The concept of rational numbers is one of the learner’s first experiences with a Mathematics concept beyond the basic skills operations on whole numbers. The personal knowledge of fractions that teachers bring to the teaching context is important because teachers mediate the conceptions that their learners construct.

This study was set up to apply Action–Process–Object–Schema theory to study primary teachers’ understanding of addition and subtraction of fractions.

The participants of this study comprised 60 undergraduate full-time students, studying to become teachers. The participants were enrolled in a foundational course in Mathematics because they had not passed Mathematics at Grade 12 level. This course was intended to help deepen their understanding of basic numeracy, allowing them to continue with further courses if they wanted to specialise in teaching primary mathematics

Data were collected using written responses of the pre-service students to two tasks that focused on operations with fractions. Ten students volunteered to be interviewed of which three are drawn upon in this article.

Many of the pre-service teachers coped well with addition and subtraction of common fractions with the same denominator. However, more than 52% struggled to carry out these operations on common fractions with different denominators, showing that their conceptions had not developed into object-level structures.

It is evident that the incorrect procedures have become embedded in the students mental schema. It is crucial that programmes for upgrading pre-service teachers should include opportunities for teachers to interrogate their personal understandings of the basic mathematics concepts.

Many learners, as well as teachers, often groan with dismay when they hear the word ‘fraction’ because it is associated with early experiences in primary school of numbers that did not make sense, where rules without reasons were applied. The introduction of fractions is one of learners’ first experiences with a math concept beyond the basic skills of whole number arithmetic (Chinnapan

Fractions are considered an essential skill for future Mathematics success but also a difficult concept to learn and to teach (Hecht, Close & Santisi

Siegler et al. (

There is consensus in the teacher education literature that a strong knowledge of the subject taught is a core component of teacher competence (Baumert et al.

Siegler et al. (

However, helping learners develop these key ideas is not easily achieved as is evident from the numerous studies focusing on students’ difficulties with fractions (Cramer et al.

Furthermore, Cramer et al. (

Hackenberg and Lee (

Many researchers have investigated what teachers know about Mathematics, teaching Mathematics and how they know it, since Shulman (

Chinnappan and Forrester (

According to Hansen et al. (

Ji-Won and Ji-Eunlee’s (

A collaborative action research project by Bruce et al. (

A study by Isik and Kar (

This study is based on APOS theory, which proposes that an individual has appropriate mental structures built up through particular mental mechanisms, to make sense of a given mathematical concept. The mental structures refer to the likely

Research based on this theory requires that for a given concept the likely mental structures be identified, detected and then suitable learning activities designed to support the construction of those mental structures. One of the major tools used in APOS-based research is genetic decomposition (GD), which is a hypothetical model of mental construction that a student may need to make in order to learn a mathematical concept (Arnon et al.

We draw upon Hackenberg and Lee’s (

How students generate and coordinate composite units is the foundation of how we understand students’ multiplicative concepts. These multiplicative concepts are the interiorised results of students’ units coordinating schemes as they progress from one concept to the next (Hackenberg & Lee

At an action level a person sees a fraction as representing a part of a whole, where the numerator is identified in terms of the number of shaded or selected equal parts while the denominator is the total number of equal parts that the whole has been divided into. At this stage the individual is able to make sense of unitising a fraction by the operation of partitioning or splitting a whole into a certain number of equal parts (Hackenberg & Lee

Fraction scheme.

As a learner continues working with the physical representations, the action of the operation of partitioning is interiorised so that individuals can recognise a fraction without having to physically count each part, that is, they are able to carry out the process of partitioning mentally. At a process level a learner is not restricted to a physical representation when working with fractions but can work with the symbolic representation of 3/5 as representing 3 parts out of a whole of 5 parts.

The process of partitioning is encapsulated into an object when the individual is able to conceive the partitioning process as fractional numbers (Steffe & Olive

The mental structures for addition and subtraction are similar to each other, so we will describe them with respect to addition, but they can be the same for subtraction as well.

To add fractions, a learner requires diagrams or physical representations to carry out the addition operation. The learner performs a single operation at a time, without thinking beyond the action of the single addition operation; for example, the addition operation 1/4 + 1/4 = 1/2 needs to be represented as fractions using a pizza diagram or any other figure, as shown in

Addition and subtraction of fractions.

If the fractions have different denominator, say ½ and ¼, then the learner would first transform the fractions into equivalent ones that share a common denominator, so that the action of putting them together can be done. They may require a physical representation to substitute one part of size 1/4, say with two parts of size 1/8, if considering the sum 1/4 + 3/8, for example.

The action of adding two fractions is interiorised into a process when a learner can carry out the addition operation without requiring physical representation. To add fractions with different denominators a learner will first transform fractions mentally or symbolically into equivalent fractions that share a common denominator so that the fractions can be added.

When a learner becomes aware of the process of adding two like fractions as a totality, realises that transformations can act on that totality and can actually construct such transformations (explicitly or in one’s imagination), then we say the individual has encapsulated the process of adding fractions into a cognitive object called a ‘sum of fractions’. For example, a learner could find multiples of a sum of fractions or explain the properties of the sum of fractions, such as showing that the sum of fractions is commutative. At this stage a learner is able to distinguish between the results of objects arising from similar processes. Hence a learner can perceive the equivalence between the two processes of adding two fractions with different denominators – for example in finding the sum

Sfard and Linchevski (

The participants of this study comprised 60 undergraduate full-time students, who were enrolled in a 3-year Bachelor of Education degree to become teachers. The 60 participants were enrolled in a foundational course in Mathematics because they had not passed Mathematics at Grade 12 level. This course was intended to help deepen their understanding of basic numeracy, including aspects such as operations on fractions with respect to meaning and use of representation. These students were all primary school pre-service teachers and would take on further primary Mathematics Education courses, having passed the basic course. This study was interpretative in nature as it recognised that individuals with their own varied backgrounds and experiences contribute to the ongoing construction of reality (Wahyuni

The research question explored in this study is: What are the pre-service teachers’ misconceptions that permeate from addition and subtraction of fractions that can be described using APOS theory? The preliminary GD presented in the section ‘Genetic decomposition for operations on fractions’ served as an analytical tool for the study.

We got the approval of the students who happen to be our student before using their script and interview extract in writing the article.

The results of the activity test, which comprised two tasks with sub questions (see

This first task (which appears in

Task 1.

Simplify the following fractions, leaving your answers as fractions. Show all your workings. (1.1)

(1.2)

Total number of participants’ responses to Task 1.

Task 1 item | Correct response |
Incorrect response |
No response |
---|---|---|---|

1.1 | 60 (100.00) | - | - |

1.2 | 29 (48.33) | 20 (33.33) | 11 (18.33) |

1.3 | 60 (100.00) | - | - |

1.4 | 24 (40.00) | 23 (38.33) | 13 (21.67) |

All participants responded correctly to Q1.1 and Q1.3, which involved the addition and subtraction, respectively, of fractions with the same denominator. This could indicate that all participants had developed at least an action conception of addition of fractions. However, the success rate dropped to 48.3% in Q1.2

Alternatively students may opt to use the lowest common denominator (LCD) addition–subtraction algorithm, requiring that they find the LCD of the two fractions. Some of the students who worked out the addition and subtraction tasks correctly may well have opted for the algorithm so it is not possible to gauge their own understanding of fractions because the students were not interviewed and their working details did not provide sufficient evidence of the method they used. The incorrect responses, however, show that for many of the participants, it was their conception of a fraction that had hampered their progress in understanding the addition of fractions. It is clear that at least 20 participants (33.3%) who provided wrong responses to Q1.2 had not progressed past seeing a fraction as two numbers separated by a line and had not developed a process (or partitive fraction). Of the 20 participants, eight participants wrote down the question but could not solve further, while the other 12 participants displayed a common misconception of using fractions as two separate whole numbers, as illustrated by T57’s response in

The response of T57 to addition of fractions with different denominators.

Participant T57 was interviewed about his approach to addition. See response in

Interview extract (T57).

A: Looking at the items, I can see that your answer is

T57: Because 2 + 1 = 3 and 3 + 5 = 8, so the result is

T57 explained that he got the answer he did considering the sum as two whole number sums: the sum of the two numerators and the sum of denominators. The response of T25, which appears in

Siegler et al. (

The written response of T25 to subtraction of fractions.

We now look in more detail at the interview with Participant T25, which provides more insight into his conception of fractions and the operations of addition and subtraction. In the interview, Participant T25 was asked to explain the

Interview extract (T25) about the idea of a fraction.

T25: Er, er, fraction is basically about knowledge about division of an object. For example, we are five in a room, and we have one orange we need to share, so we cut it in pieces so that it will accommodate all of us.

A: Must the cutting be equal?

T25: Not exactly; it depends on choice.

The extract in

The interview proceeded to the subject of

T25 interview extracts (T25) about addition and subtraction of fractions with the same denominator.

A: Now when you are given two fractions that have same denominator and you are asked to add them, what do you do? Let’s say,

T25: First I think that I have two people that should take portion out of five pieces of cake, then one person take two portion out of the five pieces of cake and the other person takes one portion out of the same five pieces of cake. Then I ask myself, how many portions did the two people take and it is three portions out of the five pieces of cake. So

A: Suppose you are given

T25: Mmm, that means one out of four plus something is equal to two out of four … I think it will be

As explained in

The participant’s explanations of

Interview extracts (T25) about addition and subtraction of fractions with different denominators.

A: Alright, what if you are asked to add two fractions with different denominators, for instance

T25: In this case, we have two people again, the first person take two portions out of three pieces of cake and the other person takes one portion out of five pieces of cake. Then, mmm, the total will be three portions out of eight pieces of cake. Therefore,

A: In a situation where you are asked to subtract one fraction from another, for instance

T25: Mmmm, still [

A: Is it possible to have three portions out of two pieces of cake?

T25: Not really [

A: What if you are given

T25: Eeeehh, the answer will be zero.

A: Why?

T25: Because 2 – 3 is not possible and 1 – 1 is zero, so the answer will be zero.

The extract in

It is clear that the participant’s understanding of fractions was embedded within the physical representation of the number of selected pieces divided by the total number of pieces, and the action had not been interiorised. The fraction was not seen as an object upon which transformations can be carried out, and the participants were unable to distinguish between the object arising from the process of addition or subtraction of fractions with the same denominator and the object arising from the process of addition or subtraction of fractions with different denominators. Because the participant did not understand the part–whole relationship between the numerator and the denominator, he treated these numbers as whole numbers. He instead constructed a pseudostructural conception to help him deal with addition or subtraction of fractions with different denominators. However, when subtracting part–whole fractions, they must attend to the unit (the number of pieces the whole has been partitioned into) before subtracting quantities. As learners transition from whole numbers to other number systems, including fractions, explicit attention to and naming of the unit is important so that they develop this understanding. Although unit fractions have not typically been a central focus in fraction teaching in South Africa, according to researchers like Pienaar (

The second task, presented in

Task 2.

Simplify, leaving your answer in fraction. Show all your workings.

2.1)

2.2)

Total number of participants’ responses to Task 2.

Task 2 item | Correct response |
Incorrect response |
No response |
---|---|---|---|

2.1 | 28 (46.67) | 24 (40.00) | 8 (13.33) |

2.2 | 25 (41.67) | 25 (41.67) | 10 (16.67) |

The response of T38 to multiplication and addition of fractions with different denominators.

Interview extracts (T38) about Q2.1 and Q2.2.

A: Looking at the items in Task 2 of the activity test, what do you do when asked to simplify the fraction

T38: Mmmmm, I think with BODMAS rule I am going to start with bracket

Step 1:

Step 2:

A: Why did you multiply the numerators and the denominators as you did in the addition?

T38: Yes ma’am, in addition and subtraction of fractions you add or subtract the denominators and the numerators, as the case may be. You know, multiplication is same as repeated addition, so in multiplication and division, which you have to change to multiplication, you also have to multiply the numerators and denominators separately.

A: So the rule is that you work on the numerators separately, using the given operation, and the denominators separately as well, using the said operation?

T38: Yes, that is the rule.

A: I can see that your answer to Item 2.2, that is,

T38: Mmmmm, I think with BODMAS rule am going to start with multiplications first:

Step 1:

Step 2:

Step 3:

A: So in addition, subtraction and multiplication you work on the numerators and denominators separately using the given operation?

T38: Yes ma, that is the rule.

BODMAS: bracket, of, division, multiplication, addition and subtraction

The response from the interview confirmed that the student had used the method of whole number addition applied to the numerators and denominators. The written response of, and interview with, Participant T38 shows how strongly embedded her pseudostructural conception of addition of fractions had become. Hence it is clear that this incorrect method had become interiorised into a pseudostructural ‘process’ to the extent that she was able to carry out transformations on this ‘process’ of addition of a fraction by multiplying it by other fractions, using the correct method. In fact she obtained the same answer to both problems using this pseudostructural rule, based on whole number addition. That is, she showed that with her rule in

This relationship is true in general for fraction multiplication and addition because the operation of multiplication is distributive over addition. The question that arises is whether this result will hold true in general if the incorrect rule is used. In other words, using this incorrect rule for fraction addition, will multiplication be distributive over this incorrect addition rule? To see this, given fractions

Now if we work out the sum of the products in Q2.2, the result is shown by

The algebraic verification shows that the participant’s consistent application of her pseudostructural rules gives rise to the same incorrect answer of

Hence, it can be seen that the correct rule for multiplication is actually distributive over the pseudostructural rule of addition. This distributive property helps us to understand how this incorrect rule has become so widely used and applied. It is closely aligned to whole number addition facts and satisfies certain properties that are satisfied by the correct rule.

This article studied 60 pre-service teachers’ responses to questions involving the basics of fractions. The questions involved operations on fractions with the same denominator and fractions with different denominators, in order to explore their mental constructions of the understanding of operations on fractions using APOS theory. The findings reveal that many teachers were able to answer those items requiring action level engagement with the addition and subtraction of fractions with the same denominator but struggled with those that required higher levels of engagement. Although all the participants answered Items 1.1 and 1.3 correctly, more than 52% of the participants could not respond correctly to Items 1.2 and 1.4, indicating that the participants had not developed object-level conceptions of addition and subtraction of fractions. The analysis of the responses and the interviews showed that a common reason for the non-encapsulation of addition and subtraction of fractions was because of their weak conceptions of what a fraction entails. For many students their notion of fractions was that of a numerator, which represents the number of selected pieces, and a denominator, which represents the total number of pieces. A key idea that was missing from this fraction conception was the notion of the equality of the pieces: a fraction is the ratio of the number of

It is likely that for these students their learning experiences neglected the action level of development of the part–whole conception of fractions. The addition of fractions could not make sense beyond working with like fractions. They were able to add fractions with the same denominators by considering the action of putting pieces together, without having to check the sizes of the pieces. However, this conception does not work for addition of fractions with different denominators, because the difference in denominators means that the sizes of the pieces are different, so putting together pieces of different sizes cannot lead to a measurement of a fraction. Instead some students, such as T38, had instead interiorised an incorrect rule for addition (and subtraction) of fractions with different denominators that arose from their idea of portions of a cake. So

There is much concern in the literature about the poor background of primary school learners in basic Mathematics, which hinders them from progressing sufficiently in higher-level Mathematics (Hckenburg & Tillema

Task 1

Simplify the following fractions leaving your answers as fractions. Show all your workings. (1.1)

(1.2)

Task 2

Simplify leaving your answer in fraction. Show all your workings.

2.1)

2.2)