An Exploration Of Learners Theorems-In-Action Used In Problems On Ratio And Proportion

http://www.pythagoras.org.za doi:10.4102/pythagoras.v36i2.252
Page 1 of 10 Original Research
Authors:
Patisizwe T. Mahlabela1
Sarah Bansilal1
Affiliations:
1
Department of Mathematics
Education, School of
Education, University of
Kwazulu-Natal, South Africa
Correspondence to:
Sarah Bansilal
Email:
bansilals@ukzn.ac.za
Postal address:
8 Zeeman Place, Malvern
4093, South Africa
Dates:
Received: 06 Jan. 2014
Accepted: 29 Sept. 2015
Published: 18 Dec. 2015
How to cite this article:
Mahlabela P.T., & Bansilal S.
(2015). An exploration of
learners’ theorems-in-action
used in problems on ratio
and proportion. Pythagoras,
36(2), Art. #252, 10 pages.
http://dx.doi.org/10.4102/
pythagoras.v36i2.252
Copyright:
© 2015. The Authors.
Licensee: AOSIS
OpenJournals. This work is
licensed under the Creative
Commons Attribution
License.
The purpose of this study is to explore Grade 9 learners’ understanding of ratio and
proportion. The sample consists of a group of 30 mathematics learners from a rural school in
the province of KwaZulu-Natal, South Africa. Data were generated from their responses to
two missing value items, adapted from the Concepts in Secondary Mathematics and Science
test set in the United Kingdom over 30 years ago. The study utilised Vergnaud’s notion of
theorems-in-action to describe the learners’ strategies. It was found that the most common
strategy was the cross multiplication strategy. The data reveal that the strategy was reduced
to identifying and placing (often arbitrarily) three given quantities and one unknown in four
positions, allowing the learners to then carry out an operation of multiplication followed
by the operation of division to produce an answer. The study recommends that the role of
the function underlying the proportional relationship should be foregrounded during the
teaching of ratio and proportion.
An exploration of learners’ theorems-in-action used in
problems on ratio and proportion
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Introduction
There have been numerous studies focused on ratio and proportion which have looked at learners’
strategies and errors (Ben-Chaim, Keret & Ilany, 2012; Chick & Harris, 2007; Jiang, 2008; Lamon,
2007; Long, 2011; Md-Nor, 1997; Misailidou & Williams, 2002; Olivier, 1992). Many researchers
agree that understanding proportional relationships is a long-term developmental process (Ben-
Chaim et al., 2012) and forms a fundamental building block for many other areas in mathematics
as well as in the sciences. Many phenomena studied in daily life, physics, chemistry, biology,
geography, agriculture, woodwork and needlework and economics are defined using proportion
(Ben-Chaim et al., 2012; Olivier, 1992). It is evident that a poor understanding of proportion can
limit a learner’s access to various concepts and subjects.
This article reports on part of a bigger study which investigated Grade 9 learners’ understanding
of ratio and proportion (Mahlabela, 2012). The bigger study used eight test items adapted from
the original Concepts in Secondary Mathematics and Science study (Hart, 1981). In this article
we report on the strategies and errors exhibited by the learners in two of these items, which
consist of missing value problems where three quantities are given and the task is to find the
fourth one.
Strategies used in solving missing value problems
To understand learner errors, one has to look at the methods or strategies that the learners use to
arrive at the incorrect solutions. Errors could be the results of incorrect strategies or the results
of incorrect use of correct strategies. Some strategies identified in the literature are now briefly
described.
The for every strategy entails finding the simplest ratio first and then multiplying by a factor
that yields the required result. The unit value strategy is similar except that the simplest ratio
is reduced to a unitary ratio. This strategy has been identified in studies conducted in various
countries (Hart, 1988; Jiang, 2008; Md-Nor, 1997; Misailidou & Williams, 2003).
The multiplicative strategy (within measure space approach) entails determining a ratio of
measures from the same space and using it as a factor (Hart, 1988; Md-Nor, 1997; Misailidou &
Williams, 2003).
The cross multiplication strategy based on setting up a proportion is also described as the rule of
three by some (Md-Nor, 1997; Vergnaud, 1998) or algebraic method (Jiang, 2008) or the use of the
formula x/a = y/b (Hart, 1984; Olivier, 1992).

In the constant difference or incorrect addition strategy or additive
strategy, the relationship within the ratios is computed by
subtracting one term from another and then applying the
difference to the second ratio (Hart, 1988; Md-Nor, 1997;
Misailidou & Williams, 2003).
According to Misailidou and Williams (2003) the additive
strategy is the most commonly reported erroneous strategy
in the research literature. Long (2011) made a similar finding
in the Trends in International Mathematics and Science Study
(TIMSS) with South African learners.
The building-up strategy involves establishing a relationship
within a ratio and then extending it to the second ratio by
addition (Md-Nor, 1997, p. 34) and is also referred to as the
addition and scaling strategy (Hart, 1988), because it involves a
multiplicative strategy combined with an additive one.
Theincompletestrategy(Misailidou&Williams,2003)involves
using the same number given for the measure space. Long
(2007) found that in TIMSS some responses were incorrect
because of ‘incomplete reasoning’. She says that learners’
‘reasoning took them part way towards the answer’ (p. 16).
An incorrect strategy is the incorrect doubling method. Since
there are problems involving ratio and proportion where
doubling is a correct strategy, the word ‘incorrect’ signals
that the use of doubling was wrongly implemented in the
setting.
A framework for understanding
ratio and proportion
Researchers (Ben-Chaim et al., 2012; Olivier, 1992; Vergnaud,
1998; Wu, 2006) point out that proportional reasoning is just
mathematical reasoning based on the concept of a linear
function without a constant term.
Proportional reasoning requires a recognition that a situation
is completely described by a function of the form shown in
Equation 1:
f x cx c
( ) ,
= for some constant [Eqn 1]
Knowing
f x
x
c
( )
= for all x implies that for any two non-zero
values x1
and x2
, Equation 2 is true:
f x
x
f x
x
( ) ( )
1
1
2
2
= [Eqn 2]
We say the four numbers f(x1
), x1
, f(x2
) and x2
form a proportion
which can be expressed in terms of ratios: f(x1
): x1
= f(x2
): x2
.
The proportional relationship can also be written as
Equation 3:
[Eqn 3]
Equivalently, it can be expressed as the ratio representation
x2
:x1
= f(x2
):f(x1
).
The relationship expressed in Equation 2 is such that the two
quantities forming the fraction are from different measure
spaces; this expression is referred to as a rate and the value
is actually the constant coefficient (Vergnaud, 1998) or the
constant of proportionality, c, from Equation 1. The term
measure spaces is used by Vergnaud to express the fact that
certain quantities belong to one measure space, which may
have particular measures of, for example, length or cost, and
others belong to a second measure space, which may have
dimension of weight, distance or length, for example.
Considering Equation 3, we note that the fraction formed
by the two quantities in each case has no dimension because
they are from the same measure space. Vergnaud (1998)
refers to this as a scalar fraction. Hence, the proportional
relationship can be expressed as an equality between two
rates (Equation 2) or as an equality between two scalar
fractions (Equation 3).
Vergnaud (1998) uses the term theorems-in-action to describe
the cognitive processes and yet-to-be concepts of a learner.
A theorem-in-action is the set of mathematical relations
considered by learners when they choose an operation or a
sequence of operations to solve a problem. These theorems
are implicitly held by learners and may not be articulated.
Sometimes these theorems-in-action are not correct, but
they can help teachers and researchers to analyse learners’
intuitive strategies and give insight into their thinking. Long
(2011, p. 109) asserts that identifying ‘theorems–in-action
provides a way to make a better diagnosis of what students
know and do not know’.
An important property of the linear function f(x) is described
by Equation 4:
f(λ1
x + λ2
x) = λ1
f(x)+ λ2
f(x) [Eqn 4]
Vergnaud (1998) draws on this property to illustrate learners’
implicit theorems-in-action when solving a problem based
on a train travelling at constant speed, which took 16 minutes
to travel 40 km from Town A to Town B. Learners were asked
for the distance between Town B and Town C, given that it
took the train 36 minutes to travel from Town B to Town C.
A learner wrote: 40 × 2 = 80; 80 + 10 = 90. Vergnaud explains that
36 minutes can be decomposed into 2 × 16 minutes + 4 minutes,
and 4 minutes = ¼ of 16 minutes. Therefore, the corresponding
distance is 2 × 40 km + ¼ of 40 km.
Note that one can manipulate Equation 4, to derive
a further property used by Vergnaud (1998), that is
f(m.1) = c(m.1) = c.m = f(1).m = m.f(1), thus leading to
f f m
m
( ) ( )
1
1
= , and hence to Equation 5:
f(1)= f(m)/m or f(m) = m.f(1), [Eqn 5]
where m is any real number
x
x
f x
f x
2
1
2
1
=
( )
( )

Vergnaud describes another way of solving proportionality
problems, which he calls the rule of three. Equation 6 follows
directly from Equation 3:
f x
f x x
x
( )
( )
2
1 2
1
= [Eqn 6]
In Equation 6, f(x1
), x1
and x2
are known and f(x2
) is required.
Methodology
This qualitative study was conducted with a group of 30
Grade 9 learners from one school in KwaZulu-Natal, South
Africa.Theschoolissituatedinaruralarea,withmostlearners
coming from impoverished backgrounds. The school was
selected because of its proximity to the authors, which can
be described as convenience sampling. Ethical requirements
were fulfilled according to the ethical procedures stipulated
by the local university.
The original larger study (Mahlabela, 2012) used eight
questions that were adapted from the Concepts in Secondary
Mathematics and Science study. However, in this article we
only report on learners’ responses to two items, because of
the variety of rich responses that emerged from these. After
an initial marking and coding of the scripts, five learners
were identified and selected for semi-structured interviews.
Learners are referenced as Learner 1 to Learner 30, according
to the order in which the scripts were collected; the numbering
does not denote any difference in ability between the learners.
The research questions that underpin this study are:
1. What strategies do learners use to solve the two test
items?
2. What are the underlying theorems-in-action associated
with these strategies?
3. What do the theorems-in-action suggest about the learners’
understanding of ratio and proportion?
The learners’ written responses were analysed by studying
their final answers and their working details to identify
the strategy that was used. In some cases we were unable
to identify a definitive strategy and this was described as
‘miscellaneous’. Interview responses helped to support our
strategy classification.
Findings
Performance on Question 1
Learner responses are analysed separately: Question 1a and
Question 1b first and the rest later. Table 1 shows how
learners responded to Question 1a and Question 1b; Table 2
shows performance in the rest of the questions.
One way of solving Question 1a and Question 1b would be to
halve the ingredients in the given recipe since soup is being
prepared for four people. More than 70% of the participants
responded correctly to both questions. Some learners used
the incorrect strategy of incomplete halving, where they just
indicated that they needed half of the ingredients given in the
recipe, but did not work out what half the actual amounts were.
Other errors were related to incorrect cross multiplication as
some learners obtained the solution of 4 in Question 1a as
shown in Figure 2.
Learner 13 omitted an equal sign between the equivalent
fractions. The ratio of people in the two recipes (given recipe
for eight people and the recipe for four people) is 8:4, which
the learner wrote as (8 people)/(4 people). The ratio of water
amounts is 2:x, where x is the amount of water needed to
make soup for four people. She correctly formulated the
two scalar fractions described in Equation 3 by comparing
quantities from the same measure space (i.e. quantities of the
same kind or elements of the same set). To find the amount
of water needed (value of x), she multiplied 8 (the number of
people given in the recipe) by 2 (the amount of water needed
to make the soup for eight people) and then divided the
product by 4 (the number of people that soup was needed
for). We refer to this error as incorrect cross multiplication. The
learner explained her method of cross multiplication and,
interestingly, provided an explanation different from the
written one appearing in Figure 2:
Interviewer: I can see that your answer to this question is
4 pints. Tell me how you got 4 pints?
Learner 13: I took 8, the number of people in the recipe and
wrote it down. I then took 2, the number of
pints needed to make soup. I wrote 8 over 4, the
number of people that I want to make the soup
for. Because I do not know the number of pints
of water needed, I wrote over x [the unknown].
TABLE 1: Learner responses to Question 1a and Question 1b (N = 30).
Question Description of
response
Correct Incomplete
halving
Incorrect cross
multiplication
1a Learner response 1 ½ 4; 16
No. of learners 23 3 4
1b Learner response 2 ½ 8
No. of learners 26 1 3
Onion soup recipe for 8 people
8 onions
2 pints of water
4 cubes of chicken soup
2 spoons of bu er
½ pint of cream
I am cooking onion soup for 4 people.
(a) How much water do I need?
(b) How many cubes of chicken soup do I need?
I am cooking onion soup for 6 people.
(c) How much water do I need?
(d) How many cubes of chicken soup do I need?
(e) How much cream do I need?
FIGURE 1: The onion soup recipe (Question 1).
FIGURE 2: Response of Learner 13 to Question 1a.

I then cross multiplied. I got 8x and 8. I divided
by x, although I am no longer sure this is how
I did the calculation [she realises that this solution
is different from the one which appears on the script].
I then wrote x and moved 8 to the other side
[as if she has the equation 8x = 8].
Learner 13 in her written response in Figure 2 seemed to
confuse cross multiplication with the rule for multiplication
of fractions, that is, multiplying the numerators of the scalar
fraction to get the numerator in the answer and multiplying
denominators to get the denominator in the answer (however
the confusion is not exhibited in her interview). A perusal of
the learner scripts revealed that the error in Figure 2 recurs in
other solutions obtained using this strategy.
One way of solving Question 1d and Question 1e would be to
halve(toobtainingredientsforfourpeople),halveagain(obtain
ingredients for two people) and then add the result to the first
halving to find the ingredients required for 6 people − called
the build-up strategy. For example, to determine the amount of
water needed to make soup for six people, learners could halve
2 pints of water (water needed to make soup for eight people)
to get 1 pint (amount of water needed to make soup for four
people). They could halve 1 pint of water again to get ½ pint
(amount of water needed to make soup for two people). The
amount of water needed to make soup for six people is then
1 pint of water + ½ pint of water = 1½ pints of water.
The problems could also be solved by the use of the for every
strategy or the multiplicative strategy. The performance of
learners is shown in Table 2.
Over half (60%) of the learners answered Question 1c
correctly, more than 66% correctly answered Question 1d,
but only 10% answered Question 1e correctly. This is not
surprising as Question 1e required halving and addition of
fractions whilst Question 1d required halving and addition
of whole numbers and Question 1c required halving of whole
numbers and addition of a whole number to a fraction.
The incomplete use of the build-up strategy (incomplete
build-up strategy) was observed in all three questions. In
Question 1c, for example, 6% of the learners either halved
once to get 1, or halved twice to get ½. Learners who halved
twice did not add the first answer to the second answer.
Many learners also used an incomplete strategy (using the
same number given for the measure space). For example,
in Question 1e it is stated that ½ pint of cream was needed
to make soup for eight people. About 13% of the learners
said that ½ pint of cream was needed to make soup for six
people too. Some strategies were based on incorrect cross
multiplication.
Other incorrect answers obtained by the learners,
categorised as ‘other’, were diverse. The solutions emanated
from the incorrect performance of either basic operations,
conversions from one unit to the other or both. For example,
in Question 1d, learners divided 24 by 8 and obtained 2 as
the quotient (computational error). In Question 1e, learners
divided (½ × 6) by 8 and obtained 0,75 (computational
error). Some learners found the product of 6 and ½ to be
7,2 (a computational error) and then some correctly divided
7,2 by 8 to obtain 0,9, whilst others divided 7,2 by 8 to
obtain an incorrect answer of 9. In Question 1d one learner
incorrectly converted ½ to a decimal as 1,5. To arrive at ½
in Question 1e, Learner 15 incorrectly answered as shown
in Figure 3; this was categorised as an incomplete strategy. It
however includes the same misconception as that displayed
by Learner 13 in Figure 2, who multiplied fractions instead
of ‘cross multiplying’ across an equal sign, because there
was no equal sign between the fractions.
Learner 15’s response in Figure 3 shows that he has modelled
the relationship correctly by mapping 8 people to 6 people and
½ pint of cream to the unknown volume of cream. However,
by not expressing the relationship as equivalent fractions, the
learner has taken this as a product of two fractions (similar
to the misconception Learner 13 displayed in Question 1a).
However, Learner 15 has furthermore ignored the unknown
and carried out the computation with numbers only. A third
error is that he has interpreted the quotient of (8 × 6) and ½
as 4 × 3, which represents two errors: taking divided by ½ as
divided by 2 and also taking pq/2 as p/2 × q/2. Thus, there
are many algebraic misconceptions in these few lines.
Performance on Question 2
The question in Figure 4, a version of Piaget’s popular
eel question, was the second question in the research
instrument.
Piaget’s eel question is concerned with the amount of
food given to eels of different lengths, the amount being
TABLE 2: Learner responses to Question 1c–e (N = 30).
Question Description of response Correct response Incomplete build-up
strategy
Incomplete strategy Incorrect cross
multiplication
Other incorrect
strategies
1c Learner response 1½ ½ or 1 2 2; 6; 24 Miscellaneous
Number of learners 18 2 3 3 4
1d Learner response 3 2 4 5; 12 Miscellaneous
1e Learner response 3/8 ¼ ½ 2/3 Miscellaneous
FIGURE 3: Response of Learner 15 to Question 1e.

proportionate to the length of the eel (Hart, 1981). Table 3
shows the performance of learners for each sub-question.
More than 50% of the learners responded correctly to each
sub-question, except in Question2d. The performance of
learners in Question 2a was outstanding as 90% of the
learners responded correctly to this sub-question.
Many errors in this question resulted from the incorrect use of
cross multiplication strategy. Figure 5 shows how Learner 6
arrived at the incorrect answer of 25 sprats in Question 2a(i).
TABLE 3: Learner responses to Question 2 (N = 30).
Question Description of response Correct response Incorrect cross
multiplication
Incorrect addition
strategy
Incorrect doubling or
halving
Other incorrect
strategies
2a(i) Learner response 4 25 – – 5
2a(ii) Learner response 6 37,5 – 8 4; 7 or 75
2b Learner response 18 12,5 14 or 17 24 2
No. of learners 17 3 4 4 1
2c Learner response 6 16,7 4 or 7 4½ (18) Miscellaneous
Number of learners 19 2 2 2 (1) 4
2d Learner response 5 125 6 or 7 8 Miscellaneous
2e Learner response 15 41,7 11 18 (36) Miscellaneous
2f(i) Learner response 4 25 6 2½ Miscellaneous
2f(ii) Learner response 6 37,5 5 or 8 5 Miscellaneous
FIGURE 5: Response of Learner 6 to Question 2a(i).
FIGURE 4: The eel question (Question 2).
(a) If eel A is fed 2 sprats,
(i) How many sprats should eel B be fed?
(ii) How many sprats should eel C be fed?
(b) If eel B gets 12 sprats, how many sprats should eel C be fed?
(c) If eel C gets 9 sprats, how many sprats should eel B get?
Three other eels X, Y and Z are fed fish fingers.
The mass of the fish fingers depends on the length of the eel.
(d) If eel X gets 2 grams of fish fingers, how much fish fingers should be given
to eel Z?
(e) If eel Y gets 9 grams of fish fingers, how much fish fingers should be given
to eel Z?
(f) If eel Z gets 10 grams of fish fingers,
(i) how much should eel X get?
(ii) how much should eel Y get?
A
B
C
5 cm long
10 cm long
15 cm long
10 cm long
15 cm long
25 cm long
X
Y
Z
Learner 6 used the equal sign incorrectly by expressing
equality between quantities from two different measure
spaces (2 sprat = 5 cm and x sprats = 10 cm). In order to apply
the cross multiplication strategy, two equivalent rates or scalar
fractions would be needed, which Learner 6 did not identify
correctly. In fact, the result was obtained by multiplying the
length of eel A (5 cm) by the length of eel B (10 cm) and then
dividing the product by 2 (the number of sprats that A is fed
with).
The responses of Learner 17 (see Figure 6) indicate the use of
the cross multiplication rule. However, the choice of which
of the available numbers should be the multipliers and
which should be the divisor seems to systematically follow a
FIGURE 6: Response of Learner 17 to Question 2.

set formula. In the first case it was that the number of sprats
for eel B =
×
length A lengthB
noof sprats foreel A
; this was similar in each case.
Learner 17 has thus consistently used the cross multiplication
rule as f x
x x
f x
( )
( )
2
1 2
1
=
×
, where x is the length of the eel and f(x) is
the number of sprats fed to the eel.
Note that at no point in these or any other solutions
which used the cross multiplication rule was there a
point where any learner expressed equality between two
scalar fractions or two rates. For example, consider the
response of Learner 14 to Question 2 in Figure 7. The
learner has modelled the situation correctly by associating
the quantities from one measure space to the next, in the
correct order, for each of the questions. She then carried
out a cross multiplication across the vertically opposite
numbers and created an equation in each case which
she then solved correctly. All of these steps were done
four times, without once expressing an equation in the
form of Equation 2 or Equation 3, without which none of
these relationships holds true. The fact that none of the
learners expressed such equations suggests that they were
influenced by teaching methods.
The incorrect addition strategy was also observed in learner
responses. Participants suggested that the eels should be
given two more or fewer sprats or fish fingers, depending on
the length of the eel. Actually the addition strategy leads to
the correct answer for Question 2a. If eel A is fed two sprats,
then eel B should be given two more sprats, because eel B
is twice the length of eel A. However, the strategy does not
work for other sub-questions. When used in Question 2b,
it yields an incorrect answer of 14 (obtained by 10% of the
participants). There were also instances where learners added
or subtracted 5 (the difference in eel lengths). For example, an
incorrect result of 17 sprats in Question 2b was obtained by
adding 5 sprats to 12 sprats.
The incorrect doubling or incorrect halving strategy was also
identified in some responses. The strategy works correctly
in Question 2a(i), but immediately fails in Question 2a(ii),
yielding an incorrect result of 8. The reason for the incorrect
answer of 4½ in Question 2c is incorrect halving.
Figure 8 illustrates correct and incorrect solutions, based on
the unitary strategy from the script of Learner 16. Learner 16
divided or multiplied the eel length by the number of sprats
or mass of fish fingers it is fed. She divided where the number
of sprats or mass of fish fingers was given for a longer eel in
order to find how much should be given to the shorter eel.
To determine how much the eel should be given, its length
should be divided by the quotient obtained during the first
division. In Question 2c the longer eel is 15 cm. The eel is fed
9 sprats; hence, Learner 16 divided 15 by 9 and got 1,67. The
length of eel B (10 cm) was then divided by 1,67. She obtained
an answer of 5,99 which she then incorrectly rounded off to
5,10. The correct answer is 6.
Her attempts at Question 2d reveal her uncertainty. She first
wrote 10 ÷ 2 = 5, followed by 25 ÷ 2 = 12,5 (instead of 25 ÷ 5,
which is the quotient of the first answer). She then struck off
the division signs, replaced them with multiplication signs
and repeated the same method: 10 × 2 = 20, then 25 × 2 = 50.
Similarly for Question 2e, she first wrote 15 ÷ 9 = 1,67 and
25 ÷ 1,67, which is correct. However, she then replaced this
with 15 × 9 = 135 and 25 × 1,67 = 41,67. Learner 16 seemed to
have confused herself about the operations that she needed
to carry out.
A scrutiny of Question 2f shows that she seems to have
worked through her confusion and presented the correct
operations and sequence of operations as she did for
Question 2c. However, she has made a slip and swopped the
two solutions. That is, her response for Question 2f(i) is the
solution to 2f(ii) and vice versa.
Additional information on learner errors was obtained
through interviews. The interview with Learner 8 is now
presented in three excerpts, each of which details a different
FIGURE 7: Responses of Learner 14 to Question 2. FIGURE 8: Response of Learner 16 to Question 2.

strategy. The dialogue below (Excerpt 1) elaborates on how
Learner 8 arrived at his solution for Question 2a:
Interviewer: You said that if eel A gets 2 sprats, then eel
B must get 4 sprats. … How did you arrive
at that?
Learner 8: Eel A is 5 cm long, right. If eel A gets 2 sprats,
then eel B gets 4 because 5 is half of 10. So if eel
A gets 2, eel B gets 4.
Interviewer: If A gets 3?
Learner 8: Then B gets 6.
Learner 8 seems to recognise proportion very well in this
problem. He realised that eel B is double the length of eel A
and therefore should get double the number of sprats given
to eel A.
The conversation (Excerpt 2) continued by looking at
Question 2b as follows:
Interviewer: If eel B gets 12 sprats how many should eel A
get?
Learner 8: If eel B gets 12 sprats, then eel A should get 6.
Interviewer: Good. If eel B gets 12 sprats, you said in
your script eel C gets 24 sprats. How did you
get 24?
Learner 8: 24? I was in a rush then, I must have said 12 × 2
and got 24. I should have said if B gets 12, C gets
30.
Interviewer: 30? … How did you get 30?
Learner 8: I said 15 × 2.
Interviewer: Why did you say that?
Learner 8 Eel C eats more than eel B.
Interviewer: More?
Learner 8: Yes.
Interviewer: Why do you multiply by 2?
Learner 8: I was in a hurry and the bell was ringing.
In Excerpt 2, Learner 8 multiplied 12 by 2 to get 24 and
multiplied 15 by 2 to get 30, showing that he has now tried
to extend the doubling strategy which yielded the correct
answer in Excerpt 1. However, doubling does not work in this
case. He seems convinced that eel C should get twice what
eel B gets. The continuation of the conversation (Excerpt 3)
now reveals a different strategy to Question 2d:
Interviewer: Let us look at this one [Question 2d]. You said
that if eel X gets 2 grams fish fingers, eel Z must
get 7 grams. How did you get 7?
Learner 8: I added 5.
Interviewer: OK. You said if eel X gets 2 grams, eel Z gets
2 + 5 grams?
Learner 8: I said 2 + 5 and got 7.
Interviewer: Where did you get 5 from?
Learner 8: I said 25/5 and got 5, then here I said 2 × 5 to get
10 [length of eel X] and if I multiply 5 by 5 I get 25
[length of eel Z] over here.
Interviewer: Here [pointing at the learner working] you
multiplied 5 by 2 and here you multiplied
5 by 5? So when you get 2 you add 5?
Learner 8: Yes, and I got 7.
The conversation with Learner 8 in Excerpt 3 reveals that he
has used an incorrect additive strategy: adding 5. However,
his explanation is interesting. He correctly identified that the
eel lengths are multiples of five with eel X being the second
multiple of 5 (2 × 5) and eel Z being the fifth multiple of 5
(i.e. 5 × 5). However, having identified that the length of eel
Z was the fifth multiple of 5, he then added the 5 to the 2
(from number of sprats that should be fed to eel X. He has
not realised that he had obtained the required answer of 5.
Note that Learner 8 has described three different strategies in
the three excerpts of the interview, showing his uncertainty
about the underlying relationships.
A conversation with Learner 27, who also used an additive
strategy, went as follows:
Interviewer: Let us look at this one [Question 2d]. Eel X is
given 2 grams of fish fingers. How did you get
the answer that eel Z should get 8 grams of fish
fingers?
Learner: If I give eel X 2 grams, then I must give eel Y
4 grams, looking at how their lengths differ
[observing a pattern of lengths]. I saw the lengths
were 10, 15 and 20. I think I did not see that eel Z
is 25; I think eel Z was supposed to get 8 grams
if its length was 20 [seems to be trying to establish
a pattern of 2, 4, 8 for eels X, Y, Z respectively]. I
was not supposed to give eel Z 8 grams. I was
supposed to give eel Z 12 grams.
Interviewer: Now your pattern is 2, 4, 8 for 10, 15, 20. Why
is it not 2, 4, 6, 8? An eel of length 20 would get
6 grams and it would make sense to give eel Z
8 grams.
Learner: Actually I think that is exactly what I did.
Learner 27 suggests that he tried to establish a pattern based
on repeated addition, that is forming an arithmetic sequence.
For eel length, he saw patterns or addition by 5. For fish
finger mass the learner saw patterns of addition by 2.
Discussion
The study found that learners used various strategies. There
was also evidence that learners shifted between strategies,
using different ones for the same question as Learner 13
and Learner 8 revealed in their interviews. This tendency
demonstrates their uncertainty about the underlying
relationships, which led them to adopt different ‘methods’
at different times because the methods seem arbitrary and
are not grounded in the properties of the proportional
relationship.
Learners often used incorrect mathematical notations such
as 2 sprats = 5 cm (e.g. Learner 6 in Figure 5). This suggests
little comprehension of what a proportional relationship

entails. Learners’ misconceptions about number and algebra
added to their confusion as was the case with Learner 15
in Figure 3, who displayed multiple errors, such as
multiplying the equivalent fractions as well as incorrectly
simplifying 8 6
1
2
4 3
1
× ×
to .
The underlying theorems-in-actions associated with the
strategies are now presented. The analysis of the underlying
theorem-in-action is a tool that can be used to check the
validity of a particular strategy and also helps us identify the
scope and limits of application of a strategy.
The use of the doubling and halving strategies
It seems as if many learners recognised that some problems
could be solved by halving or doubling or a combination
of the operations. This implies that the operations were
intuitive efforts in trying to obtain fitting answers. However,
many of them performed incorrect operations or stopped
short of completing all the steps. Most learners were able
to solve Question 1a and Question 1b, which required just
one operation of halving correctly, whilst some learners
recognised the need for halving but did not know which
quantity to halve. Doubling also led to the correct answer for
Question 2a(i), which is discussed under additive strategies.
The underlying theorem-in-action based on Equation 4 is
that f(½ x) = ½ f(x) or f(2x) = 2f(x), which can yield the correct
answer to the given proportionality problems. However,
some learners may have doubled the elements across a
measure space, such as Learner 8 in interview Excerpt 2,
who doubled the length of the sprat Y and expressed it as the
number of fish fingers required for Y. At other times learners
doubled or halved quantities in the same measure space
inappropriately such as those who used the halving strategy
incorrectly in Question 2c.
Additive and subtractive strategies
Some learners resorted to addition or subtraction, which
was incorrect in many cases. However, addition using
the build-up strategy can lead to the correct answer. For
example, Question 1e could be solved by first reducing the
given ingredients by two factors and then adding the results.
This is possible because f ( ) ( ( ) ( )) ( ) ( )
6 8 8 8 8
1
2
1
4
1
2
1
4
= + = +
f f f
(using Equation 4, where x is the number of people and f(x) is
the amount of cream needed for x people).
Another instance when addition led to a correct answer was
in Question 2a(i). The length of the eel increased from 5 cm
to 10 cm. Hence the number of sprats increased from 2 to
4; this increase could be seen as an addition of 2. Here the
function could be seen as f(x) = 2
5 x, where x is the length
of the eel X and f(x) is the number of sprats fed to eel X.
Hence f(10) = 4 = 2f(5) = 2(2) which is also equal to 2 + 2. So
this is true only because 2 + 2 = 2 × 2 in this instance. However
in general, f(2x) ≠ 2 + f(x).
For Question 2d Learner 8 used an incorrect additive
strategy, as explained in Excerpt 2. He correctly identified
that the eel lengths are multiples of five, with length of
eel X being the second multiple of 5 and the length of
eel Z being the fifth multiple of 5. Hence the number
of fish fingers for eel X would be 2 (i.e. f(x) = 1
5 x, then
f(10) = 1
5 (10) = 2) and the number of fish fingers for
eel Z would be 5 (i.e. f(25) = 1
5 (25) = 5). However, having
identified that the length of eel Z was the fifth multiple
of 5, he then added 5 to the 2. He did not realise that he
had obtained the required answer of 5. His solution can
be expressed as f(z) = f(x) + f(z), which does not hold in the
case of a linear function unless x = 0 and f(x) = 0.
Also in Question 2d, Learner 27 used an additive strategy in a
different and also incorrect way. As the lengths of the lengths
of the eels increased by 5 cm, in a corresponding manner he
increased the number of fish fingers by 2. Hence, his incorrect
solution can be expressed as f(x) = f(10) = 2, f(15) = f(10) + 2,
f(20) = f(10) + 4; f(25) = f(10) + 6, statements which are not
true.
Unitary strategy
In general there were few responses that indicated the use
of the unitary method; however, Learner 16’s responses to
Question 2c–2e are based on a version of the unitary method
(Equation 5). However, it seems that he was uncertain about
which operation to use and what the result of the operation
represented. In working out 15 ÷ 9 = 1,67 for Question 2c the
underlying theorem in action was f(1)=
f m
m
( )
( ) (Equation 5).
That is, he worked out f ( )
9
9
= 1,67 or 12
3 . That is, f(1) = 12
3 .
He then went on to work out 10 ÷ 1,67 which is actually
working out f m
f
( )
( )
1
, also using Equation 5, here where f(m) = 10,
f(1) = 1,67 and m is unknown. So his method was correct;
however, he entangled himself in trying to keep track of the
operations and divisions and multiplications whilst also
trying to identify which elements were from the different
measure spaces.
Cross multiplication strategy
Vergnaud refers to this strategy as the rule of three strategy.
Suppose that the three quantities x1
, x2
and f(x1
) are given and
f(x2
) is required. Then, using Equation 2, we can follow five
steps:
f x f x x
f x
x
f
( ): ( ):
( ) (
1 1 2 2
1
1
1
x =
=
[step ] (ratio representation)
x
x
x
f x
2
2
1
2
)
(
[step ] (converting to fraction representation)
)
)
.
( )
.
x
x x
f x
x
x x
1
1 2
2
2
1 2 3
× = × [step ] (multiplying throughout by
y the LCM)
[step ] (simplifying)
f x x f x x
f x
f
( ). ( ).
( )
(
1 2 2 1
2
4
=
=
x
x x
x
1 2
1
5
)
[step ] (simplifying)
However, as shown in this study, even those learners who
correctly carried out the strategy moved directly from step 1
to step 5, leaving all the intervening steps or transformations
out. It is those omitted steps that demonstrate why the
strategy works. In particular, it was noticed that learners

expressed step 1 as step 1*, without using any equal or ratio
sign:
f x x
f x x
( )
( )
1 1
2 2
→
→
[step 1*]
Also, none of the learners expressed step 2 in their working
details. Their strategy therefore was not based on knowledge
of ratios and proportional relationships, but on meaningless
operations of ‘cross multiply and divide’. Even the work of
Learner 14 who arrived at the correct answers show that
she moved directly from step 1* to step 5 without any of
the intervening steps. This learner was able to consistently
reproducestep5withoutincludinganyofthetransformations
between step 1 and step 5, which is quite remarkable.
However, many of the other learners’ lack of knowledge of the
specific transformations that make the cross multiplication
rule work led to mistakes such as those demonstrated by
Learner 13 in Question 1a, who formulated the correct
relationship in step 1*, but who got confused when carrying
out the operations and multiplied the scalar fractions
x
x
1
2
and
f x
f x
( )
( )
1
2
, where f(x) was the amount of water needed to make
soup for x people.
Learner 15, in Figure 3, made a similar mistake, but his
response suggested many other misconceptions besides the
one held by Learner 13.
The errors displayed by Learner 17 in Question 2 can also be
explained as incorrect application of the cross multiplication
rule. Learner 17 only produced step 5 in each case of the form
f(x2
) = x x
f x
1 2
1
×
( )
, instead of f(x2
) = x f x
x
2 1
1
× ( )
, which is the result of the
correct application of the cross multiplication rule.
These results support Olivier’s (1992) advice given more
than 20 years ago that:
any teaching strategy which merely supplies pupils with recipes
such as … cross multiplication and the unitary method which
can solve certain classes of stereotypes proportional problems …
cannot be effective. (p. 301)
Vergnaud (1998) notes in his study that only 1% of the
participants used the rule of three, explaining that this low rate
was because most learners consider that there is no meaning in
multiplying quantities from different measure spaces (f(x1
) x2
).
Hart(1981)reportsthat‘ofthe2257children…only20[15from
the same school] wrote down an equation of the form
a
b
c
d
=
and used it consistently and correctly’ (p. 89). ‘There was
little evidence that the taught rule
a
b
c
d
= was remembered
and used by children’ (p. 21). ‘Teaching an algorithm such
as
a
b
c
d
= is of little value unless the child understands the
need for it and is capable of using it. Children who are not at
a level suitable to the understanding of
a
b
c
d
= will just forget
the formula’ (p. 101).
In this study the cross multiplication method was the
most common method, even though it is not an intuitive
strategy. Of the 12 learners who showed some working
details, all displayed evidence of using this strategy in their
responses. It is therefore clear that the recipe for this method
was taught to the learners, which many teachers under
pressure may decide to do. However, not a single learner,
even those who produced correct responses, were able to
provide an explanation that showed reasoning beyond
step 1*, suggesting that they did not know why the strategy
worked. The cross multiplication strategy was clearly taught
without ensuring an understanding of when and why it
works; this effect was evident in the learners’ responses.
It is acknowledged that the dynamics of the classroom are
complex and often teachers face dilemmas about whether
to teach for conceptual understanding or to focus on getting
good results by focusing on procedures. However, teaching
procedures without understanding, as alluded to by Olivier
(1992) above, is not effective. In fact, this study shows that
learning procedures without understanding the background
does not even lead to good results.
Conclusion
In this article we analysed learners’ strategies to tackle
questions based on ratio and proportion by identifying the
underlying theorems-in-action. The identification of the
theorems-in-action provided insight into whether their
strategies were correct. This also helped us understand
whether or not some incorrect methods produced correct
answers coincidently because of the numbers that were used.
As was shown, some of these incorrect theorems–in-action
revealed procedural ways of working that contradicted
properties of the direct proportional relationship. It was also
shown that many learners opted for the cross multiplication
rule; however, many were confused about which quantities
should be the multipliers and which should be the divisor
in the rule of three. We argue that the confusion emerged
because none of the learners’ responses indicated Equation
2 or Equation 3, which are the crux of the proportional
relationship. They then carried out operations without
understanding the algorithm and why it worked. The
number of possible permutations of four numbers in four
positions is 4
P4
= 24. Hence, there are 24 possible ways of
arranging four numbers in terms of two equivalent fractions
and only eight of them correctly represent the proportional
relationship. In the absence of knowing the functional
relationship that dictates exactly which fractions are equal,
and why they are equal, learners seem to have guessed and
placed the quantities (three given and one unknown) in
arbitrary positions, allowing them to carry out the operations
of ‘cross multiplication’ and division to arrive at an answer.
The study also found that no learner mentioned the words
ratio or proportion. We can thus infer that most of these




learners are working out the problems (some doing it
successfully) without realising the meaning of equivalent
ratios, or knowing the conditions under which four quantities
form a proportional relationship. Hence they were carrying
out the procedures without engaging with the linear function
of the type y = kx, which defines the relationship between the
quantities. The role of the function is not addressed, even
in textbooks, and perhaps this omission is a large factor
accounting for learners’ struggles with solving such missing
value problems based on proportional relationships. Without
reference to the linear function y = kx learners can rely only
on memorised strategies. We therefore suggest that shortcuts
such as the cross multiplication strategy should be not be
used with secondary school learners without a concurrent
emphasis on the role of the function that sets the conditions
for the proportional reasoning.
Acknowledgements
Competing interests
The authors declare that they have no financial or personal
relationship(s) that may have inappropriately influenced
them in writing this article.
Authors’ contributions
The data collection and analysis was carried out by P.M.
(University of KwaZulu-Natal). S.B. (University of KwaZulu-
Natal) led the write-up of the article.
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