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:
- 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);
- 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
- 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
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Operations and Computation | 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
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Geometrv and Measurement | Geometrv and Measurement - Primary focus on naming geometric figures
- Memorization of equivalencies between units of measurement
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Probability and Statistics - Collection and organization of data
Exploration of chance | |
Patterns and Relationships | |
Problem Solving | Problem Solving - Use of clue words to determine which operation to use
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Instructional Practices | Instructional Practices - Rote practice
- Rote memorization of rules
- One answer and one method
- Use of worksheets
- Written practice
- Teaching by telling
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† 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 | 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
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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
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Number /Operations /Computation | 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
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Algebra. Patterns, and Functions | Algebra. Patterns, and Functions - Manipulating symbols
- Memorizing procedures and drilling on equation solving
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Statistics and Probability | Statistics and Probability |
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
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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...
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