In a recent post, I referred to a study at MIT that suggested that infants reason by mentally simulating possible scenarios in a given configuration (like different colored objects bouncing around in a container). They then figure out which outcome is most likely based on just a few physical principles (like whether the object nearest the exit of a container would bounce out first). Another MIT study was just published in the June 24 issue of Science. It reported that “16-month-old infants can, based on very little information, make accurate judgments of whether a failed action is due to their own mistake or to circumstances beyond their control.”
MIT’s own reporting on the study says the following:
Infants who saw evidence suggesting the agent [using the toy] had failed tried to hand the toy to their parents for help, suggesting the babies assumed the failure was their own fault. Conversely, babies who saw evidence suggesting that the toy was broken were more likely to reach for a new toy (a red one that was always within reach).
Much of the significance of these studies lies in the extent to which they support probabilistic inferential learning models.
Schulz says she was at first “blown away” that 16-month-olds could use very limited evidence (the distribution of outcomes across the experimenters’ actions) to infer the source of failure and decide whether to ask for help or seek another toy. That finding lends strong support to the probabilistic inferential learning model.
There is a growing interest in using probabilistic inferential models to investigate questions about cognitive development. They provide an alternative theoretical perspective on human development by finding some middle ground between the opposing views of nativists and empiricists. Nativists are of the opinion that we possess innate conceptual primitives while empiricists believe that there are only perceptual primitives and that learning uses essentially associative mechanisms. The middle ground is sometimes called rational constructivism, where learning mechanisms are thought to be rational, inferential and statistical.
All of this work is relevant to mathematics in at least two ways. One has to do with how statistical models have come to be used to predict human behavior. Probabilistic models make use of tools that were developed in statistics and computer science only over the last 20 years. But also worth noting is what is implied by the fact that these models are such accurate predictors of behavior. It seems that our bodies are employing statistical methods in perception and learning. This is a provocative idea. In a document intended for a special issue of the journal Cognition, Berkeley’s Fei Xu and Thomas L. Griffiths suggest this:
Perhaps in addition to a set of perceptual (proto-conceptual?) primitives, the infant also has the capacity to represent variables, to track individuals, to form categories and higher-order units through statistical analyses, and maybe even the representational capacity for logical operators such as and/or/all/some – these capacities enable the infant to acquire more complex concepts and new learning biases. (see Bonatti, 2009 and Marcus, 2001 for related discussions).
Reporting on the content of a workshop, Fei Xu and Tom Griffiths also make the following observation:
The hierarchical Bayesian approach provides a richer picture of learning than that assumed in many computational approaches, with the learner considering not just the solution to a particular problem but also forming generalizations about what solutions to these problems look like. In this way, a learner
can form “overhypotheses” that guide future inferences.
Variables, operators, generalizations about future solutions, all of this suggests that the seeds of mathematics can be found in how the body lives.
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