Math Education / Number

Why is learning to count so hard?

Contributed by David Barner

The origins of human mathematical practices extend back in history to the earliest moments of human culture. Early cuneiform writing systems not only arose in large part for the purposes of accounting and trade, but the presence of numerical symbols in these systems has sometimes been the wedge that allowed historians to crack their codes. How these ancient systems first evolved – and how they made the leap to natural language in the form of verbal counting and calculation systems – remains a tantalizing puzzle that researchers are only beginning to put together. But as it turns out, the answer to how humans created mathematical systems may not be locked in the evaporating traces of ancient history.

In a study published today in Cognitive Psychology, with Dr. Katie Wagner, Berkeley grad student Katherine Kimura, Wesleyan post doc Pierina Cheung, and yours truly, we presented part of the answer to this question by studying how 2- to 5-year-old children learn to count. When children learn to count they pass through developmental stages that are remarkably similar to the stages of mathematical changes found in the human anthropological record. In particular, as we argue in this paper, children appear to learn their first numbers using two distinct and somewhat independent routes. To begin, as first documented in the pioneering work of Yale psychologist Karen Wynn, children learn the meanings of the numbers “one”, “two”, and “three” in sequence over a period of many months, and in most cases, years. In the US, English-speaking children first learn “one” and can give exactly one object when asked for one. About half a year later, they learn the meaning of “two” and can give two objects when asked. Then, several months after that, they learn “three”. And then “four”. But here, amazingly, is where things stop. Just as many human languages have number words that end at 3 or 4, children in every country tested so far – including Russia, Japan, Brazil, Saudi Arabia, Slovenia, and Taiwan – appear to learn numbers one by one until they get stuck at 3 or 4. After this, a second system of numbers takes over – a memorized counting procedure – which children begin reciting at age 2, but don’t fully master until they are as old as 5 or 6. Using this system, children utter words in sequence while pointing at objects, and use the final label in the count sequence to label the final counted set.

In our new study, we sought to figure out how these two processes are related to each other – and what makes them so hard for humans to master. Our conclusion is that they are fundamentally dissimilar and probably unrelated systems and that they depend on deeply different learning processes that reflect distinct human capacities – the same capacities that allowed ancient humans to solve the problem of number thousands of years ago. To understand why numbers are so hard for kids, we tested bilingual learners and asked a simple question: When children learn number words in one language, how does this affect their knowledge of numbers in their second language?

What we found was surprising. During the first stages of number word learning, when children are learning words for 1, 2, and 3, Spanish-English and French-English bilingual children appear to learn their numbers separately in each of their two languages. Children who understood the number “three” in English often only knew “un” in French, for example. Overall, a child’s number “knower level” in one language did not predict their level in the other language for these small numbers. This result, we concluded, suggests that the long delays between learning “one”, “two”, and “three” that Wynn originally reported are the product of a largely language-specific problem of mapping sounds onto non-verbal concepts. If the long delays between stages were due to learning entirely new concepts, then some amount of transfer between languages would be expected (since the concepts learned once wouldn’t need to be learned again). But none was found.

Second, despite the fact that knowledge of small numbers did not transfer, children’s ability to use the counting procedure did. In fact, almost every child who could count and give quantities greater than 4 in one language could also do so in their second language. This is important, because it suggests that figuring out how the counting procedure works is probably a very general conceptual breakthrough – one that can be shared across languages – rather than a specialized fact about English or French. And it is exactly this breakthrough that some cultures – but not others – made when they invented tally systems, body count systems, counting boards, and other accounting systems that transcended the verbal system for small numbers.

These results shed light on the puzzle of how human number systems likely evolved thousands of years ago. Two independent systems – one rooted in the morphology and syntax of natural language and the other rooted in memorized routines and procedures – were fused into a single system for labeling cardinalities that went beyond the limits of either system taken separately. They also point to how we might intervene on learning in the classroom. Whereas past studies suggested that we might teach children to count by first teaching them the meanings of “one”, “two” and “three”, this new research suggests that these are in fact separate problems, and that the fastest way to teach children to count large sets is to develop smart training procedures that target counting, per se. This shouldn’t just target rote reciting of the list alone, or pointing at objects, but must also focus on how counting applies to collections of objects taken together. Our study also suggests that, when children are exposed to two languages, facts that are linguistically encoded – like labels for quantities, multiplication tables, or simple memorized sums – should be taught in both languages, whereas principles and procedures that do not depend on linguistic memory – like the logic of counting itself – may safely be taught once and left to naturally spread to the child’s other language.

Important bits and pieces of this puzzle remain. We know why today’s children learn to count: We teach them lists and train them. But why did ancient humans begin doing this, and how did they first figure out that counting could be used to accurately label large quantities? Without an adult model or independent way to accurately perceive large quantities, how could these earlier humans verify that their counting systems reliably reflected reality? Here, our best hope again may be a joint research project that combines observations from anthropology and developmental psychology, and that targets both how humans solved this problem in the past, and how they solve it now, as they first encounter counting and number words.

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