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:

He was trying to remember a method for calculating z. He couldn’t, so turned to the textbook, but there was no method described there. Given that K could not recall a method for solving this problem, what then?

Following recent work on mathematical questioning (e.g. here and originally here), I have introduced a set of five prompts that are always written on the board, to which I drew his attention:

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.

*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.

One could suggest that K 'should' have remembered that parallel vectors are multiples of each other, and that he just needs more practice. This problem would certainly have been easier if he had recalled this fact. Whilst work can be done to strengthen recall, it will remain an unreliable way of knowing.

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)

(b)

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

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.

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.

Inspection of the prompts above may reveal that they are based on functionings (as well as memory). I will now present two more examples:

Example 1:

Example 2:

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:*GIven that memory is a weak power of the mind, and that students will be faced with the unfamiliar, does it make sense to base a pedagogy primarily on memory?

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.

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.