The Problem With Problem Solving

The main problem with problem solving lies in the fourth element listed above: problem solving is a heuristic. Recall that a heuristic is a guideline that may or may not yield success but, unlike an algorithm, it does not depend on knowledge of the problem to be successful. Heuristic is a method of thought that does not pertain to any specific problems or content. The element is problematic because it contradicts three other elements within the theory: the definition of problem solving, successful problem solving requires a knowledge base, and problem solving enables learners to transfer knowledge. Each of these three elements implies that previously learned knowledge of the problem is necessary to solving the problem, whereas use of a heuristic assumes no knowledge is necessary.

I argue, like Peikoff (1985), that there is no way to separate thinking or problem solving from knowledge. Just like instruction and curriculum, these concepts imply one another and cannot be discussed separately for long. Likewise, to acquire knowledge, one must think. This is not to say that students cannot construct knowledge as they solve a given problem, only to say that often the problems they are presented only require

them to apply existing knowledge. From this perspective, it must be assumed that students do not construct all of the knowledge in a given curriculum. Yet problem solving as a heuristic is the most cherished aspect of problem solving because it is content-less. For example, in the preface to Mathematical Discovery, George Polya (1962), one of the foremost thinkers on problem solving says, I wish to call heuristic the study that the present work attempts, the study of means and methods of problem solving. The term heuristic, which was used by some philosophers in the past, is half forgotten and half-discredited nowadays, but I am not afraid to use it.

In fact, most of the time the present work offers a down-to-earth practical aspect of heuristic. (p. vi) Instructional textbooks sometimes play off this process versus content dichotomy: a teacher can either teach students to be critical thinkers and problem solvers or she can teach students more content knowledge. The authors of one textbook say, Too often children are taught in school as though the answers to all the important questions were in textbooks. In reality, most of the problems faced by individuals have no easy answers. There are no reference books in which one can find the solution to life’s perplexing problems. (Gunter, Estes, & Schwab, 2003, pp. 128–129) The dichotomy implies that thinking and knowledge are mutually exclusive, when in fact critical thinking and problem solving require a great deal of specific content knowledge. 

Problem solving and heuristics cannot be contentless and still be effective. Critical thinking, problem solving, and heuristics must include a knowledge base (Fredricksen, 1984; Ormrod, 1999). Including the knowledge base enables the principle cognitive function of problem solving—the application of conceptual knowledge, or transfer—to occur (Peikoff, 1985). However, the degree to which Dewey and Polya actually believed that a heuristic could be completely content-less and still be effective is not clear. Further, many instructional textbooks actually stress the importance of content knowledge in solving problems (Henson, 2004; Kauchak & Eggen, 2007; Lang & Evans, 2006).

Each of the above elements of problem solving will be reviewed again in light of the relationship between thinking and knowledge and the research base on problem solving. Element one, the definition of a problem, implies that one must have some knowledge of the problem to solve it. How can one solve a problem without first knowing what the problem is? In fact, identification of the problem is what is called for in the first two steps, Read and Explore, of the heuristic. In this step, the student first becomes aware of the problem and then seeks to define what it is or what the problem requires for its solution. Awareness and definition comprise the knowledge that is essential to solving the problem. Consider the effectiveness of students relative to their respective experiences with a given problem. The student more familiar with the problem will probably be better able to solve it. In contrast, the student new to the problem, who has only studied the heuristic, would have to re-invent the solution to the problem.

So the first two steps of the heuristic imply that one needs a great deal of knowledge about the problem to be an effective problem solver. In fact, if one wants to solve the problem for the long term, one would want to thoroughly study the problem until some kind of principles were developed with regard to it. The final outcome of such an inquiry, ironically, would yield the construction of an algorithm.

The second element, the definition of problem solving, also implies a connection between thinking and knowledge. It says that problem solving is essentially applying old knowledge to a new situation (Krulik & Rudnick, 1987). However, if knowledge or a problem is genuinely new, then the old knowledge would not apply to it in any way. Ormrod (1999) suggests that the so-called new situation is really the same as the old in principle. For example, the principle of addition a student would use to solve the problem  1+ 2 = 3 is essentially the same principle one would apply to 1 + x = 3. The form may be different but ultimately the same principle is used to solve both problems. If this is the case, then a more proper element of problem solving would be number eight, the transfer of knowledge or application of conceptual knowledge.

The third and fourth elements algorithms and heuristics are problematic. Krulik and Rudnick (1980) distinguish between algorithms and heuristics. Unlike employing an algorithm, using a heuristic requires the problem solver to think on the highest level and fully understand the problem. Krulik and Rudnick also prefer heuristics to algorithms because the latter only applies to specific situations, whereas a heuristic applies to many as yet undiscovered problems. However, an algorithm requires more than mere memorization; it requires deep thinking too. First, in order to apply an algorithm, the student must have sufficient information about the problem to know which algorithm to apply. This would only be possible if the student possessed a conceptual understanding of the subject matter. Further, even if a student could somehow memorize when to apply certain algorithms, it does not follow that he or she would also be able to memorize how to apply it (Hu, 2006; Hundhausen & Brown, 2008; Johanning, 2006; Rusch, 2005).

Second, algorithms and problem solving are related to one another. Algorithms are the product of successful problem solving and to be a successful problem solver one often must have knowledge of algorithms (Hu, 2006; Hundhausen & Brown, 2008; Johanning, 2006; Rusch, 2005). Algorithms exist to eliminate needless thought, and in this sense, they actually are the end product of heuristics. The necessity to teach heuristics exists, but heuristics and algorithms should not be divided and set against one another. Rather, teachers should explain their relationship and how both are used in solving problems.

A secondary problem that results from this flawed dichotomy between algorithms and heuristics is that advocates of problem solving prefer heuristics because algorithms only apply to specific situations, whereas heuristics do not pertain to any specific knowledge. If one reflects upon the steps of problem solving listed above one will see that they require one to know the problem to be successful at solving it. Consider the sample problem above to which the heuristic was applied. If one knows the heuristic process and possesses no background knowledge of  similar problems, one would not be able to solve the problem. For example, in the first step of the heuristic one is supposed to Read the problem, identify the problem, and list key facts of the problem. Without a great deal of specific content knowledge how will the student know what the teacher means by “problem,” “key facts,” and so on? The teacher will probably have to engage the student in several problems. Without extensive knowledge of facts, how does the student know what mathematical facts are, and how they apply to word problems, for example?

In the second step, Explore, the problem solver looks for a pattern or identifies the principle or concept. Again, how can one identify the pattern, principle, or concept without already possessing several stored patterns, principles, and concepts? Indeed, to a student with very little mathematical knowledge, this problem would be extremely difficult to solve. The heuristic would be of little help. The heuristic says to draw a diagram, presumably to make the problem more concrete and therefore more accessible to the student, but without already knowing what the concept the problem exhibits this would very difficult, if not impossible. Using the chart with the data as an example, it would require previous knowledge in mathematics to be able to construct it. It seems that the heuristic in this problem is in reality just another algorithm that the teacher will have to teach as directly and as repetitively until the students learn how and when to apply it, which is the very opposite of what advocates of problem solving want. The same is also true of step five, Review and Extend. Presumably if a student could represent this problem in algebraic form, he or she should also be able to solve the same problem without recourse to drawing diagrams,  recording data, etc. One could simply solve the problem right after step one.

The sample problem illustrates what scientists have discovered about novices and experts. In studies that examined expert and novice chess players, researchers found that their respective memories were no different in relation to random arrangements of chess pieces. When the pieces were arranged in ways that would make sense in a chess game, the experts’ memories were much better. The theory is that an expert chess player is not a better problem solver, he or she just has a more extensive knowledge base than a novice player. He or she is past the rudimentary hypothesis testing stage of learning, past the problem solving heuristic stage and is now simply applying algorithms to already-solved problems (Ross, 2006). The same could be said for students applying a heuristic to the above problem. The only ones who could solve it would be those who use an algorithm. Even if a teacher taught the heuristic to students, he or she would essentially be teaching an algorithm.

Advocates of problem solving are not solely to blame for the misconception between thinking and knowledge and between heuristics and algorithms. The misconception is likely due to teachers that have overused algorithms and never shown students how they are formed, that they come from heuristics, and that

one should have a conceptual understanding of when they should be used, not merely a memorized understanding of them. The fundamentally flawed dichotomy within problem solving probably stems from thinking in terms of “either-ors.” One side defines appropriate education as teaching algorithms by having students memorize when to use them but not why. The other side, by contrast, emphasizes that thinking for understanding is preferable to simply memorized knowledge. Perhaps what has happened in the shift from the former to the latter practices is the instructional emphasis has shifted from content to thinking so much that the knowledge base has been wiped out in the process. Ironically, eliminating knowledge from the equation also eliminates the effectiveness of problem solving.

The dichotomy between knowledge and thinking has also affected elements five and six. Number five states that problem solving connects theory and practice. At the core of this element is yet another flawed dichotomy. Many educators hold that education should prepare students for the real world by focusing less on theory and more on practice. However, dividing the two into separate cognitive domains that are mutually exclusive is not possible. Thinking is actually the integration of theory and practice, the abstract and the concrete, the conceptual and the particular.

Theories are actually only general principles based on several practical instances. Likewise, abstract concepts are only general ideas based on several concrete particulars. Dividing the two is not possible because each implies the other (Lang & Evans, 2006). Effective instruction combines both theory and practice in specific ways. When effective teachers introduce a new concept, they first present a perceptual, concrete example of it to the student. By presenting several concrete examples to the student, the concept is better understood because this is in fact the sequence of how humans form concepts (Bruner, Goodnow, & Austin, 1956; Cone 1969; Ormrod, 1999; Peikoff, 1993). They begin with two or more concrete particulars and abstract from them the essential  defining characteristics into a concept. For example, after experiencing several actual tables a human eventually abstracts the concept a piece of furniture with legs and a top (Lang & Evans, 2006).

On the other hand, learning is not complete if one can only match the concept with the particular example of it that the teacher has supplied. A successful student is one who can match the concept to the as yet unseen examples or present an example that the teacher has not presented. Using the table as an example, the  student would be able to generate an example of a new table that the teacher has not exhibited or discussed.

This is an example of principle eight, the transfer of knowledge or applying conceptual knowledge. The dichotomy between theory and practice also seems to stem from the dichotomous relationship between the teaching for content-knowledge and teaching for thinking. The former is typically characterized as teaching concepts out of context, without a particular concrete example to experience through the five senses. The latter, however, is often characterized as being too concrete. Effective instruction integrates both the concrete and abstract but in a specific sequence. First, new learning requires specific real problems. Second, from these concrete problems, the learner forms an abstract principle or concept. Finally, the student then attempts to apply that conceptual knowledge to a new, never before experienced problem (Bruner, Goodnow, & Austin, 1956; Cone, 1969; Ormrod, 1999; Peikoff, 1993).

The theory vs. practice debate is related to problem solving because problem solving is often marketed as the integration of theory and practice. I argue, however, it leaves out too much theory in its effort to be practical. That is, it leaves out the application of conceptual knowledge and its requisite knowledge base. Element six, problem solving teaches creativity, is also problematic. To create is to generate the new, so one must ask how someone can teach another to generate something new. Are there specific processes within a human mind that lead to creative output that can also be taught? The answer would depend at least in part on the definition of create. When an artist creates, he or she is actually re-creating reality according to his or her philosophical viewpoint, but much, if not all, of what is included in the creation is not a creation at all but an integration or an arranging of already existing things or ideas. So in one sense, no one creates; one only integrates or applies previously learned knowledge. No idea is entirely new; it relates to other ideas or things. The theory of relativity, for example, changed the foundational assumptions of physics, but it was developed in concert with ideas that already existed. There may be no such thing as pure creativity, making something from nothing. What seems like creativity is more properly transfer or the application of concepts, recognizing that what appears like two different things are really the same thing in principle.

On the other hand, it is possible to provide an environment that is conducive to creativity. Many problem solving theorists have argued correctly for the inclusion of such an atmosphere in classrooms (Christy & Lima, 2007; Krulik & Rudnick, 1980; Slavin, 1997; Sriraman, 2001). I only object to the claim that problem solving teaches creativity defined as creating the new. It can, however, teach creativity defined as the application of previously learned principles to new situations.

Element seven, problem solving requires a knowledge base, although not problematic is only neglected within the theory of problem solving. This is ironic given how important it is. Jeanne Ormrod (1999) says, “Successful (expert) problem solvers have a more complete and better organized knowledge base for the problems they solve” (p. 370). She also relates how one research inquiry that studied the practice of problem solving in a high school physics class observed that the high achievers had “better organized information about concepts related to electricity” (p. 370). Not only was it better organized, the students were also aware of “the particular relationships that different concepts had with one another” (Cochran, 1988, p. 101). Norman (1980) also says, I do not believe we yet know enough to make strong statements about what ought to be or ought not to be included in a course on general problem solving methods. Although there are some general methods that could be of use…I suspect that in most real situations it is…specific knowledge that is most important. (p. 101)

Finally, element eight, problem solving is the application of concepts or transfer, is also not problematic; it too is only neglected within the theory of problem solving. Norman Frederiksen (1984) says, for example, “the ability to formulate abstract concepts is an ability that underlies the acquisition of knowledge. [Teaching how to conceptualize] accounts for generality or transfer to new situations” (p. 379).   According to this passage, it is the application of conceptual knowledge and not the heuristic alone that as Frederiksen says, “accounts for generality or transfer,” (p. 379) which the advocates of problem solving so desire.

Problem solving would be more effective if the knowledge base and the application of that knowledge were the primary principles of the theory and practice. Currently, it seems that a content-less heuristic is the primary principle, which, as I have argued, is problematic because it dichotomizes thinking and knowledge into two mutually exclusive domains. In fact, in the course of solving any problem one will find themselves learning of all things not a heuristic, but an algorithm. In other words, teachers must not only teach students the heuristic and set their students free upon the problems of everyday life. Rather, teachers must, in addition to teaching students sound thinking skills, teach them what knowledge in the past has been successful at solving the problems and why.