What does knowledge look like in Maths

Written by Ben Rapley Monday, 24 June 2019

Edited and compiled by Robert J Smith @RJS2212

La Salle Education can be found on Twitter @LaSalleEd @LaSalleEd #MathsConf20 on Saturday 21st September 2019 Conference with both MA and ATM, tickets still available.

The next La Salle Ed Maths Conference is #MathsConf20 in Edinburgh

'What does 'knowledge' look like in Maths?’ ' is a blog about Ben Rapley's #MathsConf19 session/workshop at #MathsConf19.

The term 'knowledge' has become a bit of a buzz word in schools. Knowledge first teaching is championed and knowledge organisers are everywhere. Ben Rapley looks at what does 'knowledge' mean in mathematics and how we might use it in our classroom.

What Does ‘Knowledge’ Look Like In Mathematics?

“It is impossible to think well about a topic in the absence of factual knowledge about the topic.” (1) Daniel T. Willingham

Knowledge is important, but what does that ‘knowledge’ look like? Is it knowing all the facts in mathematics? The angle rules, some formula and a vocabulary glossary and you are all set? Probably not, however as soon as you start trying to think of all the things you want your students to know the list can suddenly spiral out of control.

I want to look at what is ‘knowledge’ in a subject that is very practical, knowing all of the facts around expanding double brackets does not necessarily mean you can do it. Martin Robinson, in his researching of creating the perfect curriculum talks about three forgotten classical arts; Grammar, Dialectic and Rhetoric. Grammar is described as “the art of interpreting the world through foundational knowledge” (2). This can also be described as the subject disciplines, the fundamental principles of a subject. If the subject disciplines are what we call knowledge that makes a lot more sense.

Daniel T. Willingham states “Factual knowledge precedes skill” , I think skill here is defined as other subjects use it; analyse, critique, explain. In mathematical terms we might call it problem solving. Students need to be able to complete a process before they can problem solve with it.

In my previous and current schools we have had on a focus of the Core Skills, the parts of mathematics that students need to be able to access higher order mathematical thinking and problem solving. I would argue that they form a large basis of the knowledge of our subject. We practice these skills every lesson at the start (don’t tell Mark McCourt). We then have regular low stakes quizzes that help us diagnose areas we need to improve on as a class and they will become the focus of our starters. You can read more about Core Skills here. The aim is that by transferring these processes into our long-term memory then that frees up space for students to be able use their working memory to deal with interpreting a question and solving the problem.

These processes are important as they form the chunks of information that will be stored in the long-term memory; however, these will not be transferred to schema unless connections between ideas have been made and understood. Schema can then be transferred to the long-term memory through consolidation and practice (4). Creating a knowledge rich environment can balance process and understanding, making it clear what we need student to know and practicing it at the same time as providing the information to enable students to create the connections of mathematical concepts in their head.  

Here are some of the elements we are going to be using in our curriculum to try to create a knowledge rich environment:

• Core Skills
• Clear outcomes for each topic carefully sequenced
• Clear about examples we want students to see
• Multiple representations
• Vocabulary

Clear outcomes for each topic

We are separating our schemes of learning into access, core and depth. The core outcomes are areas that I expect every student to have covered and will form the bulk of our assessments. We have spent time sequencing each of these carefully, staff are then able to divide their lesson time as needed.

Clear about examples we want students to see

There can be a lot variety between question styles and they can be quite intricate and easily missed. We aim to show students all of those different styles of questions. Regular use of worked example pairs and showing boundary examples as well as non-examples. Rather than having knowledge organisers we are moving towards creating example collections.

Multiple representations

Dual coding is highlighted as one of the six strategies for effective learning by The Learning Scientists (5), one of the ways we can do this is to use manipulatives. It is clear that manipulatives can be a very powerful tool in supporting students understanding. We are getting better at using them as a department but it is definitely where we are developing. The EEF published a guide to Improving Mathematics in KS2&3 with a section dedicated to manipulatives. It says that having a strategy to remove manipulatives is just as important as the strategy to introduce them (6). We are dedicating time within department meetings to better our understanding of how we can use them and support the transition from concrete to pictorial to abstract.


It is important to equip our students with the ability to access and understand questions, one of the ways we can do this is by explicitly teaching the vocabulary we can use. Alex Quigley says “We should avoid the assumption that the language of mathematics is simply absorbed ‘naturally’ over time by children.” (7). We can do this through exposition of a word’s etymology or morphology, or using Frayer models, amongst other methods. They can be powerful tools to help students make connections with words. We are highlighting words in our schemes where teaching the vocabulary may elevate a student’s understanding, being careful to not create cognitive overload by adding too much complexity.

Our curriculum isn’t complete and will probably continue to be an ever evolving thing, however I feel confident that we are working towards creating a knowledge rich environment that will help students thrive and understand mathematics in a way that they never thought they could.

1) Why Don’t Student’s Like School, DT Willingham, 2009, p210
2) Trivium 21 C, Robinson M, 2013, p20-25
3) Why Don’t Student’s Like School, DT Willingham, 2009, p210
4) Understanding how we learn – a visual guide, Weinstein Y, Sumeracki M & Caviglioni O, 2019, p75-76
5) http://www.learningscientists.org/blog/2016/10/6-1?rq=six%20effective
6) Improving Mathematics in Key Stages Two and Three, EEF, 2017, P10
7) Closing The Vocabulary Gap, Quigley A, 2018, 104-105

Ben Rapley spoke about "What does 'knowledge' look like in Maths?" at #MathsConf19 at the Penistone Grammar School held on Saturday 22nd June

Don't forget in July we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

About the Author

Ben Rapley

Ben Rapley tweets as @MISTERRAPLEY Ben is Head of Maths at Newfield School in Sheffield. He has a passion for using evidence based approaches in the classroom. He has previously presented at MathsConf15 on Core Skills which you can read more about on his blog misterrapley.wordpress.com

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