# Use visualisation and modelling

Encouraging students to visualise or summon up a mental image of something – seeing it in their mind – can help them to better understand it. An image may be a shape, graph or diagram, it may be a set of symbols or a procedure.

Once a student becomes comfortable visualising numeracy ideas and can consistently translate this thinking, you can then ask them to describe what they 'see' (quantities, actions, outcomes). When a student describes their thinking, they begin to recognise the relationship between language and actions. This helps them to apply the actions more easily.

You may need to provide modelling or scaffolding for some students initially. Use guiding questions such as:

- What was in your head when you were thinking of that?
- What do we do next?
- How would we get to the answer from here?

## Build numeracy vocabulary

When students can describe numeracy actions, you can begin to build their vocabulary and associate this language with symbols. For example, teach a student to say 'minus' instead of 'take away' when talking about subtraction or when using concrete materials. Have them draw or make the subtraction symbol (-) while they do this.

Gradually, students will think about these ideas in an abstract way and will be able read and represent numeracy ideas using just symbols instead of needing to explicitly link them to mental images or concrete materials.

This page uses Year 2 student Mahli as an example.

Mahli's teacher could use the following activities to teach Mahli how to read numbers up to 20.

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### Focus on demonstrating physical actions on quantities

- Arrange a set of items in a row of 10 and a single item on the next row. Have Mahli count the number of items.
- Introduce the idea of the row of 10.
- 'How many items are there up to here?' 'This is a row of 10. Here is one more item. That makes 11.'
- Repeat this with 12 items, making a row of 10 and a row of two below. Again, have Mahli count the number of items.
- Ask Mahli to point to the row of 10 and then to the two left over.
- Ask Mahli to name each part when you point to it.
- Introduce 13 items arranged in a similar way. Ask Mahli to point to the row of 10 and then count on from 10.

##
### Focus on describing the idea in language

- Introduce how to write the number for each quantity. Begin with 12.
- 'There are 12 items here. One row of 10 and two left over. I write '1' for the row of 10, and '2' for the two left over.
- Ask Mahli to say what each quantity is and then write the number in parts. Place the written number 12 on the set of items. Repeat this for the set of 13 items and others.
- Show Mahli a set of 14 items. Point to the row of 10 and ask Mahli, 'How many up to here? What do we call this?' Ask Mahli to count on from 10 with the remaining four.
- Ask, 'How many altogether?' Say, 'There are 14 altogether. There is one row of 10 and four left over.' Write each part of the number as you say it. Ask Mahli to write it. Place the written number on the set.

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### Focus on interpreting the symbols

- Point to the written numbers on each set. Say the number and ask Mahli to say each number. Repeat this at least four times. Make some quantity cards that show 11, 12, 13 or 14 items arranged in different ways. Ask Mahli to count how many items are in each, write the number that matches each, and place the written number on the matching set.
- Show Mahli a set of number cards from 10 to 20 but not in order. Ask Mahli to say the numbers in order and then arrange the cards in order. Point to one card at a time and ask Mahli to read the number. Show Mahli an incomplete number ladder and ask them to complete it.

## Teach students how to reason and think about numeracy concepts

Students need to be able to move between symbolic statements, word statements and quantities quickly and easily. All students, but particularly those with learning difficulties, need to be explicitly taught the language of numeracy and provided with opportunities to practise and consolidate this knowledge.

To form an understanding (and develop a more sophisticated understanding) of numeracy concepts, students need to think about them in particular ways. Students with learning difficulties may think about certain concepts in similar ways to younger students.

Until they develop their numeracy language and understanding these students will be unable to learn more complex numeracy knowledge and skills.

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### Numeracy

For a student understand the concept of odd and even numbers, addition and subtraction, they first need to be able to define, recognise and sort things into categories. Once they can do this they can begin to learn about odd and even numbers as categories (sorting quantity and identifying what each set has in common and how they differ).

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### Identify and complete patterns

When completing counting sequences, a student first needs to learn to recognise and analyse patterns. Initially, you could use colours or even sounds. Once they can recognise patterns easily, they can be taught what to look for in a number pattern and how to describe it (for example, 'It goes three steps up each time'). Once they have mastered these skills, they can then be taught how to extend and use this knowledge (for example, 'If I go up three steps from 18, I will get to 21').

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### Transferring ideas

Students need to be able to apply familiar operations in unfamiliar contexts. This skill comes from making comparisons with what they already know and identifying similarities. For example, when responding to the question, 'What numbers are closest to 73?', a student needs to recognise that this problem is equivalent to 73 – 1 = 72, and 73 + 1 = 74. To scaffold this transfer, you might ask a student to draw a number ladder and place 73 in the middle. Once they fill in this proforma, you can make the connection between the new task and number ladders. You can also ask them to explain how the new task is similar to what they already know or how else they might visualise this problem in future using other concrete materials (for example, wooden MAB blocks).

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### Example

Mahli needs to be taught to enumerate ten-grouped quantities, that is, establish the number of sets of items without explicitly counting them (subitising). When asked for the total number of sticks in a ten-grouped quantity of loose and bundled sticks, Mahli counted each item in each set by saying, '10, 20, 30, 40, 50. There are 50 sticks.

To think about the task differently, Mahli needs to be taught:

- that the quantity has both bundles (of 10) and loose sticks (one)
- to first count in 10s and then count in ones
- to recognise where to change the counting pattern, 'I will change counting here'. Mahli should point to where they will change and then point to the first loose stick in the set, 'I'll call this stick "one",'. Mahli should use an object such as a pencil to mark where they will change the counting pattern, as a reminder.

To consolidate their understanding, Mahli should also:

- practise applying this skill to several examples with an increasing number of bundles and loose sticks
- practise enumerating different ten-grouped quantities and loose items (for example, a string of 10 beads or 10 blocks joined together)
- practise enumerating pictures of ten-grouped quantities and loose items
- learn to describe a ten-grouped quantity in different ways (for example, 32 as 'three bundles and two sticks' and 'three 10s and two 1s')
- practise distinguishing between counting quantities that consist only of groups of 10, quantities that consist only of ones and quantities that consist of both
- say what they will do when they have a ten-grouped or mixed quantity in the future.

This sequence begins with Mahli being taught to act physically on quantities, then to describe the action verbally, to visualise applying it and then to use this thinking in a more general way. This type of sequence can be used to teach any numeracy thinking or strategy.

## Teach metacognitive strategies

Students with learning difficulties can find it challenging to reflect on their learning and direct their thinking. These metacognitive strategies need to be explicitly taught to help them to take ownership of their learning in numeracy and maths.

Teaching sentence stems will assist students to reflect on what they have learned (for example, 'I am learning about …' 'I didn't know …' 'I now know …' 'I will remember when doing … to …').

Encourage students to draw on these strategies at three separate points when working on a problem or learning activity.

At the beginning of the task: students interpret the problem using what they know and decide how it matches or doesn't fit with their existing knowledge. This includes planning how they will work through it.

While working through the task: students monitor their progress and have strategies prepared when they encounter obstacles or make errors and whether they need to change or reassess their approach.

After completing the task: students review what they have learned, acknowledging and making a note of their new understanding and how it adds to what they already know.

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### Example

Teaching metacognitive strategies can help to support Mahli's understanding of enumerating ten-grouped quantities.

At the beginning of the task:

- scaffold Mahli, helping them to define the task in their own words ('I've got to say how many sticks are here altogether.
- help Mahli to recall similar procedures they have learned previously ('Have you done something like this before?')
- encourage Mahli to explain their thinking at each step ('What do you think you should do first? What will you do next?').

While working through the task:

- remind Mahli to monitor their progress ('Do you think you are getting closer to your answer?')
- if Mahli encounters an obstacle, encourage them to think about other procedures and learning they have done before and to assess if these might be relevant
- if Mahli makes an error, ask them to explain where they think the mistake might be and to make a change. For example, if Mahli counts the loose sticks (ones) as bundles (10s) ask them to count the quantities again but this time to start counting the loose sticks first.

After completing the task:

- ask Mahli to take you through what they did and how they did it
- if they made an error or encountered an obstacle, ask Mahli to tell you what they will remember to do next time.

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### Embed positive attitudes about learning and doing mathematics

Students who have learning difficulties may be reluctant to engage in numeracy activities or have negative views about their abilities.

All students will benefit from encouragement and recognition of their progress. When students are engaging in a numeracy activity, draw on reflective questions that highlight the relevance of what they are learning or how it is adding to what they know and can do. For example:

- What do you know now that you didn't know before?
- How has doing this helped you? What can you do now that you could not before?
- How do you think you might use what you have learned in the future?

These kinds of interactions help students see that they can be successful as users of numeracy, and that their knowledge and skills change when they do maths. While reflecting on how maths works for them, they will start to see that they can be successful and why numeracy is important.

## Teach routines and provide opportunities for practise

Students will be most motivated when learning a new idea or skill for the first time. The aim is for them to eventually use this new knowledge or skill automatically. This is the result of learning, repetition and practise.

The more students practise and link this learning with what they already know, the more they will be able to use it with less effort and the more their numeracy will improve. For example, when multiplication becomes automatic through the memorisation of times tables, students use only a very small amount of their cognition allowing them to instead focus on how they need to apply the information or on other parts of the problem.

Being able to recall and use the names of written numbers, the names of the numbers in order, number facts, symbols and their meanings is crucial for all students and should form the basis of their numerical fluency.

To teach for fluency, provide tasks that are carefully graded for complexity and that ultimately require students to do more and more independently.

### Determine independence

Determining when a student is ready to demonstrate more independence and to what extent can be difficult. It is appropriate to give greater independence to a student when they consistently:

- identify and justify what word reading and/or comprehension strategies they will use to complete specific tasks
- demonstrate motivation or enthusiasm to engage in learning activities
- attempt (or express a desire to attempt) tasks by themselves and without assistance.

It's important to think carefully about the stages in which you will release responsibility for learning to a student and how you will monitor and record their progress. It can be easy to release too much responsibility too quickly and before students are completely ready.

For more information, visit Monitoring students' progress.

Last Update: 25 March 2022