Richard E. Mayer
Consider the word problems presented in Table8.1. Some people are able produce solutions to these problems, whereas others make errors, get frus- trated, and fail to generate a correct answer. What do successful mathe- matical 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 concern- ing five kinds of knowledge required for mathematical problem solving:
(1) factual knowledge, (2) conceptual knowledge, (3) procedural knowl- edge, (4) strategic knowledge, and (5) metacognitive knowledge.
Table8.2provides 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 are100cents 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 knowledge of procedures such as knowing how to add two decimal numbers. Metacognitive knowledge refers to knowledge of about how one thinks, 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.
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table 8.1. Some Mathematical Word Problems
At Lucky, butter costs65cents per stick. This is2cents less per stick than butter at Vons. If you need to buy4sticks of butter, how much will you pay at Vons?
(Hegarty, Mayer, & Monk,1995) Answer: $2.68
Typical wrong answer: $2.52
Flying east between two cities that are300miles apart, a plane’s speed is 150mph. On the return trip, it flies at300mph. Find the average speed for the round trip.
(Reed,1984).
Answer:200mph
Typical wrong answer:225mph
An army bus holds36soldiers. If1128soldiers are being bussed to their training site, how many buses are needed?
(Carpenter, Linquist, Mathews, & Silver,1983) Answer:32
Typical wrong answer:31remainder12
A car in Philadelphia starts towards New York at40miles per hour. Fifteen minutes later, a car in New York starts toward Philadelphia,90miles away, at55miles per hour. Which car is nearest Philadelphia when they meet?
(Davidson,1995)
Answer: They are equally close to Philadelphia when they meet.
Typical wrong answer: Attempt to use distance–rate–time formula and carry out computations.
definitions
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 entitledOn-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, I have expressed this idea by saying a problem occurs when “a situation is in a given state, the problem solver wants the situation to be in a goal state, and there is no obvious way of transforming it from the given state to the goal state” (Mayer,1990, p.284).
Problems may be characterized as well defined or ill defined. A well- defined problem occurs when the given state, goal state, and allowable operators are clearly specified. For example, most mathematical word problems are well defined because the given state and goal state are described in the presented problem, and the allowable operators include the rules of arithmetic and algebra. An ill-defined problem occurs when the given state, goal state, and/or allowable operators are not clearly specified.
table 8.2.Five Kinds of Knowledge Required for Mathematical Problem Solving
Name Definition Example
Factual knowledge Knowledge of facts “How much does it cost to buy 5pencils if each one costs 30cents?” requires knowing that $1.00is the same as 100cents
Conceptual knowledge
Knowledge of concepts
“Cents” is a monetary unit Knowledge of
categories
“How much does it cost to buy 5pencils if each one costs 30cents?” is based on the schema: (total cost)=(unit cost)×(number of units) Strategic knowledge Knowledge of
strategies
“John has3marbles. Pete has 2more marbles than John.
How many marbles does Pete have?” requires a solution plan such as “add 3to2”
Procedural knowledge Knowledge of procedures
The rules of arithmetic (e.g., 3+2=5) and the rules of algebra (e.g., ifX=3, then to find2+X= , add2and3) Metacognitive
knowledge
Attitudes and beliefs “I can solve this problem if I work hard on it.”
An example of an ill-defined problem is solving the energy crisis, because the allowable operators are not clear and even the goal state is not clear.
Problems can be characterized as routine or nonroutine. A routine prob- lem occurs when a problem solver knows a solution procedure, such as when a typical adult is given the problem233×567= . This is a routine problem for most adults because the problem solver knows how to carry out the procedure for long multiplication. Technically, routine problems do not meet the criteria in the definition of a problem because there is an obvious way of transforming it from the given state to the goal state. A nonroutine problem occurs when a problem solver has not had enough experience to know a solution procedure. For a6-year-old who has not mastered addition facts, for example, the problem3+5= may be a non- routine problem that is solved by thinking, “I can take one from the five and give it to the three, so I have four plus four; the answer is eight.” Whether a problem is routine or nonroutine depends on the problem solver’s prior experience, so the definition depends on who the problem solver is.
What is problem solving? Problem solving is “cognitive processing directed at transforming a given situation into a goal situation when no obvious method of solution is available to the problem solver” (Mayer, 1990, p.284). According to this definition, problem solving is (a) cognitive, that is, it occurs in the problem solver’s cognitive system and can only be inferred indirectly from behavior, (b) a process, that is, it involves apply- ing operations that cause changes in internal mental representations, (c) directed, that is, it is intended to achieve a goal, and (d) personal, that is, it depends on the existing knowledge of the problem solver. This definition is broad enough to include a wide range of cognitive activities ranging from solving mathematical word problems to writing essays to testing scientific hypotheses.
What is mathematical problem solving?Mathematical problem solving is problem solving that involves mathematical content, such as the problems in Table8.1. Mathematical problem solving generally involves well-defined operators such as the rules of arithmetic and algebra. Mathematical prob- lem solving can be broken into two main phases –problem representationand problem solution(Mayer,1992). Problem representation involves building a mental representation of the problem and includestranslating(convert- ing each sentence into a mental representation) andintegrating(integrating the information into a coherent representation of the problem, sometimes called asituation model). Problem solution involves producing an answer and includes planning(devising a solution plan) andexecuting (carrying out the plan).
Although in the United States mathematical instruction often empha- sizes the executing phase of problem solving (Mayer, Sims, & Tajika,1995), the major difficulties in problem solving tend to involve the other phases, as can be seen in the typical wrong answers reported in Table8.1. In the first problem, for example, the typical wrong answer is to subtract2cents from65cents to get63cents and multiple the result by4, yielding $2.52.
Although the computations are correct, this approach is based on an incor- rect understanding of the situation described in the problem. In the second problem, the typical wrong answer is to add the two speeds together (150 and300) and divide by2, yielding225. Again, although the computations are correct, this approach is based on an incorrect plan for solving the problem. In the third problem, the typical wrong answer is based on cor- rect computations yielding “31remainder12,” coupled with a failure to correctly interpret the implications of the numerical answer for the ques- tion asked in the problem. Finally, in the last problem in Table 8.1, the typical incorrect answer involves correctly carrying out a number of com- putations based on the formula distance=rate ×time. Problem solvers may fail to take the time to understand the situation described in the prob- lem, namely that when the two vehicles meet they are in the same spot so they are equally distant from the city they started from. Thus, in each case,
successful problem solving depends on more than being able to carry out computational procedures.
The types of knowledge listed in Table8.2support different problem- solving phases: factual knowledge is required for translating, conceptual knowledge supports integrating, strategic knowledge supports planning, and procedural knowledge supports executing. In addition, metacognitive knowledge supports the process ofmonitoring– assessing and adjusting one’s approach to each of the foregoing four processes during problem solving. In the remainder of this chapter, I explore exemplary research concerning how mathematical problem solving is related to each of the five kinds of knowledge listed in Table8.2.
role of factual knowledge in mathematical problem solving
The first kind of knowledge needed for solving mathematical word prob- lems is factual knowledge. For example, some students may have difficulty understanding relational statements involving phrases such as “two less than” or “five more than.” In a relational statement, two variables are com- pared such as, “Butter at Lucky costs two cents less per stick than butter at Vons,” or “Tim has five more marbles than Paul.”
First, consider a situation in which elementary school students are asked to listen to a problem (such as, “Paul has three marbles. Tim has five more marbles than Paul. How many marbles does Tim have?”) and then repeat it back. In a classic study by Riley, Greeno, and Heller (1982), a common error involved misremembering the relational statement in the problem such as,
“Paul has three marbles. Tim has five marbles. How many marbles does Tim have?” This type of error suggests that children have difficulty in mentally representing relational statements.
Second, consider a situation in which college students are asked to write an equation to represent statements such as, “There are six times as many students as professors at this university.” Soloway, Lochhead, and Clement (1982) found that a common incorrect answer was, “6S=P,” again suggesting that people have difficulty in mentally representing relational statements.
As a third example, consider a situation in which college students read a list of eight word problems and then recall them. The most common errors involved misremembering relational statements (Mayer,1982). For exam- ple, one problem contained the statement, “the steamer’s engine drives in still water at 12 miles per hour more than the rate of the current,”
which was remembered as “its engines push the boat at 12 miles per hour in still water.” Overall, these three pieces of evidence support the idea that people may lack the knowledge needed to understand relational statements.
Is problem-solving performance related to this kind of knowledge? To answer this question, Hegarty, Mayer, and Monk (1995) asked students to solve12word problems and later gave them a memory test for the prob- lems. Good problem solvers (i.e., students who produced correct answers on the word problems) tended to remember the relations correctly even if they forgot the exact wording. For example, if a problem said, “Gas at ARCO is5cents less than gas at Chevron,” good problem solvers would be more likely than poor problem solvers to recognize that “Gas at Chevron is5cents more than gas at Arco.” Poor problem solvers (i.e., students who made many errors on the12-item problem-solving test) tended to remem- ber the relations incorrectly even if they remembered some of the exact words. For example, they were far more likely than the good problem solvers to say they recognized factually incorrect statements such as, “Gas at Chevron costs5cents less per gallon than gas at ARCO.” Overall, these results suggest that the learner’s knowledge about relational statements correlates with being able to solve mathematical word problems.
In a more direct test of the role of knowledge of relational statements in mathematical problem solving, Lewis (1989) provided direct instruction in how to represent relational statements as part of a number line diagram.
Students who received the training showed greater improvements in their solving of mathematical word problems than did students who did not receive the training. This research is consistent with the idea that improving students’ knowledge of relational statements results in improvements in their mathematical problem-solving performance.
role of conceptual knowledge in mathematical problem solving
The second kind of knowledge needed for mathematical problem solv- ing is conceptual knowledge. For example, one important kind of con- ceptual knowledge is knowledge of problem types, such as having sepa- rate schemas in long-term memory for time–rate–distance problems, work problems, mixture problems, interest problems, and so on.
A first line of evidence that students develop problem schemas comes from a study by Hinsley, Hays, and Simon (1977) in which high school students were asked to sort a collection of problems into categories. The students were able to carry out this task with ease and with much agree- ment, yielding 18problem categories such as time–rate–distance, work, mixture, and interest. Similar results were obtained by Silver (1981) con- cerning sorting of arithmetic word problems by experienced learners and by Quilici and Mayer (1996) concerning sorting of statistics word problems by experienced learners.
Is knowledge of problem schemas related to problem solving per- formance? Silver (1981) found that successful problem solvers were more
likely to sort arithmetic word problems on the basis of their problem types than were unsuccessful problem solvers. Similarly, Quilici and Mayer (1996) found that college students who were experienced in solving statis- tics problems were much better at sorting statistics word problems based on the underlying statistical test than were students who lacked expe- rience in solving statistics problems. Across both studies, students who lacked expertise tended to sort problems based on their surface fea- tures (such as putting all problems about rainfall in the same category), whereas students who were more expert in problem solving tended to sort problems based on their semantic features (such as the solution method required).
Does problem-solving performance improve because of training in prob- lem types? Quilici and Mayer (2002) taught some students how to catego- rize statistics word problems based on whether the problems involved t test, χ2 or correlation. Students who received the training showed a strong improvement in their sorting performance, suggesting that knowl- edge of problem schemas could be taught. However, additional work is needed to determine whether students who receive schema training also show an improvement in problem-solving performance. Some encourag- ing evidence comes from a study in which students who took a course in statistics showed an improvement in their ability to sort statistics word problems into problem types (Quilici & Mayer, 1996). In a more recent study, students who received direct instruction and practice in how to sort insight word problems based on their underlying structure showed improvements on problem-solving performance as compared to students who did not receive the schema training (Dow & Mayer,2004).
Low and Over (1990; Low, 1989) developed a useful technique for schema training, in which students are given word problems and must indicate whether each problem has sufficient information, irrelevant information, or missing information. Students are given specific feed- back on each problem, aimed at helping them understand the situation described in the problem. Students who received schema training showed a strong increase in their performance on solving word problems as com- pared to students who received no schema training. Overall, this line of research shows that helping students acquire a specific kind of knowl- edge – knowledge of problem types – can improve their problem-solving performance.
role of strategic knowledge in mathematical problem solving
The third kind of knowledge involved in mathematical problem solving is strategic knowledge. For example, presenting worked-out examples is a common way to teach students the solution strategy for solving various
mathematics problems (Atkinson, Derry, Renkl, & Wortham,2000; Renkl, 2005). Worked-out examples show each solution step and explain each step.
There is consistent evidence that training with worked-out examples that clarify the problem-solving strategy can result in improved problem- solving performance. In general, students who study well-explained worked-out examples perform better on solving subsequent problems than do students who simply practice solving problems (Catrambone, 1995;
Reed,1999; Sweller & Cooper,1985).
Worked-out examples are particularly effective when learners are encouraged to engage in self-explanations, that is, in explaining the solu- tion steps to themselves (Chi, Bassok, Lewis, Reimann, & Glaser, 1989;
Renkl, 1997). For example, students who were given training in how to produce self-explanations for worked-out examples showed greater improvements in their mathematical problem-solving performance than did students who did not receive the self-explanation training (Renkl, Stark, Gruber, & Mandl, 1998). Similarly, students performed better on subsequent problem-solving tests when they were prompted to give self- explanations during worked-example training as compared to students who are not prompted to give self-explanations (Atkinson, Renkl, &
Merrill,2003).
Another way to increase the effectiveness of worked-out examples is to highlight the major subgoals in the solution strategy, either by visu- ally isolating them or giving them a label (Catrambone,1995,1996,1998).
Similarly, the major subgoals can be emphasized by presenting the solu- tion in step-by-step fashion via computer (Atkinson & Derry,2000; Renkl, 1997). In this way, the problem is broken into major modules that all fit into an overall solution strategy. Students who received worked-out exam- ples that emphasized the subgoals in these ways tended to perform better on subsequent problem-solving tests than did students who received the worked-out examples with all the steps presented simultaneously.
In general, research on worked-out examples confirms that it is possible to teach students specific solution strategies and that knowledge of these strategies improves students’ mathematical problem-solving performance.
role of procedural knowledge in mathematical problem solving
The fourth kind of knowledge required for mathematical problem solving is procedural knowledge. Although the previous sections have encouraged the idea that factual, conceptual, and strategic knowledge support mathe- matical problem solving, the most emphasis in mathematics instruction is on procedural knowledge (Mayer, Sims, & Tajika,1995; Stigler & Hiebert, 1999). For example, a common form of procedural knowledge is knowledge