Teaching mathematics always had a dilemma, not huge, do you teach maths or do you teach for exams?
I always favoured the latter because I was conscious that you could not teach students to be mathematicians, in my view they either were or not (growth mindset?). But the exams were sufficiently problematic that it was necessary to teach the former as well. Let me say that this is categorically not why I got into teaching but in the end it was what the students wanted. And the exams were not too interested in creative maths to help me design a better programme of study. I suspect now you are expected to do the same but disguise it with creative educationalism.
Let’s begin with the latter. I established a programme of study based on previous exams – especially for years 10 and 11 leaving a good amount of time for exam cramming. (Equivalent for years 12 and 13 at A level.) During the cramming I was encouraging the students to recognise the technique required, this was pure conditioning. If we consider the model:-
then the steps are:-
In this process of “technique recognition”, being aware of conditioning is rarely an issue as this process is designed to fit in with the conditioning approach of doing 1 on the board and setting 10.
However this was never a teaching process that could get top grades at A level because at A level or IB insight was required (see below). No matter how well crammed students were at that level they could never get top grades.
When I first retired I did a small section on study skills, but for maths and problem-solving the issue was finding the insight. It was this insight that separated the mathematicians from the exam successes. And the insight fits into our model as it is the conditioning moment. Although the study skills gives you additional approaches that don’t exactly fit the model, the overall process is concerned with finding the insight once you have immersed yourself in the problem. It is that insight moment, the Eureka moment, that shows you how to solve the problem. The mind needs clarity and concentration to determine the insight – staying focussed, but with that clarity for some the insight comes.
Insight cannot be taught although it can be encouraged, and there are methods for facilitating insight. But there is not 1 insight, do 10.
At one school in the Middle East I was employed with the specific purpose of getting the top grades. Previously a science teacher had become the head of maths, and he was apparently excellent at here’s 1 do 10; he was accepted at the school. But the parents complained because the students did not get top grades – fair.
Except the school was not fair, the school was dominated by the students and parents, students regularly “fired” teachers. I had a hard-working class but they were mentally stuck. They were very good at here’s 1 do 10, but they had no idea about problem-solving. And problem-solving was the grades I had been brought in to produce.
Here is where the school was not fair. The administration employed me to raise the grades, but they could not back me with the students as the rich and powerful students ran the school. Whilst my educational rationale was sound, it needed discipline. I was told I had admin backing until it was too late, and there was a clash between me and rich kids. By the time I realised this it was too late, and no amount of here’s 1 do 10 would help.
It was all sad really because there were 2 or 3 insightful mathematicians in the class but they were so used to doing 10 they never tried to apply insight.
I remember one time the President of the school, yes that’s what he called himself, came into my department training; he had been a maths teacher. He was impressed with what I was delivering, it was exactly what he had employed me for. 3 months later he let the students “fire” me.
Wow, 15 years ago now.
Blogs:- Ginsukapaapdee, Mandtao, Zandtao.