The Mathematics Educator 2007, Vol. 17, No. 2, 7–14 A Problem With Problem Solving: Teaching Thinking Without Teaching Knowledge Jamin Carson Problem solving theory and practice suggest that thinking is more important to solving problems than knowledge and that it is possible to teach thinking in situations where little or no knowledge of the problem is needed. Such an assumption has led problem solving advocates to champion content-less heuristics as the primary element of problem solving while relegating the knowledge base and the application of concepts or transfer to secondary status. In the following theoretical analysis, it will be argued that the knowledge base and transfer of knowledge—not the content-less heuristic—are the most essential elements of problem solving. Problem solving theory and practice suggest that is to know the meaning of the term problem. This thinking is more important in solving problems than theoretical framework uses the definition of problem knowledge and that it is possible to teach thinking in presented by Stephen Krulik and Jesse Rudnick (1980) situations where little or no knowledge of the problem in Problem Solving: A Handbook for Teachers. A is needed. Such an assumption has led problem solving problem is “a situation, quantitative or otherwise, that advocates to champion content-less heuristics as the confronts an individual or group of individuals, that primary element of problem solving while relegating requires resolution, and for which the individual sees the knowledge base and the transfer or application of no apparent or obvious means or path to obtaining a conceptual knowledge to secondary status. Yet if one solution” (p. 3). analyzes the meaning of problem solving, the The Definition of Problem Solving knowledge base and the transfer of that knowledge are the most essential elements in solving problems. Krulik and Rudnick (1980) also define problem solving as Theoretical Framework the means by which an individual uses previously Problem solving is only one type of a larger acquired knowledge, skills, and understanding to category of thinking skills that teachers use to teach satisfy the demands of an unfamiliar situation. The students how to think. Other means of developing student must synthesize what he or she has learned, thinking skills are problem-based learning, critical and apply it to a new and different situation. (p. 4) thinking skills, creative thinking skills, decision This definition is similar to the definition of the making, conceptualizing, and information processing eighth element of problem solving, transfer: “[w]hen (Ellis, 2005). Although scholars and practitioners often learning in one situation facilitates learning or imply different meanings by each of these terms, most performance in another situation” (Ormrod, 1999, p. thinking skills programs share the same basic elements: 348). (1) the definition of a problem, (2) the definition of problem solving, (3) algorithms, (4) heuristics, (5) the Problem Solving is Not an Algorithm relationship between theory and practice, (6) teaching One of the primary elements of this framework is creativity, (7) a knowledge base, and (8) the transfer or that problem solving is not an algorithm. For example, the application of conceptual knowledge. Krulik and Rudnick (1980) say, The Definition of a Problem The existence of a problem implies that the The first element of the theory of problem solving individual is confronted by something he or she does not recognize, and to which he or she cannot merely apply a model. A problem will no longer be Dr. Jamin Carson is an assistant professor of curriculum and considered a problem once it can easily be solved instruction at East Carolina University. He teaches the theory and practice of instruction as well as classroom management and by algorithms that have been previously learned. discipline. His primary research interest is the epistemology of (p. 3) curriculum and instruction. Jamin Carson 7 Table 1 Types of Problem Solving John Dewey (1933) George Polya (1988) Stephen Krulik and Jesse Rudnick (1980) Confront Problem Understanding the Problem Read Steps in Diagnose or Define Problem Devising a Plan Explore Problem Solving Inventory Several Solutions Carrying Out the Plan Select a Strategy Conjecture Consequences of Looking Back Solve Solutions Test Consequences Review and Extend Additionally, advocates of problem solving imply one large long table. How many of these small that algorithms are inferior models of thinking because tables are needed to seat all 24 people? (Krulik & they do not require thought on a high level, nor do they Rudnick, 1987, pp. 29–31) require deep understanding of the concept or problem. The first step, Read, is when one identifies the Algorithms only require memory and routine problem. The problem solver does this by noting key application. Further, they are not useful for solving words, asking oneself what is being asked in the new problems (Krulik & Rudnick, 1980). problem, or restating the problem in language that he Problem Solving is a Heuristic or she can understand more easily. The key words of Advocates of problem solving argue that educators the problem are small square tables, twelve couples, need to teach a method of thought that does not pertain one large table, and 24 people. to specific or pre-solved problems or to any specific The second step, Explore, is when one looks for content or knowledge. A heuristic is this kind of patterns or attempts to determine the concept or method. It is a process or a set of guidelines that a principle at play within the problem. This is essentially person applies to various situations. Heuristics do not a higher form of step one in which the student guarantee success as an algorithm does (Krulik & identifies what the problem is and represents it in a Rudnick, 1980; Ormrod, 1999), but what is lost in way that is easier to understand. In this step, however, effectiveness is gained in utility. the student is really asking, “What is this problem Three examples of a problem solving heuristic are like?” He or she is connecting the new problem to prior presented in Table 1. The first belongs to John Dewey, knowledge. The student might draw a picture of what who explicated a method of problem solving in How the situation would look like for one table, two tables, We Think (1933). The second is George Polya’s, whose three tables, and so on. After drawing the tables, the method is mostly associated with problem solving in student would note patterns in a chart. (See below.) mathematics. The last is a more contemporary version The third step, Select a Strategy, is where one developed by Krulik and Rudnick, in which they draws a conclusion or makes a hypothesis about how to explicate what should occur in each stage of problem solve the problem based on the what he or she found in solving. I will explain the last one because it is the steps one and two. One experiments, looks for a more recently developed. However, the three are simpler problem, and then conjectures, guesses, forms fundamentally the same. a tentative hypothesis, and assumes a solution. The following is an example of how the heuristic is The fourth step is Solve the Problem. Once the applied to a problem. method has been selected the student applies it to the problem. In this instance, one could simply continue Problem: Twelve couples have been invited to a the chart in step three until one reached 24 guests. party. The couples will be seated at a series of small square tables, placed end to end so as to form 8 Problem Solving Step 2: Explore. The final step, Review and Extend, is where the student verifies his or her answer and looks for Draw a diagram to represent the problem. variations in the method of solving the problem; e.g., n " 2 t = , where represents the number of tables. Or we 2 could ask for a formula to determine how many guests we can seat given the number of tables. For example, n = 2t + 2 or n = 2(t + 1). ! Problem Solving Connects Theory and Practice A perennial charge brought against education is that it fails to prepare students for the real world. It teaches theory but not practice. Problem solving connects theory and practice. In a sense this element is the same as the definitions of problem solving and transfer, only it specifically relates to applying abstract Make a chart, record the data, and look for patterns. school knowledge to concrete real world experiences (Krulik & Rudnick, 1980). Number of 1 2 3 4 . Problem Solving Teaches Creativity tables Real world situations require creativity. However, Number of it has often been claimed that traditional classrooms or 4 6 8 10 . guests teaching approaches do not focus on developing the creative faculty of students. Advocates of problem Pattern: As we add a table, the number of guests that solving, by contrast, claim that problem solving can be seated increases by 2. develops the students’ creative capacities (Frederiksen, 1984; Slavin, 1997). Successful Problem Solvers Have a Complete and Step 3: Select a Strategy. Organized Knowledge Base A knowledge base consists of all of the specific Number of knowledge a student has that he or she can use to solve 1 2 3 4 5 6 7 tables a given problem. For example, in order to solve algebraic problems, one not only needs to know Number of 4 6 8 10 12 14 16 information about numbers and how to add, subtract, guests multiply, and divide, but one must also possess the knowledge that goes beyond basic arithmetic. A Form a tentative hypothesis. Since the pattern seems to knowledge base is what must accompany the teaching be holding true for 16 guests, we can continue to add 1 of a heuristic for successful problem solving to occur.