Big numbers, hard calculations, and little heads

A phenomenon currently of great interest to those in cognitive development is the mental representations of large numbers, and the relationships between these representations.  Number is an excellent example of an overarching cognitive structure, which can be systematically broken down and studied by way of each contributing field of cognitive science.  The ability to represent large amounts (and, critically, the relationship in memory between these quantities) has driven entire theories of animal learning in behavioral neuroscience.  In even the most basic models of reasoning, estimating probability (and hence, ratios) is integral to the organisms' decisions to act or not to act, to infer or not to infer.  It is well known that for all animals, including adult humans, differing schedules of reinforcement produce differing rates of responding, which must be computed by comparing the likelihood of reinforcement with the total number of trials lapsed. The ability to compare two amounts and abstract their ratio is possibly the anchor of reasoning. In this talk, I will describe several lines of research on the capacity for numerical reasoning in early infancy, the parallels of this work with work done in animals, and possible ways to integrate these findings into our educational systems.

There appear to be two main systems that infants use for representing number- one for small numbers, and one for large numbers.  The small-number system is used for 1-4 items and is utilized mainly when an organism must track objects in their environment.  It is very exact and contains a lot of information about the featural properties of these objects (such as size, shape, and color).  As the bulk of my work (and the bulk of work done with animals) has focused on the other, large-number, system, that is what I will be discussing today.  Both the animal and adult literature suggest that the representation of large quantities is mediated by a preverbal analog model of magnitude representation.  This model was originally developed to account for perceptual and numerical competencies in rats, and subsequently proposed as underlying humans' approximate large number representation . It postulates an accumulator mechanism composed of: a sensory source for a stream of impulses, a pulse former which gates this stream of impulses to an accumulator for a fixed duration (around 200 ms) whenever an object or event is counted, an accumulator which sums the impulses gated to it, and a mechanism which moves the magnitude from the accumulator to memory when the last object has been counted.  This accumulator is subject to psychophysical laws, namely Weber's fraction limit, which states that the variability of perception of magnitude increases with the amount to be represented (i.e., representation becomes more approximate and discrimination between quantities less exact as the magnitude in the accumulator increases.)  Two processes have been hypothesized regarding this accumulator output.  Estimator processes refer to the simple production of mental representations of number (called numerons.)  Operator processes are processes such as addition and subtraction, in which multiple numerons are manipulated to produce another numeron.

This topic is not completely unexplored in human infants, as Spelke and colleagues have looked at large number estimator processes in infants and found they can discriminate 16 from 32, and 8 from 16, but not 16 from 24 or 8 from 12 (until around 9 months of age).  They conclude that this “number sense” is present in infancy, but has a long way to go before reaching adult–like levels of quantity discrimination of 1.15 : 1.  A different set of theoretically interesting questions arises when we look at the nature of the representation and the type of information moved from discrimination to storage. 

Specifically, I have studied the proficiency of operator processes across development.  I currently have research which suggests that infants can perform operations analogous to addition and subtraction over large amounts of objects. 9-month-olds look longer to incorrect outcomes of the large-number addition and subtraction problems 5+5 (=10 or 5), 10-5 (=10 or 5), 4+5 (=9 or 6), and 10-4 (=9 or 6).  Critically, the incorrect and correct outcomes differ by a particular ratio (such as 2:1 in the 10 and 5 object outcomes, and 3:2 in the 9 and 6 object outcomes.)  Interestingly, this proficiency in addition and subtraction, which accords to the ratio of the outcomes, appears to be equivalent to their ability to simply discriminate two amounts.  The operations shown to the infants appear to add little (or no) error to the final representations of the outcomes.  I am currently extending this work in two ways.  First, I am examining the “breakdown point” by giving infants operations whose outcomes are indiscriminable even in a pure comparison case (such as 12 versus 9 objects.)  If infants fail at this task, then we can take steps towards concluding that an approximate magnitude mechanism was underlying their representation of these amounts. 

Next, I am conducting an intermodal addition and subtraction study.  In this study 5 objects are moved behind a screen, and then 5 “clunks” (which indicate the addition of more objects) are heard by the infants.  If infants look longer to the incorrect outcome of 5 objects (and shorter to 10 objects), then it stands to reason that they were inferring a particular amount from the audio stimuli, and adding that amount to the visual stimuli.

Another operation (which is of great interest to me, perhaps because it has been largely ignored in the infant literature) is the seemingly complicated yet ubiquitous one of division, alluded to in the beginning of this talk.  In what is known as rate-of-return, animals in a naturalistic feeding situation are able to spontaneously compute ratios by dividing amount of food with the area in which that food is found (i.e., 50 apples per tree here, and 10 apples per tree there) and return to feed at levels similar to that ratio (around 85% of time return for the first batch, 15% for the second.) .  The relevant question in light of this finding, therefore, is not whether infants are capable of comparing two quantities of items in a discrimination task, but whether they can be “pushed up” a level to discriminating two ratios of items.  If so, this could be considered analogous to rate-of-return.  There has been no systematic study in infancy of this crucial ability, which is offered by some evolutionary theorists as the reason amount representation evolved in the first place. 

Here at Yale we have found very good evidence that infants are capable of this ratio abstraction.  When habituated to a sequence of scenes displaying either 2:1 ratios of object type x: object type y (such as 20:10, 8:4, 32:16) or 4:1 ratios of these objects  (such as 20:5, 40:20, 12:6), and then tested with either a new ratio or a new exemplar of the old ratio (so, both a set of new 2:1 and 4:1 scenes), children look differentially to these two types of test ratios and this looking time will switch as a function of habituation group.  We appear to have found evidence for these ‘higher-order' representations in the infant's memory store.  The next condition we ran controlled for extraneous perceptual cues (such as confounds with area, or contour length), and infants were still able to discriminate the two ratios presented, indicating that they were computing these ratios over an abstract representation of number itself, and not just perceptual cues associated with number of items.

There are many potential educational directions to take this area of study.  I propose a multi-step process which will strengthen children's representations of number, incorporation of carefully controlled displays to the children's texts and games, and a set of games which tap into children's inherent sense of large number and the operations that can be done with this intuition.  To strengthen the representation of number, one would pair sights and sounds with the identical number being played or displayed at the same time.  So, for instance, 5 drumbeats played as 5 trees come up on screen.  Sometimes these objects would be big, sometimes little, sometimes the same, and sometimes different.  By varying the specific perceptual aspects of both the sights and the sounds, the children may better learn that 5 means 5, no matter what it looks or sounds like, as an entity unto itself.  The games would be based on the videos used by many researchers who study nonverbal operations, with the children watching as a group and then taking turns to give a sticker to a display which showed the correct outcome (to the addition or subtraction problem) or the same ratio (after seeing many examples of this ratio before).  By incorporating the fun of games and scientifically created stimuli which taps into our natural, animal-like processes we may be able to enhance the child's educational experience.

This line of research starts to address the broader issue of how we process and organize information from our environment, how this activity changes across development, and how we can use this natural way of processing to add to our educational resources.  By informing our infancy research with established phenomena in the animal literature, we are closer to discovery of the “fundamentals” of our nature.  As this talk indicates, my passion has come to lie at ferreting out the shared junctures which reflect the development of cognition throughout evolutionary history.  The representation of number is a fascinating, prime candidate through which to do so.