## By Paul McGarr, London maths teacher

Much has been made in the media of “The National Academy” – the definite article should induce some caution.

It claims to be a collaboration of 40 teachers “plus organisations across the sector”. Well, yes. But the organisations, and funding source, include key Multi Academy Trusts and the Academy’s board has such people in ample supply. They include the chief executive of the national coordinating body for academies and MATs, and the chief executive of United Learning – and Ark and Harris are involved too.

Key figures also include Daisy Christoudolou – who made her name pushing the merits of “declarative knowledge” beloved of key Tory education figures, straight from the playbook of US thinker E.D. Hirsch.

Behind all this stands funding and support from the government.

So a bit more than a coming together of classroom teachers in a time of crisis.

Others will no doubt write more on this topic. And others too will, no doubt, look at areas of the curriculum offered at different age groups and subjects by the academy.

Here I want to limit my remarks to an area I feel more qualified to comment on— the secondary mathematics content offered.

I should make clear that, just because I disagree with the educational philosophy behind some of the key figures backing the academy does not mean, of course, that they can’t produce good material.

Education would be a poor business indeed if we always thought like that. As a maths teacher I have seen and enjoyed many lessons designed and delivered by people I strongly disagree with on many educational questions – and long may that continue.

Also I am sure that the vast bulk of the teachers involved are excellent colleagues and professionals doing their best in these difficult times – and we all need all the support we can get. The BBC and many others have weighed in with many helpful contributions and resources.

I do want, however, to raise some discussion about the approach adopted so far in the secondary maths curriculum. There are many lessons that have good elements and teachers who know their business.

But there are also some common themes running through the whole package which I want to question.

First, every lesson is almost uniformly the same. There is a pre-quiz, a video presentation with worked examples, a worksheet or similar and then a post quiz. Far be it from me to say this is not a format which can sometimes be appropriate. But to have this rigid structure for every single lesson becomes, well, just a little robotic after a while.

A second point is that the topic is, so far, identical for every year group. So years 7- 10 all get algebra this week – and much of it is virtually identical from year to year. Again I appreciate the constraints of time – but is this really the best or most appropriate way to organise a curriculum?

More serious is the approach in the lessons. They are all rigidly procedural – with some algebraic technique modelled and students expected to then practise, with very little hint of why this matters, how it connects to other areas of maths let alone the world, or the wider curriculum.

One teacher gives the game away a little when explaining the structure of every lesson and says the post test is there to “check to make sure you have consumed the knowledge”.

It is knowledge which the experts have and which you have to take on trust from them is important in some undefined way and which you, the student, just have to consume and demonstrate you have learned.

This may be some people’s idea of education. It is not mine.

Could there not be at least some reference to the historical development of algebra? Could students not, for example, be asked to research a figure like Al-Khwarizmi or watch a video from one Jim Al-Khalili’s excellent BBC series on science and Islam, to cite just one among many excellent examples?

Could there not be at least some discussion of how algebra relates to a generalisation of the basic laws of arithmetic and the structure of arithmetical calculations – and yes, you can do this in a way that is perfectly suitable for even year 7s with a little thought.

There is, in short, little attempt to develop and deepen conceptual understanding at all – and instead a narrow focus on isolated techniques and procedures which you just have to consume and replicate. It is all paint by numbers, step 1, step 2, do as I do – for reasons you aren’t even asked to think about – and now repeat after me.

I hate to be overly critical of teachers trying their best, as I am sure many of those involved are.

But were I observing these lessons and asked to give constructive feedback aimed at professional development I would feel obliged to pick up on some important mathematical and conceptual problems and missed opportunities as well.

To pick out a few, with apologies to non-subject specialists.

We are told, correctly, that 4x4x5 can be worked out as either 16 x 5 or 4 x 20. And that the reason is that multiplication is commutative – the order doesn’t matter.

This is wrong. Yes multiplication is commutative – so 4×5 = 5×4. But the example given is true because multiplication is also associative – grouping doesn’t matter – so (doing what is in brackets first) (4×4)x5 = 4x(4×5).

The two properties are fundamentally different – and this distinction becomes very important when doing algebra – the very subject which all the lessons so far focus on.

Let me leave aside the use of apples when talking about the use of the letter “a” as a variable in algebra – never a good idea – and instead look at the basic idea of using letter symbols in algebra.

We are told they are variables – things which can take any value (sometimes restricted to particular types of number). But a common misconception among students is the difference between when a letter is actually such a variable and when it is instead a particular but as yet unknown number.

Yet we are introduced to n being a variable (any number) when it represents my age and then that if my father is twice my age we can write his age as 2n.

Well yes – except then n is not a variable as the relation between my age and my father’s will not be the same next year when I am 1 year older. If I am 20 today and he is 40 then next year I will be 21 and he 41 – definitely not 2n if my age is n.

These questions may seem nit picking to some – but I think these things matter if you are teaching a subject you love.

One final point is the constant use of BIDMAS as a rule students have to learn to decide which order to do calculations in. This is an acronym for Brackets, Indices, Division, Multiplication, Addition, Subtraction. And it is constantly referred to in all the videos as a rule you have to just learn and use – though where it comes from and its justification are never addressed.

The bigger problem is that the rule is wrong and taught in this way will lead to misunderstanding and error.

What is true is that things in brackets and indices (powers like squares and cubes) must be done first. It is also true that all division and multiplications must be done before all additions and subtractions. But it is NOT true that you have to do division before multiplications or that you have to do additions before subtractions.

These pairs of operations are inverses of each other – and therefore fundamentally related – and if a calculation only has either division and multiplication or only has addition and subtraction you just work from left to right. If we want to get technical, multiplication and division are distributive over addition and subtraction – but addition is not distributive over subtraction.

Why does this matter? Because if you mechanically learn and apply the so-called rule you will misunderstand arithmetic fundamentally and make significant errors – presumably what we should be seeking to avoid in education.

To illustrate take the simple sum 1-2+4

If we follow the BIDMAS rule we should do the addition first and get 1 – (2+4) = 1-6 = -5

But this is wrong. You should just work from left to right and get (1-2)+ 4 = -1+4 = 3 which is correct.

So it matters. ‘OK’, someone may say ‘but couldn’t we just do -2+4 to get +2 and then add the 1?’

Yes, but then you are straying into even bigger problems as you have used the “-“ symbol to get your +2 so how do you then combine your answer with the 1 – unless you mysteriously suggest that there is a missing + sign which you just didn’t bother writing.

It comes down to another confusion – and a very common one which takes a lot of developing understanding among students to properly deal with: that (unfortunately) we use symbols to mean two different things in maths.

We use, for example, the “-“ sign to mean a negative number – an object on the number line, the mathematical equivalent of a noun if you will. But we also use it to mean the operation of taking away, of subtraction – something you do to two numbers to produce a third number (a binary operation to use the technical term) – the mathematical equivalent of a verb.

I am not saying these things are easy – either to learn or to teach, but they do matter and the simple learning by rote of a rule, without thinking all these things through, leads to a poorer understanding and to errors such as getting the answer -5 instead of 3 for the simple calculation above.

I could go on, but I think that will suffice to make my point. I could put up with the fact I am not enamoured of some of the forces behind the academy, and even the rigidity of the lessons.

But the flawed yet unwavering focus on simple procedure and on “consuming” knowledge, coupled with significant conceptual problems in the mathematics being taught is not something I can or will tolerate.

Why this worries me comes back to that definite article “the” national academy. Is this the model that those government figures behind the project – the trinity of Gibb, Gove and Cummings – have in mind for their vision of education after this crisis? I hope not, but if so we have a fight on our hands to defend a very different vision of education.

Paul McGarr April 2020