*Visualising and Representing is part of our Developing Mathematical Thinking **Primary** and **Secondary** collections. This page accompanies the Visualising and Representing **Primary** and **Secondary** resources.*

*This follows on from **Exploring and Noticing**, **Working Systematically** and **Conjecturing and Generalising**.*

## Show Primary webinar recording

During this webinar we had a go at the following tasks:

Three Neighbours

Make 37

We used the representations in these first two tasks to explain why four consecutive numbers add to an even number which is not a multiple of 4

Picture Your Method

We discussed careful use of language, in particular referring to 'exchanging', rather than 'borrowing', which is useful for Subtraction Surprise

## Show Secondary webinar recording

During this webinar we had a go at the following tasks:

Summing Consecutive Numbers

Slick Summing

Painted Cube

Up, Down, Flying Around

Let's assume that students are on a problem-solving journey, and have been exploring and noticing, working systematically, and have conjectured and generalised. What next...?

Mathematicians try to make sense of what they've discovered, before formalising their results with justifications and proofs.

So how do mathematicians go about this sense-making? They look for connections, and try to gain insights into mathematical structure, by turning to visual images, and numerical and algebraic representations. Carefully chosen language can also be very helpful, and metaphors can often support understanding.

If we are to offer students opportunities to make sense of mathematical ideas, then we may need to think about the following:

## Values and ethos in our classrooms

- Believing that mathematics is about relationships and interconnected ideas
- Recognising that there is never just one way to approach a problem
- Recognising the value of building on students' ideas.

## Ways of representing mathematical ideas

The tasks listed below can be used to introduce students to a variety of helpful representations.

### Visual

Eightness of Eight encourages us to picture different ways of 'splitting up' 8.

Digit Addition encourages students to deepen their understanding of place value.

Three Neighbours - Liz shows why three consecutive numbers sum to a multiple of three.

Picture Your Method offers different visual representations of calculation methods.

Make 37 encourages us to picture what happens when we add odd and even numbers.

What Numbers Can We Make? helps us see that all numbers are either multiples of 3, or one more, or two more than multiples of 3.

Picturing Square Numbers shows why the first n odd numbers add up to n^{2}.

Slick Summing encourages us to picture a useful way of arranging the numbers to be added.

Seven Squares - invites us to picture three ways of drawing seven squares and then to imagine how many matchsticks would be required to draw a million and one squares.

Always a Multiple? - Alison explains the result using Multilink cubes to represent place value.

Picturing Triangular Numbers shows why the n^{th} triangular number is equal to $ \frac {1}{2} n(n+1)$.

At Least One... demonstrates the power of tree diagrams to represent all possible outcomes.

Which is Cheaper? offers an opportunity to appreciate the power of a graphical representation.

Pair Products demonstrates how visual representations can help us explain general results.

Factorising with Multilink offers students an opportunity to discover for themselves the properties needed for a quadratic expression to factorise.

Quadratic Patterns encourages students to see the connections between multiple representations.

## Show further examples

### Algebraic

Your Number Is... sheds light on what seems like magic.

Three Neighbours - Claire shows why three consecutive numbers sum to a multiple of three.

Perimeter Expressions uses algebraic expressions to represent different perimeters.

Always a Multiple? - Charlie explains the result using algebra.

Pair Products demonstrates the power of algebra to explain general results.

Always Perfect demonstrates the power of algebra to explain general results.

Quadratic Patterns encourages students to see the connections between multiple representations.

## Show further examples

### Using commonly understood analogous contexts

Got It - students may discover that landing on the numbers 3, 8, 13 and 18 guarantees them victory. Picturing crossing a river with these numbers as stepping stones, can offer an image of forcing their opponent to land between the stepping stones and provide the 'landing place' needed to reach the next stepping stone. (See the published students' solutions.)

Seesaw Shenanigans and Number Balance use balance as an analogy for equivalence, which requires the same reasoning skills as needed for solving equations.

Adding and Subtracting Positive and Negative Numbers offers several models for understanding the 'rules'.

Fruity Totals and What's it Worth? begin to develop the reasoning skills needed when manipulating simultaneous equations.

Subtraction Surprise has video with no sound. In this task, referring to 'exchanging', rather than 'borrowing', is a much more useful description if students are going to make use of subtraction.

**You can find problems that offer opportunities for your students to use a variety of representations to make sense of mathematical ideas in our ****Primary**** and ****Secondary**** Visualising and Representing collections. **

## Follow-up resources

You may be interested in this collection of follow-up resources:

Dave Hewitt alerts us to 'the richness that can be gained by looking at a particular situation in some depth, rather than looking at it superficially in order to get a result for a table and then rushing on to the next example'.

Relational Understanding and Instrumental Understanding

Richard Skemp draws attention to the need to teach for relational understanding (whereby students know what to do and are able to explain why) rather than instrumental understanding (whereby students know rules and procedures without understanding why they work).

*Three linked articles by Dave Hewitt: *

*"If I'm having to remember..., then I'm not working on mathematics"*

Arbitrary and Necessary Part 1: A Way of Viewing the Mathematics Curriculum

Arbitrary and Necessary Part 2: Assisting Memory

Arbitrary and Necessary Part 3: Educating Awareness

Seeing is Understanding paper by Jo Boaler with Lang Chen, Cathy Williams & Montserrat Cordero

Effective Teachers of Numeracy: Report of a study carried out for the Teacher Training Agency by Askew, M. et al

Math is FigureOutAble

In his article Generic examples: seeing through the particular to the general, Tim Rowlands outlines how one carefully chosen example can help us perceive the generality