In the fifth blog in this series, Primary School Headteacher Clare Sealy looks at the struggles primary school pupils can have when implementing critical thinking skills when subject knowledge is lacking, and the effect this can have on their attempts at problem solving activities in KS2.
In the introduction to this series, I outlined how Craig Barton, in his book How I wish I’d taught maths, described how he had changed his teaching as the result of reading research around cognitive science. In the latter part of his book, the focus turns to helping pupils use what they know.
Whatever the age of the children we teach, many find it hard to transfer what they know how to do in one context to another. This is most evident when it comes to problem solving, or in the SATs reasoning papers. They know the maths, they just can’t work out which bit of maths they need in this specific circumstance.
Results From The Research: The Ability To Complete Problem Solving Activities In KS2 Does Not Come Naturally
The ability to solve problems is not a generic skill that can be taught and that children can transfer from one problem to another. While there are some metacognitive strategies that can help a bit, what is really crucial is having a very secure understanding of the actual maths – the domain specific knowledge – that lies at the heart of the problem. Craig quotes Daniel Willingham (2006)
So what do all of these studies Craig looked at boil down to?
- Critical thinking (as well as scientific thinking and other domain-based thinking) is not a skill. There is not a set of critical thinking skills that can be acquired and deployed regardless of context.
- There are metacognitive strategies that, once learned, make critical thinking more likely.
- The ability to think critically (to actually do what the metacognitive strategies call for) depends on domain knowledge and practice. For teachers, the situation is not hopeless, but no one should underestimate the difficulty of teaching pupils to think critically.
The metacognitive strategies mentioned involve reflecting on what you are doing during problem solving activities in KS2, asking yourself questions such as:
‘What am I doing?’
‘Why am I doing this?’
‘How does it help me?’
This is all very well if you have secure domain knowledge and can answer these questions. However, if you lack this knowledge, the questions are just frustrating.
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How To Help Your Pupils With Problem Solving Activities Throughout KS2
There are of course various strategies we can give pupils to help them with problem solving activities. For example, underlining the important words. However, this relies on pupils understanding what the important word are in the first place.
Often, irrelevant surface features seem important to pupils whereas we experts can see they are completely irrelevant, because our domain knowledge and experience of answering many, many questions means we can spot the deep underlying structure a mile off.
It’s the same with other strategies such as setting work out systematically (you have to know what system is likely to be helpful), working backwards (you have to know whether this is likely to be useful in this situation) or even using a bar model. Bar models can be so helpful, but you have to know whether or not this kind of question is suitable for the bar model treatment.
The Third Space Learning context
One of the reasons we asked Clare to write this series of blog posts was because the book How I Wish I’d Taught Maths reflects much of the approach we aim to take in our 1-to-1 maths lessons each week.
Tutors are trained on the importance of breaking down learning into small steps, building pupils’ metacognitive skills, and presenting questions and activities in a clear order, building one onto the next. If you’re interested in finding out more about the effectiveness of the 1-to-1 we give 7,000 UK primary pupils every week, just give us a ring on 0203 771 0095 or book a demo here.
‘Same surface, different deep’ (SSDD) problems
Maths problems usually have an arbitrary surface structure and a deep structure. The surface structure involves the context in which the problem is set and has nothing to do with the actual maths; for example, in a question about buying tickets to a funfair, the funfair and ticket are part of the surface structure.
They are but the wrapper in which the real maths is wrapped. Pupils can get fixated on this ‘wrapper’, rather than the underlying deep mathematical structure held within it.
I recall a SATs question about paving inside a greenhouse. The child thought that they couldn’t do it because they didn’t know what a greenhouse was! Whereas I immediately knew that this was going to be an area question. The surface structure was transparent to me whereas, it was thoroughly opaque to the pupil.
All the underlining, systematic working or bar modelling in the world wouldn’t get past this erroneous latching onto surface features.
How To Get Past The Surface Features
To overcome this hurdle, Craig recommends teaching children to recognise the deep structure of problems and how to identify and then disregard surface features. It should go without saying that children need to be thoroughly secure in the underlying maths before attempting problem solving.
It is a mistake to think that problem solving is a good way of consolidating learning, let alone using it in the initial knowledge acquisition phase. Problem solving is about transferring learning from one context to another.
Problem solving is about generalising.
It therefore comes at the end of learning to do something, not mid- way and definitely not at the beginning.
But what is more, if at the end of a unit on, say division, we give children a load of division problems, this will not help them work out what the deep structure is. They already know; it’s division! This is fine, but it won’t help children learn to decide whether or not a particular problem requires division or not.
As well as problem solving activities at the end of units, teachers also need to allocate separate times where children have to work out what the deep structure of a problem actually is, regardless of surface features. This means setting a range of SSDD problems sharing the same surface features – for example a shopping problem involving apples and pears – but which each have a different deep structure.
Translating this to a primary school context
Let’s return to the question about stickers from the 2017 KS2 SATs paper we considered in a previous blog:
The surface feature here is stickers.
As experts, we know straightaway that we could substitute packs of stickers with boxes of apples or packets of balloons or even a family ticket to the cinema.
In fact, in a variation of Craig’s SSDD technique for a primary context, I’d suggest also doing DSSD problems (different surface, same deep) problems too, asking children to cross out the words ‘pack of stickers’ and replace with suitable alternative, and then repeat the problem to understand that the surface features do not change the underlying maths at all.
Then I’d suggest moving on to SSDD problems. Let’s stick with stickers as our ‘same surface’. The deep structure of our original question involved knowing that you had to multiply to find the price of 12 separate stickers and then subtracting to find the difference. But we could ask mathematically different questions while keeping the context and visual look of the problem the same.
- How much does one sticker cost? (though I’d adapt the price so the division came out as a whole number of pence)
- Stickers are 8cm wide and 6cm high. Ally sticks 3 stickers in a row, without any gaps. What is the perimeter of the shape she has now made?
- Ally buys 7 packs of stickers a month, Jack buys 3 packs of stickers a month and Chen buys 5 packets a month. What is the average number of packets bought by the 3 children in one month?
- Ally buys a pack of 12 stickers. She has spent 15% of her birthday money. How much birthday money has Ally got left? (again, I would adjust the price into something more workable)
Another great way to translate problem solving into a primary context is through topical maths activities, and you can read more about that in this blog.
Extension Ideas For Problem Solving Activities In KS2
Extending both ideas, we could make a grid where the rows contained questions with a different surface structure and the columns contained questions with the same deep structure. This grid could be cut into individual boxes with pupils having to sort each box accordingly, to reconstruct the grid.
Tigers, Cake or Money? A Unique Approach To Critical Thinking
One way of helping children understand the deep structure of division problems, is to ask children if this is a tiger, cake or money sort of division question.
What this means is, could we swap the surface features of the problem we are given to one involving tigers, or cake or money?
Why these three I hear you ask?
This is because, where division problems do not divide exactly, it is really useful to:
- Be able to decide if you need to round up or down (These are the tiger questions. If you haven’t got enough cages for your tigers you might get eaten)
- Have a remainder that’s a fraction (These are the cake questions as we can each have 1 and a half cakes)
- Or have a remainder expressed as a decimal (These are the money questions as we can have £2.47 each)
An Example Of A Tiger Question
This is a great example of a tiger question. With 4 spare tigers, you need to have an extra box! Having 2/3 of a box wouldn’t work, neither would having 0.666 of a box. Rewriting this as a tiger question helps understand the deep structure.
A cage holds 6 tigers
How many cages are needed to hold 52 tigers?
How To Make A Trickier Tiger Question
Here is a slightly harder ‘tiger’ problem:
Let’s rewrite this:
A zookeeper has 7,600 tigers (!)
Cages can contain 500 tigers.
How many cages does the zookeeper need?
15.2 cages is obviously not enough to stop the keeper from being eaten.
Answers requiring a decimal answer are usually money questions already, or calculations rather than word problems. Hence they are easy to categorise.
Some children find ‘tiger’ type questions particularly hard, and give answers that don’t make sense because they haven’t rounded up or down. So in the brick example above, they give the answer as 15.2 because they haven’t recognised that doesn’t make sense.
By naming certain deep structures, children are more able to identify them when they arise, and this is a fantastic way to help children with problem solving activities throughout KS2.
Sources of Inspiration
Willingham, D.T. (2006) ‘How knowledge helps: it speeds and strengthens reading comprehension, learning and thinking’. American Educator 30 (1) p.30
This is the fifth blog in a series of 6 adapting the book How I Wish I’d Taught Maths for a primary audience. If you wish to read the remaining blogs in the series, check them out below:
Clare Sealy has also written several thought provoking pieces on primary learning and leadership. If you are interested in this, take a look at the Confessions of a Headteacher series on how she changed marking, feedback and observation.
Additional further reading: 20 maths strategies that we use in our teaching to guarantee success for any pupil.