The brain learns by forming **connections**.

**Successful learning** in maths relies upon these connections.

In training I use an approach called** ‘The Maths Sandwich’.** Instead of going straight into the focused teaching, I aim to create a vital ‘*Oh, I get what you mean….but I can’t do that yet’* moment.

**Adapting these ideas:** I’ve worked with all of the ideas below at an age-appropriate level with EYFS to Year 6. The principals of good planning remain the same throughout, particularly in terms of engaging pupils and activating existing knowledge.

**I begin my planning journey with the first slice of bread in the ‘Maths Sandwich’.**

This is the ‘**hook**‘ or context for the learning. A meaningful problem worthy of my children’s **thought** and ‘**learning stamina’**.

For example, following all of the recent festivities, many of use had to cook far larger quantities of food for our guests. This happened to me with my mince pie bites (delicious!).

The recipe was for 20 portions and, as they were very small and I had lots of visitors, I knew this wouldn’t be enough. That’s it. That’s my starting point. It’s that easy.

This is the kind of story I’d share with my class as my **hook**. It’s all true, so easy to engage with, and it’s about a scenario most children have experienced so they can begin to **reason** successfully about what I could do next.

**First steps: **Children work in groups and spend** time ****discussing the problem** and **identifying** what they know I need to do and what information they don’t have yet. This is an incredibly motivating way to begin a learning journey. It’s a problem that’s **accessible** to every child but also means I can set **the bar very high** in terms of how far some learners will be able to take this.

Next we weigh out the **real** ingredients that will make one batch of mincemeat bites to create a **strong visual reference** and also **model** accurate use of measuring equipment.

This approach enables my children work out **for themselves** that I need to make a lot more than twenty mince pie bites. Have they thought about:

- Multiplying the recipe (Multiplication as scaling without values)
- Multiplying the quantities (multiplication as scaling with values)

If they hadn’t, I would need to think about how I could **encourage** them to think in this way and if they had, I would need to **carefully question** them to assess their level of true **understanding**. Do they see this as multiplication? Can they see the **connection** with repeated addition?

The ‘filling’ strips the context (bread) away and leaves just the** pure maths** skills. Here we identify the **key tools** the children will need to sharpen and use our **‘Concrete -Pictorial- Abstract’** approach to** teach and practise** these skills.

We’ve identified that understanding and calculating using **multiplication** as scaling is key to this problem.

At this stage I get the children to use **Cuisenaire rods** to model the multiplicative structures in the problem. We develop the necessary language to accompany what is happening when we scale the recipe.

*double twice as much three times as much *

*multiply equal to equivalent*

Using **rods** allows children to focus upon the **structures and relationships** without having to calculate values. The **generalisation** that each ingredient would increase by an **equal amount** each time is fundamental to this problem.

Next we transfer this to a **bar model drawing**. We are moving here from **concrete** representations of the problem to **pictorial**.

Now I return to the actual ingredients that we measured out for one batch of mincemeat bites.

How much of each ingredient did we need? What **unit of measurement** did we use for each ingredient? What was the same and what was different?

The children now understand that the recipe needs to be **scaled** to feed more guests.

…*we made two batches? How many people would that feed? How much of each ingredient would we need need and why? *

*…people wanted two each rather than one each (as they are so delicious)*

*…what if we had x number of guests and we wanted to give them y number of mincemeat bites?*

As the children explore what they would need to do to each ingredient and focused upon explaining why, I then introduce ratio tables. Tables like this help** organise thinking,** **use known facts** and **use our multiplication facts meaningfully**. Using the tables, the children record the calculations they need to do (rather than the answers at this stage). This offers us opportunities to consider strategies rather than calculating answers. E.g. *How can I use what I know about 1 batch to generate facts about 10 batches, 5 batches, 2 batches etc.*

The children **extend their thinking** by considering how to cater for a precise number that wasn’t a multiple of the original amount. For example, if we had 15 guests and we wanted to give them two each, how much of each ingredient would you need to make an exact amount?

This is the time to remind the children that there are special **concrete** and **pictorial** resources we can use to help us **calculate efficiently**. Using concept images of **ten frames** or real ten frames will help them calculate any number using the base ten system. Adding fractions is much easier when we use **bars** or circles.

If the quantities in the recipe were ‘unfriendly’ I might adjust them to simply the calculations and build fluency and then use more challenging values. Using recipes is a great opportunity to challenge the ‘*bigger numbers are harder’* belief. We’ll often be using fractions and smaller amounts that are difficult to multiply. For example

1/4 teaspoon of bicarbonate of soda x 3

200g flour x 3

70ml water x 3

Which of these calculations is more challenging to solve and why? Is it the one with the biggest number?

So now we’re solving the problem using the tools of **multiplication** as **scaling** and **ratio**.

At this stage the children more easily set their own questions about this recipe or change recipes to something they haven’t made and go through similar processes.

Successful mathematicians fluently **transfer** their tools from **one mathematical context to another**. This is a vital stage in the ‘Maths Sandwich’ approach.

The key tools the children have been sharpening are around multiplication as scaling. This involved connecting multiplication and division and also calculating whole numbers and fractions.

At this stage, it’s likely that the children will associate these maths skills with mincemeat bites or only for cooking. Hmmm. Not what I want them to take away with them.

It’s vital that the children see their newly sharpened skills as having **multiple (if not infinite) applications**

So, what else do we scale or use scaling to make sense of?

Example 1:

- A British grass snake grows to approximately 60cm whereas an anaconda can grow to 3 metres long! I know that 100cm is equal to 1 metre and I could adjust 60cm to 50cm which would mean that an anaconda was six times as long as a grass snake!

Example 2:

- Body parts have been used as non-standard units of measurement for thousands of years and there are some very interesting equivalent parts!
- Your body is equal to 6 or 7 times your head!
- Your face is equal to four or five widths of your eye!
- There are huge potential gains in teaching in this way:
- Children apply their newly sharpened skills in a new context to they see that mathematics has wide application

The children should be able to work with a **far greater level of independenc**e so there are **rich opportunities** to carry out reliable formative assessment

The children will be connecting and using other skills such as measurement (length) which involves **decimal numbers,** **equivalence** and **conversion**.

This approach fulfils the definition of true **mastery** in every way. We are actively going **deeper**, **making connections, making learning accessible for all** and giving the children a reason to **sharpen these essential skills.** **Motivation** and **ownership** should be high and levels of **resilience** and **co-operation** maintained due to relevance.

Karen Wilding is an independent primary maths consultant offering high-quality, inspiring maths training in the UK and internationally. If you’d like to learn more about improving whole-school problem-solving using ‘The Maths Sandwich’, as well as other crucial areas of development such as ‘Number Sense’ and ‘Maths Mindset’, contact Karen at:

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