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Presentations on Day Two Capturing Patterns and Functions: Variables and Joint Variation (G. Lappan) Functions and Relations: A Unifying Theme for School Algebra in Gracles 9-12 (C. Hirsch) Micic3le School Algebra from a Modeling Perspective (G. Kleiman) Why Mocleling Matters by. GoclboIcl) Modeling: Changing the Mathematics Experience in Postseconciary Classrooms (R. Dance) Algebraic Structure in the Mathematics of Elementary-Schoo} Children (C. Tierney) Structure in School Algebra (Micic3le School) (M. van Recuwijk) The Role of Algebraic Structure in the Mathematics Curriculum of Gracles ~1-14 (G. Foley) Language and Representation in Algebra: A View from the Micic3le (R. BilIstein) Teaching Algebra: Lessons Learned by a Curriculum Developer (D. Resek) The Nature and Role of Algebra: Language and Representation (D. Hughes Hallett) ss

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Cap Aiding Patterns and Functions: Va~ialoles and Joint Variation Glenda Lappan Michigan State University East Lansing, Michigan In ordinary English context helps distinguish among possible meanings of common words. As a representation of ideas, a word stated free of some meaningful context does not communicate very well. For example, the definition of a word in the dictionary usually includes several possible meanings. The word used in context helps the listener or reader to differentiate among the various meanings and understand what the speaker or writer intends. In mathematics, we face the same dilemmas. Many mathematical words have different or at least different shades of meaning. In learning to understand both how to communicate in and how to decipher the language of mathematics, students have to determine meaning from contexts of use. Two of the key concepts in developing a deep understanding of functions are variables and joint variation. Variable is one of those mathematical words that has many meanings that must be determined from context of use. In the new curricula that have been developed as a response to the NCTM's Curriculum and Evaluation Standards for School Mathematics and Professional Teaching Standards for School Mathematics, students learn mathematics through engagement with problems embedded in interesting contexts. This means that students have to learn to interpret different kinds of contexts. Students have to negotiate the "story" of the context whether the "story" is from the real world, the world of whimsy, or the world of mathematics to find ways to mathematicize the situation, presented in the story to manipulate the representations to find solutions, to interpret the solution in the original context, and to look for ways to generalize the solution to a whole class of problems. In addition, students have to interpret the context of the mathematization of the situation. Is the meaning of variable that of a place holder for an unknown? Or is the meaning of variable that of a domain of possible values for one of the changing phenomena in the context? Or is variable used in yet another way in the mathematization? Joint variation of variables is the heart of understanding patterns and functions. As students grow in their ability to derive meaning for variables in contexts, they encounter variables that are changing in relation to each other. This pattern of change, or joint variation, becomes the object of study as certain kinds of change produce recognizable tables, graphs, and symbolic expressions. At the middle-school level, these important families of predictable joint variation are linear, quadratic, and exponential functions. In order to see how students can grow in their understanding of variable and joint variation, let us turn to a series of examples of significant stages of development in one of the reform middle-school curricula, the Connected Mathematics Project. In the first situation given on the next page, students are challenged to determine variables of interest and to find ways to represent how these variables change in relation to each other over the day. The story line is that six college students are in the process of planning a five-day bicycle tour from Philadelphia to Williamsburg in the summer as a money-making business. They are exploring the route of the tour while gathering data on each day's trip. Day five encompasses a trip from Chincoteague Island to Norfolk, Virginia. The data are presented as word notes on the trip. 57

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58 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM Malcolm and Sarah's notes: We started at 8:30 A.M. and rode into a strong wind until our midmorning break. About midmorning, the wind shifted to our backs. We stopped for lunch at a barbecue stand and rested for about an hour. By this time, we had traveled about halfway to Norfolk. Around 2:00 P.M., we stopped for a brief swim in the ocean. At around 3:30 P.M., we had reached the north end of the Chesapeake Bay Bridge and Tunnel. We stopped for a few minutes to watch the ships passing by. Since bikes are prohibited on the bridge, we put our bikes in the van and drove across the bridge. We took 7 1/2 hours to complete the day' s 80-mile trip. In this problem, the variables of time and distance traveled change in relation to each other but not in a mathematically predictable way. The elements, terrain, creature comforts, and rules of bridge use change the rates at which the distance is changing as time passes. The graph may be linear in parts, curved in parts, and constant in parts. In the second situation, below, students are challenged to solve a problem that is stated in an open-ended way. To solve the problem, students have to find ways to represent and think about what the variables are and about two pairs of variables that are changing at different rates relative to each other. Bonne challenged his older brother, Amel, to a walking race. Amel and Bonne had figured their walking rates. Bonne walks at 1 meter per second, and Amel walks at 2.5 meters per second. Amel gives his brother a 45 meter head start. Amel knows his brother would enjoy winning the race, but he does not want to make the race so short that it is obvious his brother will win. What would be a good distance for the race if Amel wants his brother to win but wants it to appear to be a close race? Here students have to decide how to represent the progress of each brother so that they can determine what a good length would be for the race. Since the rate of change between time and distance for each brother is constant, each brother's time versus distance relationship is linear. The variables of time and distance in this situation change in a predictable way. One can predict the distance traveled for any number of seconds, which is a very different pattern of joint variation from the first situation. Students can graph both functions on the same axes and see that the two lines cross. They have to figure out the significance of the point at which the lines cross and how this relates to determining a good distance for the race. Of course, students may reason from tables of data for each of the brothers. But, in any case, they have to identify the variables and make sense of which variable depends on the other and how the two pairs of variables relate to each other. Here the expression 2.5t can represent Amel's progress over t seconds. The variable stands for a whole domain of possible times. Students could write the functions = 2.5t or d = 2.5t to show the independent variable, time, and the dependent variable, distance. They might also write 2.5t = 45 + It to show the equation that must be solved to find the point of intersection. Here the variable is standing for an unknown value, the value of t that makes the equation true. In the third situation, below, students again have to deal with variables and the pattern of change between related variables, but the nature of the change or variation is different from either of the first two situations. U.S. Malls Incorporated wants to build a new shopping center. The mall developer has bought all of the land on the proposed site except for one square lot that measures 125 meters on each side. The family that owns the land is reluctant to sell the lot. In exchange for the lot, the developer has offered to give the family a rectangular lot of land that is 100 meters longer on one side and 100 meters shorter on another side than the square lot. Is this a fair trade? Here students may talk about the problem at a general level or solve the particular case by comparing areas. However, the question of whether the results are true in general for a beginning square of any size remains. Here the problem can be restated with smaller numbers as follows: What happens if you own a square piece of land that is n meters by n meters and you are offered a piece of land that is 2 meters shorter on one side and 2 meters longer on the other? How does the area of the new lot compare to the original?

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PRESENTATIONS ONDAY TWO Increase: 59 Here is a table that many groups of students make to record what is happening as the original square sides Original Square Length 3 m 4m 5 m n New Rectangle Area 9 m2 16 m2 25 m2 . n2 Length 5 m 6 m 7 m . n+2 Width 1 m 2m 3 m Change in Area Area 5 m2 12m2 21 m2 . (n+2)(n-2) n2- (n+2)(n-2) = 4 4m2 4m2 4m2 The students notice that the change in areas from the square to the rectangle seems to be constant, 4 m. Since the students also develop a symbolic expression that tells what they do each time to find the change in area between the square and the rectangle, n2- (n + 2~(n -2), the question of equivalence of expressions naturally arises. Why does n2- (n + 2~(n - 2) = 4? Students have various ways of looking at this equivalence. They can graph the data in different ways to observe the behavior of the graphs. There are different graphs that can be made from the data and that show different kinds of change. The functions, A = n2, A = (n + 2~(n - 2), where A is area and n is the length of the side of the square, show quadratic growth, and D = n2- (n + 2~(n - 2), where D is the difference between the areas, gives a constant value and hence a horizontal graph. But the students also are motivated to examine different ways to transform the symbolic expression. What is another way to express (n + 2~(n -2~? By looking at a rectangle that is n + 2 on one edge and n -2 on the other, students can use the distributive property to see that this expression is equivalent to n2 _ 4. n 2 n-2 n2 _ 2n 2n-4 Therefore, the whole expression n2 _ (n2 _ 4) is always equal to 4. The three problems I have presented here would be appropriate for different stages of a student' s development of algebraic skill, but, nonetheless, all three serve to illustrate the centrality of variable and joint variation in understanding and using function to make sense of situations.

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Functions and Relations: A Unifying Theme for School Al,gelora in Grades 9-12 Christian Hirsch Western Michigan University Kalamazoo, Michigan One of the most important transitions from middle- to high-school mathematics is the emergence of algebraic concepts and methods for studying general numerical patterns, quantitative variables and relationships among those variables, and important patterns of change in those relationships. The mathematical ideas that are central to that kind of quantitative reasoning are variables, functions, and (to a somewhat lesser extent) relations and their representations in numerical, graphic, symbolic, and verbal forms. Organizing school algebra around the study of the major families of elementary functions (linear, exponential, quadratic and polynomial, rational, and periodic) offers the opportunity to bring greater coherence to the study of algebra. Situating that study in explorations of contextual settings can provide more meaning to algebra and can provide a broader population greater access to algebraic thinking. GOALS AND APPROACHES From a functions and relations perspective, the continued study of algebra at the high-school level should enable all students to develop the ability to examine data or quantitative conditions; to choose appropriate algebraic models that fit patterns in the data or conditions; to write equations, inequalities, and other calculations to match important questions in the given situations; and to use a variety of strategies to answer the questions. Achievement of these goals would suggest that the study of algebra be rooted in the modeling of interesting data and phenomena in the physical, biological, and social sciences, in economics, and in students' daily lives. Through investigations of rich problem situations in which quantitative relations are modeled well by the type of function under study, students can develop important ideas of recognizing underlying mathematical features of problems in data patterns and expressing those relations in suitable algebraic forms. Answering questions about the situations being modeled leads to questions such as the following, some of which are at the heart of a traditional algebra program. For a given function modeling rule fix), find fix) for x = a; x so that f~x)=a; x so that maximum or minimum values offer) occur; the rate of change inf near x = a; the average value off over the interval (a,b). Early work by Fey and his colleagues at the University of Maryland (cf. Fey and Good, 1985; Fey, Held, et al., 1995) using computer utilities demonstrated the promise of such a modeling and function-based approach. The 61

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62 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM emergence of hand-held graphics calculators puts such an approach in reach of all students and teachers of high- school mathematics. Graphics calculator technology provides powerful new visual, numenc, and even symbolic approaches to answering questions such as those given on the previous page. The technology also facilitates exploration of more general properties of each family of functions and of all functions collectively; these can then be formally organized and verified at a later point in the curriculum. A UNIFYING CONCEPT Functions are a central and unifying concept of school algebra and, more generally, of school mathematics (c.f. Coxford et al., 1996~. For example, symbolic expressions for function rules provide compact representations for patterns revealed by data analysis. Fundamental concepts of statistics, such as transformations of un~vanate or bivanate data, and of probability, such as probability distnbutions, are expressed and understood through the idea of function. The function concept can be generalized naturally to mappings such as (x, y) ~ (3x, 3y) or the following, x O -! Y 1 0 'lye which describe transformations of the plane. In discrete mathematics, early experiences with recursive descriptions of linear change (NEXT = NOW + b) and exponential change (NEXT = NOW x b) lead naturally to more general modeling with difference equations. Again, matrices are linked with transformations, and matrix methods are dependent on syntax and inference rules of algebraic symbolism. Finally, the mathematics of continuous change or calculus is fundamentally the study of the behavior of functions, including rates of change and accumulation. SUMMARY Organizing school algebra around functions and their use in mathematical modeling can provide a meaningful and broadly useful path to algebra for all students. Algebra as a language and means of representation is a natural by-product of this approach. Patterns that emerge through modeling with functions and studying families of functions can motivate at a later stage a study of the structure of algebra through deductive methods. Finally, the theme of functions and relations offers a way to provide a more unified approach to the high-school mathematics curriculum. REFERENCES Coxford, Arthur F., James T. Fey, Christian R. Hirsch, Harold L. Schoen, Gail Burrill, Eric W. Hart, Ann E. Watkins, Mary Jo Messenger, and Beth Ritsema. (1996.) Contemporary Mathematics in Context: A Unified Approach. Chicago: Everyday Learning Corporation. Fey, James T., and Richard A. Good. (1985.) "Rethinking the Sequence and Priorities of High School Mathematics Curricula." In The Secondary School Mathematics Curriculum, 1985 Yearbook of the National Council of Teachers of Mathematics, edited by Christian R. Hirsch and Marilyn J. Zweng, pp. 43-52. Reston, VA: NCTM. Fey, James T., and M. Kathleen Held, with Richard A. Good, Charlene Sheets, Glendon W. Blume, and Rose Mary Zbiek. (1995.) Concepts in algebra: A TechnologicalApproach. Dedham, MA: Janson Publications.

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Middle School Algebra from a Modeling Perspective Glenn M. Kleiman Education Development Center, Inc. Newton, Massachusetts To begin, we place algebra within the very general framework shown in Diagram 1 below. ( Situation ') Extracting and Representing - - ~/ ~ Mathematical <: Representations Interpreting and Applying - ` Mathematical ,' J Diagram 1 Mathematical ~` Findings This framework emphasizes that mathematics is more than working with mathematical symbols and tools. It also includes (a) extracting information from a situation and representing that information mathematically the process of "mathematizing"; and (b) interpreting and applying mathematical findings to have meaning within specific situations. This same framework can be applied to any area of mathematics. Algebra is defined by the representations, tools, techniques it provides, and the types of problem situations it enables one to address. Some specifics, focusing on algebra grades 6-8, are given in this paper. These are expansions of each of the three corners of Diagram 1 above. The organizing theme of this framework is modeling. The other themes are incorporated within modeling. The language and representation theme is reflected in the processes of extracting and representing bringing the original situation into a mathematical form and interpreting and applying translating back from a mathematical 63

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64 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM form to the situation. As we will see below in Diagram 3, the Unctions and relations and the structure themes are reflected in the mathematical analyses and mathematicalfindings components of the diagram. First, let us expand upon possible types of situations in Diagram 2 below. Pictorial or _ Physical Physical Arrangement Mathematical Experiment . Prohl~m 1 ' ~ I Real World I Ad\ \ | Data | ~\\ \\ /~ Diagram 2 ~ .. a.... 1 id I Game or | / ,~| Puzzle | it_ ~~ The middle-school curriculum should include a wide variety of types of situations. An appropriate situation has the following characteristics: it is engaging for many middle school students; it can lead to significant mathematical explorations at an appropriate level of complexity; and it is manageable within the classroom. Next, let' s expand upon the mathematical representations for the middle-school algebra curriculum. The link across the representations in Diagram 3 below is a reminder that understanding the relationships among these representations is also important. ~ \ / My Mathematical ~~\ RepresentationsI F: J Let Pictures Tab Diagrams | ~ _ ;{ inequalities 1 Scatter Plot | | Linear 1 1 1 1 1 1 Line ~ ~Quadratic Diagram 3 - ~

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PRESENTATIONS ONDAY TWO 65 Next, in Diagram 4 below are some of the types of mathematical findings we emphasize within the middle- school algebra strand Mathematical Representations Mathematical > Analyses - Mathematical Findings Identify I Solve for I I Test If/Then I Patterns I ~ Unknowns ~Conjectures Diagram 4 Identify Functional Relationships Students should be able to use these findings to do such things as (1) use patterns to predict new cases in the situation; (2) interpret solutions of unknowns in terms of filling in missing information about the situation; (3) make if-then statements about the situation; and (4) use knowledge of functions to predict what will happen when one thing changes in the situation. To get to these four types of findings, students need a repertoire of mathematical tools and understandings. Diagram 5 on the next page shows some categories of patterns students should understand; knowledge and techniques students will need to solve equations and inequalities; some tools for testing conjectures; and types of functions that students should become able to recognize and apply to understanding situations. All of these can be introduced in the middle-school curriculum, in many cases at an informal, context-based level, that forms a conceptual base for the more formal and abstract understandings that will develop in later grades.

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84 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM The fact that t3/t2 equals t is algebra, that t3 for large t is much larger than t2 is calculus, and that 1000/100 = 10 is arithmetic. When talking about school algebra, I mean something other than mathematical algebra. Algebra in the context of school algebra is a coherent integration of elements from the three domains: arithmetic, algebra, and calculus. ALGEBRA AT THE MIDDLE GRADES Over the past five years, the Freudenthal Institute has been involved in a curriculum development project in which a complete, new mathematics curriculum named MiC has been developed for American students between 10 and 14 years-of-age. One of the content strands in this curriculum is "algebra," and it contains a collection of topics from different mathematical domains. In the algebraic structure session, I will use some examples to illustrate the philosophy and approach in this curriculum towards algebra. In this paper, I have restricted myself to outlining the philosophy and approach in general terms. ALGEBRA IN MATHEMATICS IN CONTEXT The algebra strand in MiC emphasizes the study of relationships between variables, the study of joint variation. Students learn how to describe these relationships with a variety of representations and how to connect these representations. The goal is not for students merely to learn the structure and symbols of algebra but for them to use algebra as a tool to solve problems that arise in the real world. For students to use algebra effectively, they must be able to make reasonable choices about what algebraic representation, if any, to use in solving a problem. MiC APPROACH TO ALGEBRA The MiC curriculum especially the algebra strand is characterized by progressive formalization. In other words, students rely heavily, first, on their intuitive understanding of a concept, then they work with the concept more abstractly. The realistic problem contexts support this progression from informal, intuitive understanding to a more formal, abstract understanding. Students can move back and forth from informal to formal depending on the concepts and the problem contexts. Their ability to understand and to use algebra formally develops gradually over the four-year curriculum. By the end of the four-year curriculum, students have developed an understanding of algebraic concepts and are able to work quite formally with algebraic symbols and expressions. Algebra in MiC lays a solid groundwork for mathematics at the high-school level. EXAMPLE Even and odd numbers can be visualized by dot patterns. Dot patterns also can be used to visualize and to investigate more complex (number) patterns. Symbols, expressions, and formulas (recursive and direct) can be used to describe the patterns. The formulas themselves can then become an object of study that lead to re-inventing such mathematical properties as distributivity and factorization. For example, when students investigate the structure of rectangular and triangular numbers, they can use visual representations to support finding appropriate algebraic expressions. In the algebraic structure session, I will illustrate this example with problems from the curriculum materials. NO ALGEBRAIC STRUCTURES BUT STRUCTURE IN SCHOOL ALGEBRA Algebra at the middle-grade levels builds on students' intuitive and informal knowledge of arithmetic, of symbols, patterns, regularities, processes, change, and so on as developed in the early grades. Algebra in the middle grades does not need to lead to a complete and formal understanding of (the parts of) algebra. It is not the end of students' education. High school follows, and that is the place to formalize the concepts. Algebraic structure as described by Greg Foley, for example is "number theory." We should be careful about making topics from number theory the focus of the mathematics curriculum, especially at the middle grades.

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PRESENTATIONS ONDAY TWO 85 Factoring, divisibility rules, prime factorization, manipulating symbols and expressions, and other such topics have been the focus of the algebra curriculum, and students have not then had an opportunity to develop a meaningful understanding of the underlying concepts. The focus should be a long-term learning strand in which students can re-invent the algebra themselves, with the result being a mathematical system that is meaningful to students. The MiC algebra strand serves as an example of how this goal can be achieved.

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The Role of Al,~ebraic St~nctnre in the Mathematics Cliche of Grades I 1-14 Gregory D. Foley Sam Houston State University Huntsville, Texas MATHEMATICAL THEORY VERSUS RELEVANCE The so-called "new math" movement of the 1960s brought such logic-based organizing themes as set theory and algebraic structure to the fore of school mathematics in the United States. Mathematical reasoning, axiomatic structure, and within-mathematics connections were driving forces of a reform movement motivated by American- Soviet competition and led by research mathematicians. The goal of preparing a cadre of highly capable engineers and scientists caused us to focus on the most able students. By contrast, the National Council of Teachers of Mathematics (NCTM) Standards-inspired school mathematics reform of the l990s has been driven by calls for relevance realistic applications, modeling, genuine data, and mathematics in context and by a powerful collec- tion of emerging instructional technologies. Mathematical communication, problem solving, and cross-disciplinary connections drive the current reform. The need for a generally well-educated population to remain competitive in a global economy has led us to conclude that "everybody counts" and that we need algebra for everyone. In the 1960s, we sought to motivate the mathematics; in the l990s, we seek to motivate the students. This, of course, is an oversimplification. A careful reader of the NCTM Curriculum and Evaluation Standards (1989) will notice an overarching theme of "Mathematics as Reasoning" and will see that the document says high- school students should learn about matrices, abstraction and symbolism, finite graphs, sequences, recurrence relations, algorithms, and mathematical systems and their structural characteristics, and that, in addition, college- intending students should gain facility with formal proof, algebraic transformations, operations on functions, linear programming, difference equations, the complex number system, elementary theorems of groups and fields, and the nature and purpose of axiomatic systems. The American Mathematical Association of Two-Year Colleges' (AMATYC) Crossroads in Mathematics (1995) contains similar calls for the content of introductory college mathematics. Ideally, there should be a balance between solid mathematics and relevance to the student and societal needs. The Standards documents for Grades 11-14 recognize this. TECHNOLOGY AS CURRICULAR CATALYST The influence of technology should not be downplayed. Technology is affecting the mathematics curriculum in several ways. Compared to the past, current technology gives students access to relatively advanced mathemat 87

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88 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM ical concepts and allows them to explore, descnbe, and display data with relative ease. Modern hand-held computers, such as the TI-92, have powerful features that allow students to operate with integers, rational numbers, real numbers, or complex numbers; to define, algebraically manipulate, graph, and tabulate functions of one vanahle. narametnc equations sequences, polar equations, and functions of two vanables; to solve equations, find zeros of functions, and factor and expand expressions; to define, algebraically manipulate, graph, and tabulate sequences, polar equations, and real-valued functions of two vanables; to operate on lists, vectors, and matrices whose entries are integers, rational numbers, real numbers, or complex numbers; to organize, display, process, and analyze data; to wnte, store, edit, and execute programs; and to construct and explore geometric objects dynamically and interactively. , ~ , In addition, modern technology and the related emergence of computer science make the knowledge of discrete mathematical structures more important. Technology has indirectly increased the use of statistics throughout society. It is no wonder, then, that the University of Chicago School Mathematics Project has a two-year sequence of Functions, Statistics, and Trigonometry (Rubenstein et al., 1992) followed by Precalculus and Discrete Mathematics (Peressini et al., 1992~. There is simply more appropriate content after second-year algebra in preparation for postsecondary work in statistics, discrete mathematics, calculus, and linear algebra than in past decades. Technology makes this both possible and desirable. WHAT IS THE ROLE OF ALGEBRAIC STRUCTURE? Teachers of mathematics in Grades 11-14 must understand algebraic groups, nngs, fields, and the associated theory. They need, for example, to recognize the importance of the complex numbers being a field and the significance of the fact that matrix multiplication is noncommutative and that matrices have zero divisors. They should see a loganthm~c function as an isomorphism between groups and recognize geometric transformations as forming a group under composition. Furthermore, in keeping with the NCTM curriculum standards for college- intending students, high-school teachers need to be able to convey such understanding to their upper level students. This should be reinforced, amplified, and extended in lower division postsecondary mathematics courses, especial- ly those in discrete mathematical structures and linear algebra. We must, however, be careful not to make algebraic structure the overriding focus of mathematics in Grades 11-13, except, possibly, for the most gifted and talented students. On the other hand, it is essential that, in Grades 14-16, students acquire a clear vision of the "big picture" provided by a structural understanding of algebra. While we help students acquire this vision, we continually should call their attention to the numerous specific examples of groups, nngs, and so on, as they learn the common structure and associated theory. There are abstract algebra textbooks, such as Fraleigh's (1989), that do a good job of this. Abstract algebraic structures can serve as important organizing tools for the mathematics curnculum, but we should not fall into the trap of creating a new "new math." REFERENCES American Mathematical Association of Two-Year Colleges. (1995.) Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus. Memphis, TN: Author. Fraleigh, J. B. (1989.) A First Course in Abstract Algebra, 4th Ed. Reading, MA: Addison-Wesley. National Council of Teachers of Mathematics. ( 1989.) Curriculum and Evaluation Standards for School Mathematics. Reston, VA: Author. Peressini, A. L., et al. (1992.) Precalculus and Discrete Mathematics (University of Chicago School Mathematics Project). Glenview, IL: Scott-Foresman. Rubenstein, R. N., et al. (1992.) Functions, Statistics, and Trigonometry (University of Chicago School Mathematics Project). Glenview, IL: Scott-Foresman.

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Language and Representation in Algebra: A View from the Middle Rick Billstein Director, STEM Project University of Montana Missoula, Montana Algebra can be thought of as a language, and students learn language best in their early years. We should begin teaching informal algebraic concepts in the elementary grades and continue to develop the concepts throughout the middle-school years. Algebra too often is taught as rules and tricks without an understanding of the concepts. Then the jump to the formal level is often made too quickly for the concepts to be mastered. Topics should be developed slowly and informally without symbol manipulation as the primary goal. The jump into symbol manipulation should come only after students recognize the need for it. Algebra has been described as a way of thinking about and representing many situations. Unfortunately, many textbooks confine algebra to solving equations and manipulating symbols. Other representations, such as graphs, tables, patterns, diagrams, and other visual displays, should be used as appropriate. Visual representations are powerful because they help abstract mathematical ideas to become concrete. Since different representations may provide new or fresh insights about a problem, each representation is important and plays a role in the learning of algebra. There should be many opportunities for students to make transitions between the various representations. As students mature mathematically, they learn which types of representations are most useful for which kinds of problems. Students need to describe various representations in their own words. After a representation has been used, it is important to discuss it in terms of the original context. Interaction between teachers and students is important in development of language and representation skills. At the middle-school level, students might be asked to translate between words, tables, graphs, and equations. Given any one of these representations, they could be asked to determine any of the other three. Having students work in groups and share representations makes them aware that different representations can be equivalent yet look quite different. This is a powerful experience in middle school and will pay huge benefits at the high-school level. Instead of always translating from words to representations, as in the traditional curriculum, we now ask students to translate representations to words. With the use of technology on the rise, new understandings of symbol manipulation are needed to model situations that can be entered into a computer. For example, spreadsheets can be used to analyze complex numerical data from a problem situation. Algebra becomes important because it is the language used to communicate with the technology. Spreadsheet formulas are but one example of a form of algebra. Technology allows students to experiment, to investigate patterns, and to make and test conjectures. Technology allows us to go where we could not go before because the mathematics became too "messy." Students need experiences making representations with and without technology. Students should be involved in "doing" mathematics at the middle-school level. It is important that they investigate problems and be involved in hands-on activities. "Doing" mathematics provides students with opportunities to communicate about algebra. There is little communication in a typical algebra textbook. The language that students use will develop as they become more mathematically mature. Curriculum materials not 89

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9o THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM only must contain good problems, but also the problems must be structured in such a way that solving them will help students achieve the desired learning outcomes. New materials are now becoming available in which algebraic ideas are taught throughout the curriculum. Teachers must be made aware of these materials and support must be provided to train teachers to use these materials. Teachers must understand that when we say we want to include algebra in the middle school, we are not talking about the algebra that they had when they were in school. Hugh Burkhardt of the Shell Center for Mathematics Education in England describes algebra as "inherently slippery" and has said that "having separate algebra courses is one of the United States' great self-inflicted wounds" (1997~. Most National Science Foundation (NSF)-funded middle-school curriculum developers have struggled with this and are constantly asked about the role of algebra in the materials and how their curricula fit with an Algebra I course. The "Six Through Eight Mathematics" (STEM) project response has always been that algebra should not be a separate course taught at a particular grade level but, rather, that it should be a strand taught within the mathematics curriculum at every grade level. The teaching of algebra should be integrated with the teaching of other mathematical strands, such as statistics or geometry. The traditional Algebra I course should not be a required eighth-grade course as it is in many schools because this means the sixth, seventh, and eighth-grade curriculum must be covered in only two years. Two years is not enough time to develop adequately topics in probability, statistics, measurement, discrete math, number theory, and geometry. Many of the negative feelings that develop towards mathematics as a result of an "algebra course" might be eliminated if the algebra were integrated into the curriculum as a strand. Students would no longer remember algebra as a course in manipulating expressions and solving symbolic equations. Student experiences in middle- school mathematics courses might then actually prepare them for and encourage them to take additional mathematics courses, especially if those courses were taught in the reform mode of the new NSF high-school projects. STEM has found that one way to teach algebra effectively is to make it useful to students. Real contexts that are meaningful to students play a major role in algebraic learning. Real contexts do not mean that all problems have to come from students' everyday lives but, rather, that problems must make sense to students. Algebraic abstraction is motivated by the need to represent the patterns found in the context. Algebraic thinking is more important than algebraic manipulating. To develop algebraic thinking, we need to include informal work with algebraic concepts in the middle school and not move too quickly to the abstract level. For example, being able to set up graphs or tables in various problem settings brings mathematical power and understanding to students. Experiences with graphs should include a detailed plotting of points to determine a graph as well as experiences with the overall shapes of graphs based on the information in the problem. If students are given a graph, they should be able to write a story about it. Having students communicate about mathematics is a worthwhile goal in the new middle-school curricula. Don Chambers wrote, "Algebra for all is the right goal at the right time. We just need to find the right algebra" , ~ (1994~. The NSF-funded middle school projects are taking us closer to finding the right algebra. REFERENCES Burkhardt, H. (1997.) Personal Conversation at the National Council of Teachers of Mathematics National Meeting in Minneapolis, MN. Chambers, D. (1994.) "The Right Algebra for All." Educational Leadership, 51, 85-86.

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Teaching Algebra: Lessons [earned by a C?~ric?~?~m Developer Diane Resek San Francisco State University San Francisco, California In this symposium, I am approaching the issue of teaching algebra from the perspective of a curriculum developer. I am one of the developers of the National Science Foundation's (NSF's) secondary curriculum, "Interactive Mathematics Program" (IMP). The original design for the program was based on the experiences of the developers in past curriculum projects and in their own teaching. In this new project, we have had experience teaching algebra to students in elementary school, high school, and college. The original IMP curriculum has been rewritten three times shaped by the authors' observations of the curriculum as it was taught in different classrooms, by the comments and suggestions of teachers, and by student work. What follows are some statements about what I now believe about the teaching of algebra. INTUITION SHOULD COME FIRST People often ask for evidence that the new curriculum projects "work." There is some evidence of this, but there are mountains and mountains of evidence that the traditional methods don't "work." Now, exactly why the traditional curriculum does not work is open to question. My personal belief is that the chief culprit is the teaching of manipulative skills in a way that does not allow adequate intuition to come into play about what the symbols mean and why the manipulation is valid and useful. It is not that we do not know how to teach manipulation in meaningful ways. Many projects have shown us how to do this for years. One way to develop intuition about manipulating equations symbolically is by tapping into both the students' familiarity with the fact that equations are statements about functions and the students' comfort in associating the symbolic form of functions with other representations of functions, graphical or numerical. Student familiarity and comfort must be developed over time. It is my hope that much of the work in elementary-school and middle-school algebra will be on developing student comfort in moving between representations of functions. In general, we can decide what we need to teach at various levels by looking at what is difficult to teach later on. Traditionally, we have looked at what skills we thought were needed for success at one level and then taught those skills at the lower level. I am suggesting that we look at what skills or understandings are difficult for students at one level and try to develop intuition at lower levels that might serve as a basis for those skills or understandings. UNDERSTANDING DOES NOT COME IN DISCRETE PACKAGES One difficulty with building students' use of intuition is that this requires time often several years. Teaching one concept over time conflicts with the traditional idea of organizing teaching around subject matter. Traditional- ly, one studies a chapter on linear equations at one time and that subject is then checked off. However, if we want 91

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92 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM students to understand linear functions in different representations, this must be worked on over several years. Striving for long-term exposure to subject matter creates a difficult bookkeeping problem. It becomes difficult to check off the skills that students have. As the public pushes us for accountability, they push for a neat and tidy assessment system. Unfortunately, that kind of assessment system does not match the way students learn. At any level first grade, high school, or college students do not study a deep topic and suddenly "get it." Understanding comes gradually. It develops over time. Anyone who has asked students to write about a topic knows this. Reading what they have written, we can see that there are things the student seems to understand and other things he or she has not yet come to terms with. Rarely do we get a picture of perfect understanding. Our curriculum must be structured so that students can work on tasks at different levels and so that everyone in a class will grow. I am not saying that teachers should not be accountable for students' learning or that students should not eventually master key ideas. I am saying that we need to take into account how students learn and that this is gradual. We must not let the difficulty of assessing understanding sway us from trying to teach effectively. WE CAN LEAD STUDENTS TO WATER, BUT ... Once we have decided what we want to teach and how we want to teach it, we have to wrap it in the right paper. This is not because students are lazy or do not have good taste. It is because their minds cannot actively work with material if they have no way to relate to it. Contexts and relationships to other subject matter can provide students with a door to approach new mathematics. This is not to say we should never teach mathematics without a "real life" context. I am saying that we have to introduce the mathematics in a "real life" situation that students can relate to or in the context of other mathematics that they are working with. Once they have gone into the mathematical ideas, they can and will go on to work on the "bare" mathematics. A few students do think well symbolically and do not require much of a context. Most of us here were that kind of student. In the past, success in algebra was reserved for us and others like us. Algebraic knowledge is too important to reserve for so few. It also is not clear that people who think in this way have the most to contribute even to pure mathematics. We have to open the doors to others. It is not that hard to do.

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The Nature and Role of Algebra. I~an~na~e and Representation Deborah Hughes Hallet Harvard University Cambridge, Massachusetts Students arriving at college should be familiar with verbal, symbolic, graphical, and numerical representations. They need to be able to manipulate each one and be able to convert one to another. Manipulating each of the representations requires some degree of technical skill supported by conceptual understanding. This understanding must comprise both an understanding of how the representation works in general and of each particular object being represented. To work with graphs, students need to understand how graphs are constructed. For example, they need to understand that values of inputs to a function are measured horizontally, whereas values of outputs from the function are measured vertically. This will enable students to interpret intercepts and to estimate values of limits and asymptotes. Technology is changing how much technical skill students need to have in drawing graphs by hand, but it has not changed the fact that students need to understand how the zeros and symmetries of a function appear on a graph, where to expect asymptotes, and what sort of scale will show all the features of the graph. Numerical data, usually in a table, is for many students the least familiar representation. Students need to understand how the data were generated (from an experiment, by using a formula, for example). They should be able to work with numerical data, such as rounding, interpolating, and extrapolating (where these make sense). The ability to find patterns in data, such as where values are increasing or where there are constant differences, is a useful skill. The manipulation of symbols traditionally has formed the largest part of an algebra course. It is still central. Students must be able to solve equations, collect terms, simplify, and factor. The degree of skill and the speed required may be altered by changes in technology. For example, methods of factoring higher degree polynomials are probably not as important as they used to be. However, what it means to factor a polynomial (for example, that it is not usually useful to write x2 - 2x = xtx) - 2x) is as important as ever. Experience and observation will suggest the most effective balance between paper-and-pencil work and technology. Currently, there is a wide range of opinion about the best way to develop manipulative skill, ranging from not allowing technology to allowing it to be used heavily. As we try to figure out how to teach this skill, we should be mindful of the fact that we were not very successful at teaching symbol manipulation before technology complicated the situation. It is tempting to gild the past, but weaknesses that we currently observe are probably not the result of technology. Our charge is to figure out how to fix them. Besides acquiring skill in manipulating graphs, numerical data, and symbols, students need to be able to move easily between these representations. For example, given a straight line graph, students should be able to figure out its (approximate) equation. Given a table of data from an exponentially growing population, students should be able to figure out a formula for the function. Given a data set, students should be able to make a mental sketch of the data or match data with the correct sketch. 93

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94 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM Mastery of the language of algebra requires a two-pronged approach: 1. What does it mean? 2. How do wedoit? Equal emphasis on both of these leads to students who know both what algebra means and how to use it correctly. The future is likely to change the balance between these two because the skills required to do algebra are likely to change. However, we always will need to make sure students can use graphs, tables, symbols, and verbal descriptions fluently.