Developing Meanings for the Operations:Part2

What I read
Chapter 10 of Elementary and Middle School Mathematics

Part 2 in developing meanings for the operations focuses on multiplication and division, what strategies children use, what types of modeling a teacher should use and how the teacher’s understanding of approaches can enrich students’ understanding.

Multiplication and division can both be learned as early as 1st grade. Sound ridiculous? Not if you understand the relationships between all the properties. For example, when teaching multiplication to children, one of the most used approaches is to show how multiplication is merely repeated addition. For example: Eduardo grew 8 tomato plants. Each tomato plant produced exactly 5 tomatoes each. How many tomatoes will Eduardo harvest? The problem boils down to 8 x 5. There are 8 equal groups of 5 sets. The first factor (number) counts how many parts or sets. The second factor tells the size of each set or part and is also known as the multiplicand. In order to solve this problem using repeated addition, you create 8 groups of objects with 5 objects in each group. The resulting equation: 5+5+5+5+5+5+5+5 = 40 should be written down by the student as well as 8 x 5 = 40 next to it. Students will begin to learn the relationship. Equal group problems may also be considered as rate problems.

The major conceptual hurdle to developing meaning for multiplication and division is understanding groups of things as single entities and also that each group has individual objects. Once this mental obstacle is conquered, children of just about any age can begin working with these operations.

Van De Walle talks about four classes of “multiplicative structure,” and that there are two used most often in teaching multiplication. The first is equal groups, which I have written about already. The second class is multiplicative comparison. There are some fantastic example problems of both types that Van De Walle includes in his book. Comparison problems usually compare two people who have two different amounts of something. For example, Trudy built 5 models during art class. Her friend Clarissa built 3 times as many models as Trudy. How many models did Clarissa build?

Consider the two multiplication examples I have given here. Figuring out how to solve each one requires a slightly different way of thinking and certainly a different strategy. Compare this type of thinking to solving a page of textbook multiplication problems. You will be developing meanings for multiplication, not just how to get the right answer. Understanding these meanings are a foundation for algebra as well as other types of advanced mathematics. Van De Walle suggests each math lesson should include 2-3 problems that allow students to use any technique and any math tool they like. For each problem they solve, they must explain, preferably in writing and then later in a discussion, how they solved each problem. Van De Walle also includes a great activity for finding factors in this book, assigning numbers to students and having them use counters, arrays, etc. and then write two equations, an addition and a multiplication for each factor.

Remember how multiplication and addition have a special relationship via repeated addition? A similar relationship occurs between division and subtraction. Division is repeated subtraction. For example, the problem 24/6 = 24-6-6-6-6. A hundreds chart is a useful tool for solving these problems and helping your students skip count backwards as well as forwards will help them use this strategy more efficiently. Other helpful models for practice are counters, number lines and arrays.

When teaching division, teach it at the same time as multiplication or very soon after so students may practice both types of problems and begin to see their relationships. Avoid using the term “goes into,” which carries very little meaning, and instead use “divided by.” Additionally, don’t be afraid of large numbers. Since the strategies and understanding of products and quotients are the same for small vs. large quantities, solve  2 x 4, 13 x 11, and 102 x 15, 6/3, 24/8, 200/5 and even much larger numbers should be introduced at once. To help students see the relationship between multiplication and division, think about how you will word each problem. For example, asking “6 times what is 24?” is more effective than asking “24 divided by 6 is what?”

A note on remainders. There are two methods for remainders in division problems. Method 1 is to leave the remainder as a quantity left over (i.e. 16 R1). Method 2 is to partition the remainder into fractions (i.e. 16 1/16). Van De Walle prefers method 2. In real contexts, you sometimes need a whole number answer so you must round up or down. In a problem where a certain number of cars are needed to transport students for example, using a fraction or remainder answer is useless. There are three ways to decide whether to round the number up of down. If you discard the remaining amount (nobody gets the leftover jellybean) you round the number down. A forced amount (you need an entire car for the one extra student) you must round up. A rounded number will go up or down depending on whether the remaining amount is greater or less than five (We will need 9.6, or about 10 hours to build a compost container).

Finally, when you write or re-write problems for your classroom, make up problems with different contexts. Instead of just using objects such as brownies or pencils. Be sure to use problems that use length, time and volume.

Teacher Resource
Book: Knowing and Teaching Elementary Mathematics: Teacher’s Understanding of Fundamental Mathematics in China and the United States
This book by Liping Ma was highly recommended to me. I have not read it but thought I would pass it on to you. Let me know by posting a comment, if you have read this book and what you think of it. Ma defends the belief that children need to know both arithmetic as well as theory for solid mathematical thinking. Read the editorial review on Amazon by clicking on the book link above or click here.