And this is how many children feel about maths. They know that putting some numbers into a calculation will give the desired answer, but they don't really have a clue what goes on inside the 'machine' of that procedure. This is all well and good until that child has to apply this learning - having no understanding of the mechanics of mathematics makes it very difficult to use procedures in context.

In my blog post for Third Space Learning entitled 'Maths Tricks or Bad Habits? 5 Bad Habits in Maths We're Still Teaching Our Pupils' I make several suggestions for how to use visual representations to teach good conceptual understanding of some tricky aspects of the maths curriculum, such as the ones below:

The recent EEF guidance document on improving maths in KS2 and KS3 backs up the importance of modelling good conceptual understanding in maths lessons, rather than relying on tricks that work but don't help children to have an understanding of the 'why' and the 'how':

Recommendation 4: Enable pupils to develop a rich network of mathematical knowledge

In order to teach maths well, and in order for children to succeed in maths, teachers need to make sure children understand what is going on when they carry out a mathematical procedure. A great way of developing this understanding is using manipulatives and representations:"Pupils are able to apply procedures most effectively when they understand how the procedures workand in what circumstances they are useful. Fluent recall of a procedure is important, butteachers should ensure that appropriate time is spent on developing understanding.One reason for encouraging understanding is to enable pupils to reconstruct steps in a procedure that they may have forgotten.The recommendations in this guidance on visual representations, misconceptions, and setting problems in real-world contexts are useful here."

Recommendation 2: Use manipulatives and representations

"Manipulatives and representations can be powerful tools for supporting pupils to engage with mathematical ideas.However, manipulatives and representations are just tools: how they are used is important. They need to be used purposefully and appropriately in order to have an impact. Teachers should ensure that there is a clear rationale for using a particular manipulative or representation to teach a specific mathematical concept.The aim is to use manipulatives and representations to reveal mathematical structures and enable pupils to understand and use mathematics independently.

As I wrote in the guide to Bar Modelling that I produced for Third Space Learning (click to download for free):Teachers should: Enable pupils to understand the links between the manipulatives and the mathematical ideas they represent.This requires teachers to encourage pupils to link the materials (and the actions performed on or with them) to the mathematics of the situation, to appreciate the limitations of concrete materials, and to develop related mathematical images, representations and symbols."

If we don't do this, we run the risk of allowing children to proceed in their mathematical education with misconceptions:

Recommendation 1: Use assessment to build on pupils’ existing knowledge and understanding

"A misconception is an understanding that leads to a ‘systematic pattern of errors’. Often misconceptions are formed when knowledge has been applied outside of the context in which it is useful. For example, the ‘multiplication makes bigger, division makes smaller’ conception applies to positive, whole numbers greater than 1. However, when subsequent mathematical concepts appear (for example, numbers less than or equal to 1), this conception, extended beyond its useful context, becomes a misconception.

When weIt is important that misconceptions are uncovered and addressed rather than side-stepped or ignored. Pupils will often defend their misconceptions, especially if they are based on sound, albeit limited, ideas. In this situation, teachers could think about how a misconception might have arisen and explore with pupils the ‘partial truth’ that it is built on and the circumstances where it no longer applies. Counterexamples can be effective in challenging pupils’ belief in a misconception. However,pupils may need time and teacher support to develop richer and more robust conceptions."

*do*teach children using appropriate models and images so that they understand the mathematical concepts behind the procedures (or the 'tricks'), we provide children with something that they can actually look at and explain. Explaining something that is concrete is easier than explaining an abstract concept.

In the bar modelling guide (click to download for free) I pointed out that:

By developing children's skills to represent and explain their understanding using a model, we develop their independence and motivation:

Recommendation 5: Develop pupils’ independence and motivation

Next time you plan a maths lesson question how you will ensure that children have a good conceptual understanding of the content you teach. Often, concrete or pictorial representations will be the best way to show children the inner-workings of the concepts you cover. Following Psychologist Jerome Bruner's research-based CPA (Concrete - Pictorial - Abstract) approach means that children (and adults) are more likely to understand what is going on inside the maths machine as calculations and processes take place."Teachers can provide regular opportunities for pupils to develop independent metacognition through:

encouraging self-explanation—pupils explaining to themselves how they planned, monitored, and evaluated their completion of a task; and

encouraging pupils to explain their metacognitive thinking to the teacher and other pupils."

__Further Reading and Resources:__

- These AET Mathematics plans give ideas for how to represent every single learning objective in the national curriculum from year 1 to year 11.
- My article for Third Space Learning: Maths Tricks or Bad Habits? 5 Bad Habits in Maths We're Still Teaching Our Pupils
- Articles by Mike Askew: KS1 and KS2 Maths – using visual models and Models in Mind
- Mr Bee's article: Fractions? I Love Fractions!