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 Polya and Gabor Szego'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 Polya's How to Solve It in 1945. Polya introduced the notion of heuristic strategy–a strategy that, while not guaranteed to work, might help one to better understand or solve a problem. Polya 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. Polya'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 Polya'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.

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