Teaching Mathematics using an approach that is both conceptual and procedural -
Nine keys to the hybrid conceptual-procedural approach
Richard Andrew
Constructivism unpacked (as it relates to this article)
Let me offer a few ‘feet-on-the-ground’ thoughts on constructivism.
My take is that constructivism is a description of the process by which most of us learn anything of real importance. By ‘real importance’, I mean deciding we need a new skill and so we initiate a self-directed, learning-by-doing path to master that skill. Examples? Learning to sew, to build a shed (or house), mastering the art of gardening, painting and so on.
As an example, a few years ago I built my first deck. I’d never built anything ‘that mattered’ before - my prior explorations were centred around ‘items built out of recycled wood that can be burnt if they don’t work out’. I had never concreted (vertical) posts into the ground, never constructed bearers and joists, never used a drop-saw, and didn’t know any of the building ‘tricks’ required to build a deck of the type that I could be proud of. How did I learn? Mostly via constructivism. After schooling myself up on articles and Youtube videos I talked to people who knew deck-building and the remaining 60% of learning occurred by ‘doing’. Most of the learning-by-doing occurred through solving problems. It was a wonderful period of mathematical, discovery-based learning.
So constructivism could be seen to be learning via self-initiated, practical exploration. I would say ‘Montessori’ falls under the umbrella of constructivism.
This is an intentional over-simplification- I wish to steer well clear of the heady, overly intellectual discourse that usually accompanies the term 'constructivism'.
Recalling the workshopped open-ended decimal questions example given in the unpacking of Key #3 it should be obvious that I’m not taking the constructivist path here. That example was conceptually-based and explicit. The activity was highly structured, the teacher was very much directing the learning. This is not constructivism.
Truth be known, I’m all for constructivism. As per my deck building example, clearly, I am a constructivist learner. I know of schools that require students to chart their own practically-based, student-centred learning path. I assume these schools draw heavily on constructivism. This style of education makes a lot of sense to me.
However, the reality is most teachers are required to follow a conventional, prescriptive curriculum and I would argue that constructivism is not the ideal approach when addressing a conventional, prescriptive curriculum.
My take is that constructivism is a description of the process by which most of us learn anything of real importance. By ‘real importance’, I mean deciding we need a new skill and so we initiate a self-directed, learning-by-doing path to master that skill. Examples? Learning to sew, to build a shed (or house), mastering the art of gardening, painting and so on.
As an example, a few years ago I built my first deck. I’d never built anything ‘that mattered’ before - my prior explorations were centred around ‘items built out of recycled wood that can be burnt if they don’t work out’. I had never concreted (vertical) posts into the ground, never constructed bearers and joists, never used a drop-saw, and didn’t know any of the building ‘tricks’ required to build a deck of the type that I could be proud of. How did I learn? Mostly via constructivism. After schooling myself up on articles and Youtube videos I talked to people who knew deck-building and the remaining 60% of learning occurred by ‘doing’. Most of the learning-by-doing occurred through solving problems. It was a wonderful period of mathematical, discovery-based learning.
So constructivism could be seen to be learning via self-initiated, practical exploration. I would say ‘Montessori’ falls under the umbrella of constructivism.
This is an intentional over-simplification- I wish to steer well clear of the heady, overly intellectual discourse that usually accompanies the term 'constructivism'.
Recalling the workshopped open-ended decimal questions example given in the unpacking of Key #3 it should be obvious that I’m not taking the constructivist path here. That example was conceptually-based and explicit. The activity was highly structured, the teacher was very much directing the learning. This is not constructivism.
Truth be known, I’m all for constructivism. As per my deck building example, clearly, I am a constructivist learner. I know of schools that require students to chart their own practically-based, student-centred learning path. I assume these schools draw heavily on constructivism. This style of education makes a lot of sense to me.
However, the reality is most teachers are required to follow a conventional, prescriptive curriculum and I would argue that constructivism is not the ideal approach when addressing a conventional, prescriptive curriculum.
Why are we talking about constructivism?
I'm referencing constructivism because maths educators often confuse ‘a conceptual approach’ with ‘a constructivist approach’. However, in this article when using the term conceptual, we are not referring to constructivism. We are referring to something more prescriptive, more explicit, more directed.
Acknowledgement
(A shout out to Sam Shepherd, a participant of the 'Maths-Engagement' course who, after reading an early draft, rightly suggested a reference to constructivism should be included.)
Acknowledgement
(A shout out to Sam Shepherd, a participant of the 'Maths-Engagement' course who, after reading an early draft, rightly suggested a reference to constructivism should be included.)
