We have started revising Higher maths this week. We started by working on a wide range of problems in the hope of diagnosing what areas we might need to focus on. As J and K worked on these problems, I became aware of an approach which I will describe as: ‘trying to recall a method’. An example of this happened when K was working on this problem:
- Of what is this an example?
- What do I know? (... so I also know…)
- Can I solve a simpler problem? (make it as simple as possible)
- This is unfamiliar: how can I get-a-sense-of this...?
- Say what you see: what are the key features...?
After looking at the prompts, K suggested the problem might be about gradient, and then drew a 2-d sketch. Looking at the sketch, he could see that the (6,-9) triangle he had drawn was three times bigger than the (-2,3) triangle. He then made the conjecture that z = -18.
My reason for presenting the example above is to illustrate what can happen when some fact or method is not accessible to the learner. I conjecture that:
(a) trying to remember a method got in the way of K solving this problem, and
(b) when faced with an unfamiliar problem, other ways of coming to know - beyond memory - are required.
In What We Owe Children, Caleb Gattegno suggests that exercises, homework, reviews, tests, more reviews, and more tests, are the result of a pedagogy based primarily on the 'weak power of memory'. He suggests that an alternative is to work with learner’s strengths: ways of knowing which he calls functionings. Functionings include the ability (shared by all humans, used since a very young age) to extract, transform, abstract, stress/ignore, imagine…
Gattegno suggests that a pedagogy based on such functionings (as well as memory) will lead to more efficient learning, which matches with my experience. It is my contention that the current over-emphasis on the role of memory for learning is like building a house upon sand. It has led to a requirement to introduce ever-more complex strategies for bolstering memory, the result being that other important ways of knowing are ignored.
Example 1: Neither student could determine the minimum value of f(x) = 4cos(x - pi/3) + 6. J used the prompt “Can I simplify the problem? Make it as simple as possible.” to simplify the function to f(x) = cos(x), for which he knew the minimum value. He then built back up to find the solution to the original problem.
Example 2: K used the prompt: "This is unfamiliar... How can I get a sense of this problem?" in order to realise that he could solve the following unusual problem by substituting a couple of different values for a:
Would it be more useful to develop a pedagogy that incorporates the full range of human functionings? For this to happen, a good start would be to observe very closely what learners do, how they function, in order to identify and specify exactly what is meant by a functioning, and then develop teaching approaches that might exploit that functioning.