Copyright © 2014 by 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 other formats without written permission from NCTM.

■ CHAPTER 17

Exploring Functions in Elementary School: Leveraging the Representational Context Darrell Earnest, University of Massachusetts, Amherst

T

here is no longer any question about whether elementary school children can or should be engaged in algebraic reasoning—they can be and they should be. Research has provided compelling evidence that a functions approach to elementary algebra, in which instruction builds on young children’s understanding of arithmetic through functions, supports a lgebraic reasoning in powerful ways (Carraher and Schliemann 2007; Kaput 1998). Early algebra does not mean simply spreading the content of an algebra 1 course over e arlier grades; rather, it means providing children opportunities to represent and generalize relations between sets of quantities in increasingly systematic ways through rich problem contexts (Kaput 2008). While standards have long since been revised to emphasize algebraic reasoning in the early grades (NCTM 2000; National Governors Association Center for Best Practices [NGA Center] and Council of Chief State School Officers [CCSSO] 2010), a remaining challenge in implementation is recognizing in-the-moment teaching moves that enable students to build and express generalizations systematically. The challenge reflects the difficult work of bridging domain concerns (e.g., How is this lesson related to algebra?) with developmental concerns (e.g., What are appropriate and productive teaching moves—including questions and strategic uses of representations—that authentically target learning appropriate for a young student?). I argue that the mathematics modeled in a representation is directly related to how a teacher can support elementary students building and expressing generalizations. To illustrate this, this chapter considers the role of the representational context (Ball 1993), which refers to the interactions and discourse co-constructed by the teacher and students that allow students to “explore, test, reason, and argue about” mathematical ideas (p. 160). Exploring, testing, reasoning, and arguing are of particular import when considering the

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topic of f unctions in elementary school, as these mathematical practices foster discussion and e nable students to analyze and express generalizations about quantities. The representational context construct has potential to support in-service and preservice teachers’ understanding of the mathematics and pedagogy involved in teaching and learning functions topics in elementary school. In this chapter, I present a case study of one third-grade teacher’s lesson implementation, in which students produced increasingly systematic expressions of generalizations. The case illuminates how the representational context creates opportunities for students’ mathematical work to focus on relations between two sets of quantities. I first elaborate on the focal lesson, the Dots problem (Bass, Usiskin, and Burrill 2002), which features a growing pattern that changes over time. I next describe the researcher-teacher collaborative project from which this case comes. I then present the case of one grade 3 lesson in which the representational context is used to carefully and continuously interweave domain and developmental considerations, thereby serving as an anchor for students’ systematic work to build and express generalizations.

■ The Focal Lesson The Dots problem features a growing pattern that starts with a single dot (see Bass et al. 2002 for an enactment in grade 6), with four dots added after each passing minute (see fig. 17.1). The dots representation models the function f(x) = 4x + 1. The function’s variable quantities include the number of minutes (x) and the total number of dots (f(x)), yet students must also grapple with the change (4 dots) and constant (1 dot). The functional aspect of the Dots problem is in the covarying of two sets of quantities: the number of minutes and the total number of dots; the total number of dots output depends on the number of minutes input (similar problems may use a step number instead of time). As the teacher strategically provides prompts to engage students in considering the relation between the two sets of quantities, children reason and argue about patterns that are at once obvious to them and yet challenging to record in ways that reflect the functional aspects of the growing pattern. While the problem design is relatively simple and without a real-world context, it is because of—and not in spite of—the simplicity of the dots problem that young students have access to the mathematical function. a.

b.

Start

c.

After one minute

d.

After two minutes

After three minutes

Fig. 17.1. The Dots problem during the first three minutes

The representational context creates a space for students to focus on important mathematical details involving the behavior of the two variable quantities. While students in a traditional

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algebra 1 course may be tasked to quickly find an equation for the function in the form of y = mx + b, younger students’ focus is on analyzing quantities and how those quantities covary (change in relation to each other) and expressing generalizations in increasingly systematic ways. When asked about the Dots problem after a given number of minutes (e.g., one minute or two minutes), thirdgrade students easily provide a single value for the total number of dots (e.g., five dots or nine dots). However, the total is, in a conceptual sense, not the sum of its parts, as the single value masks the covariation of the number of minutes to the total number of dots. When young students are given the opportunity to disambiguate these quantities from the singular total—that is, create a record of work that allows them to analyze the covariation, the change, and the constant—they have access to algebraic properties of the problem.

■ A Research-Practice Collaboration Data presented in this chapter come from the Early Algebra, Early Arithmetic Project (National Science Foundation grant #0310171, Algebra in Early Mathematics, http://ase.tufts.edu/ education/earlyalgebra/about.asp), a longitudinal investigation involving students from grades 3 to 5. Using a functions approach to problem design, the goal of the project was to explore the algebraic character of arithmetic through activities in which, instead of computing results of operations on specific values, students were asked to consider relations between sets of numbers and to use multiple representations for such relations. Starting from their own intuitive ways to represent verbal statements, students were gradually introduced to number lines, tables of values, graphs, and algebra notation as they considered the covariation between two sets of values. Our classroom intervention data have shown that young students can use mathematical notation not only to record their understanding but also to structure and further their ongoing thinking (Brizuela and Earnest 2008; Carraher et al. 2006; Earnest and Balti 2008). The classroom intervention took place in an urban public school with a predominantly African American population and a strong presence of immigrants from Central America and Cape Verde. A first cohort of students (n = 27) was taught by a research team member beginning in grade 3; the following academic year, the s econd cohort (n = 28), featured in this chapter, was taught by the regular classroom t eacher in collaboration with the research team. Grade 3 featured a total of fifty-one lessons taught approximately twice a week. The case discussed in this chapter focuses on a single fifty-minute, inquiry-oriented lesson from February. Each lesson followed the same three-part structure: an opening discussion in which the problem was introduced, individual or group work during which students grappled with the problem, and a final discussion. Prior to the Dots problem, students e ncountered activities focused on representing additive differences with drawings, number lines, t ables, and algebraic notation for variables. All lessons were videotaped by a research team m ember.

■ Case Study The case highlights the strategic role of the representational context (RC) in allowing students to explore, test, reason, and form arguments about the covarying quantities of the mathematical function. I identified the following three aspects of the focal lesson to illuminate how the RC enabled systematic exploration of the function, each of which I discuss in turn: (a) disambiguating the quantities, (b) exploring functions and recursions, and (c) examining records of mathematical work.

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Disambiguating the Quantities Ms. B introduced the problem with a single dot drawn on a whiteboard. As each subsequent minute passes in the context of the problem, dots appear to form an ever-extending X shape (fig. 17.1). From the beginning, Ms. B set the stage for her students to eventually compare two sets of quantities as they vary: number of minutes and total number of dots. After informing students that she was displaying what happens after one minute and soliciting observations—Connor (all student and teacher names have been changed) pointed out that the dots resembled an X, while Max observed that it grew by four dots—Ms. B asked questions to direct students’ attention to the varying quantities of the function: How many dots did we start with? How many minutes have passed? How many dots did it grow by? How many dots are there total? Various students contributed to answering these questions. After repeating these questions as the dots context reached two and then three minutes, Ms. B introduced the idea that they might be able to use the dots representation to keep track of the passing minutes. Although each set of four-dots-per-minute may appear obvious when drawn, this fleeting moment is quickly over, leaving the dots without a parsed structure reflecting the function. The visually rich representation invites children to parse (impose structure on) the dots in various ways that may or may not serve as a model of the function. In various implementations, we have seen children identify and count triangles formed by the dots, number individual dots for a total count, or use boxes to group four dots for each minute (see fig. 17.2), yet only the box strategy models the function. Ms. B challenged students to explore and test the dots representation to show the number of minutes, thereby further supporting the disambiguation of the covarying quantities and contributing to the co-construction of the RC. a.

b.

1

1 2

2

9 3

4 3 7

c.

10 4

5

8

11

6

12 13

Fig. 17.2. Parsing strategies, including (a) counting triangles, (b) assigning a unique number to each dot, and (c) using boxes for groups of four dots

Initial suggestions for how to keep track of dots reflected the challenge of parsing the representation in mathematically productive ways. Samuel suggested that they place a number beside each dot to count them all. Alexis suggested placing large circles around some of the dots, which Ms. B threw back to the class by asking exactly which dots should be connected in order to show the minutes. Connor came to the board to connect sets of four dots using the boxes strategy (see fig. 17.2c), consistent with an implementation involving sixth graders (Bass et al. 2002). Sebastian suggested that someone could count all the boxes to determine the number of minutes. Connor’s box contribution allowed other students to test this strategy for keeping track of the minutes. Anchored by the representation, the classroom discussion progressed to making connections between the dots and the underlying mathematical function.

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Exploring Functions and Recursions The Dots problem allows students to focus on either recursive properties of the function (adding on by four) or on input-output relations. Exploring the recursive function is a critical aspect of third-graders’ work, yet this is ultimately insufficient on its own to explore covariation between sets of quantities (i.e., relations between the set of inputs and corresponding set of outputs). Children working with growing patterns like the Dots problem often easily discover the recursive function. As Max initially observed, “It grew by four!” With the recursive function, students determine the total number of dots by adding four to the total for the prior minute. Using this approach, one may determine the number of dots for one hundred minutes only by first finding the total for each of the previous number of minutes, from minute one, and adding four dots each time. After students first considered the recursive properties of the function, the RC supported students’ moving toward input-output relations. Ms. B introduced a function table and asked students to help her fill it in. Her strategy is consistent with Ball’s (1993) emphasis on additional representations to further support classroom dialogue. Writing the value at the start (one dot) and after one minute (five dots), Ms. B asked the class to help fill in the number of dots after two and three minutes (see fig. 17.3). Emmanuel provided the number after two minutes in terms of the recursive function (“Five plus four equals nine”), and Angelica replicated this strategy for three minutes (“Thirteen. Because you have nine, and you have four more, and that’s thirteen”). Sebastian then used input-output relations for the first time by counting the boxes in the dots graphic: “I counted by fours because there’s four dots in each box. And after, I counted to twelve and then I added one more ’cause there’s one left over. And I got thirteen.” His contribution marked a transition that eventually allowed students to consider the function in terms of input-output relations. Minutes

Total Number of Dots

0

1

1

5

2

9

3

13

Fig. 17.3. The function table used for the Dots problem

The focus on the recursive function is mathematically productive and marks important first steps in determining input-output relations. An important note is that Ms. B treated students’ focus on the recursive function as legitimate and valuable, while at the same time she drew upon the RC to enable their consideration of the covariation between input and output. This shift to considering relations between sets of quantities is critical, because it signifies that children do not need to know the number of dots for a prior minute to add four to calculate the new total. Rather, children can determine the precise number of dots given any number of minutes. This work in the initial discussion prepared students to consider the total number of dots after ten minutes and one hundred minutes.

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Examining Records of Mathematical Work Written representations are inextricably linked to mathematical work. In this case, Ms. B relies on two representations, the dots and the function table, as models of the mathematical function. In individual work and in the final discussion, children explore and coordinate these in important ways that reflect the critical role of records of mathematical work in extending and corroborating thinking. Next I provide three examples: (1) Samuel’s doubling strategy, (2) Tyler’s multiples of four, and (3) Emma’s and Connor’s solution for one hundred minutes. First, Samuel attempted to extend the function table during individual work to determine the total number of dots after 10 minutes. Observing the function table on the whiteboard as a record for up to 5 minutes (21 dots), Samuel explained to Ms. B that, since 10 is the double of 5, the total number of dots must also double from 21 to 42 (see fig. 17.4). Mathematically speaking, Samuel has incorrectly doubled the entire function including the “+ 1” invariant (2(f(x) = 2(4x + 1)) instead of doubling just the number of minutes (f(2x) = 4(2x) + 1)). Such a strategy highlights the tension between domain and developmental concerns. As a third grader, Samuel has likely encountered pattern activities that did not include a starting amount, such as common elementary activities involving color patterns (e.g., a-b-c-a-b-c) or covering hexagon pattern blocks with triangles or trapezoids. Unlike these activities, the Dots problem required that Samuel consider the starting dot, which serves as a constant in the function (or, as the y-intercept b in y = mx + b). As a result, a mathematical strategy that is successful in other contexts is not sufficient in the Dots problem. Rather than being an issue, this presents a rich opportunity to explore the covarying quantities of the function and the +1 constant. Minutes

Total Number of Dots

0

1

1

5

2

9

3

13

4

17

5

21

5×2

21 × 2 10

10

Fig. 17.4. Samuel’s function-table doubling strategy

Recognizing that Samuel counted the starting dot twice, Ms. B suggested that he test the 42 dots in his function table by coordinating with the dots graphic, thereby capitalizing on the RC. Later, in the final discussion, Ms. B called on Samuel to respond to the question of 10 minutes without knowing if Samuel had revised his response. He reported to the class, “I

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did 21 plus 21. That’s 42 dots, but it’s 41. Because 42 doesn’t equal how many dots there are.” Reflecting on his initial solution of 42 dots, he stated, “I added one more.” Samuel coordinated the two records of mathematical work to corroborate (and disprove) his initial assertion of 42 dots for 10 minutes. Ms. B’s initial teaching moves that enabled exploration of the function eventually allowed Samuel to test for himself the quantities of the function. Tyler began individual work by writing a string of the first ten multiples of 4 to find the number of dots after 10 minutes, ending with 40 (see fig. 17.5a) instead of 41. Ms. B recognized that Tyler, like Samuel, might be ignoring the starting dot (note that Tyler’s total for 10 minutes is one dot less than 41, while Samuel’s total is one dot more). Ms. B asked him to draw for her what the dots look like after one minute; in other words, she asked him to corroborate the values in his string of multiples with the dots representation. Tyler’s resulting drawing allowed Ms. B to confirm that his thinking indeed omitted the starting dot (see fig. 17.5b) and also that the boxes were a powerful representational component for Tyler. Ms. B asked him how many dots they had before any minutes passed. Instead of using words, Tyler looked at the dots graphic on the whiteboard and then amended his own drawing to include a single dot in the middle (see fig. 17.5c). The RC enabled Ms. B to address Tyler’s idea in such a way that he generated the accurate response himself through existing records of mathematical work.

Fig. 17.5. Tyler (a) recorded the multiples of 4, (b) drew 4 dots after one minute, and (c) amended his drawing to include the starting dot.

Two students, Emma and Connor, each extended the dots graphic to determine the total number of dots after 100 minutes. Large numbers are a useful instructional strategy to promote consideration of input-output relations (Earnest and Balti 2008), particularly because d rawing all dots after 100 minutes would be cumbersome and time consuming, thereby motivating a need to generalize from input to output. Instead of drawing the dots for 100 minutes, both Emma and Connor abstracted an emerging pattern in each of the four legs of the dots representation (see fig. 17.6). Emma accurately explained during the final discussion that while she did not draw out all the dots, she knew that there would be 100 dots in each leg of the X, just as there had been 10 dots in each leg after 10 minutes. Emma captured this by saying, “100 plus 100 plus 100 plus 100 is 400, and then I added the 1, so it’s 401.” The RC supported Emma’s and Connor’s increasingly more systematic generalizations of the function’s properties. In these three examples of student interactions, the various records of mathematical work provided access for third-grade students to the function’s covariation.

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Fig. 17.6. Connor’s representation of dots after one hundred minutes

■ Conclusion The Dots problem case study illuminates how representations that model a mathematical function provide opportunities for students to explore, test, reason, and argue about relations between sets of quantities. Data from our research-practice collaboration demonstrate the utility of the representational context for the exploration of functions in elementary school. First, strategic development of the RC supports students in analyzing the covarying quantities involved in a functions-based problem for elementary students. Second, the RC provides a mechanism by which to support a transition from the recursive function to input-output relations. Third, records of mathematical work allow students to develop, express, and corroborate generalizations observed in sets of quantities. The purpose of this chapter was to highlight the supportive role of the representational context in learning and teaching the topic of functions in elementary school. The case of the Dots problem underscores the importance of making explicit for children the mathematical connections between the problem context and the representations used in a lesson. Teaching moves that leverage the representational context, like those in the featured case study, have the potential to support algebraic reasoning in elementary school.

■ References Ball, Deborah Loewenberg. “Halves, Pieces, and Twoths: Constructing and Using Representational Contexts in Teaching Fractions.” In Rational Numbers: An Integration of Research, edited by Thomas P. Carpenter, Elizabeth Fennema, and Thomas A. Romberg, pp. 157–95. Hillsdale, N.J.: Lawrence Erlbaum, 1993. Bass, Hyman, Zalman Usiskin, and Gail Burrill, eds. Appendix F in Studying Classroom Teaching as a M edium for Professional Development: Proceedings of a U.S.-Japan Workshop, pp. 157–76. Washington, D.C.: National Academies Press, 2002. Brizuela, Bárbara, and Darrell Earnest. “Multiple Notations and Algebraic Understandings: The Case of the Best Deal Problem.” In Algebra in the Early Grades, edited by James J. Kaput, David W. Carraher, and Maria L. Blanton, pp. 273–301. Mahwah, N.J.: Lawrence Erlbaum Associates, 2008. Carraher, David, and Analúcia Schliemann. “Early Algebra and Algebraic Reasoning.” In Second Handbook of Research on Mathematics Teaching and Learning, edited by Frank K. Lester, Jr., pp. 669–705. Charlotte, N.C.: Information Age Publishing, 2007.

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Carraher, David W., Analúcia D. Schliemann, Bárbara M. Brizuela, and Darrell Earnest. “Arithmetic and Algebra in Early Mathematics Education.” Journal for Research in Mathematics Education 37, no. 2 (March 2006): 87–115. Earnest, Darrell, and Aadina A. Balti. “Instructional Strategies for Grade 3 Algebra: Results of a Research Practice Collaboration.” Teaching Children Mathematics (May 2008): 518–22. Kaput, James J. “What Is Algebra? What Is Early Algebra?” In Algebra in the Early Grades, edited by James J. Kaput, David W. Carraher, and Maria L. Blanton, pp. 5–17. Mahwah, N.J.: Lawrence Erlbaum, 2008. ———. “Transforming Algebra from an Engine of Inequity to an Engine of Mathematical Power by ‘Algebrafying’ the K–12 Curriculum.” In The Nature and Role of Algebra in the K–14 Curriculum, edited by the National Council of Teachers of Mathematics, pp. 25–26. Washington, D.C.: National Academies Press, 1998. National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, Va.: NCTM, 2000. National Governors Association Center for Best Practices and Council of Chief State School Officers (NGA Center and CCSSO). Common Core State Standards for Mathematics. Washington, D.C.: NGA Center and CCSSO, 2010. http://www.corestandards.org.

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Exploring Functions in Elementary School: Leveraging the Representational Context Darrell Earnest, University of Massachusetts, Amherst

T

here is no longer any question about whether elementary school children can or should be engaged in algebraic reasoning—they can be and they should be. Research has provided compelling evidence that a functions approach to elementary algebra, in which instruction builds on young children’s understanding of arithmetic through functions, supports a lgebraic reasoning in powerful ways (Carraher and Schliemann 2007; Kaput 1998). Early algebra does not mean simply spreading the content of an algebra 1 course over e arlier grades; rather, it means providing children opportunities to represent and generalize relations between sets of quantities in increasingly systematic ways through rich problem contexts (Kaput 2008). While standards have long since been revised to emphasize algebraic reasoning in the early grades (NCTM 2000; National Governors Association Center for Best Practices [NGA Center] and Council of Chief State School Officers [CCSSO] 2010), a remaining challenge in implementation is recognizing in-the-moment teaching moves that enable students to build and express generalizations systematically. The challenge reflects the difficult work of bridging domain concerns (e.g., How is this lesson related to algebra?) with developmental concerns (e.g., What are appropriate and productive teaching moves—including questions and strategic uses of representations—that authentically target learning appropriate for a young student?). I argue that the mathematics modeled in a representation is directly related to how a teacher can support elementary students building and expressing generalizations. To illustrate this, this chapter considers the role of the representational context (Ball 1993), which refers to the interactions and discourse co-constructed by the teacher and students that allow students to “explore, test, reason, and argue about” mathematical ideas (p. 160). Exploring, testing, reasoning, and arguing are of particular import when considering the

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Using Research to Improve Instruction

topic of f unctions in elementary school, as these mathematical practices foster discussion and e nable students to analyze and express generalizations about quantities. The representational context construct has potential to support in-service and preservice teachers’ understanding of the mathematics and pedagogy involved in teaching and learning functions topics in elementary school. In this chapter, I present a case study of one third-grade teacher’s lesson implementation, in which students produced increasingly systematic expressions of generalizations. The case illuminates how the representational context creates opportunities for students’ mathematical work to focus on relations between two sets of quantities. I first elaborate on the focal lesson, the Dots problem (Bass, Usiskin, and Burrill 2002), which features a growing pattern that changes over time. I next describe the researcher-teacher collaborative project from which this case comes. I then present the case of one grade 3 lesson in which the representational context is used to carefully and continuously interweave domain and developmental considerations, thereby serving as an anchor for students’ systematic work to build and express generalizations.

■ The Focal Lesson The Dots problem features a growing pattern that starts with a single dot (see Bass et al. 2002 for an enactment in grade 6), with four dots added after each passing minute (see fig. 17.1). The dots representation models the function f(x) = 4x + 1. The function’s variable quantities include the number of minutes (x) and the total number of dots (f(x)), yet students must also grapple with the change (4 dots) and constant (1 dot). The functional aspect of the Dots problem is in the covarying of two sets of quantities: the number of minutes and the total number of dots; the total number of dots output depends on the number of minutes input (similar problems may use a step number instead of time). As the teacher strategically provides prompts to engage students in considering the relation between the two sets of quantities, children reason and argue about patterns that are at once obvious to them and yet challenging to record in ways that reflect the functional aspects of the growing pattern. While the problem design is relatively simple and without a real-world context, it is because of—and not in spite of—the simplicity of the dots problem that young students have access to the mathematical function. a.

b.

Start

c.

After one minute

d.

After two minutes

After three minutes

Fig. 17.1. The Dots problem during the first three minutes

The representational context creates a space for students to focus on important mathematical details involving the behavior of the two variable quantities. While students in a traditional

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algebra 1 course may be tasked to quickly find an equation for the function in the form of y = mx + b, younger students’ focus is on analyzing quantities and how those quantities covary (change in relation to each other) and expressing generalizations in increasingly systematic ways. When asked about the Dots problem after a given number of minutes (e.g., one minute or two minutes), thirdgrade students easily provide a single value for the total number of dots (e.g., five dots or nine dots). However, the total is, in a conceptual sense, not the sum of its parts, as the single value masks the covariation of the number of minutes to the total number of dots. When young students are given the opportunity to disambiguate these quantities from the singular total—that is, create a record of work that allows them to analyze the covariation, the change, and the constant—they have access to algebraic properties of the problem.

■ A Research-Practice Collaboration Data presented in this chapter come from the Early Algebra, Early Arithmetic Project (National Science Foundation grant #0310171, Algebra in Early Mathematics, http://ase.tufts.edu/ education/earlyalgebra/about.asp), a longitudinal investigation involving students from grades 3 to 5. Using a functions approach to problem design, the goal of the project was to explore the algebraic character of arithmetic through activities in which, instead of computing results of operations on specific values, students were asked to consider relations between sets of numbers and to use multiple representations for such relations. Starting from their own intuitive ways to represent verbal statements, students were gradually introduced to number lines, tables of values, graphs, and algebra notation as they considered the covariation between two sets of values. Our classroom intervention data have shown that young students can use mathematical notation not only to record their understanding but also to structure and further their ongoing thinking (Brizuela and Earnest 2008; Carraher et al. 2006; Earnest and Balti 2008). The classroom intervention took place in an urban public school with a predominantly African American population and a strong presence of immigrants from Central America and Cape Verde. A first cohort of students (n = 27) was taught by a research team member beginning in grade 3; the following academic year, the s econd cohort (n = 28), featured in this chapter, was taught by the regular classroom t eacher in collaboration with the research team. Grade 3 featured a total of fifty-one lessons taught approximately twice a week. The case discussed in this chapter focuses on a single fifty-minute, inquiry-oriented lesson from February. Each lesson followed the same three-part structure: an opening discussion in which the problem was introduced, individual or group work during which students grappled with the problem, and a final discussion. Prior to the Dots problem, students e ncountered activities focused on representing additive differences with drawings, number lines, t ables, and algebraic notation for variables. All lessons were videotaped by a research team m ember.

■ Case Study The case highlights the strategic role of the representational context (RC) in allowing students to explore, test, reason, and form arguments about the covarying quantities of the mathematical function. I identified the following three aspects of the focal lesson to illuminate how the RC enabled systematic exploration of the function, each of which I discuss in turn: (a) disambiguating the quantities, (b) exploring functions and recursions, and (c) examining records of mathematical work.

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Disambiguating the Quantities Ms. B introduced the problem with a single dot drawn on a whiteboard. As each subsequent minute passes in the context of the problem, dots appear to form an ever-extending X shape (fig. 17.1). From the beginning, Ms. B set the stage for her students to eventually compare two sets of quantities as they vary: number of minutes and total number of dots. After informing students that she was displaying what happens after one minute and soliciting observations—Connor (all student and teacher names have been changed) pointed out that the dots resembled an X, while Max observed that it grew by four dots—Ms. B asked questions to direct students’ attention to the varying quantities of the function: How many dots did we start with? How many minutes have passed? How many dots did it grow by? How many dots are there total? Various students contributed to answering these questions. After repeating these questions as the dots context reached two and then three minutes, Ms. B introduced the idea that they might be able to use the dots representation to keep track of the passing minutes. Although each set of four-dots-per-minute may appear obvious when drawn, this fleeting moment is quickly over, leaving the dots without a parsed structure reflecting the function. The visually rich representation invites children to parse (impose structure on) the dots in various ways that may or may not serve as a model of the function. In various implementations, we have seen children identify and count triangles formed by the dots, number individual dots for a total count, or use boxes to group four dots for each minute (see fig. 17.2), yet only the box strategy models the function. Ms. B challenged students to explore and test the dots representation to show the number of minutes, thereby further supporting the disambiguation of the covarying quantities and contributing to the co-construction of the RC. a.

b.

1

1 2

2

9 3

4 3 7

c.

10 4

5

8

11

6

12 13

Fig. 17.2. Parsing strategies, including (a) counting triangles, (b) assigning a unique number to each dot, and (c) using boxes for groups of four dots

Initial suggestions for how to keep track of dots reflected the challenge of parsing the representation in mathematically productive ways. Samuel suggested that they place a number beside each dot to count them all. Alexis suggested placing large circles around some of the dots, which Ms. B threw back to the class by asking exactly which dots should be connected in order to show the minutes. Connor came to the board to connect sets of four dots using the boxes strategy (see fig. 17.2c), consistent with an implementation involving sixth graders (Bass et al. 2002). Sebastian suggested that someone could count all the boxes to determine the number of minutes. Connor’s box contribution allowed other students to test this strategy for keeping track of the minutes. Anchored by the representation, the classroom discussion progressed to making connections between the dots and the underlying mathematical function.

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Exploring Functions and Recursions The Dots problem allows students to focus on either recursive properties of the function (adding on by four) or on input-output relations. Exploring the recursive function is a critical aspect of third-graders’ work, yet this is ultimately insufficient on its own to explore covariation between sets of quantities (i.e., relations between the set of inputs and corresponding set of outputs). Children working with growing patterns like the Dots problem often easily discover the recursive function. As Max initially observed, “It grew by four!” With the recursive function, students determine the total number of dots by adding four to the total for the prior minute. Using this approach, one may determine the number of dots for one hundred minutes only by first finding the total for each of the previous number of minutes, from minute one, and adding four dots each time. After students first considered the recursive properties of the function, the RC supported students’ moving toward input-output relations. Ms. B introduced a function table and asked students to help her fill it in. Her strategy is consistent with Ball’s (1993) emphasis on additional representations to further support classroom dialogue. Writing the value at the start (one dot) and after one minute (five dots), Ms. B asked the class to help fill in the number of dots after two and three minutes (see fig. 17.3). Emmanuel provided the number after two minutes in terms of the recursive function (“Five plus four equals nine”), and Angelica replicated this strategy for three minutes (“Thirteen. Because you have nine, and you have four more, and that’s thirteen”). Sebastian then used input-output relations for the first time by counting the boxes in the dots graphic: “I counted by fours because there’s four dots in each box. And after, I counted to twelve and then I added one more ’cause there’s one left over. And I got thirteen.” His contribution marked a transition that eventually allowed students to consider the function in terms of input-output relations. Minutes

Total Number of Dots

0

1

1

5

2

9

3

13

Fig. 17.3. The function table used for the Dots problem

The focus on the recursive function is mathematically productive and marks important first steps in determining input-output relations. An important note is that Ms. B treated students’ focus on the recursive function as legitimate and valuable, while at the same time she drew upon the RC to enable their consideration of the covariation between input and output. This shift to considering relations between sets of quantities is critical, because it signifies that children do not need to know the number of dots for a prior minute to add four to calculate the new total. Rather, children can determine the precise number of dots given any number of minutes. This work in the initial discussion prepared students to consider the total number of dots after ten minutes and one hundred minutes.

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Examining Records of Mathematical Work Written representations are inextricably linked to mathematical work. In this case, Ms. B relies on two representations, the dots and the function table, as models of the mathematical function. In individual work and in the final discussion, children explore and coordinate these in important ways that reflect the critical role of records of mathematical work in extending and corroborating thinking. Next I provide three examples: (1) Samuel’s doubling strategy, (2) Tyler’s multiples of four, and (3) Emma’s and Connor’s solution for one hundred minutes. First, Samuel attempted to extend the function table during individual work to determine the total number of dots after 10 minutes. Observing the function table on the whiteboard as a record for up to 5 minutes (21 dots), Samuel explained to Ms. B that, since 10 is the double of 5, the total number of dots must also double from 21 to 42 (see fig. 17.4). Mathematically speaking, Samuel has incorrectly doubled the entire function including the “+ 1” invariant (2(f(x) = 2(4x + 1)) instead of doubling just the number of minutes (f(2x) = 4(2x) + 1)). Such a strategy highlights the tension between domain and developmental concerns. As a third grader, Samuel has likely encountered pattern activities that did not include a starting amount, such as common elementary activities involving color patterns (e.g., a-b-c-a-b-c) or covering hexagon pattern blocks with triangles or trapezoids. Unlike these activities, the Dots problem required that Samuel consider the starting dot, which serves as a constant in the function (or, as the y-intercept b in y = mx + b). As a result, a mathematical strategy that is successful in other contexts is not sufficient in the Dots problem. Rather than being an issue, this presents a rich opportunity to explore the covarying quantities of the function and the +1 constant. Minutes

Total Number of Dots

0

1

1

5

2

9

3

13

4

17

5

21

5×2

21 × 2 10

10

Fig. 17.4. Samuel’s function-table doubling strategy

Recognizing that Samuel counted the starting dot twice, Ms. B suggested that he test the 42 dots in his function table by coordinating with the dots graphic, thereby capitalizing on the RC. Later, in the final discussion, Ms. B called on Samuel to respond to the question of 10 minutes without knowing if Samuel had revised his response. He reported to the class, “I

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did 21 plus 21. That’s 42 dots, but it’s 41. Because 42 doesn’t equal how many dots there are.” Reflecting on his initial solution of 42 dots, he stated, “I added one more.” Samuel coordinated the two records of mathematical work to corroborate (and disprove) his initial assertion of 42 dots for 10 minutes. Ms. B’s initial teaching moves that enabled exploration of the function eventually allowed Samuel to test for himself the quantities of the function. Tyler began individual work by writing a string of the first ten multiples of 4 to find the number of dots after 10 minutes, ending with 40 (see fig. 17.5a) instead of 41. Ms. B recognized that Tyler, like Samuel, might be ignoring the starting dot (note that Tyler’s total for 10 minutes is one dot less than 41, while Samuel’s total is one dot more). Ms. B asked him to draw for her what the dots look like after one minute; in other words, she asked him to corroborate the values in his string of multiples with the dots representation. Tyler’s resulting drawing allowed Ms. B to confirm that his thinking indeed omitted the starting dot (see fig. 17.5b) and also that the boxes were a powerful representational component for Tyler. Ms. B asked him how many dots they had before any minutes passed. Instead of using words, Tyler looked at the dots graphic on the whiteboard and then amended his own drawing to include a single dot in the middle (see fig. 17.5c). The RC enabled Ms. B to address Tyler’s idea in such a way that he generated the accurate response himself through existing records of mathematical work.

Fig. 17.5. Tyler (a) recorded the multiples of 4, (b) drew 4 dots after one minute, and (c) amended his drawing to include the starting dot.

Two students, Emma and Connor, each extended the dots graphic to determine the total number of dots after 100 minutes. Large numbers are a useful instructional strategy to promote consideration of input-output relations (Earnest and Balti 2008), particularly because d rawing all dots after 100 minutes would be cumbersome and time consuming, thereby motivating a need to generalize from input to output. Instead of drawing the dots for 100 minutes, both Emma and Connor abstracted an emerging pattern in each of the four legs of the dots representation (see fig. 17.6). Emma accurately explained during the final discussion that while she did not draw out all the dots, she knew that there would be 100 dots in each leg of the X, just as there had been 10 dots in each leg after 10 minutes. Emma captured this by saying, “100 plus 100 plus 100 plus 100 is 400, and then I added the 1, so it’s 401.” The RC supported Emma’s and Connor’s increasingly more systematic generalizations of the function’s properties. In these three examples of student interactions, the various records of mathematical work provided access for third-grade students to the function’s covariation.

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Fig. 17.6. Connor’s representation of dots after one hundred minutes

■ Conclusion The Dots problem case study illuminates how representations that model a mathematical function provide opportunities for students to explore, test, reason, and argue about relations between sets of quantities. Data from our research-practice collaboration demonstrate the utility of the representational context for the exploration of functions in elementary school. First, strategic development of the RC supports students in analyzing the covarying quantities involved in a functions-based problem for elementary students. Second, the RC provides a mechanism by which to support a transition from the recursive function to input-output relations. Third, records of mathematical work allow students to develop, express, and corroborate generalizations observed in sets of quantities. The purpose of this chapter was to highlight the supportive role of the representational context in learning and teaching the topic of functions in elementary school. The case of the Dots problem underscores the importance of making explicit for children the mathematical connections between the problem context and the representations used in a lesson. Teaching moves that leverage the representational context, like those in the featured case study, have the potential to support algebraic reasoning in elementary school.

■ References Ball, Deborah Loewenberg. “Halves, Pieces, and Twoths: Constructing and Using Representational Contexts in Teaching Fractions.” In Rational Numbers: An Integration of Research, edited by Thomas P. Carpenter, Elizabeth Fennema, and Thomas A. Romberg, pp. 157–95. Hillsdale, N.J.: Lawrence Erlbaum, 1993. Bass, Hyman, Zalman Usiskin, and Gail Burrill, eds. Appendix F in Studying Classroom Teaching as a M edium for Professional Development: Proceedings of a U.S.-Japan Workshop, pp. 157–76. Washington, D.C.: National Academies Press, 2002. Brizuela, Bárbara, and Darrell Earnest. “Multiple Notations and Algebraic Understandings: The Case of the Best Deal Problem.” In Algebra in the Early Grades, edited by James J. Kaput, David W. Carraher, and Maria L. Blanton, pp. 273–301. Mahwah, N.J.: Lawrence Erlbaum Associates, 2008. Carraher, David, and Analúcia Schliemann. “Early Algebra and Algebraic Reasoning.” In Second Handbook of Research on Mathematics Teaching and Learning, edited by Frank K. Lester, Jr., pp. 669–705. Charlotte, N.C.: Information Age Publishing, 2007.

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Carraher, David W., Analúcia D. Schliemann, Bárbara M. Brizuela, and Darrell Earnest. “Arithmetic and Algebra in Early Mathematics Education.” Journal for Research in Mathematics Education 37, no. 2 (March 2006): 87–115. Earnest, Darrell, and Aadina A. Balti. “Instructional Strategies for Grade 3 Algebra: Results of a Research Practice Collaboration.” Teaching Children Mathematics (May 2008): 518–22. Kaput, James J. “What Is Algebra? What Is Early Algebra?” In Algebra in the Early Grades, edited by James J. Kaput, David W. Carraher, and Maria L. Blanton, pp. 5–17. Mahwah, N.J.: Lawrence Erlbaum, 2008. ———. “Transforming Algebra from an Engine of Inequity to an Engine of Mathematical Power by ‘Algebrafying’ the K–12 Curriculum.” In The Nature and Role of Algebra in the K–14 Curriculum, edited by the National Council of Teachers of Mathematics, pp. 25–26. Washington, D.C.: National Academies Press, 1998. National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, Va.: NCTM, 2000. National Governors Association Center for Best Practices and Council of Chief State School Officers (NGA Center and CCSSO). Common Core State Standards for Mathematics. Washington, D.C.: NGA Center and CCSSO, 2010. http://www.corestandards.org.

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