Monthly Archives: October 2015

Listening to Xarchakos Koritsia ston Hlio


Mathematics teachers can teach you about x to the power of x and how to find its derivative. They tend to be unaware of the power void in the organogram in mathematics education, relating to their own responsibility on how they teach you. Hm. Say again ?

Dutch Parliament distinguishes between the What and How in education:

  • Parliament decides What is taught
  • Teachers decide How they do this.

This principle was established by the Dijsselbloem commission in 2008. Jeroen made his career by observing that too much had gone wrong in the past by former Parliaments not respecting the responsibilities of mere teachers. See my earlier report in 2013 on Dijsselbloem on money and math.

His excellency Jeroen Dijsselbloem, minister of Finance of Holland, and president of the EurogroupHis excellency Jeroen Dijsselbloem, minister of Finance of Holland,
and president of the Eurogroup (source: wikimedia)

This division of labour between Parliament and teachers sounds fine, also when Parliament decides that pupils and students should be educated on mathematics, except … when math teachers don’t teach mathematics but teach so-called “mathematics”.

To avoid over-quotation, I will write math teachers instead of “math” teachers.

Defunct teachers in a power void

When teachers are defunct then Parliament needs to be able to observe that it doesn’t get what it wants. But can it do so, when it wants to remain at some distance ? Will it wait till Hell freezes over ?

What happens when there is a power void in the world of How, with no way to settle issues except a dirty math war ?

  • Mathematician Hans Freudenthal and his minions (I did enjoy watching this movie “Minions” with my youngest son (a requirement)) have been pushing “realistic mathematics education” (RME). This has been wreaking havoc in education – and in Dutch society as a whole, in so far as it depends upon competence in math (but it may also be taken over by China).
  • Mathematician Jan van de Craats has been pushing for traditional math education. But Van de Craats has no training as a math teacher and is creating his own conundrums in mathematics education.
  • Also, RME has been exported to the USA, and now the OECD is transporting it back to Holland under the guise of “21st Century Skills”. Van de Craats may not have expected this boomerang, and may now be poised to export his own delusions to the world.

To resolve the power void on the How, my suggestion from 2008 is to have a Simon Stevin Institute (SSI), that can implement a proper mathematics curriculum and train and advise teachers on didactics. A major role in SSI is for teachers with a training on empirical methods.

The distinction between mathematics and so-called “mathematics” can be indicated by these two tables.

What & How and math & "math"

What (math or “math”) & How

Neoclassics & Classics and math & "math"

Math (neoclassic or classic) versus “math” (old or new)

Legend for the tables
  • Freudenthal was a mathematician who assumed the role of math educator for which he hadn’t been trained. He stole ideas from teacher Pierre van Hiele. See here.
  • RME textbooks tend to avoid mixed fractions, discovered Liesbeth van der Plas. If tests give you a B instead of an A then this still is above average, so don’t complain that you haven’t been taught everything.
  • Jan van de Craats is a mathematician who opposes the chaos that RME is wreaking. He has no training on education either, and is creating his own chaos. He doesn’t defend Van Hiele against Freudenthal’s fraud, likely because he doesn’t understand what that didactic issue is about. See the breach of integrity of science and the weblog entries on van Hiele and on fractions.
  • Classical math is 2 + ½ while it is confusing to write  2½  (two-times-a-half as in 2√2).
  • Pierre van Hiele suggested that fractions could be abolished, and instead he suggested using the power notation which students have to learn anyway.
  • My own suggestion is to use H = -1 so that the use of -1 can be avoided. Pupils who are still learning arithmetic and who see -1 might think that they must subtract something. Instead they learn the rule that x x^H = 1 provided that x ≠ 0. Later when powers and roots are introduced then they can see the value for H (eta).
  • See the Appendix below for the abolition of fractions by means of H. This supplements our earlier discussion.
Long division

A key role in the debate about RME is for long division. Charles Fadel, who has no degree in math education but a bachelor in electronics and an MBA in international marketing, advises the OECD on “21st Century Skills”, and wonders, see his weblog, whether kids must learn long division. My response as a teacher is that if you want kids to learn arithmetic then they must also master long division. I hope that Fadel supports the notion of a Simon Stevin Institute, which provides the environment to properly discuss his ideas, and that he stops using his elbows in an environment with a power void. Stop taking advantage of an old lady who cannot defend herself.

Traditional mathematicians insist that kids learn safe and sure algorithms, and that practice makes perfect. RME suggests that kids need to understand what they do, and that it doesn’t matter how they reach the answer as long as it is good. RME forgets that proper arithmetic algorithms also prepare for later algebra, so that it really matters how you arrive at an answer. Apparently long division got abolished as “mechanical”, and replaced by a different method, called “partial quotients“. This method is also mechanical but deviously not called so since it allows more room for time-consuming random-guessing.

My position is the neoclassical one. Something is to be said for arguments on both traditional and RME sides. There is little use in having this debate, when there is no Simon Stevin Institute that provides a level playing field for this discussion. (Find another hero (-in) for your own country.)

Organogram with the void

The key contribution of this weblog entry is this organogram of mathematics education in Holland. It shows a proper power structure for Parliament and its What. It also shows the void for the How. It is a jungle out there, with all kinds of commissions, and two clear wrongs: (a) mathematicians are meddling, because they regard math ed as their subject, which it isn’t (see also Norma Presmeg’s diagram here), (b) Parliament (deviously or desperately ?) tries to gain control over the How anyway, e.g. by the Inspection. In my analysis, the SSI must control the Inspection, instead of the current inversion.

Organisational Chart for mathematics education in Holland

Organisational Chart for mathematics education in Holland

Legend of the Chart:

  • There is a clear distinction between mathematicians and teachers of mathematics.
  • There is power base for What but a power void for How.
  • Mathematicians create noise on both What (2½) and How (RME, traditional).
  • Parliament is isolated from How by a layer of bureaucracy, but this requires quality public management, which is rather risky and not quite realistic.
  • Minister and Parliament have a tricky influence on How by means of the Inspection. In the past the Inspection imposed RME while teachers protested ! This is a wrong allocation of power. The Inspection must be under control of teachers.
  • The Simon Stevin Institute would redress this jungle and make for proper governance with shorter connections. The council of SSI would have teachers, parents, entrepreneurs, students, and also some mathematicians. See the book “Elegance with Substance” on your national equivalent.
  • NB. The meaning of the arrows is ambiguous. The meanings rather depend upon the boxes the arrows are between. The arrows from Bodies and Publishers to nowhere replace a network to other boxes. While the diagram focuses on math professionals, there are of course also pedagogues, psychologists, and other.
Nota Bene

NB 1. For Dutch readers, there is this memo. Observe that Jan van de Craats has not been supporting this 2008 proposal for a Simon Stevin Institute, but running his own sect.

NB 2. A key role is for Member of Parliament Paul van Meenen, math teacher, and member of opposition party D66. This party of “Democrats” (as they call themselves) has been founded in 1966 with the objective of having direct elections of prime minister and mayors, with a system of district voting, while also referenda were included in their “crown jewels”. However, these methods are not democratic but rather anti-democratic. The “scientific bureau” of D66 neglects and suppresses this critique. D66-founder Hans van Mierlo was rather a demagogue who was in love with the USA of John F. Kennedy and who disliked the Dutch polder methods. Math teacher Paul van Meenen should be able to show his D66 in mathematical manner that D66 can better abolish itself, since those “crown jewels” are not what D66 says they are. D66 lies to the electorate. See Voting Theory for Democracy and my political pamphlet calling for the abolition of D66 (it may be politics when you want to see respect for science). See also the relation of D66 to LibDem in the UK 2015 General Elections.

Appendix: Mathematical abolition of fractions (old notation)

Consider the crooked so-called “mathematical” division of mixed fractions: 2½ / 3⅓ .

Van der Plas (2008) reports that this operation hardly occurs in “realistic mathematics” textbooks. Traditional mathematicians like Van de Craats et al. and Hung-Hsi Wu from the USA want to see a lot of practice on this again. For students the notation however is confusing, for in handwriting it is easy to get 2 ½ / 3 ⅓ = 1 / 3 ⅓ = 1 / 9.

The mathematical meaning of the inverse x^H is that: x x^H = 1 (for x ≠ 0).

On the calculator we find a numerical approximation of x^H by: (x)^(-1).

A relation for exponents is: x = (x^H)^H.

Above expression becomes in neoclassical mathematics: (2 + 2^H) (3 + 3^H) ^ H

This is a new notation. Who is used to it may take longer strides. For now, we take small steps. A classical operation may be at least as long. Properties of H (“eta”) are stated in the bookA child wants nice and no mean numbers (2015) with some observations that might be useful for elementary school (I have no degree on that area).

Eventually, students must learn to handle exponents. The following might seem complex but eventually it will be faster and more insightful. Obviously, this is only an expectation, and it must be checked with pupils whether this expectation is corroborated. It are the pupils who determine what works. I can only hope that there will be a SSI such that this testing can be done without the mud of the current dirty math war in Holland (supported by Freudenthal Institute in Utrecht and Jan van de Craats at UvA).

Operations by steps are:


For comparison: the crooked manner of traditional so-called “mathematics”, in which the notation determines what must be done, and in which you don’t merely denote what you are doing.


This is a very serious issue. Nothing to laugh about. Which reminds me that humour is a serious issue too. One problem in this discussion about Pierre and Dina van Hiele – Geldof and Hans Freudenthal is that participants may not laugh enough. It is a dirty math war and no fun. Slinging mud and no throwing of cakes with whipcream. I consulted my source on this, professor of humour Art Buchwald. He advised: “When you try to be funny try to remember that people should laugh.”

Readers may lose sight of what the issue is. Too many texts. Do you still remember it ? Some didactics may be required. A picture tells more than a thousand words.

It was already necessary to explain that Hans Freudenthal is no demon. I never said or implied that he would be a demon, but somehow people like to demonize. It will also be useful to explain that Sacha la Bastide – van Gemert is no demoness. Both are mere fallible human beings. When both are treated with respect then all will be well. Some misunderstandings will be cleared and some errors repaired.

Below I will show the problematic email exchange with LB-VG of 2014. The exchange is in Dutch, and thus I provide an independent Google Translation too, so that you can ask your Dutch friends what the fun language actually means. Hopefully the MAA provides for independent translation.

The problem is that LB-VG said that she did not have time (“now”) to look into the inconsistency that I spotted in her thesis: yet, in 2015 she put out an English translation ? For all I know the inconsistency is still there, now for a larger audience (the world). It is not just that she took me for a walk in the woods and left me there. It is also that she provided English readers (the world) with a text that she knew (had been alerted to) was problematic. And indeed, when we look closer at the source that she had (the Euclides article of 1957), then her treatment is even more problematic: see here.

On the bright side: LB-VG did not forge data like Diederik Stapel.

It will be didactical to first look at what she didn’t do and then at what she did do.

LB-VG didn’t forge this letter

Let us first look at the letter that LB-VG didn’t forge. We can stamp it with the “Uncle Hans is watching you” poster that we discovered from some fans in an earlier weblog. Do not forget to check out the website of Raymond Johnson at the Freudenthal Institute USA, who is closely tracking issues for criticism that he wants to neglect.

Forged letter of April 1 1957

Forged letter of April 1 1957, True poster by Dr Bote September 8 2013

The importance of not-forging this letter by LB-VG is:

  1. She doesn’t have to forge letters when she can achieve the same end by including an inconsistency in her thesis, and not looking into this when she is alerted to this, and then translate it into English. The thesis may portray Freudenthal as a hero and not a crook.
  2. She may actually have looked into the question on the inconsistency, and decided that in her mind there was no inconsistency, so that she thought that she honestly could proceed with the English translation. She only forgot, or simply didn’t have time, to inform me about this check-up and decision of hers. Why should she inform me ? I am just another scientist asking a question, and there are so many other scientists and other people asking questions. (The problem with answering a question is that this may cause new questions, in particular when you don’t really answer the question.)
  3. She may have reasoned that the thesis was accepted in 2006, so that the English translation should be about that accepted thesis as it is. This would reflect the true state of affairs of 2006, with scientific backing by thesis supervisors Karel van Berkel and Jan van Maanen (and others in the committee). Any discussion about new questions could happen after putting out the translation.

Point 1 shows the didactic power of the forged letter: it captures what is happening, and how hero worship may make people blind to hero error.

Point 2 would confirm the incompetence, and also that she may not have known enough about the Van Hiele level theory (though misreading statements that are very clear). Perhaps the thesis was overambitious, with too many topics, or the lure too great of including the level theory as another feat by Freudenthal ?

Point 3 would be self-serving. It is an argument by a lawyer, not a scientist.

PM. After LB-VG stated not to have time, I did query these other people involved with the thesis. They declined interest in the same manner, see later publications of emails.

The LB-VG statement in 2014 that she didn’t have time

It will be useful to start with the shortest email. The Dutch original:

From: Bastide-van Gemert, S la
To: Thomas Cool / Thomas Colignatus
Subject: RE: N.a.v. uw proefschrift, Hoofdstuk 7, Van Hiele niveaux
Date: Tue, 19 Aug 2014

Geachte mijnheer Cool,

Dank voor uw e-mail. Ik heb nu helaas geen mogelijkheid inhoudelijk te reageren, maar wens u veel succes met uw artikel.

Met vriendelijke groet,

Sacha la Bastide

Google Translate actually gives a rather fair translation. It curiously eliminates the “now” (Dutch “nu”). In Dutch, LB-VG still leaves me hope that she will look into it later.

Dear Mr. Cool,

Thanks for your email. I have unfortunately no opportunity to respond to the content, but wish you much success with your article.


Sacha la Bastide

Observe that emails from the University Medical Center Groningen (UMCG) come with the following disclaimer, which is inappropriate to include in this kind of scientific exchange:

“The contents of this message are confidential and only intended for the eyes of the addressee(s). Others than the addressee(s) are not allowed to use this message, to make it public or to distribute or multiply this message in any way. The UMCG cannot be held responsible for incomplete reception or delay of this transferred message.” (Included in the LB-VG email)

My email of 2014 that asked about the inconsistency

Dutch original:

Van: Thomas Cool / Thomas Colignatus
Verzonden: 19 August 2014
Aan: Bastide-van Gemert, S la
Onderwerp: N.a.v. uw proefschrift, Hoofdstuk 7, Van Hiele niveaux

Geachte dr. La Bastide – Van Gemert,

Heeft u nog interesse in uw proefschrift, of bent u doorgegaan naar nieuwe terreinen, zoals de epidemiologie ?

N.a.v. de situatie in het onderwijs in wiskunde en rekenen kwam ik ertoe ook te kijken naar de invloed van Hans Freudenthal.

Relevant leken daartoe ook de herinneringen van David Tall, een Engelsman. Toen ik hem e.e.a. navroeg begon hij ook over Pierre van Hiele en zijn eigen jongste boek (2013).

E.e.a. leidde tot dit artikel, beoogd voor een tijdschrift:

Uw Hoofdstuk 7 bespreek ik op p7-8. Mijn conclusie is dat u eigenlijk inconsistent bent, wanneer de Van Hieles in 1957 in Euclides al een algemene geldigheid voor hun theorie claimen, door u geciteerd, en u tegelijkertijd stelt dat Freudenthal dat pas aanbracht. M.i. heeft u dan een roze bril t.a.v. Freudenthal gehad, en niet doorgehad wat hier allemaal gebeurde. Per saldo kom ik tot de conclusie dat niet alleen Freudenthal maar nu ook David Tall een neiging hadden / hebben om Van Hiele in het hokje van de meetkunde te plaatsen, terwijl de Van Hieles juist in relatie tot Piaget een algemene theorie presenteerden met meetkunde slechts als voorbeeld.

Ik houd me aanbevolen voor een reactie.

Met vriendelijke groet,

Thomas Cool / Thomas Colignatus
Econometrist en leraar wiskunde

Google Translate, perhaps as curious as the letter that LB-VG didn’t forge, but with the same idea.

From: Cool Thomas / Thomas Colignatus
Sent: 19 August 2014
To: Bastide-van Gemert, S la
Topic: N.a.v. your thesis, Chapter 7, Van Hiele niveaux

Dear Dr La Bastide -. Van Gemert,

If you have any interest in your dissertation, or are you moved on to new areas such as epidemiology?

N.a.v. the situation in education in mathematics and arithmetic I committed came to look at the influence of Hans Freudenthal.

Relevant to this end also seemed the memories of David Tall, an Englishman. When I queried him eea he started about Pierre van Hiele and his most recent book (2013).

E.e.a. led to this article, intended for a magazine:

Your Chapter 7, I discuss on p7-8. My conclusion is that you are actually inconsistent, when the Van Hiele in 1957 in Euclid already claiming a universal validity for their theory, you quoted, and you set it at the same time that it was only put in Freudenthal. MFI you then have a rose-tinted glasses had Attn Freudenthal, not by what had happened here. On balance, I come to the conclusion that not only Freudenthal but now David Tall had a tendency / have to Van Hiele in the box to place the geometry, while the Van Hiele’s right in relation to Piaget presented a general theory to geometry just as example.

I love to hear a response.


Cool Thomas / Thomas Colignatus
Econometrician and math teacher

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.

The report of 2014 is that academic professor Hans Freudenthal (1905-1990) committed fraud. He stole major didactic ideas from mathematics teacher Pierre van Hiele (1909-2010) with his practical experience, misrepresented those, and claimed without evidence that these falsified ways would work in the classroom.

  • The implication of this report is that he was a crook.
  • The report is not that Freudenthal was a demon and ate babies for breakfast.

On occasion, someone reacts that Freudenthal did also some good things. I don’t know why anyone deems it necessary to tell me this. Does someone think that I hold Freudenthal for a demon ? Really, if I had discovered that Freudenthal ate babies for breakfast, I would have reported this. None of that. The report is just what it is about.

Indeed, the implication is that major elements in mathematics education in the world have been built upon this fraud. When math education generates poor results, one element is that it doesn’t use the proper Van Hiele theory.

A reader may be so shocked by this, that it translates for brain cells A into: “this is so horrible that Freudenthal might as well be a demon”, so that brain cells B feel the need to correct with: “ho-ho, tut-tut, calm down: fraud isn’t the same as being a demon”, whence brain cells C get confused and fire the neurons: “it must be the messenger of the bad news who told us that Freudenthal was a demon, while we discovered that he just committed some fraud, and he really did some good things too”, and then the rest of the brain D comes to the conclusion: “let us respond in that manner and then go back to sleep”. Apparently it are even highly trained people whose brains work like this.

When you feel the urge to tell me that Freudenthal did also some good things: train some brain cells E to fire neurons: “hey, it generally pays to really read what a report says”.

Freudenthal’s theft went unnoticed for a long time, not because it was such a clever ploy, indeed it is petty theft, but because of Dutch society, with its respect for the highly learned professor and dwindling respect for the common teacher. It could have been discovered by interviewers Gerard Alberts and Rainer Kaenders in 2005 if they had listened to the answers (and asked a bit more). It could have been found by Sacha la Bastide – van Gemert (LB-VG) in her 2006 thesis if she had been alert – and known more about Van Hiele’s work. It could have been discovered by David Tall when he discovered the importance of the Van Hiele levels, when Van Hiele passed away in 2010 and this caused him to think about it. But Tall misread Van Hiele, and didn’t stop to wonder whether smart Van Hiele really hadn’t seen the general applicability himself, and thus Tall didn’t investigate the other publications by Van Hiele to check it up. Tall is an expert on mathematics education and should have read it long ago in Van Hiele, Structure and Insight: A Theory of Mathematics Education. Academic Press 1986.

In May 2015 the Dutch Association of Teachers of Mathematics (NVVW) celebrated its 90th anniversary. As part of the celebration the colleagues made an electronic archive of all editions of the journal Euclides. Thus a wealth of didactic material is available now, albeit still in the sink of Dutch language.

Let me use this archive to look at two articles in Euclides:

  • one by Pierre and Dina van Hiele at the occasion of their Ph. D. theses of 1957, and
  • one by Hans Freudenthal of 1948, to witness that he was involved with math education already back then.

For the Van Hieles, download the article from the archive. We shall have these texts below: (i) a paragraph from the archive, (ii) transformed for use for Google Translate, (iii) my translation into English, (iv) Google’s translation (with its crookedness contributing to the fun element). This seems the best way to assure the readership that there is a decent translation. Of course I invite bodies like the MAA to generate an independent translation.

Subsequently, we will see what LB-VG makes of this 1957 article.

This entry today looks at the Van Hieles only. The next entry for tomorrow is for Freudenthal. For Freudenthal’s paper I skip step (iii) since you can do the exercise as well.

Pierre and Dieke van Hiele in Euclides 1957 (iii)

It is best to start with my translation so that you as a reader know what this is all about.

To keep in mind: the purpose is to show that the Van Hieles claimed general applicability, before Hans Freudenthal and David Tall tried to take it away from them. And we don’t want to show this by a false translation but by presenting the fact. In this translation, I adopt the English preference for shorter sentences, and I insert the word “Bildung” to better express what the Van Hieles intend.

“Above, we presented a didactic approach to introducing geometry. This approach has the advantage that students experience how you can make a field of knowledge accessible for objective consideration. For such a field a requirement is that students already have command of global structures. They experience how they proceed from those to further analysis. The approach presented here for geometry namely can be used also for other fields of knowledge. Whether it will be possible to treat such a field also in mathematical manner depends upon the nature of the field. For mathematical treatment it is necessary, amongst others, that the relations do not lose their nature when they are transformed into logical relations. For students, who have participated once in this approach, it will be easier to recognize the limitations than for those students, who have been forced to accept the logical-deductive system as a ready-made given. Thus we are dealing here with a formative value (Bildung), that can be acquired by the education in the introduction into geometry.” (Translation by TC)

The confusion by Stellan Ohlsson also surfaces here. The Van Hieles move from concrete to abstract and from global to precise. While Ohlsson’s confusing terminology has from abstract to concrete.

Pierre and Dieke van Hiele in Euclides 1957 (iv)

Here is Google Translate (our fun element):

“The above-indicated way to start the geometry teaching has the advantage that the students experience, how an area of knowledge, which one global structures owned by analysis accessible to objective considerations. The here for the geometry specified path can clear are also used for other fields of knowledge. Whether there will also be possible to the field of knowledge finally mathematize, depends on the nature of the field off. Necessary for this is, after all, among other things, that the relationships are not denatured, when they are converted into logical relationships . for those who have participated once in this method are active, it will be easier to recognize the borders than for those who have had to accept given the logical deductive system as a ready-made. so we have to do here with an educational value , which can be obtained by the teaching of the start of the geometry.” (Google Translate)

La Bastide – Van Gemert’s treatment of Van Hieles 1957

LB-VG’s thesis chapter 7, Dutch original online, has this text on page 202, my translation and emphasis:

“Now the Van Hieles had thought about it themselves as well to apply the theory of levels to other subjects in mathematics education. Already in 1957 the Van Hieles indicated, in an article in Euclides about the phenomenology of education that gives an introduction to geometry, not to exclude that possibility: [quote and footnote 82]” (Translation by TC)

The curious points about this text are:

  • The Van Hieles do not limit this to mathematics education only. They speak about other fields of knowledge. It is LB-VG who puts it into the box of mathematics education only.
  • It isn’t “not excluded” but emphasized.
  • The general claim is in the theses (ceremony July 5 1957) under supervision of Freudenthal and not just the article (October 1 1957).
  • Why not quote the full paragraph ? It would show why the Van Hieles select geometry also for its ability by excellence to teach this general lesson. Pierre van Hiele’s thesis has the word “demonstration” in the title, to that the discussion of geometry is only intended to demonstrate the general applicability (in the same manner as demonstration is used in geometry itself).

LB-VG’s text is here. Below under (i) you can see the full paragraph of the quoted text.

La Bastide - Van Gemert, 2006 p202

La Bastide – Van Gemert, 2006, Ch. 7, p202

Pierre and Dieke van Hiele in Euclides 1957 (i)

The original full paragraph in Euclides reads:

Pierre and Dina van Hiele, Euclides 1957, p45

Pierre and Dina van Hiele, Euclides 1957, p45

Pierre and Dieke van Hiele in Euclides 1957 (ii)

Thanks to the 90th anniversary of NVVW we can just copy & paste. We remove the paragraph marks and double spaces, i.e. replace “^p” by a space ” “, and then replace double spaces ”  ” by single space ” ” again. We also remove some glitches that the OCR interpreted as dots, dashes and quotation marks, that make Dutch look even weirder. For fairness to Google we also replace the abbreviations and put the h in mathematiseren. If you want to, you can take this text and have Google translate it into your own language.

“De hiervoor aangeduide wijze om het meetkunde-onderwijs te beginnen heeft het voordeel, dat de leerlingen ervaren, hoe men een kennisgebied, waarvan men globale strukturen bezit, door analyse voor objektieve beschouwingen toegankelijk kan maken. De hier voor de meetkunde aangegeven weg kan namelijk ook voor andere kennisgebieden gebruikt worden. Of het daar ook mogelijk zal zijn het kennisveld tenslotte te mathematiseren, hangt van de aard van het veld af. Noodzakelijk daarvoor is immers onder andere, dat de relaties niet gedenatureerd worden, wanneer zij in logische relaties worden omgezet. Voor hen, die aan deze werkwijze eens aktief hebben deelgenomen, zal het gemakkelijker zijn de grenzen te herkennen dan voor hen, die het logisch deduktieve systeem als een kant en klaar gegeven hebben moeten aanvaarden. We hebben hier dus te doen met een vormende waarde, die verkregen kan worden door het onderwijs in het begin van de meetkunde.”

Part 2 is on Freudenthal 1948.

Tracing the confusions on Pierre van Hiele can be gratifying since you meet with all kinds of interesting research and the people doing this, but it also appears to be an arduous task since the number of possible confusions is amazingly large.

The day before yesterday, I discussed the review at MAA by Annie Selden of the book by Sacha la Bastide – van Gemert. Publisher Springer proudly quotes from this review. Indeed, the world will tend to rely upon professor Selden’s reputation. The MAA has an Annie and John Selden Prize. See also professor Selden’s impressive cv and statement at LinkedIn: “Specialties: Mathematics education,Qualitative research, Curriculum development, Editing”. Since I myself wrote a book on logic and looked at proofs for some mathematical theorems, like by Gödel and Cantor, I am looking forward to read, at some hoped for moment in the future, the Selden’s paper on teaching how proper proofs are constructed.

However: the Selden review missed Freudenthal’s fraud.

Let me restate the quote at Springer and give the two screenshots, and then discuss.


“It concentrates on the historical development of Freudenthal’s ideas on the didactics of mathematics. … it would primarily be of interest to mathematics education researchers, especially those who use RME as their theoretical framework and to those interested in the history and development of the field … .” (Annie Selden, MAA Reviews, June, 2015) QUOTED AT SPRINGER

Screenshot of the page at Springer

Screenshot of the page at Springer

Screenshot of the MAA John and Annie Selden Prize

Screenshot of the Annie and John Selden Prize at MAA


I informed professor Selden about the weblog text, and gave my estimate that it shouldn’t take too much time to confirm the fraud, once you know what to look for. I also informed her that historians would be interested in her confirmation. She replied succinctly:

Tue, 13 Oct 2015
Dear Mr. Cool,
I am not a historian of mathematics. I am also not a historian of mathematics education. I have no knowledge of the alleged fraud (on the part of Professor Freudendthal [sic]) that you mention in your email.  I cannot help you.
Also, I am very busy with my own work and concerns right now.
Sincerely yours,
Annie Selden
Math. Dept.

We should not guess a respectable lady’s age. Selden has a B.A. from 1959 and thus might have been born around 1938. We thus will not do the math but respect her need to prioritise. David Tall (1941) wrote me last year – then at age 73 – that he was in the autumn of his life and that he had to prioritise too. He considers his confusion about Van Hiele not his most urgent issue. I can only respect this. Henceforth I inform professor Mike Thomas of Auckland (NZ) who may filter points of relevance for Tall – such as the publication of the English translation of the LB-VG thesis.

Still, Selden’s reponse causes some questions, which I now transferred to the MAA Reviews editor together with the earlier ones.

  • There is no alleged fraud. I don’t allege. I don’t appreciate it when some suggests that I would allege something. I report on what I discovered and invite others to size up the evidence with the option to correct me. I can imagine that professor Selden has no time to look at the evidence but this doesn’t allow the inference that I only allege something. I presume that professor Selden is so pressed for time that she didn’t have time to think about a proper expression instead of “alleged”.
  • I didn’t ask for help but was trying to help her.
  • I never implied that professor Selden was a historian. I contacted her on her reviews (also the other on Freudenthal’s 1973 book).
  • Her review recommends the LB-VG book for “those interested in the history and development of the field”, and if professor Selden now feels that she is not qualified for such a statement, then this should be retracted from the review before the interview as a whole is removed as inadequate.
  • I did inform Selden that I copied my message to La Bastide – Van Gemert and some other historians. The fraud can be established by persons with the expertise also stated by Selden, which is also my expertise: mathematics education and its research. I am no historian either. Those who take it upon themselves to write or review the book should be able to establish it, who need not necessarily be historians. But I find it useful to involve historians as well, since they may have additional expertise, and since they write books on history. (Except Amir Alexander, who first selects the story and then selects some data that fit the story.)

This again shows how horrible fraud is. It wreaks havoc all over.

Originally La Bastide – Van Gemert spent years of her life investigating what she thought to be an important researcher on mathematics education. Last year it appeared that she was only tricked in believing such, and that she only made a record of mostly misrepresentation, nonsense and manipulation. When I informed her about this last year, she declined looking into this. Instead she proceeded with the English translation, which is deliberate neglect. It is a deliberate act with additional neglect of consequences for the publisher, the reviewers and the readers, and what they do further. The ebook comes at the hefty price of $139, and it depends upon the percentage for LB-VG whether she is cashing in on Freudenthal’s fame in the circles of mathematics education (or Van Hiele’s fame, with chapter 7).

There was already David Tall as a victim in 2014, since I asked her to look into this in that context. With Springer as a second victim, we now find a third victim in a respectable lady who thought to do well with a review, relying upon the reputation of others, relying about the “scientific process in Holland”, and who is in the autumn of her life too and who doesn’t know where to find the time to restore the error when the bad news arrives. Can we consider it mathematically proven that LB-VG is harrassing senior citizens ?

We can only hope that the MAA Reviews editor has more resources and deals with this properly.

Intermezzo: On the brighter side

In the wonderful world of Google, tracing the above generated a paper by Norma Presmeg in Fried & Dreyfus (2014), also at Springer.

First there is the following diagram, useful for who wants to judge upon expertise, such as on who would be qualified for issues of fraud in mathematics education research. I agree with the diagram, except that I propose to regard the diagram as per Van Hiele level and math topic. There could be needless confusion when researchers on higher education make statements on primary education that they are not qualified for, as well.

Diagram by Norma Presmeg p 46 with "Topic !" amendment

Diagram by Norma Presmeg in Fried & Dreyfus (2014:46) with “Topic !” amendment

Secondly, Presmeg recollects, complimenting Van Hiele but restricting it to geometry, instead of emphasizing the general applicability (like the law of conservation of energy), and the work was done in the 1950s, with the two theses by Dina and Pierre in 1957:

Norma Presmeg (2014), p52-53

Norma Presmeg in Fried & Dreyfus (2014:52-53)

Intermezzo: Return to dismay

As happy we were in discovering the Presmeg article, as great is the dismay again to again see the article by David Tall in the same volume, in which he states his confusion on page 225, as if Van Hiele didn’t state the general applicability of his theory (and focused on geometry only).

David Tall, in Fried & Dreyfus (2014:225)

David Tall, in Fried & Dreyfus (2014:225)

The Fried & Dreyfus volume was available in January 2014 and my discovery was in July 2014. See this article later in August for Tall’s misreading of his reference to Van Hiele (2002). There is no deliberate misrepresentation on Tall’s part, just a significant error. Though Tall will not correct, I still advise Springer to put a health warning there.

While Sacha la Bastide – van Gemert has the inconsistent report that both Van Hiele and Freudenthal would be the originators of the idea of the general applicability of the theory of levels, readers may now observe that David Tall claims it for himself: not only misreading Van Hiele (2002) but also unaware of Freudenthal’s earlier claim and misrepresentation. You can be famous without being read well.

Email to the MAA

Date: Thu, 15 Oct 2015
To the MAA, executive director Michael Pearson
CC Book Reviews editor [..], professor Selden, dr La Bastide – van Gemert, historians
Subject: Urgent: wrt two reviews by Annie Selden who didn’t notice the fraud by Hans Freudent[h]al

Dear dr. Pearson,

Let me urge you to do your best to preserve professor Annie Selden’s health and reputation, in what has become a sordid matter, due to fraud, incompetence and deliberate negligence in Holland.

Yesterday I informed MAA Book Reviews editor […] on this, giving him a head start. As I told him, I just reworked my message to him into a weblog text, available here:

I copy to professor Selden, but with hesitation: She stated that she has no time for this. Yet, it are her two reviews that cause a problem. Given her age she is less subject to the illusion of youngsters that there is ample time to choose what to do. I want to respect that, but, in that case the human invention of “organisation” would have to take over. I urge you to do so.

I copy to dr. Sacha la Bastide – van Gemert, whom I subsequently will report to her authorities. I discovered the existence of the 2015 English translation only a few days ago, and this puts matters of 2014 into a different light. Seeing the consequences not only for David Tall in 2014 but also Annie Selden in 2015 proves to me that this unavoidable.

I also copy to the historians who I copied to earlier, so that they can see both professor Selden’s reply to me and that I try to deal with this in the best way possible.

I also copy to two Dutch historians of science, […] and Danny Beckers. Beckers earlier wrote a review in Dutch of the Dutch thesis by LB-VG, and he missed the fraud by Freudenthal too. Given the existence of the English translation, I have asked him to translate his review into English, and also look into the Freudenthal fraud.

Earlier I copied to Gerard Alberts and Rainer Kaenders, who interviewed Pierre van Hiele in 2005. I have come to the conclusion that Kaenders may well be in breach of scientific integrity as well. While Alberts has no background in mathematics education research, Kaenders has. He should be able to spot the Freudenthal fraud when it is pointed out to him. But he doesn’t react to this. He might be on holiday now and not read his emails. But there may also be more to this. I will inform him about this problem and will copy to you too.

I will put this email into above weblog text, turning names anonymous where needed.

Sincerely yours,

Thomas Cool / Thomas Colignatus
Econometrician (Groningen 1982) and teacher of mathematics (Leiden 2008)
Scheveningen, Holland

Sacha la Bastide-van Gemert (LB-VG) has her 2006 thesis (Dutch online) about Hans Freudenthal and the didactics of mathematics translated and now published by Springer: All positive action starts with criticism (2015). The ebook has a hefty price of $139 while the Dutch thesis is online for free, but perhaps translators of didactics of mathematics are a luxury indeed.

S. la Bastide-van Gemert (2015)

S. la Bastide-van Gemert

This translation is a surprise to me and comes with mixed feelings.

The positive aspect is that there now is an independent translation. Also the important chapter 7 Freudenthal and the Van Hieles’ Level Theory is still included, at a single price of $29.95. Thus readers can check my argument and my own translations from key Dutch sentences, as given in the appendix of my paper Pierre van Hiele and David Tall: Getting the facts right (2014). A key point there is that Freudenthal suggests that Van Hiele limited his theory of levels of insight to geometry only, so that it would be Freudenthal himself who would have seen the general applicability. However, Van Hiele already stated in his thesis of 1957 that this theory has general applicability, and Freudenthal was the thesis supervisor. See here for Van Hiele’s protest against Freudenthal’s abuse. Crucial is the observation that LB-VG’s thesis contains an inconsistency here, for she allows that both Van Hiele and Freudenthal would be the originators of the general applicability. (And later David Tall claimed that too.) So now you should be able to check yourself, and also read more on the context.

The negative aspect is that I am not going to pay $139 or even $30 to check whether the independent translation fits with the original Dutch and my own decent translation. It is conceivable that LB-VG changed some phrasings, and I would be in the dark about that. I just discovered this today and sent a message today about the situation, which got the reply of a leave of unstated duration. I just report the current situation.

The double negative aspect is that I asked LB-VG in 2014 whether she could resolve that inconsistency. In 2014 she replied that she didn’t have time for this. I presumed that this was because of her new job at the University Medical Center Groningen (UMCG). Now it appears that she still had some involvement in this subject. She must have checked the English translation since she indeed is in command of English. Perhaps my comment had alerted her to this issue ? Perhaps she realised that she did not know as much about Van Hiele as she thought she knew about Freudenthal ? I tend to feel misinformed about that lack of time.

She defended the thesis in Groningen in 2006, and there was a commercial Dutch version in 2006 as well. The Alberts & Kaenders interview of 2005 with Pierre van Hiele’s protest was the preceding year. It is not in her list of references. She might have been so absorbed in other details that she missed that interview. Were the supervisors Klaas van Berkel and Jan van Maanen, and readers like Martin Goedhart excused from reading that interview and subsequently linking this to a rather obvious question about that inconsistency ? Actually, I asked them in 2014 too, and I must still document that these academics are in breach of integrity of science because of not looking into this. This thesis has a major inconsistency, and one cannot leave it just like that, misinforming the readership who might not be alert to this.

Forced to google the issue afresh, it is interesting itself to see that also a reviewer like Danny Beckers (2007) apparently neither read that interview, for he characterises Van Hiele as a “friend” of Freudenthal. There are strong “frames” that appear to cause people to overlook Freudenthal’s fraud. One should hope that Beckers now also translates his review into English, and takes along the new information about the fraud.

The triple negative (think of the negative of Belgium beer Tripel) is that, most likely, the inconsistency is stil there, and that it is broadcasted into the English world how great Freudenthal was. Young teachers and researchers are at risk of buying into the story and getting misled and brainwashed and so on, making it much harder again to explain the fraud. Who would you believe: me and the evidence, or a thesis published by Springer with the original support by Klaas van Berkel, Jan van Maanen and Martin Goedhart, who declined to look into this in 2014, and let Springer publish it in 2015 ?

Indeed, the Springer glossy is suggestive of St. Hans Superstar, and Springer quotes from the MAA review (full text), with RME = “realistic mathematics education”:

“It concentrates on the historical development of Freudenthal’s ideas on the didactics of mathematics. … it would primarily be of interest to mathematics education researchers, especially those who use RME as their theoretical framework and to those interested in the history and development of the field … .” (Annie Selden, MAA Reviews, June 5 2015)

But please, ms. Selden, RME is a fraud ! Didn’t you read my weblog entry of last year ? There are no relevant “Freudenthal’s ideas”, there are “Freudenthal’s misrepresentations of Van Hiele’s ideas”.

Selden states:

“Chapter 6 covers the period from 1950 to 1957 when Freudenthal’s national and international reputation as a mathematics educator grew enormously. Also, towards the end of the period, his mathematical-didactical ideas were greatly influenced by the pedagogical dissertation studies of Pierre and Dina van Hiele on geometry. “

I had a different reading of this. In the period before 1957 Freudenthal’s ideas on education are rather bland. LB-VG describes how they grow into RME only after 1957, when he has the theses by the Van Hieles. Again, there is the suggestion that Pierre van Hiele looked only at geometry, while he stated the general relevance, and used geometry only for demonstration (with a wink reference to the role of demonstration in geometry).

Selden states:

“Of special interest to mathematics education researchers who use realistic mathematics education (RME) as their theoretical framework is Section 7.4 titled, “Freudenthal and the theory of the van Hieles: From ‘level theory’ to ‘guided re-invention’”. According to the author, it was during this time period that Freudenthal introduced the ideas of “guided re-invention” and the “anti-didactical inversion”. These terms “did not come out of the blue. … [B]oth concepts were already mentioned before in more guarded terms. But it is the first time that Freudenthal mentioned and defined them explicitly.” (p. 195).”

This is however where intellectual theft takes place. It is amazing that Selden doesn’t observe it, but, she might not know the work by the Van Hiele’s so well. The key question is whether the notions of “guided re-invention” and “anti-didactical inversion” are deep and special. If they would be, then Freudenthal could claim major discoveries. In fact, they turn out to be simplistic rephrasings of what Van Hiele already described. Van Hiele was interested in insight, and transitions to higher levels of insight. Now, isn’t invention the phenomenon of arriving at more insight ? It is basically just another word. The same holds for the Van Hiele process from concrete to abstract, that is opposite to Euclid’s Elements that starts with proofs. It is a bland rephrasing. Education in 1957 didn’t have a refined taxonomy such that the lawyers of the Lesson Study inquisition could haggle about student A having a Van Hiele level transition and student B having a Freudenthal guided re-invention, with numbers to show that Freudenthal made the more relevant discovery. The conclusion is that these are just rephrasings, and that Freudenthal could, once he had his own terms and publications, refer to his own work rather than Van Hiele. Case closed. Abolish the Freudenthal prize at IMU / ICMI. Read my letter to them.

PM. If you like to think about the difference between cars and ideas, then there is this argument. You might suggest that new ideas are always your own. Thus Freudenthal’s new phrases would still be something of his own, and he could always claim credit for them. For cars, this would mean that if the robber puts a new paint on your car, he can keep it. It is an interesting suggestion. It would also hold when the robber puts so much paint on the radiator and exhaust that the car would hardly run, like RME doesn’t hardly work. Thus, think about it. Your dear car, stolen and turned into a wreck, with the robber dancing and prancing atop.

Mathematics as an educational task 1973

Selden also reviewed Freudenthal’s “Mathematics as an educational task” (1973). I can only regard this title as a contradiction in terms, since mathematics is abstract and education is an empirical issue. The book is basically useless since it contains “Freudenthal’s ideas” about education but he didn’t know what he was speaking about.

Selden repeats the misrepresentation that Freudenthal gave: that Van Hiele applied the levels only to geometry, and that it was Freudenthal who discovered the general applicability. This is purely false. See the weblog text on the fraud and the paper on the confusion by David Tall (who apparently missed Freudenthal’s claim and started to claim it himself).

“During the 1950s, he directed the Ph.D. dissertations of Dieke van Hiele-Geldolf and Pierre van Hiele on geometry. Pierre was inspired to create his level theory by Dieke’s observations of her lower secondary pupils’ learning of geometry (…) Freudenthal, while noting that the van Hieles deserved all the credit for their discovery, saw levels as relative rather than absolute, attributed a rise in levels to reflection, and applied the theory to other areas of mathematics learning.” (Annie Selden, MAA Reviews, June 9 2014)

Translation is tricky

All this causes the question why the Royal Dutch Society for Mathematics has never succeeded in making an English translation of Van Hiele’s thesis. For them, mathematics is no educational task either. All the effort that the translator has spent on LB-VG’s thesis would have been much more valuable for translating Van Hiele’s thesis.

But I would not suggest that the same translator would do the job, since the Van Hiele terms for didactics require an independent translation. This LB-VG translator comes with the risk of having been trained on Freudenthal’s fraudulent misrepresentations. For example, when Van Hiele invented A and Freudenthal misrepresented this into B, then a translator trained on B might translate Van Hiele’s A as B too, so that it would be proven to the English world that Van Hiele had no originality of himself. It is actually amazing in how many ways you can abuse Van Hiele.

In 2005, Gerard Alberts (mathematician, historian) and Rainer Kaenders (mathematician, educator) interviewed Pierre van Hiele (1909-2010). The interview was published in the journal of the Royal Dutch Society for Mathematics, as G. Alberts & R. Kaenders (2005), “Interview Pierre van Hiele. Ik liet de kinderen wél iets leren”, NAW 5/6 nr. 3, september, p247-251. The publication is in Dutch and I will translate some parts into English.


The introduction to the interview is:

Pierre van Hiele is the silent force in didactics of mathematics in The Netherlands. He was teacher of mathematics and chemistry and never much stepped in the floodlights. His work receives broad international recognition and one cannot think about didactics of mathematics without it. His work is still studied, amongst others Stucture and Insight. Van Hiele is ninety-six. (Dutch: Pierre van Hiele is de stille kracht van de didactiek van de wiskunde in Nederland. Hij was wiskunde- en scheikundeleraar en is nooit veel op de voorgrond getreden. Zijn werk vindt brede internationale erkenning en is tegenwoordig niet meer weg te denken uit de wiskundedidactiek. Zijn werk, waaronder het invloedrijke Begrip en inzicht, werkboek van de wiskundedidactiek wordt nog steeds bestudeerd. Van Hiele is zesennegentig jaar.)

Hans Freudenthal (1905-1990) was supervisor to the 1957 thesis in which Van Hiele presented his levels of insight – see here for their crucial role in epistemology, comparable to the law of conservation of energy in physics. In 2014 I observed that Freudenthal committed fraud by misrepresenting Van Hiele’s work and by stealing elements for his supposedly “own” approach of “realistic mathematics education” (RME). This RME is a disaster because it doesn’t stick to Van Hiele’s proper didactics. Van Hiele was a teacher of mathematics and Freudenthal was an apparently jealous abstract thinking professor in topology. (Freudenthal’s wife Suus Freudenthal-Lutter was involved with pedagogy and perhaps this inspired him to similar efforts too.)

My discovery in 2014 of Freudenthal’s fraud happened in the context of discovering David Tall’s misconception, in which Tall reduces Van Hiele’s approach to geometry only. Tall claims to have found general epistemological applicability, which is correct and a useful confirmation of what Van Hiele already stated in his 1957 thesis, but which is incorrect in that Tall should not claim this general notion for himself and reduce Van Hiele to geometry only. It might be that Freudenthal was instrumental in Tall’s misconception. See Pierre van Hiele and David Tall: Getting the facts right (2014).

In all this, we have the Dutch language sink, in which Dutch readers have access to the local lingo that other language readers are barred from, except for still inadequate Google Translate and such.

The 2005 interview is important for understanding the situation but I have been hesitating to translate this myself. When I discovered the Freudenthal fraud, I felt that readers would better be served by translations that are independent from this discovery. Readers might all too easy think that such a discovery might be based upon wrong translations. See the Van Hiele & Tall paper for such translation issues. Thus, I asked Alberts & Kaenders whether they would be wiling to try to make such a translation. Up to now, there is no indication for this however.

Also, I recently discovered that Van Hiele has no page in the MacTutor History Archive, which is somewhat contrary to above observation by Alberts & Kaenders for his importance for mathematics education and its research. Notably, Gerard Alberts wrote an excellent and inspiring history booklet about David van Dantzig (1900-1959), who has an entry in the MacTutor archive. One would hope that something similar would be possible for Pierre van Hiele (only 9 years Van Dantzig’s junior).

My history comment caused some questions from the readership. These questions now cause me to give a translation of some key passages in the 2005 interview, selected on the Van Hiele – Freudenthal relationship, with the understanding that there should be an independent translation later on. I also provide the Google Translation, if only for fun.

NB. For vectors in primary education, see the pdf online: A child wants nice and no mean numbers(2015).

Page 247 on sabotage

“My relation with Freudenthal wasn’t so good, that I would go and drink coffee with him. Besides, Freudenthal has later frequently sabotaged my work, guys.”

Google: “So good was my relationship with Freudenthal not think I was going to drink coffee with him. Besides, Freudenthal has me later often put a heel, guys.

Dutch: “Zo goed was mijn relatie met Freudenthal niet dat ik met hem ging koffie drinken. Trouwens, Freudenthal heeft mij later nogal eens een hak gezet, jongens.”

Page 251 on Hans Freudental

What role did Freudenthal play in your life? “I did not mix well with Freudenthal. From the beginning. He was a bossy person. He did cause me to get ideas. That is rather all.”

Freudenthal used different descriptions of the process of abstraction. In the Vorrede zu einer Wissenschaft vom Mathematikunterricht he presented this process in terms of comprehension and apprehension. Did he also think differently about the role of levels of insight? “Yes, I believe actually that he did not really understand much about the levels of insight.”

Freudenthal was your thesis supervisor (promotor). Did he also help you in stepping outside of the small circle – with contacts outside of The Netherlands? “The last thing definitely not. No, the situation was actually that I had to vouch for myself. For example I remember a conference in America, at which a speaker referred to my work and said: ‘Mr. Van Hiele whom I am mentioning now is actually present in this very lecture hall. Mr. Van Hiele, please rise (so that everyone can recognize you).” Someone in the audience, a German, asked where he could read about my work. I replied that there would appear a book of mine in English shortly. Then Freudenthal who was also present said: ‘You can also read about it in my book.’ Which wasn’t true. He just was sabotaging me again. Freudenthal was like this, yes.” He was sabotaging you all the time? “Actually yes. Freudenthal never was a friendly person for me, no.” Where can he have been sabotaging you? Didn’t you work in entirely different environments? “Yes, but he tried to pinch something from me all the time.”

Later you got more recognition for your work. Were you able to make peace with him then? “Well, peace? No, actually not. In that case you first would have made war. I don’t make war.”

You presume that Freudenthal did not fully understand your work. Did you understand him, conversely? “Yes, I understood what I knew of him. And I often agreed with it too. I wasn’t in constant quarrel with Freudenthal. From his side, he had very much respect for my ideas on vectors in primary education. He praised me very much for that.”

Page 251 Google Translation

What role does Freudenthal played in your life? “I did not like very much with Freudenthal. From the outset, not all. He was a bossy person. He gave me ideas. That was it really. “

Freudenthal wielded other descriptions of the process of abstraction. In Vorrede zu einer Wissenschaft vom Mathematik Unterricht he explained that process in terms of comprehension and apprehension. Did he think differently about the role of levels? “Yes, I actually believe he levels that do not understand much.”

Freudenthal was your promoter. He did he nevertheless helped to come out – outside the Netherlands? “The latter certainly do not. No, it was just that I had to defend myself. Example, I remember a conference in America, where a speaker referred to my work and said, “Van Hiele what I am now, sitting here in the audience. Mr. Van Hiele state momentarily. “Someone in the audience, a German, asked where he could read my work. I replied that soon a book of mine would appear in English. Then Freudenthal said who was also present: “You can read them with me in my book.” That was not true. He just sat cross me again. Freudenthal was so, yes. “Was he can cross all the time? “Actually, yes. Freudenthal has never been a nice guy to me, no. “Wherever he may well have been bothering you. You still worked in very different environments? “Yes, but every time he tried to steal me something off.”

Later did you find more recognition for your work. Do you then can make peace with him? “Well, peace? Not really. Then you must first make war. I do not make war. “

You assumes that Freudenthal your job did not quite understand. You knew him, conversely, is it? “Yes, what I knew of him that I understood. And I was also often agree. I was not in permanent quarrel with Freudenthal. On his side, he had a lot of respect for my ideas of vectors in elementary school. There he has given me much to praise.”

Page 251 Dutch original

Welke rol heeft Freudenthal in uw leven gespeeld? “Ik had niet zo erg veel op met Freudenthal. Van begin af aan al niet. Hij was een bazig iemand. Hij bracht me wel op ideeën. Dat was het eigenlijk.”

Freudenthal hanteerde andere beschrijvingen van het proces van abstractie. In de Vorrede zu einer Wissenschaft vom Mathematikunterricht zette hij dat proces uiteen in termen van comprehensie en apprehensie . Dacht hij ook anders over de rol van denkniveaus? “Ja, ik geloof eigenlijk dat hij van die denkniveaus niet veel begrepen heeft.”

Freudenthal was uw promotor. Hij heeft hij u toch ook geholpen naar buiten te treden — buiten Nederland? “Dat laatste beslist niet. Nee, het was juist zo dat ik voor mijzelf moest opkomen. Ik herinner mij bijvoorbeeld een congres in Amerika, waar een spreker naar mijn werk verwees en zei: ‘die Van Hiele waar ik het nu over heb, die zit hier in de zaal. Meneer Van Hiele staat u even op.’ Iemand in het publiek, een Duitser, vroeg waar hij over mijn werk kon lezen. Ik antwoordde dat er binnenkort een boek van mij in het Engels zou verschijnen. Daarop zei Freudenthal die ook aanwezig was: ‘U kunt het ook bij mij in mijn boek lezen.’ Dat was dus niet waar. Hij zat mij gewoon weer dwars. Zo was Freudenthal, ja.” Zat hij u de hele tijd dwars? “Eigenlijk wel, ja. Freudenthal is nooit voor mij een leuke man geweest, nee.” Waar kan hij u nou dwars hebben gezeten. U werkte toch in heel andere omgevingen? “Ja, maar hij probeerde iedere keer mij iets af te snoepen.”

Later hebt U meer erkenning voor uw werk gevonden. Heeft U toen vrede met hem kunnen sluiten? “Nou, vrede? Eigenlijk niet. Dan had je eerst oorlog moeten maken. Ik maak geen oorlog.”

U gaat ervan uit dat Freudenthal uw werk niet helemaal begreep. Begreep U hem, omgekeerd, wel? “Ja, wat ik van hem kende dat begreep ik. En daar was ik het ook vaak mee eens. Ik verkeerde niet in permanente ruzie met Freudenthal. Van zijn kant had hij erg veel respect voor mijn ideeën van vectoren op de lagere school. Daar heeft hij mij erg om geprezen.”

Listening to Andriopoulos & Odysseas Elutis (1984): Prosanatolismoi


Let us discuss Gerald Goldin (2003), Developing complex understandings: On the relation of mathematics education research to mathematics. I presume that the reader has checked earlier discussions on Goldin (1992), on epistemology and on Stellan Ohlsson.

The paper’s abstract is:

Goldin (2003), p171

Goldin (2003), p171

It took me a while to come to grips with this paper. Suddenly it dawned on me that the English speaking world, including Goldin, makes a distinction between science and the humanities. This is what C.P. Snow (1905-1980) called The two cultures.

For Dutch readers these categories are crooked.

  • When Goldin opposes mathematics education research (MER), which in the English world belongs to the humanities, to science including mathematics then this is the distinction between science and the humanities. But for Dutch it sounds very strange to suggest that MER would be non-science.
  • Dutch has the single word wetenschap. How can Goldin oppose things that are the same (learning) ? How can he lump together things that are different (science and mathematics) ?

Dutch convention categorises the humanities as alpha (α), science and mathematics as beta (β) and the mixture as gamma (γ): those deal with alpha subjects but use beta methods. MER would be gamma.

I am not too happy with the Dutch categories since they don’t account for the separate position of philosophy and mathematics. The better distinctions are in the next table.

Categories for general science (science and the humanities)

Categories for general science (science and the humanities)

Above table is intended to categorise whole disciplines, like physics and economics. But we can also look at sub-areas within a discipline. Since both philosophy and mathematics can run astray without some link to the external world, my suggestion is that they both take mathematics education research as some anchor to reality (but they remain what they are when they refuse to do so). They might take an example of history writing, in which most history writing uses non-experimental methods (see the history on Pierre van Hiele) but some historians will rely on the experimental sciences to recover data from the past.

We are now ready to look at the paper.

Below we will see that Goldin opposes α + Φ versus β = δ + μ, forgetting about γ, while my analysis in Elegance with Substance (EWS) (2009, 2015) (pdf online) diagnoses the problem as α + μ versus γ, while δ + Φ have run away and no longer want to take part in cleaning up the mess. Professor Hung-Hsi Wu of Berkeley calls for help by research mathematicians μ to clean up the mess in ME and MER, but in my analysis we need help from engineers and other researchers in the empirical sciences δ + γ, see here.

Goldin 2003 on the decade since 1992: integrity for the disciplines

Goldin’s paper discusses his background, and he seems very well placed to discuss mathematics, ME and MER. Goldin sees a math war and tries to bring calm by increasing complexity. His article is complex itself so that those who pass the test of reading it will understand enough of the various sides of the discussion and be less likely to vilify the other side.

Goldin’s position is that discussants on MER must respect what other discussants on MER are doing and good at. Scientific integrity tends to focus on ethical behaviour of the individual but Goldin widens this to whole disciplines. Scientists must respect the humanities. The humanities must respect science. Otherwise there is no communication and no progress.

Goldin (1992) looked back at the New Math in the 1960s and behaviourism in the 1970s. When those ‘isms’ failed to produce improvement in mathematics education, the educational departments in the humanities grabbed the opportunity to claim their way to success. Goldin would agree partly, since he in 1992 also opposed the New Math and behaviourism. The humanities however created their own ‘isms’. We can now better understand Goldin’s position w.r.t. the decade 1992-2003.

Goldin (2003), p177

Goldin (2003), p177

A key observation is that Pierre van Hiele (1909-2010) is missing in this list and that Hans Freudenthal (1905-1990) committed fraud w.r.t. the work by Van Hiele: so that Goldin has a somewhat rosy view about the “without the far-reaching dismissals, oversimplifications, and ideologies”. The reference to Leen Streefland (1998†) may highlight the ‘ism’.

A Pro Memory point is that David Tall in 2002 apparently misunderstood the Van Hiele theory, as applying only to geometry and not to epistemology in general. This doesn’t seem to be due to ideology on Tall’s part, but there seems to have been some influence of Freudenthal in the misrepresentation of Van Hiele’s work. See my paper on getting the facts right.

An example with Leen Streefland (1998†)

I have not studied Streefland’s work any deeper than the following internet links just now. Those links fit the diagnosis of sectarian behaviour of Freudenthal’s “realistic mathematics education” (RME), and thus I see no reason yet to read more. Streefland belonged to the Freudenthal sect, see this ESM 2003 issue. Pierre van Hiele suggested in 1973 to look into the abolition of fractions, but Streefland (1991) perseveres with a book on “realistic education” on fractions.

See my 2015 book, pdf online, A child wants nice and no mean numbers, also commenting on the US Common Core program and professor Hung-Hsi Wu on fractions. Professor Wu does not belong to the RME sect but his traditional answer on fractions suffers from the intellectual burying of Van Hiele, which the RME sect so effectively achieved. The ‘isms’ are not without cost.

The strategy by Hans Freudenthal and his Utrecht sect – and these are adults who know what they are doing – is to absorb elements of Van Hiele’s work, but misrepresent it to fit their own ideology – which change does not diminish the intellectual theft. They achieve two effects: (i) for an innocent audience they ride the wave of the success by Van Hiele that they are jealous about, and (ii) they exclude Van Hiele himself from the discussion since “they tell it better” – and thus Van Hiele’s protest that his work is abused will not be heard. After all, Freudenthal was a professor in Utrecht with his own Ph.D. students who later became professors, and Van Hiele remained a mere mathematics teacher doing his writings in the weekend.

In this book on fractions, Streefland (1991) p2 states the following. We can excuse authors for the uncreative use of the word “level” that pops up everywhere. The true problem lies in the ideological following of Treffers (1987) and the neglecting in 1991 of Van Hiele’s own work (not only on fractions of 1973).

Streefland (1991), p2

Streefland (1991), p2

This closes the circle: (a) Treffers (1987)’s misrepresentation of Van Hiele’s work is not only in Streefland (1991) but (b) was also copied in the 1993 MORE study, (c) critically discussed by Ben Wilbrink, here, (d) which alerted me to Wilbrink’s misrepresentation of Van Hiele. Wilbrink namely follows the RME abuse, and he also tends to include Van Hiele in the RME sect instead of saving him from it.

Thus we are back into the Dutch math war swamp, with on one side the RME sect and on the other side Jan van de Craats and others who try to save “traditional mathematics education sanity” alongside psychologist Wilbrink with his misapprehension of empirical science and Van Hiele. My position is that of Sherlock Holmes observing it all from the high ground aside.

Traditional mathematics is crooked as well. It e.g. involves torture of kids by fractions. There is every reason to desire change. It doesn’t help when mathematicians, who don’t have empirical training, team up with the humanities who don’t have empirical training either (i.e. α + μ).

Popper’s falsification

One reason why the humanities might be disrespectful of science has to do with Karl Popper’s demarcation theory to use falsification to distinguish science from non-science:

Goldin (2003), p178

Goldin (2003), p178

Goldin reminds us that the humanities are non-science, as seen from science and its experimental method. The humanities should heed the risk of turning this property into a claimed superiority.

The humanities seem to have learned that they should not claim higher wisdom, which they and only they can discover by reading old documents and watching plays by Aristophanes and Shakespeare and have reception parties afterwards to discuss the faculty gossip. But the humanities might still take the Humean skeptic position, and make fun of physics who can put electrodes upon skulls and in that manner likely will never be able to create the insights that a study of the humanities can generate (though they might actually prove some of the gossip).

Goldin’s argument: Physics can be skeptic too. Save those skeptic arguments for your autobiography, for they contribute nothing to the discussion.

My warning: Don’t make too much of falsification. See the discussion on epistemology and the definition & reality methodology.  Above δ and γ sciences rely for the empirical realm upon definitions, and a mathematician μ might well hold that definitions are non-experimental.

David Hume and Ernst von Glasersfeld

Reading a bit more about Ernst von Glasersfeld (1917-2010) was long upon my to-do-list, and Goldin’s article finally caused me to do so. Advisable is his own article Thirty Years Radical Constructivism, Constructivist Foundations 2005, vol. 1, no. 1, p9-12. It is very useful to see Von Glasersfeld’s background in mathematics (not completed because of WW 2), linguistics and cybernetics: γ rather than α. For methodological justification he might be forced to do some philosophy, but he rejects doing that.

Von Glasersfeld (1995) Radical Constructivism at ERIC is too much for now, though. I checked that he indeed discusses Hume, and also mentions the “problem” of induction (see my discussion of epistemology). Von Glasersfeld holds that the issue is not philosophy but finding mechanisms of cognition.

Comment 1: Reuben Hersh (2008) Skeptical Mathematics? Constructivist Foundations 3(2): 72, suggests that “radical constructivism” would be Humean skepticism, and I tend to agree.

Comment 2: Being a Humean skeptic is agreeable too. This (wiki-) quote by Von Glasersfeld seems accurate:

“Once knowing is no longer understood as the search for an iconic representation of ontological reality but, instead, as a search for fitting ways of behaving and thinking, the traditional problem disappears. Knowledge can now be seen as something which the organism builds up in the attempt to order the as such amorphous flow of experience by establishing repeatable experiences and relatively reliable relations between them. The possibilities of constructing such an order are determined and perpetually constrained by the preceding steps in the construction. That means that the “real” world manifests itself exclusively there where our constructions break down. But since we can describe and explain these breakdowns only in the very concepts that we have used to build the failing structures, this process can never yield a picture of a world that we could hold responsible for their failure.”

It is hard to disagree, except when you want to resort to Wigner’s magic again (see Appendix 2). But it doesn’t tell us how to design a course so that Johnny can learn arithmetic. Or how to abolish fractions.

Comment 3: Von Glasersfeld refers to Jean Piaget. Pierre van Hiele developed his theory of levels of insight, starting from Piaget as well, but eventually rejecting Piaget’s age-dependency and choosing for the logical structure that generates a general theory for epistemology.

It is a question what the contacts between Von Glasersfeld and Van Hiele were, and whether Hans Freudenthal was an interfering factor again. We find Von Glasersfeld (ed) (1991), Radical Constructivism in Mathematics Education, Kluwer, that contains a chapter by Jan van den Brink, since 1971 a member of Freudenthal’s sect in Utrecht. Searching the book generate 0 references to “Hiele”. The RME wiki on RME refers to Von Glasersfeld’s book but not to Van Hiele, even though we saw above that Streefland refers to Treffers who considered the Van Hiele levels a “pillar” of RME. Not referring saves the effort of thinking up a lie.

It is a question how the departments on education at the humanities were influenced by RME and Von Glasersfeld and others, and how they got so entangled that Goldin seems to tend to refer to them as one side of the equation (or rather imbalance). It is no key question, but something for historians of MER to be aware of (see the handbook on history of MER).

Comment 4: I started getting lost on what makes “constructivism” so special that it must be mentioned. Originally I knew about constructivism as an approach in the foundations of mathematics, as distinguished from formalism and platonism. My book Foundations of Mathematics. A Neoclassical Approach to Infinity (FMNAI) (2015) creates a ladder of degrees of constructivism (avoiding “levels”), in which the highest degree allows non-constructivist methods. When people use different approaches we should at least describe what they are doing.

But now there are all kinds of “constructivism” in education, psychology and philosophy, without authors taking the time to shortly explain what the non-constructivist opposition would entail. Fortunately, there is wikipedia that might help or contribute to confusion, here with disambiguation. and here with the general denominator in epistemology. The opposite of constructivism would be that people could know objective reality, by magic, and I wonder whether that is so useful an idea. My impression is that there is more to it. Thus authors should still specify. (And then I would not have time to read it.)

Ben Wilbrink is horribly erroneous about Pierre van Hiele and in breach of scientific integrity for not looking into it  to correct his misrepresentation, and Wilbrink can fulminate against constructivism: but at least he referred to this article by Gerald Goldin so that I found it, and he also has this page with all kinds of references on constructivism.

One book mentioned there is by Kieran Egan (2002), Getting it Wrong from the Beginning: Our Progressivist Inheritance from Herbert Spencer, John Dewey, and Jean Piaget. This fits Van Hiele’s rejection of Piaget’s theory of stages of development. But does Egan refer to Van Hiele ? Not likely, since the wikipedia portal speaks about the constructivist “idea that things (especially learning) always go from simple to complex” – and this is not how Van Hiele would phrase it: who discussed going from concrete to abstract, and who used the notion of proof to identify the highest level of abstraction.

Wilbrink also has a quote on Von Glasersfeld:

“The basic idea of The Georgia Center was to establish a community of researchers in mathematics education working on problems of interest to the community, where the experience of the researcher, conceptual analysis, and social interaction replaced the controlled experiment as “normal science.” No longer did it seem necessary to use the controlled experiment with its emphasis on statistical tests of null hypotheses and empirical generalization to claim that one was working scientifically.”

This is complex. Before you do such a costly double blind randomized trial, with the huge numbers required because of the large number of variables, variety in pupils, and sources for measurement error (see John Hattie), it is useful to have clarity on concepts, definitions, operationalisations, methods, controls, and the like. Confronted with annual unpredictable changes from the Ministry of Education, you might want to give up on such statistical ambitions, and settle for the Google “do no evil” approach. It may well be that modern MER only serves for Ph.D. students to defend a thesis, and the relevance for education may be discussed at the reception party afterwards along with the faculty gossip.

The increasingly popular Japanese Lesson Study is one promising method (tested under Japanese conditions) to deal with the data problem.

However, see the suggestion for Academic Schools modeled after the Medical School, also included in A child wants nice and no mean numbers (2015).

Goldin’s crucial blind spot

What I consider Goldin’s blind spot is that he lumps together science and mathematics, while mathematics is no empirical science but deals with abstraction and patterns.

Goldin (2003), p179

Goldin (2003), p179

Education is an empirical issue. Also mathematics education is an empirical issue. Thus the involvement of mathematicians in such education can be disastrous, when they are trained for abstraction μ and not for empirical science γ. The epitome of the abstract mathematician who got lost in this is Hans Freudenthal who invented a whole new ‘reality’ just to make sure that at least he himself understood and was happy how the world works (including the oubliette for Pierre van Hiele).

The only reason that Goldin lumps together β = δ + μ is that he is so much worried by the ‘isms’ by α + Φ that he forgets about the real problem at bottom of the case: the disastrous influence of μ in 5000 years of education of mathematics. (Fractions were already a problem for the pyramids.)

Check out this example of mathematical torture of kids on fractions. This torture is also supported by professor Hung-Hsi Wu of Berkeley for the US Common Core programme, see A child wants nice and no mean numbers.

Goldin (2003) p180 suggests a seemingly good argument for lumping together science and mathematics.

Goldin (2003) p180

Goldin (2003) p180

Thus, the abstract thinking mathematician has a special trick to describe the physical world ? Without lookin ? Like with Wigner’s magic wand ? I don’t think that we should believe this. It is physics that selects the useful model from the mathematical possibilities. Thus:

  • This misconception about the role of mathematics may help explain why Goldin (2003) does not quite see the disastrous influence of abstract thinking mathematicians upon ME and MER. Golding does make some comments that mathematicians should not think that ME is simple and can be tested as in behaviourism, but he misses the fundamental problem as discussed in Elegance with Substance.
  • The remark about mathematics and empirical modeling remains relevant for the definition & reality methodology. It supports the empirical status of say the definition / law of conservation of energy, and it supports the empirical status of Van Hiele’s theory of levels of insight (abstraction) in epistemology (and application in psychology).

(1) We can support Goldin’s conclusion and plea for eclecticism (yes, another ‘ism’).

Goldin (2003), p198

Goldin (2003), p198

(2) Since the Freudenthal Head in the Clouds Realistic Mathematics Institute (FHCRMI) in Utrecht doesn’t do MER but performs sectarian rituals, also based upon Freudenthal’s fraud, it is a disgrace to general science – including the humanities – and thus it should be abolished as soon as possible. Dutch Parliament better investigates how this could have happened and endured for so long.

Appendix 1. Kurt Gödel

W.r.t. the following I can only refer to A Logic of Exceptions (1981, 2007, 2011) (pdf online). For interesting systems the Gödeliar collapses to the Liar paradox, with no sensical conclusions.

Goldin (2003), p187

Goldin (2003), p187

Appendix 2. Philosophy of mathematics

W.r.t. the following I can refer to the discussion on Wigner on the “unreasonable effectiveness of mathematics”. Given the common meaning of “unreasonable” Wigner must refer to magic, or he didn’t know what he was writing about, as a physics professor lost in the English language. It is some kind of magic that his paper got so much attention. This discussion has also been included in Foundations of Mathematics. A Neoclassical Approach to Infinity (2015) (pdf online). Goldin uses the word “extraordinary” rather than “unreasonable”. Given that the effectiveness is ordinary for physics, he seems to take the humanities’ point of view here (whom his article is addressed at).

Goldin (2003), p188

Goldin (2003), p188