Fostering Children's Mathematical Power

An Investigative Approach To K-8 Mathematics Instruction

Arthur Baroody, Arthur J. Baroody, Jesse L.M. Wilkins, Ronald T. Coslick

• Publisher: Routledge
• Year: 1998
• eBook ISBN: 9781135674045
• Topic: Didattica
• Subtopic: Didattica generale

Teachers have the responsibility of helping all of their students construct the disposition and knowledge needed to live successfully in a complex and rapidly changing world. To meet the challenges of the 21st century, students will especially need mathematical power: a positive disposition toward mathematics (curiosity and self confidence), facility with the processes of mathematical inquiry (problem solving, reasoning and communicating), and well connected mathematical knowledge (an understanding of mathematical concepts, procedures and formulas). This guide seeks to help teachers achieve the capability to foster children's mathematical power - the ability to excite them about mathematics, help them see that it makes sense, and enable them to harness its might for solving everyday and extraordinary problems. The investigative approach attempts to foster mathematical power by making mathematics instruction process-based, understandable or relevant to the everyday life of students. Past efforts to reform mathematics instruction have focused on only one or two of these aims, whereas the investigative approach accomplishes all three. By teaching content in a purposeful context, an inquiry-based fashion, and a meaningful manner, this approach promotes chilren's mathematical learning in an interesting, thought-provoking and comprehensible way. This teaching guide is designed to help teachers appreciate the need for the investigative approach and to provide practical advice on how to make this approach happen in the classroom. It not only dispenses information, but also serves as a catalyst for exploring, conjecturing about, discussing and contemplating the teaching and learning of mathematics.

1	Fostering Mathematical Power: The Need for Purposeful, I Inquiry-Based, and Meaningful Instruction

☞ In writing, answer the following question: Why do we study mathematics?

✐The Why do we study mathematics? question can serve as the basis for a useful writing exercise. In addition to practicing language arts skills, such an exercise can provide teachers insight into their students’ view of the importance of mathematics and their reasons for studying it. Some children may think of mathematics as a tool for solving problems or for accomplishing everyday tasks. Others may see learning mathematics as a way of getting a grade. Yet others may see no purpose in studying it. The student essays could provide a basis for a class discussion, a bulletin board (as shown above), or both.

This Chapter

On the second day of class, Mr. Yant underscored the importance of mathematics by explaining, “Education is a journey in which you can acquire the tools to control more and more of your life. Mathematics is one of those tools. This tool becomes more and more useful by building an ever larger repertoire of concepts and strategies. This construction of mathematical knowledge occurs gradually through curiosity, desire, practice, and perseverance. After all, one does not become an accomplished athlete, musician, or artist overnight either.” Mr. Yant concluded by inviting his students to turn up their mathematical power. His students liked the idea of sharing in the power of mathematics.

Mathematical power implies the capacity to apply mathematical knowledge to new or unfamiliar tasks. This requires:

1. a positive disposition to learn and use mathematics (e.g., the self confidence and willingness to seek, evaluate, and apply quantitative and spatial information to solve problems and make decisions);
2. the ability to engage in the processes of mathematical inquiry (to explore, conjecture, reason logically, solve challenging problems, and communicate about and through mathematics); and
3. a deep understanding of mathematics (mathematical ideas that are well connected to other mathematical content, other subject areas, and everyday life).

Elementary-level instruction is crucial for laying a foundation for mathematical power. Experiences in these early grades shape and, in many cases, forever fix a child’s disposition toward learning and using mathematics. Early educational experiences mold and often cement habits of mathematical thinking. K-8 instruction can also help children construct a fundamental understanding of mathematical ideas needed to tackle more advanced mathematics and everyday tasks. Whether or not instruction fosters mathematical power depends on what mathematics is taught and, perhaps more importantly, on how mathematics is taught. Unfortunately, traditional instruction all too often leaves children mathematically powerless (e.g., Trafton & Shulte, 1989).

Along with chapters 0, 2, and 3, this chapter provides a general framework for the rest of the book. We examine different ways of thinking about mathematics education (Unit 1•1) and discuss a new way of teaching mathematics—an approach that can foster mathematical power (Unit 1•2). The chapter expands on the discussion of fostering a positive disposition toward mathematics begun in chapter 0. Chapter 2 will consider further the importance of focusing on the processes of mathematical inquiry such as problem solving; chapter 3, the importance of focusing on understanding. Chapters 4 through 16 will examine how the general framework can be applied to teaching specific content areas.

What the Nctm Standards Say

Founded in 1920, the National Council of Teachers of Mathematics (NCTM) is a professional association of teachers, administrators, teacher educators, and researchers dedicated to improving mathematics teaching and learning. A summary of the changes in content and emphasis suggested by NCTM (1989) are listed on pages 1–3 and 1–4.

1•1 Different Views of Mathematics Education

Summary of Changes in Content and Emphasis†

K-4 Mathematics

Increased Attention	Decreased Attention
Number Number sense Place-value concepts Meaning of fractions and decimals Estimation of quantities	Number Early attention to reading, writing, and ordering numbers symbolically
Operations and Computation Meaning of operations Operation sense Mental computation Estimation and the reasonableness of answers Selection of an appropriate computational method Use of calculators for complex computation Thinking strategies for basic facts	Operations and Computation Complex paper-and-pencil computations Isolated treatment of paper-and-pencil computations Addition and subtraction without renaming Isolated treatment of division facts Long division Long division without remainders Paper-and-pencil fraction computation Use of rounding to estimate
Geometrv and Measurement Properties of geometric figures Geometric relationships Spatial sense Process of measuring Concepts related to units of measurement Actual measuring Estimation of measurements Use of measurement and geometry ideas throughout the curriculum	Geometrv and Measurement Primary focus on naming geometric figures Memorization of equivalencies between units of measurement
Probability and Statistics Collection and organization of data Exploration of chance
Patterns and Relationships Pattern recognition and description Use of variables to express relationships
Problem Solving Word problems with a variety of structures Use of everyday problems Applications Study of patterns and relationships Problem-solving strategies	Problem Solving Use of clue words to determine which operation to use
Instructional Practices Use of manipulative materials Cooperative work Discussion of mathematics Questioning Justification of thinking Writing about mathematics Problem-solving [based] instruction Content integration Use of calculators and computers	Instructional Practices Rote practice Rote memorization of rules One answer and one method Use of worksheets Written practice Teaching by telling

† Reprinted from pages 20–21 and 70–73 of the Curriculum and Evaluations for School Mathematics, © 1989 by the NCTM, with the permission of the National Council of Teachers of Mathematics.

To get more information about the Standards, contact the NCTM at 703–620–9840 (extension 113), e-mail [email protected], or visit the NCTM website at http://www.nctm.org. NCTM documents and information can also be obtained through its fax service: 800–220–8483.

5–8 Mathematics

Increased Attention	Decreased Attention
Problem Solving. Reasoning, and Communicating Pursuing open-ended problems and extended problem solving projects Investigating and formulating questions from problem situations Representing situations verbally, numerically, graphically, geometrically, or symbolically Reasoning in spatial contexts Reasoning with proportions Reasoning from graphs Reasoning inductively and deductively Discussing, writing, reading, and listening to mathematical ideas	Problem Solving. Reasoning, and Communicating Practicing routine, one-step problems Practicing problems categorized by types (e.g., coin problems, age problems) Relying on outside authority (teacher or an answer key) Doing fill-in-the-blank worksheets Answering questions that require only yes, no, or a number as responses
Connections Connecting mathematics to other subjects and to the world outside the classroom Connecting topics within mathematics Applying mathematics	Connections Learning isolated topics Developing skills out of context
Number /Operations /Computation Developing number sense Developing operation sense Creating algorithms and procedures Using estimation both in solving problems and in checking the reasonableness of results Exploring relationships among representations of, and operations on, whole numbers, fractions, decimals, integers, and rational numbers Developing an understanding of ratio, proportion, and percent	Number/Operations/Computation Memorizing rules and algorithms Practicing tedious paper-and-pencil computations Finding exact forms of answers Memorizing procedures, such as cross-multiplication, without understanding Practicing rounding numbers out of context
Algebra. Patterns, and Functions Developing an understanding of variables, expressions, and equations Using a variety of methods to solve linear equations and informally investigate inequalities and nonlinear equations Identifying and using functional relationships Developing and using tables, graphs, and rules to describe situations Interpreting among different mathematical representations	Algebra. Patterns, and Functions Manipulating symbols Memorizing procedures and drilling on equation solving
Statistics and Probability Using statistical methods to describe, analyze, evaluate, and make decisions Creating experimental and theoretical models of situations involving probabilities	Statistics and Probability ♦ Memorizing formulas
Geometrv and Measurement Developing an understanding of geometric objects and relationships Using geometry and measurement to solve problems Estimating measurements	Geometrv and Measurement Memorizing geometric vocabulary Memorizing facts and relationships Memorizing and manipulating formulas Converting within and between measurement systems
Instructional Practices Actively involving students individually and in groups in exploring, conjecturing, analyzing, and applying mathematics in both a mathematical and a real-world context Using appropriate technology for computation and exploration Using concrete mate...

Table of contents

• Cover
• Title Page
• Copyright
• Acknowledgments
• Dedication
• Table of Contents
• Preface
• Chapter 0. Prologue: Understanding This Teaching Guide and the Role of Affect in the Teaching-Learning Process
• Chapter 1. Fostering Mathematical Power: The Need for Purposeful, Inquiry-Based, and Meaningful Instruction
• Chapter 2. Processes of Mathematical Inquiry: Problem Solving, Reasoning, and Communicating
• Chapter 3. Fostering and Evaluating Meaningful Learning: Making Connections and Assessing Understanding
• Chapter 4. Basic Mathematical Tools: Numbers and Numerals
• Chapter 5. Introducing Arithmetic: Understanding the Whole-Number Operations and Mastering the Basic Number Combinations
• Chapter 6. Understanding Base-Ten, Place-Value Skills: Reading, Writing, and Arithmetic with Multidigit Numbers
• Chapter 7. Thinking with Whole Numbers: Number Sense, Estimation, and Mental Computation
• Chapter 8. Exploring Numbers Further: Number Theory and Integers & Operations on Integers
• Chapter 9. Working with “Parts of a Whole” and Other Meanings of Rational Numbers and Common Fractions
• Chapter 10. Understanding Operations on Common Fractions
• Chapter 11. Place-Value Representations of Fractional Parts: Decimal Fractions, Decimals, and Operations on Decimals
• Chapter 12. Comparing Quantities Fairly: Ratios, Proportions, and Percent
• Chapter 13. Making Sense of Information and Using It to Make Everyday Decisions: Statistics and Probability
• Chapter 14. The Mathematics of Our Environment: Geometry and Spatial Sense
• Chapter 15. Sizing Up Things: Measurement and Measurement Formulas
• Chapter 16. The Transition from Arithmetic to Algebra: Prealgebra and Functions
• Chapter 17. Reflections on Teaching: Organizing Instruction to Enhance Mathematical Power, Professional Development, and Epilogue
• End Matter