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Arcidiacono & Perret-Clermont (2010) – The Piagetian conservation-of-matter interview, revisited

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Paper #1

In this paper, Arcidiacono & Perret-Clermont (2010) revisit the Piagetian conservation-of-matter interview in light of the theory of argumentation. The authors argue that children’s statements are co-constructed by them and their interviewers, within a specific institutional setting, i.e. the testing situation. While Piaget considered children’s statements as dependent on the cognitive level, Arcidiacono & Perret-Clermont describe children’s arguments as the result of a series of interactions with the tester and as a reaction to the tester’s framing of the interview. The authors claim that, during the Piagetian interview, adults’ interventions strongly influence the statements made by the child. Children’s thoughts do not show up as clear and distinct ideas; they are expressed in a specific social context.

This article contains a number of interesting reflections on the nature of children’s discourse within the Piagetian interview, but it leaves an important issue unresolved. Sometimes the authors seem simply to state that the specific interviewers that participated in the examples discussed did not meet the Piagetian standards for not interfering with children’s spontaneous thought and for not inducing the answers. Alternatively, they sometimes imply that the interviewers’ interference and suggestions are unavoidable because of the very nature of the Piagetian interview and of human communication in general. Which of these is the case for the authors is not clear. They claim, for example, that “the adult repeatedly diverged from the intentions of the Piagetian script and consequently induced answers to the child” and that Piaget’s intentions were “misunderstood” by the interviewers (which implies that the interviewers were not very good), and yet also claim that these diversions “might be an inevitable condition of the situation”.

In conclusion, the authors claim that, according to the Piagetian ideal, “the adult has to offer a real place for debating, so as to give epistemic agency to the child” yet they don’t make it clear whether this ideal can be achieved in the real world or not.

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

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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…