Thirty Years of Problem Solving in Mathematics Education: Policy and Promise
Samuel Otten
Michigan State University
Division of Science & Mathematics Education
Comprehensive Examination
Fall 2010
Thirty Years of Problem Solving in Mathematics Education: Policy and Promise
In a famous address that influenced the direction of mathematical research in the 20th Century, David Hilbert (1900) claimed of mathematicians, “We feel within us the perpetual call: There is a problem. Seek its solution.” Echoing this sentiment, another wellrespected mathematician, Paul Halmos (1980), stated that “the mathematician’s main reason for existence is to solve problems” (p. 519). Others have made similar remarks (e.g., Polya, 1981; Schoenfeld, 1985), but the point is clear—mathematical activity has problem solving at its center. Thus it becomes important to clarify what is meant by the term “problem.” A problem is not an exercise of some mathematical skill or procedure that is already known. Polya (1957), for example, made sure to distinguish between authentic problems and “routine problems,” which he defined as a task that “can be solved either by substituting special data into a formerly solved general problem, or by following step by step, without any trace of originality, some wellworn conspicuous example” (p. 171). In contrast, a problem of the sort that involves problem solving is “a task for which the solution method is not known in advance” (NCTM, 2000, p. 52). As Schoenfeld (1992) characterized it, problems are “problematic” (p. 338).
Although there have been calls for school mathematics to represent the discipline of mathematics in “intellectually honest” ways (Bruner, 1960; Lampert, 1990), classroom practice in the United States has typically not been focused on authentic problem solving. Instead, teachers tend to present a mathematical idea or procedure, work several examples from the front of the room, and then assign exercises in which students practice whatever has just been presented (Smith, 1996; Stigler, Fernandez, & Yoshida, 1996; Stigler & Hiebert, 1999). Recently, the Common Core State Standards Initiative (2010) has called for a shift in school mathematics instruction toward the inclusion of problem solving so that students are able to “make sense of problems and persevere in solving them” (p. 6), which they identify as a key mathematical practice that should be developed in students at all levels.
The Common Core Standards, however, are simply the most recent chapter in the story of problem solving. Reform efforts in mathematics education since the time of Hilbert’s 1900 address have included some form of call for problem solving (Stanic & Kilpatrick, 1988; Wilson, 2003). This has been especially true over the past 30 years since the publication of the National Council of Teacher’s of Mathematics’ [NCTM] Agenda for Action (1980), whose first recommendation was that “problem solving must be the focus of school mathematics” (p. 2). In this paper, I trace this call for problem solving by reviewing key policy documents of the past three decades. But first, I review some of the research and philosophical positions on problem solving in mathematics education that have led to its prominence in reform efforts. I then conclude with a brief discussion of some of the factors that may be contributing to the difficulty of achieving rich problem solving experiences in school mathematics classrooms.
Justifications for Problem Solving
The purpose for this section is to provide background that can inform the policy review in the next section. In particular, various studies and philosophical positions are reviewed that help to provide a rationale for why problem solving has been identified as a desirable feature in mathematics education. (Note that, due to the scope of this paper, this review does not deal with the large body of work in cognitive psychology on problem solving processes (e.g., Newell & Simon, 1972; Sternberg & Frensch, 1991).)
Although problem solving research has somewhat faded away in the early 21st Century (Schoenfeld, 2007), it received a great deal of attention in the 1970s and 1980s. Initially, the research was primarily quantitative in nature and was designed to identify the characteristics of difficult problems, the characteristics of successful problem solvers, and to investigate methods of training students to use problem solving heuristics (Lester, 1994). Kantowski (1977) observed that prior problem solving research seemed to focus on the product of students’ problem solving, so she set to work, as did others at that time, to better understand the process of problem solving. What processes can be observed when middle school students solve nonroutine geometry problems? How do those processes change as the students’ problemsolving abilities are developed? To answer these questions, Kantowski implemented a pretestposttest design with an intermediate phase of instruction based on Polya’s (1957, 1981) heuristics. Eight subjects were included in the study and the tests included a thinkaloud protocol for the purpose of uncovering the thought processes to an extent which is unachievable via paperandpencil tasks alone. Kantowski (1977) found, first of all, that the use of heuristics increased from the pretest to the posttest, suggesting that direct instruction of heuristics does influence the frequency of their use. Second, her results revealed a correlation between the use of heuristics and student success in solving problems, suggesting that problem solving skills are related to measurable student outcomes.
However, the frameworks used by problem solving researchers at that time, such as Kantowski, would now be considered quite narrow (SantosTrigo, 2007). Scholars, especially Alan Schoenfeld, worked to remedy this throughout the 1980s. Schoenfeld (1985) developed a theoretical framework which could be used for investigating problem solving and, more broadly, mathematical thinking. This framework comprises four domains which he claims must necessarily be addressed by any work intending to investigate mathematical problemsolving performance. These are as follows:

Resources. (The relevant mathematical knowledge—intuition, facts, algorithms, understanding—possessed by the individual.)

Heuristics. (Strategies and techniques—drawing figures, introducing suitable notation, exploiting similar problems, reformulating—for making progress on unfamiliar problems.)

Control. (Global decisions—planning, monitoring, metacognitive acts—with regard to selecting and using resources and heuristics.)

Belief systems. (The mathematical worldview—conscious and unconscious—of an individual which may determine his or her behavior.)
Schoenfeld (1985) pointed out that it is often the case that resources are assumed to be the primary determinant of success in problem solving; that is, if the requisite mathematical content for a particular problem is known, then the problem should be solvable. Schoenfeld uncovered the inappropriateness of this assumption. For instance, mathematicians with powerful heuristics and control are likely to be able to solve problems even when their resources are severely lacking, and students who possess the necessary resources may be unable to solve problems because their belief systems do not allow the connections to be made (e.g., deductive results are not called upon in empirical settings). This new perspective, reiterated by Lester (1988), can be construed as an indictment of traditional instruction that does not incorporate problem solving because merely supplying students with mathematical resources in the form of pieces of content inadequately equips them to face new situations and think mathematically. Indeed, national assessments have shown this to be the case (e.g., Silver & Kenney, 1997).
If metacognition and student beliefs are key players in problem solving, then what role is left for the knowledge of the basic mathematical facts and procedures? Resnick (1988) supplied a partial answer to this question. She studied students in the fifth grade during word problem sessions and found that insecure mathematical knowledge blocked successful problem solving from occurring. Thus, mathematical knowledge resources may be a necessary, though not sufficient, condition for successful problem solving. Furthermore, she found that there is more subtlety to the teaching of heuristics than suggested by Polya’s actual writing. For example, Resnick’s research implied that Polyalike prompting questions were often too general to provide real help. The teacher prompt “Would it help to draw a diagram?” was not helpful when the student did not know what diagram to draw.
Though the teaching of heuristics is a subtle art, there is evidence to suggest that it can be successful. Charles and Lester (1984) conducted a statewide evaluation of a problemsolving focused instructional program known as Mathematical Problem Solving (MPS). This evaluation was based on standardized tests and classroom observations, and encompassed an entire school year. They found that students in the MPS experimental classrooms were better able to understand problems, plan solution strategies, and obtain correct results than students in control classrooms. Furthermore, both students and teachers in the MPS program exhibited improved attitudes toward mathematics as measured by a survey administered at the conclusion of the study.
More recently, several school mathematics curricula were developed that emphasized problems solving as a key feature of doing (and learning) mathematics (e.g., Coxford et al., 1997; Lappan, Fey, Friel, Fitzgerald, & Phillips, 1995). Having been used since the 1990s, evaluations of their effectiveness and outcomes were collected by Senk and Thompson (2003). Overall, the findings support the notion that curricula marked by a focus on problem solving, when compared to traditional materials, are correlated with improvements in students’ success with nontrivial tasks, interpretation of mathematical representations, and conceptual understanding, while simultaneously not harming their performance on basic skills. (Although problem solving was not the only shared feature of these curricula—they also tended to support explicit reasoning, realworld contexts, and studentcentered instruction, for example—it is fair to say that problem solving opportunities and development was a unifying theme.)
The above paragraphs show that engaging in problem solving can be beneficial to students’ learning of mathematics and that equipping students with facts and procedures is not sufficient to produce competent problem solvers. Such bodies of research, however, are not the only basis of justification for the inclusion of problem solving in the policy documents reviewed in the next section. There are also important and influential philosophical arguments that have been made in favor of problem solving in mathematics education.
One such argument is based on a conception of mathematics as a “dynamic, problemdriven” discipline wherein “patterns are generated and then distilled into knowledge” (Ernest, 1988, quoted in Thompson, 1992, p. 132). From this perspective, which Ernest termed the “problemsolving view,” mathematics is a process of posing, refining, and solving problems, rather than a collection of finished products. Thompson (1992) noted that mathematics educators often adhere to this view of mathematics and parlay it into calls for instruction that aligns with it.
John Dewey provides another philosophical impetus behind the push for problem solving. Although Dewey did not often refer explicitly to problem solving, his notion of reflective thinking has been viewed as reasonably synonymous (Stanic & Kilpatrick, 1988). Dewey (1933) felt that it was the ability to think reflectively that made one human and, to him, the attitudes of openmindedness, wholeheartedness, and responsibility were more important than procedural skills or knowledge of particular facts. Moreover, techniques and skill are only truly owned by students when they are learned with understanding. Dewey maintained problem solving as a means to learning important subject matter and simultaneously as an end in itself because of its contribution to human reflective thought.
A final rationale for problem solving in school mathematics is related to the connection between school curricula and students’ lives after school. As Lesh and Zawojewski (2007) have noted recently, “there is a growing recognition that a serious mismatch exists (and is growing) between the lowlevel skills emphasized in testdriven curriculum materials and the kind of understanding and abilities that are needed for success beyond school” (p. 764). Problem solving, on the other hand, provides the creativity, flexibility and metacognitive control of thought that do align with professional and postsecondary demands. In other words, by studying problem solving in mathematics, students can become better prepared for many aspects of their lives after school (e.g., trades, professional careers, knowledgeable citizenship).
Problem Solving in Landmark Documents
In this section, I highlight the role of problem solving in several key documents since 1980. The section is divided according to the source of the documents: the first subsection focuses on documents written from within the mathematics education community, whereas the second focuses on governmentbased reports. I recognize that these categories or not mutually exclusive since the mathematics education community is often writing for the purpose of influencing governmental policy and the government reports are often prepared with input and contributions from mathematics educators. The distinction is useful, nonetheless, as an organizational tool.
