**Representation and Mathematical Structure**

### Representation and Structure

The representation needs to clearly show the concept being taught. **It exposes the structure**. In the end, the **students need to be able to do the maths without the representation. **A **stem sentence (language structure)** 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.

**Pattern and structure are related but different**: Students may have seen a pattern without understanding the structure which causes that pattern.

NCETM, 5 Big Ideas of Teaching for Mastery

Apply now for a fully funded place on a Teaching for Mastery Work Group to learn more about mathematical structure and the other four big ideas that underpin Teaching for Mastery. Contact your local maths hub for further details.

1. The representation needs to clearly show the concept being taught. **It exposes the structure**.

Dienes blocks expose the geometrical structure of base 10 and are a good visual representation of the relative sizes of digits when linked to place value.

Exploring different bases exposes the structure of place value in any base including base x.

Students are more likely to make the connection between the laws of arithmetic and apply them to base x (algebra tiles).

Dienes blocks, place value tables and algebra tiles are all representations. Making the connections reveals the **mathematical structure** and enables students to understand the concept and use the mathematics independently of the manipulatives and representations.

2. In the end, the **students need to be able to do the maths without the representation.**

3. A **stem sentence (language structure****)** describes the representation and helps the students move to working in the **abstract** (“ten tenths is equivalent to one whole”) and should be seen as a representation in itself.

**...... is divided into groups of ........**

**There are ....... groups and a remainder of .......**

4. There will be some key representations which the students will meet time and again. Guidance material is available on the NCETM website for each of the key representations. https://www.ncetm.org.uk/resources/53609

5. **Pattern and structure are related but different**: Students may have seen a pattern without understanding the structure which causes that pattern.

**5 Big Ideas of Teaching for Mastery**

**Mathematical Structure is central to the 5 Big Ideas of TfM**

## Research and Evidence

### PUPILS’ AWARENESS OF STRUCTURE ON TWO NUMBER/ALGEBRA QUESTIONS

Dietmar Küchemann and Celia Hoyles Institute of Education, University of London

Item A1 (which was familiar to English pupils) was asked in Yrs 8 and 10 (a different but parallel item was used in Yr 9). It is a standard number/algebra item involving a tile pattern, and was designed to test whether pupils could discern and describe a structure (assessed by carrying out and explaining a numerical calculation).

Spotting number patterns, no structure;

Some recognition of structure (incomplete or draws & counts);

Recognition and use of structure: specific (correct answer, eg showing 60+60+3+3);

Recognition and use of structure: general (correct answer and general rule, eg x2, +6)

Recognition and use of structure: general, with use of variables (correct answer and general rule, with naming of variables in words or letters).

### ARCHITECTURE OF MATHEMATICAL STRUCTURE

HAMSA VENKAT, MIKE ASKEW, ANNE WATSON, JOHN MASON

**structure ⇔ mathematical relationship between elements **

Emergent structure (involving analyzing/forming/ seeing **local **relationships)

Mathematical structure (involving analyzing/forming/ seeing **general **relationships)

This grid presents several formats of symbolic, iconic, graphical representations of multiplicative relationships. It also enables the user to juxtapose these to produce formats for encountering some of the structure of multiplication in the field of real number (Mason, Watson, Askew & Venkat, 2018).

### Appreciating Mathematical Structure for All

We start from the position that mastering procedures is an important component of taking advantage of opportunities to make mathematical sense, but that it is of little value to learners if it is simply a procedure, because as the number of procedures increases, the load on memory and retrieval becomes more and more burdensome. When procedures are accompanied by even a minimal appreciation of the mathematical structures which make them effective and which provide criteria for appropriateness, learning shifts to focusing on re-construction based on re-membering (literally) rather than relying totally on photographic or rote memory.

The underlying theoretical frame being used here is a distinction between different forms, states, or structures of attention (Mason, 2003, Mason & Johnston-Wilder, 2004):

• Holding wholes (gazing),

• Discerning details (making distinctions),

• Recognising relationships (among specific discerned elements),

• Perceiving properties (as generalities which may be instantiated in specific situations),

• Reasoning on the basis of identified properties.

### Designing Questions to Probe Relational or Structural Thinking in Arithmetic

Max Stephens* The University of Melbourne

### Making Connections in Mathematics

### The Connective Model

### Understanding mathematics involves identifying and understanding connections between mathematical ideas. Haylock and Cockburn (1989) suggested that effective learning in mathematics takes place when the learner makes cognitive connections. Teaching and learning of mathematics should therefore focus on making such connections. The connective model helps to make explicit the connections between different mathematical representations: symbols, mathematically structured images, language and contexts.

**https://www.babcockldp.co.uk/cms/articles/send-file/7f3bec01-7051-40af-83da-80cdacf578d5/1**

"when children are engaged in mathematical activity, they are involved in manipulating one or more of these four key components of mathematical experience: concrete materials, symbols, language and pictures”

Derek Haylock and Anne Cockburn (2008), Understanding Mathematics for young children

Symbols, language, pictures and concrete experiences can all reveal mathematical structure when planned well.

Many teachers use the concrete, pictorial, abstract (CPA) approach which is similar, but it does not stress the importance of language as an important representation of mathematical structure.

The CPA approach is often seen as a linear path from concrete to abstract which is not always appropriate. Haylock and Cockburn suggest that students are engaged in mathematical activity when they are involved in manipulating one or more of the four components.

### Using manipulatives and other representations

EEF Guidance Report (2017) *Improving mathematics in Key Stages Two and Three.*