• PART I

PLACE VALUE, ARITHMETIC MODELS & ARITHMETIC ALGORITHMS

MENTAL MATH

• PART III

PRIMES & DIVISIBILITY

• PART IV

FRACTION ARITHMETIC

• PART V

WORD PROBLEMS & MODEL DRAWING

Part 5: Word Problems/Model Drawing Introduction

Ask math teachers at any level what gives their students the most difficulty, and the answer you will hear more than any other is ‘word problems’. In fact, teachers themselves are not always comfortable with such problems, as illustrated in the following anecdote from Rafe Esquith’s book ‘There Are No Shortcuts.’

Mr. Esquith was asked to select students to represent his school on a math team (Esquith teaches at an elementary school in Los Angeles). A well-respected teacher at the school, who Esquith refers to as Mrs. Egghead, promptly recommended to him six students who she believed would represent the school brilliantly. As a screening process Esquith gave the group a set of ten sample problems. The students were to solve the problems collaboratively and turn in a single set of answers. They answered every last problem incorrectly. When Mr. Esquith informed Mrs. Egghead that her students would not be representing the school she was furious and demanded to see the sample problems. Upon reading the problems she immediately understood why her students fared poorly. She looked at the problems and said, “Well, no wonder they missed them. You gave them word problems. I haven’t taught the children how to solve such problems this year; we’ve only done math.”

This teacher is clearly missing the entire point of elementary school mathematics! The point of teaching arithmetic is not simply to foster computational skills, as Mrs. Egghead clearly thinks. As Tom Parker and Scott Baldridge state in their textbook ‘Elementary Mathematics for Teachers’ when introducing the section on word problems, “In the real world, mathematics problems seldom come as straightforward arithmetic problems; people do not run up to you in the street gasping ‘Quick! What is ?’ Instead, mathematics occurs in forms much more like word problems. It is by doing word problems that students realize the importance and applicability of mathematics.” Regardless of whether you are teaching arithmetic, algebra, or even calculus, true understanding of the material is only achieved through the ability to use computational skills to solve applied problems.

The clips in this section focus on the model drawing technique of solving word problems, which is central to the Primary Math curriculum used in Singapore. The first two clips deal with early grade problems involving addition and subtraction. The next two clips focus on problems involving multiplication and division (with whole numbers and fractions); here I introduce the concept of a ‘units sentence’ and outline the solution formatting process we will use (called a ‘Teacher’s Solution’ by Parker and Baldridge). The final two clips focus on problems involving fractions.

In my view there are two main benefits to using model drawing. First of all, it is a very powerful technique for solving word problems involving not just whole numbers, but also fractions, ratio, and percentage. In keeping with our overall theme (never teach a procedure without understanding!) the technique is not taught as simply a procedure. Instead, the process depends upon and uses previously learned models of arithmetic. Secondly, if used consistently the process can be greatly beneficial to students when they are making the transition to algebra. If students can master the concept of a units sentence, then the difficult step of translating a word problem to an algebraic equation will hopefully be far easier, as students will hopefully notice how closely a correct units sentence resembles a correct algebraic equation.

Notes:

1. Clips 5 ii through 5 v are short clips that each focus on a specific type of problem, while clip 5 vi is a much longer, classroom discussion of the entire model drawing process.