# Pierre van Hiele and Sacha la Bastide-van Gemert

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.

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.