A Comparison of Equality in Computer Algebra and Correctness in Mathematical Pedagogy

How do we recognize when an answer is “right”? This is a question that has bedevilled the use of computer systems in mathematics (as opposed to arithmetic) ever since their introduction. A computer system can certainly say that some answers are definitely wrong, in the sense that they are provably not an answer to the question posed. However, an answer can be mathematically right without being pedagogically right. Here we explore the differences and show that, despite the apparent distinction, it is possible to make many of the differences amenable to formal treatment, by asking “under which congruence is the pupil’s answer equal to the teacher’s?”.

[1]  James H. Davenport,et al.  Unit Knowledge Management , 2008, AISC/MKM/Calculemus.

[2]  James H. Davenport,et al.  The Freedom to Extend OpenMath and its Utility , 2008, Math. Comput. Sci..

[3]  James W. Thatcher,et al.  Data Type Specification: Parameterization and the Power of Specification Techniques , 1982, TOPL.

[4]  Richard D. Jenks,et al.  AXIOM: the scientific computation system , 1992 .

[5]  Cliff E. Beevers,et al.  Incorporating partial credit in computer-aided assessment of Mathematics in secondary education , 2006, Br. J. Educ. Technol..

[6]  A. Kluger,et al.  The effects of feedback interventions on performance: A historical review, a meta-analysis, and a preliminary feedback intervention theory. , 1996 .

[7]  C. J. Sangwin.  What is a mathematical question ? , 2007 .

[8]  Jean-François Nicaud,et al.  1 APLUSIX , a learning environment for algebra , actual use and benefits , 2005 .

[9]  Neil P. Strickland.  Alice Interactive Mathematics , 2002 .

[10]  Bruce W. Char,et al.  The design of maple: A compact, portable and powerful computer algebra system , 1983, EUROCAL.

[11]  James H. Davenport,et al.  Towards better simplification of elementary functions , 2002, ISSAC '02.

[12]  Henk Barendregt,et al.  The Lambda Calculus: Its Syntax and Semantics , 1985 .

[13]  脇 克志.  An Introduction to MAGMA , 1995 .

[14]  Ludovít Molnár,et al.  Analytical differentiation on a digital computer , 1970, Kybernetika.

[15]  T. L. H.,et al.  A History of Mathematical Notations , 1929, Nature.

[16]  C. Tuckey.  Examples in Algebra , 2008 .

[17]  Christopher J. Sangwin,et al.  Linear syntax for communicating elementary mathematics , 2007, J. Symb. Comput..

[18]  Chris Sangwin,et al.  STACK: making many fine judgements rapidly , 2007 .

[19]  Nicolas J. Doye.  Automated coercion for axiom , 1999, ISSAC '99.

[20]  Jacques Carette,et al.  Understanding expression simplification , 2004, ISSAC '04.

[21]  James H. Davenport.  Equality in Computer Algebra and Beyond , 2002, J. Symb. Comput..

[22]  Daniel Richardson,et al.  Some undecidable problems involving elementary functions of a real variable , 1969, Journal of Symbolic Logic.

[23]  Tim O'Shea,et al.  A self-improving quadratic tutor , 1979 .