When my son was ready to enter first grade many years ago, my wife and I visited a number of private schools to help us decide where to send him. In one of the classes we visited the students were being introduced to subtraction. The teacher had given each student some small objects they could use to help them count the totals, and she demonstrated how to use these counters to solve basic subtraction equations. The children then worked independently and came up with their papers one by one to have them checked. To the horror of the teacher, most of the calculations were incorrect. At one point she looked at us and sheepishly acknowledged, “Subtraction is killing us."
Why were the students in this class having such difficulty? The problem, I’m quite sure, was NOT with the student’s ability to grasp the concept of subtraction. The idea of removing objects from a group is something common and recognizable to six-year-olds. Moreover, all of these children were good at counting. If I had asked them to count out five gummy bears and to give me two of the gummy bears and then count how many they had left over, they all would have been able to do this task easily.
The difficulty they were experiencing in class that day lay not in the complexity of the concept of subtraction, but rather in how the teacher presented the mathematical symbolization for it. Specifically, she began by presenting an expression such as “5-2” and then demonstrated how to use the counters to find the result. But by starting with the mathematical expression, she began with something that the students were not familiar with, and so it was extremely difficult for them to make sense of the what she was showing them, even though I'm sure the teacher thought that by using the counters she was making that procedure quite explicit and clear.
The lesson would have worked much better with a very simple change in sequence. Instead of starting with the unfamiliar mathematical symbols, the lesson should should start with the familiar experience of counting various objects, such as books, or pencils, or the children themselves. After the students had counted an intitial set of objects, the teacher should remove some of the objects and then have the students tally the number of objects that remained. At each step she should show the class how each action (counting an initial set of objects, removing some of the objects, counting the objects that remained) was represented with mathematical symbols. Only after the students could easily go from performing these actions with objects to recording the action with the correct subtraction equation would the teacher then work on having the students do the work in the other direction, that is, present them with a subtraction equation and have them represent the equation with a set of counters. This last step, of course, is where the teacher we observed had started, and in doing so, she left her charges thoroughly confused.
It is important to point out that the problem with the lesson wasn’t simply that it was being executed by an inexperienced or a bad teacher per se. I believe the teacher was faithfully following the routine recommended in the curriculum the school asked her to use. Rather the problem in this lesson was a tendency all too common in math education to put the cart before the horse by rushing to present mathematical symbolization without adequately showing students what the symbols represent. I’ll have more to say about this point in the next blog post, where I will discuss how this problem bedevils not only math lessons but other areas of the curriculum as well, and how we can derive an extremely important teaching principle from it.
Epilogue: We ended up sending our son to public school, and he remained in public schools until he graduated from high school in 2011. We made the decision for several reasons. One important reason was that from what we saw, the teaching in the private schools was in general not noticeably superior to that in the public schools. Our son graduated from college with honors in 2015 and just started his second year in law school.