by Glenda Lappan, NCTM President 1998-2000
NCTM News Bulletin, January 2000
In the English language, context helps us decide which of the possible meanings of common words is meant. As representations of ideas, words stated free of some meaningful context often do not communicate very well. The definition of a word in the dictionary usually includes several possible meanings. The word used in context helps the listener or reader decide which among the various meanings the speaker or writer intends. In the language of mathematics, we also face the same dilemmas. Many mathematical words have different shades of meaning. In learning to understand how both to communicate in, and to decipher the language of, mathematics, students have to determine meaning from contextual use.
Take, for instance, the concept of variable--something students must understand as they mature mathematically. Faced with a mathematics problem, students have to find ways to use mathematics to represent the situation, manipulate the representations to find solutions, interpret the solution in the original context, and look for ways to generalize the solution to a whole class of problems. Variables play a key role in the process of mathematizing a situation. But what meaning of variable for a given situation is appropriate? Is it a placeholder for an unknown? Or is it a domain of possible values for one of the phenomena? Or is it used in yet another way? Consider this problem recently given to a sixth-grade class I observed.
A local contractor, Ms. Phillips, builds swimming pools. One model is a square pool that can be any whole number of meters on an edge. She wants to build a border around the outside edge of pools of this type. She found some very nice square tiles that are one meter on each edge. How many of the tiles will she need to make a one-meter-wide border around one of the square pools?
One student, David, drew the problem as shown at the right. In his drawing, he labeled the edge of the pool with an S. What use of variable is this S representing? Here David is using the S to represent the entire domain of whole-number lengths of edges that the builder offers in this pool type. After some thinking, David writes N = 4S + 4 as his solution. He is using letters to represent quantities in the problem, and these letters help show his representation of the problem. He is using two variables, but in this example, the N is determined by the choice of a value for S. Here a variable is a placeholder for the number we are seeking. In this simple problem, variable is used in two different ways--to represent the domain of possible pool sizes and to stand for the unknown number of tiles needed for a particular size of pool.
Interestingly for us teachers, watching how students use variables can give us insight into how those students are thinking mathematically. For instance, how is David thinking about the situation to arrive at N = 4S + 4 as a solution? What about his classmates, who came up with solutions such as N = 4(S + 1) and N = 2S + 2(S + 2)?
In addition to using variables to explain their own reasoning, students will need to be able to interpret variables in someone else's representation of a problem. Here's a problem from a research project of Jack Smith and Betty Phillips that starts in a different place from the one above by giving a string of symbols for the situation.
Suppose you turn on a pump and let it run to empty the water out of a pool. The amount of water in the pool at any time is given by the following equation: W = – 350(T – 4). Here W stands for water measured in gallons, and T stands for time measured in hours, T 4. Students are asked to answer each of the following questions and to explain how they used the equation to do so.
- How many gallons of water are being pumped out each hour?
- How much water was in the pool when the pumping started?
- How long will it take for the pump to empty the pool completely?
- What range of values makes sense for T?
- Write an equation that is equivalent to W = – 350(T – 4). What does the second equation tell you about the situation?
- Describe what the graph of the relationship between W and T looks like.
The two problems discussed above look at two different ways to use variables. In doing so, they illustrate the complexity of the language of mathematics and how context matters just as much in mathematics language as in any other.
The challenge for our students is to learn both to use that language to show their ideas, as in the first problem, and to read that language to understand the meaning of someone else's mathematical representations, as in the second problem. It is no wonder that we need carefully developed lessons to help our students gain facility with the powerful language of mathematics. Try these problems with your students and pay close attention to what they understand about the language of mathematics.