# The Craft of Word Problems

## Using Algebra Word Problems to Explore Problem Space

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## The Concept of “Problem Space”

One of the deadly sins of teaching math to which we all feel tempted at times by desperation, exhaustion or fear of approaching standardized tests is trying to give students disparate tricks or shortcuts to get answers to problems rather than engaging them in explorations and self-generated thought processes which in the long run would build better numeric and algebraic understanding. Often my students can call out an acronym (“FOIL”), phrase (“Please Excuse My Dear Aunt Sally”) or other “handy” mnemonic device that they recall from a prior math class (perhaps even from my class), but they have no idea what to do with it or no understanding of the core math ideas, which underlie it.

The “Craft of Word Problems” seminar has given me an opportunity to engage with math colleagues from several teaching levels to tease out the ways in which the language and structure we use enhances or inhibits the growth of mathematical reasoning and understanding. I teach math full time to students in grades 9-12 with a very broad range of academic ability and intellectual curiosity. The seminar has enabling me to select and design word problems more tailored to the needs of my students. In my teaching, one of the greatest challenges is getting students to read word problems for broad understanding without getting lost in details and abandoning hope. And it is here that the idea of exploring the problem space is most helpful. By crafting sets of related word problems which build around a central math idea, students can be engaged in a thought process of discovering, comparing and contrasting, refining understanding, seeing questions from several angles of view . . .NOT just crunching numbers through an abstract formula to get the answer. “The hope is that exploring the problem space can give students a better rounded view of the issues involved in solving any one of the problems. Each problem by itself may be a challenge, but solving all the problems may make the solution of each one easier and more transparent, and may create a more lasting impression than any random individual problem.”6 Word problems, well composed, ought to draw students into more subtle examination of the features of the “space” or mathematical habitat or territory where the problems reside.

Let me illustrate this point with a personal commentary:

The meaning of mathematics arises within authentic learning experiences and real life problem solving at all ages. When picking up my 4 year old grandson, Max, at his preschool one day last fall, he explained to me that, “There were 3 girls and 2 boys in our (pre) school program, but one girl left before you came to pick me up, so there are now only 2 girls and 2 boys, this many in all” (holding up 4 fingers). This is one of his real life settings of great value--he pays close attention to “who”, “what” and “how many.” As we pulled out of the parking lot to go home, he said to me, “Now there are only 2 girls and 1 boy, this many” (3 fingers). We tried to guess who would leave next and what that would do to the group remaining. (Our ensuing conversation illustrates what I mean by “exploring problem space”: a collection of related problems created around a theme to build student understanding of a mathematical idea.) I posed more questions to Max as we drove home: “If one more student leaves, how many students will be at school? How many students would have to leave to make the number of boys the same as the number of girls? How many students in all would then still be at school? Is there another possible answer that is true? How many students would be left then?” -- In this simple case, we were exploring grouping, subgroups, subtraction, the idea of equal numbers and whether “none” (zero) is a number in his 4-year-old mind. Max used his fingers as a simple calculator and “graphic organizer”. He must have had a good nap at school that day, because he stayed with the line of questioning most of the way home. Most days he falls asleep almost immediately in his car seat!

There seems to be a lot of math in social relationships at the pre school level and it is not much different in my high school, though the situations become a bit more complex! The words, numbers, and concepts used to address their lives are important for my students to understand if they are to achieve a sense of confidence and empowerment.

One of the most encouraging outlooks on teaching mathematics that we read and discussed frequently in our seminar is that of Thomas Carpenter, Megan Franke and Linda Levi in their book
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Thinking Mathematically
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(Heinemann Press, 2003): “…children throughout the elementary grades are capable of learning powerful unifying ideas of mathematics that are the foundation of both arithmetic and algebra.”7

All teachers of mathematics might benefit if the following statement were made into a poster and placed on classroom walls as a daily reminder of how active, interdisciplinary and authentic an experience the study of mathematics should be.

- Learning mathematics involves learning ways of thinking. It involves learning powerful mathematical ideas rather than a collection of disconnected procedures for carrying out calculations. But it also entails learning how to generate those ideas, how to express them using words and symbols, and how to justify to oneself and to others that those ideas are true.8

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Thinking Mathematically
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is a book that should be read by teachers of mathematics at every grade level. It does not provide a magic solution to the difficulties students and teachers experience in learning and teaching math, but it does set a framework for developing “types of problems and forms of questioning that [the authors] have found useful for eliciting children’s thinking and fostering growth in mathematical understanding.” 9