Munich Personal RePEc Archive Elegance with substance Colignatus, Thomas Samuel van Houten Genootschap 7 August 2015 Online at https://mpra.ub.uni-muenchen.de/66012/ MPRA Paper No. 66012, posted 11 Aug 2015 15:29 UTC ELEGANCE WITH SUBSTANCE 1 2 Elegance with Substance 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 Thomas Colignatus Samuel van Houten Genootschap 3 Colignatus is the preferred name of Thomas Cool in science. He is an econometrician (Groningen 1982) and teacher of mathematics (Leiden 2008). Lawful exceptions excluded, nothing of this publication may be copied or published by means of print, photocopy, microfilm, electronics or otherwise, unless a written permit has been given by the lawful owners of the copyright, and this also holds for whole or partial reworkings. The cover illustration has been taken from http://www.gutenberg.org/ebooks/24485, “A Floral Fantasy in an Old English Garden” by Walter Crane 184591,15. This illustration is for the use of anyone anywhere at no cost and with al ost no restrictions whatsoever: you ay copy it1 give it away or re9use it under the ter s of the Project Gutenberg License at www.gutenberg.net 2nd edition1 August 7 2015 > T.H.A.M. Cool http344tho ascool.eu1 cool at dataweb.nl Supported by Sa uel van Houten Genootschap Wetenschappelij2 bureau van het Sociaal Liberaal 7oru Scientific bureau of the Social Liberal 7oru The Hague ISBN ,789,4625404593 Published by MijnBestseller.nl Journal of Econo ic Literature3 JEL P16 Political econo y I20 General education A20 Econo ics education A erican Mathe atical Society3 MSC2010 ,7B10 Educational research and planning 00A35 Methodology and didactics Nederlandstalige Unifor e Aubrie2sindeling3 NUA 841 Onderwijs2unde ,18 Wis2unde alge een 780 Econo ie en bedrijf alge een (Onderwijs van wis2unde en beheer ervan1 Political Econo y. 4 Prologue to the 1st edition in 200, Our pupils and students are best treated as ladies and gentle en with elegance and substance. Providing the with eBual athe atics is our uch valued objective. Ideally athe atics would be perfect and unchanging and just be there to be discovered. Mathe atics however is as uch art as discovery. It is ade. It is a creation1 in the way that cave en carved their scores in bones and that we create virtual reality with superco puters. In the interaction between what we do and what we understand al ost all of the weight is on what we do1 which then i prints on our ind. It appears tedious and hard wor2 to go a bit in the reverse direction1 to even get where we are now1 let alone develop a notion of perfection. Given this fragile and historic nature of athe atics it should not co e as a surprise that what we currently call athe atics actually appears1 on close inspection1 to be often cu berso e or even outright irrational. Clarity and understanding are freBuently bloc2ed by contradictions and nonsense that are internal to current athe atics itself. Who has a proble astering athe atics should not be surprised. Over the years1 while teaching athe atics and writing y notes that now result in these pages1 there were any o ents that I felt frustrated and at ti es even Buite annoyed about the straightjac2et of current athe atical conventions. One is supposed to teach athe atics but it is precisely the te0tboo2 that bloc2s this prospect. 7or any pupils and students the goal is i possible fro the outset not because of their lac2 of capability but because of aw2ward conventions that only ca e about in a historical process. The flip side is that this is a Garden of Eden for didactic develop ent. What is aw2ward can be ha ered into so ething elegant. What is irrational can be turned rational and consistent. What is dar2 and nonsensical can be thrown out and replaced by clarity. There is beauty and satisfaction in redesign. This didactic reconsideration also changes what we call C athe aticsD. The interaction between what we do and what we understand shifts to a new eBuilibriu 1 a higher opti u at a ore agreeable level for both students and teacher. It will still be athe atics since it can be recogniEed as athe atics. It will be stronger and ore efficient athe atics too but it will no longer be the old one. The criterion for change lies in elegance with substance. Elegance without substance creates a dandy. Elegance ought to signal substance. Mathe atics concentrates on the elegance and specific fields of study li2e econo ics concentrate on the substance. But athe atics needs to have so e substance of itself too. The criterion is tric2y since so e people see it in the present athe atical conventions too1 where aw2wardness A plus aw2wardness B gives inconsistency C. However1 we will co pare the old ways with the suggestions of the new ways and let the criterion spea2 for itself. This should open so e eyes. Otherwise we just stay in the Garden of Eden. Which leaves e to than2 y own teachers and colleagues who trained and helped e in the old ways. A redesign starts fro so ething and when the old is replaced then this i plies that it was valuable to start with. I than2 in particular y pupils and students for what they taught e. 5 Prologue to the 2nd edition in 2015 This 2nd edition is al ost the sa e as the 1st edition. Elegance with Substance (EWS. has a strong structure1 a wealth of infor ation1 and not too any pages. It argues its case succinctly. This clarity will be lost when new issues are included. The results fro 200, generally havenFt changed. When develop ents in 200,92015 supple ent so e of the results then a footnote e0plains where ore can be found. Major new develop ents in 200,92015 have been3 • Conquest of the Plane (COTP. (2011c. presents a proof of concept for EWS and creates a primer for a re9engineered course in athe atics for highschool or first year of higher education. Gill (2012. reviewed both EWS and COTP favourably and advised to read with an open ind. At the review site of the European Mathe atical Society1 Ga boa (2011. states about COTP3 GOnce the reader beco es fa iliar with the notations and the author style1 he4she will enjoy the boo2. I a convinced that this wor2 will help the students to recogniEe what they should 2now but they ignore1 (....G • Foundations of Mathematics. A Neoclassical Approach to Infinity (7MNAI. (2015f. argues the case for having set theory and nu ber theory in the highschool progra e. A ajor deduction is that the transfinites by Georg Cantor are based upon ill9defined constructs1 co parable to AussellFs parado01 so that these can be eli inated. As a result the theory for the classroo is not as co ple0 as co only perceived. The e o What a mathematician might wish to know about my work (2013. has been included on page 125 in the Appendices. The Introduction already e0plains about y bac2ground in econo etrics and teaching athe atics1 but this e o e phasiEes again that y interest is not abstraction for the purpose of abstraction itself1 but to create scope for better ent of society. Obviously1 society will not be i proved when an applied theory is unsound1 and thus there is a role for athe atics1 but here the sounding board is not only abstraction itself but also e pirical science. 6 Contents in brief I. Introduction 13 II. Issues of notation 21 III. Opaque or confusing terms 35 IV. Breaking the chain of understanding 41 V. Like the stepmother in the fairy tale 47 VI. Redesigning mathematics itself 61 VII. Questions for evidence based education 69 VIII. Re-engineering the industry 73 I). Beating the software jung e 75 ). ,ithout time, no mora ity .1 )I. The derivative is a gebra .9 )II. 0eoc assica mathematics for the schoo s 97 )III. 1 key insight in the didactics of mathematics 103 )IV. Residua comments 109 )V. Conc usions 115 Epi ogue 117 1ppendices 121 7 Contents I. Introduction.................................................................................................... 13 1. Natural li itations to a noble art.............................................................................. 13 2. As far as the ind can reach .................................................................................. 13 7or the record ................................................................................................... 13 Definition of econo etrics ................................................................................ 14 Bea s of light (through a glass dar2ly.............................................................. 15 Understanding the ain advice ........................................................................ 15 3. An art and an industry............................................................................................. 16 The approach in this boo2 ................................................................................ 16 The industry...................................................................................................... 17 I ple enting change ....................................................................................... 18 4. Li itations to this study........................................................................................... 18 Setting1 e0perience1 anecdote1 bias.................................................................. 18 Li itations e0ist internationally ......................................................................... 1, 5. The order of discussion..........................................................................................