Seymour Papert’s 1972 paper “Teaching Children to be Mathematicians Versus Teaching About Mathematics” started with the summary statement “The important difference between the work of a child in an elementary mathematics class and that of a mathematician is not in the subject matter…but in the fact that the mathematician is creatively engaged….” Along with “creative,” a key term Papert kept using is project rather than the common notion of problem. A project is not simply a very large problem. It centrally includes a focus on sustained and active engagement. The projects in his illustrations were essentially research projects, not just multi-step, fullyprescribed, build-a-thing tasks, no matter how nice the end product might be. A mathematical playground with enough attractive destinations in it draws children naturally to pose their own tasks and projects—as they universally do in their other personal and group playgrounds—and to learn to act and think like mathematicians. They even acquire conventionally taught content through that play. Physical construction was always available, and appealed to such thinkers as Dewey, but for Papert computer programming, newly available to school, suggested a more flexible medium and a model for an ideal playground. A fact about playgrounds is that children choose challenge. In working and playing with children I’ve seen that puzzles tap some of the same personally chosen challenge that a programming centric playground offers. Children are naturally drawn to intellectual challenges of riddles (ones they learn and ones they invent) and puzzles; and adults are so lured by puzzles that even supermarkets sell books of them. So what’s the difference between real puzzles and school problems? What’s useful about creating a puzzle or posing a problem? How might puzzles and problem posing support mathematical learning? And what’s constructionist about this? This plenary will try to respond to these questions, invite some of your own responses, let you solve and create some puzzles, and explore how problem posing in programming and puzzling can support mathematics even in an age of rigid content constraints.
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Problem posing and creativity in elementary-school mathematics
> Context • In 1972, Papert emphasized that “[t]he important difference between the work of a child in an elementary mathematics class and […]a mathematician” is “not in the subject matter […]but in the fact that the mathematician is creatively engaged […]” Along with creative, Papert kept saying children should be engaged in projects rather than problems. A project is not just a large problem, but involves sustained, active engagement, like children’s play. For Papert, in 1972, computer programming suggested a flexible construction medium, ideal for a research-lab/playground tuned to mathematics for children. In 1964, without computers, Sawyer also articulated research-playgrounds for children, rooted in conventional content, in which children would learn to act and think like mathematicians. > Problem • This target article addresses the issue of designing a formal curriculum that helps children develop the mathematical habits of mind of creative tinkering, puzzling through, and perseverance. I connect the two mathematicians/educators – Papert and Sawyer – tackling three questions: How do genuine puzzles differ from school problems? What is useful about children creating puzzles? How might puzzles, problem-posing and programming-centric playgrounds enhance mathematical learning? > Method • This analysis is based on forty years of curriculum analysis, comparison and construction, and on research with children. > Results • In physical playgrounds most children choose challenge. Papert’s ideas tapped that try-something-new and puzzle-it-out-for-yourself spirit, the drive for challenge. Children can learn a lot in such an environment, but what (and how much) they learn is left to chance. Formal educational systems set standards and structures to ensure some common learning and some equity across students. For a curriculum to tap curiosity and the drive for challenge, it needs both the playful looseness that invites exploration and the structure that organizes content. > Implications • My aim is to provide support for mathematics teachers and curriculum designers to design or teach in accord with their constructivist thinking. > Constructivist content • This article enriches Papert’s constructionism with curricular ideas from Sawyer and from the work that I and my colleagues have done
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- Award ID(s):
- 1741792
- NSF-PAR ID:
- 10100698
- Date Published:
- Journal Name:
- Constructivist Foundations
- Volume:
- 14
- Issue:
- 3
- ISSN:
- 1782-348X
- Page Range / eLocation ID:
- 601–613
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Active learning—
doing —supports learning.Collaborative learning—doing
together —supports learning.Classroom discourse—focused, relevant
discussion , not just listening—supports learning.Clear articulation of one's thinking, even just to oneself, helps develop that thinking.
What this paper adds
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Continued usefulness across the grades requires developing children's sophistication and knowledge with the language; the powerful ways that children rapidly acquire facility with (natural) language provides guidance for ways they can learn a formal language as well.
Implications for policy and/or practice
Mathematics teaching can take advantage of the ways children learn through experimentation and attention to the results, and of the ways children use their language brain even for mathematics.
In particular, programming—in microworlds driven by the mathematical content, designed to minimise distraction and overhead, open to exploration and discovery
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