In 2015 the Advisory Committee on Mathematics Education (ACME) released their report *Beginning Teaching: best in class? High-quality initial teacher education for all teachers of mathematics in England*. The report found that there are many inconsistencies in the provision of maths initial teacher education (ITE) in England, in part due to the variety of routes available into the profession (including university-based postgraduate courses, School Direct, School-Centred Initial Teacher Training (SCITT) and Teach First)) but in the main due to the absence of a shared standard for maths ITE.

The ACME report arrived on the back of the *Carter review of initial teacher training*, commissioned by the DfE, which made, amongst others, the following findings:

- the ITE system “generally performs well, with some room for improvement in particular areas”
- it is generally hard to recruit sufficiently well, and those courses who have plenty of applicants often report that many are of poor quality
- there is “considerable variability across the system” in terms of expectation about the “knowledge, understanding and skills new teachers should have”
- there is unsatisfactory variability in the way subject knowledge is addressed

Why is variability in the provision of subject knowledge training unsatisfactory? To answer that question you need to be aware of the two types of knowledge essential to teaching: *knowledge of your subject* and *knowledge of how to teach your subject (pedagogy)*. Both of these things are, of course, subject-specific. What makes good maths teaching may look very different to what makes good history teaching, for instance, although there are plenty of pedagogical techniques that are highly efficient across all domains.

The Carter Review makes its first set of recommendations on subject knowledge (not pedagogical knowledge). It is quite significant that subject knowledge is addressed first – the review acknowledges that one of the most important facets of ITE is poorly addressed throughout the system. While stating that “the most effective courses address gaps and misconceptions in trainees’ core subject knowledge”, the recommendations it makes are:

Recommendation 1a: Subject knowledge development should be part of a future framework for ITT content.

Recommendation 2: All ITT partnerships should:

- rigorously audit, track and systematically improve trainees’ subject knowledge throughout the programme
- ensure that changes to the curriculum and exam syllabi are embedded in ITT programmes
- ensure that trainees have access to high quality subject expertise
- ensure that trainees have opportunities to learn with others training in the same subject

I can see why subject knowledge training can be pushed to the side, especially on school-based courses. When training in school, with teachers who are already insanely busy, it is easy for trainees to become focussed on their classroom management, or how to produce a lesson that fits the framework particular to that school. In the throes of observing other teachers and starting to plan lessons of their own, the trainee has little time or imperative to study their subject for its own sake, they are too occupied with creating worksheets and panicking about standing in front of 30 teenagers. Schools may assume that the background knowledge is already existent from previous qualifications so, in their limited time, choose to focus on pedagogy and systems and assimilating the trainee into the language and world of education.

But subject knowledge is the foundation on which great teaching is built.

Coming back to maths, the annual School Workforce Census found last year that 45.4% of maths teachers in England had a degree or higher in a relevant subject and that 26.3% had no relevant post-A level qualification. That’s a quarter of our maths teachers learning no maths past A level, and many of them training on a course with little-to-no subject knowledge input. If ever there was proof of a need for meticulous and consistent subject knowledge training across the country, that statistic could well be it. It is also counter-intuitive to note that even those with a mathematics degree often enter teaching lacking the necessary subject knowledge. Doug French wrote, in 2005, that

“it is often naively assumed that somebody who has studied at least some mathematics at degree level would be able to achieve [full marks on an exam paper] with little difficulty, although there may be the occasional topic that is unfamiliar. Again it might be assumed that somebody used to working at a higher level would be able to acquire the necessary knowledge readily. It is very clear to those who work with students aspiring to be secondary teachers that many certainly do not have this level of knowledge in relation to A level mathematics, but there are often very significant gaps and misunderstandings with much more elementary aspects of the subject.”

My own experience corroborates this statement. Going into teaching with a First degree in maths I naïvely assumed that I knew school maths. Oh, how naïve I was! I distinctly remember one of my early PGCE sessions when we were asked to differentiate between and define the terms *equation*, *identity*, *expression* and *formula*. Man, that got me thinking. And then they asked me why we “flip the second fraction over and multiply” when dividing fractions, and I genuinely did not know. This is the thing with mathematics, when you are at school you learn a lot and get good at a lot of procedures, but often don’t understand why things work. You then go to university and move onto advanced maths, you see very few numbers and hardly ever have to do any arithmetic, and so you rarely think about your school maths in any depth. You needed it, of course, because without it you wouldn’t have built up the knowledge necessary to study your degree, but you never have to think about *why* column arithmetic works, or what similar triangles have to do with trigonometry. Once you come to teach the subject, however, you need to know these things, and you need to know them really, really well. You need to be prepared for any question some bright spark throws at you, and you need to know how to challenge with deeper thought at one end of the spectrum while, at the other end, how to make the most complex of ideas (the oneness of one, how to be numerate, etc) accessible to those who genuinely struggle. You can only do this if your own subject knowledge is extremely good.

I was curious whether my experience was “normal” so last week I put a question out on Twitter. Of course this is highly unscientific, probably with a biased sampling frame, but nonetheless I found that I wasn’t alone in feeling like I didn’t know enough all those years ago.

Over the course of a day I received loads of replies, mainly from UK teachers but some from abroad too. There was a large spread of answers with a few common themes:

- Venn diagrams (set theory)
- Box plots
- Stem-and-leaf diagrams
- Circle theorems (common one here)
- Loci
- Trigonometry (
*why*rather than*how*) - Differential equations
- Geometric proofs
- nth term of quadratic sequences
- Stats
- More stats

From conversations it arose that often teachers had never learnt a topic when they were at school (box plots, set theory), and the topic never arose on their degree. Those teachers who found maths particularly easy at school and who never learnt formal methods for certain topics (like factorising a quadratic where a>1) had no idea what methods existed for such procedures. Those with higher qualifications seem to struggle with teaching the “easiest” topics, perhaps because they always found them easy themselves and never really had to think about them.

It was a very interesting thread and highlighted my concerns, and the observations of people like Doug French, that a new trainee teacher’s subject knowledge is nowhere near substantial enough to teach well. I have also observed that even teachers who have been in post for one or two years still have plenty of subject knowledge-related queries and can be thrown if you ask them the right question.

I believe that the ACME report is absolutely right in observing the need for more rigorous and consistent teacher training in England and I am convinced that a major part of that training should be the requirement that all maths teachers develop their subject knowledge to a high degree over the course of their training year. This subject knowledge should be expertly paired with pedagogical knowledge, including the common misconceptions that students hold for each topic, so that we can be confident that *all* our trainees are entering the profession with the strongest of starts, not just those fortunate enough to be on a course that already recognises the paramount importance of being an expert in your subject.

September 25, 2016 at 6:18 am

Reblogged this on The Echo Chamber.

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September 25, 2016 at 4:01 pm

Interesting ideas. In particular this phrase caught my attention: “Those teachers who found maths particularly easy at school and who never learnt formal methods for certain topics (like factorising a quadratic where a>1) …”

While never taught a “formal” method for factorization of a trinomial where a>1, there were techniques offered. I only recently found a more formal method (in an algebra textbook by Paul Foerster). There are those, however, who view factorization as “empty math” and only important as a step in solving quadratic equation, per this comment on Joanne Jacobs blog by a US teacher: “Completing the Square and Factoring Quadratics is good to show, play with for one day, then use the formula from then on. The triangle inequality thereom is another. We’re teaching lower quality topics currently. While there is value in manipulating symbols (a cross-cutting practice), the actual content leads nowhere.”

Yet, there are those who say “math is about patterns” (a statement which I find overblown, full of pedantry as well as ignorance, but let’s go with it for a moment). If we accept that math is about patterns why WOULDN’T we want students to know how to manipulate symbols to get expressions and equations in a form that allows for insight and solutions of equations. So an equation like x^4 + 5x^2 + 6 = 0 can be solved by making the substitution of y = x^2 and then we have a quadratic in the form y^2 + 5y + 6 = 0, which can be easily factored or solved by the quadratic formula. Also, factorization has a role in evaluating limits as well as simplifying rational expressions. But the prevailing view is this is empty content that “leads nowhere”. And less emphasis is given to them for the seductively deeper topics like statistics and linear algebra, which in the end don’t do very much in preparing students for the math they will encounter their first year at college.

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September 25, 2016 at 4:09 pm

It’s interesting you say this is the “prevailing view”. I don’t find that in the UK. Every maths teacher I know would disagree with that quote you give.

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September 27, 2016 at 7:14 am

Good subject knowledge is essential at all levels of teaching maths. How can anyone develop pupil’s understanding if they do not understand how and why themselves. I am priveleged in my work in maths education to work at all levels in may settings, the lack of subject knowledge is worrying.

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