Depending on your particular take, SKU might refer to ‘stock keeping unit’, the unique identifier enabling tracking of an item for stock-taking purposes; or if you are an educator trained towards the end of the National Curriculum it might mean skills, knowledge and understanding. Of course, you might just be dreadful at spelling or good at texting and associate this with skew – the mathematical term for bias.

Strange how they are all linked in my mind – with another of those student moments ‘why are we learning this (circle theorems)?’, which was also curiously linked to a passing comment both by a fellow teacher and later by a candidate who had applied to join us, neither of whom could see a reason for including circle theorems in the curriculum. My student was both cheesed off and hoping (I suspect) to wind me up! When he was in a far more amiable mood the following morning, we picked up this thread again. Although there might be no immediate use of these ideas in what he might choose to study at A-Level or university I said, I was hoping to teach him and his peers how to learn. With rapidly changing technology and working practices, they will need to be very much more independent in the future; they will need to be prepared to take risks, knowing that they will have the ability to teach themselves. The class thought about this for a while (in the meantime being challenged by lots more ‘irrelevant’ maths) and have since had a determination to understand even the toughest concepts I have thrown their way.

When I picked this GCSE class up a year before, almost every lesson was a battle. Mathematically very able students can have very limited experiences of developing understanding – some much prefer worksheets with masses of skills based questions – the repetition becomes their comfort blanket. So to them, often the learning environment I try to engender can be threatening: I expect them to teach themselves.

A typical independent lesson involves me giving them the lesson objectives, or just an idea or concept, and some lesson outcomes, which might be to produce a presentation, a PowerPoint, a revision document etc. This has to be emailed to me by the end of the lesson (which is 100 minutes long) or printed for their future use. Students support each other, explain, question, theorize, investigate, analyse and evaluate. There are now 25 ‘teachers’ in each lesson. At times, I cannot match the depth and variety of questions and problems that are a feature of MangaHigh without serious preparation! We have netbooks – every student in school has access to the internet, but I developed these types of lessons prior to internet access with textbooks and other reference resources.

Another lesson might have several types of differentiated questions on the board as a starter, whereupon students can pick or be guided to attempt those relevant to their prior learning; then developing an open question and answer session mostly driven by the students’ eagerness to interpret or discover more. They always have a list of 5 or so topics which are their individual targets, and as they become confident with one, it is replaced by another. Everyone is likely to be working on something different by the end of the lesson, even though they began doing a similar topic at the beginning.

So what do I do while they are busy beavering away teaching themselves? I can have a meaningful and probing conversation with each student at least once in the lesson; query their understanding and feedback to them relating to bookwork or progress tests. I can now give intensive one-to-one intervention with any student who has a need to develop deeper understanding. I am listening to their conversations, and I interject with finer nuanced explanations, drawing and linking a host of mathematics topics together.

So skills/ knowledge and/or understanding? A year on, they have developed into quite self-assured and resilient learners, able to tackle some demanding problems, thinking about and applying their mathematics in some novel and unexpected ways. Occasionally they will go back to an exercise for skills based practice, and then seek problems that test their depth of understanding. They value guidance and feedback and are able to build on this effectively. They are impatient to get on with their learning, and seek to do this for themselves.

It is difficult for some colleagues to loosen their grip on knowledge-based practices; they have spent a long time being reassured that their students are competent at attempting exam questions that comes from teaching to the test. Unfortunately, their students are woefully prepared when the need arises and they fail to apply their mathematical learning in unfamiliar circumstances. I am sure there has always been a debate in education about skills-based and knowledge-based learning. Loosely defined as:

- Knowledge learning predicates the use of memory as information about a subject is delivered and then generally the learners’ knowledge retention is tested.
- Skills learning develops specific expertise by practicing skills through engagement, with the use of collaboration and reflection.

The best of mathematics learning is to synthesise both skills and knowledge because students need to apply their understanding with each different problem. Mastery is paramount, but knowledge is of little use unless it can be applied. Even so, I am averse to merging mathematics with science, geography, economics or other subjects, where by being subsumed, the detail and depth of mathematics knowledge would be unlikely to ever reach the levels required at A-Level or beyond.

My learning environment has become a fusion of techniques and ideas from proponents of Accelerated Learning, Mastery Mathematics, TEEP, the National Curriculum, CIMT, MEI, the 2020 and 2040 Visions, Bloom’s and SOLO Taxonomy , AFL, and many other educational thinkers and institutions; much of what I find I try out, and cherry pick to match the needs of my learners. Currently I am developing a curriculum rethink that I hope will develop the depth our students lack and enhance their curiosity, their stamina and their appetite to learn more about mathematics beyond their time at school.

AM I skewed towards independent learning? Yep. Am I biased about the educational experiences that students receive in their mathematics lessons? This has to be the best job in the world!