Fractions
by K. Hart, Chelsea College
The CSMS (Mathematics) research team, based at Chelsea
College (University of London) from 1974-79 and funded by
the Social Science Research Council, investigated secondary
school children's understanding of 11 mathematics topics which
appear in the British secondary school curriculum. The main
purpose of the research was to describe hierarchies in each
topic. The team wrote word (or diagram) problems for each
topic to embody 'key' ideas; these were then used for interviews with about 30 children of secondary school age (about
300 children in all were interviewed). From the interviews both
correct and incorrect methods used by children to solve the
problems were identified. After class trials a written test paper
formed from the problems was issued to a representative sample
of English children. The sample was representative on IQ scores
in each age group tested for each topic, so that the children in
the sample for a test were approximately matched to a mixedability comprehensive school class. All the schools used in the
research (50 in number) were volunteers and were from both
urban and rural areas throughout England. London schools
were on the whole omitted from the paper and pencil testing,
but used for the interviews. About 10 000 children were tested;
nearly all of them took two test papers. The papers were
marked for correct and specific incorrect answers, these being
those that appeared during the interviews.
One topic investigated was 'Fractions', for which two papers
were written. The paper for the first two years in the secondary
school (ages 12-13) contained questions on the labelling of
'parts of a whole', equivalence, addition and subtraction. The
other paper, for children aged 14-15, contained multiplication
and division besides the above. First ideas for questions on
these papers came from a group of Bristol teachers attending
an in-service course, who interviewed children in their own
classes on how they did the problems. An additional feature of
the fractions test was that a parallel paper composed entirely
of computations (each intended to mirror one of the problems)
was given to the children immediately after the problem paper.
For example:
Computation 4 of 4
Problem:
Shade in I of the dotted
section of the disc. What
fraction of the whole disc
have you shaded? ..............
It was soon apparent that many of the problems were found
to be easier than the parallel computation forms, pointing to
the fact that many children were using methods other than
computational rules to solve them. The methods the children
used on interview were varied but many departed from what
the teacher would normally put forward as 'the method'. From
the test results and from talking to children it is obvious that "
(one half) is regarded as a familiar and useful entity. Questions
which had a half square as an area measuring unit were easy if
collection (adding) was all that was needed. An equivalence
involving 2 was also easier than other equivalent fractions. In
questions on Ratio (a separate test) those requiring the child to
halve or double were the easiest. Ratios such as 3: 4 when set
in a context such as 'this much for eight people, how much for
six' were predominantly solved (in the interviews) by saying
'Half of eight people is four people; half of four people is two
people; find both amounts and add'. This method was not
applied when halves could not be found. The fact that a child
understands 'one half' is no indication that he understands
fractions.
The equivalence of two fractions was tested in a number of
ways. The interviews brought to light the fact that what we as
teachers consider to be a generalisable method 'multiply top
and bottom by the same amount' is not used. The questions
which dealt with equivalences were of very different facilities.
Thus:
Question Facility (%)
1st year 2nd year
(i)
=
71.5
77.3
(ii)0 - 65.9 68.3
(iii) 2 = 59.3 58.3
(iv)
2=
_
57.3
55.0
(V) f -1-424.0
(v)
= 1024.0 23.9
Children interviewed (this is an indication of methods and
not necessarily generalisable) said for (i) above "1 added to
itself is 2, 3 added to itself is 6". This is cumbersome
when applied to '= A and some of the twos are likely to be
lost. For (ii) however they said "5 is half of 10, I want something which is half of 30" pointing again to the special nature
of one half. Question (v) was difficult mainly because the children either attempted to deal with all three fractions at once or
tried to form an equivalence between 4 and - which involved
yet another fraction. Popular substitutions for A were 21 (7%)
or 28 (14%) arising from a number pattern derived from the
denominators.
The youngest children (first years) performed better than
any older group on the addition and subtraction computations
but there was a slight increase in the facilities of the problems
as older groups were tested. Many children misremembered
rules when dealing with the computations, one of the most
popular being "to add fractions, add tops and add bottoms".
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For example
Years
1st 2nd
3rd 4th
%
(i) + 2=
(ii) + = 16
%
18.3 29.1
21.8 19.9
8.5 19.7
14.3 16.7
An interesting phenomenon was that in the problem form of
(ii) the incidence of this error was less, i.e. 'In a baker's shop
j of the flour is used for bread and I of the flour is used for
cakes. What fraction of the flour has been used?'.
12 yrs 13 yrs 14 yrs 15 yrs
problem
4.9
7.4
6.8
6.0
- problem
77.0 73.0 80.8 77.2
- computation
77.6 66.3 71.8 67.9
Giving a fraction name to part of a whole was tested on the
papers for both samples. The labelling is not necessarily
straightforward and success depends on the model presented.
For example these three presentations were successfully identified by different percentages of children:
fed to an eel 10 cm in length and asked how much a 25 cm eel
would be fed (proportionately) no child on interview multiplied
by -. Many said "take the amount for 10 cm, take it again, take
a half and do an add". Others found the food for a 5 cm eel (not
given) then for a 15 cm eel (drawn in question) by adding and
finally added the food for the 15 cm eel to the food of the 10 cm
eel to find the amount for the 25 cm eel. More difficult were
problems where this addition method included fractions other
than 1. Asked to provide the new upright when the base was
5 cm and the new figure was an enlargement of 2
3
the method involves the fact that 5 contains 3 and a bit of 3.
In this case about 40% of the 3 000 children tested opted for 5
is 2 greater than 3 so the upright must be 4 cm. Indeed many
children said "there is no way you can get from 3 to 5 except
by adding".
In measurement the application of the formula for the volume
of a cuboid when it involved dimensions which were given as
fractions, proved to be very difficult, for example note the
percentage success rate for these two items:
Find the volume
Age
12+ 13+ 14+
%
(i)
%
%
55.6 59.7 67.3
1. Shade in two-thirds of this shape.
3 cm
Age
ccm
12+ 13+
2 cm
92.7 92.5
2a. John wins + of these marbles.
Draw a ring round his marbles.
Age
(ii)
76.8 77.7
12+ 13+ 14+
21 cm K
22 cm
2 cm
2b. Jane wins J of the marbles.
14.2 % 27.9
14.2 18.7 27.9
Un interview tor question (1) a number ot chnildren counted the
squares in a layer and then added for the number of layers, so
part of the success on this item can probably be traced to naive
methods rather than the use of a formula. This form of addition
How many marbles does she win?
70.7 73.1
3. Shade in two-thirds of this shape.
61.4 66.3
cannot be applied to question (ii) unless the child has very good
perception and can ascertain the size of the blocks at the back
and even then he must add fractions.
In the CSMS investigation of Number Operations (Brown and
Kiichemann, 1976) children were asked to write stories which
illustrated computations such as 9 + 3 and 4 x 5. The division
was in nearly all cases interpreted as sharing sweets between
friends and the multiplication was seen as repeated addition or
At the other end of the scale, questions which required multiplication or division of fractions (given only to third and fourth
years) were successfully solved by very few children, e.g.
Age
(i)
Scm
'rate', e.g. 'There are 84 skirts on a rack at a clothes shop.
There are 28 racks. How many skirts altogether?' (for 84 x 28).
It seems likely that the children tested on fractions (and
decimals) tried to use these whole number models with little
success. When faced with 4 +-, a child who looks for a number
of objects and a number of friends is doomed to failure. In this
connection it is noteworthy that asked for a result to 3 + 5 many
translated the question into 5+ 3, which would match the
< length >
Area = j square centimetres
Length =
(ii) An Australian in London converts
pounds to Australian dollars when
he wishes to buy something. a1 is
equivalent to 1- Australian dollars.
If he pays a45.50 for a coat, how
much is this in Australian dollars? 10.1 11.6
In both the above, the correct operation has to be identified
then carried out; over half the sample did not attempt the
questions.
We have found when dealing with ratio and proportion
problems that multiplication by fractions is avoided at all
costs. If a solution can be found for a problem by doing addition, this will be used. For example, given the amount of food
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descriptions given of whole number division, i.e. sharing sweets
between friends. In 3 + 5 of course there are insufficient sweets
to share between five friends.
Different Responses to 3 + 5
- or .6 1 1 rem 2 5 or 1l
12 yrs 35 5.3 18.3 3.3
13 yrs 31 9.4 17.5 8.7
Between 25 and 30% of each year divided the smaller number
into the larger in some way.
Similarly when faced with 16 20 51% of the first year
(12 + ) and even 23% of the 15 year olds said the question was
impossible, even though the question paper was titled 'Place
Value and Decimals'.
Even when adding fractions it seems likely that children were
interpreting it as the addition of whole numbers. + = - is a
logical outcome of the view that the question asked is 1 + 1
cannot generalise about them and probably do not see them as
an extension of the set of whole numbers. The last denies to
and then 4 + 2, but keeping the position of the figures correct.
We require children to view fractions in a number of ways:
them the possibility of carrying out computations or working
problems which within whole numbers were impossible.
Should we perhaps make children more aware of the reasons
for fractions (and indeed for decimals and negatives) so that
they can see some point for teaching them?
As the name of a part, e.g.
(The research of the mathematics team of CSMS is fully
2. As a division of two integers, e.g. I
3. As a multiplier in ratio, e.g. 3: 4
reported in a research monograph (Hart, 1980) and a book for
teachers (CSMS team, 1980).)
4. As an equivalence class (', 6, ...).
I suggest that when teaching the topic we concentrate (certainly
in the primary school) on the 'part' aspect, and expect the
other meanings to be 'picked up'. Do we make a conscious
effort to inter-relate the different meanings and show that they
each serve a purpose?
The results of the CSMS investigation tend to show that the
majority of secondary school children avoid using fractions,
The octahedron
by Henry Lulli, John C. Fremont High School,
Los Angeles, CA 90003
B
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Fig. 3
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A
Fig. 1
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Fig. 4
B Fig. 2
s/ ~
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K
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Open the pattern and label all points on both sides of
the fold. The top and bottom strips are folded in,
forming rectangle IJNM. Fold corners J to M. Crease
OP. See Figure 3. Fold A IOJ on top of A JOP; and A
JPN under A JOP. This results in Figure 4. Point I may
not be on the mid-point crease of the side OP.
A
E
K
H F
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Research Monograph, C.S.E. Chelsea College, London University.
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11-16, John Murray, London.
Hart, K. M. (1980) Secondary School Children's Understanding of Mathematics,
H
B
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along EH, as shown in Figure 1. Crease. Open the first
fold ED. Bring edge CD up to point G, holding the.sides
parallel. Crease the fold and label it MN. It is shown in
Figure 2. Open the upper flap and crease a fold through
G by bringing the bottom MN up while keeping the
sides parallel. The resulting fold is KL, also shown in
Figure 2. Finally, bring the upper flap containing edge
AB down, creasing and labelling it IJ as in Figure 2.
E
in School, 5, 5 and 6, 1.
CSMS Mathematics Team. (1980) Children's Understanding of Mathematics,
/\8 /\a\O /
Open the sheet. Place the edge AD on the diagonal BD.
Crease and label the fold ED. Likewise, place edge BE
1
Brown, M. and Kiichemann, D. E. (1976) "Is it an 'add' Miss?" Mathematics
Fig. 5
A model of the regular octahedron may be constructed
from a square sheet of paper without the aid of
geometric tools, scissors, or glue. This is the third
Platonic polyhedron constructed using the above
technique. The references mention the others.
Start with a square sheet ABCD as in Figure 1. All
labelling is done on both sides. For a sturdier model,
stiff paper is recommended. Fold the diagonal BD.
A
References
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6 4 / / \R
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/ \ / 23
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Fold vertex 0 to mid-point of opposite side G. Open
the vertex and repeat with the other two vertices P and J.
Now open the entire sheet to reveal the triangular
tessellation of Figure 5. Due to the many folds of
paper, the four border triangles, having the common
vertex N, may be somewhat distorted. Do not readjust.
Label all points and number the polygons as shown.
Refold all creases inward. Gently tear off and discard
the shaded A BNH and A NCV. Also tear along the four
solid lines: EO, GP, PR, and US. Reverse fold KQ.
The assemble, rotate A 1, CCW about point S, on
top of A 2, so point U is on top of point R. Next, rotate
A 3 CW on top of A 4. Now, hold the figure with the
left hand outside A 4. Fold A 5 and A 6 inward, so
point I coincides with point M. Fold A MDT inward
and flap 7 on top. Flap 8 is folded on top of flap 7.
Points A and E are draped
over the adjacent sides.
Fig. 6
Now place A 9 atop flap 8.
Finally, A 10 is tucked into the
"pocket" under A 2. The result
is the regular octahedron of
Figure 6.
References
1. Lulli, H. (October 1978) "The Hexahedron." School Science and
Mathematics, 78, 518-519.
D
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2. Lulli, H. (1978-80) "Tangled Tetrahedron." Journal of Recreational Mathematics, 12, 3, 172-176.
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