Elegance with Substance

First Edition, June 2009, ISBN 978 90 3610 138 7

€ 14.95


Mathematics and its education designed for Ladies and Gentlemen

What is wrong with mathematics education and how it can be righted

On the political economy of mathematics and its education 

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Ordering at Dutch University Press
(via Rozenberg Publishers)

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Conquest of the Plane is an implementation of Elegance with Substance

(Listed in the AMS Book List, Notices Vol 58, No 11, p1474.)




Cover text

National parliaments around the world are advised to each have their own national parliamentary enquiry into the education in mathematics and into what is called 'mathematics'. Current mathematics namely fails and causes extreme social costs. 

The failure in mathematics and math education can be traced to a deep rooted tradition and culture in mathematics itself. Mathematicians are trained for abstract theory but when they teach then they meet with real life pupils and students. Didactics requires a mindset that is sensitive to empirical observation which is not what mathematicians are basically trained for. 

When mathematicians deal with empirical issues then problems arise in general. The stock market crash in 2008 was caused by many factors, including mismanagement by bank managers and failing regulation, but also by mathematicians and 'rocket scientists' mistaking abstract models for reality (Mandelbrot & Taleb 2009). Another failure arises in the modelling of the economics of the environment where an influx of mathematical approaches causes too much emphasis on elegant form and easy notions of risk but insufficient attention to reality, statistics and real risk (Tinbergen & Hueting 1991). Errors by mathematicians on realistic assumptions have important consequences for civic discourse and democracy as well (Colignatus 2007). The failure in math education is only one example in a whole range.

The discussion of mathematics in this book can be understood by anyone with a decent command of highschool mathematics. While school math should be clear and didactically effective, a closer look shows that it is cumbersome and illogical. (1) This is illustrated with some twenty examples from a larger stock of potential topics. (2) Additional shopping lists for improvement on both content and didactic method can be formulated as well. (3) Improvements appear possible with respect to mathematics itself, on logic, voting theory, trigonometry and calculus. (4) What is called mathematics thus is not really mathematics. Pupils and students are psychologically tortured and withheld from proper mathematical insight and competence. Other subjects, like the education in economics, biology or physics, suffer as well. 

Application of economic theory helps us to understand that markets for education and ideas tend to be characterized by monopolistic competition and natural monopolies. Regulations then are important. Apparently the industry of mathematics education currently is not adequately regulated. The regulation of financial markets is a hot topic nowadays but the persistent failure of mathematics education should rather be high on the list as well. It will be important to let the industry become more open to society.

When you want to understand the underlying historical processes that cause the current state of the world then this is the book for you. Mathematics education must be tackled, both as a noble goal of itself and for the larger causes. 

Thomas Colignatus (1954) is an econometrician and teacher of mathematics.


      1. Review of "Elegance with Substance" and "Conquest of the Plane" by Richard Gill, January 2012, published in Nieuw Archief voor Wiskunde, 2012 no 1
      2. A child wants nice and not mean numbers, April 2012 (short version of longer Dutch collection)
      3. Original versions of Trig Rerigged and Proportions and fractions that are now dealt with integrally in Conquest of the Plane
      4. See also Towards Mathix / Math-x, a computer algebra language that can create its own operating system, and that is amazingly user-friendly and secure (PDF) or HTML version
      5. See also Review in 2009 of Richard Fateman's "Review of Mathematica" (1992) (2009)
      6. Brain research and mathematics education: some comments (2011)
      7. Contra Cantor Pro Occam, August 2011
      8. Contra Cantor Pro Occam - PCWA, March 2012
      9. PM. Another author: See H. Wu in the AMS Notices 2011 on the situation in the USA. I don't quite agree with the analysis but EwS is an answer.
      10. In fact, this paper Neoclassical mathematics for the schools is my article on EwS and Wu.
      11. My contact with ACME in the UK: (a) Letter to Dame Julia, (b) Her reply, (c) My discussion of the exchange.
      12. What a mathematician might wish to know about my work, March 26 2013
      13. Integrity of science in Dutch research in didactics of mathematics, july 16 2014
      14. Pierre van Hiele and David Tall: Getting the facts right, July 27 2014
      15. Education, division & derivative: Putting a Sky above a Field or a Meadow, September 8-11 2014
      16. An explanation for Wigner's "Unreasonable effectiveness of mathematics in the natural sciences", January 9 2015

Dutch (more Dutch texts on mathematics education)

      1. Nieuw boek: Een kind wil aardige en geen gemene getallen, mei 2012
      2. Reacties van Nederlandse wiskundigen op "Elegance with Substance" (in Dutch)
      3. Review van "Elegance with Substance": Tijdschrift voor het Economisch Onderwijs, Jaargang 2009, Nummer 4, p34 (though not listed on this website)
      4. Review van "Elegance with Substance": Euclides, blad van de Ned. Ver. voor Wiskundeleraren, december 2010, 86-3, p130-131 (though rather slanderous)
      5. "Review" van "Conquest of the Plane", Euclides, februari 2012, 87-4, p168-170 (though quite slanderous), Reactie op deze laatste "review", 9 februari 2012, Detailkritiek op deze laatste "review", 13 februari 2012

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