Republic of Mathematics blog

Over at Parenting is Political, @northTOmom has written a recent post:  THIS MATH DEPRESSES ME.

Part of what concerns @northTOmom is what she identifies as “constructivist” math.

Here is what she writes:

“So why do I object to constructivist math? One reason is that it is, by-design, non-incremental or “spiral”: its textbooks jump around from topic to topic, never staying on a subject long enough to allow for deep understanding or competence. I also dislike reform math because it frowns upon direct instruction. Since constructivist math teachers believe children can “construct” or “discover” mathematical truths and come up with their own algorithms to solve problems, they offer students minimal guidance, and are not averse to putting the cart before the horse: e.g., assigning algebra-type problems before teaching the tools of algebra, or asking kids to divide or multiply by decimals or fractions without having first taught them how decimals and fractions work.”

I was surprised to read this because it’s not what I understand constructivist theory applied to mathematics teaching  to be.  In fact so far as I know, constructivist theory is a theory of learning and says little to nothing about teaching. It is a theory of how people learn, which might be very useful to teachers, but does not imply “discovery” learning.

JeanPiaget

As I recall, one of the fathers of constructivist thought, Jean Piaget remarked that it is absurd to imagine children can, or will, discover large parts of mathematics for themselves: there’s just simply too much of it, most of it thought of by mature adults.

It seems to me that part of the problem here is one of semantics compounded by the U.S. Bill of rights: every U.S. citizen feels free to use whatever words they choose in whatever way they choose to use them, without reference to previous thinkers and scholars. Such a polyglot leads to gross misunderstanding in communication, and eventually to misinformed practice as @northTOmom observes.

Constructivism in mathematics classes is not about letting children “discover” mathematics. What it is about, to a first approximation is students reflecting on their actions.

Piaget used the term “reflective abstraction”; the power of generalization and abstraction from reflecting on one’s actions. We see this for example when young children count – to 10, for example – and we ask them afterward: “When you were counting those 10 crackers, what if I gave you 2 more? How many would you have then?”

If the child already has experience in reflecting on their counting they may be able to solve this problem by imagining the objects in their mind. If not, they will probably look for 2 more crackers and count again using the real crackers.

John Locke

Ernst von Glasersfeld points out that Piaget’s use of reflective abstraction as a cornerstone of a constructivist theory of learning was not original with him. Already in 1690 John Locke had observed:

“the ideas it affords being such only as the mind gets by reflecting on its own operations within itself. (Locke, 1690, Book II, 4)

So, if your child’s mathematics teacher calls themselves a constructivist teacher, ask what that means. If they mention discovery learning, and do not mention reflective abstraction as a core idea, then they either do not understand constructivism, or do not understand mathematics, or both.

Piaget’s ideas spoke to a truly experimental view of knowledge: how is that we obtain new knowledge as we make our way in the world?

Reflective abstraction is a key part of his theory of how we do that. In and of itself it has little to do with teaching, yet because it is about learning a wise teacher will look for ways to encourage students to reflect on their experiences, and to abstract and generalize from them.

Reading

Constructivism in Education, Leslie P. Steffe (Editor), Jerry Gale (Editor), Routledge (1995). ISBN: 978-0805810967

Radical Constructivism in Action: Building on the Pioneering Work of Ernst von Glasersfeld, Leslie P. Steffe (Editor), Patrick W. Thompson (Editor). Routledge (2000). ISBN: 978-0750709897

Postscript

Raymond Johnson alerted me to Lee V. Stiff’s 2001 article “Constructivist Mathematics and Unicorns“. In this article Lee gives a decent summary of constructivism and points out how opponents of certain progressive classroom practices in mathematics use the term “constructivist” pejoratively.

Many thanks Raymond for this link.