Where is your Mathematical Bar?

There are many things in education, which seem taboo to discuss, of great importance when considering how to realistically improve the quality of classroom practice.

The most effective teachers are those with highly developed, subject specific pedagogical and didactical knowledge and skills, combined with excellent subject content knowledge.

This content knowledge should, to be most effective, go beyond the national curriculum. Only knowing the diet of content that students will meet, prevents one from seeing the connections in mathematics (or subject of your choice) and the place that a certain concept, idea, fact or skill takes in the grand scheme of learning the subject.

The teacher census each year tells us some of the facts. We know, for example, that of all the teachers teaching mathematics in England day in, day out, only around 24% of them have studied mathematics beyond school level. We know, for example, that many thousands of children are being taught mathematics (even at high school level) by adults trained in other subject or adults with no training whatsoever.

But these facts only give us an insight into those cold workforce and structural things, which although they matter a great deal, do not reveal what is happening on the ground.

I would like to ask you, dear reader, where your mathematical bar lies. By this, I mean at what level of mathematics are you completely comfortable and confident in being able to walk into a room of students and teach a concept, skill or fact off the cuff? This might sound like a silly question at first, but my experience in visiting many, many schools across the country is that many, many children are being taught lessons by a teacher who is, in lots of cases, only one hour ahead in learning the topic themselves. I don’t offer this as a criticism, but as a sincere discussion point. I think it would be a helpful thing to acknowledge, discuss and address. I think that subject specific content knowledge is of such great important, yet is so rarely the focus of CPD.

Maths teachers will, on the whole, happily discuss these issues when they are talking about, say, the highest modules on Further Pure Mathematics or preparing bright sixth form students for a STEP paper or Cambridge Entrance Exam. It is not uncommon to hear teachers ask for help with understanding some of the content in ‘Mechanics 2’, for example.

Initiatives, such as the ‘Further Maths Support Programme’, have done great work in helping teachers to improve their subject knowledge, but I do wonder if it should be okay for teachers to be just one lesson ahead. Should someone teaching Further Pure Mathematics have their bar set at, at the very least, second or third year undergraduate mathematics? Should they be comfortable and confidence in teaching, say, fluid dynamics, abstract algebra or vector calculus, so that they can see where the topics they are teaching these bright teenagers are heading?

Where is your bar?

Teachers will often tell me it is at GCSE level and that they are not happy to teach A Level maths. Some teachers will say, Key Stage 3 is where they stop and that the idea of having to teach 3D trigonometry or statistical methods to a top set Year 11 makes them break into a cold sweat.

Where is your bar?

It is tempting to believe that the bar is only an upper limit, but, in fact, the lowest level that your bar can go to is equally as important.

Recently, during some talks I have been giving to secondary and primary school mathematics leaders, I have asked the audience whether they know, understand and can explain what counting, addition, decimals and place value are. Not surprisingly, all said they could indeed. We then did some (hopefully interesting and amusing) activities to reveal that, actually, very few of the delegates did have a full grasp of these seemingly simple areas of the curriculum.

I regularly visit schools and hear of ‘those kids’ who will ‘never get’ maths. They are, let’s say, 15 years old and can’t number bond to 10. Yet, when challenged, it is very often the case that their teacher does not know and cannot deploy strategies and didactics that would lead to the student being able to grasp the topic.

This is an exceptionally common issue and one that makes perfect sense. All of the real intellectual demand of teaching school level mathematics happens at the early concepts. It is far more demanding of a teacher for them to master how to get a child to understand the sense of a number or how to add or what multiplication, place value, a base system, or a whole host of other early fundamentals actually are, than it is to get a child to be able to solve a quadratic equation.

So, teachers have an upper and lower bar. Where do yours lie?

More importantly, how do we go about making the range larger between these bars? How can we get teachers to improve their subject specific knowledge when all around them they are faced with mindless initiatives and policies that take up all of their time, but have no impact on the quality of teaching?

I think that one important step to take is like a mass coming out. It can be difficult for a teacher to say they are not completely secure in their subject knowledge. I recall entering the profession myself, having had a mathematical career to date that focused almost exclusively on pure mathematics, mechanics and modeling. My knowledge of statistics was pretty dire, yet I was asked from day one to teach A Level statistics. When I told the head of maths that I was garbage at stats, he told me I shouldn’t own up to things like that. I thought this terribly odd, having come from industry (where if you don’t know something, you tell someone and they help you to get it sorted). My subject knowledge of the lowest level concepts was also weak. After all, I was a mathematician by trade, had spent years working with very high level maths and it seemed like an age since my mind had ever had to give any thoughts whatsoever to things like ‘what is a number?’

I have loved expanding the range of my subject knowledge, upwards into advanced statistics (to plug the gap) and right down to the earliest concepts (to ensure I can plug any gaps children might have).

I now try to help teachers expand the range of their subject knowledge, through the events I run and my termly MathsConf.

Subject specific knowledge and subject specific pedagogy and didactics are the single most important ingredients in improving the quality of teaching. So, even if not for public discussion or in the comments below, I do feel it is helpful if we all ask ourselves where our bars lie and how we might push them a little bit each year.

If you are willing to share your stories, please do comment below - it would be great to collate some examples of teachers coming out of the subject content knowledge closet.