Encuesta semiestructurada

Peter Chifamba University of Kwazulu Natal [email protected]

This study was an attempt to understand some of the learning obstacles encountered by students when learning the concepts of differentiability and continuity in Real Analysis at undergraduate level. The study is qualitative in nature and takes on the form of a case study of a group of 13 B. Ed in -service teachers majoring in mathematics at Great Zimbabwe University. The framework used for the design of the study and the analysis was the APOS theoretical framework. The research instrument used in this study was an assessment task with questions on continuity and differentiation. . The study found that most students showed that they did not have appropriate mental structures for the concepts at process, or object level. It is recommended that students should be given opportunities of working with different representations of the functions to facilitate their understanding of these abstract concepts Key words: Continuity, Differentiability, APOS Theory Framework

Introduction

The mathematics curriculum for Bed In -service honours program at Great Zimbabwe university has 21 modules in mathematics and mathematics education. Real analysis is offered as a second year module after having covered linear algebra, abstract algebra and advanced calculus at part one level, while complex analysis is done at part three. Most of the concepts in other areas of mathematics are an extension of real analysis ideas such as limits, continuity and differentiability. A strong background of these concepts is necessary in any undergraduate mathematics curriculum. Experience and work by other researchers show that learners have difficulties with some concepts in real analysis and that they tend to memorise lecture notes and reproduce theorems in order to pass the exam. This research looks at how students understand the concepts of continuity and differentiability. Similar research has been done in calculus for differentiation by Maharaji (2013), Orton (1983), Uygur and Ozdas (2005) who noted that learners have certain difficulties with the derivative concept and that students cannot explain why rules of differentiation work. Hlankioniemi (2004) noted that students’ understanding of the derivative improves if several kinds of representations are used, while Zandieh (2000) noted that learners prefer graphical representations when dealing with properties of functions. This research will consider learning difficulties of the concepts of differentiability and continuity in real analysis where conceptual knowledge should be demonstrated.

Literature Review

Research points out that undergraduate students have difficulties in understanding certain concepts in real analysis Tall (1991). A majority of students at university level simply memorize proofs and pass the examination (Sawyer 1987). If the students are asked to

perform the same tasks a few weeks after the examination, they may have forgotten. Artigue (1987, cited in Tall, 1991) reported that learning analysis presents problems that are due to some of the following factors, the highly sophisticated level of the concepts e.g. sequences and functions, the formalization of analytic definitions to provide rigor in the course, and difficulties posed in learning specific technical terms in the course such lower and, upper bounds and axiom of completeness.

Several abstract and theoretical topics have to be developed over a very short period of time such as a semester, the concepts are closely related by the limit concept which most learners have difficulties with, The use of ε – δ definition of some concepts such as limits and continuity may not be easy to comprehend as this approach involves a lot of rigor Also the ideas were developed over a long period of time and most of them were a result of providing answers to specific classical problems and yet undergraduate students are expected to

understand them in a very short period of time such as a one hour lecture.

The conceptual field of analysis is wide and at elementary level is structured around real numbers, functions, limits of sequences and functions, continuity, differentiation and the Riemann integral. A satisfactory discussion of the main concepts of analysis such as convergence, continuity, differentiation and integration must be based on an accurately defined number concept. Rudin R. (1976)

The concepts of continuity and differentiability are at the center of the course.

Integration is ant -differentiation, and applications of differentiation are very wide e.g. rates of change, velocity and acceleration and partial derivatives .The relationship between the two concepts of differentiability and continuity is central in certain theorems. Differentiability implies continuity and a counter example can be used to prove that the reverse is not true. Certain results e.g. Rolles’ theorem and the Mean Value theorems are centered on a function that is both differentiable and continuous on an interval.

Rolles’ theorem states that suppose a function f is continuous on [a ,b] and differentiable on (a ,b) and suppose f(a) = f(b), then there exist c in (a.b) such that f(c) =0. Rolles’ theorem leads to the Mean Value Theorems.

The concept of continuity is defined in terms of limits: A function f(x) is continuous at x0 iff given any ε >0, there exist δ > 0 such that |x – x 0 | < δ implies |f (x) – f(x0)| < ε. This definition is an abstraction of the definition of continuity in terms of limit f (x) as x approach x0 = f (x0). These two definitions are equivalent. A function is continuous on a set A if it is continuous at each point in A. The notion of continuity has several applications and leads to some of the following results: uniform continuity, differentiation and integration. Every differentiable function is continuous.

Differentiability has been characterized in terms of: 1. Limits e.g. f/(x) = lim

ℎ→0

𝑓(𝑥+ℎ)−𝑓(𝑥)

ℎ i.e. a function f(x) is differentiable at a point x0 if

this limit exist at x = x0 .

2. Left hand and right hand derivatives i.e. a function f(x) is differentiable at x = a if the left hand derivative is equal to the right hand derivative at the point.

Both continuity and differentiation are concepts expressed in terms of limits of functions. Vuma (1998) asserts that in complex analysis the concepts of continuity and differentiability are defined in the same way as in real analysis and different authors use different terms to describe functions that are differentiable in the complex sense such as regular, holomorphic, analytic and entire. The limit concept plays a very important role in mathematical analysis.

The mathematical concept of a limit is a particularly difficult notion, typical of the kind of thought required in advanced mathematics. It holds a central position which permeates the whole of mathematical analysis-as a foundation of the theory of approximation, of

continuity, and of differential and integral calculus. (Corn, 1991, as cited in Tall, 1991, p. 133)

Bezuidenhout (2001) also observed that first year university students held some misconceptions with limits and continuity in a calculus course.

Various views have been expressed on what constitutes mathematical understanding. To understand something means to assimilate it into an appropriate schema Skemp 1971:46

The same authority identified two types of understanding namely relational and

instrumental understanding Relational understanding is knowing what to do and why while instrumental understanding is possession of rules and the ability to use them correctly. Skemp (1976) Relational understanding involves deriving concepts from first principles and calls for a high level of understanding why a procedure works while instrumental understanding is ability to carry out computational skills. For the concept differentiation a learner who has can differentiate the function f(x) = sinx from first principles i.e. using limits may be considered to have a relational understanding of the derivative of the function while the learner who only know that the derivative of sinx is cosx from a given list of formulas has an instrumental understanding of the concept. Instrumental understanding appears easier to achieve but students may easily forget what they think they have learnt, while relational understanding enables the learner to perform new tasks and concepts are easily remembered. Most topics in real analysis are theoretical and demand the latter type of understanding.

Other researchers suggest that mathematical knowledge held by an individual is either procedural or conceptual. (Heibert & Leiverve, 1986). These researchers define procedural knowledge as competence in carrying out mathematical tasks while conceptual knowledge is knowledge that is rich in connections. This classification of knowledge is similar to Skemps’ notions of types of understanding. Nickerson (1985) suggests some characteristics that show that a learner has understood which are, agreement with experts, being able to see deeper characteristics of a concept, looking for specific information quickly and the ability to see connections within several concepts. It may not be easy to have one type of understanding without the other or a certain type of knowledge without the other, but most researchers seem to prefer instruction that promote why procedures work and to show connections between concepts as this promotes deeper understanding of mathematics concept and future learning. Basic notions in real analysis require the learners to have deep insights about them. The concepts of differentiability and continuity are also very abstract in nature and students should have correct concept images and concept definitions. A concept definition gives precisely instances and non-instances of a concept while a concept image is an individuals’ perception of the concept. The correct concept image may be formed if the learner is given several experiences of the concept. Dreyfus & Vinner (1989) suggests that concept images are not formed by definitions alone but through experiences.

Statement of the Problem

This research will consider how APOS theory based research can be used to detect the level of understanding of the concepts of continuity , differentiation and their relationship at undergraduate level. It will also suggest possible causes of learning obstacles to the two concepts.

Research Questions This research will be guided by the following questions:

Major question. What is the state of students’ understanding the concepts of continuity and differentiability?

Sub-questions: (a) What are the causes of learning obstacles in the concepts of continuity and differentiability? (b) How can APOS theory be used to overcome the problems?

Research Methodology Research Design

The case study design will be used because of the nature of the problem under investigation. A case study is an empirical inquiry that investigates a contemporary

and context are not clearly evident Yin (1981). A case study is a method of gathering and analyzing data as a way of investigating a conjecture in order to understand a real life problem. A case study can be about a target group of people, individuals or events. The case study is appropriate for this research since it is investigating a particular group of learners’ problems in an undergraduate course in real analysis. Case study results are usually suitable for publication and such results relate very well with other researchers in a specific area of research.

Population and Sample

The research was done on a group of 13 in-service Bed students who were majoring in mathematics since the class was very small all students were involved in the research. The students had completed a diploma in education at teachers college level with mathematics as their major subject over a period of 3 years. In addition more than half of the group had an ‘A’ level pass in Mathematics and the program done at teachers college had also prepared them to teach mathematics up to ‘o’ level. The teachers college curriculum also has some ‘A’ level topics such as functions, basic differentiation, integration and trigonometry.

Research Instruments

A test was used to collect data from the group about the level of understanding of the concepts of continuity, differentiability and applications after teaching the concepts. The test had 3 tasks on the concepts.

The tasks assessed their ability to sketch the graph of a function, prove continuity on a set and check differentiability at a particular point. This task is usually taken to be the

standard counter example proof that continuity does not imply differentiability. Task: Let f(x) = |x| on R.

i. Sketch the graph of f(x).

ii. Prove that f(x) is continuous on R.

iii. Prove that f(x) is not differentiable at (0,0). Suggested Solutions:

Sketch of graph of the function f (x) = |x|.

(ii) Let ε >0 be given. We want to find δ >0 such that |x- xo| < δ implies |f(x) – f(x0)|< ε. Note that for any real numbers a and b, ||a| - |b|| ≤ |a – b|. Therefore | f(x) – f(x0)| = ||x| - |x0|| ≤ |x- x0|. So choose δ = ε. Since the δ is a function of ε only this is uniform continuity. (iv)The left hand derivative at x=0 is -1, while the right hand derivative is +1. Since these two are not equal the function f(x) is not differentiable at x = 0.

Also note that 𝑓(𝑥) = {−𝑥, 𝑥 < 0_{𝑥, 𝑥 ≥ 0} which may result that f/(x) ={_{−1, 𝑥 ≥ 0}1 , 𝑥 < 0 Theoretical Framework

This research was guided by APOS theory (Dubinsky and McDonald ;2001). The theory is an extension of Piagets’ work on reflexive abstraction, which is another framework that describes the construction of mathematical structures by an individual during the course of cognitive development. According to Dubinsky (2010) APOS theory and its applications is based on the assumption that an individuals’ mathematical knowledge is his/her ability to respond to mathematical problem situations and solutions by reflecting on them in a social context and constructing or reconstructing to use in dealing with the situations. It also has the hypotheses that an individual does not learn mathematics concepts directly. He/she applies uses mental structures to make sense of a concept. The mental structures refer to the likely actions, processes, objects and schemas needed during the learning of a concept. The definitions of the mental structures given below have were given by Weller et al (2009). Related examples on continuity have be suggested.

Actions. An individuals’ first experience with a concept is external and may be taken to be an action. A learner who can substitute a value in a function to check continuity of a function at that point has an action understanding of the concept of continuity.

Process. After repeating the action several times a learner may interiorize the action as a process. A learner who can substitute several values of the function near x=a to check

continuity of the function f(x) at x=a has understood the continuity idea as a process. Object. A learner may encapsulate the process into an object. A learner who can draw the correct graph of the function to determine its continuity properties has an object

understanding of continuity.

Schema. A mature schema is a final mental image or representation of the concept and other previously developed schemas. It is the final desirable outcome of the collection of actions, processes, objects and other existing schemas. A learner who can who can use the definition of continuity such as use of limits to prove continuity has the appropriate schema of continuity. The main mechanisms for building mental structures action, processes, objects and schema are called interiorisation and encapsulation Dubinsky (2010), Weiller

(2009).Piaget and Gracia (1983/1989,1996) explained schema development in terms of the triad.

Researches based on APOS require the instructor to help the students to identify the appropriate mental structures for learning a particular concept and then provide effective learning. Asiala et al. (1996) proposed a framework for APOS theory based research and the framework has 3 components namely theoretical analysis, design, and implementation of instruction and data analysis. In theoretical analysis the researcher predicts the likely mental structures for the concept. This is usually done relative to the researcher’s knowledge about the concept. After identifying the likely mental structures appropriate learning materials and activities have to be suggested. After teaching the concept a task can be given to check the level of performance of the learners. Questions in the task may be able to detect the level of performance at action, process, object and schema level. APOS based research has already been used by successfully by other researchers for undergraduate mathematics learning. e.g. Cotrill (1999), the chain rule and its relation to composite functions, Clarke etal (1997) students understanding of the chain rule.

Results 1. Sketch of the graph of f(x) = |x|.

This question was meant to check the learner’s knowledge of curve sketching of the graph of f(x = |x|. . Two of the students could not give the correct sketch of the graph and one of them did not attempt the question. These individuals had no idea of basic curve properties of f(x) = |x|. The two learners who gave incorrect graphs of the function did not even realize that the graph passes through the origin. This shows that the students had a very weak

background of graphs of functions and yet functions are central to the concepts under consideration. The incorrect graphs are shown below:

Continuity

2. Prove that f(x) = |x| is continuous on R. 9 candidates out of 13 representing 69.2% of the students had incorrect solutions to the task and 4 of these 9 candidates checked continuity at origin and not on R and these learners can be considered to be at the action level of

understanding the concept. The four did not understand the difference between continuity at a point and continuity on a set. However 4 students used the ε, δ definition correctly demonstrated the process, object and schema understanding of the concept. Most of the group showed that they did not have appropriate mental structures at process object and schema level of the continuity concept on a set. They had the concept of continuity at a point and not on an infinite set or had no idea at all i.e. Pre action level. One student used a table of values, assuming that the real line is a finite set and the individual demonstrated the process level of understanding the concept of continuity.

Differentiability

6 out of 13 candidates (46%) had an idea of differentiability as a technique of computing gradient functions(procedural knowledge) and also at action stage but could not apply the use of left hand and right hand derivatives ( object or schema stage) to check whether a function is differentiable or not. The concept of differentiability in analysis is conceptual and not procedural. Learners are expected to prove differentiability of a function and not to differentiate the function. Individuals who showed difficulties with the task demonstrated that they were not aware of these two contexts. Use of left hand and right hand derivatives was correctly done by 5 learners out of the whole group (38%). Some used the converse theorem: differentiability implies continuity, which is incorrect.

One student did not attempt the task on differentiability showing that the individual had no idea. He/she had no appropriate mental structures of the concept showing that the learner

was at the pre- action level. 6 students (46%) showed some mix up of the concept of