How do children think about numbers on the number line? Is the number line only “sparsely populated” with the counting numbers or does it include fractions and decimals? If so, how many fractions are there between any two integers? Ultimately, learning to measure physical quantities goes hand in hand with the development of a broader concept of rational number; the concept of rational number, in turn, depends upon understanding that numbers can be divided as well as added or subtracted.
In our Granularity of Number Task, students judge how many numbers fall between two integers. Approximately 9 of 10 Grade 3 students, from both the treatment and control groups, only note the counting numbers spontaneously. Many more students acknowledge (sometimes after a prod from the interviewer) that there are fractions amidst the counting numbers. But almost without exception they treat these as a few isolated cases, for example, the case of "halves" or "fourths". Again, there were no differences between treatment and control students.
We also decided to assess students' understanding of division of numbers in order to compare it to the questions about the repeated halving of a piece of clay. Would students treat these cases as essentially the same or different? Would repeated having of a piece of clay eventually terminate, leaving no clay remaining? Would the repeated division of a number by 2 always leave a non-zero result or would the process stop at zero?
We were somewhat surprised to learn that third grade students were often unfamiliar with the idea of dividing one integer by another. At the beginning of Grade 3, fewer than 1 in 5 students correctly identified the result of (a) eight divided by four and (b) one half of one. Only one of every two students realized that one-half of two equals one.
By the end of grade three, a much greater portion of students seemed to know what is meant by dividing by two, although many students still did not realize that half of 1 is 1/2. However, the treatment as well as the control students overwhelmingly believed that the repeated division by two would terminate, reaching zero. (For many it terminates rather quickly after one of two divisions.) It is unclear why they think so, but we will be examining several alternatives as we continue to analyze further questions in the section on number.
The data indicate that for many students, the problem is that they are confusing division with repeated subtraction of a constant value, underscoring the fact that division is not yet well understood in the context of number and is psychologically quite different from (and more difficult than) understanding of repeated division of physical quantity (assessed in Task 1). In fact, almost half of the 3rd grade pre-Treatment and Control children judge that there is always a remainder when one repeatedly divides a piece of clay; almost two-thirds of the post-Treatment grade 3 children do so.
Perhaps learning about fractions will make the interminability of repeated division of number more accessible; after all, each successive division merely doubles the size of the denominator. Significantly, some (of the few) students who showed insight about the repeated division of number, talked about “cutting” and alluded to the similarity of the two problems (dividing physical quantity and dividing number.) In any case, we will follow with eagerness developments in their thinking and we hope to relate their increased awareness about number with their notions about the repeated divisibility of matter. Ultimately, numerical division and division of matter part ways: the former never ends, but the later does. One wonders about the interaction between these opposing points of view at different points of learning and development.