I worked with a 7th grader the other day who was solving a “proportion problem”. He had set it up as and then solved it by reasoning that 30 was 5 times bigger than 6, so the scale factor was 5, and applying the scale factor to 4 gave him . He used his calculator to check the scale factor, and to check the scaling up of 4 to 20. Perfectly fine reasoning, but I decided to probe him a little further, and I asked if he had any way to check his answer. What I expected him to do, was to use his calculator to divide 4 by 6, getting a decimal number, and comparing that decimal number to the one he got from dividing 20 by 30. Alternatively, he might have simplified both fractions, getting in both cases.
Instead, he sat there for quite a while, unsure as to how to proceed to check his answer. I then asked if he could turn both sides of the “=” into single numbers and then compare them. He said no. I asked if there was any key on his calculator that matched the horizontal line separating the 4 and the 6. He said no – the horizontal line meant “over”, 4 over 6, and there was no “over” key on the calculator. For him, the equation doesn’t show up as an equation, and the horizontal lines don’t show up as fractions. It appears that he wrote it that way because only because he knew he was supposed to, and at least it allowed him to keep track of four quantities in a particular pattern. But what was the pattern that he was responding to?I believe it quite possible – likely even – that his mental model for the situation was something like the picture above: four numbers arranged in a grid, with part of the grid representing the votes for Joe, another part of the grid representing all the votes, and in the other direction, part of the grid representing the sample and the other part representing the entire class. The proportion he set up, as he was taught, to answer the question: “how many votes can we expect for Joe in the entire class if Joe got 4 votes in a random sample of 6 class mates,” might for him still be just a way to capture the diagram. In the diagram, there is the logic of scaling and proportionality, not the logic of fractions. He responded to my questions consistently with the logic of the diagram, not consistent with the logic of fractions.
His insistence that the horizontal lines didn’t represent fractions, and that they had nothing to do with division was unusually pointed. And yet, it continues to surprise me how many kids, at very different grade levels, see fractions as these disconnected things, having little or nothing to do with division. If you ask them how much is, they typically don’t respond by dividing 20 by 4, they respond by using the tool set they know from working with fractions: they look for common divisors, “simplify the fraction” by writing it as or and then either stop or write it as 5. The horizontal stripe of the fraction, which they correctly never confuse for a subtraction sign, is also seen as completely separate and distinct from the symbol for division, whether they use the diagonal stripe “/” or the horizontal stripe with a dot above and below “÷”. Even in a world of calculators, where all operations are naturally embedded in the domain of (finite precision) decimals, kids see fractions as existing wholly outside of that. I believe that this is an avoidable artifact of how we typically teach fractions.