The Mental Number Line

Cognitive scientists have established over the past few decades, pretty conclusively, that as human beings we are 'born to numerate'. Some simple but brilliant work with preschoolers shows that they develop and practice basic, key numerical skills before the age of 4 or 5, spontaneously. The development of these skills is reminiscent of the way children learn language: there seems to be an innate module in our brains that clicks into action given a 'minimum' environmental input. The essential accomplishment of the early years is the proper understanding and use of a mental number line (MNL), used in the act of counting. This is no trivial thing! When a toddler or preschooler counts a set of objects, she invokes no less than five crucial principles.

1. There has to be a one-to-one correspondence between each object and a number name. For eg., you cannot assign more than one object the number 'four'.

2. Yet the number names do not belong to the objects in any way; they can be reassigned. On a recount, you can change all the assignments!

3. Number names are always to be spoken in the same, invariant order. In fact, many toddlers have the wrong number order—one, two, three, five, seven, eight, nine, ten!—but they use it invariantly (till they eventually correct themselves, of course).

4. The final number spoken is always the size of the set.

5. Counting is something you can do with any set of objects, from pins to people.

In time, they use this MNL to compare two numbers to say which is larger, and soon begin to do simple addition using a method called 'counting on'. That is, to add 4 and 2, they start with the larger number 4 on the MNL, and move two units to the right to reach 6. Five year olds have been observed to invent this sophisticated method spontaneously, as they learn to combine their skills of comparison with counting on the MNL.

When children begin formal schooling, they typically should have this informal number knowledge available to

understand anything new that is taught. But they do not all begin school equal; studies show (as does any teacher's experience) that in first standard, children vary in their levels of number knowledge. Some students have mastered several number facts, which means that they can quickly recall from memory facts such as '4+2=6' without having to actually re-perform the sum. They also are more likely to use strategies such as counting on, more efficiently, when faced with new problems. Other students are at a disadvantage, not having proper representations of counting on the MNL, therefore not having invented certain strategies, and therefore not having enough number facts at their disposal.

Several reasons have been proposed to explain these differences, but psychologists are also working on how to close the gap sooner rather than later, to help weaker students build the foundation they need. The most obvious suggestion is to include explicit instruction of the MNL and its properties in first standard curricula, since it is not generally taught that way. Interestingly, one of the strongest correlates of these initial differences in numerical ability is the socioeconomic status (SES) of the child entering school. Children from lower SES are at a significant disadvantage compared to middle or high SES children at number knowledge when they begin school. Unaddressed, this gap only widens with time. Developmental psychologists Robert Seigler and Geeta Ramani offer one interesting reason for the difference: lower SES children do not have access to the kinds of board games other children routinely play with. The main element of many simple board games (such as Snakes-and-Ladders or Ludo) is a series of numbered spaces, linearly arranged. You move your token along a certain number of spaces, counting as you go, one number for each move. Playing this game, they say, gives preschoolers the right stimuli to develop correct understandings of the MNL.

In a recent study, Siegler and Ramani worked with a large number of children from lower- and middle-SES

backgrounds in the U.S. They first replicated the general finding that the lower-SES children perform significantly worse on the following simple numerical magnitude estimation task: given a line with 0 at one end and 100 at the other end, place a third number (say, 37) correctly on the line. Second, they provided the lower-SES children with around 30 brief sessions of playing a very basic board game with just ten linearly arranged spaces. The total time of intervention was only around two hours, and yet post-tests revealed a virtual 'catch-up' of these children! This study needs replication in India, of course. But given the importance of the MNL for early arithmetic, and given the simplicity of the intervention, it is definitely worth investigation.