About the Author: Dr. Connie S. Schrock is an Associate Professor of Mathematics and Computer Science at Emporia State University in Emporia, Kansas. She can be reached via email at schrockc@emporia.com.

What do you think of when someone mentions problem solving? Does it bring to mind visions of algebra word problems? For example: If Hayley is 4 years older than John is and John is twice as old as Matt, how old is Hayley now? Think about the reasonableness of this problem. "Doesn't Hayley know how old she is?" What about the problem where $5,000 is invested, part of it at 3% and part of it at 8%? At the end of the year you know how much you have in interest, but you don't know how much you had invested at each rate. Why didn't you invest it all at 8% anyway? Students look at problems like this and ask why? Many students and teachers equate problem solving with mathematics. While it is true there is a great deal of problem solving done in mathematics, problem solving is much larger—it is a life skill.

When students first encounter word problems they also hear the words "problem solving." However, much of what they are doing is simply exercises designed to reinforce mathematical skills. I believe that many of the word problems in mathematics books fall into that category. There is nothing wrong with doing exercises and they are important for skill development, but exercises must not be confused with problem solving. One of the early methods students use to solve word problems is to find the numbers and do the operation the teacher was doing yesterday. Students find success with this strategy and continue to use it. We must help them expand beyond these ideas to see problem solving in a broader light. Instead of just trying to find the right answer, students need to be encouraged to interpret the situation and actually solve the problem.

Definition of a Problem

One of the first steps to improve problem solving is to be sure that both teachers and students can identify authentic problem solving tasks. Here is a formal definition of a problem: "A problem is a situation, quantitative or otherwise, that confronts an individual or group of individuals, that requires resolution, and for which the individual sees no apparent or obvious means or path to obtaining a solution." (Krulik & Rudnick, 1987) In life we say, "I have a problem" only when we do not know what to do. Many different definitions may be found for problem solving, but there are at least three criteria that each problem must meet. First, the student must accept that he or she will be involved in the problem. Secondly, the student must have blockage and not possess a method for immediately solving the problem. Finally, he/she must actively explore the problem in attempting a solution. Problem solving involves both dispositions and abilities. If students say, "I can't do it," they will be right. The disposition must allow the student to attempt to solve the problem. Creating a positive environment in which to solve problems is critical. Teachers and students must learn to explore a variety of methods to approach the problems. Some of these methods include lateral thinking, heuristics, journal writing, and employing problem-solving strategies.

Lateral Thinking

The type of thinking that a student uses to approach a problem must be enhanced. Schools often emphasize vertical thinking (de Bono, 1970). When we use vertical thinking we move from one logical step to the next, always moving toward the correct answer. From kindergarten age, we teach children to be vertical thinkers. Students enter the classroom and sit down; when the signal is given a child may get a center necklace, put it on and move to the center, etc. We need to create order in the classroom and vertical thinking is a way to do that. Vertical thinking selects a solution by excluding other solutions. Lateral thinking does not select but generates as multiple paths to the solution. With lateral thinking, students seek to generate as many approaches as they can. Using lateral thinking opens the students to creativity and insight. Ideas that might have been discarded now become possible. Students who have better lateral thinking skills attack problems differently and remove the artificial constraints that they had previously applied to problem solving.

Think about this lateral thinking exercise or conundrum: quot;A police officer saw a truck driver clearly going the wrong way down a one-way street but did not try to stop him. Why not?" (Sloane & MacHale, 1994). Do you know the answer? Would it help if I said, "Read it again or think a little harder?" Sometimes the things we say to students when they have trouble do not help them. If this were truly problem solving to you at this point, you would have blockage and not know what to do. That is not a pleasant feeling and students feel uncomfortable at this point as well. They need help to begin to explore and try to find solutions.

Heuristic

A heuristic is a plan of attack. Students need to know how to approach a problem. A heuristic is designed to help problem solvers approach, understand, and attempt to solve a problem. One possible heuristic is:

1. Read and visualize the situation.
2. Explore ideas.
3. Choose a strategy.
4. Find a solution.
5. Check to see if it solves the problem.

Students think that we make up that last step just to annoy them, but in the real world when we solve a problem, we always look back to make sure it satisfies the situation. Another heuristic that may be better in a different setting is:

1. What is the problem?
2. What are the alternatives?
3. What are the advantages or disadvantages?
4. What is the solution?
5. How well is it working?

The same components are there, but sometimes the change in wording is helpful for different disciplines. Logan Avenue Elementary School (Emporia, Kansas) called their heuristic LAPS (Logan Avenue Problem Solving). The teachers had a diagram created and posted in each classroom so that they could remind students of the steps for problem solving. (See Figure 1)

Figure 1 Logan Avenue Problem Solving Heuristic

When I work lateral thinking problems with students, I encourage them to ask yes or no questions that focus on key words in order to help them solve the problem. In the example with the truck driver a student might ask, "Did he drive a dump truck?" or "Was it a real one way street?" I praise the students for every question even if I know it will not lead to the solution. They must expand on mindsets that are created for them in the problem. When students figure out the answer it is powerful, and they seem to know immediately.

The key to doing this with an entire class is to make sure that no one shouts out the answer. I encourage students to ask a subtle question that lets me know that they know the answer but does not give it away to the rest of the class. What happens when someone gives the correct answer to any question that a teacher asks in class? The rest of the class turns off. The teacher now has the answer and the other students believe that the activity is done. They think it is no longer necessary for them to continue to think about the problem even though that is exactly what we need them to do. For the earlier lateral-thinking problem, one question to ask might be "Was there an emergency?" Another question the students often ask is "Were they friends?" The answer to both questions is no. It is important to encourage their questions and to allow them to leave the room not knowing the answer. They will often continue to think about the problem. Perseverance is an important part of problem solving, and it is very difficult to teach. We must dispel the myth that any problem that cannot be solved in five minutes is unsolvable.

Journals

Using the lateral thinking exercises in journals encourages students to solve the problems for themselves and removes some of the initial pressure to solve the problem. I present the conundrum and then ask only that students write three "yes" or "no" questions. This encourages the students to make steps toward the solution. I answer "yes" or "no" and have the students write three more questions. Using a journaling process allows students to proceed toward the solution at their own pace. If a student is having difficulty, the teacher can write a question and also answer it to help move the student along. Journals provide an avenue for integrating problem solving and writing. The questioning skills and lateral thinking skills learned in journal exercises will transfer to other problem solving activities.

Strategies

An important component within the heuristic is the selection of a problem solving strategy. Students must be taught problem solving strategies and be provided with good settings in which to apply these strategies. William Allen White Elementary School (Emporia, Kansas) focused on the teaching and learning of problem solving strategies as an intervention. They emphasized different skills at different grade levels.

Kindergarten: emphasized the use of manipulatives, "act it out", and draw a picture

First grade: emphasized guess and check, make a table, and list or chart

Second grade: emphasized "look for a pattern" and use logical reasoning

Third grade: focused on the "work backwards" strategy

Each year the strategies taught in the previous years were reinforced and new ones were introduced. That doesn't mean that when textbooks introduced or used a strategy that was not a focus for that grade level that the teachers skipped it. It just means that the students worked to become proficient with the targeted strategies. Good problems are those that may be solved using multiple strategies. Students need to recognize that it is not necessary to use a particular strategy for a given problem. Each discipline will also have certain strategies that are more beneficial than others, but most strategies are applicable to a variety of problems.

Vocabulary

In mathematics and the sciences students often hear the words "problem solving," but they get left out of the vocabulary in many disciplines. That does not mean that problem solving is left out, it is simply called something else. If a teacher assigned a paper an English class, he or she might say, "I want you to write a paper that . . . ." But if the teacher were focusing on the problem solving aspects of the paper he or she might say, " I have a problem that I would like you to solve, and I want you to do it by writing a paper with . . ." When given a writing task, many students will not know where to start (blockage) and experience the same questions that come up in any problem solving activity. In a social science class a teacher may have a great worksheet that asks students to compare data given in charts and texts to reach conclusions. Instead of saying, "I have a worksheet for you to do," the teacher should say, "I have some problems I want you to work." Mathematics and problem solving are not the same thing. Both are important subjects and can be useful tools for many other disciplines. Teachers need to help students recognize the difference between the two.

Evaluation

Problem solving lends itself to a wide variety of assessment methods. The traditional test is not always a good tool for assessing students' problem solving abilities. If the task is a real problem then the blockage a student experiences may last longer than the test allows.

Observation is a good tool when students are working in groups. A teacher can assess the students' perseverance, self-confidence, interest, cooperation, contribution and much more. I was using a middle school classroom to try assessment by observation. I had given the students a few problems to work, in groups, and I was going to observe and write down what I learned from that day. After I passed out the problems I spent the rest of the period running from group to group answering the same questions. The next day I decided to try something new. I made question coupons saying "Present this coupon when you ask for help on a problem." I explained to each group they would have three coupons; and when they wanted to ask a question, they needed to use a coupon. The problems were distributed and I waited. Jeff raised his hand and I went over to his group. He said, "I have a question." I asked for the coupon, but before he could give it to me Amie said, "Don't ask her, ask us." What a concept! Talk to the people in your group before you ask the teacher. The rest of the period went well, and at the end of class I had great evidence about some of their problem solving skills. The students did ask a question I had not thought about, "What do we get with our leftover coupons?"

Metacognitive journals help students to think about their own thinking and improve their reasoning and problem solving processes. One technique in journaling is to have students work a problem and on the next page describe their own thinking as they proceed through the solution. Another possibility would be to have them write a paragraph after they have completed a problem-solving task. The assignment can be open-ended or it can be guided with questions about the process. Portfolios are also a good place to display works the student selects to demonstrate his/her best effort.

Graphic organizers can help the student organize his/her work and also make it helpful to grade. In Figure 2, each section of the diagram provides a small working space that helps remind the student to follow each step. When asked to identify the problem, I prefer that it be stated in the student's own words and not simply copied from the statement he/she was given. In the section labeled "Find the Facts", pertinent facts can be listed as the student identifies them. The "Strategies" area encourages the student to state the strategy he/she has selected to use. "Solve" is where the student presents the solution. The lower portion of the figure (the triangle entitled "Does it Work?" stresses the importance of demonstrating that the proposed solution should work. Each part of the organizer is important and students should be given credit for each part that is completed correctly. The center rectangle has five sections so that each aspect of problem solving may be scored according to the teacher's grading rubric. The points from all areas will be totaled, and the student will know the total score as well as the value of each component. This encourages the student to move through all of the steps of the heuristic.

Figure 2 Graphic Organizer for Problem Solving

Conclusion

Problem solving involves risk. Both teacher and student must be willing participants. A positive attitude and repeated exposure is critical for success. Remember, a real problem presents a challenge that cannot be resolved by mere recall, and an individual must accept the challenge. An exercise is a situation that involves practice to reinforce a learned skill. Teachers must include exercises, but exercises should not be confused with problem solving. When a teacher asks his/her students to apply a new map skill to make decisions in a social studies classroom, the first time the student completes the task it is problem solving. If he/she is repeatedly asked to apply the same skills and strategy the problems will become an exercise. Problems often develop into exercises as skills are learned. In the same classroom some tasks will be problems for a few students, exercises for others, and mere recall for several. There is no way to avoid this. We must simply remember to challenge all students and provide repeated exposure to problem solving throughout the year. Oh, and in case you have not yet solved the conundrum, the truck driver was walking.

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