With my sophisticated high-tech math teaching device, the quadrille pad, I have been posing some 7th graders the following problem:
“Where is the middle between A and B?”
I have yet to meet a student who cannot solve this. Most use a two-handed approach, with one hand marking the left and the other marking the right, and moving in unison towards the center. When their fingers meet, they have found the center. Since the line is marked with blue vertical markers, you will see the movement of their hands be jerky, not smooth, moving in hops. Most students hop one unit at a time, some hop in larger units, but rarely larger than two. In the particular problem shown, many students will stop their movement when their hands are still one unit apart and then say that the middle is “right there”, pointing at it with their head or their glance. If I ask them to be more precise, they’ll take a pencil and make a mark in the center between two of the blue delineations. Some students don’t use the two-handed approach, but count the number of spaces between A and B, take half of that number, and then use this smaller number to count off from either A or B, towards the center. None of the many students I’ve done this with have ever grabbed their ruler and measured the distance (in whichever units) and calculated half the distance. It appears that the blue delineations are quite important to how the students think of this problem. (For all you classical geometry teachers out there, it probably won’t surprise you that none of them used compass-and-[unmarked]-ruler approaches to find the center.)
By posing the question as a number line problem, the purely geometric nature of the question is further de-emphasized: “What number is exactly half way between 0 and 23?”
Many of the seventh graders I’ve done this with will take the same approach as the “middle between A and B” problem above. But more of them will say things like the following: “the distance between 0 and 23 is 23, so the middle is going to be half of 23, which is eleven and a half.” And then they point to a place between 11 and 12 on the number line.
My next move is to ask them for the middle between 8 and 13. Almost all of them use the two-handed approach for this, even those that dealt with the middle between 0 and 23 by getting a number for the distance and cutting it in half. Yet when I ask for the middle between 3 and 23, far fewer use the two-handed approach. Apparently there is some fairly sophisticated judging going on as to what approach will turn out to be easiest. They judge 8 and 13 to be close enough so that the two-handed approach, apart from being simple and certain, will also be quick. With 3 and 23 they seem to anticipate a longer process.
Those students that choose a numeric approach will figure out the distance by subtracting the two numbers (largest number minus the smallest number) and then calculate half of this distance. This gives them the distance from the end to the center, and they will add this distance to the smallest number. Sometimes they add it to the largest number, and catch themselves since the result doesn’t make sense to them. If they arrive at their center number, and I ask them if they are sure – or if I ask them if they have a way to check their result – they will sometimes subtract the distance to the center from the largest number, showing that they arrive at the same center number this way.
I had cycled through well over a dozen seventh graders who approached the problem numerically before I stumled onto one who computed the average: 3 + 23 is 26; half of 26 is 13; 13 is the middle between 3 and 23. My first thought on seeing Janey do it that way was – and I apologize deeply – “finally a smart student!” Yet I did have the wherewithal to ask Janey “how does that get you the middle?” Upon which she responded that her sixth grade teacher had told her to do it that way. “How does it work to get you the middle?” Janey responded that it just does.
The notion of average as getting you the middle, the center, of a set of numbers is beloved by teachers, but doesn’t appear to make much sense among students. And yet they are perfectly clear what it means to find the middle – they have their own approaches, and those approaches work.
After Janey, I started to ask other seventh graders more directly – after they had found the middle in their own way – whether they could have found it also by taking the average. I asked Jason, who told me, patiently, “no, no – to get the average you have to add the numbers.” Jason’s view made sense to me. He’s got a model for the difference between 23 and 3 – that’s the distance between the 3 on the number line and the 23 on the number line. Conversely, the sum of 3 and 23 has no obvious meaning in the problem, nor in the picture of the number line.
I’ve asked students like Jason to actually compute the average and compare that with the number they had found for the middle themselves. They are surprised that the numbers match, but it has generated no insights. I’ve also asked some students who had more than usual skill in manipulating formulas to show that where represents the distance from end to center and represents the average. Jeremy was one of the few students who didn’t get totally bogged down in this, but still found absolutely no insight from doing this.
With Jason, I tried the following. Once he had figured out the middle between 3 and 23, I then asked him to find the middle between 2 and 24. He saw very quickly that the middle would come out the same as the one he had already found for 3 and 23. I then asked for the middle between 1 and 25. Yup, he saw it would be the same. When we came to 0 and 26, he had a glimpse of something. These numbers, 0 and 26, had the same middle as 3 and 23, and he now had some way to relate to the number 26 as meaningful to the problem. Yes, 26 was the sum of 3 and 23 – but now the 26 had a role to play that it didn’t play before. After this exercise, Jason was still far from a full understanding of why the average gives you the middle of two numbers on the number line.
For myself, I now think of finding the middle like the following picture:
The number line is shown, with 0, A and B marked. The “two-handed approach” of the students is shown above the line. As you go higher and higher, the fingertips get closer and closer together, till they meet in the middle. This is a nice static representation of what the students did with their two hands. The invariant relationship is that at each moment in time, the middle they’re looking for is the exact middle between where their hands are at the moment. The alternative problems I posed to Jason are shown below the line. You could think of this as the hands moving further and further apart. Though on first glance this doesn’t help, it does maintain the same invariant relationship. When our left hand has moved “A” to the left, arriving at 0, our right hand must have moved “A” to the right, arriving at A+B. The middle between A and B is the same as the middle between 0 and A+B. Hence, the middle is at half of A+B, which is the number that we normally think of as the average of A and B. Now, at least, I have a model for why I would ever bother to add the two numbers to find the middle! I haven’t tried this out with Jason yet, though.
The two-handed approach isn’t one I would ever want to discourage the students from using. It generalizes very nicely to locating the median of a set of numbers. I expect to get back to medians, and also to weighted averages, in a subsequent post.