What Does Problem Solving Involve? (page 2)
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.
Types of Problems
In general, problems can be classified as either real-world application or process problems. The former refers to the type of problems that require adaptive reasoning in authentic, everyday situations (Silver, Kilpatrick, & Schlesinger, 1990). An example is the following:
Nate and Matt are friends at school. Nate likes to play at Matt’s house in the afternoon. Matt lives nine miles from the school. How far does Nate travel back to his house when he goes from school to Matt’s house and then back to his house, if he lives five miles on the other side of the school?
In contrast, a process problem is considered nontraditional. It describes a situation in which the solution path is not as readily discernable from the story as the application problem presents. Examples include:
- If an 8-inch pizza serves two people, how many should two 12-inch pizzas serve?
Find two consecutive numbers whose product is an even number.
- Arrange the digits 1–6 in six slots on the perimeter of a triangle so that all three sides add up to the same sum. 3
Both types of problems require the solver to apply some type of heuristic. The word heuristic derives from the Greek word for discover and means a general strategy used to guide the problem solver in finding a solution (Charles & Lester, 1982). Typical heuristics include:
- drawing a picture
- using manipulatives
- working backward
- acting it out
- making a table or list
- using smaller numbers
- looking for a pattern
- guessing and checking
Unlike algorithms, which prescribe specific procedures that yield a result, the application of heuristics does not guarantee career solutions. One could draw a diagram or construct a table, for example, and not necessarily arrive at a logical or correct result. However in the absence of knowledge of exactly what to do, a heuristic helps students begin to solve the problem—the more heuristics with which students are familiar, the greater the likelihood of finding a solution.
© ______ 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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