Elegance with Substance

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

€ 14.95  Ordering  -   Advised: Buy this in print and read it regularly

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 

Download the PDF from MPRA      

Download PDF from Zenodo


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


Check out the "Redesign of Mathematics Education" page at Zenodo

How to support this


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.


About the book

      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. See on the situation in the USA this other author: H. Wu in the AMS Notices. Wu wants involvement of research mathematicians, but those lack empirical experience, and we would rather see involvement by engineers. EwS is an answer.
      3. My contact with ACME in the UK: (a) Letter to Dame Julia, (b) Her reply, (c) My discussion of the exchange.
      4. What a mathematician might wish to know about my work, March 26 2013 - included in EWS en FMNAI
      5. Membership dues and Open Access Publishing, October 12, and Excel sheet, October 16 2016

New theory

      1. Education, division & derivative: Putting a Sky above a Field or a Meadow, September 8-11 2014, Zenodo 292244
      2. Draft paper: Sketch of influence by G. Mannoury (1867-1956) on his students P.M. van Hiele (1909-2010) and A.D. de Groot (1914-2006), and the continued importance of this influence for a general theory of epistemology (2015), didactics of mathematics (2008) and methodology of science (including the humanities) (1990), February 18 2016
      3. Some remarks w.r.t. Henk Boonstra's weblog, elementary school and math & arithmetic, March 16 2016
      4. An algebraic approach to the derivative, (old link), August 14 2016, NAW referee report October 3, my reply October 3 2016, weblog discussion, and the documentation of November 21 2016 on the breach in integrity of science. Updated version February 16 2017, Zenodo 292631.
      5. Trig Rerigged 2.0, draft, September 4 2016, Zenodo 292213 - summary in weblog
      6. Session C6. 1 Redesign and Re-engineering of Mathematics Education, workshop sheets, November 2 2016
      7. Meta-cognitive skills: Implementing Polya "How to solve it", November 15 2016, with explanatory chart (pdf, doc) and blank chart (pdf, doc)
      8. Algebraic approach to the derivative and continuity, November 30 2016, Zenodo 292253 
      9. Finally a sound interpretation for differentials, December 8-10, 2016
      10. A potential relation between the algebraic approach to calculus and rational functions, February 4 2017, Zenodo 292247
      11. Comparing two algebraic approaches to calculus: WIC Prelude and COTP, February 12 2017, Zenodo 292250   
      12. Comparing two approaches to calculus: "direct" and algebra, February 13 2017

Computer algebra

      1. 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
      2. Review in 2009 of Richard Fateman's "Review of Mathematica" (1992) (2009)
      3. Computer algebra is a revolution. But 21st century skills ? With protest against misrepresentation by Koeno Gravemeijer, December 5 2015

Scientific integrity in mathematics education research (MER)

      1. Integrity of science in Dutch research in didactics of mathematics, july 16 2014
      2. Pierre van Hiele and David Tall: Getting the facts right, July 27 2014, update August 2014
      3. Pierre van Hiele, David Tall and Hans Freudenthal: Getting the facts right, November 3 2015 (replacing the 2014 version)
      4. Letter to NRO, April 15 2016. (The national organisation for research in education), 2nd letter to NRO, July 12 2016
      5. Letter to VOR and directors of institutes for training of teachers, May 9 2016, 2nd letter to them, July 15 2016
      6. Letter to President KNAW and Directorate CPB, May 17 2016, with Supplement May 25 2016
      7. Email exchange in May 2016 with Kool, Noteboom and Tijdeman on KNAW 2009 on arithmetic education, May 25 2016
      8. Letter to board of NVvW, the Dutch association of mathematics teachers, June 28 2016
      9. Submission to the integrity commission at Leiden university, September 30 2016
      10. Preliminary evaluation of "mathematical think-activities" (MTA), October 31 2016
      11. Draft book BHRM

Scientific integrity in research mathematics, partly infringing upon MER

      1. Breach by Jan van de Craats and Ben Wilbrink w.r.t. scientific integrity, September 15 2015
      2. Labor omnia vicit, improbus et duris urgens in rebus egestas. Breach by H.C.M. de Swart, February 23 2016
      3. Labor omnia vicit, continued. Letter to F.A. Muller, March 2 2016
      4. Some experiences with four Dutch mathematicians: Their path of destruction, March 15 2016
      5. Labor omnia vicit, third document. Breach by F.A. Muller, March 18 2016
      6. No good deed goes unpunished, case of Jan Bergstra, May 6 2016
      7. Breach of integrity of science by the editors of Nieuw Archief voor Wiskunde (NAW) w.r.t. the 2016 article about the algebraic approach to the derivative, November 21 2016
      8. Letter to Mastermath: Your training programme for "Zij-instromers" for teacher of mathematics, January 8 2017

Dutch Parliament

      1. Letter to the House of Representatives: Onderwijs2032 and the letter of April 22 2016 by state secretary Sander Dekker on "Voorstel vervolg Onderwijs2032", his reference 927957, September 27 2016

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", door Alfons F. Mantel: 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). My protest against this "review" that the editors however did not print in the journal.
      5. Mention in NVOX, p351, journal for physics education, September 2011
      6. "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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