2020-04-04T22:34:05Z
https://zenodo.org/oai2d
oai:zenodo.org:1241405
2020-01-25T07:27:25Z
software
user-re-engineering-math-ed
Colignatus, Thomas
2018-05-05
This notebook presents tables for addition and subtraction that have a better use of the place value system. The method is already used in Holland for addition in levels but this notebook extends for addition in differences and subtraction in levels and differences. The method is only intended for an intermediate stage in teaching before addition and subtraction are mentally fully automated. For example 99 + 21 can be added per digit position. To keep digits in the range [0, 9] we remove underflow or overflow. We work from right to left, since the numbers come from India and Arabia. Then we get 99 + 21 -> {9, 9} + {2, 1} = {11, 10} = {11, 10} + {1, -10} = {12, 0} = {0, 12, 0} + {1, -10, 0} = {1, 2, 0} -> 120. Compared to existing methods: (1) This method does not change the original sum. (2) The workflow is into a single direction. (3) Allowing positions to have values outside the [0 , 9] range focuses attention upon the place value. (4) There is a unity of approach to both addition and subtraction. The method fits within the US Common Core when we tell kids that a step with {1, -10} represents the subtraction 1 * 10 - 10 * 1 = 0. It would be more fundamental to adapt the curriculum for negative numbers though. Routines to create such tables for addition and subtraction are available in a package. Obviously pupils in elementary school must master the method by hand, but the package allows for examples and checking.
This is a Mathematica notebook that includes a package in an initialisation cell. See the PDF at https://zenodo.org/record/1241350
https://zenodo.org/record/1241405
10.5281/zenodo.1241405
oai:zenodo.org:1241405
eng
url:https://zenodo.org/record/1241350
url:https://zenodo.org/record/1241383
url:https://zenodo.org/record/291974
doi:10.5281/zenodo.1241404
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
mathematics education
elementary school
Mathematica
Wolfram language
Tables for addition and subtraction with better use of the place value system
info:eu-repo/semantics/other
software
oai:zenodo.org:292213
2020-01-20T17:02:01Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2016-09-07
Trigonometry can be re-engineered by taking the plane itself as unit of measurement. The angular circle has circumference 1. There are variables {X, Y} on the unit circle such that X^2 + Y^2 = 1. The functions xur and yur relate the angles on the angular circle to the points {X, Y}. This notion was in "Trig rerigged" in 2008 and was adopted in the books "Elegance with Substance" 2009 and "Conquest of the Plane" 2011. The update in 2016 concerns the disk with the area rather than the circumference. This is still a draft text but gives a relevant overview.
This is still a draft text but gives a relevant overview.
https://zenodo.org/record/292213
10.5281/zenodo.292213
oai:zenodo.org:292213
doi:10.5281/zenodo.291974
doi:10.5281/zenodo.291972
doi:10.5281/zenodo.291979
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
trigonometry, education, didactics, mathematics, angle, circle, disk, hook, xur, yur, tur, sin, cos, tan, archi, pi
Trig Rerigged 2.0
info:eu-repo/semantics/workingPaper
publication-workingpaper
oai:zenodo.org:292244
2020-01-20T14:52:08Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2014-09-23
The sky is a tentative suggestion for extension group theory, that has supervariables with flexible domains, that would allow the formulation of dynamic rational functions, such that division by zero is prevented by manipulation of the domain. This would allow an algebraic approach to the derivative. Meadows are no alternative to such an approach. The world needs Academic Schools in which teaching is merged with empirical research on didactics.
See 10.5281/zenodo.292247 "A potential relation between the algebraic approach to calculus and rational functions", February 2017, for an improved understanding of the notion of "rational function". This new paper on rational functions seems to require an edit of the paper on the "sky".
https://zenodo.org/record/292244
10.5281/zenodo.292244
oai:zenodo.org:292244
doi:10.5281/zenodo.291972
doi:10.5281/zenodo.291848
doi:10.5281/zenodo.292247
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
division, quotient, dynamic quotient, simplify, algebraic approach to the derivative, calculus, supervariables, field, meadow, division by zero, abstract data types, computer algebra, rational functions, Pierre van Hiele, levels of abstraction, didactics of mathematics, mathematics education research, research mathematics, academic school
Education, division & derivative: Putting a Sky above a Field or a Meadow
info:eu-repo/semantics/workingPaper
publication-workingpaper
oai:zenodo.org:292631
2020-01-20T14:29:52Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2017-02-16
There is a new approach to the derivative, an algebraic one, different from using infinitesimals or limits. The approach originated in research on mathematics education and has been developed to the stage that it can be tested there. For mathematics research there is scope for further development of foundations. The paper shortly reviews the approach and the literature about it.
Related website: http://thomascool.eu/Papers/Math/Index.html
https://zenodo.org/record/292631
10.5281/zenodo.292631
oai:zenodo.org:292631
doi:10.5281/zenodo.291974
doi:10.5281/zenodo.291972
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
algebraic approach to derivative, mathematics eduction, didactics, re-engineering, calculus, algebra, analytic geometry, differentials
An algebraic approach to the derivative
info:eu-repo/semantics/workingPaper
publication-workingpaper
oai:zenodo.org:1241383
2020-01-25T19:22:58Z
software
user-re-engineering-math-ed
Colignatus, Thomas
2018-05-23
H = -1 is an universal constant. H represents a half turn along a circle, like complex number i represents a quarter turn. Kids know what it is to turn around and walk back along the same path. H creates the additive inverse with x + H x = 0 and the multiplicative inverse with x x^H = 1 for x != 0. Pronounce H as "ehta" or "symbolic negative one". The choice of H is well-considered: its shape reminds of -1 and even more (-1). Pierre van Hiele (1909-2010) already proposed to use y x^-1 and drop the fraction bar y / x with its needless complexity. Students must learn exponents anyway. The negative exponent might confuse pupils to think that they must subtract something, but the use of an algebraic symbol clinches the proposal. Also 5/2 can be written as 2 + 2^H, so that it is clearer where it is on the number line. This approach also causes a re-evaluation of the didactics of the negative numbers. The US Common Core has them only in Grade 6 which is remarkably late. The negative numbers arise from the positive axis x by rotating or alternatively mirroring into H x. Algebraic thinking starts with the rules that a + H a can be replaced by 0 and that H H can be replaced by 1. Subtraction a - b >= 0 may be extended into a - b < 0 with its present didactics, e.g. 2 - 5 = 2 - (2 + 3) = 2 - 2 - 3 = 0 - 3 = -3, but there is an intermediate stage with familiar addition 2 + 5 H = 2 + (2 + 3) H = 2 + 2 H + 3 H = 0 + 3 H = 3 H, that does not require (i) the switch at the brackets from plus to minus and (ii) the transformation of binary 0 - 3 to number -3. The expression a - (-b) involves (scalar) multiplication which indicates why pupils find this hard, and a + H H b is clearer. The use of H would affect the whole curriculum. There appears to be a remarkable incoherence in mathematics education and its research w.r.t. the negative numbers, which reminds of the problems that the world itself had since the discovery of direction by Albert Girard in 1629 and the introduction of the number line by John Wallis in 1673. This notebook provides a package to support the use of H in Mathematica. The notebook and package are intended for researchers, teachers and (Common Core) educators in mathematics education. Pupils in elementary school would work with pencil and paper of course.
This page gives the notebook with packages. The PDF is here: https://zenodo.org/record/1251687. This notebook with packages is also available at (same link, major update of notebook and packages): https://www.wolframcloud.com/objects/thomas-cool/MathEd/2018-04-02-Arithmetic-with-H.nb and http://community.wolfram.com/groups/-/m/t/1313302
https://zenodo.org/record/1241383
10.5281/zenodo.1241383
oai:zenodo.org:1241383
eng
url:https://zenodo.org/record/1251687
url:https://zenodo.org/record/291979
url:https://zenodo.org/record/291974
doi:10.5281/zenodo.1241382
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
mathematics education
kindergarten
elementary school
highschool
Common Core
H = -1
negative number
fraction
mixed number
power
exponent
Mathematica
Wolfram language
programming
package
MSC2010: 97M70 Mathematics education. Behavioral and social sciences
additive inverse
multiplicative inverse
Arithmetic with H = -1: subtraction, negative numbers, division, rationals and mixed numbers
info:eu-repo/semantics/other
software
oai:zenodo.org:1244009
2020-01-25T07:26:29Z
software
user-re-engineering-math-ed
Colignatus, Thomas
2018-05-09
Kids in kindergarten live in a world of sounds, so that the pronunciation of numbers is important. When they start learning to read and write, the co-ordination of (i) sounds, (ii) words and (iii) numerals is important. We already have the place value system fully in the numerals but not yet in written words and pronunciation. It appears that we can provide for a pronunciation of the integers with the full use of the place value system. The definitions should become an ISO standard, though the notebook and package still are quite simple. This notebook and package provide an implementation for English, German, French, Dutch and Danish.
The package is in the initialisation cell in the notebook. See the PDF at https://zenodo.org/record/1244064. This notebook with package is also available at: http://community.wolfram.com/web/cool
https://zenodo.org/record/1244009
10.5281/zenodo.1244009
oai:zenodo.org:1244009
eng
url:https://zenodo.org/record/292984
url:https://zenodo.org/record/291979
url:https://zenodo.org/record/1244064
doi:10.5281/zenodo.1244008
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
mathematics education
place value system
pronunciation
Common Core
Mathematica
Wolfram language
programming
package
97M70 Mathematics education. Behavioral and social sciences
Pronunciation of the integers with full use of the place value system
info:eu-repo/semantics/other
software
oai:zenodo.org:291848
2020-01-20T17:23:54Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2011-03-15
A LOGIC OF EXCEPTIONS provides the concepts and tools for sound inference. Discussed are: (1) the basic elements: propositional operators, predicates and sets; (2) the basic notions: inference, syllogism, axiomatics, proof theory; (3) the basic extra’s: history, relation to the scientific method, the paradoxes. The new elements in the book are: (4) a logic of exceptions, solutions for those paradoxes, analysis of common errors in the literature, routines in Mathematica. The book is intended to be used in the first year of college or university. The last two chapters require a more advanced level that is worked up to.
Logic is used not only in science and mathematics but also in business and sometimes in politics and government. Logic and inference however can suffer from paradoxes such as the Liar paradox “This sentence is false” or the proof-theoretic variant by Gödel “This statement is not provable” or the Russell set paradox of “The catalogue of all catalogues that don’t mention themselves”. This book explains and solves those paradoxes, and thereby gives a clarity that was lacking up to now. The author proposes the new approach that a concept, such as the definition of truth or the notion of proof or the definition of a set, also reckons with the exceptions that may pertain to its very definition. The approach to keep exceptions in the back of one’s mind is a general sign of intelligence.
A quote from this book: “Since the Egyptians, mankind has been trying to solve the problem of bureaucracy. One frequent approach is the rule of law, say, that a supreme law-giver defines a rule that a bureaucracy must enforce. It is difficult for a law however to account for all kinds of exceptions that might be considered in its implementation. Ruthless enforcement might well destroy the very intentions of that law. Some bureaucrats might still opt for such enforcement merely to play it safe that nobody can say that they don't do their job. Decades may pass before such detrimental application is noticed and revised. There is the story of Catherine the Great regularly visiting a small park for a rest in the open air, so that they put a guard there; and some hundred years after her death somebody noticed that guarding that small park had become kind of silly. When both law-givers and bureaucrats grow more aware of some logic of exceptions then they might better deal with the contingencies of public management. It is a long shot to think so, of course, but in general it would help when people are not only aware of the rigour of a logical argument or rule but also of the possibility of some exception.”
You can benefit from this book also when you don’t have the software. However, with the software, you will have an interactive environment in which you can test the propositions in this book and your own deductions. The software is included in The Economics Pack - by the same author - which is an application of Mathematica, a system for doing mathematics with the computer. The Pack has users in many countries in the world. The Pack is available for Windows XP, Macintosh and Unix platforms and requires Mathematica 8.0.1 or later. It can be freely downloaded, but you need a licence to run it.
ISBN 978-90-804774-7-6. The book website is: http://thomascool.eu/Papers/ALOE/Index.html
https://zenodo.org/record/291848
10.5281/zenodo.291848
oai:zenodo.org:291848
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
logic, liar paradox, Gödeliar, three-valued logic, theory of levels, proof theory, Cantor diagonal argument, division by zero
A Logic of Exceptions
info:eu-repo/semantics/book
publication-book
oai:zenodo.org:1248953
2020-01-25T07:26:19Z
software
user-re-engineering-math-ed
Colignatus, Thomas
2018-05-17
Kids in kindergarten and Grade 1 live in a world of sounds so that the pronunciation of numbers is important. When they are learning to read and write, the co-ordination of (i) sounds, (ii) numerals and (iii) written words is important. We already have the place value system fully in the numerals but not yet in pronunciation and written words. The following provides for this. The definition should become an ISO standard, though the notebook with package is quite simple because of the nature of the issue. The notebook with package provides an implementation and transliteration for English, German, French, Dutch and Danish, while other languages might employ Mathematica's IntegerName and WordTranslation without transliteration. Four levels in the curriculum are recognised for which routines are provided: (1) sounds, codified by words, (2) learning the numerals, (3) advanced: numerals in blocks of three digits, such that 123456 = {1 hundred, 2 ten, 3} thousand & {4 hundred, 5 ten, 6}, with the comma pronounced as "&" too, and (4) accomplished: 123 thousand 456 pronounced in above place value manner. The traditional pronunciation has level -4.
The package is in the initialisation cell in the notebook. See the PDF at https://zenodo.org/record/1244063. This notebook with package is also available at: http://community.wolfram.com/web/community/groups/-/m/t/1334793
https://zenodo.org/record/1248953
10.5281/zenodo.1248953
oai:zenodo.org:1248953
eng
url:https://zenodo.org/record/774866
url:https://zenodo.org/record/291979
url:https://zenodo.org/record/1244063
doi:10.5281/zenodo.1244008
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
mathematics education
place value system
pronunciation
Common Core
Mathematica
Wolfram language
programming
package
MSC2010: 97M70 Mathematics education. Behavioral and social sciences
Pronunciation of the integers with full use of the place value system
info:eu-repo/semantics/other
software
oai:zenodo.org:1299784
2020-01-25T07:26:31Z
software
user-re-engineering-math-ed
Colignatus, Thomas
2018-06-28
Kids in kindergarten and Grade 1 live in a world of sounds so that the pronunciation of numbers is important. When they are learning to read and write, the co-ordination of (i) sounds, (ii) numerals and (iii) written words is important. We already have the place value system fully in the numerals but not yet in pronunciation and written words. The following provides for this. The definition should become an ISO standard, though the notebook with package is quite simple because of the nature of the issue. The notebook with package provides an implementation and transliteration for English, German, French, Dutch and Danish, while other languages might employ Mathematica's IntegerName and WordTranslation without transliteration. Four levels in the curriculum are recognised for which routines are provided: (1) sounds, codified by words, (2) learning the numerals, (3) advanced: numerals in blocks of three digits, such that 123456 = {1 hundred, 2 ten, 3} thousand & {4 hundred, 5 ten, 6}, with the comma pronounced as "&" too, and (4) accomplished: 123 thousand 456 pronounced in above place value manner. The traditional pronunciation has level -4.
The package is in the initialisation cell in the notebook. See the PDF at https://zenodo.org/record/1244063. See there also for version history. (An earlier version of this notebook with package is also available at: http://community.wolfram.com/web/community/groups/-/m/t/1334793 )
https://zenodo.org/record/1299784
10.5281/zenodo.1299784
oai:zenodo.org:1299784
eng
url:https://zenodo.org/record/774866
url:https://zenodo.org/record/291979
url:https://zenodo.org/record/1244063
doi:10.5281/zenodo.1244008
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
mathematics education
place value system
pronunciation
Common Core
Mathematica
Wolfram language
programming
package
MSC2010: 97M70 Mathematics education. Behavioral and social sciences
Pronunciation of the integers with full use of the place value system
info:eu-repo/semantics/other
software
oai:zenodo.org:1270381
2020-01-20T17:02:35Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2018-06-04
These are the sheets of a presentation on June 8 2018, at the conference of Dutch and Flemish political science. These sheets give an overview, and see the full article for precision: https://zenodo.org/record/1228640
The Politicologenetmaal 2018 has this link: https://politicologenetmaal.eu See the full article at https://zenodo.org/record/1228640
https://zenodo.org/record/1270381
10.5281/zenodo.1270381
oai:zenodo.org:1270381
eng
url:https://zenodo.org/record/1228640
doi:10.5281/zenodo.1270380
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
elementary statistics
correlation
cosine
sine-diagonal inequalty / disproportionality (SDID)
political science on votes and seats
redesign of statistics education
An overview of the elementary statistics of correlation, R-Squared, cosine, sine, Xur, Yur, and regression through the origin, with application to votes and seats for parliament (sheets)
info:eu-repo/semantics/lecture
presentation
oai:zenodo.org:292933
2020-01-20T14:54:27Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2016-11-05
Sheets of the presentation held at the Dutch Teachers of Mathematics Day, November 5 2016.
Some passages are in Dutch but most is in English, and the intended readership could follow the formulas.
Related website is: http://thomascool.eu/Papers/Math/Index.html
https://zenodo.org/record/292933
10.5281/zenodo.292933
oai:zenodo.org:292933
doi:10.5281/zenodo.291974
doi:10.5281/zenodo.291979
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
mathematics eduction, didactics, re-engineering, fractions, mixed fractions, negative numbers, H = -1, subtraction, decimal notation, place value system, Pierre van Hiele,
Session C6. 1 Redesign and Re-engineering of Mathematics Education
info:eu-repo/semantics/lecture
presentation
oai:zenodo.org:1269392
2020-01-20T17:38:53Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2018-06-04
These are the sheets of a presentation on June 8 2018, at the conference of Dutch and Flemish political science. These sheets give an overview, and see "Voting Theory for Democracy" (VTFD) for precision. Arrow's theorem is that four axioms would be reasonable and morally required each by themselves, but together they result into a contradiction. The deduction stands but the interpretation can be rejected. Arrow confuses voting and deciding. The axiom of "pairwise decision making" can be rejected - and Arrow's label "independence of irrelevant alternatives" is distractive. A method that many would find interesting is Borda Fixed Point.
This is an update of the sheets of my presentation on March 16 2001 for the Social Choice group in Tilburg, The Politicologenetmaal 2018 has this link: https://politicologenetmaal.eu See Voting Theory for Democracy at https://zenodo.org/record/291985
https://zenodo.org/record/1269392
10.5281/zenodo.1269392
oai:zenodo.org:1269392
eng
url:https://zenodo.org/record/291985
doi:10.5281/zenodo.1269391
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
Social Choice
Arrow's Impossibility Theorem
Borda Fixed Point
Cordorcet
Voting Theory for Democracy
The solution to Arrow's difficulty in social choice (sheets)
info:eu-repo/semantics/lecture
presentation
oai:zenodo.org:291974
2020-01-20T16:14:27Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2015-08-07
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.
ISBN 978-946254045-3. The book website is: http://thomascool.eu/Papers/Math/Index.html . MPRA website: https://mpra.ub.uni-muenchen.de/66012 . The book discusses mathematics education both w.r.t. didactics and the economics of the industry. The submission to Zenodo serves the community interested in didactics of mathematics. The submission to MPRA serves the community of economists of education.
https://zenodo.org/record/291974
10.5281/zenodo.291974
oai:zenodo.org:291974
doi:10.5281/zenodo.291972
doi:10.5281/zenodo.291848
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
education, didactics, re-engineering, mathematics, economics, school, college, university, training, skill, human capital policy, human development, computer algebra, textbook publishing, learning, teaching, calculus, fractions, dynamic quotient, trigonometry, voting theory, parliamentarian enquiry
Elegance with Substance
info:eu-repo/semantics/book
publication-book
oai:zenodo.org:1241350
2020-01-20T15:16:41Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2018-05-05
This notebook presents tables for addition and subtraction that have a better use of the place value system. The method is already used in Holland for addition in levels but this notebook extends for addition in differences and subtraction in levels and differences. The method is only intended for an intermediate stage in teaching before addition and subtraction are mentally fully automated. For example 99 + 21 can be added per digit position. To keep digits in the range [0, 9] we remove underflow or overflow. We work from right to left, since the numbers come from India and Arabia. Then we get 99 + 21 -> {9, 9} + {2, 1} = {11, 10} = {11, 10} + {1, -10} = {12, 0} = {0, 12, 0} + {1, -10, 0} = {1, 2, 0} -> 120. Compared to existing methods: (1) This method does not change the original sum. (2) The workflow is into a single direction. (3) Allowing positions to have values outside the [0 , 9] range focuses attention upon the place value. (4) There is a unity of approach to both addition and subtraction. The method fits within the US Common Core when we tell kids that a step with {1, -10} represents the subtraction 1 * 10 - 10 * 1 = 0. It would be more fundamental to adapt the curriculum for negative numbers though. Routines to create such tables for addition and subtraction are available in a package. Obviously pupils in elementary school must master the method by hand, but the package allows for examples and checking.
This page concerns the PDF. See the supplementary files there for the Mathematica notebook and package. This notebook with package is also available at: https://www.wolframcloud.com/objects/thomas-cool/MathEd/2018-04-02-AdditionTable-and-SubtractionTable.nb (same link but update notebook and package) and http://community.wolfram.com/groups/-/m/t/1313408 (though the latter displays the earlier version in html and a short community title)
https://zenodo.org/record/1241350
10.5281/zenodo.1241350
oai:zenodo.org:1241350
eng
url:https://zenodo.org/record/1241405
doi:10.5281/zenodo.1241349
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
mathematics education
addition
subtraction
place value system
negative numbers
Common Core State Standards
Mathematica
Wolfram language
programming
package
Tables for addition and subtraction with better use of the place value system
info:eu-repo/semantics/workingPaper
publication-workingpaper
oai:zenodo.org:1251687
2020-01-20T17:37:29Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2018-05-23
H = -1 is an universal constant. H represents a half turn along a circle, like complex number i represents a quarter turn. Kids know what it is to turn around and walk back along the same path. H creates the additive inverse with x + H x = 0 and the multiplicative inverse with x x^H = 1 for x != 0. Pronounce H as "ehta" or "symbolic negative one". The choice of H is well-considered: its shape reminds of -1 and even more (-1). Pierre van Hiele (1909-2010) already proposed to use y x^-1 and drop the fraction bar y / x with its needless complexity. Students must learn exponents anyway. The negative exponent might confuse pupils to think that they must subtract something, but the use of an algebraic symbol clinches the proposal. Also 5/2 can be written as 2 + 2^H, so that it is clearer where it is on the number line. This approach also causes a re-evaluation of the didactics of the negative numbers. The US Common Core has them only in Grade 6 which is remarkably late. The negative numbers arise from the positive axis x by rotating or alternatively mirroring into H x. Algebraic thinking starts with the rules that a + H a can be replaced by 0 and that H H can be replaced by 1. Subtraction a - b >= 0 may be extended into a - b < 0 with its present didactics, e.g. 2 - 5 = 2 - (2 + 3) = 2 - 2 - 3 = 0 - 3 = -3, but there is an intermediate stage with familiar addition 2 + 5 H = 2 + (2 + 3) H = 2 + 2 H + 3 H = 0 + 3 H = 3 H, that does not require (i) the switch at the brackets from plus to minus and (ii) the transformation of binary 0 - 3 to number -3. The expression a - (-b) involves (scalar) multiplication which indicates why pupils find this hard, and a + H H b is clearer. The use of H would affect the whole curriculum. There appears to be a remarkable incoherence in mathematics education and its research w.r.t. the negative numbers, which reminds of the problems that the world itself had since the discovery of direction by Albert Girard in 1629 and the introduction of the number line by John Wallis in 1673. This notebook provides a package to support the use of H in Mathematica. The notebook and package are intended for researchers, teachers and (Common Core) educators in mathematics education. Pupils in elementary school would work with pencil and paper of course.
This page shows the PDF. See this link for the Mathematica notebook with the packages: https://zenodo.org/record/1241383
https://zenodo.org/record/1251687
10.5281/zenodo.1251687
oai:zenodo.org:1251687
eng
url:https://zenodo.org/record/1241383
url:https://zenodo.org/record/291974
url:https://zenodo.org/record/291979
doi:10.5281/zenodo.1251686
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
mathematics education
kindergarten
elementary school
highschool
Common Core
H = -1
negative number
fraction
mixed number
power
exponent
additive inverse
multiplicative inverse
Mathematica
Wolfram language
programming
package
MSC2010: 97M70 Mathematics education. Behavioral and social sciences
Arithmetic with H = -1: subtraction, negative numbers, division, rationals and mixed numbers
info:eu-repo/semantics/workingPaper
publication-workingpaper
oai:zenodo.org:292931
2020-01-20T14:22:47Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2013-11-11
Sheets of the presentation held at the Dutch Teachers of Mathematics Day, November 9 2013.
NB. A video is here: http://youtu.be/gn_BKZaDa-o
Translated in English and recreated with video capture. The original talk was 30 minutes with discussion. The video does not contain that discussion but adds 15 minutes to the talk. My apologies for the sound recording (including its speed).
Related website: http://thomascool.eu/Papers/Math/Index.html
https://zenodo.org/record/292931
10.5281/zenodo.292931
oai:zenodo.org:292931
doi:10.5281/zenodo.291974
doi:10.5281/zenodo.291972
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
algebraic approach to derivative, mathematics eduction, didactics, re-engineering, calculus, algebra, analytic geometry
Session D5. The algebraic approach to the derivative
info:eu-repo/semantics/lecture
presentation
oai:zenodo.org:291982
2020-01-20T17:38:14Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2015-07-26
Foundations of Mathematics. A Neoclassical Approach to Infinity is for (1) students interested in methodology and the foundations of mathematics – e.g. studying physics, engineering, economics, psychology, thus a broad group who use mathematics – and (2) teachers of mathematics who are sympathetic to the idea of bringing set theory and number theory into mathematics education.
The book presents:
(A) Constructivism with Abstraction, as a methodology of science.
(B) Particulars about infinity and number theory, within foundations and set theory.
(C) Correction of errors within mathematics on (B), caused by neglect of (A).
Other readers are (3) research mathematicians, who would benefit from last correction, but who must mend for that they are not in the prime target groups.
Set theory and number theory would be important for a better educational programme:
(i) They greatly enhance competence and confidence.
(ii) They open up the mind to logical structure and calculation also in other subjects.
(iii) They are fundamental for learning and teaching themselves.
The axiomatic system for set theory ZFC is shown to be inconsistent. Mathematics has been in error since Cantor 1874 because of neglecting above methodology of science.
ISBN 978 94 625422-0-4. The book website is: http://thomascool.eu/Papers/FMNAI/Index.html
https://zenodo.org/record/291982
10.5281/zenodo.291982
oai:zenodo.org:291982
doi:10.5281/zenodo.291848
doi:10.5281/zenodo.291974
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
foundations of mathematics, infinity, Cantor, Occam, diagonal argument, ZFC, consistency, neoclassical approach, bijection by abstraction, Paul of Venice, education, didactics, re-engineering
Foundations of Mathematics. A Neoclassical Approach to Infinity
info:eu-repo/semantics/book
publication-book
oai:zenodo.org:1434693
2020-01-20T16:58:07Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2018-09-14
Mathematics education is a mess. In primary education the development in arithmetic, geometry and other math topics is greatly hindered.
Mathematicians are trained for abstraction while education is an empirical issue. When mathematicians meet in class with real life pupils then they solve their cognitive dissonance by holding on to tradition. But tradition and its course materials have not been designed for empirical didactics. Pupils suffer the consequences.
The West reads from left to right but the numbers come from India and Arabia where one reads from right to left. In English 14 is pronounced as fourteen but it should rather be ten & four. 21 is pronounced in proper order as twenty-one, but is better pronounced as two·ten & one, so that the decimal positional system or place value system is supported by pronunciation too.
This is just one example of a long list. An advice is to re-engineer mathematics education for all age-groups. This book looks at primary education. Parliaments around the world are each advised to have their own parliamentary enquiry to investigate the issue and make funds available for change.
The author is an econometrician and teacher of mathematics.
This is the 2nd edition, ISBN 978-946367275-7. The book website: http://thomascool.eu/Papers/NiceNumbers/Index.html
https://zenodo.org/record/1434693
10.5281/zenodo.1434693
oai:zenodo.org:1434693
eng
doi:10.5281/zenodo.291974
doi:10.5281/zenodo.774866
doi:10.5281/zenodo.1244063
doi:10.5281/zenodo.1251686
url:https://mpra.ub.uni-muenchen.de/88810/
doi:10.5281/zenodo.774272
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
primary education, arithmetic, numbers, number sense, left to right, right to left, pronunciation, brain research, didactics, re-engineering, parliamentarian enquiry
A child wants nice and no mean numbers
info:eu-repo/semantics/book
publication-book
oai:zenodo.org:291979
2020-01-20T17:02:04Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2015-08-20
Mathematics education is a mess. In primary education the development in arithmetic, geometry and other math topics is greatly hindered.
Mathematicians are trained for abstraction while education is an empirical issue. When mathematicians meet in class with real life pupils then they solve their cognitive dissonance by holding on to tradition. But tradition and its course materials have not been designed for empirical didactics. Pupils suffer the consequences.
The West reads and writes from left to right but the numbers come from India and Arabia where one reads and writes from right to left. In English 14 is pronounced as fourteen but it should rather be ten·four. 21 is pronounced in proper order as twenty·one, but is better pronounced as two·ten·one, so that the decimal positional system is also supported by pronunciation.
This is just one example of a long list. An advice is to re-engineer mathematics education for all age-groups. This book looks at primary education. Parliaments around the world are advised to have each their own parliamentary enquiry to investigate the issue and make funds available for change.
The author is an econometrician and teacher of mathematics.
ISBN 978-946318970-5. The book website: http://thomascool.eu/Papers/NiceNumbers/Index.html
https://zenodo.org/record/291979
10.5281/zenodo.291979
oai:zenodo.org:291979
doi:10.5281/zenodo.291974
doi:10.5281/zenodo.774272
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
primary education, arithmetic, numbers, number sense, left to right, right to left, pronunciation, brain research, didactics, re-engineering, parliamentarian enquiry
A child wants nice and no mean numbers
info:eu-repo/semantics/book
publication-book
oai:zenodo.org:292253
2020-01-20T17:44:29Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2016-11-30
Continuity is relevant for the real numbers and functions, namely to understand singularities and jumps. The standard approach first defines the notion of a limit and then defines continuity using limits. Surprisingly, Vredenduin (1969), Van der Blij (1970) and Van Dormolen (1970), in main Dutch texts about didactics of mathematics (journal Euclides and Wansink (1970, volume III)), work reversely for highschool students: they assume continuity and define the limit in terms of the notion of continuity. Vredenduin (1969) also prefers to set the value at the limit point (x = a) instead of getting close to it (x → a). Their approach fits the algebraic approach to the derivative, presented since 2007. Conclusions are: (1) The didactic discussions by Vredenduin (1969), Van der Blij (1970) and Van Dormolen (1970) provide support for the algebraic approach to the derivative. (2) For education, it is best and feasible to start with continuity, first for the reals, and then show how this transfers to functions. (3) The notion of a limit can be defined using continuity. The main reason to mention the notion of a limit at all is to link up with the discussion about limits elsewhere (say on the internet). Later, students may see the standard approach. (4) Education has not much use for limits since one will look at continuity. The relevant use of limits is for infinity.
A website on re-engineering mathematics education is: http://thomascool.eu/Papers/Math/Index.html
https://zenodo.org/record/292253
10.5281/zenodo.292253
oai:zenodo.org:292253
doi:10.5281/zenodo.291972
doi:10.5281/zenodo.292250
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
algebraic approach to calculus, derivative, limit, continuity, approximation, mathematics education, didactics, re-engineering
Algebraic approach to the derivative and continuity
info:eu-repo/semantics/workingPaper
publication-workingpaper
oai:zenodo.org:346001
2020-01-20T17:23:03Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2017-03-06
The mathematical "theory of expressions" better be developed in a general fashion, so that it can be referred to in various applications. The paper discusses an example when there is a confusion between syntax (unevaluated) and semantics (evaluated), when substitution causes a contradiction. However, education should not wait till such a mathematical theory of expressions is fully developed. Computer algebra is sufficiently developed to support and clarify these issues. Fractions y / x can actually be abolised and replaced with y xH with H = -1 as a constant like exponential number e or imaginary number i.
https://zenodo.org/record/346001
10.5281/zenodo.346001
oai:zenodo.org:346001
doi:10.5281/zenodo.291979
doi:10.5281/zenodo.292244
doi:10.5281/zenodo.292247
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
Expression, fraction, syntax, semantics, substitution, numerator, denominator, H, computer algebra, Mathematica, mathematics education
Some steps towards a theory of expressions
info:eu-repo/semantics/workingPaper
publication-workingpaper
oai:zenodo.org:1228640
2020-01-20T17:17:05Z
openaire
user-re-engineering-math-ed
Thomas Colignatus
2018-04-25
The correlation between two vectors is the cosine of the angle between the centered data. While the cosine is a measure of association, the literature has spent little attention to the use of the sine as a measure of distance. A key application of the sine is a new "sine-diagonal inequality / disproportionality" (SDID) measure for votes and their assigned seats for parties for Parliament. This application has nonnegative data and uses regression through the origin (RTO) with non-centered data. Textbooks are advised to discuss this case because the geometry will improve the understanding of both regression and the distinction between descriptive statistics and statistical decision theory. Regression may better be introduced and explained by looking at the angle between a vector and its estimate rather than looking at the Euclidean distance and the sum of squared errors. The paper provides an overview of the issues involved. A new relation between the sine and the Euclidean distance is derived. The application to votes and seats shows that a majority of the electorate in the USA and UK, that have District Representation (DR) and not Equal or Proportional Representation (EPR), still tends to have "taxation without representation".
This version of the paper is included here for its relation to the education in mathematics and statistics. Update: In the former version 3.0 on Zenodo, the didactics subsection 2.3 was erroneously not included in the table of contents. Some minor edits.
https://zenodo.org/record/1228640
10.5281/zenodo.1228640
oai:zenodo.org:1228640
eng
url:https://mpra.ub.uni-muenchen.de/86307/
url:https://mpra.ub.uni-muenchen.de/84482/
url:https://zenodo.org/record/291985
url:https://zenodo.org/record/291974
doi:10.5281/zenodo.1227328
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
social choice and social welfare
election and contest
sine-diagonal inequality / disproportionality
representation and parliament
R-squared
regression and correlation, regression through the origin
trigonometry
Xur and Yur
statistics education, statistics ethics
political science and Brexit
An overview of the elementary statistics of correlation, R-Squared, cosine, sine, Xur, Yur, and regression through the origin, with application to votes and seats for parliament
info:eu-repo/semantics/workingPaper
publication-workingpaper
oai:zenodo.org:1227329
2020-01-20T17:18:04Z
openaire
user-re-engineering-math-ed
Thomas Colignatus
2018-04-22
The correlation between two vectors is the cosine of the angle between the centered data. While the cosine is a measure of association, the literature has spent little attention to the use of the sine as a measure of distance. A key application of the sine is a new "sine-diagonal inequality / disproportionality" (SDID) measure for votes and their assigned seats for parties for Parliament. This application has nonnegative data and uses regression through the origin (RTO) with non-centered data. Textbooks are advised to discuss this case because the geometry will improve the understanding of both regression and the distinction between descriptive statistics and statistical decision theory. Regression may better be introduced and explained by looking at the angles relevant for a vector and its estimate rather than looking at the Euclidean distance and the sum of squared errors. The paper provides an overview of the issues involved. A new relation between the sine and the Euclidean distance is derived. The application to votes and seats shows that a majority of the electorate in the USA and UK, that have District Representation (DR) and not Equal or Proportional Representation (EPR), still tends to have "taxation without representation".
This version of the paper is included here for its relation to the education in mathematics and statistics. Update: A small error in the table of contents. See the update https://zenodo.org/record/1228640
https://zenodo.org/record/1227329
10.5281/zenodo.1227329
oai:zenodo.org:1227329
eng
url:https://mpra.ub.uni-muenchen.de/86307/
url:https://mpra.ub.uni-muenchen.de/84482/
url:https://zenodo.org/record/291985
url:https://zenodo.org/record/291974
doi:10.5281/zenodo.1227328
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
social choice and social welfare
election and contest
sine-diagonal inequality / disproportionality
representation and parliament
R-squared
regression and correlation, regression through the origin
trigonometry
Xur and Yur
statistics education, statistics ethics
political science and Brexit
An overview of the elementary statistics of correlation, R-Squared, cosine, sine, Xur, Yur, and regression through the origin, with application to votes and seats for parliament
info:eu-repo/semantics/workingPaper
publication-workingpaper
oai:zenodo.org:1246684
2020-01-20T17:42:47Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2018-05-14
Current English for 14 is fourteen but mathematically it is ten & four. Research on number sense, counting, arithmetic and the predictive value for later mathematical abilities tends to be methodologically invalid when it doesn't measure true number sense that can develop when the numbers are pronounced in mathematical proper fashion. Researchers can correct by including proper names in the research design, but this involves some choices, and when each research design adopts a different scheme, also differently across languages, then results become incomparable. A standard would be useful, both ISO for general principles and national implementations. Research may not have the time to wait for such (inter ) national consensus. This article suggests principles of design and implementations for said languages. This can support the awareness about the need for a process towards ISO and national consensus, and in the mean time provides a baseline for research.
The update is to use "&" as the connective of the place value positions. Thus also for higher positions as e.g. 657 = six·hundred & five·ten & seven. See the developed software and http://thomascool.eu/Papers/NiceNumbers/Index.html
https://zenodo.org/record/1246684
10.5281/zenodo.1246684
oai:zenodo.org:1246684
eng
doi:10.5281/zenodo.291979
doi:10.5281/zenodo.1244063
doi:10.5281/zenodo.774866
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
number sense, counting, arithmetic, mathematical ability, invalidity, design, standards, language, pronunciation, metastudy, number processing, numerical development, inversion effects, language-moderated effects, Google Translate, Child, Child Development, Educational Measurement, Humans, Intelligence, Longitudinal Studies, Mathematics/education, Mathematics/methods, Mental Processes, Students
The need for a standard for the mathematical pronunciation of the natural numbers. Suggested principles of design. Implementation for English, German, French, Dutch and Danish
info:eu-repo/semantics/workingPaper
publication-workingpaper
oai:zenodo.org:1299679
2020-01-20T17:42:08Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2018-06-28
Current English for 14 is fourteen but mathematically it is ten & four. Research on number sense, counting, arithmetic and the predictive value for later mathematical abilities tends to be methodologically invalid when it doesn't measure true number sense that can develop when the numbers are pronounced in mathematical proper fashion. Researchers can correct by including proper names in the research design, but this involves some choices, and when each research design adopts a different scheme, also differently across languages, then results become incomparable. A standard would be useful, both ISO for general principles and national implementations. Research may not have the time to wait for such (inter ) national consensus. This article suggests principles of design and implementations for said languages. This can support the awareness about the need for a process towards ISO and national consensus, and in the mean time provides a baseline for research.
(a) May 14 2018: The update is to use "&" as the connective of the place value positions. Thus also for higher positions as e.g. 657 = six·hundred & five·ten & seven. The ampersand better is 1 fontsize smaller than the other text. See the developed software and http://thomascool.eu/Papers/NiceNumbers/Index.html (b) June 28 2018: This research finds support at https://zwanzigeins.jetzt/infos/normung-der-aussprache
https://zenodo.org/record/1299679
10.5281/zenodo.1299679
oai:zenodo.org:1299679
eng
doi:10.5281/zenodo.291979
doi:10.5281/zenodo.1244063
doi:10.5281/zenodo.1244008
doi:10.5281/zenodo.774866
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
number sense, counting, arithmetic, mathematical ability, invalidity, design, standards, language, pronunciation, metastudy, number processing, numerical development, inversion effects, language-moderated effects, Google Translate, Child, Child Development, Educational Measurement, Humans, Intelligence, Longitudinal Studies, Mathematics/education, Mathematics/methods, Mental Processes, Students
The need for a standard for the mathematical pronunciation of the natural numbers. Suggested principles of design. Implementation for English, German, French, Dutch and Danish
info:eu-repo/semantics/workingPaper
publication-workingpaper
oai:zenodo.org:292257
2020-01-20T17:13:04Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2017-02-12
The books COTP and FMNAI have a time stamp with their ISBN. Over time there arise some comments that might be useful for a 2nd edition. These comments are included in this document.
These reading notes are first available on the books' website before they might be uploaded to Zenodo. See: http://thomascool.eu/Papers/COTP/Index.html
https://zenodo.org/record/292257
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handle:10.5281/zenodo.291972
handle:10.5281/zenodo.291982
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
re-engineering mathematics education, conquest of the plan, foundations of mathematics, analytic geometry, calculus, infinity
Reading Notes on "Conquest of the Plane" and "Contra Cantor Pro Occam"
info:eu-repo/semantics/workingPaper
publication-workingpaper
oai:zenodo.org:291985
2020-01-20T17:34:34Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2014-05-04
VOTING THEORY FOR DEMOCRACY provides the concepts and tools for democratic decision making. Voting is used not only in politics and government, but also in business - and not only in the shareholders' meetings but also in teams. Voting however can suffer from paradoxes. In some systems, it is possible that candidate A wins from B, B from C, and C from A again. This book explains and solves those paradoxes, and thereby it gives a clarity that was lacking up to now. The author proposes the new scheme of 'Pareto Majority' which combines the good properties of the older schemes proposed by Pareto, Borda and Condorcet, while it adds the notion of a (Brouwer) 'fixed point'. Many people will likely prefer this new scheme over Plurality voting which is currently the common practice.
The literature on voting theory has suffered from some serious miscommunications in the last 50 years. Nobel Prize winning economists Kenneth Arrow and Amartya Sen created correct mathematical theorems, but gave incorrect verbal explanations. The author emphasises that there is a distinction between 'voting' and deciding. A voting field only becomes a decision by explicitly dealing with the paradoxes. Arrow and Sen did not solve the paradoxes and used them instead to conclude that it was 'impossible' to find a 'good' system. This however is a wrong approach. Once we understand the paradoxes, we can find the system that we want to use.
This book develops the theory of games (with Rasch - Elo rating) to show that decisions can change, even dramatically, when candidates or items are added to the list or deleted from it. The use of the fixed point criterion however limits the impact of such changes, and if these occur, they are quite reasonable. Groups are advised, therefor, to spend time on establishing what budget they will vote on.
You can benefit from this book also when you do not have the software. However, with the software, you will have an interactive environment in which you and your group can use the various voting schemes, or test them to decide which scheme better fits your purposes. The software is included in The Economics Pack - by the same author - which is an application of Mathematica, a system for doing mathematics with the computer. The Pack has users in many countries in the world. The Pack is available for Windows XP, Macintosh and Unix platforms and requires Mathematica 8.0.1 or later. It can be freely downloaded, but you need a licence to run it.
80% of this book is at an undergraduate level and 20% requires an advanced level.
Developed, supported and published by:
Thomas Cool Consultancy & Econometrics
http://thomascool.eu
ISBN 978-90-804774-9-0. The book website is: http://thomascool.eu/Papers/VTFD/Index.html
https://zenodo.org/record/291985
10.5281/zenodo.291985
oai:zenodo.org:291985
doi:10.5281/zenodo.291974
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
voting, democracy, paradox, Arrow's Impossibility Theorem, Borda, Condorcet, fixed point, plurality, majority, education, didactics, re-engineering
Voting Theory for Democracy
info:eu-repo/semantics/book
publication-book
oai:zenodo.org:292247
2020-01-20T17:21:39Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2017-02-13
This paper considers: (1) Colignatus (2011), "Conquest of the Plane" (COTP), with its new algebraic approach to calculus, and (2) the theory of rational functions (RF).
This paper assumes that most readers will come from a background with RF. They might be interested whether there might be some (re-) design by using some notions from COTP.
The derivative df / dx = f ' [x] concerns the slope of the function. If df is polynomial then we have a rational function r = df / dx. The theory of rational functions at the fundamental level (RF-FL) recognises domains and singularities. Conventionally singularities must be resolved by using limits. For polynomials there is the possibility of factoring, df = f ' [x] dx. Multiplicative factoring can be proven by use of coefficients only, which leads to Ruffini's Rule.
The major conceptual issue w.r.t. factoring is whether the multiplicative form df = f ' [x] dx can still be recognised as the slope df / dx (since a slope is given by the tangent in trigonometry). Ruffini's Rule factors and solves df / dx by "synthetic division", but to what extent is "synthetic" also proper division, so that "eliminating" the factor dx generates a result that can be understood as the slope of the function at that point ?
This conceptual problem is resolved as follows. We better state explicitly that the domain must be manipulated. Let y // x be the following process or program, called dynamic division or dynamic quotient, with numerator y and denominator x:
y // x ≡ { y / x, unless x is a variable and then: assume x ≠ 0, simplify the expression y / x, declare the result valid also for the domain extension x = 0 }
The algebraic definition of the derivative then follows directly:
f ’[x] = {Δf // Δx, then set Δx = 0}
This implies that the expression "df / dx" only has proper meaning as an operator "d / dx" applied to f, without proper division. This also means that we finally have a sound interpretation for differentials. These would not be infinitesimals. The differentials df and dx are better seen as variables, so that, when f ' [x] has been found by other methods of algebraic manipulation, we can define df = f ' [x] dx for the incline (tangent) to f.
This gives: df // dx = f ' [x] dx // dx = f ' [x]
The group theory approach to rational functions (RF-GT) (the version that we looked at) appears to have limited value, because of the assumption that these "functions" don't have domains. If its results are to be useful, they must be translated, and domains and singularities come into consideration anyhow. The notion of an equivalence class relies on limits and continuity, and the manipulation of the domain is not explicit enough.
An algebraic approach to calculus is possible that relies on algebra and expressions only, and that manipulates the domain to find the slope of the function. The formal continuity given by the expression is sufficient, and there is no need for numerical continuity and limits.
https://zenodo.org/record/292247
10.5281/zenodo.292247
oai:zenodo.org:292247
doi:10.5281/zenodo.291848
doi:10.5281/zenodo.291972
doi:10.5281/zenodo.292244
doi:10.5281/zenodo.292250
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
derivative, calculus, rational functions, algebra, algebraic approach, domain, dynamic quotient, mathematics education, didactics, re-engineering
A potential relation between the algebraic approach to calculus and rational functions
info:eu-repo/semantics/workingPaper
publication-workingpaper
oai:zenodo.org:1246653
2020-01-20T17:42:36Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2018-05-14
Current English for 14 is fourteen but mathematically it is ten & four. Research on number sense, counting, arithmetic and the predictive value for later mathematical abilities tends to be methodologically invalid when it doesn't measure true number sense that can develop when the numbers are pronounced in mathematical proper fashion. Researchers can correct by including proper names in the research design, but this involves some choices, and when each research design adopts a different scheme, also differently across languages, then results become incomparable. A standard would be useful, both ISO for general principles and national implementations. Research may not have the time to wait for such (inter ) national consensus. This article suggests principles of design and implementations for said languages. This can support the awareness about the need for a process towards ISO and national consensus, and in the mean time provides a baseline for research.
The update is to use "&" as the connective of the place value positions. See the developed software and http://thomascool.eu/Papers/NiceNumbers/Index.html
https://zenodo.org/record/1246653
10.5281/zenodo.1246653
oai:zenodo.org:1246653
eng
doi:10.5281/zenodo.291979
doi:10.5281/zenodo.1244063
doi:10.5281/zenodo.774866
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
number sense, counting, arithmetic, mathematical ability, invalidity, design, standards, language, pronunciation, metastudy, number processing, numerical development, inversion effects, language-moderated effects, Google Translate, Child, Child Development, Educational Measurement, Humans, Intelligence, Longitudinal Studies, Mathematics/education, Mathematics/methods, Mental Processes, Students
The need for a standard for the mathematical pronunciation of the natural numbers. Suggested principles of design. Implementation for English, German, French, Dutch and Danish
info:eu-repo/semantics/workingPaper
publication-workingpaper
oai:zenodo.org:1400351
2020-01-20T17:40:20Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2018-08-20
Current English for 14 is fourteen but mathematically it is ten & four. Research on number sense, counting, arithmetic and the predictive value for later mathematical abilities tends to be methodologically invalid when it doesn't measure true number sense that can develop when the numbers are pronounced in mathematical proper fashion. Researchers can correct by including proper names in the research design, but this involves some choices, and when each research design adopts a different scheme, also differently across languages, then results become incomparable. A standard would be useful, both ISO for general principles and national implementations. Research may not have the time to wait for such (inter ) national consensus. This article suggests principles of design and implementations for said languages. This can support the awareness about the need for a process towards ISO and national consensus, and in the mean time provides a baseline for research.
(a) May 14 2018: The update is to use "&" as the connective of the place value positions. Thus also for higher positions as e.g. 657 = six·hundred & five·ten & seven. The ampersand better is 1 fontsize smaller than the other text. See the developed software and http://thomascool.eu/Papers/NiceNumbers/Index.html (b) June 28 2018: This research finds support at https://zwanzigeins.jetzt (c) August 20 2018: Better expression of the distinction between simplified 11 = ten & one for everyday use and non-simplified 11 = one.ten & one.one for clarification of the place value system.
https://zenodo.org/record/1400351
10.5281/zenodo.1400351
oai:zenodo.org:1400351
eng
doi:10.5281/zenodo.291979
doi:10.5281/zenodo.1244063
doi:10.5281/zenodo.1244008
doi:10.5281/zenodo.774866
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
number sense, counting, arithmetic, mathematical ability, invalidity, design, standards, language, pronunciation, metastudy, number processing, numerical development, inversion effects, language-moderated effects, Google Translate, Child, Child Development, Educational Measurement, Humans, Intelligence, Longitudinal Studies, Mathematics/education, Mathematics/methods, Mental Processes, Students
The need for a standard for the mathematical pronunciation of the natural numbers. Suggested principles of design. Implementation for English, German, French, Dutch and Danish
info:eu-repo/semantics/workingPaper
publication-workingpaper
oai:zenodo.org:292984
2020-01-20T17:42:19Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2015-09-09
Current English for 14 is fourteen but mathematically it is ten—four. Research on number sense, counting, arithmetic and the predictive value for later mathematical abilities tends to be methodologically invalid when it doesn't measure true number sense that can develop when the numbers are pronounced in mathematical proper fashion. Researchers can correct by including proper names in the research design, but this involves some choices, and when each research design adopts a different scheme, also differently across languages, then results become incomparable. A standard would be useful, both ISO for general principles and national implementations. Research may not have the time to wait for such (inter ) national consensus. This article suggests principles of design and implementations for said languages. This can support the awareness about the need for a process towards ISO and national consensus, and in the mean time provides a baseline for research.
See http://thomascool.eu/Papers/NiceNumbers/Index.html
https://zenodo.org/record/292984
10.5281/zenodo.292984
oai:zenodo.org:292984
doi:10.5281/zenodo.291979
doi:10.5281/zenodo.774866
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
number sense, counting, arithmetic, mathematical ability, invalidity, design, standards, language, pronunciation, metastudy, number processing, numerical development, inversion effects, language-moderated effects, Google Translate, Child, Child Development, Educational Measurement, Humans, Intelligence, Longitudinal Studies, Mathematics/education, Mathematics/methods, Mental Processes, Students
The need for a standard for the mathematical pronunciation of the natural numbers. Suggested principles of design. Implementation for English, German, French, Dutch and Danish
info:eu-repo/semantics/workingPaper
publication-workingpaper
oai:zenodo.org:1246715
2020-01-20T17:42:13Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2018-05-14
Current English for 14 is fourteen but mathematically it is ten & four. Research on number sense, counting, arithmetic and the predictive value for later mathematical abilities tends to be methodologically invalid when it doesn't measure true number sense that can develop when the numbers are pronounced in mathematical proper fashion. Researchers can correct by including proper names in the research design, but this involves some choices, and when each research design adopts a different scheme, also differently across languages, then results become incomparable. A standard would be useful, both ISO for general principles and national implementations. Research may not have the time to wait for such (inter ) national consensus. This article suggests principles of design and implementations for said languages. This can support the awareness about the need for a process towards ISO and national consensus, and in the mean time provides a baseline for research.
The update is to use "&" as the connective of the place value positions. Thus also for higher positions as e.g. 657 = six·hundred & five·ten & seven. The ampersand better is 1 fontsize smaller than the other text. See the developed software and http://thomascool.eu/Papers/NiceNumbers/Index.html
https://zenodo.org/record/1246715
10.5281/zenodo.1246715
oai:zenodo.org:1246715
eng
doi:10.5281/zenodo.291979
doi:10.5281/zenodo.1244063
doi:10.5281/zenodo.774866
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
number sense, counting, arithmetic, mathematical ability, invalidity, design, standards, language, pronunciation, metastudy, number processing, numerical development, inversion effects, language-moderated effects, Google Translate, Child, Child Development, Educational Measurement, Humans, Intelligence, Longitudinal Studies, Mathematics/education, Mathematics/methods, Mental Processes, Students
The need for a standard for the mathematical pronunciation of the natural numbers. Suggested principles of design. Implementation for English, German, French, Dutch and Danish
info:eu-repo/semantics/workingPaper
publication-workingpaper
oai:zenodo.org:1418480
2020-01-20T17:40:48Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2018-09-14
Current English for 14 is fourteen but mathematically it is ten & four. Research on number sense, counting, arithmetic and the predictive value for later mathematical abilities tends to be methodologically invalid when it doesn't measure true number sense that can develop when the numbers are pronounced in mathematical proper fashion. Researchers can correct by including proper names in the research design, but this involves some choices, and when each research design adopts a different scheme, also differently across languages, then results become incomparable. A standard would be useful, both ISO for general principles and national implementations. Research may not have the time to wait for such (inter ) national consensus. This article suggests principles of design and implementations for said languages. This can support the awareness about the need for a process towards ISO and national consensus, and in the mean time provides a baseline for research.
(a) May 14 2018: The update is to use "&" as the connective of the place value positions. Thus also for higher positions as e.g. 657 = six·hundred & five·ten & seven. The ampersand better is 1 fontsize smaller than the other text. See the developed software and http://thomascool.eu/Papers/NiceNumbers/Index.html (b) June 28 2018: This research finds support at https://zwanzigeins.jetzt (c) August 20 2018: Better expression of the distinction between simplified 11 = ten & one for everyday use and non-simplified 11 = one.ten & one.one for clarification of the place value system. (d) September 14 2018: The hyphen is used for common expressions like twenty-one.
https://zenodo.org/record/1418480
10.5281/zenodo.1418480
oai:zenodo.org:1418480
eng
doi:10.5281/zenodo.291979
doi:10.5281/zenodo.1244063
doi:10.5281/zenodo.1244008
doi:10.5281/zenodo.774866
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
number sense, counting, arithmetic, mathematical ability, invalidity, design, standards, language, pronunciation, metastudy, number processing, numerical development, inversion effects, language-moderated effects, Google Translate, Child, Child Development, Educational Measurement, Humans, Intelligence, Longitudinal Studies, Mathematics/education, Mathematics/methods, Mental Processes, Students
The need for a standard for the mathematical pronunciation of the natural numbers. Suggested principles of design. Implementation for English, German, French, Dutch and Danish
info:eu-repo/semantics/workingPaper
publication-workingpaper
oai:zenodo.org:1248957
2020-01-20T17:40:11Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2018-05-17
Kids in kindergarten and Grade 1 live in a world of sounds so that the pronunciation of numbers is important. When they are learning to read and write, the co-ordination of (i) sounds, (ii) numerals and (iii) written words is important. We already have the place value system fully in the numerals but not yet in pronunciation and written words. The following provides for this. The definition should become an ISO standard, though the notebook with package is quite simple because of the nature of the issue. The notebook with package provides an implementation and transliteration for English, German, French, Dutch and Danish, while other languages might employ Mathematica's IntegerName and WordTranslation without transliteration. Four levels in the curriculum are recognised for which routines are provided: (1) sounds, codified by words, (2) learning the numerals, (3) advanced: numerals in blocks of three digits, such that 123456 = {1 hundred, 2 ten, 3} thousand & {4 hundred, 5 ten, 6}, with the comma pronounced as "&" too, and (4) accomplished: 123 thousand 456 pronounced in above place value manner. The traditional pronunciation has level -4.
Major revision. This is the PDF of the Mathematica notebook and package provided at https://zenodo.org/record/1244008. The notebook and package are also available at http://community.wolfram.com/web/community/groups/-/m/t/1334793
https://zenodo.org/record/1248957
10.5281/zenodo.1248957
oai:zenodo.org:1248957
eng
url:https://zenodo.org/record/1244008
url:https://zenodo.org/record/774866
url:https://zenodo.org/record/291979
doi:10.5281/zenodo.1244063
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
mathematics education
place value system
pronunciation
Common Core
Mathematica
Wolfram language
programming
package
MSC2010: 97M70 Mathematics education. Behavioral and social sciences
Pronunciation of the integers with full use of the place value system
info:eu-repo/semantics/workingPaper
publication-workingpaper
oai:zenodo.org:1299781
2020-01-20T17:41:25Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2018-06-28
Kids in kindergarten and Grade 1 live in a world of sounds so that the pronunciation of numbers is important. When they are learning to read and write, the co-ordination of (i) sounds, (ii) numerals and (iii) written words is important. We already have the place value system fully in the numerals but not yet in pronunciation and written words. The following provides for this. The definition should become an ISO standard, though the notebook with package is quite simple because of the nature of the issue. The notebook with package provides an implementation and transliteration for English, German, French, Dutch and Danish, while other languages might employ Mathematica's IntegerName and WordTranslation without transliteration. Four levels in the curriculum are recognised for which routines are provided: (1) sounds, codified by words, (2) learning the numerals, (3) advanced: numerals in blocks of three digits, such that 123456 = {1 hundred, 2 ten, 3} thousand & {4 hundred, 5 ten, 6}, with the comma pronounced as "&" too, and (4) accomplished: 123 thousand 456 pronounced in above place value manner. The traditional pronunciation has level -4.
May 17 2018: Major revision. This is the PDF of the Mathematica notebook and package provided at https://zenodo.org/record/1244008. The notebook and package are also available at http://community.wolfram.com/web/community/groups/-/m/t/1334793 June 28 2018: (a) discovery of Gothic "tigus", (b) support for this research at https://zwanzigeins.jetzt/infos/normung-der-aussprache
https://zenodo.org/record/1299781
10.5281/zenodo.1299781
oai:zenodo.org:1299781
eng
url:https://zenodo.org/record/1244008
url:https://zenodo.org/record/774866
url:https://zenodo.org/record/291979
doi:10.5281/zenodo.1244063
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
mathematics education
place value system
pronunciation
Common Core
Mathematica
Wolfram language
programming
package
MSC2010: 97M70 Mathematics education. Behavioral and social sciences
Pronunciation of the integers with full use of the place value system
info:eu-repo/semantics/workingPaper
publication-workingpaper
oai:zenodo.org:1244064
2020-01-20T17:42:09Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2018-05-09
Kids in kindergarten live in a world of sounds, so that the pronunciation of numbers is important. When they start learning to read and write, the co-ordination of (i) sounds, (ii) words and (iii) numerals is important. We already have the place value system fully in the numerals but not yet in written words and pronunciation. It appears that we can provide for a pronunciation of the integers with the full use of the place value system. The definitions should become an ISO standard, though the notebook and package still are quite simple. This notebook and package provide an implementation for English, German, French, Dutch and Danish.
This is the PDF of the notebook and package provided at https://zenodo.org/record/1244009. The notebook and package are also available at http://community.wolfram.com/web/cool
https://zenodo.org/record/1244064
10.5281/zenodo.1244064
oai:zenodo.org:1244064
eng
url:https://zenodo.org/record/1244009
url:https://zenodo.org/record/292984
url:https://zenodo.org/record/291979
doi:10.5281/zenodo.1244063
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
mathematics education
place value system
pronunciation
Common Core
Mathematica
Wolfram language
programming
package
97M70 Mathematics education. Behavioral and social sciences
Pronunciation of the integers with full use of the place value system
info:eu-repo/semantics/workingPaper
publication-workingpaper
oai:zenodo.org:292250
2020-01-20T17:15:38Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2017-02-13
Michael Range (2016) "What is Calculus?" (WIC) and its chapter "Prelude" and Thomas Colignatus (2011) "Conquest of the Plane" (COTP) are proofs of concept, that show how one might implement a course in calculus starting with an algebraic approach and avoiding limits as long as possible. A proof of concept comes with notes for instructors and discussion of didactics, but not all is explained, since the idea is to show how it works. Thus, evaluations by others are useful to highlight not only the explicit explanations but also the actual (implicit) implementations that only transpire from following the method step by step. In the present discussion, teachers and other readers will find information about WIC Prelude that cannot be found in the WIC "Notes for instructors".
This comparison concerns the algebraic approach to calculus and thus concerns the WIC Prelude and not the main body of WIC. WIC shows an approach without awareness or reference to COTP. As author of COTP I may have a bias but I will try to evaluate WIC Prelude as unbiasedly as possible. This discussion should highlight aspects of COTP as well. The reader is invited not to mistake this highlighting as sign of bias.
WIC claims this readership: "Undergraduates, high school students, instructors and teachers, and scientifically literate readers with special interest in calculus and analysis." This would be too ambitious. The WIC Prelude relies on (group theory) notions of "rational function" and "polynomial theory" that will only fit the matricola of science students and up. On the other hand, COTP is a primer and thus targets teachers and researchers of didactics. It only relies on notions for non-mathematics majors in matricola and highschool and thus can support such students as well.
A website on re-engineering mathematics education is: http://thomascool.eu/Papers/Math/Index.html
https://zenodo.org/record/292250
10.5281/zenodo.292250
oai:zenodo.org:292250
doi:10.5281/zenodo.291848
doi:10.5281/zenodo.291972
doi:10.5281/zenodo.292247
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
algebraic approach to calculus, derivative, education, didactics, re-engineering, rational functions, incline, tangent, double root
Comparing two algebraic approaches to calculus: WIC Prelude and COTP
info:eu-repo/semantics/workingPaper
publication-workingpaper
oai:zenodo.org:291972
2020-01-20T17:28:23Z
openaire
user-re-engineering-math-ed
Colignatus, Thomas
2011-03-15
CONQUEST OF THE PLANE provides:
an integrated course for geometry and analysis
a didactic build-up that avoids traditional clutter
use of only the essentials for good understanding
proper place for vectors, complex numbers, linear algebra and trigonometry
an original and elegant development of trigonometry
an original and elegant foundation for calculus
examples from physics, economics and statistics
integration within the dynamic environment of Mathematica
considered didactic explanation in a 'meta' section.
The book is a primer. A primer is not a textbook but this is a primer in textbook format. The first four parts have been written with students in mind. The fifth meta part explains the didactics to teachers and students in didactics of mathematics. Good teachers will enjoy the innovations. As a textbook it gives a fast track introduction for students at advanced highschool or the first year of college and university and their parents who want to help with homework
For the professions that apply mathematics like physics, engineering, biology, economics and evidence based medicine, the book provides documentation to judge on the proposal to let the national parliaments look into mathematics education, as explained in the book Elegance with Substance by the same author.
Calculus can be developed with algebra and without the use of limits and infinitesimals. Define y / x as the outcome of division and y // x as the procedure of division. Using y // x with x possibly becoming zero will not be paradoxical when the paradoxical part has first been eliminated by algebraic simplification. The Weierstrasz epsilon and delta and its Cauchy shorthand with limits are paradoxical since those exclude the zero values that are precisely the values of interest at the point where the limit is taken. Much of calculus might well do without the limit idea and it could be advantageous to see calculus as part of algebra rather than a separate subject. This is not just a didactic observation but an essential refoundation of calculus.
Didactic issues in trigonometry concern the opaque names of sine and cosine and the cluttering of questions with or 360 degrees whereas a simple 1 suffices. Basically the plane itself gives that unit of 1, and angles are mere sections. The solution is to use the unit turn or unit of measurement (meter) around (UMA) as the yardstick for angles. This gives the Xur and Yur functions, defined on the circle with unit circumference and generating the {X, Y} co-ordinates on the circle with unit radius. Finally students will understand what Cos and Sin are. The common term 'dimensionless' appears to confuse 'no unit of measurement specified' (with a metric, in planimetry and trigonometry) with 'no dimension' (a pure number, in number theory). The relevant mathematical constant is Archi = 2 Pi (capital theta, reminiscent of a circle) rather than Pi and it comes into use much less when we use UMAs instead of radians. The sine and cosine remain relevant the derivative, and that can also be shown in an elegant manner.
You can benefit from this book also when you don't have the software. However, with the software, you will have an interactive environment in which you can test the propositions in this book and your own deductions. The software is included in The Economics Pack - by the same author - which is an application of Mathematica, a system for doing mathematics with the computer. The Pack has users in many countries in the world. The Pack is available for Windows XP, Macintosh and Unix platforms and requires Mathematica 8.0.1 or later. It can be freely downloaded, but you need a licence to run it.
ISBN 978-90-804774-6-9. The website of the book is: http://thomascool.eu/Papers/COTP/Index.html. Comments w.r.t. the book have been collected in the Reading Notes, see 10.5281/zenodo.292257
https://zenodo.org/record/291972
10.5281/zenodo.291972
oai:zenodo.org:291972
doi:10.5281/zenodo.291848
doi:10.5281/zenodo.292257
url:https://zenodo.org/communities/re-engineering-math-ed
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
analytic geometry, calculus, primer, trigonometry, archi, xur, yur, dynamic quotient, re-engineering mathematics education
Conquest of the Plane
info:eu-repo/semantics/book
publication-book