Excerpt: In the classroom, I find myself more clinician than theorist, making spur-of-the moment decisions that are more art than science. But I also find a use for theories of how people learn. They help me think about my teaching when I have the time and luxury to do so. Constructivism is one of those theories. This essay is neither a sales pitch for constructivism, nor a critique of it – just a clarification. In fact, these days, the word, itself, is “out,” at least in the U. S., too much of a red flag. Theorize as you wish, but don’t ask and don’t tell. So, let’s just understand the theory of learning, and forget the “ism.”

Abstract: Crafting constructionist supplements to enrich curriculum is not easy; crafting a full set of constructionist-designed materials for day-to-day use by students and teachers is downright hard; both are possible. If one chooses to build in programming, decisions about what computer language has the “ideal” characteristics may depend on the specific subject matter or purpose to which that language will be applied. Mathematics, even for young children, imposes demands on that programming language - among them, the ability to create and compose functions - that other expressive purposes may not.

Open peer commentary on the article “Roles and Demands in Constructionist Teaching of Computational Thinking in University Mathematics” by Chantal Buteau, Ana Isabel Sacristán & Eric Muller. Abstract: This is a caution to producers and consumers of educational theory and research to recognize places where, in an honest effort to distinguish the core of an idea from the incidental trappings, we inadvertently say more than we would defend scientifically.

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. Key words: Problem posing, puzzles, mathematics, algebra, computer programming.