Category Archives: math

Kanngiesser on children’s application of the merit principle

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Kanngiesser, P., & Warneken, F. (2012). Young Children Consider Merit when Sharing Resources with Others. PLoS ONE, 7(8).

This is a great paper. It tackles the classic problem of merit as a principle of fairness (or of distributive justice): rewards should be distributed according to how much someone contributed to a task.

Kanngiesser and Warneken did two studies about children’s application of the merit principle. They made children play against a puppet at a game of collecting (“fishing”) coins that were later exchanged for rewards. They varied the work-contribution of both partners by manipulating how many coins each partner collected. Three- and five-year-olds kept, on average, significantly more stickers for themselves in the more-work condition than in the less-work condition. Children, in other words, kept fewer stickers in trials in which they had contributed less than in trials in which they had contributed more than the partner, showing that they took merit into account. Therefore, it seems that three- and five year- old children already use merit to share resources with others, even when sharing is costly for the child.

Although this appears to show that children take merit into account to calibrate their responses, it should also be noted that children almost never give away more than half of the stickers when the partner had worked more. “Even though children were clearly able to consider different work contributions, this tendency was constrained by a self-serving bias.” Thus, merit-based sharing is also mixed with or calibrated by the egotistic, self-serving bias documented by Rochat and many others.

The paper also presents a similar, second study, that shows that children’s sharing behavior is not just determined by their own absolute work-effort. Rather, children appear to take into account their own and their partner’s relative contributions when allocating resources. (Therefore, there  is some kind of elemental proportional or relational thinking here). “Young children can use comparisons between work-contribution to allocate resources.”

Dialogue of the deaf

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Dialogue of the deaf

I had a stimulating discussion with a neuroscientist the other day. I tried to explain to her that my interest in children’s cognitive development is linked to my interest in epistemology, that is, to what I refer to in this blog as the normativity of thought.

For example, I argue that researchers who try to explain children’s knowledge of math from a nativist point of view, can only explain the starting point of cognitive development. The starting point is innate mathematical knowledge, which is mostly implicit, and basically consists in an ability to identify the numerosity of collections of objects found in the outside world. In other words: researchers have shown that animals (humans included) have the innate ability to assess the size of a collection of perceived objects (for example, they can notice that a collection of 15 pebbles is greater than a collection of 10 pebbles). They can also discriminate among exact quantities, but only when dealing with small sets (two, three, and perhaps four objects). Also, some animals and human babies can perform elementary arithmetic operations on small sets (adding two plus one, subtracting one from two, etc.) I am referring here to studies by Dehaene (2011), Izard, Sann, Spelke, & Streri (2009), Spelke (2011), and many others.

This basic capacity is certainly different from fully-fledged “human math.” The latter involves, at the very least, the symbolic representation of exact numbers larger than three. We (humans) can represent an exact number by saying its name (“nine”), or by using a gesture that stands for the number in question (depending on the culture, this might be done by touching a part of one’s body, showing a number of fingers, etc. – see Saxe ( 1991) and also http://en.wikipedia.org/wiki/Chinese_number_gestures). And, of course, we can write down a sign that represents the number (for example, with using the Arabic numeral “9”).

Scholars agree on the fact that advanced math is explicit and symbolic, and that it builds on (and uses similar brain areas to) its precursor, innate math. Once they operate on the symbolic level, humans can do things like: performing operations (addition, subtraction, multiplication, division, and others), demonstrating mathematical propositions, proving that one particular solution to a mathematical problem is the correct one, etc. To sum up: our symbolic capacities allow us to re-describe our intuitive approach to math on a precise, normative, epistemic level.

Now, here’s when it gets tricky. I argue that the application of algorithms on the symbolic level is not merely mechanical. Humans are not computers applying rules from a rule book, one after the other (like Searle in his Chinese room). Rather, as Dehaene (2011) argues, numbers mean something for us. “Nine” means nine of something (anything). “Nine plus one” means performing the action of adding one more unit to the set of nine units. There is a core of meaning in innate math; and this core is expanded and refined in our more advanced, symbolic math.

When executing mathematical operations (either in a purely mental fashion, or supported by objects) one gets a feeling of satisfaction when one arrives to a right (fair, correct, just) result. Notice the normative language we apply here (fair, correct, right, true, just). We actually experience something similar to a sense of justice when both sides of an equation are equal, or when we arrive to a result that is necessarily correct. (Note to myself: talk to Mariano S. We might perhaps do brain fMRIs and study if the areas of the brain that get activated by the “sense of justice” in legal situations, also light up when the “sense of justice” is reached by finding the right responses in math. If a similar region gets activated, that might suggest that there is a normative aspect to math that corresponds to the normative aspect of morality).

For me, then, the million dollar question is: how do humans go from the implicit, non-symbolic, automatic level to the explicit, symbolic, intentional and normative level? What is involved in this transition? What kind of biological processes, social experiences and individual constructions are necessary to achieve the “higher,” explicit level? (These are interesting questions both for the field of math and for the field of morality). And my hypothesis is that this transition necessarily demands the intervention of a particular type of social experience, namely, the experience of the normative world of social exchanges and rules of ownership (I’ve talked a little about such reckless hypotheses in other posts of this blog).

Now, when I try to explain all this to the neuroscientist, I lose her. She doesn’t follow me. For her, human knowledge is the sum of a) innate knowledge and b) learning from the environment. Learning is the process by which our brain acquires new information from the world, information that was not pre-wired, that didn’t came ready to use “out of the box.” Whether such learning involves a direct exposure to certain stimuli that represent contents (a school teacher teaching math to his or her students) or a more indirect process of exposure to social interactions is not an interesting question for her. It doesn’t change her basic view according to which there are two things, and two things only: innate knowledge and acquired knowledge. What we know is the result of combining the two. And this is the case both for humans and for other animals. Period.

Something similar happens when I talk to her about the difference between “cold processing” and “hot processing.” We were discussing the research I am conducting right now. I interview children about ownership and stealing. In my interview design, children watch a movie where one character steals a bar of chocolate from another, and eats it. The interviewer then asks the child a series of questions aimed at understanding her reasoning about ownership and theft. Now, the movie presents a third person situation. This means that the child might be interested in the movie, but he or she is not really affected by it. Children reason about what they see in the movie, and sometimes they seem to say what they think it’s the appropriate thing to say, echoing adults’ discourse. Because, after all, the movie is fiction, not the real world.

I believe that normativity emerges not from absorbing social information that comes from external events (watching movies, attending to teachers’ explanations) but from children’s real immersion in first person, real world, conflictive situations. When a child is fighting against another for the possession of a toy, there are cries and sometimes there even is physical violence. These encounters end up in different ways; sometimes children work out a rule for sharing the scarce resource, sometimes they just fight, and sometimes an adult intervenes and adjudicates in the conflict. The child’s reactions during these events is not dictated by cold reasoning but by deeper impulses. It is in these situations where we should look for the emergence of our basic normative categories, such as reciprocity (both social and logical, or “reversibility”), ownership (or the relationship between substance and its “properties”), quantity (used to implement equity and equality), etc.

But, again, my biologist friend does not feel that the distinction between the impulsive, intense, hot reactions we experience when involved in real conflicts and the kind of third person reasoning that is triggered by movies and artificial stimuli is an important one. In both cases, she argues, it’s the same cognitive system that is at work. What we think about third person characters is probably similar to how we reason about ourselves (thanks to our capacity for empathy, our mirror-neurons, etc.)

I don’t know who’s right and who’s wrong here.

 

Dehaene, S. (2011). The Number Sense: How the Mind Creates Mathematics, Revised and Updated Edition. The number sense How the mind creates mathematics rev and updated ed (p. 352). Oxford University Press, USA. Retrieved from http://www.amazon.com/dp/0199753873

Izard, V., Sann, C., Spelke, E. S., & Streri, A. (2009). Newborn infants perceive abstract numbers. Proceedings of the National Academy of Science, 106(25), 10382–10385.

Saxe, G. B. (1991). Culture and Cognitive Development: Studies in Mathematical Understanding. Hillsdale: Lawrence Erlbaum Associates.

Spelke, E. S. (2011). Quinian bootstrapping or Fodorian combination? Core and constructed knowledge of number. Behavioral and Brain Sciences, 34(3), 149–150.

 

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

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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.