What Does Problem Solving Involve?
The document Principles and Standards of School Mathematics (NCTM, 2000) defines problem solving as “engaging in a task for which the solution method is not known in advance. In order to find a solution, students must draw on their knowledge, and through this process, they will often develop new mathematical understanding” (p. 52). Further, “solving problems is not only a goal of learning mathematics but also a major means of doing so. Students should have frequent opportunities to formulate, grapple with, and solve complex problems that require a significant amount of effort and should then be encouraged to reflect on their thinking” (p. 52).
Clearly, problem solving is a process of thinking mathematically. The term has also been defined as “strategic competence,” which describes the “ability to formulate, represent, and solve mathematical problems” (Kilpatrick, Swafford, & Findell, 2001, p. 5).
Problem solving, as a goal, process, and skill, is contrasted with completing computational exercises with which students are already familiar. To solve problems, one must combine and recombine known algorithmic procedures in a new way. Students often think they “can’t do” problem solving because they are unsure of ways to apply their own reasoning to questions. Yet, “good problems give students the chance to solidify and extend what they know and, when well chosen, can stimulate mathematics learning.” (NCTM, 2000, p. 52). Because the purpose of mathematics is to solve problems, students and their teachers should not view computation as an end in itself, but as a means to problem solving.
Consider the following question:
How many cookies would Craig have if he had three cookies and got five more?
This situation is really an addition computation exercise, if addition has already been introduced. The words do little to stimulate creative thinking or encourage reasoning. Students most likely add three and five because they see the numbers, know the previous questions were all addition types, and think little about the action really involved. If many number stories follow that example, each requiring the combination of two addends to rename the sum, there is little novel, adaptive thinking going on in the classroom.
However, students could be asked:
Craig is supposed to buy cookies for his class party. He has $5.00 to spend for himself and 25 classmates. The students want three kinds of cookies: 9 chocolate chip, 8 oatmeal, and 9 sugar. Cookies are sold in this way:
|Chocolate chip:||$.25 each|
Craig can also buy the cookies in packages this way:
|Chocolate chip:||1-pound bag for $1.75|
|Oatmeal:||1-pound bag for $1.50|
|Sugar:||1-pound bag for $1.00|
Each bag contains 12 cookies. What is the most economical way for Craig to buy the cookies? How much did he spend? Explain your reasoning.
Solving this problem requires reasoning, number sense, computation, and combining that information with knowledge of currency to find the result. If that process is unique in the problem set, then students are not merely doing algorithms that have words in them, but are, in fact, solving problems.
Problem solving, rather than the memorization of facts, should be considered the “basics” of mathematics. Understanding how to find results in a wide variety of settings, as well as the purpose of their work, helps to prevent mathematics from becoming a ritualized chore for students. Problems must be valued, not only as a purpose of learning mathematics, but also as the primary means of doing so.
© ______ 2009, Allyn & Bacon, an imprint of Pearson Education Inc. Used by permission. All rights reserved. The reproduction, duplication, or distribution of this material by any means including but not limited to email and blogs is strictly prohibited without the explicit permission of the publisher.
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