Problem solving-Elementary level
The focus of this entry is on use of problem solving in the elementary and middle school level classrooms, mainly for mathematics. Chris Snodgrass - EPSY 400 (Anderson)
Problem solving, in any academic area, involves being presented with a situation that requires a resolution. Being a problem-solver requires an ability to come up with a means to resolve the situation fully.
Problem: A question proposed for solution; a knotty point to be cleared up. Solve: To explain; to make clear; to unravel; to work out. (New Expanded Webster's Dictionary, 1988)
In Mathematics, problem solving generally involves being presented with a written out problem in which the learner has to interpret the problem, devise a method to solve it, follow mathematical procedures to achieve the result and then analyze the result to see if it is an acceptable solution to the problem presented. A version of these steps are addressed in the book "How to Solve It" written by G. Polya and in the 4-Step Plan described below.
Typical problem solving we see in elementary-level classrooms involve word problems that present the learner with simulated situations which they can correlate to actual real-life situations. The goal is for the learner to be able to apply the strategies used to resolve the simulated situation to real-life.
Instruction should cultivate the discovery of relationships, conceptual learning, and thinking skills as well as mastery of basic facts and procedures. ...the focus of instruction should be on meaningful learning and problem solving. (Baroody, 1989)
Description of the 4-Step Plan
Explore -Determine what information is given in the problem and what you need to find. (Glencoe/McGraw-Hill, 2001)
In the "explore" step, students should be encouraged to read the problem carefully and determine what information is needed to solve the problem. This also requires the student to decide if there is information included that is irrelevant to solving the problem.
Plan -After you understand the problem, select a strategy for solving it. (Glencoe/McGraw-Hill, 2001)
In the "plan" step, the students should devise a strategy or strategies to find a solution to the problem. This may require using several mathematical methods of computation or setting up an equation. Student should also be encouraged at this point to form an estimate of the solution. The estimate will help in determining whether the final answer is reasonable.
Solve -Solve the problem by carrying out your plan. (Glencoe/McGraw-Hill, 2001)
In the "solve" step, the students will perform the mathematical computations necessary to determine an answer. In many cases the answer may not be acceptable at the first attempt, so the students should realize that they may have to perform their computations more than once in order to achieve the desired result. In addition, the students may find that the methods of computation they chose will not work toward the solution, therefore alternate methods may have to be used.
Examine -Finally, examine your answer carefully. See if it fits the facts given in the problem. (Glencoe/McGraw-Hill, 2001)
In the "examine" step, the students need to analyze their solution to see if it is an acceptable answer to their presented problem. Students should look at the estimate they formed in the "plan" step to see if it is similar to their calculated outcome. Often students should re-read the original problem to be sure that they interpretted it correctly the first time. If it is determined that the solution is not acceptable, they should return to the "plan" step and re-solve the problem.
How I Teach It...
When teaching the "4-Step Plan" to my sixth and seventh grade students, I ask them if they are familiar with the "plan" first. In most cases, they have studied the "plan" in a previous textbook or class. Then I ask the students if they use the "plan" to solve word problems outside of the lesson that teaches it, encouraging them to be honest. In most classes the majority of the students say they do not use the "plan". I then break the plan down by asking the following questions...
1) Do you read the word problem?
2) Do you pick out the information you need to solve the problem?
If so, you just did the "explore" step.
3) Do you determine a method to solve the problem?
4) Do you form an estimate in your head? (most kids say no to this one)
If so, you just did the "plan" step.
5) Do you do the math to find your answer?
If so, you just did the "solve" step.
6) Do you make sure that your answer fits the problem?
If so, you just did the examine step.
Several of my students then decide that they do use the "4-Step Plan", at least partially. I feel that the "plan" is the basic thought process that is required to effectively solve problems. In the next section, I will connect this with Polya's 4 steps.
Polya's Four Steps "How To Solve It"
(Polya, 1957, p.xvi)
Polya's problem solving plan is not geared for interpretation by elementary students, but has the same basic steps as the "4-Step Plan" above. Rather than give Polya's version verbatim, I will comment on each step and include his commentary when needed. I would encourage getting the book.
Understanding the Problem (Polya, 1957)
"You have to understand the problem." (Polya, 1957)
In this step, the solver is encouraged to find the unknown, gather the data and separate the data into parts.
Devising a Plan (Polya, 1957)
"Find a connection between the data and the unknown. ...You should obtain eventually a plan of the solution." (Polya, 1957)
In this step, the solver is encouraged to make connections to previously solved problems.
Carrying out the Plan (Polya, 1957)
"Carry out your plan." (Polya, 1957)
In this step, the solver is encouraged to check each step along the way and think of ways of proving it's accuracy.
Looking Back (Polya, 1957)
"Examine the solution obtained." (Polya, 1957)
In this step, the solver is encouraged to check the result, think of other methods to solve the same problem and decide if the strategy could be used for other problems.
I feel encouraged by teaching the 4-Step Plan because of the connection that can be made to Polya. The same strategies are used in both, but the 4-Step Plan is easier for the elementary learner to understand.
Types of Math Problems Used in Problem Solving
Make a chart:
Connections to Psychology
Problem solving strategies, as stated in the sections above, are important to improving a student's ability to solve problems.
Knowledge of a general problem-solving strategy also improves performance, even among children. (Bruning, Schraw & Ronning, 1999, p.211)
In my reading while investigating problem-solving I encountered discussion on whether problem-solving skills should be taught as an individual course or included in the curriculum areas. Students themselves would likely benefit from both. A general problem-solving class could serve as an entry level course introducing students to basic problem-solving skills before they are encountered in the individual curriculum areas. A follow-up problem-solving course, toward the end of elementary-level teaching could serve as a way of compiling the knowledge the students have aquired and aiding in their ability to make connections to other situations.
Instead of relying solely on general problem-solving courses, every subject matter will incorporate teaching of relevant cognitive skills. (Mayer, 2003, p.425)
Problem solving web-sites
Mathematical Problem Solving
General Problem Solving
New Expanded Webster's Dictionary, edited by R.F. Patterson, M.A., D.Litt. Copyright 1988, P.S.I. & Associates, Inc., 10481 S.W. 123rd Street, Miami, Florida 33176.
Glencoe/McGraw-Hill, 2001, Glencoe MATHEMATICS Applications and Connections, Course 1, 2 & 3. McGraw-Hill Companies, Inc., 8787 Orion Place, Columbus, Ohio 43240.
Arthur J. Baroody, 1989, A Guide to Teaching Mathematics in the Primary Grades. Allyn and Bacon, a division of Simon & Schuster, 160 Gould Street, Needham Heights, Massachusetts 02194.
G. Polya, 1957, How to Solve It. Copyright 1945 by Princeton University Press, Princeton, New Jersey. 2nd Edition copyright 1957 by Princeton University Press.
Richard E. Mayer, 2003, Learning and Instruction. Pearson Educaton, Inc., Upper Saddle River, New Jersey 07458.
Roger H. Bruning, Gregory J. Schraw and Royce R. Ronning, 1999, Cognitive Psychology and Instruction, 3rd edition. Prentice-Hall, Inc., Simon & Schuster/A Viacom Company, Upper Saddle River, New Jersey 07458.
When I taught problem solving to my junior high students, I always made my students explain HOW they solved their problems. This is required when students take the ISAT test. I wanted them to explain their process and what worked and did not work when solving the problem. I thought that this activity made the students think about problem solving. Nichole Jessup
As Gifted Coordinator, it is my responsibility to oversee the special education services provided to the district's gifted population. I work very hard to give these students problem solving exercises that go beyond the typical math text word problem. If we are to encourage problem solving and critical thinking among our youth it is important that we stimulate them with problems and activities that are both intellectually challenging and conceivably possible. In real life the division problem doesn't always work out evenly and fractions don't always reduce as we would like them to. Stacy Borkgren
Problem solving as a method to learn is a wonderful idea. I have used this in the library, when students are doing research. Most often they would like me (the librarian) to find the resources and the exact information that they may be looking for. As a joke, I often ask if they would like me to write their paper for them. Some would gladly allow that to happen. My point is that the research is the most exciting part of writing a paper. Students need to learn problem solving methods to gain the greatest amount of resources and knowledge of the topic they are studying. M. Youngblood
I use a problem solving book in my class called Daily Mathematics which gives the students a problem of the day to solve. I require them to not only solve the problem but also to share how they solved the problem. I like that it exposes the students to so manyu different types of problems. But I wonder how effective it is to practice these problems in such a random nature. - E. Remington