Dear Matt,
My eighth grade students seem to “glaze over” whenever we generalize a process by creating a formula, such as the laws of exponents. One of my colleagues said something about students having trouble with the concept of a variable, but didn’t say what to do about it. Can you give me more details about how to help students understand that a variable represents a number?
Response:
Dear Eighth Grade Teacher:
First, let’s take a look at the idea of variable. This may help us understand why students are often confused about this idea. Until 1650 AD, variables were only used as a placeholder for a specific unknown quantity. Since then, they have been used in many different ways. This table lists a few of these.
Use of variable |
Example |
In a formula |
A = lw |
To state a property |
a + b = b + a;
b = an |
As a placeholder or an equation to solve |
15 = x – 5 |
To indicate a relationship, in this case, linear |
y = mx + b |
Note that in several of these equations, there are several different variables, and they have different meanings. For example, in the last example, the m and b are identified for particular lines, while the y and x vary for every line. There are other examples where the variable represents an angle (sine x) or a statement (If p then q). So it is not surprising that students are confused about variables, since we use them widely and with multiple meanings, often without distinguishing very carefully among these meanings.
So now that we understand that formulas, such as the laws of exponents are only one of the many ways we use variables, maybe we can find some ways of planning instruction so that students make sense of these generalizations and are able to use and apply them. The way we may have been instructed in these laws went through a sequence that goes something like this: give an example of a number – usually 2 or 3 – multiplied by itself several times, state the symbolic way to represent this, show what happens when you multiply say four 2s by three 2s, giving seven 2s, generalize this to the first law of exponents, then practice some problems and assign homework.
Let’s look as an alternative at the recent work of Robert Moses. Moses, who was well known as a leader during the Civil Rights movement of the 1960s, and is now devoting his time and energy to helping students learn algebra. He calls algebra the new Civil Right, since it is the gatekeeper to all high wage employment in today’s economy. Moses believes that students learn algebra with understanding through classroom activities that lead them through the following steps, which I summarize in a table with some examples:
Step in learning algebraic concepts |
Example of step |
1) A physical event, an activity in a context that makes sense to students |
Measure a box or container |
2) Represent this situation with a picture or model |
Use cubes to model filling the box or container |
3) Talk about the situation with informal “people talk” |
Discuss methods of counting the cubes and finding how many it takes to fill the container |
4) Moving to more structured and mathematical language |
Discussion leads to terms like dimension, and area, and stating formula in words |
5) Representing the ideas using symbols |
Develop a generalization for finding the volume for the given figure, finding formula in symbols |
When you plan a rich activity incorporating these ideas, students can progress through these steps, perhaps over several class periods. Though it seems to be time-consuming, both for planning and instruction, it allows students to make the idea their own and remember it. The students’ statement of the generalization – in your case the first law of exponents – may not be in the same format as a textbook, it will be remembered if has meaning to students. Some students may not be ready to use symbols; if they can generalize using language, they will still be able to remember, use and apply the formula when called for.
Finally, remember that generalization is one of the important reasoning processes of mathematics, one with which your students should be very familiar, which will come from multiple experiences. So:
- Give experience before formulas
- Let students develop the formula rather than the teacher
- Generalize away and take that glaze out of their eyes!
Good luck!
Matt |
Matt Mentor, a
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