19 A*s

I have called this blog 19 A*s for judicious reasons. My 27 top set maths GCSE students achieved 19 A*s and 8 As this summer so the first reason is to show off. A*s are a high value currency in schools, so the other reason is that I hope the title will catch the eye of those involved in education, and they end up reading it; it’s a marketing device. Effectively, this is about pedagogy, and although its fairly lengthy, I recommend sticking with it to the end, there’s a fair amount of gold here. 

I am going to discuss how to teach, well, how I think one should teach. I welcome disagreement, indeed I think it’s one of the best ways to improve teaching practice: to have reasonable disagreements and conversations on what the best way to teach is. I welcome comments to this blog even if they disagree with everything I am saying.

I am talking about teaching, not just learning. It is possible to learn all kinds of things by oneself, for example, many teenagers learn to play the guitar or DJ or rap to a high level by learning for themselves, but this is not what I want to discuss; this blog is about what happens in schools.

I contend that all teaching practice can be encapsulated in two simple principles:

  1. Have the students learned what I wanted them to learn?
  2. How could I improve on this?

I can draw out important themes inherent in these principles, for me there are five:

  • The Curriculum (the substance of what I want students to learn)
  • Expectations (how what I want them to learn varies with different students)
  • The Mediation of the Curriculum with the Students by the Teacher (the teacher’s role)
  • Assessment (what have/ haven’t they learned? How can I use this information to inform my teaching, and my feedback for , and dialogues with, students?)
  • Teacher Improvement (what could I do differently to improve the students’ learning?)

I will discuss these themes with reference to my subject, maths. I believe that the thoughts I am presenting are relevant to all school subjects.

The Curriculum

By its nature, the school curriculum is a sample of the subject to be studied. For example, the maths National Curriculum (NC) contains much about Functions and nothing about Number Bases other than 10. Some thought can be given to what is and isn’t put into the school curriculum and the reasoning behind these decisions. If a school is not required to follow the NC, as some aren’t, they are still required to teach what is required for the public exams; the curriculum is determined by the GCSE exam.

It is not just the choices of subject content that is contentious, Daisy Christodoulou observes that in ‘the KS3 [NC], last revised in 2007, there is a deliberate reduction, and in some cases complete removal, or subject content … [the history] KS3 [NC] prescribes not knowledge but instead a list of skills.’ There is a fundamental dichotomy in curricula between those based on knowledge, and those based on skills.

In her book, Seven Myths About Education, Christodoulou recruits narratives from literature to illuminate her arguments about the unhelpfulness of a skills based curriculum as opposed to a knowledge based one. For example, she notes how the perception of Dickens’ character Gradgrind represents and perhaps encourages the foolhardy notion that teaching facts has a pernicious effect on students:

‘Comparing a teacher or anyone involved in education to Gradgrind is an insult, suggesting that the teacher is both emotionally stunted and doing great emotional damage to their pupils.’

Recently, reading Engleby by Sebastian Faulks I came upon these thoughts of the eponymous protagonist:

‘the 1970s schoolteachers who decided, for some perverse political reason, to withhold knowledge from our schoolchildren. The first generation thus deprived are now themselves the teachers, so it’s less treasonous for them: they don’t have the knowledge to hold back.’

Albeit that Engleby is an unreliable narrator, indeed a murderous psychopath, it certainly seems that the dichotomy is firmly entrenched in our culture. I am firmly on the side of a knowledge based curriculum, I’m with Gradgrind, and in this I have been persuaded by the excellent writing of Joe Kirby.

Moving back to the classroom, I was much taken by the principle I was taught for planning my teaching on my PGCE at King’s College London: ‘what is the maths?

I think this is a helpful way to cut through the noise that can be generated by the differing approaches to curricula. For example, if I am teaching Trigonometry, I start by mapping out what maths is involved in the topic, in this case: ratio, similarity, wave functions, and so on. I can draw a distinction with this from different, and I believe unhelpful approaches to planning such as:

  • what relevance does this have to the students’ lives?
  • how does this relate to real life?
  • what do the students need to know to pass the exam?

I call this ‘teaching maths within the discipline of maths’ and it ties in with a knowledge based approach to the curriculum. I think an essential part of planning is the recontextualisation of one’s own subject knowledge in order to find a consistent knowledge based narrative that will drive the learning of the topic.

I think it also feeds in to what I think should be the basis of all school curricula: ‘the best that has been thought and said’. I teach maths because it is a wonderful, rich subject to study. My end goal is for students to leave school with a wonderful, deep knowledge of maths. Helpfully, in my experience of working in a number of comprehensive schools, it is the substance of this esoteric world of maths and the effort to master it that is of most motivation to students of every ability level.



I recall that the first essay I wrote for my PGCE, before we had even started the course, was one about our experience of education thus far. I chose to open my essay with a quote from a belle & sebastian song called Expectations:

‘And the head said that you always were a queer one from the start
For careers you say you went to be remembered for your art’

The line captures a mood of low expectations and a lack of understanding of children by schools that chimes with my own schooldays. I have a visceral aversion to arguments that seek to limit the potential of groups of students, and it is always a shock to me when I read an organisation such as Maths Mastery having to explicitly state as one of their key 6 principles ‘High Expectations’.

For me, high expectations are a sine qua non of teaching. In my experience, it is probable that the proportion of students able to achieve a good GCSE in maths (at least a C or equivalent) approaches 100%. There may well be reasons that a certain number of students cannot reach this level, however I am not qualified to write about those students and it is not helpful to this discussion.

It seems reasonable to me to have a stated aim within every school for all students to achieve at least a C grade or equivalent at GCSE. However, I think perhaps a covert aim of all students achieving an A or A* grade or equivalent makes more sense. This aim incorporates the high achieving students as well, and focuses pedagogy on an excellent education for all as opposed to focusing on a group of threshold students. In short, it gives all teachers involved in the school the very highest expectations of all students. Clearly, not all students will get an A or A*, but it is a different and better mind-set to have a proportion of your students failing to reach the high level you would like them to meet but still getting a good grade rather than a proportion just overcoming the low hurdle you have set them.

I have heard it said that great exam results are necessary but not sufficient for a great education, and I wholeheartedly agree with this. I consider the attitude that results are the be-all and end-all of a school’s purpose to be a function of low expectations. I think it is far preferable to look at exam results as part of a broader view of education: if students are taught the best that has been thought and said in maths, and taught this well, then outstanding exam results can be a product of this teaching. In students’ five secondary school years before the GCSE exam, I propose that the first four years are focused on mastering maths, with the final year bringing a shift in focus the exams, using that mastery in maths to ensure fantastic exam results.


The Mediation of the Curriculum with the Students by the Teacher

Mediation is a word I use advisedly and for two main reasons. First, it is not straightforward for students to learn the curriculum. Were it so, we could just give students a series of books to read, and they could all ace the exam after a few years. This does not happen and mediation is necessary. The nature of schools is that teachers act as a conduit for ensuring that students learn the curriculum, and with increasing expertise of the teacher, the mediation gets smoother, and learning increases and is enhanced.

Second, there is a hazardous aspect of teacher’s role in transmitting the curriculum to the students. Mediation does not always work. Indeed, we might all have experienced, or know someone who has experienced a teacher who made it harder to learn a subject. For example, if I, as an inexpert teacher, decided to teach GCSE Trigonometry by saying that all students needed to do was to remember the mnemonic SOHCAHTOA, then I might get them to pass certain exam questions on this topic, but it would compromise their ability to understand other maths associated with Trigonometry, such as the proof of the cosine rule, and would certainly inhibit their chances in maths at A-Level. It seems to me that there all kinds of pedagogical decisions such as this that could lead to a failure in the mediation.

There is a dichotomy in the way that one chooses to teach, the two schools of thought are Traditional and Progressive, and this seems to me to be the most contentious issue in contemporary educational discourse.

Before stating my position, I think it is helpful to look at the analogy of one’s political views. Here there is a dichotomy between Left and Right, and I will look again to Faulks’ Engleby for an illumination:

‘I don’t really understand British politics… you’d have thought that nowadays most people would want some sort of market economy to get the motor turning vigorously, then buckets of free health care from the resulting tax take…. No. As of May 1987, a true Brit wants either a) socialism with as few deviations as possible from a command economy (Kinnock); or (b) a Malthusian free-for-all, in which the survival of the fittest takes on a quasi-moral dimension (Thatcher)’

Engleby describes the pre-Blair politics of the late 1980s as effectively being a choice between the Left and the Right. Faulks is writing with the benefit of hindsight, and he brings Engleby up to speed later in the novel, when in the 21st century he observes that ‘all British politicians are Social Democrats now, but back then holding such beliefs was derided as having no policies.’

We have lived for most of the last 20 years with Prime Ministers in Blair and Cameron who occupied the centre ground of politics, choosing some tenets from the Left and some from the Right. Indeed, the majority of people in the country vote for parties that have fundamentally capitalist views on trade, and fundamentally socialist views on healthcare. Pure ideologues such as Jeremy Corbyn seem anachronistic and extremely unlikely to win large amounts of public support.

Back to the Traditionalists and Progressives, I would say it is still common to find teachers who favour one or the other in its purest form, the ideologues. However, I think the general and helpful trend is for increasing numbers of teachers to occupy the centre ground, and take some beliefs from the Traditionalists, such as behavioural conformity being an essential part of classroom norms, and some from the Progressives, for example a belief that every student can succeed regardless of background and that it is the responsibility of teachers to strive to ensure this happens.

Perhaps the archetypal Progressive teacher is John Holt, who held that ‘learning is not the product of teaching, learning is the product of the activity of learners.’ Holt wrote anecdotal books about his experiences as a teacher as opposed to researched ones, so his writing is not part of institutional educational discourse. However, Daisy Christodoulou writes polemically about Progressive educational researchers who she considers to have had a strong and pernicious influence on the NC and education in Great Britain in general: ‘just as with Dewey, Rousseau and Freire, the rhetoric used in the NC documents also reveals unease about facts and knowledge. In the explanation of the intended outcomes of the curriculum, the words ‘knowledge’ and ‘facts’ were not used once’. She goes on to say:

‘Rousseau, Dewey and Freire were wrong to see facts as the enemy of understanding. All the scientific research of the last half-century proves them wrong. The modern bureaucrats and educationalists who base policy and practice on their thinking are wrong too, and with less excuse, as they have been alive when evidence that refutes these ideas has been discovered. Rousseau was writing in the eighteenth century; Dewey at the turn of the twentieth; Freire in the 1970s. Research from the second half of the twentieth century tells us that their analyses of factual learning are based on fundamentally faulty premises.’

There is no shortage of critiques of Progressive education in the same timbre. I am not well-read enough to quote them all, but David Didau, Greg Ashman and Joe Kirby all write excellently on the misapprehensions apparent in educational discourse stemming from a Progressive outlook. I am persuaded by their excellent blogs and arguments, and as a result I do think that my teaching has improved; I wonder why it took me so long to even hear about some of the Traditional techniques. So, does that make me a Traditionalist?

I think that four or more years ago you could have accurately described me as a Progressive, and I have changed hugely since then. Ashman describes here the signs that one might be a Progressive educator, and certainly I believed that students learned best by doing rather than being told. My thinking was in line with a lot of John Holt’s views. Now here’s the thing, I wasn’t a bad teacher then, my classes still did well, and I had many positive comments from both students and colleagues. It is my view that a more important educational dichotomy is one between teachers who are reflective, and those who are dogmatic in their approaches.

My position on the Traditional/ Progressive dichotomy is that I am in the centre ground, perhaps slightly towards the Traditionalists. I believe in what Ashman calls explicit instruction, and I think that worked examples, silent working on numerous questions, and elements of Engelmann’s ideas about teaching a new concept by using examples of what it is and what is isn’t are all good, important aspects of classroom practice. I think my teaching has improved a great deal since I started using more Traditional ideas. So, why am I not more of a Tradionalist?

I think that a lot of Progressive ideas have value. It seems to me that a lot of the critiques of Progressive education are critiques of interpretations, often government interpretations, of these ideas and not the ideas themselves. If we think for a moment of what a government roll out of a Traditional perspective would look like, then I think it’s fair to say that a lot of the nuance would be lost in bureaucracy and administration, leaving ideas like explicit instruction ripe for blunt critiques.

The Progressive ideas I like are to do with how students learn. Here, I pay respect to the advances of cognitive psychology and have to admit that I do not know enough as I should about them. I am reading books by Daniel Willingham and David Didau on the subject in order to remedy this. I judge that there is in broad terms, an alignment with cognitive psychology and the Traditionalists, and sociology and the Progressives. I turn my attention to the sociology of education.

I see no reason why sociology should not stand alongside psychology in the analysis of how students learn. I heard a leading thinker in Education recently say that some teaching ideas were stuck in the 1980s. Presumably this points at a lack of incorporation of advances in psychology in the thinking of the teachers. I would argue that how students learned in the 1980s is not so different from how students learn now. Perhaps some ideas in sociology have now been proved false, and intellectual progress must refine and reject ideas as time marches on, I assume this will be the case for cognitive psychology as well.

Jeremy Burke cites the example of Mike Ollerton offering ‘a range of starting points and extension ideas’ as one possible strategy for mathematics pedagogy. On a personal level, firstly, Jeremy Burke was the leader of my PGCE course, and secondly, I have used Ollerton’s 100 Ideas for Teaching Mathematics extensively, especially in my early years as a teacher. The time I used them the most often was when I worked as the Maths Leader at a Pupil Referral Unit for students at GCSE age who had been permanently excluded from mainstream school. I remember the Headteacher telling me how one of the students, Sly Wun, had said that ‘it had touched his heart’ when he saw me planning lessons using that book. Empirically, this approach to teaching maths worked with those students. I had many comments from the students saying that I have reinvigorated their interest in maths, or who had found they had liked maths for the first time, and the results were the best they had ever been when I was there. Now, there are many factors other than my use of Ollerton’s ideas that contributed to this. So, I will try to analyse what it is in these Progressive ideas that worked for me.

Burke describes an idea of Ollerton’s called ‘Max Box’, and identifies it as an example of a Situation-Composition-Question pedagogical strategy. This seems to be to be a Progressive way of teaching, and I think my point of view in the conversation I had with David Didau about the best way to teach a new concept runs along this line of thinking. Burke identifies more strategies, amongst them: Demonstration-Drill-Correction, which may be a Traditional approach, and Exposition-Problem-Explanation, which I think has a foot in both camps, and perhaps provides entry to the false dichotomy argument.

I will now set out my argument as to why I think that Situation-Composition-Question is a helpful pedagogical strategy. In the conversation about teaching a new concept mentioned above, I talk about teaching Congruence by first giving the students a mathematical situation in which a simple question is posed, and it requires very little mediation to introduce. The students can explore that mathematical terrain in order to answer the question, but little guidance is given by me. In my observations of this happening, on numerous occasions, the question always comes up of whether one can class certain shapes as the same or different. Fundamentally, this is what congruence is about, so I am able to explain it to students when they are in a receptive state to hear the information. I have heard it argued that it is impossible to think about something if you know nothing about it, and yet I regularly hear my students discussing congruence without any prior knowledge of what it is. A number of interesting mathematical ideas arise from this activity, and I can compose a series of questions that address Transformations of the 2D plane based on it. The key idea here is that it is a mathematical domain that the students are working on and their mathematical vista is not limited by my mediation. Now, I think it takes a tremendous amount of expertise and experience to use this strategy effectively, but I see no reason for that to mean that it has no merit. In short, I can choose my pedagogical strategies from the Progressive or Traditional tradition as I see fit on a topic by topic basis.



I relish two comments attributed to Dylan Wiliam about assessment. First, Tom Sherrington, in a typically brilliant blog, paraphrases Wiliam as saying ‘the total time spent by teachers on marking costs taxpayers £billions for very little benefit, making it the most expensive PR exercise in history’; second, I’ve heard Daisy Cristodoulou say that Wiliam would have preferred Assessment for Learning to be called Responsive Teaching.

I have a visceral hatred of pointless marking. When a member of SLT, at a school I worked in previously, justified the elaborate marking policy as being simply a tick-box exercise for Ofsted, I was apoplectic. The draining hours of writing comments that I know are not helpful to the students, because I have been told I have to, is demoralising.

My teaching responds to the learning of the students. Through sequence planning and not lesson planning, I am flexible in what happens during lessons, and I react to what I observe is the learning in the lesson. One good way I have found to effectively assess the learning of the class is the judicious use of Multiple Choice Questions (MCQ). A well designed MCQ can give me feedback on who understands the topic, and can draw out misconceptions and who has them.

Alongside MCQs, I think it is useful to spend time talking to the students on a one-to-one basis each lesson. I am in the process of attempting to ensure 15 minutes silent work every lesson in which I can talk to 5 students for 3 minutes each giving them personalised feedback on their work. Whether this feedback needs to be written down is debateable, what is not in doubt is that this type of one-on-one contact is both rare and helpful to students.

My plan for assessment this year is: MCQs in lessons and midway through a topic; one-on-one feedback; end of topic assessments with one-on-one feedback, and then revisiting topics after one month, 3 months, and 6 months.


Teacher Improvement

As I said at the beginning of this piece, I think one of the best ways to improve one’s practice is to engage in constructive discourse involving disagreements about pedagogy. I like the title (as well as the content) of David Didau’s book What if everything you knew about education was wrong? Giving consideration to views that clash with one’s own can be difficult and frustrating. Often teaching practice is informed by a teacher’s deep rooted experiences of their own education and what has or hasn’t worked for them. The strong beliefs that arise from this can be an emotive issue and Didau thinks that it is exactly these beliefs that could bear the most fruit if we are prepared to challenge them. As he puts it ‘ask yourself: what is it I believe most strongly? Those areas may well be where the gold is, if only you’re prepared to dig.’

I have adjusted some of my most treasured beliefs after constructive disagreement, and this has been to the benefit of my students. Reading, discussing, attending events such as ResearchEd, and crucially, being a reflective teacher as opposed to a dogmatic one are what I think are the most important factors in teacher improvement. And what to reflect on? My two principles of teaching:

  1. Have the students learned what I wanted them to learn?
  2. How could I improve on this?

This is where I stand in terms of pedagogy at this moment in time, September 2016. Now, tell me where you think I’m wrong.



Paper by Jeremy Burke is ‘Being told or finding out or not: a sociological analysis of pedagogic tasks’ (2014).

Engleby by Sebastian Faulks was published in 2007 (Hutchinson)

Seven Myths About Educations by Daisy Cristodoulou was published in 2014 (Routledge)

What if everything you knew about Education was Wrong by David Didau was published in 2015 (Crown House Publishing)

The belle & sebastian song Expectations is written by Stuart Murdoch and appears on the album Tigermilk initially released in 1996 (Electric Honey)


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