Learning and Mathematics

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Mathetics - Papert (1993)

In Chapters Five, Six, and Seven of his book, The Children's Machine, Seymour Papert examines the art of learning, a topic that he contends has been widely ignored by educational researchers and practitioners. In Chapter Five he introduces the concept of 'mathetics,' which he defines as the art or act of learning, and discusses the issues that surround mathetics - in the school setting, theoretically, and in light of his own experiences. Chapter Six presents a series of case studies that build upon and draw from his discussion of mathetics, and which demonstrate the utility of computers in promoting flexible, personal, and connected learning. Chapter Seven contains a more theoretical discussion of instructionist versus constructionist viewpoints, as well as a defense of concrete knowledge and thought in the face of educational trends that favor abstract reasoning. Overall, Papert stresses support for personal variation in learning styles, and for the increased acceptance by schools of the ability of children to learn without assistance.


Papert, S. (1993). The Children's Machine: Rethinking School in the Age of the Computer. New York: Basic Books.

Quotes and Comments:

Students frequently don't realize that the principles of real learning (mathetics) also include heuristics, the art of intellectual discovery, and 'taking the time' to learn. As Papert writes, "Attempting to apply heuristic rules checks students in the rush to get done with a problem and get on with the next. It has them spend more time with the problems, and my mathetic point is simply that spending relaxed time with a problem leads to getting to know it, and through this, to improving one's ability to deal with other problems like it. It is not using the rule that solves the problem; it is thinking about the problem that fosters learning" (p. 87).

"A central tenet of mathetics is that good discussion promotes learning, and one of its central research goals is to elucidate the kinds of discussion that do most good and the kinds of circumstances that favor such discussions. Yet in most circles talking about what really goes on in our minds is blocked by taboos as firm as those that inhibited Victorians from expressing their sexual fantasies. These taboos are encouraged by School, but go far beyond it, and point to ways in which our general culture is profoundly 'antimathetic'" (p. 89). (This emphasis on the importance of discussion in the classroom, and especially in the field of mathematics, is echoed by other prominent theorists such as Schoenfeld; Brown, Campione, Reeve, Ferrara and Palincsar; Resnick; Lampert; Collins, Brown, and Newman; and Cocking and Chipman, all of whom have been the focus of previous Forum discussions.)

Papert also offers reasons for including the concept of the 'taboo' in his discussion of mathetics. He writes, "Exaggerated or not, the suggestion of a taboo is intended to state emphatically that getting people to talk about learning is not simply a matter of providing the subject matter and the language. The lack of language is important. But there is also an active resistance of some kind. Thus advancing toward the goal of mathetics requires more than technical aids to discussion. It also requires developing a system of psychological support. The simplest form of support system I can imagine is to adopt the practice of opening oneself by freely talking about learning experiences" (p. 92).

Papert continues, "On a pragmatic level, 'Look for connections!' is sound mathetic advice, and on a theoretical level the metaphor leads to a range of interesting questions about the connectivity of knowledge. It even suggests that the deliberate part of learning consists of making connections between mental entities that already exist; new mental entities seem to come into existence in more subtle ways that escape conscious control. However that may be, thinking about the interconnectivity of knowledge suggests a theory of why some knowledge is so easily acquired without deliberate teaching. In the sense in which it is said that no two Americans are separated by more than five handshakes, this cultural knowledge is so interconnected that learning will spread by free migration to all its regions. This suggests a strategy to facilitate learning by improving the connectivity in the learning environment, by actions on cultures rather than on individuals" (pp. 104-105).

We might there understand the use of discussion techniques in mathematics as providing the means to more connections. This is not to suggest, however, that all mathematics is best taught through discussion. Discussion can be used to bring an issue to mind and can then be followed by direct instruction; It can also be used following instruction to enable students to consolidate their "real" understanding of what has been presented. Discussion can also be used in evaluating how students have synthesized their understanding of a new skill in combination with prior skills.

In Chapter Six, Papert presents six case studies, each of which has a learning moral. The first story again emphasizes the need for connectedness, and for tying new learning to interests and knowledge structures that are already in place. The second looks at people who use math informally to adjust recipes. At the end of this story, Papert summarizes the morals: "The central epistemological moral is that we all used concrete forms of reasoning. The central mathetic moral is that in doing this we demonstrated we had learned to do something mathematical without instruction -- and even despite having been taught to proceed differently" (p. 115).

The third study shows how students must be allowed to cross the gender/cultural divide in their own time and manner. As Papert writes, "Knowing that one can exercise choice in shaping and reshaping one's intellectual identity may be the most empowering idea one can ever achieve" (p.123). The fourth story stresses the need for humor in learning, and demonstrates how jokes can be used as learning tools. The fifth story emphasizes the need for 'bricolage' or tinkering in learning -- namely, giving children the opportunity to explore problems and projects freely, and to develop their goals as they work. The last story differentiates between 'clean' (unconnected) and 'dirty' (connected) learning, and discusses the school's preference for 'clean' learning.

Papert argues for a constructionist philosophy that will promote teaching "in such a way as to produce the most learning for the least teaching" (p. 139). He contrasts this view with that of instructionism, in which "the route to better learning must be the improvement of instruction" (p. 139). According to Papert, constructionism is tied to mathetics in the sense that children can often learn without the benefit of schooling, and, if given the incentive to learn it independently, will even learn a subject better than they would have learned it in school.

As Papert writes, "On some level we know that if we become really involved with an area of knowledge, we learn it -- with or without School, and in any case without the paraphernalia of curriculum and tests and segregation by age groups that School takes as axiomatic. We also know that if we do not become involved with the area of knowledge, we'll have trouble learning it with or without School's methods. In the context of a School-dominated society, the most important principle of mathetics may be the incitement to revolt against accepted wisdom that comes from knowing you can learn without being taught and often learn best when taught least" (p. 141).

It should be noted, however, that instituting discussions or moving toward more student-directed classes requires that teachers be ready to take up questions to which they do not know the answers, come prepared with a variety of resources, and in general, be well-versed in the content that they are teaching.

"The important mathetic skill is that of constructing concrete knowledge" (p. 143). Papert builds on this statement by stressing the importance of bricolage (tinkering) in creating concrete knowledge. "The basic tenets of bricolage as a methodology for intellectual activity are: Use what you've got, improvise, make do" (p. 144). He also brings computer use into the picture: "The computer imply, but very significantly, enlarges the range of opportunities to engage as a bricoleur or bricoleuse in activities with scientific and mathematical content" (p. 145).

Finally, Papert emphasizes the importance of concrete thinking at all ages, and argues against popular trends that favor abstract reasoning over concrete knowledge. He writes, "My strategy is to strengthen and perpetuate the typical concrete processes even at my age. Rather than pushing children to think like adults, we might do better to remember that they are great learners and to try harder to be more like them. While formal thinking may be able to do much that is beyond the scope of concrete methods, the concrete processes have their own power" (p. 155).

-- summarized by Jane Ehrenfeld

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