Between the ages of one and two, toddlers typically rapidly expand their vocabularies. Tots seem to suddenly go from babbling hesitantly to confidently chatting up a storm. But it turns out the leap from mama to precocious follows a simple mathematical pattern: the bell curve.

Cognitive scientist Bob McMurray of the University of Iowa set up a relatively simple mathematical model of word learning on a commonly available spreadsheet, assessing the potential to learn each of some 200 words. He set a numerical threshold at which a given word would be considered learned, operating under the assumption that it would take kids time to learn each word, they could pick up multiple words at the same time, and that some words were more difficult to process.

A simple analogy would be jelly bean jars: Every time a word is heard, a jelly bean goes into a jar representing that word. More difficult words require bigger jars with more beans in order for the words to be considered learned, but once that jar is full, the word is in the toddler's vocabulary, McMurray explains.

Running the actual mathematical model, McMurray found that toddlers inevitably experienced an explosion in word learning after a set period of time, no matter what. Initially, the most common words are learned relatively slowly followed by a rapid jump in vocabulary. In other words, a little girl might know only the most often heard words at 12 months but a short four months later be able to command several hundred words that she only hears occasionally.

The shape of this learning curve is a simple, normal statistical distribution, more commonly known as the bell curve. Scaling up the model to include 10,000 words showed the same pattern as did varying the underlying variables (for example, assuming that a child could only learn one word at a time). "It's a little bit slower but it's still the same vocabulary explosion," McMurray says.

The finding casts doubt on the necessity of any particular special function acquired during childhood that enables kids to master language. Many suggestions for such an innate mechanism have been made: from syntactic bootstrapping (using a new word's context to determine its meaning) to mutual exclusivity (using known words to learn new ones by process of elimination). "These specialized mechanisms are working on the margins," McMurray says. But "to explain the big picture, it's much, much simpler. … Anytime you have more difficult than easy words [the learning curve] will have this property."

The curve should also apply to children learning other things, such as facts or physical skills, and drops off precipitously, perhaps in late adolescence. "It has not been clearly established but it's almost certainly happening," McMurray says.

The curve may explain why some children learn language faster than others, mainly because of differences in the amount of words to which they are exposed or differences in short-term memory abilities. "This is a big complex behavior that isn't in the genes and it's only dimly in the environment. Really it's in the math of parallel learning," McMurray says. "That's a different way to think about development."