Pierre van Hiele and Ben Wilbrink

To my surprise, today gives more on psychology. Since highschool I denote this as Ψ. I appreciate social Ψ (paper 1996) but am not attracted to other flavours of Ψ.

Last week we looked at some (neuro-) Ψ on number sense, and a few days ago at some cognitive Ψ. Dutch readers may look at some comments last year w.r.t. the work by Leiden Ψmetrist Marian Hickendorff who explains that she is no expert on math education but still presents research on it.

Today I will look at what Dutch Ψist and education researcher Ben Wilbrink states about the work by math education researcher Pierre van Hiele (1909-2010). I already observed a few days ago that Wilbrink didn’t understand Van Hiele’s theory of levels of insight. Let me become more specific.

ME and MER are a mess, but Ψ maybe too

The overall context is that math education (ME) and its research (MER) are a mess. Mathematicians are trained for abstraction and cannot deal well with real existing pupils and the empirical science of MER.

When Ψ has criticism on this, it will be easy for them to be right.

Unfortunately, Ψ appears to suffer from its own handicap. Ψ people namely study Ψ. They do not study ME or MER. Ψists invent their own world full of Ψ theories alpha to omega, but it is not guaranteed that this really concerns ME and MER. We saw this in (neuro-) Ψ and in cognitive Ψ in above weblog texts. It appears also to hold for Wilbrink. Whether Ψ is a mess I cannot judge though, since I am no Ψist myself.

Ψ itself has theories about how people can be shortsighted. But we don’t need such theory. A main element in the explanation is that Ψists tend to regard mathematicians as the experts in ME, while those are actually quite misguided. A mathematician’s view on ME tends put the horse behind the carriage. Then Ψ comes around to advise ways to do this more efficiently.

When Pierre van Hiele criticises conventional MER, then Wilbrink comes to the fore to criticise Van Hiele:

  1. for not knowing enough of Ψ,
  2. and for doing proposals that other mathematicians reject.

Welcome in the wonderful world of Kafka Ψ.

This has become an issue of research integrity

I have asked Ben Wilbrink to correct some misrepresentations. He refuses.

He might have excellent reasons for this. My problem is that he doesn’t state them. I can only guess. One potential argument by Wilbrink is that he does Ψ. Perhaps he means to say that when I would get my third degree in Ψ too then I might better understand his misrepresentations. This is unconvincing. A misrepresentation remains a misrepresentation, whatever the amount of Ψ you put into it. Unless Wilbrink means to say that Ψ is misrepresentation by itself. Perhaps.

But: Wilbrink’s refusal to provide answers to some questions turns this into an issue in research integrity.

Wilbrink (1944, now 70+) originally worked on the Ψ approach to test methodology (testing people rather than eggs). See for example the Item Response theory by Arpad Elo and Georg Rasch, also discussed in my book Voting Theory for Democracy. The debate in Holland on dismal education in arithmetic causes Wilbrink to emphasize the (neglected) role of Ψ. He also tracks other aspects, e.g. his website lists my book Elegance with Substance (EWS) (2009), but he makes his own selection. Perhaps he hasn’t read EWS. At least he doesn’t mention my advice to a parliamentarian enquiry into mathematics education. All this is fine with me, and I appreciate much of Wilbrink’s discussions.

However, now there is this issue on research integrity.

Let us look at the details. The basic evidence is given by Wilbrink’s webpage (2012) on Pierre and Dina van Hiele-Geldof (retrieved today).

1. Having a hammer turns everything into a nail (empirics)

If you want to say something scientifically about mathematics education (ME), then you enter mathematics education research (MER).

  • When you meet with criticism by people in MER that you overlook some known results, then check this.
  • Ben Wilbrink overlooks some known results.
  • But he refuses to check those, even when asked to.

In particular, he states that the Van Hiele theory of levels of insight would not be empirical.

But my books and weblog texts, also this recent one, explain that it is an empirical theory. I informed him about this. Wilbrink must check this, ask questions when he doesn’t understand this, and give a counterargument if he does not agree. But he doesn’t do that. What he does, is neglect MER, and simply state his view, and neglect this criticism. Thus:

  • he misrepresents scientific results,
  • he assumes a professional qualification that he doesn’t have,
  • and he misinforms his readership.
2. Having a hammer turns everything into a nail (Ohlsson)

Wilbrink (here, w.r.t. p233 ftnt 38) adopts Ohlsson’s inversion of the learning direction from concrete to abstract, and then rejects Van Hiele’s theory. However, proper understanding of Van Hiele’s theory shows that Ohlsson’s inversion is empirically untenable.

  • Wilbrink doesn’t react to the explanation how Van Hiele’s theory (how learning really works) shows Ohlsson’s theory empirically untenable.
  • As a scientist Wilbrink should give a counterargument, but he merely neglects it.
3. Having a hammer turns everything into a nail (Freudenthal)

A third case that Wilbrink (here, w.r.t. p233 ftnt 38 again) shows that he doesn’t understand the subject he is writing about, is that he lumps Van Hiele and Freudenthal together, i.e. on the theory of levels. But their approaches are quite different. Van Hiele has concrete versus abstract, Freudenthal has pure versus applied mathematics. Freudenthal’s conceptual error is not to see that you already must master mathematics before you can do applied mathematics. You will not master mathematics by applying it when you cannot apply it yet. Guided reinvention is a wonderful word, like sim sala bim.

It is a huge error by Wilbrink to not see this distinction. Wilbrink doesn’t know enough about MER. This turns from sloppy science into an issue of research integrity when he does not respond to criticism on this.

Remarkably, Wilbrink (here, on Structure and Insight) rightly concludes that Van Hiele is critical of Freudenthal and doesn’t actually belong to that approach. Apparently, it doesn’t really register. Wilbrink maintains two conflicting notions in his mind, and doesn’t care. (See also points 10 and 14 below.)

4. Having a hammer turns everything into a nail (Kant)

Wilbrink looks at ME and MER from the angle of Ψ. This looks like a valuable contribution. He however appears to hold that only Ψ is valid, and MER would only be useful when it satisfies norms and results established by Ψ. This is scientifically unwarranted.

  • There are cases in which Ψ missed insights from MER. See above. I have noted no Ψist making the observations that can be found in Elegance with Substance.
  • The Van Hiele theory is a general theory in epistemology (see here), and thus also Ψ must respect that. When Wilbrink doesn’t do that, he should give an argument.

A conceivable argument by Wilbrink might be that Van Hiele did not publish a paper in a journal on philosophy (my notation Φ) so that the sons and daughters of Kant could have hailed it as a breakthrough in epistemology. The lack of this seal of approval might be construed as an argument that Ψ and Wilbrink would be justified to neglect it. This would be an invalid argument. When Wilbrink studies MER and Van Hiele’s theory of levels, and reads about Van Hiele’s claim of general epistemological relevance, then every academic worth his or her salt on scientific methodolgy, and especially Ψists, can recognise it for what it is: a breakthrough.

5. Having a hammer turns everything into a nail (testing validity)

Wilbrink’s question whether there has been any testing on validity on Van Hiele’s theory at first seems like a proper question from a Ψist, but neglects the epistemological status of the theory. He would require from physicists that they “test” the law of conservation of energy, or from economists that they “test” that savings are what remain from income after consumption. This is quite silly, and only shows that Wilbrink did not get it. Perhaps his annoyance about Freudenthal caused him to attack Van Hiele as well ? Wilbrink should correct his misrepresentation, or provide a good reason why being silly is good Ψ.

6. Having a hammer makes you require that everyone is hammering

Wilbrink suggests that Pierre and Dina Van Hiele – Geldof performed “folk psychology”. This runs counter to the fact that Pierre studied Piaget, and explicitly rejected Piaget’s theory of stages. His 1957 thesis (almost 60 years ago) has three pages of references that include also other Ψ. Perhaps Wilbrink requires that they should have studied more of Ψ. That might be proper when the objective was to become a Ψist. But the objective was to do MER. Dina did the thesis with Langeveld, a pedagogue, and Pierre with Freudenthal, mathematician and not known yet for the educational theories that he stole from Pierre (and distorted, but it remains stealing).

If the Ψists would succeed in presenting a general coherent and empirically corroborated theory, that every academic can master in say a year, then perhaps Ψists might complain that this is being neglected. Now that Ψists however create a wealth of different approaches, then researchers in MER are justified in selecting what is relevant for their subject, and proceed with the subject.

Wilbrink’s suggestion on “folk psychology” is disrespectful and slanderous.

7. Having a hammer makes you look for nails at low tide (pettifoggery)

Wilbrink reports that Dina van Geldof mentions only the acquisition of insight and does not refer to the relevance of geometry for a later career in society. Perhaps she doesn’t. Her topic of study was acquisition of insight. Perhaps Wilbrink only makes a factual observation. What is the relevance of this ? It is a comment like: “Dollar bills don’t state that people also use them in Mexico.” Since Wilbrink reports this in the context of above disrespectful “folk psychology”, the comment only serves to downgrade the competence of Dina van Geldof, and thus is slanderous. As if she would not understand it, when Pierre explained to her that his theory of levels had general epistemological value.

8. Having a hammer makes you look for nails in 1957

Wilbrink imposes norms of modern study design and citation upon the work of the Van Hieles in 1957 (when Pierre was 48). The few references in Pierre’s “Begrip en inzicht” (2nd book, not the thesis, also translated as “Stucture and insight”) cause Wilbrink to hold, in paraphrase,

“by not referring, Van Hiele reduces his comments to personal wisdoms, by which he inadvertedly downgrades them.”

This is a serious misrepresentation, even though the statement is that Van Hiele’s texts were more than just personal wisdoms.

(a) It is true that Van Hiele isn’t the modern researcher who always refers and is explicit about framework and study design. What a surprise. The observation is correct that norms of presentation of results have changed. Perhaps authors in the USA 1957 already referred, but this need not have been the case in Europe. (See a discussion on this w.r.t. John Maynard Keynes.)

(b) The suggestion as if Van Hiele should have referred is false however. In that period the number of researchers and size of literature were relatively small, and an author could assume that readers would know what one was writing about. Some found it also pedantic to include footnotes.

Thus: (i) The lack of footnotes does not in any way reduce Van Hiele’s comments to “personal wisdoms”. Wilbrink is lazy and if he is serious about the issue then he should reconstruct the general state of knowledge in that period. (ii) The comment must be rewritten in what is factually correct, and the insinuation must be removed.

9. Having a hammer makes you put nails in other people’s mouths

Wilbrink refers to an issue on fractions. He quotes Van Hiele’s suggestion to use tables of proportions, which has been adopted by the Freudenthal Institute, and quotes criticism by modern mathematicians Kaenders & Landsman that those tables block insight into algebra.

This is a misrepresentation.

This is an example of that a Ψist quotes mathematicians as authorities, and regards their misunderstanding as infallible evidence. A student of MER however would (hopefully) see that there is more to it.

The very quote by Van Hiele contains his suggestion to look at multiplication. Indeed, the book “Begrip en Inzicht” chapter 22 contains a proposal to abolish fractions, and to deal with that algebraically – what Kaenders & Landsman may not know about.

The true criticism is that the Freudenthal Head in the Cloud Realistic Mathematics Institute mishandled Van Hiele’s work: (a) selected only an easy part, and (b) did not further develop Van Hiele’s real approach.

A proposal how Van Hiele’s real approach can be developed is here. I agree with Kaenders & Landsman to the extent that presenting only such tables is wrong, and that also the algebraic relation should be specified. The student then has the option to use either, and learn the shift.

Curiously, Wilbrink comments on this chapter 22 with some approval. Thus he should have seen that he provided a false link between Van Hiele on tables of proportion and the critique by Kaenders & Landsman.

10. Having a hammer makes you hate who refuses to be a nail

Wilbrink discusses Structure and Insight (not the thesis) here. He quotes Van Hiele from p. viii:

“Many original ideas can be found in this book. I came upon them in analyzing dubious theories of both psychologists and pedagogues. It is not difficult to unmask such theories: simply test them in practice. Often this is not done because of the prestige of the theory’s proponents.”

Wilbrink’s judgement (my translation):

“The quoted opinion is incredibly arrogant, lousy, or how do you call such a thing. Van Hiele is mathematician, and makes the same error here as Freudenthal made in his whole later life: judging the development of psychological theory not in the context of psychology, but in the context of one’s own common sense. This clearly gives gibberish. Thus I will continue reading Van Hiele with extraordinary suspicion.”

My comments on Wilbrink:

  • Van Hiele was a mathematician but also a teacher, with much attention for the empirics of education. This is quite in contrast with Freudenthal who lived by abstraction. (Freudenthal did not create a professorship in math education for Van Hiele, but took the task himself.)
  • Van Hiele does precisely what Wilbrink requires: look at Ψ and look at empirics (in this case: practice). The only thing what happens is that Van Hiele then rejects Ψ, and this is what Wilbrink doesn’t swallow. While Van Hiele does MER, Wilbrink redefines this as Ψ, and then sends Van Hiele to the gallows for not sticking to some Ψ paradigm.
  • It is useful to mention that Van Hiele does the same thing in the preface of his thesis. He states that Ψ theories have been shown inadequate (his references are three pages) and that he will concentrate on the notion of insight as it is used in educational practice. He opposes insight to rote learning, and mentions the criterion of being able to deal with new situations that differ from the learning phase.
  • It is incorrect of Wilbrink to distinguish only the categories of either Ψ or “one’s own common sense” or “folk psychology”. It is quite obvious why Van Hiele cannot find in books on Ψ what he is looking for and actually does: He presents his epistemological theory of levels. Those aren’t in those books on Ψ. If Van Hiele would do what Wilbrink requires, then he cannot present his theory of levels, since Wilbrink’s strict requirements would force him to keep on barking up the wrong tree. It beats me why Wilbrink doesn’t see that.
11. Having a hammer turns your foot into a nail

Wilbrink also quotes from viii:

“Some psychologies lay much stress on the learning of facts. The learning of structures, however, is a superior goal. Facts very often become outmoded; they sink into oblivion because of their lack of coherence. In a structure facts have sense; if part of a structure is forgotten, the remaining part facilitates recall of the lost one. It is worth studying the way structures work because of their importance for the process of thinking. For this reason a considerable part of my book is devoted to structures.”

Wilbrink’s comment on this is (my translation):

“For me this is psychological gibberish, though I rather get what Van Hiele intends (…)”

By which it is established that Wilbrink understands gibberish and may call gibberish what he understands.

12. Having a hammer makes that you run in a loop of nails

Wilbrink’s subsequent quote from Structure and Insight:

“In this book you will find a description of a theory of cognitive levels. I show you how levels of thinking demonstrate themselves, how they come into existence, how they are experienced by teachers and how by pupils. You will also see how we can take account of those levels in writing textbooks.”

Wilbrink (my translation):

“You cannot simply do this. At least Van Hiele must show by experiment that intersubjective agreement can be reached about who when what level has demonstrated by operational achievements (because we cannot observe thoughts directly). (…) Indeed, at least for himself it is evident. Can this idea be transferred to others ? Undoubtedly, for other people have invited him to make this English translation of his earlier book. But that is not the point. The crucial point is: does his theory survive empirical testing?”

My comment: It is a repetition of the above, but it shows that Van Hiele’s repeated explanation about the epistemological relevance of his theory for educational practice continues, time and time again, to elude Wilbrink’s frame of mind.

Of course, statistical science already established before 1957 that the golden standard of experimental testing consists of the double blind randomized trial. Instead, Van Hiele developed his theory over the course of years as teacher in practice. Though he mentions didactic observations already from his time as a student in highschool. But we are back in a repetitive loop when we must observe that it is false to require statistics for Van Hiele’s purposes.

13. Having a hammer makes you avoid number 13 for fear that it might make you superstitious
Hermann von Helmholtz, on the law of conservation of energy (source: wikimedia commons)

Hermann von Helmholtz, on the law of conservation of energy (source: wikimedia commons)

14. Having a hammer makes you miss a real nail

Wilbrink (2012) refers to the MORE study of 1993 that defined realistic mathematics education (RME) as consisting of:

  • Van Hiele’s theory of levels
  • Freudenthal’s didactic phenomenology
  • the principle of progressive mathematizing according to Wiskobas (JStor).

It is actually nice that Van Hiele is mentioned in 1993, for at least since 2008 he isn’t mentioned in the Freudenthal Head in the Clouds Realistic Mathematics Institute wiki on RME (retrieved today). His levels have been replaced by Adri Treffer’s concept of “vertical mathematization”. Wilbrink might be happy that he doesn’t have to criticise the levels at FHCRMI anymore. It is now a vague mist that eludes criticism.

Wilbrink’s criticism of Freudenthal’s didactic phenomenology and Wiskobas are on target. It is indeed rather shocking that policy makers and the world of mathematics teaching went along with the nonsense and ideology. The only explanation is that mathematicians made a chaos with their New MathIf Pierre van Hiele had been treated in scientific decent fashion, his approach would have won, but Freudenthal was in a position to prevent that.

Wilbrink apparently thinks that Van Hiele belongs to the Freudenthal group, even though he observes elsewhere that Van Hiele rejects this. Wilbrink assumes both options, and his mind is in chaos.

Wilbrink doesn’t see that the Freudenthal clique only mentions Van Hiele to piggyback on his success, to manoeuvre him out, and later create some matching phrases so that Van Hiele doesn’t have to be mentioned anymore.

The following is a repetition of point 5, but it can be found on this particular page & section by Wilbrink, and may deserve a comment too. Namely, regarding Van Hiele as a pillar of realistic mathematics education, Wilbrink states (my translation):

“Okay, I can infer that the theory of levels can be found in Van Hiele’s thesis, but that thesis is of a conceptual nature, and it doesn’t contain empirical research. Van Hiele doesn’t deny the latter, see the passage on his pages 188-189; but that is really rather sensational: everyone parrots his theory of levels, without looking for empirical support. Every well-thinking person, who has read his Popper for example, can see that you can do just anything with that ‘theory of levels’: It is in the formulation by Van Hiele 1958 [article following the 1957 thesis ?] a theory that excludes almost nothing. I return to this extensively on the Van Hiele page.”

My comments for completeness:

  • Van Hiele’s theory is as empirical as the law of conservation of energy or the economic principle that savings are the remainder of income after consumption. This is not pure mathematics but it applies to reality. Thus Van Hiele’s theory is hugely empirical. See the former weblog text.
  • Van Hiele’s thesis p188-189 indeed mentions the subsequent relevance of statistical testing to ground out details. This is something else than testing on falsification. What Van Hiele states is not quite what Wilbrink suggests. The fact of the lack of statistical testing is correct. But Van Hiele does not subscribe to Wilbrink’s criterion of “empiricism”.
  • Van Hiele does not expect that there will be much statistical development of the levels. Therefor he judges that his theory will tend to be of more value for teachers in practical teaching.
  • You can do with the theory of levels as much as with the law of conservation of energy. A bit, but a crucial bit. Who has read Popper will see that the idea of falsification must make an amendment on definitions.

Thus, if Wilbrink had had an open mind on epistemology, he could have nailed the FHCRMI for producing nonsense and abusing the wonderful theory by Van Hiele. He missed.

But the key point is that his also misinforms his readership, and refuses to correct after he has been informed about it.

15. Having a hammer makes that only masochist nails like you

Wilbrink’s discussion of Van Hiele’s thesis chapter 1 (here, “Wat is inzicht?”) shows a lack of understanding about the difference between a theorem and a proof. Euclid turns in his grave.

Wilbrink makes a distinction between “mathematics and psychology of mathematics”, without explanation or definition, perhaps in the mood of writing for Ψists who will immediately smell the nest and cheer and be happy.

Wilbrink writes “Brrrrr” (check the r’s) when Van Hiele distinguishes insight based upon inference and insight based upon non-inference. Wilbrink does not explain whether his Brrrrr is based upon inference or non-inference.

Wilbrink fears that Van Hiele will base his didactic insight upon “reason” instead of “theory with empirical testing”. He does not explain what is against reasoning and teaching experience and reading in the literature, for developing a new theory. Perhaps Wilbrink thinks that true theories can only be found in books of Ψ ?

Wilbrink’s final judgement on Van Hiele’s thesis chapter 1 is that it is a “tattle tale”. It is a free world, and Wilbrink may think so and put this on his website. But if he wants to be seen as a scientist, then he should provide evidence. In this case, Van Hiele clearly stated that he found the Ψ theories useless, so that he returned to the notion of insight in educational practice. His discussion of what this means is clarifying. It links up with his theory of levels. Overall it makes sense. As an author he is free in the way how he presents his findings. He builds it up, from the concrete to the abstract. Wilbrink does not respect Van Hiele’s judgement, but provides no other argument than Brrrrr or the spraying with the label of Ψ or invoking the spell of the double blind randomized trial.

16. Having a hammer doesn’t make you a carpenter

Wilbrink (2012) doesn’t comment on Van Hiele’s thesis’s final chapter XVIII about the relevance of the theory of levels for epistemology. An ostrich keeps its head in the sand, where it is warm and dark, like in the womb of its egg.


Originally, I saw some of Ben Wilbrink’s texts on Van Hiele before, and appreciated them for the discussion and references, since there is hardly anyone else in Holland who pays attention to Van Hiele. However, Wilbrink’s reaction to Ohlsson, to the effect that Van Hiele would be wrong about the learning direction of concrete to abstract, caused me to make this evaluation above.

Wilbrink maltreats Van Hiele’s work. Wilbrink doesn’t know enough about mathematics education research (MER) to be able to write about it adequately. He misinforms the public.

I have asked Wilbrink to make adequate corrections, or otherwise specify his (reply) arguments so that I could look into those. He refuses either. This constitutes a breach in the integrity of science.


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