I’ve been wondering for a while what the fundamental difficulties are that kids encounter in mathematics, at the middle school level.
I used to think that the fundamental difficulty had to do with the introduction of variables, and with expressions and formulas now used to reason about arithmetic rather than doing arithmetic. In elementary school 4+5 = has a clear interpretation (similar to what a calculator makes of these marks), “take the number 4 and take the number 5 and add them now, and what matters is the number that results: the answer“. In middle school kids see x+5=y and yet there is no obvious adding to be done, let alone now, and it isn’t clear what an answer would look like. Though “+” and “=” look the same, they no longer trigger actions the same way – and teachers are often oblivious about this aspect, locating the obvious difficulty the kids have in the newness and strangeness of the “x” and “y”.
More recently, I’ve come to think that an even more fundamental difficulty, one that cuts across the other one, has to do with the understanding of multiplication as distinct from addition. I’m not talking about whether kids know their multiplication facts, and I’m not talking about whether kids can memorize different sets of rules for things additive as for things multiplicative.
The conclusion I’m coming to is that the kids who have the deepest trouble with middle school math are those without a clear and rich set of models for what multiplication is and how it is different from addition. In the absence of such models, the kids tend to rely increasingly on memorizing rules and how-to’s, and their grasp is tentative and fragile. They have given up trying to make sense of what they are doing, and instead rely on their quite well-honed skills of guessing what the teacher wants. To my surprise, this includes many of the kids who have decent grades, and who can get by quite well with symbolic manipulation of formulas that may hold very little meaning to them.
As an example, I ran into the following situation with an eighth-grader the other day:
Presented with the figure on the left, Joy knew exactly what to do. She filled in the boxes as shown on the right and then wrote down separately: . She offered that this was the expanded form. She knew that she was also supposed to produce the factored form, but here is where some cracks started to show. She tried but this didn’t look right to her, and then she tried .
For the purposes of this post, I’m not going to follow her particular thinking or how it could be used as a starting point for her learning something new about mathematics. Right now, I would just like to venture that on a normal day you would consider that somebody who filled out the picture on the right was displaying some decent understanding about algebra and multiplication. Instead, it turned out to be something she had learned to do as a performance, as something to please the teacher.
Joy had no trouble expanding and but couldn’t see any of those as relevant to expanding or see it as in any way related to what was happening in the picture. Joy is one of those students where your first diagnosis would be that her problem is in understanding variables.
The seventh-grade counterparts to Joy are working on a math unit where their skill and understanding in adding, subtracting, multiplication and division of positive and negative numbers is assumed. This unit comes right after a unit in which multiplication and division of positive and negative numbers was explained and practiced with. So the kids do indeed have sufficient facility with multiplying and dividing positive and negative numbers. However, the ability to add and subtract positive and negative numbers seems to be gone for a sizable fraction of them. That unit, the one that explained and practiced adding and subtracting positive and negative numbers, is already gone with nary a trace. These kids passed the test, they could do the performance, they could please the teacher long enough. But the memory cells they used for remembering how to add positives and negatives seem to have been re-used. They seem to have been re-used to remember multiplying and dividing positive and negative numbers. So now, when they encounter 6 + (-5) they mumble something about a positive times a negative and come up with -1. These kids have given up expecting math to make sense, and they have settled for trying to remember the rules long enough to pass the test.
And in our teaching, we still insist and move on to the next unit, even if we kind of know full well that the understanding isn’t there in the previous unit. We do see the same thing year after year.
I think diagnosing what is going on is important, since the set of solutions I would propose if I thought the issue was with variables is different from the future I see if I think the issue has to do with kids being unclear about what multiplication is. In any case, I encourage people in similar environments to share their observations on exactly what it is that kids are having trouble with – and conversely, exactly what it is that kids understand richly and deeply. As you look back over the many posts in this blog, you will see that I’ve brought examples of both. I believe a rich body of examples is an important resource, more so than a set of quick-quick solutions and tricks.