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.

There have been numerous studies focused on ratio and proportion which have looked at learners’ strategies and errors (Ben-Chaim, Keret & Ilany,

This article reports on part of a bigger study which investigated Grade 9 learners’ understanding of ratio and proportion (Mahlabela,

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

The

The

In the

According to Misailidou and Williams (

The

The

An incorrect strategy is the

Researchers (Ben-Chaim et al.,

Proportional reasoning requires a recognition that a situation is completely described by a function of the form shown in

Knowing _{1} and _{2},

We say the four numbers _{1}), _{1}, _{2}) and _{2} form a proportion which can be expressed in terms of ratios: _{1}): _{1} = _{2}): _{2}.

The proportional relationship can also be written as

Equivalently, it can be expressed as the ratio representation _{2}:_{1} = _{2}):_{1}).

The relationship expressed in

Considering

Vergnaud (

An important property of the linear function

Vergnaud (

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

where

Vergnaud describes another way of solving proportionality problems, which he calls the

In _{1}), _{1} and _{2} are known and _{2}) is required.

This qualitative study was conducted with a group of 30 Grade 9 learners from one school in KwaZulu-Natal, South Africa. The school is situated in a rural area, with most learners 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,

What strategies do learners use to solve the two test items?

What are the underlying theorems-in-action associated with these strategies?

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.

Learner responses are analysed separately: Question 1a and Question 1b first and the rest later.

Learner responses to Question 1a and Question 1b (N = 30).

Question | Description of response | Correct | Incompletehalving | 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 |

The onion soup recipe (Question 1).

Response of Learner 13 to Question 1a.

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

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:

I can see that your answer to this question is 4 pints. Tell me how you got 4 pints?

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

Learner 13 in her written response in

One way of solving Question 1d and Question 1e would be to halve (to obtain ingredients for four people), halve again (obtain ingredients for two people) and then add the result to the first halving to find the ingredients required for 6 people – called the

The problems could also be solved by the use of the

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

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

Learner 15’s response in

The question in

Piaget’s eel question is concerned with the amount of food given to eels of different lengths, the amount being proportionate to the length of the eel (Hart,

Response of Learner 15 to Question 1e.

Learner responses to Question 1c–e (

Question | Description of response | Correct response | Incomplete build-up strategy | Incomplete strategy | Incorrect cross multiplication | Other incorrect strategies |
---|---|---|---|---|---|---|

1c | Learner response | A or 1 | 2 | 2; 6; 24 | Miscellaneous | |

Number of learners | 18 | 2 | 3 | 3 | 4 | |

1d | Learner response | 3 | 2 | 4 | 5; 12 | Miscellaneous |

Number of learners | 20 | 3 | 0 | 2 | 5 | |

1e | Learner response | 3/8 | A | A | 2/3 | Miscellaneous |

Number of learners | 3 | 7 | 4 | 2 | 14 |

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

The eel question (Question 2).

The responses of Learner 17 (see

Response of Learner 6 to Question 2a(i).

Response of Learner 17 to Question 2.

Learner responses to Question 2 (

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 |

Number of learners | 27 | 2 | 0 | 0 | 1 | |

2a(ii) | Learner response | 6 | 37,5 | - | 8 | 4; 7 or 75 |

Number of learners | 24 | 1 | 0 | 2 | 3 | |

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 | 4/½ (18) | Miscellaneous |

Number of learners | 19 | 2 | 2 | 2 (1) | 4 | |

2d | Learner response | 5 | 125 | 6 or 7 | 8 | Miscellaneous |

Number of learners | 11 | 2 | 5 | 8 | 4 | |

2e | Learner response | 15 | 41,7 | 11 | 18 (36) | Miscellaneous |

Number of learners | 17 | 3 | 1 | 4 | 5 | |

2f(i) | Learner response | 4 | 25 | 6 | 24 | Miscellaneous |

Number of learners | 15 | 1 | 2 | 1 | 11 | |

2f(ii) | Learner response | 6 | 37,5 | 5 or 8 | 5 | Miscellaneous |

Number of learners | 17 | 1 | 4 | 5 | 3 |

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

The

The

Responses of Learner 14 to Question 2.

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.

Response of Learner 16 to Question 2.

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 strategy. The dialogue below (Excerpt 1) elaborates on how Learner 8 arrived at his solution for Question 2a:

You said that if eel A gets 2 sprats, then eel B must get 4 sprats. … How did you arrive at that?

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.

If A gets 3?

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:

If eel B gets 12 sprats how many should eel A get?

If eel B gets 12 sprats, then eel A should get 6.

Good. If eel B gets 12 sprats, you said in your script eel C gets 24 sprats. How did you get 24?

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.

30? … How did you get 30?

I said 15 × 2.

Why did you say that?

Eel C eats more than eel B.

More?

Yes.

Why do you multiply by 2?

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:

Let us look at this one

I added 5.

OK. You said if eel X gets 2 grams, eel Z gets 2 + 5 grams?

I said 2 + 5 and got 7.

Where did you get 5 from?

I said 25/5 and got 5, then here I said 2 × 5 to get 10

Here

Yes, and I got 7.

The conversation with Learner 8 in Excerpt 3 reveals that he has used an

A conversation with Learner 27, who also used an

Let us look at this one

If I give eel X 2 grams, then I must give eel Y 4 grams, looking at how their lengths differ [

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.

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.

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

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.

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

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

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

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.

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

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 (

Vergnaud refers to this strategy as the rule of three strategy. Suppose that the three quantities _{1}, _{2} and _{1}) are given and _{2}) is required. Then, using

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:

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 reproduce step 5 without including any of the transformations 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

Learner 15, in

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

These results support Olivier’s (

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 (_{i}) _{2}).

Hart (

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 (

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 _{4}P_{4} = 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

The authors declare that they have no financial or personal relationship(s) that may have inappropriately influenced them in writing this article.

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.