Pierre van Hiele and Jan van de Craats

Jan van de Craats (University of Amsterdam)  wrote the textbook All you need in maths!, using the UK “maths” instead of the USA “math”. The book need not fit a national curriculum and is presented as a book with exercises. The idea is to counter the trend in Freudenthal’s realistic mathematics education that forgets about decent practice and exercise.

I sent the following email to Van de Craats cc some other people involved in the Dutch discussion on mathematics education. The email speaks for itself. I take the liberty to include some weblinks for outsiders to the discussion. The original email contained fully stated URLs, but for readability on a web page I transform these in linked labels. The sections are made clearer. Some typo’s have been corrected. This weblog text closes with a comment that was not in the email.

The email

Date: Sun, 06 Sep 2015
To:     “Craats, Jan van de” (UVA)
From: Thomas Cool / Thomas Colignatus
Subject: Inadequacy, maltreatment and abuse w.r.t. the work by Pierre van Hiele (1909-2010)
Cc: Persons mentioned below

Dear professor Van de Craats,

You are an informal leader of the movement amongst Dutch mathematicians to correct the so-called “didactics” of the Freudenthal Institute, which didactics [is] scientifically proven invalid but nevertheless dominates Dutch education in mathematics including arithmetic.

In the Dutch situation there is inadequacy, maltreatment and abuse w.r.t. the work by Pierre van Hiele (1909-2010). My intention is to inform you about this, because this helps for understanding the situation w.r.t. the Freudenthal Institute and mathematics education, and for identifying the direction for improvement.


Last year, 2014, the Dutch Academy of Sciences (KNAW) had a conference on education in arithmetic. I asked Jan Bergstra (UvA), secretary of the mathematics section at KNAW to read Van Hiele’s “Structure and Insight” (in the Dutch original “Begrip en Inzicht”). I also asked him to support at Academic Press that they put out a new edition of this, and to fund an English translation of Van Hiele’s thesis. It took a while, but Bergstra now has reported that he read the book, and can do little with it. He seems to refer to his own interest in fractions (and division by zero), but that wasn’t the question. I expect a decent discussion at the KNAW math section about the crucial importance of Van Hiele’s work for math education, internationally. It is inadequate and a maltreatment that this section doesn’t have this discussion and evaluation, or did not report back to me so that I could see the quality of the argumentation. I cc to Jan.


I asked Nellie Verhoef (TU Twente) what information she gave to David Tall (United Kingdom) about sources in Dutch about Van Hiele’s work. I already spotted one crucial mistranslation w.r.t. the meaning of “realism” in “realistic mathematics education”. Verhoef refuses to answer. David Tall appears to think that Van Hiele limited his theory of levels to geometry only. It would be David Tall who saw that they apply in general. This is a misconception, since Van Hiele indicated the general applicability already in his thesis of 1957. It is important however that Tall confirms the general value. Tall’s book still requires a correction. It is crucial to know what information Nellie Verhoef gave him. It is a breach of the integrity of science that she refuses to disclose this information. I copy to Verhoef. I copy to Harrie Broekman (UU) who is connected to this issue. I reported the issue to Jan Bergstra in his capacity at KNAW, but he seems to neglect it. I copy to professor Mike Thomas [in New Zealand], so that he can check whether this email is relevant for David Tall (given his age and interest).

These two links give more information about the issue.


The thesis by S. la Bastide-van Gemert about Freudenthal contains some curious passages that Freudenthal took the theory of levels from Van Hiele and that Freudenthal himself was the inventor. I asked La Bastide what to make of this, and what her diagnosis about the origin was. She stated not to have time for this, in her current work at the Groningen Medical Center. Subsequently, I posed the same question to the thesis supervisors and readers, still at Academe so that it can be regarded as their work. I did this one by one, so not to overburden all. I informed each about the rejection by the predecessors. Each rejected to look into this. They neither fully and openly confirmed the inconsistency. But this is a breach in the integrity of science too. There is an inconsistency in a thesis, which one should not accept. There is all indication that Freudenthal stole the concept from Van Hiele, which is important to understand the full situation. It is unacceptable that this issue is covered up. I copy to La Bastide, thesis supervisors Klaas van Berkel en Jan van Maanen, en reader Martin Goedhart, all in Groningen. I reported the issue to Jan Bergstra in his capacity at KNAW, but he seems to neglect it.

The issue is documented in the appendix of my paper on [Van Hiele] and Tall, cited above.

The thesis by La Bastide is [here].


There is the issue of retired psychologist Ben Wilbrink who discussed Van Hiele’s theory of levels. I have asked Wilbrink to correct his misrepresentation, but he refuses to do so, and, what turns this into a breach of scientific integrity, refuses to explain why. Since Wilbrink is retired, I asked him whether he could mention a mediator who he would be willing to listen to. See my email to him below.

I have documented the issue [here].

In sum, it is established beyond reasonable doubt that there is inadequacy, maltreatment and abuse in Holland w.r.t. the work by Pierre van Hiele (1909-2010).

Perhaps the problem is being caused by the “many hands” phenomenon, that there are many people involved and each individual is not aware of the impact of the sum total, but, still, if each maintained proper adherence to the rules of science, then there would have been no reason for this email.

One may hold that each case is an issue for the commissions of integrity at the separate universities, but my experience is that these don’t function well, see how they treated the slander w.r.t. my book Conquest of the Plane, and see my letter to KNAW-LOWI on the collective breach on integrity:

I copy to the board of the KNAW section on mathematics, excluding Johan van Benthem, who maltreated my work on logic when I was a student in econometrics in Groningen around 1980 and when I had a course in logic by Van Benthem. I kindly ask chairman Broer to forward this email to professores emeriti Van der Poel and [Zandbergen] for whom I cannot find an email address.

I copy to the president of KNAW, professor Van Dijck.

I will put this email on my weblog.

Kind regards,

Thomas Cool / Thomas Colignatus
Econometrician and teacher of mathematics
Scheveningen, Holland

Date: Sat, 05 Sep 2015
To: “Ben Wilbrink”
From: Thomas Cool / Thomas Colignatus
Subject: Kun je een bemiddelaar voorstellen ? (…)

Dag Ben,

At 2015-09-04, Ben Wilbrink wrote:

Ik wil dit niet, Thomas. Ik ga er niet op in.

Je zult gemerkt hebben dat ik een zeer tolerant persoon ben. Je negeert al jarenlang mijn kritiek op het onderwijs in wiskunde, en ik heb er weinig van gezegd. Ik respecteer ook je kennis en bijdragen.

Maar […] t.a.v. je behandeling van Van Hiele maak ik nu groot bezwaar op grond van wetenschappelijke deugdelijkheid. Bij andere psychologen heb ik al opgemerkt dat ze te weinig van didactiek van wiskunde weten, en t.a.v. jou kan ik geen uitzondering maken.

Mijn tekst hierover:


Een oplossingstraject is dat je een bemiddelaar voorstelt, en ik kijk of ik akkoord ga.

Iemand voor wie je wel respect hebt en die jou hopelijk kan uitleggen in termen die je wel begrijpt dat deze zaken zijn op te lossen.

Met groet,


Closing statement of this weblog entry w.r.t. the email

Van de Craats wrote the book with Rob Bosch (Netherlands Defense Academy). Bosch was member of the Social Choice Theory group that used false arguments to block my invited presentation in 2001 at the 37th Dutch mathematics conference (NMC), and discussion with Donald Saari. Bosch is also member of the team of editors of the journal Euclides for Dutch math teachers, that maltreated my books EWS and COTP, see here. I haven’t looked at the contents of All you need in maths!, but it is reasonable to expect that it doesn’t contain the didactic improvements suggested by EWS and COTP (and neither refers to those). Yes, when conventional math formats are crummy then you need more exercises to master them. While the true objective is to understand the math and not merely solve the sums.

Jan van de Craats and his book All You Need in Maths (source: website)

Jan van de Craats and his book All You Need in Maths! (source: his website)


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