Boost Students’ Knowledge of Fractions with Questions they can Fully Understand
We need to set tasks that require pupils to make sense of what is being asked of them, rather than simply reproducing a procedure, says Mike Askew…
- by Mike Askew
Here are two calculations from the 2017 KS2 maths test:
37.8 – 14.671 = [ ] ¼ + ⅓ = [ ]
Which do you think pupils were more successful on?
If you think this is a trick question, you are right. The second question was on the paper, but not in that form. The form it was presented in was to say how much of this circle was not shaded if a ¼ and 1/6 had been shaded.
If you have your wits about you, that basically boils down to finding the sum of ¼ and ⅓. On the subtraction of decimals, 68% of pupils got this correct. In contrast, only 39% could answer the fractions question.
It is also informative to look at the proportions of pupils not attempting each question. Only 4% of learners omitted to try to answer the decimal subtraction: the vast majority of pupils clearly thought they could answer it, even if they got it wrong.
Compare that with the 11% of pupils not attempting the fraction problem: over one in 10 pupils thought they could not do this question.
No doubt the Nick Gibbs of this world will think it impressive that over two-thirds of pupils leaving primary school can subtract 14.671 from 37.8. But is that really a success story?
I can think of no time in my life, outside of school, when I have had to carry out a calculation that comes anywhere close to a subtraction like this (I invite you to try and put this calculation into a realistic problem context) and although I am confident that I can do this calculation using paper and pencil, I am equally confident that were I to find myself ever needing the answer I would be reaching for my phone’s calculator to avoid a slip-up (setting aside the fact that it would also be quicker).
Playing with figures
Another question on the reasoning paper provides further insights into pupils’ understanding of fractions and decimals. A boy is reported as saying ‘0.25 is smaller than 2/5’ and the test taker has to explain why this is correct.
It’s a nice problem as there are a range of approaches one can take.
If you know that 0.25 is ¼, then putting ¼ and 2/5 over a common denominator of 20 is not too challenging. If you think of 0.25 as 25/100, you’ll see that 1/5 is 20/100, so 2/5 is 40/100.
I think a mathematically elegant answer would be to note that 0.25 is a quarter, and that four lots of 2/5 would result in 6/5, which is clearly greater than one, so 2/5 must be greater than a quarter.
Despite the variety of ways that pupils might have reasoned here – none of them requiring terribly sophisticated understanding of fractions and decimals – only 38% came up with a correct reason, with 12% omitting to try and find an answer.
While preparing to write this piece (OK, you might call it procrastinating), I found this headline from a UK newspaper, reporting on a survey from the USA: ‘Autism rates spiked 15% last year’.
Now, a ‘rocketing’ (as the report goes on to say) of 15% sounds dramatic. The actual figures were, however, that the number of diagnoses had risen from 1.5% in 2014, to 1.7% in 2017. That is about two more children per thousand (from 15 to 17).
A medical website reporting on the findings describe this as a ‘slight’ rise, putting it down largely to better rates of diagnoses, rather than to a change in the overall proportion of the population being on the autism spectrum.
In these times of access to vast swathes of news, much of which plays with figures in ways like this report did, surely we have a duty to help learners reason through what is behind the dramatic headlines?
And isn’t being able to reason about comparing 0.25 to 2/5 going to be more helpful in this than being able to subtract two abstruse decimals?
Working on reasoning
The challenge when it comes to teaching is that it seems easier to teach a procedure for subtracting than it is to teach pupils to reason about fractions and decimals. But if that latter goal is more difficult, that’s still no reason not to try.
Working on reasoning means setting tasks that require pupils to think and make sense of what is being asked of them (as opposed to reproducing a procedure). It also means placing less emphasis on getting right answers and more on talking about the thinking behind arriving at them.
One thing is clear from research: being good at ‘pure arithmetic’ is not a precursor to getting good at reasoning, and we do not have to ‘wait’ until pupils are fluent at calculating with fractions before we can engage them in reasoning about the sorts of situations that involve fractions.
A big reasoning idea that pupils need to encounter, talk about and work on is that fractions express a relationship, rather than name objects.
We can help pupils understand this by always talking about the whole to which a fraction is related: this is half of one whole apple; six grapes are half of a total of 12 grapes. It only makes sense to compare fractions when the whole is the same. Let’s look at some activities that might promote this sort of reasoning.
KS1 activities
Odd one out
Talk about which image here pupils think is the odd one out and why. There is no one right answer here, but the key thing to bring the talk around to is why the circle and the square both show halves, even though these look very different, and equally important, why the rectangle has not been halved.
Party time
Set up the context of going to a children’s party and noticing that half of the children there were boys. Set up pupils to work in pairs and ask them to come up with several different solutions for the total number of children who could have been at the party.
Draw up a class list of their solutions and talk about doubling and halving. Discuss why there couldn’t have been 15 children at the party.
Strawberry picking
Set up this pair of problems:
Mitch bought a bag of strawberries. Mitch ate half the bag of strawberries. There were nine strawberries left in the bag. How many strawberries were in the bag to start with?
Elsie bought a bag of strawberries. Elsie ate nine of the strawberries. There were three strawberries left in the bag. What fraction of the bag of strawberries did Elsie eat?
Pupils are likely to need counters to help them find the solution to each problem. When both problems have been solved, talk about whether the pupils agree with these statements:
Mitch and Elsie both ate the same number of strawberries, so they must each have eaten the same fraction of the bag.
Elsie ate ¾ of her bag of strawberries, but Mitch only ate ½ of his bag, so she ate more strawberries than he did.
LKS2 activities
Another, another
Provide pupils with a number line with zero and one fraction marked, for example:
Ask children to mark and label another fraction on the line, then another and another. How many fractions can they mark? It is likely that some pupils will think that only three other fractions can be marked – ⅕, ⅖ and ⅗. Encourage them to mark ‘1’ on the line and think about other fractions they can then add in.
Bags of marbles
Set up the following:
This shows 0.25 of Sameera’s bag of marbles.
This shows 2/5 of Corin’s bag of marbles.
Who has more marbles in their bag, Sameera or Corin?
Bars of chocolate
Explain to pupils that in the picture below, a fraction of each bar of chocolate is shown. The rest of each bar is under a piece of cardboard. Which whole bar is bigger? Challenge them to explain how they know.
UKS2 activities
Fair shares
Set up this problem:
Tom bought two bars of the same chocolate as he had on Saturday. He shares both bars equally with Dick and Harry. Tom says he ate ⅔ of a bar of chocolate. Dick says he ate ⅓ of the chocolate. Harry says Tom must have eaten more chocolate than Dick because ⅔ is greater than ⅓.
Discuss in pairs whether or not Tom, Dick and Harry are each correct. Share explanations as a class.
Shopping
This is a variation on the problem above.
Laurel and Hardy go out shopping. Laurel spends 0.25 of his money and Hardy spends 2/5 of his money. Hardy says he spent more money because 2/5 is greater than 0.25. Laurel says they spent the same because 0.25 = 2/5.
Ask pupils to explain why they are both wrong.
Saving
Pose this problem:
Last month, Max saved 2/5 of his £10 pocket money. He also saved 0.25 of his £10 birthday money. Did he save more pocket money or birthday money? How much did Max save altogether?
Bringing it together
A key point in working on activities like these is that everyone – teacher and pupils – have to set aside expectations that there is a single correct way of answering each one. It is through the experiences of sharing different arguments – and agreeing on whether or not they are correct – that pupils’ reasoning and understanding deepens.
Mike Askew is adjunct professor of education at Monash University, Melbourne. A former primary teacher, he now researches, speaks and writes on teaching and learning mathematics. Find out more at mikeaskew.net and follow him on Twitter at @mikeaskew26.
