# The Assessment Regime and Criteria for the Award of the MWTC

Warwick Teaching Certificate (Mathematics Variant)

Reading List

As teachers, we are all expected to think about how people learn mathematics and how we can influence the process. It is part of our professional responsibility to question what we do and to become more effective at doing it. The reading matter suggested below is meant to stimulate both your theoretical and practical thoughts on the subject.

Books

- The second edition of Steven Krantz’s book How to Teach Mathematics (American Mathematical Society 1999, ISBN 0821813986).
*I like this opinionated book. Krantz is a real mathematician and is not shy about giving his personal take on the nuts and bolts of lecturing. At the end of the second edition Krantz is big enough to include critical counter-views from 10 colleagues.* - John Mason's
*Mathematics Teaching Practice: a guide for university and college lecturers*(Albion/Horwood Publishing House 2002, ISBN 1898563799).*Mason is also a mathematician by training and usually has some thoughtful and helpful things to say about mathematics education and teaching. Also he has recently co-edited the following book, a copy of which has been ordered for the Institute Library.* *Fundamental Constructs in Mathematics Education*edited by John Mason and Sue Johnston-Wilder. (RoutledgeFalmer: Open University, 2004, ISBN0415326974 (hardback) and 0415326982 (paperback)).- Derek Holton's edited collection
*The Teaching and Learning of Mathematics at University*

Level (Kluwer Academic, ISBN 0792371917) is worth a look -- see review at - A more general introduction to teaching in Higher Education can be found in Preparing to Teach by Graham Gibbs and Trevor Habeshaw. It is published by Technical and Educational Services Ltd, Fifth Reprint 2001, ISBN 0947885560 and copies are given to all participants of the generic version of the WTC.

Articles

The articles on this list have been chosen from suggestions made by colleagues who work in mathematics education in response to my request that they should be accessible and of interest to mathematicians as well as educationalists. Feedback on the value of the articles would be greatly welcomed by me,

Trevor Hawkes, (), or

David Mond ().

It is my intention to put copies of those articles that are not locally available in a folder in the Maths Education section (QA11) of the Institute Library.

- Alcock, Lara: 2004, “Mathematicians’ Perspectives on the Teaching and Learning of Proof” (Preprint accepted for publication in Research in Collegiate Mathematics Education.)
*This article is based on interviews with mathematicians and has some entertaining anecdotes, but also some collective insight into what students actually need to learn in order to become good at proving.* - Alcock, Lara: 2004, “Proof Validation in Real Analysis: Inferring and Checking Warrants” (Preprint submitted to the Journal of Mathematical Behavior.)
*A paper about what students need to do in order to read proofs - the data is from Warwick.* - Alcock, L.J. & Simpson, A.P., 2001, “The Warwick Analysis Project: practice and theory” in D. Holton, (Ed.),
*The teaching and learning of mathematics at university level*, Kluwer, Dordrecht:99–111. - Dorier, J.-L., Robert, A. and Rogalski, M.: 2003, “Some comments on “The Role of Proof in Comprehending and Teaching Elementary Linear Algebra‚”” by F. Uhlig”” Educ. Studies in Math.51, 185–191.

The two previous articles give quite different views of linear algebra teaching (one relatively traditional and from a mathematician’s perspective, one less traditional and from a maths education perspective). - Edwards, B.S. & Ward, M.B. (2004) "Surprises from mathematics education research: student (mis)use of definitions", American Mathematical Monthly111, 411-424.
- Epp, Susanna S.: 2003, “The Role of Logic in Teaching Proof”, American Mathematical Monthly 110, 886–899.
- Kitcher, Philip: 1984
*The nature of mathematical knowledge.*Oxford University Press, ISBN: 0195035410. - Leron, U. and Dubinsky, E.: 1995, “An abstract algebra story”, American Mathematical Monthly 102, 227–242.
*This is aimed at mathematicians and focuses on some of the problems with teaching some abstract algebra and gives an alternative approach based on a rather old package called ISETL. Interesting nonetheless. There are more interesting newer papers on abstract algebra from the same research tradition, but they’re less accessible.* - Nardi, Elena: 1998 “The Novice Mathematician’s Encounter with Formal Mathematical Reasoning: A SAMPLE OF FINDINGS FROM A CROSS-TOPICAL STUDY.” TaLUM Newsletter No 9, Published by the Mathematics Association.
- Thurston, William P.: 1994 “On Proof and Progress in Mathematics”, Bulletin of the AMS 30(2),161-177.
- Uhlig, F.: 2002, “The role of proof in comprehending and teaching elementary linear algebra”, Educ. Studies in Math. 50, 335-346.
- Weber, K. 2001, “Student difficulty in constructing proofs: The need for strategic knowledge.” Educational Studies in Mathematics, 48(1), 101-119.
*A nice paper on two approaches to proving in an abstract algebra context. Not too much eduspeak, but pretty strong methodologically.* - Wigner, Eugene: 1960 “The unreasonable effectiveness of mathematics in the natural sciences”,
*Communications in Pure and Applied Mathematics*, 13(1), 1-14.

**Web Collections **

- The
*Learning and Teaching Support Network*(LTSN) for Mathematics, Statistics, and Operational Research (MSOR) has a searchable database of reviewed educational research. It was prepared by Adrian Simpson (of Warwick) and Lara Alcock, who have annotated the entries with brief helpful comments You can search the databse by author, by title/keywords, or choose between three useful categories defined by level and approach. It can be found at

- David Tall was a student of Michael Atiyah’s and began his Warwick career in the Mathematics Department. When his research interests moved towards education, he joined the Warwick Institute of Education and gradually established an international reputation as an original thinker in the field of mathematics education. A collection of his research papers, many of them PDF downloads, can be found at

- The educational issues and concerns of physicists are often close to those of mathematicians. For this reason, you might like to peruse the
*Annotated Bibliography of Research into the Teaching and Learning of the Physical Sciences at the Higher Education Level,*prepared by David Palmer for the LTSN in Physical Sciences. It can be downloaded from

Trevor Hawkes

January 2005