Tammy Hall
April 16, 1999
Math 498
Problem Solving
In the field of education it has become increasingly more important to stress higher order thinking for good reason. The skills that students obtain when using higher order thinking carry over into everyday situations. A prime example is problem solving in the mathematical setting. Students can learn fundamental skills to solve any problem. The National Council of Supervisors of Mathematics has decreed that "learning to solve problems is the principal reason for studying mathematics" (Fernandez, Hadaway, & Wilson, 1993, p. 57).
First, an examination of the history of problem solving in mathematics education is relevant. In the 1980s mathematics education was pronounced "in crisis." But, this was not the first time. In 1957, the Soviet Union triumphantly began the space race by launching the satellite Sputnik. This action raised panic among science and mathematics educators across America. The fear was that the Soviets had reached "technological and military supremacy" (Schoenfeld, 1992, p. 336). In response, a new type of education was demanded. Therefore, in the 1960s the "new math" began a more abstract approach. Students were now learning about set theory and group properties (Schoenfeld, 1992, p. 336). High school students were required to prove theorems in both algebra and geometry. This program was short lived and soon seen as a failure. The belief was that the students learning under the new system were failed in two ways. First, they did not obtain the basic knowledge that students before them had gained, and secondly, they failed to acquire the skills of the "new math." The new system was forgotten, and American students went back to learning the "basics." The thought at that time was students should master the basics in order to have a foundation for higher order thinking problems. The "back-to-basics" theme continued to be the center of mathematics curriculum for the 1970s. By the end of the decade it became apparent that mistakes have been made. The students who had completed the basic curriculum were unable to perform well on critical thinking and problem solving exercises. It was also discovered that these students did not perform any better than the "new math" students on basic skills exercises. Obviously, a reform was needed. As a result, the focus of mathematics education was shifted to problem solving and became the theme of the mathematics curriculum for the 1980s (Schoenfeld, 1992).
Why is problem solving important? Problem solving curricula allow students the opportunity to "study mathematics as an exploratory, dynamic, evolving discipline rather than as a rigid, absolute, closed body of laws to be memorized" (Schoenfeld, 1992, p. 335). Problem solving differentiates rote learning from meaningful learning. Richard Mayer defines these terms in his essay "The Psychology of Mathematical Problem Solving." Mayer states that rote learning occurs when the student gives a memorized response without understanding. He defines meaningful learning as the act of problem solving by correlating it with other knowledge and gaining an understanding of the material (1982). The principle goal of any educational instruction is to "develop skills, knowledge, and abilities that transfer tasks not explicitly covered in the curriculum" (Fernandez, Hadaway, & Wilson, 1993, p. 58). Most mathematics students will seldom use higher mathematics in everyday situations. But, these same students will need to solve quantitative problems that are present in day to day situations. These situations include anything from figuring out how much to tip a waiter to how much fence to buy to enclose your yard. In most situations, quantitative questions occur as problems that require solving. In this perspective, "mathematics is problem solving!" (Krulik & Rudnick, 1980, p. 5). Alan Schoenfeld, a leading researcher in the field of problem solving education, states that "learning mathematics is empowering." Students who possess this power are "quantitatively literate" (Schoenfeld, 1992, p 335).
The next logical step is to define a problem and problem solving. First, 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 obtain the solution" (Krulik & Rudnick, 1980, p. 3). According to Krulik and Rudnick, in order to classify a situation as a problem it must fulfill three criteria. The criteria are acceptance, blockage, and exploration. First, the student must read the problem and accept it as a problem. This requires personal involvement influenced by internal or external motivation. After the student accepts the problem, blockage occurs. Blockage requires that the student's original effort in solving the problem is unsuccessful. The student's initial instinct in solving the problem is fruitless, and any prior strategies the student has learned do not help the student find an easy solution. Next, the student undergoes exploration. The student must find new techniques for solving the problem. If these criteria are satisfied, then the situation is considered a problem (Krulik & Rudnick, 1980). Schoenfeld implies that classifying a problem is relative to each student (1992). What is perceived as a problem for one student may be a routine exercise for another student (Fernandez, Hadaway, & Wilson, 1993).
Problem solving is defined as a process. When an individual engages in problem solving, he uses "previously acquired knowledge, skills, and understanding to satisfy the demands of an unfamiliar situation" (Krulick & Rudnick, 1980, p. 4). The student applies schema (i.e. knowledge that is previously acquired and useful to the learner only in certain contexts) to the new situation. Two of the categories of mathematical knowledge are information and facts and the ability to use information and facts. The latter is fundamental to successful problem solving (Krulik & Rudnick, 1980).
The dilemma with problem solving in education is the difficulty and disparity that occurs when trying to define it. In the past, problem solving has had many definitions ranging from completing routine exercises to doing mathematics professionally (Schoenfeld, 1992). Many people mistakenly see problem solving as completing word problems. Often times, the student is using an algorithm rather than solving a problem. An algorithm is a "technique that applies to a single case of 'problems' and that guarantees success if mechanical errors are avoided" (Fernandez, Hadaway, & Wilson, 1993, p. 63). Examples of algorithms include mixture problems, river problems, and frame problems (Mayer, 1982). Once the student has mastered solving the word problem, he can solve any problem of that nature successfully. Most often when working with algorithms, students are given a set of problems with one worked through as an example. From this, the students can decipher the solution to all of the problems by using the algorithm (Schoenfeld, 1992). The process of completing an algorithm is not problem solving. On the other hand, creating an algorithm can be (Fernandez, Hadaway, & Wilson, 1993).
Stanic and Kilpatrick identify three main ideas in their historical review of problem solving (1988). The first is "problem solving as context" (Schoenfeld, 1992, p. 338). This theme deals more with the algorithmic approach to solving problems. Here, problem solving is used to supplement other curricular needs as opposed to existing as a curricular goal itself. Stanic and Kilpatrick offer five parts that problems play in achieving other curricular goals. The roles are as a justification for teaching mathematics, to provide specific motivation for subject topics, as recreation, as a means of developing new skills, and as practice. These are the principle functions that problems play in the context area. The next theme is "problem solving as skill" (Schoenfeld, 1992, p. 338). The belief of this theme is that problem-solving skills learned in the mathematics domain will transfer into other domains. This belief is the fundamental goal in any area of instruction. The skill theme is similar to the first in that it does not make problem solving a curricular goal, but uses it to obtain other objectives. The final theme identified by Stanic and Kilpatrick is "problem solving as art" (Schoenfeld, 1992, p. 338). This theme is different from the other two because it endorses teaching problem solving as a skill to be mastered. Schoenfeld explains this theme as the belief that "real problem solving is the heart of mathematics." Real problem solving includes problems that are complex and perplexing. Many mathematicians believe that these types of problems are what define mathematics and people who possess the skill of successfully solving problems are true mathematicians (Schoenfeld, 1992).
The next topic to introduce in problem solving is heuristics. Heuristics are methods for solving problems. Synonyms for heuristics include strategies or techniques. Heuristics are the theories that are "enlarged to incorporate classroom contexts, past knowledge and experience, and beliefs" (Schoenfeld, 1992, p. 352). Essentially, heuristics is a game plan. Whereas rote problems have algorithms, meaningful problems require heuristics to achieve a solution. There are many different heuristics available. The teacher must find one that best suits the needs of the students (Schoenfeld, 1992). Over the years, task specific and general heuristics have been developed for the purpose of aiding students in problem solving. Three examples are offered with an explanation accompanying each.
One example is offered by Krulik and Rudnick in their book Problem Solving: A Handbook for Teachers (1980). Their set of heuristics includes reading, exploring, selecting a strategy, solving, and reviewing and extending. The student should first read the problem and look for key words. In reading the problem, he/she should also familiarize himself with the setting of the problem, determine what the problem is asking, and restate the problem in his own words. After this is complete, the student should explore the problem further by drawing a diagram, making a chart, or making a model. While exploring, the pupil will record data and search for patterns in the problem. Next, the problem solver will select a strategy for solving the problem. During this phase, the solver's ability to make smart guesses is required. The student will experiment and look for a simpler problem. Then, he/she will make a guess and assume a solution. As the fourth step, the student will attempt to solve the problem by carrying out his solution. Finally, the student will review and extend. This includes checking the answer and looking for variations of the problem (Krulik & Rudnick, 1980).
Another example is offered in the Glencoe Algebra I textbook. The authors suggest that the student should first explore the problem. Similar to Krulik and Rudnick's strategy, the student should examine the problem and determine what it is asking. Next, the student should plan the solution. After exploring the student should then decide on a strategy for solving the problem. The next step is for the student to solve the problem. Solving entails putting the plan of action to work to obtain the solution. The final step is to examine the solution. Once the student has reached a solution, he should inspect his solution to verify if it makes sense and if any alterations can be made to make the problem more formidable. The authors also give a list of problem solving tactics. These include checking for hidden assumptions, using a graph, eliminating possibilities, looking for patterns, identifying subgoals, and solving a simpler or related problem (Collins, Cuevas, et al., 1998).
Most sets of heuristics are similar in their steps with few variations. This is because most heuristics are based on the ones created by Polya. Polya is a mathematician best known for his contributions in creating and examining heuristics that students can use to solve problems. Polya is credited with first using the term "modern heuristics" (Schoenfeld, 1992, p. 352). Polya's How to Solve It was published in 1945, and his strategies and research have become the basis for problem solving in education today (Schoenfeld, 1992). Polya's main approach to problem solving was to make sense out of the problem. The first step in the heuristic Polya is known for is to understand the problem. The student should examine the situation and decide if it is a problem and if so, what it is asking. The second step is devising a plan or strategy. Once this is complete, the student moves on to carrying out the plan. The student will verify that there are possible solutions or that no solution exists. The final step is looking back. If no solution exists, the student must modify the problem, or if the solution is obtained, the student should alter the problem to make it more challenging (Fernandez, Gonzales, & Knecht, 1996).
Most educators believe that the looking back process is the most important factor in problem solving and the most difficult to accomplish with students. The looking back process allows the student to "learn from the problem" (Fernandez, Hadaway, & Wilson, 1993, p. 64). Fernandez, Hadaway, and Wilson identify extending problems, solutions, and processes as three of the most important activities used in helping students to learn from the problem solving process (1993).
The next question to examine is what makes a good problem solver. Students must have a desire to solve problems and be willing to face the challenge. The student's willingness is controlled by their attitude, which is part of a person's affective domain. Good problem solvers persevere when failure occurs. They simply try another approach. They must also possess the ability to make smart guesses (Krulik & Rudnick, 1980). More importantly, the student must be able to self-monitor their own progress. During Plato's era this was referred to as mental discipline. The belief was that persons trained in mathematics would be good thinkers similar to those trained in exercise would have good bodies. Good problem solvers possess the mental discipline to self-monitor and self-manage their own progress in solving a problem. The ability to self-regulate is called metacognition. Metacognition is the act of evaluating what one is thinking (Schoenfeld, 1992). Self-reflection is also fundamental to being a good problem solver (Fernandez, Hadaway, & Wilson, 1993). Schoenfeld believes students learn most from problem solving when they can focus their "attention to a set of guidelines for reflecting on the problem solving activities for which the students were engaged" (1992, p.346). In order for problem solving instruction to be effective, it must offer the students "the opportunity to reflect, in a systematic and constructive way, during problem solving activities" (Fernandez, Hadaway, & Wilson, 1993, p. 64). Also, to be an effective problem solver one must "construct some decision mechanism" that allows you to select methods of solving problems and to create new methods as the need arises (Fernandez, Hadaway, & Wilson, 1993, p.64). Students will learn the most if they are able to examine their own thought processes while solving problems.
Teaching problem solving can be a difficult process. The teacher must be energetic and interested in the problem solving process. The teacher should also be aware of the students' needs and interests (Krulik & Rudnick, 1980). One difficulty with teaching problem solving is the misconceptions students have about the procedure. Many students believe that math problems have only one right answer and there is only one way to find that answer. Numerous students also think that once they understand the lesson studied, they should be able to apply an algorithm to any problem in order to solve it. Students that have these beliefs typically quit a problem after attempting it many minutes without any success. Problem solvers of this nature are called novices. Novice problem solvers will commonly read the problem, immediately choose a method of attack, and then proceed with attempting to solve the problem. They will continue their method even when it is apparent that their efforts are unsuccessful (Schoenfeld, 1992).
On the other hand, expert problem solvers will read the problem then analyze it carefully. Next, they will decide on a plan and implement it. The distinction between an expert and novice problem solver is that the expert has the metacognitive ability that clues them to an incorrect choice of action. Unlike the novice, the expert has the ability to recognize when the original plan is unsuccessful and when to try a new one. Therefore, if the expert sees that the first plan is futile, he will reanalyze the problem and try a new approach. The expert continues this process until the problem is solved successfully (Schoenfeld, 1992).
Problem solving is imperative to the study of mathematics. Educators should realize its fundamental function in studying mathematics and apply it in the classroom. Students who are well trained in problem solving skills have the tools for solving any mathematical problem, be it rote or meaningful.