The Nature and Role of Algebra in the K-14 Curriculum Proceedings of a National Symposium (1998) / Chapter Skim
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Appendix E: A Framework for Constructing a Vision of Algebra: A Discussion Document
Appendix E: A Framework for Constructing a Vision of Algebra: A Discussion Document
Pages 145-190
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Appendix E A Framework for Constructing a Vision of Algebra: A Discussion Document Working Draft This document has been adapted from the "Algebra in the K-12 Curriculum: Dilemmas and Possibilities," submitted in March 1995 by the Algebra Working Group to the National Council of Teachers of Mathematics. This document reflects the comments and suggestions on the original document by the National Council of Teachers of Mathematics Board of Directors and reviewers from the mathematics community.
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Copyright 1997 by The National Council of Teachers of Mathematics, Inc. 1906 Association Drive, Reston, VA 22091-1593 All rights reserved.
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Lindquist, Board Liaison National Council of Teachers of Mathematics Reston, Virginia 147 Elizabeth Phillips Michigan State University East Lansing, Michigan Merrie L Schroeder Price Laboratory School Cedar Falls, Iowa Jacqueline Stewart Okemos Public Schools Okemos, Michigan Lee V
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CONTENTS Preface Introduction A Promising Practice Critical Issues A Framework Building a Dynamic View of Algebra Embedding Algebraic Reasoning in Contextual Settings Bringing Coherence to the Algebra Curriculum Organizing Themes Summary Examples from Contextual Settings Bringing Meaning to the Framework Example 1: From the Contextual Setting of Growth and Change Example 2: Contextual Settings Within Size and Shape Example 3: Contextual Settings Within Number Using the Framework Bibliography 149 151 153 155 158 160 160 161 164 165 165 166 173 179 185 187
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This document entitled A Framework for Constructing a Vision of Algebra was presented to the National Council of Teachers of Mathematics Board of Directors in 1995 as a final report of the Working Group. It has been circulated since that time upon request to many groups and individuals interested in questions about school algebra.
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also has provided further elaboration and discussion of how a K-12 focus on algebra and algebraic thinking might be formulated. This document contributes further to the ongoing examination and work of shaping the school algebra curriculum, largely through a proposed framework for organizing discussion about algebra in the K-12 curriculum.
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What might such early introduction of key algebraic concepts and processes mean for the revision of the traditional secondary school algebra curriculum? Currently, there is a strong trend toward algebra for all in the nation' s eighth grades and secondary schools.
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Recipe A: 2 cups cranberry juice 3 cups water Recipe B: 4 cups cranberry juice 8 cups water Recipe C: 3 cups cranberry juice 5 cups water Recipe D: 1 cup cranberry juice 4 cups water Discussion: The interaction that ensued was very lively. Students interpreted the question of which recipe has the most cranberry juice in a variety of ways.
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A closer examination of the reasoning process reveals that students were also developing understandings of proportions, rates of change, and linear relationships. Recipe A: This group seemed to have the following organization in mind: Cups of Cranberry Juice Cups of Water Total Cups Punch in One Recipe 2 ?
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Their reasoning focused on the relationship between the cranberry juice (CJ) and the water (W)
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What are the implications of the differing views of school algebra? Should algebra be a course?
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The paper describes differing views of algebra, argues that algebra should be more than a course, and gives some examples that illustrate how algebra might look in a K-12 setting. These issues must be resolved by various groups as they participate in discussions and construct their own vision of algebra for all students in grades K-12.
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The Punch Problem provides a context for algebraic thinking, and it is important to have some organizing theme to develop this thinking across the grades. The development of algebra and algebraic reasoning is embedded in contextual settings; these contextual settings are connected by organizing themes that bring coherence to the curriculum.
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Bringing Coherence to the Algebra Curriculum Organizing Themes Contextual settings help bring meaning to important mathematical ideas. There is an intrinsic difficulty, however, in any attempt to build a mathematics curriculum out of students' experiences with a range of concrete contexts, in that such a curriculum may be perceived by students as essentially disconnected and not coherent.
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In high school or college, examples might include the study of the derivative of a function and its applications. The relationship between the amount of cranberry juice and sparkling water used to make punch is a direct proportionality, a special case of the larger family of linear functions.
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· How can these themes bring coherence to a K-12 curriculum? · Are some themes more appropriate at different grade levels?
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Summary The "Framework" proposes a way to develop algebraic reasoning by exploring a variety of contextual settings that are connected by organizing themes. By serving as organizers, themes help students recognize important ideas and make connections.
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Thus, the examples not only illustrate how the themes and contextual settings interact within a situation, they also reflect the kind of thinking that may occur in a particular grade level or across grade levels. While each problem is posed initially at one level, it connects to earlier and later levels.
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Eventually, it is necessary for students to be able to use all of these representations, from an informal and personally meaningful representation to the most economical, abstract, and symbolically powerful. The two problems in this section illustrate how a mathematical idea, in this case exponential growth, can be developed from elementary through high school, thus demonstrating one way to achieve coherence in a K-12 algebra curriculum.
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Discovering that the number of regions depends on the number of folds brings the theme offunctions to the forefront. As students reason about the patterns, they look for appropriate language and representation to capture their reasoning.
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Initially, it is sufficient to express the relationship found using words, models, pictures and either addition or multiplication sentences to represent exponential growth. While middle school students are studying a unit on growth and change or one on exponents, they might revisit the problem and attempt to generalize by extending the pattern beyond the possible number of actual folds.
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These questions also develop understandings of what it means to solve equations by observing the connections between the various representations. To find the number of regions for 20 folds,
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Comparing Two Patterns of Growth Students working with situations of exponential growth quickly see that quantities that grow exponentially get very large very rapidly. However, while this is generally an accurate description of exponential growth, it does not capture the essential mathematical qualities.
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As the students begin to make tables, they are able to describe the difference in the way the amount changes for each plan. They say things like: "In Plan A, the jumps in the money are always just $100.
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Students can contrast linear and exponential growth with quadratic and cubic, developing an understanding of the behavior of other functions and of the power of the exponential by considering rate of change as well as other important features. 9000 8000 7000 6000 o .> 8 43 o 5000 4000 3000 2000 1 000 o 1 l ~3 8 10 12 14 16 18 20 22 Age (years)
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Example 2: Contextual Settings Within Size and Shape Questions about size and shape confront people daily. How big is something?
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How many white border tiles are used? Can you make a square with 12 blue tiles?
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1 1 1 to To L] _ _ 4X2 + 4 Some students draw a square with six tiles per side and color in the middle: If w is the number of white tiles on the edge of the border, they express their rule as w2 - (w- 2~2 for the total number of white border tiles; (w- 2~2 for the blue tiles.
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Others may need to see this relation clearly laid out in a table. Figure Blue tiles Border tiles Total tiles 1 1 8 9 2 4 12 16 A discussion of the patterns in the table will bring out recursive thinking: If the number of white border tiles is 12, what happens next?
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Students who are facile with symbol manipulation attempt to show that their symbolic expressions are equivalent. Other students who are less facile with symbol manipulation use measurements from specific examples to verify the equivalence of expressions.
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Using the distributive property, the area of trapezoid ABCD can be written as the equivalent expression, 2 hubs + big. This expression leads some students to wonder if there is another way to cut and rearrange the trapezoid to get a triangle whose base is (b' + b2)
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Example 3: Contextual Settings Within Number Another setting people experience is that of number. People use number to order and categorize their world.
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Other organizing themes, however, are also present as students develop and apply their understanding. Understanding the distributive property allows students to · think about different characteristics of the same situation: · represent, generalize, and confirm conjectures; · gain new information about a situation from an equivalent representation.
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The following example illustrates how the distributive property provides a link between different ways of thinking. The Telephone Network Problem: There are many different houses in a particular region.
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Those who try to find a more direct rule may think very differently and use different symbolic representations. One link that shows these different ways of thinking are mathematically equivalent is the distributive property.
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has area x2 + My + y2 and reinforces use of the distributive property. x x + Y 2 my Y _ Vet The geometric model reinforces the need for the cross products fly.
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The distributive property also plays a role in operating on and understanding relationships in systems where symbols represent objects other than number. Scalar multiplication distributes over addition of matrices; if the dimensions are aligned, matrix multiplication distributes over matrix addition.
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· Is there a hierarchy to the themes? · What contextual settings can be fruitful grounds for exploring algebraic concepts?
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186 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM understand the reasons for and nature of a different view of algebra. Reconfiguring algebra as a K-12 endeavor will take time, commitment, and deep thinking on the part of the entire mathematics education community to make a successful algebraic experience a reality for all students.
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(1993~. "The linear algebra curriculum study group recommendations for the first course in linear algebra." The College Mathematics Journal, 24, 4-46.
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(1989~. "School algebra for the year 2000," in S
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National Council of Teachers of Mathematics (1980~. An Agenda for Action: Recommendations for School Mathematics of the 1980s.
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(1988~. "Conceptions of school algebra and uses of variables," in A.F.
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