Top tips for deepening students’ understanding in maths

Wednesday 21st February 2018
Mathematics Mastery

Helen DruryDr Helen Drury is a pioneer of teaching and learning for mastery in the UK. She is the founder and director of Ark Venture Mathematics Mastery, a non-profit school improvement programme aiming to transform mathematics education in the UK. Helen is also the author of ‘How to Teach Mathematics for Mastery’ recently published by OUP. 


We’ve all been there. You’re teaching equivalent fractions. Feeling confident as you assess their learning, you throw in a question your students haven’t encountered before. You’re met with a sea of bewildered faces. Then, it hits you – your students have not yet moved beyond a surface-level understanding of this mathematical concept.  

So how do we enable students to deepen their understanding of mathematical concepts? Especially when these concepts are highly abstract? 

One simple and powerful approach, is to use visual representations in your classroom. 

Using diagrams, pictures, manipulatives and other types of representations known as the Concrete Pictorial Abstract approach (CPA) helps learners visualise concepts. This supports them to carry out effective calculations. The approach also facilitates reasoning, problem solving and allows students to demonstrate proof. 

To help you introduce the CPA approach in your classroom, here’s three tops tips:

  1. Place manipulatives on every table. This encourages students to get into the habit of using them. Dienes blocks, bead strings and multilink cubes not only help to make lessons practical, engaging and fun but also help learners to understand abstract concepts.
  2. Use pictorial representations. These can be as simple as a number line to help with positive or negative integers or bar modelling for arithmetic, proportional reasoning or algebraic problems. Bar models are particularly useful in helping students to make sense of word problems - great for helping exam technique. 
  3. Try virtual representations. ICT-supported representations can aid understanding of graphs, variables, functions and the modelling process. If your students enjoy learning with the aid of ICT, why not give this a try?

Incorporating visual representations into your lessons will help your students grasp concepts, as well as offer stretch and challenge to your higher attainers. Remember that the journey to mastery takes time, so be patient when trying out this new approach. But most of all, have fun with it!

If you’d like to develop your teaching and find more classroom ideas, sign up for a free trial of the Mathematics Mastery programme at