Teaching using manipulatives is emphasised, especially in the early grades, to help learners conceptualise operations on whole numbers. Therefore, teachers’ competencies in using manipulatives is the key in helping learners master these basic operation skills.
Drawing from the literature on using manipulatives to improve learners’ performance in mathematics, this study recounts foundation phase preservice teachers’ conception of using manipulatives to enhance their competencies and reasoning skills to model the solution in number operations.
Data presented here was collected from 31 participants. These preservice teachers either passed mathematics or mathematical literacy with 40% at the grade 12 level.
Data was collected from participants’ written work (e.g. classroom tasks, homework, tests and examinations) and during class discussions. Interviews were conducted with some students. We analysed their conception guided by the APOS theory, namely, ActionProcess ObjectSchema.
We observed improvement in the conception of using manipulatives among preservice teachers. In the first semester, most students display action conception of using manipulatives to either represent or model a solution. However, in the second semester, most students either display process or object conception as explained in the genetic decomposition. We attributed the improvement to change of instruction in the second semester as we taught in accordance with the APOS theory.
It is evident that there are a number of contributing factors to preservice teachers’ conception of mathematical concepts, and teacher educators need to pay particular attention to these to help preservice teachers master the concepts they would teach at school.
This article focusses on analysing foundation phase preservice mathematics teachers’ evolution of their conception of using manipulatives in number operations. The literature has long advocated for the use of manipulatives to improve learners’ understanding of mathematical concepts (Ball
In his article entitled ‘What it means to understand mathematics’, Usiskin (
‘Mathematical manipulatives are physical objects that are designed to represent explicitly and concretely mathematical ideas that are abstract’ (Moyer
Extracted from Foundation Phase Curriculum Assessment Policy Statement document.
Content area  Weight of content areas 


Grade 1 (%)  Grade 2 (%)  Grade 3 (%)  
Number, operations and relationships  65  60  58 
Patterns, functions and algebra  10  10  10 
Space and shape (geometry)  11  13  13 
Measurement  9  12  14 
Data handling  5  5  5 
Note: In Grade R – 3, it is important that the area of numbers, operations and relationships is the main focus of Mathematics. Learners need to exit the foundation phase with a secure number sense and operational fluency. The aim is for learners to be competent and confident with numbers and calculations. For this reason, the notional time allocated to number operations and relationships has been increased. Most of the work on patterns should focus on number patterns to consolidate learners’ number ability further.
Although a certain percentage is specified, the truth is that number operations are embedded in all the topics. Therefore, it is imperative that teachers’ conception of number operations are intact. The literature that pays particular attention to teaching and learning for understanding mathematics still emphasises the use of concrete materials and modelling mathematical concepts (Van de Walle
Studies on the use of manipulatives had shown that manipulatives are important in enhancing the understanding of abstract mathematics (e.g. Ball
This study was conducted according to a specific framework for research and curriculum development in undergraduate mathematics, which guided the systematic enquiry of how students cognitively construct mathematical knowledge. The framework consists of three components, namely, theoretical analysis, design and implementation, and observation and assessments of student learning, as proposed by Asiala et al. (
Framework for research.
Under theoretical analysis, this study used ActionProcessObjectSchema (APOS) theory to describe and analyse preservice teachers’ evolution of their conception of using manipulatives in number operations. By using manipulatives in number operations, in this study we refer to preservice mathematics teachers’ ability to compute and model the solution by means of concrete objects and diagrammatic representation to show the conceptualisation of place value. The genetic decomposition explaining the cognitive constructs associated with using manipulatives in number operations is provided in the ‘Key concepts in ActionProcessObjectSchema theory’ section that serves as the analytic tool to analyse students’ responses in relation to APOS theory. To ascertain the evolution of preservice teachers’ conception of using manipulatives, assessment tasks were designed and implemented and responses were analysed by means of the genetic decompositions that allow for the categorisation of responses, as shown in the ‘Results’ section.
ActionProcessObjectSchema theory is deemed useful for explaining student conception of mathematical concepts. In APOS theory, conception refers to individual understanding and concept refers to collective understanding of that content by community of mathematicians (Arnon et al.
The design of genetic decomposition is premised on three factors, namely, researchers’ mathematical understanding of the concept; researchers’ experience of teaching a particular concept; and research on students’ thinking about the concept and historical perspectives on the development of the concept. As there has been limited research focussing on preservice teachers’ thinking of the concepts and historical perspectives on the development of the concept among preservice teachers, this genetic decomposition is premised on the researchers’ understanding of the use of manipulatives and experiences in teaching this concept to preservice teachers.
An action (Arnon et al.
[
In this study, when an individual represents numbers using concrete objects, his or her reasoning is considered to be at an action level. This includes the knowledge of constructing or deconstructing numbers. For example, given 123 an individual uses manipulatives to deconstruct the number to its components. An individual understands that 7 can be constructed in various ways such as 2 + 5 or 1 + 6 and can build these numbers using different manipulatives. However, the place value of each digit is not considered.
As actions are repeated and reflected on, an individual moves from relying on external cues to having internal control over them. ‘This is characterised by an ability to imagine carrying out the steps without necessarily having to perform each one explicitly’ (Arnon et al.
‘This occurs when an individual applies an action to a process that sees a dynamic structure as a static one to which actions can be applied’ (Arnon et al.
To add whole numbers, an individual requires an understanding of the differences between unary and binary operations. An individual would first use standard algorithm to perform the computation process, and the action of representing the solution is done separately. Because the representation of numbers diagrammatically is not yet fully conceived, even the solution is not accurately represented diagrammatically.
The action of performing binary operation is interiorised into a process when an individual performs the computation without relying on standard algorithms. An individual not only performs operations but also is able to make choices of appropriate mental illustrations to use and consider the efficiency of alternatives (Saka & Robert
The process of modelling the solution using manipulatives is encapsulated into an object when an individual transforms the solution to its components and uses appropriate mathematical language to explain the process.
This study is underpinned by an interpretive paradigm as it strives to inquire participant’s conception of using manipulatives in number operations. With respect to approach, this study used qualitative approach as it allows for the voice of the participant to be heard and allows for more diversity in responses (Flick
A cohort of 98 preservice teachers enrolled for the undergraduate fulltime course to study towards becoming primary mathematics teachers. However, the data presented here are from 31 preservice teachers who consented to participate in this study. The cohort of 98 students were divided into tutorial groups. The categorisation into tutorial groups was not academically based. Those who consented to take part in this study were put into one tutorial group, and fortunately, they were of diverse academic performance. Data were collected in two semesters by means of written work, video recordings and interviews. Interviews were audio recorded and later transcribed. Written work includes tutorial tasks, homework, tests and examination. During tutorial sessions, students engage in group discussions. These discussions were video recorded to capture students’ thought process as they talk about their solutions. These discussions were transcribed and analysed together with the written work by means of the genetic decomposition, as presented in
The aim of the interviews was to probe and interrogate students’ conceptions. Before the interviews, students were given time to reflect on their written work and where necessary video clips from tutorial discussions were played. Reflections were an hour long, followed by interviews that also lasted approximately an hour. Reflections were done with the whole group, but interviews were conducted with students purposefully selected.
In this study, we draw upon three students purposefully selected, and attention was on those students where we noticed evolution of their reasoning in using manipulatives in number operations.
The transcription of video and audio recordings to textual data allowed us to use the genetic decomposition to analyse students’ conception. Using various methods of data collection allowed for triangulation of the data captured ensuring trustworthiness of our findings. Moreover, students were given time to reflect on their written work and class discussions before being interviewed to verify their responses and that the work they were commenting on is theirs.
Ethical clearance was obtained from the Humanities and Social Sciences Research Ethics Committee of the University of KwaZuluNatal (reference number: HSS/0050/016).
In this section, we used extracts to explain our observation of preservice teachers’ conception of using manipulatives in number operations. A variety of activities were administered, and a sample of questions and responses is presented below.
The data presented in
Level of conceptualisation of the students’ use of manipulatives to represent whole numbers.
Mental constructs  Description  Number of correct responses in assessment tasks 


Item 1  Item 2  
Action  Students use concrete models to represent numbers; however, no reasoning about the meaning of place value is evident in the response.  20  20 
Process  The concept image of the number is interiorised. This is observed as students represent numbers using accurately drawn diagrams as representation of concrete models.  4  4 
Object  Students have some intuitive understanding of place value. Understand that the two digits of a twodigit number represent amounts of tens and ones.  7  7 
Level of conceptualisation of the students’ use of manipulatives to model solution.
Mental constructs  Description  Number of correct responses in assessment tasks 


Item 3  Item 4  
Action  Students understand binary operations but cannot perform it using manipulatives and cannot infuse mathematical language to explain.  16  20 
Process  Students make choices of appropriate manipulatives to use and model the solution, and even in the absence of concrete objects students can construct necessary representations to perform the necessary calculations.  8  10 
Object  Students perform actions on objects and infuse appropriate mathematical language. For example, 75 – 36 – see each twodigit number as a whole entity and apply action using concrete models or diagram to transform it. Knowledge of trading one ten for ten ones is constructed.  7  1 
The sample of responses selected for discussion here were those that all students attempted to answer. In
(a, b & c) Sample of students’ written responses to items 1 and 2.
When students were asked to model the solutions using manipulatives, as shown in the sample of responses below, we observed that the majority of them could use manipulatives to illustrate or represent inputs of binary operations, however, they could not model the solution. As observed in the extracts below, the majority determined the solution using standard algorithms, thus showing that their conception was restricted to the action level (
Sample of students’ written responses to item 3.
In the interview, Kadizo said:
I did not use the standard algorithm, but I used a calculator. The arrows were used to show how I subtracted, e.g. 70 – 30 = 40; 5 – 1 = 4; then add 40 + 4 = 44. (Kadizo, male, student)
Although Kadizo tried to argue that he did not use standard algorithm, he agreed that he did not use manipulatives. From his interview response, we assumed that he considered standard algorithm to be the vertical column method and because his solution was not structured vertically, he argued that he did not use the standard method. However, the arrows and his explanation, during the interview, focus on subtracting from left to right strategy, suggesting that he used the standard method. It was observed that in the absence of concrete objects, Kadizo had challenges in modelling the computation using diagrams as representation of concrete objects. It was captured in the video clip that Kadizo could not recall using number builder cards even as a learner or he was never exposed to using manipulatives to perform operations. His exposure to manipulatives was for counting purposes. The extract taken from video clips suggested that the lack of previous knowledge in the use of manipulatives to compute hindered Kadizo’s concept development. In the learning of mathematics, previous knowledge plays a crucial role in the construction of new knowledge, it seems the lack of previous knowledge therefore hindered Kadizo’s conception of using manipulatives (
Sample of students’ written responses to item 4.
Again, in this extract, Kadizo did not only struggle with modelling the solution but he also had difficulties with using mental images to represent digits, thus confirming that the action of using physical objects has not been interiorised into a process. This we observed as he represents tens and hundreds by the same diagrams, suggesting that he has not conceived the place value concept of each digit in twodigit or threedigit numbers.
Similarly, Azinga, as is evident in
Sample of students’ written responses to item 4.
However, she was able to use mental images to represent the inputs of the binary operation, but not the solution. During the interview, she was given different numbers to compute and was asked to model the solution. He first used standard method and then used Dienes block to represent the solution, as shown in
Urmilla’s written response suggested that she was also still restricted to the action level. The response below was extracted from the video clip taken during tutorial discussion (
Sample of students’ response to item 5.
Urmilla like Azinga used the standard algorithm to determine the solution and only used the concrete objects to represent the solution. However, her explanation during the interview revealed the evolution of the process conception, although not yet fully constructed. This observation was based on her attempt to infuse appropriate mathematical language, explaining that nine plus eight gives 17 ones and traded ten ones for one ten to get six tens, as shown in
Sample of students’ response to item 6.
Students’ performances in semester 1.
Categories of responses in terms of ActionProcessObject  Item 1  Item 2  Item 3  Item 4 

Action  20  20  20  16 
Process  4  4  0  8 
Object  7  7  10  7 
From
Students’ development process of using manipulatives in number operations in semester 2.
Categories of responses in terms of ActionProcessObject  Item 7  Item 8  Item 9  Item 10  Item 11 

Action  4  7  4  6  7 
Process  10  14  20  15  14 
Object  17  10  7  10  10 
From
In
Sample of students’ written responses to item 7.
Similarly, during the group discussion, we observed evolution of students’ thinking process in using manipulatives to model the solution. Kadizo’s group could use diagrams to represent inputs of the binary operation 47 + 76 and use the physical objects to model the solution 123 and show the process of trading ten tens for one hundred.
Furthermore, when performing subtraction sums, as shown in
I will make 7 tens from Dienes blocks and 5 ones which makes 75. Thirty one has 3 tens and one one. Since I am subtracting, I am not building 31 but I will take it away from 75. First take away one unit from five ones, I am left with 4 ones – then take away 3 tens from 7 tens, I am left with 4 tens. 4 tens is 40 + 4 ones = 44. (Kadizo, male, student)
(a & b) Sample of students’ written responses to item 8.
(a & b) Sample of students’ response to item 9.
The above findings support the suggestion by Ball (
In item 10, students were asked to use diagrams to represent a different set of numbers. Students were not restricted to use any particular model. They could either use Dienes blocks, number builder cards or abacus. Most students used Dienes block; unlike in semester 1 students take cognisance of proportionality when representing different digits, as shown in Azinga’s response (
Sample of students’ response to item 10.
Although still not accurately drawn, the proportionality is taken into account to differentiate tens from hundreds.
Item 11 was the same as item 4 in semester 1. This question was poorly answered in semester 1, therefore, we wanted to see if students could now use manipulatives to model the solution. In
4 tens plus 7 tens equals eleven tens but eleven tens is made up of ten tens which is same as hundred. 6 ones plus 7 ones equals 13 ones but 13 ones is made of ten ones and 3 ones so tens ones is the same as one ten. Putting it all together I will have one hundred, two tens and 3 ones and that is what I was trying to show here. (Azinga, female, student)
Sample of students’ response to item 11.
In item 11 in semester 2, we saw an evolution of preservice teachers’ conception of using manipulatives from action to process.
In the quest to answer our perennial question about preservice teachers’ conception of using manipulatives and exploring the contributing factors that enable or hinder their conception, we concluded that indeed student conception of using manipulatives to represent whole numbers and model solution in addition and subtraction of whole numbers gradually evolved. This we observed as we saw from the above data in semester 2 that many students showed the interiorisation of action into a process and further encapsulating process into objects, thus showing conceptualisation of the concept. While it can be argued that there could be many contributing factors associated with development in their cognitive growth, we observed that as students engage in structured cooperative learning and engage effectively in mathematics talk, students’ reasoning improved. These findings, to a certain extent, concur with findings by Ubah and Bansilal (
In the three cases of students we analysed, we observed evolution of the thought processes when using manipulatives in number operations, mainly the concept of modelling the solution and the use of appropriate mathematical language, leading to the development of place value notion. There is much concern in the literature about preservice mathematics teachers’ subject matter knowledge of school mathematics (Ball et al.
The authors would like to thank PrimTed project for the support in this study.
The authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article.
Z.A.N., as a lecturer of the module, was responsible for coordinating the research project and writing up of the article. L.C. assisted with the data collection and structuring of the article. Z.A.N. has also assisted L.C. as a young researcher to present preliminary findings at the AMESA regional conference in 2016.
This research received no specific grant from any funding agency in the public, commercial or notforprofit sectors.
Data sharing is not applicable for this article as no new data were created or analysed in this study.
The views and opinions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of any affiliated agency of the authors.