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:
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?
The children should be able to work with a far greater level of independence 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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‘One of our EYFS Teachers has just attended this training and came back totally inspired and singing your praises. She said that she learnt more on this course than any course she has ever attended!’
C Magnocavallo Head of Maths, Chesham Preparatory School, Bucks
Thank you for showing us HOW to change. So many courses just tell us what isn't working but not how to go about addressing this. Your approaches make so much sense!
Just to say thank you again for 3 really brilliant talks at SGIS. We're a small school near Basel and we’d be interested in anything you’re doing nearby (Zurich way) so please let us know!