Enriching Education - Conceptual Maths

What is maths? Interestingly, although it has been an integral part of the school curriculum for the best part of the last 75 years, there is little consensus on the answer to this question amongst maths teachers.

Some people will say that maths is a body of connected knowledge, others that it is a way of behaving and making sense of the world, whilst others believe that it is a collection of theorems based on fundamental axioms.

To truly learn ‘about’ maths, one has to go deeper, beyond the procedures, and into the structure of its different concepts. After all, concepts are ultimately at the heart of the study of mathematics – they are the ideas that remain constant whenever they are encountered, but that combine and build upon each other to create the mathematical universe. The structure of each concept is what gives rise to the procedures and processes that are used in calculation and problem-solving.

Peter Mattock, an accredited maths professional development lead with over 15 years’ teaching experience, believes that exposing learners to mathematical structures can ensure that they achieve both procedural fluency and conceptual understanding, whether teachers’ preferred pedagogy is to teach such structure explicitly or to offer learners activities to enable them to discover it through inquiry.

Now, in his stunning new book, Conceptual Maths, Peter provides a detailed description of concepts, starting with an examination of the concept of number, including its generalisation into algebra. He then moves on to looking at the standard numerical operations in three parts: addition/subtraction, multiplication and multiples, and finally division and factors. This is followed by an examination of ideas around equivalence and equality before shifting attention to proportionality and then functionality. From here, the book explores concepts in the realm of geometry, including measures, accuracy, shape and transformation, before finishing in the realms of chance, data and graphing/charting.

In reading this book, teachers will become familiar with the underlying structure for the key concepts in school-level mathematics and will then be able to use this knowledge to support learners in making sense of the content they study. Whether teachers support learners in constructing that sense for themselves or explicitly teach good ways of making sense of concepts, ensuring that learners can make sense of mathematical concepts puts them in a much better place to see the connections between the topics they study.

The point about making connections is important. In the last few years, cognitive science has had an increasing influence on teaching practice. One of the key ideas in cognitive science is that of a schema – a schema represents our pool of knowledge in a particular area and how it is connected. If we wish to learn something new in that area, we need to connect it to our existing schema. By teaching about the structure of concepts, these connections become much easier to highlight because the concept is recognisable every time it appears.

Ultimately, Conceptual Maths aims to act as a guide for teachers to support learners in recognising the structure behind different mathematical concepts and help them assimilate that new knowledge of the concept into their schema.

Suitable for maths teachers across UK primary(4-11), secondary(11-16) and post-16 years.

Peter Mattock has been teaching maths for over 15 years. He is a specialist leader of education (SLE) and an accredited secondary maths professional development lead, who regularly presents at conferences across the country.

Peter also develops teaching for mastery in the secondary school classroom, having been part of the first cohort of specialists trained in mastery approaches by the National Centre for Excellence in the Teaching of Mathematics (NCETM).