Giving Prominence to Pedagogical Issues

David Carraher, TERC

Analúcia Schliemann, Tufts University

Introduction

One of the key challenges of using computer technology in mathematics education is to give proper attention to pedagogical issues, questions of teaching and learning.

Consider, for example, the following issues:

- Why are many students puzzled by the fact that multiplication does not always increase a quantity nor does division necessarily diminish it? Does it help to think about how integers work? Or should rational numbers be introduced without appealing to intuitions about whole numbers?
- How should we bridge arithmetical and algebraic understanding? Or should they be treated, as they often are, as distinct forms of reasoning best learned at different stages in schooling?
- Can advanced mathematical ideas be grounded on what students know about quantities? Or is quantitative thinking essentially an impediment to the learning of abstract concepts, a crutch that must be relinquished as soon as possible?
- Elementary mathematics focuses heavily on numbers as counts (cardinality of sets). How can we introduce the idea of numbers as ratios, even though representations of ratios use sets of (countable) items? (Is it no wonder when students prefers 4/6 of a bag of candy to 2/3 of a bag of candy?—The numerator tells them how many items they get!)

Clearly, these sorts of issues will not be clarified by placing computers in the classroom and hoping for the best. They involve thorny questions of how learning occurs and on how new knowledge builds and relies on former knowledge. Fortunately, research has made significant headway in these matters. For example, authors observed years ago that fractions can be introduced in ways consistent with what students already knew about integers (Braunfeld & Wolfe, 1966; Weckesser, 1970). Teaching studies carried out in the former Soviet Union (e.g. Bodanskii, 1991) provide many insights into how to bridge arithmetical and algebraic understanding; they demonstrate that elementary schoolchildren can understand and solve algebra problems that most educators would not dare to hand to students before their ninth year of schooling. Researchers and educators of many persuasions have given special attention to the role of quantitative thinking in mathematical understanding (Bell et al, 1984; Freudenthal, 1983, Fridman, 1991; Kobayashi, 1988; Piaget et al, 19 ; Schwartz, 1995; Vergnaud, 1982). We have begun to understand how to separate issues of cardinality from relative magnitude (Carraher & Schliemann, 1991; Carraher, 1996).

We would like to present here some initial research that has been inspired by such progress on the pedagogical front. First we will describe a software environment, called *The Visual Calculator*, that was designed to help students and teachers engage in grounded discussions about concepts related to the above issues. Then we will look at an example of middle school students using the software. It concerns how a 5^{th} grade class solved a problem posed by two members of the class. We conclude with brief remarks on the role of technology in supporting the discussion.

A Visual Calculator

Operations

*The Visual Calculator*^{TM} was developed to help students visualize and discuss what happens to quantities when subjected to arithmetical operations of multiplication, division, addition, and subtraction (see Carraher, 1993, 1996, and Carraher & Schliemann, 1993). The software makes use of students' deeply rooted intuitions about the meaning of operations. For example, adding is represented in the software as joining. Subtraction can be thought of as removing or taking away. But even such basic intuitions must undergo adjustments when operations are modeled. For example, subtraction is not a distinct operation. It is composed of two operations: inverting (a subtrahend quantity) and joining (with the minuend quantity). If may not be immediately clear to the reader whether subtraction should be represented in this non-conventional way; however there are advantages to viewing the operations "-C + A" and "A-C" as equivalent operations rather than as addition and subtraction, respectively, as conventional wisdom would dictate. But admittedly this entails questions of when to introduce children to negative quantities and vectors. We have clearly opted for an early introduction of such concepts instead of presenting more simplified models that must later be undone.

Multiplication and division can be introduced in ways that build upon the intuitions that multiplying makes bigger/division makes smaller. If we multiply a quantity by 3 and then divide the result by 7, we obtain a quantity smaller than the original quantity. Multiplying made the quantity bigger, the division made the resultant quantity smaller. The overall effect of the two operations is to produce a quantity smaller than the original one (see Figure below). Now if we go back and multiply the original quantity by 3/7, the result is the same length as (read: "equal to") that produced by the two integer operations. In fact, multiplying by 3/7 can be thought of as shorthand for the two operations of "times 3"and "divided by 7".

This may be obvious or even trivial to those of us who have developed sturdy intuitions about rational operations. However, there is no reason to suppose that middle school students will quickly come to the same conclusions as we do. The important thing is that students can be given conditions to mull over and discuss these very issues, to compare set up tests and carry out the steps in their own reasoning. How they do this, how they coordinate their understanding of integers with their emerging understanding of rational operators, merits careful research.

Multiple Units

A classic adage for teaching about fractions has been that "you shouldn't compare different units". This advice is motivated by the desire to avoid potentially confusing descriptions such as "2/3 = 1/2", where the first term refers to one unit and the second term refers to another unit (50% greater than the first unit). Unfortunately, this constraint, and the convention to not specify the unit in notation, hide from view important connections between units of measure, variables, and functions. These connections are much more salient when one adopts a more robust and explicit notation, such as "2/3 A = 1/2 B". The resemblance to algebraic equations is not mere coincidence. We are consciously attempting to use descriptions of relations among physical quantities to serve as precursors to relations among variable quantities and, ultimately, mathematical variables. The uncanny similarity between this notation and algebra can prove to be very useful for placing arithmetical and measurement situations in an algebraic context.

A unit of measure is any quantity that a student has decided to name (normally by a letter). We can represent multiple units of measure in the *Visual Calculator* and call them by their respective names rather than referring to them indiscriminately as "units". At first a name refers to a particular line segment. However, since line segments can be replicated and will keep the same name as the original, the name can quickly take on the role of a measure in a given unit.

When a student decides to give a quantity a name, the software automatically creates a "ruler" in units of that magnitude. As instantiated in the software, a ruler looks more like a grid that cuts vertical lines throughout the space; it is not restricted to a number line sitting on the x-axis. This feature holds the potential for uniting number line and part-whole models of fractions, which research has shown to be very difficult for students to reconcile (Booth, 1984). In addition, multiple rulers can be simultaneously present on the screen. This encourages students and teachers alike to refer to how the different units relate to one another.

In an interview with two 5^{th} grade students, Pilar and Sharmin, they created units of P (Pilar) and S (Sharmin) and displayed both rulers at once.

The students were struck by the fact that the numbers from one ruler occasionally aligned horizontally with numbers from the other ruler. (We have often found 5^{th} and 6^{th} year students to be intrigued by the patterns produced by multiple rulers. ) It was not immediately obvious to them what these patterns meant, but by thinking about and discussing what the lines stood for, they formulated a description of how the two units of measure were related. Sharmain wrote out her discovery as

"P x 5 equals the same as S x 3".

Pilar expressed the same relation, in writing, as

"5P = 3S".

These descriptions were different from the notation they had been taught in school. And yet they seemed to flow forth naturally in this setting where units were being given identifying names. It seems to us that these algebraic-like statements, once written, helped the students clarify for themselves (and for us, the researchers), the patterns they noticed among the grids. The transition from notation, on the one hand, as a means of registering operations and actions to be carried out, and on the other as a means for describing relations among objects is an important one. It evokes the important distinction between process and object that other authors have recently drawn attention to (Sfard, 1991, Dubinsky, 1991).

Determining an Unknown Ratio

We will summarize here how a fifth year class solved a problem of determining an unknown ratio of two quantities.

A fifth grade teacher asked us to show the Visual Calculator to her class one day, since not all of the students had been able to participate in our out-of-class interviews using the software. We showed how the basic operations were represented on screen. We talked about how to describe the relations among diverse units that the students had created. The teacher asked the students to determine whether 3 units of measure would eventually line up together (The units were 2.5, 3.5 and 1 unit, respectively.)

In the second half hour we set up a problem situation in which there would be problem posers and solvers. We asked two students to select a pair of numbers between 1 and 10. They would pose the problem, that is, choose an unknown ratio. The rest of the students would attempt to discover the ratio chosen by exploring two quantities constructed from that ratio. The following screen dumps display the visual information students had at their disposal as they discussed the problem.

The first suggestion comes from Ben who states that "the top [line] is two and I think the bottom is four and a half". When the teacher clarifies that the numbers were whole number he changes to "The S is four and the T is nine, because I think this (the space between the dotted lines produced by the S ruler) are four, so there is four and then four and then a little bit to make nine".

The lines for the T ruler are then displayed. A few students take turns in attempting to prove that Ben is right. One of them explains that "Because in the beginning of the T one of those looks like one fourth of two Ss... Its one fourth past two. So that means it has to be either 4 or 8 or 16". Another says that "I said, 2 times 4 is 8, and it's right here, and I think this part is one more, so that's 9".

When the examiner counterargues that the little bit that is left over could be "one fifth and not one fourth", another student states that she thinks that "the top one is 3 and the bottom one is 7". The class is thus faced with two competing possibilities for the ratio.

After some discussion, Rachel volunteers to walk up and present her argument in support of Ben's proposal that one line is 4 and the other is 9. Our interpretations are bracketed.

Rachel: This [the piece of T that projects beyond the second S-line] has to be one fourth, because, and then that would be one half [i.e. the second T-line falls halfway between the 4^{th} and the 5^{th} S-lines] and that would be three fourths [the third T-line falls ¾ of the way between the 6^{th} and the 7^{th} S-lines] and this would be... if there were one more [T-unit], this would be a whole. If you look at these lines, you see, this is one fourth and then this is in the middle between these two and so that would be a half, and then this would be three fourths...

D: Show me the middle again. Where is the middle?

Rachel [pointing to the T-line falling at 4.5 S]: Here.

D: … So you're saying that second line is in the middle of two of the other ones?

Rachel: Yes, well, no, in the middle of the Ss.

D: In the middle of these two [pointing to the 4^{th} and 5^{th} S-Lines with the mouse]?

Rachel: Yes, and then this one [at 6.75 S] is over a little, and this one is... they're both the same [ie the 4^{th} T-line aligns with the 9^{th} S-Line]. They both line up.

Teacher: So you're thinking that this is a whole unit and you've got this much...

Rachel: Yes, I'm saying...

Teacher: Put your fingers like this, between the 2 and the 3 so people know what you're talking about. So, you're basically spanning the distance between 2 and 3, and then you're showing that two lines...

Rachel: Yes, this line is in the middle of these two... and this is over to the right and this is three fourths.

D: Look at that last one, let's go carefully on that one. What happened there?

Rachel: They're both at the same.

D: What are the numbers that are there?

Rachel: It's a 4 and a 9.

Several members of the class (with surprise, as they figure it out): Oh.....

This visual method of proof was convincing to the class, including the girl who had put forth the idea that the ratio was 3:7.

When the teacher asked them what number would serve as a multiplier to make the shorter segment, S, equal in length to the larger segment, T, a boy suggested that they use the number, 2.25. When they carried out the multiplication they saw that the result (2.25S and T equal in length) consistent with the prediction. Although technically not a proof of the relation, it provides fairly convincing evidence that the students were correct.

Concluding Remarks

This paper only begins to scratch the surface of how technology can exploit pedagogical issues that have repeatedly emerged from research and teaching. We are still at a beginning stage of investigating how children can make sense of traditionally difficult concepts in this computer supported learning environment. It is clear that the software does not work on its own but depends heavily on the spirit of inquiry that teachers or researchers are able to establish together with students. Given the proper social climate, which should not be presumed to exist but must be achieved, the software allows students and teachers to hold discussions about ideas and mathematical objects that are difficult to talk about in normal circumstances, without the aid of such technology.

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