I’ve been reading some constructivist stuff lately, so…

What resources does this student have for refining their understanding of fractions?

(Thanks for the submission, Dionn!)

• First, I’d encourage the student to learn with fractions (proper, then improper) before working with mixed numbers again — the mixed number may be what is causing them to “subtract all the numbers the same way”. Converting mixed numbers to improper fractions may be temporarily necessary. Second, a sense of units is needed for the student to understand they can’t add or subtract unlike denominators — they are, however, able to correctly work problems when there is a like denominator (#1, #3, #6 was a near miss). Student doesn’t seem to know that 0/3 = 0; once they learn this the idea can be brought to bear on problems like #12. Student may also have some issues with positive and negative numbers, given the “always positive” result of their subtractions, but it could be an artifact of their fraction understanding gap.

I don’t know any resources I would recommend, except to say I think the selection of problems by “Math-Drills.com” is highly suspect. There is no difficulty curve — #2 and #5 are ridiculous compared to the rest of #1-7 (and are the first two examples where the second fraction is larger), and #9 is almost as ugly. #3 is a very poor selection. Also, placing #5 to the right of #1 with #2 below #1 makes me … unwell. That might be the biggest mistake of all here…

• Amy

I recommend Do the Math, by Marilyn Burns, and K-5 Math Teaching Resources (online). Both resources promote the use of models to solve and understand addition and subtraction of fractions. I notice that someone did draw a model on the bottom left corner of the drill sheet. At this stage in understanding, this student should be using a model every time s/he operates on fractions.

• Andy

I like how the student drew a representation on the bottom of the page. I am curious to see how they used it. I wonder how they visualize what is happening.

• Andy: I’m pretty sure that representation was drawn by someone else, perhaps the teacher; you can tell by the completely different “4” used.