Developing Meanings for the Operations:Part 1

What I read
Chapter 10 of Elementary and Middle School Mathematics

Part 1 is a synopsis of Chapter 10 of John A. Van de Walle’s book, how to help children develop an understanding for the meanings of addition & subtraction and how they relate. Part 2 will discuss multiplication & division.

There are two main methods for teaching addition and subtraction. The first method is to use contextual problems. The second method is to use multiple models such as counters, drawings and number lines. Using these two methods will help children construct a rich understanding of the addition and subtraction operations.

The problems presented should be structured like Cognitively Guided Instruction (CGI) if you are familiar with that. In short, instead of “adding” and “subtracting,” you “join,” “separate,” use “part-part-whole,” and “compare.” Any of the quantities you work with (initial/starting amount, change amount, resulting amount) can be the unknown. For example, here is a part-part-whole problem with the initial amount unknown:

Jayden had some jelly beans. She ate 13 of them on Monday morning. One Monday afternoon she ate the remaining 12. How many jelly beans did Jayden begin with?

Contextual problems must be somehow connected to the children’s lives. You could write your own problems easily by reading some samples presented in the book. Problems could be about a recent experience the class had together such as a field trip, a discussion in another content area, or your read-aloud (or another book you’ve all read).

A typical development for students to make in early math is as follows:

1. Uses counters and counts everything seen
2. Counts on from the first set given
3. Counts on from the largest set
4. Uses facts retrieved from memory and relies less on counters

Part of what will help children progress to the next stage of mathematical development is to teach with models, showing what to do every step of the way, being sure to use a variety of kinds [Designs, picture "stories," unifix cubes, 2-colored counters (or other object), or part-part-whole mat] in order to give students different ways to see the problem and to model how to use each kind of model.

When teaching these first two operations, teach addition and subtraction simultaneously. If you ask what 7 – 3 is, you must also ask what 5+3 is  so students will learn the pattern. Subtraction must be taught as “Think-Addition,” instead of “Count What’s Left.” So instead of subtracting 6 counters from a student’s pile of 9 counters and asking them for the answer by counting what remains or asking what is 9 – 6 , help your students ask the question 6 and what make nine? As kids practice with each other in pairs, they say and write the equations that match with their problems. This is actually the beginnings of algebra.

Lastly, something we often don’t think of because it seems so obvious to us as adults. When teaching addition and subtraction, help children understand the “Order Property” (usually known as the commutative property) and the “Zero Property.” Using understanding of the order property, you learn that when adding two numbers together, the order of the addends don’t affect the result. Using the zero property, you learn that any number plus or minus zero is that number. Van de Walle here says that some students really do get confused because they connect the idea of addition with a number getting larger and subtraction with a number getting smaller, so to some, 12 – 0 should be less than 12, so some students may write an answer of 11.

Teacher Resource
http://www.aaamath.com/
This is a nice little site that offers tutorials and definitions of 25 different math topics for K-8 grades. You can sort by grade or subject. After each tutorial there is an interactive practice application that produces problems and automatically checks answers. I advise you to check out the practice prior to assigning it to a student because I tried the fraction subtraction practice (that’s a mouthful!) and it wouldn’t accept reduced answers. For example, it wanted the answer 2/2 and not “1.” This is reasonable, just something the student should know.

NEXT: PART 2: MULTIPLICATION & DIVISION