In many countries, students have come to experience school mathematics as cold, hard, and unapproachable, a mysterious activity quite distinct from their everyday lives and reserved for people with special talents. After repeated failure in school mathematics and estrangement from the discipline, students often assume a view similar to what a student once expressed to the first author: ‘‘mathematics is something that you do, not something that you understand.’’
Similar views emerge from other students’ school experiences. A considerable proportion of such students become excluded from meaningful participation in academic mathematics. This is particularly true of students who are members of socially excluded sectors of their societies, lacking in privileged economic or social capital, to use Bourdieu’s (1986) categories. As Zevenbergen (2000) notes, ‘‘aspects of pedagogy and curriculum. . .can exclude students . . . [since] patterns of language, work, and power are implicated in the construction of mathematics, it becomes [important] to understand how we can change our practices in order
that they become more accessible and equitable for our students’’ (p. 219). To contribute toward making mathematics more accessible and equitable or less exclusionary and, thereby, more inclusive, this chapter posits the use of mathematical tasks that have particular characteristics. Even further, in addition to the social function of inclusion, such tasks have important psychological and cognitive consequences. This will explicate how engaging students in solving challenging mathematical problems can lead them to construct effective and important problem-solving schemas. The pedagogical goal is to engage students with different mathematical backgrounds in different settings so that they can further develop their mathematical ideas, reasoning and problem solving strategies, as well as enjoy being mathematical problem solvers.
A paramount goal of mathematics education is to promote among learners effective problem solving. Mathematics teaching strives to enhance students’ ability to solve individually and collaboratively problems that they have not previously encountered. To discuss the role of schemas in achieving this goal, we first discuss our understanding of problem solving and then that of schemas. The meaning of mathematical ‘‘problem solving’’ is neither unique nor universal. Its meaning depends on ontological and epistemological stances, on philosophical views of mathematics and mathematics education. For the purposes of this chapter, we subscribe to how Mayer and Wittrock (1996) define problem solving and its psychological characteristics:
Problem solving is cognitive processing directed at achieving a goal when no solution method is obvious to the problem solver (Mayer 1992). According to this definition, problem solving has four main characteristics. First, problem solving is cognitive—it occurs within the problem solver’s cognitive system and can be inferred indirectly from changes in the problem solver’s behavior. Second, problem solving is a process—it involves representing and manipulating knowledge in the problem solver’s cognitive system. Third, problem solving is directed—the problem solver’s thoughts are motivated by goals. Fourth, problem solving is personal—the individual knowledge and skills of the problem solver help determine the difficulty or ease with which obstacles to solutions can be overcome. (p. 47)
Coupled with these cognitive and other psychological characteristics, problem solving also has social and cultural features. Some features include what an individual or cultural group considers to be a mathematical problem (D’Ambrosio 2001, Powell and Frankenstein 1997), the context in which an individual may prefer to engage in mathematical problem solving, and how problem solvers understand a given problem as well as what they consider to be adequate responses (Lakatos 1976). In instructional settings, students’ problem solving activities are strongly influenced by teachers’ representational strategies, which are constrained by cultural and social factors (Cai and Lester 2005). An attribute that distinguishes expert mathematical problem solvers from less successful problem solvers is that experts have and use schemas—or abstract knowledge about the underlying, similar mathematical structure of common classes of problems—to form solutions to problems. In general terms a problem schema, as Hayes (1989) characterizes it ‘‘is a package of information
about the properties of a particular problem type’’ (p. 11).
The role of schemas in mathematical problem solving has been investigated by psychologists and cognitive scientists, as well as mathematics education researchers. Below is a summary of this research (Schoenfeld 1992): Experts can categorize problems into types based on their underlying mathematical structure, sometimes after reading only the first few words of the problem (Hinsley et al. 1977, Schoenfeld and Hermann 1982). Schemas suggest to experts what aspects of the problem are likely to be important. This allows experts to focus on important aspects of the problem while they are reading it, and to form sub-goals of what quantities need to be found during the problem-solving process (Chi et al. 1981, Hinsley et al. 1977).
Schemas are often equipped with techniques (e.g. procedures, equations) that are useful for formulating solutions to classes of problems (Weber 2001). To illustrate the notion and utility of schemas for problem solving, consider the following problem: Two men start at the same spot. The first man walks 10 miles north and 4 miles east. The second man walks 4 miles west and 4 miles north. How far apart are the two men? In discussing a similar problem, Hayes (1989) notes that when experienced mathematical problem solvers read this statement, it will evoke a ‘‘right triangle schema’’ (problems in which individuals walk in parallel or orthogonal directions to one another can often be solved by constructing an appropriate right triangle and finding the lengths of all of its sides). A technique for solving such problems involves framing the problems in
terms of finding the missing length of a right triangle, setting as a sub-goal finding the lengths of two of the sides of the triangle, and using the Pythagorean theorem to deduce the length of the unknown side.
In the mathematics and mathematics education literature, no universally accepted definition exists for the mathematical terms ‘‘task’’, ‘‘problem’’, or ‘‘exercise’’ and for the appellation ‘‘challenging’’ when describing a mathematical task or problem. In this chapter, as a starting point, we use Hayes’s (1989) sense of what a problem is: ‘‘Whenever there is a gap between where you are now [an initial situation] and where you want to be [an adequate response], and you don’t know how to find a way [a sequence of actions] to cross that gap, you have a problem’’ (p. xii).
In other words, ‘‘a problem occurs when a problem solver wants to transform a problem situation from the given state into the goal state but lacks an obvious method for accomplishing the transformation’’ (Mayer and Wittrock 1996, p. 47). For something that may or may not be a problem, to talk about it, we use the generic term ‘‘task’’. To complete a mathematical task, a problem solver needs to apply a sequence of mathematical actions to the initial situation to arrive at an adequate response. Even before applying mathematical actions, the problem solver will have to represent the gap virtually or physically—which is to say, to understand the nature of the problem (Hayes 1989).
The definition provided by Hayes as well as that provided by Mayer and Wittrock suggest grounds to distinguish between two closely related tasks: exercises and problems. Distinguishing these terms cannot be done without consideration of the problem solver. A mathematical task is an exercise to an individual learner if, due to the individual’s experience, the learner knows what sequence of mathematical actions should be applied to achieve the task (such as knowing what equation into which to insert givens). In contrast, solving a mathematical problem involves understanding the task, formulating an appropriate sequence of actions or strategy, applying the strategy to produce a solution, and then reflecting on the solution to ensure that it produced an appropriate response.
A mathematical problem may present several plausible actions from which to choose (Schoenfeld 1992, Weber 2005). We call a mathematical problem challenging if the individual is not aware of procedural or algorithmic tools that are critical for solving the problem and, therefore, will have to build or otherwise invent a subset of mathematical actions to solve the problem. For instance, most proofs in high school geometry are problems, and sometimes difficult ones, since the prover needs to decide which theorems and rules of inference to apply from many alternatives (Weber 2001). However, proofs that require the prover to create new mathematical concepts or derive novel theorems would make these proofs challenging problems. To solve challenging mathematics problems, learners build what are for them new mathematical ideas and go beyond their previous knowledge.
From a cognitive perspective, through meaningful engagement over time with problems within a strand of mathematics, students build effective and important problem-solving schemas. They develop insights into the mathematical structure of related problems and this knowledge becomes schematized. Moreover, students need to develop flexible schemas since rigid ones may inadvertently cause a problem solver to choose a non-optimal or inadequate solution method or approach. Resolving inconsistent and contradictory propositions or paradoxes can support the development of flexible schemas. Most research on schema construction has been done using traditional psychological paradigms, investigating how and (more often) to what extent individuals can construct and apply schemas in a short period of time. They believe that this change in perspective radically altered the nature of their findings. If their research participants were given straightforward problems, they would not have had the need to develop the useful representations for these problems that were critical for their schema construction.
If they were only given a short period of time to explore these problems, the schemas also would likely not have been constructed. In fact, students initially did not see the deep connections between the various problems on which they worked. Hence, looking at the processes that individuals use to form and use schemas in relatively short periods of time is looking at only a subset of the processes used in this regard. The work of Weber et al. (2006) demonstrates that studying the way that students solve challenging strands of problems over longer periods of time provides a more comprehensive and useful look at how students can construct and use problem-solving schemas.
Challenging mathematics problems are suitable for a range of audiences and didactical situations. They are apt as interview questions for entrance into university mathematics programs to obtain windows into how students think mathematically; as investigations for teacher candidates to further develop their own mathematical understanding and to acquire insight into how learners learn mathematics; as supplements to or material integrated throughout a course; as a means to reinsert marginalized students into mathematics, providing them with a context with which to entertain their minds; and, by placing mathematical challenges in a university’s daily or weekly newspaper, as vehicles to popularize and create interest in mathematics among students studying the subject in a language other than their own. Challenging mathematics problems can be instruments to stimulate creativity, to encourage collaboration and to study learners’ untutored, emergent ideas. We have also shown that they are appropriate for secondary and post secondary students as well as for high-achieving and low-achieving learners.
From a didactical perspective, it is important that the problems require little specific background and generally can be attempted successfully by students of varying mathematical backgrounds. Economic and social capital need not be markers of who can participate in mathematics. In Fioriti and Gorgorio´ (2006), the authors indicate how it is possible to engage socially excluded youngsters with challenging mathematics problems so that they are reinserted into school settings and thereby widen their possible social and academic participation in their society. Clearly, there are a host of socioeconomic realities that need to be addressed to truly democratize academic and social participation. However, engaging students of diverse backgrounds in challenging mathematics problems contributes to this larger goal. Making mathematics less exclusionary and more inclusive depends on shifting from traditional pedagogies and procedural views of mathematics learning (Boaler and Greeno 2000). It requires reversing a common belief among teachers that higher-order thinking is not appropriate in the instruction of low achieving students (Zohar et al. 2001). If challenging mathematics problems were used in settings such as formal classrooms and other informal arenas,
learners might begin to recognize mathematics as accessible and attractive (cf. Zohar and Dori 2003). They would have opportunities to build mathematical ideas and reasoning over time, develop flexible schemas and inventive problem solving approaches, and become socialized into thinking mathematically. As Resnick (1988) suggests: ‘‘If we want students to treat mathematics as an ill-structured discipline [that is, one that invites more than one rigidly defined interpretation of a task]—making sense of it, arguing about it, and creating it, rather than merely doing it according to prescribed rules—we will have to socialize them as much as to instruct them. This means that we cannot expect any brief or encapsulated program on problem solving to do the job. Instead, we must seek the kind of long-term engagement in mathematical thinking that the concept of socialization implies’’. (p. 58)
If mathematics educators and teachers adopt a long-term perspective on the development of problem-solving schemas, then a paramount goal of mathematics education—to further learners’ effective problem solving—would be more achievable.
No comments:
Post a Comment