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Friday, July 10, 2009

The Dreaded Word Problem

All across this great country of ours, math students grapple to the death with their arch nemeses- the dreaded word problems.

Regardless of how they are dressed up or repackaged, most math textbooks are still pretty much the same. A particular lesson or skill set is explained, several examples are given, and then two to three pages of exercises follow. At the very end of these practice problems, buried in the back so they are easily ignored are the much maligned word problems. While the future mathematicians relish with excitement the chance to challenge themselves with these rhetorical abstractions, the average students find ways to conveniently skip over them, like peas being pushed around an otherwise empty plate.

Despite our efforts to reassure the students, to give them problem solving techniques and confidence boosters, we must admit that we are sending mixed signals. Anyone who truly understands mathematics realizes that word problems are not only a key part of math, they are the only part of math. Mathematics is a way of thinking about our world. Seldom does one find themselves confronted by a floating quadratic function demanding to be solved at gunpoint. Instead, we encounter normal, everyday questions or problems that can be illuminated using the tools of mathematics. This means translating the idea into language and translating that language into a mathematical construct. Thus, by hiding these problems at the end, we allow our students to skip the only problems that they really ought to attempt at all.

Rather than work umpteen practice problems, already laid out in clearly defined mathematical language exactly mirroring the guided examples, the students should skip directly to the world problems. Too often I hear my pupils tell me that they understand everything but the word problems. I politely respond that if they don't understand the word problems, they don't understand anything. Instead, they are confusing familiarity with understanding. They think because they can generate the expected answers with a series of repetitious algorithms, that they are preparing themselves for the exam and beyond. There curriculum is a mile wide and an inch deep.

What I would prefer is this: fewer practice problems investigated with greater depth. I would like the answers to come in narrative form, where the student explains to me and to themselves exactly why they made the decisions they made and what axioms of mathematics allow them to employ the techniques they chose. Take the following example:

An Internet service provider charges $9.95/month for the first 20 hours and
$0.50 for each additional hour. Write an expression representing the charges for
h hours of use in one month when h is more than 20 hours. What is the charge for
35 hours?

I don't want to see 9.95 + 0.5(h-20) = Cost (h) ; Cost (35) = 17.45. I want to see the following:

The Internet service provider is offering 20 hours a month of internet usage for
a flat rate of $9.95. This means that regardless of how many hours we use, our
bill will be at least $9.95. In other words, this value is a constant. If we go
over our allotted 20 hours, we will have to pay an overage fee of $0.50 for each
additional hour. Since the number of hours we use will change each month, we can
represent that value by the variable h. (We label it h out of convenience, since
hours starts with the letter h.) It is important to note that we only pay
overage fees on the hours we use beyond 20. An expression for those extra hours
would be h – 20. Therefore, our total bill will be $9.95 plus $0.50 for every
extra hour, or in algebraic terms,

Total Cost = 9.95 + 0.50(h – 20)

To evaluate the expression where h = 35 hours, we simply plug 35 in for h and solve.

Total Cost = 9.95 + 0.50(35 – 20)

9.95 + 0.50(15) = 9.95 + 7.50 = $17.45

That, ladies and gentlemen is how you solve a word problem, and until our students are able to clearly explain every step, they aren't mathematicians. They are walking, talking abacuses.

1 comment:

Zeno said...

I try to persuade my students that word problems are the most "real" part of our math classes, but they regard that notion with horror. I can make a pretty good case: "So when do you think your boss is ever going to come into your office and demand that you solve a quadratic equation?" They laugh at that, because it confirms their prejudice that math classes are just an arbitrary obstacle placed in the way of their graduation. Obviously no one actually uses that stuff. But then I dip into my trove of word problems that model actual situations that faced my brother (a farmer), a former student (a theater arts set designer), a previous employer (a state legislator), a newspaper (a science story whose numbers didn't pencil out and needed correction), and others in that vein.

Sure, some eyes glaze over, but others widen as they assimilate the notion that word problems actually happen. They're not just fairy tales cooked up for textbooks.

But it sure is an uphill battle.