DOCSLIB.ORG

Teaching Math to Young Children

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Teaching Math to Young Children EDUCATOR’S PRACTICE GUIDE WHAT WORKS CLEARINGHOUSE™ Teaching Math to Young Children NCEE 2014-4005 U.S. DEPARTMENT OF EDUCATION The Institute of Education Sciences (IES) publishes practice guides in education to bring the best available evidence and expertise to bear on current challenges in education. Authors of practice guides combine their expertise with the findings of rigorous research, when available, to develop specific recommendations for addressing these challenges. The authors rate the strength of the research evidence supporting each of their recommendations. See Appendix A for a full description of practice guides. The goal of this practice guide is to offer educators specific, evidence-based recommendations that address the challenge of teaching early math to children ages 3 to 6. The guide provides practical, clear information on critical topics related to teaching early math and is based on the best available evidence as judged by the authors. Practice guides published by IES are available on our website at http://whatworks.ed.gov. IES Practice Guide Teaching Math to Young Children November 2013 Panel Douglas Frye (Chair) UNIVERSITY OF PENNSYLVANIA Arthur J. Baroody UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN AND UNIVERSITY OF DENVER Margaret Burchinal UNIVERSITY OF NORTH CAROLINA Sharon M. Carver CARNEGIE MELLON UNIVERSITY CHILDREN’S SCHOOL Nancy C. Jordan UNIVERSITY OF DELAWARE Judy McDowell SCHOOL DISTRICT OF PHILADELPHIA Staff M. C. Bradley Elizabeth Cavadel Julia Lyskawa Libby Makowsky Moira McCullough Bryce Onaran Michael Barna MATHEMATICA POLICY RESEARCH Marc Moss ABT ASSOCIATES Project Officers Joy Lesnick Diana McCallum INSTITUTE OF EDUCATION SCIENCES NCEE 2014-4005 U.S. DEPARTMENT OF EDUCATION This report was prepared for the National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences under Contract ED-IES-13-C-0010 by the What Works Clearinghouse, which is operated by Mathematica Policy Research. Disclaimer The opinions and positions expressed in this practice guide are those of the authors and do not necessarily represent the opinions and positions of the Institute of Education Sciences or the U.S. Department of Education. This practice guide should be reviewed and applied according to the specific needs of the educators and education agency using it, and with full realization that it represents the judgments of the review panel regarding what constitutes sensible practice, based on the research that was available at the time of publication. This practice guide should be used as a tool to assist in decisionmaking rather than as a “cookbook.” Any references within the document to specific education products are illustrative and do not imply endorsement of these products to the exclusion of other products that are not referenced. U.S. Department of Education Arne Duncan Secretary Institute of Education Sciences John Q. Easton Director National Center for Education Evaluation and Regional Assistance Ruth Neild Commissioner November 2013 This report is in the public domain. Although permission to reprint this publication is not necessary, the citation should be: Frye, D., Baroody, A. J., Burchinal, M., Carver, S. M., Jordan, N. C., & McDowell, J. (2013). Teaching math to young children: A practice guide (NCEE 2014-4005). Washington, DC: National Center for Education Evaluation and Regional Assistance (NCEE), Institute of Education Sciences, U.S. Department of Educa- tion. Retrieved from the NCEE website: http://whatworks.ed.gov What Works Clearinghouse practice guide citations begin with the panel chair, followed by the names of the panelists listed in alphabetical order. This report is available on the IES website at http://whatworks.ed.gov. Alternate Formats On request, this publication can be made available in alternate formats, such as Braille, large print, or CD. For more information, contact the Alternate Format Center at (202) 260-0852 or (202) 260-0818. Table of Contents Teaching Math to Young Children Table of Contents Overview of Recommendations . .1 Acknowledgements . 3 Institute of Education Sciences Levels of Evidence for Practice Guides . .4 Introduction to the Teaching Math to Young Children Practice Guide . 7 Recommendation 1. Teach number and operations using a developmental progression . 12 Recommendation 2. Teach geometry, patterns, measurement, and data analysis using a developmental progression . 25 Recommendation 3. Use progress monitoring to ensure that math instruction builds on what each child knows . 36 Recommendation 4. Teach children to view and describe their world mathematically . 42 Recommendation 5. Dedicate time each day to teaching math, and integrate math instruction throughout the school day . 47 Glossary . 57 Appendix A. Postscript from the Institute of Education Sciences . 59 Appendix B. About the Authors . 61 Appendix C. Disclosure of Potential Conflicts of Interest . 64 Appendix D. Rationale for Evidence Ratings . 65 Endnotes . 132 References . 152 List of Tables Table 1. Institute of Education Sciences levels of evidence for practice guides. 5 Table 2. Recommendations and corresponding levels of evidence . .11 Table 3. Examples of a specific developmental progression for number knowledge . 13 Table 4. Common counting errors . 19 Table 5. Examples of vocabulary words for types of measurement . 32 ( iii ) Table of Contents (continued) Table 6. Using informal representations . 43 Table 7. Linking familiar concepts to formal symbols . 44 Table 8. Examples of open-ended questions . 45 Table 9. Integrating math across the curriculum . 51 Table 10. Examples of tools that can be useful in each math content area . 52 Table D.1. Summary of studies contributing to the body of evidence, by recommendation . 67 Table D.2. Studies of early math curricula that taught number and operations and contributed to the level of evidence rating . 72 Table D.3. Studies of comprehensive curricula with an explicit math component that taught number and operations and contributed to the level of evidence rating . 76 Table D.4. Studies of targeted interventions that taught number and operations and contributed to the level of evidence rating . 81 Table D.5. Studies of interventions that taught geometry, patterns, measurement, or data analysis and contributed to the level of evidence rating . 94 Table D.6. Studies of interventions that used a deliberate progress-monitoring process and contributed to the level of evidence rating . 104 Table D.7. Studies of interventions that incorporated math communication, math vocabulary, and linking informal knowledge to formal knowledge and contributed to the level of evidence rating . 112 Table D.8. Studies of interventions that included regular math time, incorporated math into other aspects of the school day, and used games to reinforce math skills and contributed to the level of evidence rating . 121 List of Examples Example 1. The Basic Hiding game . 16 Example 2. The Hidden Stars game . 18 Example 3. The Concentration: Numerals and Dots game . 22 Example 4. The Shapes game . 29 Example 5. Creating and extending patterns . 31 Example 6. The Favorites game . 34 Example 7. The flow of progress monitoring . 39 Example 8. Progress-monitoring checklist . 40 Example 9. Linking large groups to small groups . 49 Example 10. Snack time . 50 E x a m p l e 11. The Animal Spots game . 54 List of Figures Figure 1. Modeling one-to-one counting with one to three items . 17 Figure 2. Sample cardinality chart . 20 Figure 3. Sample number list . 21 Figure 4. Combining and separating shapes . 28 Figure 5. Moving from simple to complex patterns . 30 Figure 6. The repetitive nature of the calendar . 30 Figure 7. An example of a math-rich environment in the classroom . 53 ( iv )( ) Overview of Recommendations Recommendation 1. Teach number and operations using a developmental progression. • First, provide opportunities for children to practice recognizing the total number of objects in small collections (one to three items) and labeling them with a number word without needing to count them. • Next, promote accurate one-to-one counting as a means of identifying the total number of items in a collection. • Once children can recognize or count collections, provide opportunities for children to use number words and counting to compare quantities. • Encourage children to label collections with number words and numerals. • Once children develop these fundamental number skills, encourage them to solve basic problems. Recommendation 2. Teach geometry, patterns, measurement, and data analysis using a developmental progression. • Help children to recognize, name, and compare shapes, and then teach them to combine and separate shapes. • Encourage children to look for and identify patterns, and then teach them to extend, correct, and create patterns. • Promote children’s understanding of measurement by teaching them to make direct comparisons and to use both informal or nonstandard (e.g., the child’s hand or foot) and formal or standard (e.g., a ruler) units and tools. • Help children to collect and organize information, and then teach them to represent that infor- mation graphically. Recommendation 3. Use progress monitoring to ensure that math instruction builds on what each child knows. • Use introductory activities, observations, and assessments to determine each child’s existing math knowledge, or the level of understanding or skill he or she has reached on a develop- mental progression. • Tailor instruction to each child’s needs, and relate
Load more

Recommended publications
  • 2.1 the Algebra of Sets
    Chapter 2 I Abstract Algebra 83 part of abstract algebra, sets are fundamental to all areas of mathematics and we need to establish a precise language for sets. We also explore operations on sets and relations between sets, developing an “algebra of sets” that strongly resembles aspects of the algebra of sentential logic. In addition, as we discussed in chapter 1, a fundamental goal in mathematics is crafting articulate, thorough, convincing, and insightful arguments for the truth of mathematical statements. We continue the development of theorem-proving and proof-writing skills in the context of basic set theory. After exploring the algebra of sets, we study two number systems denoted Zn and U(n) that are closely related to the integers. Our approach is based on a widely used strategy of mathematicians: we work with specific examples and look for general patterns. This study leads to the definition of modified addition and multiplication operations on certain finite subsets of the integers. We isolate key axioms, or properties, that are satisfied by these and many other number systems and then examine number systems that share the “group” properties of the integers. Finally, we consider an application of this mathematics to check digit schemes, which have become increasingly important for the success of business and telecommunications in our technologically based society. Through the study of these topics, we engage in a thorough introduction to abstract algebra from the perspective of the mathematician— working with specific examples to identify key abstract properties common to diverse and interesting mathematical systems. 2.1 The Algebra of Sets Intuitively, a set is a “collection” of objects known as “elements.” But in the early 1900’s, a radical transformation occurred in mathematicians’ understanding of sets when the British philosopher Bertrand Russell identified a fundamental paradox inherent in this intuitive notion of a set (this paradox is discussed in exercises 66–70 at the end of this section).
    [Show full text]
  • The Five Fundamental Operations of Mathematics: Addition, Subtraction
    The five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms Kenneth A. Ribet UC Berkeley Trinity University March 31, 2008 Kenneth A. Ribet Five fundamental operations This talk is about counting, and it’s about solving equations. Counting is a very familiar activity in mathematics. Many universities teach sophomore-level courses on discrete mathematics that turn out to be mostly about counting. For example, we ask our students to find the number of different ways of constituting a bag of a dozen lollipops if there are 5 different flavors. (The answer is 1820, I think.) Kenneth A. Ribet Five fundamental operations Solving equations is even more of a flagship activity for mathematicians. At a mathematics conference at Sundance, Robert Redford told a group of my colleagues “I hope you solve all your equations”! The kind of equations that I like to solve are Diophantine equations. Diophantus of Alexandria (third century AD) was Robert Redford’s kind of mathematician. This “father of algebra” focused on the solution to algebraic equations, especially in contexts where the solutions are constrained to be whole numbers or fractions. Kenneth A. Ribet Five fundamental operations Here’s a typical example. Consider the equation y 2 = x3 + 1. In an algebra or high school class, we might graph this equation in the plane; there’s little challenge. But what if we ask for solutions in integers (i.e., whole numbers)? It is relatively easy to discover the solutions (0; ±1), (−1; 0) and (2; ±3), and Diophantus might have asked if there are any more.
    [Show full text]
  • Section 4.2 Counting
    Section 4.2 Counting CS 130 – Discrete Structures Counting: Multiplication Principle • The idea is to find out how many members are present in a finite set • Multiplication Principle: If there are n possible outcomes for a first event and m possible outcomes for a second event, then there are n*m possible outcomes for the sequence of two events. • From the multiplication principle, it follows that for 2 sets A and B, |A x B| = |A|x|B| – A x B consists of all ordered pairs with first component from A and second component from B CS 130 – Discrete Structures 27 Examples • How many four digit number can there be if repetitions of numbers is allowed? • And if repetition of numbers is not allowed? • If a man has 4 suits, 8 shirts and 5 ties, how many outfits can he put together? CS 130 – Discrete Structures 28 Counting: Addition Principle • Addition Principle: If A and B are disjoint events with n and m outcomes, respectively, then the total number of possible outcomes for event “A or B” is n+m • If A and B are disjoint sets, then |A B| = |A| + |B| using the addition principle • Example: A customer wants to purchase a vehicle from a dealer. The dealer has 23 autos and 14 trucks in stock. How many selections does the customer have? CS 130 – Discrete Structures 29 More On Addition Principle • If A and B are disjoint sets, then |A B| = |A| + |B| • Prove that if A and B are finite sets then |A-B| = |A| - |A B| and |A-B| = |A| - |B| if B A (A-B) (A B) = (A B) (A B) = A (B B) = A U = A Also, A-B and A B are disjoint sets, therefore using the addition principle, |A| = | (A-B) (A B) | = |A-B| + |A B| Hence, |A-B| = |A| - |A B| If B A, then A B = B Hence, |A-B| = |A| - |B| CS 130 – Discrete Structures 30 Using Principles Together • How many four-digit numbers begin with a 4 or a 5 • How many three-digit integers (numbers between 100 and 999 inclusive) are even? • Suppose the last four digit of a telephone number must include at least one repeated digit.
    [Show full text]
  • A Quick Algebra Review
    A Quick Algebra Review 1. Simplifying Expressions 2. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Exponents 9. Quadratics 10. Rationals 11. Radicals Simplifying Expressions An expression is a mathematical “phrase.” Expressions contain numbers and variables, but not an equal sign. An equation has an “equal” sign. For example: Expression: Equation: 5 + 3 5 + 3 = 8 x + 3 x + 3 = 8 (x + 4)(x – 2) (x + 4)(x – 2) = 10 x² + 5x + 6 x² + 5x + 6 = 0 x – 8 x – 8 > 3 When we simplify an expression, we work until there are as few terms as possible. This process makes the expression easier to use, (that’s why it’s called “simplify”). The first thing we want to do when simplifying an expression is to combine like terms. For example: There are many terms to look at! Let’s start with x². There Simplify: are no other terms with x² in them, so we move on. 10x x² + 10x – 6 – 5x + 4 and 5x are like terms, so we add their coefficients = x² + 5x – 6 + 4 together. 10 + (-5) = 5, so we write 5x. -6 and 4 are also = x² + 5x – 2 like terms, so we can combine them to get -2. Isn’t the simplified expression much nicer? Now you try: x² + 5x + 3x² + x³ - 5 + 3 [You should get x³ + 4x² + 5x – 2] Order of Operations PEMDAS – Please Excuse My Dear Aunt Sally, remember that from Algebra class? It tells the order in which we can complete operations when solving an equation.
    [Show full text]
  • 7.2 Binary Operators Closure
    last edited April 19, 2016 7.2 Binary Operators A precise discussion of symmetry benefits from the development of what math- ematicians call a group, which is a special kind of set we have not yet explicitly considered. However, before we define a group and explore its properties, we reconsider several familiar sets and some of their most basic features. Over the last several sections, we have considered many di↵erent kinds of sets. We have considered sets of integers (natural numbers, even numbers, odd numbers), sets of rational numbers, sets of vertices, edges, colors, polyhedra and many others. In many of these examples – though certainly not in all of them – we are familiar with rules that tell us how to combine two elements to form another element. For example, if we are dealing with the natural numbers, we might considered the rules of addition, or the rules of multiplication, both of which tell us how to take two elements of N and combine them to give us a (possibly distinct) third element. This motivates the following definition. Definition 26. Given a set S,abinary operator ? is a rule that takes two elements a, b S and manipulates them to give us a third, not necessarily distinct, element2 a?b. Although the term binary operator might be new to us, we are already familiar with many examples. As hinted to earlier, the rule for adding two numbers to give us a third number is a binary operator on the set of integers, or on the set of rational numbers, or on the set of real numbers.
    [Show full text]
  • Rules for Matrix Operations
    Math 2270 - Lecture 8: Rules for Matrix Operations Dylan Zwick Fall 2012 This lecture covers section 2.4 of the textbook. 1 Matrix Basix Most of this lecture is about formalizing rules and operations that we’ve already been using in the class up to this point. So, it should be mostly a review, but a necessary one. If any of this is new to you please make sure you understand it, as it is the foundation for everything else we’ll be doing in this course! A matrix is a rectangular array of numbers, and an “m by n” matrix, also written rn x n, has rn rows and n columns. We can add two matrices if they are the same shape and size. Addition is termwise. We can also mul tiply any matrix A by a constant c, and this multiplication just multiplies every entry of A by c. For example: /2 3\ /3 5\ /5 8 (34 )+( 10 Hf \i 2) \\2 3) \\3 5 /1 2\ /3 6 3 3 ‘ = 9 12 1 I 1 2 4) \6 12 1 Moving on. Matrix multiplication is more tricky than matrix addition, because it isn’t done termwise. In fact, if two matrices have the same size and shape, it’s not necessarily true that you can multiply them. In fact, it’s only true if that shape is square. In order to multiply two matrices A and B to get AB the number of columns of A must equal the number of rows of B. So, we could not, for example, multiply a 2 x 3 matrix by a 2 x 3 matrix.
    [Show full text]
  • 180: Counting Techniques
    180: Counting Techniques In the following exercise we demonstrate the use of a few fundamental counting principles, namely the addition, multiplication, complementary, and inclusion-exclusion principles. While none of the principles are particular complicated in their own right, it does take some practice to become familiar with them, and recognise when they are applicable. I have attempted to indicate where alternate approaches are possible (and reasonable). Problem: Assume n ≥ 2 and m ≥ 1. Count the number of functions f :[n] ! [m] (i) in total. (ii) such that f(1) = 1 or f(2) = 1. (iii) with minx2[n] f(x) ≤ 5. (iv) such that f(1) ≥ f(2). (v) that are strictly increasing; that is, whenever x < y, f(x) < f(y). (vi) that are one-to-one (injective). (vii) that are onto (surjective). (viii) that are bijections. (ix) such that f(x) + f(y) is even for every x; y 2 [n]. (x) with maxx2[n] f(x) = minx2[n] f(x) + 1. Solution: (i) A function f :[n] ! [m] assigns for every integer 1 ≤ x ≤ n an integer 1 ≤ f(x) ≤ m. For each integer x, we have m options. As we make n such choices (independently), the total number of functions is m × m × : : : m = mn. (ii) (We assume n ≥ 2.) Let A1 be the set of functions with f(1) = 1, and A2 the set of functions with f(2) = 1. Then A1 [ A2 represents those functions with f(1) = 1 or f(2) = 1, which is precisely what we need to count. We have jA1 [ A2j = jA1j + jA2j − jA1 \ A2j.
    [Show full text]
  • A Case Study Exploring the Nature of How Prekindergarten Teachers’ Mathematics Instruction Decisions Relate to Their Mathematics Pedagogical Content Knowledge
    Old Dominion University ODU Digital Commons Teaching & Learning Theses & Dissertations Teaching & Learning Spring 2020 A Case Study Exploring the Nature of How Prekindergarten Teachers’ Mathematics Instruction Decisions Relate to Their Mathematics Pedagogical Content Knowledge Raleta Summers Dawkins Old Dominion University, [email protected] Follow this and additional works at: https://digitalcommons.odu.edu/teachinglearning_etds Part of the Early Childhood Education Commons, Science and Mathematics Education Commons, and the Teacher Education and Professional Development Commons Recommended Citation Dawkins, Raleta S.. "A Case Study Exploring the Nature of How Prekindergarten Teachers’ Mathematics Instruction Decisions Relate to Their Mathematics Pedagogical Content Knowledge" (2020). Doctor of Philosophy (PhD), Dissertation, Teaching & Learning, Old Dominion University, DOI: 10.25777/63kg-gz86 https://digitalcommons.odu.edu/teachinglearning_etds/61 This Dissertation is brought to you for free and open access by the Teaching & Learning at ODU Digital Commons. It has been accepted for inclusion in Teaching & Learning Theses & Dissertations by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. A CASE STUDY EXPLORING THE NATURE OF HOW PREKINDERGARTEN TEACHERS’ MATHEMATICS INSTRUCTION DECISIONS RELATE TO THEIR MATHEMATICS PEDAGOGICAL CONTENT KNOWLEDGE by Raleta Summers Dawkins B.S. December 2004, North Carolina Agricultural & Technical State University M.A. Ed. December 2008,
    [Show full text]
  • Useful Relations in Permutations and Combination 1. Useful Relations
    Useful Relations in Permutations and Combination 1. Useful Relations - Factorial n! = n.(n-1)! 2. n퐶푟= n푃푟/r! n n-1 3. Pr = n( Pr-1) 4. Useful Relations - Combinations n n 1. Cr = C(n - r) Example 8 8 C6 = C2 = 8×72×1 = 28 n 2. Cn = 1 n 3. C0 = 1 n n n n n 4. C0 + C1 + C2 + ... + Cn = 2 Example 4 4 4 4 4 4 C0 + C1 + C2 + C3+ C4 = (1 + 4 + 6 + 4 + 1) = 16 = 2 n n (n+1) Cr-1 + Cr = Cr (Pascal's Law) n퐶푟 =n/퐶푟−1=n-r+1/r n n If Cx = Cy then either x = y or (n-x) = y. 5. Selection from identical objects: Some Basic Facts The number of selections of r objects out of n identical objects is 1. Total number of selections of zero or more objects from n identical objects is n+1. 6. Permutations of Objects when All Objects are Not Distinct The number of ways in which n things can be arranged taking them all at a time, when st nd 푃1 of the things are exactly alike of 1 type, 푃2 of them are exactly alike of a 2 type, and th 푃푟of them are exactly alike of r type and the rest of all are distinct is n!/ 푃1! 푃2! ... 푃푟! 1 Example: how many ways can you arrange the letters in the word THESE? 5!/2!=120/2=60 Example: how many ways can you arrange the letters in the word REFERENCE? 9!/2!.4!=362880/2*24=7560 7.Circular Permutations: Case 1: when clockwise and anticlockwise arrangements are different Number of circular permutations (arrangements) of n different things is (n-1)! 1.
    [Show full text]
  • UNIT 1: Counting and Cardinality
    Grade: K Unit Number: 1 Unit Name: Counting and Cardinality Instructional Days: 40 days EVERGREEN SCHOOL K DISTRICT GRADE Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Counting Operations Geometry Measurement Numbers & Algebraic Thinking and and Data Operations in Cardinality Base 10 8 weeks 4 weeks 8 weeks 3 weeks 4 weeks 5 weeks UNIT 1: Counting and Cardinality Dear Colleagues, Enclosed is a unit that addresses all of the Common Core Counting and Cardinality standards for Kindergarten. We took the time to analyze, group and organize them into a logical learning sequence. Thank you for entrusting us with the task of designing a rich learning experience for all students, and we hope to improve the unit as you pilot it and make it your own. Sincerely, Grade K Math Unit Design Team CRITICAL THINKING COLLABORATION COMMUNICATION CREATIVITY Evergreen School District 1 MATH Curriculum Map aligned to the California Common Core State Standards 7/29/14 Grade: K Unit Number: 1 Unit Name: Counting and Cardinality Instructional Days: 40 days UNIT 1 TABLE OF CONTENTS Overview of the grade K Mathematics Program . 3 Essential Standards . 4 Emphasized Mathematical Practices . 4 Enduring Understandings & Essential Questions . 5 Chapter Overviews . 6 Chapter 1 . 8 Chapter 2 . 9 Chapter 3 . 10 Chapter 4 . 11 End-of-Unit Performance Task . 12 Appendices . 13 Evergreen School District 2 MATH Curriculum Map aligned to the California Common Core State Standards 7/29/14 Grade: K Unit Number: 1 Unit Name: Counting and Cardinality Instructional Days: 40 days Overview of the Grade K Mathematics Program UNIT NAME APPROX.
    [Show full text]
  • Extended Instruction in Business Courses to Enhance Student Achievement in Math Lessie Mcnabb Houseworth Walden University
    Walden University ScholarWorks Walden Dissertations and Doctoral Studies Walden Dissertations and Doctoral Studies Collection 2015 Extended Instruction in Business Courses to Enhance Student Achievement in Math Lessie McNabb Houseworth Walden University Follow this and additional works at: https://scholarworks.waldenu.edu/dissertations Part of the Other Education Commons, and the Science and Mathematics Education Commons This Dissertation is brought to you for free and open access by the Walden Dissertations and Doctoral Studies Collection at ScholarWorks. It has been accepted for inclusion in Walden Dissertations and Doctoral Studies by an authorized administrator of ScholarWorks. For more information, please contact [email protected]. Walden University COLLEGE OF EDUCATION This is to certify that the doctoral study by Lessie Houseworth has been found to be complete and satisfactory in all respects, and that any and all revisions required by the review committee have been made. Review Committee Dr. Calvin Lathan, Committee Chairperson, Education Faculty Dr. Jennifer Brown, Committee Member, Education Faculty Dr. Amy Hanson, University Reviewer, Education Faculty Chief Academic Officer Eric Riedel, Ph.D. Walden University 2015 Abstract Extended Instruction in Business Courses to Enhance Student Achievement in Math by Lessie M. Houseworth EdS, Walden University, 2010 MS, Indiana University, 1989 BS, Indiana University, 1976 Doctoral Study Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Education Walden University March 2015 Abstract Poor achievement on standardized math tests negatively impacts high school graduation rates. The purpose of this quantitative study was to investigate if math instruction in business classes could improve student achievement in math. As supported by constructivist theory, the students in this study were encouraged to use prior knowledge and experiences to make new connections between math concepts and business applications.
    [Show full text]
  • Combinatorics
    Combinatorics Problem: How to count without counting. I How do you figure out how many things there are with a certain property without actually enumerating all of them. Sometimes this requires a lot of cleverness and deep mathematical insights. But there are some standard techniques. I That's what we'll be studying. We sometimes use the bijection rule without even realizing it: I count how many people voted are in favor of something by counting the number of hands raised: I I'm hoping that there's a bijection between the people in favor and the hands raised! Bijection Rule The Bijection Rule: If f : A ! B is a bijection, then jAj = jBj. I We used this rule in defining cardinality for infinite sets. I Now we'll focus on finite sets. Bijection Rule The Bijection Rule: If f : A ! B is a bijection, then jAj = jBj. I We used this rule in defining cardinality for infinite sets. I Now we'll focus on finite sets. We sometimes use the bijection rule without even realizing it: I count how many people voted are in favor of something by counting the number of hands raised: I I'm hoping that there's a bijection between the people in favor and the hands raised! Answer: 26 choices for the first letter, 26 for the second, 10 choices for the first number, the second number, and the third number: 262 × 103 = 676; 000 Example 2: A traveling salesman wants to do a tour of all 50 state capitals. How many ways can he do this? Answer: 50 choices for the first place to visit, 49 for the second, .
    [Show full text]