Experimentation on the Effects of Mathematical Diversity

Abstract

We examine, with in-depth teaching recordings and interviews, how tenth grade (15–16-year-old) students react when confronted with an ancient cuneiform clay tablet. The question is whether mathematical diversity can produce new questions (to be further used by teachers) linked to area and measure concepts. We observed conceptual changes with regard to mathematics, but it was difficult for students to make them explicit. In terms of “nature of science” aspects, we were able to document a change in debate content, and we also formulated some precautions. We provide a methodological reflection. We are attentive to the consequences of historical constraints on making links between ancient and current mathematics.

Appendix—Interviews’ Content

• I. Questions: mathematical part

Could you tell me how you calculate today the area of a square? (Individual reflection time)

• 1.

  • If the student has given a formula: Do you know why this formula works? How would you explain it to someone?

For these first two questions, we do not assume that the results will be very different between the control and HSG groups. We expect some “grid” type of explanations and a majority of students with difficulties to answer, “I do not know” or students who simply give the formula. We can imagine, however, that the HSG may raise more spontaneous remarks such as “I wonder why the formula works” (see a priori analysis).

• 2.

  • If the student has only given a formula, not applied to an example: Can you give me an example with a square the size of your choice? (Individual reflection time)

This question will allow us to analyze the space given to units of measurement in the application of the formula and their choice.

• 3. In your opinion, what are all the steps involved in calculating a square area, if you were to give them something like a kitchen recipe? (Individual reflection time)

This question has a key role; it will allow us to analyze a possible change in the ability of the students of the HSG to distinguish the mathematical steps and objects involved in each step of the algorithm after meeting with a different algorithm. I recall that we make the hypothesis of a difficulty distinguishing between steps (at least for the control group), with a tendency to summarize the area calculation in the single multiplication step.

• 4. In your formula, what is the “3” (which is multiplied)?

I recall that we make the hypothesis of a difficulty in explaining what is multiplied, especially for pupils unable to mobilize the idea of “grid,” as well as a difficulty in distinguishing number and numerical value associated with a magnitude value. We expect a few answers like “a number of tiles” and a majority of answers like “length of the side,” on which we could counter and ask for clarification, on why “it works.”

This may be an opportunity to see a difference between the control and HSG groups. The latter could, for example, make spontaneous remarks such as “ah, but it is true that today it is both” (number multiplied and measure of length), while in Mesopotamia it is not the same (SPVN and length measurement numerical system).

• 5. In multiplication, what is multiplied?

Same remarks as in the previous question

• 6. Why do we put “cm²” as a unit of measurement here?

I recall that we make the assumption of a general difficulty in explaining the choice of the unit of measurement, due to an automatic use of the area unit of measurement “corresponding” to the length unit of measurement.

This question may be the occasion to note the emergence of spontaneous questions in the HSG, due to the encountering with area units of measurement unrelated to lengths units of measurement, like: why do our units of measurement bear the “same name”?

• 7. What is “cm²” to you?

We a priori expect answers of the type “a tile” and answers of the type “an area unit of measurement;” without distinction between the two groups (control and HSG).

• 8. Here, there are length measurements and there, surface measurements; when did we go from one to the other?

This question, although asked in a “partially mathematically legitimate” way, will allow us to draw students’ attention to different type quantities at stake in the algorithm. Here we expect a difference between control and HSG groups. We believe that control group will respond “at the time of multiplication,” giving a key role to the multiplication that seems to do a “magical” transformation. We believe that the HSG, which has found the existence of SPVN numbers and metrological tables, will be divided and hesitant. Perhaps spontaneous questions will emerge at this moment, like: “in the tablet we have made a correspondence between measure of length and numbers, then between numbers and area measure; I do not know how we do it today/I do not know where this step is today.”

Some pupils in each group may use the idea of a “grid” to answer that length measurement does not really matter, and that it is actually a matter of knowing the number of tiles (the number of cm²), with multiplication operating on the number of tiles per row and column. Then the length measurement simply gives the number of tiles on a line. It could be said that there is an implicit mapping of the length measurement from one side to the number of tiles on the side.

• 9. If I have a square (20 cm sided) pool, and if I want to tile it with tiles (squares) of 1 cm side, how many tiles do I need?

I recall (see previous analysis) that we make the hypothesis of a general difficulty in mobilizing the grid, that hence bears no meaning anymore, in the memorized formula. This question will show which method is preferred by the students: to count tiles or to calculate the area and then divide by the area of a tile. We assume a majority of use of this latter method, in both groups. We think the question may allow some of the students to remobilize the “grid.”

• 10. Can you explain why the formula works?

We want to see if the students who “remobilized the grid” change their explanations.

• II. Questions: nature of sciences

[The first six questions based on the cuneiform tablet (historical, mathematical) are omitted because they were asked only to the HSG.]

• 1. Do you like math? Yes/no, why?

This question shall be used to investigate the relationship of students with history of science sessions, based on their relationship to mathematics, and a possible explicit effect (or not) of the sessions on their tastes, by comparison with the control group.

• 2. In your opinion, what is important when presenting a historical text?

This very direct question will help us to know whether or not students are able to make explicit their relationship to the historical discipline; and to see possible changes between control group and HSG.

• 3. What is history of sciences, for you?

This very direct question will help us to know whether or not students are able to make explicit their relationship to the history of sciences; and to see possible changes between control group and HSG.