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Studies in the teaching and learning of fractions reveal that learners have considerable difficulty in understanding the relationship between a rational number and its decimal expansion. This qualitative study investigated forty four pre-service students understanding of this relationship at a South African university. Mathematics performance of high school pre-service students was observed after the implementation of activity sheets based on APOS Theory and carried out using the ACE Teaching Cycle. The class of students was divided into groups and worked collaboratively. The findings showed that the APOS designed worksheets positively influenced their answering of the question whether = 1.
Keywords: APOS theory, ACE Teaching Cycle, Recurring Decimal Number
Some studies, for example (Weller, 2009; Conradie, 2009, Maharaj et al., 2006, 2007) analysed student mathematical learning on fractions. Our study replicated the study carried out by Weller et al. (2009) in the United States of America. We employed ideas from that study in a South African context. In another South African study, Conradie et al. (2009) asked prospective mathematics teachers to explore whether was equal to 1. Unsurprisingly, the majority of students initial response was that was less than 1. However, after discussions some students came up with = 1. The procedure adopted by that study was to provide four arguments that show that a recurring decimal like equals a rational number like 1. Our study on the other hand decided to design certain tasks that would lead students to deciding on the question: Is ? The activity sheets are based on APOS Theory.
Knowledge Base for Educators
One of the expectations of the Norms and Standards for Educators (DoE, 1999) is that the educator be well grounded in the knowledge relevant to the occupational practice. She/he has to have a well-developed understanding of the knowledge appropriate to the specialism. Many mathematics educators find themselves in a position requiring them to implement the syllabus, which includes certain topics they are unfamiliar with. According to Adler (2002), educators with a very limited knowledge of mathematics need to develop a base of mathematical knowledge. They need to relearn mathematics so as to develop conceptual understanding. Taking this into account we attempted to make certain that trainee-teachers leave with a base of knowledge relevant to their occupational needs. Mwakapenda (2004) concurs when stating that a significant concern in school mathematics is learning with understanding of mathematical concepts. The National Curriculum Statement (NCS) emphasises a learner-centred, outcomes-based approach to the teaching of mathematics to achieve the critical and developmental outcomes (DoE, 2003). The following question guided our inquiry into pre-service students' understandings of the relationship between and 1:
How does the implementation of an "APOS theory designed activity sheet" and the "ACE Teaching Cycle" facilitate students' learning process with regard to accepting a relationship between a rational number and its decimal expansion?
The main intention of the study was to observe how learning of mathematics content, whether effective or not, took place under these circumstances.
This study is based on APOS Theory (Dubinsky & McDonald, 2001). APOS Theory proposes that an individual has to have appropriate mental structures to make sense of a given mathematical concept. The mental structures refer to the likely actions, processes, objects and schema required to learn the concept. Research based on this theory requires that for a given concept the likely mental structures need to be detected, and then suitable learning activities should be designed to support the construction of these mental structures.
Asiala et al. (1996) proposed a specific framework for research and curriculum development in undergraduate mathematics education which guided our enquiry into how students acquire mathematical knowledge and what instructional interventions contribute to student learning. The framework consists of the following three components: theoretical analysis, instructional treatment, and observations and assessment of student learning. According to Asiala et al. (1996), APOS Theory functions according to the paradigm illustrated in Figure 1.
Design and implementation of instruction
Collection and analysis of data
Figure 1: Paradigm: General Research Programme
In this paradigm, theoretical analysis occurs relative to the researchers' knowledge of the concept in question and knowledge of APOS Theory. This theoretical analysis helps to predict the mental structures that are required to learn the concept. For a given mathematical concept, the theoretical analysis informs the design and implementation of instruction. These are used for collection and analysis of data. The theoretical analysis guides the latter, which Figure 1 indicates could lead to a modification of the initial theoretical analysis of the given mathematical concept.
Description of the APOS/ACE Instructional Treatment
APOS Theory and its application to teaching practice are based on two general hypotheses developed to understand the ideas of Jean Piaget. In some studies (see, for example, Weller et al., 2003), these ideas were recast and applied to various topics in post-secondary mathematics. Piaget investigated the thinking of adolescents and adults, including research mathematicians. Those investigations led him to discover common characteristics, specifically certain mental structures and mental mechanisms that guide concept acquisition (Piaget, 1970). According to Dubinsky (2010), APOS theory and its application to teaching practice are based on the following assumptions on mathematical knowledge and hypothesis on learning mathematics.
ï‚· Assumption on mathematical knowledge: An individual's mathematical knowledge is his/her tendency to respond to perceived mathematical problem situations and their solutions by [a] reflecting on them in a social context, and [b] constructing or reconstructing mental structures to use in dealing with the situations.
ï‚· Hypothesis on learning: An individual does not learn mathematical concepts directly. He/she applies mental structures to make sense of a concept (Piaget, 1964). Learning is facilitated if the individual possesses mental structures appropriate for a given mathematical concept. If appropriate mental structures are not present, then learning the concept is almost impossible.
The above imply that the goal for teaching should consist of strategies for: [a] helping students build appropriate mental structures, and [b] guiding them to apply these structures to construct their understanding of mathematical concepts. In APOS Theory, the mental structures are actions, processes, objects, and schemas. In the following each of these are briefly described. Then the ACE Teaching Cycle; which constitutes the pedagogical strategies used to follow the hypothesis and the implication for teaching; is described.
After these general considerations, the assumption on mathematical knowlege is focused on by making an APOS analysis of the cognitive relation between an integer or a fraction and its decimal expansion(s). The result of this analysis is called a genetic decomposition. A genetic decomposition of a concept is a structured set of mental constructs which might describe how the concept can develop in the mind of an individual (Asiala, et. al., 1996). So, a genetic decomposition postulates the particular actions, processes, and objects that play a role in the construction of a mental schema for dealing with a given mathematical situation.
The main mental mechanisms for building the mental structures of action, process, object, and schema are called interiorization and encapsulation (Dubinsky, 2010; Weller et al., 2003). Action, process, object, and schema constitute the acronym APOS. The theory postulates that a mathematical concept develops as one tries to transform existing physical or mental objects. The descriptions of action, process, object and schema; given below; are based on those given by Weller, Arnon & Dubinsky (2009).
Action: A transformation is first conceived as an action, when it is a reaction to stimuli which an individual perceives as external. It requires specific instructions, and the need to perform each step of the transformation explicitly. For example, if a student requires an explicit expression to think about a function and can do little more than substitute for the variable in the expression and manipulate it, then such a student is considered to have an action understanding of functions.
Process: As an individual repeats and reflects on an action, it may be interiorized into a mental process. A process is a mental structure that performs the same operation as the action, but wholly in the mind of the individual. Specifically, the individual can imagine performing the transformation without having to execute each step explicitly. For example, an individual with a process understanding of function will construct a mental process for a given function and think in terms of inputs, possibly unspecified, and transformations of those inputs to produce outputs.
Object: If one becomes aware of a process as a totality, realizes 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 into a cognitive object.
Schema: A mathematical topic often involves many actions, processes, and objects that need to be organized and linked into a coherent framework, called a schema. It is coherent in that it provides an individual with a way of deciding, when presented with a particular mathematical situation, whether the schema applies.
Explanations offered by an APOS analysis are limited to descriptions of the thinking of which an individual might be capable. It is not asserted that such an analysis describse what "really" happens in an individual's mind, since this is probably unknowable. Further, the fact that an individual possesses a certain mental structure does not mean that he or she will necessarily apply it in a given situation. This depends on other factors, for example managerial strategies, prompts and emotional states. The main use of an APOS analysis is to point to possible pedagogical strategies. Data is collected to validate the analysis or to indicate that it must be reconsidered. For more details, see Asiala et al. (1996) and Dubinsky and McDonald (2001).
The ACE Teaching Cycle
This pedagogical approach, based on APOS Theory and the hypothesis on learning and teaching, is a repeated cycle consisting of three components: (A) activities, (C) classroom discussion, and (E) exercises done outside of class. Although variations exist, based on the particular topic and local conditions, each iteration of the cycle, in most implementations, takes about one week. The students do all of their work in cooperative groups.
The activities, which form the first step of the cycle, are designed to foster the students' development of the mental structures called for by an APOS analysis. In the classroom the teacher guides the students to reflect on the activities and their relation to the mathematical concepts being studied. Students do this by performing mathematical tasks. They discuss their results and listen to explanations, by fellow students or the teacher, of the mathematical meanings of what they are working on. The homework exercises are fairly standard problems. They reinforce the knowledge obtained in the activities and classroom discussions. Students apply this knowledge to solve standard problems related to the topic being studied.
The implementation of this approach and its effectiveness in helping students make mental constructions and learn mathematics has been reported in several research studies. A summary of early work can be found in Weller et al. (2003).
Our study analysed teacher-trainees' understanding, after they carried out investigations first individually and then in a collaborative manner. This is to address the learner-centred approach which underpins Curriculum 2005 (DoE, 2003). We report on an investigation based on the use of worksheets and group-work to construct concepts. To collaborate is to work with another or others. In practice, collaborative learning has come to mean students working in pairs or small groups to achieve shared learning goals (Barkley et al., 2005). Vidakovic (1996, 1997) used APOS theory in the context of collaborative learning. Those investigations focused on the differences between group and individual mental constructions of the inverse function concept.
A genetic decomposition of a repeating infinite decimal and its relation to a rational number can be fairly simple. At the action level, the student can only list the first few digits of the decimal and may or may not begin to see a repeating cycle. A process understanding of a repeating decimal emerges as the student can imagine writing out all of the digits of the decimal and see that there is a finite sequence of digits that, from some point on, repeats forever to form an infinite string. At the object level, the student sees this string as a totality, and can perform mental or written actions on it (Weller et al., 2009).
Design and implementation of instruction
The method adopted four stages: (a) APOS design of activity sheet, (b) Facilitation of control group learning, (c) facilitation of experimental group learning and (d) Interviews. The data collection relied to a large extent on what students could say or write about their learning experiences. The tasks were completed over four double periods, each of one and a half hour duration.. This included the individual work by students, the discussions in the groups, the group class presentations and the final discussion involving the tutor. The interviews were done with individuals later during the triangulation stage when analysing the data.
Background of students
The students involved were undergraduate teacher trainees from the University of KwaZulu-Natal. They pursue a module on Real Analysis in their final year. This module, which included elementary topology of the real line, involves the learning of concepts in set theory, relations and functions, cardinality, countability, denseness, convergence and other related ideas. The forty five students were divided into a control group comprising twenty four students and the experimental group with twenty one students.
Facilitation of group-work
The students were arranged according to their previous semester grades and were matched accordingly. In this way the two groups were comprised of members with similar ability levels. The experimental lot had twenty one students and the control lot had twenty four students. The control group was taught in a traditional manner with the lecturer not involved with the design of activity sheets based on APOS Theory. The experimental group was taught by another lecturer who was involved in the APOS Theory design activity sheets and implemented these in this group. Students in the experimental group were divided into seven groups and engaged with the activities individually for approximately fifteen to twenty minutes before coming together into their respective groups. This was to allow students to make contributions when working in a group setting. Each group, after discussing and reaching a collective decision, presented their mathematical ideas to the class. The student facilitators reported on the collective ideas or thoughts of their groups. The students were given time limits set by the facilitator to encourage them to focus on the task at hand. The groups were similar in that they had members with a spread of ability levels. At the end of the group presentations, an intensive classroom discussion including responses from the lecturer led to students establishing the expected responses to the tasks. This paper reports on learning in the experimental group only. We analyse the written responses of students in this group only. We, in subsequent stages of this research, will triangulate the responses with semi-structured interviews.
This section focuses on [a] Pre-knowledge, and [b] Design of worksheet.
At the beginning of the sessions on this investigation, we found it necessary to recall concepts/techniques that were necessary for the activities leading to answering the question "Is = 1?". At the outset we wanted the students to know the difference between rational and irrational numbers. We insisted that they provide examples to illustrate these numbers. Also, we consolidated the meanings of terminating decimal number, recurring decimal number, a string of a decimal number, the length of a string and a cycle. Students were also introduced to the notation to be used in class by the researcher and the activity sheets. Conversion of rational to decimal numbers and the reverse conversion were dealt with. The tasks actually encountered by students appear in Appendix One. This work was done in a ninety minute session. The last question (task eleven of Appendix One) was originally designed to gain responses so as to carry out a comparison at the end of this investigation.
Design of worksheet
Worksheets were designed in accordance with ideas postulated by Asiala et. al (1996). In this theoretical paradigm we devised four activity sheets (see appendix two), each keeping in mind the mental constructions we felt necessary for successful development so as to conclude that = 1. Each activity was done in a ninety minute session. We outline the learning outcome and the APOS outcome below.
Outcome of activity: To conceive an infinite string of digits comprising a repeating decimal.
APOS outcome: To help students interiorise the action of listing digits to a mental process.
Question two of activity one allows the students to work with repeating decimal numbers. They are expected to find the first nine digits for different recurring decimal numbers. This is done in a repeatable manner so that students consolidate their understanding at an action level of APOS. However, questions three, four and five expected mental procedures on the recurring numbers. This was to assist the students to interiorise these actions. Question six was designed to help the students generalise the concepts learned in this activity by using symbolic representation, for example where and are single digit natural numbers.
Outcome of activity: To perform operations on strings.
APOS outcome: To help students encapsulate infinite digit strings which they conceived as processes into mental objects to which actions could be applied.
In question one of this activity the students were expected to rewrite expanded notation of recurring decimal numbers in compact form. The principle of reversal was used. Reversal we described as the ability to reverse thought processes of previous interiorised processes (Brijlall & Maharaj, 2010). The second part of question one allows for the four operations on the decimal numbers. In question two, they were required to find a repeating decimal number between and . This required them to apply actions on mental objects of and . Also arranging in ascending and descending order meant that they derive a schema for working with recurring decimal numbers.
Outcome of activity: To establish a relation between an infinite digit string and it's corresponding fraction or integer.
APOS outcome: To help students coordinate processes into objects.
In these question we provided opportunity for more operations on recurring decimals. The difference here, however, was to create a platform for "seeing" the operated entity as an object in it's own right. So, a + b the sum of two recurring decimal numbers must be looked upon as a single decimal number having similar properties. Question three is a crucial one in this activity as it allows for perceiving as a sum of two different recurring decimal numbers.
Outcome of activity: To establish a relation between an infinite digit string and it's corresponding fraction or integer and locate it on the number line.
APOS outcome: To help students organize actions, processes and objects into a coherent schema that allows for the unification of and 1.
This activity uses a number line model to allow for student identifying the location of decimal numbers. We intentionally work from locating 0,9 on the interval [0; 1] and then 0,99 on [0,9; 1]. The idea here was to then complete question 3 and hopefully deduce that as we subtracted smaller numbers from 1 the number was approaching . We hoped finally that the conclusion to question 3 in Activity 4 would lead them to answering the question whether = 1.
Analysis of data and written responses
We asked the question Is = 1 in the pre-knowledge stage and at the end of activity four. We obtained the following results:
Table 1: Data illustrating post- and pre-activity sheets engagement
Prior to APOS interrogation
After APOS interrogation
Just over fifty percent of the students now indicated that = 1 as compared to one in twenty one before the implementation of the worksheets. This could mean that the worksheets had a positive overall improvement in mathematical correctness. However, many students still felt that this equality was not true. After the implementation of the worksheets there were students who were now unsure of the result. We shall analyse some episodes of written work for each of these categories.
Those who responded "Yes" to the question
In this category we found that there were students who indicated that = 1. However, not all students could justify their conclusion. The following is an example (see Figure 1) of such a case:
Figure 1: An example of a student's response with unreasonable justification
This reasoning appeared often and it is clear that these students thought that rounding off (in this case to any number of decimal places) would result in 1. However, it is a problem that these students will now move on to teach in schools with the misconception that a number is equal to the rounded off number when it is actually an approximation to the said number of places. We notice that this type of "approximation" misconception arises in other learning experiences ,as well. One such situation is when students use as an approximation for Ï€. Obviously these two entities are not equal but students inherently accept this equality as instructors say/write to calculate the area of a circle of radius 2 cm. There were students in this category who provided mathematically correct reasons to why = 1. An example of this follows in Figure 2.
Figure 2: An example of a mathematically sound reason
Here the individual reflected on operations (multiplication and subtraction) applied to a particular process. He/she is aware of the process as a totality, and constructed such transformations to see and 1 as individual entities and being identical. She/he is hence thinking of this process as an object.
Those who responded "no" to the question
Figure 3 illustrates a response where the student rejected the equality of 1 and .
Figure 3: Student who believed = 1 is false
This student seems to know that rounding off leads to an approximation and on approximating she/he gets 1. But, she/he thinks that they are not identical. These mental structures could have developed whilst the student attempted the tasks in the activities. Activity 1 (see Appendix One) stimulated the student's move from a decimal entity to a whole one. The equality of these entities could have developed during Activity 4 (see Appendix Two) which intended the development of the establishment a relation between an infinite digit string and it's corresponding fraction or integer and locate it on the number line. Here the student has applied specific mental structures to make sense of the equality of and 1. Another example supporting this thought is indicated in Figure 4.
Figure 4: An example of equality and approximation
Those who were unsure
Whenever students used the words "maybe" or "sometimes" we classified the response as unsure as a degree of doubt is conveyed. An example of this is illustrated in Figure 5.
Figure 5: An example of a response with doubt whether = 1
This response indicates an error and a misconception. The misconception of approximation and equality arose in the second part of the written response. No clear explanation occurred for the first part of . This would need further investigation. The pilot study hence finds it necessary to carry out interviews in the further study. This would allow for such a response to ascertain reasonable explanation to such a response.
The findings of this study showed that some students demonstrated the ability to make use of technique from the worksheets (see Figure 2, for instance). We also found that a greater number of students concluded that = 1, after the implementation of the worksheets. It seemed that the APOS designed worksheets had a positive influence on their answering that question, based on the assumption that an individual does not learn a mathematical concept directly, but applies mental structures to make sense of the concept. However, we need to validate the responses and we are currently involved with a series of interviews to verify our findings.