Consider the word problems presented in Table 8.1. Some people are able to produce solutions to these problems whereas others make errors, get frustrated, and fail to generate a correct answer. What do successful mathematical problem solvers know that less successful mathematical problem solvers do not know? This seemingly straightforward question motivates this chapter.
A review of research on mathematical problem solving supports the conclusion that proficiency in solving mathematical problems depends on the domain knowledge of the problem solver (Kilpatrick, Swafford, & Findell, 2001). In this chapter, I examine the research evidence concerning five kinds of knowledge required for mathematical problem solving: factual knowledge, conceptual knowledge, procedural knowledge, strategic knowledge, and metacognitive knowledge (Mayer, 2008, 2011, 2013).
Table 8.2 provides definitions and examples of each of the five kinds of knowledge relevant to mathematical problem solving. Factual knowledge refers to knowledge of facts such as knowing that there are 100 cents in a dollar. Conceptual knowledge refers to knowledge of concepts such as knowing that a dollar is a monetary unit, and knowledge of categories such as knowing that a given problem is based on the structure, (total cost) = (unit cost) × (number of units). Strategic knowledge refers to knowledge of strategies such as knowing how to break a problem into parts. Procedural knowledge refers to knowldege of procedures such as knowing how to add two decimal numbers. Metacognitive knowledge refers to knowledge of about how to assess and adjust one's thinking, including attitudes and beliefs concerning one's competence in solving mathematics word problems. In this chapter, after reviewing definitions of key terms, I examine research evidence concerning the role of each of these kinds of knowledge in supporting mathematical problem solving.
An important first step is to define key terms such as problem, problem solving, and mathematical problem solving.
What Is a Problem?
In his classic monograph entitled “On-Problem Solving,” Duncker (1945, p. 1) eloquently wrote that a problem arises when a problem solver “has a goal but does not know how this goal is to be reached.” More recently, we have expressed this idea by saying “a problem occurs when a problem solver wants to transform a problem situation from the given state to the goal state, but lacks an obvious method for accomplishing the transformation” (Mayer & Wittrock, 2006, p. 288).