Hans Freudenthal citations (27 citations)

Source: The Concept and the Role of the Model in Mathematics and Natural and Social Sciences (1961), p. 79; Part of the article "Models in applied probability", published earlier in Synthese, 12 (1960), p. 204-210.

“The present book is not a methodology of mathematics in the sense that I will systematically show how some teaching matter should taught; it is not even a systematic analysis of subject matter. I hardly ever refer to well-organized classroom experiments evaluated by statistical methods, nor do I cite experimental results of developmental psychology or the psychology of learning. Maybe the most striking feature is that this book contains few quotations. I will try to justify all these features.”

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Hans Freudenthal

Source: Mathematics as an Educational Task (1973), p. v;As cited in: Ben Wilbrink (2013)

“Learners should be allowed to find their own levels and explore the paths leading there with as much and as little guidance as each particular case requires.”

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Hans Freudenthal

Source: Didactical Phenomenology of Mathematical Structures (1983), p. 47

“No doubt once it was real progress when developers and teachers offered learners tangible material in order to teach them arithmetic of whole number… The best palpable material you can give the child is its own body.”

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Hans Freudenthal

Source: Mathematics as an Educational Task (1973), p. 75-76; As cited in: Anne Birgitte Fyhn (2007, p. 6)

“Science should be distinguished from technique and its scientific instrumentation, technology. Science is practised by scientists, and techniques by ‘engineers’ — a term that in our terminology includes physicians, lawyers, and teachers. If for the scientist knowledge and cognition are primary, it is action and construction that characterises the work of the engineer, though in fact his activity may be based on science. In history, technique often preceded science.”

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Hans Freudenthal

Hans Freudenthal (1978). Weeding and Sowing. Preface to a Science of Mathematical Education; As cited in: Ben Wilbrink (2013) " Hans Freudenthal Aantekeningen bij zijn publicaties http://www.benwilbrink.nl/literature/freudenthal.htm".

“Euclid defines the angle as an inclination of lines…he meant halflines, because otherwise he would not be able to distinguish adjacent angles from each other… Euclid does not know zero angles, nor straight and bigger than straight angles…Euclid takes the liberty of adding angles beyond two and even four right angles; the result cannot be angles according to the original definitions…Nevertheless one feels that Euclid’s angle concept is consistent.”

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Hans Freudenthal

Source: Mathematics as an Educational Task (1973), p. 476-477

“The urn model is to be the expression of three postulates: (1) the constancy of a probability distribution, ensured by the solidity of the vessel, (2) the randomcharacter of the choice, ensured by the narrowness of the mouth, which is to prevent visibility of the contents and any consciously selective choice, (3) the independence of successive choices, whenever the drawn balls are put back into the urn. Of course in abstract probability and statistics the word " choice " can be avoided and all can be done without any reference to such a model. But as soon as the abstract theory is to be applied, random choice plays an essential role.”

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Hans Freudenthal

Source: The Concept and the Role of the Model in Mathematics and Natural and Social Sciences (1961), p. 80; Cited in: Lev D. Beklemishev (2000) Provability, Computability and Reflection. p. 9

“Even measured by Wiener's standards Cybernetics is a badly organised work — a collection of misprints, wrong mathematical statements, mistaken formulas, splendid but unrelated ideas, and logical absurdities. It is sad that this work earned Wiener the greater part of his public renown, but this is an afterthought. At that time mathematical readers were more fascinated by the richness of its ideas than by its shortcomings.”

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Hans Freudenthal

Norbert Wiener (2008)

“Geometry is grasping space. And since it is about the education of children, it is grasping that space in which the child lives, breathes and moves. The space that the child must learn to know, explore, conquer, in order to live, breathe and move better in it.”

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Hans Freudenthal

Source: Mathematics as an Educational Task (1973), p. 403

“No statistician present at this moment will have been in doubt about the meaning of my words when I mentioned the common statistical model. It must be a stochastic device producing random results. Tossing coins or a dice or playing at cards are not flexible enough. The most general chance instrument is the urn filled with balls of different colours or with tickets bearing some ciphers or letters. This model is continuously used in our courses as a didactic tool, and in our statistical analyses as a means of translating realistic problems into mathematical ones. In statistical language " urn model " is a standard expression.”

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Hans Freudenthal

Source: The Concept and the Role of the Model in Mathematics and Natural and Social Sciences (1961), p. 79; Partly cited in: Norman L. Johnson and Samuel Kotz (1977) Urn Models and Their Application: an. Approach to Modern Discrete Probability Theory http://dis.unal.edu.co/~gjhernandezp/sim/hide/Urn%20Models%20and%20Their%20Application%20-%20An%20approach%20to%20modern%20discrete%20probability%20theory_Norman%20L.Johnson(Wiley%201977%20413s).pdf, John Wiley & Sons.

“Angles are measured by arcs, such that 360° and 2π correspond to each other.”

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Hans Freudenthal

Source: Mathematics as an Educational Task (1973), p. 477

“Grasping spatial gestalts as figures is mathematizing of space. Arranging the properties of a parallelogram such that a particular one pops up to base the others on it in order to arrive at a definition of parallelogram, that is mathematizing the conceptual field of the parallelogram.”

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Hans Freudenthal

Source: Mathematics as an Educational Task (1973), p. 133

“Horizontal mathematising leads from the world of life to the world of symbols.”

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Hans Freudenthal

Source: Revisiting Mathematics Education (1991), p. 41

“No mathematical idea has ever been published in the way it was discovered. Techniques have been developed and are used, if a problem has been solved, to turn the solution procedure upside down, or if it is a larger complex of statements and theories, to turn definitions into propositions, and propositions into definitions, the hot invention into icy beauty. This then if it has affected teaching matter, is the didactical inversion, which as it happens may be anti-didactical. Rather than behaving anti-didactically, one should recognise that the learner is entitled to recapitulate in a fashion of mankind. Not in the trivial matter of an abridged version, but equally we cannot require the new generation to start at the point where their predecessors left off.”

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Hans Freudenthal

Source: The Concept and the Role of the Model in Mathematics and Natural and Social Sciences (1961), p. ix

“Vertical mathematising is the most likely part of the learning process for the bonds with reality to be loosened and eventually cut.”

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Hans Freudenthal

Source: Revisiting Mathematics Education (1991), p. 68

“[Guided reinvention is] striking a subtle balance between the freedom of inventing and the force of guiding, between allowing the learner to please himself and asking him to please the teacher. Moreover, the learner’s free choice is already restricted by the “re” of “reinvention”. The learner shall invent something that is new to him but well-known to the guide.”

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Hans Freudenthal

Source: Revisiting Mathematics Education (1991), p. 48; As cited in: Anne Birgitte Fyhn (2007, p. 14)

“A true aphorism legitimates itself; whoever feels the need to legitimate an aphorism, admits that it is illegal. The surface of an aphorism should conceal profound truth. The claim that everybody can learn everything is superficial, but is as wrong as it can be. As a matter of fact, it is no aphorism but an advertising slogan, and the excuse that it is an aphorism, is a mere wink: in advertising you cannot do without exaggerating. But even as a wink it does not become more true.”

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Hans Freudenthal

Hans Freudenthal (1977), Weeding and Sowing: Preface to a Science of Mathematical Education, p. 56

“Space and the bodies around us are early mental objects… Name-giving is a first step towards consciousness.”

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Hans Freudenthal

Source: Mathematics as an Educational Task (1973), p. 63; As cited in: Anne Birgitte Fyhn (2007) Angles as Tool for Grasping Space http://munin.uit.no/bitstream/handle/10037/994/thesis.pdf?sequence=1. p. 2

“The classic instrument to measure drawn angles and to draw angles of a given measure is the protractor — essentially half a circular ring, subdivided by ray segments into 180 degrees. For reasons I was unable to find out, this instrument has recently been superseded by an isosceles right triangle — called geo-triangle, solid, transparant, made of plastic — with an angular division radiating from the midpoint of the hypotenuse to the other sides. Well, inside the triangle half a circle with the midpoint of the hypotenuse as its centre is indicated, and from the position of the degree numbers it becomes clear that it is the semicircle that really matters. One is inclined to say "an outrageously misleading instrument"…”

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Hans Freudenthal

Source: Mathematics as an Educational Task (1973), p. 363

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Hans Freudenthal

Rather than behaving anti-didactically, one should recognise that the learner is entitled to recapitulate in a fashion of mankind. Not in the trivial matter of an abridged version, but equally we cannot require the new generation to start at the point where their predecessors left off.
Source: The Concept and the Role of the Model in Mathematics and Natural and Social Sciences (1961), p. ix