The Centroid
IN THIS ISSUE: • Bite Sized Mathematics: The Case of the Radical • Knowing When to Which: Addressing the Calculator Quandary • Strategies to Support Mathematical Understanding of LEP Students • Butterfly Sticker & Homework Recording Sheet • 2005 State Math Fair Winners • 2005 State Math Contest Winners • 2005 Logo Contest Winners
OFFICIAL JOURNAL OF THE NORTH CAROLINA COUNCIL OF TEACHERS OF MATHEMATICS VOLUME 31 • NUMBER 2 • FALL 2005
The Centroid is the official journal of the North General articles are welcome, as are the following special categories of articles: Carolina Council of Teachers of Mathematics (NCCTM). Its aim is to provide information and • A Teacher’s Story, ideas for teachers of mathematics—pre-kindergarten • History Corner, through teacher education. The Centroid is published • Teaching with Technology, in January and August. Subscribe by joining • It’s Elementary! NCCTM; see the Membership Form on the last page. • Math in the Middle, and • Algebra for Everyone. Submission of Manuscripts We invite the submission of news, announcements, Editorial Board and articles useful to school mathematics teachers or mathematics teacher educators. In particular, K-12 Editors teachers are encouraged to submit articles describing Holly P. Hirst, Appalachian State University teaching mathematical content in innovative ways. Brian H. Felkel, Appalachian State University To be considered for inclusion in an issue, news and announcements must be received by November 1 for Board Members Deborah A. Crocker, Appalachian State University the spring issue and by June 1 for the fall issue. Anita N. Kitchens, Appalachian State University Manuscripts that have not been published before and are not under review elsewhere may be submitted Jill T. Richie, Appalachian State University at any time to the address below. Submit one elec- Pamela W. Schram, Appalachian State University tronic copy via e-mail attachment (preferred) or disk- Solomon Willis, Gaston Day School ette in Microsoft Word or rich text file format. To Women and Minorities Column allow for blind review, the author’s name and contact Sarah J. Greenwald, Appalachian State University information should appear only on a separate title page. Manuscripts should not exceed 10 pages dou- Problems To Ponder Editor ble-spaced with one-inch margins. Figures and other Gregory S. Rhoads, Appalachian State University pictures should be included in the document in line with the text (not as floating objects). Scannable pho- About the Cover tos are acceptable and should be large glossy prints The Centroid logo is based on the following theorem: mailed to the editor or minimum 300 dpi tiff files The limit of the sequence of midtriangles of a triangle emailed to the editor. Proof of the photographer’s is the centroid of the triangle. permission is required. For photos of students, parent or guardian permission is required. Manuscripts should follow APA style guidelines Copyright from the fifth edition of the Publication Manual of Educators are granted general permission to photo- the American Psychological Association (2001). Ref- copy material from The Centroid for noncommercial erences should be listed at the end of the article, and instructional and scholarly use. Contact the author(s) should also follow APA style, e.g., concerning other copying. Bruner, J. S. (1977). The process of education (2nd ed.). Cambridge, MA: Harvard University Press. Contact Information National Council of Teachers of Mathematics. Address all correspondence and submissions to (2000). Principles and standards for school The Centroid mathematics. Reston, VA: Author. c/o Dr. Holly Hirst, Editor North Carolina Department of Public Instruction. Department of Mathematical Sciences (1999). North Carolina standard course of 121 Bodenheimer Drive study: Mathematics, Grade 3 [On-line]. Avail- Appalachian State University able: http://www.ncpublicschools.org/ Boone NC 28608-2092 curriculum/mathematics/grade_3.html or send email to
OFFICIAL JOURNAL OF THE NORTH CAROLINA COUNCIL OF TEACHERS OF MATHEMATICS VOLUME 31 • NUMBER 2 • FALL 2005
Articles
7 Bite Sized Mathematics: The Case of the Radical Cos D. Fi 10 Knowing When to Which: Addressing the Calculator Quandary Carrie S. Cutler 13 Strategies to Support Mathematical Understanding of LEP Students David K. Pugalee, Ian Brailsford, and Theresa Perez 23 Problems to Ponder Gregory S. Rhodes 27 Butterfly Sticker & Homework Recording Sheet June L. Blackwell News & Information
2 From the Editor
3 Presidents’ Messages
9 NCCTM Materials Marketplace
12 Mini-Grants from NCCTM
17 2005 State Math Fair Winners
19 2005 State Math Contest Winners
21 2005 Logo Contest Winners
29 Rankin Award Nominations
30 Innovator Award Nominations
31 NCCTM Trust Fund Scholarships
From the Editor
The Transition to SCS 2003 Holly Hirst Appalachian State University Boone, North Carolina
Another summer is over, and we are starting which” seemed timely. Anyone who has a new school year. Here at the Centroid, looked at census data for North Carolina Brian and I are reading the submissions we will understand the rationale for the article received in the last three months in prepara- on teaching LEP students. Finally, we in- tion for the review process this fall. We are clude a handy worksheet designed to reduce pleased that you are sending us submissions. the time it takes teachers to pull grades to- We need more! Please consider submitting, gether at the end of a term. especially if you have a cool classroom ac- As always, we encourage you to con- tivity to share. We have also changed the re- sider assisting with The Centroid by: quirements for submission—one electronic • submitting a manuscript – general arti- copy via email or on a disk in the US mail is cles are welcome, as are the following sufficient. special categories of articles: A In her message, President Jeane Joiner Teacher’s Story, History Corner, Teach- describes fall as a time of renewal. This fall ing with Technology, It’s Elementary!, is also a time of implementation. The new Math in the Middle, and Algebra for NC math curriculum for grades 9 through 12 Everyone. will be “officially” taught this year, with • becoming a reviewer – please send e- both old and new curricula in effect through mail to me if you are interested in help- the transition documents for algebra 1, ge- ing in this way. ometry, and algebra 2. As an interested ob- server in discussions about these documents, Contact information. Feel free to contact I appreciate the work required of teachers in us at any time with submissions, news items, these courses; good luck! questions, or concerns. We hope that you will enjoy the articles Holly Hirst
NCTM in the News
One way to keep up with education news is to read the news clips mentioning NCTM and mathematics education. Many are posted on the NCTM website.
2 The Centroid
Presidents’ Messages
State President ers at every level, preK through higher edu- Jeane Joyner cation. The bottom line, I believe, is In conventional wisdom, spring is the season NCCTM will be only as effective as you of renewal, but this is not necessarily true make it. for teachers. Fall is a time of new and excit- If you are like me, attending an NCCTM ing beginnings. New groups of students fall conference is always a joy. I see old come to us to spend a year learning mathe- friends, learn different ideas and strategies, matics under our guidance. Whether a examine materials, meet new people, and of- teacher, administrator, or educational leader ten leave with something intriguing to think in some other capacity, fall brings us oppor- about. But quality conferences do not just tunities to begin anew with the challenges happen. They come together through hun- students always present. dreds of hours of work by numerous volun- NCCTM also faces challenges. We must teers. Conference and program committee decide which services our professional or- members donate their professional time to ganization, run by volunteers, should pro- put together a program and sort through de- vide to help members “recharge” and con- tails to make the conference run smoothly. tinue to be learners. How do we best help Presenters spend hours preparing talks. Tee- novice and veteran teachers prepare for shirts with new math logos result from ever-changing student populations? How do teachers and their students submitting logo we identify, recognize, and utilize outstand- ideas, volunteers sorting and judging, and ing teachers of mathematics? What strate- colleagues completing paper work, ordering gies do we employ to help our members put shirts, and manning a booth. For example, research into practice, make wise decisions NCCTM has a Marketplace with donated about curricular materials, and become less materials that are sold for miniscule amounts intimidated by the pressures of accountabil- to support new mathematics teachers. ity testing? In what ways can we, as an or- In her May 12 email message to Na- ganization, help to address the real problems tional Council of Teachers of Mathematics that face schools today–problems such as a members, Cathy Seeley, NCTM President, shortage of qualified mathematics teachers, wrote an increasingly diverse student population, It is oversimplified, unrealistic, and unfair to and accountability pressures that may hinder try and raise students’ achievement in rather than support quality teaching and mathematics simply by putting pressure on learning? How do we convince those who teachers to “try harder.” To assume that control budgets and disperse funds to sup- teachers aren’t already “trying hard enough” port teachers’ participation in professional is grossly inaccurate. Across the board, teachers want students to achieve at high development through NCCTM? levels, and they do whatever they can to If we are to respond to these questions in help them learn. positive and powerful ways, we need to work together to make wise decisions. Anyone who has ever examined the NCCTM is your professional organization hours of time donated by so many educators as teachers of mathematics. Because we are in working with the various levels of the dedicated to serving our members and, State Mathematics Contest or sat in the through them, students across North Caro- classrooms of those we honor as outstanding lina, we can be most effective when we teachers of mathematics or participate in the make specific efforts to involve many teach- programs funded through NCCTM’s mini-
Vol. 31, No. 2 • Fall 2005 3
grants knows that North Carolina teachers hope that you will help me in sustaining the are working very hard to help their students strength of our organization by continuing to find success in mathematics. Look at our be active participants and by providing me math fairs, this professional journal, the and the rest of the board with suggestions scholarships we award! In so many ways, for activities that will meet your professional mathematics educators in North Carolina needs. should be cited as models for others to copy. We have a wonderful group of officers We cannot, however, rest on our past throughout the state and we are very excited achievements. In order to continue as a vital about all that we hope to accomplish. organization all of us must bring fellow NCCTM board members for the Eastern Re- mathematics educators into the organization gion are: and take an even more active role in NCCTM’s endeavors. If each member in- Elementary Vice President: Carolann Wade vites and involves one more mathematics Middle Grades Vice President: Lucy Kay educator each year, think about what we Secondary Vice President: June Blackwell could do together! We have wonderful men- College Vice President: Gail Stafford Secretary: Elizabeth Murray tors in our midst. The expertise among re- tired and veteran NCCTM members is there waiting for you to tap into. The 2005 Spring Conference at Meredith Would you like to get involved right College in Raleigh was so great that we have away? Consider applying for a mini-grant, decided to do it again! Thank you to every- involve your class in the logo contest, have one who played a part in creating and con- your students participate in a mathematics ducting such a wonderful staff development contest, or create a project for the math fairs. and networking opportunity. The Central Ask your principal or supervisor for support and Eastern Regions are planning another and attend the NCCTM fall conference. joint Spring Conference to be held at Mere- There will be several different conference dith at the end of February; let’s hope that opportunities this year that you can read the weather cooperates. about in the preliminary program. Contact I hope that you have had a relaxing and me
4 The Centroid
grams (perhaps you would like to make a for details and updates on math related ac- presentation). tivities. Make it a great school year! I look forward to working with the new Central Region vice presidents, Sylvia Central Region President Davis, Karen Ellis, Elaine Tucker, Cos Fi, Emogene Kernodle and secretary, Angela Flowers. We want to Greetings from the Central Region. For serve you. Please let us hear from you to let most, this summer has been a little longer us know your needs and interests. for relaxation and reflection of the past school year(s). Western Region President We would like to thank our outgoing of- Carmen Wilson ficers, Vicki Moss, Tina McSwain, Patty Greetings! I hope that you enjoyed the Schram, Barbara McGill, Susan Friel, and summer months and that you are ready now Amy Travis for their great leadership for the to begin an exciting new school year. past two years. They have truly shown their Thanks to everyone who helped with the dedication and commitment to our organiza- Western Region Math Fair and Western Re- tion. We will continue to look to them for gion Spring Conference in 2005. The West- ern Region Math Fair was held on March their guidance and support. th The State Mathematics Conference will 19 at Appalachian State University. One be October 13-14 at the Koury Convention hundred seventy-nine students, representing Center in Greensboro. Central Region mem- 33 schools in our region, participated in this bers Dot Doyle and Maria Hernandez, pro- event. Over 350 in-service and pre-service gram chairs, have built a great program and teachers attended the Spring Math Confer- ence held at the Hickory Metropolitan Barbara McGill, conference chair, has th worked hard to make the conference a Higher Education Center on February 26 . meaningful educational event. We appreci- The event was really two conferences at one ate the work of other Central Region mem- site. Pre-service teachers attended sessions bers who give of their time and effort to that were themed “Getting Started in the make the annual conference a success. Teaching Profession.” They took away some The joint regional conference in the great ideas as well as some handy door spring of 2004 with the Eastern Region was prizes. Sessions for experienced teachers deemed a success. We have planned to join were about “Assessment and the Newly Re- them again on February 25, 2006 at Mere- vised Standard Course of Study in Mathe- dith College. Mark your calendar now! matics.” It was great to see so many teachers Not too long ago someone asked me the participating in this conference. benefits of joining NCCTM. Two benefits We are planning the 2006 Western Re- gion Spring Conference already. The date for NCCTM members are the scholarship th program for those who are pursuing gradu- will be Saturday, February 18 and the loca- ate studies in mathematics education and the tion will be the campus of Appalachian State Mini-Grant awards for excellence in University. Mark your calendars now and mathematics education. For students, watch for more information to be coming NCCTM sponsors the Math Logo Contest out soon. If you would like to assist in the and Mathematics Fairs. You will find infor- conference planning or at the site, please mation about these and other activities here contact me
Vol. 31, No. 2 • Fall 2005 5
plications are due by September 15th. The Masters program, pursue National Board state conference is in Greensboro on Octo- Certification, participate in Teacher Action ber 13th and 14th. Plan now to attend, and get Research, or incorporate some new technol- ready to be energized by the exciting new ogy into your classroom. Whatever you de- ideas your colleagues will be sharing in the cide to undertake this school year, best many sessions and workshops. wishes to you and your students! Finally, as you nurture your students and watch them grow, why not grow along with them? Perhaps this is the year to begin a
NCCTM Fall Conference
35th Annual Conference of the North Carolina Teachers of Mathematics
Building Student Success Through Rich Mathematics
October 13 and 14, 2005
Joseph S. Koury Convention Center Sheraton Greensboro Hotel at Four Seasons Greensboro, North Carolina
Registration Fees: $55 ($45 for members, $5 for students)
Registration: Forms will be available at www.ncctm.org or at the conference Wednesday evening, or Thursday and Friday all day.
The conference program will be posted on the NCCTM website a month prior to the conference. Contact Cyndy Davis
Hope to see you there!
6 The Centroid
Bite Sized Mathematics: The Case of the Radical
Cos D. Fi1 University of North Carolina at Greensboro Greensboro, NC
Studying subject matter in relation to subject Questions α and β matter pedagogy helps teachers be more ef- fective. Since by definition, the radical ( ) repre- sents the principal square root of the argu- In this article, we will explore the meaning ment, and the principal root is always posi- of the radical ( ) and explore its use in re- tive, the following holds for question α: solving simple quadratic equations. In ac- x = 4 if and only if x = 4 if and only if cordance with the Ball, Bass, and Hill x = 2. This shows that (2004) quote above and with the recommen- dations from the MET report (CBMS, 2001), x = 4 if and only if x = 2. this article represents an attempt to enga ge Hence the solution set to question α is just middle and secondary school mathemati cs teachers in the exploration of the nuanced {2}, which also answers question β. That is, complexities of important mathematics con- 4 = 2, and nothing more. cepts, namely solving simple quadratics us- ing radicals. Question χ
Quadratics are important in many con- This question is different from the two pre- texts such as area, trajectory (path) of an o b- vious questions in fundamental ways. In par- ject thrown at an angle between 0 and 90º, ticular, we will show that x = 2 is not and the height of an object dropped from 2 equivalent to x = 4 at the conclusion of our above ground, to name a few applications. Radicals appear in the solution of many exploration of question χ. In many mathematics classrooms, teach- quadratics, and there is sometimes confusion ers and students engage in the following as to what the radical symbol ( ) actually represents. Textbooks and teachers are mathematical wizardry: known to explicate that the radical repre- If x 2 = 4, then x = ±2. sents the principal square root. What does that mean? Why do radicals sometimes seem First of all, the conclusion in this state- to give one answer and other times two? Let ment actually represents the two equations us unpack the meaning of the radical ( ) by x = 2 or x = !2. If the original question had examining the following questions: asked for the solution to the equation x 2 = 4, then providing another equation (α) Suppose x = 4, is the square root of x does not fully resolve the original equation. just 2 or both 2 and –2? In fact, the correct solution ought to be the (β) What is the value of 4 ? Is it just 2 or set {!2,2}. Thinking about solutions as sets
both 2 and –2. demonstrates that the equation x = 2, which (χ) What is the solution of x 2 = 4? Is it just has the solution set {2}, is not equivalent to 2 or both 2 and –2.
1 Dr. Fi is an assistant professor of mathematics education in the Department of Curriculum and Instruction at
UNC-G and serves as the College Vice President for the Central Region of NCCTM.
Vol. 31, No. 2 • Fall 2005 7
the equation x 2 = 4, which has the solution The previous discussion shows that we set {!2,2}. can help students understand the mathemati- The major problem with the solution cal “short hand” method that yields x = ±2 through wizardry 2 if x = 4, then x = ±2, is that the underlying mathematical proc- esses and concepts remain a mystery. by unpacking the meaning and structure of Through this wizardry, students may be led the radical using equivalent equations, the to the beli ef that the radical ( ) can actually def inition of the absolute value function, and produce both positive and negative numbers. properties of equalities. By focusing on We ought to challenge this belief with sense making and learning with understand- mathematical reason and help students un- ing, teachers can help students appreciate derstand that mathematics makes sense and the logical integrity of mathematics. More that mathematics coheres beautifully. importantly, students might begin to appre- 2 ciate the meaningfulness of mathematics and Reconsider the equation x = 4 . Let us retain the definition that the radical is the the nature of mathematical truths as based principal root and can only produce positive on reason and not just because the teacher numbers. The following argument unpacks said so. Such pedagogy promises to promote the mystery and substantiates the wizardry students’ agency: Students will begin to on sound mathematical arguments and con- view themselves as doers of mathematics structs. and not just consumers of mathematics or The properties of equality and the defini- outsiders to mathematics. tion of the radical ( ) as the principal I would be remiss if I did not highlight a possible misinterpretation of the previous square root support the following argument: mathematical argument that can result from x 2 = 4 if and only if x 2 = 4 . an uninvolved look at the applied processes. It seems as if I have produced a negative Moreover, since the radical produces number (-2) from an argument that claimed 2 only positive values, x must be a positive that x 2 is a positive quantity. It is true that quantity and so the following holds: x is a positive value and x 2 = x , since x 2 = 4 if and only if x = 2. the radical ( ) produces only positive val- ues. However, x itself can be either positive We need to apply the definition of the or negative. A similar argument is equally absolute value function to complete t he important in generating the inverse for problem. If x is negative, then x = !x , since 2 f (x) = x . Note also that ! produces a –x is positive when x is negative. If x is posi- negative value and should not be confused tive then x = x . Hence, we get the results as with or (negative value) . shown in Table 1 from the equation x = 2 .
Other questions for curriculum depth x = 2 and x is negative x = 2 and x is positive • In what other ways can you convince ! -x = 2 with –x >0 ! x = 2 ! x = -2 your students that x yields only posi- The solution set is {!2} The solution set is {2} tive values? How will you distinguish from ? Hence the sol ution set to the equation x = 2 • ! • How will you help students understand is {!2,2}. Thus the solution set to the equation that (negative value) is not a negative num- x 2 = 4 is also !2,2 . { } ber? 2 Table 1. x = 2 and its equ ivalence to x = 4 • How do these arguments affect the proc- ess of finding the inverse of f (x) = x 2?
8 The Centroid
References Mathematics, Science, and Technology Educa- tion, Cape Town, South Africa. Ball, D. L., Bass, H., & Hill, H. (2004, January). Conference Board of the Mathematical Sciences Knowing and using mathematical knowledge in (CBMS). (2001). The mathematical education of teaching: Learning what matters. Paper pre- teachers. Providence, RI: American Mathemati- sented at Southern African Association of cal Society.
The Materials Marketplace at the NCCTM Conference
As those of you who attended the Conference in Greensboro last year know, NCCTM started a new program: The Materials Marketplace. Preservice and new inservice teachers were invited to come and purchase at rock bottom prices all sorts of materials—textbooks, technology, sup- plies, etc.—to start building their resource base. It was wildly successful! There seemed to be a lot of stuff to sell, until the marketplace opened. The materials were gone in record time, and all involved raved about the success of this venture. The Marketplace will be back this fall, and the organizers need your help! Please consider donating materials to the marketplace. We are looking for new or gently used supplies such as manipulatives, posters, books, professional development materials—anything that would be use- ful to new teachers. Please contact coordinators Kim Aiello and Shana Runge
NCTM Issues New Position Statement Computation, Calculators, and Common Sense
Question: Is there a place for both computation and calculators in the math classroom?
NCTM Position: School mathematics programs should provide students with a range of knowl- edge, skills, and tools. Students need an understanding of number and operations, including the use of computational procedures, estimation, mental mathematics, and the appropriate use of the calculator. A balanced mathematics program develops students’ confidence and understanding of when and how to use these skills and tools. Students need to develop their basic mathematical understandings to solve problems in and out of school. For the full Position Statement go to
NCTM Positions and Position Statements define a particular problem, issue, or need and de- scribe its relevance to mathematics education. Each statement defines the Council's position or answers a question central to the issue. The NCTM Board of Directors approves all positions and position statements. The newly released Position on calculators is in addition to the two pre- vious Position Statements on the subject.
Vol. 31, No. 2 • Fall 2005 9
Knowing When to Which: Addressing the Calculator Quandary
Carrie S. Cutler1 University of Houston Houston, Texas
Should an elementary school teacher always When to Which make calculators available to students? How To engage teachers, parents, and students in does a teacher know when the use of a cal- a discussion about the judicious use of cal- culator is appropriate? More importantly, culators, I begin by asking about their pre- how can we strengthen our students’ dis- ferred computation strategies—those ap- cernment about when to use a calculator, proaches they find most efficient and effec- mental math, paper and pencil, or physical tive. We list the strategies on the board and manipulatives? group them into four general categories: The calculator serves as a valuable tool manipulative (concrete), paper and pencil, to enhance exploration, problem solving, mental math, and calculator (technological). and real-world applications. Nonetheless, as I then divide the participants into groups a former classroom teacher and current in- of five and assign one person to use the cal- structor of preservice elementary mathemat- culator, another to use mental math, another ics teachers, I witness firsthand the confu- to use paper and pencil, and another to use sion surrounding the prudent use of calcula- base ten blocks to represent and solve com- tors in the elementary classroom. Often, I putation items. The fifth member of the hear comments such as: group serves as the judge and recorder, re- I don’t think there is any room for calcula- sponsible for certifying who solved the tors in the elementary classroom. —Parent problem most quickly and keeping a tally of I don’t know when to let my students use the results for each problem. calculators, so I usually just keep them in Next, I display one arithmetic problem at the cupboard. —Classroom teacher a time on the overhead (Figure 1) and re- You ask any sixth grader on the street, and mind the participants to solve the problem they can’t tell you what two plus two is. with their assigned tools. They won’t learn the basics if they have a Following the exercise (and a flood of calculator. That’s all there is to it.— giggles and groans from the groups), we dis- Preservice Teacher cuss our results. Parents, teachers, and stu- The National Council of Teachers of dents often comment on their surprise in Mathematics’ Principles and Standards for seeing that mental math is not the quickest School Mathematics (2000) espouses com- approach for every arithmetic problem. Of- putational fluency, but encourages balance ten, those who are compelled to punch a between that fluency and conceptual under- memorized math fact into the calculator or standing. The document equally advocates lay out an absurd number of manipulatives both efficient computation and the ability to express frustration with the unwieldiness of wisely select strategies and tools. their assigned method. Participants note that selecting a tool or strategy requires a great
1 Ms. Cutler has taught both first and fourth grades in Utah and Idaho and is completing her doctorate this year in Curriculum and Instruction – Mathematics Education at Houston, where she teaches mathematics methods courses.
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deal of judgment—of knowing when to Another student disagreed, “No. First which. We generally conclude the activity we need to know how much cheese goes on the pizzas.” with a consensus that students need a quiver full of strategies from which to select wisely The students pursued multiple avenues as well as plenty of opportunities to hone in solving the problem. Calculators enabled their decision-making skills. the students to use the actual numbers in- volved—rather than less realistic, but more 55 ! 2 = 19 ! 0 = computationally feasible, lower digits. The students found personally satisfying results, especially when they involved the cafeteria 100 323 staff and the principal in their wrap-up pizza !63 !217 party. Whether the activity focuses on process or product influences a personal rationale 624.27 16 ! 5 = for calculator (Thompson & Sproule, 2000). +97.14 A process-oriented activity involves students in problem solving or mathematical explora- tion. Here the calculator can help students 1,295 437 "189 = focus on what calculations to perform, not "375 just how to perform those calculations, thus enhancing exploration, problem solving, and Figure 1. Arithmetic problems for overhead. real-world applications (Reys & Arbaugh,
2001). A product-oriented activity, on the Personal Rationales for Calculators other hand, most highly values the determi- Teachers must develop personal rationales nation of a computational result. In this case for their decisions regarding the use of cal- the calculator serves as a mere computa- culators in their classrooms. They should tional tool, limited to figuring the correct begin by asking: Is the calculator essential numerical solution. In the pizza problem, for maintaining the quality of the task? valuable mathematical processes were the Would the calculator allow for the use of focus rather than the accurate computation real-world data or connections that would of the amount of cheese—the product. otherwise be prohibitive? (Thompson & Many teachers and students believe that Sproule, 2000). using a calculator guarantees speed and ac- Recently, I watched a primary class ex- curacy of computation, but little else. Pro- tend a picture book about pizza into a rich ponents of calculator use, however, note its mathematical task facilitated by the use of unique potential for developing number calculators. The students expressed great cu- sense and positive dispositions toward riosity about a pizza’s ingredients. They mathematics (Charles, 1999). Indeed, a my- wanted to know how much cheese the cafe- opic view of a calculator as a quick-fix for teria staff would need to make enough piz- arithmetic ignores its value for allowing stu- zas for everyone in the school to have a dents to easily explore mathematical con- slice. Students energetically engaged in dis- cepts and efficiently solve problems using a cussion about the problem, offering their variety of creative approaches. Students may suggestions and conjectures, and eventually investigate place value, discover new phases reasoning through the mathematics involved of skip counting, search for number patterns, in the investigation. and examine number magnitude while using “First we need to know how many people the calculator to visualize very large and are in the school. I think we should add up very small numbers. the first graders, then the second graders, and start there,” proposed one student.
Vol. 31, No. 2 • Fall 2005 11
Just as teachers refine their individual ra- ing, students in turn develop judgment about tionales for calculator use, students should, when to which. A teacher noted: “I can see through experience with process-oriented that, when I make calculators available and tasks, begin to develop their own sense of students choose to use them, the students are when to which, thus observing the calcula- more autonomous and motivated. Liberated tor’s utility in solving real-world problems from time-consuming computation, they can that require rigorous reasoning and problem focus on exploring the rich mathematics in- solving. volved in the task.”
Conclusion References Should an elementary school teacher always Charles, R. I. (1999, May/June). Calculators at the make calculators available? How does a elementary school level? Yes it just makes sense! teacher know when to which? The calculator Mathematics Education Dialogues, 2(3), 11. is essential when the focus of instruction is a National Council of Teachers of Mathematics. process-oriented task involving real-world (2000). Principles and standards for school mathematics. Reston, VA: Author. numbers that may be beyond the scope of Reys, B. J., & Arbaugh, F. (2001). Clearing up the students’ computational ability. The use of confusion over calculator use in grades k-5. the calculator expedites the problem-solving Teaching Children Mathematics, 8(2), 90-94. process, allowing students to move beyond Thompson, A. D., & Sproule, S. L. (2000). Deciding arithmetic to the “meat” of mathematics— when to use calculators. Mathematics Teaching reasoning, conjecture, and so forth. Through in the Middle School, 6(2), 126-129. continual experience with rich, process- oriented tasks that foster wise decision mak-
Mini-Grants from NCCTM
NCCTM is pleased to be able to offer Mini-Grants to teachers who need some support to imple- ment an innovative idea at their schools. There are no preconceived criteria for projects except that students should benefit from the grant. Possible projects for consideration include math clubs, field days, contests, workshops for parents, math activities, math laboratories, and research topics. Each of the three NCCTM state regions has $5000 to award to teachers in their area. The av- erage mini-grant is about $600 but some have been awarded for as little as $100 or as much as $2000. Applications will be accepted only from persons who are NCCTM members as of 1 Sep- tember 2005. Don’t let the application process intimidate you! See the sample application on the website and use it as a guide! Completed applications must be received by 15 September 2005 to be considered. For more information and submission guidelines, see the website
12 The Centroid
Strategies to Support Mathematical Understanding of LEP Students
David K. Pugalee1 University of North Carolina at Charlotte Charlotte, NC
Ian Brailsford2 University of North Carolina at Charlotte Charlotte, NC
Theresa Perez3 University of North Carolina at Charlotte Charlotte, NC
Language has traditionally taken the back- proximately 494 percent over the previous seat in mathematics classrooms. This is es- decade. The educational attainment of our pecially problematic for students with Lim- nation’s Hispanic population is especially ited English Proficiency (LEP), and inhibits troubling. According to a recent study (Se- the type of learning environment indicative cada et. al, 1998): of developing mathematical understanding. • Nearly one in five of our nation’s His- According to the 2000 United States census, panics between the ages of 16 and 24, a language other than English is spoken in who have been enrolled in a United approximately 18% of households, with States school at one time, left school Spanish being the spoken language in more without a high school diploma or equiva- than 10% of these homes. As of 2002-2003 lent; slightly more than 10% of all U.S. students • The Hispanic dropout rate is estimated at are limited English proficient (Callahan, a staggering 30%; 2005). In North Carolina, a language other • Hispanics account for 56% of all U.S. than English is spoken in more than 8% of immigrants but account for nearly 90% the homes, with Spanish as the spoken lan- of all immigrant dropouts. guage in approximately 5%. According to the NC Department of Public Instruction, In order to address the needs of ESL stu- there were more than 78,000 LEP students dents, mathematics must be a targeted con- enrolled in public schools in the 2003-2004 tent area. Analysis of the 1996 National As- academic year, about 5.9% of the total en- sessment of Educational Performance found rollment thus representing an increase of ap-
1 An associate professor specializing in mathematics education in the department of middle, secondary, and K-12 education at UNC-C, Dr. Pugalee coordinates the PhD program in curriculum and instruction. 2 Mr. Brailsford just completed requirements for the M.Ed. with an emphasis in Teaching English as a Second Lan- guage from UNC Charlotte. He currently works with Central Piedmont Community College. 3 Dr. Perez is a professor in the department of middle, secondary, and K-12 education and specializes in bilingual and multicultural education.
Vol. 31, No. 2 • Fall 2005 13
that non-LEP students performed better on Instruction involving the language of the mathematics test than LEP students, and mathematics should not be separate from these differences held even after controlling math (MacGregor & Moore, 1995). Students for students’ reading levels (Abedi, Lord, & need to be proficient in all four areas of Hofstetter, 2001). LEP students can benefit English, i.e. reading, writing, listening, and from teachers’ explicit actions addressing speaking. Whereas most math classrooms their limited language skills. This paper will utilize reading and listening skills, little time provide some general guidelines for address- is devoted to speaking and writing. The de- ing the needs of limited English proficient velopment of a knowledge base must in- students in studying mathematics and some clude emphasis on mathematics content as specific strategies that can improve all stu- well as all four of these areas of language. dents’ mathematical understanding (Mac- The use of small groups provides an effec- Gregor & Moore, 1995; Lanou, 2000). tive, efficient, and engaging arrangement for The role of language needs to be better students to develop facility with language integrated into the math classroom in order while developing important mathematical to promote a better understanding of the understanding. concepts that are presented. There are three Building on prior knowledge is an essen- main areas to address in approaching lin- tial strategy for teaching any subject. This guistically diverse learners. The knowledge can be done through the use of pictures to area consists of choosing relevant contexts elicit meanings (Lanou, 2000). Drawing ex- and building on prior knowledge. The vo- ercises that emphasize students’ facility with cabulary area consists of identifying key mathematical ideas and expressions are words and having students express concepts powerful experiences that assist students in in their own words. The concept area con- making important connections. For example, sists of careful planning for presenting ab- students are asked to describe a graph with- stract concepts and the use of manipulatives. out showing the picture to the other mem- bers of their group. The other students then The Knowledge Area sketch the graph and write a representative It is important to choose relevant contexts function from the given verbal description. when presenting mathematical problems. Students then rotate, describing and drawing Ordinary words may have a specialized their respective graphs. Finally, students meaning that we, as Americans, take for compare their graphs and discuss any dis- granted as general knowledge. Lily Wong crepancies. This allows students to reinforce Fillmore (2002) gives an excellent example the relationship between the graphical, ver- in presenting the idea of pizza size in inches. bal, and symbolic representations. To some it is obvious that a size of 12′′ in a y word problem refers to the diameter. For 10.00 those not familiar with American culture, this measurement could very well refer to the radius or circumference of the pizza! Additionally, although the use of standard –20.00 20.00 x English units for measurement is useful, an emphasis on metric units will provide a fa- y = (1/5)x + 3 miliar context for many students from coun- tries where the metric system is common- –10.00 !x = 5.00 place. The metric system of measurement !y = 2.00 will also be important for future learning Figure 1. Example of a graph for student experiences, such as in chemistry or physics descriptions. classes.
14 The Centroid
The Vocabulary Area strategies. Data suggests that if teachers The teaching of key word vocabulary is learn some math-related vocabulary in Span- helpful not only for LEP students. Maintain- ish, or reinforce math vocabulary with Span- ing vocabulary lists helps all students to or- ish equivalents, they might be able to assist ganize data in a logical fashion (Lanou, a majority of their LEP students in learning 2000; MacGregor & Moore, 1995). Cloze concepts. exercises are a good way of reinforcing lan- Having students express concepts in guage in the math classroom. In cloze exer- their own words is another strategy that can cises students select from a list of words, be used. This may take place through infor- frequently words that have conceptual den- mation sharing or cooperative problem solv- sity in mathematics, to complete statements, ing (MacGregor & Moore, 1995). It is also as in Figure 2. important that students have the opportunity to develop concepts in not only English, but also their native language. Conceptual un- derstanding is easier if it is first learned in students’ native language and followed by its application in the English-speaking class- room (Wong Fillmore, 2002). For class- rooms where bilingual resources are avail- able, students benefit if they can access in- formation in their native language as a pre- cursor to developing those concepts in Eng- Rectangle Triangle Pentagon Three Four Five Six Ten Twelve Hypotenuse Area lish. This includes providing students oppor- Perimeter Circumference Length tunities to discuss concepts in their native language with others. This shape is a ______. It has a base of ______centimeters. The Concept Area ½ of the base times the height gives an area of 2 Presenting abstract concepts has always ______cm . been a difficulty for math instructors. A The ______is 12 centimeters. good approach to overcome this challenge is Figure 2. Cloze exercise for reinforcing vocabulary to first develop concrete examples using and concepts manipulatives. This will make abstractions easier to understand. The notion of abstract However, it is important to remember concepts should be presented to children that the teaching of academic language, i.e. while they are still young. At around age 7, key words, in itself is not enough. Students most children are both physically and men- need to think about the ways in which the tally able to start comprehending logical rea- language is used. soning. Therefore it is important for teachers Going over word associations, such as to start modeling early. synonyms, is an effective technique for de- Manipulatives are important tools from a fining new vocabulary and also increases the developmental perspective as students move students’ speaking and writing skills (Mac- from concrete to abstract ideas. Cummins Gregor & Moore, 1995). Labeling and card (1981) argues that visual supports make lan- matching exercises are also useful tech- guage more comprehensible. Students who niques for connecting words with numbers “see” concepts through physical manipula- and equations. Students benefit when teach- tives can understand those concepts more ers of mathematics consult with the rest of readily than if the ideas are only explained the faculty in order to coordinate learning abstractly using vocabulary that is not en- activities and plan/develop instructional tirely familiar to the students.
Vol. 31, No. 2 • Fall 2005 15
Examples of manipulatives include ronment for all students, not just those with Cuisenaire rods and problem reconstruction limited English proficiency. Thus success in using cards. (Lanou, 2000; MacGregor & communication has gone from a seemingly Moore, 1995). The use of rulers, protractors, impossible to an obtainable goal and the re- calculators, and computers are also effective alization of mathematical power for all stu- manipulative tools. Graduated cylinders may dents is closer to attainment. be used to develop concepts related to vol- ume. Graduated cylinders, flasks, and beak- References ers available in many science classrooms Abedi, J., Lord, C., & Hofstetter, C. (September, can be used to illustrate volume concepts al- 2001). Impact of selected background variables lowing students to develop proficiency in on students’ NAEP math performance. Washing- associating volume with a specific numeri- ton, DC: National Center for Education Statis- tics, Office of Educational Research and Devel- cal quantifier. Such visual, hands-on experi- opment, Department of Education. [Publication ences are more effective than the typical il- NCES 201111] lustrations of volume provided in many Callahan, R. M. (2005). Tracking and high school textbooks. Fraction bars can also be an ef- English learners: Limiting opportunity to learn. fective means of relating visual models to American Educational Research Journal, 42(5), part/whole relationships. Students can be 305-328. asked to do basic operations using the strips Cummins, J. (1981). The role of primary language development in promoting educational success or make comparisons such as “How many for language minority students. In Schooling and fourths are in one half?” “Which is larger ¾ language minority students: A theoretical or 2/3?” framework (pp. 3-49). Office of Bilingual Bicul- tural Education, California State Department of 1/2 1/2 Education, Sacramento. Los Angeles: Evalua- 1/3 1/3 1/3 tion, Dissemination and Assessment Center, 1/4 1/4 1/4 1/4 California State University. Figure 3. Sample Fraction Bars Lanou, E. (2000). Instructional strategies: Ridding math word problems of language barriers. The Link, 19(2). [Online at http://www.ael.org/link/, Conclusion accessed October, 2001] As the reader may have already surmised, McGregor, M. & Moore, R. (1995). Teaching the most effective strategy for approaching mathematics in the multicultural classroom: A the LEP population lies in an integration of resource for teachers and teacher educators. various strategies. Language needs to be bet- Adelaide, Australia: Australian Association of ter integrated into the math classroom in or- Mathematics Teachers. North Carolina Department of Public Instruction. der to promote conceptual understanding. (2004). 2003-2004 Facts and Figures. Strategies for approaching such instruction [http://www.ncpublicshools.org/fbs/]. include choosing relevant contexts that build Secada, W. G. et. al. (February, 1998). No more ex- on prior knowledge; identifying key words cuses: The final report of the Hispanic dropout and having students express concepts in project. Washington, DC: U.S. Department of their own words, and careful planning for Education. Wong Fillmore, L & National Association for Bilin- presenting abstract concepts using manipula- gual Education. (2002). Teachers teaching or tives. These strategies may be used in isola- teachers testing? A question of priorities [video- tion, but the combination of these various tape]. NABE Star Series Volume 1. Washington, strategies creates an effective learning envi- DC: NABE.
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2005 State Math Fair Winners
Primary Division, Grades K-2
1st Place: Nicholas Kowalski 2nd Place: Mrs. Barb Frea’s Class “The Probability of Being Like Me” (reps: Julie Fulk & Andrew Holmes) South Greenville Elementary School “What Kind of Truck Would You Like to Ride In?” Greenville, NC Pinnacle Elementary School Pinnacle, NC 3rd Place: Ethan Walters “Measuring It Up” Codington Elementary School Wilmington, NC Honorable Mentions Alec Laning Livi Guglielmi “Pulleys” “My Little Pony Parade” Lake Norman Elementary School Lake Norman Elementary School Mooresville, NC Mooresville, NC
Alex Suggs & Dane Hedgepath Samantha Freeman “Put Your Right Foot In, Do the Measuring, “Lake Norman Elementary School That’s What It’s All About” Supply Fundraiser” Snow Hill Primary School Lake Norman Elementary School Snow Hill, NC Mooresville, NC
Elementary Division, Grades 3-5
1st Place: Steven Holloway 2nd Place: Eli Stutzman & Daniel Monzel “Mathematics: The Cornerstone of Architecture” “Traveling Salesman Problem” Little River Elementary School Davidson Elementary School Durham, NC Davidson, NC
3rd Place: Max Copeland & Alec Castillo “How Many Trees Are In the Forest?” Lake Norman Elementary School Mooresville, NC Honorable Mentions Brittany Smith & Taylor Parks Keyshon Coston & Emmy Mathias “Pig Versus Cow” “Dessert, Anyone?” D.F. Walker Elementary School D.F. Walker Elementary School Edenton, NC Edenton, NC
Vol. 31, No. 2 • Fall 2005 17
Honorable Mentions (continued) San Coston Katie Moses “On the Road Again” “The Pony That Eats Like a Horse” D.F. Walker Elementary School Lakeshore Elementary School Edenton, NC Mooresville, NC
Mrs. Debbie Sheron’s Class (reps: Lia Johnson & Kaitlyn Mason) “How Is the Pisgah Covered Bridge Mathematical?” Franklinville Elementary School Franklinville, NC
Middle School Division, Grades 6-8
1st Place: Rebecca Lynn Gregory 2nd Place: Abdul-Rahman Eljabaly “The Thrill of the Ride” “How Much Water is in the Pond?” Camden Middle School Al-Iman School Camden, NC Raleigh, NC
3rd Place: Katie Shuford & Kristen Leger Honorable Mention: Melissa Sawyer & “Sounds of Pi” Jeanine Koontz H.M. Arndt Middle School “Which Fabric Dries the Fastest?” Hickory, NC Camden Middle School Camden, NC
High School Division, Grades 9-12
1st Place: Maressa McCall 2nd Place: Erica Wallace & David Johnson “The Power of One: An Exploration “How Will You be Remembered” of Weighted Voting Systems” Victory Christian Center School A.C. Reynolds High School Charlotte, NC Ashville, NC
3rd Place: Bobby Schultz Honorable Mention: Rebecca Barket & Hannah Cherry “Anim-Matrices” “Vedic Mathematics” Raleigh Charter High School Hertford County High School Raleigh, NC Ahoskie, NC
NCTM 2006 National Meeting
The next national meeting will be held in St. Louis, Missouri on April 26 through 29, 2006. Pre- registration will open in early November. See the NCTM website
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State Math Contest Winners
John Goebel Durhan Academy Durham, North Carolina
The finals for the State Mathematics Con- To qualify for one of the other divisions, a test in the Comprehensive Division were student must place in the top ten percent. held at the North Carolina School of Sci- This year 123 students qualified for the ence and Mathematics in Durham on April Comprehensive finals, while 86, 87, and 28th, 2005. A week later, the State Finals 82 students participated respectively in in Algebra Two, Geometry, and Algebra Algebra I, Geometry, and Algebra II. In One were held at three regional sites: the comprehensive finals, the top 40 stu- UNC-Asheville, UNC-Greensboro, and dents receive awards. In addition, many NC Wesleyan College in Rocky Mount. colleges and universities in North Carolina Before students can compete in these offer scholarships to the top students. state finals, they compete at a regional The top 10 students in the Comprehen- qualifying site. To qualify for the Com- sive Division and the top five students in prehensive Finals, a student must place in each of the other divisions are below. the top seven percent at one of these sites.
COMPREHENSIVE: Place Student School Grade Teacher 1 Jeremy Diepenbrock Enloe High 12 John Noland 2 Arnav Tripathy East Chapel Hill 10 Jay Wilson 3 Derrick Sund Home School 11 Mary Sund 4 Yakov Berchenco-Kogan Broughton High 10 Michael Clinkscales 5 John Pardon Durham Academy 10 Verle Regnerus 6 Jeremy Hahn East Chapel Hill 9 Virginia Johnson 7 Noah Blach East Chapel Hill 10 Jay Wilson 8 Gregory Filpus East Chapel Hill 12 Pete Shaginau 9 Amy Wen NCSSM/Early College 11 Maria Hernandez at Guilford 9 Jim Ouyang East Chapel Hill 12 Jay Wilson 10 Paul Huang NCSSM /Durham 12 Dot Doyle School of the Arts ALGEBRA I: Place Student School Teacher 1 Jenny Chen E.B. Aycock Middle Ruth Hicks 2 Cara Lewis A.C. Reynolds Middle William Punshon 3 Michelle E. Phillips North Davie Middle School Steve Rareshide 4 Chung-hyun Ma Martin Middle School Mrs. Slate 5 Michael Chang Cape Fear Center for Inquiry Elizabeth Woodside
GEOMETRY: Place Student School Teacher 1 HoSeok Lee Guy B Phillips Middle Angela Short 2 Kevin Li Ligon GT Magnet Middle Pat Heald 3 John Halvorsen T.C. Roberson High School Ms. Wood 4 Kevin Ji Durham Academy Jim McGivney 4 Jorge Ballesteros Aycock Middle Carolyn Warren
Vol. 31, No. 2 • Fall 2005 19
ALGEBRA II: Place Student School Teacher 1 William Schlieper III Green Hope High School Kathy Snapp 2 Andrew Tibbits David Butler High Mrs. Humphrey 3 Megan Heysham T.C. Roberson High School Ms. Hallman-Morris 4 James Finley T.C. Roberson High School Ms. Hallman-Morris 5 Brandon Reid Western Guilford High Catherine Miller
Out of these talented students, the Ken Twing, (Freedom High), Kathy Hill teams that represented North Carolina at (retired: Athens Drive), and David Mer- the American Regions Mathematics min (Duke Student and former ARML League (ARML) Meet at Penn State Uni- Team Member). versity on June 4th were selected. The top This year the A team placed seventh twenty students from the Comprehensive out of 29 teams in their division while the competition are automatically invited for B team placed 24th out of 65 teams. Arnav the A team, as are other top-performing Tripaty placed third in the Individual younger students for the B Team. The Round out of over 1400 students. The ARML Meet is the only national on-site ARML Competition consists of four parts. contest with teams from the US, Canada, Three of the events are team events: the and occasionally teams from countries like Team Round, Power Round, and Relay. Canada, China, Russia, and Bulgaria in at- The final event consists of eight individual tendance. The Duke Energy Foundation questions. The A team had a perfect score has given us a generous contribution for on the Team Round, tied for first on the the past two years to send our teams to Power Round, and was in second place go- Pennsylvania at no cost to the students. ing into the final Relay Round. On June 3rd, 34 students and five Students who would like to compete coaches traveled to Penn State University on the NC ARML teams should take the to compete in the ARML meet. Head AMC and AIME test, compete in the State Coach Archie Benton from North Bun- Mathematics Contests, and attend the combe did an outstanding job selecting the ARML practice sessions that will be teams, organizing practice sessions and posted on the State Math Contest Website problems, and recruiting the other coaches,
Check it out! The National Science Digital Library
The National Science Foundation created the NSDL to support innovations in teaching and learning in science, technology, engineering, and mathematics. The project has cataloged many excellent resources, and offers several search techniques. In addition to being a good place to find things, educators may sign up for the free newsletter and suggest resources for inclusion into the Library.
< http://nsdl.org/>
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NCCTM Math Logo Contest Winners
From a field of approximately 3,600 entries, twelve logos submitted by the following stu- dents were judged to be the winners in the 2005 Logo Contest.
State Winner Chris Gyori, Grade 11 Sanderson High School Raleigh, NC 27609 Teacher: June Blackwell
Eastern Regional Finalists Mary Lindsay West, Grade 2 Lacy McClure, Grade 8 Snow Hill Primary Daniels Middle School Snow Hill, NC 28580 Raleigh, NC 27608 Teacher: Carol Taylor Teacher: Gena Cherry
Merry Nguyen, Grade 5 Chris Gyori, Grade 11 Forestville Road Elementary Sanderson High School Knightdale, NC 27545 Raleigh, NC 27609 Teacher: Meredith Weipert Teacher: June Blackwell
Central Regional Finalists Coral Young, Grade K Callie Ricketts, Grade 8 Glendale Acres Elementary Anson Middle School Fayetteville, NC 28304 Wadesboro, NC 28170 Teacher: Kathy Denman Teacher: Patricia McQueen
Merlee Purkerson, Grade 5 Jamie Melvin, Grade 12 Lindley Park School South View High School Asheboro, NC 27203 Hope Mills, NC 28348 Teacher: Linda Isbell Teacher: Candy Phipps
Western Regional Finalists Caeley Riordan, Grade 2 Samantha Moody, Grade 8 Mount Pleasant Elementary Cane Creek Middle School Mount Pleasant, NC 28124 Fletcher, NC 28732 Teacher: Ms. Pat Freeman Teacher: Mary Gasaway
Justin L. Merrill, Grade 3 Leanna Watts, Grade 10 Brevard Elementary School North Lincoln High School Brevard, NC 28712 Lincolnton, NC 28092 Teacher: Ms. Yaxley Teacher: Libby Fletcher
Vol. 31, No. 2 • Fall 2005 21
2006 NCCTM Math Logo Contest
The Celebrate Mathematics Committee will once again sponsor a Math Logo Contest. The win- ning logo will be the mathematics logo for NCCTM for the 2006-2007 school year. This logo will be used on a poster to promote interest in mathematics and as the basic design for NCCTM’s 2006 T-shirt. A professional graphic artist will prepare the final art of the winning entry for printing. Though the illustration is an important part of the logo, entries are judged on idea or concept conveyed. Use of copyrighted work (clip art, cartoon characters, etc.) will automatically disqualify the entry. The words of the slogan should not be on top of or touch the design. Please duplicate the following label and include on the back of each entry. Entries submitted with incomplete information may be disqualified.
Student’s Name: Grade: Home Address:
School Name: LEA: School Address:
Teacher’s Name: *NCCTM Region: Eastern Central Western (circle one) *See NCCTM Regional Map Below
Submit the entries on 8 1/2 x 11 paper in black ink or black marker only. Entries must be postmarked by 1 March 2006. Mail to: Rebecca Caison, MATH LOGO CONTEST, 101 E. Laramie Drive, Mebane, NC 27302
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Problems to Ponder
Fall 2005 Problems Gregory S. Rhoads Appalachian State University Boone, North Carolina
Grades K–2 The Smith children are counting their pennies to see if they can buy an ice cream cone. Sallie has 12 pennies. Susie has half as many as Sallie, and Paul has one third as many as Sallie. If they combine all of their pennies together, do they have enough to buy an ice cream cone that costs 20 cents?
Grades 3–5 A quantity and its half added together become 24. What is the quantity? This problem is similar to the problems found on the ancient Egyptian Rhind Papyrus, circa 1650 B.C.
Grades 6–8 “A square and 10 roots are equal to 39”—or, in modern terms, the square of a quantity plus 10 times that quantity equals 39—what is the number? Can you find the negative solution also? This problem is from Hisab al-jabr w’al-muqabala,by al-Khwarizmi, 9th century A.D., thought to have been born in Baghdad.
Grades 9–12 A square, walled city measures 200 paces on each side. Gates are located at the centers of each side. If there is a tree 15 paces from the east gate, how far must a man travel out of the south gate to see the tree? This is from the ancient Chinese text, Nine Chapters on the Mathematical Art; the date is unknown, but commentary by Liu Hui on this text dates to 263 A.D. according to The History of Mathematics (Fifth Edition) by David Burton, McGraw Hill, 2003.)
Directions for submitting solutions sion. Keep in mind that proper acknow- 1. Neatly print the following at the top of ledgement is contingent on legible informa- each solution page: tion and solutions.
• Your full name (first and last) Send solutions by 30 October to: • Your teacher’s name Problems to Ponder Your grade c/o Dr. Greg Rhoads • Dept. of Mathematical Sciences • Your school Appalachian State University 2. Submit one problem per page. Boone, NC 28608
Students who submit correct solutions will As these problems are intended to stimulate inde- pendent thinking, it is expected that a submitted solu- be recognized in the next issue of The Cen- tion indicates that the student completed a significant troid. We publish creative or well-written part of the work. Please try to have the students use solutions from those submitted. If you complete sentences when they write up their solutions would rather not have your solution pub- to promote effective communication of their ideas. lished, please so indicate on your submis-
Vol. 31, No. 2 • Fall 2005 23
Erratum: In the Spring 2005 Problems to Ponder column, the students Rhiana Balmer, Caitlin Drake, Jonathan Jackson, Kayla Krender, Stephen Pearsall, Taylor Rogister, and Shae Scott from Moyock Elementary correctly solved the 3-5 problem and not the K-2 problem as reported. We apologize for the error.
Solutions for Problems from the Spring 2005 Issue
Grades K-2
Peter asked John how many pieces of fruit he had. John said the numbers 4, 8, 5, and 2, and then he said the largest of those numbers was how many apples and the smallest of those numbers was how many oranges he had. How many apples and oranges total did John have?
Solution: Alivia Gaither, 1st grade, Harmony Elementary (teacher: Ms. Walker)
Correct Solutions were received from Issac Boulter, Noah Gavenus, Andrew Hall, Amber Hensley, Irsie Hilbert, Ernesto Leco, Payton Peterson, Amanda Robertson, and Zeb Snyder from Burnsville Elementary, Hannah Perry from Conn Elementary, Maria Joe M. Sanqui from Hardin Park Elementary, Autumn Atwood, Jacob Phil Beck, Payton Carther, Tanner Church, Allison Cullen, Tyler Duncan, Wendy Escobar-Cesar, David Ferretiz, Dupre Folks, Alivia Gaither, Karina Garcia, Zack Gryder, Tyler Harris, Lydia Hopkins, Jordan Hughes, Tyrek Kearney, Kyle Laws, Elliott Martinez, Colby McAlister, Noah Miethe, Darby Mitchell, Hope Morrison, Arianna Munson, Trista Mykins, Amelia Peck, Austin Rash, Victoria Reavis, Jordan Reid, Anna Mae Roark, Olivia Roop, Donald Shoffner, Seth Singleton, Anna Smith, Courtney Stroud, Macey Turant, Anna Wagoner, Zachary Walker, Makayla Wellman, Ethan Yount, and Nehemiah from Harmony Elementary, Ryan Gandy and Kay Lauren Rhea from Sandhills Classical Christian School, Chloe Carroll from Shining Light Academy, Da- kota Griffin from Star-Biscoe Elementary, and Rebecca Hettrick from W.D. Williams Elemen- tary.
Grades 3-5
How many different whole numbers can be made with the digits 1, 6, and 9? The whole number may have 1, 2, or 3 digits in it, but a digit may not be repeated in any number.
Solution: Rebecca Landon, 5th grade, North Ridge Elementary (teacher: Mrs. Byrne).
Correct Solutions were received from Tyler Hatch from Alamance Elementary, Dylan Hulon from Galberry Farms Elementary, Brian Barnes, Aljeanae Broadnax, Johnathan Jackson, Kayla Kreuder, Kristen Murphy, Stephen Pearsall, Daniel Prescott, Taylor Rogister, and River Wil- kinson from Moyock Elementary, Devin Ayscue, Donovan Collins, Logan Knox, and Rebecca Lan- don from Northridge Elementary, Brittany
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Dove, Carlie Estabrook, Madison Lambkin, and Rebekah Martin from Park View Elementary, Anne Williams from Rocky Mount Academy, Rebekah Cross, Johnny Gandy, Ben Middleton, William Proulx, and Joshua Rhea from Sandhills Classical Christian School, Carrie Ann At- kins, Dylan Brown, Hayleigh Clapp, Ethan Cole, Erin Hayes, Nathan Kiser, Cheyenne Madden, Dillon McNeill, Melissa Morales, Jose' Osorio, Jessica Woodell, Amy Wright, and Austin Yow from Seagrove School, Corbin Carroll from Shining Light Academy, and Nakayla Adams, Soloman Banks, Elizabeth Bautista, Dewayne Brown, Patrick Caffferello, Kaleel Chesson, Me- gan Daniel, Kristy Eakes, Brittney Evans, Chelsea Farrar, Diane Garcia, Tyra Jenkins, Bennett Jethro, Johnson Little, Andrew Pofahl, Sarah Quinn, Ryan Singh, Ryan Slade, Khyree Stokes, Dylan Styron, Brittney Teel, Erica Whitehurst from South Greenville Elementary.
Grades 6-8
A container holds 48 marbles of various colors (yellow, red, blue, and white). There are twice as many yellow as red marbles and twice as many blue as white marbles. There are also six more white marbles than red marbles. What is the probability of randomly drawing a white marble out of the container?
Solution: Adam Folleman, 7th grade, Sandhills Clas- sical Christian School, (teacher: Ms. Rhea).
Correct Solutions were received from Kyle Arnold, Tom Jackson, Anne Lucas, Liddy Roer, and Katherine Smith from Canterbury School, Courtney Anderson, Michael Laverick, Sabrina Mason, and Miriam Rosen from Cary Academy, Faith Beamon, Jacob Bitinas, Sarah Paige Fletcher, Sydney McCauley, Berkeley Tate, and Grace Williams from Currituck Middle, Brandi Baker, Aurelio Cazares, Trei Jolly, Melinda Justice, Joseph Martin, April Russell, Aaron Tallman, Caleb Underwood, Matthew Vaughn, and Leah Woods from East Alexander Middle, Nicholas Lacombe and Chantrel Reynolds from Eastern Wayne Middle, Kate Currin, Ali Denny, Betsy Dickerson, Cody Evans, Chris Hill, Allison May, Ashley McRae, Seth Newton, Chelsea Parker, Gracen Pittard, Lynwood Roberts, Latricia Taylor, Shaniqua Taylor, Patrick Warren, Sara Watkins, Marianna Whitt, Gray Williams, and Kristen Williford from Northern Granville Middle, Amanda Johnson, Kevin Powell, and Jacob Ridlon from North- west Middle, Jayme Gillespie, Cameron Smith, Chort Thao, and Bau Yang from River Bend Middle, Justin Dearman, Adam Fogleman, Ashley Kidd, and Sami Middleton from Sandhills Classical Christian School, Meredith Nichter from Shelby Middle, Chance Cockrell from Southeast Guilford Middle, Dequante Andrews, Jasmine Barnes, Brianna Barrymar, Chris'ta Bazemore, Tamary Bazemore, Tristian Bazemore, Kayana Belfield, Joseph Bond, Larry Bond, Shakeila Bond, Yashonti Bridges, Chaniece Brown, Trenesa Brown, Angel Bryant, Fontasia Cherry, Norman Cherry III, Quanesha Cooper, Lauren Cuttino, Courtney Dunlow, Jarvis Gil- liam, Akiana Glenn, Nikki Harrell, Zachary Harris, Kiara Heckstall, Vincent Holley, Kirk Hutchinson, Chris Jernigan, Jaelyn Johnson, Tameakia Lee, Christian Lilley, Lester Miller, Donna Mizelle, Terri Morris, Zachariah Newkirk, Tiffany Newsome, Jack Nicholson II, Riketta Norfleet, Anthony Outlaw, Jamie Outlaw, Jorie Parker, Quanisha Perry, Sha'Tauga Pugh, Anto- nio Roscoe, Niki Ruffin, Jyrell Shepperson, Jadale Smallwood, Robert Smallwood, Brittany Smith, Lakeisha Speller, Tyriek Spivey, Equalia Vines, Pearl Vines, Jasmine Ward, Darrius Wesson, Kennedy Wesson, Jerron Wiggins, and Quadedra Williams from Southwestern Mid- dle, Alex Averbook, Matthew Friesen, Charlie Johnson, Christopher Mayner, Maura Thornton, and Alex Zhang from The O’Neal School, Austin Allen, Brittany Bell, Robert Brown, Brandon
Vol. 31, No. 2 • Fall 2005 25
Candy, Sarah Dalton, Rianda Evans, Caroline Gaither, Lucy Goodwin-Johansson, Caleb Hasel- ton, Anna Hill, Grant Huffman, Sara Hurdle, Maddie Johnson, Jin Kim, Joshua Krafft, Justin Lea, Trinh Ngo, Keegan Pace, Travis Poole, Helen Powell, Cassandra Rigsbee, Wahab Sheikh, Ashely Sochor, Laural Strausbaugh, Kayla Trollinger, and Eric Turrentine from Turrentine Middle, and Matthew Draelos from Weslyan Christian Academy.
Grades 9-12
Find the area of the region satisfying both y ! x and x 2 + y 2 !1.
Solution: Jayson Rivera and Damaris Squires, 10th grade, Jones Senior High (teacher: Ms. Sri- vatsa)
Correct Solutions were received from Jayson Rivera and Damaris Squires from Jones Senior High, Bolu Adeyeye, Kam Horvath, Marion Mogk, and Megan Trull from Northwest Cabarrus High, and Will Blair and Luke Michael Taylor from Ragsdale High.
DPI Math Resources on the Web
The website you are accustomed to for DPI mathematics resources is moving. Please check the new site
26 The Centroid
Butterfly Sticker & Homework Recording Sheet
June L. Blackwell1 Sanderson High School Raleigh, North Carolina
Do you love to check and grade your student’s The next part of the recording sheet is daily homework? Do you love to decide the Quarter Hall Pass. I like for students to whether a student should receive a 1, 2, 3 or take care of their bathroom and locker check, check plus, check minus on daily needs before or after my class. This cou- homework? Do you have a way to get students pon puts the responsibility on the student. to become more responsible for their work and Each box under the Quarter Hall pass sec- actions? If you answered yes to any of these tion is worth 0.5 points, so if a student questions, on the next page is the time saving needs to use his/her coupon, then I hole recording sheet for you! punch the 1 pt box leaving the 0.5 pt box. First, you need to decide a sticker or If a student needs to use the pass a second stamp that you like to see because trust me, time, then the 0.5 box is punched and no you will see it often! I like butterflies, so that additional points are available to the stu- is what I use. At the beginning of each class, I dent for the grading period. walk around the room with my butterfly rub- Each year I tell my students I am sure ber stamp and inkpad. Students who have that they have come across a teacher who completed their homework get a full stamp on does not waiver when a student’s grade is their sheet. If they have partially completed a few decimal points away from the next their homework, I tilt the stamp and give them higher grade. For instance, a student’s half of a stamp, and if they have not done their grade might be 84.2 and a B starts at 85. homework, they do not receive a stamp. By using the coupon method, if a student’s The next section of the recording sheet is coupon is un-punched then they can re- for positive re-enforcement. I place a small ceive the additional decimal points that butterfly sticker on my students’ papers when they need to push them up to the next they make 85 or better on a quiz or test. I also grade. Most teachers I know would proba- use stickers as prizes in review games and for bly give the student the benefit of the good behavior. The students place the sticker doubt and the next higher grade, but by us- on their recording sheet by covering up the ing the coupon method students become number. This provides a quick reference to see more responsible for taking care of their how many stickers they have received when I personal needs. collect their sheet. At the end of the quarter, I All three of these recording and class- collect the students’ pages, and I add the num- room management techniques are con- ber of sticker points to their lowest quiz grade. tained on one sheet of paper, so that when This helps to teach the students to work on I collect the sheet at the end of each grad- their grade along the way instead of waiting ing period, I have numerous pieces of in- for the last minute, to miraculously improve formation at my finger tips for each stu- their grade. dent. Each year, I tweak and revise my Butterfly Sticker & Recording Sheet. This
1 Ms. Blackwell, aka the Butterfly Lady, is a National Board Certified teacher and Kenan Fellow who teaches at Sanderson High School. She also serves as NCCTM Eastern Regional VP for Seconday Schools.
Vol. 31, No. 2 • Fall 2005 27
year, I added additional pieces of information classes, so I decided to use the same tech- like a section to record grades, Tardy Warn- nique in my high school classes. ing, Character Education traits, and SAT I hope this sheet of ideas is helpful and dates. I often put a grade recording section on useful in starting the new school year off my class schedule for my Community College on a good “wing” – ha ha ha.
Butterfly Sticker & Recording Sheet Name Ms. June L. Blackwell Character Education Traits - The Core of Education and Life! Courage Self-Discipline Perseverance Integrity Respect Responsibility Kindness Good Judgment Date Sticker Grade Recording Section EC / Quarterly Hall Pass 1 UT Quiz Project Stickers
2
You May Use This Punch Card Pass for up to Two (2) 3 Trips During a Quarter:
Bathroom Locker Water Other 4
Un-punched boxes may be redeemed for up to 1 pt
5 added to your Quarter Grade.
1.0 6
7 0.5
This pass May Not be sold, traded or given Away! 8 The Management reserves the right to deny you a pass!!!
9
10
SAT Dates Total 11/1, 12/6, 1/24, 3/27, 5/1, 6/5 Write the Daily Homework Assignment & Date in each box. 1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 Total Number of Butter- fly Homework Stamps
Tardy Warning Hole Punch
28 The Centroid
Rankin Award Nominations
The Rankin Award is designed to recognize and honor individuals for their outstanding contribu- tions to NCCTM and to mathematics education in the State. Presented in the fall at the State Mathematics Conference, the award, named in memory of W.W. Rankin, Professor of Mathe- matics at Duke University, is the highest honor NCCTM can bestow upon an individual.
If you have nominated someone in the past who has not received the award to date, or if you would like to nominate someone now, please submit as much of the following information as possible. Nominations are accepted at any time.
Nominations should include the following information. Use as many typewritten pages as needed. If possible, attach a vita of the nominee.
Name of the nominee
Current position
Your relationship to the nominee (e.g. principal, co-worker, etc.)
The nominee’s contributions to mathematics education, NCTM, NCCTM, etc. (Please include information on specific offices held and honors received by the nominee.)
Any information about contributions to the community, teaching, and education that would be of value to the Rankin Award Committee in its deliberations
Other relevant information
Letters of endorsement from other colleagues may be included.
Date of nomination
Nominator* Name Current position Business or educational institution Preferred mailing address Preferred telephone number
*The Rankin Award Committee reserves the right to use portions of nomination information in the presentation of the award if the candidate is selected.
Send to: Dr. Ralph DeVane P. O. Box 1762 Cullowhee, NC 28723
Vol. 31, No. 2 • Fall 2005 29
Innovator Award Nominations
The North Carolina Council of Teachers of Mathematics accepts nominations for the Innovator Award at any time. The purpose of this award is to recognize and reward individuals or groups who have made an outstanding and noteworthy contribution to mathematics education and/or NCCTM by having founded, initiated, pioneered, or developed some program in mathematics education of service to a geographic region of the state or the entire state. Further, this program must have been sustained for a period of at least three years. A number of organizations have made significant contributions to mathematics education in North Carolina; the Committee en- courages the nomination of organizations as well as individuals. Any NCCTM member may submit nominations by sending in the form below. Nominations will be retained in the active file for at least three years.
Nomination Form
Name of Nominee______
Present Position______
Outstanding contributions to mathematics education in North Carolina which serves as the basis for this nomination:
Additional information that would be of value to the selection committee:
Signature: Date:
Name (print/type):
Position:
Business or Institution:
Address:
Phone: Business: Home: Email:
Send to: Phillip Johnson, Math and Science Education Center, ASU Box 32091 Boone, NC 28608-2091
30 The Centroid
NCCTM Trust Fund Scholarship
$500 scholarships are available from NCCTM to financially support North Carolina teachers who are enrolled in graduate degree programs to enhance mathematics instruction.
Applicants must be: • Currently employed as a pre-K-12 teacher in North Carolina; • Currently an NCCTM member (for at least one year) at the time of submitting this application; • Currently enrolled in an accredited graduate program in North Carolina; • Currently enrolled in a mathematics or mathematics education course, or have completed a mathematics or mathematics education course within the previous four months of the application deadline.
Applications will be reviewed biannually, and the deadlines for applications are: • March 1 • October 1
Send completed applications to: Direct inquiries to: NCCTM Trust Fund Chairperson Bill Paul, Chairperson P.O. Box 121 Phone: (828) 297-3839 Sugar Grove, NC 28679 E-mail: [email protected]
(Please print all information.)
PERSONAL INFORMATION:
Name:
Home address:
Home phone: Home e-mail:
NCCTM membership number:
EMPLOYMENT INFORMATION:
How many years of teaching experience?
Currently employed in what school system?_
School name:
School address:
School phone: School e-mail:
Current teaching assignment:
Principal’s name:
Vol. 31, No. 2 • Fall 2005 31
COURSE INFORMATION:
Institution of higher education:
Graduate degree program in which you are currently enrolled:
Course name: Course number:
Dates of enrollment: (circle one) Fall semester Spring semester Summer session Year:
Name of course instructor:
PROFESSIONAL ACTIVITIES WITHIN PAST 5 YEARS:
BRIEF STATEMENT OF FUTURE PROFESSIONAL GOALS:
REQUIRED SIGNATURES:
Applicant’s Signature: Date:
Principal’s Signature: Date:
Instructor’s Signature (if currently enrolled): Date:
REQUIRED ATTACHMENTS: Please attach a copy of verification of acceptance and enrollment in accredited graduate program in North Carolina.
NOTE: Applications must be complete to be considered. If your application is approved, an official course grade report must be submitted to verify successful completion of the course before scholarship funds will be issued.
32 The Centroid NORTH CAROLINA COUNCIL OF TEACHERS OF MATHEMATICS BOARD OF DIRECTORS Office State Eastern Central Western President Jeane Joyner Julie Kolb Emogene Kernodle Carmen Wilson Raleigh Raleigh Elon Deep Gap Past President Jan Wessell Kathryn Hill Vickie Moss Betty Long Wrightsville Beach Raleigh Asheboro Boone Secretary Tery Gunter Elizabeth Murray Angela Flowers Cindy Robinson Durham Leland Sophia Boone Vice President Susan Friel Gail Stafford Cos D. Fi Terry Rose Colleges Chapel Hill Rocky Mount Greensboro Knoxville (TN) Vice President Wendy Rich Carolann Wade Sylvia Davis Debbie Ross Elementary Asheboro Cary Asheboro Asheville Vice President Sandra Childrey Lucy Kay Karen Ellis Kim Tyson-Aiello Middle Grades Cary Raleigh Greensboro Granite Falls Vice President Tony Sapp June Blackwell Elaine Tucker Kim Daily Secondary Swannanoa Raleigh Pinehurst Swannanoa
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