17 Questions with Peter Mattock: Making Mathematics Visible, Connected and Meaningful

Peter Mattock is an experienced maths teacher, leader and author. In these 17 questions, he discusses effective maths teaching, problem-solving, teacher development and the importance of helping pupils make sense of mathematics.

Question 1

How has your career in maths education developed over time?

It started much the same way as any other. I completed my PGCE in 2006 and started teaching secondary maths that year. I moved up to second in charge and then head of department over the next five years.

It was a couple of years into my first tenure as head of maths that, I guess, things took a different turn. I was successful in securing a contract to film some CPD videos that were featured on the Pearson ActivTeach platform. They wanted the use of representations to be included in the videos. Prior to that, like most secondary teachers of maths, I hadn’t really used representations in my teaching, so I had to do quite a lot of research myself around it. This started my journey with the use of manipulatives and representations.

Shortly after I moved to my most recent school in 2014, I was lucky enough to be part of the first cohort of secondary mastery specialists trained by the NCETM. I combined these experiences to lead to the production of my first book, Visible Maths. I started using all of my spare time and energy to support the development of mathematics teaching through my writing, my websites, and attending and speaking at numerous conferences – following my passion to make maths education in this country as good as it can be – as well as continuing to read and, most importantly, think. Teaching is, for me, ultimately a thinking profession – as teachers, we should be immensely curious about learning and committed to our own intellectual development as much as our pupils.

This then led to the publication of my second book, Conceptual Maths, which contains approaches to teaching every idea in the school maths curriculum, at least as it stands, and, more importantly, highlights the connections between them. It is these connections that are often lacking in pupils’ experience of mathematics learning, but that I think are crucial in ensuring more pupils make sense of mathematics. At that time, I also decided to write Leading Maths so that others could benefit from my and some of my colleagues’ leadership experience.

In 2021, I was promoted to a whole-school leadership role in charge of data and assessment, intervention and timetabling in the school where I had worked since 2014. Whilst I was good at my job, I began to realise that it didn’t instil in me the same enjoyment and passion as when I was working day-to-day in maths education. When I went on to complete my master’s degree in Educational Leadership, it drove home to me that I wasn’t interested in moving further up the leadership ladder – I didn’t want to become a headteacher, as the role would only serve to move me further from my passion. Fortunately, at that time, the role of National Secondary Maths Lead for Twinkl Educational Publishing became available, which I hoped would provide me with the opportunity to continue to have a positive impact on maths teaching across the country as a full-time job.

Question 2

What first inspired you to become a maths teacher?

Interestingly, I wasn’t inspired to become a maths teacher until several years into my career teaching mathematics.

When I was younger, I was a senior member of the Army Cadet Force, and so spent a lot of time instructing more junior cadets and found I was quite good at it. However, I didn’t aim to make a career in teaching. Having been disqualified from my first-choice career as an Army officer on medical grounds, I intended to follow my passion for physics into a career in research or academia. I started a theoretical physics degree at the University of Reading but, being completely uninterested in experimental physics, I quickly fell behind in my lab work. So, in my first year, I swapped to a mathematics degree, with no clear idea as to what the future held. During university, I got involved with the cadets again briefly as an instructor, and that experience reminded me of the pleasure I found in teaching. So, I participated in the Student Associates Scheme during my later years at university and successfully applied to do my PGCE.

However, at this point, I wasn’t inspired to become a maths teacher so much as to be a teacher who happened to teach the subject I did my degree in. I wanted my pupils to do well, of course, but that was more about getting them to do well in exams, whether they really learnt any maths or not. I think I was first inspired to become a maths teacher – by which I mean really care about the mathematics making sense to pupils – during my later years teaching in Oxford and early years teaching in the Midlands, when I started doing more reading and research, attending conferences and immersing myself in the world of maths education beyond my school.

Question 3

What do you enjoy most about teaching mathematics?

The most joyful thing for me is seeing pupils make sense of an idea. Seeing the light in their eyes and hearing the pleasure in their voices when things click into place. The phrase I most enjoy in the classroom is, ‘Oh, that’s just like when we did…’ because then I know an idea has stuck – it has been understood and connected to the bigger concept that we are studying.

Question 4

Why do some pupils find maths difficult or intimidating?

I think there are, broadly speaking, three reasons why pupils can find maths tricky:

Due to the nature of the curriculum and pressures of accountability, much of the content pupils encounter is either rushed or premature – by which I mean the prerequisite ideas are not secure enough, either due to gaps in prior knowledge or simply because pupils haven’t had enough time to mature with an idea.

The connections I talked about before are not made explicit for pupils, which leads to maths being viewed as a collection of disparate facts and processes that need to be remembered. When maths becomes a memory game like this, there will always be winners and losers.

As mathematics learning progresses into secondary school, which is when enjoyment of maths learning drops off, it necessarily becomes more abstract and removed from pupils’ concrete experience – particularly when concrete manipulatives and pictorial representations are not used as frequently as in primary school. Whilst, for me, this move towards abstract generalisation is what makes maths interesting and important, many pupils struggle to bridge this gap, and their lack of ‘success’ and the lack of apparent relevance have a demotivating effect.

Question 5

How can teachers help pupils understand maths rather than simply remember a method?

I think there are a couple of factors at play here.

First are pupils’ early experiences of formal mathematics teaching, which are often focused on speed and correctness. This starts the journey for many pupils, of whom girls are over-represented, towards viewing maths as something you are simply ‘good at’ or not, and believing that, if you are not immediately successful, the subject isn’t worth pursuing. We need to change what we value in early – and, to be fair, all – mathematics education so that speed and correctness are a by-product of depth of learning and sense-making, rather than the goal in and of themselves.

Secondly, as I mentioned before, the connections between and within concepts are rarely made explicit. The considered and deliberate use of models and metaphors, with manipulatives and representations being used to give insight into those models, can help to highlight these connections – when the same model and the same representation are encountered, it makes it clear that the same concept is at play. Unfortunately, the curriculum documentation that many schools work with, including the national curriculum, does not do enough to make teachers aware of these connections – rarely do you see things like congruence linked to trigonometry or constructions, for example.

Question 6

What is the value of using visual representations in maths lessons?

As I said above, manipulatives and representations are important in supporting pupils to make sense of the models we use for abstract concepts. Even something like ‘3’ is an abstract concept – it only makes sense when we learn to link that symbol to collections of three objects. This use of concrete and visual representations is ingrained in that early learning of mathematics, but is quickly lost as pupils move through school, rather than being maintained as a ‘normal’ way of learning about mathematics.

Question 7

What is one mathematical concept that is often taught poorly?

There are many that can be taught poorly, but if I had to pick one, it would probably be early introductions to algebra. However, that is because the prerequisite generalisations in pupils’ numerical thinking are often not built. When pupils see that 3 + 2 = 5, 30 + 20 = 50, 3 × 12 + 2 × 12 = 5 × 12 (no BIDMAS here), and 3/7 + 2/7 = 5/7 are all governed by the same mathematical structure, then 3x + 2x = 5x becomes the logical extension, not some mysterious thing to be learned as a separate fact.

This comes back to the focus on speed and accuracy talked about earlier – when the focus in the calculation 3 × 12 + 2 × 12, for example, is on getting to the answer quickly rather than understanding the structure of calculations of this type, then these connections get lost.

Question 8

How can teachers help pupils become confident problem-solvers in maths?

Part of this is what we have been talking about – to solve problems in mathematics, pupils need that deep, connected knowledge of mathematical ideas.

Alongside this, pupils need exposure to a wide array of examples and non-examples of problems to engage with. Using things like ‘same surface, different deep’ (SSDD) problems, more-same-less grids and other task structures that provide opportunities for all pupils to reason and problem-solve ensures that pupils have a chance to apply and further develop their skills.

I would add to this the use of goal-free problems. Cognitive science, specifically the goal-free effect, tells us that removing the goal allows pupils to focus on the mathematics they are using, rather than on whether they have solved the problem or not.

Finally, one of the things that I talk about with my ITE trainees is the importance of a teacher modelling not just the strategy, but the thinking behind it as well. Giving pupils access to the thought processes of experts – why they are choosing a particular method or how they decide which mathematics to employ in a particular situation – is an essential part of developing problem-solvers.

Question 9

What makes a really effective secondary maths lesson?

The first point to make is that a ‘lesson’ – as in a 50-, 60- or whatever-minute period pupils spend learning mathematics in a single sitting – is not a useful unit of time for judging learning, and therefore the effectiveness of the teaching. Some pupils take longer than others, and some mathematical ideas are more intricate and need more work than others.

That being said, in order to learn about a mathematical idea or to carry out a mathematical process, there are aspects of the experience we can offer to pupils that make it more likely that pupils will master what we want them to. These include:

Careful choice of a key learning point – often the focus chosen by the teacher is too broad, for example, ‘solving one-step equations’. These could be equations of the form x + a = b, x − a = b, ax = b, xa = b. Focus on one of these at a time.

Exemplification of the concept or strategy, with the considered use of examples and non-examples, models, manipulatives and representations.

Guiding early practice, with variation theory used to design the questions so that mathematical structure is carefully revealed and immediate feedback is used to correct errors and misconceptions, working towards pupils securing a high success rate with the mathematics they are learning and the gradual removal of scaffolds.

Independent practice that provides opportunities for pupils to think about the mathematics they have learned in different ways, including interweaving other content with the current object of study, interleaving other concepts so that pupils have to decide whether the recently studied mathematics is appropriate or not, and the aforementioned opportunities for reasoning and problem-solving.

The retesting of previously learned knowledge over time to maintain and improve retrieval and storage strength. Interleaving helps with this, but further opportunities are generally required.

Question 10

What inspired you to write your first book, Visible Maths?

As I mentioned, I started my journey looking at representations and manipulatives several years before Visible Maths. During my reading and research, I encountered several excellent blogs and books about different aspects of representations and manipulatives, or particular representations and manipulatives, but what was lacking was a more complete picture of how these came together across different areas of the curriculum and how these areas link together.

The other thing I noticed was that everything I read or heard focused on where the manipulative or representation worked. But all models, and therefore all manipulatives or representations used to bring life to those models, break down at some stage. So, I wanted not only to provide more guidance for teachers on choosing appropriate models and manipulatives, but also to highlight the situations where they fail so that teachers can plan for how and when to transition to a new model or away from the manipulative.

Question 11

What can teachers expect to learn from your latest book, Practising Maths?

Practice is the lifeblood of learning. However well something is explained or modelled, however well connected to prior learning it is, pupils won’t achieve that important deep, connected knowledge without practice. However, a lot of practice doesn’t do the job it is supposed to do – developing learners’ confidence in applying the maths they learn.

So, what I wanted to do with Practising Maths was twofold. First, to highlight the different approaches to practising mathematics that exist – everything from the use of conceptual and procedural variation in carefully sequenced questions through to the use of open-ended, rich tasks. Similarly to Visible Maths, I wanted to provide a single place where all of these approaches were explored, to help guide teachers as to when and at which stage of learning these different types of practice might be useful and where they might not be, as well as the important things to consider if designing that sort of practice for themselves.

Secondly, there are a huge number of sources for these tasks out there, and some are better than others. Teachers are very busy people and will find their tasks from everywhere and anywhere if it saves them time. So, in Practising Maths, I wanted to highlight some of the better sources of tasks that are out there to save teachers time if they wanted to incorporate different practice types into their lessons.

Question 12

What common mistakes do pupils make when practising mathematics?

I mean, there are common errors in all areas of mathematics, far too many to name. When it comes to general practice, the main mistake that pupils make is to try to avoid challenge. But that isn’t really their fault – as humans, we are generally hard-wired to find the path of least resistance to our goal. This makes the real mistake not one in practice, but one in attitude – working towards the wrong goal. For many pupils, the goal is to get the right answers and get the work done quickly. The goal should be to gain a deep understanding of the object of study, to become really educated. However, it is for teachers and schools, supported by parents and carers, to establish this culture in their schools, to ensure they value – both explicitly and through their approach to education – hard work and depth of knowledge. Schools need to maintain an expectation that all pupils can achieve this depth and provide the support for all pupils to work towards attaining it.

Question 13

How can schools improve the transition from primary to secondary maths?

Fundamentally, transition is about communication. Secondary schools and departments need to communicate frequently and regularly with their primary counterparts about difficulties, approaches taken and everything concerning how pupils are progressing with their mathematics. That conversation then needs to go the other way, with secondary schools feeding back to primary schools about issues that have arisen when pupils join their secondary school, so that primary schools can adapt their practices to ensure the same things don’t arise for future cohorts. In general, sharing practice in both directions helps to ensure that the journey through mathematics is as coherent as possible for pupils and helps to make sure that secondary school teachers can build properly on what has come before.

Question 14

What advice would you give to a new or aspiring head of maths?

For new heads of maths, my biggest piece of advice would be to take your time over anything substantial. Yes, you might need to take immediate action to address underperformance in outcomes or practice, but big jobs like wholesale curriculum or practice change need time to set out the change and ensure staff understand the rationale behind it, train staff in the new approach or practice, monitor and provide feedback, and continue tweaking and embedding things.

For aspiring heads of maths, I would say to look at as many job descriptions and person specifications as you can, even for jobs in areas of the country you have no intention of applying to. Measure your experience, skills and knowledge against these and take steps to address any gaps.

I set out both these pieces of advice, and more, in my book Leading Maths.

Question 15

What should schools prioritise when supporting early-career maths teachers?

Early-career maths teachers lack the automated responses that more experienced maths teachers have, so schools need to prioritise supporting them in building good habits. The focus should be on granular decision-making – picking out specific things and rehearsing them, watching them in action, using feedback to improve them and, ultimately, embedding them. What these things are will depend on the ECT and what they bring to the classroom already. It might be selecting pupils to answer questions, a particular routine for checking for understanding, part of an approach to modelling or exemplification, or anything else. The important thing is to discuss and agree this with the ECT, building their sense of professional agency and collaborative working so the support is done with them, not done to them.

Of course, the biggest thing that ECTs are often concerned about is behaviour management, but that is no different. They still need support in developing good habits, focusing on particular responses and routines.

Question 16

What change would you most like to see in maths education?

The biggest change I would want to see is a fundamental shift in what we value from a mathematics education. As I mentioned earlier, the current experience of too many pupils is that mathematics is about getting things right quickly. I want to see teachers focus on depth and connection rather than speed and process.

However, this needs support at all levels: a national curriculum and assessment system that focuses on these things; more training for teachers to understand these links and help them be confident in the depth required; better curriculum documentation to highlight the connections and provide guidance as to how they can be made clear to pupils; access to lesson materials that focus on these aspects of maths learning; and a change in ITE provision to ensure that the new generation of teachers coming through understand these goals for a mathematical education.

Question 17

What is the best piece of advice you have received during your career?

I think the one that resonates most strongly with me is about the importance of securing buy-in from your staff, so that all of you are working towards the same goal. Building a united sense of purpose across the team ensures that staff are motivated and leads to them being active participants in delivering your goals.