I was reading an article on fractions, waiting for students to show up for after-school tutoring. One of them asked me what I was reading, so I told him. He groaned. I asked him what his least favorite topic in mathematics was, and he told me it was fractions. I nodded, saying that I reckoned that if I took a poll of every student in the school about their least favorite subject in mathematics, “fractions” would be the number one answer.

I have some thoughts about why, although they’re still somewhat rudimentary. As a high school teacher, I didn’t expect I’d find so many students with such deep misunderstandings of fractions, so I’m still getting my bearings on the subject. I did take the opportunity to talk with this student about his understanding.

First I asked him what a fraction meant to him. He couldn’t tell me. I tried a specific example: “What is seven over five?” He didn’t have an answer. “How would you write it?” I asked. He didn’t know.

Okay, time for a different tack: “How are you at division?” Personally, I see fractions two ways, one of them being a division problem. He tells me he’s pretty good there. So: “What’s seven divided by five?”

He said, “Well, first we have to write the problem.” He wrote: \(\frac{7}{5} ?\) So while he could not interpret “seven over five” as a phrase representing a fraction that he could represent, he knew how to write “seven divided by five” as a division problem that was visually identical to a fraction.

He then wrote \(5 \overline{)7}\), completed the division, and concluded that the answer was \(1 \frac{2}{5}\). I asked him to convert that to decimal, and he offered \(1.25\). I asked him to explain that, and he gave a confused response that suggested that he thought he needed to rearrange the given numbers into a single line.

Next I asked him about seventy divided by five. “Fourteen”, he told me. I asked him to write it down, which he did, doing long division and subtracting 70 from 70 to get a remainder of zero. I asked him how he’d solved it, and he told me that he knew that there were twelve fives in sixty, and that he needed two more, so that would be fourteen.

I then asked him if that helped him complete the earlier problem. No. I pointed to the two problems on the paper. No. I suggested he write “.0” after the 7 in the first problem, which he did. He brought down the 0 and finished the problem, then excitedly asked for another.

Since I hadn’t prepped for this, I just gave him nine over five. No problem, 1.8. Then I gave him 36 over 25. He set it up as he’d just learned: \(25\overline{)36.0}\). First step, subtract 25. That makes 1, bring down 11. Bring down the 0 to make 110. There are four 25s in 110, so that makes 1.4, but “what do I do with the 10?”. I suggested he add another zero to the end, which he did, and finished the problem. 1.44.

New problem: 3600 over 25. He told me that there are fourteen 25s in 360, so that’s 350 with a remainder of 10. What he didn’t do: Notice that this was the same basic problem that he’d just completed. What he did do: Quickly multiply two two-digit numbers mentally to find a maximal value.

Personally, I don’t think I would ever approach a problem in the fashion that he did. It strikes me as too mentally inefficient to try to find two-digit multiplicands like that when it’s easier to just do long division step-by-step. It suggests to me that at some point he’s been overexposed to mental math shortcuts and “friendly” re-groupings that are getting in his way.

He finished up correctly, and gave me an answer of 144. I pointed out that this was basically the same answer as before, and he seemed surprised. We then discussed place value briefly, before the tutoring session ended.

During this time, I asked another student to solve \(\frac{36}{25}\) on his own. He worked quietly by himself, first setting it up as \(36\overline{)25}\) and working from there. I knew he was doing it backwards, but rather than stop him I decided to let him go and see where he’d go. After a few minutes, he came up with .69. So he did the long division correctly (discarding the remainder), but he reversed the division. I pointed that out to him, and he fairly quickly returned with 1.44.

I’ve noted the complaint in the past that “36 divided by 25” (or something analogous) is backwards when written as long division: It’s not \(36\overline{)25}\) but rather \(25\overline{)36}\). While the student did the division itself correctly, he lacked the number sense to see that 0.69 was not a logically acceptable solution for \(\frac{36}{25}\).