Category Archives: ideas for experiments

Clarification on the purpose of my planned experiment on “practical math”

What follows is the response to some questions my friends Philippe and Samar raised about the experiment I describe here (previous post).

1) How is the normative context you are proposing different from a school math context?

I try to embed math problems in narratives that remind children of everyday, familiar situations that involve observance or transgressions of exchange and distribution rules. Such narratives, I believe, will awaken children’s sense of justice and motivate them to balance a situation that they see as unbalanced or unfair (“A gave a present to B but B didn’t make a matching present to A”, “A stole something from B”, etc.) Such narrative contexts should remind children about the institutions and rules or reciprocity that govern exchange and distribution. So, this is very different from the formal, instructional school context.

I’m not primarily focused on the educational applications. My questions are theoretical. I’m interested in mapping the social aspects of human cognition. If my work gets the desired results, then the educational applications might follow… but that’s not a primary goal for me. The experiment aims at proving a theoretical point.

2)  Do you think that the social/normative context of math problems would boost children numerical competence?

Yes, my hypothesis is that the social-normative context of these math problems will probably improve children numerical competence. But I would not expect any deep or long term effects from just one session. My idea is as follows: if we can use this one session to show just a local effect of the narrative context on how children construe and solve these problems, this is relevant enough. This would prove that social meanings are transferred to the mathematical domain and have an impact on children’s performance. I think that proving such local effect is much simpler than doing a longitudinal study (which might be a second step in the research). I also proposed to “do some standardized numeracy tests (perhaps those used by Opfer & Siegler, Dehaene, Piagetian conservation tests, etc.) right after the main task in order to evaluate if each of these normative contexts has “sensitized” the child to quantities in a special way.” In other words, we would not be testing for any lasting effects, but we would test numerical competence and/or quantity conservation right after the main experiment, to see whether this “sensibility” to number gets transferred to different problems. So this would only test for immediate effects, but we are interested in the child’s performance in a second, apparently unrelated problem, in the domain of math, to see if there is a “spill-over” from one situation to the other.

3) Why should normative and social context as provided in the narrative improve children’s performance?

Math problems that involve some kind of “equalization” between different parties are social in nature. This type of math was created historically to deal with such social problems (barter and purchase, paying back, getting even, managing debt). The history of math seems to go hand in hand with the history of human exchange and distribution systems. For example, the popularization of coins and the establishment of a class of merchants seems to happen at the same time as (and probably facilitate) the emergence of formal arithmetic. Calculus (developed simultaneously by Newton and Leibniz) is invented at a time when the first stock exchanges are being created.

We are not merely providing children with a social metaphor in this experiment, we are re-embedding math problems in their original social context. It’s the meaningfulness of the situation that should impact on children’s performance. This is the idea I want to test.

4) Where’s the novelty of your approach? 

Most current researchers (Dehaene, Opher, Siegler, Spelke, Lourenco, among many others) are interested in the innate, Approximate Number System (ANS) that humans share with other animals. Although there are differences among authors in the details, there is consensus that such a system is a pre-condition for the development of symbolic number and arithmetic (which are unique to humans). These authors show that symbolic number builds upon such innate capacity but they don’t provide good explanations about how we go beyond the ANS and up to human math. They mention “culture” but they treat culture as a mere collection of arbitrary conventions, technologies and techniques. In the case of number, culture is seen as providing a more or less fast and effective set of arbitrary procedures to perform calculations.

So, again, my immediate aim is not so much to discover the best strategy for training kids or to improve academic performance in the long term, but to prove a theoretical point about the social nature of math.

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Thinking about an experiment on “practical math” in normative contexts

I am trying to think about an experimental situation that would allow me to test how normative-institutional contexts impact on children quantitative reasoning. Ideally, it has to be an easy experimental task that can be tested quickly with children from different cultures. What follows is a half-baked draft. Your feedback and criticism is most welcome.

So this is the idea… Children (ages 4 to 7) are interviewed individually. During the interview, they are shown a series of very short puppet plays. After each play children are questioned about the best way to solve a problem that arose in the play. Children are required to offer quantitative answers to such problems; for example, “how much money does character A have to pay character B to get even?” or “How many blocks does character A need to add in order to complete the building?”, etc.  The narratives are different in nature. Some narratives provide a social and normative context to the problem, in the sense that they highlight certain social rules children need to take into account in order to respond appropriately to the situation. Other narratives, by way of contrast, highlight “technical” or “engineering” problems, and involve means-ends reasoning. They problems they involve are similar to the normative problems in their mathematical content, yet the narrative context is markedly different.

Examples:

A1: “Negative reciprocity and reparation”. Character A has a bag with three candy bars. Character A shows the bag to character B and tells her that she loves candy bars and that she plans to eat them with her friends the next day. Character A goes to sleep. Character B steals the bars and eats them. Character A wakes up and finds character B stole the candy, and asks character B to return them. Character B says she doesn’t have the candy anymore but that she can offer character A some money to make up for the stolen candy. She opens a purse and drops some coins and bills on the table. The child is asked to choose the coins and bills character B has to hand over to character A in order to get even. They child is questioned about how she made that decision; and how she calculated how many bills and coins that character B must give character A.

A2: “Destruction and reconstruction”. The child is shown a tower formed by six big blocks. The child is told that a powerful storm and strong winds hit the building during the night and broke the three upper stories of the building. She’s then given a number of smaller blocks of different sizes and is asked to rebuild the tower so that it is as high as it was before the storm. The child is questioned the criteria she used to select the blocks and to decide how many blocks to use.

B1. “Positive reciprocity”. Character A visits character B and shows up with a present: a stack of stickers or trading cards. Each character returns to her own home. Then character B says that character A was really nice and that she would also like to give her a present to “get even”. The child is asked to help character B prepare her present. She is shown a cup and a collection of marbles and is told to fill the cup up until there are enough for A’s present. The child is also asked about how she decided how many marbles to give; i.e. to justify her decision.

B2. “Bridging the gap”. The child is shown a model of a river. On the river there is half bridge built with legos. The bridge starts on one shore and goes only half-way over the river. The child is asked to pick the lego pieces that she would need to build the other half of the bridge. The set of lego pieces the child can choose from have a different size than the ones used to build the first half of the bridge.

All four situations involve some kind of addition and subtraction of different units; they also involve compensating different dimensions of problems (values of the goods exchanged, sizes of different objects, etc.) A1 and B1 are “social” and “normative”: they involve the concept of justice; A2 and B2 are “technical”: they involve a kind of means-ends reasoning.

One possibility is to give situations A1-B1 to one group and A2-B2 to a different group. One could then compare the reasoning and argumentation of children who are given a “normative” vis- à-vis a “technical” narrative. To this end, one might use the theory of argumentation and other tools of discourse analysis. One could also do some standardized numeracy tests (perhaps those used by Opfer & Siegler, Dehaene, Piagetian conservation tests, etc.) after the main tasks in order to evaluate if each of these normative contexts has “sensitized” the child to quantities in a special way; i.e. if the children who just completed the “technical” problem perform better or worse than the children who did the “social-normative” problem.

Another possibility is to give the same children all four situations so as to compare the features of quantitative thinking in technical vs. normative contexts in the same children.

Still thinking…