One of our readers emailed us asking if there has ever been research on whether kids’ understanding of numbers — especially large numbers — differs from adults. Greta did a little poking around and found a fascinating study on second- and fourth-graders.

In the U.S. (and I suspect around the world), kids this age are usually taught about numbers using a number line. In first grade, they might be introduced to a line from 0 or 1 to 10. In second grade, this is typically expanded up to 100. But what happens when second-graders are asked to place numbers on a line extending all the way up to 1,000?

This happens:

I’ve placed the 500 (in gray) on these lines for reference; in the real study, conducted by John Opfer and Rober Siegler, the kids used lines with just 0 and 1000 labeled. They were then given numbers within that range and asked to draw a vertical line through the number line where each number fell (they used a new, blank number line each time). The figure above represents (in red) the average results for a few of the numbers used in the study. As you can see, the second graders are way off, especially for lower numbers. They typically placed the number 150 almost halfway across the number line! Fourth graders perform nearly as well as adults on the task, putting all the numbers in just about the right spot.

But there’s a pattern to the second-graders’ responses. Nearly all the kids (93 were tested) understood that 750 was a larger number than 366; they just squeezed too many large numbers on the far-right side of the number line. In fact, their results show more of a logarithmic pattern than the proper linear pattern. This chart of the results makes the pattern clear:

Both the logarithmic and linear curves fit the data for their population quite closely. What’s more, the second-graders’ performance didn’t change over time as they worked on the problems. It appears that they simply have a different conception of how numbers are distributed over the number line compared to older kids and adults.

So what happens between second grade and fourth grade to cause this difference? How do kids learn the relationship between numbers and their proper place on the line? In a second experiment, Opfer and Siegler worked with just second graders to see what it took to teach them to place the numbers correctly.

In their first experiment, they had found that about a third of the second-graders matched the performance of fourth-graders. So working just with the 61 second-graders whose results best fit with a logarithmic pattern, they repeated the study with a key difference. This time, on certain key trials, the kids were given feedback: If they were off by more than 10 percent, the experimenter told them where the number should go on the number line. If they were correct, they were asked how they knew to put the number in the right spot. At the end of the study, which took less than an hour, the second graders’ results looked remarkably like the fourth-graders’: accurate, linear placement of the numbers on the line.

What’s more, most of the kids followed the same pattern in learning: When they got it, they got if right away. Take a look at this graph:

The experiment was divided into four blocks, with feedback given before each block. The graph defines “0” as the block at which point the function that best explained their answers switched from logarithmic to the correct linear function. The gray line shows the portion of the results were best explained by a logarithmic function, while the black line shows the portion best explained by a linear function. As you can see, once the linear form is learned, the transformation is quick, and permanent.

Interestingly, even if students were given feedback only for higher numbers or lower numbers, they still learned the correct linear form for the entire length of the number line. They learned fastest when they were given feedback for numbers around 150, which represented the largest typical error, but nearly all the students who were given feedback eventually understood the concept.

Opfer and Siegler argue that a “rapid and broad” change like this can occur in a variety of different types of knowledge, ranging from fractions to biology. They claim that there are certain concepts which, when grasped, open up a whole different understanding or representation of how a system works.

One final thought: I wonder if even adults would display a similar pattern to second graders, if the numbers were large enough. I’d be interested to see a study on adults using numbers, say, from zero to a billion. How many adults would properly place the value of one million just a third of a millimeter from the left end of a 30-centimeter line?

Opfer, J., Siegler, R. (2007). Representational change and children’s numerical estimation. Cognitive Psychology, 55(3), 169-195. DOI: 10.1016/j.cogpsych.2006.09.002