# Mathematics Learning

## Complex Problem Solving

In April 2000, the National Council of Teachers of Mathematics (NCTM) published Principles and Standards for School Mathematics, a document intended to serve as "a resource and a guide for all who make decisions that affect the mathematics education of students in prekindergarten through grade 12," and that represented the best understandings regarding mathematical thinking, learning, and problem solving of the mathematics education community at the dawn of the twenty-first century. It also reflected a radically different view from the perspective that dominated through much of the twentieth century.

Principles and Standards specifies five mathematical content domains as core aspects of the curriculum: number and operations, algebra, geometry, measurement, and data analysis and probability. These content areas reflect an evolution of the curriculum over the course of the twentieth century. The first four were present, to various degrees, in 1900. Almost all children studied number and measurement, which comprised the bulk of the elementary curriculum in 1900. Algebra and geometry were mainstays of the secondary curriculum, which was studied only by the elite; approximately 10 percent of the nation's fourteen-year-olds attended high school. Data analysis and probability were nowhere to be seen. Over the course of the twentieth century, the democratization of American education resulted in increasing numbers of students attending, and graduating from, high school.

Curriculum content evolved slowly, with once-advanced topics such as algebra and geometry becoming required of increasing numbers of students. The study of statistics and probability entered the curriculum in the 1980s, and by 2000 it was a central component of most mathematics curricula. This reflected an emphasis on the study of school mathematics for "real world" applications, as well as in preparation for mathematics at the collegiate level.

While content changes can thus be seen as evolutionary, perspectives on mathematical processes must be seen as representing a much more fundamental shift in perspective and curricular goals. Given equal weight with the five content areas in Principles and Standards are five process standards: problem solving, reasoning and proof, communication, connections, and representation. All of these are deeply intertwined, representing an integrated view of complex mathematical thinking and problem solving. Problem solving might be viewed as a "first among equals," in the sense that the ultimate goal of mathematics instruction can be seen as enabling students to confront and solve problems–not only problems that they have been taught to solve, but unfamiliar problems as well. However, as will be elaborated below, the ability to solve problems and to use one's mathematical knowledge effectively depends not only on content knowledge, but also on the process standards listed above.

Solving difficult problems has always been the concern of professional mathematicians. Early in the twentieth century, problem books were viewed as ways for advanced students to develop their mathematical understandings. Perhaps the best exemplar is George Pólya and Gabor Szegö's Problems and Theorems in Analysis, first published in 1924. The book offered a graded series of exercises. Readers who managed to solve all the problems would have learned a significant amount of mathematical content, and (although implicitly) a number of problem-solving strategies.

The idea that one could isolate and teach strategies for problem solving remained tacit until the publication of Pólya's How to Solve It in 1945. Pólya introduced the notion of heuristic strategy–a strategy that, while not guaranteed to work, might help one to better understand or solve a problem. Pólya illustrated the use of certain strategies, such as drawing diagrams; "working backwards" from the goal one wants to achieve; and decomposing a problem into parts, solving the parts, and recombining them to obtain a solution to the original problem. Pólya's ideas resonated within the mathematical community, but they were exceptionally difficult to implement in practice. For example, while it was clear that one should draw diagrams, it was not at all clear which diagrams should be drawn, or what properties those diagrams should have. A problem could be decomposed in many ways, but it was not certain which ways would turn out to be productive.

Means of addressing such issues became available in the 1970s and 1980s, as the field of artificial intelligence (AI) flourished. Researchers in AI wrote computer programs to solve problems, basing the programs on fine-grained observations of human problem solvers. Allen Newell and Herbert Simon's classic 1972 book Human Problem Solving showed how one could abstract regularities in the behavior of people playing chess or solving problems in symbolic logic–and codify that regularity in computer programs. Their work suggested that one might do the same for much more complex human problem-solving strategies, if one attended to fine matters of detail. Alan Schoenfeld's 1985 book Mathematical Problem Solving (and his subsequent work) showed that such work could be done successfully. Schoenfeld provided evidence that Pólya's heuristic strategies were too broadly defined to be teachable, but that when one specified them more narrowly, students could learn to use them. His book provided evidence that students could indeed learn to use problem-solving strategies–and use them to solve problems unlike the ones they had been taught to solve. It also indicated, however, along with other contemporary research, that problem solving involved more than the mastery of relevant knowledge and powerful problem-solving strategies.

One issue, which came to be known as metacognition or self-regulation, concerns the effectiveness with which problem solvers use the resources (including knowledge and time) potentially at their disposal. Research indicated that students often fail to solve problems that they might have solved because they waste a great deal of time and effort pursuing inappropriate directions. Schoenfeld's work indicated that students could learn to reflect on the state of their problem solving and become more effective at curtailing inappropriate pursuits. This, however, was still only one component of complex mathematical behavior.

Research at a variety of grade levels indicated that much student behavior in mathematics was shaped by students' beliefs about the mathematical enterprise. For example, having been assigned literally thousands of "problems" that could be solved in a few minutes each, students tended to believe that all mathematical problems could be solved in just a few minutes. Moreover, they believed that if they failed to solve a problem in short order, it was because they didn't understand the relevant method. This led them to give up working on problems that might well have yielded to further efforts. As Magdalene Lampert observed, "Commonly, mathematics is associated with certainty; knowing it, with being able to get the right answer, quickly. These cultural assumptions are shaped by school experience, in which doing mathematics means following the rules laid down by the teacher; knowing mathematics means remembering and applying the correct rule when the teacher asks a question; and mathematical truth is determined when the answer is ratified by the teacher. Beliefs about how to do mathematics and what it means to know it in school are acquired through years of watching, listening, and practicing"(p. 31).

Lampert argued that the very practices of schooling resulted in the development of inappropriate beliefs about the nature of mathematics, and that those beliefs resulted in students' poor mathematical performance. Given the link between students' experiences and their beliefs, the necessary remedy was to revise instructional practices–to create instructional contexts in which students could engage in mathematics as an act of sense-making, and thereby develop a more appropriate set of knowledge, beliefs, and understandings.

Research on mathematical thinking and problem solving conducted in the 1970s and 1980s established the underpinnings for the first major "reform" document, Curriculum and Evaluation Standards for School Mathematics (1989), also published by NCTM. The climate was right for change, for the nation was concerned about its students' mathematical performance. Reports such as A Nation at Risk (1983) had documented American students' weak mathematical performance in comparison to that of students from other nations, and there was a sense of national crisis regarding the nation's mathematical and scientific capacities. In the early 1990s the U. S. National Science Foundation began to support the development of curricular materials consistent with emerging research on mathematical thinking and learning. The first wave of curricula developed along these lines began to be adopted in the late 1990s.

Many of the new curricula call for students to work on complex problems over extended periods of time. In some cases, important mathematical ideas are introduced and developed through working on problems, rather than taught first and "applied" later. Either way, the fundamental idea is that students will need to have opportunities to develop both the content and process understandings described in Principles and Standards. As indicated above, this calls for changes in classroom practices. The best way for students to develop productive mathematical dispositions and knowledge is for them to be supported, in the classroom, in activities that involve meaningful mathematical problem solving. Given a complex problem, students can work together, under the guidance of a knowledgeable teacher, to begin to understand the task and the resources necessary to solve it. This can help them develop productive mathematical dispositions (i.e., the understanding that complex problems will yield to sustained, systematic efforts) and analytic skills. Complex problems may span mathematical areas or be drawn from real-world applications, thus helping students make mathematical connections.

Understanding and working through such problems calls for learning various representational tools–the symbolic and pictorial languages of mathematics. Tasks that call for explaining one's reasoning (i.e., asking students to make a choice between two options and to write a memo that justifies their choice on mathematical grounds) can help students develop their skills at mathematical argument. They also reinforce the idea that obtaining an answer is not enough; one must also be able to convince others of its correctness. Teachers can help students understand that there are standards for communicating mathematical ideas. The arguments students present should be coherent and logical, and ultimately, as students develop, formalizable as mathematical proofs. In these ways, complex problem solving becomes a curricular vehicle as well as a curricular goal.

## BIBLIOGRAPHY

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LAMPERT, MAGDALENE. 1990. "When the Problem Is Not the Question and the Solution Is Not the Answer: Mathematical Knowing and Teaching." American Educational Research Journal 17:29–64.

LESTER, FRANK. 1994. "Musings about Mathematical Problem-Solving Research: 1970–1994." Journal for Research in Mathematics Education 25 (6):660–675.

NATIONAL COMMISSION ON EXCELLENCE IN EDUCATION. 1983. A Nation at Risk: The Imperative for Educational Reform. Washington, DC: U.S. Government Printing Office.

NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS. 1989. Curriculum and Evaluation Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.

NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS. 2000. Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.

NEWELL, ALLEN, and SIMON, HERBERT A. 1972. Human Problem Solving. Englewood Cliffs, NJ: Prentice-Hall.

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SCHOENFELD, ALAN H. 1985. Mathematical Problem Solving. Orlando, FL: Academic Press.

SCHOENFELD, ALAN H. 1992. "Learning to Think Mathematically: Problem Solving, Metacognition, and Sense-Making in Mathematics." In Handbook of Research on Mathematics Teaching and Learning, ed. Douglas A. Grouws. New York: Macmillan.

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ALAN H. SCHOENFELD