# How to scaffold algebraic thinking in teaching integers

By | April 14, 2011

One way to make algebra make sense to students is to show where those mathematics objects (e.g. algebraic expressions/equations/formula) come from. It will even makes sense more to students if they themselves can generate those objects. As John Mason puts it,

The mechanics of algebra (algebraic manipulation) are concerned with studying the effects of combining, undoing and otherwise relating expressions. These make little or no sense unless learners have themselves gained facility in generating expressions so that they know how they arise.  – from Actions and Objects by John Mason.

Generating expressions helps develop algebraic thinking. There are many ways of of embedding this in your teaching. For example with equations, you can ask the students to find as many equations given a solution. Read the post on how to teach the properties of equality on how this can be done. My post about teaching algebraic expressions also shows an example of a task that generates several equivalent algebraic expressions from the same problem situation. Generating formulas by deriving it from other expressions can also be a good activity.  Examples of these is deriving the formula of the area of the triangle from parallelogram/ rectangle and then from these deriving the area of trapezoids. These type of activities help develop students algebraic thinking skills.

Now, how can we do this generating expressions in earlier grades? Let me describe a lesson I taught to a class of year 6 students. This lesson is a continuation of the lesson on teaching integers via the number line with a twist. In that lesson, instead of asking student to arrange numbers, I asked them to arrange number expressions. From there we were able to extend the numbers they know (whole numbers) to now include the negative numbers. The main aim of that lesson is to extend the students’ concept image of negative number from a number that can be used to represent situation (see post on a problem solving approach for introducing integers) to a number that results when you take away a bigger number from a smaller number.

In this lesson with negative numbers, zero, and positive numbers on the number line, I can now proceed to defining integers or perhaps compare integers. But what will the students learn from that except that they are called integers? If I ask them to compare the numbers what good is that at this point? Where will they use that knowledge? So the task that I gave  them in this lesson was to make as many number expressions whose answer corresponds to the numbers in the number line. Note that this task is an open-ended problem solving task. Below is a sample student solution.

When I asked the class to share their answers I was surprised that they did not restrict themselves to addition and subtraction operation.

With the data shown on the board (pardon my handwriting) I asked them to make some observations and generalizations. They gave the following:

1. It is easy to make number expressions when the answer should be positive.

2. You always get a negative if you subtract a bigger number from a smaller number.

3. You always get a zero if you subtract equal numbers. (If you think this knowledge easily transfer to negatives, you’re wrong. I did try my luck when I asked them “is it also true to -7 – (-7)?”. I got blank stares. Clearly the expression it is still beyond them.

If these are not powerful mathematics to you especially #2 and well, #3, I don’t know what they are. Anyway, the point of my story here is that it is good practice to ask students to generate expressions. It is like asking them to think of a problem given the solution, a highly recommended  mathematics teaching practice.