How can technology support and promote thinking mathematically? In broad strokes, what appear to be the richest loci of potential cognitive and motivational support of technologies for math education? We can think of two sides to the educational practices of mathematics learning and ask how software can help. The first side is the personal side-will students choose to commit themselves to learning to think mathematically? Mathematics educators have to some extent neglected the concepts of motivation and purpose (e.g., McLeod, 1985): that neglect may help explain girls' and minorities' documented lack of interest in mathematics. What students learn also depends on the cognitive support given them as they learn the many problem-solving skills involved in thinking mathematically.
My perspective on the functions necessary for cognitive technologies thus has two vantage points. First, students are purposive, goal-directed learners, who have the will (on any given occasion or over time) to learn to think mathematically or not. Then once they have embarked on mathematical thinking, they may be aided by technologies in mathematical thinking. For simplicity of exposition. we thus divide function types between: (a) those which promote PURPOSE-engaging students to think mathematically; and (b) those which promote PROCESS-aiding them once they do so.
What lies at the heart of cognitive technologies that help make mathematical thinking purposeful and help commit the learner to the pursuit of understanding? Cognitive technologies that accomplish these goals are based on a participatory link between self and knowledge rather than an arbitrary one. This organic relationship was central to John Dewey's pedagogical writings and integral to Piaget's constructivism: We must build on the child's interests, desires and concerns, and more generally, on the child's world view. But what exactly does this mean? The key idea behind purpose functions is that they promote the formation of promathematics belief systems in students and thus ensure that students become mathematical thinkers who participate in and own what is learned. Students benefiting from purpose functions are no longer mere storage bins for or executors of "someone else's math." The implication is that technologies for mathematics education should be tools for promoting the student's self-perception as mathematical "agent," as subject or creator of mathematics (Papert, 1972, 1980). For example, Schoenfeld (1985a, 1985b) argues that the belief systems an individual holds can dramatically influence the very possibilities of mathematical education:
Students abstract a "mathematical world view" both from their experiences with mathematical objects in the real world and from their classroom experiences with mathematics. . . . These perspectives affect the ways that students behave when confronted with a mathematical problem. both influencing what they perceive to be important in the problem and what sets of ideas, or cognitive resources, they use (Schoenfeld, 1985, p. 157).
Although Schoenfeld's focus is broader than the point here, the student's mathematical world view includes the self: What am I in relation to this mathematical behavior I am producing? If students do not view themselves as mathematical thinkers, but only as recipients of the "inert" mathematical knowledge that others possess (Whitehead, 1929), then math education for thinking is going to be problematical-because the agent is missing. In the prototypical educational setting, we often erroneously presuppose that we have engaged the student's learning commitment. But the student rarely sees significance in the learning; someone else has made all the decisions about scope and sequence, about the lesson for the day. The learning is meant to deal not with the student's problem or a problematic situation the teacher has helped highlight, but with someone else's. And the knowledge used to solve the problem is someone else's as well, something that someone else might have found useful at some other time. Even that past utility is seldom conveyed: students are almost never told how measurement activities were essential to building projects or making clothes, or how numeration systems were necessary for trade (McLellan & Dewey, 1895).
According to Dewey's (1933, 1938) scheme for the logic of inquiry, the prototypical system of delivering mathematical facts leaves out the necessary 3rst step in problem solving: the identification of the problem, the
tension that arises between what the student already knows and what he or she needs to know that drives subsequent problem-solving processes. It is interesting that Polya (1957) also omits this first step; in other respects his phases of problem solving correspond to Dewey's seminal treatment: problem definition, plan creation, plan execution, plan evaluation, and reflection for generalization of what can be learned from this episode for the future (cf., Noddings, 1985). Perhaps the expert mathematician takes this first step for granted: For who could not notice mathematical problems? The world is full of them! But for the child meeting the formal systems that mathematics offers and the historically accrued problem-solving contexts for which mathematics has been found useful, the first step is a giant one, requiring support. Purpose functions that help the student become a thinking subject can be incorporated into mathematically oriented educational technologies in many ways. Here, we go beyond Dewey to suggest other component features of mathematical agency:
1. Ownership. Agency is more likely when the student has primary ownership of the problem for which the knowledge is needed (or secondary ownership, i.e., identification with the actor in the problem setting, in
an "as if it were me" simulation). A central pedagogical concern is to find ways to help people "own" their own thoughts and the problems through which they will learn. Kaput (1985) and Papert (1980) have provided suggestive examples from software mathematics discovery environments where the "epistemological context" is redefined: Authority for what is known must rest on proof by either the student or the teacher; it must not rest exclusively with the teacher and the text. Students can offer new problems to be solved, and they can also create new knowledge.
2. Self-worth. It is hard for students to be mathematical agents if they view opportunities for thinking as occasions for failure and diminished self-worth. Student performance depends partly on self-concept and selfevaluation (Harter, 1985). Research on the motivation to achieve by Dweck and colleagues (e.g., Dweck & Elliot, 1983) indicates that students tend to hold one of two dominant views of intelligence, and that the one held by each particular student helps determine his or her goals. On one hand, if the child views intelligence as an entity, a given quantity of something that one either has or has not, then the learning events
arranged at school become opportunities for success or occasions for failure; if the child looks bad. his or her self-concept is negatively affected. On the other hand, if the child views intelligence as "incremental," then these same learning events are viewed as opportunities for acquiring new understanding. Although little is known about the ontogenesis of the detrimental entity view, it is apparent that this belief can hinder the possibility of mathematical agency and that software or thinking practices that foster an incremental world view should be sought.
3. Knowledge for action. A third condition for promoting mathematical agency is either that the mathematical knowledge and skills to be acquired have an impact on students' own lives or future careers or that knowledge actually facilitates their solution of real-world problems. New knowledge, whether problem-solving skills or new mathematical ideas, should EMPOWER children to understand or do something better than they could prior to its acquisition. That this condition is important is clear from research on the transferability of instructed thinking skills such as memory strategies (e.g., Campione & Brown, 1978). This research indicates easier transfer of the new skills to other problem settings if one simply explains the benefits of the skill to be learned, that is. that more material will be remembered if one learns this strategy. Technologies for mathematical thinking that incorporate these Purpose functions should make clear the impact of the new knowledge on the students' lives.
To Summarize: In characterizing the general category of Purpose functions for cognitive technologies. I have focused on the importance of linking the child-as-agent with the knowledge to be acquired instead of on
the alleged motivational value (e.g.. Lepper. 1985; Malone, 1981) of mathematical educational technologies. 1 have done so because it is inappropriate to think about technologies as artifacts that mechanistically induce motivation. That perspective has led to the extrinsic motivation characterizing most current learning-game software: bells and whistles are added that serve no function in the student's mathematical thinking. Furthermore, these extrinsic motivational features are not proagentive in the sense described earlier. Incorporating the purpose functions I have described into educational technologies could help strengthen intrinsic motivation. This can be done by building educational technologies based in specific types of functional and social environments.
No comments:
Post a Comment