Connecting Rate Using Dynamic Representational Technologies

The New Jersey Mathematics Teacher, December 2013, Vol. 71, No. 2 13

Connecting Rate Using Dynamic Representational Technologies

Dr. Steven Greenstein Dept. of Mathematical Sciences

Montclair State University [email protected]

Abstract

Dynamic representational technologies (DRTs) in mathematics education are software environments that allow users to observe and interact with multiple and connected representations of concepts. This article makes a case for their use to teach rate concepts, which historically have been difficult for teachers to teach and for students to learn. Introduction

“Well, what we used to do in math class, we would, we would look at graphs and write the information down from them. But in [Mathworlds], when you do graphs, you like, you dissect it, like, you look at each part, what it means, what it’s showing, how to explain it.”

This quote from a middle school student suggests the advantages of using an approach to learning about the concept of rate that is enabled by a particular technology. Students in elementary and middle school historically have had difficulties learning rate concepts (Kouba, Zawojewski, & Strutchens, 1997). In particular, Kouba and her colleagues suggest that “students will need considerably more experience with justifying and explaining their mathematical work than they have had in the past” (p. 138). The study in which the student quoted above was a participant found significant student achievement gains (Roschelle et al., 2010), indicating that as a “dynamic representational technology,” MathWorlds could be helpful in this regard. In this article I describe how it helped students make sense of rate by justifying relationships they identified between representations of rate.

Dynamic Representation Technologies

MathWorlds is a kind of “dynamic representational technology” (DRT) that is typically used along with other curriculum materials to teach students about rate and proportionality. DRTs are software environments that allow users to observe and interact with multiple and connected representations of mathematical concepts. These multiple representations may range from animations of experiences such as moving elevators, traveling vehicles, and swimming creatures, to more formal representations such as tables of values, graphs of functions, and algebraic equations. These representations are connected in the sense that the manipulation of any one of them has an immediate, observable, logical consequence for the others.

Figure 1 shows the connected representations of MathWorlds. In clockwise order beginning at the top left are the “world” where two fish swim, and the graphical, algebraic, and tabular representations of their motion. The arrows indicate that all representations are connected to each other. Students can control the motions of the fish by modifying mathematical functions in either of their graphical or algebraic representations. After editing the functions, students can press a play button to see the

14 The New Jersey Mathematics Teacher, December 2013, Vol. 71, No. 2

corresponding animation. For example, a student can change the coefficient of x in the algebraic representation, press play, and then observe the effect of the change on Teacher’s motion in the world. In the

curriculum materials that accompany MathWorlds, students are often asked to make up stories that correspond to the functions and animations.

Figure 1: The connected representations of MathWorlds

MathWorlds provides a context for an investigation of rate in the form of speed. Students can observe speed represented in four ways: in the motions of two fish, in the algebraic and graphical representations of their motion, and in the tabular representations of their distances at particular times. What the realistic situation of a race between two fish provides is a context that students find engaging. What the connectedness of representations provides is a unique opportunity for students to make sense of the concept of “rate as speed,” as illustrated in Figure 2.

Figure 2: The concept of speed and its four

representations in MathWorlds. The unlabeled connections are relationships that students could

identify and make sense of using the software.


Connecting Representations

The Standards for Mathematical Practice of the Common Core rest upon “important ‘processes and proficiencies’” (CCSSO, 2010, p. 6) including NCTM’s Connections and Representations Process Standards. These process standards suggest that instruction enable all students to “recognize and use connections among mathematical ideas” and to “translate among mathematical representations to solve problems” (NCTM, 2000). Research shows that students can develop their understanding of concepts if teaching attends explicitly to representations of those concepts and to connections among them (Hiebert & Grouws, 2007). Therefore, because it features connected representations of rate, MathWorlds can be a particularly useful tool for supporting the development of students’ understanding of rate. The connections that the software illuminates can be seen as a resource for students’ sense-making. As students move between representations, interpreting the consequences of their manipulations of one representation on another, they can make sense of the consequences of these actions. And by making sense of these consequences, they are making sense of the mathematics embedded in these representations. This is the essence of mathematical activity. Coincidentally, the strategic use of tools like MathWorlds and the construction of viable arguments reflect the essence of the Common Core’s Standards for Mathematical Practice (CCSSO, 2010, pp. 6-7).

Speed in MathWorlds is embodied in four connected representations. The manipulation of each representation has a logical consequence for the others. That is, there’s a good reason for these consequences, but that reason must be discovered. Determining these reasons is the

essence of sense-making in this environment. For example, by changing the coefficient of x in the algebraic representation and then observing the corresponding change in the steepness of a line, a student may realize there is a connection between the graphical and algebraic representations, and wonder, “What happens to the graph if I change this number?”, “What if instead of increasing it, I decrease it?”, and “What if I make it zero?” MathWorlds is designed to be exploratory in order to support this kind of inquiry-oriented activity. Students have the control to pursue “What if?” questions – the kind that Eleanor Duckworth would call “occasions to wonder” (1995) – by making changes to the graphical or algebraic representations of motion and investigating the consequences of these changes on the other representations.

New Tools Make New Things Possible

I observed several middle school math classrooms as part of a study in which an instructional approach using MathWorlds was investigated (Empson, Greenstein, Maldonado, & Roschelle, 2012). In order to illustrate what the approach may make possible, I will describe a sequence of classroom events that we found to be useful in supporting the development of students’ understanding of rate. I’ll refer to a composite of these teachers as “Ms. Newell” and to a composite pair of students as “Mollie and Enrique.”

Ms. Newell wanted to engage her

students in “What-if” questions like the ones above to develop their understanding of rate. Specifically, this was her goal for instruction: The steeper of two lines on a position-time graph corresponds to the greater speed, because the steeper line represents a greater distance traveled (than


the less steep line) in the same amount of time.

This goal includes a connection between the motion of the fish and the steepness of the graphs of their motion, as well as a justification for this relationship. Ms. Newell’s students already knew how to graph a linear equation and to calculate (constant) speed by dividing the total distance traveled by the total time elapsed. An instructional approach featuring MathWorlds along with the following task helped many of her students make sense of rate in the context of the motion of two swimming fish named Teacher and Fishy. 1. Teacher and Fishy are having a 20-

meter race. Find two ways for Fishy to win.

a. Looking at the two ways you solved the problem, what’s alike about them? What’s different?

b. Looking across some of the ways that you and your classmates solved the problem, what do you notice? What do

you wonder about? What’s alike about them? What’s different? Do any of the solutions stand out as unique? Mollie and Enrique began by pressing

play in MathWorlds. They mentioned that both fish swam for 10 seconds and that Teacher won the race. Enrique’s first step was to change Fishy’s graph so that both fish would swim 20 meters. He moved the endpoint of Fishy’s graph up to the 20-meter mark and then pressed play. Both fish swam 20 meters in 10 seconds. Mollie responded, “Now we just need Fishy to get there faster. How do we do that?” Enrique replied, “That just means Fishy has to do it in under 10 seconds.” He moved the endpoint again, this time to the 7-second mark. He pressed play and smiled. Fishy won the race by swimming 20 meters in 7 seconds. Teacher finished in 10 seconds.

Figure 3 shows Mollie and Enrique’s two solutions to the problem. Fishy is the green fish and Teacher is the red fish.

Figure 3: Mollie and Enrique's two solutions to the problem.

A single solution to the problem could not allow Mollie and Enrique to observe that the steeper of the two lines represents the movement of the faster fish. It was only after they constructed a second solution to the problem that they felt as if they were onto something. Here’s what they said:

Mollie: We discovered that the speed has something to do with the steepness of the line, because the faster fish always had the steeper line. Enrique: We also noticed that when we make a line steeper, it makes the fish swim faster.


Mollie and Enrique noticed in both of their solutions that the line representing Fishy’s motion was steeper than Teacher’s. It was then that they began to wonder whether it was indeed the case that the steeper line represents the faster fish. This tendency for students to begin to generalize from one or two cases is powerful mathematical activity. Mathematicians engage in the same behavior, because they realize that doing so lends itself to discovering what might be true. Mollie and Enrique’s suspicions about the relationship between speed and steepness were further confirmed by looking across some of the solutions produced by their classmates. This identification of a common feature across all solutions – “the faster fish always had the steeper line” – allowed them to establish a connection between the motion of the fish in the world, in their graphs, and in their speeds (Figure 4).

Figure 4: The connection Mollie and Enrique

discovered between the motion of the fish in the world, their graphs, and their speeds.

Once Mollie and Enrique and other students in the class discovered that “steeper means faster,” they shared their findings in whole-class discussion. Unfortunately, some of the classes I observed concluded their investigation at this point. I say this is unfortunate, because – as Ms. Newell seemed to realize – “steeper means faster” is insufficient evidence of conceptual understanding. The phrase alone identifies a connection between speed and steepness, but it does not explain the relationship. Ms. Newell pressed her students to explain that relationship by justifying their claim.

Ms. Newell: So you said that the faster fish had the steeper line, right? And I see that in

your solutions. Can you tell me why the faster fish seems to always have the steeper line? Mollie and Enrique responded by calculating the speed of each fish using the table, which provided them with the total time it took for each fish to swim 20 meters. Other students made connections to the graph, and used each of the endpoints to find the total times and distances. Mollie: The faster fish had the greater speed. Ms. Newell: How did you figure that out? Mollie: We calculated Teacher’s speed and Fishy’s speed, and Fishy’s was more than Teacher’s. Ms. Newell: OK, so we know that the faster fish has the greater speed. But how do we know that the steeper line has the greater speed? Enrique: Well, it’s a 20-meter race, so both fish swam the same distance. But Fishy did it in less time than Teacher. See? Fishy’s line is steeper, because Fishy swam the race in less time. Ms. Newell: And what about Teacher’s line? Mollie: It goes farther, I mean, it’s longer. So it can’t be as steep, because they both had the same distance. That’s why Teacher is slower; her time was longer. In both of Mollie and Enrique’s solutions, the steeper line was associated with the faster fish, and the faster fish swam the same distance as the slower fish but in less time. By making sense of the connection between the motion of the fish in the world and the graphical representations of their speed, Mollie and Enrique were able to justify why the steeper line represents the faster fish.

The diagram in Figure 5 maps the connections Mollie and Enrique verbalized as they completed the problem. Their use of


MathWorlds to make sense of the consequences of their manipulations of the algebraic and graphical representations provided them with opportunities to understand why it is that “steeper means faster.” The labels are evidence of Mollie and Enrique’s sense-making, and they provided their teacher with opportunities to formatively assess their thinking as they participated in thought-revealing mathematical activities such as talking about

their ideas, testing their conjectures, and comparing their findings.

Ms. Newell helped her students use

MathWorlds with the task above to understand why it is that the steeper line corresponds to the faster fish. The task was particularly useful in that students used the patterns they noticed across multiple

Figure 5: The labeled connections are relationships Mollie and Enrique made sense of as they used MathWorlds

to complete the problem.

solutions to make sense of the connections they identified. Tasks like these have also been useful to teachers for supporting the development of their students’ understanding of rate: 2. Find two ways for Fishy to win by

swimming at twice Teacher’s speed. 3. Find two ways for Fishy to win in half

the time in takes Teacher to finish. 4. Find two ways for Fishy and Teacher to

tie. 5. Find two ways for Fishy to pass Teacher

at some point during the race.

Conclusion

MathWorlds offers students a meaningful context for thinking about rate. Furthermore, it provides connections between an animation of rate and other more formal representations. When students engaged in collaborative, inquiry-oriented problem solving using the technology along with tasks yielding multiple solutions (Stroup, Ares, & Hurford, 2005), they used these solutions as a resource for whole-class discussion focused on making sense of these connections. The teacher’s role was critical in supporting these outcomes and cannot be


emphasized enough. Posing “What if” questions is a student practice that needs to be nurtured, and the skill of facilitating whole-class discussion with the purpose of using students’ work to further support sense-making is a skill that takes considerable time to develop (Stein, Engle, Smith, & Hughes, 2008).

The concept of rate is both difficult to teach and difficult to learn. And teachers have many choices of instructional routes they could take. This is what one teacher had to say about the route she took: “I like that — going from table data to the real world [in MathWorlds] and making those connections. So, question by question, [the students] were being led toward certain conclusions, rather than, ‘Hey, here is what you are going to see,’ or ‘Let me show you an example and now you mimic what I just did…’” The implications of these findings suggest that using dynamic representational technologies along with tasks that yield multiple solutions could be a viable instructional approach for other teachers, as well.

Resources Follow this link to download SimCalc MathWorlds and a variety of curriculum units correlated with NCTM and Common Core Standards: http://www.kaputcenter.umassd.edu/products/software/smwcomp/download/

References

Council of Chief State School Officers

(CCSSO). (2010). Common Core State Standards for Mathematics. Washington, D.C.

Duckworth, E. (1995). "The Having of Wonderful Ideas" and Other Essays

on Teaching and Learning. New York: Teachers College Press.

Empson, S., Greenstein, S., Maldonado, L., & Roschelle, J. (2012). Scaling Up Innovative Mathematics in the Middle Grades: Case Studies of “Good Enough” Enactments. In S. Hegedus & J. Roschelle (Eds.), The SimCalc Vision and Contributions: Democratizing Access to Important Mathematics (pp. 251-270): Springer.

Hiebert, J., & Grouws, D.A. (2007). The effects of classroom mathematics teaching on students’ learning. In F. K. Lester, Jr. (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 371-404). Charlotte, NC: Information Age Publishing.

Kouba, V., Zawojewski, J., & Strutchens, M.E. (1997). What Do Students Know about Numbers and Operations? In P. A. Kenney & E. A. Silver (Eds.), Results from the Sixth Mathematics Assessment of the National Assessment of Educational Progress (pp. 87-140). Reston, VA: NCTM.

National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM. Retrieved from http://www.nctm.org/standards/content.aspx?id=322.

Roschelle, J., Schectman, N., Tatar, D., Hegedus, S., Hopkins, B., Empson, S., . . . Gallagher, L.P. (2010). Integration of Technology, Curriculum, and Professional Development for Advancing Middle School Mathematics: Three Large-Scale Studies. American Educational Research Journal, 47(4), 833-878.

Stein, M.K., Engle, R.A., Smith, M.S., & Hughes, E.K. (2008). Orchestrating


Productive Mathematical Discussions: Five Practices for Helping Teachers Move Beyond Show and Tell. Mathematical Thinking and Learning, 10, 313-340.

Stroup, W., Ares, N., & Hurford, A. (2005). A Dialectic Analysis of Generativity: Issues of Network-Supported Design in Mathematics and Science.

Mathematical Thinking and Learning, 7(3), 181-206.

The material presented here is based on work supported by the National Science Foundation under Grant Number 04-37861 to SRI International (PI: Jeremy Roschelle). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Professor Greenstein is an Assistant Professor of Mathematics/Education at Montclair State University. His primary research interest is children’s mathematical thinking. He is also interested in the design of exploratory playgrounds for learning mathematics; contextually situated, culturally resonant pedagogy; and issues of education and social justice.

Connecting Rate Using Dynamic Representational Technologies

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