A recent webinar with Mike Askew explored the connection between reasoning, problem solving and fluency. This blog post summaries the key takeaways from this webinar.
Using reasoning to support fluency and problem solving
You’ll probably be very familiar with the aims of the National Curriculum for mathematics in England: fluency, problem-solving and reasoning. An accepted logic of progression for these is for children to become fluent in the basics, apply this to problem-solving, and then reason about what they have done. However, this sequence tends towards treating reasoning as the icing on the cake, suggesting that it might be a final step that not all children in the class will reach. So let’s turn this logic on its head and consider the possibility that much mathematical reasoning is in actual fact independent of arithmetical fluency.
What does progress in mathematical reasoning look like?
Since we cannot actually ‘see’ children’s progression in learning, in the way we can see a journey’s progression on a SatNav, we often use metaphors to talk about progression in learning. One popular metaphor is to liken learning to ‘being on track’, with the implication that we can check if children going in the right direction, reaching ‘stations’ of fluency along the way. Or we talk about progression in learning as though it were similar to building up blocks, where some ideas provide the ‘foundations’ that can be ‘built upon’.
Instead of thinking about reasoning as a series of stations along a train track or a pile of building blocks, we can instead take a gardening metaphor, and think about reasoning as an ‘unfolding’ of things. With this metaphor, just as the sunflower ‘emerges’ from the seed, so our mathematical reasoning is contained within our early experiences. A five-year-old may not be able to solve 3 divided by 4, but they will be able to share three chocolate bars between four friends – that early experience of ‘sharing chocolate’ contains the seeds of formal division leading to fractions.1
Of course, the five-year-old is not interested in how much chocolate each friend gets, but whether everyone gets the same amount – it’s the child’s interest in relationships between quantities, rather than the actual quantities that holds the seeds of thinking mathematically.
The role of relationships in thinking mathematically
Quantitative relationships refer to how quantities relate to each other. Consider this example:
I have some friends round on Saturday for a tea party and buy a packet of biscuits, which we share equally. On Sunday, I have another tea party, we share a second, equivalent packet of the biscuits. We share out the same number of biscuits as yesterday, but there are more people at the table. Does each person get more or less biscuits?2
Once people are reassured that this is not a trick question3 then it is clear that if there are more people and the same quantity of biscuits, everyone must get a smaller amount to eat on Sunday than the Saturday crowd did. Note, importantly, we can reason this conclusion without knowing exact quantities, either of people or biscuits.
This example had the change from Saturday to Sunday being that the number of biscuits stayed the same, while the number of people went up. As each of these quantities can do three things between Saturday and Sunday – go down, stay the same, go up – there are nine variations to the problem, summarised in this table, with the solution shown to the particular version above.
Before reading on, you might like to take a moment to think about which of the other cells in the table can be filled in. (The solution is at the end of this blog).
It turns out that in 7 out of 9 cases, we can reason what will happen without doing any arithmetic.4 We can then use this reasoning to help us understand what happens when we do put numbers in. For example, what we essentially have here is a division – quantity of biscuits divided between number of friends – and we can record the changes in the quantities of biscuits and/or people as fractions:
So, the two fractions represent 5 biscuits shared between 6 friends (5/6) and 5 biscuits shared between 8 (5/8). To reason through which of these fractions is bigger we can apply our quantitative reasoning here to see that everyone must get fewer biscuits on Sunday – there are more friends, but the same quantity of biscuits to go around. We do not need to generate images of each fraction to ‘see’ which is larger, and we certainly do not need to put them both over a common denominator of 48. We can reason about these fractions, not as being static parts of an object, but as a result of a familiar action on the world and in doing so developing our understanding of fractions. This is exactly what MathsBeat does, using this idea of reasoning in context to help children understand what the abstract mathematics might look like.
By structural relationships, I mean how we can break up and deal with a quantity in structural ways. Try this:
Jot down a two-digit number (say, 32)
Add the two digits (3 + 2 = 5)
Subtract that sum from your original number (32 – 5 = 27)
Do at least three more
Do you notice anything about your answers?
If you’ve done this, then you’ll probably notice that all of your answers are multiples of nine (and, if like most folks, you just read on, then do check this is the case with a couple of numbers now).
This result might look like a bit of mathematical magic, but there must be a reason.
We might model this using three base tens, and two units, decomposing one of our tens into units in order to take away five units. But this probably gives us no sense of the underlying structure or any physical sensation of why we always end up with a multiple of nine.
If we approach this differently, thinking about where our five came from –three tens and two units – rather than decomposing one of the tens into units, we could start by taking away two, which cancels out.
And then rather than subtracting three from one of our tens, we could take away one from each ten, leaving us with three nines. And a moment’s reflection may reveal that this will work for any starting number: 45 – (4 + 5), well the, five within the nine being subtracted clears the five ones in 45, and the 4 matches the number of tens, and that will always be the case. Through the concrete, we begin to get the sense that this will always be true.
If we take this into more formal recording, we are ensuring that children have a real sense of what the structure is: a structural sense, which complements their number sense.
Decomposing and recomposing is one way of doing subtraction, but we’re going beyond this by really unpacking and laying bare the underlying structure: a really powerful way of helping children understand what’s going on.
So in summary, much mathematical reasoning is independent of arithmetical fluency.
This is a bold statement, but as you can see from the examples above, our reasoning doesn’t necessarily depend upon or change with different numbers. In fact, it stays exactly the same. We can even say something is true and have absolutely no idea how to do the calculation. (Is it true that 37.5 x 13.57 = 40 x 13.57 – 2.5 x 13.37?)
Maybe it’s time to reverse the logic and start to think about mathematics emerging from reasoning to problem-solving to fluency.
Mike Askew: Before moving into teacher education, Professor Mike Askew began his career as a primary school teacher. He now researches, speaks and writes on teaching and learning mathematics. Mike believes that all children can find mathematical activity engaging and enjoyable, and therefore develop the confidence in their ability to do maths.
Mike is also the Series Editor of MathsBeat, a new digitally-led maths mastery programme that has been designed and written to bring a consistent and coherent approach to the National Curriculum, covering all of the aims – fluency, problem solving and reasoning – thoroughly and comprehensively. MathsBeat’s clear progression and easy-to-follow sequence of tasks develops children’s knowledge, fluency and understanding with suggested prompts, actions and questions to give all children opportunities for deep learning. Find out more here.
You can watch Mike’s full webinar, The role of reasoning in supporting problem solving and fluency, here. (Note: you will be taken to a sign-up page and asked to enter your details; this is so that we can email you a CPD certificate on competition of the webinar).
Solution to Changes from Saturday to Sunday and the result
1 If you would like to read more about this, I recommend Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books.
2 Adapted from a problem in: Lamon, S. (2005). Teaching Fractions and Ratios for Understanding. Essential Content Knowledge and Instructional Strategies for Teachers, 2nd Edition. Routledge.
3 Because, of course in this mathematical world of friends, no one is on a diet or gluten intolerant!
4 The more/more and less/less solutions are determined by the actual quantities: biscuits going up by, say, 20 , but only one more friend turning up on Sunday is going to be very different by only having 1 more biscuit on Sunday but 20 more friends arriving.