Assessing theoretical knowledge in real analysis

This seminar will be based on the invited lecture I gave at the conference "Didactics of Higher Mathematics as a scientific discipline", organised by KDHM at Schloss Herrenhausen, Hannover, Dec. 2015.

Common introductory courses on calculus and linear algebra often focus on computational techniques, and are given in a variety of educational programs, from engineering to business. The average calculus text book is dominated by explanations of calculation methods and (especially) worked examples; they reflect a focus on technical knowledge, with standard techniques to be mastered and applied in a variety of tasks. Students get credit on the course if they demonstrate such technical knowledge (e.g. on finding extrema for a functions of two variables), usually during written tests with exercises that are “similar” to worked examples or exercises from the course. Such tests can be graded with a high level of consistency, and while the educational value of skills such as the one mentioned is open to debate, the assessment practice is well aligned with common teaching practices in such courses. Students in pure mathematics may well take such “calculation oriented” courses at the beginning of their studies, but eventually they get more theoretical courses with titles like “abstract algebra”, “topology” and “real analysis”. In such courses, theoretical structures, built by definitions, theorems and proofs, form the core of the study material (textbooks, lecture notes etc.). There is little research on the common ways of assessing students’ work with theoretical structures, on their effects, and on possible alternatives (cf. e.g. Grønbæk, Misfeldt and Winsløw, 2009). In some institutions, closed-book written exams are reported to be the norm, with questions like “State and prove X’s theorem” (cf e.g. Conradie and Frith, 2000, 225). In other institutions, such as Danish universities, the common form of assessment for theoretical knowledge in mathematics is the oral exam, based on questions of the same type (questions drawn at random from a list known in advance, with some preparation time between drawing and the actual examination). The two forms share major potential drawbacks, including a disproportionate effort by students to memorize proofs, and difficulties to provide and practice transparent criteria for grading. To counter these challenges, alternative ideas have been proposed for written and oral examinations in theoretical mathematics, focusing on students’ proof comprehension (Cnop and Gransard, 1994; Conradie and Frith, 2000; Mejia-Ramos et al., 2012) or even proof production (Grønbæk and Winsløw, 2007). In this abstract, we outline a number of specific and interrelated challenges which we encountered while investigating and developing the alignment between student work and examination practices in a first course on real analysis at the University of Copenhagen.

References

Conradie, J. & Frith, J. (2000). Comprehension tests in mathematics. Educational Studies in Mathematics, 42, 225-235.
Cnop, I. & Grandsard, F. (1994). An open‐book exam for non‐mathematics majors. International Journal of Mathematical Education in Science and Technology, 25(1), 125-130.
Gravesen, K. (2015). Forskningslignende situationer på et førsteårskursus i matematisk analyse. M.Sc. dissertation, University of Copenhagen (in Danish).
Grønbæk, N., Misfeldt, M. & Winsløw, C. (2009). Assessment and contract-like relationships in undergraduate mathematics education. In O. Skovsmose et al. (eds), University science and Mathematics Education. Challenges and possibilities, pp. 85-108. New York: Springer Science.
Grønbæk, N. & Winsløw, C. (2007). Developing and assessing specific competencies in a first course on real analysis. In F. Hitt, G. Harel, & A. Selden (Eds.), Research in collegiate mathematics education VI, pp. 99-138. Providence, RI: American Mathematical Society.
Madsen, L. & Winsløw, C. (2009). Relations between teaching and research in physical geography and mathematics at research intensive universities. International Journal of Science and Mathematics, Education 7, 741-763.
Mejia-Ramos, J., Fuller, E., Weber, K., Rhoads, K. & Samkoff, A. (2012). An assessment model for proof comprehension in undergraduate mathematics. Educational Studies in Mathematics 79, 3-18.
Winsløw, C., Barquero, B., de Vleeschouwer, M., and Hardy, N. (2014). An institutional approach to university mathematics education : from dual vector spaces to questioning the world. Research in Mathematics Education 16(2), 91-111.