Last week I had the opportunity to visit a 7th grade math class in an alternative school. Lots of school districts have at least one school like this, in which all the students are there because they've had some sort of difficulty at their "regular" school. Often the kids have been suspended for behavioral reasons, and many struggle academically because of diagnosed learning disorders of one kind or another.

In this particular class there were six students, all of whom were practicing problems that involved adding and subtracting fractions with unlike denominators. You may recall that this requires finding a common denominator, converting one or both fractions into an equivalent fraction that uses the common denominator, and then performing the addition or subtraction operation. Straightforward enough once you understand how to do it but not necessarily obvious, especially if you already have other misconceptions involving fractions.

I noticed one student, TJ, struggling with this problem: 62/9 - 3/2. He had written the following on his paper: 31/18 - 27/18. I asked him if he could explain to me what he was doing and he said, "These are hard because [the numbers] are bigger," and he showed me another problem, 2/3 - 1/6, which he had solved correctly the previous day. So I asked him to explain what he did to solve that problem. He couldn't remember, so he turned to a classmate sitting nearby and asked him to explain it to me (which he did).

I continued asking TJ questions:

Me: Even though these numbers are bigger, are you still using the same process that you used with the smaller numbers?

TJ: Yeah.

Me: So how did you get 31?

TJ: That's 62 divided by 2.

Me: So you're dividing here?

Just in case it's not clear, what I was trying to do was ask questions that would focus TJ's attention on a specific part of his work. I was trying to get him to realize:

- He already knew the correct procedure (though perhaps not too well, and he may not have understood why the procedure works);
- The procedure should work with big numbers as well as small numbers;
- He applied the procedure incorrectly with the big number problem, using division instead of multiplication.

I suspected that because 62 x 2 = 124, and 124 is a much bigger number than the other numbers he'd been working with, TJ was suspicious that that number was incorrect. I also suspected that he might be remembering another procedure he had used with fractions that involved dividing instead of multiplying, and he thought that that might be the correct procedure to use here, and since 62/2 = 31, and 31 is smaller, and therefore less suspicious than 124, he thought 31 was probably correct so he went with that.

Keep in mind, I'd only been talking with TJ for about 5 minutes, so I could have been wrong about his misconceptions. There was a lot I didn't know about his understanding of these problems, but I was hoping to find out more. (TJ seemed reasonably willing to answer my questions, which indicated to me that perseverance was a skill he might possess in some abundance; that was a good sign.)

It was at this point that one of the two teacher assistants wandered over. (Counting me, there were actually four math teachers in this class of six students.) She listened to our conversation for a few seconds, then interrupted saying, "It's just like I showed you yesterday, whatever you multiply on the bottom, you multiply the same thing on the top..." and then proceeded to tell him, step-by-step, how to work the problem correctly. I stayed and listened for another few minutes, but left soon after the following exchange:

TA: Do you understand?

TJ: Yes.

TA: So then what would [this answer] be?

TJ: I don't know.

TA: Well, why did you say you understood if you don't understand?

As tempting as it is to just straight up *tell* a confused student how to work the problem, research shows that this approach doesn't work very well and that, as suggested by the Mathematics Assessment Project, we can better "help students to make further progress by asking questions that focus attention on aspects of their work."

Many often ask, why, why don’t students just remember the procedures they have been taught and practiced? The fact is, that approach works fine in the short term but, as every teacher knows, if procedural knowledge is not underpinned by conceptual understanding, students will quickly forget “how to do it”. Research underlines this. For example, a detailed study of subtraction in arithmetic found that the students who made errors did so in many different ways – not too surprising. But the research also found that many students who got their calculations correct did so in different ways, not only the way they had been taught.

Key research result: Those who are effective at mathematics don’t remember exactly how to do things; they remember roughly how, and can check and correct (debug) their own procedures. [pdf]

This is something that many teachers struggle with, and it's only partly due to impatience; it's true that helping students by asking questions takes longer than just telling them how to do it, and teachers are very aware of how much material they're expected to cover in the courses they teach. But it's also true that teachers want desperately to help their students; that's a big reason why we became teachers in the first place. And it can be really difficult to watch a student struggle in confusion when you already know what's confusing him and you could just tell him the answer.

Unfortunately for us teachers, just telling our students how to get the answer is not an effective way to help them learn. If we want to help them learn, we have to learn more effective techniques.

Photo Credit: Student Writing Center via Compfight cc

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