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An APOS analysis of Bachelor of education mathematics students performance in assessment task on limits of functions and their applications

By Peter Chifamba

Focus of the study

 The study seeks to analyse cognitively Bed in service

honours Mathematics students ‘ level 2 understanding

• f limits of sequences and limits of two variable

functions.

Statement of the problem

 Globally, researchers noted that students have

problems with limits of functions and their applications.(Cornu,1992;Davis&Vinner,1986; Williams,1991;Bezuidenhout,2001;Parameswaran,2006; Maharaj,2010;Gucher,2012 ,Aydin & Mutlu,2013,Mrdja,Romano &Zubac 2015))

 During academic Board of Examiners at a university in

Zimbabwe, Bed students in Mathematics level 2 where the limits of functions and applications is one of the concepts students reported performed badly.

Statement of the Problem cont.

 During lectures and tutorials in calculus students

displayed difficulties in solving problems in limits and its applications.

 The study considers the use of APOS theory to

measure the performance of 4 Bed students on tasks

• n limits of sequences and limits of several variables

Motivation

 My personal experience as a mathematics

educationist, I discovered that concept of limits and applications are taught from high school, colleges and universities with more importance.

 The concept of limits and applications pervades

several disciplines in universities curricula in Zimbabwe and is developed from single variable functions to several valued functions.

 During the lectures students displayed challenges in

solving concepts in limits and applications in contexts

• f sequences and continuity.

Motivation cont..

 ”The mathematical concept of a limit is a particularly

difficult notion, typical of the kind of thought required in advanced mathematical thinking. It holds a central position which permeates the whole of mathematical analysis as a foundation of the theory of approximations

• f continuity, differential and integral calculus.”( Cornu in

Tall 1991, p53)

 These observations motivated the researcher to analyse

cognitively undergraduate Mathematics students’ understanding of limits of sequences and limits of several valued functions at a university in Zimbabwe context.

Literature review

 Several studies have been done on the learning of the

limit concept (Moru,2008,Maharaj,2010;Gucha,2012;Aydin & Mutlu,2013; Mrdja,2015,)

 Not much has been done on limits of sequences.  Not much has been done on calculus concepts of

several valued functions. (Trigueros & Planell,2009; Dorko & Weber,2014; Mcgee &Moore Russo,2014).

Literature review cont

 Some studies observed that students do not have

appropriate mental structures when learning limits of

• functions. (Maharaj 2010;Aydin & Mutlu,2013)

 Other researchers assert that students do no use the

correct technologies when learning limits of functions

 (Mrdja.Romano & Zubac,2015)  Epistemological obstacles in historical development

• f limits have been noted (Cornu,1991)

Literature review cont: limits

 At undergraduate level the limit concept covers the

following aspects

 Intuitive definition and finding limits of single

variable functions Theorems on limits

 Analytic definition  Limits of sequences and continuity.  Analytic definition for several variable functions

Theoretical Framework

1.The theoretical framework will be used is called Action- Process-Object – Schema acronmyed APOS. (Arnon,2014) 2.Chief proponent: Dubinsky and others (1991,1996,2001). 3.Basic assumptions: 1.students should have appropriate mental structures needed to learn a mathematical concept. .Instructors should assist students to develop appropriate mental structures. 4 Asiala etal (1996) identified a framework for APOS based research and curriculum development.

• 5. ACE teaching cycle

6.Used successfully in other studies: concept of two variable functions (Trigueros & Planell,20o9), Limits of functions (Maharaj,2010) , Matrix Algebra,(Ndlovu & Brijlall, 2015).

Genetic Decomposition for limits of sequences and functions

Action level

 At action level a student when confronted with the limit of a

function can do little more than substitute the value of x = a or values of x close to a in the expression f(x) In the case of sequence large values of n are substituted into Un . He or she will make deductions based on patterns he/she may see. Process level

 As the student repeats the action of substitution several times it

may be interiorized into a single process of inputs and outputs in which the sequence Un approaches a limit L as n approaches infinity or as x approach a approach a. in the case of functions At this stage a student may be able to judge whether a sequence converges by checking whether the limit of the sequence is unique. In the case of functions, he/she may find limits on different paths.

Genetic Decomposition cont..

Object level

 The student sees the string as a totality and can perform

mental actions on limits of the functions along paths and is aware of theorems on limits and the process understanding is encapsulated to an object. Use of theorems to find limits

• f functions The student is able to provide proofs related to

properties and existence of limits. A student is comfortable with using the ε, δ definition hence can easily extend to limits of several variables. Schema level

Assessment task

Genetic Decomposition cont..

Schema level

 Schema Actions processes and objects are organized into a

coherent framework of a limit and the student can use several approaches including the definition, existence and non existence to all types of functions.

Genetic Decomposition cont..

Object level

 The student sees the string as a totality and can perform mental

actions on one sided limits of the function and the process understanding is encapsulated to an object. The student is able to provide proofs related to properties and existence of limits. A student is comfortable with using the ε, δ definition hence can easily extend to limits of several variables. Schema level

 Schema Actions processes and objects are organized into a

coherent framework of a limit and the student can use several approaches including the definition, existence and non existence to all types of functions.

Research questions

This study will be guided by the following questions. What insights would APOS analysis of students’ understanding of limits of functions reveal ?

 How should the teaching of functions and applications

limits of functions be approached ?

Methodology

An assessment task was given to 4 students after having attended lectures ant tutorials on relevant concepts for two weeks. The pedagogical approach used followed the ACE teaching cycle Students were allowed to respond to questions on limits of sequences and limits on several variable functions as (x,y) approach (0,0)

Selection of Participants

 The researcher will use purposive sampling technique

basing on level 2 Bachelor of Education in-service mathematic students.

 The selection will be based on mathematics major  The group will consists of4 students both males and

females.

 The students were following a part 2 level compulsory

course in calculus

Data generating methods

An assessment on problems on limits of sequences of functions and limits of two variable functions task was given to students after two weeks lectures. Students were allowed to respond freely to the task.

Data Analysis Sequences

 Data analysis will be informed by the theoretical

framework (APOS).

 Students performed badly on problems on sequences.

They found limit Un approach 0 instead of limit Un as n approach infinity. All students applied a wrong schema of the limit of a sequence in this question1.

 In question 2 students were suppose to use excessive

algebraic manipulations to the given term of the sequence and they were not successful.

Two Variable functions

 All students successfully did the first question because

they had done a similar question in classroom activities.

 All students could not do the question which required

them to use standard limits.

Ethical Issues

 All ethical issues will conform to the ethical

requirements of the University of kwaZulu Natal ethical committee. Names of participants will not be disclosed

Limitations

 This is a case study research it cannot be generalised

but it can be used for decision making and instruments are transferable.

References

 Arnon I (2014) APOS theory: A framework for research and

curriculum development in mathematics. Volume1;2014

 Aydin,S..& Mutlu,C (2013). Students’ understanding of the

concept of the limit of a function in vocational high school

• mathematics. TOJSAT: The online journal of science &

technology, Volume3 issue1.

 Dorko,A & Weber,E (2014) generalising calculus ideas from

two dim to three: how multvariate calculus students think about domain and range. Research in mathematics education,:Routledge.

References cont

 Gucher, B. (2012). Examining the discourse on the limit concept in

a beginning level calculus classroom. Dordrecht :. Springer Science + Business .

 Maharaj, A. (2010). An APOS Analysis of Students’ Understanding

• f the Concept of a Limit of a Function. Pythagoras, 71, 41-

52.

 Ndlovu, Z., & Brijlall, D. (2015. Pre-service Teachers mental

construction of concepts in matrix algebra. African journal of research in mathematics, science and technology of education, 19(2), 156-171.

End of Presentation

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