The two faces of Division

W hen I teach beginning primary school
teachers about division, I usually begin by
asking them to come up with a ‘story’ for
the calculation 12 ÷ 4. They tend to respond, as my
secondary beginning mathematics teachers do also,
with a story about sharing. Cakes and sweets are
enormously popular. This response is also typical of
secondary age schoolchildren in my experience and
that of the Concepts in Secondary Mathematics and
Science (CSMS) investigators, (Hart et al, 1981).
I have a suspicion that this reveals something about
the mental pictures, and movement schemes,
people carry about division and I think that the root
of a significant difficulty is hidden inside this mental
picture. There are two forms of division which are
distinct enactively, we act them out differently when
we learn about them and when we ‘do’ them in our
early lives, but equivalent numerically. That is that
the ‘answer’ is the same number. This is why the
distinction is hidden.
The two forms of division are sometimes named
partition and quotition but are more commonly
referred to as sharing and grouping. Think about
sharing out twelve sweets, one by one, amongst a
group of four friends, “one for you, one for you, one
for you, one for me”, repeated until you run out of
sweets. It is very similar to dealing cards. This is the
division we call sharing and is the one almost all of
the stories above are based on. I would keep dealing
out sweets until I ran out, possibly checking near
the end to see whether the resulting quotas, sweets
per person, were ‘fair’ or whether there is some
remainder. Remaining sweets might be cut up, to give
fair shares (quotas) which are not an integer number
of sweets.

The other form is grouping. It is seen in contexts like
boxing up items, filling carriages on fairground rides
or setting up teams for games. If I want my twelve
pupils to get into groups of four, I might ask, “Albie,
Bonnie, Clint and Dilys, can you work together, Edie,
Frank, George and Hannah, you can be a group,”
and so on. Notice, here I pick pupils in fours. I might
well be more aware that I am subtracting four each
time (repeated subtraction) from my total class of
pupils (my dividend). Any remainder here cannot
meaningfully be split. I might need one group of three
children for my group-work task.
















The three quantities involved in a division each have
their own proper names. Each also has its own unit.
You will notice that the units also tend to show us
whether a division is sharing or grouping.









Young children happily expand their vocabularies on
a daily basis, and I would suggest that these words
could be taught to give them a language to use in
order to describe sharing and grouping situations
with clarity and thus make the mental distinctions and
patterns clearer.
Note that in sharing the unit of the divisor is simple,
children, whilst that of the quotient is compound. In
this case, sweets per child.
Contrast this with grouping, where the divisor has a
compound unit, pupils per group, leading the quotient
to have a simple unit, the number of groups.

So far we have seen that the hidden duality of division
has its roots in the distinctions between sharing and
grouping. My sense is that these two aspects are very
rarely made explicit in early secondary mathematics
teaching in England and I guess that this is also true in
many primary classrooms. My conjecture is that this
hiddenness contributes strongly to learners’ difficulty
with division. This difficulty may be compounded by
our tendency to teach algorithms early. The order of
writing may confuse the learner about which is the
dividend and which the divisor and also signal one
of the forms of division when in fact the other form is
being attempted.
For example, in sharing scenarios like £158 ÷ 5
people, the ‘bus-stop’ algorithm is often read left-toright
as “fives into 158” which linguistically signals
grouping and places the dividend, the number
inside the bus-stop second in the spoken phrase as
opposed to first in the ‘linear’ version.

This algorithm can also be interpreted as a sharing,
that is:
100 to split between five people, how many
hundreds does each get, none; use the hundred to
make ten tens, so now fifteen tens to split between
five people giving three tens each. Finally eight
ones split between five people gives one, one
each and three ones remainder.
Or as a grouping:
How many fives in one (hundred; less than one
(hundred fives) so call it zero and recycle that
hundred, how many fives in 15 (tens); three (tens
of fives); how many fives in 8, just one and three
remainder.
In both cases the algorithm relies on division being
distributive over addition, allowing the dividend to be
partitioned on place-value lines, another element of
complexity. Chunking algorithms on the other hand
tend to signal grouping. It is more obvious what the
dividend is, but re-constituting the quotient may be
demanding. Here I have switched to a grouping
problem:

Up to now, our dividends and our divisors have all
been integers, suggesting discrete data. When
we introduce non-integer values, possibly from
continuous data, another hurdle presents itself. For
example, 158 ÷ 1.5 is hard to imagine as a sharing
problem, certainly we cannot have 1.5 people.
We could perhaps think of sharing 158 apples
between 1.5 wooden crates, a full sized crate and
a half sized mini-crate, yielding 105 1/3 apples per
full, whole, crate. But most often, the contexts we
can imagine will involve grouping, for example 158
litres of lager ÷ 1.5 litres per stein yielding 105 1/3
full steins.
In order to understand division further, pupils need
to consider the relationships which exist between
dividend and quotient (direct proportion) and between
divisor and quotient (inverse proportion), perhaps
helped by considering the units involved. This can
be linked to developing families of facts from a given
trio of related quantities and can help with definitions
of reciprocal and support general multiplicative and
proportional reasoning.

There follows two extracts from classroom dialogues
and visual representations shared by my colleague
Kathryn. As you read them, can you identify which
form of division problem the children were discussing?
Do you think it would have been possible to teach
them the terms dividend, divisor, quotient, remainder
as part of these episodes?


Example 1
Accurate use of language is at the heart of
understanding any mathematical concept. In one
primary classroom, the dialogue between the teacher
and pupils is around interpretation of the problem in
the first instance. This example demonstrates how the
pupils interpreted the problem and then discovered
The hidden faces of division
18 December 2017 www.atm.org.uk
The hidden faces of division
the type of division they were attempting to solve
through real life situations. There was an element
of role play followed by visualisation techniques to
ensure understanding. The conversation around it
was as follows:
Mr Green the Grocer has had a delivery of 52
oranges. He has 4 empty boxes on his shelf in
which to put them. How many oranges should he
put in each box?
T: What kind of a problem is this?
P: It’s a division one.
T: How do you know?
P: Because there are some oranges and we have to
put them into four boxes, so we’re going to give them
out between the boxes. We have to share them out.
T: Does that seem right? Is it true? Shall we divide? Is
this a sharing problem?
P: Yes, we can share them out.
T: Have we done a division like this before? Tom says
that we are sharing… have we shared before?
P: Yes, when Lucy brought her sweets in for her
birthday, she shared them between all thirty-one of
us.
T: So what’s the same and what’s different about this
new problem?
P: Giving the sweets out to each person is like giving
out the oranges. There are only four boxes though,
but there were thirty-one children.
T: OK, so what will our equation look like? (Teacher
scribed as pupil answered)
52 ÷ 4 =
T: Let’s read the equation together.
P: Fifty-two oranges divided into four boxes equals
……….. oranges in each box.
T: What does the fifty-two represent in this equation?
P: The fifty-two represents the number of oranges Mr
Green had delivered at the start. That is the whole.
T: That’s right. What does the four represent?
P: The four represents the number of boxes to put the
oranges into. They are the parts.
T: So, what does the represent?
P: The represents the number of oranges in each
box at the end.
Part ? Part ? Part ? Part ?
The ‘whole’ is 52 oranges.
T: How shall we solve the problem?
P: We need to start sharing the oranges.
The children proceeded to tackle the problem
independently using Dienes apparatus. They started
by putting single ‘ones’ cubes into each box, calling
each cube an orange, but very quickly one child
discovered that it was slow and said, “We could put
two at a time into the boxes”. Teacher then guided the
pupils into efficient use of the equipment.
T: Is there a way of being even more efficient?
Everyone place in front of you five ‘tens’ and two
‘ones’. I can see that Amy did that. Are we able to
share the tens equally between the 4 boxes?
P: Yes.
T: How many ‘tens’ will each box contain?
P: One ‘ten’ in each box.
T: What shall we do next?
P: There is still a ‘ten’ stick left, so we need to split it
into ‘ones’.
T: Ok, everyone do that. We can regroup the ten. So,
what can you see now?
P: There are twelve ‘ones’ now. We can share them.
T: Agreed. How many ‘ones’ can be put into each
box?
P: Three ‘ones’. That means there are thirteen
oranges in each box.
T: Are there any oranges left to share?
P: No, the boxes have an equal number of oranges
with none left.
The teacher revisited the question and asked the
pupils to speak the ‘number sentence’ or equation
again, this time including the result of the calculation.
P: Fifty-two oranges divided into four boxes equals
thirteen oranges in each box.
T: What did the fifty-two represent again?
P: The fifty-two represented the number of oranges
Mr Green had delivered at the start.
T: What did the four represent?
P: The four represented the number of boxes to put
December 2017 www.atm.org.uk 19
the oranges into.
T: What did the thirteen represent?
P: The thirteen represented the number of oranges in
each box in the end.
Example 2
Problem: A baker has completed making a batch
of 96 pies. Once they have cooled, he will need to
put them into boxes ready to sell. Each box can
hold 8 pies. How many boxes will the baker need?
The teacher observed the pupils tackling this problem
on whiteboards. There was a mixture of recording
strategies, such as those children attempting an
answer through mental arithmetic and one pupil who
used recall of division facts to record an answer.
However, it was unclear how that pupil had interpreted
the problem. Another method involved drawing.
T: What kind of a problem is this?
P: Another division one.
T: How do you know?
P: Like the oranges we have to share them out.
P: It’s not the same though because we have to sort
out eight pies and pack them, then do it again and
again until they are all gone.
T: Who is right? Are they both right? Are neither right?
P: They are both right that it is dividing, but I agree
with the sorting into eights.
T: What kind of dividing is that?
P: Its grouping, like the time we had to put donuts
into boxes of four. Every box had to contain four
doughnuts.
T: That’s right. We have to carry out the problem in a
different way in real life.
What will the equation look like this time?
P: 96 ÷ 8 =
T: Ninety-six pies divide into boxes of eight pies
equals ………… boxes needed.
Try this problem with your Dienes apparatus again,
but this time, what are we calling each ‘one’ cube?
P: Each ‘one’ cube will be one pie.
T: OK, what are we trying to find again?
P: How many boxes does the baker need for his pies?
The pupils immediately started to gather nine ‘tens’
this time and six ‘ones’. It was only when a child
pointed out that the cubes in tens didn’t really help to
create boxes of eight pies that the thinking changed.
The pupils started to gather ninety-six ‘ones’ instead.
It was apparent that the pupils were used to using
the whiteboards for helping to record their maths
because there were drawings appearing on them
each time a box was full.


T: Have we succeeded in creating groups?
P: Yes, every box is a group.
T: So, how many boxes have we filled? How many
groups are there?
P: There are twelve boxes of pies, twelve groups.
T: Let’s revisit the equation. 96 ÷ 8 = 12. How shall we
make sense of it?
P: Ninety-six pies divided into boxes of eight pies in
each box equals twelve boxes needed.
T: Is there a way to check the calculation?
P: We can multiply the number of boxes by the
number of pies in each box.
T: What is that called and what will that show?
P: It is using the inverse operation and it should tell us
how many pies the baker started with.



I expect you’ll have found this all a bit complicated, I
certainly do, but really, that is my point. We too often
present division to pupils as a simple thing, just the
inverse of multiplication, so we are surprised by how
often pupils do not divide 5x by 5 when solving for x, or
they guess what number 5 needs to be multiplied by
to give 158 instead of actually dividing. In reality, there
is a lot going on beneath the surface to be grappled
with. As a teacher, I need to grasp these ideas so
that I do not present mathematics to my pupils in a
contradictory and confusing way. As a pupil, a better
grasp of these ideas, through this language, will help
me to reason with and to comprehend some of the
most fundamental concepts in secondary maths.



References
Hart, K. (Ed.) (1981), Children’s Understanding of
Mathematics 11-16. London: John Murray.

Mark Simmons is a tutor on the mathematics PGCE at
the University of Nottingham.