Secondary Teaching for Mastery Big Idea: Representation and Structure Five Big Ideas Big Idea: Representation and Structure

Key Messages 1. The representation needs to clearly show the concept being taught, and in particular the key difficulty point. It exposes the structure. 2. In the end, the students need to be able to do the maths without the representation. 3. A stem sentence describes the representation and helps the students move to working in the abstract (ten tenths is

equivalent to one whole) and could be seen as a representation in itself. 4. There will be some key representations which the students will meet time and again. 5. Pattern and structure are related but different: students may have seen a pattern without understanding the structure which causes that pattern.

Concrete, Pictorial, Abstract Bruners theory focuses on three models of representation of mathematical ideas: Enactive model Iconic mode Symbolic mode. Learners can access abstract ideas through manipulating objects to derive enactive

knowledge. Then diagrams help them acquire iconic knowledge to act as a bridge to formal symbolic knowledge. What? Representation refers to a particular form in which mathematics is presented.

NCETM glossary (2014) What? Examples of different representations include: two fractions could be represented on a number line a quadratic function could be expressed

algebraically or presented visually as a graph; and a probability distribution could be presented in a table or represented as a histogram. NCETM glossary (2014) Manipulatives and representations can be powerful tools for supporting pupils to

engage with mathematical ideas. EEF Improving Mathematics in Key Stages Two and Three Guidance Report (2017) How? Algebra tiles 1 1

Show me three Adding and subtracting directed numbers Represent these calculations with unit squares How?

Algebra tiles 1 1 1

How? Algebra tiles

Represent the following expressions using algebra tiles Factorise, using algebra tiles How can you use algebra tiles to solve linear equations? solve quadratic equations?

Key considerations Ensure that there is a clear rationale Enable pupils to understand the links between the manipulatives and the mathematical ideas they represent. Try to avoid pupils becoming reliant on manipulatives

Manipulatives should act as a scaffold, which can be removed Manipulatives can be used to support pupils of all ages. EEF Improving Mathematics in Key Stages Two and Three Guidance Report (2017) Big Idea: Representation and

Structure Key Messages 1. The representation needs to clearly show the concept being taught, and in particular the key difficulty point. It exposes the structure. 2. In the end, the students need to be able to do the maths without the representation. 3. A stem sentence describes the representation and helps the

students move to working in the abstract (ten tenths is equivalent to one whole) and could be seen as a representation in itself. 4. There will be some key representations which the students will meet time and again. 5. Pattern and structure are related but different: Students may have seen a pattern without understanding the structure which causes that pattern.

Connect with the NCETM and your local Maths Hub Join the NCETM at ncetm.org.uk for full access to the website plus a monthly emagazine the Secondary Round-up sent straight to your inbox. Find your local Maths Hub via mathshubs.org.uk/find-your-hub. Click

through to your local hubs page to subscribe to their mailing list and follow them on Twitter. Connect with the NCETM and your local Maths Hub References and more info Education Endowment Foundation (2017) Improving Mathematics in Key

Stages Two and Three Guidance Report https://educationendowmentfoundation.org.uk/public/files/Support/Links/C ampaigns/Maths/KS2_KS3_Maths_Guidance_2017.pdf Hill, C. (2018) Using tokens to teach directed numbers, Mathematics Teaching 264:13-16 Mattock, P. (2019) Visible Maths: Using representations and structure to enhance mathematics teaching in schools, Crown House NCETM (2014) Mathematics glossary for teachers in Key Stages 1 to 3

www.ncetm.org.uk/files/19226855/National+Curriculum+Glossary.pdf Images from MathsBot.com https://mathsbot.com/manipulatives/tiles Willingham (2017) Do Manipulatives Help Students Learn? www.aft.org/sites/default/files/periodicals/ae_fall2017_willingham.pdf References and more info.. Blogs Will Emeny

www.greatmathsteachingideas.com/2015/04/04/algebra-tiles-from-counting-to-co mpleting-the-square/ Podcast Mr Barton Maths Podcast, Bernie Westacott: Teaching maths with visuals and manipulatives www.mrbartonmaths.com/blog/bernie-westacott-teaching-maths-with-visuals-andmanipulatives/ Twitter

Follow @MccreaEmma, @EmathsUK, @Maths_Master, @MrMattock, @berniewestacott & @StudyMaths Resources Online algebra tiles https://mathsbot.com/manipulatives/tiles Class set () https://completemaths.com/teaching-tools/physical-manipulatives