The ultimate goal of this project is to provide a novel model of the cognitive and neural basis of numerical cognition, and to use this knowledge to guide the development of new training methods that could improve mathematical abilities in children. The project is a collaboration among investigators at the University of Rochester, Johns Hopkins University, and Cold Spring Harbor Laboratories. Recent research suggests that acuity of numerosity judgments is predictive of success in formal mathematics education, and that similar cognitive processes can be trained by specific kinds of domain-general experience. The core idea is that the firing of neurons encodes a probability distribution, thereby representing simultaneously the most probable sample from the distribution and the variance (i.e., confidence) of the estimate.
This project will develop and test a formal Bayesian model that has the unique feature of naturally accounting for a number of metacognitive factors, a critical but undertested factor in the acquisition of expertise. The primary advantages of this Bayesian approach are its ability to provide a natural description of: 1) how the confidence of a learner relates to the precision of their number knowledge; 2) how a learner can combine information from multiple sources of information about number; 3) how intuitive preferences (also known as prior belief) predict learners' errors; and 4) how improvements in probabilistic inference may benefit the precision of the number sense.
The goal of this research was to gain a deeper understanding of humans' ability to understand and represent numbers. Our experiments focussed on the Approximate Number Sense: the widespread ability of humans to estimate and manipulate numbers even without formal mathematical training. A deeper understanding of the Approximate Number Sense may allow us to develop better educational methods for teaching mathematics in school. In addition, a deeper understanding has the potential to be informative about how the brain processes complex information and uses it to make a decision. This research was part of a collaborative effort among five investigators. The experimental design and analyses described here were enhanced by discussions with all collaborators, but my efforts were most closely linked with a thoeretical collaborator, Dr. Alexandre Pouget. We collected behavioral data from human volunteers who were trained to make judgements about the number of stimulus events that were presented to them during each of a series of trials. Stimuli could be auditory clicks, visual flashes, or sometimes both together (multisensory trials). Subjects' decisions about multisensory trials were especially informative about the brain's representation of number. Specifically, they revealed whether subjects represented numbers as single scalars (Fig 1, right) or, by contrast, as probability distribtuions (Fig 1, left). Our data strongly suggested the latter: that subjects represent probability distributions when estimating number. Probabilistic representations have the advantage that they automatically encode the reliability of the stimulus. In the probabilistic representation shown in figure 1, the broad distribution might correspond to uncertainty about the true number. Such uncertainty might result if the individual elements that a subject is estimating are hard to see or blurry. An estimate of uncertainty can be thought of as "knowing what you know," a critical piece of information when making quantitative estimates. We presented this work at the 2013 Society for Neuroscience Meeting where it was highlighted in the Press Book as a high profile result. We have since submitted the work for publication. Once the work is published, we intend to make the behavioral data freely available on the laboratory website so that other reserarchers may download and further analyze it. This will allow other researchers to test theories we have not yet considered, opening the possibilty that other important insights may be gained from our data. We also plan to publicly share the analyses we used in the paper to evaluate possible strategies that the subjects used to estimate number. Our analyses went beyond those typically used in multisensory experiments and will likely benefit other labs who work on multisensory processing.
