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Mastery of mathematical induction among JC students
Author
Chow, Ming Kong
Supervisor
Edge, Douglas Richard Montgomery
Abstract
Mathematical induction is part of the mathematics syllabus required for the Singapore-Cambridge General Certificate in Education Advanced Level Examinations, and is a topic that students find difficult to understand. The aim of this study was to pursue an answer to the following general questions:
1. What is the level of mastery of mathematical induction among junior college students?
2. What learning difficulties do students experience when they learn mathematical induction?
A review of selected literature showed that mathematical induction has attracted the attention of many researchers. Ernest (1984) provided a comprehensive analysis of this proof technique. The contributions from others, such as Avital and Libeskind (1978), Baker (1996), Dubinsky and Lewin (1986), and Movshovitz-Hadar (1993), informed our specific understanding of some of the difficulties encountered in terms of procedural and conceptual knowledge.
For this study, the data collection involved the use of proof-writing tasks in questionnaires to determine students' performance, levels of understanding and to identify students' learning difficulties. The questionnaire included five proof-writing questions on identities involving (a) indices, (b) the summation symbol, (c) sequences, (d) differentiation of one variable and factorial symbol, and (e) formulating a conjecture and then proving it. Thirty questionnaires were selected from students for data analysis. After a preliminary analysis of the data collected from the diagnostic test, seven students were interviewed to clarify and elaborate their written answers. Data collected from the interview also provided information on students' level of understanding of mathematical induction and students' learning difficulties.
With respect to students' performance, students in this study did well with proof involving indices and summation. Proofs involving sequences, differentiation and factorial symbol were not as easy for the students. The most difficult question involved formulating a conjecture and then proving it using mathematical induction. From the proof-writing tasks, the researcher also noticed that one critical step in determining students' performance in the induction step was the writing down of the key equation that linked the propositions Pk to Pk+1. Many students had difficulty writing this key equation to link Pk to Pk+1 for questions involving sequences and differentiation.
As for levels of understanding, the students focused on the procedural aspect of mathematical induction far more often than on the conceptual aspect. Critically, the junior college students had significant difficulties with the proof technique, mathematically, conceptually or technically. Most of the students in the study encountered difficulty when the basis case is not n=1. A primary source of difficulty was attributed to a lack of mathematical content knowledge. Good students were observed to have acquired the following skills: (a) identify common factors, (b) use commutative law to rearrange algebraic fractions, (c) simplify target expression in Pk+1, (d) isolate the constants, and (e) observe for pattern.
The study concluded with a discussion on implications for teaching, limitations of the study and implications for further research.
1. What is the level of mastery of mathematical induction among junior college students?
2. What learning difficulties do students experience when they learn mathematical induction?
A review of selected literature showed that mathematical induction has attracted the attention of many researchers. Ernest (1984) provided a comprehensive analysis of this proof technique. The contributions from others, such as Avital and Libeskind (1978), Baker (1996), Dubinsky and Lewin (1986), and Movshovitz-Hadar (1993), informed our specific understanding of some of the difficulties encountered in terms of procedural and conceptual knowledge.
For this study, the data collection involved the use of proof-writing tasks in questionnaires to determine students' performance, levels of understanding and to identify students' learning difficulties. The questionnaire included five proof-writing questions on identities involving (a) indices, (b) the summation symbol, (c) sequences, (d) differentiation of one variable and factorial symbol, and (e) formulating a conjecture and then proving it. Thirty questionnaires were selected from students for data analysis. After a preliminary analysis of the data collected from the diagnostic test, seven students were interviewed to clarify and elaborate their written answers. Data collected from the interview also provided information on students' level of understanding of mathematical induction and students' learning difficulties.
With respect to students' performance, students in this study did well with proof involving indices and summation. Proofs involving sequences, differentiation and factorial symbol were not as easy for the students. The most difficult question involved formulating a conjecture and then proving it using mathematical induction. From the proof-writing tasks, the researcher also noticed that one critical step in determining students' performance in the induction step was the writing down of the key equation that linked the propositions Pk to Pk+1. Many students had difficulty writing this key equation to link Pk to Pk+1 for questions involving sequences and differentiation.
As for levels of understanding, the students focused on the procedural aspect of mathematical induction far more often than on the conceptual aspect. Critically, the junior college students had significant difficulties with the proof technique, mathematically, conceptually or technically. Most of the students in the study encountered difficulty when the basis case is not n=1. A primary source of difficulty was attributed to a lack of mathematical content knowledge. Good students were observed to have acquired the following skills: (a) identify common factors, (b) use commutative law to rearrange algebraic fractions, (c) simplify target expression in Pk+1, (d) isolate the constants, and (e) observe for pattern.
The study concluded with a discussion on implications for teaching, limitations of the study and implications for further research.
Date Issued
2002
Call Number
QA14.S55 Cho
Date Submitted
2002