When Ofsted look for continuous progress in maths lessons, what subject is it that they are thinking of? When SLT make 10 minute snap observations in which they expect to see students making progress, are they talking about the subject we know as mathematics? In the introduction to the Horizon episode on the proof of Fermat’s last theorem, Wiles describes the process of doing mathematics very beautifully (YouTube Link). Martin Gardiner described that wonderful moment when you realise you have found a solution as an Aha Moment (Article). Cleary, Ofsted and SLT are not talking about this subject, because they would know that moving forward mathematically requires an indeterminate period of struggle, followed by a dramatic shift. Progress modeled not by a linear function, but by a step function.
I think as maths educators we need to stop pretending that the subject that is examined in our schools (and increasingly in our Universities) is same subject as the one we call mathematics. Godfrey hardy railed against the Cambridge tripos exams of the late 19th century as being in opposition to mathematical learning. Indeed, the principle tutors for the exam were not the academic of the University, but private tuition services offering exam training. Only when the exam was completed could students get on with the business of doing mathematics. So, Hardy campaigned to get the exam abandoned.
In schools, we have 5 years to apprentice young mathematicians and train them to pass an exam. That is surely enough time if we focus on the two courses we have to teach. Let’s say a four year course in mathematics (years 7 to 10) followed by a one year course in passing a GCSE exam, which by then will be easy as we will have created thoughtful mathematicians who are adepts at discussing and describing their work and explaining their reasoning and all of those things mathematicians do.