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

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

[3]  Chris Sangwin,et al.  An analysis of undergraduate core material in the light of hand-held computer algebra systems , 2003 .

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

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

[6]  J. Davenport,et al.  A Comparison of Equality in Computer Algebra and Correctness in Mathematical Pedagogy , 2009, Calculemus/MKM.

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

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

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

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

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

[12]  John Seely Brown,et al.  Diagnostic Models for Procedural Bugs in Basic Mathematical Skills , 1978, Cogn. Sci..

[13]  第六高等学校,et al.  Examples in Algebra , 1902, Nature.

[14]  Richard M. Young,et al.  Errors in Children's Subtraction , 1981, Cogn. Sci..

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

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

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

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

[19]  S. Klai,et al.  Using Maple and the Web to grade mathematics tests , 2000, Proceedings International Workshop on Advanced Learning Technologies. IWALT 2000. Advanced Learning Technology: Design and Development Issues.

[20]  Bruce R Miller.  An expression formatter for MACSYMA , 1995 .

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

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

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

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

[25]  José Meseguer,et al.  Order-Sorted Algebra I: Equational Deduction for Multiple Inheritance, Overloading, Exceptions and Partial Operations , 1992, Theor. Comput. Sci..

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

[27]  Mirina Grosz,et al.  World Wide Web Consortium , 2010 .

[28]  S. R. Czapor,et al.  Computer Algebra , 1983, Computing Supplementa.

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

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

[31]  Johan Jeuring,et al.  Canonical Forms in Interactive Exercise Assistants , 2009, Calculemus/MKM.

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

[33]  Stephen M. Watt,et al.  Mathematical Markup Language (MathML) Version 3.0 , 2001, WWW 2001.

[34]  J. Calmet.  Computer Algebra , 1982 .