Results from a Survey of Abstract Algebra Instructors across the United States: Understanding the Choice to (Not) Lecture

In the United States, there is significant interest from policy boards and funding agencies to change students’ experiences in undergraduate mathematics classes. Abstract algebra, an upper division undergraduate course typically required for mathematics majors, has been the subject of reform initiatives since at least the 1960s; yet there is little evidence as to whether these change initiatives have influenced the way abstract algebra is taught. We conducted a national survey of abstract algebra instructors at Master’s- and Doctorate-granting institutions in the United States to investigate teaching practices, to identify beliefs and contextual factors that support/constrain non-lecture teaching practices, and to identify commonalities and differences between those who do and do not lecture. This work provides insight into how abstract algebra is taught in the United States, factors that influence pedagogical decisions, and avenues for how to approach and better support those are interested in implementing non-lecture teaching approaches.

[1]  Philip J. Davis,et al.  The Mathematical Experience , 1982 .

[2]  William C. Kyle.  Editorial: The imperative to improve undergraduate education in science, mathematics, engineering, and technology , 1997 .

[3]  T. Fukawa-Connelly,et al.  Analyzing the teaching of advanced mathematics courses via the enacted example space , 2014 .

[4]  Sean Larsen,et al.  Designing and scaling up an innovation in abstract algebra , 2013 .

[5]  Natasha M. Speer,et al.  Mathematics Teaching Assistant Preparation And Development , 2005 .

[6]  R. Sitgreaves.  Psychometric theory (2nd ed.). , 1979 .

[7]  Michelle K. Smith,et al.  Active learning increases student performance in science, engineering, and mathematics , 2014, Proceedings of the National Academy of Sciences.

[8]  Stephanie Treffert-Thomas,et al.  Characterising the teaching of university mathematics: a case of linear algebra , 2009 .

[9]  Chris Rasmussen,et al.  Pedagogical Content Tools: Integrating Student Reasoning and Mathematics in Instruction , 2006 .

[10]  J. Calderhead.  Teachers’ beliefs and knowledge , 2019, Problem Solving for Teaching and Learning.

[11]  David R. Hershey.  America's Academic Future: A Report of the Presidential Young Investigator Colloquium on U.S. Engineering, Mathematics and Science Education for the Year 2010 and Beyond The National Science Foundation , 1992 .

[12]  Investigating students' levels of engagement with mathematics: critical events, motivations, and influences on behaviour , 2016 .

[13]  Barbara Jaworski,et al.  A Spectrum of Pedagogical Awareness for Undergraduate Mathematics: From "Tricks" to "Techniques" , 2005 .

[14]  Elena Nardi,et al.  Amongst Mathematicians: Teaching and Learning Mathematics at University Level , 2007 .

[15]  Tommy Dreyfus,et al.  Advanced Mathematical Thinking Processes , 2002 .

[16]  Edwin E. Moise,et al.  II. The Problem of Learning to Teach , 1975 .

[17]  Daniel Chazan,et al.  Studying the Practical Rationality of Mathematics Teaching: What Goes Into “Installing” a Theorem in Geometry? , 2011 .

[18]  Keith Weber.  Mathematicians’ perspectives on their pedagogical practice with respect to proof , 2012 .

[19]  Alan H. Schoenfeld,et al.  Toward a theory of teaching-in-context , 1998 .

[20]  M. Burgan.  In Defense of Lecturing , 2006 .

[21]  Estrella Johnson,et al.  Can Math Education Research Improve the Teaching of Abstract Algebra , 2016 .

[22]  John S. Caughman,et al.  Implementing inquiry-oriented curriculum: From the mathematicians’ perspective , 2013 .

[23]  H. Tunney,et al.  To Lecture or Not to Lecture , 1930 .

[24]  John Paul Cook,et al.  The emergence of algebraic structure: students come to understand units and zero-divisors , 2014 .

[25]  James Maxwell,et al.  Statistical Abstract of Undergraduate Programs in the Mathematical Sciences in the United States: Fall 2015 CBMS Survey , 2018 .

[26]  C. Rasmussen,et al.  It's about time: the relationships between coverage and instructional practices in college calculus , 2016 .

[27]  G. Edgar Parker,et al.  The Moore Method: A Pathway to Learner-Centered Instruction , 2009 .

[28]  Keith Weber,et al.  Factors mathematicians profess to consider when presenting pedagogical proofs , 2014 .

[29]  Keith Weber,et al.  Traditional instruction in advanced mathematics courses: a case study of one professor’s lectures and proofs in an introductory real analysis course , 2004 .

[30]  Melissa H. Dancy,et al.  Impact of physics education research on the teaching of introductory quantitative physics in the United States , 2009 .

[31]  Timothy Patrick Fukawa-Connelly,et al.  A case study of one instructor’s lecture-based teaching of proof in abstract algebra: making sense of her pedagogical moves , 2012 .

[32]  Uri Leron,et al.  Learning abstract algebra with ISETL , 1993 .

[33]  Natasha M. Speer,et al.  Beyond mathematical content knowledge: A mathematician's knowledge needed for teaching an inquiry-oriented differential equations course , 2007 .

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

[35]  F. B. Jones.  The Moore Method. , 1977 .

[36]  Keith Weber,et al.  Lectures in Advanced Mathematics: Why Students Might Not Understand What the Mathematics Professor Is Trying to Convey. , 2016 .

[37]  Melissa H. Dancy,et al.  Barriers to the use of research-based instructional strategies: The influence of both individual and situational characteristics , 2007 .

[38]  C. Henderson,et al.  Facilitating change in undergraduate STEM instructional practices: An analytic review of the literature , 2011 .