Elegance with Substance

2nd Edition, August 2015, ISBN 978-946254045-3

€ 14.95  Ordering

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 

2nd edition soon


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Companion books are:

(1) Conquest of the Plane is a "proof of concept" of Elegance with Substance

(2) Foundations of Mathematics. A Neoclassical Approach to Infinity shows that it is feasible to treat set theory and number theory in highschool (if not earlier)

(3) A child wants nice and no mean numbers on mathematics in primary education

(4) A Logic of Exceptions gives the underlying approach in logic


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 (DeLong 1991, 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. 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.
      3. 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
      4. See also Review in 2009 of Richard Fateman's "Review of Mathematica" (1992) (2009)
      5. My contact with ACME in the UK: (a) Letter to Dame Julia, (b) Her reply, (c) My discussion of the exchange.
      6. What a mathematician might wish to know about my work, March 26 2013 - included in EWS en FMNAI
      7. Integrity of science in Dutch research in didactics of mathematics, july 16 2014
      8. Pierre van Hiele and David Tall: Getting the facts right, July 27 2014, update August 2014
      9. Education, division & derivative: Putting a Sky above a Field or a Meadow, September 8-11 2014
      10. Breach by Jan van de Craats and Ben Wilbrink w.r.t. scientific integrity, September 15 2015
      11. Pierre van Hiele, David Tall and Hans Freudenthal: Getting the facts right, November 3 2015 (replacing the 2014 version)

Dutch (more Dutch texts on mathematics education)

      1. 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
First Edition of EWS, June 2009, ISBN 978 90 3610 138 7 
Dutch University Press (via Rozenberg Publishers)
Download the PDF from MPRA (2009)   
Listed in the AMS Book List, Notices Vol 58, No 11, p1474.

English - legacy papers
  1. Legacy versions of Trig Rerigged and Proportions and fractions that are now dealt with integrally in Conquest of the Plane
  2. Legacy: Brain research and mathematics education: some comments (2011) - see now FMNAI (2015)
  3. Legacy: Contra Cantor Pro Occam, August 2011 - here still with some comments on non-standard analysis and history
  4. Legacy: Contra Cantor Pro Occam - PCWA, March 2012  - see now FMNAI (2015)
  5. Legacy paper Neoclassical mathematics for the schools (2011) is my article on EwS and Wu - see now FMNAI (2015)
  6. Legacy: An explanation for Wigner's "Unreasonable effectiveness of mathematics in the natural sciences", January 9 2015 - see now FMNAI (2015)

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