About Constructivism

First submitted to Math Forum:

Most of us associate the word "constructivism" with Piaget, who studied how children develop and internalize a model of reality in stages. Not every concept is equally accessible in each age range. There's a progression, and a sense of prerequisites, i.e. a child needs layer A to build layer B and so on. In other words, children need to "construct" their own understanding of the world in stages and this is a dynamic activity involving manipulation of the environment, getting feedback (reality checks) from others, including one's peers (learning as a social activity).

Constructivism was in some ways a response to a more Prussian vision of education as a well oiled machine in which students sat in rows and columns and quietly absorbed (received) the teachings coming from the one authority in the room. Students were discouraged from communicating with one another or teaching their peers. Teamwork and group activities were strictly verboten.

At some layer in cultural archeology, questioning that system comes across as "rebellious" whereas in others it's more common sense that children learn through play, discovery, activity, as do adults i.e. the authoritarian model isn't even remembered, so there's nothing to rebel against. Almost all education in the USA would be considered "constructivist" by some of these older European standards.

What's important to remember is constructivism is not a synonym for "undisciplined" and the play is not without structure or "scaffolding". Rather, the model constructivist is a research mathematician exploring new areas, developing new mathematics. This implies focus, concentration, devotion to the work, not simply recitation of known material or rote memorization of facts i.e. there's inventiveness involved, hence the word "constructivism".

Constructivism might have been called inventivism (not really a word), with students encouraged to develop those habits of mind they will need in order to contribute new thinking, not simply pipeline what's not original with them -- although there's nothing wrong with pipelining others' materials, teaching it to peers (that's also part of the constructivist model: teaching others, not just leaving that role to "the teacher").

Where constructivism was perhaps most successful was in the reform of science education. The experimental sciences are so obviously based in hands-on activities with actual equipment, involve discovery, inducing the rules or patterns in nature from especially designed tests. A key figure in bringing constructivist models to science teaching was Robert Karplus. I've been working in cahoots with one of his main students, Dr. Bob Fuller, University of Nebraska, on various aspects of the physics curriculum, and so have developed some appreciation for his outlook and approach.

Another key figure, in mathematics this time, was Caleb Gattegno. Gattegno took those Cuisenaire rods, colorful rectilinear blocks of various pre-defined lengths, developed by the Belgian mathematician for whom they are named, and built a constructivist algebra curriculum around them. His approach was to label the blocks with letters and introduce the four arithmetic operations using these letters, with students in discovery mode, deducing relationships. This forms a nucleus of the AlgebraFirst curriculum, with proponents at Stanford and the UK.

Since the PC and open source revolutions starting in the 1980s, constructivism has received another boost, as working with computers is clearly an exploratory or investigative activity, a kind of play, although again we're talking about developing self discipline for a later career.

Seymour Papert rebranded his thinking, based around the Logo language, as "constructionism", yet the constructivist legacy was clear. His focus on including machine executable math notations (as Kenneth Iverson called computer languages) fed into a movement that has been gathering momentum ever since, although not always within the USA, or at least unevenly therein.

The twin goals of self study and peer teaching have been increasingly achieved with the aid of the Internet, wherein many subcultures have formed into "teaching communities" each intent on perpetuating a critical mass of skilled users of whatever tools -- the process by which civilizations persist from one generation to the next. A lot of these tools are mathematical in nature, so the next step may be to move more concertedly beyond calculators and more into executable math notations, as is already happening in some schools. This would be consistent with the USA administration's goal of providing a world class education and bridging the so-called "digital divide".

Mathematics as a controller for technology is an old theme, tracing back to Leibniz and Pascal. What has changed is the ubiquitous nature of this technology, as well as our level of mastery and control i.e. electronic computers are no longer regarded as "too esoteric" (or "too expensive") to share with children. Indeed, a youthful demographic is trailblazing the new curriculum, following in the footsteps of some of the aforementioned elders or ancestors.

Kirby

Endnotes: Re Karplus; Re Gattegno; Re Iverson