An
insight into Year 4 children’s fluency with addition and subtraction
Introduction:
The 2014 national curriculum for mathematics
outlined three key aims. To ensure that all pupils:
- Become fluent in the fundamental of mathematics
- Can reason mathematically
- Can solve mathematical problems
These three aims – fluency, reasoning and problem
solving – are the foundation of our maths curriculum in school, which is built
on the White Rose scheme of learning. In 2015-16 we developed fluency across
the school, but with a focus on written methods. Previously, each teacher
taught calculation in a way they preferred – for example half of Year 5 learned only the grid method for long multiplication and the parallel class learned only the
traditional ‘long multiplication’ method. To combat this, we introduced a
calculation policy and set out prescribed formal written methods for the ‘big
four’ operations – addition, subtraction, multiplication and division. This led
to big improvement in arithmetic across school.
In 2016-17 and 17-18 we focused on reasoning
and problem solving in school, with much success. Work in lessons was more
challenging and scores on the reasoning papers of the KS1 and KS2 NC tests
improved. However, in 2018, improved in arithmetic tests stalled at around 75%
(as in average % correct on the arithmetic test at KS2 was 75%). This meant
that for 2018-19 I decided to focus on fluency again, but I knew something
needed to be different if we were going to improve attainment.
I had noticed, in both lessons and assessments,
there was an overreliance on formal written methods. I’d observed children dividing
a three digit number by 1 using a written method, and even using a column method
to do 57 + 10. These children got the correct answer, but wasted a lot of time.
I also noticed in this year’s KS2 NC assessment that the question 0.5x28, not one child noticed that multiplying by 0.5 was
equivalent to halving a number. A truly fluent child would notice that, and
reach an answer of 14 very quickly. Most of our children embarked on the formal
written method of long multiplication, and most got it correct, but it again
took a considerable amount of time.
Clearly, there was a difference between the
ability to correctly calculate an answer and ‘notice’ something about the
numbers that may lead to a ‘shortcut’. After reading up on the concept of fluency,
I found an article by Russell (2000) which shaped my thinking for this year’s
development.
Russell (2000) defines mathematical fluency as
being comprised of three concepts:
- Efficiency
- Accuracy
- Flexibility
I had made the mistake previously of
considering fluency as primarily about accuracy (getting the answer correct) to
detriment of children’s flexibility and efficient with number. I decided to
conduct a very small scale piece of research and interview some children about
their approach to addition and subtraction, with the hope to gain an insight
into how efficient, accurate and flexible our children are at arithmetic. To do
this, I interviewed two children at random from Year 4 – one who achieved ‘Greater
Depth’ at the end of KS1 and one who achieved the ‘Expected Standard’. They
were given each question one at a time and asked to explain to me how they
would work each one out.
Methodology:
The questions had been designed so that they
could credibly be solved in a number of different ways by a fluent child. When
I wrote the questions, I considered how a fluent child in September of Year 4
may solve them efficiently.
a) 547
+ 312 this could be done
mentally, as no bridging 10 is required
b) 547
+ 400 again,
this could be done mentally by adding the hundreds only
c) 547
+ 298 298 could be added by
adding 300 then subtracting 2
d) 547
+ 378 this involves bridging
and would likely be done using a formal written method
e) 547
+ 453 this uses number bonds
and children may recognise the answer as 1000.
f) 824
– 413 this could be done
mentally, as no bridging 10 is required
g) 824
– 324 children may recognise
that this is 500 as the Tens and Ones are the same
h) 824
– 199 199 could be subtracted
by subtracting 200 then adding 1
i) 824
– 578 this
involves bridging and would likely be done using a formal written method
j) 824
– 807 the numbers are close
and may be thought of as a difference (counting up)
Child
M: (GDS at end of KS1)
M was able to do a) and b) mentally, and
explained his thinking. For a), he added the hundreds, then the tens and
finally the ones to reach 859. For b), he did 500+400=900 then “just added on the
47”. He tried c) mentally, but became confused. When prompted with “how else
could you solve it?” he chose the column method and found the correct answer,
which he did again for d). As soon as e) was revealed, he said “that’s just
1000 – I know straight away!”. When I asked how he knew that, he explained he
recognised 47+53 as a number bond to 100 and saw that 500+400 made 900, meaning
he had 1000 altogether.
The subtraction questions saw a similar
pattern. M solved f) and g) mentally, by subtracting the hundreds, then the
tens then the ones. For h) and i) he used the column method, but consistently
made the same mistake when ‘borrowing’ hundreds, meaning that his answers were
both incorrect. For j), he recognised 800-800 to be 0, so wrote out a column
for 24-7 which he correctly calculated as 17.
M showed good flexibility throughout the task,
employing a variety of different strategies for different questions. Generally,
his efficiency was good – knowing which questions he could do mentally and
which he would need a written method for. However, he wasn’t able to manipulate
199, for example by adding 200 and subtracting 1.His level of accuracy was
high, with 8 out of the 10 questions correct, however he has a consistent
misunderstanding with the column subtraction method.
Child
J: (EXS at end of KS1)
J solved a) using column addition and then
solved b) mentally, and used a column method again for c) and d).
Interestingly, J got c) and d) incorrect because he had forgotten to ‘carry’
the hundred over, although he had carried then ten over from the ones. This
suggests that he as a misunderstanding around the formal written method. As
soon he saw e), J said, “it’s gonna be about 1000” – showing that he was
mentally estimating an answer to his calculations. When pressed how he knew
that, he said, “47 + 53 is a number bond to 100, and then 500 + 400 = 900 … no
wait, it is 1000 exactly”. He was confident recognising number bonds and using
these to calculate.
For the subtraction questions, J did f) in his
head correctly, and correctly used the column method for g) – failing to spot
that the tens and the ones were the same in the subtrahend. He attempted h) in
his head, but got answer of 775 (subtracting 2 from 9 and 4 from 9, to avoid
‘borrowing’). He made a similar mistake with i), showing a misconception around
subtraction and the commutative law. He did solve j) mentally by recognising
800-800 was 0 and then subtracting 7 from 24 to get 17.
J showed flexibility in the task, moving from
written and mental method to try to be efficient. However, J wasn’t very accurate,
as he had two major misconceptions – one around the written method for addition
and one around subtraction and the fact subtraction is not commutative.
Conclusions:
Children who achieved GDS at the end of KS1
may generally be more accurate than those who achieved EXS but not necessarily
more flexible or efficient – both children were able to calculate with
large number in their heads using their own strategies (to various degrees of
success).
The children don’t have an over-reliance on
formal written methods – they are able to recognise some calculations which
could be done more efficiently using a mental method – but they don’t always
select the most efficient mental method.
Children are less secure with subtraction than
addition. Neither child thought of 824 – 807 as a ‘difference’; both talked
about ‘taking away’. We need to do more overt teaching on the three subtraction
structures:
- Take away
- Difference
- Partition
Children are not fully secure in using column
methods when it comes to “borrowing and carrying”. Children made slips
with addition but showed fundamental misunderstanding with column subtraction.
Children seem to be able to use number facts to
calculate quickly (for example 5 tens plus 4 tens is 9 tens) without using
counting on or fingers. Some children are also able to automatically recognise
number bonds, even to 100 or 1000. It would be worth investigating whether
children could use ‘nearly’ bonds to calculate – for example “73 + 28 = 101
because I know 73 + 27 = 100 and 28 is one more”. However the next finding does
suggest they are possible not able to yet…
Neither child was able to use a compensation
strategy (or ‘nearly number’) – for example to add 299 by adding 300 and
subtracting 1. There may be value in explicitly teaching this strategy across
school, using appropriate numbers.
In the future I’ll be speaking to more children
in different year groups and trying to gain an understanding of their efficiency,
flexibility and accuracy, with a view to gaining an insight into how we can
develop their fluency as a whole.
