# Teach Everything Correctly First Time

In 1995, the government in England embarked upon a major project in an attempt to raise standards in primary school mathematics. Following a general election in 1997, the National Numeracy Project was continued by the subsequent government, growing into the National Numeracy Strategy. A sharp increase in attainment quickly followed and the perceived success of the Strategy saw it extended into secondary education. The National Strategy continued until 2010, making it one of the most significant, large-scale and costliest initiatives to improve mathematics teaching that the world had ever seen. However, despite the positive and much-lauded improvements in primary mathematics, the effects did not appear to continue through to give similar gains in secondary school examinations.

A wide-reaching, large-scale professional development programme was designed and deployed across the entire country in order to train primary teachers in approaches to teaching mathematics. These approaches were comprehensively supported by extensive teaching and learning resources and even scripted and timed lessons.

It appeared baffling, then, that following such extensive support and the expected improvements in attainment at primary, the Strategy was having no impact on the critical exams that pupils would take at the end of secondary school.

One key reason for this strange outcome lay in the very training and support that all primary teachers were required to engage with. The approaches to teaching key mathematical ideas were very often sufficient for handling the level of mathematics that pupils encountered at the end of primary school, but did not hold true or did not generalise readily for use with higher-level mathematics. Many of the methods pupils were being taught were ‘backward facing’, suitable only to take them to a known end point rather than ‘forward-facing’ approaches to mathematics that plan for later mathematical ideas. This profound flaw in the design and implementation of the National Strategy resulted in a neglect for what is needed for mathematical progression.

An effective mathematics education system is one that focuses on teaching approaches that maximise subsequent progression, rather than a pursuit of short-term, superficial success.

Arranged as a coherent, long-term and unified journey through mathematics, the curriculum can drive a responsibility at each stage to prepare for the next. It is not the job of the primary teacher to get pupils to pass a primary mathematics test; rather, it is their moral duty to ensure that the mathematics they teach enables their pupils to continue to learn mathematics in secondary school. The secondary school teacher, too, has the same moral burden upon them – they must ready their pupils to be able to continue to study the subject at a higher level should they choose to do so.

In a conveyor belt system, it is all too understandable why, when faced with a pupil who is struggling, teachers often show the pupil a quick trick, rule or method that they can apply there and then in order to perform and ‘get through’ the lesson. These backward facing methods often give a quick win – allowing the pupil to feel comfortable and perform – but they seriously hinder the pupil when, later, they are faced with a problem that requires a proper understanding of an approach rather than the limited trick they were shown some years back by a teacher under the erroneous belief that their job was to get through a curriculum rather than to create young mathematicians.

We must seek to identify the reference framework for the ideas pupils will meet on their journey through a discipline. This framework must be known by all teachers at all stages so they understand what knowledge a new idea sits upon and the reasons for laying the new idea down – it will become the firm foundation of some other idea later in the curriculum. We must always refer back to the reference framework every time we introduce a new idea and then build the knowledge and the connections to other ideas gradually and unforgettably.

Ensuring that all approaches are always forward facing requires the explicit stating of a coherent, carefully sequenced curriculum, covering the entire journey through the subject. A curriculum planned for progression; a curriculum that ensures everything is taught correctly, first time. It is, therefore, incumbent upon all schools to ensure the curriculum they have at hand is not one limited to a year group or key stage, but is the full curriculum for the entire journey. The stated curriculum must be common to primary and secondary schools so that the forensically planned sequencing is not broken.

Some educators – particularly those who do not or have not ever taught in real classrooms – persistently object that approaches to teaching an idea must be personal to the teacher, that the teacher should be empowered to write their own curriculum and always make their own resources. This is an objection that holds back the teaching profession and prevents other professions from taking the education sector seriously. There are no heart surgeons in training who are told, ‘Hey, you come up with your own way of doing this surgery – anything goes, as long as it makes you feel good.’ A serious profession, with a canon of professional knowledge, extensively trains new entrants in the tried-and tested methods that have been shown to work by the entire profession over many years. Of course, as teachers become expert, they too will contribute to the professional canon and it is undoubtedly an important professional skill to be able to design tasks and, perhaps, curricula. But this should not be used as a false empowerment agenda that leaves thousands of teachers flailing and hunting around for ideas as they try to reinvent perfectly good wheels.

In mathematics, we ask pupils to grip around 320 mathematical ideas during their entire time at school (assuming they are to learn everything on school level mathematics). Surely, then, it should simply be the expectation that every single person who wishes to teach mathematics is instructed in the very best ways our profession knows of teaching every one of those 320 ideas. They should be instructed in the models, metaphors and examples that can be used to bring about meaning-making. They should be taught several approaches to each of those ideas such that they have at their disposal a selection of explanations, allowing them to vary those explanations expertly so that every single pupil has the moment of enlightenment. They should have shared with them the effective questions, prompts, tasks, investigations and activities that we, as a profession, have created and know to be impactful. They should be instructed in the known misconceptions that arise in pupils’ minds when encountering each of the ideas and have explained to them the ways in which they can best mitigate those misconceptions. And they should be required to know the mathematics.

Imagine what a difference could be made if, instead of a teacher-training route that is obsessed with pedagogy, teacher training was a serious discourse about the mathematics that pupils will meet and the effective ways to explain it **with meaning**.

Imagine what a difference could be made if we took the view that every single child can grip every single idea, not because of some chance ‘gift’ of ability but because their teacher is an expert at communicating all of those mathematical ideas.

Teacher training that subscribes to the cult of pedagogy without didactics sells our new entrants short and leaves them with little intellectually rigorous knowledge on how to teach their subject. I am completely uninterested in the teacher who can teach well. What matters is that they can teach *mathematics *well.

Teaching makes absolutely no sense without considering what is being taught. Yet, across Western countries in particular, over the past 30 years we have created a system that rewards the teacher who is the exquisite performer – the teacher who has the funkiest resources and games, the card sorter, the teamwork facilitator, the teacher who orchestrates the ‘active’ classroom by asking pupils to do times-tables questions whilst running around bean bags, the teacher with the big personality. A regime of inspection and observation that fetishises pedagogy has seen the performing teacher elevated over the teacher who is expert at enabling pupils to grip the difficult new ideas they need to get their heads around if they are to gain anything meaningful from their schooling, the teacher who knows that thoughts and ideas matter more than feelings, the teacher who demands effort and the truly purposeful grappling with tricky ideas.

The cult of pedagogy has almost entirely wiped out the central importance of didactics and reduced the assessment of the teaching process to one that focuses entirely on style without even a nod to substance.

Pedagogy is, of course, important – it is concerned with the general frameworks, styles, models and assumptions of education – but it is not sufficient. Pedagogy without didactics is a key contributor to the problems faced in many Western education systems.

Didactics is concerned with technical details. Pedagogy and didactics combined strengthen the focus of education beyond simply performing, beyond the style of social interactions in the classroom, to the details of specifically how best an idea can be taught, the prerequisite ideas on which it must build and whether those connected ideas are readily retrievable by the pupil. Pedagogy with didactics moves beyond simple exercise to the design of specific exercises that we know will more likely lead to the pupil making sense of and gripping the novel idea at hand. It moves beyond repetitive practice to carefully varied practice with specific sequences of questions or tasks that extend a pupil’s thinking and demands that they grapple with the underlying principles and structures of an idea. Pedagogy with didactics – the technical details that underpin a mathematical idea – forensically brings into existence an associated web of ideas, with correctness and reason, that forces the pupil to consider a novel idea in a broader framework, including problems beyond their current grasp such that they identify the need for further learning and purposeful work.

Mathematics teaching is about generating a long-term change in a pupil’s understanding of, and progress with, specific mathematical content. It is not good enough that the teacher plans a lesson on, say, fractions. What specific aspect of fractions should be examined and what specific models, examples and metaphors should the teacher use to explain that specific aspect? And on what specific questions and tasks should the pupils work? And why choose those specific questions – what do they specifically lead to and how will working on those specific questions define the associated web of ideas? Importantly, how will these specific experiences result in the pupils’ robust preparedness for subsequent specific content in their journey through learning mathematics?

Getting to grips with the didactics – the technical details – of the mathematics being learnt means that pupils have correct and efficient techniques which they can deploy at any time.

There are those who protest that mathematics education should not be about techniques but instead should focus on problem solving. This is a false dichotomy and the argument that wages eternally between mathematics factions is unhelpful. The reality is that if we want to prepare pupils for a continuing love of and desire to study mathematics, we must integrate the requirement that they become fully proficient with techniques with the requirement to solve problems.

Solving problems makes no sense as a separate activity, but only as an integral part for all pupils of learning to use each technique in its full complexity – which is what is needed if those techniques are to support subsequent mathematics.

There are many subject areas that have, for a long time, truly embedded a mastery approach to schooling. In art, music, sport and dance, it is the norm for the teacher to carefully and deliberately progress the pupil through the subject. The sport teacher does not demand the pupil who can leap only 80 centimetres to suddenly jump 2 metres. The music teacher knows that the pupil must go through the arduous process of getting to grips with scales as a step towards playing a piece with ease. The outcome of successful art teaching may well be a beautiful and unique sculpture or painting, but the art teacher knows that the pupil must invest many years in perfecting technique before those techniques can come together into something magical. Consider the eloquent professional ballet dancer, who moves with apparent ease in a stunning and profound performance. How wonderful it would be to be able to replicate their achievements – but those achievements are the result of thousands of committed hours rehearsing micro-techniques over and over again until they become behaviours.

Mathematics, too, comes together in an eloquent dance. All of those basic techniques that once stood alone and were practised repeatedly and with purpose form a complex, integrated schema of techniques, understanding and behaviours that combine into something truly beautiful as the mathematician dances their sublime dance. It is only possible because the pupil has been exposed to techniques in a carefully scheduled journey and they have worked hard to perfect those techniques with great accuracy. Each of the techniques that they meet along the way is a doorway to profound and awe inspiring mathematics.

The didactics of mathematics analyses how the content of elementary mathematics can be arranged into a carefully sequenced succession of ideas and how each idea can be introduced and organised so that all pupils understand its underlying principles and can fluently work with all of its related processes. The didactics of mathematics also examines the main barriers to learning the new idea, meaning the teacher can design better-informed instructional approaches.

Although the focus above is largely on my own subject area, mathematics, the central point holds for all subjects: pedagogy without didactics is simply not good enough. We should all be aiming for pedagogy with didactics.