Transition Toward Algebra

Transition Toward Algebra M A R I A L. F E R N A N D E Z A N D C Y N T H I A O. A N H A L T ucators must consider the range of mathematics and the continued development of mathematics as a science of pattern and order. Informal explorations involving physical models, data, graphs, and symbols are vital to a solid foundation for students’ transition into algebra. Also important is developing a deep understanding of such concepts as decimals, fractions, ratio and proportion, measurement, integers, functional relationships, and variables. As students in elementary school and middle school work with number and operation and other mathematics strands, the focus should be on informal discussion and investigations that lead students to build, describe, represent patterns, develop and apply relationships, make and verify rules or generalizations, and explore mathematical properties (NCTM 1989, 2000). We worked from these premises with mathematics teachers of grades 5–9 during the two-year Transition Toward Algebra project (T 2A), funded by the Eisenhower Mathematics and Science Education Program. T 2A was developed to help teachers experience and think through effective teaching strategies that promote a rich and positive environment to maximize opportunities for students’ learning and transition toward algebra. This article presents some of the mathematics tasks used to help teachers expand their views of experiences that support students’ transition toward algebra and development of algebraic thinking. MARIA FERNANDEZ, mariaf@u.arizona.edu, teaches at the University of Arizona, Tucson, AZ 85721-0069. She is interested in algebraic thinking and the use of technology in school mathematics. CYNTHIA ANHALT, anhalt@u .arizona.edu, is a graduate student at that same institution, following ten years of teaching in the public schools and three years as a clinical faculty member in mathematics education. 236 PHOTOGRAPH BY MARIA FERNANDEZ; ALL RIGHTS RESERVED I N PREPARING STUDENTS FOR ALGEBRA, ED- Project Design THE FORTY PARTICIPANTS IN THE TWO-YEAR project included twelve fifth-grade, five sixth-grade, nine seventh-grade, five eighth-grade, and nine ninth-grade teachers. The first project year began with twenty teachers enrolled in a month-long summer institute, followed up during the academic school year with monthly meetings, classroom collaborations, school site visits by project staff, and videotaped feedback for teachers. During the second year, these teachers completed a week-long summer institute and led sessions involving a new MATHEMATICS TEACHING IN THE MIDDLE SCHOOL Copyright © 2001 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM. group of twenty teachers participating in activities paralleling those of the first year. An important characteristic of the project was that teams of teachers from schools were sought to promote cooperation, collaboration, and support among colleagues (Fullan 1990). Another feature of the project was placing the teachers in the position of students as they solved problems and shared their thinking, strategies, and results. The teachers used a variety of tools, including manipulatives and technology, to enhance their understanding and use of mathematical representations. In turn, the teachers’ deeper understanding would enable them as “algebra is solving equations with unknowns.” An important project goal was to expand the teachers’ and, in turn, their students’ views of algebra and algebraic thinking. T 2 A presented three views of algebra for teachers to explore, discuss, and reflect on: (1) as the study of patterns and relationships, (2) as a tool for problem solving, and (3) as generalized arithmetic. Many of the explorations, problems, and activities overlapped all three views. The following paragraphs describe some tasks that represent the kinds of explorations that the program presented. Algebra as the study of patterns and relationships The Many Mirrors Company problem in figure 1 was one task used to highlight algebra as the study of patterns and relationships. This task emphasized both inductive reasoning based on numerical patterns and deductive reasoning based on logical arguments involving visual models. The problem also accentuated the importance of engaging students in making connections between both forms of reasoning. The teachers approached this problem by assigning different colors to the different types of tiles and drawing diagrams of the square mirrors. The teachers collected information about the number of tiles of each type needed and began organizing their data into tables, as shown in table 1. Initially, a few teachers began collecting tile data arbitrarily; however, through collaboration, they realized the value of using an organized list to study the relationships. Using their tables, the teachers looked for patterns and relationships among the dimensions of the square mirrors and the number of each type of tile needed to complete the mirrors. They observed that the number of two-beveled-edge tiles needed was always four. They could easily defend this numerical pattern, noting that logically, for any size mirror, only the four corners required tiles with two beveled to encourage their students to use and transfer information among different representations. The teachers translated and made connections among five representations, described by Lesh, Post, and Behr (1987): (1) language, (2) contextual, (3) concrete-manipulative, (4) semiconcrete-pictorial, and (5) abstract or symbolic. Expanding Teachers’ Views of Algebra THROUGH BASELINE DATA COLLECTED BEFORE the project, we found that T 2 A teachers’ initial views of algebra were often limited to such beliefs TABLE 1 Teachers’ Table to Solve the Mirror-Tiling Problem SQUARE MIRROR SIDE LENGTH 2 BEVELED EDGES 2 ft. × 2 ft. 3 ft. × 3 ft. 4 ft. × 4 ft. 5 ft. × 5 ft. . . . n×n 2 ft. 3 ft. 4 ft. 5 ft. . . . n 4 4 4 4 . . . 4 1 BEVELED EDGE 0 BEVELED EDGES 0 0 4 1 8 4 12 9 . . . . . . 4n – 8 or 4(n – 2) (n – 2)(n – 2) V O L . 7 , N O . 4 . DECEMBER 2001 237 Fig. 1 The mirror-tiling problem Many Mirrors Company The Many Mirrors Company makes square mirror tiles that can be placed on walls as tiles are placed on floors. There are three types of tiles, as shown below: plain tiles, one-beveled-edge tiles, and twobeveled-edge tiles for corners. The outer tiles of any tiling will have beveled edges for safety. Your task is to determine the number of mirror tiles with two beveled edges, one beveled edge, and no beveled edges needed to make square mirrors of varying sizes. Each mirror tile is 1 square foot in size. The dimensions of completed square mirrors will be in feet (i.e., 2 ft. × 2 ft., 3 ft. × 3 ft., and so on). Find a way to determine the number of 1 ft. × 1 ft. tiles of each type needed to make a square mirror of any dimension. Defend and explain your solution. 0 beveled edges (inner tile) 1 beveled edge (outer tile) 2 beveled edges (corner tile) Extensions: Graph the data gathered for each type of tile, and compare the graphs. When are the number of tiles for two or more of the types the same? Which tile types should the company manufacture the most of? PHOTOGRAPH BY MARIA FERNANDEZ; ALL RIGHTS RESERVED edges. The data for tiles with no beveled edges suggested a squaring pattern. By seeking a relationship between the length of a side of the mirror and the values of the squared data, the teachers observed that the numbers being squared were always 2 less than the corresponding mirror side length. This relationship for no beveled edges was more difficult to defend using deductive arguments than the pattern for tiles with two beveled edges (see fig. 2 for deductive arguments). Finally, the numerical relationship of the length of the mirror side to the number of tiles with one beveled edge was most difficult for teachers to arrive at inductively. A guess-check-andrevise strategy for determining the relationship proved time-consuming and futile for many. A few teachers noticed the constant increases in mirror lengths and tiles with one beveled edge; they graphed data points and determined equations, such as 4n – 8 or 4(n – 2), for the linear relationship. Because of the limited success that they had with other methods, the teachers recognized the value of reasoning deductively to justify the relationship. Once prompted, they argued logically that the mirror had four sides that need beveled edges. Because each side has two corner tiles, the number of tiles needed for each side was 2 less than the mirror side length (i.e., n – 2). Thus, the number of tiles needed with one beveled edge was 4(n – 2) for any mirror length n (see fig. 2). The teachers recognized that students could equate 4n – 8 and 4(n – 2) by graphing both relationships, that is, y = 4x – 8 and y = 4(x – 2), or by using the distributive property. The teachers also used graphs to compare the growth patterns of each type of tile. The graphs were used to examine when the number of onebeveled-edge tiles needed was the same as the number of zero-beveled-edge tiles needed for a 6 foot × 6 foot mirror and to discuss which tiles the company should manufacture most. This aspect of the task emphasized the importance of developing students’ understanding and interpretation of graphs as they progress toward algebra. Algebra as a tool for problem solving To foster algebra as a problem-solving tool, the T 2A program presented the Rope around the Earth problem: 238 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL The teachers made predictions before solving the problem. The majority of teachers predicted that an ant could walk under the rope, a few guessed a snake, and some believed that the creature would be smaller than an ant, maybe an ameba. Modeling this question with string and different-sized balls, including a Ping-Pong ball, a basketball, and a large beach ball, clarified the problem. Using the models demonstrated that for any size ball, the six-foot overlap causes a separation between the ball and the Two Beveled Edges 4: One on each corner of any square mirror, 4 corners. One Beveled Edge 4(n – 2): n – 2 on each edge of the square, which has 4 edges; n – 2 on each edge because we need to subtract the two corner tiles along each edge. In the 3 × 3 example below, the number of one-beveled-edge tiles is 3 – 2 corner tiles along each edge, so the number of one-beveled-edge tiles is 4(1) = 4. Zero Beveled Edges (n – 2)2: (n – 2)2 because we need to subtract two rows of tiles along opposing edges to determine the length of each dimension of the square with zero beveled edges found in the center of each mirror. In the 4 × 4 example below, the dimension of the inner square mirror is (4 – 2) along each dimension of the square; thus, the number of tiles with zero beveled edges is 2 × 2. Fig. 2 Deductive argument to solve the mirror-tiling problem PHOTOGRAPH BY MARIA FERNANDEZ; ALL RIGHTS RESERVED If we could wrap a rope around the Earth’s equator with an overlap of exactly six feet of rope, how far off the ground would the rope be if we lifted it the same distance at all points around the Earth so that the ends connected? What is the tallest living thing that could walk underneath the rope: ant, snake, rabbit, collie, person, horse? string that is about the same, slightly less than a foot. That this same separation would exist for a rope around the Earth is incredible. To solve this problem, students must understand the relationships among the circumference, radius, and diameter of circles. Some of the T 2A teachers did not recall these relationships. We engaged all the teachers in tasks that they could use to develop their students’ understanding of these relationships with the Rope around the Earth problem as a context. Using round candies and other circular objects of different sizes, the teachers gathered and graphed data for the circumference and diameter of different-sized circles. Using tables and linear graphs of the candy data, the teachers conjectured that for all circles, the circumference divided by the diameter was approximately 3.1, or π, and developed the generalizations C/d = π and C = πd, where C is circumference and d is diameter. Then, using a piece of rope to define a radius from a given point, the teachers, working in small groups, gathered circumference and radius data. One teacher held the end of a piece of rope; another teacher, holding the rope taut a given distance away, walked in a circle around the first teacher. The teachers gathered circumference data for various radii, such as 2 feet, 2.5 feet, and 3 feet. From these data, they developed the relationships C/r = 2π, C = 2πr, and d = 2r, where C is circumference, r is radius, and d is diameter. We then returned to the Rope around the Earth problem. Some teachers requested the value of the Earth’s circumference, which is approximately 25,000 miles, or its diameter; others did not need any measurements. The teachers used diagrams and the V O L . 7 , N O . 4 . DECEMBER 2001 239 lengths, that is, slightly less than 1 foot. Individuals who need more concrete results could complete the calculations using actual ball measurements. PHOTOGRAPH BY MARIA FERNANDEZ; ALL RIGHTS RESERVED Algebra as generalized arithmetic circumference relationships to determine the amount of separation between the rope and the Earth. A few teachers solved the problem abstractly without using measurements of the Earth. Figure 3 shows one of these generalized solutions. The teachers determined that the separation between the rope and the Earth is slightly less than 1 foot, which would allow a rabbit to hop under the rope. After they found the generalized solution, the teachers discussed why the earlier explorations with the string and the balls seemed to produce the same distance between the balls and their corresponding string Helping T 2A teachers develop the view of algebra as generalized arithmetic was important in helping them realize the value of having students explore and generate relationships for numerical operations. The circumference investigations that led to the generalizations for C/d and C/r helped foster this view. Explorations of arithmetic operations involving a variety of representations with whole and rational numbers were used to discuss ways to develop generalized rules for these operations. Results of the project Through engaging in and discussing these mathematics experiences, T 2A teachers have expanded their views of algebra and algebraic thinking, as well as their understanding of the importance of engaging students in problem solving and exploration of mathematical ideas through a variety of representations. Before the project, approximately 85 percent of the teachers’ definitions of algebra focused on the manipulation of symbols or variables. They described algebra as “solving equations” and “the use of numbers and letters.” Through this project, all the teachers have expanded their views of algebra, often using more than one view to define the discipline. Seventy percent described algebra as the study of relationships or patterns; 55 percent, as a tool for problem solving; and 50 percent, as generalizing arithmetic. Figure 4 shows a fifth-grade teacher’s written statement that captures this change in perspective. In connection with the broadened views, the ex- I learned to look at algebra differently than I had in previous years. Whereas I felt algebra was the substitution of variables for numbers (without a lot of understanding). Now I see that algebra has to deal with the relationships of numbers and objects; the generalization of mathematical problems; and the study of patterns. I also developed an understanding that I should have students problem solve first then explain a process (something I did just the opposite of). Fig. 3 Generalized solution for the Rope around the Earth problem 240 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL Fig. 4 A fifth-grade teacher’s expanded view of algebra amples given by the teachers to represent algebra problems also changed greatly. Before the project, the majority of teachers gave examples focusing on symbolic manipulation, such as solving 2x + 5 = 13 or simplifying 3x – 5 + 5x. Through this project, the teachers’ examples changed to include problems and contexts that involve finding and generalizing a relationship, as well as word problems that involve algebraic thinking. Their later examples often used multiple representations. Figure 5 shows one teacher’s examples before and after participating in the project. Example before the project: 3x 2 + 4x + 8 = 44 Example after participation in the project: 1 2 3 How many blocks are in the 10th figure? In the “nth” figure? Fig. 5 A seventh-grade teacher’s examples of algebra problems The teachers’ expanded views have helped them understand the importance of involving their students in discovering relationships, translating among representations, and engaging in problem solving. Project teachers have begun to emphasize problem solving and students’ discovery of relationships and patterns in their lessons. Some are working in teams to plan, share, or teach these lessons. Recognizing the connections and fundamental nature of these activities to students’ study of algebra was very important for the elementary school and middle school teachers. All the teachers are encouraging their students to use and translate among mathematical representations that they had not addressed in the past. In particular, project funds provided the teachers with money to purchase manipulatives, such as fraction bars, color tiles, base-ten blocks, 1-inch cubes, and algebra tiles, that the teachers have begun to incorporate in their lessons. Overall, the T 2A project is having a positive influence on teachers’ knowledge and practices. We encourage other teachers to seek out similar grantfunded programs in their areas or write to the authors for more information on the T 2A project. References Fullan, Michael G. “Staff Development, Innovation and Institutional Development.” In Changing School Culture through Staff Development, edited by Bruce Joyce, pp. 3–25. Alexandria, Va.: Association for Supervision and Curriculum Development, 1990. Lesh, Richard A., Thomas R. Post, and Marilyn J. Behr. “Representations and Translations among Representations in Mathematics Learning and Problem Solving.” In Problems of Representations in the Teaching and Learning of Mathematics, edited by Claude Janvier, pp. 33–40. Hillside, N.J.: Lawrence Erlbaum Associates, 1987. National Council of Teachers of Mathematics (NCTM). Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: NCTM, 1989. ———. Principles and Standards for School Mathematics. Reston, Va.: NCTM, 2000. C Statement of Ownership, Management, and Circulation Statement of ownership, management, and circulation (Required by 39 U.S.C. 3685). 1. Publication title: Mathematics Teaching in the Middle School. 2. Publication number: 1072-0839. 3. Filing date: September 2001. 4. Issue frequency: September–May, monthly. 5. Number of issues published annually: 9. 6. 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