Hans Freudenthal: petty crook, no demon (2)

Part 1 above was on the Van Hieles 1957. This Part 2 looks at nine years earlier: 1948, three years after the end of WW 2. The source used for this weblog entry is Euclides, the journal of the Dutch teachers of mathematics, namely issue 24, for the year 1948-1949, no 3. (Download this issue.) This may help readers to verify that Freudenthal was no demon, as perhaps some people think after his fraud.

My advice is to have independent translations into English. I will be hesitant to provide my own now. I will give brief summaries and Google Translate results. Let me warn about putting too much information into a translation. Translating from Dutch 1948 is different from translating from Dutch 2015. It will not do to translate the 1957 Van Hiele theory of levels into Freudenthal’s text of 1948. Once you have the 1957 theory it may actually be of no interest anymore what Freudenthal said in 1948, except for historical reasons. In the interview in 2005:

“I believe actually that he did not really understand much about the levels of insight.” (Pierre van Hiele)

To get a feel for the time: Dr. L. Bunt on page 89 refers to the head start in Anglo-Saxon countries with statistical methods and with journals such as Econometrica, Biometrika and Psychometrika. He mentions that already 13 issues of Psychometrika have appeared and that none of these are available in any of the 43 public scientific libraries included in the Dutch Central Catalog of 1948. Bunt needs many words to explain that mathematics has a role in quantitative science, which still may be news to mathematicians used to axiomatics:

“By now you will have asked yourself the question: what does this collecting, ordering and processing of quantitative data have to do with “clear thinking”. And indeed, this way of conceiving issues is fundamentally different from the commonly given approach in the so-called exercise of thinking. It is the purpose, here, to clarify that mathematical thinking plays a fundamental role among other things when one wishes to give a clear image to oneself and others of the conclusions, to which a particular collection of quantitative data leads on close consideration.” (Translated by TC)

There is a discussion about logarithmic tables that is almost the same as the current discussion of using computers (page 133). The teachers rely on their practice rather than on statistical significance to decide what is the best didactic approach:

“At the end there arises a discussion about to what extent students are allowed to work mechanically. Mr. Van Hiele had rejected it. Dr. Mooy however recommends it. It saves us energy. The students must for example be able to search in a log-table, provided it is done without error. Others reject the exclusively mechanical operation too. When students have learned to work mechanically with a log-table or slide rule, then they quickly forget how it is done, and they later cannot reproduce it anymore. Mechanical work thus is lost time. When they have learned to do it with understanding, then they will be able to reproduce it later themselves if needed.” (Translated by TC)

Hans Freudenthal (1905-1990), then 43, and Pierre van Hiele (1909-2010), then 39 and teaching for 12 years, had already met before. L.E.J. Brouwer had invited Freudenthal from Germany to Amsterdam as an assistant, and Freudenthal had become one of the teachers of Van Hiele, four years his junior. It may be seen as remarkable that Freudenthal already is involved in math education. However, training teachers seems to have been part of his job from the outset.

Of this issue of Euclides (download):

  • Pages 122-133 contain a discussion paper by Pierre van Hiele to formulate guidelines for didactics of mathematics, and a subsequent discussion. I leave that be.
  • Pages 106-121 contain a discussion by Hans Freudenthal on the algebraic and analytical views on the concept of number in elementary mathematics (i.e. school mathematics, non-university education). This we will look into.

Freudenthal 1948 on the concept of number in school math

Precursor to Van Hiele levels ?

It is not necessary that I translate all following lines. Freudenthal indicates the difference between numbers and algebra (group theory). Important elements in this quote are:

  • The phrase: far advanced stage of abstraction (“ver gevorderd stadium van abstractie”).
  • This stage: has the possibility of proof  (“dat voor een bewijs vatbaar is”).
  • This stage of abstraction requires a particular mental attitude (“geesteshouding“), different from the “primitive” (Primitive Man rather than the child, since he mentioned the development of mankind in the former paragraphs).

It seems as if we suddenly see Van Hiele’s levels of insight.

The key question is: why aren’t these the Van Hiele levels ?

Freudenthal presents rather a historical and not a didactic view. In so far as there is education involved, the view that he presents in 1948 is actually rather conventional for that period. Primitives and children use concrete apples and numbers, and mathematicians do the abstract proofs. We are so much used to Van Hiele’s theory that we may no longer see the originality in its design and advancement above the conventional view that Freudenthal states here.

Van Hiele started from Piaget (the age factor) rather than from Freudenthal, and then: (i) clearly defines the level distinctions and properties, (ii) presents the notion that students need the lower level before they can proceed to the next level, so that Euclid’s top-down method doesn’t work, (iii) observes the language issue: that the same words mean something else depending upon the level, (iv) provides the didactics required for the level transitions, (v) formulates the general applicability for fields of knowledge.

For Freudenthal it might still be conceivable to teach from abstraction to concreteness, i.e. like Euclid. It is mankind that reached the higher level of abstraction. He compares a child that counts to a mathematician applying numerical succession (“mathematical induction”). Below we will see him teaching a child, one of his sons, and how he translates his own abstractions into a form that works for his son.

From page 106:

Freudenthal, 1948:106

Freudenthal, 1948:106

Dutch original that you may offer to Google Translate:

“Dat 3 + 4 = 4 + 3 is, aanvaardt de wiskundige even grif als het kind, voor wie de getallen hoeveelheden blokken of vingers zijn, maar dat a + b = b + a is, is voor hem een wiskundige stelling, waarmee de algebra-leerling zonder aarzelen instemt, en die pas in een ver gevorderd stadium van abstractie erkend wordt als iets, dat voor een bewijs vatbaar is. Hiervoor is namelijk een geesteshouding vereist, die van de primitieve zeer verschilt: men moet – hebben begrepen, dat het er niet alleen op aankomt, materieel nieuwe waarheden te ontdekken, en men moet hebben geleerd, problemen te zien en te zoeken, waar – oppervlakkig bekeken – van geen problematiek sprake is.”

Google Translate:

‘This 3 + 4 = 4 + 3 accepts the mathematician as readily as a child, for whom the numbers are quantities blocks or fingers, but that a + b = b + a is, for him is a mathematical theorem, that the algebra pupil consents without hesitation, and that is recognized only in an advanced stage of abstraction as something that is amenable to proof. Good faith presupposes an attitude of primitive very different: one must – have understood that it comes not only on equipment to discover new truths, and one should have learned to see problems and look where – superficially views – of no problem exists.” (Google Translate)

High theory, home practice, unkind joke

The following quote from p108-109 contains the unkind joke: that students are not familiar with knocking on his door to ask questions, but wait till the exam to show that they didn’t get it. He also jokes that his own practical experience with teaching is not “the class” but his study at home. It is one of his young sons knocking at the door who works with negative numbers and who doesn’t get it. Indeed, Freudenthal taught at university and, as far as I know, not at other levels. He argues that teachers need a higher point of view (“standpunt“) – in mathematics and pedagogy-psychology – to think about elementary issues again. This higher point of view isn’t (yet) a higher Van Hiele level: only a vantage point.

Freudenthal, 1948:108-109

Freudenthal, 1948:108-109

Dutch original that you may offer to Google Translate:

“telkens weer doordenkt vanuit hoger standpunt (waarbij ik dan toegeef, dat dit hoger standpunt niet altijd het wiskundige, maar vaak ook het paedagogisch-psychologische is): Ik wil het met een voorbeeld uit mijn eigen praktijk toelichten. Die praktijk is dan niet ,,de klas”, maar mijn eigen studeerkamer, waar op de deur wordt geklopt door iemand, die ,,er weer niets van snapt” – ik bedoel niet een student, want studenten zijn aan dergelijke farniliariteiten nog niet toe en wachten liever het tentamen af, om te demonstreren, dat ze het niet gesnapt, hebben. Het proefkonijn nu is een van mijn zoontjes, die zijn eerste schreden in de algebra moet doen, en deze schreden leiden volgens geijkte methoden naar de negatieve kant van onze getallenrij”

Google Translate

“again and again by thinking from a higher point of view (which I admit that this higher position is not always the mathematical, but often it is paedagogisch psychological): I want to explain it with an example from my own practice. This practice is not,, class, “but my own study, which is a knock at the door by someone who,, again nothing understand” – I do not mean a student because students are such farniliariteiten not yet and would rather wait for the examination to demonstrate that they do not get caught in it have. The guinea pig is now one of my sons who are first steps in algebra to do, and these steps according to traditional methods lead to the negative side of our sequence” (Google Translate)

Mechanical method and no context on negative numbers

The question of his son was: why 5 – (-3) = 8. Freudenthal describes that this caused him to consider theory (numerical succession, “mathematical induction”) to resolve it, and then to transform it (“didactic induction”) into the world of his son, with the tables of addition (“naive induction”). He argues that this scheme should work in general.

Freudenthal came up with this ploy, after which his son indicated that he now got it. Or understood that his father could not help him.

Freudenthal, 1948:109

Freudenthal, 1948:109

Unfortunately, Freudenthal doesn’t further check what it means, that his son states that he got it. Presumably, the son now can do similar sums. But the didactic method consists of stating a mechanical ploy. Freudenthal associates abstract mathematical understanding of numerical succession with mechanical understanding of counting. Would the son also understand “negative debt” and “subtraction of coldness” ? Questions like these:

  • are now asked under the label of “transfer”.
  • have been abused under the label of “realistic mathematics education” (RME): thus, RME doesn’t use such mechanical schemes but uses contexts, such as negative debt and such, and lets kids discover meanings and methods in “guided re-invention”.
Structure and insight

This quote gives us two words that Van Hiele also used in titles of his books: structure and insight (“inzicht in een structuur“). These words may have been in use already for centuries. The use of the same words allows Van Hiele to indicate what his 1957 theory applies to. That Freudenthal uses these same words in 1948 does not mean that he already understood the later theory of levels.

  • Freudenthal states that more complex methods are in line with elementary methods. This reminds of the Van Hiele levels, but also reminds of the possibility that multiplication is an extension of addition. Freudenthal specifies it as “greater exactness”. Google Translate turns this into “accuracy” but in Dutch the natural sciences are called the “exact” sciences, which refers to measurement and reproducibility (objectivity). Freudenthal links exactness to the notion of proof – which Google translates as “evidence”. Proof is to generate more insight into a structure. This fits the highest Van Hiele level, but Van Hiele applies “insight” also to the lower levels. One might also argue that those lower levels of insight aren’t really insight that we are looking for (at least in mathematics).
  • Freudenthal states that the historical development shouldn’t be merely copied in didactics. Perhaps in 1948 this was a major insight.
Freudenthal 1948:110

Freudenthal 1948:110

Dutch original, dropping a dash for Google Translate, and my emphasis:

“Ik geloof, dat dit een algemeen vruchtbaar paedagogisch beginsel is. De volledige inductie is een inkleding in streng geformaliseerde taal van de naieve inductie, die niet slechter hoeft te zijn dan haar grote zuster. En zo – komt het mij voor – is het overal in de schoolwiskunde. Haar bewijzen moeten in de taalkundige formulering slordiger zijn dan die der hogere wiskunde – in wezen hoeven ze er niet van te verschillen. Alleen zijn er zeer veel nuances van exactheid – niet alleen het verschil tussen elementaire en hogere wiskunde, maar nuances in het schoolonderwijs zelf, waarvan men zich bewust moet bedienen, om de grotere exactheid in concentrische kringen te benaderen, tot de exactheid der hogere wiskunde toe, die er als het ware de apotheose van is. Het doel van elk bewijs is helderheid van inzicht in een structuur, en de verscherping van exactheids eisen mag enkel het gevolg zijn van het doorzien van problemen, die die helderheid hebben vertroebeld.
Onze gehele hogere wiskunde ligt in het verlengde van elementaire werkwijzen. We moeten nu niet de historische ontwikkeling klakkeloos in onze didactiek overnemen, maar de lijn vanuit de hogere wiskunde weer terugverlengen, om de juiste didactische methoden te vinden. Om te laten zien, dat dit de goede weg is, zou ik de hele schoolwiskunde onder het mes moeten nemen, maar dit kan mijn taak niet zijn. Ik wil bij enkele stadia van de ontwikkeling van het getalbegrip blijven staan.”

Google Translate:

“I believe that this is a general principle paedagogisch fruitful. The mathematical induction is a garb in strictly formalized language of naive induction, which does not have to be worse than her big sister. And so – it seems to me – it is everywhere in the school mathematics. Her evidence should be sloppy in the linguistic formulation than that of higher mathematics – in essence, they do not like to differ. Only there are a lot of nuances of exactness – not just the difference between elementary and higher mathematics, but nuances in school education itself, which one must operate aware, to approach the greater accuracy in concentric circles to the accuracy of higher mathematics to who as it were, is the apotheosis of. The goal of any evidence clarity of insight into a structure, and the sharpening of exactheids requirements can only be the result of seeing through issues that have clouded the brightness.
Our entire higher mathematics is an extension of basic methods.
We must not take the historical development indiscriminately in our didactics, but the line from the higher mathematics extend back again, to find the right teaching methods. Order to show that this is the right path, should I take the whole school mathematics under the knife, but this can not be my task.
I want to remain at some stages of the development of the concept of number.” (Google Translate)

Real numbers and the decimal expansion

Here, Freudenthal has finished his didactic review and proceeds with the technical issues of real numbers. He proposes to use decimal expansions in school mathematics (e.g. a real number 1 / 3 = 0.333…), and to forget about Dedekind cuts.

  • I am a bit amazed that he really mentions Dedekind cuts. He probably sticks to the rule that one has to give argumentation even if it forces one to state the obvious.
  • I fully agree that decimal expansions must be treated well. My book Foundation of Mathematics. A Neoclassical Approach to Infinity (2015) proposes the same. Thus, we both oppose the sloppy use of real numbers, as now is the case in 2015. School mathematics should discuss that 1 = 3 * 1 / 3 = 3 * 0.3333….. = 0.9999…. and such issues.
Complex numbers

There are at least three ways to use complex numbers in school mathematics: symbol i, expression √(-1) and co-ordinates {x, y}. Freudenthal prefers the first. I prefer the truth, i.e. the co-ordinates, with a decent explanation of the rules, and not the caricature that Freudenthal makes of it. See the earlier discussion of complex numbers, with hopelessly confused mathematics professor Edward Frenkel who prefers √(-1).

Freudenthal, 1948:119

Freudenthal, 1948:119

Dutch original that you may offer to Google Translate:

“geschoolde leerling met een i, waarvan het kwadraat -1 is, met even weinig gemoedsbezwaren als met een √2, waarvan het kwadraat 2 is gesteld. Ik zie dus geen aanleiding tot paedagogische aarzelingen, om het getal i op deze naieve wijze in te voeren. Men zet dan slechts dat voort, wat men vroeger is begonnen, toen men met √2 formeel ging rekenen; men bereidt de leerling door dit formalisme voor op formele methoden, zoals men er meer in de wiskunde kent, en men doet bovendien iets wat van een hoger standpunt van exacheid geheel verantwoord is. Ik weet wel, dat er paedagogen zijn, die reeds op school de complexe getallen willen introduceren als paren van reële getallen, die aan een stel rare rekenwetten voldoen. Ik ben het hiermee niet eens. Wil de leerling er de zin van vatten, dan zou men eerst zijn naieve vertrouwen in de onfeilbaarheid van de formele methoden grondig moeten doorzeven – met misschien als gevolg, dat hij het geloof niet alleen in i maar ook nog in √2 verliest. Zonder deze critische houding is de waarde van al die finesses op zijn minst zeer problematisch. Waarom zou men ook in het schoolonderwijs hoger eisen van exactheid stellen, wanneer men de imaginaire getallen invoert, dan bij het invoeren der gehele getallen, der gebroken getallen, der worteluitdrukkingen en der reële getallen?”

Google Translate:

“skilled apprentice with an i, whose square is -1, with as little scruples as a √2, whose square is placed second. I therefore see no reason to pedagogic hesitations, to enter the number i in this naive way. They just put it on, which was begun earlier, when they started calculating formally with √2; it prepares the student through this formalism for formal methods, such as one knows it more in mathematics, and it also does something from a higher point of exacheid fully accountable. I know that there are pedagogues, already in school to introduce the complex numbers as pairs of real numbers, which conform to a set of strange arithmetic laws. I do not agree. If the student grasp the meaning of it, one would first have to be naive confidence by seven thoroughly in the infallibility of the formal methods – with perhaps as a result, he lost faith not only I but also in √2. Without this critical attitude is the value of all the subtleties at least very problematic. Why could also set higher standards of rigor in the school education when they enter the imaginary numbers, then when entering the integers of broken numbers of root and expressions of real numbers?” (Google Translate)

Training of teachers

The article is presented at a conference with a discussion afterwards. One topic is whether the education for mathematician should be split between research mathematicians (RM) and teachers (MT). A comparison is made with medical doctors who after graduation still aren’t allowed to practice, and need additional courses. The MT object to different curricula since they would not be able to understand RM anymore. Freudenthal holds that a training for mathematician is only interesting for 2 out of 15 students. He advises for teachers: 2 years of university, 1 + ½ years practice, 1 + ½ years back to university. “Didactic lessons have no value when they cannot be tested in practice.”

  • He doesn’t care that highschools don’t have half years.
  • Denoting 1 + ½ as 1½ risks confusion with multiplication: compare 2√2. See Elegance with Substance.
Freudenthal, 1948:120

Freudenthal, 1948:120


Freudenthal doesn’t discuss level transition and thus is still a distance from the notion of the Van Hiele levels.

He discusses how he lets his son construct a table until the penny drops and his son gets it. This still isn’t the Van Hiele level transition. Van Hiele levels are a more complex phenomenon, that students perform at some plateau and cannot proceed until a major transition occurs. Merely not getting it is not enough. There must be a system in not getting it.

Like there is a system in that IMU / ICMI doesn’t get it yet that Freudenthal committed his fraud, after I reported it in September 2014.


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