A leading figure in mathematics education, Mark McCourt has led many large-scale government education initiatives, both in the UK and overseas. Mark was a Director at the National Centre for Excellence in the Teaching of Mathematics (NCETM) and has also been a school leader, an Advanced Skills Teacher, a school inspector and a teacher trainer. He was a founder and was Chairman of the Teacher Development Trust. Mark has extensive experience of mathematics teaching and learning across all age and ability groups, having taught students from age 3 to PhD!

I am acutely aware of the ridiculousness of the title – there is no way to cover such a topic in a short blog, but I shall attempt to briefly outline some of the key issues to consider if one wishes to bring about learning in the maths classroom.

Teachers need more than mathematical knowledge
Many teachers of mathematics enter the profession with high levels of mathematics content knowledge. This knowledge is connected to, but not the same as, mathematics pedagogical knowledge. Knowing how to bring about learning is complex and requires many years of professional learning to acquire. Some of this knowledge can be studied, reading the best evidence (propositional knowledge), some of it can be acquired through hearing about practice, perhaps a teacher giving a presentation at a CPD event (case knowledge), and, most importantly, some of this knowledge only comes about by teachers experiencing events themselves (strategic knowledge). This strategic knowledge involves teachers thinking about and considering propositional and case knowledge, which they then develop further based on actual practice in real classrooms.

It takes a long time to become an expert teacher. The first answer to the question of how to teach mathematics is to become an expert teacher. This means continuing to be engaged in professional learning throughout one’s career.

Mathematics is not about answers
Many children are conditioned to believe that mathematics is about wading through questions and getting right of wrong answers. This is an odd situation given that those of us who work as mathematicians do not do this. We know (see, for example, Nunes 2009) that strong mathematical reasoning is a greater predictor of future success in mathematics than, for instance, computational skills. Mathematical confidence comes about when pupils conjecture, test and receive confirmation (or can reason why confirmation did not happen!) Mathematicians are involved in considering scenarios, asking their own questions about those scenarios and following lines of enquiry. Children are not mathematicians, but we do need to give them opportunities to behave mathematically if we are to expect any of them to choose to become mathematicians.

Understanding understanding mathematics
Teachers must understand what it means to understand mathematics if they are to plan for and recognise mathematical understanding. We can say that a mathematical idea has been understood if it sits in the network of mathematical ideas with strong connectivity.

“The mathematics is understood if its mental representation is part of a network of representations. The degree of understanding is determined by the number and strength of its connections. A mathematical idea, procedure, or fact is understood thoroughly if it is linked to existing networks with stronger or more numerous connections” (Hiebert and Carpenter, 1992)

The obvious implication here is that we, as teachers, must create experiences in the classroom that bring about the creation and strengthening of these mental representations. This can be best achieved by including multiple and varied matophors, models and examples of mathematical ideas. This could be a book in itself, but by way of a short example, would include the long term and sustained use of concrete materials as a way of representing reason.

Learning mathematics is not linear. There is no single question that indicates a pupil has learnt an idea because understanding a mathematical idea is not something that simply comes about by experiencing a unit of work. Rather, learning mathematics is an ever evolving process (and includes unlearning and forgetting!). As pupils meet more mathematical ideas, they build new and stronger internal schema, which allow them to see previously encountered ideas in a new light and make new connections and gain more understanding.

Imagining the whole domain of mathematics as a complex, interconnected web of ideas, we can think of understanding as being the strength of the reasoning that pupils can make regarding how a new idea is connected to other, known ideas. The process of learning about an idea never ends, since there are always new connections that can be made as pupils mature. In learning mathematics, maturation matters! It is crucial, therefore, that all pupils have opportunities to repeatedly encounter mathematical concepts are they learn more. The strength of the reasoning between the ideas is driven by the number of representations that a pupil encounters and can justify.

The importance of attending
The teaching process is about creating changes in the pupil’s long term memory. The only known way to achieve a change in the long term memory is through attending. In learning mathematics, this means giving attention to mathematical structure, connections, patterns and meaning. Too often, mathematical problems are presented in twee contexts, which serve to distract pupils. A set of percentages problems about pupils’ favourite TV shows, for instance, results in pupils attending to the context rather than the mathematics!

At the point of teaching new ideas, strip problems of unnecessary contexts.

Behaving mathematically
As ideas, knowledge, concepts and skills are learnt, teachers must then provide opportunities for pupils to behave mathematically. Mathematicians conjecture, they create questions, follow lines of enquiry – including many that results in unexpected results (surely a better name than ‘wrong answers’) – they draw on their knowledge of other mathematical ideas, they model and, most crucially, they generalise and prove. These processes necessarily result in having to justify and reason. This reasoning strengthens understanding (by creating more or improved and refined connections in the nodes of the complex web of mathematics).

Teach mathematics lessons
This sounds blindingly obvious, of course. But, a large number of ‘mathematics lessons’ that pupils attend are not mathematics lessons at all. Teaching necessarily results in learning, which we take to mean a change in the long term memory that adds to a pupils mathematical understanding. If learning did not occur, nor did teaching. It was just presentation. An ‘objective led’ approach to teaching mathematics has resulted in the widespread practice of teachers ‘getting through’ a curriculum, rather than concentrating on whether or not learning has occurred. And, finally, a suggestion: if pupils did not have the opportunity to generalise, the lesson was not a mathematics lesson!

Clearly, this short blog was not really meant to answer the question of the title. Here, I simply wanted to raise some issues. These issues point to one important implication: the need for teachers of mathematics to continue to develop their expertise. We do this through reading, through research, through working with colleagues in professional learning networks and, critically, through real classroom practice.

For more thoughts on teaching mathematics, please see my blog at https://emaths.co.uk/

Further Resources and Reading
We know an enormous amount about teaching mathematics and are lucky in the UK to be blessed with some of the most informed and important maths education thinkers and projects in the world. I’d suggest that all teachers of mathematics should be familiar with

Shell Centre for Mathematical Education http://mathshell.org/
Centre for Innovation in Mathematics Teaching http://www.cimt.org.uk/

The collective thinking of the profession is best embodied in the mathematics teaching subject associations, particularly

The Mathematical Association http://www.m-a.org.uk/
The Association of Teachers of Mathematics https://www.atm.org.uk/

Notable books
Thinking Mathematically (Mason et al, 1982)
Collaborative Learning in Mathematics (Swan, 2006)
Learning Mathematics: The Cognitive Approach to Mathematics Education (Davis, 1984)
Understanding in Mathematics (Sierpinska, 1994)
Probing Understanding (White and Gunston, 1992)

CPD Sessions
I run a series of CPD sessions across the country that address the issues above, more details and course descriptions can be found here: https://completemaths.com/cpd