This article focuses on learners’ understanding and their descriptions of the concepts of area and perimeter, how learners solve problems involving area and perimeter and the relationship between them and misconceptions, and the causes of these misconceptions as revealed by learners when solving these problems. A written test was administered to 30 learners and clinical interviews were conducted with three of these learners, selected based on their responses in the test. This article shows that learners lack a conceptual understanding of area and they do not know what a perimeter is. Learners also hold misconceptions about the relationship between area and perimeter. It appears that inadequate prior knowledge of area and perimeter is the root cause of these misconceptions. This article provides suggestions on how to deal with the concepts of area and perimeter.

Research in the field of mathematics education, locally and internationally, often reveals poor understanding of the concepts of area and perimeter (Gough,

This article is based on the study by Machaba (

How do Grade 10 learners describe the concepts of area and perimeter?

How do Grade 10 learners solve problems involving area and perimeter and the relationship between them?

What misconceptions are evident?

What might be at the root of these misconceptions?

In this article, I argue that learners lack conceptual understanding of the concept of area and they do not know what a perimeter is. It further appears that inadequate prior knowledge of area and perimeter is the root cause of these misconceptions.

Firstly, the location of this study in the curriculum will be discussed. I will then explain and describe the concepts of area and perimeter and the thought processes around area and perimeter, drawing from a variety of publications. After describing the theoretical framework of the study, the methodology used and the analysis of the data, the conclusion of the study once again focuses on the research questions and the answers yielded by an analysis of the data. Finally, I will make practical suggestions on how educators and textbook writers or curriculum designers can improve learners’ ability to deal with the concepts of area and perimeter.

The National Curriculum Statement for Mathematics, Grade 7–9, determines that learners should be able to describe and represent the characteristics of and relationships between 2-D figures and 3-D objects in a variety of orientations and positions (Department of Education,

The Gauteng Institute for Curriculum Development (

Dickson (

The mathematics education literature reports that many learners and even adults adhere to the view that figures with the same perimeter must have the same area (e.g. Outhred & Mitchelmore

On the other hand, learners can also establish a relationship between area and perimeter that rectangles with the same area have dimensions that are factors of the fixed area. When the difference between the dimensions of a rectangle with a fixed area is the smallest, you will have the smallest perimeter. When the difference between the dimensions of a rectangle with a fixed area is the largest, you will have the largest perimeter. Given a fixed perimeter, the rectangle with the largest area will be the one with the dimensions that are closest together (a square). Given a fixed perimeter, the rectangle with the smallest area will be the one with the dimensions farthest apart.

The literature, as stated above, indicates that there are many misconceptions for learners and adults about the complex relationship between area and perimeter. Learners need to have experiences in which they are manipulating the spaces that they are measuring, to construct deep understanding. Because of this, it is important to use a variety of manipulatives to develop the concepts. If these are not used, learners would view the relationship between area and perimeter as the result of the application of the intuitive rule ‘Same A – Same B’ (Same perimeter – Same area; Same area – Same perimeter). This kind of mysterious connection between perimeter and area is further discussed below.

Olivier (

Olivier (

Olivier (

Another common misconception has been researched by Tsamir and Mandel (

Nunes et al. (

In my study I was interested to see if learners in South Africa do the same as reported by Dickson (

This study is informed by Piaget’s theory of constructivism, which is about learners assimilating new learning into their existing schema. Constructivism holds that learning occurs efficiently and effectively when new knowledge is linked to existing or prior knowledge (Hatano,

Skemp (

The integrated networks of connections between ideas referred to as cognitive schemas are the product of constructing knowledge and the tools with which additional new knowledge can be constructed (Skemp,

Understanding at this rich and strongly interconnected end of the continuum can be referred to as ‘relational understanding’, while that at the other end of the continuum, where the ideas are completely isolated, can be referred to as ‘instrumental understanding’ (Skemp,

The research methodology used in this study was qualitative and was organised around a written test administered to 30 learners as well as a clinical interview carried out with a sample of six learners. The interviews were conducted after all the learners had written the test of six questions (

The written test on area and perimeter.

Grade 10 learners were chosen from a secondary school in Soshanguve. The reason why I chose this school was because I was not teaching in this school. This assisted to minimise researcher bias that might emerge from familiarity with the learners and with the school concerned. Below is the written test that was given to learners to write on the concept of area and perimeter.

This question tests learners’ conceptual understanding, without using formulae, and their ability to express these ideas or concepts in words.

This question was also aimed at testing learners’ understanding of the concepts of area and perimeter, without being given numbers or measurements. It tests whether learners are able to count square units to determine the area of a figure. It also tests whether learners know that calculating the area of a rectangle by multiplying the number of square centimetres in a row by the number of rows is the same as multiplying the number of square centimetres in the length by the number of centimetres in the breadth.

This question tests the learners’ ability to calculate the perimeter and area of an irregular figure when given measurements.

This question tests learners’ understanding of the concept of area and whether they can measure the area of an irregular figure.

This question tests learners’ understanding of the relationship between area and perimeter and whether they apply the intuitive ‘Same A – Same B’ rule, finding that when increasing the length of two opposite sides of a square by a given factor and reducing the length of the other two remaining sides by the same factor, the perimeter and the area will remain the same.

This question tests learners’ understanding of the relationship between area and perimeter and their application of the intuitive ‘Same A – Same B’ rule, finding that when increasing or multiplying the length of two opposite sides of a square by a given factor and reducing or dividing the length of the other two remaining sides by the same factor, the perimeter and the area will remain the same.

To reiterate, the purpose of the study was to investigate the insights and misconceptions that some Grade 10 learners in one school in Soshanguve have with regard to the concepts of area and perimeter. A written test was administered to 30 learners in a classroom and an interview was conducted with a selected six of these learners. However, for this article I report on the results of only three of the learners with whom the interviews were conducted, because only from them had I obtained saturated data, so I was forced to omit the data obtained from the other three learners. The results of the interviews and the written test will be reported simultaneously.

Permission to conduct the research was granted by the Gauteng Department of Education, the district and the school where this research was conducted. As this study was not conducted at my school, I wrote a letter to the principal of the school, describing the required grade (Grade 10) and the purpose and the rationale of the study. Grade 10 learners were informed of the study so that they could decide whether or not to participate in the study.

Learners who agreed to participate in the study were guaranteed anonymity and confidentiality. At both school and individual levels, participants’ anonymity and confidentiality were maintained by use of pseudonyms (e.g. L1, L2 and L3). Learners were informed that their real names would not be used in the study and whatever they said would be kept confidential. I developed a rapport with them so that they would not perceive me as an evaluator or judge, that is, as somebody who wanted to detect their learning flaws or faults that could be used to determine their promotion. Rather, I intended to be perceived as one who was interested in how they think and reason mathematically. Participants were informed that they would be provided with the report of the study.

Data were collected by me using both the instrument (test) and interviews. To ensure reliability of the instrument I initially collected pilot data and then tested an instrument to see if it would be interpreted in the same way. Since some of the questions had not yet been used in any research before, I thought it would be imperative to find out whether the test items were appropriate and tested my critical questions through piloting. I involved five Grade 10 learners in piloting. I gave test items to each and they spent 45 minutes on average answering the questions. I marked their test and chose two learners for an interview. The selection of the two was based on how they had answered the test items. I selected one whose performance was good in the test and one who performed poorly.

Furthermore, I presented the test at conferences and postgraduate meetings, where it underwent rigorous peer reviewing before taking its final form. Because I interviewed few learners, one cannot generalise the findings beyond the studied cases. This is the nature of case studies. However, consistent with the objective of the study, the findings could provide principles for dealing with learners’ misconceptions.

Summary of findings (

Question | Correct (%) | Partially correct (%) | Incorrect (%) |
---|---|---|---|

1a | 10 | 23 | 67 |

1b | - | 10 | 90 |

2a | - | 10 | 90 |

2b | 63 | - | 37 |

3 | - | 17 | 83 |

4 | - | 27 | 73 |

5a | - | 17 | 83 |

5b | - | 23 | 77 |

6a | - | 47 | 53 |

6b | - | 17 | 83 |

The findings as summarised in

It was therefore imperative to find out during the interview why learners were unable to define area and perimeter without using the formulae. It is evident from

Of course, the table does not show how the learners solved the problems, or how during the interviewing process some were able to obtain the correct solutions. The discussion that follows is an analysis of the responses of the three selected learners to each of the questions in the test and during the interview.

I will refer to the three learners that were interviewed as L1, L2 and L3, without implying through the labelling the order in which they were interviewed. The researcher will be referred to as the ‘interviewer’ in the transcript.

The major finding was that the learners held the same misconceptions that had been identified by other researchers. This claim is based on both the test responses and the follow-up interviews.

The learners cited the ingrained, formalised method of multiplying length by breadth to get the area. This indicates a lack of conceptual understanding of area as a surface and perimeter as the distance around the edge of the figure. The learners described both the area and the perimeter in terms of a formula. For example, with regard to Question 1a and Question 1b, L1 responded as follows:

It is the answer that you get after multiplying both the length and the breadth.

Why did you define by using the formula, when you had been forbidden to use the formula?

Mmm. … I use the formula because there was no other way I can define the area without using the formula.

Perimeter is the sign of showing that it is a cm, km or m.

What do you mean? Can you say more on that?

I mean like in a ruler [

Similarly, L2 responded as follows:

[

Have you ever heard about the word perimeter in your life?

No.

The learners overgeneralised when moving from working with rectangles to working with non-rectangles. They thought that the formula

No, the leaf does not have an area because there is no length and breadth.

I do not think the leaf has an area, because the leaf is not a rectangle and does not have length and breadth.

The learners claimed that when the length of two opposite sides of a square were increased by a given number of centimetres and the length of the other two sides were decreased by the same number of centimetres, both the perimeter and the area would remain the same. Similarly so if the length of two sides is multiplied by a certain factor and the other sides are divided by the same factor.

The response to Questions 6 was as follows:

You said your answer would be ‘equal to’ in your solution of 6(a) and 6(b). Can you give a reason why you said so?

Sir, I think if you lengthened these two sides of the square, nee! … by 6 nee!! … mmm … and shortened the other two sides by 6 again [

Which means, what you are saying is the perimeter of the rectangle will be equal to the perimeter of square?

Yes, Sir.

What about the area?

The area of the square will also be equal to the area of rectangle.

Why are you saying so?

Because, Sir, you add 6 and subtract 6.

It is clear that L1 uses the intuitive ‘Same A – Same B’ rule when solving the problem. L2 did likewise:

No, I think the perimeters of the two diagrams are equal, Sir.

Because the two opposite sides of the square are increased by 6 cm and the other two have been also decreased by 6 cm.

What about the area of the two diagrams?

I think, are also equal because of the 6 cm, which was added and subtracted.

The application of the intuitive ‘Same A – Same B’ rule confirms findings by (Tsamir & Mandel,

In this section, I shall return to my research questions and answer them on the basis of the analysis of my data. I will treat each question as a subheading of this section when stating my findings. I shall state explicitly what I have found. Lastly, I will discuss the implications of my findings and reflect on them.

The critical questions that guided this study were: How do Grade 10 learners describe the concepts of area and perimeter? How do Grade 10 learners solve problems involving area and perimeter and the relationship between them? What misconceptions are evident when learners are solving these problems? What might be the cause of these misconceptions?

Learners had problems defining the concept of area without using the formula

I found that learners do not know what the concept ‘perimeter’ entails. None of the learners could correctly define perimeter. One of the learners defined perimeter as units, for example km, m, cm, while others defined perimeter as ‘the length and breadth’. Dickson (

In this study learners were able to calculate area when given measurements, but were unable to determine the area when measurements were not given on the figure. They did not know that the area could be determined through counting square centimetres. Their failure to make a connection between the figure with square centimetres and the one with measurements leads us to conclude that they also lack a conceptual understanding of area. The lack of the integrated network of connections between ideas (cognitive schema) was the product of being unable to construct new knowledge based on existing knowledge. Learners who have difficulty translating a concept from one representation to another have difficulty solving problems and understanding computation (Van de Walle et al.

Learners were unable to calculate perimeter, which shows that they do not have an understanding about the concept of perimeter. In their research report, Kilpatrick et al. (

The first misconception that these learners displayed was overgeneralisation. They thought that the formula

The second misconception that was found was the application of the intuitive ‘Same A – Same B’ rule. Learners claimed that when increasing the lengths or adding a certain figure to the lengths of two opposite sides of a square and reducing the lengths or subtracting the same figure from the lengths of the other two remaining sides, the perimeter and the area would remain the same. This finding resonates with the findings of Tsamir and Mandel (

Tsamir and Mandel (

Learners also wrongly concluded that if the perimeter of the original square in Question 5a is equal to the perimeter of the created rectangle, the perimeter of the original square in Question 6a would also be equal to that of created rectangle and that if the area of the original square in Question 6b is equal to the created rectangle, the area of the original square in Question 5b would be equal to that of the created rectangle. This erroneous conclusion is another form of overgeneralisation which indicates a lack of knowledge of the two concepts: perimeter and area. Tirosh and Starvy (

It was evident from their incorrect definitions of area and perimeter that all three of the interviewed learners lacked prior knowledge of area and perimeter and that they had no conceptual understanding of perimeter as a distance. Furthermore, none of them mentioned the square centimetre grid in their discussion with the interviewer. Learners could not add new knowledge to the existing knowledge by making sense of what is already inside their heads. They could not organise, structure and restructure their experience in the light of available schemes of thought (Van de Walle et al.

In the light of the above findings I can say that in dealing with the concepts of area and perimeter, learners have the following problems:

They lack conceptual understanding of area as a surface. This became evident when they described area as length multiplied by breadth.

They overgeneralise, in other words, they assume that the formula

They use the intuitive ‘Same A – Same B’ rule when dealing with area and perimeter and therefore obtain only partially correct (false positive) results. They believe that when the size of two opposite sides of a square are increased by a given factor and then the size of the other two remaining sides is decreased by the same factor, the perimeter and the area would remain the same.

This study, like other similar studies (e.g. Dickson,

Teachers should distinguish between the concepts of area and perimeter, yet emphasise the relationship between them, so that learners will not see them as isolated concepts. In their progress map, the Gauteng Institute for Curriculum Development (

If teachers and textbook authors could, in the lower grades, emphasise the fact that the area is the size of a surface and the perimeter is the size of the edge of a figure, many misconceptions could be avoided. The extent of such misconceptions was evident when L2 and L3 responded in Question 5 that a leaf does not have an area, because there is no length and breadth.

Teachers should also be aware of the role that the intuitive rule plays in the concepts that learners form. In other words, when designing problems, teachers should consider whether they might elicit the use of an intuitive rule or counter it. This also implies that teachers should not be satisfied with the correct answers alone, but probe further to be certain that the learners are not just applying the intuitive ‘Same A – Same B’ rule (Olivier,

The author declares that he does not have financial or personal relationship(s) that may have inappropriately influenced him in writing this article.