZ? F5 Ffi f— A lgebra Calc O ther PrgwIO C lear a - z .- a a a ■ a a a a a ■ a -a a ■b-b-b-b'b-b-b b' aaabbbb a^b^ [MrtlN DEQ mUTO FUNf Ç/Î9 Figure 2.15: Examples of integer exponentation using the TI-92 Plus CAS After some discussion, students may be encouraged to look at examples that extend their initial understanding of exponents. Students examine problems involving multipication with terms that initially contain exponents. Examples such as those depicted in Figure 2.16 encourage students look for patterns in calculator output. F I T ^ Y R ? F3^ Y FH-*- Y F5 7 F6 fllgebra|Calc|OtheriPrgm10[Clear a-: a2 a= a aS a6 a-2-alG a> aG a5 a a ^ -a2 a iMmiN PEG AUTO FUNC 5/30 Figure 2.16: Numerous examples of exponent multiplication encourage students to generalize patterns they see 22 A teacher and her students may enter exponent expressions such cr ■ cf into the TI-92 CAS. While examining the output (i.e. with students, the teacher may ask the following questions: • How can we write the expressions cr and o' without exponents? How many a 's would we need in order to write out both expressions without exponents? • Does anyone see a pattern between the exponents of the expression cr ■ cP and the expression cP^. In a similar manner, patterns involving division with exponents may be explored by students using CAS. Some examples are provided in Figure 2.17. F Ï ^ T F3v Y FHv F 5 FS ▼ ir— Algebra Calc Other PrgnlO Clear a-z... b " b ^ 1 IMAIN PEG flUTD F U N C s m 1 Figure 2.17: Examples of exponent division encourage students to generalize patterns they see CAS proves particularly helpful when introducing the notion of negative integer exponents to intermediate algebra students. After examining a handful of CAS- generated input/output pairs, hypotheses such as the following may be generated by- students: 23 For all a^O, and any integer n, = a" The hypothesis is suggested by examples in Figure 2.18. n - T m Q Y FZ^ Y F3V Y FS'*- F5 , FG ▼ t — Algebra Calc Other PrgmIO Clear a-z... -2 -3 -2 PEG AUTO FUNC 3 /3 0 Figure 2.18: Examples of negative integer exponents encourage students to generalize patterns illustrated by input/output pairs Linear Equations from Multiple Perspectives Solving the equation 2x+ 7=3 provides an example in which CAS may transform traditional algebra instruction. The equation has a host of different solution strategies — each of which provide the novice algebra student with “connections” to different types of algebra knowledge. Below, the equation 2x+7=3 is examined from 3 distinct perspectives — by viewing the equation (a) as a boolean object; (b) as a set of simultaneous functions; (c) as a pair of graphs. Equations as boolean objects The computer symbolic algebra utilities on the Tl-89 enable students to view equations as boolean objects that are either true or false. This view of equation has traditionally been explored by students when “checking answers” or when solving 24 equations using trial and error methods. Viewing equations as boolean objects remains relevant as one checks answers with the TI-89. problem: 2x + 7 = 3 rri^T F2^ rF3-T FH^T FS T FS» T ) |T « o 1s{A13« b ro |C o 1c io th « r { P r 4 mID|C1« a n Q f| | 2X + 7-7 = 3-7 ■2-x + 7 = 3 2-x + 7 = 3 2x = -4 ■C2-x + 7 = 3)-7 2-x = -4 2x = -4 _ 2-x= -4 2 x= -2 2 2 ansCl)x2I MAIN DCGAUTO ru N C X = -2 |T»#1%|AU4brc|Ca1c|0th*r|Pr)ml0|c1$qmr FI-T rZ', TF3-T Fh~T F5 X U»| 1 '| | check: 2(-2) 4-7 = 3 -4 4-7=3 '2--2 + 7 = 3______true «■■■■ ■■ ■■ _ 3 = 3 true I MAIN______OEGmUTO FUMC 1 / 3 0 I Figure 2.19: Equations represented as procedures and functions on the TI-89 Equations as a set of simultaneous functions Alternatively, a student may be encouraged to look at each side of an equation as separate function. For instance, in the example 2x4-7= i, one may view 2x+ 7 as a function and i as a function. In solving the equation, one finds input values that produce the same output in both functions. This is accomplished with the calculator’s tab le utility. 25 r Fi- FZ- F3 FE- FS- f: ( Fl- F2 rî T rs fzi 1 IToolîZoomEdit ✓ A17Style -.. Tools Setup *rLUi> X yl y2 ''yl=2-x + 7 -3. 1. 3. 3. 3. y4= -1. 5. 3. Q. 7. 3. gi= 1. 9. o. y3(x)= |x= 2. 1 MAIN PEG AUTO FUNC IMAIN PEG AUTO FUNC 1 Figure 2.20: The expressions 2x+7 and 3 viewed as functions of x within the TI-89 table utility As Figure 2.20 illustrates, the functions 2x+ 7 and 3 provide equivalent output when ,r = —2. Equations as graphs In the secondary school curriculum, students are provided with much time to explore graphs of linear equations. Viewing simple equations as a set of simultaneous graphs provides students with a visual interpretation of equation solving. Interestingly, students may not make the connection between equations and graphs on their own. Providing simultaneous graphs o f y=2x+7 andy=i require students to link previously understood content (graphing lines) a novel way to solve equations. Figure 2.21 illustrates this idea. 26 F l - FZ, F3 FH F 5 - Ffi-r F?- Tools Zoom Troco RtQrcph Moth Drew K«n Intersection xc:~2. / yc: 3. MAIN DEG HUTD FUNC Figure 2.21: Solving equations by locating intersections of functions on a graphing calculator The solution to the equation 2x+7=3 occurs at the intersection of the individual functions 2x+ 7 and 3. The Nature and Limitations of Computer Algebra Although the above examples do much to highlight the power of both graphing and computer symbolic algebra utilities, limitations of the technology currently exist. Such limitations are more easily understood with background knowledge of the design of symbolic manipulators. In the article “Computer Algebra,” Pave lie, et al. (1981) describe several data structures that CAS use to store algebraic expressions. An application such as Derive initially reads user-entered expressions as a collection of characters (commonly called a string). Employing a technique called parsing, the user input is placed into a binary tree structure such as the one shown in Figure 2.22.. For example, a parser might convert the expression 2-(x+5) into the following logical format in computer memory. 27 Figure 2.22: A binary tree structure for 2(x + 5) In the above example, positions of the tree containing operators {•, +} and operands {2, X , 5} are called nodes. The above tree contains 5 nodes. Descriptions of trees make frequent use of family-oriented terminology. For instance, 2 above is called the left child o f the parent node•. Likewise, + above is called the right child o f the parent node•. The topmost node of the tree is called a root. A procedure which begins at the root of a tree and recursively visits the left child, then the right child, and lastly the parent node will eventually visit all nodes of a tree. Such an algoritlim converts the data stored in any binary tree into postfix form. The tree shown in Figure 2.22 is converted into the postfix form 2 .r 5 + - by the algorithm described above. Postfix form is easily evaluated by computer applications (Binstock & Rex, 1995). The tree structure allows programs to perform order of operations more efficiently because postfix form automatically embeds operations without the need for parentheses. Because the tree structure may be extended to include an almost limitless number of branches, such a scheme enables users to evaluate expressions of almost any length. This method is not without its drawbacks, however. Unlike their human counterparts, computer programs parse algebraic expressions in rigid At times,ways. this lack of 28 flexibility causes CAS utilities to produce unpredictable results - that while algebraically correct — may certainly confuse novice algebra students. For example, imagine the following pencil and paper style problem. Factor completely: y —y Perhaps the most obvious solution procedure involves recognition that each term has y as a factor. Hence y should be factored out. y - y =y-{^-y) Unfortunately, due to programming limitations, the above problem is calculated in the following manner on the TI-89. ' F I - F 2- F3- F S - FS Ffi- Tools AlStbKO C alc O th tr PrSMlO Cltan Up factorCy - -y (y - 1) f actor (y-y-^2; MAIN DEGftUTD FUNC 1 /3 0 Figure 2.23: An unexpected answer from the TI-89 resulting from limitations of CAS design Although the above solution is technically correct, it obscures the fact that, in this instance, factoring is simply the distributive property reversed. The example suggests that CAS may be less than helpful in a variety of teaching situations. This view is further supported by examples which illustrate the TI-89’s tendency to simplify expressions automatically. 29 < FI- F2- F?- FH- FEFS- 'I Fl- FZ- F3- FS- F5 F 6 - Tools A13«br ■ x5 ■C2x)^ 8 x^ . i d x 2 ■ sin(9G -x) cos(x) X MAIN DEGAUTU FUNC 3/30 1 MAIN DEG AUTO FUNC 1/30 1 Figure 2.24: Limitations of CAS design are evidenced by the TI-89* s tendency to skip intermediate steps and ignorance of domain restrictions Although CAS tools are quite powerful and a capable of solving a wide variety of problems traditionally taught in the secondary mathematics curriculum, it remains a fact that no implementation of CAS will provide pedagogically useful results in all situations. At times, results that the calculator provides may prove bewildering to novice algebra students. Unlike a classroom teacher, the TI-89 is not trained to think flexibly — the calculator answers all questions as it was programmed to do. It is useful to recall that CAS-based tools were originally developed as a research tools for mathematicians — not conceptual aides for secondary mathematics students with fragile understanding of symbolic manipulation. 30 Summary This chapter has provided the reader with a general overview of Computer Algebra Systems (CAS) from several different perspectives. First, the reader was provided with a definition of CAS and computer symbolic algebra. Example screens from CAS and non-CAS calculators were provided to illustrate the major differences which exist between the two types of calculators. Next, the historical evolutions of graphing calculators and CAS were examined to illustrate the ways in which these technologies have impacted (and will continue to impact) algebra reform at the secondary level. Sample screens from several CAS served to illustrate both the power and limitations of computer algebra software. In the next chapter, CAS is examined from a more theoretical perspective. Several educational theories are examined that are of particular relevance to computer algebra utilization in secondary schools. For instance, Vygotsky’s Zone of Proximal Development provides useful models with which to construct classroom activities. Perhaps most importantly, a sound theoretical rationale for introducing computer symbolic algebra utilities into secondary classrooms is provided. 31 CHAPTERS THEORETICAL FRAMEWORK All my best thoughts were stolen by the ancients. Ralph Waldo Emerson Introduction At first glance, few observers would liken a university researcher’s job with that of a newspaper reporter. Yet upon further examination, one sees that the two vocations share many similarities. Several occupational duties that journalists and researchers have in common include: • Analyzing and describing real-life events, situations, and outcomes • Explaining and predicting phenomena • Reporting data to readers Although journalists and university researchers do perform similar tasks, Caliendo and Kyle (1996) remind us that the two jobs are not the same. Unlike journalists, scholarly researchers are obligated to include a “structured theoretical, methodological, or analytical framework” (p. 225) in their writings. Caliendo and Kyle also stress that 32 scholars who “fail to explicate or offer a rationale for their theory and methodology” undermine “the essence of [their] scholarly work” (p. 225). In a sense, a research article without a well-defined theoretical component reads more like a long news story than a piece of scholarly writing. As Romberg (1992) notes, “too often, researchers report only procedures and findings, not the model or world view. The findings of any particular study are interpretable only in terms of the world view. If it is not stated, readers will undoubtedly use their own notions to interpret the study” (p. 53). Unlike a newspaper article, the chapters that follow represent a scholarly study of technology in secondary mathematics classrooms. This dissertation is more than a journalistic enterprise — it is a piece of scholarly research. A central question which has guided this research and the development of its theoretical basis is rather simple: How do students learn mathematics with technology? Basic themes and assumptions that emerge throughout this study include the following: • Student knowledge consists of hierarchies of concepts — with new understandings built upon prior knowledge. Meaningful learning is facilitated through social interaction and interactions with “expert others.” • Activities and discovery aid in student understanding. Students employ mathematical algorithms in two fundamental ways — as “black box” processes and as “white box” processes. Discussion of the theoretical underpirmings of this study draw heavily upon a number of educational theories, most notably: Bruner (1960) and his theories of 33 Discovery Learning and Vygotsky’s Zone o f Proximal Development (Van Oers, 1996). In addition, Buchberger’s (1990) white box/black box principles are discussed to provide background information relevant to pedagogical issues in CAS-rich classrooms. The theories assist this study in several ways: • They help in the generation of CAS-based course materials. • They help to inform instructional practice. • They help provide possible explanations for data collected during this study. The Nature of Knowledge Knowledge is Constructed To solve mathematical tasks meaningfully, students first need to restate problems in terms that make sense to them. Typically, these meanings differ from student to student. Some variation may be explained by the differing experiences students bring with them to the classroom. Variation in understanding may also be related to students’ ability (or inability) to connect new problem situations to old ones. For instance, a student with a naive view of algebra might reconstruct the problem Solve: 2x + 5 = 7 as simply “Take 2x + 7 = 3 and get x by itself.” A common strategy employed by students working at this level is to “take away numbers repeatedly to get x by itself.” Certainly such a conceptualization may be useful in certain situations. For instance, this type of procedural logic leads to the equivalent expressions: 34 (2,r+7)-7 = (3)-7 or 2x= —4 The student simply takes away (i.e. subtracts) constants that lie on the same side of the equal sign as x. Unfortunately, this type of understanding is brittle in the sense that it “works” only under a very narrow range of circumstances. For instance, it leads to an error, common among many beginning algebra students: 2,r—2 = -4 —2 or x = -6 In this instance, the student applied the same logic used in the previous step to “take the 2 away from 2x.” On a purely semantic level, taking the symbol 2 away from the expression 2x does leave us with x. The idea is logically sound, however it is (obviously) not mathematically correct. On the other hand, a student possessing a deeper understanding of the problem might reconstruct it as “find a value of x which makes 2x + 7 equal to i.” A student with this more conceptual view of equation solving is less likely to make errors such as the one illustrated above. The intermediate step: 2.r = -4 is translated into the problem “find a value x that when doubled equals -4.” The mature algebra student recognizes x=-2 as the solution to this problem. The differences in student understanding illustrated above suggest a major role of technology in secondary mathematics instruction — namely, helping students reconstruct their algebraic understandings into more mature forms. 35 In the article Mathematical Processes and Symbols o f the Mind, Tall (1992) points out that mathematical symbols are commonly used to represent both processes and conceptual objects. The dual role that mathematical symbols assume creates confusion for many learners. Weaker students are often unable to view symbols as concepts, preferring to look at them procedurally. On the other hand, stronger students are able to move back and forth between conceptual and procedural interpretations. Computer symbolic algebra utilities - such as those found on the Tl-89 graphing calculator — may encourage weak students to examine algebraic expressions from a more conceptual point of view, enabling students to build more mature notions of equation solving. This idea is illustrated with an example from the text Solving Linear Equations with the TI- 92 (Kutzler, 1997). The equation 2x + 7 = J is solved using the Tl-89 CAS. Figures 3.1 through 3.5 highlight the calculator’s ability to represent equations as both procedures and as mathematical objects. First the user types an equation to solve on the Tl-89 and presses e n te r. The result of entering the expression is shown on the bottom right of the screen. f F l - FZ- F3 , F5 F6^ To*ls A13«br 2-X + 7 = 3 2 X + 7 = 3 MAIN PCS HUTU FUNC 1 /3 0 Figure 3.1: Entering an equation to solve on the TI-89 36 The TI-89 simultaneously treats any algebraic equation as an object and a process. Procedurally, the calculator views the equation 2x + 7 = i as the following series of steps: 1. For any value o f x, multiply 2 and x; 2. Add 7 to the previous calculation; 3. Decide whether or not this final calculation equal to 3. If so, return true. If not, return false. Figure 3.2 illustrates the result of intepretting the equation 2x + 7 = 5 in such a manner when X = I . F l - F2» FT-r FH, F5 F fi, ' Tools A lS o tra CoTc Dth«r FrJroin C1«on Up 2-x + 7 = 3 2 X -t- 7 = 3 2-x-t-7 = 3 I X = 1 false 2x-*-7=3 I x= 1 MAIN RAD EXACT FUNC 2 /3 D Figure 3.2: Using the TI-89 to promote a procedural view equations On the other hand, the TI-89 treats algebraic equations as objects that may be manipulated symbolically. For instance, using binary operators, one may combine various algebraic expressions with 2x+7=5. Conveniently, the TI-89 stores results of the calculator’s most recent calculation in a variable called ans(l). With ans(l) equal to 2x-r 7= 3, an operation such as ans(l) - 7 subtracts 7 from each side of the equation. 37 As illustrated in Figure 3.3, the TI-89 returns the equation 2x= -4 when 7 is subtracted from the original equation. The parentheses in the calculation (2x+ 7= 3)-7 emphasize that the calculator views equation 2x+ 7=3 as an object to be manipulated algebraically. r Fi- F3- F3-T FM- F5 F 6 - Tools AMoOro CoTc|Oth«r Premia C lto n (If I 2 X + 7 = 3 2 X + 7 = 3 I (2 X + 7 = 3) - 7 2 - X = ~4 MAIN PEG AUTO FUNC 2 /3 0 Figure 3.3: Using the TI-89 to promote an object-based view of equations Many students mistakenly subtract 2 from each side of the equation 2x=-4 to “get x by itself.'’ Unfortunately, the results of such an operation does not “cancel out” multiplication by 2. This is shown in Figure 3.4. F I - F 2 - F 3 -T F I - F5 F S - T o o ls AWt&KOC o lc |O th tr P re m ia C1«on UP " 2 X + 7 = 3 2 X + 7 = 3 "(2 x + 7 = 3)-7 2 X = -4 "(2 X = -4)— 2 2-x- 2 = -6 ans< 1 ) - MHIN PESaU TD FUNC 3 /3 0 Figure 3.4: Using the TI-89 to highlight an error common among novice equation solvers Using the delete key with the TI-89, a student is able to clear off this incorrect step and try again. 38 r Fi- F 2 - F I - F5 F fi- IT oolJ A 13«bro ICoTclnih«r FK3mlDa « o n Up ■2-x + 7 = 3 2 X-l-7 = 3 ■(2-x -1-7=3 ) - 7 2 - X = -4 _ 2 - X = -4 X = "2 2 1 IMAIN BEQAUTD FUNC 3 /3 0 1 Figure 3.5: Solving a linear equation successfully using the TI-89 CAS Such techniques illustrate the use of computer symbolic algebra as a pedagogical tool, assisting students in constructing sound conceptual imderstandings that underlie symbolic manipulation. Used responsibly, computer algebra may encourage students to view algebraic expressions as both objects and processes. The above example suggests that computer symbolic algebra may be utilized by novice algebra students learning to solve equations. The Importance of Prior Knowledge in Mathematics Learning As one uses CAS utilities with young learners, questions (and concerns) soon arise regarding possible negative effects of the technology on student imderstanding. From first hand experience, researchers know that a sizable number of students can leam to solve equations by hand — after all, the majority of math educators learned how to solve equations using paper and pencil techniques. For the most part, a traditional training has served math educators well, providing them with an adequate foundation on which to build subsequent mathematical knowledge (regrettably, the same statement cannot be 39 made for all mathematics students). It is too early to tell whether or not a technology- based mathematics curriculum will provide an adequate mathematical foundation for today’s novice algebra student. Research does indicate that technology may promote student misconceptions regarding functions and graphs. When attempting to extinguish faulty mathematical notions, researchers have found that students’ prior knowledge plays a critical role (Goldenberg, 1988). To illustrate the importance of prior knowledge, we consider the graphs of the following functions: / { x ) = sin(2 - x'j yCx) = sin(50 - x) With a simple visual inspection of the equations, it is evident that the period of each function is different. Yet, when these functions are graphed in TRIG mode on a TI-83, overlapping graphs are produced. Ploti Plots \ViBsin(2X) \YzBsin(50X) \V3=B n Vh = \ y V V V \Yg= n V? = Figure 3.6: Graphs of non-equivalent functions that appear to be equivalent Students without an adequate knowledge of trigonometric functions believe that the above functions are equivalent. To complicate matters further, the calculator’s table of values will support this view. 4 0 X Vi V2 0 0 .ZGIB s s .Hu603 .86603 .7H5H 1 1 1.0172 .86603 .86603 1.309 SS 13708 0 0 X=0 Figure 3.7: Table of values suggest that non-equivalent functions are equivalent What is to be done in situations such as these? How can such faulty thinking be remedied or prevented in technologically-rich classrooms? Which theories of learning may assist us in eliminating such unfortunate misconceptions? Knowledge is a Hierarchy of Concepts Figures 3.6 and 3.7 suggest that an understanding of how students build mathematical knowledge remains important in technology-rich classrooms. Without an understanding of the ways in which students build prior knowledge, educators cannot hope to eradicate learners’ faulty reasoning. In this study, it is assumed that mathematical knowledge is hierarchical — that new understandings build on existing knowledge structures. To help students reduce mathematical errors, educators need tools to ensure that students have strong mathematical backgrounds. Students who lack prerequisite knowledge often struggle to succeed in mathematics class. In the following paragraphs, we explain ways in which technology — particularly CAS utilities — may help students to overcome such deficits. The following discussion is based in large part on the work of Kutzler (1996) and his notion of “scaffolding didactics.” 41 In the text Improving Mathematics Teaching with DERIVE (1996), Kutzler compares mathematical knowledge building to the construction of a house. Mathematical knowledge builds on top of prior knowledge in much the same way that upper levels of a house are built upon lower levels. A weakness in the foundation of a house compromises the structural integrity of the entire building. Similarly, weak arithmetic skills often hamper mathematics students in later courses. Kutzler describes the “house of mathematics” of a typical high school student graphically (Kutzler, 1996, p. 23). E quations Elementary Algebra Arithmetic •: Figure 3.8: Kutzler’s “house of mathematics” of a typical high school student Kutzler points out that in an ideal teaching situation, a high school student will have no mathematical misconceptions at any stage below his or her current level of study. This situation may be depicted pictorially in the following manner: Ba E quations Elementary Algebra A rithm etic Figure 3.9: The house of mathematics of a student possessing a solid foundation on which to build advanced mathematical concepts 42 Unfortunately, many high school students possess incomplete mathematical understanding more closely represented in Figure 3.10. Holes in mathematical knowledge affect subsequent levels of understanding. M*r Figure 3.10: Mathematical understanding of typical high school students contains holes and misconceptions Kutzler maintains that students weak in algebra are unlikely to attain robust understandings of trigonometry. In other words, a situation such as that depicted below seems improbable. lementary Algebr H thnieti Figure 3.11: Success in basic trigonometry without knowledge of arithmetic, algebra, and equations is improbable Kutzler (1996) describes the dilemma facing teachers and students in the following manner. It would be ideal for every student to receive some special tutoring to fill the gaps and to plug the holes in their learning. For such utopianism there is, unfortunately, not time; the teacher must move on to the next subject in the syllabus. As to how the teachers go on, the question 43 remains as to how to proceed without losing the students, especially those with exceptionally large and structurally important gaps (p. 30). Kutzler (1996) suggests that teachers and students may use computer symbolic algebra and CAS as cognitive supports. For instance, a student with misconceptions regarding equation solving can still study trigonometry meaningfully using CAS. As equations are revisited in the context of trigonometry, the student has additional opportunities to leam concepts of equation solving — but remains able to solve equations using CAS. E q uation s Figure 3.12: CAS enables students to study more advanced concepts while resolving misconceptions with earlier material The calculator provides extra time for the student to leam procedural equation solving skills without the immediate threat of failure. Leaming is a Process In recent years, increasing numbers of educators have begun to re-evaluate the effectiveness of traditional teaching techniques in secondary mathematics classrooms. Reformers argue that traditional mathematics teaching has placed undue emphasis on concept acquisition and teacher-led instmction while ignoring the development of students’ flexible thinking skills. Viewing leaming as a process (as opposed to a product), these educators contend that a primary aim of education should be the teaching of general knowledge-getting and problem-solving skills. Such views are 4 4 embraced the National Council of Teachers of Mathematics Curriculum and Evaluation Standards (National Council of Teachers of Mathematics, 1989). Alternative methods of instruction will require the teacher’s role to shift from dispensing information to facilitating leaming, from that of director to that of catalyst and coach. The introduction of new topics and most subsumed objective should, whenever possible, be embedded in problem situations posed in an environment that encourages students to explore, formulate and test conjectures, prove generalizations, and discuss and apply the results o f their investigations (p. 128). Current efforts to increase emphasis on student discovery and problem solving have their historical roots in the writings of Bruner (I960; 1966). The article Psychology and Mathematics Education (Schulman, 1970) highlights a teaching activity developed by Bruner and Dienes (Bruner, 1966) that captures the spirit of discovery leaming. The lesson highlights the use of base-ten blocks to teach a basic algebra lesson while illustrating several important aspects of Bruner’s leaming theories. A Discovery Leaming Example'. The following example is typical of Bruner’s approach to teaching. By allowing the child to manipulate hands-on materials, Bruner leads the child through three distinct levels of understanding. The following levels are described by Shulman (1970, p. 29). • Enactive level: This is the most basic level of student understanding. To represent ideas and solve problems, the child must manipulate materials directly. • Ikonic level: At this intermediate level of understanding, students are able to represent ideas and solve problems mentally without relying on direct manipulation of materials. The child uses mental images of physical objects at this stage. • Symbolic level: At this advanced stage of understanding, students are able to solve problems by strictly manipulating symbols that represent concrete objects. Mental images and manipulatives are no longer required. 45 Bruner and Dienes use blocks to examine fundamental algebraic principles. Basically, the blocks come in three different dimensions: a 1 unit by 1 unit square (referred to as /), a 1 unit by x unit rectangle (referred to as x), and an x unit by x unit square (referred to as X square). These are shown in Figure 3.13. 1 Figure 3.13: The three fundamental base ten blocks described by Bruner and Dienes At the beginning of the activity, students are provided with an opportunity to explore the shapes — to play with them, to rearrange them, and to combine them together in whatever manner they wish. After some experimentation, Bruner asks the children to discover ways to combine these pieces to form larger squares with no overlap. After some time, the learners produce the following types of combinations. Figure 3.14: Several possible squares constructed with base ten blocks 46 Bruner then asks the children to describe the combinations of blocks they used for their constructions. A student building the middle square might respond by saying “I took one X square, fourx’s, and four / ’s to make my square.” After the students feel comfortable describing their squares in this manner, Bruner encourages them to represent the squares’ constructions with symbols — replacing the word “and” with “+ ” and the phrase “x square" with “x*^”. For instance, the three squares above may be represented symbolically from left to right as: x*^ + 2x + /; x°+ 4x + 4\ and xP+ 6x + P. By proceeding in this manner, Bruner guides students through enactive, ikonic, and symbolic levels of understanding. Topics such as factoring and completing the square follow directly from the exercise. For instance, the above squares may also be described in terms of their side lengths as illustrated below. Components Length of Alternate of squares side name xO +2x + 1 x + 1 (x + 1) s q u a re or (x + 1 )0 x P +4x + 4 X + 2 (X + 2) s q u a re or (x + 2)0 xO +6x + 9 X + 3 (x + 3) sq u a re or (x + 3)0 Figure 3.15: Equivalent algebraic names for squares generated by Bruner’s students Students are challenged to move from enactive modes of thought to ikonic and symbolic modes in a natural progression. At later stages of the lesson, students are encouraged to link symbols back to more concrete forms. Instead of introducing x to students as a 47 symbol representing an unknown quantity (a somewhat mysterious notion), Bruner provides students with a concrete representation of x — namely the length of a shape. The daunting task of binomial expansion is replaced with concrete activities involving blocks. Incredibly, Bruner used the above lesson successfully with eight year-old students. In an often quoted passage from The Process o f Education (1960), Bruner writes the following. We begin with the hypothesis that any subject can be taught effectively in some intellectually honest form to any child at any stage of development (p. 33). The above quote (and the preceding activity) illustrate an important part of Brimer's theory — the spiraling curriculum. Bruner advocates that students be encouraged to study the same mathematical topics repeatedly, with each presentation matching students’ current stage of intellectual developmental. For instance, eight year olds might study binomial squares in a very concrete manner using base ten blocks. When the students are a few years older, the topic may be revisited with an in-depth study using symbols. In middle school, students may plot binomial squares as functions ofx. Meaningful Teaming is facilitated through interactions with Learned Others For practical reasons (e.g. standardized testing requirements, attention span of young students), the practice of many classroom teachers falls somewhere between the content-driven, teacher-centered environments of traditionally taught classes and the 48 loose, exploration-based environments encouraged by Bruner. Many teachers successfully use a combination of lecture and discovery to help students leam important mathematical ideas. For instance, teachers may first provide direct instruction regarding specific mathematical topics, then follow this instruction with challenging group exploration activities. Similarly, exploration may come before direct instruction to capture learners interest in the topic. To foster an active problem solving environment in their classrooms, teachers may provide problems that are just out of students’ cognitive grasp - difficult enough to encourage collaboration, and straightforward enough to be solved while working with others. Vvgotskv’s Zone of Proximal Development The notion that good teaching aims somewhere above a student’s current level of understanding is commonly attributed to Vygotsky, a Russian theorist whose work was originally published in the early part of the 20^ century. Vygotsky speaks of two distinct ability levels which help to describe student intellectual performance: (1) a level of actual development, and (2) a level of potential development. A ctu a l development refers to the highest level of performance attainable for the student while working unassisted. On the other hand, potential development refers to the highest level of performance attainable by the student when he/she interacts with a teacher or a more capable peer. Vygotsky states that “what children can do with assistance today they will be able to do by themselves tomorrow” (Jones & Thornton, 1993, p. 20). 49 Potential Development ZPD Actual Development Figure 3.16: A graphical interpretation of Vygotsl^’s ZPD Vygotsky referred to the region between a child’s actual development and potential development as the zone o f proximal development (ZPD). As Taylor (1993) points out, “Vygotsky initially developed the notion of ZPD to describe the knowledge acquisition of 9 to 12 year-olds” (p. 6). Older students were not mentioned in his work. Nevertheless, the construct has proved successful in a variety of other settings, including those involving high school students (Rachlin, 1986) and adult learners (Wertsch, 1985). In this study, use of technology as an “expert other” is explored. Below, a short vignette illustrates how technology may be used to enable students to solve problems above their actual level of development but within Vygotsky’s zone of proximal development. The vignette illustrates how computer symbolic algebra and CAS may function as an “expert other” in the algebra classroom. 50 Suppose that students in an algebra class have a strong grasp of linear and quadratic equations, but have no experience solving polynomial equations of higher order. The classroom teacher provides her class with the following exercise: Solve: jr'-t-2jr=l2 In groups of two, students attempt to solve the equation with the help of CAS-equipped calculators. Student A: I’ve never done any problems like this one before! Student B: Neither have 1! What do you want to do first? Student A: Well, to me it looks almost like a quadratic equation. W hat do you mean?Student B: What do you mean?Student Student A: Well, if the x'* were to the second power and the were to the first power, then the equation would be a quadratic. Student B: I see what you mean. If we could get the equation to be quadratic, we could complete the square to find answers for x. Student A: Let’s try something like this out (Student A types the equation into the calculator). ■ FZ-r ■ ' ' ' FH-r ' FS ■' Fgv Y Algebra Calc Other PrgnlO Clean Up + 2-x^ = 12 x"* + 2 x 2 _ 12 [m a i n rad AUTO FUNC 1/3» Figure 3.17: Student A enters a higher order polynomial equation into the TI-92 CAS 51 Student B: Let’s try completing the square. We want the constant term on the left side to be 1, right? Student A: Uh.. .Why? Student B: Don’t you remember? Take half of the middle term and square it. Student A: Oh yeah! You mean two divided by one squared equals 1. Let’s add 1 to each side of the equation. (Student A types + 1 into calculator and presses enter). |[^ïmiJ:braTc:l^l0t:A;rïPrG% 10Ïcie% r Up! I ■ x^ + 2 x2 _ j2 x ‘*^ +2- x2 = 12 " ( x ^ + 2 x 2 = 12] + 1 + 2 x 2 +1 = 13 Im AIN RAD RUTD FUNC 2 /30 Figure 3.18: Students add I to each side of equation using TI-92 CAS Student B: Now each side should factor nicely. Student A: (Student A uses factor command to factor each side o f the equation). The equation factors! FZ? Y Y Y FS Y FS^ Algebra Calc Other PrgmIO Clean Up x"* +2x2 _ ^2 (x'^ + 2x2= 12) + 1 x"* + 2x2 +1 = 13 ■ factorCx^^ + 2-x2 + 1 = 13) (x 2 + l) = isj factor Figure 3.19: Students factor each side of equation using TI-92 fa c to r command 52 Student B: We seem to be on the right track. With completing the square, we always ended up with one side of the equation as a single squared term. Student A: We can take the square root of each side. Let’s try square rooting like we did with quadratics. Student B: Okay. (Student B types in s q r t (ans (1) ) into calculator). f r ÿ m f Î 2 ^ Y F3V Y FHV Y FS Y Ffv Algebra CalG Other PrgnIO Clean Up x* + 2x2 = 12 x ‘^ + 2-x 2 = ( x “^ + 2-x2 = 1 2 )+ 1 x*^ + 2 - x ^ + 1 = 13 factorCx*^ + 2-x^ + 1 = 13) (x^+l) =13 i x^ + l] = 13 x2+ 1 =iÏ3 IMAIN RAD RUTD FUNC V 3 0 1 Figure 3.20: Using the TI-92 CAS, students calculate the square root of both sides of an equation Student A: Awesome! This looks like a simple quadratic! Our method appears to work! Student B: Yeah! So now we all we need to do is solve the equation j r + 1 = V u . Let’s get X by itself. Subtract 1 from each side of the equation. Student A: Sounds good. (Student A subtracts 1 from each side o f equation). So now we just square root each side again. We’ll have two solutions. Student B: Yeah. (Student B applies square root to each side o f equation). So the solutions for X must be ±1.61417200926. kl-MHn ' F3^ ' ' FÎ-*- ' ' F4-» ” FS FEV 1 Algebra Calc Other P rgn 10 Clean Up | ■ i ■ X — T j. X ■ factorfx"^ + 2-x^ + 1 = 13) (x2+ l)^= 13 ■ J(x^ + l) = 1 3 x^ + 1 =fl3 ■(x2+ 1 =JT3) - 1 x 2 = J Ï 3 - 1 ■ Jx^ =fÏ3 — 1 |x| =1.61417200926 ■MftlN Figure 3.21: Student finds solutions for x by subtracting 1 and taking square root of each side of equation 53 Student A: Let’s graph the original equation to check our work. Student B: Okay. I’ll graph the functions y = + 2jt and y=12 and see where the intersect. (Student B graphs Junctions on calculator). F3 Y FH n? ^Yî=: ■ Zoom T ra c e R e g ra p h M ath D raw K k VJ Intersection xc:1.614172 y c : 12. MAIN RAD AUTO FUNC Figure 3.22: Student checks algebraic solutions using graphing capabilites of the TI-92 CAS Student A: The graphs intersect at two points. These points correspond to our two values of x. Student B: The .v-coordinate of the right-most intersection matches one of our symbolic answers. We did the problem correctly! The above vignette illustrates how technology can assist students as they solve unfamiliar problems. Several points are important to mention with regard to the example. First of all note that computer symbolic algebra was exploited extensively. Secondly, note that student understanding involving the solution of a higher-order polynomial equation (an unfamiliar activity) was linked back to a more familiar situation (solving quadratics by completing the square). The calculator played the principal role as “expert other” in this example, guiding students in a step-by-step fashion as they attempted to solve the problem. The “answer checking” capabilities of 54 the calculator provided closure for the students - after graphing intersections, they knew that their solutions were correct. Buchberger’s white box/black box Principle In many ways, the computing power of today’s hand-held algebra manipulators are reminiscent of the first scientific calculators. The early devices could calculate the square root of any number with a single key press. Likewise, CAS-equipped calculators are able to solve equations with the press of a button. Today, students are no longer taught how to compute square roots by hand (although at one time this was a popular topic in high school math classes). It simply isn’t important to know how to calculate square roots by hand anymore — innovations in technology made such skills obsolete. In the near future, will the same argument be made with regard to equation solving? Will the intermediate steps required to solve a linear equation remain a worthwhile algebra topic? Such considerations form the basis of Buchberger’s (1990) white box/black box methodologies. Note that symbolic manipulation capabilities of the TI-89 (Texas Instruments, 1997) allow students to solve equations in a single step — all underlying mathematical processes may be hidden from the user. 55 ' F t- F2» F5- FH^ F5 FS-.- T**lJ A13«Dr ■ solveCl.25 + m -.25 = 10, n) n = 35. s o lv e ( 1. 25+rn+. 25= 10, m' MftIN ROD ftUTD FUNC i/3 0 Figure 3.23: Using the so lv e function on the TI-89 CAS to determine the solution for a basic linear equation Because all intermediate steps are hidden when using the s o lv e function, it is commonly referred to as a “black box” process. Conversely, solution strategies which do not hide intermediate calculations are referred to as “white box” activities (Buchberger, 1990). Algebra students (and teachers) using CAS-equipped calculators have the ability to choose between white box and black box techniques when solving problems. When calculator users are naive algebra students, the appropriateness of various solution techniques becomes critical. The methods chosen may impact the learner’s conceptual understanding of a host of mathematical processes. For instance, beginning algebra students using black box strategies exclusively to solve equations may do so at the expense of their conceptual understanding of equation solving. Although math teachers may feel comfortable relegating time-consuming pencil and paper tasks to the calculator as black box exercises, the effects of such decisions on student cognition remain unclear. 56 Buchberger was among the first to propose general guidelines for appropriate calculator use in CAS-equipped classes. Although Buchberger’s recommendations do not solve all problems associated with teaching mathematics using computer symbolic algebra utilities, his theories provide a starting point from which to formulate ideas regarding the appropriate use of CAS. In the paragraphs below, the basic features of Buchberger’s principles are described along with sample screens from the TI-89 graphing calculator. The notion of white box/black box was first proposed by Buchberger (1990) in the article “Should students leam integration rules?”. In the article, Buchberger designates various mathematics problems and topics as “trivialized” due to the existence of "feasible, efficient, tractable algorithm[s] that can solve any instance of a problem from this area” (p. 10). The article identifies the following as trivialized branches of mathematics: • Arithmetic on the natural numbers • Integrating functions described by elementary transcendental expressions (trivialized by Risch’s algorithm) • Geometrical theorem proving (trivialized by Wu’s algorithm or by the Grobner basis algorithm) (p. 11) Using Buchberger’s definition of trivialized, the following secondary mathematics topics may be added to his list. • Solving linear equations • Factoring polynomial expressions 57 • Rationalizing denominators of improper fractions • Calculating maxima and minima of polynomial flmctions It is natural to wonder whether classroom instructors should be teaching “trivialized” areas of mathematics to students. For instance, should students study factoring when the calculator’s f a c t o r feature will do much of this work for them? Buchberger answers such questions in the following manner: I think it is totally inappropriate to answer such a question by a strict “yes” or “no.” Rather, the answer depends on the stage o f teaching [emphasis added] area X (p. 13). Buchberger identifies two distinct, chronological stages of learning — each with a distinct technology protocol. The first stage, referred to by Buchberger as the white box stage, occurs while students are learning a mathematical concept or technique for the first time. In this stage, “the student needs to do relevant operations by hand” (Drijvers, 1995, p. 4). Buchberger has the following to say about the white box stage: In the first stage . . . the mathematical theory (definitions, theorems, proofs) must be developed on which the (algorithmic) solution to the problems studied . . . is based. This is the stage where mathematical insight and new mathematical techniques are acquired. It would be disastrous for the future of mathematics if the insights and techniques that can be taught and learned in this stage would be ignored because the area is “trivialized” [emphasis added] (p. 13). After a student understands a given topic suitably well, the learning process of the concept enters a black box stage. At this time, it is appropriate for students to use technology to solve these types of problems. Below is a flowchart depicting the evolution of a mathematical process from the white box stage to tlie black box stage. 58 Define concepts related to the process Study theorems Verify techniques related to the related to the process process Apply new by-hand methods (white-box) and old routines (black-box) to produce solution No, student needs more practice Is the solution routine? Yes Trivialize Process Figure 3.24: A flowchart depicting Buchberger’s white box / black box principle Application of Buchberger’s white box / black box principle are commonplace. A basic example is Kutzler’s (1996) solving of linear equations. When using Kutzler’s CAS techniques to solve an equation, basic arithmetic (i.e. adding and subtracting real numbers) is a black box activity, whereas repeated applications of inverse functions to isolate a variable are white box activities. 59 More recently a number of CAS advocates have proposed that introducing new mathematical ideas as black box processes may actually facilitate student understanding. As Drijvers (1995) notes: This exploration phase (the black box phase) can lead to interesting discoveries and forms the basis of the explanation phases. In the later phase the results of the explorations will be sorted and proved, or they will lead to the development of new concepts (the white box phase) (p. 5). In contrast to Buchberger, researchers such as Dijvers point out that symbolic manipulation utilities enable students to explore processes, look for patterns, and form conjectures about unfamiliar functions. Emergent Theory of CAS Usage in the Secondary Mathematics Classroom Below, the works of several educational theorists — Buchberger, Bruner, Kutzler, and Vygotsky — are synthesized to form a conceptualization of symbolic algebra usage in secondary mathematics classrooms. The model is depicted pictorially with two diagrams. 1. One which depicts classroom interactions between a CAS-equipped calculator and a single student 2. One which depicts a global view of classroom interaction among teacher, CAS- calculators, and an entire class 60 Basicm gonometry ,— Equations Elementary Algebra Arithmetic w v Kn swiedge Bui ding an< I Reconstruc tion concept hypothesis building building ^ ^ Student ^^2%^ decoding scaffolding^ expert n expert feedback feedback step-by-step solution generation hypothesis testing white black box box CAS Calculator Figure 3.25: Edwards’s model of student/calculator interaction The diagram in Figure 3.25 describes the interaction between an individual student and a CAS-equipped graphing calculator. The student’s current level of mathematical understanding of various topics (basic trigonometry, equations, elementary algebra, and arithmetic) is illustrated by rectangles at the top of the diagram. The rectangles are not fully formed - suggesting incomplete or immature understanding. As the student interacts with the CAS-equipped calculator (depicted as a rectangular box at the bottom of the diagram), the student actively builds and reconstructs his or her knowledge of mathematics. The diagram illustrates that the 61 calculator may be used as either a “black box” (shown as a black rectangle on the right side of the calculator) or as a “white box” (shown as a white rectangle on the left side of the calculator). The directed line segment connecting “student” to “white box” represents interactions in which the student uses the calculator to perform algebraic processes in a step-by-step manner. The result of such an interaction, denoted by a directed line segment connecting “white box” back to “student,” provides the student with expert feedback prompting the student for further input. In other words, the white box interaction provides cognitive scaffolding for the student. Figure 3.25 suggests that this type of interaction fosters student conceptual understanding. On the other hand, the directed line segment connecting “student” to “black box” represents interactions in which the student uses the calculator in a manner which hides intermediate algebraic steps. In this instance, the calculator is used as an “answer generator.” The result of such an interaction, denoted by a directed line segment connecting “black box” back to “student,” provides the student with expert feedback that, when performed repeatedly, may help students see algebraic patterns and generate hypotheses. This notion is suggested by the words “hypothesis building” which appear to the right of “student” in the diagram. The horizontal line segment labeled “decoding” represents a potential barrier to successful interaction between student and calculator. Before meaningful interaction can take place between student and calculator, students must be able to pose questions 62 to CAS in a syntactical manner that the calcualtor is able to understand. Similarly, the calculator must be able to express answers in a manner familar to the student for CAS output to be readily understood by the user. Until the user types in commands that a calculator is able to interpret, technology is of little benefit to students or teacher. In the context of an entire class, one may wish to “zoom out” from the model provided in Figure 3.25 and describe interactions among students, calculator, and teacher in a technology rich classroom. Such interactions are suggested by the model shown in Figure 3.26. □ CAS Calcutatof Classroom Teacher ^ S tu d en t S tu dent Stu - c -1 am CAS Calculator CAS Calculator CAS Calculator Figure 3.26: Interactions in a CAS-equipped classroom 63 The top of Figure 3.26 depicts interaction between a classroom teacher and a CAS- equipped calculator. Line segments with two arrowheads denote two-way interactions between the calculator and the user. For instance, a teacher (or student) may type the command so lv e (y=3x+4 -22, x ) into the TI-92 CAS - prompting the calculator for an answer. Likewise, the calculator may prompt the user to type in information (such as a list where data is stored or an initial starting value for a table). Three students using CAS-equipped calculators are represented at the bottom of Figure 3.26. Note that although Figure 3-26 suggests that classroom teacher continues to provide students with expert feedback, the role of “expert other” is now shared between calculator and instructor. Considerable classroom dialog also takes place between pupils as they investigate questions posed by the teacher and classmates. Note that in the above models, problems are posed both by peers and by the teacher. While the models in Figures 3.25 and 3.26 do not capture every facet of human- calculator interaction, they provide a valuable starting point for examining classroom dialog in a more thoughtful manner. 64 Summary In this chapter, several learning theories were elucidated in an effort to provide insight regarding possible uses for computer symbolic algebra utilities in secondary schools. A careful examination of the works of Bruner (1960; 1966), Buchberger (1990), Kutzler (1996), and Vygotsky (Jones & Thornton, 1993; Taylor, 1993) illustrate that one’s beliefs regarding the nature of knowledge profoundly impacts one’s views of educational practice. Through a variety of examples, this chapter has proposed that symbolic algebra utilities are tools that support both the process goals of Vygotsky and Bruner and the content-based goals of Kutzler with equal effectiveness. This study blends ideas from each of the above theories in an effort to address differing learning styles of individual students. Regardless of teaching style, teachers that use symbolic algebra tools with students must take special care to ensure that student misconceptions are addressed. A balanced calculator teaching style supporting student hypothesis generation (i.e. black box activities) as well as process specific goals (i.e. white box activities) is recommended. 65 CHAPTER 4 REVIEW OF THE LITERATURE The answers you get from literature depend upon the questions you pose. Margaret Atwood, Canadian Author Introduction This chapter explores the state of school algebra in the United States at the beginning of the twenty-first century. Of particular interest are the algebra reform efforts originating in the early 1980s with the publication of documents such as A Nation at Risk: The Imperative for Educational Reform (NCEE, 1983) and Curriculum and Evaluation Standards (NCTM, 1989). Also relevant are initial findings from the Third International Mathematics and Science Study (NCES, 1997), which provide the reader with a snapshot of the mathematical performance of United States high school seniors during the past decade. Background knowledge of pupil performance and reform efforts allow the reader to more fully appreciate the possible impact of CAS in American high schools. 66 After a general discussion of school algebra, the reader is provided with an overview of research regarding the use of CAS in secondary schools and universities. In broad terms, this chapter aims to explore the following questions. What is the nature of reform efforts for school mathematics and algebra in the United States? • What teaching methods and philosophies characterize algebra classes in the United States at the turn of the 2U' century? • What does research/anecdotal evidence suggest about the effective use of CAS in schools? Mathematics in the Information Age Global economies are in the midst of an information revolution. Computer networks and software engineers have largely replaced the assembly lines and factory workers of previous decades. Robotics and computer automation have dramatically reduced the demand for manual labor in today’s economy. Increasingly, businesses seek workers with technical know-how to manage day-to-day operations of companies. The Mathematical Sciences Education Board expresses a growing need for a mathematically literate society in the reform document Reshaping School Mathematics (MSEB, 1990). As the economy adapts to information-age needs, workers in every sector — from hotel clerks to secretaries, from automobile mechanics to travel agents — must leam to interpret intelligently computer-controlled processes. Most jobs now require analytical rather than merely mechanical skills, so most students need more mathematical power in school as preparation for routine jobs. Similarly, the extensive use of graphical, financial, and statistical data in daily newspapers and in public 67 policy discussions compels a higher standard of quantitative literacy for effective participation in a democratic society (p. 2). In contrast to earlier times - in which rigorous mathematical training was reserved for a relative handful of elite students — society in the 1990s and beyond requires mathematical know-how from all students. The focus of school mathematics is shifting from a dualistic mission — minimal mathematics for the majority, advanced mathematics for a few — to a singular focus on a significant common core of mathematics for all [emphasis added] students (MSEB, 1990, p. 5) Publications such as Algebra for Everyone (Edwards, 1990) and countless journal articles (Harvey, Waits, & Demana, 1995; Held, 1995; Usiskin, 1980) advocate curricular shifts to include the participation of all students in rigorous mathematical activity. As schools attempt to adapt to the many cultural changes brought about by a revolution of information, mathematics educators have posed important questions regarding the teaching of algebra in schools. For instance, • How will the participation of more students in high school algebra courses affect school algebra content? • How will technological innovations, such as hand-held symbolic manipulators and graphing calculators, affect the way in which school algebra courses are taught? • What should the nature of school mathematics be? An applied science? A study of logical structures and proof? A combination of the two? While educators and policy-makers continue to debate answers to questions such as these, a number of authors have recommended that the algebra curriculum begin to 68 stress more realistic problems — settings which require the use of technology to collect and interpret data and to solve equations. As Taylor (1990) states: Along with high expectations comes the need to give the algebra curriculum new focus to offer students instruction that is relevant to their future and eliminate content that is irrelevant and unnecessary. The algebra curriculum should emphasize understanding of algebraic concepts; applications of algebra in science, business, and other fields; and relationships between algebra and geometry. In the future, most symbol manipulation will be done electronically [emphasis added], so topics like factoring and simplification of rational expressions can be deemphasized (p. 51) In the following section, I describe key historical events — the development of new technologies and publications of various reform documents — which have fostered an increased support for school algebra reform such as that described by Taylor. The Beginnings of Algebra Reform As many of the examples provided in this document have suggested — calculators and computers have brought into question the usefulness of by-hand computational techniques such as factoring and root-finding. Not surprisingly, both calculators and computers have encouraged the publication of a variety documents advocating reform of secondary mathematics instruction. The following section includes an examination of several of the more important reform documents of the past two decades. • An Agenda fo r Action (NCTM, 1980) “What Should Not Be in the Algebra and Geometry Curricula of Average College- bound Students” (Usiskin, 1980) 69 • A Nation at Risk: The Imperative for Educational Reform (NCEE, 1983) • Curriculum and Evaluation Standards for Secondary School Mathematics (NCTM, 1989) • References to various calculator-based reform articles While by no means exhaustive, the following discussion is intended to provide the reader with a better understanding of the origins of current algebra reform efforts. Examples of reform-minded exercises are provided to help one to appreciate the impact of various reform philosophies on the day-to-day practice of school mathematics teachers and students. An Agenda for Action Written in the era of the programmable scientific calculator and at the dawn of personal computing. An Agenda for Action (NCTM, 1980) was the “first significant policy document published by the NCTM in a thirty-five year period” (Wagner, 1995). Anticipating profound technological advancements of the coming decade. An Agenda fo r Action (NCTM, 1980) sought to shift the emphasis of algebra instruction in schools away from drill-and-practice techniques, advocating a more problem-centered school curriculum. An Agenda fo r Action (NCTM, 1980) paved the way for subsequent reform documents, most notably the NCTMs Curriculum and Evaluation Standards (NCTM, 1989), calling for “basic skills in mathematics [to] be defined to encompass more than computational facility” (p. 1). The document was among the first to advocate that 70 “mathematics programs take full advantage of the power of calculators and computers at all grade levels” (p. 1). Usiskin’s Vision of School Algebra Subsequent mathematics reform efforts — particularly with regard to algebra — were influenced by Usiskin’s (1980) article “What Should Not Be in the Algebra and Geometry Curricula of Average College-bound Students.” In this article, Usiskin comments that the algebra curriculum is “quite overcrowded and there is no alternative but to take some things out” (p. 413). Usiskin proposes the following criteria to help determine the relative worth of various high school mathematics topics. • Is the topic important in understanding or coping in society? • Is the topic important for future work in mathematics? Is the topic important in understanding what mathematics is about? (p. 415) As Figure 4.1 suggests, these questions become critical with the introduction of computer symbolic algebra calculators in mathematics classrooms. Symbolic manipulation utilities threaten to make many of the by-hand calculation techniques taught in schools obsolete. This is illustrated in Figure 4.1. If F1» [ F2* I F3*T FH^ f F5 f F6- [ 1 ilT4o1skl5«hr4|C4Tc|Oth«r|Fr^mtO|Cl«4n ll»j | (1 3 + 1 )-J2 ■ co s(1 5 î 4 ■ expand((2 x + 5) 8 -x ^ + 60 x ^ 4 • 150-x-t- 125 r ...... M»IN DCG EXACT TUNC Z/3* Figure 4.1: Monomial expansions and exact trigonometric values are routine tasks with the TI-89 71 Usiskin’s writing has influenced a variety of algebra reformers see school mathematics as more than the teaching of symbolic manipulation. As Demana and Waits (1998) note: We expect that problem solving and proof (or giving convincing arguments) will play a more important role [in algebra classrooms], and paper-and-pencil computation will occupy a smaller share of a balanced, modem curriculum. We do not mean to suggest that the time spent on these features should be the same. However, computation should not take up 85% of class time as it did in the past (and in the present in most high school classrooms) (p. 6). Others agree with Usiskin’s emphasis on the solving of realistic applications. Authors such as Kutzler (1996) point out that the ability to convert problem situations into a format that calculators can solve becomes a valuable skill in technology-rich environments. Kutzler suggests that this may create difficulties for weaker students. Calculating skills are a safety net for some students, often being a guarantee of passing an examination. In an examination situation, many students immediately go to those problems that involve only the (mechanical) skills of differentiating, transforming, or simplifying an expression. What should happen to these students? Some will achieve a better understanding of mathematics because the teacher will have more time to concentrate on the teaching and training of concepts necessary for problem solving. For others, a new safety net must be constructed (pp. 43-44). As a principal editor of one of the most successful reform-minded textbook series (the University of Chicago School Mathematics Project), Usiskin’s views regarding algebra reform remain influential. 72 A Nation at Risk One of the most influential publications of the 1980s — one that would have a far- reaching impact on educational policy in this country — was the document A Nation at Risk: The Imperative for Educational Reform (NCEE, 1983). Appealing to feelings of nationalism and patriotism, and to the issue of national security, A Nation at Risk alarmed policy makers and the public about an apparent decline of secondary education in the United States. The ominous tone of the report is evidenced in its first pages. We report to the American people that while we can take justifiable pride in what our schools and colleges have historically accomplished and contributed to the United States and the well-being of its people, the educational foundations of our society are presently being eroded by a rising tide of mediocrity that threatens our very future as a Nation and a people. What was unimaginable a generation ago has begun to occur — others are matching and surpassing our educational attainments (p. 5). A Nation at Risk was the end-product of a study conducted by the National Commission on Excellence in Education (NCEE), a group organized by Secretary of Education T. H. Bell to “help define problems afflicting American education and to provide solutions, not search for scapegoats” (p. iii). The NCEE consisted of eighteen individuals from a variety of backgrounds: university presidents and professors, state and local school administrators, state governors, foundation presidents, school teachers, and corporate executives. The committee collected data for its study from a variety of sources, including the following. Reports and papers commissioned firom experts • Testimony at various government hearings regarding educational issues 73 • Existing reports and analyses of educational programs • Letters firom citizens, teachers, and administrators • Descriptions of existing educational programs deemed to be of high quality by experts in the field Similar to other reform documents of the time, the report emphasized the need for a well-educated work force in an age increasingly dependent on computers and technology. The document places strong emphasis on the impact of education on the economy of the United States. Knowledge, learning, information, and skilled intelligence are the new raw materials of international commerce ... learning is the indispensable investment required for success in the “information age” we are entering (p. 7). The document points out that service-oriented jobs such as health care, food processing, and construction are undergoing “radical transformations” in the wake of high technology. According to the NCEE, jobs of the future will require significant computer and analytical skills at a time when the United States is “raising a new generation of Americans that is scientifically and technologically illiterate” (p. 10). In light of this situation, the report proposed the following expectations for school mathematics programs. The teaching of mathematics in high school should equip graduates to: (a) understand geometric and algebraic concepts; (b) understand elementary probability and statistics; (c) apply mathematics to everyday situations; and (d) estimate, measure, and test the accuracy of their calculations. In addition to the traditional sequence of studies available for college-bound students, new, equally demanding curricula need to be developed for those who do not plan to continue their formal education immediately (p. 25). 74 Citing factors such as declining math SAT scores, weak college admission requirements, watered-down mathematics and science textbooks, and low teacher expectations of students — the report sent shockwaves through the mathematics education establishment. The NCEE provided policy-makers with recommendations for improving the quality of mathematics education in United States schools. Noteworthy among these were: National standardized tests in mathematics Higher expectations in teacher education programs Updated textbooks for mathematics and science classes An expanded academic year (200-220 days) Higher teacher salaries Three years of math and science instruction for all high school students Although A Nation at Risk did not provide detailed curriculum recommendations for school mathematics, the document did contribute to public fear regarding the quality of education in the United States, leading to a renewed public interest in the importance of mathematics education. Its publication inspired a multitude of other educational commissions and reform documents, notably the National Council of Teachers of Mathematics Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989). As we shall see, the Standards repackages and consolidates various visions of 75 reform though a mathematics education lens - paving the way for subsequent reform efforts that continue to the present. The Standards and Algebra Reform Arguably the most significant (and controversial) mathematics reform document of the last twenty years. Curriculum and Evaluation Standards (NCTM, 1989) provides a- broad vision for instructional reform. Detailing expectations for teachers and students in an age of calculators and computers, the document provides a blueprint for mathematics instruction for grades K - 12. Primary aims of the document include the following. • [To] create a coherent vision of what it means to be mathematically literate both in a world that relies on calculators and computers to carry out mathematical procedures and in a world where mathematics is rapidly growing and is extensively being applied in diverse fields. [To] create a set of standards to guide the revision of the school mathematics curriculum and its associated evaluation towards this vision (p. I). Influenced by earlier documents such as An Agenda for Action (NCTM, 1980) and A Nation at Risk: The Imperative for Educational Reform (NCEE, 1983), the Standards combines the notion of a common set of curricular expectations for all students — which it refers to as a “core curriculum” (p. 129) — together with ambitious goals regarding the integration of technology into school mathematics programs. New priorities for algebra. The Standards identifies four curricular “strands” which constitute the core of current secondary mathematics programs — (1) algebra; (2) geometry; (3) trigonometry; and (4) functions. The Standards provides specific 76 recommendations regarding the priority of content associated with each strand. For instance, the Standards recommends that the following algebra topics receive increased emphasis. • The use of real-world problems to motivate and apply theory • The use of computer utilities to develop conceptual understanding • Computer-based methods such as successive approximations and graphing utilities for solving equations and inequalities • The structure of number systems • Matrices and their applications (p. 126) On the other hand, the Standards calls for other algebra topics to receive less attention. These include: Word problems by type, such as coin, digit, and work The simplification of radical expressions The use of factoring to solve equations and to simplify rational expressions Operations with rational expressions Paper-and-pencil graphing of equations by point plotting Logarithmic calculations using tables and interpolation The solution of systems of equations using determinants Conic sections (p. 127) 77 Several of the above topics - in particular, factoring and simplifying radicals - have historically occupied prominent positions in the high school mathematics curriculum. The Standards provides a rationale for prioritization and possible elimination of these topics from the existing algebra curriculum. A Question of Balance. Although the Standards sets forth general recommendations for instructional practices from grades 9 to 12 that include “decreased attention to paper-and-pencil manipulative skill work” and “increased attention to the use of calculators and computers as tools for learning and doing mathematics” (p. 129), the document does recognize the importance of “balance” regarding the use of technology with young students. Calculators and computers for users of mathematics, like word processors for writers, are tools that simplify, but do not accomplish, the work at hand . . . Similarly, the availability of calculators does not eliminate the need for students to learn algorithms. Some proficiency [emphasis added] with paper-and-pencil algorithms is important, but such knowledge should grow out of the problem situations that give rise to the need for such algorithms (p. 8). As Fey (Kaput, 1992) points out, the fact that “a student can leam and do more mathematics with a computer than without is moot” (p. 518). Nonetheless, the Standards view of instructional technology has aroused considerable debate within the mathematics education community. Particularly troublesome for educators are terms such as balanced or appropriate use. For instance, what does the phrase “some proficiency with paper-and-pencil algorithms” actually mean? Does paper-and-pencil proficiency imply several weeks of factoring polynomials by hand before allowing a 78 calculator to do it? Or do several days of factoring constitute proficiency? The Standards do not directly address how much pencil-and-paper drill is required of students prior to instruction using technology. It is not clear whether the authors of the Standards anticipated the rapid development of hand-held CAS utilities. Regardless of their intentions, the vagueness of the Standards becomes increasingly critical as inexpensive, hand-held symbolic algebra tools become widely available to secondary school students. Problem solving and the Standards It may be argued that the Standards have indirectly addressed issues such as “appropriate use of technology” by emphasizing the role of problem solving and realistic problem contexts in the study of mathematics. Despite this emphasis, advocates of the Standards contend that algebra continues to be viewed primarily as “a generalization of arithmetic and a study of techniques and algorithms for solving certain kinds of problems” (Harvey, et al., 1995, p. 79). Describing the state of school algebra, Usiskin’s (1980) comments remain relevant. The standard [algebra] course seems to turn off more students than it turns on. It gives a view of mathematics as a mechanical process devoted to the solution of equations, simplification of expressions, and recitation of rules with the sole rationale for these processes being an axiom system that itself is mysterious to most students (p. 415) Such insight may provide a partial explanation for the potentially controversial nature of computer symbolic algebra in the United States. Symbolic manipulation utilities directly challenge what algebra is to most people — namely the “symbolizing (of) 79 general numerical relationships and mathematical structures and . . (the) operation on those structures” (Kieran, 1992, p. 391). With regard to algebraic manipulation, the Standards state that: The proposed algebra curriculum will move away from a tight focus on manipulative facility to include a greater emphasis on conceptual understanding, on algebra as a means of representation, and on algebraic methods as a problem solving tool (p. 150). With statements such as this, the Standards have encouraged the introduction of symbolic algebra utilities into American high schools. Unfortunately, at the same time the document has failed to provide a clear vision of the “appropriate use” of such tools in secondary school classrooms. The Standards-hsiSQd curriculum has raised the suspicions of parents, teachers, and researchers alike. Namely they wonder — is this good mathematics? Also, do students who solve problems o f this sort — as opposed to more traditional paper and pencil activities — know as much mathematics as other students? In the next sections, we investigate questions such as these with a discussion of research involving symbolic algebra utilities and mathematics teaching. Research Findings In general, studies of CAS and computer symbolic algebra may be divided into three primary categories: (1) studies of high school-aged students; (2) investigations 80 involving remedial college students; and (3) studies involving university students (typically calculus students). Most studies have investigated CAS from a broad context — examining the symbolic algebra component of CAS as well as graphical and tabular representations. Studies that focus entirely on symbolic algebra features of CAS, at the exclusion of graphical representations, are limited. In addition, studies typically contain both quantitative and qualitative components, seeking to discover changes in student achievement as well as changes in student attitudes. Because a large portion of American research consist of doctoral theses, many studies of CAS are short — typically lasting between 5 and 12 weeks. Many reports from foreign countries are strictly qualitative — providing the reader with sample problems used in classrooms or participant attitudes regarding CAS. In the following paragraphs, the results of several CAS research studies are described. The following discussion does not attempt to highlight all CAS studies from the past fifteen years. Rather, the purpose is to provide a general overview of CAS research — highlighting major studies, as well as those having particular relevance to this work. Studies o f CAS in Secondary Schools We begin our discussion with an examination of studies using CAS in secondary school settings. Unfortunately, research of American secondary students is virtually non-existent, with the majority of studies taking place in Europe (particularly Austria, Finland, Britain and France). Nonetheless, results of these studies are important 81 because they provide the reader with information regarding the use of symbolic algebra utilities with novice algebra students. Hunter. Marshall. Monaghan, and Roper tl993'). Of particular interest is a study by Hunter, Marshall, Monaghan, and Roper (1993). Hunter, et al. investigated the pencil- and-paper manipulation skills and attitudes of two groups of 14 and 15 year-old algebra students (n=55) in a British secondary school. The students were split into control and experimental groups. The experimental subjects learned about linear and quadratic equations in a student-led approach using Derive (Soft Warehouse, 1995). Meanwhile, the control group studied identical course materials using traditional, teacher-led pencil- and-paper style activities. Hunter, et al. collected a variety of different data for the study, including written pretests and posttests of algebraic skills, videotapes of classroom activities, and interviews of students. On the pretests and posttests, neither group was allowed to use CAS of any kind. The study found no significant differences between the paper-and-pencil skills of the two groups of students. But as Mayes (1997) notes: The researchers found that although the students were initially motivated by the use of the technology, they often became frustrated with mastering the intricacies of the technology and in interpreting the outputs given by CAS . . . The overall resulting attitude was appreciation for the time saved of drawing graphs, but doubt about the usefulness of CAS in doing mathematics (p. 187). The researchers concluded that for CAS to be implemented successfully with younger students, considerable time needs to be spent teaching students about the CAS — in 82 particular, students need instruction regarding scaling and zooming of graphing windows. The researchers also recommended direct instruction regarding the interpretation of numerical outputs provided by CAS. Videotaped class sessions revealed that students worked more effectively with the CAS in pairs than they did independently. When working in pairs, students were better able to interpret confusing output and discuss various problem-solving strategies. Wain. Hunter. Marshall. Monaghan, and Roper (19931 In a follow up to the above study. Wain, Hunter, Marshall, Monaghan, and Roper (1993) report on an experiment xxsmg fo u r groups of 14 and 15 year-old algebra students. Two pairs of middle-ability and high-ability classes from two different schools were studied. In each school, one class served as a control group, while the other group was taught using Derive (Soft Warehouse, 1995). Figure 4.2 illustrates the design of the study. S ch o o l A High Middle Ability Ability S ch ool B High Middle Ability Ability 1 I Control Group Figure 4.2: Study design of Wain, et al. (1993) 83 Note that in one school, the high ability class served as the control group. In the other school, the middle ability class served as the control group. The experiment lasted three weeks, with each class studying linear functions for one week and quadratic functions for two weeks. The experimental groups studied functions using Derive (Soft Warehouse, 1995); the control group was taught using traditional pencil-and-paper techniques. The experimental students in the Wain, et al. (1993) study used palmtop computers (more akin to a calculator) instead of the larger desktop computers used in the Hunter, et al. (1993) treatment. Four different types of data were collected in the study: pretests and posttests, videotaped lessons, student papers and computer files, and student interviews. The pretests and posttests were modified versions of the Chelsea diagnostic algebra and graphs test. A second posttest of linear and quadratic functions was also administered. As Monaghan (1994) reports: The graphs test revealed little difference between the groups other than the fact that the control group appeared to be better able to interpret scales. . . The linear and quadratic test results indicated that students perform better when they leam without Derive though this is on items where traditional methods may be expected to produce better results. When the experimental group re-sat the test with Derive available there was an increased performance in algebraic items that could be answered by simple menu commands but no improvement in items that called for conceptual understanding (p. 4). The British studies encourage hesitancy for those desiring to incorporate computer symbolic algebra utilities into introductory algebra courses. It should be kept in mind, however, that the above studies do not involve calculators equipped with CAS utilities. 84 Students were not allowed to take the technology home with them, so they were unable to use the technology on homework problems. Because Derive (Soft Warehouse, 1995) was run on desktop computers, students could not investigate mathematics problems in their regular classroom. Instead, they went to a computer lab to work on math problems using CAS. Design constraints of early CAS studies suggest that initial conclusions regarding the effectiveness of the utilities must be viewed with caution. The availability of more convenient, powerful handheld technologies such as the TI-92 (Texas Instruments, 1995) and TI-89 (Texas Instruments, 1997) resolve many of the problems that impacted earlier studies. Lesh 119871. In the article “The Evolution of Problem Representations in the Presence of Powerful Conceptual Amplifiers,” Lesh (1987) summarizes research involving the effectiveness of various instructional strategies that utilize CAS-based tools. Two groups of ninth grade students - a “utilities” group and a “computation” group - were provided with access to a computer-based CAS refened to as “SAM” (p. 203). While solving simultaneous systems of linear equations, students in the utilities group initially used SAM in a white-box fashion - which Lesh refers to as “magic mode” (p. 203). In magic mode, each step in the equation-solving process is revealed to students both graphically and symbolically. As Lesh notes: In SAM’s “magic mode,” if a student correctly translated a problem into algebraic setences, then SAM would step through the equation-solution steps, with each step in the equation solving process being accompanied by graphs . . . During the first session in the magic mode, the student’s 85 task was simply to write a “correct pair of equations” to describe each problem. SAM judged the correcmess of these descriptions and gave hints if the descriptions were not correct... During the second and third sessions, SAM did not judge the correctness of the problem descriptions ... the solution steps were enacted and graphed one at a time (p. 203). During the final sessions of the study, students in the utilities group used SAM in a black-box fashion - which Lesh refers to as “command mode.” In command mode, SAM no longer generated solution steps for students. As Lesh notes “students were given a set of primitive commands (e.g., multiply, combine, simplify) and were told that SAM would enact and graph the results of any command” (p. 204). Unlike students in the utilities group, students in the computation group used SAM to enhance traditional by-hand manipulation techniques. Students in the computation group provided SAM with equations to solve. SAM provided students with a list of transformations to apply to solve the equations. In each case, actual calculations of transformations were left as pencil-and-paper exercises for students. SAM did not perform any symbolic manipulations for students in the compuation group. Lesh describes the computation group in the following manner. After a few examples, the computer gave correct procedural commands and the [computation group] students were asked to carry out each step, one at a time, until a solution was reached. In a sense, the students in the computation group performed the exact role that SAM had performed in the utilities group; and the computer performed the role that the utilities group students had performed (p. 204). At the conclusion of the study, identical final examinations were administered to all students in both groups. Half of the examination items were constructed to measure 86 student abilit>' at solving applications-oriented problems. The remaining items were computational in nature - with no direct application involved. Neither group used SAM on any part of the final examination. Lesh notes that students in the utilities group outperformed their computation group counterparts on all types of final examination items. Not surprisingly, the utilities group significantly outperformed the computation group on the applications part of the final examination. More surprising was the fact that they also outperformed their (originally comparable) peers on the computation half of the test (p. 205). In his concluding remarks, Lesh points to four characteristics of CAS that make the tools worthy teaching tools: (1) Their “expression checking” capabilities; (2) Their ability to generate “solution traces” in a step-by-step fashion; (3) Their multi- represenational capabilities; and (4) Their ability to heighten awareness of “nonanswer- giving” phases of equation solving. Lesh comments that “because SAM is not simply an answer-giver, it goes beyond being a tool for thinking to become a powerful tool to think about thinking’' (p. 205). The Austrian DERIVE Project One of the first projects to study of the effects of CAS in secondary schools was the Austrian DERIVE project. Two major events in the late 1980s and early 1990s made the study of computer algebra systems possible in Austria. First of all, in 1985 the Austrian Ministry of Education made computer science (knowm as informatics) a compulsory subject in the country. By requiring all schools to have up-to-date 87 computers to support programming software, the hardware required to support CAS systems was already in place. Secondly, a general license of Derive (Soft Warehouse, 1995) was purchased for all Austrian secondary schools in 1991, enabling all Austrian teachers and students access to the program. In 1993, the Austrian Ministry of Education funded a research project to investigate the potential impact of CAS on secondary mathematics teaching and learning. The project involved 34 math classes, 28 teachers and 700 pupils from three Austrian provinces — Lower Austria, Upper Austria and Salzburg. The schools involved in the study were each college-preparator}: institutions. A handful of reports from tlie study have been translated to English and are the focus of articles contained The International Derive Journal (Volume 3, Number 1) which are described in the following sections. Klinger (19941. Students in Klinger’s (1994) study were enrolled in the 3rd form in a typical Austrian grammar school (comparable to 7th grade in the United States). Figure 4.3 illustrates the curriculum that typical 3rd and 4th form students encoimtered at the time of the studv. 88 3rd Form Content Areas Examples of Specific Topics basic arithmetic area calculations integers and rational numbers volumes basic algebra similarity proportions statistics 4th Form Content Areas Examples of Specific Topics real numbers Pythagorean Theorem intermediate algebra circles linear functions cylinders, cones, spheres linear equations statistics Figure 4.3: Third and Fourth form curriculum in Austrian schools All together, 147 beginning algebra students took part in the study. The primary goal of the project was to measure Derive's (Soft Warehouse, 1995) effect on students’ ability to solve symbolic algebra problems. At the beginning of the 1992 school year, students were divided into three groups — a control group (n=90), a Derive supported group (n=31), and a Derive user group (n=26). For an entire school year, the control group learned about symbolic manipulation using traditional pencil-and-paper methods. The Derive user group learned about algebraic manipulation both traditionally and in a laboratory setting, using Derive “as a tutor for structural interpretations of terms” (Klinger, 1994, p. 126). Meanwhile, the Derive supported group explored manipulation exclusively through computer-based methods. As Klinger notes, the Derive supported group 89 “systematically used Derive for interpretation and argumentation of algebraic structures” (1994, p. 126). At the beginning of the next school year, after a full year of instruction under one of the three methods, students were given a six-item test designed to measure basic algebraic skills. None of the groups were allowed to use CAS utilities on the test. Test items are reproduced from Klinger (1994) in Figure 4.4. Test 1 : Calculate each example on the left side and give reasons for the method you chose on the right side. Try to use exact mathematical language. Example 1) Example 2) 6a • 3b - Sab + b • 2a a’'- 3b • 3a + 5b* Example 3) Example 4) 4 ( 3a + 5 ) - 4a - 7* 3 Example 5) Example 6) X* xy* 5r-2s ^ 7s + 3r + 5 ^ 2s - 6r + 8 Figure 4.4: Posttest from Klinger (1994) Klinger reports the results of the test in the form of group percentages with no analysis of the effects of teacher or school differences. As Klinger (1994) himself comments, “This interpretation is no scientific way of evaluation. It is only an intuitive way of seeing — the circumstances of the forms, the differences between children, the different 90 teaching methods between different persons are not taken [into] consideration” (p. 126). Klinger (1994) found that the Derive user group performed at higher levels on the final test than either the pencil-and-paper or technology-only groups, as a group answering over 80 percent o f test questions correctly. Based on his experiences and the above test results, Klinger draws several hypotheses regarding the use of CAS with novice algebra students. Among these conjectures include the following: • Using Derive is useful for understanding structures of expressions • Learning new algebraic content is well supported by Derive • With computer supported mathematics, teachers and students may exercise and consolidate algebraic structures in a better way • The curriculum aims are well-supported by Derive As is the case with much of the translated Austrian work, the above study provides anecdotal evidence supporting the use of CAS with young students. Likewise, the study raises important questions for further research. Unfortunately, no caution has been taken to control for group or instructor differences. French Computer Algebra Studies As Hirlimann (1996) notes, “two countries stand out for conducting extensive experiments in the use of computer algebra systems at school and beyond: Austria and France.” In fact, an entire edition of the International DERIVE Journal (1996, Vol. 3, No. 3) discusses results from French studies of CAS in the early 1990s. The French reports are qualitative — describing in general terms types of problems students solved 91 with CAS, changes in pedagogy resulting from introduction of computer symbolic algebra, and student attitudes regarding the technology. Below, two studies from the project are discussed: (I) A study conducted by Michel Rousselet (1996) describes a three-year experiment with young students (ages ranging from 11 to 15 years) using CAS; (2) A study conducted by Gilles Aldon (1996) examines a year-long experiment with upper level high school students (between 16 and 18 years of age). Both articles discuss pedagogical issues that apply directly to the teaching of novice algebra students. Rousselet and Lehning Cl 9961. This study is of particular interest because of the age of the students involved — the youngest only 10 years old. The study focused on algebraic problem solving and the role of CAS with regard to paper-and-pencil activities. As Hirlimann, et al. (1996) note, the French curricula “are established nationally. They focus on problem solving, developing mathematical ability, proving and explaining more than on acquiring techniques” (p. 10). Nevertheless, Rousselet (1996) concedes that young students may still need to develop pencil and paper skills before using CAS. “Between the ages of 10 and 14, our pupils will still have to leam basic manipulations: expanding, factoring, solving equations” and so forth (p. 11). In the study, “DERIVE [was] only included [in] teaching and learning — it was not used in examinations” (p. 10). Three basic types of CAS problems are identified by Rousselet (1996): (1) Problems solved directly with CAS; (2) Problems solved indirectly using CAS; and (3) Problems not solvable with CAS. Problems solved directly with CAS are those that have 92 traditionally received significant attention in American school classrooms — mechanical, manipulation-style problems. With regard to these exercises, Rousselet states that “the monotonous repetition of technical exercises which lack interest will virtually disappear” (p. 12). The types of activities he found best suited for CAS were those that “allow students to focus on the meaning, approach to, and the relevant equations of the problem” (p. 7). Figure 4.5 illustrates an example o f one such problem well-suited for advanced algebra students. A B We know that CD = 5 miles. We want to build a bridge that spans the motorway. On which point M shall we place the bridge in order that AM = MB? D M Figure 4.5: Algebra problem from French CAS study (Rousselet, 1996) Rousselet notes the last step of the problem requires students to prove their answers using arithmetic manipulation-style arguments. In such situations, Rousselet (1996) found Derive “essential” (p. 7). In a related French study, Lehning (1996) describes a problem that is not solvable with CAS that is suitable for study with young students. He examines the following. Simplify: 2 , 93 Lehning (1996) notes that when Derive (Soft Warehouse, 1995) is set to provide exact . , . , 228826127-102334155-Vs ^ , answers, it evaluates the above expression to equal ------= 7. In “approximate mode,” the software approximates the expression to equal 4.37020 x 10'^ 1 -Vs = 0. If the user notes that < 1, then “it is obvious that only the first result can be true” (p. 40). In such cases, an understanding of algebra without technology is critical for problem solving. With regard to arithmetic and algebra content, observations from the Lehning included the following. • CAS requires teachers to develop a “spirit of criticism” in students (p. 12) • CAS does not replace the need to leam basic manipulation skills (p. 11) • CAS appears to provide a widening notion of number in young students (p. 6) • CAS appears to provide students with a better understanding of mathematical operations (p. 7) Aldon 119961. Because Aldon (1996) worked with older students (ranging in age from 16 to 18), it is interesting to note similarities and differences between his study and the one conducted by Rousselet and Lehning. Aldon focuses attention primarily on building problems suited for study with CAS. In the study, two types of problems are identified. • Short term problems (ones that take an evening to solve) • Long term problems (ones that are revisited periodically over the course of several months) 94 Because symbolic algebra utilities perform routine by-hand calculations, problems developed by Aldon are typically conceptual in nature. Figure 4.6 illustrates an example of a long term problem developed by Aldon for the study (p. 16). Find two functions that best model the shape of the car below: Figure 4.6: An example of a “long term” problem by Aldon After examining student work collected regarding the above problem, Aldon notes that his students “used a lot of relevant mathematics” to solve the problem. He also notes that the exercises “appeared to be very useful in students’ construction of mathematical knowledge and teachers’ awareness of their construction” (p. 18). In the study, data regarding students’ problem solving is collected in a variety of ways. • Videotaped interviews of 7 students were produced Questionnaires were given to an entire class of CAS students Written work of all students was collected and analyzed Informal results of the study included the following observations. • Students using Derive moved back and forth from algebraic to graphical representations of algebraic structures in a natural way • No two students used the technology in precisely the same way. The technology enhanced creativity of students 95 • Students “realize that the first thing to do, before using Derive, is to understand the problem and develop a solution strategy” (p. 18) • “CAS problems are more time consuming” (p. 14) • Students need to be able to take CAS home for true effects of technology to be seen. “Research carried out without this condition is likely to miss out important aspects o f students’ development” (p. 14) Statisticallv formal studies. Statistically more formal studies of novice algebra students do exist — particularly at the post-secondary level. The experiences of college algebra students do not mirror the experiences of high school students since the university students have typically seen similar mathematics content in previous courses. Nevertheless, such research proves useful for this study because course content is similar and valid statistical designs are included with the reports. We turn our attention to several of these projects below. Studies Of Remedial College Algebra Students Concordia University. In the article “Basic functions through the lens of computer algebra systems,” (1992), Hillel, et al. describe the introduction of Maple into a remedial functions course at Concordia University in Montreal, Canada. The authors note that although remedial students are not considered “mainstream” mathematics students by many mathematics faculty, these students “constitute a substantial part of undergraduate enrollment in many universities” with “31.4 percent of the total undergraduate enrollment in mathematics in PhD-granting US and Canadian 96 Universities” made up of remedial students taking the college equivalent of high school math classes (Hillel, et al., 1992, p. 126). The authors contend that many of these remedial students take the courses to fulfill university requirements — not because they are planning to major in mathematics related fields. Often the students that are able to pass these classes find little meaning in them. Few of the "survivors” actually make any sense at all of their mathematics experience. Rather they view it as a kind of ritualistic hurdle to be overcome before they can get into their chosen field of study. Attempts by our faculty to improve matters are usually of a “patch-up” nature — a slight shift of emphasis here, dropping a topic there, changing texts, but basically failing to address the real pedagogical challenge, that of having students make some sense of the mathematics they are learning . . . Our hope was to create, by means of CAS, intellectually and mathematically honest activities which were less dependent on the formal and the symbolic. (Hillel, et al., 1992, p. 127- 8). With a failure rate in the remedial functions course “hovering around the 50% level,” the mathematics department decided that the class could benefit from the use of CAS. It was thought that the technology would enable students to see more useful applications of mathematics while freeing them from the heavy demands that the traditional curriculum placed on their shaky algebraic skills. Prior to the introduction of CAS, the content and teaching methods of the functions course was very traditional. Hillel, et al. (1992) describe it as a “multi-section lecture- style course” covering: • coordinates and graphs • distance formula 97 • circles • formal definition of function • function composition and inverses • linear, quadratic, exponential, logarithmic and trigonometric functions (p. 133) The researchers note that the introduction of CAS changed the topics covered in the course, the sequence in which they were presented, as well as the instructional style of the class. “Students were introduced to the computers and Maple in the first week of class and they regularly spent one of the two 75-minute weekly classes in the Maple lab” (p. 133). In order to assess the effectiveness of the new course, the mathematics department decided to collect data regarding the various qualitative and quantitative features of the class. Three observers shadowed a group of students throughout the thirteen-week semester, taking notes on students as they worked through the various lab activities and homework assignments. At the end of the semester, the research team wrote complete profiles of the observed students and conducted interviews with them to gain a better understanding of their thoughts about the course. In addition, all class handouts and detailed teacher notes were compiled along with student data from the course’s three- hour final examination. The research team describes the course final examination in the following manner. The first three questions were completely standard fare involving solving logarithmic and exponential equations, solving a triangle and evaluating composition of a quadratic with a linear function and evaluating the inverse of a linear function (hence comparable with questions posed for 98 the non-experimental group). However the remaining seven questions were conceptually more demanding and all of them involved deriving properties of functions from one representation and describing them in another form (Hillel, et al., 1992, p. 137-8). The research team assessed the results of the project treating all of the students in the two sections of the experimental functions class as a single group. They note that “there would have been little sense using a traditional functions class as a control group — too many aspects of the course were changed in the experimental class” (p. 137). Although they did analyze student performance on the final exam, their final report concentrated primarily on qualitative issues of the course. The researchers comment that the Maple group “performed just as well on some and much better on most of the items in the first three (more technical questions) which had very similar counterparts in the final of the other classes” (p. 138). Some of the more positive observations that the research team gleaned from the data include the following. • “The lab presented a picture which differed substantially from that of students passively sitting and waiting to copy notes of the blackboard” (p. 153). Students were honestly engaged in many of the lab activities. • Students were more willing to explore problems motivated by problems on the worksheets but which were not strictly part of the assignments. In other words, the lab sparked the mathematical curiosities of some students. • More students “stayed in the ball game” in the Maple course (p. 156). Some less positive observations of the course included the following. • Observers noted that it was difficult to capture student attention in the lab. This led to a “rethinking” regarding the role of organized discussion in the lab. 99 “Some students just did the minimum possible . . skipping the activities requiring exploration altogether and went directly to question requiring more precise answers” (p. 153). “We observed students simply by-passing the stage which required reflection on their part and going directly to Maple for the answer” (p. 153). • The graphical data generated in Maple created difficulties for many students. The intention of the mathematics department was to create a course that was mathematically meaningful, freeing remedial students from some of the druggery required of paper-and-pencil calculations — allowing them to see more applications of mathematics. The researchers conclude that many of these goals were met by the course “not spectacularly, but reasonably well” (p. 156). They note that it is too early to evaluate CAS as a whole until more courses adopt the technology in meaningful ways. West Virginia University. Mayes (1995; 1997) describes a study involving seven sections of an introductory algebra course offered in the in the Spring of 1991. Four of the courses were randomly selected to be experimental classes taught with Derive (Soft Warehouse, 1995). The remaining three courses were utilized as control groups. To avoid student self-selection in the study, the courses were not designated as “experimental” until the semester began. Unfortunately, roughly 25 percent of the students who had originally enrolled in the experimental course dropped within the first two weeks of class — citing computer anxiety and additional required lab time as reasons for leaving the course. Complete data were collected for 137 subjects, with 61 in the experimental group and 76 in the control group. 100 The experimental courses were taught by graduate students with no previous exposure to the Derive (Soft Warehouse, 1995) treatment. Mayes conducted frequent teaching seminars for the experimental instructors, “providing explicit training in using the CAS as a tool to teach from a constructivist perspective” (Mayes, 1997, p. 185). The study had several basic objectives. The first objective was to improve students’ understanding of function by means of multiple representations — graphical, symbolic, and tabular. Mayes (1995; 1997) conjectured that instruction with CAS would enhance students’ ability to explore algebra from these various perspectives in a more “connected” fashion. A second goal of the study was to enhance students’ problem solving abilities through the investigation of real-world applications of algebra. Lastly, Mayes was interested in collecting data regarding the impact of CAS on students’ attitudes towards mathematics. Data collections for the study consisted of the following: • A five item problem solving ability test — two problems involving inductive reasoning, two problems involving graphical analysis, and one application. A mathematics placement test was used as a pretest of algebraic manipulative skill. The course-wide algebra final exam was used as a post-test of algebraic manipulative skill. • A 38 item Likert-scale attitude inventory was administered to measure student feelings regarding five main topics: (1) the role of ability versus effort in mathematics class; (2) extrinsic versus intrinsic motivation; (3) thoughts regarding Derive (Soft Warehouse, 1995) in math class; (4) the importance of conceptual versus procedural understanding; (5) active and passive learning (p. 166) 101 A pair of analysis of variance tests indicated no significant difference between the experimental and control groups with regard to the algebra placement test or the finall exam. Using ANOVA, significant differences favoring the Derive group with regard to the problem solving test. Analysis of the attitude inventory was accomplished with a Wilcoxon Signed Ranks test run on each of the five total scores. As Mayes (1997) reports, “only one of the five categories was significantly different, the use of a CAS, and it was in the negative direction” (p. 186) This was attributed to the fact that “students felt overburdened byv using the CAS and covering the same amount of material as the control group” (p. 186). Overall, Mayes concluded that CAS was a useful tool to build problem solving skills in students. However, effective use of the technology suggests that some content shoulcS be eliminated — otherwise, students may feel too stressed to complete traditional course activities as well as building competence with the CAS. Studies of University Calculus students As noted in the introduction to this section, the majority of university research» regarding CAS have been .dissertations written by graduate students of Education- Unlike the studies highlighted above, which have investigated the impact of CAS on» high school students and algebra students, the majority of dissertation studies have: investigated the role of computer algebra in calculus classrooms. Due to the limitecS applicability of these studies to this project, I present their results in a table form. Thee interested reader is encouraged to read the Mayes article for a thorough description off 102 the calculus studies. Mayes (1997, p. 174) provides the reader with a useful table of CAS dissertations in the article “Current state of research into CAS in mathematics education.” I reproduce the table in its original form in Table 4.1. 103 Author, Year Math Area CAS Used How used? AMOR Galinda-Morales Calculus M athem atica LAB 0 0 1995 Padgett Calculus Maple LAB 1995 Porzio Calculus Mathematica LAB 1995 Keller Calculus Maple/Theorist LAB - f- 1994 Klein D.E. Mathematica DEMO 0 0 1994 A lexander Algebra Derive CAI 0 + + 1993 Coons Calculus Maple LAB 1993 Melin-Gonejeros Calculus Derive HW 0 0 0 + 1993 Park Calculus Mathematica LAB + 0 + 1993 Trout Algebra Toolkit DEMO/HW + 1993 Crocker Calculus Mathematica LAB + 1992 Cunningham Calculus True Basic LAB 0 1992 Smith Calculus Derive LAB/DEMO 0 0 1992 Schrock Calculus Maple LAB + 0 + 1990 Judson Calculus Maple HW 0 0 0 1988 Key: A=Student Attitude, M=Manipulation skills, C=Conceptual skills, P=Problem solving. A plus (+) denotes a significant gain with CAS use, (0) signifies no gain, (-) signifies a significant difference in favor of non-CAS. Table 4.1: Summary of CAS dissertation findings 104 Summary This project attempts to combine the strengths of individual CAS research in a single study. In this regard, the study proposed in the following chapters is unique. None of the studies which currently exist combine all of these elements into one study: • A long treatment period. This study examines student achievement over the course of an entire school year as opposed to a single academic quarter. • High school subjects. This study examines the introduction of CAS to algebra students during their initial exposure to basic symbolic manipulation, as opposed to remedial algebra students (Hillel, et al., 1992; Mayes, 1995) or calculus students (Judson, 1990; Palmiter, 1991; Porzio, 1995). • Daily use o f calculator in class. Many investigations have examined the use of CAS in a laboratory setting removed from the mathematics classroom (Porzio, 1995). Specific methodologies employed in this study are elucidated in Chapter 5 of this document. 105 CHAPTERS METHODOLOGY Determine that the thing can and shall be done, and then we shall find the way. Abraham Lincoln Introduction To assist in the introduction of this chapter, I provide a hypothetical discussion between myself and a student during the first days of this project. Student: Hey, Mr. Edwards. You’re here after school again'} Mr. Edwards: Yeah. Pretty crazy, huh! Student: I’ll say. What is it that you type here each day, anyway? Mr. Edwards: Well, do you remember how I told class about the study I’m doing here at the school? Student: Oh yeah. Something about calculators? Mr. Edwards: Yeah. I want to see if calculators help people learn algebra. Student: That’s why you made us take all those tests? Mr. Edwards: Ha! Well, that’s part of it. I want to use those test scores to find out whether or not calculators affect student learning. Student: So we’re gonna be a bunch of test scores in some fancy report? 106 Mr. Edwards: Well, yes, I suppose. But you and your classmates represent more than a collection of test scores to me. I don’t think scores tell the whole story of you and our class. Student: Huh? Mr. Edwards: It’s like this. Have you ever taken a test and felt like you should have done better? Like your test score didn’t really describe how much you understood? Student: Yeah. Like on Ms. Smith’s history exam. Man, was that tough! Mr. Edwards: Okay. It’s the same way with my study. I feel that test scores paint a very narrow picture of your class as students. Therefore, I’m including a lot of other data in my report — student reflections about technology, calculator screenshots, even excerpts from a journal that I write in each day. This data helps me explain the entire story of my project to people who can’t be here to see it for themselves. Student: Wow. Sounds like a lot of work! Mr. Edwards: It is. But it’s worth it, because the data I collect will help strengthen the conclusions I reach at the end of my study. My research may help influence decisions of other teachers and even policy-makers. Student: That’s actually pretty cool, Mr. Edwards! It sounds like you’re making a difference. Mr. Edwards: I think I am! Student: So what are you typing now? Mr. Edwards: Right now I’m working on the methodology chapter of my dissertation. It describes what we just talked about — the methods of data collection used in my study as well as a rationale for the techniques I’ve chosen. In the third chapter of this manuscript, a theoretical basis for the study of computer symbolic algebra in secondary school settings was proposed. In particular, the role of 107 the calculator as an “expert other” was discussed in terms of Vygotskian scaffolding and ZPD (Jones & Thornton, 1993; Taylor, 1993). Specific classroom teaching strategies - such as Buchberger’s (1990) White Box/Black Box methodologies were elucidated within this context. Chapter four summarized current CAS research in education, providing the reader with a sense of the existing literature in this area. Secondary and university studies of various computer algebra systems — Mathematica (Wolfram Research, 1997), Derive (Soft Warehouse, 1995), Maple (Waterloo Maple, 2000) — were discussed with emphasis placed on important themes emerging from these studies. Citing a lack of research of computer algebra tools at the secondary level in the United States, a rationale was provided for this project. In a sense, this chapter combines ideas from both Chapters 3 and 4 while describing data collection techniques of this study. For instance, the underlying philosophies summarized in Chapter 3 guided the manner in which data was collected for this report. Likewise, the summary of existing research from Chapter 4 shed light on gaps present in the current literature, while providing research questions to pursue in this study. The main ideas addressed in this chapter include: • The setting of this project • The subjects involved in the study • The types of data collected • A rationale for a mixed-methodological study 108 • Limitations of this investigation First, the reader is provided with rich a description of Midvale* High school — its students, its teachers, and the community of which it is a part. Such details provide a context for this study and help remind the reader of the principal focus of this document — people using technology in the teaching and learning o f mathematics (with special attention paid to ways in which technology impacts the ability of students to perform mathematically). Next, class activities — course content, textbooks, methods of assessment — are discussed along with the pedagogical terrain of classes — presentation methods utilized, educational and professional background of teachers, and instructional climate. Such details enable the reader to better understand the conditions present at the time that symbolic algebra utilities were first introduced at Midvale High school. Lastly, specifics of data collection and analysis are described — instrumentation, data collection techniques, and data analysis issues are elaborated. A rationale for a mixed qualitative/quantitative methodology is provided as the most comprehensive means of addressing the questions initially posed at the outset of this study. It is my hope that individuals conducting further research regarding computer algebra will use the information provided in this chapter to better inform their own work. A principal goal of this project is the furtherance of research in an area of critical 1 A pseudonym. 109 importance to the future of school mathematics - the responsible use of computer algebra tools with novice algebra students at the secondary school level. Description of Setting The School and Community The setting for this study was Midvale High school — a Midwestern, public high school with a student population of approximately 1,600. Located in an affluent Ohio suburb, Midvale High school is minutes away from both the state capitol and one of the largest land-grant research universities in the country. The convenient location of the school made it an ideal site from which to conduct educational research. For instance, the school is within a ten minute drive of an ERIC center and the Eisenhower National Clearinghouse for School Science and Mathematics. Through partnerships with neighboring colleges, the high school has played an active role in a variety of mathematics education studies, including several involving technology and mathematics education. For instance, the first CASIO graphing calculator — the fx-7000G — was field tested at the school in the mid-eighties. In addition, several technology-based textbooks were piloted at the school. Throughout this study, the administrators at Midvale welcomed interaction between university researchers and classroom teachers — as a means of enhancing teacher morale and as a way of encouraging professional development of the faculty. Throughout the study, I enjoyed considerable community backing for the calculator project. 110 In general terms, students at Midvale High school were well-motivated with regard to the study of mathematics and the sciences. As evidenced in Table 5.1, passage rates on statewide mathematics proficiency exams were among the highest in the region. Approximately ninety percent of seniors went on to pursue some type of post-secondary education during the year of data collection. As noted in the Midvale Schools 1996- 1997 Annual report, “Midvale student scores on the ACT and SAT college entrance exams typically exceed the national averages; SAT scores typically are in the top 5 percent in the U.S.” (Upper Arlington City Schools, 1997, p. 3). Subj ect District Average % State Average % Citizenship 94 . 4 77.3 Mathematics 90. 9 63. 6 Reading 97. 8 85.2 Writing 95.1 80.1 Science N/A N/A (Upper Arlington City Schools, 1997) Table 5.1: Comparison of statewide 9th grade proficiency test score averages with scores of students at Midvale High Background of Students The subjects of this study were students enrolled in one of two year-long, advanced algebra courses offered at Midvale High school — both classes taught by myself (my dual role as teacher and researcher is discussed later in the section Limitations). The students, typically sophomores or juniors, had completed a year-long introductory 111 algebra course and a year-long geometry course during the previous two academic years. Because students at the school were not required to purchase graphing calculators prior to enrollment in the advanced algebra course, their calculator skills varied considerably at the beginning of the academic year. In writing prompts assigned to students throughout the year, many indicated that they had never used a graphing calculator before their involvement in the study. Others had used an older sibling’s calculator to download games from the internet — but were largely unaware of the machine’s powerful computational capabilities. On the other hand, a handful of students were able to graph functions, find intercepts and plot data on the first days of class. None of the students indicated that they had used computer symbolic algebra utilities such as those found on the TI-92 or TI-89 in their previous mathematics courses. This diversity posed both challenges and opportunities for me as a classroom teacher and researcher — for instance, I wondered how I would approach the varied technological backgrounds of my students. Because a primary directive of this study was to measure technology’s effect upon students’ mathematical performance levels, precautions were taken account for possible discrepancies in student backgrounds regarding both initial mathematics knowledge and overall calculator skills. These precautions are discussed in the section Data Collection of this chapter. 1 1 2 The Curriculum As previously stated, Midvale High school enjoyed a strong academic tradition during the time that data was collected for this study. This tradition benefited me and my students in many ways — it helped me maintain rigorous academic standards throughout the school year; it motivated students to work diligently throughout the study; and it provided strong parental support for the program throughout the academic year. During my tenure at Midvale High School, the mathematics department saw a need to move away from the “pencil-and-paper” approach which dominated the school curriculum for countless years. The change in philosophy was fueled in part by recent policy changes armoimced by College Board examiners. The 1998-99 calculator policies for College Board math/science tests that allow or require a calculator (PSAT/NMSQT, SAT I, SAT II Math Level IC & Level IIC, AP Calculus, AP Statistics, AP Chemistry, AP Physics) will not change—therefore the Casio CFX-9970, TI-73, and TI-89 will be permitted for use on those tests since graphing calculators without QWERTY (typewriter-like) keyboards are allowed. The Casio CFX- 9970, TI-73, and TI-89 have been added to the list of approved graphing calculators for AP Calculus (College Board, 1998). A healthy step towards reform was the adoption of University o f Chicago School Mathematics Project (UCSMP) materials for grades 7 through 11. The UCSMP texts incorporate technology into lessons in meaningful ways. In the words of the UCSMP authors: State-of-the-art technology enhances mathematical understanding and strengthens problem solving skills. Applications using calculators, graphics calculators, and computers are incorporated throughout the text (Usiskin, et al., 1990, p. T9). 113 Several years prior to the data collection phase of this project, the mathematics department purchased classroom sets of TI-92 and TI-83 graphing calculators. Unfortunately, during my time at the school, the power of these machines was not exploited in a systematic fashion. For instance, no department-wide policy regarding the appropriate use of technology in individual courses existed at the time of the study. Decisions regarding the use of technology were made exclusively on a teacher-by- teacher basis. As this study began, teachers at the school did not share a common vision regarding the role of technology in mathematics classrooms. Some teachers considered calculators as powerful tools for teaching mathematics, while others referred to the devices as “toys in the classroom” (Teacher Journal, October 20). Needless to say, an important aim of this study was the examination of the effects of calculator symbolic algebra on student and teacher beliefs. As this study began, I found myself asking the following questions. Q l. What sorts of problems and activities will students investigate using hand-held technology? Which problems are well-suited for investigation with CAS? Which (if any) problems are not well-suited for investigation with CAS? Q2. What attitudes do intermediate algebra students have regarding graphing calculators and symbolic manipulation utilities? Do their attitudes change as they gain experience with the utilities? How do these attitudes differ among CAS and non-CAS students? Answers to such questions are addressed in Chapter 7, Qualitative Data Analysis. Despite any shortcomings of the study site (e.g. reluctance to incorporate technology into mathematics classes in a systematic fashion), Midvale High School afforded me 114 significant flexibility with regard to the teaching of course content. Although “scope and sequence” guides were adopted for each mathematics course offered by the school, individual teachers were allowed to supplement (or even replace) text materials with outside resources as they saw fit. Although mathematics faculty members were required to administer departmental semester and final exams, most math teachers wrote their own quizzes and tests, using the UCSMP materials as guides. Many composed their own homework assignments for students. A creative atmosphere pervaded many classrooms and teacher work areas throughout the building. Pedagogical Features of Classroom At first glance, the physical setting of the study appears rather traditional. Throughout the school year, desks were arranged in rows with my desk situated at the front of the class. Figure 5.1 illustrates the classroom used in this study. 115 c h a lk b o a rd § Teacher Desk doorway 5 overhead projector § student desks "Oc ^9 ) JCO) c o o I ■ D chalkboard chalkboard Figure 5.1: Physical features of Room 202 of Midvale High school The physical appearance of the classroom fails to indicate non-traditional aspects of the classrooms involved in the study. In the paragraphs below, some of the instructional features that set both classes apart from more traditionally taught classes are described. Teacher Background. At the beginning of the study, I had completed four years of teaching at the study site — having taught geometry two of these years, precalculus courses for two years, computer science courses for three years, and advanced algebra for two years. My teaching experience extended beyond Midvale High school. I began my teaching career as a graduate teaching associate at Ohio University in Athens, Ohio. 116 During this time, I taught three years of remedial algebra courses while earning two master’s degrees — one in pure mathematics and another in secondary mathematics education. Prior to the study, 1 had taught algebra in a variety of settings — including a state prison — and developed a blend of different approaches for helping students leam algebraic content. Needless to say, my experiences had a profound influence on the teaching methods 1 employed during this study. Lecture-based instruction. Although much course content was delivered to students via an overhead projector, chalkboard, and worksheets — 1 feel that this surface description fails to capture what was unique about my teaching during the data collection phase of this study. In each class, emphasis was placed on the social aspects of learning mathematics. During the study, this process was fostered as students worked collaboratively on many warmup problems and small group projects. Course topics were presented to all groups using a mixture of instructional approaches. Certainly some class sessions were taught in a traditional fashion, in a manner described by Loveless (1998). At the beginning of class, I instruct students to get out their homework from the previous night. A transparency with solutions is placed on an overhead projector as I walk around the room, checking individual students’ homework. This is typically a quick glance, since there are so many students in the room and only 50 minutes of class time. After all the homework is checked, 1 move to the overhead and ask students if they have any questions. If the students have questions, 1 answer them on the overhead with the assistance of student volunteers. The interaction occurring in class is directed by me. If students have no questions, 1 provide a lecture regarding new material while the students sit in their seats. Occasionally the lecture involves students taking turns reading aloud from the text or from handouts I’ve designed. Such 117 activity typically lasts about twenty minutes. At the end of the class, a homework assignment covering the new lecture material is handed out to the students. I give them the time that remains to begin their homework (personal conversation, 1998). Certainly the above passage does not describe everyday interactions that took place in my classes during the study — although it does describe one of many strategies that I used to reach students. I employed other teaching methods in the classroom with equal frequency. Alternative teaching methods are described below. Investigation Worksheets / Warm Up Problems. I found in-class investigation worksheets particularly useful for encouraging student discovery with the calculator. During approximately the first half hour of class, students worked in pairs answering questions which encouraged them to generalize algebraic patterns that they discovered. Often, the patterns would be difficult for students to discover on their own — particularly in 30 minutes. Using calculators frequently in a “black box” style (Buchberger, 1990), questions on the worksheets helped lead students in the right direction — modeling Vygotskian notions of scaffolding (Jones & Thornton, 1993; Taylor, 1993) discussed in Chapter 3 of this document. During the last 15 or so minutes of the period, class typically reconvened — with my providing closure for that day’s activities. Such a design promoted social interaction, group learning, and active participation of all students. Presentation Problems. Typically as class approached the end of a particular topic, I compiled a list of 10 or so challenging problems for students to present in front of class. 118 The problems tended to be more difficult than those found in the student text, requiring students to synthesize information from their textbooks in novel ways. Often students chose a partner and presented problems collaboratively their classmates. Students were given one or two class periods to work on their group’s problem. Then we spent one or two more days observing class presentations. Ultimately, students were responsible for all problems that their peers presented. In this way, students were compelled to ask questions of their classmates, present problems clearly, and work cooperatively. Such activities afforded students an opportunity to share mathematical ideas with individual class members and refine their technical presentation skills. Furthermore, the activities helped students gain an appreciation of how difficult teaching mathematics can be. Design Features of Studv In an attempt to answer questions posed in Chapter 1 of this document, a combination of qualitative and quantitative methods were utilized. In this study, data obtained from two separate groups of advanced algebra students were examined. Characteristics of the two student groups are described in greater detail below. Because a major goal of this report was to document changes that occured in secondary mathematics classrooms when CAS utilities were introduced, I found it useful to compare various aspects of a class taught without CAS-equipped calculators witli features of a CAS-enhanced class. In this study, I refer to the group taught without CAS-based utilities as the non-CAS group. Likewise, I refer to the group taught with 119 symbolic manipulation utilities as the CAS group. Although the terms “experimental” and “control” are used at times to describe the non-CAS and CAS groups, I recognize that the study is not a true experiment. Unlike a scientific laboratory, the classes investigated contained many uncontrolled variables (see section Rationale for Mixed Methodology Study for a more detailed discussion). Nevertheless, because I wished to investigate change — it was worthwhile to examine classes as they existed prior to the use of symbolic manipulators (the non-CAS group) and with symbolic manipulation utilities (the CAS group). In a sense, the classes involved in this investigation did retain some of the features of a classic pretest/posttest study in mathematics education. For instance, a number of variables remained fixed. Both groups shared the same instructor (i.e. myself), attended the same school, used the same textbook. Furthermore, the existence of identical departmental midterm and final examinations for both groups necessitated coverage of the same general course topics. Functions Matrices Systems of Equations Logarithms Inverse Functions Trigonometry Polynomials Sequences and Series However, the introduction of symbolic algebra utilities influenced some of the teaching methods I employed. I acknowledge that technology altered classroom practice, and 120 rather than ignore this fact, I sought to investigate these changes in a well-organized, thoughtful, and rigorous fashion. Calculator accessibility. Graphing calculator use was encouraged in both the CAS and non-CAS classes. However, devices equipped with symbolic algebra utilities were not utilized in any systematic fashion in the non-CAS group. This is not to say that the control group was denied access to technology of all forms. Calculators with graphing utilities were required (and used daily) with all students in both classes. However, only the students in the CAS group were taught algebraic concepts using symbolic manipulation utilities. Course Content. I was interested in gathering preliminary data regarding the effectiveness of simply “adding on" the use of symbolic manipulators into an existing high school classroom. Because such a strategy is likely to be employed by many teachers and curriculum developers, I aimed to discover if such a plan was ill-advised. Data were collected to examine whether or not CAS utilities enable students to grasp algebraic concepts more readily without major shifts in emphases from the existing curriculum. Before the study, it was unclear if a resequencing (or deletion) of certain topics would better facilitate CAS usage in secondary mathematics classrooms. The latter approach is supported by findings in previous studies (Heid, 1988; Judson, 1990). Although significant curriculum resequencing (and content deletion) was not employed in this study, results of the investigation may motivate further secondary school research such an area. 121 Data Collection Because the effects of the introduction of symbolic manipulators were largely unknown at the secondary level at the time of this study, data from a variety of sources — both qualitative and quantitative — were collected. A rationale for a mixed- methodology is discussed later in this chapter. The data collection methods included: • Statistical pretest/posttest and Multivariate analysis of variance (MANOVA) designs to measure the effect of CAS on student mathematical performance. • Student surveys and sample writings to measure student attitudes regarding the use of symbolic manipulators in algebra classes throughout the school year. An in-depth teacher journal to document possible promises and pitfalls of symbolic algebra utilities in secondary school classrooms. • A year-long appraisal of student problem solving methods as observed on class tests and quizzes. Table 5.2 illustrates the principal research questions addressed in this study together with corresponding data collection techniques. As the table indicates, this study included a year-long examination of advanced algebra instruction. Because it took some time for students to feel comfortable with a new calculator (particularly those that had never used a graphing calculator of any kind before), a study of any shorter duration was ill-advised. I describe each data collection instrument in greater detail below. Descriptions center around research questions originally proposed in Chapter 1 of this document. 122 Research Questions Data Collection Methods Timeline How does CAS affect Pretest/Posttest of algebraic Fall 1999, Spring 2000 overall student algebraic knowledge understanding? How does CAS affect Comparison of group Entire school year student algebraic performance levels on 1999-2000 understanding in chapter tests and quizzes specific content areas? How do students use CAS Analysis of student work on Entire school year utilities? various algebra assessments 1999-2000 (e.g. tests, quizzes, collected work) What are some factors Likert-style attitudinal surveys Twice during school year that contribute to this (Fall and Spring) CAS use? Analysis of teacher journal Year long journal Analysis of student essays Approximately once and writing assignments each quarter (each ten weeks) How does CAS impact Analysis of teacher journal Year long journal course content in secondary school math classrooms? Analysis of class handouts, Entire school year (1999- class activities, calendars, and 2000) homework Table 5.2: Research questions and data collection methods Measurement of algebraic understanding Algebraic Skills Pretest/Technology Literacy Pretest. At the beginning of the school year, two pretests were administered to all students involved in the study (copies are available in Appendix B of this document). The tests consisted of the following: An Algebra Skills Pretest consisting of 24 final exam items designed by the authors of UCSMP Algebra (Usiskin, et al., 1990). 123 • A Technology Literacy Pretest consisting of 15 questions designed to measure students’ prior knowledge of Texas Instruments graphing utilities. The Algebraic Skills Pretest component measured student proficiency in seven major content strands: 1. Number sense 2. Algebra 3. Geometry 4. Measurement 5. Logic and Reasoning 6. Statistics/Data analysis 7. Patterns and Functions (Usiskin, et ah, 1990). The questions were taken directly from the UCSMP Algebra Assessment Sourcebook (Usiskin, et al., 1990) which was designed to accompany students’ Algebra I text. Because these items were taken from materials from a previous course (namely Algebra I), it was hoped that none of the questions from the Algebraic Skills Pretest would appear alien to students. Unlike the UCSMP questions, the Technology Literacy Pretest questions were designed to be less familiar to students. The Technology Literacy Pretest questions were broadly arranged into two basic lyqjes: (1) Non-CAS calculator questions and (2) CAS (TI-89/92) specific questions. It was anticipated that students with little experience using hand-held solvers would experience difficulty with these problems. A copy of the Technology Literacy Pretest is provided in Appendix B of this document. Both the Algebraic Skills Pretest and the Technology Literacy Pretest instruments were designed to measure baseline student knowledge as they began advanced algebra 124 instruction. Both pretests were administered to students exactly once — at the beginning of the school year, before any fom^al training with symbolic algebra utilities began and before the introduction of any new course content. To simplify scoring, all items on both tests were graded as either “correct” or “incorrect”, with one point awarded for correct solutions and zero points awarded for incorrect solutions. A multivariate analysis of variance (MANOVA) was used to determine whether to reject the following null hypotheses. HI. The mean scores of the CAS and non-CAS groups on the Algebraic Skills Pretest was not significantly different (with significance measured at a .05 alpha level). H2. The mean scores of the CAS and non-CAS groups on the Technology Literacy Pretest was not significantly different (with significance measured at a .05 alpha level). Student Attimdinal Survev. A 4-point Likert-type attitudinal scale (agree strongly = 1; agree = 2; disagree = 3; disagree strongly = 4) was administered to all students both at the beginning of the study and at the end of the school year (a copy is available in Appendix B of this document). The survey contained 16 items, with each item appearing exactly twice on the test — once as a “negatively-worded” item and once as a “positively-worded” item, for a total of 32 survey questions. The scale avoided space error by placing the statements in random order, with “negative” and “positive” items appearing randomly throughout the test. Items on the questionnaire were designed to measure student attitude regarding the usé of calculators in mathematics class. The items addressed five different subtopics: 125 Calculator as authority General calculator attitude Building concepts with calculators Building skills with calculators Learning mathematics with calculators Below, in Figures 5.2 through 5.6, calculator-specific item pairs are presented. Positive items are denoted with a (P). Negative items are denoted with an (N). The number to the left of the statement denotes its relative position on the questionnaire. Attitude 22. Graphing calculators 5. Graphing calculators don't increase my desire to increase my desire to do do mathematics. (N) mathematics. (P) 7. Graphing calculators do 23. Graphing calculators not help me on exams. (N) help me when working on exam s. (P) Figure 5.2: Attitude-specific items from Student Attitudinal Survey Authority 26. Even when I type in 11. When the data I type in correct data, there is a is correct, then I can fully possibility that the calculator trust the graphing calculator's will give me an incorrect output. (P) answer. (N) 31. The graphing calculator 16. The graphing calculator isn't useful for checking the is a useful tool to check my result of a calculation. (N) work. (P) 3. If my calculator gives me 20. If my calculator gives me no answer, then the problem an answer, then I reexamine has no solution. (N) the solution without a calculator. (P) Figure 5.3: Authority-specific tems from Student Attitudinal Survey 126 Building 21. Graphing calculators aren't 4. Graphing calculators are C oncepts useful for discovering algebraic a useful support for discovering rules. (N) algebraic rules. (P) 24. Graphing calculators don't 8. Graphing calculators help help me understand me understand mathematics. mathematics. (N) (P) 27. The graphing calculator 12. The graphing calculator really doesn't help me to get helps me to get an idea of the an idea of the result of a result of a calculation before calculation before I do it. (N) doing it. (P) Figure 5.4: Concept-specific items from Student Attitudinal Survey Building 2. When using a graphing 19. When using the graphing Skills calculator in class, I do not calculator in class, I still have need to learn how to to know how to compute compute because the because the calculator won't calculator does everything do everything for me. (P) for me. (N) 6 . 1 get lost doing 13. The graphing calculator calculations on the helps me solve problems graphing calculator. (N) without getting lost in calculations. (P) 17.1 am dependent on 32. I am not dependent on the graphing calculator the graphing calculator for for doing simple arithmetic doing simple arithmetic (such as adding two (such as adding two single digit numbers single digit numbers together). (N) together). (P) 30. The graphing calculator 15. The graphing calculator doesn't help with lengthy especially helps with lengthy and boring calculations. (N) and boring calculations. (P) Figure 5.5: Skill specific items from Student Attitudinal Survey 127 Learning 1. Graphing calculators are 18. Graphing calculators are Mathematics complicated and don't help easy to use and help me in with a me in learning mathematics. learning mathematics. (P) Calculator (N) 28. Calculations are no easier 10. Calculations are easier with a calculator. (N) with a graphing calculator. (P) 25. People who have difficulties 9. Graphing calculators help with algebra have the sam e people who have difficulties difficulties - even with a with algebra still be able to calculator. (N) do mathematics. (P) 29. The graphing calculator is 14. The graphing calculator is confusing because there is useful because it allows me to more than one way to look at look at the same problem in every problem (for example, more than one way (for example, as a graph or symbolically). (N) as a graph or symbolically). (P) Figure 5.6: Items from Student Attitudinal Survey dealing specifically with learning/doing mathematics with a calculator The content validity of the questionnaire was evaluated by a panel of three faculty members from Ohio State University and one teaching colleague from Midvale High School. The faculty members — two from the College of Education and one a Mathematics professor emeritus — have extensive experience using technology in secondary school settings. For instance, one of the faculty members was a co-founder of the Texas Instruments’ Teachers Teaching with Technology (i.e. T-cubed) — an international program which provides classroom teachers with instruction regarding the use of graphing calculators with school students. Another faculty member coordinated a large study of secondary school students’ interactions with desktop computers — the Apple Classroom o f Tomorrow (ACOT) program — as a program planner and software developer. In addition, the classroom teacher and third faculty member had previous 128 experience facilitating technology presentations at international conferences in China and the United Kingdom. The members of the panel were asked to review all items on the questionnaire. Each member provided suggestions for improving the clarity of particular test items. Several items were removed from the test because they were deemed unclear or redundant. Others were reworded or placed in different positions on the test. Ultimately, the panel agreed that the items were appropriate for measuring student opinion of mathematics and the role of calculators in mathematics teaching and learning. Before analyzing the test results, student responses to positive items were reverse scaled (so that a higher responses indicated a more positive attitude). Internal consistency estimates of reliability were computed for the attitudinal survey using Cronbach’s Coefficient Alpha with SPSS version 10 for Windows (SPSS Inc., 1999). The results of the Cronbach test for both Fall and Spring are shown in Table 5.3. Cronbach' s Group n Alpha Caiculat:or Items (FALL) 45 0.7 3 62 Calculator Items (SPRING) 44 0.8199 Table 5.3: Reliability coefficient of instrument: Student Attitudinal Survey As Carson notes: The widely-accepted social science cut-off is that alpha should be .70 or higher for a set of items to be considered a scale, but some 129 use .75 or .80. That .70 is as low as one should go is reflected in the fact that when alpha is .70, the standard error of measurement will be over half (0.55) a standard deviation (Garson, 2000). The high level of reliability of the calculator items on the survey are reflected in the agreement of “negatively stated” and “positively stated” test items in both the Fall and Spring administrations of the test. Fall questionnaire results are reported in Appendices C and D of this document. Spring results are reported in Appendices G and F. Agreement between “negativelv stated” and “positivelv stated” Fall test items. On the Fall test, group means for “negatively stated” items tended to agree with responses for their “positively stated” counterparts across groups. In other words, if the mean response to a “positively stated” item indicated student agreement toward the item, then student responses also indicated student disagreement with the item’s “negatively stated” counterpart. Exceptions to this were mean responses of the non-CAS group on item pair 11/26. The non-CAS group generally agreed with item 11 from the questionnaire — “When the data I type in is correct, then I can fully trust the graphing calculator’s output” received a mean response of 2.7273. However non-CAS students also tended to agree with item 26, its negatively stated counterpart — “Even when I type in correct data, there is a possibility that the calculator will give me an incorrect answer” received a mean response of 2.2727. Ideally, the mean response of items 11 and 26 would be closer together, with both greater than 2.5. 130 As was the case with the non-CAS group, CAS student responses to “negatively stated” and “positively stated” item pairs tended to agree with one another. Exceptions to this were the mean responses of the CAS group on item pairs 4/21 and 9/25. The CAS group generally agreed with item 4 from the questionnaire — “Graphing calculators are a useful support for discovering mathematical rules” received a mean response of 3.1153. However they also tended to agree with item 21, its negatively stated counterpart — “Graphing calculators aren’t useful for discovering mathematical rules” received a mean response of 2.360. Similarly, the CAS group generally agreed with item 9 from the questionnaire — “Graphing calculators help people who have difficulties with algebra to still be able to do mathematics.” received a mean response of 2.8846. However, item 25, its negatively stated counterpart — “People who have difficulties with algebra have the same difficulties — even with a calculator” received a mean response of 2.3846, also denoting agreement. Agreement between “negativelv stated” and “positivelv stated” Spring test items. As was the case for student responses on the Student Attitudinal Sw-vey in the Fall, group means for “negatively stated” items tended to agree with responses for their “positively stated” counterparts across groups in the Spring. In other words, if the mean response to a “positively stated” item indicated student agreement with the item, tlren student responses tended to indicate student disagreement with its “negatively stated” counterpart. Exceptions to this were mean responses of the non-CAS group on item 131 pairs 9/25 and 11/26. Neither of these pairs were a non-match on the Fall administration of the survey. The non-CAS group agreed with item 9 from the questionnaire - “Graphing calculators help people who have difficulties with algebra to still be able to do mathematics” received a mean response of 2.9545. However non-CAS students also tended to agree with item 25, its negatively stated counterpart — “People who have difficulties with algebra have the same difficulties — even with a calculator” received a mean response of 2.3182 (recall that negatively stated items were not reverse scaled). Ideally, the mean response of items 9 and 25 would be closer together, with both greater than 2.5. Similarly, the non-CAS group generally agreed with item 11 from the questionnaire — “When the data 1 type in is correct, then I can fully trust the graphing calculator’s output” received a mean response of 2.9091. However they also tended to agree with item 26, its negatively stated counterpart - “Even when 1 type in correct data, there is a possibility that the calculator will give me an incorrect answer” received a mean response of 1.8636 (recall that negatively stated items were not reverse scaled). As was the case with the non-CAS group, CAS student responses to “negatively stated” and “positively stated” item pairs tended to agree with one another. Exceptions to this were the mean responses of the CAS group on item pair 9/25. The CAS group agreed with item 9 from the questionnaire — “Graphing calculators help people who have difficulties with algebra to still be able to do mathematics” received a mean response of 3.0909. However non-CAS students also tended to agree with item 25, its 132 negatively stated counterpart — “People who have difficulties with algebra have the same difficulties — even with a calculator” received a mean response of 2.2727 (recall that negatively stated items were not reverse scaled). The null hypotheses that the survey investigated included the following. H3. Before the study, there was not a significant difference, at the 0.05 level, between the general attitude of CAS students towards calculator use and the general attitude of non-CAS students towards calculator use. H4. After the study, there was not a significant difference, at the 0.05 level, between the general attitude of CAS students towards calculator use and the general attitude of non-CAS students towards calculator use. H5. The change in attitude of CAS students regarding calculator use from Fall to Spring was not significant at the 0.05 level of significance. H6. The change in attitude of non-CAS students regarding calculator use from Fall to Spring was not significant at the 0.05 level of significance. These hypotheses are investigated in Chapter 6 of this document. Final Examination. At the end of the school year, identical 50-item multiple choice exams were administered to all students in both the CAS and non-CAS groups. The exam was designed by the mathematics department at Midvale High School and was administered to all students enrolled in advanced algebra (A copy of the exam is provided in Appendix B of this dissertation). To simplify grading, and to ensure scoring fairness, all items were graded as either “correct” or “incorrect” — with correct student responses receiving one point and incorrect responses receiving zero points. In this document, student scores are reported as the percentage of items answered correctly. 133 Table 5.4 provides the reader with a better understanding of the types of problems encountered on the exam. 134 TOPIC ITEM(S) TOPIC ITEM(S) Quadratic Forms Exponents/Radicals Vertex Form 41 Ordering 8, 10, 39 Discriminants 47,48 Equivalent Forms 9, 19, 20, 37, 38 Applications 50 Applications 25, 33, 40 Solving Equations Trigonometry Exponential 17 Simplifying 12, 13 Logarithmic 21, 22 Properties 14, 16, 34, 45, 46 Quadratic 27 Solving Triangles 15. 35 Radical 18 Applications 11 Logarithms Complex Numbers Equivalent Forms 23 Equivalent Forms 28, 29, 32 Properties 36. 42, 43 Roots 30 Graphs Sequences Finding equations 6, 26, 49 Identifying type 7 Functions Composition 1.2 Definition 4 Domain/Range 5, 24 Inverses 3 .3 1 .4 4 Table 5.4: Items on Advanced Algebra Final Examination organized by topic Table 5.4 highlights nine core topics covered by the Advanced Algebra Final Examination — functions, solving equations, graphical understanding, sequences, logarithms, exponents and radicals, trigonometry, complex numbers, and quadratic forms. Item numbers corresponding to questions on the test appear underneath each core topic. Figure 5.7 illustrates the relative frequency with which various core topics appeared on the Advanced Algebra Final Examination. 135 Distribution of core topics on Advanced Algebra end-of-year examination Graphical Sequences 2% Understanding^ — ______ Logarithms Trigonometry Complex Æ Numbers Quadratic Forms Exponents/Radicals Equation Solving Functions Figure 5.7: Distribution of core topics on Advanced Algebra Final Examination Note that the core topic “Exponents / Radicals” accounted for 22 percent of all test items — the most of any topic. Trigonometry items accounted for 20 percent of all test questions. Figures 5.8 and 5.9 provide the reader with a better sense of typical questions that appeared on the Advanced Algebra Final Examination. Figure 5-8 illustrates two examples of “Functions” test items. Figure 5-9 illustrates a “Logarithm” test item. In problems 1 and 2, use the functions f(x) = 2x^- 3x - 5 and g(x) = 3x - 2. 1. Find f(g(3)) (a) 72 (b)4 (c)10 (d) 170 2. Find f(g(x)) (a) 18x2- 9x + 9 (b) 6x2- 9x - 17 (c) 18x2- 33x + 9 (d) 6x2- . j Figure 5.8: Items I and 2 from the Advanced Algebra Final Examination 136 43. Suppose I want to graph y = logg(x) on my graphing calculator. Which of the following expressions would allow me to do this? = (c)y = log(|.) (d)y=Jl|a (e) none Figure 5.9: Item 43 from t\iQ Advanced Algebra Final Examination The data collected from the Advanced Algebra Final Examination provided information regarding the effects of hand-held CAS devices on student ability to solve problems typically encountered in a second-year algebra class. The administration of a combined Algebraic Skills Pretest and Technology Literacy Pretest together with a common final examination formed the basis of a quasi-experimental design implemented in this study. o X CAS O o X Non-CAS O Figure 5.10: Schemata of Quasi-Experimental Design An analysis of covariance (ANCOVA), using the General Linear Model, was used to determine, at the 0.05 significance (alpha) level, whether to reject the following null hypothesis. H7. There was no significant difference in Advanced Algebra Final Examination scores between CAS and non-CAS groups. 137 The Algebraic Skills Pretest and Technology Literacy Pretest scores were used as separate covariates to strengthen the power of the statistical analysis, by reducing within-group (error) variance. As previously noted, the CAS and non-CAS groups shared many similarities. For instance, each group was taught by the same instructor at the same school. Furthermore, each used the same textbook, was expected to possess the same content knowledge at the end of the school year (as mandated by the school’s Scope and Sequence objectives), and took identical semester and final examinations. The groups’ primary differences were directly related to the use of symbolic manipulation utilities in their classes. By examining two similar groups of students with a non-equivalent control group design, concerns regarding a lack of internal validity were greatly reduced. As Stanley and Campbell (Campbell, Stanley, & Gage, 1963) note: The more similar the experimental and the control groups are in their recruitment, and the more this similarity is confirmed by the scores on the pretest, the more effective this control becomes. Assuming that these desiderata are approximated for purposes of internal validity, we can regard the design as controlling the main effects of history, maturation, testing, and instrumentation, in that the difference for the experimental group (if greater than that for the control group) cannot be explained by main effects of these variables (p. 48). Hence, assuming that both groups were approximately equal as measured by pretest scores, major sources of internal validity were checked. Nevertheless, threats to internal validity remained and are discussed in the Limitations section at the end of this chapter. 138 To further study performance levels of students on the year-end Advanced Algebra Final Examination, students from each class were split into three distinct groups based on their initial performance on the pretests. • Low performing students — those whose combined pretest scores fall below the lower one-third quantile of their class • Middle performing students — those whose combined pretest scores fall above the lower one-third quantile and below the upper one-third quantile of their class • High performing students — those whose combined pretest scores fall above the upper one-third quantile of their class Using these groupings, the following hypotheses dealing with student performance on the Advanced Algebra Final Examination were tested. H8. The difference in performance on the Advanced Algebra Final Examination of low-performing CAS students and a similar group of low-performing non-CAS students was not statistically significant. H9. The difference in performance on the Advanced Algebra Final Examination of middle-performing CAS students and a similar group of middle-performing non- CAS students was not statistically significant. HIO. The difference in performance on the Advanced Algebra Final Examination of high-performing CAS students and a similar group of high-performing non-CAS students was not statistically significant. Measurement of student CAS usage Examination of student test and quiz items. Typically after students complete the study of a chapter in UCSMP Advanced Algebra (Usiskin, et al., 1990), teachers at Midvale administer a chapter exam covering material from the unit. My classes were 139 no different in this regard. The tests that I administered to my students were constructed from a variety of sources. • Homework questions from UCSMP Advanced Algebra (Usiskin, et al., 1990) text • Examples provided by students in class discussions • Problems from UCSMP Assessment Sourcebook (Usiskin, et al., 1990) • Alternate reform publications (e.g. textbooks, journal articles) • Questions motivated by discussions with teaching cohorts On each chapter test, one or two items were included that allowed students significant latitude in terms of problem-solving technique. Solution strategies available to students included: Graphical techniques (with calculator or by hand) Symbolic manipulation (with calculator or by-hand) Tabular representations Hand-drawn pictures or diagrams A combination of the above strategies Coding of student work. Student work on test items was collected from all smdents during the entire school year. Items were coded based on the dominant solution strategy(ies) employed by each student: (a) Graphical (calculator or by-hand); (b) By hand (calculator or by-hand); (c) Tabular; (d) Hand drawn pictures or diagrams; (e) Other. The items were coded by myself and another advanced algebra teacher to help ensure that scoring accurately reflected the solution strategies employed by students, 140 with an inter-rater reliability established between graders (these results are reported in Chaper 6 - Quantitative Data Analysis). Descriptive statistics were used to describe solution strategies employed by control and experimental groups. Data analysis of the items focused on the following hypotheses. H ll. When given the opportunity to use various solution strategies on test items, the CAS group will rely less heavily on graphs than the non-CAS group. H12. When given the opportunity to use various solution strategies on test items, the CAS group will rely more heavily on symbolic manipulation techniques to solve problems than will the non-CAS group. H13. When given the opportunity to use various solution strategies on test items, CAS students will rely less heavily on tables to solve problems than students in the non-CAS group. At the time this study began, very little was known about ways in which students interact with CAS-equipped devices. A number of quantitative studies had examined whether or not symbolic manipulators enhance student performance on exams — but had generally not probed deeper to uncover possible explanations for differences in student performance. For this reason, this study also included measures of student attitude throughout the school year. These measures are described in the next section. Measurement of CAS impact on course content Teacher Field Notes. A primary source of qualitative data was a daily journal that I kept during the study. The journal served as an open forum for ideas, observations, concerns and personal reflections regarding the use of computer algebra utilities in secondary school classrooms. As Glesne and Peshkin (1992) note: 141 The field notebook or field log is the primary recording tool of the qualitative researcher. . . after each day of participant observation, the qualitative researcher takes time for reflective and analytic noting. This is the time to write down feelings, work out problems, jot down ideas and make flexible short- and long-term plans for the days to come (p. 47- 49). The majority of my field notes describe day-to-day class instruction emphasizing student use of CAS — with particular attention paid to potential student difficulties and new problem solving strategies developed by students using CAS. Nevertheless, the field notes are not tightly structured. Discussions with fellow high school mathematics teachers and mathematics education faculty as well as informal talks with parents and students comprise a significant portion of the journal. Extended time frame. A principal strength of this study was its extended time frame. The gathering of data over an entire school year enabled me to uncover patterns in student cognition that would certainly have gone urmoticed in a shorter length of time. Significant time was required for students to become conversant with new problem solving tools and calculator-based methods. The initial amount of instructional time I invested helping students leam how to use the technology seemed steep. A ten week study involving technology, especially with novice students, was inadequate for examining anything more than superficial changes in student understanding. As Glesne and Peshkin (1992) note: One always needs to allow extra time for data collection . . . your observations should take account of different phases of the cycle, as well as of the different occasions. This does not mean that observations need to occur every day, but it does mean that time, as well as places and people, must be sampled. Findings made from classroom observations 142 during the first quarter of a school year are likely to differ from those made in the fourth quarter. Classroom observations made only on Mondays may present a very different picture from observations made on other days of the week (p. 29-30). Because of my role as a full-time teacher at Midvale High school, I possessed the luxury of time. My double identity as researcher and teacher afforded me the opportunity to present a more naturalistic examination of an introduction of computer algebra utilities into a secondary school setting. Rationale for Mixed-Methodology Study As noted at the beginning of this chapter, this study employed both qualitative and quantitative research methods. A rationale for this mixed-methodological design was based upon my personal experiences as an educator as well as the thoughts of a growing number of educational researchers who felt that test scores paint a very narrow picture of student understanding. Classrooms are unlike scientific laboratories in the sense that a myriad of uncontrolled variables confront students and teachers each day. Peer pressure, unstable home lives, and drug use are just a few variables that make secondary school teaching an unpredictable enterprise to say the least. This unpredictability makes scientific experimentation difficult. Even in circumstances where variables are adequately controlled, purely quantitative methods pose a threat to the educational researcher. As Newman and Benz (1998) note: Because true experimental designs require tightly controlled conditions, the richness and depth of meaning for participants may be sacrificed. As 143 a validity concern, this may be a limitation of quantitative designs (p. 19). For these reasons, qualitative research methodologies were incorporated meaningfully into this project. Written student reflections, calculator screenshots, teacher field notes, test scores and student attitudinal surveys helped explain the entire story of my research in a manner that purely qualitative or purely quantitative methods could not. In the text Qualitative-quantitative research methodology: Exploring the interactive continuum (1998), Newman and Benz point out that “both qualitative and quantitative strategies are almost always involved to at least some degree in every research study” (p. 14). Although quantitative data gathering techniques have dominated educational research in mathematics up to the present, increasingly educators have come to recognize the inadequacy of purely quantitative research designs. In a discussion of the content of journal articles published in the Journal for Research in Mathematics Education, Lester (1994) notes: Whereas in 1973 the predominant methodology was statistical in nature (mostly using hypothesis testing and regression analysis designs), by 1983 about one third of all research reports in the journal were nonstatistical and a few of the essentially statistical studies used statistics for descriptive purposes only. The trend away from reliance on statistical methods has continued. In 1993, only three eighths of all manuscripts submitted used statistical methods exclusively, one half use various non-statistical methods, and the remaining one-eighth used some combination of quantitative and qualitative methods (p. 2-3). 144 As researchers continue to discover the complementary strengths of quantitative and qualitative methodologies, it seems clear that the above trend towards mixed methodological approaches in educational studies will persist. Limitations of this Study As an educator, I am committed to the promotion of teaching strategies that enhance student understanding and desire to leam mathematics. In no instance would I knowingly advocate the use of technology in mathematics instruction in a way that would harm students or impedes the furtherance of their mathematical education. However, it is equally true that in my experiences as an educator - I have foimd calculators to be extremely useful and powerful tools for helping young people leam mathematics. Certainly, the presence of symbolic manipulation utilities in my classroom affected my teaching style. For this reason, differences in student achievement (as measured by a pretest/posttest design) may have actually been influenced more by teaching differences than by differences related to CAS. Although my dual role as teacher and researcher impacted this study in many positive ways, it did bring into question the objectivity of the study findings. In addition to researcher bias, 1 also had to consider possible problems regarding uncontrolled noise factors when examining statistical results of this study. Possible sources of noise included: 145 Differences in Teaching Methods: In a pilot study of CAS-based tools during the 1998-1999 school year, 1 found that the introduction of symbolic manipulators encouraged more discovery-based learning activities. It is possible that differences in student achievement may be attributed to differences in instructional delivery and not because of differences in hand-held calculator usage. • Uncontrolled Student Access to CAS Tools: It is simply not possible (and not ethically sound) to prevent students in control classes from using CAS utilities outside of class. Based on their discussions with friends from the CAS class, several non-CAS students decided to purchase CAS-equipped calculators. Although non-CAS smdents received no formal training regarding the use of such calculators, they could not be prevented from using them merely for convenience purposes of my study. For reasons cited above, it seemed obvious that over-reliance on statistical measurement (while attempting to ignore possible sources of data contamination) to measure algebraic understanding of students was ill-advised. As noted previously, it is important to note that 1 was both the primary researcher and also the classroom teacher in this project. Although this may be construed as a limitation, such a design strengthened the study in many ways. • Differences in teacher expertise regarding the use of CAS were controlled. • Pace and sequencing differences of course topics were largely eliminated. • Homework assignments, tests and quizzes were made identical for both groups. 146 Nonetheless, I brought particular biases regarding technology to the classroom. Because this study may impact curriculum decisions which, in turn, affect students, I made every effort to teach both classes in as similar a manner as possible. Because this study was implemented using two intact classes — with subjects not randomly assigned to groups — the results of this study are not generalizable to large populations of high school students. However, because research in the use of handheld CAS devices in American high schools was virtually non-existent at the time of the study, it was my belief that the research, whether generalizable or not, would provide anecdotal evidence supporting continued research into this new area of mathematics education. 147 CHAPTER 6 QUANTITATIVE DATA ANALYSIS A problem is a chance for you to do your best. Duke Ellington Introduction In this chapter, I summarize data collected during a year-long study of calculator use in two secondary school advanced algebra classes. This chapter focuses on quantitative data obtained from two groups of students — a group taught without CAS-based calculators and a group that used CAS in their day-to-day class activities. As noted in the methodology chapter of this document, the CAS group and non-CAS group shared much in common during the study period. For instance, both studied identical content from identical mathematics textbooks. In addition, students in both classes completed identical tests and quizzes authored by the same classroom teacher. The primary difference between the two classes was the systematic use of calculator-based algebraic manipulation utilities in the CAS group. Several types of quantitative instruments were 148 used to measure students' attitudes and knowledge throughout the school year. Among these were: • An Algebraic Skills Pretest designed to measure student algebraic knowledge prior to formal advanced algebra instruction • A Technology Literacy Pretest designed to measure student calculator knowledge prior to formal advanced algebra instruction • Various algebra assessments, including tests, quizzes, and collected homework • A Likert-style Student Attitudinal Survey designed to measure students' attitudes regarding calculator usage in mathematics classes. The survey was administered to students twice during the school year — once in the Fall and again at the conclusion of the study in the Spring • A fifty-item multiple choice Advanced Algebra Final Examination covering all material studied throughout the school year Watchful for differences between the mathematical performance of the two groups, a variety of statistical methods were employed to examine the student data. I report the results of these statistical tests in this chapter. Unless specified otherwise, all tests were performed using the SPSS version 10 for Windows [SPSS Inc., 1999] software package. Specifically, data was analyzed to test the following hypotheses. HI. The mean scores of the CAS and non-CAS groups on the Algebraic Skills Pretest were not significantly different (with significance measured at a .05 alpha level). H2. The mean scores of the CAS and non-CAS groups on the Technology Literacy Pretest were not significantly different (with significance measured at a .05 alpha level). H3. Before the study, there was no significant difference, at the 0.05 level, between the general attitude of CAS students towards calculator use and the general attitude of non-CAS students towards calculator use. 149 H4. After the study, there was no significant difference, at the 0.05 level, between the general attitude of CAS students towards calculator use and the general attitude of non-CAS students towards calculator use. H5. The change in attitude of CAS students regarding calculator use from Fall to Spring was not significant at the 0.05 level of significance. H6. The change in attitude of non-CAS students regarding calculator use from Fall to Spring was not significant at the 0.05 level of significance. H7. There was no significant difference in Advanced Algebra Final Examination scores between CAS and non-CAS groups. H8. The difference in performance on the Advanced Algebra Final Examination o f low-performing CAS students and a similar group of low-performing non-CAS students was not statistically significant. H9. The difference in performance on the Advanced Algebra Final Examination of middle-performing CAS students and a similar group of middle-performing non- CAS students was not statistically significant. HIO. The difference in performance on the Advanced Algebra Final Examination of high-performing CAS students and a similar group of high-performing non-CAS students was not statistically significant. H ll. When given the opportunity to use various solution strategies on test items, the CAS group relied less heavily on graphs than the non-CAS group. H12. When given the opportunity to use various solution strategies on test items, the CAS group relied more heavily on symbolic manipulation techniques to solve problems than the non-CAS group. H13. When given the opportunity to use various solution strategies on test items, CAS students relied less heavily on tables to solve problems than students in the non- CAS group. 150 Analysis of Pretest Data Algebraic Skills Pretest Descriptive measures During the first week of the school year, a twenty-four item Algebraic Skills Pretest was administered to all subjects in both classes. The Algebraic Skills Pretest was designed by the authors of the University of Chicago School Mathematics Project (UCSMP) (Usiskin, et al., 1990) as an end-of-year cumulative exam for first-year algebra students. In this study, the test was used to measure baseline algebraic understanding of students prior to formal study of advanced algebra. In accordance with the author’s recommendations, all participants were granted access to non-CAS calculators with graphing capabilities (e.g. TI-82 or TI-83) to complete the test. To minimize grading bias, test items were graded either as “correct” or “incorrect” — with correct responses earning one point and incorrect responses earning zero points. Final scores were then tabulated as the percentage of items answered correctly. Descriptive measures of student performance in both the CAS and non-CAS groups are provided in Table 6.1. S t d . G roup N R ange M in Max M ean D ev. CAS 25 .542 . 333 .875 .54833 .12993 non-CAS 22 .583 .333 . 917 .61932 .15067 Table 6.1: Descriptive statistics oi Algebraic Skills Pretest data 151 As noted in Table 6.1, the range of student performance on the pretest appeared similar across groups. For instance, low scorers in both the CAS and non-CAS group answered 0.33 of the items correctly. The highest overall score was achieved by a student in the non-CAS group with a score of 0.92. The high scorer in the CAS group answered 0.88 of all test items correctly. Note that the non-CAS group obtained a higher mean score (0.62) than the CAS group (0.55). An Anderson-Darling test confirmed that the pretest scores of the non-CAS and CAS groups were normally distributed (p = 0.119 and p = 0.218, respectively). Technology Literacv Pretest Descriptive measures As previously noted, the Algebraic Skills Pretest was designed to measure differences in student algebraic understanding as opposed to differences in calculator experience. Since tliis study examined the impact of technology on student mathematical understanding, the administration of a second pretest measuring initial student calculator proficiency was warranted. To better describe each group's knowledge of hand-held graphers prior to the treatment period, a fifteen item Technology Literacy Pretest was administered to all students immediately following the Algebraic Skills Pretest. Items included on the Technology Literacy Pretest were o f two basic types. • Ten standard, non-symbolic graphing calculator questions (which could be answered correctly by students familiar with standard graphing calculators such as the TI-83) 152 Five CAS-based calculator questions (which require some knowledge of symbolic algebra routines on the TI-92/89 graphing calculators) Unlike the Algebraic Skills Pretest, the Technology Literacy Pretest was a "no calculator" test. Students were not allowed to use calculators while completing the test's fifteen items. Student performance on the pretest is presented in Table 6.2. As before, all scores are described in terms of percentage of items correct. non-CA S CAS B o th n 22 25 47 Minimum 0.200 0. 133 0.133 Maximum 0 . 667 0. 600 0. 667 Mean 0. 379 0 . 371 0. 375 SD 0.145 0 . 114 0 .128 Table 6.2: Descriptive statistics of Technology Literacy Pretest data The results of the Technology Literacy Pretest indicate that student knowledge regarding graphing calculators varied markedly among students within a given group at the beginning of the study. However, the performance by the overall groups appear quite similar. For instance, scores ranged from 0.20 to 0.67 correct in the non-CAS group and from 0.13 to 0.60 correct in the CAS group. Variation in students' knowledge of calculators may be partially explained by the differing calculator experiences of students in earlier grades. An Anderson-Darling test confirmed that the pretest scores of the non-CAS and CAS groups were normally distributed (p = 0.769 153 and p = 0.319, respectively). Student descriptions of calculator usage in previous courses are provided in Chapter 7 {Qualitative Data Analysis) of this document. Equivalence of group means for all pretest data A one-way multivariate analysis of variance (MANOVA) was conducted to determine the effect of student group membership (i.e. CAS or non-CAS) on the two independent variables — Technology Literacy Pretest scores and the Algebraic Skills Pretest scores. Specifically, the following hypotheses were tested: HI. The mean scores of the CAS and non-CAS groups on the Algebraic Skills Pretest were not significantly different (with significance measured at a .05 alpha level). H2. The mean scores of the CAS and non-CAS groups on the Technology Literacy Pretest were not significantly different (with significance measured at a .05 alpha level). With regard to the Algebraic Skills Pretest, no significant differences were found between CAS and non-CAS groups on the dependent measures, F(l,47) = 3.00, p = 0.096, T|^ = 0.063. Thus, data did not support the rejection of the null hypothesis that mean scores of CAS and non-CAS groups on Algebraic Skills Pretest were not significantly different (i.e. we failed to reject HI). With regard to the Technology Literacy Pretest, no significant differences were found between CAS and non-CAS groups on the dependent measures, F(l,47) = 0.046, p = 0.830, 0.01. Thus, data did not support the rejection of the null hypothesis that mean scores of CAS and non- CAS groups on Technology Literacy Skills were not significantly different (i.e. we failed to reject H2). Hence, the experimental and control groups were considered 154 similar in calculator abilities prior to participation in the study. Figure 6.1 summarizes the performance of CAS and non-CAS groups on both pretests. 0.8 £ 0.6 0o " 5 c 0.-4 u« 1 0.2 0.0 CAS Group Figure 6.1: Distributions of all pretest scores by group Student Attitudinal Surveys Twice during the school year — once in the Fall and once in the Spring — students in both CAS and non-CAS classes were administered a Likert-style Student Attitudinal Survey (a copy of the test is included in Appendix B of this document). The survey contained 16 items, with each item appearing exactly twice on the test - once as a “negatively-worded” item and once as a “positively-worded” item, for a total of 32 test questions. Students were given the opportunity to respond to each of the test items in the following ways: (1) Agree strongly; (2) Agree; (3) Disagree; and (4) Disagree strongly. Items on the questionnaire were designed to measure students’ attitudes 155 regarding the use of calculators in mathematics class. The items were further subdivided into five different subtopics. 1. Calculator as Authority 2. General Attitude 3. Building Concepts with Calculators 4. Building Skills with Calculators 5. Learning Mathematics with Calculators Internal consistency estimates of reliability were computed for the attitudinal survey using Cronbach’s Coefficient Alpha. Results of the Cronbach Test are reported in Table 5.3 of Chapter 5. Before analyzing Student Attitudinal Survey test results, student responses to positive items were reverse scaled (so that higher responses indicated a more positive attitude). Each item from the Student Attitudinal Survey is listed in Figures 5.2 through 5.6 in the Methodology chapter and again in Appendix B of this document. Fall questionnaire results A presentation of group mean responses for individual items on the Fall administration of the Student Attitudinal Survey is provided for the non-CAS group in Appendix C and for the CAS group in Appendix D. Since the expected value of any particular questionnaire item was 2.5, a group mean higher than 2.5 denoted a general positive attitude toward the item. Likewise, a group mean lower than 2.5 denoted a general negative attitude toward the item. A group mean higher than 3.5 for a particular item was considered an indication of “strong agreement” by the group for the given statement. 156 Differing responses between groups. The CAS group generally disagreed with the statement “Graphing calculators increase my desire to do mathematics” while they agreed with its negative counterpart “Graphing calculators don’t increase my desire to do mathematics” (these items received mean responses of 2.4615 and 2.3076, respectively). On the other hand, the non-CAS group responded to these items in the opposite manner. “Graphing calculators increase my desire to do mathematics” received a mean response of 2.5455, while its negative counterpart received a mean response o f2.5000 from the non-CAS group. Similaritv between group responses. Of the sixteen question pairs, the mean responses of the CAS and non-CAS groups agreed on twelve items on the Fall administration of the test. All agreement was in a positive fashion. The CAS and non- CAS groups did not share negative responses on any item pairs. The CAS and non- CAS groups had differing responses on one item. Responses for the remaining three items could not be directly compared since the mean responses didn’t match within a particular group. Both groups strongly agreed to question pair 2/19 — “When using a graphing calculator in class, I still have to know how to compute because the calculator won’t do everything for me” / “When using a graphing calculator in class, I do not need to learn how to compute because the calculator will do everything for me.” Comparison of Fall attitudes bv group. The sum of all calculator item responses for a given student, referred to as the student’s calculator attitude, was used as an index for 157 measuring the student’s attitude towards calculator use in mathematics class. Because of the manner in which items were rescored (1 = strongly disagree, 2 = disagree, 3 = agree, 4 = strongly agree), a student with a high calculator attitude score was considered to have a strong positive attitude regarding calculators. Table 6.3 provides descriptive statistics for the overall calculator attitude for both the CAS and non-CAS groups. I te m G roup N M ean SD M in Max Calculator non-CAS 22 98 . 0000 8.9496 78.00 112.00 Attitude CAS 26 94.4615 6.3450 84.00 105.00 Table 6.3: Overall student calculator attitudes (Fall) Figure 6.2 summarizes this information in a graphical manner. 120 . 2 110 uo CO "O=J 100 90 - 70 - non-CAS CAS Group Figure 6.2: Distributions oi Student Attitudinal Survey scores 158 Analyses of variances (ANOVA) on each dependent variable were conducted to test the following hypothesis: H3. Before the study, there was no significant difference, at the 0.05 level, between the general attitude of CAS students towards calculator use and the general attitude of non-CAS students towards calculator use. Table 6.4 presents the results of the one-way ANOVA on student calculator attitude. Source S3 df MS F p Calculator Treatment 149.205 1 149.205 2.553 0.117 Attitude Error 2588.462 4 6 58.445 Total 2837.667 47 Table 6.4: ANOVA for Calculator Attitude (Fall Student Attitudinal Survey) The ANOVA comparing mean calculator attitude scores was non-significant, F(l,47) = 2.553, p = 0.117. Hence, the data did not support the rejection of the null hypothesis that there was no significant difference, at the 0.05 level, between the general attitude of CAS students towards calculator use and the general attitude of non-CAS students towards calculator use (i.e. we failed to reject H3). Spring questionnaire results A presentation of group mean responses for individual items on the Spring administration of the Student Attitudinal Survey is provided for the non-CAS group in Appendix E and for the CAS group in Appendix F. As was the case on the Fall survey, the expected value of any particular questionnaire item was 2.5. Hence, a group mean 159 higher than 2.5 for a particular survey item denoted a general positive feeling regarding the item. Likewise, a group mean lower than 2.5 denoted a general negative feeling toward the item. A group mean higher than 3.5 for a particular item was considered an indication of “strong agreement” by the group for the given statement. Differing responses between groups. Unlike the results from the Fall administration of the Student Attitudinal Survey, the CAS and non-CAS groups did not produce mean responses that opposed each other. In other words, there was no item on the questionnaire for which the CAS group agreed but the non-CAS group disagreed (or vice versa). In fact, of the sixteen question pairs, the mean responses of the CAS and non-CAS groups agreed on fourteen items (as compared to twelve items from the Fall administration). Responses for the remaining two items could not be directly compared since the mean responses didn’t match within a particular group Comparison of spring class attitudes. As was the case in the Fall administration of the survey, the sum of all calculator item responses for a given student, referred to as the student’s calculator attitude, was used as an index for measuring the student’s attitude towards calculator use in mathematics class. Because of the manner in which items were rescored ( I = strongly disagree, 2 = disagree, 3 = agree, 4 = strongly agree), a student with a high calculator attitude score was considered to have a strong positive attitude regarding calculators. Table 6.6 provides descriptive statistics of overall calculator attitude for both the CAS and non-CAS groups. 160 N M in Max Mean SD Calculator non-CAS 22 83.00 116.00 99.5000 8.4389 Attitude CAS 22 77 .00 113.00 9 5 .1 3 6 4 9 .3 8 7 4 Table 6.6: Overall student calculator attitudes (Spring) Analyses of variances (ANOVA) on each dependent variable were conducted to test the following hypotheses. H4. After the study, there was no significant difference, at the 0.05 level, between the general attitude of CAS students towards calculator use and the general attitude of non-CAS students towards calculator use. H5. The change in attitude of CAS students regarding calculator use from Fall to Spring was not significant at the 0.05 level of significance. H6. The change in attitude of non-CAS students regarding calculator use from Fall to Spring was not significant at the 0.05 level of significance. Table 6.7 presents the results of the one-way ANOVA comparing calculator attitude of the Spring CAS and non-CAS students. Source S3 df MS F p Calculator Treatment 209.455 1 209.455 2.629 0.112 Attitude Error 3345.091 42 79.669 Total 3555.545 43 Table 6.7: ANOVA for calculator attitude (Spring Student Attitudinal Survey) The ANOVA comparing mean calculator attitude scores was nonsignificant, F(l,43 ) = 2.629, p = 0.112. Hence, the null hypothesis that after the study there was no significant difference between the general attitude of CAS students towards calculators 161 and the general attitude of non-CAS students towards calculators was not rejected at the .05 level of significance (i.e. we failed to reject H4). Analysis of attitudinal change during school year. Figure 6.3 illustrates the change in calculator attitude of CAS and non-CAS students from Fall to Spring. 100 99 0) 9 8 ■ o 3 (0 9 7 oL. ra U3 9 6 COa c 3 9 5 CLASS s - - non-CAS 9 4 CAS 9 3 Fall TIME Spring Figure 6.3: Plot of mean calculator attitude of CAS and non-CAS students during Fall and Spring As Figure 6.3 suggests, the mean calculator attitude of the non-CAS students was more positive at the end of the school year than it was when the study began. On the other hand, the mean calculator attitude of the CAS students was less positive at the end of the school year. 162 To determine whether changes in calculator attitude for either group were statistically significant, one-way analyses of variance were conducted to evaluate the relationship between the calculator attitude of CAS and non-CAS students before and after the treatment period. For all analyses — the independent variable, time o f year, included two levels: (1) Fall and (2) Spring. The dependent variable was calculator attitude -measured as the sum of all calculator-based responses on the Algebra Attitudinal Survey. Table 6.8 presents the results of the one-way ANOVA on change in student calculator attitude for the CAS group. Source SS df MS F p Calculator Treatment 3.527 1 3.527 .063 .804 Attitude Error 2420.251 43 56.285 Total 2423.778 44 Table 6.8: ANOVA for change in calculator attitude (CAS Group) As Table 6.8 suggests, the ANOVA was not significant, F(l,47) = 0.063, p = 0.804. Hence, the data did not support the rejection of the null hypothesis that there was no significant difference, at the 0.05 level, between the general attitude of CAS students towards calculators at the beginning of the school year and at the end of the school year (i.e we failed to reject H5). Table 6.9 presents the results of the one-way ANOVA on change in student calculator attitude for the non-CAS group. 163 S o u rc e SS d f MS F p Calculator Treatment 24.750 1 24.750 0.327 .570 Attitude Error 3177.500 42 75.655 Total 3202.250 43 Table 6.9: ANOVA for change in calculator attitude (non-CAS group) As Table 6.9 suggests, the ANOVA was not significant, F(l,43) = 0.327, p = 0.570. Hence, the data did not support the rejection of the null hypothesis that there was no significant difference, at the 0.05 level, between the general attitude of non-CAS students towards calculators at the beginning of the school year and at the end of the school year (i.e. we failed to reject H6). Analysis of Posttest Data Final Examination Descriptive statistics regarding Advanced Algebra Final Examination. Descriptive measures of student performance in both the CAS and non-CAS groups on the Advanced Algebra Final Examination are provided in Table 6.10. Recall that all scores are reported in terms of percentage of items answered correctly. S td . N R ange Min Max Mean Dev CAS 25 .400 .460 . 860 . 687 . 129 non-CAS 22 .440 .500 . 940 .765 . 113 Table 6.10: Descriptive statistics of Advanced Algebra Final Examination 164 As noted in Table 6.10, the mean score of the non-CAS group was decidedly larger on the Advanced Algebra Final Examination. On average, non-CAS students answered approximately 76.6 percent of items correctly compared to only 68.7 correct for CAS students. On the other hand, the spread of student performances appeared similar across groups — with a standard deviation of 0.113 for non-CAS scores and a standard deviation of 0.129 for CAS scores. Appendix G lists each item on the A dvanced Algebra Final Examination along with the percentage of correct responses by class. The items in Appendix G are further organized by test topic. Appendix G indicates that non-CAS students outperformed CAS students on all items involving exponents, radical notation, and graphical understanding. Performance was more evenly distributed on topics such as trigonometry, equation solving and quadratic forms. Hvpothesis tests involving Advanced Alsebra Final Examination. The data collected from the Advanced Algebra Final Examination provides information regarding the effects of hand-held CAS devices on student ability to solve basic algebra problems. The administration of a combined Algebraic Skills Pretest and Technology Literacy Pretest together with a common final examination allowed the testing of the following hypothesis. H7. There was no significant difference in Advanced Algebra Final Examination scores between CAS and non-CAS groups. A one-way analysis of covariance (ANCOVA) was conducted. The independent variable, student group, included two levels: (1) non-CAS and (2) CAS. The dependent 165 variable was the percentage of items answered correctly on the Advanced Algebra Final Examination. Student performance on the Technology Literacy Pretest and the Algebraic Skills Pretest were used as two separate covariates to control for any differences between the two groups at the beginning of the study period. Table 6.11 describes the results of the analysis of covariance (ANCOVA). E t a S o u rc e SS n MS F S ig . S g u a re d Model 24.729 4 6.182 397.354 . 000 . 974 Skills 0.002 1 0.002 0. 098 .756 . 002 Tech <0.000 1 < 0.000 0.003 . 957 . 000 Class 1.192 2 0.596 38.301 < .000 . 640 Error 0. 669 43 0.015 Total 25.398 47 Table 6.11: ANCOVA comparison of treatment, Technology Literacy Pretest scores, and Algebraic Skills Pretest scores As Table 6.11 illustrates, the ANCOVA was significant, F(2,43) = 38.301, MSE = 0.596, p < 0.000. The strength of the relationship between class membership (i.e. CAS or non-CAS) and performance on the Advanced Algebra Final Examination was very strong, as assessed by a partial T(^, with class membership accounting for 64% of the variance of the dependent variable, holding constant student performance on the Technology Literacy Pretest and Algebraic Skills Pretest. Hence we reject the null hypothesis that there was no significant difference in Advanced Algebra Final Examination scores (i.e. posttest scores) between the two treatment groups, and conclude that the use of CAS (or lack thereof) did contribute to statistically significant 166 differences in group performance levels favoring students that did not use CAS utilities as part of their mathematics instruction. The non-CAS group significantly outperformed the CAS group on the Advanced Algebra Final Examination (i.e. we rejected H7). As this study began, it was unclear how the use of CAS-based calculators might impact the mathematical understanding of low, middle, or high performing students. For instance, in his discussion of a student’s “house of mathematics,” Kutzler (1996) suggests that the pedagogical use of CAS might allow weaker students to remain engaged with classroom activities. On the other hand, Mayes (1996) notes that learning how to use the TI-92 while simultaneously learning new mathematics content may prove burdensome for weaker students. To gain a better understanding of CAS’s impact on student mathematical performance, students from both the CAS and non- CAS classes were split into three distinct groups. Membership in each group was based on the percentage of items answered correctly on both the Technology Literacy Pretest and the Algebraic Skills Pretest. • Low performing students — those whose combined pretest scores fall below the lower one-third quantile of their class • Middle performing students — those whose combined pretest scores fall above the lower one-third quantile and below the upper one-third quantile of their class • High performing students — those whose combined pretest scores fall above the upper one-third quantile of their class 167 Lower and upper third quantiles on combined Technology Literacy Pretest and Algebraic Skills Pretest scores are highlighted in Table 6.12 for both groups. Lower Third Upper Third Group N Quantile Quantile non-CAS 22 .462 .564 CAS 25 .436 .513 Table 6.12: Lower third and upper third quantiles for Combined Pretest scores The quantiles shown in Table 6.12 were used as cut-off points for membership in each of the performance groups. For instance, students in the non-CAS class who answered 46.2 percent or fewer of the items from the combined Algebraic Skills Pretest and Technology Literacy Pretest were identified as “Low Performing Students.” Figure 6.4 displays the number of students identified in each performance group for both the non- CAS and CAS classes. 168 Performance groups by pretest scores Low Performance Middle Performance High Performance non-CAS CAS GROUP Figure 6.4: Performance groups by class as determined by Combined Pretest scores Membership in initial performance groups allowed for a more detailed analysis of student Advanced Algebra Final Examination scores. In particular, the groupings allowed examination of the following hypotheses. H8. The difference in performance on the Advanced Algebra Final Examination of low-performing CAS students and a similar group of low-performing non-CAS students was not statistically significant. H9. The difference in performance on the Advanced Algebra Final Examination of middle-performing CAS students and a similar group of middle-performing non- CAS students was not statistically significant. HIO. The difference in performance on the Advanced Algebra Final Examination of high-performing CAS students and a similar group of high-performing non-CAS students was not statistically significant. 169 After student data were arranged into one of three performance groups, one-way analyses of covariance (ANCOVA) were conducted for each of the hypotheses H8, H9 and HIO. For each ANCOVA, student group was considered the independent variable and included two levels: (I) non-CAS and (2) CAS. The percentage of items answered correctly on the Advanced Algebra Final Examination was considered the dependent variable, with scores on Algebraic Skills Pretest and Technology Literacy Pretest considered as individual covariates. Table 6.13 provides descriptive statistics regarding performance on the Advanced Algebra Final Examination by performance group in both CAS and non-CAS classes. Low Middle High Group n Mean SD n Mean SD n Mean SD Non-CAS 9 .727 .1 1 2 6 .740 .100 7 .837 .103 CAS 11 .596 .1 1 0 7 .711 .072 7 .806 .093 Table 6.13: Descriptive statistics of Advanced Algebra Final Examination by performance group Results of the analysis of covariance (ANCOVA) for low-performing students are described in Table 6.14. 170 E t a S o u rc e SS d f MS F P S q u a r e d M odel 8.715 4 2.179 202.491 0.000 0. 981 Algebra 0. 019 1 0.019 1.736 0.206 0.098 Technology 0. 038 1 0.038 3.562 0.077 0 .182 Class 0. 096 2 0 . 048 4.455 0.029 0 . 358 Error 0.172 16 0 . 017 Total 8 . 888 20 Table 6.14: ANCOVA comparison of treatment and Combined Protest Scores for low performing students As Table 6.14 illustrates, the ANCOVA was significant for low-perfoerming students, F(2,16) = 4.455, MSE = 0.048, p = 0.029. The strength of the relati.onship between class membership (i.e. CAS or non-CAS) and performance on the Fimal Examination was strong, as assessed by a partial ri“, with class membership accouniting for 36% of the variance of the dependent variable, holding constant student perfo: rmance on both the Technology Literacy Pretest and Algebraic Skills Pretest. Hence we rejected the null hypothesis that non-CAS students who initially achieved at low-perrformance levels on the Algebraic Skills Pretest and Technology Literacy Pretest did not significantly (at the .05 level) outperform a similar group of low-performing CAS students on the Advanced Algebra Final Examination and conclude that the use ozf CAS (or lack thereof) did contribute to statistically significant differences in group performance levels among low-performing students favoring students who did not use CAS utilities as part of their mathematics instruction (i.e. we rejected H8). 171 Results of the analysis of covariance (ANCOVA) for middle-performing students are described in Table 6.15. E ta S o u rc e S3 d f MS F P S q u a re d M odel 6 . 867 4 1.717 359.619 0 . 000 0 . 994 A lg e b r a 0 . 024 1 0.024 4 . 990 0 . 052 0 . 357 T e c h n o lo g y 0.031 1 0.031 6.396 0.032 0.415 Class 0 . 049 2 0.024 5 . 088 0.033 0 . 531 Error 0 . 043 9 0 . 005 Total 6. 910 13 Table 6.15: ANCOVA comparison of treatment and Combined Pretest Scores for middle performing students As Table 6.15 illustrates, the ANCOVA was significant for middle-performing students, F(2,9) = 5.088, MSE = 0.024, p = 0.033. The strength of the relationship between class membership (i.e. CAS or non-CAS) and performance on the Advanced Algebra Final Examination was very strong, as assessed by a partial T|^, with class membership accounting for 53% of the variance of the dependent variable, holding constant student performance on the Technology Literacy Pretest and Algebraic Skills Pretest. Hence we rejected the null hypothesis that non-CAS students who initially achieved at average performance levels on the Algebraic Skills Pretest and Technology Literacy Pretest did not significantly (at the .05 level) outperform a similar group of middle-performing CAS students on the Advanced Algebra Final Examination and conclude that the use of CAS (or lack thereof) did contribute to statistically significant differences in group 172 performance levels among middle-performing students favoring students who did not use CAS utilities as part of their mathematics instruction (i.e. we rejected H9). Results of the analysis of covariance (ANCOVA) for high-performing students are described in Table 6.16. E ta S o u r c e SS d f MS F P S q u a re d Model 9.496 4 2.374 337.537 0. 000 0. 993 Algebra 0 . 002 1 0.002 0.258 0. 622 0.025 Technology 0 . 021 1 0.021 3.043 0.112 0.233 Class 0 . 012 2 0.006 0.832 0.4 63 0.143 Error 0 . 070 10 0 . 007 Total 9.566 14 Table 6.16: ANCOVA comparison of treatment and Combined Pretest Scores for high performing students As Table 6.16 indicates, the ANCOVA was not significant for high-performing students, F(2,10) = 0.832, MSE = 0.006, p = 0.463. We fail to reject the null hypothesis that CAS students who initially achieved at high performance levels on the Algebraic Skills Pretest and Technology Literacy Pretest will not significantly (at the .05 level) outperform a similar group of high-performing non-CAS students on the year-end Advanced Algebra Final Examination. We conclude that the use of CAS (or lack thereof) did not contribute to statistically significant differences in group performance levels among high-performing students (i.e. we failed to reject HIO). 173 Analysis of Student Tests and Quizzes Teacher-generated tests served as a primary means of evaluating student conceptual understanding throughout the year. Test scores were also used to assign quarterly grades to students. Individuals in both the CAS and non-CAS classes took identical tests throughout the school year. Typically the tests were administered after students had read and investigated all of the key ideas from a given chapter from the UCSMP Advanced Algebra (Usiskin, et al., 1990) text. Unlike tests, chapter quizzes were typically used as a means of assessing student strengths and /or weaknesses regarding a given topic and were administered prior to the completion of a chapter from the text. Although quizzes were used as an evaluation tool to help determine a student’s grade, they were also used to help guide instruction and provided students with feedback regarding their understanding of particular course concepts. Tests and quizzes were constructed from a variety of sources. Homework questions from UCSMP Advanced Algebra (Usiskin, et al., 1990) text Examples provided by students in class discussions Problems from UCSMP Assessment Sourcebook (Usiskin, et al., 1990) Alternate reform publications (e.g. textbooks, journal articles) Questions motivated by discussions with teaching cohorts At the time this study began, very little was known about ways in which students interacted with CAS devices. Although a number of quantitative studies had examined whether or not CAS enhanced overall student performance, few had examined possible 174 effects of symbolic manipulator usage on student performance with regard to specific mathematics topics. By examining mean test and quiz scores of both CAS and non-CAS groups over an extended period, while covering a wide range of mathematical content, I was able to investigate the usefulness of CAS-based tools with regard to a wide range of high school mathematics topics. Mean test and quiz scores for all tests and quizzes for both the CAS and non-CAS classes are provided in Table 6.17. All scores are expressed in terms of percentage of items answered correctly. For each test or quiz item, an analysis of variance (ANOVA) was administered to determine whether differences between the performances of the CAS and non-CAS groups were significant. The column “Sig.” in Table 6.18 refers to the p-value for the corresponding analysis of variance. 175 Group Means Test/Quiz Topic W eek non-CAS CAS Sig. Functions (Quiz) 2 .8120 .7920 . 510 Functions (Test) 4 . 8398 .8156 . 436 Variations(Quiz) 7 ,8176 .7815 . 405 Variations (Test) 9 , 9741 . 9496 . 440 Linear Relations (Test) 11 , 8527 .8413 . 677 Geometrical Matrices (Quiz) 13 , 9286 .8943 . 094 Geometrical Matrices (Test) 15 7682 .8130 . 346 Systems and Matrices (Quiz) 16 , 8591 .8660 . 873 Systems and Matrices (Test) 19 8530 .7900 .293 Semester Exam 21 8100 . 7480 . 063 Quadratics(Quiz) 23 7333 .7420 . 833 Quadratics(Quiz) 24 7848 .8040 .724 Quadratics (Test) 26 8180 .7670 . 420 Powers / Exponents (Quiz) 29 7864 .8027 .778 Powers / Exponents (Test) 31 7420 . 7505 . 852 Inverses / Radicals (Quiz) 32 8295 .8640 .299 Inverses / Radicals (Test) 34 8644 .8093 . 135 Logarithms (Test) 38 9053 .8407 . 066 Trigonometry (Test) 39 8250 . 9070 . 119 Table 6.17: Class test and quiz means with respect to various mathematics topics The same data is depicted graphically by a plot of the mean score of each test or quiz for both groups with respect to time in Figure 6.5. 176 TEST AND QUIZ PERFORMANCE 1.0 0.9 £ uo CO 3 a 0.8 sI CAS mean 0.7 non-CAS mean 0.6 2 4 7 9 11 13 15 16 19 21 23 24 26 29 31 32 34 38 39 41 WEEK Figure 6.5: Plot of test and quiz performance of CAS and non-CAS groups with respect to time (in weeks) Although none of the mean differences are significant at the .05 level, several trends are suggested by the data. • For the first 13 weeks of class, the non-CAS group outperformed the CAS group on every test or quiz. • The CAS group fared better than the non-CAS group on all tests involving power functions and exponents as well as trigonometry. • The non-CAS group outperformed the CAS group on all tests involving functions, variations, logarithms, and linear relations. • It is unclear whether symbolic manipulation utilities were useful to students studying matrices or quadratics, since the CAS group fared well in comparison to the non-CAS group on some (but not all) tests of these topics. 177 Measurement of student CAS Usage. Throughout the school year, students were encouraged to explore alternate solution techniques while working on various class exercises. A number of test and quiz items were designed specifically to measure student solution preferences. At the end of data collection, these specially designed items were coded based on four possible solution strategies available to each student. 1. Graphical techniques (with calculator or by hand) 2. Symbolic manipulation 3. Tabular representations 4. Hand-drawn pictures or diagrams The test and quiz items were coded by myself and Jeff Reinhardt, an advanced algebra teaching colleague, to help ensure accurate scoring. To code each test or quiz question, we answered each of the five questions in a “yes” or “no” fashion. • Were graphical techniques used to solve the problem? Was symbolic manipulation (either by-hand or on the calculator) used to solve the problem? Were tabular representations used to solve the problem? • Were hand-drawn pictures or diagrams used to solve the problem? • Was no strategy apparent? If a student used more than one of the four problem-solving methods to complete a particular problem, each method was tabulated for the item. 178 Table 6.18 illustrates the inter-rater reliability established between Jeff Reinhardt and Todd Edwards. Number of Percentage Number of Percentage Raters Items Agreement Codings Agreement Teaching Cohort /Researcher 375 86.4 1875 92.8 Table 6.18: Percentage of agreement between raters Of the 375 test and quiz items, our codings matched identically on 324 occasions (for a percentage of agreement of 86.4 percent). Of the 1875 individual codings (recall there were 5 codings per item), our tabulations matched 1741 times (for an overall percentage of agreement of 92.8 percent). Each of the coded test and quiz items are listed in Appendix H. Descriptive statistics regarding the popularity of various solution strategies are provided in Table 6.19. Data analysis of student problem-solving strategies focused on the following hypotheses. H ll. When given the opportunity to use various solution strategies on test items, the CAS group will rely less heavily on graphs than the non-CAS group. H12. When given the opportunity to use various solution strategies on test items, the CAS group will rely more heavily on symbolic manipulation techniques to solve problems than will the non-CAS group. H13. When given the opportunity to use various solution strategies on test items, CAS students will rely less heavily on tables to solve problems than students in the non-CAS group. 179 Popularity of Methods Ite m W eek G roup n G S T D N one Item 3, Quiz 1 2 non-CAS 22 2 5 4 15 1 CAS 23 11 12 4 1 4 Item 6, Test 1 4 non-CAS 11 11 0 0 0 0 CAS 14 8 6 0 0 0 Item 4, Test 1 4 non-CAS 14 0 14 0 0 0 CAS 23 4 20 0 0 0 Item 4, Quiz 2 7 non-CAS 16 3 10 5 0 1 CAS 24 12 9 13 0 3 Item 2, Test 3 11 non-CAS 20 1 11 11 0 0 CAS 25 2 19 7 0 0 Item 4, Quiz 6 23 non-CAS 22 4 19 1 0 0 CAS 25 1 25 0 0 0 Item 2, Test 6 26 non-CAS 20 18 3 0 0 0 CAS 21 21 18 2 0 0 Item Ir Test 7 31 non-CAS 23 11 3 12 0 0 CAS 25 13 12 8 0 0 Item 8, Test 9 38 non-CAS 22 13 17 6 0 0 CAS 25 14 16 4 0 0 G=graphical, S=symbolic, T=tabular, D=hand drawn diagrams, None=no strategy apparent Table 6.19: Popularity of various solution techniques on various test and quiz items Figure 6.6 provides a plot of symbolic manipulation usage (as a percentage of overall problem solving techniques) with respect to time. 180 1.2 1.0 - c o a 0.8 - S. CLASS 0.2 - non-CAS CAS 0.0 2 4 7 11 23 26 31 38 WEEK Figure 6.6: Plot of symbolic manipulation usage (as a percentage of overall problem solving techniques) with respect to time by class An analysis of solution strategies across all tests and quizzes reveals that the popularity of symbolic manipulation — when measured as a percentage of overall solution strategies utilized — was consistently lower for the non-CAS class. In other words, non- CAS students appeared to rely less heavily on symbolic manipulation to answer quiz and test questions relative to alternate solution strategies (such as graphing or tables). With the exception of weeks 7 and 38, the popularity of symbolic manipulation strategies was higher for the CAS group. Figure 6.7 provides a plot of table usage (as a percentage of overall problem solving techniques) with respect to time. 181 1.2 1.0 • 9. 0 .8 ■ «j 0 .4 ■ CLASS non-CAS CAS WEEK Figure 6.7: Plot of table usage (as a percentage of overall problem solving techniques) with respect to time by class Figure 6.7 reveals that the popularity of tables — when measured as a percentage of overall solution strategies utilized - was generally lower for the CAS class. On only two occasions (out of eight testing situations) were tables more popular with the CAS group. Figure 6.7 indicates that CAS students relied less heavily on tables to answer quiz and test questions relative to alternate solution strategies (such as graphing or symbolic manipulation). Figures 6 .8 and 6.9 summarize the overall popularity of various solution strategies for the nine test or quiz items originally identified in Table 6.19. The popularity of any particular method was calculated as a ratio of the total number of instances that the technique was used to the total number of instances of all techniques used. 182 Home screen 1 % Diagrams Table Symbolic Methods Graphical Methods Figure 6.8: Overall popularity of solution strategies (non-CAS group) Home screen 3% Table Symbolic Methods Graphical Methods Figure 6.9: Overall popularity of solution strategies (CAS group) Note that each of the following conclusions appear to be supported by Figures 6 .8 and 6.9. When given the opportunity to use various solution strategies on test items, the CAS group relied more heavily on symbolic manipulation techniques to solve problems than will the non-CAS group. On the items coded, the CAS group relied on symbolic techniques 50.9 percent of the time, as compared to 40.8 percent for the non-CAS group (i.e. H ll appears to hold). 183 • When given the opportunity to use various solution strategies on test items, the CAS group will rely less heavily on tables than the non-CAS group. On the items coded, the CAS group relied on table-based techniques 14.1 percent of the time, as to 19.4 percent for the non-CAS group (i.e. H12 appears to hold). Note that the following hypotheses do not appear to be supported by the figures. • When given the opportunity to use various solution strategies on test items, the CAS group will rely less heavily on graphs than the non-CAS group. On the items coded, the CAS group relied on graphical techniques 32 percent of the time, as compared to a nearly equal 31.3 percent for the non-CAS group (i.e. H13 does not appear to hold). Summary In an effort to address fundamental questions regarding the nature of CAS use at the secondary school level, the primary focus of this chapter was the examination of quantitative data collected from a variety of sources: • An Algebraic Skills Pretest designed to measure student algebraic knowledge prior to formal advanced algebra instruction • A Technology Literacy Pretest designed to measure student calculator knowledge prior to formal advanced algebra instruction • Various algebra assessments, including tests, quizzes, and collected homework • A Likert-style Student Attitudinal Survey designed to measure students' attitudes regarding calculator usage in mathematics classes. The survey was administered to students twice during the school year — once in the Fall and again at the conclusion of the study in the Spring. A fifty-item multiple choice Advanced Algebra Final Examination covering all material studied throughout the school year. Analysis of data supported each of the following hypotheses: • The mean scores of the CAS and non-CAS groups on the Algebraic Skills Pretest were not significantly different (with significance measured at a .05 alpha level). 184 • The mean scores of the CAS and non-CAS groups on the Technology Literacy Pretest were not significantly different (with significance measured at a .05 alpha level). Before the study, there was no significant difference, at the 0.05 level, between the general attitude of CAS students towards calculator use and the general attitude of non-CAS students towards calculator use. • After the study, there was no significant difference, at the 0.05 level, between the general attitude of CAS students towards calculator use and the general attitude of non-CAS students towards calculator use. • The change in attitude of CAS students regarding calculator use from Fall to Spring was not significant at the 0.05 level of significance. • The change in attitude of non-CAS students regarding calculator use from Fall to Spring was not significant at the 0.05 level of significance. • The difference in performance on the Advanced Algebra Final Examination of high- performing CAS students and a similar group of high-performing non-CAS students was not statistically significant. • When given the opportunity to use various solution strategies on test items, the CAS group relied more heavily on symbolic manipulation techniques to solve problems than the non-CAS group. • When given the opportunity to use various solution strategies on test items, CAS students relied less heavily on tables to solve problems than students in the non- CAS group. On the other hand, the hypotheses listed below were not supported by data collected in this study: • There was no significant difference in Advanced Algebra Final Examination scores between CAS and non-CAS groups. • The difference in performance on the Advanced Algebra Final Examination of low- performing CAS students and a similar group of low-performing non-CAS students was not statistically significant. 185 The difference in performance on the Advanced Algebra Final Examination of middle-performing CAS students and a similar group of middle-performing non- CAS students was not statistically significant. When given the opportunity to use various solution strategies on test items, the CAS group relied less heavily on graphs than the non-CAS group. In the next chapter of this document — Chapter 7, Quantitative Data Analysis — data from a variety of alternate sources are examined in an attempt to provide a more complete picture of CAS use during the data collection period of this study. Of particular interest in Chapter 7 are student writings and teacher entries from a year-long journal. The concluding chapter of this dissertation — Chapter 8, Summary and Conclusions - discusses themes present in all study data and provides possible explanations for findings. 186 CHAPTER? QUALITATIVE DATA ANALYSIS Listening to both sides of a story will convince you that there is more to a story than both sides. Frank Tyger Introduction In this chapter, as in the last, I summarize data collected during a year-long study of calculator use in two secondary school advanced algebra classes. This chapter focuses on qualitative data obtained from two groups of students — a control group taught without symbolic manipulators and an experimental group that used symbolic manipulators in their day-to-day class activities. In the previous chapter of this document, students in the CAS and non-CAS classes were compared using quantitative measures — test and quiz scores, attitudinal survey results, algebraic and technology- based pretest scores, and performance on a common final exam. In order to present a more complete picture of the events that took place within my classroom, I now focus attention on qualitative aspects of the data collection. Recall that several types of 187 qualitative instruments were used to document students' attitudes and knowledge throughout the study period. Among these were: • Student writings regarding the use of symbolic manipulators in algebra classes • An in-depth teacher journal documenting possible promises and pitfalls of symbolic algebra utilities in secondary school classrooms In this chapter, I discuss the data collected from these instruments. Watchful for themes present in the data, I address the following questions originally posed at the beginning of this study: Ql. What sorts of problems and activities did students investigate using hand-held technology? Which problems seemed well-suited for investigation with CAS? Which (if any) problems were not well-suited for investigation with CAS? Q2. What attitudes do intermediate algebra students have regarding graphing calculators and symbolic manipulation utilities? Do their attitudes change as they gain experience with the utilities? How do these attitudes differ among CAS and non-CAS students? Throughout this chapter, answers to the above questions are guided by an analysis of student writing assignments. A chronological examination of student writings helps to capture a sense of the growth and attitudinal change that study participants — both teacher and students - underwent while working with CAS utilities. Because student writings often refer to specific class exercises, I describe numerous class activities in this chapter. Additionally, I provide my reactions to the student writings with passages from a year-long teacher journal. The combination of these sources — student writings, example problems and solutions, and entries from a teacher 188 journal — attempt to capture the rich and varied experiences we had with technology during the year-long study period. Analysis of Student Essays Throughout the school year, students in both classes wrote a series of short responses to questions regarding their mathematical experiences. Students completed writing assignments approximately once every ten weeks during the study period. Table 7.1 lists the essays collected for this study. Number T i t l e D ate(W eek ) o f Ite m s Solving Linear Equations 1 0 3 Systems Essay 16 2 Calculator Algebra Questionnaire1 26 6 End o f Year Short Answer Essays 39 4 Table 7.1 : Student writing assignments Student essay questions addressed the primary study questions described in Chapter 1 of this document. Writing prompts were assigned approximately every ten weeks in order to examine trends and changes in student attitude throughout the school year. Student responses are examined chronologically in the paragraphs below. 189 Solving Linear Equations Essay Collected during the tenth week of class, the Solving Linear Equations Essay was the first student writing assignment that dealt specifically with calculator use. The assignment required students to employ several different solution strategies to find the equation of a line passing through the points (2,5) and (7,12). Questions prompted students to determine solutions in two distinct ways — with ( 1) calculator-based methods and (2) by-hand methods. After the students calculated a solution to the problem using both by-hand and calculator-based methods, a writing prompt asked them to identify a preferred solution strategy, justifying their preferences in several complete sentences. All of the items from the Solving Linear Equations essay are illustrated in Appendix I. The Solving Linear Equations Essay was administered to students after they had spent several days examining similar problems in collaborative settings. Both groups of students were exposed to regression-based solution techniques and by-hand manipulation methods prior to the essay assignment - with such strategies proving popular in both classes. As illustrated below, the varied complexity of calculator-based methods provided a rich backdrop for the examination of student problem solving preference in both classes. Regression on the TI-83 To perform a linear regression on a TI-83 calculator, a student creates two lists — one containing x coordinates of data points and another containing corresponding y 190 values. To create lists, a student first presses the STAT button on the TI-83 and is subsequently presented with a series of options. CHLC TESTS it... :SortA( 3 !Sortp< SsSetUplditor Figure 7.1: Statistical options on the TI-83 Students may then enter data points into the calculator’s statistics editor. The student enters x coordinates into one list (in this case Li) and y coordinates into a second list (in this case L 2). LI L2 L3 2 2 S ? A&m i L2(ï) = Figure 7.2: Data entry using TI-83 After all data is entered into the calculator, the student presses the STAT button again and selects a specific regression to perform. For instance, a student selects linear regression (i.e. LinReg) to construct a least-squares regression line to data stored in lists L, and L 2. 191 EDIT HSIii TESTS is1-Var Stats 25 2-Var Stats 35 Med-Med gPLinRe9 Figure 7.3: Regression options available to TI-83 users The student presses ENTER and the regression is performed. The results of this step are shown in Figure 7.4. The r-squared value indicates that the regression fits the points exactly. LinRe9 y=ax+b a=1 .4 ^ ï î i ^ r=l Figure 7.4: Calculations resulting from a linear regression on the TI-83 Regression on the TI-92 The tasks required to perform linear regression on a TI-89/92 calculator are similar to those employed on a non-CAS calculator, however several additional steps are needed. As was the case with the TI-83, the student must first create two lists. On the TI-92, the student must first invoke the calculator’s data/matrix editor application. This is accomplished first by pressing the APPS button on the TI-92, then selecting the data/matrix editor option. This option is illustrated in Figure 7.5. 192 OFFLICATIDN* r Ffi I 1 ■ * ■ 1: Home 2:V= Editor 3 : Window E d ito r /Program Editor ► 8;Geometry ► 9:Text Editor ► 1 M DIN RHD RUTD FUNC 0 / 3 0 Figure 7.5: Option to invoke data matrix editor on the TI-92 The TI-92 data/matrix editor is similar to most spreadsheet applications found on a personal computer. The data/matrix editor first requires students to create a file — a step which requires students to leam several additional steps not required of TI-83 users. Students select the New option from the data/matrix editor submenu. RPPLiCATIDNF r Fs I • f— 1: Home 2:V= Editor 3 :Window E d ito r 1: Current I /^Program toitor ► 2 :0 p ig T ^ ^ I 8: Geometry *• 9:Text Editor ► 1 M OIN ROD RUTD FUNC 0 / 3 0 Figure 7.6: Creating a new data file on the TI-92 Next, the user must provide a name for the spreadsheet file. The new file dialog box, shown in Figure 7.7, allows users to choose a name for the data file. As shown in Figure 7.7, the name lin e d a ta is chosen for the file. 193 iPlot Setup|Ce 11 Header Calc U til S tat Type: Data-»- Folder: nain^ Uariable:|linedata , <'Enter=OK Figure 7.7: Creation of data file lin e d a ta using the TI-92 From this point, the steps required to perform a regression on the TI-92 are similar to those required on the TI-83 calculator. After entering a name for the spreadsheet file, a list editor is visible on the calculator’s home screen. Students then enter poircts into the editor. As Figure 7.8 illustrates, the student enters x coordinates into one list and y coordinates into a second list. rÆTpiot^=betupC?uÏHe^‘’derfâcIul?lïsïitl Dnin c l c2 c3 C4 c5 1 2 5 2 7 12 3 4 5 6 7 i*3c2= MAIN RAD AUTO FUNC Figure 7.8: Entering data into lists using the TI-92 After all data is entered into the calculator, the student presses the CALC me=nu option (found underneath F5). The student must select a specific regression to perform from within the CALC dialog box. Figure 7.9 illustrates the selection required to jperform a linear regression on the TI-92. 194 ^ mqinS1in««lata Calculate ■ \ yl] y Calculation Type. 1 :OneVar X# 2:TuoU ar 3:C ubicR eg T 2 v:: _____ 3 Use Freq and Catego 7:MedMed 4 8:PowerReg b 9:QuadReg A AzQuartReg 7 X: C ::v trO-:=’ CESC=CANCELD r (Enter=SmUE) M AIN RAD AUTO FUNC Figure 7.9: Menu option to perform linear regression on TI-92 Unlike the TI-83, the TI-92 linear regression utility requires students to specify where x and y data are stored (on the TI-83, the default location of x data is L[ and the default location of y data is L?). As Figure 7.10 illustrates, the student must specify that x coordinates are stored in column 1 (i.e. Ci) and y coordinates are stored in column 2 (i.e. cz). Calculation Type. LinReg^ X...... I d 1 y ...... Ica 2 store RegEQ to ... none-» 3 Use Freq and Categories? NO-» 4 5 Z6 ÎA. Figure 7.10: Dialog box in which regression parameters are entered on the TI-92 The student presses ENTER and the regression is performed. As Figure 7.11 illustrates, the r-squared value indicates that the regression fits the points exactly. 195 FZ y=a * x+b c l =1.4 = 2.2 R2c o r r MflIH Figure 7.11: Calculations resulting from a linear regression on the TI-92 Comments from earlier entries in my teaching journal indicate that the extra steps required of CAS students — saving data as a file and specifying lists for regression — initially discouraged students from using the calculator. The average student at our high school has little familiarity using spreadsheet programs. Because our curriculum is basically college preparatory, we don’t offer courses in accounting or basic computer skills for most students (a curriculum decision which puzzles me to no end) — so many students have never used applications such as Microsoft Excel prior to my class. This makes regression on the TI-92 much more difficult for students . . . since the data matrix editor is similar to a spreadsheet application in the sense that students must save file names, open applications, and use function keys to select regression techniques. I think this is a principal reason that the by-hand methods seem to be more popular with TI-92 users (September 21, 1999). Because regression proved overly complicated on the TI-92, CAS students were exposed to alternatives to regression prior to administration of the essay. I illustrate a manipulation-based calculator strategy below. Using svmbolic manipulators to complete the Solving Linear Equations problem In the CAS class, we investigated the linear equation problem from a more traditional symbolic manipulation viewpoint — but used the calculator to avoid common 196 manipulation-based errors. First students typed in a formula for the slope of a line passing through points (xi,yi) and (xz, yz)- Although most students were familiar with slope as “change in y over change in x” or less formally as “rise over run,” they experienced initial difficulties solving the slope problem on the calculator. Their use of the familiar slope expression m = —— — resulted in an error since y i is considered a X, - Xt function by the TI-92. (TI-T^CiY F Z ^ Y F Î - - ■ ■ F 4 - - Y F S __ ' ■ FG l-f—lAlgebralCalc OtherlPrgmIO Clear a-z_ 2 EKRDR > Too few arguments m = < v2-w l >/' Figure 7.12: The TI-92 does not allow use of familiar slope formula 1/ — On the other hand, the less familiar form m = ------works nicely with the calculator. X— X, As Figure 7.13 illustrates, this is the formula I used with students in the CAS class. rfvSBlT Y Fî-' Y f s Y" FG l-rf—IfllgebralCalc Other|PrgnIO|C C lear a-z y - y2 y - y2 X - x2 X - x2 n= Figure 7.13: A modified formula for slope works successfully on the TI-92 197 After entering the formula into the calculator, students used it as a template to calculate the slope of the line through the points (2,5) and (7,12). Y F 3 ^ Y F H V f FË Y F 6 Algebra Calc Other PrgmIO Clear a- y — y2 ' X — x2 5 - 1 2 I n =- 2 - 7 m = 7 /5 n=<5-12>/<2-7> MAIN ______AAD AUTD Figure 7.14: A numerical value for slope obtained by substituting values into the into a general slope formula on TI-92 In the original formula, students replaced m with — and (x%, 72) with the coordinates of a given point to obtain a linear expression containing only x and y. This strategy was similar to methods non-CAS students implemented when performing the steps by hand. T F Î-»- T f ' v ' T f5 r fs i l-t—iFUgebralCalclotherlPrgnlOlClear a-z_ 1 , „ - y - y2 y - y2 x - x 2 x - x 2 m = 7 /5 " 7 / 5 = 7 /5 = ^ _ | IM « IN RMO AUTO FUNC 3 / 3 0 1 Figure 7.15: Replacing variables with constants using the TI-92 s command line editor Next, CAS students used Kutzler’s methods to convert the expression into an equivalent expression in standard Ax + By = C form. For instance, some students multiplied each 198 side of the equation by (x -2 ). Using the TI-92, this calculation is performed by multiplying ans ( 1 ) by {x—2). 1 gIbraTcïïcTotherTPrgn I oTc 1 ear* a-z_ 1 y - yz p , - y ■ x - x 2 " x - x 2 m = 7 /5 ■■»= 2 - 7 " 7 ^ = x - 2 2) - 5 MAIN ItKO AUTQ FUNC S / 3 0 Figure 7.16: First step of solving a linear equation from the general slope formula using Kutzler’s methods on the TI-92 Using tlie EXPAND function, students applied the distributive property to algebraic expressions. This is illustrated in Figure 7.17. f S I n 1 glbraTca 1 cTotherTPrgn I oTc 1 ear* a-z_ r 7 Cx - 2) A ■ expand]^ ^ - y s j ^ ^ — 14/5 = y - 5 SS] MAIN RAO AUTO FUNC S / 3 0 Figure 7.17: Application of the distributive property using EXPAND function on the TI-92 From this point, adding and subtracting terms from each side of various equations allows students to obtain a linear equation in the desired form Ax + By = C. As illustrated in Figure 7.18, students may use Kutzler’s methods to generate the final Ix -11 answ er y =----- . 5 5 199 F3^T ry FS Y FS Algebra Calc Other PrgnlO Clear a-z I expandj^— = y - 5 7 X - 1 4/5 = y — 5 ■ - 14/5 = y - s j + 14/5 - ^ = y - l l / 5 ' ^ = y - 11/5^ - y - y = -11/ ans <1>—y Figure 7.18: Using TI-92 to manipulate a linear equation into standard form Students were able to check their answers by plugging either of the original ordered Ix —11 pairs (2,5) or (7,12) into their final final answer —— y = —^- This is shown in Figure 7.19. FZ^ Y F3-r Y Y FS FS 1^4— Algebra Calc|Other|PrgnIO C le a r a- 7 X ' ["^5^ = y - 1 l / s j - y - y =■11/5 7 7 5 - 12= - 11/5 tr u e 7 2 - 5- - 5 = - 11/5 tr u e 7*2/5-5=11/5 RAD ftUTD Figure 7.19: Verification of linear equation solution using boolean algebra capabilities of TI-92 Comments from my teaching journal indicate that although symbolic manipulation utilities proved to be a useful teaching technique with this problem, students in the CAS class remained uncomfortable with the approach because they were not able to use familiar symbolic representations when solving the problem. Our text states that “if a line contains (jr,,>i)and has slope m, then it has equation {y — yi)=m-{x— ).” I pointed out to students that this idea is really just a rehash of the slope formula (an algorithm that ever^mne is familiar with). In the CAS class, we typed in the point/slope formula and manipulated it. Unfortunately the calculator won't allow us to use the 200 variable^ (since this is seen as a function by the calculator), so we had to use>', instead. Some of the CAS students had a difficult time with this since they want to follow the book closely until they feel more comfortable with the formulas. The situation is unfortunate, since the students are really just learning the material for the first time and really do rely on the book to review ideas at home. The fact that the calculator doesn’t facilitate using formulas firom the book discourages students from using symbolic manipulation on the calculator (November 2, 1999). In constrast, I explained the same idea to the non-CAS students with an overhead projector with very little concern or controversy. In retrospect, I think that an argument on the overhead projector (in the non-CAS class) was more effective than the presentation which used the TI-92 symbolic manipulation utility because the overhead didn't care which names I gave to variables. The manipulations were not overly difficult by hand, and (using the overhead) students were allowed to model representations provided in homework problems in our textbook (November 2, 1999). The difficulty posed by the TI-92 calculator on the problem may have influenced preferred solution methods of students in each class. As Table 7.2 indicates, students in the non-CAS group were more likely to use calculator-based techniques than their counterparts in the CAS class. Solution, Preference Class n Calculator (%) By-hand. (%) Neither (%) Non-CAS 21 13(61.9) 6(28.6) 2(9.5) CAS 23 6(26.1) 15(65.2) 2(8.7) Table 7.2: Student responses to item 3 of Solving Linear Equations Essay 2 0 1 Approximately 65 percent of CAS students identified by-hand methods as the method of choice for determining a linear equation passing through two points. On the other hand, a similar proportion of non-CAS students (61.9 percent) preferred calculator- based methods. When asked to explain why they preferred a particular problem solving method, students in the CAS class typically responded that by-hand methods were less complicated than calculator-based methods. A number of students in the CAS class expressed reservations about using a calculator that they recognized would not be used in future classes at the high school. 1 prefer the first way (the by-hand method) because I can just as easily figure it out by hand as I can on the calculator. Besides, in the future, I will not have a calculator this sophisticated (Alison Myser, CAS group). Conversely, students in the non-CAS class typically preferred using the calculator because they found it easy to use. Many students responded favorably to the multi- representational aspects of the graphing calculator. In addition, students mentioned that they were motivated to use the calculator because they knew it would be utilized in future classes. (I prefer the) calculator, although both (calculator and non-calculator methods) are just as easy. I prefer the calculator because I don’t have to write it down and I should always try and take the opportunity to get to know the TI-83 better because I will be using it more in the future (John Beige, non-CAS group). Another student from the non-CAS class made the following comments. I like the calculator method. The reason for this is that I believe that it is much easier to do that way. Also, I believe that your equation is much 202 more clear and it is easier to see on the calculator because you are able to graph it to see if it fits (Chris Archer, non-CAS group). Teacher Response to Solving Linear Equations Essavs After reading student responses to the Solving Linear Equations essay, it became apparent to me that CAS students would not realize many of the potential benefits of symbolic manipulator use unless compelling reasons for using the TI-92 were explicitly presented. Because many students sensed that symbolic manipulators would not be used in subsequent mathematics classes at the high school, they initially saw few rewards in taking the time to learn features of the TI-92. During the four weeks immediately following the Solving Linear Equations essay, much class time was spent investigating problems designed to encourage students to use symbolic manipulation utilities or graphical methods. The following comments were made in my teacher journal immediately after the Solving Linear Equations Essays were collected. I’m somewhat concerned that I’ve given students in the CAS class a less than ideal introduction to the TI-92 and calculators in general. It seems like we’ve spent more time in the CAS class pushing buttons than we have with the TI-83 (group), and this worries me. I want to provide students with a balanced view of technology. Certainly there are some problems that are best solved using traditional by-hand techniques. However, many problems are well-suited for the calculator. In the next few weeks, I think it is really critical for me to find problems that illustrate the idea of balance to students. Otherwise it seems unlikely . . . that the CAS students will ever embrace the TI-92 as anything other than a fancy “answer checker.” The first grading period was a critical time to establish the calculator as a useful device (November 5, 1999). Equation solving proved a fruitful springboard for the development of problems that would use the TI-92 and TI-83 in such a manner. 203 To better facilitate the reader’s understanding of the types of problems developed after the Solving Linear Equations essay, and to provide a context in which to place subsequent student questions and responses, I illustrate examples of two problems that were given to students in the days following the first essay. The first problem, shown in Appendix J, highlights an example of a skills-based problem not well-suited for by-hand manipulation by typical advanced algebra students (I refer to this as the Skills-Based Warm-up Problem^. The second problem, also shown in Appendix J, illustrates an application problem that encourages students to look beyond manipulation as the principal goal of secondary mathematics instruction (I refer to this as the Systems Warm-up Problem). In the paragraphs below, I discuss solution strategies that students used when solving my Skills-based Warm-up Problem and Systems Warm-Up Problem. Skills-based warm-up problem Working in pairs, students in both classes were provided with approximately 20 minutes of class time to work on the problem. The assignment was not collected — but instead was administered to provide students with an opportunity to practice solving systems while discussing the relative merits of various equation solving strategies. After students discussed the problem at length, they regrouped — with several volunteers from each class copying their work on a dry erase board in the front of the classroom (the results of the volunteer work were compiled in the teacher journal). Initially, 1 did not provide students with my solution to the system of equations. 204 Instead, encouraged students to discuss their own strategies and judge for themselves the accuracy of the answers provided by their classmates. Student bv-hand manipulation strategies. Figures 7.20 through 7.23 illustrate the steps that one student volunteer in the CAS class employed to obtain a solution. Like the majority of her classmates, the student decided to solve the problem using by-hand manipulation techniques. Note that although the student’s choice of transformation in each step was correct, a simple arithmetic error undermined her ability to construct a correct solution. Specifically, the student first multipled one of the system equations by ' 200 on both sides. This was a reasonable first step. '-Zp® (x-“F ' -Z Figure 7.20: A student’s first step at solving a system of equations using by-hand methods Unfortunately, the student applied the distributive rule incorrectly — introducing an error to her solution. Figure 7.21: Mechanical errors involving the distributive property introduces error into the student’s solution 205 As Figures 7.21 through 7.23 illustrate, subsequent student steps are correct. For instance. Figure 7.21 shows the student’s decision to use a linear combination method to eliminate the variable x. Figure 7.22 shows how the student divided each side of an equation by 353.995 to solve for y . Although these steps are correct, earlier miscalculations provide the incorrect solution shown in Figures 7.22 and 7.23. 3 5 J. tfX Figure 7.22: An incorrect solution for variable y found with by-hand methods As Figme 7.23 illustrates, the student took the value she found for y and substituted it back into one of the original equations of the system. This left her with a single equation in terms of x, which she was able to solve using familiar by-hand techniques. ,X + :.x = Figure 7.23: An incorrect solution for variable x found with by-hand methods Figures 7.20 through 7.23 illustrate that students may have a strong theoretical understanding of equation solving, yet arrive at incorrect solutions stemming firom arithmetic errors. Student graphical solution strategies. Several students firom each class attempted to solve the skills-based warmup problem using graphical techiques. For instance, 206 students in the CAS class first solved both equations in terms of y and typed the expressions into the TI-92’sy= editor, as shown in Figure 7.24. ' F 2 V •’ n ' FH ' Zoom E d it fill ..f L D T J P lo t 4: P lo t 3! P lo t 2: P lo t 1: . . 300 — 200 - X “ 0Ô5 ■ - v 3 (x )= MAIN RAB EXACT FUNC Figure 7.24: Both equations of the system typed into the y= editor of the TI-92 Next, students chose the ZOOMFIT menu option — a calculator utility that generates a viewing window that displays graphs of all active equations on the same screen. This is illustrated in Figure 7.25. 1 :ZoonBox P lo t 2:ZoomIn P lo t 3 :ZoomOut P lo t 4 :ZoonDec P lo t 5: ZooptSqr 6 :ZoonStd ✓yl=: 7:ZoonTrig 8 :ZoomInt ✓y2=- 9:ZoonData B:Memory F C:SetFactors... w i f i M«IN (too EXACT FUNC Figure 7.25: The TI-92 ZOOMFIT menu option Figure 7.26 illustrates the graph that ZOOMFIT produces. Students in both classes found the screen confusing, since only one curve is seemingly visible. 207 RAD EXACTMAIN FUNC F ig u re 7.26: Misleading results produced by TI-92 ZOOMFIT utility Although most students realized that solutions to a system of equations exist wherever graphs of the two equations intersect, their inexperience with the calculator hindered their ability to find solutions for the Skills-based Warm-up Problem. Because the slopes of the curves were dramatically different (one close to zero, the other quite large), finding a window that clearly displays the intersection proved difficult for students. Initial Student Reactions to Problems. Student response to the activity was similar in both the CAS and non-CAS classes, with both groups expressing frustration regarding the difficulty of the problem. Student dissatisfaction provided an opportunity to emphasize the importance of utilizing multiple solution methods. In my view, the Skills-based Warm-Up problem was not well-suited to any single solution strategy, but was relatively easy to solve when several techniques were combined together. Combination of methods in CAS class. In the CAS class, a combination of symbolic manipulation and graphing provided students with compelling reasons for using calculator-based solution strategies. Figures 7.27 through 7.32 provide the reader with a better understanding of the methods my students and I eventually used to solve 208 the problem. Figure 7.27 illustrates a first step with the calculator. Initially we entered the system of equations onto the TI-92’s home screen. ' F t ? f s Y" Rlgebra|Calc|Other[PrgnIO|C C lea r a - I 200 X — . 005 ' y — 300 200 * x ** • 005 * y — 300 I X + 3 5 4 -y = 12 x + 354 y = 12 x + 3 5 4 y = 1 2 Figure 7.27: System of equations entered directly onto TI-92 home screen Next, using CAS, we multiplied equation .r+354^=12 by '200 and added the result to the equation 200^-0.005y=300. This approach, depicted in Figure 7.28, is based upon ideas proposed by the CAS student volunteer. F Z ? Y F3^ T Ft'*’ Y FS Y FS |fllgebra|Calc|Other|Prgi»iIO|Clear a-z ■ 200 X — . 005 • y = 300 200 x — . 005 • y = 300 ' X + 354 y = 12 x + 354 y = 12 I -200 (x + 3 54-y = 12) +(200 x - .0 0 5 -y = 3'k ______-70800. - y = -2100 2 G 0 * a n s < 1 > + an s <2 > Figure 7.28: Isolating y variable using traditional by-hand procedures within TI- 92 environment When working through these steps, it was emphasized that the calculator is able to perform mechanical algebraic manipulations, but it is not able to select appropriate transformations to apply to equations. In other words, the calculator does not do all of the work involved in solving a system of equations. Rather, the calculator performs routine steps while allowing students to focus on more theoretical aspects of equation 209 solving. As Figure 7.29 indicates, adding the two equations together on the TI-92 results in a single equation in terms of one variable, y, namely 70800-y =—2 1 0 0 . Using the calculator, the student solves for y by dividing each side of this equation by y 's coefficient. This is illustrated in Figure 7.29. K t — fllgebralC alc O th er PrgftIO C lea r a- I 200 • X — . 005 - y = 300 200 • x — . 005 • y = 300 I X + 354-y = 12 x + 3 54-y = 12 I -200 (x + 354 y = 12) +(200 -x - . 005-y = 3i> -70800. y = -2100 -70800.005 y = -2100 -70800 1. -y = 236 ans(l)/-70800 Figure 7.29: Solving for y using traditional by-hand procedures on the TI-92 To solve for .y, we used strategies originally employed by the unsuccessful student — taking the value we found for y and substituting it into 200x—0.005_y = 300(i.e. the first equation of the original system). This results in a single equation in terms of x, 21 namely = 12 which is solveable in a step-by-step fashion on the calculator. |V?2ïfllglbraTcarcTot%rTPrgnIoTciear^ a-z_ 1 " X + 354-y = 12 X + 354-y = 12 -X-K354 -236 -12 X + 2 1 /2 = 12 " (x + 21/2 = 12) - 2 1 / 2 x = 3 /2 MAIN RAD AUTO FUNC T /3 0 Figure 7.30: Solving for jc using traditional by-hand procedures on the TI-92 To check work, we graphed both equations. However, instead of relying on the calculator’s ability to generate a meaningful window, we constructed window settings 2 1 0 based on symbolically-derived answers. We wanted xm in and xiaax to be centered around our solution for x, hence 0 and 3 were logical choices. The ymin and ymax were chosen in a similar fashion. Our choices are shown in Figure 7.31. ^ f —iZoon xnin=0. xnax=3. Figure 7.31: Window settings for graph based on CAS-derived solutions The intersection of the two curves are clearly visible using the window settings shown in Figure 7.31. As Figure 7.32 illustrates, the intersection of the two equations are now easily found. Intersection xc: 1 .5 y c : .029661 Figure 7.32: Graphical depiction of system solution using TI-92 Combining unsuccessful problem solving strategies to obtain correct solutions surprised many students, as I noted in my teacher journal. Today, I think that students in both classes were taken aback to see me use ideas proposed by unsuccessful students to construct a correct answer. The students seem to view problem-solving as a rigidly defined affair. After choosing a method for solving a problem, they seem 211 reluctant to combine other approaches with their initial idea — in their minds a method is either correct or incorrect. Such a point of view discourages multiple solution strategies (since only one strategy is the correct one). I think the examples today reassured students that their ideas are not entirely wrong, even when they make mistakes. More importantly, I think I was able to convince a few students in the CAS class to give the calculator another chance. It was apparent to many that the calculator could be used to enhance one’s ability to solve equations — by helping them to identify which transformations to apply to equations to solve for variables (November 10, 2000). Combination of methods in non-CAS class. As was the case in the CAS class, several students in the non-CAS class attempted to use graphical methods to solve the system of equations. Unfortunately, ZOOMFIT works no better on the TI-83 than it does on the TI-92. Figure 7.33: Simultaneous graphs of both equations using ZOOMFIT reveals little information regarding the solution of the system on the TI-83 In the non-CAS class, the students and I made use of the table feature of the TI-83 to reclaim graphing as a viable solution strategy. By determining where the y-values of both equations were approximately equal, we were able to type in window settings that made the intersection of the equations clearly visible. 212 M«tL Mot2 Mo» \YiB(12-X)/354 \YzB<300-Z00X)/- .005 \Y ] = >-Vh = xYs = \Yg = Figure 7.34: Equations for both equations of the system typed into the TI-83 y= editor Instead of directly graphing the equations, I suggested that an examination of the y- values of both equations from the TI-83’s table might help us determine suitable window settings. A screen shot of the table is highlighted in Figure 7.35. X Yi Yz -50000 L .03107 -30000 t .03135 30000 .035H3 50000 H .0335 100000 5 .01977 IhOOOO fi .01595 150000 X=0 Figure 7.35: Values of various x-y pairs that satisfy the system of equations as shown on the TI-83 table Several students from the non-CAS class noticed that along the interval (1,2) the values o f y change sign. Since all values of y, hover near 0, students suggested that we pick xmin and xmax to be 1 and 2, respectively. The ymin and ymax values were chosen based on the values of y, since its graph was so close to 0 and we wanted to see it distinctly from the x-axis. 213 WINDOW Xmin=l Xmax=Z X&cl=0 Vnin=-02825 Ymax=.03107N Vscl=l Xres=l Figure 7.36: Window values suggested by non-CAS students By examining values of both equations before jumping to create a graph, students were able to find the solution of the system quickly. Figure 7.37 shows a graphical solution on the TI-83. V=.029iflQ2 Figure 7.37: Graphical depiction of system solution using TI-83 The Skills-based warmup problem is representative of activities assigned to students during the month immediately following the Solving Linear Equations essays. Students in both classes spent much time examining problems which encouraged them to look beyond paper and pencil manipulation as the sole means of solving problems while investigating a wide range of solution strategies. Svstems Warm-up Problem The teaching strategies employed with the Systems Warm-up Problem were similar to those implemented with the Skills-based warmup. The problem, which has students compare phone billing plans offered by two companies, is shown in Appendix J. At the 214 beginning of class, each student chose one study partner. Working in pairs, students spent roughly 15 minutes attempting to solve the problem. The assignment was not collected — instead it was administered to help assess student comfort level with various solution strategies and as a vehicle for generating discussion regarding different solution methods. Overall, students in both the CAS and non-CAS classes seemed more content to solve the Systems Warm-up Problem using calculator-based techniques than they did when solving the Skills-Based Warmup Problem (see Appendix J). Entries in my teacher journal reveal that student perception regarding problem type appeared to impact their choice of solution strategies. I was somewhat surprised today by the variety of techniques employed to solve the phone bill problem[5y5’te/w5’ Warm-up Problem']. In particular, the CAS group’s use of calculator-based strategies was unexpected, considering their general reluctance to use calculator-based techniques to solve problems. As I walked aroimd the (CAS) class, I noticed that roughly half of the CAS group attacked the problem with the calculator. After the class regrouped, I expressed my surprise to them. One student commented that he solved the phone bill problem using the calculator because 'it was a story problem, not a math problem.’ The prevailing sentiment in the CAS class was that ‘algebra problems’ require algebraic manipulation but ‘story problems’ may be solved using a variety of methods (November 17, 1999). In both the CAS and non-CAS classes, we spent the remainder of class time sharing various strategies to solve the problem. Several popular strategies are highlighed in the paragraphs below. 215 Numerically-based Tnon-CAS^ Calculator Methods. Since the equations describing the monthly cost of each plan were easily constructed in terms of the dependent variable cost, students in both classes tended to construct the following equations (in both classes this was typically done by-hand). cost o f Plan A = 12.46 + 0.13 (number of calls) cost o f Plan B = 24.50 Students using calculator-based strategies then entered these equations into the y= editors of their calculators, replacing number o f calls with x. This technique is similar on both the TI-83 and TI-92, so I illustrate it using only screenshots from the TI-83 calculator. Plotl Plots Plots nViB12.46+.13X \V2824.50 vVs=i nVs= vVe= vVfi= sV? = Figure 7.38: Cost functions typed into y= editor of TI-83 After the cost functions were entered into the calculator, solutions were typically constructed in one of two ways: (1) by tabular methods or (2) by graphing. The ta b le feature of both CAS and non-CAS calculators provided students with a convenient way of comparing the relative merits of both plans. Initially, students set up the intial number of local phone calls displayed to be 0, with the number of calls incremented by 1 in the table. This set up is shown in Figure 7.39. 216 TABLE SETUP T b lS ta r t= 0 a T b l= l IndpnL: B Œ Ask Depend: [SPMB Ask Figure 7.39: TI-83 TBLSET (i.e. table set) utility The initial table generated by students represented various levels of local calls in a column labeled x. Costs associated with calling Plan A were highlighted in column Y i of the table, while costs associated with Plan B were shown in column Y 2. The table of values shown in Figure 7.40 suggests that Plan A is a better choice for consumers who make relatively few local phone calls. X V1 V2 1S.HG 2H.5 1 12.59 2H.5 Z 12.72 24.5 5 12.85 24.5 H 12.98 24.5 5 13.11 24.5 G 13.2H 24.5 X=0 Figure 7.40: An initial table generated by students using the TI-83 consisting of prices of both phone plans Students recognized that as the number of local calls increased, so did the cost of Plan A. By either scrolling down the table or by resetting the initial number of calls displayed in the TABLE SETUP utility, students were able to uncover the point at which Plan B became a better value than Plan A. 217 X Vi V2 B7 Z i . 7 7 2H .5 BB 2 3 .9 2H.5 B9 2H .03 2H.5 90 2H J.fi 2H.5 91 2H .29 2H.5 m i 15:1 X=93 Figure 7.41: A table suggesting that Plan B is a better option for consumers making 93 or more local calls per month Although table-based methods were popular among students in both classes, graphical techniques for solving the problem were also employed both CAS and non-CAS students. Students using graphical methods faced two basic obstacles: (1) determining suitable window settings to view the intersection and (2) interpreting the meaning of the intersection point. As illustrated in Figure 7.42 below, the TI-83 calculator finds an intersection point at (52.615385, 24.5). InteKïftctîon X=9Z.6153aS Y=2H.5 Figure 7.42: The graphs associated with the costs of calling plans A and B intersect between 92 and 93 calls Students using graphical techniques are required to recognize that making 92.615385 phone calls is simply not possible. A number of students in both classes failed to observe that any number of local calls greater than 92 will generate total costs that make Plan B a less expensive plan. 2 1 8 CAS-based calculator methods. Two additional methods for solving the problem were discussed in the CAS class. These strategies were originally described by students using a TI-92 overhead projection panel in class. One student prefaced the explanation of her solution strategy by commenting that she typically “didn’t use the calculator because it was confusing” but that “this way of solving the problem made more sense to her than graphing.” (November 19, 1999). Utilizing the TI-92’s ability to define functions, the student illustrated her use of the TI-92 as an “educated guess-and-checker” while suggesting an understanding of the Intermediate Value Theorem. The student first defined a cost function for calling Plan A on her calculator using the define feature of the TI-92. friT^Y FZ? y Y FI**' Y F S Y F G ^ Y ' 1^^—|fllgebra|Calc|other PrgnlOlClean Up| I" Define c(x) =12,46 + . 13 % Done define c Figure 7.43: Definition of cost function using DEFINE command on TI-92 Next, the student made an initial guess regarding the number of phone calls required to make C(x) larger than 24.5. Initially, her guess — 30 phone calls — was too low, so she repeated this process using 100 phone calls (see Figure 7.44). 219 ffPÎBY Y FH'*- Y TS Y ' fllgebrajCalc OtherlPrgnlOlClean Up I Define c(x) =12.46 + . 13 x Done ' c(3G) 16.36 ' ctlQQ) 2 5 .4 6 MAIN RAD AUTO FUNC 3 / 3 0 Figure 7.44: Evaluation of student-defined cost function using TI-92 with various number of minutes Starting with 90 calls, the student plugged values into c(x) until the cost exceeded $24.50. As illustrated in figure 7.45, this occurred when the number of calls equaled 93. f Ft? Y Y Y fs Y FÂ? Y RIgebra|Calc|Other[PrgnIO|Clean Up| Il ■ Define c(x) =12.46 + . 13 x Done! ■ c(3G) 16.361 ■ c(lGG) 25.461 ■ c(9G) 24.161 ■ c(91) 24.29; ■ c(92) 24.42j 0(93) 24.551 ; , MAIM RAD AUTO FUNC 7 / 3 0 Î Figure 7.45: A solution to the phone call problem using evaluation of cost function on TI-92 Using a white-box equation solving technique, another student in the CAS class illustrated a method for solving the problem. The student discussed the fact that he “wanted to find out when Plan A and Plan B would cost the same." To do this, he first typed an equation representing this situation on the home screen of the TI-92 calculator. This idea is depicted in Figure 7.46. 220 FT*- Y F3^ Y Y FS Y FS-*- IfUgebralCalclotherlPrgnlolciean Up ■ 12.46 + .13 x = 24.5 .13-X+ 12.46 = 24.5 MAIN RAO AUTO rU N C 1 / 3 0 Figure 7.46: Student-built equation representing two phone call plans with equal costs The student used methods discussed by Kutzler (1996) to solve the equation. Plans A and B have equal costs when x=92.6154. In practical terms, this means that individuals making more than 92 local calls in a month should choose Plan B. These steps are illustrated in Figure 7.47. Y Y ft'» Y FS X re Rlgebra|Calc|Other|PrgnIO|Clean n Up I I 12.46 + .13-x = 24.5 .13-X+ 12.46 = 24.5 13 x+ 12.46 = 24.5)- 12.46 .13 x= 12.04 .13 x= 12.G4_ x = 92.6154 . 13 EB MWN ______RflP ftUTfl Figure 7.47: Equation solving on the TI-92 for phone call problem After spending roughly four weeks of class time exploring alternatives to by-hand manipulation with both CAS and non-CAS students, I felt encouraged by the progress that students were making with regard to problem analysis. It seemed that students were becoming more comfortable with alternative problem solving methods. When pencil-and-paper approaches provided students with a straightforward means for solving problems, they used by-hand methods. When solutions were difficult with traditional 221 methods — or when problems were more applications based — I found more students tackling problems with graphs, tables, or calculator-symbolic approaches. This feeling is reflected in my teacher journal. 1 am sensing that students are starting to recognize that the calculator is an option when solving problems. I don’t think that they are ready to abandon traditional pencil-and-paper methods for calculator-based methods. However, it seems apparent in my conversations with students that using the calculator as an initial method for solving problems seems more reasonable to them now than it did two months ago. The process of getting students to even consider using the calculator is a long one — far longer than I had envisioned before this study began. Nevertheless, I am optimistic that by year’s end, students (and I) will have a better understanding regarding the appropriate use of technology in the study of mathematics. I think that all of my students will be better equipped to solve a wide range of problems than students who know only pencil-and- paper-based problem solving methods. I anticipate that, while many students in both classes will continue to find paper-and-pencil instruction necessary for learning purposes, many will also concede that calculators may also play a useful role in this process. (November 29, 1999). During the sixteenth week of class, students were administered a second set of essays — the Systems Essay assignment — to ascertain whether or not my perceptions regarding student calculator receptivity were accurate or merely a by-product of wishful thinking. Svstems Essav Approximately six weeks after the adminstration of the Solving Linear Equations essay, students again answered questions regarding calculator use — this time within the context of solving linear systems. The first part of the Systems Essay required students solve systems using three distinct methods; (1) linear combinations; (2) substitution, and (3) graphical techniques. A pair of follow-up questions asked students to reflect on 222 their use of calculators during the school year. The follow-up questions are listed in Appendix K. Responses to the writing prompts, referred to as the Systems Essay assignment, were collected during the sixteenth week of class. The questions were assigned after students had spent considerable time solving problems with non-traditional, graphing calculator- based procedures. Additionally, students in the CAS class had been exposed to step-by- step symbolic manipulation procedures on the calculator prior to the answering the essay questions. Student responses to Item 4 of Svstems Essay As Table 7.3 suggests, at the time of the Systems Essay, most CAS and non-CAS students agreed that significant pencil-and-paper practice is needed to learn algebra well. "To learn algebra well, a student needs lots of practice with pencil and paper." Mention need for both by-hand and calculator-based Class n Agree(%) Disagree(%) instruction(%) Non-CAS 21 20 (0.952) 1 (0. 048) 10 (0. 476) CAS 24 20 (0.833) 4 (0.166) 5 (0.208) Table 7.3: Summary of student responses to item 4 of Systems Essay assignment Non-CAS students were more likely to mention a need for balance between pencil-and- paper and calculator-based methods. With regard to the statement “To leam algebra 223 well, a student needs lots of practice with pencil and paper,” the following response was typical among non-CAS students. I agree with the above statement. However, in a changing world where technology can make learning better, I feel that learning on the calculator is important as well . . . It’s too hard on a student for them to be dependent on only one of either a calculator or the old-fashioned way with a pen and paper (Marvin Waters, non-CAS student) One popular justification for continued emphasis on by-hand manipulation offered by students was its usefulness at illustrating individual algebraic steps visually. Despite the graphing calculator’s ability to represent symbolic expressions as graphical objects, many students in both the CAS and non-CAS sections favored by-hand methods to graphing or computer symbolic manipulation. The following quotes are representative of the thoughts of many non-CAS students. I am a very visual [emphasis added] learner, and it really helps me to write things out and see [emphasis added] them on paper. I constantly leam from my mistakes and the best way to see my mistakes is when I am writing out an answer. I think it is very important to leam how to write things out onto paper ... it really helps me to see [emphasis added] each part of the equation. It helps me to break things down and evaluate each section of the problem. I think without this skill I would have more trouble checking my answers as well as finding my mistakes. In my opinion a calculator is great after you already know how to solve a problem by hand. (Slovak Dennis, non-CAS student). Another non-CAS student expressed the need for pencil-and-paper work in the following manner: Algebra calls for lots of practice. Not mental practice, but on paper. Visual learning with a pencil so that you can leam from your mistakes and erase them and start anew. (Perry Downer, non-CAS student) 224 As was the case with non-CAS students, the CAS class described the usefulness of paper-and-pencil methods with regard to visualization. When you are learning algebra, you want to practice with pencil and paper to see what you are doing visually [emphasis added] (Kristy Phillips, CAS student). Another student &om the CAS class made the following comments: By doing the problem by hand first, you will be able to see all steps written out [emphasis added]. So, if you make a mistake, you can go back (Larry Lane, CAS student). In general, students in the CAS class were more likely than their non-CAS counterparts to respond negatively to calculator use. The typical CAS student stated that pencil-and- paper techniques were the de facto methods for learning about algebra. Unlike non- CAS students, several CAS members mentioned that they felt more confident when working out problems by-hand. 1 am more confident with my own work than a calculator because sometimes they [calculators] get a little tricky. However, if I knew more about the calculator and was better with it, I think I’d rather use it. (Don Sugarman, CAS student) The above passage suggests that CAS students’ reluctance to use the calculator was partially attributable to the relative complexity of the TI-92. 225 Student responses to Item 5 of Svstems Essav The open-ended nature of item 5 of the Systems Essay (see Appendix K) offered students significant latitude with regard to their responses. Several themes were evident in student writings. Students in both groups agreed that calculators were used regularly in advanced algebra class. CAS students were more likely than non-CAS students to describe “overuse” of the calculator or “calculator dependency” in their descriptions of class activities. • All students in the non-CAS group described the graphing calculator as a useful device for solving numerical and algebraic problems. In particular, they cited the speed with which the calculator generated answers and the calculator’s graphing capabilities as a powerful problem-solving features. The opinions of CAS students were more mixed, with fewer students describing the calculator as useful for solving problems or for grasping mathematical concepts. • Comparable numbers of students from the CAS and the non-CAS groups perceived graphing calculators as expert others that were helpful as “answer checkers.” • Students from both CAS and non-CAS classes described the calculators as helpful for building mathematical self-confidence. Frequencv of calculator use in class. Both non-CAS and CAS students tended to agree that the calculator was used more extensively during the study period than at any other time of their formal schooling. A sizeable number of students in both classes commented that their lack of previous experience with the calculator caused some initial anxiety, with this anxiety diminishing as the school year progressed. The following passages are representative of the opinions of students in the non-CAS class. Calculators have been used a lot this year, and at the beginning it was extremely overwhelming because none of my past teachers had really used them at all. So, of course, at first I was very confused and upset and 226 angry with calculators, but now I have found that they usually help me more than they hinder (Marvin Waters, non-CAS student). Another non-CAS student commented that: Fve, personally, really learned a lot about calculators this year. Last year we barely used them and when we did, I didn’t really understand what was going on. This year we’ve really focused a lot on calculators and I really understand it now. Also, this year I’ve learned certain programs or certain ways of solving problems on the calculator which makes math a lot easier (Alex Fiscus, non-CAS student). The non-CAS reaction to the calculator was overwhelmingly positive — with many students commenting that they had benefited from their knowledge of graphing tools. Although many non-CAS students commented that calculators were used frequently in class, none complained about excessive use. On the other hand, a number of CAS students felt that the calculator was used too frequently. In particular, CAS students believed that symbolic manipulation calculators would not be used in future classes at the high school, thus they perceived calculator-based instruction on the TI-92 of dubious worth. Because we were given the TI-92 to use only for this year [emphasis added], everything we learned we learned using those. Next year, when we don’t have those, what are we going to do? This process on the TI-92 is useless to us next year (Andy Shoeman, CAS student). Furthermore, many CAS students perceived the TI-92 as the dominant means of instruction — not merely an alternative to pencil and paper based methods. Just for the record, I think maybe we, as a class, are too focused on the calculator, and lately we have used it less, and I believe more people have benefitted. For the beginning of the year, the calculator was used to show how to do it. It was always the first example shown. From the experience of sitting among my classmates, I know many were confused 227 and the calculator made it worse. Although lately it seems we have been learning a variety of ways to do things, including methods by hand. I feel that if we leam the methods by hand first, it will prove we really know how to do it (Abby Miller, CAS student). Such a point of view differed from feelings of typical non-CAS students who noted that the calculator was used in a manner that complemented (but did not replace) more traditional paper-and-pencil based instruction. For instance, one student noted that: I think this year I have learned more about my calculator than I had ever imagined. In class we learn how to do things without a calculator and with a calcidator [emphasis added] (Ken Streak, non-CAS student) Another non-CAS student noted: Calculators have been a central part of my math class this year. We’ve learned calculator methods which allow us to supplement [emphasis added] concepts we’re studying (Sammy Funkmeister, non-CAS student). In the next section, we examine differences in CAS and non-CAS student opinion regarding the use of the calculator as a device to aid in conceptual understanding. Usefulness of calculator as a problem solving tool / conceptual aide. Non-CAS students tended to agree that the calculator had been a useful tool for performing mathematical tasks, although opinion was more divided regarding the usefulness of the calculator as an aide in building conceptual understanding. Students who viewed the calculator as a conceptual tool t>'pically mentioned the calculator’s ability to represent information in a visual manner. Additionally, many students from both classes commented that the calculator was useful for solving problems because it generated solutions more quickly than possible with by-hand methods. 228 The calculator more than anything speeds up the process that my brain normally takes. It helps me get my answers out quicker . . . graphs have also helped me tremendously because I am such a visual learner. It helps me to be able to picture the equation even before I solve it [emphasis added] (Slovak Dennis, non-CAS student). A similar point of view was described by another non-CAS student. I feel calculators have aided my understanding of algebra because . . . I can examine algebra concepts visually [emphasis added]. [For instance] by viewing graphs, I can analyze functions through determination of domain, range, etc. 1 can also see examples of non-function graphs (Sammy Funkmeister, non-CAS student). Similar comments were made by a number of CAS students. Additionally, CAS users tended to favor graphing utilities over symbolic manipulation. This calculator makes advanced algebra so much more interesting. When learning the matrixes (sic) it came in handy. I was able to punch them in as well as see the graph. It was so helpful. I think the graph is the most helpful part o f the calculator [emphasis added] (Meg Barfuss, CAS student). Although a number of CAS students agreed with their non-CAS counterparts regarding the usefulness of the TI-92, a considerable portion of CAS students did not find the TI- 92 useful in any respect. Several CAS students commented that the TI-92 “did too much work” for them. Others found the TI-92’s commands too difficult to leam. It [the TI-92] is a useful tool but in some cases it just does the work for you and you don’t leam anything. Sometimes, like when we solve equations for x or y on the calculator it is just a hassle. I find that I can do better with my algebra without the calculator and it saves a lot o f time by doing it by hand [emphasis added] (Bill Wend, CAS student). Unlike the non-CAS students, a number of CAS students complained that the use of the TI-92 made concepts more complicated and more difficult to understand. 229 The CAS students’ views are partially supported by in-class examples from my teaching journal. The journal describes a number of examples in which pencil and paper methods appear more readily understandable than CAS-based solutions. Consider the following problem, a chapter review item from UCSMP Advanced Algebra (Usiskin, et al., 1990). k x" If y = and both x and z are multiplied by any non-zero constant c, then how is y affected? In the CAS class, we typed the given expression for y into the TI-92, then multiplied both X and z by c. As shown in Figure 7.48, the TI-92 CAS is unable to “cancel out” the common c" found in numerator and denominator. f^^jSl^glbralca^cTotherTprgn I olci e a r ‘ a -z T l I y = - y = k -z " x " z " k-(c-x)*^ I y = - y = k-(c-z) " (c x)" ( c - z ) " y= Figure 7.48: Calculation highlighting inability of TI-92 to simplify fractional expressions with common factors The October 28, 1999, journal entry discribes the effect that the example had on student attitude regarding the worth of the TI-92 at solving such problems. Notice that the TI-92 was initially did not "recognize" that c " and c" cancel each other out! Because my students have not seen negative exponents before (or have forgotten their significance), the TI-92 output meant little to them ... I can't find any function that will cancel out the c terms here! After looking at the same problem by hand, the class (and I) concluded that this problem was more easily solved with by-hand methods (October 28, 1999). 230 Problems that weren’t easily simplified by the TI-92 — particularly textbook problems which occurred early in the school year — added weight to the popular notion among CAS students that symbolic manipulation on the TI-92 was less useful than standard pencil-and-paper work. Calculator as “expert other.” CAS and non-CAS students alike felt that the calculator was useful as an “answer checker” that helped build confidence with by-hand calculations. The following passages represent the views of many non-CAS students. They [calculators] have been extremely helpful... by aiding in checking answers and actually seeing what the graph would look like. On this particular test, the calculator was very helpful because I was very unsure of my original answers. The calculator confirmed that I was right which was very relieving [emphasis added] (Marvin Waters, non-CAS student). Similar views were echoed by CAS students. The calc[ulator] has come in handy because I get the answer on the calc[ulator] then on paper. 1 know the correct answer so it helps me figure out the steps to find the solution on paper (Mack Nester, CAS student). Another CAS student noted; This semester the calculator . . . has helped me to check my answers and find them as well (Meg Barfuss, CAS student). Adthough CAS students found the graphing capabilities of the calculator helpful, for the most part CAS students did not find the TI-92 useful for teaching algebraic concepts. The passage below is typical of CAS students. Calculators have been useful in any situation we needed to graph, but they haven’t been so useful in teaching concepts. I think this because in the learning and understanding new concepts, we understand it better 231 with pencil and paper methods [emphasis added] (Marge Herman, CAS student). Calculator Algebra Questionnaire Approximately ten weeks after the administration of the Systems Essay assignment — during the twenty sixth week of this study — students in both CAS and non-CAS classes were asked to complete a set of six short follow-up questions dealing with calculator use. This questionnaire — which is referred to as the Calculator Algebra Questionnaire — was unlike previous essays that the students had completed. Most notably, the questionnaire attempted to uncover student motivation for graphing calculator use. Appendix L lists all items from the questionnaire. Student responses to Calculator Algebra Questionnaire items are discussed in several sections below. Student responses to Item 1 of Calculator Algebra Questionnaire Item 1 of the Calculator Algebra Questionnaire asked students to identify the model of calculator that they used most often in class. The actual questionnaire item is shown in Appendix L. Item 1 addressed two primary concerns of this study. • How many CAS students identified non-CAS calculators as their preferred calculator? • Were non-CAS students using CAS calculators? If so, how many non-CAS students identified CAS calculators as their preferred calculator? Because non-CAS calculators had been introduced formally into the CAS class several weeks prior to the administration of the Calculator Algebra Questionnaire, concerns existed regarding the number of CAS students that had discontinued use of the TI-92 in 232 favor of the TI-83. Likewise, a concern existed regarding the number of non-CAS students using symbolic manipulation tools in class. Table 7.4 reports student response to item 1. Group n TI-92 TI-89 TI-83 TI-85 other Non-CAS 22 0 5 17 1 1 CAS 22 17 0 6 0 0 Table 7.4: Student response to Item 1 of Calculator Algebra Questionnaire A substantial number of students in the non-CAS class identified a CAS equipped calculator as their preferred hand-held grapher. Specifically, three of the twenty-two non-CAS students identified the TI-89 as their primary class calculator. Two non-CAS students specified both the TI-83 and TI-89 as their regular class calculator. Findings were similar for the CAS group, with five of the twenty-two students identifying non- CAS calculators (the TI-83 in each case) as their preferred calculator. One CAS student specified both the TI-83 and TI-92 as his preferred calculator. Student responses to Item 6 of Calcidator Algebra Questionnaire Item 6 presented students with an opportunity to discuss their general feelings regarding mathematics class during the first twenty-six weeks of the school year. The specific essay item is illustrated in Appendix L. As was true in earlier assessments, CAS students generally expressed a desire to use the TI-92 less often in class. Typically, CAS students anticipated that TI-83 methods and by-hand methods would be used in future classes - hence, they wanted to leam 233 more about these methods. On the other hand, since the TI-92 and TI-89 were not used by other teachers at the high school, CAS students seemed less interested in learning about features specific to these calculators. The emphasis on instruction with the TI-92 calculator left a number of CAS students feeling underprepared. Also I really wish that while I was learning how to do things on the TI- 92, I would have learned how to do them on the TI-83 as well. I am very worried about the fact that next year without the TI-92 I am going to be lost. I think I rely on the TI-92 way too much and I don’t know how to do a lot of these things by hand (Marybeth Short, CAS student). Another CAS student commented in the following manner: I would of (sic) loved to use the calc[ulator] less. Actually, we use it for everything. I would also of (sic) liked to use the TI-83 much more instead of the TI-92 because in any other class and tests like ACT, SAT you can’t use the TI-92. So I’d of (sic) rather learned everything on the TI-83. I think the teaching techniques are good, but you should only use the TI-83 (Mike Badloss, CAS student). In general, the CAS students expressed more dissatisfaction with class compared to the non-CAS group. Student satisfaction levels are shown in Table 7.5. Group change less go study- More TI- U se o n l y 1 nothing calculator f a s t e r m o re 92 83 c a l c u l a t o r CAS 6 11 1 1 1 5 2 non-CAS 13 4 1 5 2 1 0 Table 7.5: Student responses regarding possible class changes as a response to item 6 of Calculator Algebra Questionnaire Of the twenty-two non-CAS students that returned the questionnaire, roughly 60 percent said that they would change nothing about the class. Nearly 23 percent 234 commented that they would have taken the class more seriously and worked more diligently at the beginning of the school year. Approximately 18 percent of non-CAS students commented that they felt that the calculator was used too frequently in class. Typical comments from non-CAS comments were as follows: I would prefer to use the calculator the same amount. I like having an equal background in both calculator methods and by-hand methods. I like using the TI-83. It’s advanced enough for my classes, but I can use it for more basic applications as well (Sammy Funkmeister, non-CAS student). On the other hand, over 70 percent o f the CAS students commented that they wish class had been structured differently. For instance, roughly a quarter of students would have preferred to use the TI-83 instead of the TI-92. Half of the respondents commented that they would prefer to learn more by-hand methods — particularly at the beginning o f the school year. Many of the CAS students commented that the TI-92 was a confusing calculator with which to start the year. In the beginning of the year I would have liked if we did more by hand and then worked up to the calculator ... by doing the calculator method first I was really confused (Larry Lane, CAS student). Another CAS student commented in the following fashion: In the beginning it was frustrating because I didn’t understand what I was doing and people kept asking the same questions over and over again. I would have prefered to use the calculator less and by hand work a lot more. I like how now we are learning to use both calculators (the TI-83 and TI-92) but it was difficult switching back and forth (Bess Nunn, CAS student). 235 Students using CAS equipped calculators in the non-CAS class experienced initial difficulties because features of this calculator were never explicitly discussed in class. I think the first half went pretty good. The TI-83 was used a lot and it was hard to follow because I had a TI-89, but once I learned how to use the TI-89,1 was fine (Kris Cupid, non-CAS student). Another non-CAS student using the TI-89 made the following comments: If I could change anything about the first half of the year, I would want you to help the people with the TI-89’s more because not many people in our class have them, but there are a few, so it can be frustrating when you don’t teach the TI-89 people also (Todd Sizemore, non-CAS student). Entries from a year-long teaching journal indicate that I often worried at the beginning of the school year about the level of expectations I had set for students — with regard to both their knowledge of the calculator and their understanding of basic algebra. Sometimes I feel like I am continually assigning homework and problems that students cannot do, then going back and "repairing" my instructional errors of judgement. It can be really nerve-wracking. Every time I've taught this course, the students have a hard time with my treatment of chapter I. No matter how much I try to "go slow" or "make content easier,” I feel that I probably have too high of expectations for the students at the beginning of the year. It seems to take me about 10 weeks to really get a feel for "where the students are at" cognitively speaking (September 13, 1999). When teaching students about linear equations, I made the following comments in the teaching journal. In hindsight, I think it was a mistake to diverge from the textbook this much. Finding equations for lines (when given particular points on lines) presupposes that students have a pretty solid understanding what linear equations really are (lines consist of all points that satisfy the line's equation). In addition, students need to be able to imderstand multiple representations of lines (graphs, ordered pairs, and equations). I think I 236 was expecting a little too much of them at this point. The class ended today with students somewhat frustrated (September 8, 1999). The teacher journal and student comments suggest that learning a how to use a new technology while simultaneously covering new course content made for a difficult Fall quarter for both teacher and students. Lack of calculator experience caused initial stress - particularly with CAS students. However, student discomfort appeared to dissipate with repeated calculator use and perserverance. Student responses to item 2(a) help to support this general conclusion. In the paragraphs below, student responses to item 2(a) and 2(b) of the Calculator Algebra Questionnaire are examined. Smdent responses to Item 2 of Calculator Algebra Questionnaire Item 2 of the Calculator Algebra Questionnaire consists of two parts — both concerned with student comfort level with calculators. Part (a) asked students to describe their comfort level. Part (b) asked students whether or not they used the calculator outside of the mathematics classroom (see Appendix L). Additionally, part (b) asked students whether or not they had shown their calculator to family members. Positive responses to any portion of part (b) suggest higher confidence levels with regard to the calculator. Table 7.6 illustrates responses to item 2(a). 237 TI-92/89 TI-8x users users very not veiry not Group comfort comfort comfort comfort comfort comfort CAS 10 8 0 3 3 0 non-CAS 2 3 0 3 15 1 Table 7.6: Student responses regarding comfort level with calculator Note that approximately 56 percent of students in the CAS class felt very comfortable using the TI-92 calculator, with the remaining TI-92 users describing a comfortable level of expertise. Results were similar for students in the CAS class that used non- CAS calculators (the TI-83 was the only other calculator used by students in the CAS class). Half of the students using non-CAS calculators in the CAS class described their level of expertise as very comfortable, with the remaining TI-83 users feeling comfortable with the calculator. None o f the students in the CAS class reported feeling uncomfortable with their calculator, although several echoed experiencing frustration with calculators at the beginning of the school year. Table 7.7 summarizes CAS student comfort levels with calculators. 238 Very Comfortable comfortable Table 7.7: CAS Student comfort level with calculator. On the other hand, note that fewer non-CAS students reported feeling very comfortable with the TI-83 - the only calculator used for classroom instruction. Of the 18 non-CAS students that described themselves as TI-83 users, only 3 described their use with the calculator as very comfortable. Roughly 83 percent of non-CAS students reported feeling comfortable with the TI-83. One non-CAS student reported feeling uncomfortable with his calculator — an older TI-82 model not used in class instruction. Results were similar for TI-89/92 users in the non-CAS class. Two of the five students using TI-89 calculators in the non-CAS class described their level of expertise with the calculator as very comfortable, with the remaining three CAS users feeling comfortable with the calculator. Overall, only one of the students in the non-CAS class felt discomfort with the calculator. As was the case with the CAS group, several students 239 from the non-CAS section mentioned initial frustration with calculators at the beginning of the school year — with frustration gradually replaced by increased confidence. It has been a gradual development, but I have become as comfortable with the calculator as I am with [calculations generated by] myself. I understand how and when to use the calculator, so 1 am more confident with myself (Slovak Dennis, non-CAS student). A CAS student made the following comments: Gradually, I’m becoming more and more comfortable, learning each process . . . keeping them straight can be difficult though (Kelly McMaran, CAS student). Table 7.8 highlights overall student comfort levels of non-CAS students with calculators. 240 Uncomfortable Very comfortable Comfortable Table 7.8: Non-CAS Student comfort level with calculator To gain further insight into student confidence with calculators, the second part of item 2 asks students to describe calculator use outside of mathematics class. Furthermore, item 2(b) asks students whether or not they “have . . . ever shown any of your family members how to use the calculator.” Positive responses to these items suggest that students did in fact feel comfortable with their calculator. Table 7.9 displays the results of student responses on item 2(b). Do you use the Have you ever shown any o f outside o£ ma.th members how to use the c l a s s ? calculator? Class Yes No No Response Yes No No Response Non-CAS 16 6 0 12 9 1 CAS 17 3 2 11 10 1 Table 7.9: Student responses regarding item 2(b) of Calculator Algebra Questionnaire 241 Student responses to item 2(b) of the Calculator Algebra Questionnaire suggest that student confidence with calculators is similar across groups. Approximately 73 percent of non-CAS students reported using their calculator outside of math class. A similar portion of CAS students made such claims — with over 77 percent of CAS students reporting calculator use outside of mathematics class. Students most commonly used calculators outside of class to complete science homework or to calculate their grades. Several students reported using calculators to teach mathematical concepts to younger brothers or sisters or to compute tax return forms with parents. Student responses indicated significant familial participation with the calculator at home. I have a seventh grade brother in algebra, and I will often explain to him easier and better calculator methods. I show my dad all the new things that I can leam in class with my calculator, because he is often very interested (Slovak Dennis, non-CAS student). Describing his interactions with his mother using the calculator, a non-CAS student made the following comments. Yes, I use it outside of school. In fact, this weekend I used it to help my mother figure out which plan to use for her cell phone. We figured out how much she spent last year and how much she could have spent on a different plan. The calculator was very helpful (Michael Burboski, non- CAS student). The following comments are typical of CAS students: Sometimes I use the calculator outside of math class. I use it once or twice a week for science class . . . I haven’t ever shovm my calculator to any of my family members besides for my sister who I often help so she can finish her homework (Don Sugarman, CAS student). 242 The fact that sizeable numbers of CAS and non-CAS students used calculators outside of class — both as a study tool and as a means for solving real-world problems — suggests that students felt comfortable with their calculators by the twenty-sixth week of the study. Student responses to Item 5 of Calculator Algebra Questionnaire Item 5 of the questionnaire asked students to describe specific ways in which they used their calculators. The actual prompt is shown in Appendix L. Since CAS-based calculators include symbolic manipulation features not typically found on standard graphing calculators, 1 anticipated that CAS students would mention by-hand manipulation more often in their responses to item 5. For instance, because considerable time had been spent with the CAS group solving equations in a step-by- step manner on the TI-92, it seemed reasonable to suppose that CAS students would mention “completing the square” or “factoring expressions” more frequently when describing scenarios in which the calculator was useful. Similarly, I anticipated that non-CAS students would mention graphical features of their calculators - “finding intercepts” or “graphing functions” — more often than CAS students, since more class time had been devoted to graphing in the non-CAS section. Actually, students in both classes tended to answer the question in terms of their general use of the calculator without specific mention of problems well-suited for calculator use. As Table 7.10 notes, equal numbers of students in the CAS and non- CAS classes described the calculator most useful as an “answer checker” — with many 243 students mentioning the confidence-enhancing ability of the calculator. Furthermore, students in both CAS and non-CAS classes specifically mentioned the calculator’s ability to confirm the correctness of by-hand work. I’m never 100% sure of my by-hand work; so it’s reassuring -when I check the answer on my calculator [emphasis added]. The calc[ulator] also provides options for solving a problem that can be solved several different ways (Millie Stroud, non-CAS student). Another non-CAS student noted that: I find the calculator most useful when checking my work — both homework and test work. I feel more confident in my answers when a calculator check confirms them [emphasis added] (Sammy Funkmeister, non-CAS student). CAS students made similar comments. I always use the TI-92 when I take tests or do homework. I think the calculator really helps because it assures me on my answers. / fin d the calculator most useful when checking problems [emphasis added]. It was really useful when we multiplied matrices earlier this year (Mack Nester, CAS student). As Table 7.10 illustrates, equal numbers of students in the CAS and non-CAS classes (ten students in each group) noted "checking answers” as the most useful function of their calculators. check equation pretty hand Class work graphs solver print work matrices tables CAS 10 7 3 1 5 5 0 non-CAS 10 10 1 0 1 3 1 Table 7.10: Student responses regarding uses of calculator 244 As noted in Table 7.10, students in the CAS group were more likely to mention the usefulness of their calculator with regard to manipulation, with five CAS students commenting that the calculator was an aid in symbolic manipulation activities, compared to only one non-CAS student. Approximately 23 percent of CAS students mentioned by-hand work, as compared to 5 percent of the non-CAS group. One CAS student made the following comments. I also like how the calculator shows all of the steps, so if you made “hand-made” mistakes, you could find where you went wrong (Faye Dickinson, CAS student). One of the non-CAS students mentioned that his TI-83 helped him with by-hand work. 1 often forget the way to find the solution to a problem by hand, but remember it on the calculator. After doing it on the calc[ulator], it often jogs my memory and I remember how to do it by hand (Michael Burboski, non-CAS student). Conversely, students in the non-CAS group were slightly more inclined to mention graphing aspects of their calculators when compared to their non-CAS counterparts. Ten of the twenty-two non-CAS students mentioned graphing capabilites as the most useful purpose of the calculator. A somewhat smaller portion of CAS students — eight of twenty-two — mentioned the benefit of graphical techniques on their calculators. I had a lot of trouble with the ball dropping/throwing problem and velocity, etc. and could not understand when doing the problems by hand. After doing the problem on my calculator, however, I could actually see what the ball’s height looked like, and it all finally clicked (Marvin Waters, non-CAS student). Another non-CAS student expressed similar opinions about graphing. 245 Graphing has become one of my favorite parts of quadratic equations. I love the fact that with the calculator I always have at least two options. Learning it has been very worth my time (Slovak Dennis, non-CAS student). In general terms, students in both groups tended to find the calculator most useful as an “error checking device” that served as a useful supplement — not a replacement — for by hand manipulative work. Graphical techniques on the calculator were popular among all students, with more students mentioning graphing than by-hand capabilities in either class. Student responses to Item 3 and Item 4 of Calculator Algebra Questionnaire Items 3 and 4 of the Calculator Algebra Questionnaire required students to examine the relevance of mathematics and calculator-based instruction within the context of higher education and career training. Because the students in both the CAS and non- CAS groups were sophomores or juniors at the time of the study, their career plans typically included 3 or 4 different occupations. Student career preferences were tabulated and are described in Table 7.11. Majors that typically require some advanced mathematical study (at or beyond the calculus level) are highlighted by an asterisk in the table. 246 College Major Number of Number o non-CAS studer students archaelogy 1 0 architecture* 2 2 business* 5 3 computers* 1 1 criminology 1 0 engineering/math* 2 0 fashion design 1 4 fine arts 1 I j ournalism 1 1 law 3 0 medicine* 4 5 music 2 0 nurse 0 1 pilot* 0 1 political science 2 2 psychology 3 1 science* 3 1 ski instructor 0 1 speech pathology 1 0 teacher education 2 3 unknown 1 3 total 36 31 fields requiring 17 13 calculus Note: Major fields that typically require calculus or above are indicated with an asterisk (*) Table 7.11: Student career interests of non-CAS students as reported in items 3 and 4 of Calculator Algebra Questionnaire In general, all students in both the CAS and non-CAS classes spoke of pursuing some type of post-secondary education. In general, a larger percentage of the non-CAS selections were fields requiring some sort of advanced mathematical study. Thirteen of the thirty-one CAS fields (i.e. 42 percent) typically require a sequence of college 247 calculus. On the other hand, seventeen of the thirty-six fields selected by the non-CAS group (i.e. 47 percent) typically require calculus training. End of the Year Short Answer Essays Approximately thirteen weeks after the administration of the Calculator Algebra Questionnaire — during the thirty-ninth week of the study — students once again were called upon to answer a series of essay questions dealing with calculator use. The End of Year Short Answer Essay consisted of four open-ended items that encouraged students to reflect upon their calculator experiences during the past school year. In an attempt to detect possible changes in student attitude during the study period, the items from the End of Year Short Answer Essay were designed to expand upon previous writing prompts. Appendix M illustrates each of the items from the End o f Year Short Answer Essay. During the last ten weeks of the school year, instructional emphasis in both CAS and non-CAS classes shifted away from calculator use. Although students continued to use calculators, much of their classwork emphasized more traditional paper-and-pencil based activities. The decision to refocus instruction was deliberate and was influenced by concerns regarding readiness of students for future mathematics courses. These concerns are noted in my teaching journal. A number of students in the CAS group have expressed anxiety regarding their ability" to perform by-hand manipulation. Others in the class have expressed a desire for more practice with the TI-83. Because all of my students will continue taking mathematics courses in high 248 school and in college, in many cases with instructors who resist technology, I feel compelled to “ease back” into more traditional modes of teaching at the end of the school year. During the last ten weeks, I plan to use the graphing calculator less frequently — opting to employ by hand methods more frequently. I still plan to use the calculator in both classes, but more as an “answer checker” and less as a conceptual teaching device (April 3, 2000). In the following paragraphs, student responses to the four items are discussed within this context of decreased calculator use. Student response to item 1 of End o f Year Short Answer Essay Balanced perception of calculator use. While none of the non-CAS students mentioned a de-emphasis of calculator-based methods, several CAS students did note such a change. In responses to item 1 of the End o f Year Short Answer Essay, one CAS student made the following observation. The calculators played a big part of our math class this year. We used them for checking over homework problems, graphing equations, unit circle, and setting up tables. They were heavily used more towards the front of the year, but were less used toward the end [emphasis added] (Paige Burke, CAS student). Another CAS student attributed improved performance levels to the de-emphasis of the calculator. At the beginning, 1 wasn’t understanding things which reflected on my tests and in turn resulted in a low grade. At the end o f the year, I am concentrating on pen and paper rather than my calculator, and my grades have rapidly increased [emphasis added] (Mack Nester, CAS student). Increased emphasis on by-hand methods noticeably influenced student perception in the CAS class. For the first time, a number of CAS students remarked that calculator 249 methods and by-hand methods were used in a balanced fashion in class activities. This point of view was a significant departure from CAS students’ earlier contentions that “calculators were used for everything” (Mike Badloss, CAS student). Balanced instruction. At year’s end, CAS and non-CAS students remarked that calculators had been used in a balanced fashion — with pencil-and-paper methods and calculator methods assuming equal status in class activities. Complaints of calculator dependency continued to be rare among non-CAS students. Due in large part to the answer checking abilities of graphing calculators, non-CAS students perceived the calculator as an aid complementing, but not replacing, by-hand methods. The usage of calculators this year did not impair my by-hand method calculation abilities. I still used by-hand method calculations on tests, and the calculator was a quick way to check my answers (Mick Iceberg, non-CAS student). For the first time, similar points of view were expressed by CAS group members. As was the case with non-CAS students, CAS users found the TI-92 useful for verifying by-hand methods. Even though we would use by-hand methods for FOILing, it is also nice to be able to do it on the calculator. This also lets us know if we have the correct answer after trying the problem by hand (Betsy Priestly, CAS group). Another CAS student echoed similar feelings. In math class this year we have used calculators a lot. Even though we used them a lot, I don’t think we used them to an extent where we were dependent on them. Their main purpose mostly was to check work or discover new ways to solve problems (Manny Fank, CAS student). 250 However, unlike the non-CAS class, a number of students continued to complain about their over-reliance on calculator-based methods. I feel that I have become way too dependent on them (calculators) and I am kind of afraid for math next year. The only main reason why I feel that the calculators were not all that helpful for me is just because I really don’t work well with them and I learn much more if I am forced to do the problem by hand (Sonny June, CAS student). Overall, students in both classes recognized that calculators were useful tools to solve problems. Response in the non-CAS class to calculators was overwhelmingly positive — with no students complaining about over-reliance on the calculator by year’s end. By the thirty-ninth week of class, many CAS students shared the views of their non-CAS counterparts, however a number of CAS students continued to describe themselves as “calculator dependent.” Positive student view of calculators. Despite any student misgivings regarding calculator dependency, CAS and non-CAS students generally agreed that calculators had provided them with positive learning experiences during the school year. While non-CAS students continued to speak highly of the calculator, student opinion of the calculator was noticeably more positive in the CAS section at the end of tlie school year than in earlier prompts. Abby Miller, a vocal critic of calculator-based instruction, made the following comments at year’s end. In the beginning of the year the calculator hurt my learning of math because I looked at it negatively. I think we used it too much and forgot to focus on the math concepts (this is when we studied functions). Eventually I became used to using the calculator and I realized how it aided me in learning math instead of just doing it for me (Abby Miller, CAS student). 251 In several instances, student attitude regarding the calculator shifted dramatically during the final weeks of class. Marybeth Short, a CAS student who several months earlier complained that “1 think 1 rely on the Tl-92 way too much, and 1 don’t know how to do a lot of these things by hand,” made the following comments about the calculator at year’s end. Overall, 1 think the calculator has played a huge role in my learning this year in math class. 1 am actually very thankful to have gained knowledge about the calculator. 1 now do not find the calculator to be so intimidating (Marybeth Short, CAS student). Descriptions of calculator usage by non-CAS students were universally positive, with many students describing the calculator in relevatory terms. This year’s more in-depth look into the many functions of the TI-83 calculator will change the way I perform and mentally process mathematics for the rest o f my life [emphasis added]. The calculator has not only boosted my trend in mathematics grades, but also my respect and applicable use of mathematics in the everyday world . . . 1 can’t stress the use of calculator more to others simply because they have helped me so much. Currently, 1 wish calculators would have been of more liberal use in the school years prior to my sophomore year. 1 would suggest that all teachers, no matter what grade-level, stress the use of calculators to their students to keep them ahead in the fast growing world of today (John Beige, non-CAS student). Another non-CAS student made the following observations about the TI-83. Almost everything 1 learned this year was in a calculator form. Keep using them!!! 1 feel sorry for teachers who are against using them. They (calculators) really advocate the whole learning experience (Jeff Jewell, non-CAS student). Overall, CAS and non-CAS students agreed that the calculator played an important role in their learning of mathematics. The difference between the two groups lie in their 252 beliefs regarding the proper emphasis of the calculator. While calculators were universally accepted in the non-CAS group - with no mention of calculator over reliance from non-CAS students - a number of CAS students continued to express anxiety regarding calculator dependency, despite the fact the calculator was used rarely during the last ten weeks of instruction. Prior classroom experiences and their relationship to calculator motivation. As previously noted, CAS and non-CAS students enjoyed using their calculators as “answer checkers.” Specifically, many students solved problems using by-hand manipulation, then verified their solutions using graphical, tabular, or calculator- manipulation methods. Students cited the graphing calculator’s ability to represent solutions in multiple representations as a primary motivation for using the device. I think calculators helped me because the calculator helps people and me look at the math problems in numerous ways, such as graphically, tables, equations, etc. (Todd Sizemore, non-CAS student). CAS students used the TI-92 in a similar fashion. With a calculator I can find most answers and also check them. On test and quizzes they are great to have because you can check all your work to make sure you got the right answer. You can also use the solve function to find the answer for most equations. It’s also very helpful to look at tables and graphs to see where lines intersect (Don Sugarman, CAS student). At year’s end, many students focused on the calculator primarily as an answer checker. This highlighted a popular belief that “finding correct answers” was the primary goal of mathematics instruction. 253 A number of student responses suggested that mathematics instruction in previous classes dealt primarily with finding solutions to manipulation-oriented problems. Hence, the importance of “finding answers” continued to be of principal importance to my students in advanced algebra. Generally, calculators were not presented to students as a tool to attain conceptual understanding, but rather as a way to find answers without much understanding. For instance, with regard to prior mathematics instruction, one CAS student noted: Prior to this class, I had hardly used the calculator to do anything other than basic calculations. To be honest, I don’t think any of my other math teachers have had the knowledge that Mr. Edwards has about the calculator. All of my teachers have seemed to have been afraid that their students will depend too heavily upon the calculator. It was almost as if it was considered to be a way to “just get by” or “cheat” in math class without really doing the work. This year my teacher seemed greatly enthusiastic about the calculator. He used the calculator not only as a way to check work, but as a way to learn new concepts as well (Marybeth Short, CAS student). Several students in the CAS class commented that previous math teachers stressed that conceptual understanding was not related to calculator knowledge. As one CAS student noted: Most of my other teachers have stressed that the calculator was not as important as “pencil and paper” work. They usually taught all new ideas on the overhead or board, instead of on the calculator . . . They always told us we needed to just understand the concepts, and not worry as much about the calculator (Laura Parisi, CAS student). Despite a popular view among students and classroom teachers that calculators are not conceptual tools, a number of students in both classes recognized the usefulness of the 254 calculator as a means of learning concepts and generating hypotheses. For instance, a non-CAS student made the following observation regarding the TI-83: Calculators have played a supplemental role in math this year. In other words, they’ve been used to reinforce concepts our teacher has taught us. For example, our understanding of functions was enriched by graphing equations on the calculator. This helped us to really visualize the concepts of domain and range, etc. Also, we used calculators to ‘discover’ patterns (and therefore algebraic rules) for certain lessons (like logarithms). This ‘hands-on’ approach allowed us to become more independent learners. (Sammy Funkmeister, non-CAS student) Similar points of view were expressed by CAS students. Calculators have played a significant role in math this year. We used them a lot in class to graph equations and other similar things. This type o f learning was new and different to me because it was more conceptual and graphs than anything [emphasis added] (Bill Wendf CAS student). Along the same lines, a number of students in both classes remarked on the calculator’s ability to generate specific examples that were useful in generating general algebraic rules. Recall that the idea of calculator as hypothesis generator formed the basis of many activities in both classes. Calculators have helped me very much in learning math this year. They mostly help me when I can’t really remember a specific rule, but I have an idea about what it could be. I just plug examples into the calculator until I can remember what the rule is by looking at the answers (Tony Dennis, non-CAS group). Student response to item 2 of End o f Year Short Answer Essav Item 2 of the End of Year Short Answer Essay asked students to describe experiences with calculators in their previous mathematics classes. Table 7.12 describes calculator experience levels of students prior to the study period. 255 Used calculator No previous for simple Used calculator Class es^erience calculations graphs/tables Other CAS 3 13 4 3 non-CAS 2 11 5 4 Table 7.12: Student Responses regarding previous calculator use Responses were coded into three broad categories: (1) no previous experience with graphing calculators; (2) graphing calculators used only for simple arithmetic; and (3) graphing calculators used to generate graphs and/or tables. Responses that didn’t fit into one of these three categories were coded as “other.” Responses to item 2 were strikingly similar for both CAS and non-CAS students. Several recurring themes resonated in the responses of students: • Although students were aware of graphing calculators prior to advanced algebra (and in many cases owned graphers prior to my class), they had only used calculators to perform simple arithmetic tasks. • The overwhelming majority of students had never graphed anything on the calculator before — despite the fact that the UCSMP Algebra text includes content on graphing linear and quadratic equations. Quotes from students serve to illustrate these themes clearly. First I provide a response to the prompt from a student in the non-CAS group. My teachers never used the calculator to try new concepts. We would always leam by hand, make graphs that take so long to make, that once it was finished I couldn’t remember the initial purpose of composing it. . . I can only recall two times when my 8'*’ grade teacher ever showed us a TI-83 or “GRAPHING” calculator. The first time we watched as he played around with it, and the second time I got to touch it, but it was never clearly explained to me what it could do (Perry Downer, non-CAS student). 256 CAS students had similar classroom experiences. Prior to this year I did not even own a TI-83. I never had a math teacher who placed any emphasis on the calculator. It was only to be used to check answers. Loveless never even had a calculator in the front of the room with him . . . I think because I never relied on a calculator before, I went into calculator shock at the beginning of this year (Abby Miller, CAS student). In light of the fact that a majority of students had learned basic algebra using by-hand manipulation methods, and because a view of the calculator as a means of “getting by” or “cheating” was popular among many at the school, it seemed unlikely that many students would consider eliminating pencil-and-paper methods from the mathematics curriculum. Yet, because a number of students in both classes recognized the calculator as a conceptual teaching tool capable of finding solving problems quickly and accurately, questions regarding the obscelence of paper-and-pencil algorithms are legitimate. Students responded the statement “Calculators will eventually make pencil and paper mathematics obsolete” in item 3 of the End o f Year Short Answer Essay. Student response to item 3 of End o f Year Short Answer Essav Generally speaking, smdents in both classes tended to agree that calculators would not replace pencil and paper mathematics — particularly in the near future. Of the two groups, CAS students were more likely to agree that calculators would eventually eliminate the need for pencil-and-paper computation, although more CAS students saw a continued need for by-hand manipulation than did not. Table 7.13 illustrates student reaction regarding the obsolence of paper-and-pencil mathematics. 257 Calculators will e v e n tu a lly make pencil and paper math em atics o b s o le te . C l a s s Y es No Not Sure non-CAS 5 15 2 CAS 7 11 4 Table 7.13: Student response regarding obsoiesence of paper-and-pencil mathematics Students from both classes provided similar explanations for the continued reliance on pencil-and-paper based methods. Rationale for the study of paper-and-pencil methods included the following. • Paper and pencil methods require less expensive hardware than calculator-based techniques. • Certain problems are more easily solved by-hand than with a calculator. • Concepts are generally more easily understood using by-hand methods. • Teachers resist using calculator-based methods in class. Most continue to prefer by hand methods. By-hand manipulation was popular among students because it enabled them to to see individual steps more clearly than possible with the calculator when solving equations. CAS and non-CAS students equated by-hand competence with conceptual understanding. One non-CAS student highlighted the continued need for by-hand methods with the following example. I truly won’t understand why that’s the answer without being taught how to do it with the pencil/paper method. For example, finding the inverse 258 of the equation j/= 3 jc^. This can be easily found with the calculator, but I did not understand why until we were taught how to solve by-hand to switch the x and y-values . . . without by hand methods, I think math is somewhat hollow (Millie Stroud, non-CAS student). Although CAS students had repeatedly solved equations in a step-by-step, manipulative fashion on the calculator — many CAS students agreed that paper-and-pencil calculations displayed “how a problem was done” more vividly than possible on the calculator. I think most knowledge about math is learned through hand-written work. Hand-written work gives the student a visible and mental track of what work was done and how the problem is solved. Calculators don’t always show the individual steps to solving equations (Manny Fank, CAS student). Another CAS student echoed these comments, while suggesting that paper-and-pencil work served as a better evaluative tool of student understanding. Paper and pencil is more thorough and shows to a teacher knowledge and absolute certainty that the student knows how to do the problem on his or her own. I like calculators because they are quicker, but I think you leam much better on pencil and paper (Mike Badloss, CAS student). Several non-CAS students provided political reasons justifying the continued use of pencil-and-paper methods. These included the relatively high cost of calculators and teacher resistance. I believe that in the workplace, calculators will run pencil/paper out of town. Companies have to make money and save time. If they need a calculation, they will take the quickest way out and use a calculator or computer or basically technology. However, in school, calculators will never make them [by-hand methods] obsolete. The school board will not allow it. Students will always have to use by-hand techniques and a calculator to check their work. The curriculum will never allow an all calculator class (Kris Cupid, non-CAS student). 259 Similarly, another non-CAS student points to the stuggle to precipitate change in school settings. The only thing that’s stopping calculators from being used instead of by hand method ways is because there are too many math teachers who are against calculators and all about the old ways (Todd Sizemore, non-CAS student). Students predicting the eventual obsoiesence of paper-and-pencil math envisioned a mathematics curriculum similar to that proposed in the text Improving Mathematics Teaching with DERIVE (Kutzier, 1996). For instance, CAS and non-CAS students that predicted a decreased emphasis on pencil and paper methods described the following features of future mathematics classes. • Mathematics will be more visual • Students will be able to solve traditional problems more quickly and accurately • There will be a greater emphasis on conceptual understanding of mathematics • Standard mathematics problems will be less accessible to paper-and-pencil methods As one CAS student noted: This year I used the calculator more than using my pencil and paper. Not only is the calculator a faster way to get an answer, but it is a great tool so we can see our work visually [emphasis added] (Meg Barfuss, CAS student). With regard to course content, non-CAS student John Beige made the following comments. In advanced algebra all year long we have been fitting quadratic models (lines) to specific given data. Doing an operation like this on paper would take tons of time and more teaching and instruction would be 260 necessary. Instead of finding proper models now we just put in x and y values (in STAT EDIT) and let the calculator find a sufficient model for the data. The reason I brought up this particular operation was because I don’t know how to find models from given data very well at all; maybe linear lines of data, but certainly not quadratic regressions or cubic regressions. Here, in this instance, paper and pencil were immediately thought to be inefficient or way too complex to complete these operations so we didn’t bother with pencil and paper and went right to using just calculators. This is what I feel will probably happening in the future with school math class because the curriculum always gets more complex and less “do-able” with a pencil and paper (John Beige, non- CAS student). Student response to item 4 of End o f Year Short Answer Essav The final item from the End of Year Short Answer Essay revisited student satisfaction with calculator use in mathematics class. Recall that during the twenty- sixth week of class students were provided with an opportunity to discuss possible instructional changes regarding the calculator. At that time, a majority of CAS students desired to use the calculator less frequently. Meanwhile, most non-CAS students were pleased with instruction (refer to Table 7.5 for details). The intent of item 4 of the End of Year Short Answer Essay was to re-examine student satisfaction following a de emphasis TI-92 use in the CAS class. 261 Group change l e s s go s tu d y More TI- Use only 1 nothing calculator f a s t e r m o re 92 83 c a l c u l a t o r CAS 11 11 0 0 0 8 4 non-CAS 19 3 1 1 0 0 0 Table 7.14: Student Responses regarding calculator satisfaction as indicated on item 4 of End of Year Short Answer Essay Table 7.14 summarizes student satisfaction at the end of the school year. As Table 7.14 illustrates, half of the CAS students continued to feel that calculators had been used too often in classroom instruction, while half of the CAS students believed that calculators had been utilized an appropriate amount during the school year. The responses of the CAS group demonstrated increased satisfaction levels. Recall that nearly 70 percent of CAS students suggested changes in instruction only thirteen weeks earlier. While many CAS students agreed with the ftequency with which calculators were used in class, many felt concerned about the type of calculator used. More specifically, eight CAS students mentioned that instruction with the TI-83 would have been preferred to CAS-based instruction. Typically, students expressed fear that subsequent high school courses would not use the TI-92. It’s not that I wish we used the calculator less often or even more often. I wish we learned more about the TI-83 and not the TI-92. Honestly, I still don’t really understand the TI-83. I am still sort of intimidated by it. What has left me really concerned is that I have become way too dependent on the TI-92. I am worried about how I am going to do next year without it (Marybeth Short, CAS student). 262 Several CAS students commented that they would have prreferred to have not used the TI-92 at all. The calculators without a doubt are a good and heelpful idea — but one regret I have is that I wish we would have used thes TI-83’s all year and not even had the 92. I think this is because we’re nmot allowed to use the 92 on any standardized tests like SAT or ACT. I tihink we should have used the calculator as much as we did — just not. the 92. I think the choice to use calculators was perfectly fine, it’s Just: the type of calc, that we used is what I wish was different (Mike Badloss,, CAS student). Negative impressions of the TI-92 calculator extended to stiudents in the non-CAS class. One student who had never used the TI-92 shared the following comments about the calculator. I am confortable with my TI-83. It can do everythming that I need to do and is not too difficult. . . Also I have seen the (calculators that other classes have used, and they are so big with so many numbers that even before I am given a problem just looking at that calculator makes me feel overwhelmed (Perry Downer, non-CAS student). As many of the CAS students reflected back upon their czalculator experiences during the past school year, it was common for them to describe initial anxiety regarding the TI-92. Yet, during the last quarter of the school year, as ccalculator methods were de emphasized, many CAS students began to change their : minds about the calculator. Abby Miller, a student in the CAS class, describes her evolution in thought in the following terms. I was very upset with the amount of emphasis on the 92 early this year. I did not understand functions or any other junk wve did for an entire quarter. I am a very by-hand person, and I did not want to be reliant on the calculator. When we began to use the calculateur less, I realized how it can be used to check or graph, but it was a sugpplement to by-hand work. I enjoyed the use of the calculator as the yeair progressed because 263 I saw how it helped us, while not being the only way to solve problems (Abby Miller, CAS student). Another CAS student, Larry Lane, made similar comments. In the beginning of the year I didn’t like the calculators, but as the year moved on and I understood how to use the calculator more the calculator became very helpful... I think that the calculator was very helpful when it came to graphs and solving equations (Larry Lane, CAS student). In a similar fashion, non-CAS students also described initial feelings of frustration with the calculator. However, an overwhelming majority of non-CAS students felt satisfied with calculator use by year’s end. Unlike their CAS counterparts, non-CAS students recognized that the calculator skills they acquired in advanced algebra would serve them well in future classes. As one non-CAS student noted: There were times when I felt they [calculators] were a waste of time, but further into the year I realized how much they would help me. Next year, I am confident that I will go into the classroom well prepared with mental and calculator work. I hope to even be able to teach other classmates who didn’t leam about the calculator as much (Ken Streak, non-CAS student). Another non-CAS student made the following comments. I think we used calculators just the right amount this year. We used them almost every class, but they weren’t the main focus when solving problems. We had them when we needed them and that was good. At the beginning of the year I might have said that we used them too much, but now I realize how helpful they actually were (Allison Fiske, non- CAS student). Overall, the non-CAS group felt that their experiences with calculators were beneficial. They recognized that mathematics — not the calculator — remained the primary focus of instruction. Furthermore, they recognized that the skills they had acquired during the 264 school year would be utilized frequently in more advanced classes at the high school. On the other hand, CAS student opinion was more mixed with regard to calculator usage. Since it was unlikely that the TI-92 calculators would be used in subsequent courses, considerable numbers of CAS students were worried about their ability to succeed in future mathematics classes. 265 Summary This chapter investigated two primary questions regarding the use of CAS tools with secondary school students. Q l. What sorts of problems and activities did students investigate using hand-held technology? Which problems seemed well-suited for investigation with CAS? Which (if any) problems were not well-suited for investigation with CAS? Q2. What attitudes do intermediate algebra students have regarding graphing calculators and symbolic manipulation utilities? Do their attitudes change as they gain experience with the utilities? How do these attitudes differ among CAS and non-CAS students? Answers to these questions were explored with an examination of qualitative data from two primary sources. • Student writings regarding the use of symbolic manipulators in algebra classes • An in-depth teacher journal documenting possible promises and pitfalls of symbolic algebra utilities in secondary school classrooms. Findings are summarized in the paragraphs below. CAS Activities Students investigated a wide range of problems with calculators while participating in the study. As one student in the CAS class commented, “calculators were used for everything" (Mike Badloss, CAS student). Student writings and entries from a teaching journal suggested that although the TI-92 was often helpful when checking work or graphing fimctions, a number of problem types were not well-suited for use with the TI- 266 92 CAS. In particular, CAS students found table and regression tools difficult to use - largely because of the Tl-92's awkward user interface. Additionally, formulas involving subscripted variables (such as y,) or simplification of algebraic expressions .r) involving exponents such as —^ -----— often resulted in unexpected results. Initial (-T difficulties with such exercises discouraged many CAS students from manipulating algebraic expressions on the Tl-92. As one CAS student noted — “sometimes, like when we solve equations for x or y on the calculator it is just a hassle. 1 find that 1 can do better with my algebra without the calculator and it saves a lot of time by doing it by hand” (Bill Wend, CAS student). In fact, throughout the study, CAS students showed a preference for by-hand manipulation over calculator-based manipulation. Activities that used graphical capabilities of calculators were popular among all students. That graphing features of the Tl-92 were more popular than manipulation utilities may be explained by students’ preference for visual data over symbolic data and the relative simplicity of graphing functions on the Tl-92. Students in both classes referred to themselves as visual learners on many occasions. In fact, several CAS students identified visualization as a primary motivation for solving equations using by hand methods. Although CAS students solved equations using step-by-step techniques on the calculator, many students made comments such as the following: “when you are learning algebra, you want to practice with pencil and paper to see what you are doing visually” (Kristy Phillips, CAS student). 267 In general, activities that encouraged a combination of by-hand and calculator-based methods and application-style problems were more likely to seen as “calculator problems” by CAS students. Traditional, by-hand manipulation problems were less likely to be construed as CAS-appropriate problems. Student Attitudes Student responses to various writing prompts indicated that students had very little experience using graphing calculators prior to the study period. Although students were aware of graphing calculators prior to advanced algebra (and in many cases owned graphers prior to my class), they had only used calculators to perform simple arithmetic tasks. Roughly two thirds of study participants indicated that they had never created a graph or used a table on the calculator prior to participation in the study. Lack of calculator experience caused initial frustration with students in both classes. As one CAS student noted: “In the beginning of the year I would have liked if we did more by-hand and then worked up to the calculator . . . by doing the calculator method first I was really confused” (Larry Lane, CAS student). In comparison to their non- CAS counterparts, CAS students seemed slower to adapt to the heavy calculator usage they experienced in advanced algebra class. For this reason, CAS students were far more likely than their non-CAS counterparts to express dissatisfaction with instruction during the first part of the school year. For instance, during the 26^ week of instruction, over 70 percent of the CAS students commented that they wish class had been 268 structured differently, with half of the respondents commenting that they would have prefered to leam more by-hand methods. On the other hand, roughly 60 percent of non- CAS students said that they would change nothing about class. Part of the CAS students’ frustration with CAS calculators stemmed from the fact that such calculators would not be used in subsequent mathematics courses at the high school. As one CAS student noted; It’s not that I wish we used the calculator less often or even more often. I wish we learned more about the TI-83 and not the TI-92. Honestly, I still don’t reaUy understand the TI-83. I am still sort of intimidated by it. What has left me really concerned is that I have become way too dependent on the TI-92. I am worried about how I am going to do next year without it (Marybeth Short, CAS student). Despite an emphasis on non-CAS calculators — and numerous attempts to highlight similarities among CAS and non-CAS calculators — many CAS students continued to worry about their preparedness for future mathematics courses at the high school at the end of the school year. Nevertheless, student opinion improved noticeably in the CAS group during the final weeks of class. By year’s end, fifty percent of CAS students said they “would change nothing with the course” with fifty-four percent of CAS students describing their comfort level with the TI-92 as “very comfortable.” Abby Miller, a student in the CAS class, described her evolution in thought in the following terms. I was very upset with the amount of emphasis on the 92 early this year. I did not understand functions or any other junk we did for an entire quarter. I am a very by-hand person, and I did not want to be reliant on the calculator. When we began to use the calculator less, I realized how it can be used to check or graph, but it was a supplement to by-hand work. I enjoyed the use of the calculator as the year progressed because 269 I saw how it helped us, while not being the only way to solve problems (Abby Miller, CAS student). In general, student attitude improved as they gained experience with CAS utilities — although time required to feel comfortable with the TI-92 was significantly longer and was marked by fears of inapplicability with regard to future mathematics courses. 270 CHAPTERS SUMMARY AND CONCLUSIONS Out of the strain of the Doing, Into the peace of the Done. Julia Louise Woodruff S u m m a ry At the dawn of the twenty-first century, we live in a world marked by rapid technological change. As computers continue to redefine ways in which we construct and share knowledge, affordable hand-held computer algebra systems (CAS) allow students with little knowledge of by-hand symbolic manipulation techniques to solve a variety of traditional, skills-based algebra problems with relative ease. Powerful CAS- equipped calculators provide students with the ability to solve equations, plot functions, factor polynomials, and simplify expressions without the aid of paper and pencil-based methods. Because hand-held symbol manipulators bring into question the value of traditional algebraic methods (and the value of traditional manipulation-oriented exercises), CAS have been the source of great debate among mathematics educators. For instance, do traditional paper-and-pencil methods remain relevant in a technological world? Should students continue to solve equations or factor polynomial expressions 271 using by-hand techniques when calculators can perform the same tasks more quickly and more accurately? What role should CAS-equipped calculators play in the teaching and learning of school algebra? Up to this point, little research has investigated the relative worth of CAS as a teaching tool with secondary school students. Much of the existing CAS literature has focused on the utilities in the following settings. • With secondary school students, examining pupil performance on a handful of specific problems over a time frame of ten weeks or less (typically European studies). • With post-secondary mathematics students, describing effects of CAS usage on general manipulative skills (typically a single pretest / posttest design) or student attitude. The need for long-range, detailed research regarding the use of CAS with secondary school students formed the basis for this report. Two groups of suburban high school students participated in this study. The students, typically sophomores or juniors, shared the same instructor, attended the same school, used the same textbooks, completed the same homework assignments, and took the same tests and quizzes for an entire school year. The primary difference between the two groups was the systematic use of CAS-based utilities (specifically the TI-92) with the CAS group and the absence systematic use of such tools with the non-CAS group. While investigating calculator use with both groups of students, this study aimed to shed light on five fundamental research questions. 1. How does CAS affect overall student algebraic understanding? 212 2. How does CAS affect student algebraic understanding in specijfic mathematical topics? 3. How do students use CAS utilities? Does this use differ from non-CAS use? 4. What attitudes do intermediate algebra students have regarding graiphing calculators and symbolic manipulation utilities? Do their attitudes chan^ge as they gain experience with the utilities? How do attitudes differ among CAS and non-CAS students? 5. Which types of problems are well-suited for use with CAS-eq^uipped utilities? Which are not? To investigate these questions in a thorough, balanced maimer — a mixture of quantitative and qualitative data collection techniques were employe=d. In particular, data collection included: • A statistical design using analysis of covariance (ANCOVA) proceedures to measure the effect of CAS on overall student mathematical performance. The performance of CAS and non-CAS groups were compared on a year-end Advancred Algebra Final Examination. To adjust for initial group differences involving calculator know-how or algebraic knowledge. Technology Literacy Pretest and Algebruiic Skills Pretest scores were used as separate covariâtes. • Additional applications of the analysis of covariance (ANCOWA) procedure to measure the effect of CAS on mathematical performance of How-performing, middle-performing, and high-performing student groups. Studentis were placed in performance groups based on their initial scores on the Technology^ Literacy Pretest and Algebraic Skills Pretest. Chapter test and quiz results of both CAS and non-CAS groups t:o investigate the usefulness of CAS-based tools with regard to a year’s worth of high school mathematics content. Mean scores were analyzed using anal^^sis of variance (ANOVA) procedures to determine whether differences between flthe performances of the CAS and non-CAS groups were statistically significant. • A year-long appraisal of student problem solving methods as obserwed on class tests and quizzes. 273 A Likert-style Student Attitudinal Survey to measure student attitude regarding calculators both before and after the study period. The surveys were evaluated for change in attitude using analysis of variance (ANOVA) statistical procedures. • Student essays and short writing items to measure student attitudes regarding the use of symbolic manipulators in algebra classes throughout the school year. • An in-depth teacher journal to document possible promises and pitfalls of symbolic algebra utilities in secondary school classrooms. Results derived from the collected data are summarized below. First, a list of statistical hypotheses are enumerated along with research outcomes. Next, the Discussion section of this chapter summarizes both quantitiative and qualitative research findings within the context of the five research questions noted above. Hvpotheses tested and summarv outcomes HI. The mean scores of the CAS and non-CAS groups on the Algebraic Skills Pretest were not significantly different (with significance measured at a 0.05 alpha level). Outcome: HI could not be rejected at the 0.05 level of significance. H2. The mean scores of the CAS and non-CAS groups on the Technology Literacy Pretest were not significantly different (with significance measured at a 0.05 alpha level). Outcome: H2 could not be rejected at the 0.05 level of significance. H3. Before the study, there was no significant difference, at the 0.05 level, between the general attitude of CAS students towards calculator use and the general attitude of non-CAS students towards calculator use. Outcome: H3 could not be rejected at the 0.05 level of significance. 274 H4. After the study, there was no significant difference, at the 0.05 level, between the general attitude of CAS students towards calculator use and the general attitude of non-CAS students towards calculator use. Outcome: H4 could not be rejected at the 0.05 level of significance. H5. The change in attitude of CAS students regarding calculator use from Fall to Spring was not significant at the 0.05 level of significance. Outcome: H5 could not be rejected at the 0.05 level of significance. H6 . The change in attitude of non-CAS students regarding calculator use from Fall to Spring was not significant at the 0.05 level of significance. Outcome: H 6 could not be rejected at the 0.05 level of significance. H7. There was no significant difference in Advanced Algebra Final Examination scores between CAS and non-CAS groups. Outcome: H7 was rejected at the 0.05 level of significance. H8. The difference in performance on the Advanced Algebra Final Examination of low-performing CAS students and a similar group of low-performing non-CAS students was not statistically significant. Outcome: H 8 was rejected at the 0.05 level of significance (n=20). The low-performing non-CAS group significantly outperformed the low-performing CAS group. H9. The difference in performance on the Advanced Algebra Final Examination o f middle-performing CAS students and a similar group of middle-performing non- CAS students was not statistically significant. Outcome: H9 was rejected at the 0.05 level of significance (n=13). The low-performing non-CAS group significantly outperformed the low-performing CAS group. HIO. The difference in performance on the Advanced Algebra Final Examination of high-performing CAS students and a similar group of high-performing non-CAS students was not statistically significant. Outcome: HIO could not be rejected at the 0.05 level of significance (n=14). 275 H ll. When given the opportunity to use various solution strategies on test items, the CAS group relied less heavily on graphs than the non-CAS group. Outcome: HI I was not supported by descriptive data analysis. H12. When given the opportunity to use various solution strategies on test items, the CAS group relied more heavily on symbolic manipulation techniques to solve problems than the non-CAS group. Outcome: H12 was supported by descriptive data analysis. H13. When given the opportunity to use various solution strategies on test items, CAS students relied less heavily on tables to solve problems than students in the non- CAS group. Outcome: H13 was supported by descriptive data analysis. Discussion Initial algebraic and calculator knowledge of CAS and non-CAS groups A primary objective of this study was to investigate the impact of the systematic use of hand-held symbolic algebra utilities (or the lack thereof) on the mathematical performance of two groups of intermediate algebra students. Failure to reject Hypothesis 1 and Hypothesis 2 indicates that, with respect to basic algebraic skills and initial calculator knowledge, no significant differences existed between the two student groups at the beginning of the study. Although the probability that the mean Algebraic Skills Pretest scores of CAS and non-CAS groups were taken from the same population of students was somewhat low (p = 0.096), with non-CAS students outperforming CAS students, this value did not warrant a rejection of Hypothesis 1. On the other hand, the 276 probability that the mean Technology Literacy Pretest scores were taken from the same population of students was convincing (p = 0.830). Comparison of groups on Advanced Algebra Final Examination The administration of the Algebraic Skills Pretest and Technology Literacy Pretest together with a common final examination formed the basis of a quasi-experimental design implemented in this study. At the end of the academic year, identical Advanced Algebra Final Examination tests were administered to all students in the CAS and non- CAS sections. In fact, the Advanced Algebra Final Examination was administered to all advanced algebra students at Midvale High School. Using an analysis of covariance (ANCOVA), the mean Advanced Algebra Final Examination scores of the CAS and non-CAS groups were compared. To compensate for any initial differences between the groups. Algebraic Skills Pretest and Technology Literacy Pretest scores were used as covariates. As the rejection of Hypothesis 7 indicates, the non-CAS group significantly outperformed CAS group on the exam — suggesting that the use of CAS utilities negatively affected student performance on the Advanced Algebra Final Examination. In an effort to uncover possible explanations for discrepancies in performance between the CAS and non-CAS sections on the final exam, students from each class were assigned membership in one of three distinct performance groups — low performing, middle performing, and high performing — based on the overall percentage of items they answered correctly on the Technology Literacy Pretest and the Algebraic Skills Pretest. To examine the impact of CAS on particular groups of learners, mean 277 Advanced Algebra Final Examination scores of each performance group were compared using analyses of covariance (ANCOVA) procedures. As rejection of Hypothesis 8 indicates, the mean Advanced Algebra Final Examination scores of low-performing CAS and low-performing non-CAS students were significantly different. This difference favored non-CAS students. Likewise, rejection of Hypothesis 9 reveals that the mean Advanced Algebra Final Examination scores of middle-performing CAS and middle-performing non-CAS students were significantly different, also favoring non- CAS students. The only group unaffected by CAS use in a significant manner was high-performing students. As the failure to reject Hypothesis 10 indicates, differences in final exam averages of high-performing CAS and non-CAS students were not deemed statistically significant. The fact that year-long CAS use had a significant, negative impact on the Advanced Algebra Final Examination scores for low- and middle-performing students may be partially explained by considering the added cognitive burdens placed on the CAS group during the last months of the study. In addition to their responsibilities to learn new mathematical content, students in the CAS group were also formally introduced to the TI-83 calculator during the second half of the school year. Topics were discussed using a mixture of by-hand techniques, TI-92 CAS specific techniques, and TI-83-based strategies. It seems plausible to suppose that the burden of following discussions using two types of calculators, coupled with student anxiety regarding the use of calculators in future coursework, overwhelmed all but the highest achieving students in the CAS 278 group. The negative attitude that the CAS group held for the TI-92 early in the Fall may have hindered the algebraic development of all but tlie highest performing for the remainder of the school year. Comparison of CAS and non-CAS group performance bv topic At the beginning of this study, little was known about the possible impact of CAS- equipped calculators on student performance with regard to specific mathematical topics — for instance, how does CAS influence student understanding o f trigonometry^ By examining mean test and quiz scores of both CAS and non-CAS groups over an entire school year, while covering a wide range of mathematical content, it was possible to examine the usefulness of CAS-based tools with regard to a wide range of high school mathematics topics. For the first 13 weeks of the school year — during the study of functions, direct and indirect variation, linear relations, and matrices — the non-CAS group outperformed the CAS group on all chapter tests and quizzes. However, analyses of variance (ANOVA) revealed that none of these differences were statistically significant. As the school year progressed, the non-CAS group continued to outperform the CAS group — obtaining higher mean scores on all tests and quizzes involving functions, variations, logarithms, and linear relations. On the other hand, the CAS group did fare better on all tests and quizzes dealing with powers/exponents and trigonometry. Results were mixed with matrices and quadratics - with CAS students outperforming non-CAS students on some (but not all) assessments in these topics. Of the nineteen tests and quizzes administered 279 to students throughout the school year, the non-CAS group outperformed the CAS group on eleven of the assessments, with none of the differences statistically significant. Student calculator use in CAS and non-CAS classes As Kutzler (1996) suggests, technology shifts the emphasis of classroom discourse from finding the answer to a comparison of various problem-solving strategies. Because CAS-based tools provide students numerous ways to solve problems — with tabular, graphical, and symbolic solution methods existing for almost any problem — an investigation of student problem-solving strategies was central to this study. Throughout the school year, test and quiz items well-suited for multiple solution techniques were earmarked for further analysis. At the end of data collection, the earmarked items were coded based on solution strategies employed by each student: (a) graphical; (b) symbolic manipulation (either calculator or by-hand); (c) tabular; or (d) hand-drawn diagrams. When more than one method was used to complete a problem, each method was tabulated. Descriptive statistics were used to compare solution strategies employed by students in the control and experimental groups. As the rejection of Hypothesis 11 indicates, students from both classes used graphing tools with comparable frequency throughout the study. With regard to earmarked problems, students in the CAS group employed graphical methods 32 percent of the time, while non-CAS students used similar strategies 31 percent o f the time. 280 On the other hand, CAS students used symbolic manipulation to solve earmarked problems more frequently than their non-CAS counterparts. Overall, CAS students employed manipulation-style solution strategies 51 percent of the time as compared to non-CAS students’ 42 percent use. This pronounced difference warranted the rejection of Hypothesis 12. Lastly, CAS students were somewhat less likely to use tables or regression features to solve earmarked problems, with CAS students using table-based methods 14 percent of the time to non-CAS students’ 18 percent. The fact that CAS students were less likely to use tables or regression to solve problems is reflected in the rejection of Hypothesis 13. Student attitudes regarding calculators In an effort to understand student preference for particular problem-solving methods, student responses to various writing prompts were collected throughout the school year. The prompts, which were assigned approximately once every ten weeks, required students to write about their use of calculators. Student writings revealed several characteristics common among CAS students. Some characteristics were shared by non-CAS counterparts, others were not. • CAS students had limited experience with graphing calculators before the study (this was also true of non-CAS students). • CAS students were dissatisfied with emphasis on calculator-based methods in class (this differed from attitude of non-CAS students). • CAS students desired to leam more by-hand manipulation techniques (this differed from attitude of non-CAS students). 281 • CAS students possessed a general desire to work more with TI-83 (non-CAS) calculators and less with the TI-92. In their responses to the End o f Year Short Answer Essays (administered during the 39'*' week of class), roughly two thirds of study participants indicated that they had never created a graph or used a table on the calculator prior to advanced algebra. Lack of calculator experience caused initial hrustration for students in both classes. As one CAS student noted: “In the beginning of the year I would have liked if we did more by hand and then worked up to the calculator ... by doing the calculator method first I was really confused” (Larry Lane, CAS student). Although both groups found the emphasis on calculator-based methods somewhat frustrating during the first weeks of the study, CAS students seemed slower to adapt to routine calculator use than non-CAS students. As they experienced frustration with the TI-92, CAS students’ dissatisfaction with class grew more pronounced. For instance, during the 26'^ week of instruction, over 70 percent of CAS students indicated a desire for modifications to the existing instructional delivery methods, with half of the respondents commenting that they desired more instruction dealing with by-hand problem-solving methods. On the other hand, in response to the same survey roughly 60 percent of non-CAS students said that they would change nothing regarding instructional delivery. CAS attitude improved somewhat as students gained experience with CAS utilities — although the time required to feel comfortable with the TI-92 was significantly longer 282 than with non-CAS calculators. By year’s end, fifty percent of CAS students said they “would change nothing with the course” (up firom the 30 percent approval rating only thirteen weeks earlier) with fifty-four percent of CAS students describing their familiarity with the TI-92 as “very comfortable.” Nevertheless, half of CAS students stated that they would have preferred less emphasis on calculator-based methods. Additionally, 36 percent of CAS students desired less instruction with the TI-92 and more instruction with the TI-83. Several CAS students commented that they would have preferred to have not used the TI-92 at all. In addition to their essay work, students completed a Student Attitudinal Survey twice during the school year - once in the Fall at the beginning of the study and again in the Spring at the study’s conclusion. The Likert-style surveys were designed to measure student attitude with regard to calculator use in mathematics class. In general, findings firom the survey data strengthen conclusions drawn firom the student essays. The fall administration of the Student Attitudinal Survey indicated that mean calculator attitude of CAS students was initially lower than the attitude of non-CAS students. Although failure to reject Hypothesis 3 indicates that this difference in attitude was not statistically significant, the more negative attitude of the CAS students foreshadowed essay responses depicting CAS students as reluctant calculator users. Although CAS and non-CAS responses tended to agree with one another on the Fall administration of the survey, with the mean responses of the CAS and non-CAS groups agreeing on twelve of the survey’s sixteen items, the groups differed with regard to 283 several important questionnaire items dealing with calculator use. For instance, CAS and non-CAS students expressed differences of opinion regarding the worth of technology as a motivator. Unlike the non-CAS group, CAS students disagreed with the statement “Graphing calculators increase my desire to do mathematics” on the Fall questionnaire (these results were identical for the spring questionnaire). As was the case in the Fall, mean calculator attitude was lower for CAS students than for non-CAS students on the Spring administration of the survey. Again, as evidenced by the failure to reject Hypothesis 4, this difference was not statistically significant. An inspection of survey results from fall to spring suggests a more positive overall calculator experience for non-CAS students. In particular, the mean calculator attitude of the non-CAS students was more positive at the end of the school year than it was when the study began. The opposite result held for CAS students, whose mean calculator attitude dropped during the school year. Although failure to reject Hypotheses 5 and 6 indicate that changes in calculator attitude were not significantly different for either group, the fact that non-CAS calculator attitude improved while CAS attitude worsened is consistent witli negative reaction to CAS-equipped calculators in student essay responses of CAS students. Suggested problem types for use with CAS Generally speaking, students in the CAS group preferred to tackle traditional skill- oriented algebra problems without the aid of the calculator, utilizing by-hand methods 284 more frequently than calculator methods to solve equations or simplify algebraic expressions. By-hand manipulation was popular among all students. Many claimed that by-hand methods enabled them to to see individual steps more clearly than possible with calculator-based methods. Typically, CAS and non-CAS students equated by-hand competence with conceptual understanding. Although CAS students had repeatedly solved equations in a step-by-step, manipulative fashion on the calculator - many CAS students agreed that paper-and- pencil calculations displayed “how a problem was done” more vividly than possible on the calculator. Notes in the teaching journal indicated that students in the CAS class were likely to favor CAS solution strategies over traditional by-hand methods when working on “application-style problems” where the mechanical aspects of algebra were not the primary focus of student activity. With regard to skill-based problems (those in which mechanical skills were the focal point), CAS students indicated that the calculator diminished opportunities for conceptual understanding. Students investigated a wide range of problems with calculators while participating in this study. As one student in the CAS class commented, “calculators were used for everything” (Mike Badloss, CAS student). Student writings and entries from a teaching journal suggested that although the TI-92 was often helpful when checking work or graphing functions, a number of problem types were not well-suited for use with the TI- 92 CAS. In particular, CAS students found table and regression tools difficult to use — 285 largely because of the TI-92's awkward user interface. For instance, regression required TI-92 users to save file names, open applications, and use function keys tto select regression techniques - steps not required on the TI-83. Additionally, students -were not always able to use familiar symbolic representations when solving problems writh CAS. Formulas involving subscripted variables (such as yO or simplification of salgebraic Æ(c- xY expressions involving exponents such as — -----^ often resulted in unexpectecd results. (c-zY Initial difficulties with such exercises discouraged many CAS studemts from manipulating algebraic expressions on the TI-92. These difficulties help to exphain CAS students’ preference for by-hand manipulation by the study’s end. On the other hand, activities that used graphical capabilities of calculatoors were popular among all students. That graphing features of the TI-92 were more popular than manipulation utilities may be partially explained by students preference fTor visual data over symbolic data and the relative simplicity of graphing functions on thme TI-92. Students in both classes referred to the importance of visualization through*out their responses to various writing prompts. In general, activities that encouraged a combination of by-hand and calculattor-based methods and application-style problems were more likely to seen as “calculator problems” by CAS students. Traditional, by-hand manipulation problems \were less likely to be construed as CAS-appropriate problems. 286 Conclusions On the basis of the analyses of both qualitative and quantitative data, six conclusions can be drawn for the study. • CAS-based teaching methods appear to negatively impact overall algebraic understanding o f low- and middle-performing students, while negligibly impacting the overall algebraic understanding o f high performing students. At the end of the school year, the Advanced Algebra Final Examination was adrninistered to students in both CAS and non-CAS classes. The test, which consisted of fifty multiple choice items, covered a wide range of advanced algebra content. Strong emphasis was placed on Trigonometry, Exponents/Radicals, and Functions, with approximately 58 percent of test items falling under one of these three broad categories. Mean test scores indicated that the non-CAS group outperformed the CAS group decisively on the final exam — 76.5 percent to 68.7 percent. Using an analysis of covariance (ANCOVA) procedure, it was determined that this difference in performance was statistically significant, F(4,43) = 397.354, MSB = 6.182, p < 0.005. To further examine performance differences between CAS and non-CAS groups on the Advanced Algebra Final Examination, the entire sample (n = 47) was split into three distinct groups based on the percentage of items answered correctly on both the Technology Literacy Pretest and the Algebraic Skills Pretest. Students whose combined pretest scores fell below the lower one-third quantile of their class were designated as low-performing students. Similarly, those whose combined pretest scores fell above the lower one-third quantile and below the upper one-third quantile of their class were 287 deemed middle-performing students. High performing students were those whose combined pretest scores fell above the upper one-third quantile of their class. These groupings enabled a comparison of CAS and non-CAS performance on the Advanced Algebra Final Examination by intial student performance level. Three separate analyses of covariance (ANCOVA) were performed. The tests compared mean final exam mean scores CAS and non-CAS students from each of the three performance groups. For the low-performing students, the ANCOVA was significant, F(2,16) = 4.455, MSE = 0.048, p — 0.029. These results confirm that the use of CAS did contribute to statistically significant differences in group performance levels among low-performing students favoring students that did not use symbolic manipulation utilities as part of their mathematics instruction. Because final exam scores are an indicator of overall student understanding of advanced algebra content, these results suggest that CAS-based teaching methods appear to negatively impact overall algebraic understanding of low-performing students. Similarly, for middle-performing students the ANCOVA was significant, F(2,9) = 5.088, MSE = 0.024, p = 0.033. These results confirm that the use of CAS did contribute to statistically significant differences in group performance levels among middle-performing students favoring students who did not use symbolic manipulation utilities as part of their mathematics instruction. Because final exam scores are an indicator of overall student understanding of advanced algebra content, these results 288 suggest that CAS-based teaching methods appear to negatively impact overall algebraic understanding of middle-performing students. Results differed with students whose combined pretest scores fell in the upper one- third quantile of their class. In the case of high-performing students, the ANCOVA was not significant, F(2,10) = 0.832, MSE = 0.006, p = 0.463. These results confirm that the use of CAS did not contribute to statistically significant differences in group performance levels among high-performing students. Because final exam scores are an indicator of overall student understanding of advanced algebra content, these results suggest that CAS-based teaching methods do not appear to negatively impact overall algebraic understanding of high-performing students. • Data regarding the effectiveness o f CAS-based instruction on student understanding o f specific content is inconclusive. Measurement of student understanding with regard to specific advanced algebra content was accomplished in two distinct ways: (1) With a comparison of group performances on chapter tests and quizzes throughout the school year; (2) With an item-by-item analysis of group performances by problem-type on the Advanced Algebra Final Examination. Throughout the school year, identical chapter tests and quizzes were administered to students in both CAS and non-CAS classes. Teacher-generated tests served as summative assessments of student conceptual understanding. Typically the tests were administered after students had read and investigated all of the key ideas from a given 289 chapter from our textbook — UCSMP Advanced Algebra (Usiskin, et al., 1990). On the other hand, chapter quizzes were typically used as a means of assessing student strengths and /or weaknesses regarding a given topic and were administered prior to the completion of a chapter from the text. Chapter quizzes were primarily a formative assessment tool, used to help guide instruction and to provide students with feedback regarding their understanding of particular coiurse concepts. By examining mean test and quiz scores of both CAS and non-CAS groups over an extended period, while covering a wide range of mathematical content, I was able to investigate the usefulness of CAS-based tools with regard to a wide range of high school mathematics topics. Of the nineteen tests and quizzes administered during the study, separate analyses of variance (ANOVA) reveal that none of the differences in performance between CAS and non-CAS groups were statistically significant. However, a simple comparison of group means reveals that the CAS class outperformed the non-CAS class on eight ocassions (42 percent of class tests and quizzes). Closer scrutiny of chronological group performance reveals that non-CAS students outperformed CAS counterparts until the fifteenth week of class — thus strengthening the notion that the TI-92 posed greater challenges for CAS students at the beginning of the school year. When one excludes data from the first fifteen weeks of classroom instruction, the CAS class actually outperforms the non-CAS class 62 percent of the time. Specifically, the CAS group outperformed non-CAS students on all tests and quizzes involving the following topics: (1) Exponents and powers and (2) Trigonometry. On the other hand, the non-CAS 290 group outperformed the CAS group on all tests and quizzes involving four separate topics: (1) Functions; (2) Direct and Inverse Variations; (3) Linear Relations; and (4) Logarithms. CAS usage does not clearly favor either group in terms of the study of matrices or quadratics, since the CAS group fared well in comparison to the non-CAS group on some (but not all) tests and quizzes dealing with these mathematical topics. Student responses on the Advanced Algebra Final Examination were also used to assess the effectiveness of CAS-based instruction on student understanding of specific topics. For each class, the number of correct responses was tabulated for each of the exam’s 50 multiple choice items. These values were divided by the number of respondents to obtain a class average for each test item. Appendix G lists class averages for each item, with items arranged by topic. Appendix G indicates that the non-CAS class had a higher group mean than the CAS class on all items involving exponents, radical notation, and graphical understanding. Performance was more evenly distributed on topics such as trigonometry, equation solving and quadratic forms. CAS students outperformed non-CAS students on all items involving complex numbers. Week-by-week comparisons of group performance on tests and quizzes contradict findings from the Advanced Algebra Final Examination. For instance, on the final exam, CAS students received lower marks on all exam items involving exponents and radical notation (although they outperformed non-CAS students in these areas during the school year). Furthermore, whereas CAS students outperformed their non-CAS counterparts on tests and quizzes involving trigonometry, performance was evenly 291 divided on final exam items covering the same content. Because final exam results contradict weekly performance results of the two groups, data regarding the effectiveness of CAS-based instruction on student understanding of specific mathematical topics is deemed inconclusive in this study. • Fears regarding the applicability o f CAS-based solution strategies in subsequent mathematics courses may interfere with students ' acquisition o f knowledge under calculator-based methods Throughout the study, but particularly at the conclusion of the school year, CAS students voiced concerns regarding the relative worth of CAS-based solution strategies in subsequent mathematics courses at the high school. In a writing assignment administered during the tenth week of instruction, CAS and non-CAS students were asked to solve linear equations using a variety of techniques - including calculator-based methods and by-hand methods — and then identify a preferred solution method. Results from the essay indicated that CAS students preferred by-hand methods over calculator-based methods. Overall, 26 percent of CAS students preferred calculator-based methods as compared to 62 percent of non-CAS students. A number of students in the CAS class expressed reservations about using a calculator that they recognized would not be used in future classes at the high school. 1 prefer the first way (the by-hand method) because I can just as easily figure it out by hand as I can on the calculator. Besides, in the future, I will not have a calculator this sophisticated (Alison Myser, CAS group). As the school year continued, CAS students continued to voice opposition to calculator methods based on the usefulness of such methods in future coursework. For instance, 29 2 during the sixteenth week of class, while responding to Systems Essay writing prompts, a CAS student made the following comment. Because we were given the TI-92 to use only for this year [emphasis added], everything we learned we learned using those. Next year, when we don’t have those, what are we going to do? This process on the TI-92 is useless to us next year (Andy Shoeman, CAS student). The emphasis on instruction with the TI-92 calculator left a number of CAS students feeling underprepared. Also 1 really wish that while 1 was learning how to do things on the TI- 92, I would have leamed how to do them on the TI-83 as well. I am very worried about the fact that next year without the TI-92 I am going to be lost. I think I rely on the TI-92 way too much and I don’t know how to do a lot of these things by hand (Marybeth Short, CAS student). During the twenty-sixth week of class, while responding to questions on the Calculator Algebra Questionnaire, over 70 percent of the CAS students commented that they wish class had been structured differently. Roughly a quarter of students commented that they would have preferred to use the TI-83 instead of the TI-92. Conversely, students in the non-CAS class mentioned that they were motivated to use the calculator because they knew it would be utilized in future classes. (I prefer the) calculator, although both (calculator and non-calculator methods) are just as easy. I prefer the calculator because 1 don’t have to write it down and 1 should always try and take the opportunity to get to know the TI-83 better because I will be using it more in the future (John Beige, non-CAS group). An overwhelming majority of non-CAS students felt satisfied with calculator use by year’s end. Unlike their CAS counterparts, non-CAS students recognized that the 293 calculator skills they acquired in advanced algebra would serve them well in future classes. As one non-CAS student noted: There were times when I felt they (calculators) were a waste of time, but further into the year I realized how much they would help me. Next year, I am confident that I will go into the classroom well prepared with mental and calculator work. I hope to even be able to teach other classmates who didn’t leam about the calculator as much (Ken Streak, non-CAS student). Overall, the non-CAS group felt that their experiences with calculators were beneficial. They recognized that mathematics — not the calculator — remained the primary focus of instruction. Furthermore, they recognized that the skills they had acquired during the school year would be utilized frequently in more advanced classes at the high school. • CAS based equation solving does not appear to support conceptual understanding to the same extent as traditional by-hand equation solving. The awkwardness o f the TI-92 output as well as the calcidator ’s tedency to perform “too many steps ” automatically may have contributed to students ’preference for by-hand methods. One popular justification for continued emphasis on by-hand manipulation offered by students was its usefulness at illustrating individual algebraic steps visually. Although CAS students were shown equation solving techniques on the TI-92 that allow users to review previous steps on the calculator’s home screen, many students continued to favor by-hand methods to graphing or calculator-based manipulation. Often, students cited visualization as a primary advantage of pencil and paper methods. When you are learning algebra, you want to practice with pencil and paper to see what you are doing visually [emphasis added] (Kristy Phillips, CAS student). Another student from the CAS class made the following comment. 294 By doing the problem by hand first, you will be able to see all steps written out [emphasis added]. So, if you make a mistake, you can go back (Larry Lane, CAS student). Comments regarding the visual aspects of by-hand manipulation were also common among non-CAS students. I am a very visual [emphasis added] learner, and it really helps me to write things out and see [emphasis added] them on paper. I constantly leam from my mistakes and the best way to see my mistakes is when I am writing out an answer. I think it is very important to leam how to write things out onto paper ... it really helps me to see [emphasis added] each part of the equation. It helps me to break things down and evaluate each section of the problem. I think without this skill I would have more trouble checking my answers as well as finding my mistakes. In my opinion a calculator is great after you already know how to solve a problem by hand. (Slovak Dennis, non-CAS student). The TI-92’s tedency to perform “too many steps” automatically may partially explain CAS students’ preference for by-hand manipulation. For instance, one student made the following observation in response to a question posed during the sixteenth week of the study. It (the TI-92) is a useful tool but in some cases it Just does the work for you and you don’t learn anything [emphasis added]. Sometimes, like when we solve equations for x or y on the calculator it is just a hassle. I find that I can do better with my algebra without the calculator and it saves a lot of time by doing it by hand (Bill Wend, CAS student). Because the TI-92 has a tendency to combine “like terms” and simplify expressions involving exponents automatically, students may be deprived the opportunity to think about equivalent expressions when solving equations on the calculator. Figure 8.1 - X X ' illustrates what happens when a student enters the equation = — into the TT-92. x + l X 295 ti'iB jf FÎ-»- m-r rs M? ' f—— Algebra C alc O ther PrgmIO C lean Up x^-x x^ x + 1 x-(x- 1) = x^ x)/< x+ l > = Figure 8.1: Steps automatically performed upon entering equation into TI-92 CAS As Figure 8.1 illustrates, the calculator automatically performs thie following tasks: 1. re-expresses — .r as .r- {^.xr — ij 2 . re-expresses — l) as (.r + 1) - (x - ij (.r+ 1) 3. re-expresses 7 f as 1 (,r+l) Y 4. re-expresses — as .r X As shown in Figure 8.2, the calculator continues to perform ttasks automatically on subsequent steps of the problem. After a student decides to subltract ,xr from each side of the equation, the calculator automatically performs these steps: : 296 1. Expands as x^—x 2. Simplifies —jrj — jr a s —.r The calculator performs each of these steps instantaneously — the moment that the student presses ENTER on the calculator — without pause for student thought or instructor guidance. fTfTHOY FZ? Y F3-- Y Y FS Y F6? '| I^ f — imigebra|CaIclotheriPrgnlOIciean Up| I x^-x x^ x (x - 1) = x^ X + I X ■[x-(x- l) = x^] -x ^ -X = 0 ans <1>-x^2 Figure 8.2: More calculations automatically performed by TI-92 CAS Figures 8.2 and 8.3 serve to illustrate the awkwardness of algebraic output on the TI-92 home screen. As Figure 8.3 suggests, TI-92 output is read from left to right, then from top to bottom (like sentences in a book). This representation differs from conventional mathematical text, in which algebraic steps are written one below the next. x^-x x^ X + 1 X “► x-(x - 1) = x^ l(x (x - 1) = x~) - x ^ ans Figure 8.3: Algebraic output is read like “sentences in a book” on the TI-92 home screen 297 Arguably less standard is the algebraic notation employed by the TI-92 CAS. Array notation such as ans(l) is unlikely to be found in any secondary mathematics textbook. Throughout their middle school and high school years, the participants in this study had been conditioned to rely heavily on school textbooks to leam mathematics (and nearly every other school subject). Although the mathematics teachers at the study site exhibited a great deal of creativity and energy with regard to the teaching of mathematics, they also relied on heavily textbooks and ancillary materials to provide homework and test problems for their students. Because mathematics teachers at the site typically taught five classes each day (roughly 150 students daily), little time existed to create original activities for students. As teachers relied on the text for assignments, students came to rely on the text for sample solutions and explanations to problems. Futhermore, students looked to the text as a means of studying for chapter tests and quizzes. Perhaps at another school, where students were required to take fewer classes each day and where teachers were responsible for fewer students (e.g. a “block scheduled” environment), heavy reliance on textbooks would have been lessened - and the non-standard notation provided by CAS would have proven less of a concern with students. Students were also confused by the TI-92’s “equation as object” notation - in which parentheses are written across an equal sign (e.g. [x- (jr-l) = ^)~ -T). It appears that 298 this awkwardness coupled with the calculator’s tedency to automatically perform “too many steps” may have contributed to students’ preference for by-hand methods • Problems involving regression, tables, or subscripted variables appear to be ill- suitedfor use with TI-92 CAS As illustrated in the previous result, the TI-92 CAS occasionally uses non-standard notation as students solve algebraic problems. In a related manner, the TI-92 occasionally does not allow students to use notation typically encountered in secondary school textbooks. For instance, as the CAS students reviewed linear relationships using the TI-92 CAS, they experienced initial difficulties solving problems with the standard slope formula m = ——— . As Figure 8.4 illustrates, the use of subscripted variables resulted x , - x , in an error on the TI-92 since y i is considered a function — and is not a valid variable name on the calculator. rri-'m r rz^ r f j -»- r r t — r r s r r e l-f— |Rlgebra|Calc|Other|PrgnIO|Clear a-z_ r ERHOR > Too few arguments CESC=CmNCEL') m= Figure 8.4: The TI-92 does not allow use of familiar slope formula 299 On the other hand, the less familiar form m = ——— caused no problems for the TI-92. X— This is illustrated in Figure 8.5. r FÎ->- Y Fl-^ Y F? 1 fZ re ■«•f— |fllgebra|CaIc|Qther|PrgnIO C le a r a-'-z-1 y - y2 X - x2 X — x2 m= Figure 8.5: A modified formula for slope works successfully on the TI-92 CAS students felt uncomfortable with the approach because they were not able to use familiar symbolic representations when solving problems. Some of the CAS students had a difficult time with this since they want to follow the book closely until they feel more comfortable with the formulas. The situation is unfortunate, since the students are really just learning the material for the first time and really do rely on the book to review ideas at home. The fact that the calculator doesn’t facilitate using formulas from the book discourages students from using symbolic manipulation on the calculator (November 2, 1999). In constrast, I explained the same idea to the non-CAS students with an overhead projector with very little concern or controversy. As noted in a year-long teaching journal: I think that an argument on the overhead projector (in the non-CAS class) was more effective than the presentation which used the TI-92 symbolic manipulation utility because the overhead didn't care which names I gave to variables. The manipulations were not overly difficult by hand, and (using the overhead) students were allowed to model representations provided in homework problems in our textbook (November 2, 1999). 300 The difficulty posed by the TI-92 calculator on the problem may have influenced preferred solution methods of students in each class. In addition to problems with subscripted variables, CAS students found table and regression tools difflcult to use — largely because of the TI-92’s awkward menu-based interface. Unlike non-CAS calculators (such as the TI-83), the TI-92 often behaved more like a personal computer. For instance, regression required TI-92 users to save file names, open applications, and use function keys to select regression techniques — steps not required on the TI-83. Comments from earlier entries in my teaching journal indicate that the extra steps required of CAS students — saving data as a file and specifying lists for regression — initially discouraged students from using the calculator. The average student at our high school has little familiarity using spreadsheet programs. Because our curriculum is basically college preparatory, we don’t offer courses in accounting or basic computer skills for most students (a curriculum decision which puzzles me to no end) — so many students have never used applications such as Microsoft Excel prior to my class. This makes regression on the TI-92 much more difficult for students . . . since the data matrix editor is similar to a spreadsheet application in the sense that students must save file names, open applications, and use function keys to select regression techniques. I think this is a principal reason that the by-hand methods seem to be more popular with TI-92 users (September 21, 1999). Based on the data collected for this study, problems involving regression, tables, or subscripted variables appear to be ill-suited for use with TI-92 CAS. 301 Implications Implications for Practice Care should be taken when using CAS-based utilities with secondary school students. Results from the end-of-year Advanced Algebra Final Examination indicate that students taught without CAS utilities performed at significantly higher levels those taught with CAS-equipped calculators. Student writings and class performances on chapter tests and quizzes suggest that students using the TI-92 intially experience more difficulties with calculators than their non-CAS counterparts. The burden of grasping specifics of the TI-92 while simultaneously learning generic graphing calculator features initially proved daunting for many CAS students. This finding is similar to that of Mayes (1997). The researchers found that although the students were initially motivated by the use of the technology, they often became frustrated with mastering the intricacies of the technology and in interpreting the outputs given by CAS . . . the overall resulting attitude was appreciation for the time saved of drawing graphs, but doubt about the usefulness of CAS in doing mathematics (Mayes, 1997, p. 187). Ideally, a student's first experiences with graphing calculators should occur before second-year algebra. Rather than introducing graphing utilities to students in an intermediate algebra class, it seems more reasonable to incorporate hand-held graphers gradually throughout their mathematics training — providing students with opportunities to familiarize themselves with the tools at various levels of the secondary school curriculum. Ideally, calculators should be introduced as part of a well-planned, tightly 302 integrated course of study in which technology is an integral part. Without previous knowledge of handheld graphers, the CAS students failed to recognize similarities among CAS and non-CAS calculators — thus questioning the worth of CAS-based solution methods in subsequent secondary mathematics courses. As teachers begin to use CAS utilities with secondary mathematics students, care should be taken regarding the tendency of symbolic manipulators to automatically perform algebraic steps. If a classroom teacher intends to use CAS to enhance student understanding of algebraic manipulation, the instructor should choose examples that leave algebraic steps to the user. Otherwise, one is advised to begin the study of equation-solving with an emphasis of by-hand techiques. Once students feel comfortable with basic algebraic transformations, the tendency of the calculator to perform steps automatically presents fewer cognitive risks to students. As one CAS student noted: “In the beginning of the year I would have liked if we did more by-hand and then worked up to the calculator . . . by doing the calculator method first I was really confused” (Larry Lane, CAS student). In this study, students in the CAS group commented that symbolic manipulation techniques were more clear when performed by hand — for young students, the TI-92 CAS does not clearly illustrate individual steps taken to solve a problem. Focusing student attention on the theory of equation solving is the real promise of symbolic manipulation with CAS. Pedagogically sound CAS enable students to choose transformations to apply to equations without worrying about simple arithmetic errors. 303 Until CAS utilities exist that allow students to choose transformations for themselves, use of CAS as a primary means of teaching equation solving is not recommended by this study. Because the primary features that recommend CAS use — graphing equations, defining functions, calculating maximums and rninimums — are more readily accessible to students with non-CAS calculators, it is recommended that teachers weigh the need for symbolic manipulation with tlie added cost and time investments required with TI-92 use. 304 Theoretical Implications TI-92 CAS inadequate as a cognitive scaffolder for secondary algebra students At the beginning of this study, it was expected that the TI-92 CAS would be utilized as an "expert other” — a learning tool that would provide immediate, helpful feedback to students as they solved problems in advanced algebra class. Vygotsky originally proposed the concept of Zone of Proximal Development to describe the gap that separates a student’s actual knowledge from his or her potential knowledge. I anticipated that the calculator’s ability to manipulate symbolic expressions would facilitate greater symbolic exploration among students — enabling them to actively bridge the gap between their actual and potential development. As the vignette in Chapter 3 illustrates, a dutiful “expert other” leads two students through a problem in a step-by-step fashion — at a level just beyond the learner’s conceptual grasp but within reach with assistance. In the case of the TI-92 CAS, the calculator was an expert at producing algebraic results — but less adept at providing output tailored to students’ current level of conceptual development. As illustrated throughout this document, on too many occasions the calculator simply did steps for students without pause or explanation. The result was considerable student frustration in the CAS group, as students longed for instruction better suited to their current levels of understanding. Support for Buchberger’s White-Box Methodology Attempts were made to capitalize on the TI-92’s tendency to simplify algebraic expressions automatically by using the calculator as a “black box.” CAS students used 305 the calculator to generate mathematical hypotheses as they examined simplified output generated by the calculator. However, use of the calculator in this fashion met with mixed results. Results on class tests and quizzes reveal that CAS students performed at lower levels than non-CAS students on many topics well-suited for such investigation — exponent laws, rules of logarithms, matrix multiplication to name a few. Results of this study suggest that Buchberger’s call for a “white box first” teaching approach — one in which step-by-step algebraic processes are well understood before resorting to use of calculator as black-box — are better-suited for the conceptual level of understanding of the typical secondary school mathematics student. CAS suggests new Computer Symbolic level of abstraction In essay writings, students from both classrooms indicated paper-and-pencil methods were preferred to the calculator when learning new mathematical concepts. One non-CAS student highlighted the continued need for by-hand methods with the following example. I truly won’t understand why that’s the answer without being taught how to do it with the pencil/paper method. For example, finding the inverse of the equation y = 2>jr. This can be easily found with the calculator, but I did not understand why until we were taught how to solve by-hand to switch the x and y-values . . . without by-hand methods, I think math is somewhat hollow (Millie Stroud, non-CAS student). It may be argued that the calculator adds an additional layer of abstraction to algebraic symbolism. Instead of manipulating symbols directly by hand, students press buttons to manipulate symbols. Furthermore, when using CAS, a new layer of understanding is 306 required of students. To use the tools properly, students need to form ideas which are not only mathematically correct, but are formed in a manner that the calculator can understand. After problems are posed to the calculator in this fashion, students must be able to interpret calculator-generated output correctly to use it. As was shown by example throughout this document, a one-to-one correspondence does not exist between the syntax used by the TI-92 CAS and standard mathematical notation. This contributed to confusion of students in the CAS class when using the calculator. One might describe the additional layer of abstraction required by the TI-92 CAS by adding a level to Bruner’s three levels of understanding. • Enactive level: This is the most basic level of student understanding. To represent ideas and solve problems, the child must manipulate materials directly. • Ikonic level: At this intermediate level of understanding, students are able to represent ideas and solve problems mentally without relying on direct manipulation of materials. The child uses mental images of physical objects at this stage. Symbolic level: At this advanced stage of understanding, students are able to solve problems by strictly manipulating symbols that represent concrete objects. Mental images and manipulatives are no longer required. • Computer Symbolic level: At this expert stage of understanding, students are able to control symbolic-manipulation devices effectively in a manner that enables the devices to manipulate symbols representing concrete objects. Mental images, manipulatives, and handwritten symbols no longer required at this level of understanding. Results of this study indicate that students had little experience working with calculators prior to their work in advanced algebra. Deficits in calculator understanding made students’ operation at the computer symbolic level difficult, since work at this advanced 307 level requires students to control symbolic manipulation devices in an effective manner. Figures 8.6 and 8.7 provide examples which help distinguish Bruner’s Symbolic level of understanding from the more advanced Computer Symbolic level proposed in this study. cx-ir [ x - i ) - X’- -lx-lx+1. - X ^ + I Figure 8.6: An example of student work illustrating Bruner’s Symbolic level of understanding ^ Y F3V Y FSv Y FS Y M ▼ f— Algebra Calc Other PrgmIO Clear a-z... ■foil[Cx- 1)^) foilCCx- 1)^) ■ expand((x - 1)^) - 2 X + 1 I m h in d e c auto FUNC 2/99 1 Figure 8.7: An example of student work illustrating Calculator Symbolic level of understanding The work in Figure 8.6 shows steps a student has taken to calculate the area of a square with a side having length x —\ units. In her work, the student has not drawn a picture of a square to solve the problem. Furthermore, it is unlikely that the student needed to 308 construct mental images of a square to calculate the area. The student has solved the problem by strictly manipulating symbols that represent concrete objects. Note that in Figure 8.6, the student has manipulated symbols directly using pencil and paper — there is no physical separation between the learner and the symbols the learner manipulates. On the other hand, the work in Figure 8.7 shows the steps another student has taken to calculate the area of the same square while using the TI-92 CAS. Here the student does not manipulate symbols directly. Rather, the student types commands that instruct a machine to manipulate symbols. To function successfully at this level of understanding, the student not only needs to understand how to manipulate symbols but also how to communicate successfully with a symbolic manipulator. Although a teacher or other students may understand a student’s intentions when FO IL ((x —l)^) is typed into the TI-92, the calculator does not understand such a request. It is only when the student types in a command such as expand ((.r—l)’) that he or she able to successfully solve the problem using the TI-92. Because this interaction with the TI-92 requires an additional layer of understanding beyond the direct manipulation of symbols, it suggests a new level of understanding beyond Bruner’s Symbolic level. I refer to this new level of understanding as the Computer Symbolic level. Previous calculator knowledge plavs significant role in student understanding Kutzler refers to the importance of previous knowledge in his work — noting that mathematical knowledge is built on top of prior knowledge in much the same way that 309 upper levels of a house are built upon lower levels. Although Kutzler does not mention prior calculator knowledge in his “house of mathematics,” results of this study suggest that previous calculator knowledge is just as important to student success as prior mathematical knowledge. Kutzler proposes that CAS may be used as a “cognitive scaffold,” enabling students to study more advanced concepts successfully despite deficiencies in with earlier mathematical content. For instance, a student with misconceptions regarding equation solving can still study trigonometry meaningfully using CAS. As equations are revisited in the context of trigonometry, the student has additional opportunities to leam concepts of equation solving — but remains able to solve equations using CAS. Equations Equations Figure 8.8: CAS enables students to study more advanced concepts while resolving misconceptions with earlier material As Kutzler notes, the calculator provides extra time for the student to leam procedural equation solving skills without the immediate threat of failure. A natural question then becomes the following: What scaffolding exists when calculator knowledge is diffident? Figures 8.9 through 8.11 illustrate the need for both strong calculator-based skills and strong algebraic skills in order to provide a stable foundation on which to build future mathematical understanding. 310 by-hand calculator skills skills Figure 8.9: Optimal conditions for acquiring new mathematical knowledge are not met when one has weak by-hand manipulative skills One’s capacity to derive maximal educational benefits from technology-rich mathematics classrooms depends both on strong calculator and procedural skills. In Figures 8.9 and 8.10, an individual student’s by-hand manipulative skill level is represented by the left-most rectangle. Similarly, the student’s calculator skill-level is represented by the right-most rectangle. The height of either rectangle is drawn relative to the student’s strength in a given area (i.e. short rectangle = weak skills). by-hand calculator skills skills Figure 8.10: Optimal conditions for acquiring new mathematical knowledge are not met when one has weak calculator skills 311 As Figure 8.9 depicts, a student may have strong calculator skills, but have a low capacity for learning new mathematical content when by-hand skills are not sufficiently developed. The foundation on which to build new knowledge structures is unstable. Likewise, a student possessing strong by-hand manipulative skills without strong calculator knowledge will experience difficulty learning new content in technology-rich settings. This scenario is depicted in Figure 8.10. When one has strong by-hand manipulative skills and strong calculator skills, a stable surface is created on which to build one’s “house of mathematics.” This is illustrated in Figure 8.11. by-hand calculator skills skills Figure 8.11: Optimal mathematical knowledge-building is possible in technology- rich classrooms only when students have strong by-hand manipulative skills and strong calculator skills In the spirit of Kutzler’s “House of Mathematics,” I refer to the structure on which student algebraic understanding is built as the student’s “foundation of mathematics” - the structure on which a future “House of Mathematics” may be built. 312 Recommendations for Future Research Based both on conclusions from this study, and from a lack of exisiting research, it is apparent that more smdies are needed to investigate various aspects of CAS usage with secondary school smdents. Some factors involving this have been explored by this study, while many other possible areas of research have been identified which necessitate further investigation. The following areas are recommended to the reader for future research: 1. Design a study that investigates the use of a CAS that provides students with opportunities to leam content in a “white box” fashion (as opposed to “black box” methods encouraged by the TI-92). How do such utilities impact students’ conceptual and/or procedural algebraic understandings? 2. Design a study of TI-92 CAS in a setting where graphing calculators were used meaningfully prior to the use of the TI-92. Do students with previous graphing calculator experience react more favorably to CAS-equipped calculators than was the case in this study? 3. Compare the attitudes of first year algebra students with the attitudes of second year algebra students when CAS is introduced in mathematics classes. 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Advanced Algebra students in my classes will have an opportunity to take part in a research study which examines the use of calculator-based symbolic manipulators (CSA) in secondary school classrooms. Throughout the year, data generated by student participants will be analyzed in an attempt to answer the following questions: • How does the use of CSA utilités affect algebraic understanding of students? • How do students use CSA utilities? • How does CSA impact course content in secondary school mathematics classrooms? In the past, much time has been spent teaching students by-hand techniques for manipulating symbols (e.g. solving equations, factoring algebraic expressions) with limited success. Calculators equipped with symbolic manipulation utilities are able to do many tasks traditionally taught with paper-and-pencil methods. The growing popularity o f CSA-based tools Although calculators equipped with symbolic manipulation utilities have been available commercially since 1995, little (if any) systematic research has investigated the implications of their use in high school classrooms (at least in the United States). Because the tools are now allowed on mathematics advanced placement exams (e.g. the AP Calculus exam) and college entrance exams (e.g. SAT), the calculators are becoming more popular with classroom teachers and students. I feel that the use of such tools should be informed by research suggesting guidelines for the technology’s proper use. Differences between two groups My study examines two groups of Advanced Algebra students. The chief goal of the study is to compare various aspects of a “traditionally taught” algebra classroom with the features of a “CSA enhanced” classroom. It is important to note that both groups use the same textbook, are assigned similar homework problems, and take similar tests. Furthermore, the existence o f identical departmental midterm andfinal exams for both groups of students necessitates coverage of the same general course topics. The “traditionally taught” group is taught no differently than any other “regular” advanced algebra class at Upper Arlington High School. In no way are these students deprived of any course materials or instructional activities. Graphing calculators are permitted in both classrooms. However, CSA utilities are not utilized in any systematic fashion in the “traditional” class. On the other hand, the CSA-enhanced group receives addtional instruction regarding the use of calculators equipped with CSA capabilities. Because the use of such tools is allowed on a battery of standardized tests (e.g. SAT and AP Calculus tests), the “CSA enhanced” students will benefit from their exposure to the tools. 325 Data collected Data will be collected from a variety of sources: • Student pretests and post-tests to measure the effect of CSA on mathematical performance • Student questionnaires to measure participant attitudes regarding the use of calculators • Year long appraisal of student problem solving methods as observed on class tests and quizzes • Audiotaping of in-class problem solving activities • Concept mapping activities to investigate student understanding of basic algebraic concepts To participate in the study, two permission forms must be signed by both student and parent/guardian. (1) The Data Usage Permission Slip (see below) and (2) the Consent Form for Participation in Social and Behavioral Research (HS-027) (which is attached to this document). Data Usage Permission Slip 1 consent to participating in (or my child’s participation in) research entitled: S tudy o f h an d h eld calculators in algebra 1 have read the above description of the study. I understand the purpose of the study, the procedures to be followed, and the expected duration of my child’s participation in the study. Possible benefits of the study have been described, as have alternative procedures. My signature below indicates that I give Mr. Edwards permission to: • Use data obtained from my (my child’s) classwork. This includes homework, tests, quizzes, in- class written work and class audiotapings • Analyze any of the above mentioned work for research purposes • Use grade information derived from the above mentioned work Note: none of the information collected will be recorded in such a manner that human subjects can be identified, either directly or through identifiers linked to the subject. 1 acknowledge that 1 have had the opportunity to obtain additional information regarding the study and that any questions 1 have raised have been answered to my full satisfaction. Furthermore, 1 understand that I am (my child is) free to withdraw consent at any time and to discontinue participation in the study without prejudice to me (my child). Data will no longer be collected from students who do not participate in the study. However, non-participants may remain enrolled in class and may still freely take part in all class activities. Date:______Signed: ______Signed: ______Parent / guardian Student / participant 326 Calculator Loan Agreement The parties whose signatures appear below assume full responsiblity for the TI-92 graphing calculator provided to them by Mr. Edwards. By signing this agreement, the below individuals are granted permission to use the TI-92 calculator in the following manner; The calculator may be used on all Advanced Algebra homework The calculator may be used on all Advanced Algebra tests and quizzes The calculator may be used on all Advanced Algebra in-class work Prohibited uses of the calculator include: Playing games in any class during regular school hours Storing answers/written text in calculator on closed book tests and/or quizzes By signing the agreement below, all parties agree that they have been provided with a calculator that was in satisfactory condition at the time of their first possession. A calculator is considered to be in satisfactory condition if and only if each of the following conditions are satisfied: The calculator contains four AA batteries that adequately power the calculator When the “on” button is pressed, the TI-92 home screen is clearly visible While turned “on,” the calculator is capable of performing mathematical calculations (i.e. the screen is not “frozen”) All keyboard keys are operational The output screen contains no cracks The calculator includes a cover which fits snuggly over the calculator The individuals whose signatures appear below agree to return the calculator in working condition (as defined in the above paragraph) at the end o f the school year, upon discontinuing participation in the handheld calculator study, or by violating the terms of acceptable use listed above. If the TI-92 calculator issued to the parties below is not returned in working condition, the parties below agree to replace the calculator with a new one. Note: TI-92 calculators may be purchased at Long’s Bookstore or Staple’s Office Supply Store. Retail price of a new TI-92 is approximately S 189.00. I have read the above contract and agree to its conditions: Student name Parent or Legal Guardian name Today’s date (please print) (please print) Student signature Parent or Legal Guardian signature 327 Appendix B Pre- and Post-Tests 328 Algebraic Skills Pretest Name: In 1-24, multiple choice. Give the letter of the correct answer. 1. What is the solution to the system j ^ ? Lx —y= 4 (a) (-2,2) (b) (1,-2) (c) (2,-2) (d) There are no solutions. 2. LetX = { U, C, D, F, T, H } and5 = { U, I , N, C, D, G, Q }. Find A r\B . (a) {U,C,N,D,G,Q,T} (b) {U,C,D,T} (c) {U,C,D,F,T,H,N,G,Q } (d) {U,C,D,F,T,H,G} 3. Simplify 10' + 7-10“ + 8-10'+7-10 + 3. (a) 178,073 (b) 70,873 (c) 17,873 (d) 170,873 4. The pattern x~ ■ is true for (a) X = 1 and x = 0 only. (b) x = -1 only (c) X = 2 only (d) all real numbers 5. Which of the following triples of numbers could not be the lengths of the sides of a triangle? (a) 8,12,19 (b) 3,7,8 (c) 14,20,32 (d) 5,9,16 a . :z 6. What is the length of AB in the graph at the right? (a) 4 Æ (b) 2V3 4 (c) VÏ4 (d) -2Vs4 329 7. Factor 4jr + 6 j:+ 2 completely. (a) (2 ,r+ 1) - 2(jr+1) (b) (4.r+ 2) - (.r+ 2) (c) (2x+ 3) - (2%+ 2) (d) (2JT-1) - (2 x - 2) 8. If a glacier is currently 200 feet from a cabin and advances 4 inches each year, in how many yeais will it be within 50 feet of the cabin? (a) 150 years (b) 300 years (c) 450 years (d) 600 years 9. The formula gives the volume F of a cone with height h and a base of J radius r. What is this formula solved for r? (c) 10. What is the domain of the function _y= J— ? (a) The set of all real numbers (b) The set of all real numbers except 0 (c) The set of all nonnegative real numbers (d) The set of all positive real numbers 11.Solve the equations 2>w+5 = —25 and 4/7—2=42. What is the sum of the solutions? (a) 1 (b) ^ (c) -5 (d) 0 330 12. If — = w #0, and z# 0, which of the following is not true? (a) ^ = Z ( b ) Z = Z w z W X (c) xz= wy (d) - = - z y 13. How many five-digit numbers can be made from the integers 1 through 9 if each integer can only appear once? (a) 9' (b) 5! (c) 9 8 -7 -6 -5 (d) 9-h8 + 7 + 6 + 5 ^ > 0 14. Identify the graph of the following system: ' x>0 (a) (b) (d) 331 15. The table below shows the average tuition for private four-year universities for selected years. What was the rate of change in the cost of the tuition for private four-year universities from 1980 to 1992? Year Tuition (S) 1980 3,811 1990 10,348 1991 11,379 1992 12,192 (a) $688 per year (b) $698.42 per year (c) 653.70 per year (d) $813 per year 16. The formula for the surface area of a sphere is S=Ak/^. If we assume the earth is a sphere, the surface area of the earth is approximately 1.97 ■ 10® square miles. What is the approximate radius of the earth? (a) 3,960 miles (b) 2,230 miles (c) 1.57 10^ miles (d) 1,120 miles 17. What is the domain of the function in the graph at the right? (a) The set of all real numbers. (b) The set of all nonnegative real numbers. (c) The set of all integers. (d) The set of all positive real numbers. 18. What are the solutions to the equation x- {2x+ 3) • (5x—2) = 0? , . « 3 —2 (a) .r= 0 orx=— orx= — 2 5 (b) x=0 ov x= —ov x= — 3 2 (c) x=Oor,r = — or.r = — 2 5 (d) x=Q or .r=6 or.r=—10 332 19. Which of the following is a graph o f the line x = -1 ? I Y.I J L X X 1 [ 11 1r (a) (b) KJL Yik 1L X X 1r 1r 1f (c) (d) 20. When x is a large number, the graph of which equation rises the fastest? (a) 3 - = / (b) 1" = ^ (c) 3" = ;, (d) 3x= y 21.Theresa’s bank account had B dollars in it. She deposited D dollars every two weeks and withdrew W dollars every month. Assuming the account earned no interest, which of the following expressions represents the amount in Theresa’s account after 6 months? (&) B+13D-6W Qo) B+UW-6D (c) 13W-6D + B (d) B -6D + 13W 333 22. Which of the following graphs is not a graph of a function? (b) (c) 23. Kim enlarged a 5-in. by 7-in. photograph to a 12.5-in. by 17.5-in. photograph. What was the size-change factor of the enlargement? (a) 2.5 (b) 2.0 (c) .4 (d) 1.5 h I 24. The graph at the right shows the height h 100 of a ball in feet seconds after being f 90 \ dropped from a height of 100 feet above \ I- the surface of the moon. About how Ï « \ many seconds after being dropped is the ball 75 feet above the ground? 4 Ô S 10 Tine isQcondsi (a)l (b)3 (c)2 (d)4 334 Technology Literacy in Mathematics 1 . 5 0 . 5 0 .2 5 Student Pretest 0 . 7 5 PlotOBtttx, Tl, . l-l>. ■ De-f'ine f(x)-2-ain(> 0 - 1 ■zerosCfCiO.xl ^2-—’Œ nl*x4r^ - — Z*attl*n + Directions: The test which follows is designed to measure your previous background using a TI graphing calculator or symbolic algebra manipulator (such as the TI-92). This test consists of 15 mulitiple choice items. For each question, circle the best response. All Items are judged as either "correct" or "incorrect." You may not use your graphing calculator on this test. The results of this test will provide educational researchers with important data regarding the proper use of technology in classrooms, so it is important that you do your best work on this test. Please do not begin until you are instructed to do so by your teacher. 335 1. What event has just occurred with the calculator pictured below: 12+X 12 (a) The calculator has solved an equation for x and has determined that x = 12. (b) The user has placed a value of 12 Into the variable x. (c) The user has defined a ray starting at coordinate (12,0). (d) None of these are correct. 2. What will be printed to the screen the next time the user presses enter? 2 .5 2 .5 flnsC2> (a) FALSE - since 2 < 2.5 (b) The product of 2 times A times n times s. (c) ERR:SYNTAX (I.e. this command won't m ake sense to the calculator). (d) 5.0 (e) None of these are correct. 3. What types of data does the user enter in this screen? rioti Plow (a) y values for a table of values (b) Functions In terms of x. (c) Data points to plot. (d) None of these. 336 4. Suppose the contents of LI and L2 are as shown below: LI LZ L3 2 LinRe9(ax+b) 0 1 L2<3) = Which Is a possible output for the command LinReg(ax+b) shown above? LinR eS L inR eg y=ax+b y=ax+b a = l a = l b= "2 b=2 ■ ■ (a) (b) x=o x=o 1Y=2 (c) (d) 337 5. Suppose that the expression 2x+S is typed into Yl. What will happen if I press 2nd/Tabie at this screen? TABLE SETUP T b lS t a r t =0 û T b l= l In d e n t: fSHMg Ask ueeend: I.WIm Ask Ça) A table of values identical to this will be created: X Vi 5 1 7 z 9 1 11 H 13 S IS 6 17 X=0 Cb) A table of values identical to this will be created: X Yl 7 z 9 3 11 H 13 S 19 e 17 7 19 X=1 Cc) A graph of the expression 2x4-5 will be created : yi=2X»9 / X=0 ^ V=9 Cd) A syntax error will occur. The user will see a screen like this: ERR:SYNTAX HBQuit zTGoto 338 6. What will happen once enter is pressed? 5^X 5 X2+2X-5I (a) The number 30 will appear below. (b) The word "POSITIVE" will appear. (c) The word "NEGATIVE" will appear. (d) The words "UNKNOWN OPERATION' will appear. (e) none of these 7. When I want to change a scatterplot to a bar graph what option must be chosen on my calculator at some time? (a) STAT PLOT (b) Y= (c) GRAPH (d) VARS (e) none of these 8. On a TI-83, the following function is graphed: y = 2x + 7. Below is a picture of the graph and its viewing window: Y1=2X»10 WINDOW X n in = -. 1 X n a x = .1 X s c l= l Yrnin= -50 Vnax=50 V s c l= l X r e s= l From the graph, I may safely conclude that: (a) The slope of th e line 2x + 7 is 0. (b) The slope of the line 2x + 7 is infinite. (c) The function y = 2x + 7 has been entered incorrectly. (d) A poor viewing window has been selected for displaying the function. (e) none of these 339 9. Consider the following screen and viewing window, What is the result when GRAPH is pressed? Motl riotz MoO WINDOW xYi = -2X'^2+2X X n in = “10 'vYz=-2(X2-X) Xmax=10 W zB Y z-Y i X s c l= l n Y h = Y n in = -1 0 \Ys= Ymax=10 \Yfi= Yscl=l xYz= X r e s= l (a) Nothing is graphed because a poor viewing window has been selected. (b) The horizontal line y=0 is graphed. (c) The vertical line x=0 is graphed. (d) The user will get a SYNTAX ERROR because Y3 is not a valid equation. 10. The lines graphed below are: Motl Metz ZloO W lB iX +.5 xYzB1.99X+.75 vVj = vVh = n Vs= \Vfi= \Vz= (a) parallel (b) perpendicular (c) intersect (d) none of the above 11. What is a reasonable output for the following command on a TI-92? F5 Tfi V |R 1 g e b r a |Ca 1 c O th er PrgmIO Clear a-z... 2*x+5+3*vl v=3l am . OCG WFFIinX (a) 2x + 3y + 5 (b) FALSE (c) 2x + 14 (d) 18x + 5 (e) none of these 340 12. Which command Is equivalent to soive(3x+5y=4,x) on a TI-92? (a) zeros(3x+5y-4=0,x) (b) csolve(3x+5y-4=0,x) (c) factor(3x+5y-4) (d) expand(3x+5y-4) (e) none of these 13. What will happen when enter is pressed? V A— lAlgebra Calc Other PrgnlO Clear a-z... ■Define f(x)=4.5 x + 22.5 Done ( f (yc:N—ir/•-< 5 ) - f NX N am . DFfi *r»i>ny (a) "4.5" will be output (b) the symbols "f(4)/4" will be output (c) the symbol "f" will be output (d) FALSE will be output (e) none of the above 14. What will happen when enter is pressed? rf FHv F5 F6 ■Ifil ge b r a C a lc O ther P rg n 10 Clear a-z... 2 x /2 ) d DCG ARM tQX rUMCam. (a) "1" will be output (b) "0" will be output "x2" vvill be output [5!"2x/2x" will be output 341 15. What will happen when enter is pressed? FZv F3V Y FHv Y F5 Y Ffi - r — A lg e b r a Calc otheriPrgn10|C1 ear a-z... ■ 3-x + 4-y = 2 3-x + 4 y = 2 ans (a) 3x = 2 - 4y will be output (b) 3x = 2 will be output (c) X = 1.5 will be output (d) none of the above 342 Attitudinal Survey Directions: Please make certain that you are using a #2 pencil. Read each of the statements below. Then, based on your opinion of each statement, fill in the corresponding bubble on your SCANTRON sheet: C D O O C D 1 2 3 4 AGREE AGREE DISAGREE DISAGREE STRONGLY STRONGLY There is no "right" or "wrong" answer. You will not be penalized for any answer that you choose. However, your responses form an important part of a university study designed to improve the quality of mathematics teaching in high schools. So it is important that you answer the following questions as truthfully and thoughtfully as possible. 1. Graphing calculators are complicated and don't help me in learning mathematics. 2. When using a graphing calculator in class, I do not need to learn how to compute because the calculator does everything for me. 3. If my calculator gives me no answer, then the problem has no solution. 4. Graphing calculators are a useful support for discovering algebraic rules. 5. Graphing calculators increase my desire to do mathematics. 6. I get lost doing calculations on the graphing calculator. 7. Graphing calculators do not help me when working on exams. 8. Graphing calculators help me understand mathematics. 9. Graphing calculators help people who have difficulties with algebra to still be able to do mathematics. 10. Calculations are easier with a graphing calculator. 11. When the data I type in is correct, then I can fully trust the graphing calculator's output. 12. The graphing calculator helps me to get an idea of the result of a calculation before doing it. 13.The graphing calculator helps me solve problems without getting lost in calculations. 343 14.The graphing calculator is useful because it allows me to look at the same problem in more than one way. 15. The graphing calculator especially helps with lengthy and boring calculations. 16.The graphing calculator is a useful tool to check my work. 17.1 am dependent on the graphing calculator for doing simple arithmetic (such as adding two single digit numbers together). 18. Graphing calculators are easy to use and help me in learning mathematics. 19. When using a graphing calculator in class, I still have to know how to compute because the calculator won't do everything for me. 20. If my calculator gives me no answer, then I reexamine the solution without a calculator. 21. Graphing calculators aren't useful for discovering algebraic rules. 22. Graphing calculators don't increase my desire to do mathematics. 23. Graphing calculators help me when working on exams. 24. Graphing calculators don't help me understand mathematics. 25. People who have difficulties with algebra have the same difficulties - even with a calculator. 26. Even when I type in correct data, there is a possibility that the calculator will give me an incorrect answer. 27.The graphing calculator really doesn't help me to get an idea of the result of a calculation before I do it. 28. Calculations are no easier with a graphing calculator. 29.The graphing calculator is confusing because there is more than one way to look at every problem (for example, as a graph or symbolically) 30. The graphing calculator doesn't help with lengthy and boring calculations. 31. The graphing calculator isn't useful for checking the result of a calculation. 32.1 am not dependent on the graphing calculator for doing simple arithmetic (such as adding two single digit numbers together). 344 Advanced Algebra Name: ______June 2000 Semester Exam In problems I and 2 use the functions f{^x) — 'ljr-Z x—S and g(x) = 3x-2. 1. Find y(^(3)) (a) 72 (b)4 (c) 10 (d) 170 2. Find /'(^(x)') (a) 1 8 .r-9 .r+ 9 (b) 6 Y -9 .r- 1 7 (c) 1 8 ^ -3 3 x + 9 (d) 6 jr -3 x -7 3. A function is given by the formula /"(x) = Sx—7. Find a formula for . (a)/-'(jr) = 7jr+8 (b )/-'W = |jr + | (c)(jr) = i . r + | (d) = 4. Which of the following formulas is not an example of a function? (a) 2 x -3 y = S (b) y =2" (c) _K=log,(-r) (d) jx z=x—5 5. The domain of the function y(x) = V .r-2 is which o f the following? (a )-r> 2 (b) —2< x < 2 (c) All reals (d) All positive reals 345 6. Which of the following screen shots depicts the graph of f{_x) = |jr— 2[? (a) (b) (c) (d) 7. Given the sequence 25, -5, 1, -0.2,... This sequence is which of the following? (a) Arithmetic (b) Geometric (c) Neither arithmetic or geometric (d) Fibonacci 8. Assuming that 0 < 6 < 1, which of the following is arranged from least to greatest? (a) b~, b'~, b, 6 '^ (b) b'^, 6'^, 6, b~ (c) b~, b, 6'^, 6'^ (d) 6'^, b, b’’^ 9. Which of the following is not equivalent to y ? (a) (b) (c) / (d) 4 7 10. Which of the following is smallest? (a) -5 -3 (b) -5 1/3 (c) (d) 5^ 11. An airplane begins a final descent to the runway from an altitude of 3500 feet. The horizontal distance from the plane to the runway is 20,000 feet. At approximately what angle of depression will the plane descend? (a) 10.1 degrees (b) 70.1 degrees (c) 9.9 degrees (d) 0.173 degrees 346 12. sin(240°)=. (c) - (d) 13. Give the exact value for cos(-30°): 1 (b) < • > # 14. If sin(6)=0.2 and 0° < 0 < 360° and 0 is obtuse, then cos(0)= (a) 0.2 (b) -0.8 (c) -0.98 (d) 0.98 15.Using the triangle below, find the measure of angle A to the nearest tenth of a degree (you may assume that C is a right triangle): C 20 A (a) 24.6° (b) 42.3° (c) 47.7° (d) 65.4° 16. Which of the following is not true? (a) sin(30°) + cos(30°) = 1 (b) sin(55°} = cos(35°) (c) sin(0) = sin(180° - 0) (d) sin^(40°) + cos^(40°) = 1 17. The solution to the equation 25‘ '"^' =125is which of the following values? (a) 2 (b)-2 (c) 1/3 (d) 1/4 347 18. Solve the following equation for x: \lx+ 1—9 = 16. (a) 27 (b) 64 (c) 15624 (d) 81 (e) 15625 19. Simplify the following expression by “combining” terms: ------—— 2 jy - (a) (b) -16- Y ' (c)-4-■ y'~ (e) none of these ^4 20. Assuming that ai^O, then is equivalent to which of the following? a (a) cr (b) (c) a~ (d) - 21. Solve for x: logx(16) = 4 (a) 2 (b) 1/2 (c) 0.25 (d) 2.7725877 22. Solve for x: 2 - ln(8) + ln(3) = In(.r) (a) 2.944439 (b) 19 (c) 5.2574954 (d) 192 23. Write as an exponential equation: log^ (3)5-*=^ (b) (-4)==^ (c) 625 = (-5)‘ (d) = 5 625 625 ,6 2 5 / 24. The domain of the exponential function y = 10^ is: (a) all positive integers (b) x (c) non-negative real numbers (d) all real numbers 348 25. Uranium X 2 has a half-life of 1.18 minutes. If a sample is measured to be one gram, and then is remeasured 10 minutes later, how much Uranium X 2 would remain? (a) 0.92146 grams (b) 0.5626 grams (c) 0.00028 grams (d) 0.00281 grams 26. Choose the equation that best fits the graph: (4,2) C2J) (a) j'=logix) (b) y = 2e'' (c) y = 2 " 27. Solve for t: / - (/ -i- 5) = 40 (a) t = 5 or 40 (b) / = -5±Vl85 (c) /= ^~V --(d) t = 8 or 5 28. The expression ^ is equivalent to which of the following? j-t-/ (a) (b) (c) (d) i 4 0 / 29. The expression is equivalent to which of the following? (a) (b) 3V2 (c) VÎÔ (d) 3/ V2 349 30. Which is not a square root of -9? (a) -3i (b) 3i (c) -3 (d) 31. Which of the following is the graph of a function whose inverse is also a function? (a) (b) (c) (d) 32. Rewrite the following in a-\- b - /form: (—7 + 2/) - (—7—2/3 (a) 45 + 0 / (b) 53 + 0 / (c) 53-4-/ (d)-14 + 0-/ 33. At what rate of interest compounded continously would you have to invest your money so that it would triple in 8 years? (a) about 6% (b) about 7.3% (c) about 13.7% (d) about 24% 34. Suppose that sin(0)=O.853 for some angle 0, where 0 is between 0 and 360 degrees. Which of the following are correct values for 0? (a) 58.54° only (b) 58.54° and-58.54° (c) 58.54° and 301.46° (d) 58.54° and 121.46° (e) none of these 350 35. Find the length of side AB below. You may assume that ZC = 90°, CB = 19 units, and Z A = 28°: C (a) 8.92 units (b) 40.47 units (c) 16.78 units (d) 21.5 units 36. If 5 < log (a) < 6, then which number below could equal a? (a) 387,000 (b) 55 (c) 56,900 (d) 3,261,000 37. Which expression below is equivalent to c"? (a) S' ■ à ■ ^a -S (b) Û - S ■ yfâ (c) S' ■ à ■ (d) none of these 38. Simplify 3- a° (a) 1 (b) 0 (c) 3 (d) a 39. If X > 1, then x~~'^. Which symbol makes this a mathematically correct statement? (a) > (b)< (c) = (d) L J 351 40. You are saving for college. When you were 5 years old, your grandparents put $2000 in an accounting 9% interest compounded monthly. After 10 years, how much money was in the account? Assume that no deposits or withdrawls were made to the account. (a) $3,800 (b) 3,920.14 (c) $5,021.56 (d) $4,902.71 (e) none of these 41. Which of the following is the vertex form of the equation: ^ = (,r +1) • (jr— 5) ? (a) jr —4x —5 (b) ^=(.r+2)'—9 (c) j/ = (x —2)~ —9 (d) y=Çr-2)--5 (e) j^=(x-h2)~+ 9 42. Consider the function: _y= log^(5 "'). What is the slope of the line created by this equation? (a)lo&;(3) 0») (0 k)g(15) (d) k%K5/3) (e) none of these 43. Suppose I want to graph log;(,r) on my graphing calculator. Which of the following expressions would allow me to do this? ( a ) , = ^ (b),=i5i^ (c),= Iog(£) (d ), = i 2 ^ log(x) log(5) 5 5 (e) none of these 44. Suppose I want to determine whether or not two functions / ’(x) = 2- x —5 and 5 ^(x) = —-— are inverses of each other. Which of the techniques listed below provide useful information? (a) Graph f(x), g(x) and the line y = -x. If f(x) and g(x) appear to be reflections of each other over the line y = -x, chances are good they are inverses. (b) Graph f(x) and l/g(x). If the two graphs overlap, the functions are inverses. (c) Calculate the composite functions f(g(x)) and g(f(x)). If both are equivalent to 1, the functions are inverses of each other. (d) Calculate the composite functions f(g(x)) and g(f(x)). If both are equivalent to X, the functions are inverses of each other. (e) None of these provide useful information regarding inverse functions. 352 45. Consider the graph of the unit circle below. Assume that the point (1,0) has been rotated A degrees around the center of the circle to obtain the point (0.1,y). Find the value of A. (a) 0.10 (b) 5.73 (c) 84.26 (d) 88.62 (e) none of these 46. In problem 45, how could one find the value of the unknown y coordinate? (a) Calculate A, then find sin(A). The value sin(A) = y. (b) Calculate A, then find cos(A). The value cos(A) = y. (c) Use the pythagorean theorem with the equation 0.1" + = I- Solve for y. (d) Both (a) and (c) are correct methods. (e) Both (b) and (c) are correct methods. 47. Suppose that a quadratic function g(x) has a discriminant with value 14. Which of the following conclusions may be safely assumed using no other information? (a) The function intersects the x axis exactly twice. (b) The function is an “upward opening” parabola. (c) The function is a “downward opening” parabola. (d) The function has no real roots (only imaginary ones). (e) None of the above may be safely assumed. 48. When the discriminant of a quadratic function is positive, i.e. ér —4ac> 0, which of the following must be true of the graph of that quadratic? (a) It intersects the x-axis once, (b) It intersects the x-axis twice. (c) It doesn’t “touch” the x-axis. (d) Not enough information to know. 353 49. Suppose that f(x) is a power function of the form yC-r) = y . If the point (6,279936) lies on a graph of the function, find n. (a) n = 3 (b) n = 5 (c) n = 7 (d) n = 6 (e) none o f these 50.Suppose a calculator is dropped from a 30 foot tall building. After how many seconds will the calculator hit the ground? (Note: assume wind resistance is negligible). (a) 2.647 seconds (b) 1.369 seconds (c) 2.011 seconds (d) 3.701 seconds 354 Appendix C Student attitudinal survey results by item (Fall, non-CAS group) 355 Mean Std Dev Graphing Calculators are a useful support for discovering algebraic rules.(4) 2.7727 1.0203 Graphing calculators aren't useful for discovering algebraic rules.(21) 3.0000 0.8728 Graphing calculators increase my desire to do mathematics.(5) 2.5454 0.8578 Graphing calculators don' t increase my desire to do mathematics. (22) 2.5000 0.8591 Graphing calculators help me understand mathematics. (8) 2.7272 0.7 672 Graphing calculators don't help me understand mathematics. (24) 3.0000 0.8728 Graphing calculators help people who have difficulties with algebra to still be able to do mathematics. (9) 2 .7 2 7 2 0.8270 People who have difficulties with algebra have the same difficulties - even with a calculator. (25) 2.5454 0.9116 Calculations are easier with a graphing calculator. (10) 3.045 0.7222 Calculations are no easier with a calculator. (28) 3.5909 0.5032 When the data I type in is correct, then r can fully trust the graphing calculator's output. (II) 2.7272 0.7025 Even when I type in correct data, there is a possibility that the calculator will give me an incorrect answer. (26) 2.2727 0.8270 The graphing calculator helps me to get an idea of the result of a calculation before doing it. (12) 3.0000 0.6900 The graphing calculator really doesn' t help me to get an idea of the result of a calculation before I do it. (27) 2.9090 0.8111 The graphing calculator helps me solve problems without getting lost in calculations. (13) 2.7272 0.6310 I get lost doing calculations on the graphing calculator.(6) 2.5909 0.8540 The graphing calculator is useful because it allows me to look at the same problem in more than one way. (14) 3.3181 0.7162 The graphing calculator is confusing because there is more than one way to look at every problem (for example, as a graph or symbolically). (29) 2.6818 0.7162 The graphing calculator especially helps with lengthy and boring calculations. (15) 3.4545 0.5096 The graphing calculator doesn' t help with lengthy and boring calculations. (30) 3.4090 0.7341 The graphing calculator is a useful tool to check my work. (16) 3.5909 0.5903 The graphing calculator isn't useful for checking the result of a calculation. (31) 3.4545 0.8578 I am dependent on the graphing calculator for doing simple arithmetic (such as adding two single digit numbers together)(17) 3.4545 0.6709 I am not dependent on the graphing calculator for doing simple arithmetic (such as adding two single digit numbers together) . (32) 3.3181 0.7798 356 Graphing calculators are easy to use and help me in learning mathematics. (18) 2.9090 0.7501 Graphing calculators are complicated and don't help me in learning mathematics. (1) 3.0909 0.6101 When using a graphing calculator in class, I still have to know how to compute because the calculator won ' t do everything for me. (19) 3.5909 0.5903 When using a graphing calculator in class, I do not need to learn how to compute because the calculator does everything for me. (2) 3.4545 0.5096 If my calculator gives me no answer, then I reexamine the solution without a calculator. (20) 3.3181 0.7798 If my calculator gives me no answer, then the problem has no solution. (3) 3.7727 0.4289 Graphing calculators help me when working on exams. (23) 3.3181 0.6463 Graphing calculators do not help me when working on exams.(7) 3.1818 0.5010 357 Appendix D Student attitudinal survey results by item (Fall, CAS group) 358 Mean Std Dev Graphing Calculators are a useful support for discovering algebraic rules.(4) 3.1153 0.7114 Graphing calculators aren't useful for discovering algebraic rules. (21) 2.3600 0.7000 Graphing calculators increase my desire to do mathematics.(5) 2.4615 0.8114 Graphing calculators don't increase my desire to do mathematics. (22) 2.3076 0.6793 Graphing calculators help me understand mathematics. (8) 2.8461 0.8338 Graphing calculators don' t help me understand mathematics.(24) 2.7307 0.6667 Graphing calculators help people who have difficulties with algebra to still be able to do mathematics. (9) 2.8846 0.8161 People who have difficulties with algebra have the same difficulties - even with a calculator. (25) 2.3846 0.7524 Calculations are easier with a graphing calculator. (10) 3.2692 0.7775 Calculations are no easier with a calculator. (28) 3.3461 0.6287 When the data I type in is correct, then I can fully trust the graphing calculator's output. (11) 2.4230 0.6433 Even when I type in correct data, there is a possibility that the calculator will give me an incorrect answer. (26) 2.2307 0.9080 The graphing calculator helps me to get an idea of the result of a calculation before doing it. (12) 2.7600 0.5972 The graphing calculator really doesn' t help me to get an idea of the result of a calculation before I do it. (27) 2.5000 0.8602 The graphing calculator helps me solve problems without getting lost in calculations. (13) 2.8846 0.5159 I get lost doing calculations on the graphing calculator.(6) 3.0400 0.7895 The graphing calculator is useful because it allows me to look at the same problem in more than one way. (14) 3.2692 0.8744 The graphing calculator is confusing because there is more than one way to look at every problem (for example, as a graph or symbolically) . (29) 2.8076 0.8494 The graphing calculator especially helps with lengthy and boring calculations. (15) 3.3846 0.4961 The graphing calculator doesn' t help with lengthy and boring calculations. (30) 3.2692 0.7243 The graphing calculator is a useful tool to check my work. (16) 3.5384 0.7060 The graphing calculator isn' t useful for checking the result of a calculation. (31) 3.2307 0.8629 I am dependent on the graphing calculator for doing simple arithmetic (such as adding two single digit numbers together)(17) 3.1153 0.8 638 I am not dependent on the graphing calculator for doing simple aritlimetic (such as adding two single digit numbers together). (32) 2.884 0.9089 359 Graphing calculators are easy to use and help me in learning mathematics. (18) 2.9615 0.5276 Graphing calculators are complicated and don't help me in learning mathematics. (I) 3.3076 0.5491 When using a graphing calculator in class, I still have to know how to compute because the calculator won't do everything for me. (19) 3.5769 0.5038 When using a graphing calculator in class, I do not need to learn how to compute because the calculator does everything for me. (2) 3.3461 0.4851 If my calculator gives me no answer, then I reexamine the solution without a calculator. (20) 3.0769 0.6275 If my calculator gives me no answer, then the problem has no solution. (3) 3.3076 0.6793 Graphing calculators help me when working on exams. (23) 3.3846 0.6373 Graphing calculators do not help me when working on exams. (7) 2.7692 0.8 629 360 Appendix E Student attitudinal survey results by item (Spring, non-CAS group) 361 Mean Std Dev Graphing Calculators are a useful support for discovering algebraic rules.(4) 3.3536 0.7895 Graphing calculators aren't useful for discovering algebraic rules. (21) 2.7272 0.7672 Graphing calculators increase my desire to do mathematics. (5) 3.0454 0.4857 Graphing calculators don' t increase my desire to do mathematics. (22) 2.6818 0.8387 Graphing calculators help me understand mathematics. (8) 3.1818 0.6644 Graphing calculators don' t help me understand mathematics. (24) 2.9545 0.8438 Graphing calculators help people who have difficulties with algebra to still be able to do mathematics. (9) 2.9545 0.6529 People who have difficulties with algebra have the same difficulties - even with a calculator. (25) 2.3181 0.6463 Calculations are easier with a graphing calculator. (10) 3.318 0.4767 Calculations are no easier with a calculator.(28) 3.2272 0.8125 When the data I type in is correct, then I can fully trust the graphing calculator's output. (11) 2.9090 0.7501 Even when I type in correct data, there is a possibility that the calculator will give me an incorrect answer. (26) 1.8636 0.8335 The graphing calculator helps me to get an idea of the result of a calculation before doing it. (12) 3.3181 0.4767 The graphing calculator really doesn't help me to get an idea of the result of a calculation before I do it. (27) 2.8181 0.8528 The graphing calculator helps me solve problems without getting lost in calculations. (13) 2.9090 0.7501 I get lost doing calculations on the graphing calculator. (6) 2.8131 0.7326 The graphing calculator is useful because it allows me to look at the same problem in more than one way. (14) 3.7727 0.4289 The graphing calculator is confusing because there is more than one way to look at every problem (for example, as a graph or symbolically). (29) 3.2272 0.7516 The graphing calculator especially helps with lengthy and boring calculations. (15) 3.3636 0.5810 The graphing calculator doesn't help with lengthy and boring calculations. (30) 3.1818 0.6644 The graphing calculator is a useful tool to check my work. (16) 3.8181 0.3947 The graphing calculator isn' t useful for checking the result of a calculation. (31) 3.5454 0.5958 I am dependent on the graphing calculator for doing simple arithmetic (such as adding two single digit numbers together)(17) 2.9545 0.8438 I am not dependent on the graphing calculator for doing simple arithmetic (such as adding two single digit numbers together). (32) 3.0454 0.7854 362 Graphing calculators are easy to use and help me in learning mathematics. (18) 3.2272 0.6119 Graphing calculators are complicated and don' t help me in learning mathematics. (1) 3.2272 0.6853 When using a graphing calculator in class, I still have to know how to compute because the calculator won't do everything for me. (19) 3.4090 0.5903 When using a graphing calculator in class, I do not need to learn how to compute because the calculator does everything for me. (2) 3.3 636 0.7267 If my calculator gives me no answer, then I reexamine the solution without a calculator. (20) 3.2727 0.5504 If my calculator gives me no answer, then the problem has no solution. (3) 3.5454 0.5958 Graphing calculators help me when working onexams. (23) 3.2727 0.7672 Graphing calculators do not help me when working on exams. (7) 2.8635 0.8335 363 Appendix F Student attitudinal survey results by item (Spring, CAS group) 364 Mean S td Dev Graphing Calculators are a useful support for discovering algebraic rules.(4) 3.4091 0.5903 Graphing calculators aren't useful for discovering algebraic rules. (21) 2.5909 0.7954 Graphing calculators increase my desire to do mathematics. (5) 2.8182 0.5645 Graphing calculators don't increase my desire to do mathematics. (22) 2.5000 0.5726 Graphing calculators help me understand mathematics. (8) 3.1818 0.3948 Graphing calculators don't help me understand mathematics. (24) 2.9091 0.9211 Graphing calculators help people who have difficulties with algebra to still be able to do mathematics. (9) 3.0909 0.5102 People who have difficulties with algebra have the same difficulties - even with a calculator. (25) 2.2727 0.9351 Calculations are easier with a graphing calculator. (10) 3.1818 0.7327 Calculations are no easier with a calculator. (28) 3.0455 0.8439 When the data I type in is correct, then I can fully trust the graphing calculator's output. (11) 2.4545 0.9117 Even when I type in correct data, there is a possibility that the calculator will give me an incorrect answer. (25) 2.1818 0.795 The graphing calculator helps me to get an idea of the result of a calculation before doing it. (12) 3.0455 0.4857 The graphing calculator really doesn' t help me to get an idea of the result of a calculation before I do it. (27) 2.7273 0.7025 The graphing calculator helps me solve problems without getting lost in calculations. (13) 3.0909 0.5254 I get lost doing calculations on the graphing calculator.(5) 2.8535 0.8888 The graphing calculator is useful because it allows me to loolc at the same problem in more than one way. (14) 3.3535 0.7895 The graphing calculator is confusing because there is more than one way to loolc at every problem (for example, as a graph or symbolically). (29) 2.7273 0.9847 The graphing calculator especially helps with lengthy and boring calculations. (15) 3.3182 0.4767 The graphing calculator doesn' t help with lengthy and boring calculations. (30) 3.1354 0.7102 The graphing calculator is a useful tool to chec)c my wor)c. ( 15) 3.5455 0.5958 The graphing calculator isn't useful for checking the result of a calculation. (31) 3.3535 0.7895 I am dependent on the graphing calculator for doing simple arithmetic (such as adding two single digit numbers together)(17) 2.5354 0.9021 I am not dependent on the graphing calculator for doing simple arithmetic (such as adding two single digit numbers together). (32) 2.7727 0.9223 365 Graphing calculators are easy to use and help me in learning mathematics. (18) 3.0909 0.7502 Graphing calculators are complicated and don' t help me in learning mathematics. (1) 3.2727 0.8270 When using a graphing calculator in class, I still have to know how to compute because the calculator won't do everything for me. (19) 3.2273 0.7516 When using a graphing calculator in class, I do not need to learn how to compute because the calculator does everything for me. (2) 2.8182 0.9580 If my calculator gives me no answer, then I reexamine the solution without a calculator. (20) 3.3182 0.4767 If my calculator gives me no answer, then the problem has no solution. (3) 3.4545 0.8004 Graphing calculators help me when working on exams. (23) 3.2273 0.6853 Graphing calculators do not help me when working on exams. (7) 2.5 0.9129 366 APPENDIX G Item analysis of Advanced Algebra Final Examination by class and topic 367 Percent Correct R e s p o n s e s P e r c e n t Ite m non-CAS CAS D i f f e r e n c e T o p ic 15 100 72 28 Trigonometry 11 77.3 56 21.3 Trigonometry 34 77.3 56 21.3 Trigonometry 12 86.4 76 10.4 Trigonometry 13 95.5 92 3.5 Trigonometry 45 95.5 92 3.5 Trigonometry 16 81.8 84 - 2 .2 Trigonometry 35 72.7 80 -7.3 Trigonometry 14 13.6 24 -10.4 Trigonometry 46 31.8 44 -12.2 Trigonometry 21 100 92 8 Solving Equations 18 86.4 84 2.4 Solving Equations 22 81.8 88 -6.2 Solving Equations 27 81.8 96 -14.2 Solving Equations 17 100 100 0 Solving Equations 7 31.8 60 -28.2 Sequences 50 77.3 56 21.3 Quadratic Forms 41 40.9 28 12.9 Quadratic Forms 47 68.2 72 -3.8 Quadratic Forms 48 77.3 88 -10.7 Quadratic Forms 23 95.5 80 15.5 Logarithms 42 40.9 36 4.9 Logarithms 43 90.9 92 -1.1 Logarithms 36 90.9 100 -9.1 Logarithms 6 95.5 80 15.5 Graphs 26 95.5 84 11.5 Graphs 4 9 90.9 88 2 .9 Graphs 44 68.2 32 36.2 Functions 31 81.8 60 21.8 Functions 1 100 84 16 Functions 3 95.5 80 15.5 Functions 5 95.5 80 15.5 Functions 2 59.1 52 7.1 Functions 4 50 44 6 Functions 24 68.2 72 -3.8 Functions 10 81.8 44 37.8 Exponents/Radicals 20 86.4 80 6.4 Exponents/Radicals 25 59.1 32 27 .1 Exponents/Radicals 40 90.9 68 22.9 Exponents/Radicals 37 77.3 56 21.3 Exponents/Radicals 39 68.2 52 16.2 Exponents/Radicals 8 54.5 44 10.5 Exponents/Radicals 19 86.4 76 10.4 Exponents/Radicals 33 77.3 68 9.3 Exponents/Radicals 9 72.7 64 8.7 Exponents/Radicals 38 90.9 84 6.9 Exponents/Radicals 28 86.4 68 18.4 Complex Numbers 29 72.7 68 4.7 Complex Numbers 30 54.5 56 -1.5 Complex Numbers 32 86.4 88 -1.6 Complex Numbers 368 APPENDIX H Test and quiz items designed for multiple solution strategies 369 3. In your own words, explain the Idea of function. You may want to use terms such as "inputs", "domain" and/or "range" in your definition. Include an example of a function and an example of a non-function to strengthen your explanation. (3 points for correct definition, 2 points for example, 2 points for non-example) Figure A.1: Question 3 from Quiz 1 covering functions (week 2) 6. Solve the same equation using either graphical or symbolic methods on the calculator. Write down all steps that you used. Be as specific as possible. Write out keys that you pressed and menu options selected. (5 pts. for correct explanantion, 5 pts. for correct solution for x) Figure A.2: Question 6 from Test 1 (part 1) covering functions (week 4) 4. Solve for x. Show all work (6 points) 2X + 17 — 8------ Figure A 3: Question 4 from Test 1 (part 2) covering functions (week 4) 4. Consider the following data: Speed traveled (s) Breaking Distance (D) 0 0 10 20 20 80 30 180 50 500 (a) Find a formula that describes distance in terms of speed (i.e. find an equation that fits this data with distance considered the dependent variable). Explain how you found this formula. You may use prior knowledge of distance and breaking distance to help you (3 pts. for correct formula with work, 2 points for explanation. No work = no credit.) Figure A 4: Question 4 from Quiz 2 covering direct variation (week 7) 370 2. A service technician charges S36.25 for 5 hours of work and $65.25 for 9 hours of work. Assuming a linear relationship between the charge and the number of hours worked, what will be the charge for 15 hours of work? EXPLAIN HOW YOU FOUND THIS SOLUTION! SHOW ALL WORK! (4 pts.) Figure A.5: Question 2 from Test 3 covering linear relations (week 11) 4. (a) Provide an algebraic definition for absolute value; (b) Solve the following equation using a method of your choice - either by hand or calculator based methods. If using by-hand methods, show all steps. If using calculator methods, explain all calculator steps in detail. I 2 x - 7 1 = 2 0 Figure A.6: Question 4 from Quiz 6 covering Quadratics (week 23) 2. Consider the parabolic equation f(x) = 2x^ + 12x + 3. (a) Rnd the vertex of this parabola. You may use calculator or algebraic methods (please show all work in either case). Figure A.7: Question 2 from Test 6 covering Quadratics (week 26) 371 1. Suppose an nth power function is graphed below. Write an equation for the function. SHOW ALL WORK. Explain your reasoning clearly in at least several complete sentences. (3,2187) Figure A.8: Question 1 from Test 7 covering power functions and exponents (week 31) 8. Verify that y = log^(x) and y = 7>^ are inverses of each other using whatever method you wish. Show all work and/or explain steps (include any graphs, tables, or equations used). Explanations must be written in several complete sentences. Figure A.9: Question 8 from Test 9 covering logarithms (week 38) 372 APPENDIX I Selected items from Solving Linear Equations Essay 373 Consider a line passing through points (2,5) and (7,12). 1. Using the point slope formula or the slope intercept formula, construct a linear equation of the form Ax + By = C that passes through the two points. SHOW ALL WORK CLEARLY. Figure A.10: Item I from Solving Linear Equations essay 2. Using calculator-based methods, construct a linear equation of the form Ax + By = C that passes through the two points (i.e. verify that your solution in 1 is correct). SHOW ALL WORK / EXPLAIN ALL STEPS CLEARLY. Figure A.11: Item 2 from Solving Linear Equations essay 3. Which method do you prefer? Why? Please explain in 3 or 4 complete sentences (I am very curious to know your opinions. Please take several minutes and answer this one thoughtfully.) Figure A.12: Item 3 from Solving Linear Equations essay 374 APPENDIX J Systems warm-up and skills-based warm-up activities 375 Skills-based Warm-up Problem 3. Solve the following system using whatever method you wish. If you decide to use calculator-based methods, please write out your steps carefully, including menu options chosen, window settings, etc. 200x - O.OOSy = 300 x + 354y= 12 Figure A. 13: Skills-based manipulation problem involving systems of equations (November 10, 1999) Systems Warm-up Problem Your parents have decided to let you have your own phone line. They will pay the set-up fees, but you will have to pay your own bill. The telephone company gives you a choice of two plans. Plan A charges $12.46 per month plus 13 cents for each local call. Plan B charges $24.50 per month with no additional charge for local calls. Consider the following steps to determine which plan is the better value: 1. For plans A and 8, write equations that represent the total monthly bill in terms of the basic rate plus the number of phone calls. 2. Determine under what circumstances each plan would be best. Be specific about the number of phone calls where each plan would be preferable. Figure A.14: Phone bill application problem (November 17, 1999) 376 APPENDIX K Systems Essay writing prompts 377 4. hi a well-formed paragraph, respond to the following statement: "To learn algebra well, a student needs lots of practice with pencil and paper." Figure A.15: Item 4 from Systems Essay assignment 5. Describe the manner in which calculators have been used in your math class this year. Do you feel that calculators have aided in your understanding of algebra concepts? Explain in a well-formed paragraph. Please include examples of problems in which the calculator has been useful (or not so useful). Figure A.16: Item 5 from Systems Essay assignment 378 APPENDIX L CalculatorAlgebra Questionnaire writing prompts 379 3. In a paragraph or two, describe any future career plans that you may have at this time. Do you anticipate that your future career plans will involve you taking mathematics-intensive classes in college? Please comment. Figure A.17: Item 3 from CalculatorAlgebra Questionnaire 4. Do you feel that your ability (or inability) to use calculators will affect your future career aspirations AND college career? How does the calculator fit into the larger picture of future career goals? Figure A. 18: Item 4 from Calculator Algebra Questionnaire 2(b). Do you use the calculator outside of math class? Have you ever shown any of your family members how to use the calculator? Please comment. Figure A. 19: Item 2(b) from Calculator Algebra Questionnaire 1. Which type of calculator is the model which you use most frequently in math class? (circle one) (a) TI-92 (b) TI-89 (c) TI-83 (d) TI-85 (e) None of the above Figure A.20: Item 1 from Calculator Algebra Questionnaire 6. If you could have the first half of the year over again, what would you change about the WAY you've learned math this year. For example, would you prefer to use the calculator more or less? Prefer to use a different calculator? Prefer different teaching techniques? Please include specific examples in your response. Figure A.21: Item 6 from CalculatorAlgebra Questionnaire 2(a). Describe your comfort level with the calculator. Figure A.22: Item 2(a) from CalculatorAlgebra Questionnaire 380 5. Characterize your use of calculators when doing math homework AND test work. Under what circumstances do you find the calculator most useful? Provide specific examples from this school year. Figure A.23: Item 5 from CalculatorAlgebra Questionnaire 381 APPENDIX M End o f year short answer essay writing prompts 3 8 2 1. Describe the role that calculators have played in your learning of mathematics this year. Please provide examples from class to strengthen your descriptions. Figure A.24: Item 1 from End of Year Short Answer Essay 2. Describe your use of calculators in math class prior to this school year. Did your math teacher use calculators to teach new concepts? Was the calculator used primarily as a way to check homework? Explain in 4 or 5 sentences. Be specific! Figure A.25: Item 2 from End of Year Short Answer Essay 3. Consider the following opinion: "Calculators will eventually make pencil and paper mathematics obsolete." Do you agree with this statement? Explain why or why not. Give specific examples from advanced algebra that help support your argument. Figure A.26: Item 3 from End of Year Short Answer Essay 4. How do you feel about the way calculators were used this year? Do you wish we would have used the calculator less often? more often? Do you think that your knowledge of calculators will help you next year (or not?) Please comment in at least 5 or 6 sentences. Specific details will help me model instruction for future classes (so please answer with sufficient details / explanations!) Figure A.27: Item 4 from End of Year Short Answer Essay 383