Using Technology to Promote and

Examine Students’ Construction of Ratio-as-Measure

Joanne E. Lobato San Diego State University lobato@saturn.sdsu.edu

Eva Thanheiser San Diego State University evat@sunstroke.sdsu.edu

This paper examines students’ construction of ratio as a measure of speed, in the context of a teaching experiment, and discusses the role of computer software in the ratio-as-measure process.

Introduction & Theoretical Framework

Much of the research on ratios and proportions has focused on numeric strategies, like unit rate and factor of change methods (Cramer, Post, & Currier, 1993). Simon & Blume (1994) argue that the construction of a ratio as the appropriate measure of an attribute (which they call “ratio-as-measure”) has been inadequately addressed. In an effort to greater understand the ratio-as-measure process, Lobato & Thanheiser (1999) and Lobato (in preparation) conducted two studies of 17 high school Algebra 1 students. None of the students were able to successfully create a ratio as a way to measure the steepness of a wheelchair ramp, the “proteinness” of nutrition bars, or how fast a mouse travels. The researchers identified two previously unreported sources of difficulty in the ratio-as-measure process: 1) isolating attributes (e.g., steepness was often conflated with attributes like “work required to climb” or “materials required to construct” in the wheelchair ramp situation), and 2) identifying which quantities affect an attribute and in what ways (e.g., whether or not the number of steps taken affects how fast one walks). Furthermore, even those students who understood how changing the height and the length of a ramp affect the steepness of a ramp or how distance and time affect speed, still did not appear to view the relationship between the quantities in these situations as proportional in nature.

Consequently, the teaching experiment described in this paper was prompted by our interest in helping students construct ratios as appropriate measures of attributes. We hypothesized that re-conceiving static situations (like the wheelchair ramp situation) as dynamic, perhaps with the help of computer software, might help students determine when it is sensible to construct a ratio between two quantities. This hypothesis developed, in part, from a disagreement in the literature regarding the distinction between ratio and rate.

For Thompson (1994) and Kaput & West (1994) a rate signifies a whole structure where the two quantities co-vary dynamically in a constant ratio, and a ratio is a static instance of a rate. In contrast, Confrey & Smith (1995) reject ratio as an instance of a relationship between quantities, claiming instead that ratios are constructed “by objectifying and naming that which is the same across proportions," i.e., to construct a ratio, one needs to first identify what is the same across more than one instance (p. 74).

On the one hand, it is clear that one can conceive of a static multiplicative comparison or ratio without the ratio continuing across more than one instance, e.g., a 35-year old father is 5 times as old as his 7-year-old son, but this ratio does not hold as both individuals age. On the other hand, Confrey and Smith's idea of perceiving sameness across multiple instances as instrumental to the construction of ratio may be more applicable to a second way of constructing ratio, namely the creation of a composition of two composite numbers (Lamon, 1995), which we call a “two-number.” For example, when buying 3 lbs of candy with $5, one can form a composite “three-five” number (which differs from the multiplicative comparison of 3 as 3/5 of 5). However, it is difficult to assess whether a student has constructed a "two-number" as opposed to thinking of "two numbers" (e.g., conceiving of $5 for every 3 lbs of candy, not simply $5 and 3 lbs) unless one sees evidence of the student iterating or partitioning the new composite number. Furthermore, we hypothesize that constructing a “two-number” ratio might be linked to conceiving the feasibility of the extension of that ratio.

A critical reader might argue that one cannot construct a family of ratios without first forming a single ratio. However, computer environments might allow students to generate a “family of values” with a given attribute (e.g., same speed) by guessing and checking, without mentally constructing a ratio. Subsequent discussion of these values might support the creation of ratios. The use of families of values representing an attribute that is visually part of a dynamic software environment permits a middle ground position between "ratio as static" and "rate as dynamic.”

Purpose. This study will examine the ratio-as-measure process while students engage in computer activities that are hypothesized to be propitious for the construction of ratio-as-measure.

Method of Inquiry

A teaching experiment was conducted during the summer in a university computer lab for about 30 hours over two weeks. Nine average-performing students (i.e., those who earned Bs or Cs) were recruited from 8^th-10^th grade math classes. The authors team-taught the course. All sessions were videotaped. Two "family of related values" tasks were used (see Figure 1). Prior to each of these tasks, students worked on isolating an attribute (i.e., motion or steepness) and identifying quantities that affected the attribute (e.g., time and distance). Due to space limitations, only results from the “same speed” task will be reported. Students used the SimCalc Mathworlds software (see Figure 2) to enter a time and distance for two animated characters and then ran the computer simulation to see if the characters walked at the same speed.

1. “Same Speed” Task. The clown travels 10 cm in 4 seconds. Find as many different ways as you can to make the second character, the frog, travel at the same speed as the clown by entering a distance (other then 10 cm) and a time (other than 4 seconds) for the frog.

2. “Same Steepness” Task. Make as many ramps as your can (using Geometer’s Sketchpad) that have the same steepness as ramp with a height of 2 cm and a length of 3 cm.

Figure 1. Two “Family of Related Values” Tasks

Figure 2. Screen capture from SimCalc Mathworlds

Results and Discussion

In the individual interviews conducted during the first day of the teaching experiment, no subject provided evidence of the creation of a ratio as a measure of the steepness of a wheelchair ramp, and all but two students showed serious proportional reasoning problems. In this section we present an example of the construction of a ratio as the measure of speed during a class discussion of the “same speed” activity. Three findings follow:

1. Students’ numeric strategies may project a misleading image of proportional reasoning ability. The “same speed” task was difficult for all students, as evidenced, in part, by the incorrect entries that each student recorded in his/her time and distance chart while working individually at a computer. Most students relied on a "guess and check" strategy (e.g., entering 15 cm and 8 sec and then adjusting the time until arriving at 15 cm in 6 sec). Four students found numeric patterns: three used factor of change strategies like doubling the distance and time, and one student used a unit rate strategy. However, three limitations to the numeric strategies were found. First, when the first author walked around to each computer station and questioned students, no one was able to explain why their numeric patterns worked.

Second, during the hour-long class discussion that followed the computer work, numeric explanations were not understood by other students. For example, Brad shared his “solution” of 90 cm in 35 seconds (as a “same speed value” as 10 cm in 4 sec). Terry disagreed, arguing that “10 goes into 90 nine times and 4 goes into 35 eight times and a little bit left over.” But the other students said that they couldn’t follow Terry’s explanation.

Third, students’ illustrations indicated a lack of connection between the numeric strategies and the quantities involved in the situation. For example, the teacher asked the students to draw a picture to explain why the doubling strategy worked, i.e., why walking 20 cm in 8 sec was the same speed as walking 10 cm in 4 sec. Terry represented the distances of the two characters without attempting to show that the frog’s distance was double the clown’s distance (see Figure 3). In fact, Terry asked whether he was working on the 20 cm or the 90 cm problem after he had represented the frog’s distance with a line. He explained that for both characters to have the same speed, they would need to walk 10 cm in 4 sec at the same time. Neither his verbal explanation nor his visual representation included frog’s distance and time after the initial 10 cm in 4 seconds. He relied on calculations, stating that “if you want frog’s distance to be 20, then you have to multiply 10 x 2 to get 20; since you multiplied 10 by 2, you also need to multiply 4 (the time it took the clown) by 2 to get 8,” without explaining why time and distance need to be doubled or how multiplying by two could be represented in his drawing. The next student to go to the board, Jim, offered a numeric explanation almost identical to Terry’s. The discussion appeared to stall, when suddenly Brad had a new idea that he seemed anxious to share.

Figure 3. Terry’s first diagram

2. Ratio-as-measure construction appears to involve an understanding of covariation and relationships between quantities in the situation. A breakthrough occurred when Brad appeared to construct a “two-number” ratio. Brad explained that doubling works “because the clown is walking the same distance; it’s just that he's walking the distance twice… he’s walking it once, going li, li, li, li, li, li, [Brad retraced the line Terry drew, up to 10 cm and drew a vertical mark], all the way here. Four seconds. OK. He’s going to walk it again. Another four seconds, li, li, li, li, li, li, li, li. Another ten centimeters in four seconds. He’s done.”

Brad’s explanation involved three elements lacking in both Terry’s and Jim’s work, suggesting a greater understanding of the covariation between distance and time. First, his picture illustrates what happens to the frog character after the initial 10 cm in 4 seconds by noting that the frog walks another 10 cm in 4 seconds. This observation may be closely tied to what permits the construction of ratio in this walking situation, namely an understanding that if one walks at a constant pace for x cm in y sec and then repeats the exact action, then one will not go faster or slower but will walk at the same speed for both journeys, as well as for the combined journey. Second, Terry seems to pick one quantity, namely 20 cm and then produces the other related quantity of 8 sec. In contrast, Brad’s work is consistent with a more sophisticated image of distance and time varying simultaneously, or at least in 10 cm in 4 sec “chunks.” Finally, Brad appears to coordinate the quantities of time and distance by using sounds to represent time while he retraces a line segment to represent distance, an important component of covariation.

3. A “two-number” ratio can be iterated and partitioned to form other ratios that represent the same speed. The construction of the “10 cm in 4 sec” unit was adopted by other students and combined with iterating and partitioning to create additional “same speed” values. For example, Denise added another 10 cm in 4 sec section onto Brad’s drawing, concluding that 30 cm in 12 sec also works. Later on, Terry explained why walking 2.5 cm in 1 cm was the same speed as walking 10 cm in 4 sec. He partitioned the “10 cm in 4 sec” unit into four segments, formed a new “2.5 cm in 1 sec” segment (as indicated by the section in Figure 4 that Terry circled), and then iterated the “2.5 cm and 1 sec” unit four times to end up with 10 cm and 4 seconds. He stated that “it would be like he's walking one fourth of the 10 and 4; it's like one fourth of each thing” [meaning 1/4 of the 10 cm and 1/4 of the 4 seconds]. Terry’s idea can form the basis for a very powerful generalization that if one travels x cm in y sec at a constant speed, then if one goes a/b of this journey, one will travel (a/b)•x cm in (a/b)•y sec. Although more work remains for students to fully develop an equivalence class of ratios and to see distance and time as flowing quantities, this approach suggests a promising avenue for further research.

10

2.5 2.5 2.5 2.5

1 1 1 1

4

Figure 4. Terry’s second diagram

Reflections on the Role of Technology

The computer environment, combined with a whole-class discussion, helped support the construction of ratio as a measure of speed. By allowing students to test same speed values, the software seemed to support an image of other distance and time pairs that would produce the same speed as the initial 10 cm in 4 sec value. By asking students to explain why their values worked, a condition in which ratio-as-measure could be constructed was supported. However, it is unlikely that students would have progressed beyond guess-and-check strategies or numeric patterns without the class discussion.

The software also afforded three unintended actions or conceptions. First, students developed a practice of checking to see whether the characters walked at the same speed by running the simulation and then looking to see if the characters walked “neck-and-neck” for only the duration of the shorter journey. This might explain why students like Jim and Terry did not initially account for the time and distance of the character that kept walking after the first character stopped. Second, throughout the discussion, despite the teacher’s efforts to focus on explanations and reasoning as ways to settle mathematical disagreements, the students seemed to consistently view the computer simulation as a highest authority. Finally, the students were unable to visually distinguish which character was going faster if the ratio is close, e.g., a character traveling 11 cm in 4.5 sec appears to be going the same speed as a character traveling 10 cm in 4 sec, though no one raised this issue as worrisome; students simply accepted these values as correct.

Acknowledgements

Preparation of this paper was supported in part by the National Science Foundation, #REC-9733942. Any opinions expressed herein are those of the authors and do not necessarily reflect the views of NSF. The authors would like to thank the following members of the Generalization of Learning Research Group for their contributions to the project: Dan Siebert, Tony Brumfield, Misty Bailey, and Ricardo Muñoz. We gratefully acknowledge Jeremy Roschelle and Janet Bowers for writing this special Mathworlds script for our project.

References

Confrey, J. & Smith, E. (1995). Splitting, covariation, and their role in the development of exponential functions. The Journal of Research in Mathematics Education, 26 (1), 66-86.

Cramer, K., Post, T., & Currier, S. (1993). Learning and teaching ratio and proportion: research implications. In D.T. Owens (Ed.), Research ideas for the classroom: Middle grades mathematics (pp. 159-178). New York: MacMillan.

Kaput, J. & West, (1994). In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 181-234). Albany, NY: SUNY Press.

Lamon, S. J. (1995). Ratio and proportion: Elementary didactical phenomenology. In J.T. Sowder & B.P. Schappelle (Eds.) Providing a foundation for teaching mathematics in the middle grades (pp. 167-198). Albany, NY: SUNY Press.

Lobato, J. (in preparation). Reconceiving transfer involves reconceiving mathematical content domains: the case of slope.

Lobato, J. & Thanheiser, E. (1999). Re-thinking slope from quantitative and phenomenological perspectives. Proceedings of the Twenty-first Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 1, 291-297.

Simon, M. A., & Blume, G. W. (1994). Mathematical modeling as a component of understanding ratio-as-measure: A study of prospective elementary teachers. The Journal of Mathematical Behavior, 13(2), 183-197.

Thompson, P.W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 181-234). Albany, NY: SUNY Press.