Going Beyond the Right Answer
By Kathy Richardson

Right answers aren't the only thing to look for when examining our students' work in mathematics. Looking at how children solve computation problems can give us insight into their thinking and help us determine what math skills they already know and what they still need to learn.

For example, Suzie solves the problem 24 + 98 = 122 and explains how she arrived at her answer: "Four plus eight is 12, so you put the one at the top and the two at the bottom. Then two plus nine is 11, but the one at the top is a plus, so the answer is 12."

Though Suzie is able to get the right answer, there's no evidence that she truly understands the underlying structure of the numbers.

These colored cubes that represent tens and ones can help to heighten children's understanding of number relationships.

Limited learning. When children learn to add, subtract, multiply and divide as a set of procedures, but do not also understand what's happening to the numbers structurally, they may be able to get correct answers, but their learning is limited. They solve every two-digit addition problem in the same way whether the strategy is efficient for a particular problem or not. They make no connections between one type of problem and another, so what they know about addition is of no help when subtracting or multiplying.

However, if children learn computation as part of the study of number relationships and the composition and decomposition of numbers, they'll be competent in dealing with all kinds of computational problems – and they will have also learned more mathematics.

Use what you know. When children are asked to solve problems in whatever way they can, their work will reveal the level of their thinking and guide our instruction. Here's an addition problem – 29 + 12 – and the very different ways in which three children approach it.

Adam, whose work is shown below, needs to learn that numbers are composed of tens and ones. His task is to find a way to organize the people into tens so it's easy for him to keep track of them. With this and other similar experiences, Adam will begin to see for himself that numbers are composed of tens and ones.

Rosa, whose work is shown below, understands that numbers are composed of tens and ones and combines the tens a few at a time to find out how many there are altogether. She doesn't yet realize that it's possible to know, without counting, that four groups of 10 are 40.

Ricky, whose work is shown below, understands that adding two-digit numbers requires finding all the tens possible. He takes the 10 out of the 12 by decomposing it into 10 and two. He adds the tens in a most efficient way by adding 10 without breaking 29 into tens and ones.

One activity, many uses. When we look at what children know and not simply at the answers they get, we can identify the mathematics each child is ready to learn and provide experiences that meet a range of needs to help them learn it.

I call these experiences "expandable tasks." The children are given the same set of tasks, but the mathematics on which they focus within that task will depend on what they're ready to learn.

"Paper Shapes" is an example of the kind of task that can be used to meet a variety of needs. Each child estimates the number of tens and ones it would take to cover a particular shape and then uses connecting cubes to find out the actual number.

When using the "Paper Shapes" activity with a child like Ricky – who understands that numbers are composed of tens and ones and combines them with ease – we can challenge him by asking him to compare two shapes to determine the difference between them.

Deeper understanding. If we look only at whether our students' answers are correct, we'll miss many important things about what and how they're learning. We'll be sure children are learning the mathematics they need to know when we look closely at what their problem-solving processes tell us about their understanding. Once we address – and augment – that understanding with activities, students' mathematical knowledge will grow.

Kathy Richardson is Program Director of Math Perspectives Teacher Development Center in Bellingham, WA. She is also the author of many professional development videos and teacher resource books, including the Developing Number Concepts Series.

January, 2004, Vol.34, No.4