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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. Permission to photocopy materials from this document is granted to individuals and groups who want to use it for discussion purposes. May, 1997 145
Copyright 1997 by The National Council of Teachers of Mathematics, Inc. 1906 Association Drive, Reston, VA 22091-1593 All rights reserved. Printed in the United States of America
NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS THE ALGEBRA WORKING GROUP Gail F. Burrill University of Wisconsin-Madison Madison, Wisconsin Jonathan Choate Groton School Groton, Massachusetts Joan Ferrini-Mundy University of New Hampshire Durham, New Hampshire Steven Monk University of Washington Seattle, Washington Beatrice Moore-Harris Fort Worth Public Schools Fort Worth, Texas Mary M. 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. Stiff North Carolina State University Raleigh, North Carolina Erna Yackel Purdue University-Calumet Hammond, Indiana
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
PREFACE In 1994, the National Council of Teachers of Mathematics Board of Directors created an Algebra Working Group and charged it to produce a document that: Expands the vision of algebra for all that begins with experiences in early elementary school and extends through secondary school; elaborates this vision by including example, practical ideas, and promising practices, and helps school systems raise questions about the process of change. The Working Group met in the summer and fall of 1994 and developed a draft document that was circulated broadly within the mathematics, mathematics education, and school community for review and comment. 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. In anticipation of the May, 1997 National Council of Teachers of Mathematics (NCTM)/Mathematical Sciences Education Board (MSEB) symposium on the nature and role of algebra in the K-14 curriculum, the document has been revised and updated to serve as a discussion and background document for the symposium. Gail Burrill Joan Ferrini-Mundy Algebra Working Group Members May, 1997 151
INTRODUCTION The release of the National Council of Teachers of Mathematics' Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989) marked a new era in K-12 mathematics education. The Curriculum Standards call for a rethinking of the mathematical goals and emphases of school mathematics. In particular, the document outlines ways in which the subject matter of algebra can be organized as a strand occurring throughout the K-12 grade span, rather than confining algebra to the typical two courses in high school. The Patterns and Relationships standard for grades K-4, for example, calls for students to · recognize, describe, extend, and create a wide variety of patterns; · represent and describe mathematical relationships; and · explore the use of variable and open sentences to express relationships (NCTM, 1989, p. 60~. The standards for grades 5-8 include a standard called Patterns and Functions and an Algebra standard. For grades 9-12, there is an Algebra and Functions standard as well as a Mathematical Structure standard. Taken as a whole, these five standards offer one sketch of a K-12 algebra strand. Curriculum developers, textbook authors, and others have elaborated such conceptualizations of K-12 algebra in their publications and materials. The NCTM through its publications (Algebra for Everyone; the Addenda Series for grades K-6, Making Sense of Data and Patterns; for grades 5-8: Dealing with Data and Change and Patterns and Functions; for grades 9-12: Algebra in a Technological World and Data Analysis and Statistics across the Curriculum; and the February, 1997 special issues of Teaching Children Mathematics, Mathematics in the Middle Grades, and the Mathematics Teacher) 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. We also offer extended examples of how algebraic reasoning might be developed and encouraged across the grades. A number of current pressures contribute to the need for ongoing examination of the algebra dimension of the school curriculum. Problems in the workplace, in industry, and in everyday life involve algebraic concepts. Fundamental mathematical ideas in the areas of growth and change, patterns and regularity, quantity, size, shape, and data are often expressed with the tools and symbols of algebra. Increasingly sophisticated technology opens a wide range of possibilities of rethinking the emphases that have been traditional in school algebra, and raises a set of serious questions. Fundamental issues about the type and amount of symbol manipulation and procedural activity that is appropriate for students can now be examined and debated within the context of heretofore unavailable technological tools. New conceptualizations of "symbol 153
154 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM sense" (Arcavi, 1994) and "function sense" (Eisenberg, 1992) have emerged within the general discussion of algebra teaching and learning. In addition to considering how the available technologies might be used to help students understand the concepts of algebra and the procedures of algebra, there is now the dual question of how these technologies might themselves necessitate changed emphases and new additions to the content of school algebra. Research and practice provide compelling evidence that children engage in significant mathematical reasoning at early ages, and that algebraic thinking can be nurtured and encouraged early in the curriculum (Bastable & Schifter, in preparation; Kaput, in preparation). How can the school algebra curriculum be formulated to develop across the grades and to capitalize on these understandings? 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. Yet, various sources of evidence indicate that, for many students, their experiences with algebra in middle and secondary schools are not leading to high levels of understanding or proficiency (Beaton et al., 1996; Reese et al., 1997~. Compounding the situation, curriculum and instructional materials currently available provide a wide, and sometimes confusing, array of distinct possibilities of how the algebra curriculum might be organized. The context and climate around algebra as a K-12 element of the mathematics curriculum is ready for discussion. The framework and examples that follow are intended as a contribution to the process of continuing deep discussion about this important area of mathematics education.
A PROMISING PRACTICE Many people assume that "algebra" means working with symbols, but in recent years there has been a great deal of discussion among the mathematics education research community on reasoning that promotes understand- ing of important algebraic concepts at early levels (Bastable & Schifter, 1997; Confrey, 1995; Confrey & Smith, 1995; Harel & Confrey, 1994; Thompson, 1995~. The following discussion parallels an episode that occurred mid- year in a heterogeneous sixth grade class.* The students were studying a unit on rational numbers, and the intent of the problem was to develop understanding of and methods for comparing ratios. The problem illustrates how students' reasoning can be used to develop understanding of algebraic ideas. Description of the Problem: The school is hosting a lunch for the senior citizens in the area, and a class is asked to test four different recipes for mixing punch made from sparkling water and cranberry juice. First, they have to decide which of the four has the most cranberry juice, and then determine how many cups of juice and water are needed to mix 120 cups of punch. 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. Some looked at the absolute number of cups of juice in each recipe. Others looked at the ratio of juice to water. Still others used the part-whole relationship of the number of cups of juice to the total number of cups in the recipe. After comparing methods and discussing choices, the class split into four groups to decide how to adapt each recipe to make enough punch for 120 cups. The groups reasoned as follows. *This vignetteisbasedon an episode from a sixth grade class taughtbyMaryBouck, who was piloting materials from the Connected Mathematics Project (CMP), a National Science Foundation funded middle school mathematics curriculum project (Fey et al., 1995). 155
156 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM Group With Recipe A: We thought each recipe would make five cups: 2 of juice and 3 of water. So to make 120 cups, it would take 120 divided by 5 or 24 batches of our recipe. To get the number of cups of juice, multiply 2 by 24 to get 48 cups of juice. To get the number of cups of water, multiply 3 by 24 to get 72 cups of water. So 48 cups of juice plus 72 cups of water gives 120 cups of punch, and it is still the same recipe because the ratio of juice to water is still 2 to 3 since 48/72 = 2/3. Group With Recipe B: We saw there were twice as many cups of water as juice, so we divided 120 cups into three equal parts: 40 + 40 + 40 cups. Since there is twice as much water, two of the parts must be water. So there are 40 + 40 or 80 cups of water and then 40 cups of juice. This gives 120 cups of punch, and the ratio of juice to water is still 1/2. Group With Recipe C: It took us a long time. We tried to double the recipe, and this was not enough. We added another batch, and this still did not get us 120 cups. We finally figured out a pattern: juice/water = 3/5 for 8 cups, 6/10 for 16 cups,9/15 for 24 cups, 12/20 for 32 cups.... We kept on with the pattern until we got to 120 cups it is 45/75. We need 45 cups of juice and 75 cups of water to make 120 cups of punch, and the ratio will still be 3/5. (Later, in class discussion, this group observed they could have gone directly from 3/5 to 45/70 by multiplying the "top and bottom" by 15, because they needed to use the recipe 15 times.) Group With Recipe D: We tried various numbers. First we tried 20 cups of juice. This meant we needed 4 times as much water or 80 cups of water. But this only gave us 100 cups of punch. So we tried 30 cups of juice, which needed 4 times as much water or 120 cups, but this was too much punch. We then tried 25 cups of juice. This needed 100 cups of water, but this made 125 cups of punch, which is close but too much. So we tried 24 cups of juice, which needed 96 cups of water. This worked! We had 120 cups of punch, and the ratio of juice to water is 1 to 4. (One of the students noticed that the 125 cups contained one too many batches of the recipe, so she subtracted one batch, that is she took away one cup of juice and 4 cups of water. She came up with the same amount as the rest of the group 24 cups of juice and 96 cups of water.) During the discussion directed by the teacher, the class thought about the similarities and differences among the Some concluded that the number of each recipe needed for 120 cups of punch could strategies used by each group. be found by using the multiple necessary to obtain an equivalent ratio with a denominator of 120. Where Is the Algebraic Reasoning? r . CUpS OI JUlCe total cups of punch While the students did not use symbols, they did develop their own representational schemes to organize their thinking and build a sense of variable. 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 ? 5 ? 120 By reasoning about quantities, they took any number of recipes, R. and found the total number of cups of punch produced, Tp, using Tp = 5R (a linear equation). For every increase of one recipe, there will be 5 cups more punch (a constant rate of change). They also used this idea to reason about the amount of the individual ingredients; the
APPENDIX E 157 total number of cups of cranberry juice, Ted, is equal to the number of cups of cranberry juice in a recipe times the number of recipes, R. Thus, Tear= 2R. Similarly, TWa~Cer= 3R, where T a- is the total number of cups of water. waler Recipe B: This group also appeared to be reasoning about quantities. They thought of the whole as having three equal parts. Their reasoning focused on the relationship between the cranberry juice (CJ) and the water (W). SO 120 = two parts W+ one part CJ. Recipe C: This group created a pattern using pairs of values (or ratios) of cups of cranberry juice, Ted, and cups of water, TWa~,er: (3, 5), (6, 10), (9, 15), ..., (Ted, TWa~Cer). They inspected the pairs to determine when TO + TWa~er = 120. (The students solved the equation by reasoning about how the two quantities were related, knowing that TCI TWaier = 3/5~) Recipe D: This group used trial and error to reason about their problem under the constraints: TCJ + TWater = 120 and Twater = 4TCJ TcJ (Total cups of CJ) TWater (Total cups of water) Tp (Total cups of punch) 20 30 25 24 80 120 100 96 100 (too little) 150 (too much) 125 (too much) 120 (right amount) The students in this class were engaged with a problem that was posed in a context that made sense to them. Allowed to explore the problem on their own, students found several different ways to reason about the problem and the underlying ideas of rate of change, linear relations, and proportional reasoning. In situations such as this it is critical, however, that the teacher recognize and understand the concepts being developed, not, at this level, to teach the abstract manipulation, but to help make appropriate decisions about the conversation and follow-up activities, to focus and direct the discussion in useful ways when the opportunity arises, to use the words that position the algebraic concept (each extra recipe produces 5 CUPS more punch), and to leave room in students' minds for extending their understandings. How can similar kinds of experiences that foster algebraic reasoning be created at all grade levels? How can such experiences contribute to an understanding of algebra in grades K through 12? What knowledge of mathematics and pedagogy does a teacher need to implement rich experiences in algebraic reasoning?
CRITICAL ISSUES As different groups address the implications for change, there are debates about the nature, history, future goals, appropriate curriculum, assessment, and pedagogy of school algebra, as well as its relationship to the mathematical discipline called algebra. As these groups focus on research, design curriculum, prepare teachers, teach students, or educate the public, however, several critical issues arise: How can algebraic experiences be designed to promote success for all students? What is algebraic reasoning? Howis it related to other mathematical reasoning? What are the implications of the differing views of school algebra? Should algebra be a course? How can coherence be brought to a K-12 algebraic experience? What is the role of symbols in a K-12 algebraic experience? What is the relation between the development of conceptual understanding and acquisition of algebraic skills? How much symbol manipulation is needed to develop symbol sense? · What knowledge about algebra is needed to use technology effectively? How does technology enhance algebraic reasoning? What other tools can help? · How will algebraic reasoning and algebraic understanding be assessed? · What teacher preparation is needed to implement a coherent algebraic experience in a K-12 curriculum? · What are the implications of state-mandated algebra courses? · What is the relationship between school algebra and research areas of mathematics? While these issues are not meant to be exhaustive, they are representative of those that commonly arise when educators consider the nature of algebra and what it should encompass. Some of these issues have been discussed in earlier works of the Council, in particular, Research Issues in the Learning and Teaching of Algebra (1989) and the 1988 yearbook, The Ideas of Algebra K-12. Many of the issues are common to mathematics in general as well as to algebra. Changes in the algebra curriculum cannot take place in isolation from the rest of mathematics, and in fact, rethinking algebra is only one phase of rethinking the entire K-12 mathematics sequence. If the suggestion that algebra is a K-12 enterprise is realized, the results will force a reconceptualization of the scope and sequence of the entire mathematics curriculum. A rich K-8 algebra experience will be of little value if students are placed in an algebra course that ignores the knowledge and understanding they bring from such a background. These changes can, in fact, be observed in the large-scale curriculum projects funded by the National Science Foundation. Some of the issues have been the object of prior research and investigation: pedagogy (Rachlin,1982~; interpreting graphs (Russell et al., 1995~; quantitative reasoning (Thompson, 1995; Tierney & Monk, in press); formal symbolism (van Reeuwijk, in press). Other issues, particularly the relationship between the development of conceptual understand 158
APPENDIX E 159 ing and acquisition of algebraic skills, are part of a much needed research program. It is not the intent of this paper to address each of the issues, but rather to provide a framework for thinking about algebra that can be used as educators wrestle with their own responses to the issues. 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.
A FRAMEWORK BUILDING A DYNAMIC VIEW OF ALGEBRA Algebra has its roots in a variety of human activities, from commerce to the study of numbers, to the elaboration of other branches of mathematics. The centrality and importance of algebra in mathematics reflects its power to express compactly and efficiently mathematical relationships within these activities and to reveal structures and patterns common to them. The usefulness and vitality of algebra are generally seen as arising from its concepts and its patterns of reasoning as they have been adapted in the attempt to solve problems in ongoing human activities. A framework for thinking about algebra should capture the notion that "algebra is a symbol system of unparalleled power for communicating quantitative information and relationships; it is a training ground in careful rule-governed reasoning; its development is a significant thread in the history of mathematics; and its theoretical structure is based on concepts and principles that generalize to provide organizing schema in nearly every other branch of mathemat- ics" (Fey, 1989, p. 207~. Such a framework might be thought about in two parts: contextual settings and organizing themes. 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. Embedding Algebraic Reasoning in Contextual Settings Historically, algebra was primarily the domain of an elite where only the privileged took part. It essentially consisted of solving equations using problems that were artificial and until the 18th century that were based primarily on other topics in mathematics, such as geometry (Katz, 1995~. A course called algebra gradually worked its way into the high school curriculum during the 19th century. And, until recently, except for a brief time during the era of "new math," algebra was essentially a course or two composed of learning skills needed to solve equations, manipulating symbolic expressions, and solving "word problems." This course was often preceded by "pre algebra," a course that focused on honing students' arithmetic skills in some simple algebraic situations. Views about what algebra should encompass that went beyond this perspective were expressed by some prior to the 1990's, and the advent of technology and symbol manipulation systems forced many others to reexamine the nature of what was called algebra. Common to these views is the belief that the core meaning of algebra extends beyond the routines and procedures of manipulating symbols. Among the important goals of the NCTM Standards is that mathematics be viewed as problem solving, reasoning, communication, and making connections. Since these thinking and reasoning processes must be about something, students' understanding of concepts should be developed through activities embedded in specific contextual settings. Contextual settings such as growth and change, size and shape, data and uncertainty, number, and patterns are productive sources to focus mathematical thinking. In particular, this collection of contextual 160
APPENDIX E 161 settings, similar to those used in On the Shoulders of Giants (NRC, 1990), provides an opportunity for students to learn and to use algebra as well as make connections between algebra and other disciplines. For example, the study of population growth is a context within the setting of growth and change; finding the number of buses used to transport a large group to a fair is a context within the setting of number; and deciding how to establish a formula to rank athletes in a given sport is a context within the setting of data. In such settings, the context can give meaning to the representation and allow students to recreate the meaning when necessary for understanding. These contextual settings also help students organize and interact with the world around them: ideas like growth and change can be realized in contexts as varied as a growing bacteria colony, a moving car, or expanding a recipe for punch. The commonality among these situations is how change in one quantity is related to change in another. It is critical, however, to respect the nature and structure of the context and the relationship to the underlying mathematics. Contextual settings can be based on applications, mathematics itself, or mathematics from a historical perspective. Describing the relationship between the zeros of a polynomial, its factors, and its graph is an exploration within mathematics. Generating an algebraic analog of the geometric reasoning and conclusions made by early mathematicians such as Fibonnaci can be fascinating for many students. Trisecting an angle or construct- ing a circle with a given area using a straight edge and compass are two historical geometry problems that led to a need to find solutions to polynomial equations. Students build concepts and develop ways to think in pursuit of activities that engage them in different contextual settings; such settings help students make sense of the algebra they are studying. 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. If a student is deeply immersed in one stimulating contextual problem and then another, this student may not see ideas common to both problems. Even though teachers or curriculum designers might see particular thinking patterns or concepts common to a range of problems, there is a danger that students might not see any of these commonalities because of the very richness of the problems themselves. In the promising practice problem involving recipes for punch, students were thinking about relationships between quantities. Yet thinking about unit rate, proportionality, or linear functions can connect this problem to other problems that at first appear to be of a very different nature. It is the ability to generalize, extend, apply, and connect ideas throughout many different situations and across several grades that enables students to make sense of mathematics and to think of it as a set of unifying ideas rather than a set of disjointed, unrelated problems. One of the ways in which the various efforts at reforming algebra differ most strongly is in the organizing themes or conceptual organizers that have been selected to give coherence to the subject by those developing curricula, implementing curriculum, or conducting research. As school algebra evolved over time, the themes used to organize the subject have changed. The "new math" of the 60's was based on structure. This was followed by a "back to the basics" movement, based heavily on thinking of algebra as a language. In the 90's, with the emergence of the graphing calculator, the concepts of function and modeling have become central points for conceptualizing algebra. To be effective in helping students make connections and understand the big ideas, teachers and cur- riculum writers must allow a variety of themes to emerge. Some of the themes that have been often discussed and realized in written curricula are described below. · Organizing Theme: Functions and Relations The central mathematical concept of a relationship underlies the equations, tables, and graphs so common to algebra. Representing a relationship using the concept of variable provides organized ways to think about an enormous variety of mathematical settings. The study of functions as the focus of algebra has been advocated by Fey and Good (1985) and Yerushalmy and Schwartz (1991~. Functions can be expressed in virtually all of the representational systems found in algebra and can serve as the basis for exploring interesting problems. In studying population growth, students might represent the function with a table, graph, or symbols. In a situation involving a geometric transformation, students
162 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM might represent the transformation with matrices. Recursive definitions of functions, because of their importance in computer methods, play a role in problems in science and economics. Change and variation are important ideas within the concept of a function and can be realized in contexts throughout the curriculum. Examples such as what happens if a pizza must be shared among larger and larger groups of picnickers are appropriate for elementary students. 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. There are other situations where a relation describes the situation. Consider, for example, the Py- thagorean Theorem that relates the lengths of the sides of a right triangle or Euler's Formula that relates the number of vertices, the number of edges, and the number of faces of a geometric figure. In the elementary grades, students could look for number relationships in their class, such as "the number of noses and ears is equal to three times the number of students in the class." The ideas that naturally arise in the contexts noted above can stimulate valuable mathematical activities for students in virtually all grade levels. At the same time, these activities can all be seen as fitting within the development of a function-and-relationship theme. · Organizing Theme: Modeling Modeling as a theme for thinking about algebra is evident in the work of Freudenthal (1983) as well as others. Many complex phenomena can be modeled by relatively simple algebraic relationships. Con- versely, the phenomena can be seen as embodiments of the relationships, so that engaging in the complex processes of modeling the phenomena will lead to grappling with the relationships. In the early grades, students can investigate the relationship between the number of shoes and the number of people by using counters to represent the number of shoes. Reasoning with these quantities helps children find patterns and make predictions about the number of shoes and the number of people. Middle grade students can explore the relationship between the stretch in a bungee cord and the weight of the jumper using a rubber band and fishing weight. By varying the factors in the experiment and recording the results, students find that an underlying linear relationship is present. Older students can generate the symbolic form for the relationship, look at the limits of the situation compared with the model, and create a spreadsheet to compare the predictions with the actual results. In other situations, such as the amount of a prescription drug present in the body, recursion relation- ships can be used to model the amount of drug in the body, taking into consideration the dosage, metabolism rates, and other factors. In yet other situations, matrices or systems of equations can be used to model ecological networks or economic decisions. The power of a model lies in its accessibility and utility. A simple function, such ash = Ma - bx), might represent two entirely different situations: the area of rectangles with a fixed perimeter where P = 200 = 2L + 2W and A = L(100 - L) or the revenue collected when selling a product for which the demand depends on the price. If the demand is given by D = 200 - 3p, where p is the price, then the revenue is given by R =p(200 - 3p). Knowledge gained in one context can inform a totally different application of the same model. Viewing algebraic relations in terms of the phenomena they model is an effective way of giving life to the situations and bringing to the study of algebra the richness of experience all students carry with them. · Organizing Theme: Structure Through the efficient and compressed symbol systems of algebra, deep yet simple structures and patterns can be represented. It does not matter in which order a column of figures is added; it does matter in which order two numbers are subtracted. This is often a source of puzzlement to students in the early grades. By examining these phenomena across many number systems and many operations, the commu- tative property emerges as a way to capture the essential nature of the systems. One of the most puzzling features of matrix multiplication is that it is not a commutative operation, yet a characterization of the formal properties of matrix multiplication can be used to manage a wide variety of problems. For some, structure is the core of school algebra (Kiernan, 1989; Cuoco, in press), building on generalized arithmetic, the formulation and manipulation of general statements about numbers. This approach requires students to move away from context-dependent problems and to develop the ability to
APPENDIX E 163 generalize and follow procedures in a systematic way. Organizing algebra around structure implies thinking about how systems operate; for instance, what enabling characteristics of a system allow fractions to be combined or equations to be solved? The power of mathematics lies in its ability to abstract and generalize the common features from a system and then to apply and extend these ideas to other systems. The search for primes by looking at the decomposition of whole numbers reveals information about the structure of the real numbers. Understanding the structure of integers allows us to seek and recognize similar structures in other systems, such as the ring of polynomials: the set of integers and the set of polynomials have a similar algebraic structure. The study of structure is a means of focusing on common aspects of many mathematical situations; a consideration of structure allows us to look beneath the more superficial aspects of the particular notational system and beyond the concepts arising from particular contexts. . Organizing Theme: Language and Representation Algebra can be seen as a language with "dialects" of literal symbols, graphs, tables, words, diagrams, and other visual displays. Thinking about algebra as the "language of arithmetic" has been an ongoing part of the discussion (Usiskin, 1988~. Euler's 1757 Introduction to Algebra summarized all of the previous work on solving equations, but used no diagrams; his analysis consisted of the manipulation of numerical quantities. Although that view of algebra still permeated the majority of the content in algebra texts for most of the 1900's, the graphic capability of computers and calculators has opened the door to new and easily accessible geometric and tabular representations. The relation between y = x2 _ 4 and y = (x - 2~(x + 2) and the corresponding graphical interpretation enables students to picture concepts in new and enriching ways. A discussion of the information gained and lost as one moves between representations reinforces understanding of the concepts, while changing parameters and observing the impact add a new dimension to student understanding. Different representations provide students with the ability to express and reconfigure quantities in a wide spectrum of situations. A table of the number of students present in a third grade classroom over a month is a particular way of describing and analyzing the characteristics of that classroom. A graph showing the number of grains of rice given each day in the "King's Chessboard" story enables students to describe and analyze the characteristics of that interesting situation." The same idea lies behind the use of spreadsheets to analyze a number of numerical phenomena in more complex situations. Developing ease in setting up and manipulating graphical and tabular representations of data, regardless of the particular situation being represented, can be a source of considerable mathematical power for students. The same is true of the symbol system of literal variables, expressions, and equations. To represent a situation in familiar terms is often the gateway to understanding the situation. Thus, algebra is a way of thinking about and representing many situations. It has a language and a syntax, along with tools and procedures, that promote this thinking and modeling. By giving students opportunities and time to experience this way of thinking, a more powerful approach to algebra can be built throughout the K-12 curriculum. · What are the essential understandings of each theme? · How can these themes bring coherence to a K-12 curriculum? · Are some themes more appropriate at different grade levels? · Is there a hierarchy to the themes? Mathematics is powerful, and algebra is one of the important mathematical tools. No single organizing theme, however, captures the reality of what it means to "know and be able to do algebra." The theme of language and representation, for example, is familiar to many teachers, because it reflects their own school experiences. Its central feature is the use of symbols to translate and represent situations. Students can become fluent in the use of symbols in limited contexts, without realizing that they can be used to model diverse situations, such as maximizing the fin this story, the king rewards his faithful servant by fulfilling his request, which is the amount of rice that can be put on a chessboard in the following way: 1 grain on the first square, 2 grains on the second square, 4 grains on the third square, 8 grains on the fourth square, etc.
164 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM number of ships passing through the Suez Canal, determining the amount of medicine in one's bloodstream, or predicting an Olympic record time. In addition, the view of algebra as language and representation has been shifted and broadened by technology. It is now necessary not only to think in the language of algebraic representation but also to extend that language to communicate effectively with computers and calculators. In addition, students who have modeled specific problems, translating and representing important aspects of the problem, can be challenged to raise their understanding to a new level by considering specific strategies, justifying why the strategies work and under what conditions. Learning to ask questions about calculations as they occur across various problems or in isolation from context is an important aspect of algebra one that leads to a search for common structures. Thus, aspects of language, modeling, and structure complement each other. Similar reflections on each of the other organizing themes establish that no single theme is sufficient as a base for algebra in the future. These themes are not necessarily parallel or disjoint. However, taken as a collection, the organizing themes can support a dynamic evolution of school algebra. Such a collection provides teachers with multiple entry points into the ways children can think about algebra. In the Punch Problem, the point of entry could be ratio or rates (structure of rational numbers); reasoning about the same situation at a later grade level can lead to students' study of functions or representations. Using a variety of organizing themes will allow more insightful assessment of students' understandings and provide alternatives to help students use algebra. The particular themes highlighted above were chosen because they reflect diverse perspectives, yet have utility and promise for being relevant in many contextual settings. Other themes might be used, but no one organizing theme can generate an algebra curriculum that will adequately serve all students for a multitude of post-high-school options. Over time, multiple organizing themes should become part of a student's understanding of algebra. 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. Contextual settings are the ground on which these themes play out. They provide the substance from which and about which to reason. The settings and themes suggested above are only a sample and are not intended to be an exhaustive list. Such a framework for curriculum provides opportunities to cross traditional and arbitrary boundaries resulting in a stronger and more useful algebra experience. The school algebra curriculum must be reconsidered from the ground up, rather than just tinkering with the present curriculum. The challenge is to build a connected and coherent algebra strand by introducing important algebraic ideas at the appropriate grade level and point in the curriculum sequence and to build on this foundation throughout the rest of the K-12 curriculum. Algebra becomes manifest through exploration of the compelling ideas within the contextual settings, while the themes bring a logic and organization to the discipline.
EXAMPLES FROM CONTEXTUAL SETTINGS Bringing Meaning to the Framework The Framework, around which to organize thinking and discussion related to the teaching and learning of algebra, can best be understood by studying examples that bring its meaning into focus. Time spent moving between the general framework and particular examples can result in fruitful deliberations about the intersection of the organizing themes and contextual settings in terms of curriculum, students and their [earnings, and teachers and their practice. The discussions are intended to promote continued thinking about the nature of algebra and its role in the K-12 mathematics curriculum. Each of the following examples was selected to illustrate development of algebraic concepts and reasoning within a particular contextual setting. The problems in the examples show that algebraic reasoning can be developed from the exploration of compelling ideas about our world as well as in mathematics itself. The examples are not in any way meant as a complete catalogue of algebraic topics or as models to be transported directly into the classroom. Rather, they represent important contextual settings for understanding the world around us, generating necessary discussion, and leading eventually to generalization of the underlying concepts. In these examples, ideas of measurement (within the setting of shape and size), exponential growth (within the setting of growth and change), and the distributive property (within the setting of number) are explored. In some examples, one theme may dominate, while in others, several themes may play parallel roles. Some of the problem situations emphasize algebra as a language for the expression and communication of ideas, while in others, algebra as structure helps to further develop understanding of the problem. In still other problems, functions, relations, and modeling are equally prominent. These themes overlap and enhance each other. A broad view of algebra embedded across grades is needed to make decisions about curriculum and to guide the development of students' reasoning. 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. When abstract symbols are used, they are not intended for all levels in many cases they are meant as a possible generalization at a later stage. The reader is expected to play an active role in deciding what age and experience level is appropriate. To keep the discussion moving, some activities are presented briefly. While reading the examples, it is important to realize that in the classroom: Students need time to · explore problems; · grapple with some uncertainty about ways to solve the problem; 165
166 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM · develop representation systems, even nonhelpful ones, as an important first step in appreciating the power of representation; · discuss and communicate their reasoning as a means of helping their thinking. Teachers need to · assess prior knowledge and experience (or if the activity was part of a curriculum, then some prior knowledge and experience could be assumed); · facilitate the classroom exploration and discourse; provide a balanced and appropriate pedagogy; build on experiences and turn difficulties into new perspectives and misconceptions into insights. The paragraphs with the heading, Where Is the Algebra?, discuss how the example fosters algebra as a system of concepts and a way of thinking. Such paragraphs show how the example might evoke algebraic reasoning and describe how this relates to aspects of algebra familiar to readers. Questions in boxes help readers reflect on concepts, connections, understandings, reasonings, pedagogies, and the choice of context illustrated by each problem. Some questions will also raise issues concerning balance, coherence, use of themes or contexts, or contrasting views. These questions can serve as a basis for discussions about the teaching and learning of algebra. The reader is further challenged to participate in the discussion by reflecting on the following more general questions while reading the examples. What organizing themes are prominent? What other contexts or settings could be used to develop the same ideas? What other algebraic concepts and reasoning could be developed using this example? · Can a similar idea be used at another level of understanding? . What issues arise? How might they be addressed? Example 1: From the Contextual Setting of Growth and Change Phenomena of growth and change are pervasive in our daily lives. Students bring a wealth of experience, intuition, and curiosity to such everyday events as their own physical growth, the movement of objects in space, patterns of successive change such as repeated doubling, and the growth of a bank account balance with compound interest. Understanding the relationship between two quantities, and in particular how one changes with respect to the other, is critical in areas such as economics, biology, and politics. For example, how fast is the population growing? What is the pattern of growth of the national debt in relation to the size of the population? How does the rate of radioactive decay of certain materials affect our environment? These are important contexts for the development of mathematical ideas. Furthermore, the mathematics developed within these contextual settings can help people understand and make informed decisions about these events. Situations of growth and change can be represented in a variety of ways: through tables, formulas, graphs, and various types of visual display. Technology has advanced the use of multiple representations, allowing students to investigate the relations among the various representations and to develop a broader understanding of growth and change in its different forms. Each representation can be regarded as a kind of symbol system that has its own particular form of mathematical power for expressing ideas. 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. The problems, money, and folding paper, also demonstrate the diversity of situations in which a single mathematical idea can be realized. In addition, the themes, functions and relations, modeling, and language and representation are intrinsically involved in working through the problems. Understanding Growth Using Multiple Representations Informed decisions about phenomena, such as bacteria growth or the national debt, require an understanding of
APPENDIX E 167 the essential elements of change. An important part of this understanding occurs when students recognize and represent the fundamental patterns of change. The following problem can begin in early elementary grades and be extended throughout the grades by adjusting the context or the nature of the questions that are asked. The original problem might be posed when students are studying the even/odd property of numbers in the context of folding a piece of paper to form regions. 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. The Paper Folding Problem: Suppose you want to fold a sheet of paper in half repeatedly to create a region for the name of each person in the class. Only one name may be written in each region there may be some blank regions. How many folds do you need to make? Questions to ask: What did the fold line (crease) do to the paper? How many folds did you make? How many regions resulted? How many regions did you have when you had no folds? One fold? Two folds? Three folds? Draw a picture of the paper after each fold. As you go from one fold to the next, how is the number of regions increasing? Does the number of regions increase by the same amount each time? Looking into the classroom: During a preliminary discussion of the problem, one student notices that there is only one region before the folding starts. Other students comment on how the folding is to be done. Number of Folds The students make a table of the number of folds, the number 0 of regions for each piece of paper, 1 and a sketch of the paper after each fold. 2 Number of Regions Model 1 2 4 8 m ~ I l l l l ~I' ' I Looking at their recorded data, students describe the patterns in different ways. One student notices that, after the first, all the numbers are even, but some even numbers, such as 6 and 10, do not occur. Others claim that the number of regions is growing fast. Some notice that the number of regions created doubles each time you add a fold the number of regions is twice the number of the previous regions. If you fold the paper in half, it cuts every region in half so you have twice as many. Some say that you add 1, then add 2, then add 4, etc. Where Is the Algebra? From an organized display of the folded pieces of paper, students can reason about the growth patterns in different ways. Some think in terms of doubling, while others think in terms of adding.
168 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM This can lead to a discussion of how these operations are related. Convincing others, for instance, that doubling a number is the same as adding the number to itself can be an important experience in reasoning about numbers. The pattern of repeated multiplication by a given number is a central one in algebra and underlies exponential growth. Finding the number of new regions by multiplying the number of the previous regions by the constant 2 is an example of a recursive process. Students can use a calculator to extend the pattern recursively, and to investigate the increase in the number of regions, from one fold to the next. At the upper elementary level, students can begin to use the terminology of power and exponent in a natural way, moving towards the theme of language and representation. Most students will recognize that there are a limited number of folds possible with any size paper. This leads to a conversation about the relation between the pattern and the physical situation. Students might be encouraged to find other situations that have the same pattern but that extend beyond the possible outcomes of folding a piece of paper. As the discussion continues students are able to focus on the salient features of exponential growth, where the growth pattern is generated by the product of a constant and the previous result. 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. · How does changing the question from counting the number of regions to analyzing how the number of regions is growing promote an understanding of exponential growth? · What are the variables in this situation? What are some of the underlying relationships that students might observe and reason about? · What do student responses indicate about their early understanding of the structure of the number system? Paper Folding Problem Extended: Suppose you have a piece of paper that can be folded indefinitely. If you could fold the paper 10 times, how many regions would be formed? Will you have enough regions to write down names for everyone in the school? Questions to ask: Suppose you could fold the paper 20 times, how many regions are formed? What about 100 times? How many folds will it take to produce 64 regions? 2,048 regions? How about 100 regions? 200 regions? Looking into the classroom: Since it is impossible to literally continue the folding process beyond a certain number of folds, students look for another way to reason about the growth patterns. Some students extend the table, recursively. Some begin to reason as follows: for 0 folds there is 1 region, for 1 fold there are 2 regions, for 2 folds there are 4 regions or 2 x 2 regions, for 3 folds there are 8 regions or 2 x 2 x 2. So for 10 folds you multiply 2 by itself 10 times. In response to the second question, some students suggest continuing the table, while others describe the process of multiplying by 2's 20 times. The number of times 2 is used as a factor is equal to the number of folds. For 100 folds, 2 x 2 x ... x 2 100 twos. Some students use tree-like diagrams to show the multiplica- tive structure. To find the number of folds needed to produce 64 regions, some students continue to fold the paper; some students extend the table. Others begin to reverse their reasoning patterns for the number of regions: "I got 64 regions by multiplying 2's together, and the number of 2's is the same as the number of folds. So I must divide by 2's until I get to one. It will take 6 folds. For 2,048 regions, it will take 11 folds." To find the number of folds necessary to find 100 regions, some students suggest using a table and observe that 100 is not an entry. It would have to be between 6 and 7 folds, which is impossible. Other
APPENDIX E 169 students use a graph to find the number of folds. Others reason with the powers of 2: 26 = 64 and 27 = 128. So it is impossible to find the number of folds using the concrete model for"folding in half." Thus, it is impossible to produce exactly 100 regions by repeatedly folding a piece of paper in half. How does extending the activity beyond the limits of physicalfolding demonstrate an important aspect of modeling? Where Is the Algebra? Some students are ready to discuss patterns in the number of regions without making the actual folds. Others may feel that the certainty of counting is worth more than the labor of repeated folding. At some point, however, most agree that the exploration should be continued without concrete actions. This can be an important first step toward algebraic reasoning about patterns. Noticing, describing, and recording a pattern can lead to two other actions understanding the mathematics in the pattern enough to make generalizations and using a common mathematical language to describe the situation. Such reasoning also provides understanding and appreciation for a mathematical model. At this stage, students may use various ways to describe their reasoning. Some use tables. Number of Number of A Shorthand Folds Re~ons Model V ersi ~ ~.. . ~ O 1 ~20 1 m 21=2 1 2 1~2 2 4 lE~x2 22=4 3 8 EE] 23 ~x2x2x2 ~ Some use trees; the opportunity to generalize with words can be a basis for generalizing to a symbolic language at a later stage: 20 times R = 2 x 2 x 2 x 2 x . . . x 2 or R = 2t, where R = number of regions, and L = number of lines. 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,
170 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM students can compute 220 or use the table. To find the number of folds it would take to find 64 regions, students are solving for L in the equation, 2t = 64. Students can reverse their reasoning process or use the table. Some students will use a graph to find the number of folds it would take to produce 64 regions or 2,048 regions. _ _ ~ ~ ~ _ _ A graphing calculator representation of y = 2X. If students use a graphing calculator to produce a graph of y= 2X, they may observe that the calculator plots points for non-intener values of x. Students may be asked to consider the meaning of such points and real- ize that it is possible to fold paper only a whole number of times. Such ex- ploration can encourage deeper thinking about concepts of range and domain and about the limits inherent in any algebraic model of a given situ- ation. Students should make connections among the entries in the table, points on a graph, and the values of a function. Students can consider how the rule y = 2x models the situation and the limitations the physical situation imposes. They can think about a way to give meaning to the function for values other than positive integers. They can reason from the input to the output, or in reverse, determine the input that leads to a given output. Asking for the number of folds that would produce 100 or 200 regions also highlights an important characteristic of exponential functions that the changes between outputs grow larger and larger. In this case, the process misses many numbers that might be expected to be outputs, setting the stage for logarithms. Students can focus on the sequence formed, its defining characteristics, how to recognize such a sequence, and how to generate the elements either recursively or for any given term using a closed form. Students can explore similar situations looking for generalizations across the situations. For example, they might think about a genealogy question, such as the number of parents or number of grandparents, and its connection to the problem. A parallel development for exponen- tial decay can be made concentrating on the area of the regions obtained by folding paper, leading to a decreasing exponential pattern. · How does this situation promote the habit of looking for generalizations in patterns? · What are some other ways students might represent their reasoning? · How can students determine if different forms of representation are equivalent? · How is thegrowth rate revealed in the various representations? Whatdoes each representation add to the understanding of exponential growth? · What understandings about functions are essentialfordevelopingthe concepts of growth and change? 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. These can best be described in terms of concepts that emerge in the theme of Unctions and relations dealing with the increment of change of related variables. One way students can
APPENDIX E 171 begin to appreciate the different kinds of change is to compare two situations, one in which a variable is changing exponentially and the other in which a variable is changing linearly. In the next example, a problem appropriate for middle school students is discussed. The problem is from the setting of growth and change the context is money. The Gift Problem: Janine's uncle has decided to give her an amount of money every year. He told her that she could have the money in one of two different ways and asked which way she prefers. The two ways he suggests are: Plan A: She receives $1 ,000 on her ninth birthday; $1 ,100 on the birthday after that; $1 ,200 on the birthday after that, and so forth. That is, she starts with a gift of $1 ,000 and then the gift she gets increases by $100 every year. Plan B: She receives $1 on her ninth birthday. (That's right!) Then on the following birthday she receives $2, and on the next it will be $4, and on the next after that it will be $8, and so forth. That is, she starts with a gift of $1 and every year the gift is doubled. Which would you choose if you were Janine? Questions to ask: How much will Janine get on her birthdays for the next several years under each of the two plans? Decide which plan you think is best, give your reasons and state any assumptions you made. Does it make any difference how long you think Janine's uncle will give her these gifts? Suppose that Janine's uncle were to change Plan B so that the first gift was 50¢ instead of $1; would this change your answer? How about 1 ¢? Looking into the classroom: Most students begin with the opinion that a pattern that starts with $1, $2, $4, etc., could not possibly ever turn into much money. By computing further values, they begin to notice that, even though the sizes of the gifts are still very, very small for the first 5 or 6 years, the pattern by which they change is somehow different from the "add $100" pattern in Plan A. They are motivated to investigate more values. 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. They never change. But, in Plan B. the jumps keep getting bigger and bigger." They begin to make guesses about how much money Janine would re- ceive "if she got really, really old." The students realize that, once Janine is 21, Plan B leads to larger gifts. However, some notice that the total amount of money she would have at that age is more for Plan A than Plan B. They begin to make a new table that also has columns for calculating the total amount of money she received since she started and find that Plan A is the better one up to age 23. As the students begin to work on the question of what they would do if the initial gift were 50¢, there is Age 9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 Plan A $1,Ooo $1,100 $1,200 $1,300 $1,400 $1,500 $1,600 $1,700 $1,800 $1,900 $2,000 $2,100 $2,200 $2,300 $2,400 Plan B $1 $2 $4 $8 $16 $32 $64 $128 $256 $512 $1,024 $2,048 $4,096 $8,192 $16,384
172 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM a strong difference of opinion as to whether they need to make all new tables or they can work the problem from the table they already have. One student points out that, if the first gift is 50¢, then the next one will be $1, and the next $2, and so forth. "It will be the same thing except just one year later." Where Is the Algebra? This problem involves students in a number of activities that promote algebraic thinking. The table is a kind of symbol system (although not a literal symbol system) in that, once a table has been formed with one set of figures and a set of operations on it mastered by the students, the given input values can be replaced with other sets of input values (such as different size gifts, different constant multiplier in Plan B. or different constant adder in Plan A) and the same reasoning process carried out. They can observe that the Plan B overtakes Plan A 11 years after the first gift. Using an expanded table helps students understand that there are two different functional relationships: the sizes of the gifts and the accumulated gift received by Janine from the beginning to any particular time. As students move to thinking about the general growth patterns, they may use a graph to represent these functions. Students can study the graph of a discrete situation and a continuous situation and discuss what is different and why. If the situations represent continuous functions, students can be asked to estimate the point of intersec- tion of the graphs of the two functions. This encourages students to develop algorithms for making approximations, reasoning from the table or graph to refine their guesses in a systematic way. Finally, this problem is an occasion to reflect on and analyze two of the most basic types of functional relationships, linear and exponential, and to understand them through an analysis of change as the variable goes from to fin + 1~. Students can make generalizations and begin to write statements such as y = 2n or y = 1000 + (n - 1~100. In later grades, this problem can be revisited with literal variables in terms of exponential and linear functions, including a table of differences to contrast the rate of change in the two functions. Students should use notation that allows them to compare and abstract: y = abX and y = ax + b, and recognize how changes in a or b will affect the table and graph and the reverse, how changes in the graph will affect a and b. 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) A comparison of Plans A and B
APPENDIX E · What are some other ways that students might reason about the problem? · What are some othersituations that can be used to compare exponentialgrowth to growth that is not exponential? · How does understanding one model promote understanding of another model? Reversing the Question Finding the Initial Population in an Exponential Growth Setting 173 There are many questions in settings of exponential growth that are intrinsically appealing to students and that can be approached either by repeated arithmetic steps or by formulas involving literal variables. One such question is a "backward question" of the standard population growth problem. Ordinarily, we start with a given population, with a given growth rate, and ask what its size will be at some future time. In the reverse version, we start with the size of the population at a later time and ask what the size should be now in order to arrive at the given size. Students who answer such questions by repeated arithmetic are likely to become lost in the problem. At the same time, many students who simply substitute values into a formula are unlikely to ever see the important ideas behind it. However, intermediate notational systems are available for problems of exponential growth that enable students to carry out computations while maintaining an overall grasp of the problem. Such notational systems can also promote a use of literal variables that leads to a use of formulas with genuine understanding. Summary The problems have been selected to show how exponential growth can serve as a contextual setting for helping students to develop their algebraic thinking. The problems or a variation can be used from the earliest grades to late high school and illustrate three of the four organizing themes used in the Framework: functions and relations, modeling, and language and representation. Each of these problems sets out a concrete situation with a puzzling question that invites the student to become engaged with the situation and to solve a genuine problem. As students begin to work on the problem, alone and with others, questions, notational schemes, and suggestions can be provided to help them express, clarify, refine, and reflect on their thinking about the situation and the problem to be solved. Such activities can be important and valuable means for promoting students' understanding of algebra. Example 2: Contextual Settings Within Size and Shape Questions about size and shape confront people daily. How big is something? Will this shape or size fit into that place? How do shapes fit together? What happens to other attributes of a shape if one measurement is altered? Answers to these questions are typically thought of as geometric in nature. This contextual setting demonstrates the interrelationship between geometric and algebraic reasoning, clarify- ing how algebraic reasoning can be useful to solve problems posed geometrically. For example, problems involving perimeter, area, and volume are situations for thinking about relationships. Depending on the grade level, students may be asked to find ways to record relationships symbolically to enhance communication or for use on a computer or calculator. These representational schemes can themselves move to the foreground and become the object of discussion and reflection. In some cases, symbolic reasoning provides connections and understandings that might otherwise elude the student. Through discussion, the complementary algebraic and geometric reasoning can evolve. Relationships that appear in a problem can develop and change when the problem is extended and explored in other 1- . Dimensions. Reasoning about Patterns In this activity, designed for the early primary grades in a unit on counting or geometry, students investigate various patterns in a series of squares built from two different colored tiles. Using concrete materials makes the Using concrete materials activity appropriate for very young children. The same activity using grid paper instead of tiles is appropriate for
174 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM older children. The problem focuses on laying the foundations for dimensions, perimeter, and area, just one subset of the landscape of size and shape, as children search for patterns in counting. The Borders and Blues Problem: The 3 squares were built using white tiles for the border and blue tiles for the interior. Build the squares and count the number of white and blue tiles in each. Can you make the same kind of square using 36 tiles, some white and some blue? Questions to ask: How many blue tiles are used in each square in this continuing pattern? How many white border tiles are used? Can you make a square with 12 blue tiles? ~1 1 l3~g 1 1 1 1 1 [~] k+.~.~.~.~.d I T I 1 1 1 ;'' E+~ ~e T T I a] Looking into the classroom: Students build the squares using tiles and record the number of each color. Some forget to label which color is represented by the number they wrote down and need to be reminded. Others draw a picture of the figure and write the numbers next to the picture. To make a square with 36 tiles, some use all white to build the square, then replace the interior tiles with blue. Some build a square with six tiles on the bottom boarder because they recognize the pattern 3, 4, 5, 6, .... Others struggle with different numbers of blue and white tiles, creating rectangles not squares, and are reminded by their classmates about the characteristics of a square. By shifting the nature of the questions to fractions or to area and perimeter, the problem can be appropriate for upper elementary students. What fraction of the square's area is blue in each figure? What fraction of the area is white? What fraction of the area of the fifth figure in the sequence will be blue? Will the fraction of the area that is blue ever be more than one-half of the area? How does the ratio of border tiles to blue tiles change? What is the perimeter of each figure? Will the perimeter and the area ever be the same number? Compare the perimeter of the blue portion of the figures to the perimeter of the border. Connecting Symbols and Situations Situations involving shape can be good contexts for finding patterns and relationships that connect clearly to symbolic representations. By building or drawing the figures, middle school students can use the way they constructed the shape to reason algebraically about patterns, in a representation that is neither graphical nor symbolic in the traditional algebra sense. Some students reason directly from tables of data from the problem. The Borders and Blues Problem: The 3 squares have been made using white tiles for the border and blue tiles for the interior. Build the squares and think about how you put the blue and white tiles together. Find a general rule for the number of blue and the number of white tiles for any square.
APPENDIX E 175 +~. o. , .~ I , .~ 1 .~3 1 1 T1 Questions to ask: How can you build the square with six tiles on each edge of the border? Does your rule work for the square with five tiles on a side? How many blue and border tiles will be in the 1 00th square? How are the formulas related? Will there be a square where the number of white border tiles equals the number of blue tiles? How can the formulas help you answer this question? In the classroom: Students notice relationships between the number of blue and border tiles as they build. Some see the relation as the number of border tiles is four times the number of blues on a side, plus four (for the four corners). 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. Other students draw or build the squares differently; their understanding is reflected in the relationships they notice. Still other students choose to organize their data in a table or a graph. Some notice that the number of border tiles will always be divisible by four. Some students question whether every multiple of four is a border tile total. Where Is the Algebra? The focus of this problem at each grade level is on understanding and working with patterns. The theme of representation and language is at the forefront. Explaining precisely, in words, the relationships in this situation becomes cumbersome, and eventually the need to express patterns in symbols leads to a generalization and a coding scheme for the sequence. In the nth blue square, the number of blues is n2 and the number of white border tiles is 4n + 4. The total number of tiles used in the nth square is (n + 2~2. This can provide motivation for verifying the identity (n + 2~2 = (4n + 4) + n2. The students that generate the rule without using the sequence, as the students above did, may use n2 as the rule for any square and neglect the position in the sequence. Making the correspondence between the first figure and the number of each color, the second figure and
176 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM the number of each color and so on is an important precursor to working with sequences and series in precalculus. Some students may struggle with notation to express this relationship using symbols such as Pi, P2, P3, but from there it is only a short step to more standard notation. 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? The theme of function can also form a lens for the problem where, for each figure, students graph (number of blue edge tiles, total number of blue tiles) and (number of blue edge tiles, total number of white tiles) and compare the graphs. Students can reason about the functions to explore when the number of blue tiles overtakes the number of white border tiles; they can find the quotient of two functions to examine the behavior of the ratio of border tiles to blue tiles. The problem extends into three dimensions, where students can investigate relationships between unit cubes needed to enclose successive central cubes. In the three-dimensional situa- tion, discussions of "border" and "center" engage students in development of clear descriptive language. Students will have diverse ways of thinking about how to find the total number of "border" cubes for a given cube. The patterns in this situation include linear, quadratic, and cubic functions. · What are some other ways that students might reason about the problem? · Why is it important for students in early grades to reason about patterns? · Are exercises in symbol manipulation made more meaningful if the symbols are about a familiar concrete situation? How does the presence of a model help students reason algebraically? · Does the context help bring meaning to the symbolic representation? Exploring Complementary Geometric and Algebraic Reasoning Directly measuring geometric attributes of some shapes is difficult. Indirect measurement, involving use of formulas, is often much simpler; for example, it is easy to compute the area of a triangle given its base and altitude. Developing such formulas involves much more than reasoning about specific measurements of any one exemplar of the shape. Rather, it involves general reasoning about the shape itself. The following problems suitable for middle school or high school demonstrate how algebraic notation is useful to express such reasoning and how symbolic manipulation is useful (needed) to compare various solutions with each other and with the standard formula. Trapezoid Problem: Explore different ways to find the area of a trapezoid. A diagram of one trapezoid is shown to help you think about the problem. Find ways to express your methods of finding the area of the trapezoid in terms of the height, h, and the lengths of the bases, by and b2. A bit B // em\ ah D b2 C
APPENDIX E D h Questions to ask: What different methods did you find? What symbolic expressions did you use to express your reasoning? Are your symbolic expressions equivalent to each other? How can you decide? Can you show that the expressions are (are not) equivalent by symbol manipulation? Looking into the classroom: Some students use grid paper and subdivide the trapezoid. They find the areas of specific triangles and use these to figure out how they might reason more generally. Other students subdivide the trapezoid into regions in various ways and reason from the diagram. Several of these students create diagrams such as those shown below and develop the corresponding symbolic expressions. 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. Still others use a symbol manipulator to demonstrate equivalence or dynamic geometry software to investigate the situation. A B - _ \C A b1 B D . \ b2 Where Is the Algebra? Area of trapezoid ABCD= area of triangle ABD + area of triangle BCD. C b2-b1-~ = Area of trapezoid ABCD= area of left triangle + area of rectangle + area of right triangle. - 2hx+b1h+ 2h~b2 -b1 -x) --hx + b1h +-hb2 --hb1 --hx 2 2 2 2 = 1 hb1 +-hb2 2 2 Partitioning a trapezoid into subregions and writing symbolic expressions for those regions demonstrates how algebraic notation can be used to express geometric reasoning. The process of expressing a geometric idea using algebraic symbols and transforming the symbols into a new statement that has a different geometric interpretation gives added meaning to manipulation of algebraic symbols. The symbols describe the geometric reasoning that directs the process. Here, the power of algebra is in the transition between reasoning with symbols and interpreting in the context of the problem. Questions of proof and what constitutes proof arise when equivalence of expressions is verified numerically for specific trapezoids. The Trapezoid Problem can be extended when students are familiar with similarity and can use proportional reasoning. 177
178 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM How does use of a symbol manipulator impact students' learning to reason algebraically? What do students need to understand aboutsymbols and symbol manipulation to use a symbol manipulator? Whatlevel of ability to manipulate symbols is prerequisite to this understanding? What are some other ways to think about the area of a trapezoid? How do the symbolic and geometric representations model the way students reasoned about the area of a trapezoid? · When are students able to appreciate the need for proof? At what stage is formal proof appropriate or necessary? How do they learn what constitutes an acceptable proof? Can problems like these help students understand and appreciate the power and elegance of algebra? Sam's Problem: To solve the Trapezoid Problem, Sam extended the lines AD and BCto form triangle DPC. He found the area of the trapezoid by subtracting the area of triangle APB from the area of triangle DPC. Sam used the similarity of the two triangles to express the difference in their areas only in terms of the height, h, of trapezoid ABCD and the lengths of its bases, b' and b2. Will Sam's method lead to the standard form for the area of a trapezoid? p D Looking into the classroom: Some students will use symbols to express their thinking and reasoning as follows: area of trapezoid ABCD = 2 (x + hubs - 2 xb' Similarity of the triangles gives x = ~ b2 - b' ~ ~ he Therefore, area of trapezoid ABCD = 2 (b ~3 b + h)b2 - 2 (b ~ b )b~ Where the student goes from here depends on availability of a symbol manipulator or facility with symbolic manipulation. 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) and height is h. Where Is the Algebra? Symbolic manipulation can be used to simplify complicated expressions using the properties of real numbers. With experience, students learn both when to attempt to simplify an expression and how to tell if what they get is, in some sense, simpler. Students who observe that the expression
APPENDIX E Summary 179 for the area of a trapezoid can be written as 2 hobo + b2) and who wonder if there is a different way to find the area are at a different level of symbolic reasoning. They are using equivalent expressions to look for other patterns in the problem. Both the Trapezoid Problem and Sam's Problem can be extended to three dimensions for high school students. For example, students might explore different ways to find the volume of the frustum of a pyramid using Cavalieri's Principle and the fact that the volume is one-third the area of the base times the height, expressing their methods symbolically in terms of the dimen- sions of the frustum. They can generalize Sam's method to three dimensions to find the volume of the frustum. Reasoning in three dimensions is challenging for many students. They can reflect on the methods they developed in two dimensions and use them as the basis for reasoning in three dimensions. Extending results to higher dimensions can lead to reasoning about what is involved in generalizing a result. Algebraic notation can help direct the reasoning. The problems in this example are obviously geometric in character, but are embedded within the theme of representation. Assigning variables to represent carefully chosen measurements can clarify relationships found in geometric shapes. An early example of this may be when students realize that the length and width of a rectangle always control the area. Being able to generalize this relationship in a formula allows students to works with the mathematical model to develop further relationships, from the simple A = I x w to the closely related A = 2 b x h, to the various forms of the area of a trapezoid and other shapes. In some problems, such as the Border Problem, students connect numerical patterns to an underlying geometry to develop a representation. In addition to working with algebraic models to investigate shapes, students can also explore relationships among the zero-, one-, two-, and three-dimensional characteristics of the same geometric shape. Using tables and graphs to represent the problem, and finding patterns and relationships within and reasoning from tables and graphs, facilitates such investigations. Students may be searching for relationships across one line of a table, or they may be using recursive patterns. Symbolic expressions not only represent relationships among attributes of a geometric shape, but also, along with tables and graphs, provide efficient ways of exploring, describing, and verifying patterns. This is particularly useful when students track and record sophisticated ideas. (What results do changes in linear dimensions of a shape cause in two- and three-dimensional measurements? Can you predict the elements of the formula for the frustum of a pyramid from your experience with Sam' s method for a trapezoid?) Using all three methods of representing problems in the size-and-shape setting raises the question of which is more effective for a particular problem or if other methods (visualization, for example) are preferred. In many cases, the ability to represent a problem in several ways may lead to complementary insights. The problems given above are intended as the beginning point for further questions about where algebra can be found and what purpose it serves in the setting of size and shape. Algebra was used as a way to describe rela- tionships and to represent problems. Algebraic reasoning was used to find equivalent solutions or to verify observations. Underlying all the problems is also the theme of structure, how algebra complements geometric reasoning. Concrete objects, pictures, diagrams, and symbols were some of the ways used to represent the situations. While this example was posed in the setting of size and shape, it also intersected with ideas from growth and change and number. Example 3: Contextual Settings Within Number Another setting people experience is that of number. People use number to order and categorize their world. Through number, they carry out the computations necessary for their work and their daily routine. Children's understanding of even/odd allows them to make judgments about sharing, choosing teams, and taking turns. As children discover that they can count forever, they develop a sense of magnitude. The study of numbers includes many important topics (structure of numbers, number patterns, operations, relations, properties of numbers, and methods of counting, ordering, and computing) that underlie different organizing themes. Number is essential to understanding the behavior of functions and relations. The order and characteristics of the real number system often provide the first example children have of mathematical structure.
180 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM The search for generalizations and methods of representation have many origins in the setting of number. As students reason about calculations, they may observe numerical patterns or represent quantities pictorially or geometrically; with advances in technology reasoning about number has broadened to include use of spreadsheets ~ ~ . ~., . ~ ~ and sets of Instructions for computers. Ultimately, reasoning about calculations produces a need for symbolic notation, both for representing the process and for modeling relations. The focus of instruction should not, however, be on the manipulation of symbols but rather on the conceptual understanding of the meaning of the symbolic representation. Choosing number as a setting provides opportunities to illustrate how student understandings of any of the themes can be developed singly or in conjunction with others. The properties of the real number system allow us to rewrite expressions in an equivalent form, to represent a situation, and then to work within that representation free from context. A focus on the distributive property as one illustration of how number is incorporated into the framework highlights the structural aspect of algebra. Other properties of number could serve the same purpose. 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. With the distributive property, students can think about the calculation embedded in a situation as a sum of quantities, each of which has two or more factors, ab + ac, or they can think of the same situation as a product of quantities, each of which may be a sum of two or more addends, am + c). Generalizing and gaining new information from a representation are equally possible for many of the properties of number. The following problem situations illustrate these different aspects of the distributive property. However, because these outcomes are not disjoint, the problems are not separated according to specific goals. Instead, the problems are grouped according to developmental level of students. They are intended to serve as a springboard for thinking about the role of the distributive property in reasoning algebraically and for further discussion of the experiences necessary for students as they construct a view of algebra from their work with number. [saying the Foundation Early childhood experiences that help children understand the setting of number include investigations into patterns and regularities. Reasoning about the relationship between quantities and about efficient ways to link two mathematical representations involves using properties such as the distributive property as a matter of course. Using situations that have two equivalent interpretations will help build understanding of the distributive property. Experiences with the distributive property can occur in many situations in the elementary grades: simplifying computation 3 x lo + 3 x 9 = 3~10 + 9) or using an area model to show how to express multiplication of two-digit numbers. 10 9 311111111111 L411 11113 10 + 9 C_ _ _ _ _ I _ _ _ _ _ 3 _ _ _ _ _ l _ _ _ _ _ I I I r T l _ _ _ _ _ I _ _ _ _ _ _ ~ T T I I I I = 3 x 10 + 3 x 9 =3x (10+9) A visual representation allows students to assimilate the equivalence of the two ways to think. · Why is it important for students to have alternative ways to calculate?
APPENDIX E 181 · What does this area model add to student understanding of the multiplication algorithm for whole, fractional, and decimal numbers? Building on the Foundation As students in the middle grades become more familiar with expressing relationships symbolically, individual students may create different but equivalent expressions, depending on how they reasoned about the problem. 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. How many different telephone paths are necessary if each house is directly connected to every other house? Looking into the classroom: Ann and Juan decide to begin with some small examples and see if they can find a pattern. Ann begins with 4 and then 5 houses and draws the number of connections; Juan uses 6 and then 7. They make a table and discover they can tell how many connections they need for 8 houses, then 9, then 10. They can see a way to get to the NEXT entry in the table, but they cannot see how to get any general rule. Number Number of Houses of Paths 4 5 6 8 6 10 15 21 NEXT = previous Number of Houses previous Number of Paths Another student writes, "If there are N houses, you won't connect a house to itself, A ~ D HE ELF 1 1 A TIC ME \G - , etc. so each will have to be connected to one less or n- 1 houses. This means there will be ntn - 1 ) connections. For example, for 8 houses... If house A is connected to house B. though, it is the same as if house B is connected to house A, so my answer should be divided by 2. There will be ntn - 1~/2 connections." Another pair of students makes a chart where each "1" represents a connection between the houses and a "O" represents no connection. A B C o o D... n A B C 1 ... 1 , ... . O 1 1 n 1 1 1 1 ... 0 0 = no connection ~ = connection The chart is a square, so if there are n houses, there will be n 2 connections in the chart.
182 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM But the diagonal is not helpful because that shows a house is connected to itself. There are n houses on the diagonal, so the number of connections is n 2 - n. The connections on each half of the diagonal are the same (A to B is the same as B to A), so really there are only in 2 - nils connections. Tina rewrote the rule as 1x nx (n-1~/2 She feels sure that this is equivalent because she recognizes she can apply the distributive property to n 2_ n. She comments that the formula for the area of a triangle is 2 be and wonders if this relates to the matrix she and her partner made. Questions to ask: Do all of the rules generate the same answers as Ann's NEXT rule? Are the rules equivalent to each other? How do you know? One of the rules looks like the formula for the area of a triangle, but where is the triangle? Would a graph facilitate understanding of different reasoning processes? Where Is the Algebra? Searching for recursive patterns can provide a solution but is not efficient in many cases. Relations are often difficult to describe in closed form. 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. Tina's contribution is somewhat different in nature. She has deliberately applied the distributive prop- erty and linked the form of the result to something else she knows. Students can try to connect her new representation to something concrete in the problem. Some representations of this problem will have a triangular aspect because the numbers generated are the triangular numbers. Some students may recognize these from Pascal's triangle. Students may reason from the sequence of numbers or the ordered pairs, as follows, Number of Number of Ordered houses paths pairs 1 0 (1,0) 2 1 (2,1) 3 3 (3,3) 4 6 (4,6) and think about what relation operating on the first would give the second. Others may reason from a graph of the ordered pairs. They should see that the resulting graph is like the graph of y = x2 but with a smaller rate of change and different vertex. This is an occasion to discuss which of the two representations, y = x2 -x ory = Ox - 1), gives the most insight on estimating answers. For very large numbers, the graphs of y = x2 and y = x2 - x are almost identical. · How do the representations capture the reasoning process of the students? What is the . . . advantage of having different representations? How are the suggestions qualitatively different from each other? How can you systemati- cally build on such thinking? At the high school level, the problem can be extended to a counting problem where there are n ways to make the first choice, n - 1 ways to make the second choice, and so ntn -1) ways to choose two things. Dividing by 2 reduces the repetition. The triangular numbers that result, 1, 3,
APPENDIX E 183 6, 10, ..., can be related to the binomial theorem, (x + yin as the coefficients of the terms in expanded form, a powerful application of the distributive property. When students learned to multiply two-digit numbers, a geometric model of the distributive property can provide them with an understanding of the algonthm. For example, a student might write 3~27) = 3~20 + 7) = 60 + 21 = 81. When the computational process is extended to 27~42), the distributive property leads to (20 + 71~40 + 2), or the product of two binomials. Initially, some students may say symbolically that (x + y)2 = X2 + y2 but drawing an area model can convince them the square (x + y) 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. The use of (x + y) as a factor being distributed as an entity, (x + Ax + (x + ply, is an extension of the thinking established in the early grades and helps explain symbolically why the sum of the two squares as an answer to (X + y)2 iS insufficient. Students should be able to think and reason about the process and the algorithm they use. Expanding a tnnom~al or cube enables students to think of reasonable ways to extend their geometric model both in two and three dimensions and allows them to begin to develop patterns that will later lead to the binomial expansion theorem (X+y~n=xn+nxn-ly+ + yn Physical models can help students understand a situation, but students should become more and more comfortable with symbols as ways to represent situations, as well as ways to generalize anthmetic. The follow-up questions deliberately focus on structural issues: When does this property apply? What is the pattern in the terms? The comparative usefulness of the geometric model or the symbolic representation highlights choices of representations. · How much emphasis should be placed on structure at each level? What essential under- standings do students need? How are these to be developed in contexts that motivate? · How much exposure and over what length of time does it take for students to recognize and be able to use the distributive property? Within the setting of number and the organizing theme of structure, it becomes important to reflect on certain patterns and behaviors in very general ways. Some situations seem to share certain charactenstics. These common attributes allow you to generalize and treat in similar ways systems that, on the surface, seem to have little in common. The distributive property is actually a specific instance of a more general structural property. College- bound students might investigate the following: The Function Problem: Think about the functions you have studied. Which of these functions have the property Pa + by = f(a) + f(b)? Looking into the classroom: Students check the functions they recall. For example they write "Is sinta + by = sin a + sin b?" Some check the validity of each conjecture by making substitutions. Others discuss the nature of the function: "The value of sin x is never
184 Summary THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM greater than 1, but sin a + sin b could be." The students continue to investigate other functions. Questions to ask: What do the functions you chose have in common? Would the property, ha * by = f(`a) * f(`b), apply to operations other than adding for the functions you identified? Where Is the Algebra? Students may conclude that Ma + b) =;ff~a) + fibs applies only to linear functions such as those of the form y = kx. Some students find a geometric argument convincing: f (a+b) f (b) - f (a) 7 I f(n) f(~) a b a+b Those who have studied abstract algebra will recognize the general question: what operations are preserved by a mapping from one algebraic structure to another? For the linear function, y = kx with k fixed, each real number x is mapped onto the real number kx. Understanding what is preserved under what conditions allows flexibility in analyzing situations; in some cases, it is easier to analyze the range elements after the transformation, in others, to begin with the original domain elements then make the transformation. This study of what is preserved under different mappings leads to the concept of homomorphic functions, those for which Ma * b) = fiat * fibs. The distributive property of multiplication over addition is a particular example of a homomor . . phi mapping. What are the connections between the distributive property of multiplication over addition and linearity? While the setting for this example is number, the theme of structure comes to the forefront. 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. To use the structural properties of an algebraic system, students have to be introduced to these properties in thoughtful and meaningful ways in the early grades, using problem settings that are familiar to both students and teachers. The ability to generalize relationships is at the heart of algebra, and the structural nature of the system allows relationships to be expressed in different forms, some more useful in particular situations than others. Each different representation can add to the knowledge gained from one of the other representations. The use of symbols and the properties governing the behavior of operations with those symbols allow students to represent a situation in symbols, to manipulate those symbols temporarily free from the situational meaning to gain insights and informa- tion about the situation, and then to return to the situation to make sense of the symbols. · What is the interface between technology and structure? · How much symbol manipulation is necessary to function effectively with symbolic representations?
USING THE FRAMEWORK Groups from the mathematics education community can use the ideas in this document to organize their thinking and discussion of school algebra to develop a vision of algebra in a K-12 curriculum and as a means to move toward that vision. Possible questions might include the following: · What is the essential nature of each theme? · How do themes help organize ideas? · Are some themes more appropriate at different grade levels than at others? · Is there a hierarchy to the themes? · What contextual settings can be fruitful grounds for exploring algebraic concepts? · How do the themes focus algebra in grades 9-12? · What are examples of curricula that provide a coherent and balanced algebra curriculum in grades K-12? · How can adequate articulation and continuity be built into a K-12 algebraic sequence? · What should be done to help students develop depth in their algebraic understanding? · How and when should algebraic understanding be assessed? · What are some characteristics of algebraic reasoning at different developmental levels? The demands of a fast-changing society and the presence of technology require a vision of school algebra that is dynamic and fluid enough to keep pace with future needs of society, yet retains the essential aspects of algebra that have made it so significant in the history of mathematics. The workforce needs citizens who can adapt to new technologies, identify problems, reason about problems, and communicate their findings using symbols, graphs, tables, pictures, and words. Studies from other countries demonstrate that students can learn to reason algebraically much earlier than grade 9 and that all students can do so. Developing and implementing a coherent and balanced algebra curriculum for grades K-12 requires a complete rethinking of the entire mathematics curriculum, a task that is already underway by some involved in curriculum development. The success of implementing any vision of school algebra ultimately lies in creating conditions, policies, assessment, curriculum materials, and support that enable teachers to provide the kind of algebra experience that is essential for all students. Discussions and policies on "who takes algebra when" must have the full participation of teachers who will be responsible for enacting the changes. The algebra that is called for in this document is quite different from the algebra that most teachers have been taught or have been teaching. A framework such as this one can support teachers in a critical and reasoned review and adaptation of new curriculum materials purporting to exemplify a rethinking of algebra. Long-term professional development activities and preservice programs must be examined in light of a framework and the issues raised while equal efforts must be made to help "lay people" 185
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.
BIBLIOGRAPHY This bibliography will be a useful resource for anyone interested in the nature and role of algebra in the K-12 curriculum. Arcavi, A. (1994~. "Symbol sense: Informal sense-making in formal mathematics." For the Learning of Mathematics, 14 (3), 24-35. Bastable, V., & Schifter, D. (1997~. "Classroom stories: Examples of elementary students engaged in early algebra," in J. Kaput (Ed.), Employing Children's Natural Powers to Build Algebraic Reasoning in the Content of Elementary Mathematics. Beaton, A.E., Mullis, I.V.S., Martin, M.O., Gonzalez, E.J., Kelly, D.L., & Smith, T.A. (1996~. Mathematics Achievement in the Middle School Years: IEA's Third International Mathematics and Science Study. Chestnut Hill, MA: Center for the Study of Testing, Evaluation, and Educational Policy, Boston College. Birkhoff, G. (1973~. " Current trends in algebra." American Mathematical Monthly, 80, 760-782. Birkhoff, G., & MacLane, S. (1992~. " A survey of modern algebra: The fiftieth anniversary of its publication." Mathematical Intelligencer, 14 ~ 1), 26-31. Blais, D.M. (1988~. "Constructivism A theoretical revolution for algebra." Mathematics Teacher, 81, 624-631. Burrill, B., & Burrill, J.C. (1992J. Data Analysis and Statistics Addenda Series, Grades 9-12. Reston, VA: NCTM. Carlson, D., Johnson, C., Lay, D., & Porter, A. (1993~. "The linear algebra curriculum study group recommendations for the first course in linear algebra." The College Mathematics Journal, 24, 4-46. Chambers, D.L. (1994~. "The right algebra for all." Educational Leadership, 51 (6), 85-86. Cipra, B. (1988~. "Recent innovations in calculus instruction, " in L. Steen (Ida., Calculus for a New Century (pp. 95-103~. Washington, DC: Mathematical Association of America. Cobb, P., Wood, T., & Yackel, E. (in press). "Learning through problem solving: A constructive approach to second grade mathematics," in E. von Glasersfeld (Ed.), Constructivism in Mathematics Education. Dordrecht, The Netherlands: Reidel. Coburn, T.G. (1993~. Patterns Addenda Series. Reston, VA: NCTM. Conference Board of the Mathematical Sciences (1983~. The Mathematical Sciences Curriculum K-12: What Is Still Fundamen- tal and What Is Not. Report to the NSB Commission on Precollege Education in Mathematics, Science, and Technology. Washington, DC: Author. Confrey, J. (1995~. "Student voice in examining splitting as an approach to ratio, proportions, and fractions," in L. Miera & D. Carraher (Eds.), Proceedings of the l 9th Annual Conference for the Psychology of Mathematics Education, Vol. 1, pp.3-29. Recife, Brazil. Confrey, J., & Smith, E. (1995~. "Splitting, covariation, and their role in the development of exponential functions." Journalfor Research in Mathematics Education, 26, 66-86. Confrey, J. (1994~. "Splitting, similarity, and rate of change: A new approach to multiplication and exponential functions," in G. Harel & J. Confrey (Eds.), The Development of Multiplicative Reasoning in the Learning of Mathematics. Albany: SUNY Press. 187
188 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM Coxford, A.F., & Shulte, A.P. (Eds.) (1988~. Ideas of Algebra, K-12. Reston, VA: NCTM. Cuoco, A., & LaCampagne, C.B. (submitted to the Notices of the AMS.) "Department of Education launches algebra initiative." Cuoco, A. (in press). "Early algebra and structure of calculations, " in J. Kaput (Ed.), Employing Children's Natural Powers to Build Algebraic Reasoning in the Content of Elementary Mathematics. Cuoco, A. (1993~. "Action to process: Constructing functions from algebra word problems." Intelligent Tutoring Media, 4 (3/4), 118-127. Day, R.P. (1993~. "Solution revolution." Mathematics Teacher, 86 (1), 15-22. Edwards, E.L. (Ed.) (1990~. Algebra for Everyone. Reston, VA: NCTM Eisenberg, T. (1992~. " On the development of a sense for functions," in G. Harel & E. Dubinsky (Eds.), The Concept of Function: Aspects of Epistemology and Pedagogy. (MAA Notes, Vol. 25, pp. 153-174~. Washington, DC: Mathematical Association of America. Euler, Leonard (1984~. Elements of Algebra. (Translated by John Hewlett.) NY: Springer-Verlag. Ferrini-Mundy, J. & Johnson, L. (1994~. "Recognizing and recording reform in mathematics: New questions, many answers." Mathematics Teacher, 87 (3), 190-193. Fey, J.T., Fitzgerald, W.M., Friel, S.N., Lappan, G.T., & Phillips. E.D. (1994~. Bits and Pieces, Part I. Connected Mathematics Project. (Limited circulation pilot edition.) East Lansing, MI. Fey, J.T. (1989~. "School algebra for the year 2000," in S. Wagner & C. Kiernan (Eds.), Research Issues in the Learning and Teaching of Algebra (pp. 199-213~. Reston, VA: NCTM. Hillsdale, NJ: Erlbaum. Fey, J.T., & Good, R. (1985~. "Rethinking the sequence and priorities of high-school mathematics curricula." In C. Hirsch (Ed.), The Secondary School Mathematics Curriculum. 1985 Yearbook of the National Council of Teachers of Mathemat- ics. Reston, VA: NCTM. Freudenthal. H. (1983~. Didactical Phenomenology of Mathematical Structures. Dordrecht, The Netherlands: Reidel. Goldenberg, E.P. (1988~. "Mathematics, metaphors, and human factors: Mathematical, technical, and pedagogical challenges in the educational use of graphical representation of functions." Journal of Mathematical Behavior, 7, 135-173. Grouws, D.A. (Ed.) (1992~. Handbook of Research on Mathematics Teaching and Learning. Reston, VA: NCTM. New York: Macmillan. Hawkins, B.D. (1993~. "Math: The great equalizer Equity 2000 and QUASAR, improving minority standing in gatekeeper courses." Black Issues in Higher Education, 10 (6), 38-41. Harel, G., & Confrey, J. (Eds.) (1994~. The Development of Multiplicative Reasoning in the Learning of Mathematics. Albany: SUNY Press. Heid, M.K. (Ed.) (1995~. Algebra in a Technological World Addenda Series, Grades 9-12. Reston, VA: NCTM. Heid, M.K. (1988~. The Impact of Computing on School Algebra: Two Case Studies Using Graphical, Numerical, and Symbolic Tools. Proceedings of ICME-6, Theme Group 2, Working Group 2.3. Budapest, Hungary. Hiebert, J., & Wearne, D. (1991~. "Methodologies for studying learning to inform teaching," in E. Fennema, T.P. Carpenter, & S.J. Lamon (Eds.), Integrating Research on Teaching and Learning Mathematics (pp. 153-176~. Albany: SUNY Press. Johnston, W.B., & Packers, A.E. (1987J. Work Force 2000: Work and Workers for the Twenty-First Century. Indianapolis: Hudson Institute. Kaput, J. (Ed.) (in preparation). Employing Children's Natural Powers to Build Algebraic Reasoning in the Context of Elementary Mathematics. Kaput, J. (forthcoming). Integrating Research on the Graphical Representation of Functions. Hillsdale, NJ: Erlbaum. Kaput, J. (in press). "Democratizing access to calculus: New routes using old roots," in A. Schoenfeld (Ed.), Mathematical Thinking and Problem Solving. Hillsdale, NJ: Erlbaum. Kaput, J. (1987~. "Representation systems in mathematics," in C. Janvier (Ed.), Problems of Representation in the Teaching and Learning of Mathematics (pp. 19-26~. Hillsdale, NJ: Erlbaum. Karpinski, L.C. (1917~. "Algebraical development among the Egyptians and Babylonians." American Mathematical Monthly, 257-265. Katz, V. (1995~. "The development of algebra and algebra education," in C. LaCampagne, W. Blair, & J. Kaput (Eds.), The Algebra Initiative Colloquium (Vol. 1). Washington, DC: U.S. Department of Education. Kieran, C. (1994~. "A functional approach to the introduction of algebra Some pros and cons," in J.P. da Porte & J.F. Matos (Eds.), Proceedings of the Eighteenth International Conference on the Psychology of Mathematics Education, 1, 157-175. Lisbon, Portugal. Kieran, C. (1992~. "The learning and teaching of school algebra," in D.A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 390-419~. New York: Macmillan. Kieran, C. (1989~. "The early learning of algebra: A structural perspective," in S. Wagner & C. Kieran (Eds.), Research Issues in the Learning and Teaching of Algebra (pp. 33-56~. Reston, VA: NCTM. Hillsdale, NJ: Erlbaum. Kleiner, I. (1989~. "Evolution of the function concept: A brief survey." College Mathematics Journal, 20 (4), 282-300. Kline, M. (1972~. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press.
APPENDIX E 189 Leinhardt, G., Zaslavsky, O., & Stein, M.K. (1990~. "Functions, graphs, and graphing: Tasks, learning, and teaching." Review of Educational Research, 60 (1), 1-64. Leitzel, J.R.C. (Ed.) (1991~. A Callfor Change. Recommendations for the Mathematical Preparation of Teachers of Mathemat- ics. Washington, DC: Mathematical Association of America. Leiva, M.A. (Ed.) (1991~. Curriculum and Evaluation Standards for School Mathematics Addenda Series, Grades K-6. Reston, VA: NCTM. Michigan Council of Teachers of Mathematics (1990~. Algebra Activities, K-9. Lansing, MI: Author. Moses, B. (1993~. "Algebra, the new civil right." Paper presented at the SUMMAC II Conference. Cambridge, MA. National Commission on Excellence in Education (1983~. A Nation at Risk: The Imperative for Educational Reform. Washing- ton, DC: U.S. Government Printing Office. National Council of Teachers of Mathematics (1994~. "Board Approves Statement on Algebra." NCTM News Bulletin. National Council of Teachers of Mathematics (1992~. "Algebra for the Twenty-First Century." Proceedings of the August 1992 NCTM Conference. Groton, MA. National Council of Teachers of Mathematics (1991~. Professional Standards for Teaching Mathematics. Reston, VA: Author. National Council of Teachers of Mathematics (1990~. E.L. Edwards, Jr. (Ed.), Algebra for Everyone. Reston, VA: Author. National Council of Teachers of Mathematics (1989~. Curriculum and Evaluation Standards for School Mathematics. Reston, VA: Author. National Council of Teachers of Mathematics (1980~. An Agenda for Action: Recommendations for School Mathematics of the 1980s. Reston, VA: Author National Research Council (1990~. Steen, L.A. (Ed.), On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: National Academy Press. National Research Council (1989~. Everybody Counts: A Report to the Nation on the Future of Mathematics. Washington, DC: National Academy Press. National Science Board Commission on Precollege Education in Mathematics, Science, and Technology (1983~. Educating Americansfor the Twenty-First Century: A Plan of ActionforImproving Mathematics, Science, and Technology Education for All American Elementary and Secondary Students So That Their Achievement Is the Best in the World by 1995. Washington, DC: National Science Foundation. Nesher, P. (1986~. "Are mathematical understanding and algorithmic performance related?" For the Learning of Mathematics, 2-9. Pelavin, S., & Kane, M. (1988~. Minority Participation in Higher Education. Prepared for the U.S. Department of Education. Phillips, E. A., Gardella, T., Reely, C., & Steward, J. (1991~. Patterns and Functions Addenda Series, Grades 5-8. Reston, VA: NCTM. Rachlin, S. (1982~. "Processes used by college students in understanding basic algebra." Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education (SE 036 097~. Reese, C.M., Miller, K.E., Mazzeo, J., & Dossey, J.A. (1997~. NAEP 1996 Mathematics Report Card for the Nation and the States. Washington, DC: National Center for Education Statistics. Reys, R.E., & Nohda, N. (1994~. Computational Alternatives for the Twenty-First Century: Cross-Cultural Perspectives from Japan and the United States. Reston, VA: NCTM. Russell, S., et al. (1995~. Investigations of 3rd/4th Grade Interpreting Graph Units. Palo Alto, CA: Dale Seymour Publications. Schifter, D. (Ed.) (in press). Voicing the New Pedagogy: Teacher Narratives and the Construction of Meaning for the Rhetoric of Mathematics Education Reform. New York: Teachers College Press. Schoenfeld, A.H. (1985~. "Metacognitive and epistemological issues in mathematical understanding," in E.A. Silver (Ed.), Teaching and Learning Mathematical Problem Solving: Multiple Research Perspectives (pp. 361-379~. Hillsdale, NJ: Erlbaum. Secretary's Commission on Achieving Necessary Skills (1990~. SCANS Report. Washington DC: U.S. Department of Labor. Sfard, A., & Linchevski, L. (1994~. "The gains and pitfalls of reflection: The case of algebra." Educational Studies in Mathematics, 26, 191 -228. Sfard, A. (1991~. "On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin." Educational Studies in Mathematics, 22, 1-36. Silver, E.A. (1994~. "Dilemmas of mathematics instructional reform in the middle grades: The case of algebra." QUASAR Occasional Paper. Steen, L.A. (1992~. "Does everybody need to study algebra?" Basic Education, 37 (4), 9-13. Swan, M. (1982~. "The teaching of functions and graphs," in G. van Barneveld & H. Krabbendam (Eds.), Proceedings of the Conference on Functions (pp. 151-165~. Enschede, The Netherlands: National Institute for Curriculum Development. Thompson, P.W. (1995~. "Quantitative reasoning, complexity, and additive structure." Educational Studies in Mathematics.
190 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM Thompson, P.W. (1994~. "The development of the concept of speed and its relationship to concepts of rate," in G. Harel & J. Confrey (Eds.), The Development of Multiplicative Reasoning in the Learning of Mathematics (pp. 181-234~. Albany: SUNY Press. Thorpe, J.A. (1989~. "Algebra: What should we teach and how should we teach it?," in S. Wagner & C. Kieran (Eds.), Research Issues in the Learning and Teaching of Algebra (pp. 11-24~. Hillsdale, NJ: Erlbaum. Tierney, C. & Monk, S. (in preparation). "Children's reasoning about change over time," in J. Kaput (Ed.), Employing Children's Natural Powers to Build Algebraic Reasoning in the Content of Elementary Mathematics. U.S. Department of Labor (1987~. Work Force 2000: Work and Workers for the 21st Century. "Executive Summary." Washington, DC: U.S. Government Printing Office. University of Wisconsin-Madison & U.S. Department of Education (1993~. In T.A. Romberg, E. Fennema, & T.P. Carpenter (Eds.), Integrating Research on the Graphical Representation of Functions. Hillsdale, NJ: Erlbaum. University of Wisconsin-Madison & U.S. Department of Education (1991~. In E. Fennema, T.P. Carpenter, & S.J. Lamon, Integrating Research on Teaching and Learning Mathematics. Albany: SUNY Press. Usiskin, Z. (1988~. "Conceptions of school algebra and uses of variables," in A.F. Coxford & A.P. Shulte (Eds.), Ideas of Algebra, K-12 (pp. 8-19~. Reston, VA: NCTM. van Reeuwijk, M. (in preparation). "Algebra and realistic mathematics," in J. Kaput (Ed.), Employing Children's Natural Powers to Build Algebraic Reasoning in the Content of Elementary Mathematics. Vergnaud, G. (1994~. "Multiplicative conceptual field: What and why?," in G. Harel & J. Confrey (Eds.), The Development of Multiplicative Reasoning in the Learning of Mathematics. Albany: SUNY Press. Wagner, S., & Kieran, C. (Eds.) (1989~. Research Issues in the Learning and Teaching of Algebra. Reston, VA: NCTM. Yerushalmy, M., & Schwartz, J.L. (1991~. "Seizing the opportunity to make algebra mathematically and pedagogically interesting," in E. Fennema, T.P. Carpenter, & S.J. Lamon (Eds.), Integrating Research on Teaching and Learning Mathematics (pp. 41-68~. Albany: SUNY Press. Zawojewski, J.S. (1991~. Dealing with Data and Change Addenda Series, Grades 5-8. Reston, VA: NCTM.