Students’ Reasoning With Reversible Multiplicative Relationships

Abstract

In an 8-month teaching experiment, I investigated how 4 sixth-grade students reasoned with reversible multiplicative relationships. One type of problem involved a known quantity that was a whole number multiple of an unknown quantity, and students were asked to determine the value of the unknown quantity. To solve these problems, students needed to produce a fraction of the known quantity that could be repeated some number of times to make the known, rather than repeat the known quantity to make the unknown quantity. This aspect of the problems involved reversibility because students who do not make a fraction of the known quantity tend to repeat the known quantity (Norton, 2008; Steffe, 2002). All four students constructed schemes to solve such problems and more complex versions where the relationship between known and unknown quantities was a fraction. Two students could not foresee the results of their schemes in thought—they had to carry out some activity, review its results, and then carry out more activity in order to solve the problems. The other two could foresee results of their schemes prior to implementing them; their schemes were anticipatory. One of these two also constructed reciprocal relationships, an advanced form of reversibility. The study shows that constructing anticipatory schemes requires coordinating three levels of units prior to activity, a particular whole number multiplicative concept. The study also reveals that even students with this multiplicative concept will be challenged to construct reciprocal relationships. Suggestions for further inquiry on student learning in this area, as well as implications for classroom practice and teacher preparation, are considered.

ACKNOWLEDGMENTS

The research reported in this article was part of the author's doctoral dissertation completed at the University of Georgia under the direction of Leslie P. Steffe. I am indebted to Les Steffe, Erik Tillema, and Andy Norton for their generous and helpful comments on earlier drafts of the article. I also gratefully acknowledge Sean Larsen's feedback on an earlier, shortened version of the article that I presented at the Twenty-eighth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education in November 2006. I also presented portions of this article at the annual Research Pre-session of the National Council of Teachers of Mathematics in March 2007.

Notes

Some might prefer the word “inverse” here. I use “reverse” to be consistent with Piaget and Inhelder's (1958) discussion of reversibility.

I emphasize that these problems are labeled from my perspective of what is involved to solve them, not according to how I infer that students view them.

Technically partitioning refers only to dividing a unit into parts, not equal-sized parts. Equi-partitioning refers to making equal-sized parts, and Steffe (2002; Steffe & Olive, 2010) regards equi-partitioning as a scheme. However, for brevity, and because the students discussed in this article conceived of partitioning as equi-partitioning, I will use partitioning to refer to making equal-sized parts.

Foresight and anticipation are not necessarily synonymous. According to standard dictionary definitions, foresight involves envisioning what might be possible or knowing ahead of time what is to happen. Anticipation tends to have connotations of hope in looking forward to what might happen. However, for the purposes of this article, I will consider cognitive foresight and cognitive anticipation to point to the same idea.

This repetition of what has functioned before is part of how people build up records of experience to be used in assimilation.

For Simon and colleagues, activity refers to mental activity and effects refer to what actors isolate in their experience following activity (cf. Tzur, 2004).

The other two functions are a duplicator partition-reducer function and a multiplier and divisor function. I do not use the duplicator partition-reducer function as an example here because in the literature I have found it used only for taking a fraction of a whole number. It can be adapted to taking a fraction of a fraction, but the adaptation requires including operations not discussed by Behr and colleagues. I do not use the multiplier and divisor function because the stretcher-shrinker function is in fact one example of it.

I distinguish between the problematic situation as posed by a teacher-researcher, and the assimilated situation. The assimilated situation is what a student (or solver) makes of the problematic situation.

They can operate this way because their units of one are iterable (Steffe, 2002).

It is common for students to solve this problem by iterating the given stick five times; if a student persistently solves problems like the Stick Problem in this way, it is a contraindication that he or she has constructed a splitting operation.

During this same selection process another researcher aimed to select four students, a pair who had not yet interiorized two levels of units but could coordinate two levels of units in activity, and a pair who were pre-multiplicative (Steffe, 1992). So, the eight students represented a range of multiplicative reasoning. A witness-researcher was also present at all selection interviews; he, the other researcher, and I formed the original team for the research, which was part of a planned 3-year project. Over the next two years, additional researchers joined the team at various points.

There was no intention to pair according to gender; based on cognitive evaluation the students ended up paired with another student of the same gender. Note that all names are pseudonyms.

Please see footnote 1.

In this episode Michael also determined the length of the new collection, 11 and 2/3 inches (Hackenberg, 2005). In solving RMR problems, the students usually determined the length of the unknown quantity. In order to present activity from all four students, in this article I focus primarily, but not exclusively, on how they made the unknown quantity.

In the data excerpts, M stands for Michael, C for Carlos, D for Deborah, B for Bridget, T for the teacher-researcher (the author), and W for a witness-researcher. Comments enclosed in brackets describe students’ nonverbal action or interaction from the teacher-researcher's perspective. Ellipses (…) indicate a sentence or idea that seems to trail off. Four periods (….) denote omitted dialogue.

To solve this problem, Michael partitioned each of the halves into six equal parts, and then colored the 12-part bar to identify thirds.

Carlos had also learned, in the first two episodes of the experiment, to split composite units where the number of units in the composite unit was a multiple of the number of parts to be made. That is, he learned to solve Type 1 RMR problems (see Table 3).

One reason for this stipulation was to open possibilities for determining the length of the resulting bar out of the partitioning activity (i.e., for Carlos to determine, out of reasoning, that 1/3 of 2 feet must be 2/3 of 1 foot).

It is also possible that Carlos knew his goal was to make thirds of the 2-foot bar and he could see that he had two parts, so he responded two-thirds. In this case, his response would indicate to me that he had relied on part-whole meanings of fractions to determine a length that, coincidentally, was correct.

In contrast with Michael, Deborah did not use the smallest number of parts necessary. A possible explanation for her partitioning the 2/3-bar into precisely 12 parts total (instead of six or some other multiple of three) is that she was not entirely sure how many parts to aim for, but she knew she was dealing with thirds and fourths. So, using 12 parts total was a “safe bet”—12 could be divided into three or four parts. Thus, rather than coordinating two and three based on a goal to make a 2-unit bar into three equal parts, she may have coordinated three and four based on a goal of making thirds and fourths in the same bar.

The girls actually used D to represent the length of the room. Since that letter is the same as the initial letter of Deborah, for readability I have changed the letter representing the length of the room from D to L.

Carlos determined this length with considerable coaching (Hackenberg, 2005).

In the interaction, Michael said “thirty-two” not “thirty-two thirds,” but the context of his comments and gestures indicates that the “thirty-two” referred to the 32/3-centimeter bar.

Both Michael and Deborah began to use this scheme in situations when the known quantity was segmented into fractional parts, as in Type 5 RMR problems, although only glimpses of Deborah's work on Type 5 problems are shown in this article.