Independent Study Workbooks for Proofs in Group Theory

This paper describes a small-scale research project based on workbooks designed to support independent study of proofs in a first course on abstract algebra. We discuss the lecturers’ aims in designing the workbooks, and set these against a background of research on students’ learning of group theory and on epistemological beliefs and study habits in higher education. We organise our analysis of student responses around three emerging themes: 1) structured support provided by the workbooks, 2) productive forced study of lecture notes, and 3) engaging with proofs. Discussion of our data in terms of these themes suggests several considerations for the design of tasks for independent study of advanced mathematics.

[1]  John T. E. Richardson,et al.  Gender, subject and context as determinants of approaches to studying in higher education , 1995 .

[2]  Yvette Solomon,et al.  Not belonging? What makes a functional learner identity in undergraduate mathematics? , 2007 .

[3]  M. Inglis,et al.  Expert and Novice Approaches to Reading Mathematical Proofs , 2012 .

[4]  Magnus Österholm,et al.  Do students need to learn how to use their mathematics textbooks? : The case of reading comprehension , 2008 .

[5]  Bob Burn.  Mathematics Teaching Practice, a guide for University and College Lecturers Mathematics Teaching Practice, a guide for University and College Lecturers Author: John H Mason Publisher: Horwood Publishing , 2002 , ISBN: 1-898563-79-9 , £30 , 2002 .

[6]  A. Mji,et al.  A three-year perspective on conceptions of and orientations to learning mathematics of prospective teachers and first year university students , 2003 .

[7]  J. Boaler.  Multiple Perspectives on Mathematics Teaching and Learning , 2000 .

[8]  Annie Selden,et al.  Unpacking the logic of mathematical statements , 1995 .

[9]  F. Marton,et al.  ON QUALITATIVE DIFFERENCES IN LEARNING: I—OUTCOME AND PROCESS* , 1976 .

[10]  Marie-Pierre Moreau,et al.  Balancing paid work and studies: working (‐class) students in higher education , 2006 .

[11]  Etienne Wenger,et al.  Situated Learning: Legitimate Peripheral Participation , 1991 .

[12]  Graham Gibbs,et al.  Teaching students to learn : a student-centred approach , 1981 .

[13]  Uri Leron,et al.  An Abstract Algebra Story. , 1995 .

[14]  Bret J. Benesh,et al.  Undergraduate Students' Self-Reported Use of Mathematics Textbooks , 2012 .

[15]  F. Marton,et al.  On qualitative differences in learning , 2013 .

[16]  Juan Pablo Mejía-Ramos,et al.  Mathematics majors’ beliefs about proof reading , 2014 .

[17]  P. Ramsden.  Improving Learning: New Perspectives , 1988 .

[18]  Teaching and Testing Mathematics Reading , 1991 .

[19]  P. Pintrich,et al.  The Development of Epistemological Theories: Beliefs About Knowledge and Knowing and Their Relation to Learning , 1997 .

[20]  Joaquim Barbé,et al.  Didactic Restrictions on the Teacher’s Practice: The Case of Limits of Functions in Spanish High Schools , 2005 .

[21]  P. Ramsden.  Learning to Teach in Higher Education , 1991 .

[22]  Proof Validation in Real Analysis: Inferring and Checking Warrants , 2006 .

[23]  Sean Larsen,et al.  Teacher listening: The role of knowledge of content and students , 2012 .

[24]  Uri Leron,et al.  Students' Use and Misuse of Mathematical Theorems: The Case of Lagrange's Theorem , 1996 .

[25]  L. Alcock,et al.  The Warwick Analysis Project: Practice and Theory , 2001 .

[26]  Michelle Zandieh,et al.  Proofs and refutations in the undergraduate mathematics classroom , 2008 .

[27]  Mary D. Shepherd.  ENCOURAGING STUDENTS TO READ MATHEMATICS , 2005 .

[28]  Orit Hazzan,et al.  Reducing Abstraction Level When Learning Abstract Algebra Concepts , 1999 .

[29]  C. Snow,et al.  Academic Language and the Challenge of Reading for Learning About Science , 2010, Science.

[30]  D. Kember.  Interpreting Student Workload and the Factors Which Shape Students' Perceptions of Their Workload. , 2004 .

[31]  Annie Selden,et al.  Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? , 2003 .

[32]  Ed Dubinsky,et al.  Development of students' understanding of cosets, normality, and quotient groups , 1997 .

[33]  Keith Weber,et al.  An assessment model for proof comprehension in undergraduate mathematics , 2012 .

[34]  Ghislaine Gueudet,et al.  Investigating the secondary–tertiary transition , 2008 .

[35]  Douglas A. Grouws,et al.  Handbook of research on mathematics teaching and learning , 1992 .

[36]  Variations in student learning and perceptions of academic quality , 2007 .

[37]  J. Vermunt.  The regulation of constructive learning processes , 1998 .

[38]  Jackie Nicholas,et al.  Conceptions of mathematics and how it is learned: The perspectives of students entering university , 1994 .

[39]  First-year engineering students’ use of their mathematics textbook - opportunities and constraints , 2012 .

[40]  Sean Larsen.  A local instructional theory for the guided reinvention of the group and isomorphism concepts , 2013 .

[41]  Uri Leron,et al.  On learning fundamental concepts of group theory , 1994 .

[42]  R. Burn,et al.  Numbers and Functions: Questions for student investigation and discussion , 2000 .

[43]  Matthew Inglis,et al.  Self-explanation training improves proof comprehension , 2014 .

[44]  Yvette Solomon,et al.  Deficit or Difference? the Role of Students' Epistemologies of Mathematics in their Interactions with Proof , 2006 .

[45]  Keith Weber,et al.  Student difficulty in constructing proofs: The need for strategic knowledge , 2001 .

[46]  W. G. Perry.  Forms of Intellectual and Ethical Development in the College Years: A Scheme. Jossey-Bass Higher and Adult Education Series. , 1970 .

[47]  The Perry Development Scheme: a Metaphor for Learning and Teaching Mathematics , 2014 .

[48]  N. Hoffart.  Basics of Qualitative Research: Techniques and Procedures for Developing Grounded Theory , 2000 .

[49]  N. Entwistle,et al.  Understanding Student Learning , 1983 .

[50]  John Mason.  Mathematics teaching practice , 2002 .

[51]  Clive Lawless,et al.  Using Learning Activities in Mathematics: Workload and study time , 2000 .

[52]  Uri Leron,et al.  Learning group isomorphism: A crossroads of many concepts , 1995 .

[53]  J. Frith,et al.  Comprehension tests in mathematics , 2000 .

[54]  Lourdes Díaz Rodríguez,et al.  The epistemological beliefs, learning approaches and study orchestrations of university students , 2006 .

[55]  Mary D. Shepherd,et al.  Reading mathematics for understanding—From novice to expert , 2014 .

[56]  Catherine E. Snow,et al.  The Challenge of Academic Language , 2009 .

[57]  Chris Rasmussen,et al.  Social and sociomathematical norms in an advanced undergraduate mathematics course , 2000 .

[58]  Jackie Nicholas,et al.  University mathematics students' conceptions of mathematics , 1998 .

[59]  David Kember,et al.  Characterising a teaching and learning environment conducive to making demands on students while not making their workload excessive , 2006 .

[60]  L. Alcock,et al.  USING WARRANTED IMPLICATIONS TO UNDERSTAND AND VALIDATE PROOFS , 2012 .

[61]  Nadia Hardy,et al.  Students’ perceptions of institutional practices: the case of limits of functions in college level Calculus courses , 2009 .

[62]  Mary D. Shepherd,et al.  University Students' Reading of Their First-Year Mathematics Textbooks , 2012 .

[63]  D. Bligh.  Learning to teach in higher education , 1993 .

[64]  K. Trigwell,et al.  How approaches to teaching are affected by discipline and teaching context , 2006 .

[65]  Ed Dubinsky,et al.  A Framework for Research and Curriculum Development in Undergraduate Mathematics Educationjpeg , 2009 .

[66]  A. Schoenfeld.  Learning to Think Mathematically: Problem Solving, Metacognition, and Sense Making in Mathematics (Reprint) , 2009 .

[67]  Krista R. Muis.  Personal Epistemology and Mathematics: A Critical Review and Synthesis of Research , 2004 .

[68]  D. Kember.  The intention to both memorise and understand: Another approach to learning? , 1996 .

[69]  John T. E. Richardson,et al.  Reliability and replicability of the Approaches to Studying Questionnaire , 1990 .