Anatomy of Holland

Listening to Beauty in red


The Scottish MacTutor history of mathematics archive contains a webpage on Hans Freudenthal (1905-1990). It is always useful to have views from outsiders.

They don’t have a webpage on Pierre van Hiele (1909-2010) yet.

I have found that Freudenthal committed fraud w.r.t. the work by Van Hiele.

Being erased from history is not so bad. What is bad is being misrepresented.

Recently, the math war in Holland reached a new low point, when a psychologist who rejects Freudenthal’s “realistic mathematics education” also started attacking Van Hiele, rather than saving him. See my letter to Jan van de Craats.

In other words, Freudenthal so massively abused Van Hiele’s work, that people may see neither Van Hiele’s real contribution nor the abuse: and then some people bunch his work together with the errors by Freudenthal.

David Tall in the UK thinks that he himself now invented what Van Hiele already had invented, see here. What will the history books later say ?

I wondered whether the MacTutor history website only concerned mathematicians with results in mathematics, or also those looking at mathematics education. It appears that they also do a bit of the latter, e.g. by discussing Emma Castelnuovo (1913-2014).

Van Hiele isn’t mentioned on Castelnuovo’s MacTutor page. A google didn’t show yet whether Castelnuovo refers to work by him. This google did recover the Karp & Schubring (ed) Handbook on the History of Mathematics Education (2014) in which they both are mentioned of course.

Freudenthal however is mentioned on her MacTutor page. Van Hiele has declared that Freudenthal misinformed others about his work and what it was really about. Thus if Castelnuovo depended upon Freudenthal for her interpretation of Van Hiele’s work, then there would be a problem.

For example, the page on Castelnuovo contains a confusion between the distinction of mathematics versus applied mathematics (Freudenthal’s “realism”) and the distinction between concrete versus abstract (Van Hiele). See here. See also Research Italy’s interview with Nicoletta Lanciano.

A major reason why Van Hiele is important for mathematics itself is that you need the Van Hiele theory on levels of insight (abstraction) to understand what mathematics is about, actually. See this discussion on epistemology.

Indeed, you can read a novel without actually knowing what a novel is. (wikipedia) Similarly, mathematicians may do mathematics without quite knowing what it is. But it helps to be aware of what you are doing.

For historians it also helps to be aware what history writing is.

PM 1. Check that Amir Alexander doesn’t know what history writing is. PM 2. For those who like irony: Freudenthal wrote on history too. PM 3. The following is not a simpleton’s reaction but the result of seven years of patience that reaches its endpoint: Jan van de Craats refused to properly answer to that letter, and now is in breach of scientific integrity himself, see here. Check how Van de Craats supports mathematics education that tortures kids with fractions.

Screenshot of MacTutor History of Mathematics Archive

Screenshot of MacTutor History of Mathematics Archive

Listening to Just like river water in the spring


Professor Jan van de Craats (University of Amsterdam, now emeritus) is in breach of integrity of science. In an email to me in 2008 he confirms some of my criticisms on mathematics education, but since then he has been effectively neglecting this and refusing to discuss matters. He founded and now advises a foundation SGR for better education in arithmetic, and they employ dubious methods, including neglect and refusal to discuss and refer to criticism. Their criterion on “good” must also contain “keep a closed mind”.

SGR was founded in 2008 and has a Committee of Recommendation. Perhaps that list requires a date, or must be updated, since SGR now supports a particular commercial product, the education method Reken Zeker at a particular publisher, and at least two persons on the list have joined the national council on education that is supposed to be impartial (Maassen van den Brink en Van der Werf at Onderwijsraad).

Let me given an indication how Van de Craats’ breach of scientific integrity also causes bad mathematics education. Let me take two screenshots from two instruction videos from this SGR website.

Two screenshots of videos at SGR

The first video discusses a division of mixed numbers, and the second video discusses the conversion of a square meter into square decimeters. The screenshots are such that you don’t need to understand Dutch. The issues are clear enough. The didactic problem lies in the presentation. An invitation to you is:

Assignment: Spot the problem in didactics of mathematics.

If you cannot spot the problem, try to draw the inference: that you need to brush up on your awareness of didactics, and that you ought to read my book Elegance with Substance, (EWS) 2009, 2nd edition 2015 (with pdf online since 2009, so that you don’t have the excuse of a paywall either).

Thus, if you hate to read EWS, and hate to drag professor Van de Craats to the courts of justice and have him hanged or drawn & quartered, to remain with the subject of fractions, then you will be encouraged to really think and spot the didactic problem that arises from comparing these two images. Clicking on the screenshots will bring you to the videos in Dutch, but only these screenshots are relevant now. Please scroll the computer window in such a way that you don’t see the discussion of the solution below till you have formulated your solution or give up.

Division by two mixed numbers at SGR (Source: website SGR)

Division by two mixed numbers at SGR (Source: website SGR)

Conversion of a square meter to decimeters at SGR (Source: SGR website)

Conversion of a square meter to square decimeters at SGR (Source: SGR website)


The didactic problem with these two screenshots

In the second screenshot 1m or 1 m represents multiplication, or 1 × m, without writing the multiplication sign. In the first screenshot 2 + ⅓ is written as 2⅓ = 2 × ⅓ = ⅔.

One might hold that it is “1 m” with a space and “2⅓” without a space, so that the notations are well defined. This is difficult to maintain in handwriting, especially for kids. It still is needlessly confusing, and thus didactically wrong.

One might also hold that the form a b/c can be recognised as a “number next to a fraction” so that kids should be able to spot the fraction b/c, and then understand that the whole expression would mean a + b/c. This is dubious. If you agree that 10 dm = m so that dm = m / 10, then above example gives a m / 10, so that kids would need to understand this as a + m / 10. Is that really your reasoning ?

If your response now would be that dimensions like m and dm must be treated differently, so that dm = m / 10 is wrong and must be dm = 1/10 m, then you are changing mathematics and introducing a second arbritrary rule just for the reason that you don’t want to admit that you were wrong. It means that you already tortured kids and don’t mind to torture more if it helps to maintain your ego and investments in textbooks full of errors.

The notation for mixed numbers was invented at some time deep in the past, but without proper didactic considerations, and the only reason to maintain it is that mathematicians don’t mind torturing kids.

See Elegance with Substance (EWS) (2009, 2015). I discuss this in 2008, Van de Craats refers to it in his email of 2008, and it could have been solved in 2009, so that it could have been in all methods that were put on the market in 2010, not only Reken Zeker.

In his other own “remedial book” Van de Craats prefers 5/2 over 2½ with the stated reason “because 5/2 is easier to calculate with”, which is a misrepresentation of the real didactic issue.

PM. The first video stops at 49/66, which might be justified since it cannot be simplified anymore or written in mixed number format. The small supplementary problem is that this should be checked and mentioned, which is’t done. The algorithm thus isn’t fully discussed. This is not the key issue here. It just surprises me since SGR puts such an emphasis on algorithms.

Van de Craats and Wilbrink on Pierre van Hiele

Van de Craats also refuses to look into and to refer to criticism w.r.t. the manner how psychologist Ben Wilbrink abuses the work by Pierre van Hiele, even though he has an extensive section with links to the site of Wilbrink. See my discussion of Van de Craats’ breach again.

One of Van Hiele’s suggestion was that fractions can be abolished. See the discussion here. Thus, SGR spends a lot of time on teaching kids fractions that can actually be abolished. Perhaps kids at some stage, when they understand the inverse of multiplication, must be instructed that old-fashioned people write mixed numbers in another fashion. But this is a short explanation. This would not obstruct the whole learning process of mastering arithmetic.


We spotted another case of the elementary sick Dutch mindset that requires a decent boycott.

In this case it is mathematics again. The key issue is that mathematicians are trained for abstract thought and not for empirical science. This is world problem.

The combination of this Dutch mindset with mathematics is especially disastrous.

The appeal to boycott Holland is targeted at the censorship of economic science since 1990 by the directorate of the Dutch Central Planning bureau. This example of the Dutch mindset confirms the analysis on the need of a boycott.

PM. For Dutch readers:

This is a petition on having a parliamentary enquiry into the censorship of economic science.

This is a petition on having a parliamentary enquiry into mathematics education.

How SGR teaches children fractions (Source: wikimedia commons on Dieric Bouts (1415-1475))

The medieval method how Van de Graats and SGR teach children fractions (Source: wikimedia commons on Dieric Bouts (1415-1475))

Listening again to Girls of Ali Mountain


I had some fun today with Google Translate. For other people this is serious research and business, but a lay translator may be excused to play a bit. Unfortunately, play causes questions, it isn’t a free lunch.

Google Translate and the pronunciation of numbers

We discussed the pronunciation of numbers in English, German, French, Dutch and Danish before. Here is a suggestion to develop a standard.

Kids of age 4-6 live and think in spoken language before they learn reading and writing. Thus proper pronunciation of numbers will help them mastering the written number system and arithmetic. A first phase of reading is reading aloud, a later phase is subvocalisation (i.e. become silent), and perhaps later the latter may disappear. Thinking would still be much in “silent spoken language”, while only later the formulas like 1 + 1 = 2 would benefit from thinking in forms (symbol sense).

Ms. Sue Shellenbarger in the Wall St. Journal September 15 2014 discussed The Best Language for Math. Confusing English Number Words Are Linked to Weaker Skills”. 

Hence I wondered how Google Translate deals with this, with their pronunciation icon, and, whether they could support the development of such a standard.

  • When you type in 11, and ask for the pronunciation, then you get eleven.
  • When you type in ten one then you get ten one.
  • Ergo, it would be feasible to create a language tab English-M so that 11 gives pronunciation ten one. (And normal English again for not-numbers.)
Speech examples

When you type in 1111  then Google speech gives eleven eleven, which is wrong. Please do not alert them on this, because I want to keep the example intact. Only 1,111  generates spoken one thousand, one hundred and eleven, which it also should be for 1111. Except that English-M  would give thousand, one hundred, ten one.

Numbers also occur in full sentences. For example translate I will give you 11 dollars into Dutch. Again eleven and elf. Now suddenly 1111 is spoken correctly, perhaps because it are dollars ?

A switch between language and language-M

It might be a single option to select mathematical pronunciation, for all languages. But the tab would need to show English-M and Dutch-M to prevent confusion. Also, at one time, one might wish for a translation from English-M to traditional Dutch. Best could be a selector icon in the row of language tabs that allows you to switch between traditional and mathematical pronunciation.

Google Translate is already prim on the distinction between UK and US English. There is only one English tab, and the translation of say Dutch strengheid gives both rigor and rigour. But this is a spelling issue. Mathematical pronunciation of numbers isn’t spelling reform but an enrichment of language. And it is neither the difference between Oxford English and Cockney. There may be more sites explaining dialects than Oxford English.

Indeed, when we try to translate Me want money from English to English, to remove grammatical or spelling errors, with the options I want money or We want money, then Google Translate doesn’t allow this. It just doesn’t permit translation from English to English. The translation to Dutch selects the Me  I option. “Mij wil geld” is a literal translation but Google corrects into proper grammer “Ik wil geld”. One would however feel that crummy English should be translated as crummy Dutch.

A bit of greater fun is that Google Translate accepts spoken 1 plus 1 = 3, but refuses the input of 1 +1 = 2, perhaps because they think that + is no accepted sign in the English language, or perhaps because they think that it doesn’t need translation.

Language research

Google Translate acknowledges use of results by numerous scientists around the world. A key source is WordNet. (In Holland Piek Vossen is involved in this.) When you look at what they are doing, it is huge and impressive.

By comparison, the pronunciation of the numbers is trivial. Let us start with the 20% of effort that generates 80% of results. It is a suggestion for WordNet and Google Translate to look into this.

Thus the WordNet research group might consider supporting the development of this standard for the pronunciation. Developing the standard might take some time, given the need for consensus to develop. Likely there will be stages: first in education, then in law.

The resources and energy of Google Translate might also make a difference for practical developments, notably by providing example implementations. Formation of English-M need not wait for French-M.

Eventually, Google Translate may develop into Google Language, with checkers on spelling and grammar, thesaurus, rhyme, and what have you. Some users might want writing support, like a warning message that a text is too abstract and that an example is required.

It shouldn’t be too difficult either to make an app how to pronounce numbers in English-M, but this weblog isn’t about commerce.

Pierre van Hiele and the levels of insight

Pierre van Hiele presented a theory of levels of insight as a general theory for all epistemology. Geometry was where he started, and what he used as his key example case. Many people didn’t listen well and assumed that he thought that the levels apply only to geometry. See the error on wikipedia that I just linked to, or the misconception by David Tall, who thinks that he was the first one to discover the generality, but who at least supports the notion.

A consequence for language

A consequence of the theory of levels is that students speak different languages.

They use the same English words but mean something else. There will generally be great confusion in the classroom and lecture hall, except for the teacher, who can mediate between students at different levels of insight, including those who are making the shift.

Thus, depending upon the particular field F ∈ {mathematics, physics, biology, economics, …} Google Translate ought to have English-F-1, …, English-F-n. Mathematics would have the highest level because of the notion of formal proof. Perhaps that the majority of fields F might work with only three levels: novice, verbally fairly competent but reproductive, and reasoning informally.

These would also be the levels required for wikipedia-1, …, wikipedia-n. Wiki-articles on math topics are dominated by MIT students who copy their textbooks, which produces gibberish for novices, which isn’t quite the purpose of an encyclopedia. (And some students think they know it better anyway, see here.)

When Google Translate could translate English-M-2 to English-M-1 (as far as possible), then Google Translate would turn into a teacher’s assistent.

Language spaghetti

It may be that current translators, say from English to Spanish, might not be aware of the Van Hiele levels. The issue might not be quite urgent.

  • When translators focus on “words only” then they might translate English words into say Spanish words, and then let others deal with what those words mean to them.
  • Speakers of English-4 might use sentences that contain a few words that users of English-3 don’t use much – e.g. the very word “proof” – so that the translation from English-4 to Spanish-4 would tend to work.

Other cases might simply be spaghetti that perhaps might be neglected.

For example, users of English-2 could use terms from English-4, that they actually don’t understand. They may translate into Spanish-4 – e.g. “I got a proof” becomes “Tengo una prueba”. They wouldn’t understand either of those – since they don’t understand the notion of proof yet – so that this might not be a great loss.

It is a wary notion that Google Translate will perhaps be mostly busy in translating what people don’t understand anyway. Perhaps an exam needs be taken before you offer something to be translated. But we live in a fast world.

It remains valuable to be aware of levels

The upshot is that it would still be a valuable idea to identify Van Hiele levels. Words that seem the same have different meanings, because of those levels.

Wikipedia already uses the disambiguation. They seem to regard it as the minimal word that isn’t ambiguous itself, and take quite some space to explain it so that misunderstandings are excluded. I still wonder about the Van Hiele levels. A novice would only be aware that the same word has different uses (A. Einstein might also be Alfred Einstein), while a more experienced wiki disambiguator would see ripe fruits everywhere.

Google Translate already knows about different communities – say, bubble originates in the soap industry but is used metaphorically (a form of abstraction) in economics (stock market bubble). The word translates nicely into Spanish burbuja, and Google already indicates that also the Spanish speaking world would be aware of the notion of living in a bubble – check here. But perhaps we are missing some higher levels of abstraction here, like 1 bubble + 1 bubble can have all kinds of outcomes, sometimes 0, 1,2, 3, … bubbles. Not only in reality, but also in economics, and perhaps some topological models, or when a man in a bubble meets a woman in a bubble. For some a bubble is just a word, for others a world.


Your level of fun may increase by maintaining a lay level of insight.

The earlier discussion on Stellan Ohlsson brought up the issue of abstraction. It appears useful to say a bit more on terminology.

An unfortunate confusion at wikipedia

Wikipedia – no source but a portal – on abstraction creates a confusion:

  1. Correct is: “Conceptual abstractions may be formed by filtering the information content of a concept or an observable phenomenon, selecting only the aspects which are relevant for a particular purpose.” Thus there is a distinction between abstract and concrete.
  2. Confused is: “For example, abstracting a leather soccer ball to the more general idea of a ball selects only the information on general ball attributes and behavior, eliminating the other characteristics of that particular ball.” However, the distinction between abstract and concrete is something else than the distinction between general and particular.
  3. Hopelessly confused is: “Abstraction involves induction of ideas or the synthesis of particular facts into one general theory about something. (…) Bacon used and promoted induction as an abstraction tool, and it countered the ancient deductive-thinking approach that had dominated the intellectual world since the times of Greek philosophers like Thales, Anaximander, and Aristotle.” This is hopelessly confused since abstraction and generalisation (with possible induction) are quite different. (And please correct for what Bacon suggested.)

A way to resolve such confusion is to put the categories in a table and look for examples for the separate cells. This is done in the table below.

In the last row, the football itself would be a particular object, but the first statement refers to the abstract notion of roundness. Mathematically only an abstract circle can be abstractly round, but the statement is not fully mathematical. To make the statement concrete, we can refer to statistical measurements, like the FIFA standards.

The general statement All people are mortal comes with the particular Socrates is mortal. One can make the issue more concrete by referring to say the people currently alive. When Larry Page would succeed in transferring his mind onto the Google supercomputer network, we may start a philosophical or legal discussion whether he still lives. Mutatis mutandis for Vladimir Putin, who seems to hope that his collaboration with China will give him access to the Chinese supercomputers.

Category (mistake) Abstract Concrete
General The general theory of relativity All people living on Earth in 2015 are mortal
Particular The football that I hold is round The football satisfies FIFA standards
The complex relation between abstract and general

The former table obscures that the relation between abstract and general still causes some questions. Science (Σ) and philosophy (Φ) strive to find universal theories – indeed, a new word in this discussion. Science also strives to get the facts right, which means focusing on details. However, such details basically relate to those universals.

The following table looks at theories (Θ) only. The labels in the cells are used in the subsequent discussion.

The suggestion is that general theories tend to move into the abstract direction, so that they become universal by (abstract) definition. Thus universal is another word for abstract definition.

A definition can be nonsensical, but Σ strives to eliminate the nonsense, and officially Φ has the same objective. A sensible definition can be relevant or not, depending upon your modeling target.

(Θ) Aspects of scientific theories (Σ) Science (Φ) Philosophy
(A) Abstract definition (developed mathematically or not) (AΣ) Empirical theory. For example law of conservation of energy, economics Y = C + S, Van Hiele levels of insight (AΦ) Metaphysics
(G) General (GΣ) Statistics (GΦ) Problem of induction
(R) Relation between (A) and (G) (RΣ) (a) Standards per field,
(b) Statistical testing of GΣ,
(c) Definition & Reality practice
(RΦ) (a) Traditional epistemology,
(b) Popper,
(c) Definition & Reality theory

Let us redo some of the definitions that we hoped to see at wikipedia but didn’t find there.

Abstraction is to leave out elements. Abstractions may be developed as models for the relevant branch of science. The Van Hiele levels of insight show how understanding can grow.

A general theory applies to more cases, and intends to enumerate them. Albert Einstein distinguished the special and the general theory of relativity. Inspired by this approach, John Maynard Keynes‘s General Theory provides an umbrella for classical equilibrium (theory of clearing markets) and expectational equilibrium (confirmation of expectations doesn’t generate information for change, causing the question of dynamic stability). This General Theory does not integrate the two cases, but merely distinguishes statics and its comparative statics from dynamics as different approaches to discuss economic developments.

Abstraction (A) is clearly different from enumeration (G). It is not impossible that the enumeration concerns items that are abstract themselves again. But it suffices to assume that this need not be the case. A general theory may concern the enumeration of many particular cases. It would be statistics (GΣ) to collect all these cases, and there arises the problem of induction (GΦ) whether all swans indeed will be white.

Having both A and G causes the question how they relate to each other. This question is studied by R.

This used to be discussed by traditional epistemology (RΦ(a)). An example is Aristotle. If I understand Aristotle correctly, he used the term physics for the issues of observations (GΣ) and metaphysics for theory (AΦ & GΦ). I presume that Aristotle was not quite unaware of the special status of AΣ, but I don’t know whether he said anything on this.

Some RΦ(a) neglect Σ and only look at the relation between GΦ and AΦ. It is the price of specialisation.

Specialisation in focus is also by statistical testing (RΣ(b)) that only looks at statistical formulations of general theories (GΣ).

The falsification theory by Karl Popper may be seen as a philosophical translation (RΦ(b)) of this statistical approach (RΣ(b)). Only those theories can receive Popper’s label “scientific” that are formulated in such manner that they can be falsified. A black swan will negate the theory that all swans are white. (1) One of Popper’s problems is the issue of measurement error, encountered in RΣ(b), with the question how one is to determine sample size and level of confidence. Philosophy may only be relevant if it becomes statistics again. (2) A second problem for Popper is that AΣ is commonly seen as scientific, and that only their relevance can be falsified. Conservation of energy might be relevant for Keynes’s theory, but not necessarily conversely.

The Definition & Reality methodology consists of theory (RΦ(c)) and practice (RΣ(c)). The practice is that scientists strive to move from the particular to AΣ. The theory is why and how. A possible intermediate stage is G but at times direct abstraction from concreteness might work too. See the discussion on Stellan Ohlsson again.


Apparently there exist some confusing notions about abstraction. These can however be clarified, see the above.

The Van Hiele theory of levels of insight is a major way to understand how abstraction works.

Paradoxically, his theory is maltreated by some researchers who don’t understand how abstraction works. It might be that they first must appreciate the theory before they can appreciate it.

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)

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.

Mathematics education research (MER) not only looks at the requirements of mathematics and the didactics developed in the field itself, but also at psychology on cognition, learning and teaching in general, at pedagogy on the development of pupils and students, and at other subjects, such as physics or economics for cases when mathematics is applied, or general philosophy indeed. The former weblog text said something about neuro-psychology. Today we have a look at cognitive psychology.

Stellan Ohlsson: Deep learning

Stellan Ohlsson (2011) Deep Learning: How the Mind Overrides Experience may be relevant for mathematics education. One teaching method is to get students to think about a problem until the penny drops. For this, Ohlsson discusses a bit more than the distinction between old and new experience:

“(…) the human mind also possesses the ability to override experience and adapt to changing circumstances. People do more than adapt; they instigate change and create novelty.” (cover text)

“If prior experience is a seriously fallible guide, learning cannot consist solely or even primarily of accumulating experiences, finding regularities therein and projecting those regularities onto the future. To successfully deal with thoroughgoing change, human beings need the ability to override the imperatives of experience and consider actions other than those suggested by the projection of that experience onto the situation at hand. Given the turbulent character of reality, the evolutionary strategy of relying primarily on learned rather than innate behaviors drove the human species to evolve cognitive mechanisms that override prior experience. This is the main theme of this book, so it deserves a label and an explicit statement:

The Deep Learning Hypothesis

In the course of shifting the basis for action from innate structures to acquired knowledge and skills, human beings evolved cognitive processes and mechanisms that enable them to suppress their experience and override its imperatives for action.” (page 21)

Stellan Ohlsson's book (2011) (Source: CUP)

Stellan Ohlsson’s book (2011) (Source: CUP)

Definition & Reality methodology

The induction question is how one can know whether all swans are white. Even a statistical statement runs into the problem that the error is unknown. Skepticism that one cannot know anything is too simple. Economists have the question how one can make a certain general statement about the relation between taxation and unemployment.

My book DRGTPE (2000, 2005, 2011) (PDF online) (though dating from 1990, see the background papers from 1992) proposes the Definition & Reality methodology. (1) The model contains definitions that provide for certainty. Best would be logical tautologies. Lack of contrary evidence allows room for other definitions. (2) When one meets a black “swan” then it is no swan. (3) It is always possible to choose a new model. When there are so many black “swans” that it becomes interesting to do something with them, then one can define “swan2”, and proceed from there. Another example is that in one case you must prove the Pythagorean Theorem and in the other case you adopt it as a definition for the distance metric that gives you Euclidean space. The methodology allows for certainty in knowledge but of course cannot prevent surprises in empirical application or future new definitions. The methodology allows DRGTPE to present a certain analysis about a particular scheme in taxation – the tax void – that causes needless unemployment all over the OECD countries.

Karl Popper (1902-1994) was trained as a psychologist, and there met with the falsification approach by Otto Selz (1881-1943). Popper turned this into a general philosophy of science. (Perhaps Selz already thought in that direction though.) The Definition & Reality methodology is a small amendment to falsificationalism. Namely, definitions are always true. Only their relevance for a particular application is falsifiably. A criterion for a scientific theory is that it can be falsified, but for definitions the strategy is to find general applicability and reduce the risk of falsification. In below table, Pierre van Hiele presented his theory of levels of insight as a general theory of epistemology, but it is useful to highlight his original application to mathematics education, with the special property of formal proof. Because of this concept of proof, mathematics may have a higher level of insight / abstraction overall. Both mathematics and philosophy also better take mathematics education research as their natural empirical application, to avoid the risk of getting lost in abstraction.

Addendum September 7: The above assumes sensible definitions. Definitions might be logically nonsensical, see ALOE or FMNAI. When a sensible definition doesn’t apply to a particular situation, then we say that it doesn’t apply, rather than that it would be untrue or false. An example is an econometric model that consists of definitions and behavioural equations. A definition that has no relevance for the topic of discussion is not included in that particular model, but may be of use in another model.

(Un-) certainty Definitions Constants Contingent
Mathematics Euclidean space Θ = 2π ?
Physics Conservation of energy Speed of light Local gravity on Earth
Economics Savings are income minus consumption Institutional (e.g. annual tax code) Behavioural equations
Mathematics education Van Hiele levels of insight Institutional Student variety

To my great satisfaction, Ohlsson (2011:234) adopts basically the same approach.

“The hypothetical process that supposedly transforms particulars into abstractions is called induction and it is often claimed to operate by extracting commonalities across multiple particulars. If the first three swans you ever see are white, the idea swans are white is likely to come to mind. However, the notion of induction is riddled with problems. How are experiences grouped for the purpose of induction? That is, how does the brain know which experiences are instances of some abstraction X, before that abstraction has been learned? How many instances are needed? Which features are to be extracted? How are abstractions with no instances in human experience such as the infinite, the future and perfect justice acquired?”

Definition of abstraction

There is an issue w.r.t. the definition of abstraction though. Compare:

  • My definition of abstraction is leaving out aspects, see here on this weblog, and see FMNAI. My suggestion is that thought itself consist of abstractions. Abstraction depends upon experience since experience feeds brain and mind, but abstraction does not depend upon repeated experience.
  • Ohlsson (2011:16) takes it as identical to induction, which explains the emphasis upon experience in his title, rather taken as repetition: “Memories of individual events are not very useful in themselves, but, according to the received view, they form the raw material for further learning. By extracting the commonalities across a set of related episodic memories, we can identify the underlying regularity, a process variously referred to as abstraction, generalization or induction.” For Ohlsson, thoughts do not consists of abstractions, but of representations (models): “In the case of human cognition – or the intellect, as it would have been called in the 19th century – the relevant stuff consists of representations. Cognitive functions like seeing, remembering, thinking and deciding are implemented by processes that create, utilize and revise representations.” and “Representations are structures that refer to something (other than themselves).” (page 29)

Ohlsson has abstraction ⇔ induction (commonality). For me it is dubious whether induction really exists. The two pathways are too different to use equivalence. (i) Comparing A and B, one must first abstract from A and then abstract from B, before one may decide whether those abstractions are the same, and before one can even say that A and B share a commonality. (ii) An abstract idea like a circle might cause an “inductive” statement that all future empirical circles will tend to be round, but this isn’t really what is meant by “induction” – which is defined as the “inference” from past swans to future swans.

For me, an abstraction can be a model too, and thus would fit Ohlsson’s term representation, but the fact that he chooses abstraction ⇔ induction rather than abstraction ⇔ representation causes conceptual problems. Ohlsson’s definition of abstraction seems to hinder his understanding of the difference between concrete versus abstract as used in mathematics education research (MER).

Concrete versus abstract

Indeed, Ohlsson suggests an inversion of how people arrive at insight:

“The second contribution of the constraint-based theory is the principle that practical knowledge starts out general and becomes more specific in the course of learning. There is a long-standing tradition, with roots in the beginnings of Western philosophy, of viewing learning as moving in the opposite direction, from particulars to abstractions. [ftnt 38 e.g. to Piaget] Particulars are given in perception while abstractions are human constructions, or so the ancient story goes.” (p234)

“The fundamental principle behind these and many other cognitive theories is that knowledge moves from concrete and specific to abstract and general in the course of learning.” (Ohlsson 2011:434 that states ftnt 38)

If I understand this correctly, and combine this with the earlier argument that general knowledge is based upon induction from specific memories, then we get the following diagram. Ohlsson’s theory seems inconsistent, since the specific memories must derive from specific knowledge but also presume those. Perhaps a foetus starts with a specific memory without knowledge, and then a time loop starts with cumulation over time, like the chicken-egg problem. But this doesn’t seem to be the intention.

Trying to understand Ohlsson's theory of knowledge

Trying to understand Ohlsson’s theory of knowledge

There is this statement on page 31 that I find confusing since now abstractions [inductions ?] depend upon representations, while earlier we had them derived from various memories.

“The power of cognition is greatly increased by our ability to form abstractions. Mathematical concepts like the square root of 2 and a four-dimensional sphere are not things we stumble on during a mountain hike. They do not exist except in our representations of them. The same is true of moral concepts like justice and fairness, as well as many less moral ones like fraud and greed. Without representation, we could not think with abstractions of any kind, because there is no other way for abstract entities to be available for reflection except via our representations of them. [ftnt 18]”

Ftnt 18 on page 402: “Although abstractions have interested philosophers for a long time, there is no widely accepted theory of exactly how abstractions are represented. The most developed candidate is schema theory. (…)”

My suggestion to Ohlsson is to adopt my terminology, so that thought, abstraction and representation cover the same notion. Leave induction to the philosophers, and look at statistics for empirical methods. Then eliminate representation as a superfluous word (except for representative democracy).

That said, we still must establish the process from concrete to abstract knowledge. This might be an issue of terminology too. There are some methodological principles involved however.

Wilbrink on Ohlsson

Dutch psychologist Ben Wilbrink alerted me to Ohlsson’s book – and I thank him for that. My own recent book A child wants nice and no mean numbers (CWNN) (PDF online) contains a reference to Wilbrink’s critical discussion of arithmetic in Dutch primary schools. Holland suffers under the regime of “realistic mathematics education” (RME) that originates from the Freudenthal “Head in the Clouds Realistic Mathematics” Institute (FHCRMI) in Utrecht. This FHCRMI is influential around the world, and the world should be warned about its dismal practices and results. Here is my observation that Freudenthal’s approach is a fraud.

Referring to Ohlsson, Wilbrink suggests that the “level theory by Piaget, and then include the levels by Van Hiele and Freudenthal too” (my translation) are outdated and shown wrong. This, however, is too fast. Ohlsson indeed refers to Piaget (stated ftnt 38) but Van Hiele and Freudenthal are missing. It may well be that Ohlsson missed the important insight by Van Hiele. It may explain why Ohlsson is confused about the directions between concrete and abstract.

A key difference between Van Hiele and Freudenthal

CWNN pages 101-106 discusses the main difference between Hans Freudenthal (1905-1990) and his Ph.D. student Pierre van Hiele (1909-2010). Freudenthal’s background was abstract mathematics. Van Hiele was interested from early on in education. He started from Piaget’s stages of development but rejected those. He discovered, though we may as well say defined, levels of insight, starting from the concrete to the higher abstract. Van Hiele presented this theory in his 1957 thesis – the year of Sputnik – as a general theory of knowledge, or epistemology.

Freudenthal accepted this as a thesis, but, mistook this as the difference between pure and applied mathematics. When Freudenthal noticed that his prowess in mathematics was declining, he offered himself the choice of proceeding his life with the history of mathematics or the education of mathematics. He chose the latter. Hence, he coined the phrase realistic mathematics education (RME), and elbowed Van Hiele out of the picture. As an abstract thinking mathematician, Freudenthal created an entire new reality, not caring about the empirical mindset and findings by Van Hiele. One should really read CWNN pages 101-106 for a closer discussion of this. Van Hiele’s theory on knowledge is hugely important, and one should be aware how it got snowed under.

A recent twist in the story is that David Tall (2013) rediscovered Van Hiele’s theory, but wrongly holds (see here) that Tall himself found the general value while Van Hiele had the misconception that it only applied to geometry. In itself it is fine that Tall supports the general relevance of the theory of levels.

The core confusion by Ohlsson on concrete versus abstract

The words “concrete” and “abstract” must not be used as absolutely fixed in exact meaning. This seems to be the core confusion of Ohlsson w.r.t. this terminology.

When a child plays with wooden blocks we would call this concrete, but our definition of thought is that thinking consists of abstractions, whence the meanings of the two words become blurred. The higher abstract achievement of one level will be the concrete base for the next level. The level shift towards more insight consists of compacting earlier insights. What once was called “abstract” suddenly is called “concrete”. The statement “from concrete to abstract” indicates both the general idea and a particular level shift.

Van Hiele’s theory is essentially a logical framework. It is difficult to argue with logic:

  1. A novice will not be able to prove laws or the theorems in abstract mathematics, even informally, and may even lack the notion of proof. Having achieved formal proof may be called the highest level.
  2. A novice will not be able to identify properties and describe their relationships. This is clearly less complex than (1), but still more complex than (3). There is no way going from (3) to (1) without passing this level.
  3. A novice best starts with what one knows. This is not applied mathematics, as Freudenthal fraudently suggested, but concerns the development of abstractions that are available at this level. Thus, use experience, grow aware of experience, use the dimensions of text, graph, number and symbol, and develop the thoughts about these.

Van Hiele mentioned five levels, e.g. with the distinction between informal and formal deduction, but this is oriented at mathematics, and above trident seems sufficient to establish the generality of this theory of knowledge. A key insight is that words have different meanings depending upon the level of insight. There are at least three different languages spoken here.

Three minor sources of confusion are

  • Ohlsson’s observation that one often goes from the general to the specific is correct. Children may be vague about the distinction between “a man” and “one man”, but as grown up lawyers they will cherish it. This phenomenon is not an argument against the theory of levels. It is an argument about becoming precise. It is incorrect to hold that “one man” is more concrete and “a man” more abstract.
  • There appears to exist a cultural difference between on one side Germans who tend to require the general concept (All men are mortal) before they can understand the particular (Socrates is mortal), and the English (or Anglo-Saxons who departed from Germany) who tend to understand only the particular and to deny the general. This cultural difference is not necessarily epistemological.
  • Education concerns knowledge, skill and attitude. Ohlsson puts much emphasis on skill. Major phases then are arriving at a rough understanding and effectiveness, practicing, mastering and achieving efficiency. One can easily see this in football, but for mathematics there is the interplay with the knowledge and the levels of insight. Since Ohlsson lacks the levels of insight, his phases give only part of the issue.

I have looked only at parts of Ohlsson’s book, in particular above sections that allow a bit more clarity on the relevance w.r.t. Van Hiele’s theory of levels of insight. Please understand my predicament. Perhaps I read more of Ohlsson’s book later on, but this need not be soon.

  • In mathematics education research (MER) we obviously look at findings of cognitive psychology, but this field is large, and it is not the objective to become a cognitive psychologist oneself.
  • When cognitive psychologists formulate theories that include mathematical abstraction, as Ohlsson does, let them please look at the general theory on knowledge by Pierre van Hiele, for this will make it more relevant for MER.
  • Perhaps cognitive psychologists should blame themselves for overlooking the theory by Pierre van Hiele, but they also should blame Hans Freudenthal, and support my letter to IMU / ICMI asking to correct the issue. They may work at universities that also have departments of mathematics and sections that deal with MER, and they can ask what happened.
  • When there is criticism on the theory by Van Hiele, please look first at the available material. There are summary statements on the internet, but these are not enough. David Tall looked basically at one article and misread a sentence (and his misunderstanding still was inconsistent with the article). For some references on Van Hiele look here. (There is the Van Hiele page by Ben Wilbrink, but, as said, Wilbrink doesn’t understand it yet.)

The preceding weblog text considered the pronunciation of numbers in English, German, French, Dutch and Danish.

There better be a general warning about invalidity of current research on number sense.

Update Sept 3: There now is also this proposal on developing an international standard for the mathematical pronunciation of the natural numbers.

Warning 1. The object of study concerns a chaotic situation

Research on how children learn numbers, counting and arithmetic, is mostly done in the context of the current confusing pronunciations. This is like studying people walking a tightrope while saying the alphabet in reverse order. This will not allow conclusions on the separate abilities: (a) dealing with arithmetic, (b) dealing with a confusing dialect.

In methodological terms: common studies suffer from invalidity. (Wikipedia.) They aren’t targeted at their research objective: number sense. Perhaps they intend to, but they are shooting into a fog, and they cannot be on target.

A positive exception is this article by Lisser Rye Ejersbo and Morten Misfeldt (2015), “The relationship between number names and number concepts”. They provide pupils with the mathematical names of numbers and study how this improves their competence. This reduces the chaos that other studies leave intact.

It is insufficient to state that you want to study “number sense in the current situation”. When you grow aware that the current situation seriously hinders number sense, then you ought to see that your research objective is invalid, since the current situation confuses number sense. If you still want to study number sense in the current situation, hit yourself with a hammer, since apparently this is the only thing that will still stop you.

Warning 2. Results will be useless

Results of studies within the current chaos will tend to be useless: (a) They cannot be used w.r.t. mathematical pronunciation, since they don’t study this. (b) Once the mathematical pronunciation is implemented, results on number sense within the current chaotic situation are irrelevant.

Validity and reliability (source: wikimedia commons)

Validity and reliability (source: wikimedia commons)

Warning Sub 2. Don’t be confused by a possible exception

There seems to be one exception to warning 2: the comparison of English, which has low chaos in pronunciation, to other situations with higher chaos (Dutch, German, French, Danish). This presumes similar setup of studies, and would only be able to show that mathematical pronunciation indeed is better. Which we already know. It is like establishing over and over again that drinking affects driving. The usefulness of this kind of study thus must be doubted too. One should not be confused in thinking that it would be useful.

Indeed, we might imagine a diagram with a horizontal axis giving skill in addition with outcomes in the range 10-20 and a vertical axis giving skill in addition with outcomes in the range 20-50, both giving the ages when satisfactory skills have been attained, and then plot the results for English, German, French, Dutch and Danish. We would see that English has lower ages, and French might actually do better than German, since the strange French number names are for 70-99. It might make for a nice diagram, but the specific locations don’t really matter since we already know the main message.

For example, Xenidou-Dervou (2015:14) states:

“Increasingly more studies are suggesting that this inconsistency between spoken and written numbers can have negative effects on school-aged children’s symbolic processing (e.g., Helmreich et al., 2011).”

Compare this with our earlier observation that professor Fred Schuh of TU Delft already proposed  on these grounds a reform of pronunciation in Dutch in 1943, 1949 and 1952 … Parliament in Norway (their “Storting”) decided in July 1950 to rename the numbers above 20 in English fashion.

It is not only problematic that Xenidou-Dervou isn’t aware of this, but also that she doesn’t see that the current chaotic situation invalidates her own research setup.

She remarks (2015:14) that the logical clarity (Schuh’s insight) has not been subjected to statistical testing. This may be true. When you don’t understand that drinking affects driving, then you might require statistics. Doing such tests is as relevant as statistical research on verifying that drinking affects driving. She states (my emphasis):

“To the best of our knowledge, the effect that the language of numbers can have in the development of a core system of numerical cognition such as children’s symbolic approximation skills [using Arabic numbers], controlling for their nonsymbolic approximation skills [using representations like dots but apparently not fingers] has not been previously addressed.”

Thus, the statistics on drunk driving are corrected for the performance when drunk riding a bicycle. It might be suggested that nonsymbolic number sense would be independent from language, and we might readily accept this for numbers smaller than 10, but to properly test this for 11-99 we need a large sample of Kaspar Hausers who are unaffected by language. Xenidou-Dervou’s correction does not remove the contamination by language.

Statistical tests may indeed be used to establish that large males tend to have a higher tolerance for drinking than small females, and to test legal standards. But questions like these are not at issue in the topic of number sense.

The relevant points are:

  • It is already logically obvious that a change to mathematical pronunciation will be beneficial. There is no need for statistical confirmation, e.g. by comparing English with other language situations. To suggest that such research would be necessary is distractive w.r.t. the real scientific question (see next).
  • The study of number sense can only be done validly in a situation with mathematical pronunciation, without the noise of the current chaotic situation of the national language dialects.

(PM. This is inverse of the case that there was statistical information that smoking was highly correlated with lung cancer, but that the tabacco industry insisted upon biological evidence. This analogy might arise when researchers would have stacks of statistical results proving that weird pronunciation is highly correlated with slow acquisition of mathematical understanding and skill, while there would be a strong lobby for maintaining national pronunciation who insist upon biological evidence. Thus do not confuse these statistical situations.)

Curiously, the press-release on Xenidou-Dervou’s promotion event and publication of the thesis of January 7 2015 states that she ‘discovered’ something which was already well known to Fred Schuh in 1943, 1949, 1952, if not some present-day teachers and children themselves:

“From age 5 the influence of teaching is larger than of natural abilities. What hinders Dutch children is the way how numbers are pronounced in Dutch. These relations have been found by Iro Xenidou-Dervou (…)”

“One of the teachers in the researched schools could confirm this with an anecdote from practice. She had heard one pupil telling another pupil doing a calculation: “Do it in English, that is easier.””

“Xenidou-Dervou thus suggests to start in Holland with education in symbolic calculation [with Arabic numbers] already before First Grade [age 6].”

Perhaps we might already start with Arabic numbers before First Grade indeed. Some children already watch Sesame Street. It would be more advisable to do something about pronunciation however. It is perhaps difficult to maintain common sense when you are in a straight-jacket of thesis research.

Warning 3. Such studies will not discover the true cause for the current chaotic situation

The barrier against the use of mathematical pronunciation doesn’t lie with the competences of children but with the national decision making structure. Thus, most current studies on education and number sense will never discover, let alone resolve, the true problem.

That the mathematical pronunciation will be advantageous is crystal clear. Of course it helps when you are allowed to first walk the tightrope and only then say the alphabet in reverse. Thus we have to look at the national decision making structure to see why this isn’t done.

Of key importance are misconceptions about mathematicians. Policy makers and education researchers often think that mathematicians know what they are doing while they don’t. Education researchers may be psychologists with limited interest in mathematics per se. Few are critical of what children actually must learn.

We may accept that psychology is something else than mathematics education, but when a psychologist researches the education of mathematics then we ought to presume that they know about mathematics education. When they don’t understand mathematics education then they should not try to force it into their psychological mold, and go study something else.

Two relevant books of mine on this issue are:

Warning 4. Mathematics education research has breaches of scientific integrity

Current research on education and number sense assumes that there is an environment with integrity of science. However, there is a serious breach by Hans Freudenthal (1905-1990) w.r.t. the results of his Ph. D. student Pierre van Hiele (1909-2010). Van Hiele discovered the key educational relevance of the distinction between concrete versus abstract, with levels of insight, while Freudenthal interpreted that as the distinction between applied and pure mathematics, and henceforth used his elbows to get Van Hiele out of the way.  Freudenthal was an abstract thinking mathematician who invented his own reality. There now exists a Freudenthal “Head in the Clouds Realistic Mathematics” Institute in Utrecht. Its employees behave as a sect, reject criticism, will not look into Freudenthal’s breach of integrity of science, and will not undo the damage. See my letter to IMU / ICMI. Other researchers tend not to know about this, and tend to accept “findings” from Utrecht assuming that it has a “good reputation”.

This warning holds in general

Just to be sure: this warning on invalidity of research on number sense is general. We might for example think of issues discussed in the Oxford Handbook of Numerical Cognition (2015), edited by Ann Dowker. Or think about issues discussed by Korbinian Moeller et al. (2011), or E. Klein et al. (2013). But, this weblog is about a major problem in Holland, and thus it might help to make some remarks concerning the anatomy of Holland.

Comment w.r.t. the Dutch MathChild project

The Dutch MathChild project can be found here, with contacts in Belgium, UK and Canada. Its background is in psychology and not in mathematics education.

The Amsterdam thesis by Iro Xenidou-Dervou (2015) is not fully online and it should be.

There is the full thesis by Ilona Friso-van den Bos (2014). She did the thesis at the dept. of education & pedagogy in Utrecht, but now she is at the Freudenthal “Head in the Clouds Realistic Mathematics” Institute (FHCRMI). I looked at this thesis only diagonally. Issues quickly become technical and this is secondary to the first question about validity. At first glance the thesis does not show sect behaviour (allowing for contagion from FHCRMI to other places at Utrecht University). The names of Freudenthal and Van Hiele are not in the thesis. The thesis has a neuro-psychological setup with a focus on working memory, which suggests some distance from mathematics education.The scheme of the thesis is that you define a test for number sense, a test for working memory, and a test for mathematical proficiency (try to imagine this without number sense and working memory), and then use children to see what model parameters can be estimated. Criticism 1 is that “mathematics achievement” is in the title and used frequently (see also the picture on p282), and taken for Holland as the CITO score (p160), which has a high FHCRMI content (so we find contagion indeed). Criticism 2 is that working memory belongs to the current fashion in neuro-psychology but is less relevant for mathematics education. For ME it is important to get rid of Freudenthal’s misconceptions and to look at Van Hiele levels of insight. Thus, get proper use of working memory, rather than train it to become a bit larger to do crummy FHCRMI math.

Criticism 3 concerns our present issue: the handling of the pronunciation of numbers. The thesis gives:

“(…) a difference between participants from linguistic backgrounds in which number words are inverted (e.g., saying six-and-twenty instead of twenty-six), because these inversions have been suggested to be a source of difficulty in number processing (Klein et al., 2013), and that errors related to inversion can be associated with central executive performance (Zuber, Pixner, Moeller, & Nuerk, 2009).” (p82)

“Publication year and inversion of number words did not play a role in the prediction of effect sizes.” (p97)

On p197-198 we find, my emphasis:

“An alternative explanation for the deviation in findings between previous studies (e.g., Barth & Paladino, 2011) and the current study is that in all previous studies, children were taught in English, in which the number system is more uniform than the Dutch number system. Dutch number words include the ones before the tens, instead of tens before ones (e.g., instead of saying thirty-five, one would say five-and-thirty), which is inconsistent with the order of written numerals. This may make it more difficult for young children to gain insight into the number system, and might explain the large number of children being placed in the random group during kindergarten, leading children to prevail in using less mature placement strategies and skipping the strategy with three reference points to inform number line placements in favour of the most advanced strategy, which is making linear placements. This hypothesis, however, rests under the assumption that children make placements through interpretation of verbal number words, either by transcoding the written number or by listening closely to the experimenter reading the numbers out loud. A study by Helmreich et al. (2011) indeed suggested that inversion errors may be of influence on number line placements in primary school children, although an important difference with the current study was that no numbers were read out loud by the experimenter, making the chance of inversion errors larger. More experimental studies are needed to investigate similar differences in findings and manipulate strategy use through variations in instruction in various groups.”

Criticism 3 thus generates the sub-criticisms:

  1. It is not only problematic that Friso-Van den Bos doesn’t give the earlier reference to professor Fred Schuh of TU Delft in 1943, 1949 and 1952, but also that she doesn’t see that the current chaotic situation invalidates her own research setup. Yes, we do see that she makes a correction at times, but the point is that the proper correction is that the thesis as a whole is shelved, since the situation that she studies cannot render the data that she needs.
  2. It is curious that she states that “more experimental studies are needed”. Compare this with a study of drunken driving in London, Paris, Oslo, Athens, … to test whether there are differences … I cannot understand how an educator can observe the crooked pronunciation of numbers, and not see immediately how important it is to remove the bottleneck rather than further research it. This is like finding a cancer and not remove it but argue that it needs more study. One might say that it is “only a Ph. D. study”, but the idea of a dissertation is that it shows that one can do scientific research by oneself individually. A researcher should be able to spot issues on validity. (Perhaps most Ph. D. students are too young or perhaps standards are too low given current academic culture.)
Concluding on the responsibility of educators of mathematics

As in the earlier weblog text, the main responsibility lies with Parliament: to investigate the issue.

It will still be the educators of mathematics who have the responsibility to re-engineer the mathematical pronunciation of numbers, to be used in education, and subsequently also in society and courts of justice. As a teacher of mathematics, I have presented my suggestions in the earlier weblog text, see here.

The Dutch government wants to determine the national research agenda to 2025. Not only the minister of education and science but also the minister of economic affairs expressed an interest in this. These ministers set up a “knowledge coalition” consisting of some research institutes and users of science like an organisation of employers. This “coalition” formed a “steering group”, under the joint chair of Alexander Rinnooy Kan (ARK) (1949) and Beatrice de Graaf (BdG) (1976).

ARK & BdG thought it a good idea to allow all Dutch people to send in their research questions. This caused 11700 questions. Also using text recognition software by Piek Vossen of Vrije Universiteit (VU), these were reduced to 252 umbrella questions, except for some 2000 that were not reduced. It is not guaranteed that this “wisdom of the crowd” will generate anything useful. Hence there is a “phase of dialogue” till the end of October, in which the mandarins of the “knowledge coalition” discuss what they really want to do. Perhaps the 11700 questions make for interesting wallpaper but it is not unlikely that the final report will give some evaluation of the entire exercise.

My own 14  questions are here, and I am wondering whether I am in an open society or in a maze.

My first contribution to the dialogue was a debunking of some questions on religion studies, see this PDF in Dutch, or see below. My second contribution to the dialogue are the following comments.

Linking up and down

A compliment for the people at the research agenda project is that they have linked the umbrella questions to the underlying separate questions, and vice versa. This is handy.

It increases the feeling that you are in a maze but it at least you can see where you are in there. Now we don’t need to discuss a question but only whether it has been allocated to the right umbrella.

This linking might actually also be done with internet pages of individual scientists. They normally state their research interests, and these might be processed in similar manner. The advantage of this particular ARK & BdG project is the common format: title, 200 words, keywords, use of Dutch. There is a “complaint” that some scientists have been “abusing” the “crowd sourcing” to advocate their own research, but I would rather have that input.

One modest question arises too: I am wondering what would happen when my 14 questions were taken as umbrella questions: would the total increase to 252 + 14 or would it reduce to say 250, with less than 2000 left-overs ? These would be marginal changes in terms of software results, but the advantage would be that the discussion could also focus on those 14 questions, instead of hiding and dispersing them all-over. Perhaps other people feel the same about their submissions. Results of course depend upon the software rules.

K.P. Hart on mathematics

Klaas Pieter Hart of TU Delft apparently was struck by the occurrence of lay questions on mathematics, and started a weblog on “math questions that have been answered already”. He frequently refers to wikipedia pages, and one indeed wonders why the questioners did not look there first. My impression is that Hart makes too much of the matter. We can also regard it as very kind of him to take more time to explain that such questions have already been answered.

A less kind interpretation is that mathematical arrogance is at play again. Hart’s weblog has the attitude that lay people do not understand mathematics, and that more explanation should close the gap. This is a strange attitude. People have been getting education in mathematics for ages 6-18. This would not be enough to settle the basic questions that Hart discusses ? This education is not enough to clarify to people that they should first study the question e.g. on wikipedia before submitting it to the national research agenda ? My diagnosis is that something is wrong with math education. It is strange that Hart doesn’t arrive at the same diagnosis. He puts the error with people, I put the error with him and his fellow mathematicians.

There is more to it. Let us look at some issues.

(1) There is my question on the training of teachers of mathematics. Hart hasn’t written on this yet. Perhaps it is safe to conclude that this is not “answered already”. But we must wait till Hart stops his weblog to be sure. This makes for a difficult dialogue. It would be more efficient if he could state ahead what he will be writing about.

My question has also been moved under the umbrella question on future education, with a total of 10 sub-questions. I wonder whether that is a useful allocation. Mathematics education is such a core issue that I feel that it deserves umbrella status by itself. For the diagnosis of the other 252 umbrella questions it is important to grow aware that many (social) problems are being caused by bad education in mathematics.

(2) On Pi, Hart misses the opportunity to point to Archi = 2 Pi (check the short movie).

Imagine shoe shops selling only single shoes instead of pairs of shoes. Or builders selling only half houses instead of whole houses. Mathematicians however don’t mind making life difficult for you. It is your problem that you don’t get it, and they will be kind enough to explain it again. Well, perhaps it isn’t kindness: what’s in it for them is that they can feel superior in understanding. This psychological reward system works against the student.

(3) On the sine function, Hart writes that sine and cosine might also be identified by the co-ordinates of the endpoint of the arc on the unit circle. [Dutch: “Eigenlijk hadden we ook kunnen zeggen: cos(t) en sin(t) zijn respectievelijk de x- en y-coördinaat van het eindpunt van het boogje.”]

This is precisely my suggestion: to use functions xur and yur, see my paper Trig rerigged (2008), seven years ago, and books Elegance with Substance (2009, 2015) and Conquest of the Plane (2011). (Best link to the 2nd edition EWS 2015.) Thus, there exists a didactically much better presentation for sine and cosine than mathematicians have been forcing down the throats of students for ages. Hart mentions it in passing, but it is a core issue, and an important piece of evidence for my question on the training of math teachers.

Note also that wikipedia today still has no article on xur and yur. Neither on my proposal to take the plane itself as the unit of account for angles, even though Hart refers to that wikipedia page in his text on “90 graden“. The mathematics pages on wikipedia tend to be run by students from MIT who tend to copy what is in their textbooks, and who lack training on keeping an open mind. See here how wikipedia has been disinforming the world for some years now.

In that weblog article Hart confirms that sine and cosine can be defined by some criteria on the derivatives. This is the approach followed by Conquest of the Plane (2011). It is nice to see agreement that this is the elegant approach indeed. I think that it is advisable that Hart reads COTP and writes a report on it, so that he can also check that the slanderous “review” by his TU Delft colleague Jeroen Spandaw is slanderous indeed – see here.

(4) On numerical succession – a.k.a. mathematical induction but see my discussionHart holds that this does not exist for the real numbers. This is essentially the question whether there is a bijection between the natural numbers N and the real numbers R. Please observe:

  • Since 2011-2012 I present the notion that there is a “bijection by abstraction” between N and R. See my book FMNAI (2015).
  • In his research, Hart has a vested interest in that there is no such bijection.
  • Hart has been partly neglecting the new alternative argument, partly sabotaging its dissemination, partly disinforming his readership about the existence of this new approach. This is a breach in scientific integrity. See my documentation of Hart’s malpractice.
  • It is okay when Hart wants to say that this issue has been “answered already” in his view, e.g. by traditional mathematics. There is freedom of expression. However, as a scientist, he is under the obligation to give all relevant information. This he doesn’t do.

Overall: Hart’s weblog is biased and unscientific.

Religion studies

One of my questions for the research agenda is how to arrive at a deconstruction of Christianity. I also observed that the Dutch national research school on religion studies and theology NOSTER had submitted four questions. I looked at those questions and their 200 word summaries, and found them scientifically inadequate. See my discussion of this (in Dutch). I informed the research agenda secretariat of my finding, in an email of June 10. To my surprise, one of these inadequate questions from NOSTER has been selected as an umbrella question S14, and my own question that debunks it has become a sub-question for it !

The software for doing this has been developed at originally religious Vrije Universiteit (VU), but I presume that this heritage has nothing to do with this.

Censorship at CPB

My question on the censorship of economic science since 1990 by the directorate of the Dutch Central Planning Bureau (CPB) has not been allocated to an umbrella question yet. I suppose that this doesn’t disqualify the question.

Economic Supreme Court

My question on the Economic Supreme Court (ESC) has been allocated to umbrella question S06, that looks at maximizing national sovereignty in an international legal order. Beware though, since the phrasing of S06 might also be read as a ploy by international lawyers to get more attention for international law. The umbrella contains 29 sub-questions, many of which indeed deal with globalisation.

Admittedly, when each nation has its own ESC then international co-operation will be enhanced, since the ESCs will exchange information in scientific manner. This is part of my analysis e.g. on money.

However, the ESC is also very relevant for national issues, e.g. in umbrella question G7 on social-economic institutions. Such institutions could differ with or without an ESC. Findings from this realm would provide evidence in support of an ESC, and thus be relevant for parliament that must decide on amending the constitution to create an ESC.

These matters of CPB and ESC affect one’s view on the social order. This would warrant umbrella status for them.

Democracy and voting theory

I would hold that it is important to have a clear view on democracy and voting theory. My question for the research agenda on this was how one could convince the editors of that their website contains a misleading article. I have been trying for years to get this corrected but they will simply refuse to look into this. This question has been moved to umbrella S31 on issues of privacy and database monopolies. I don’t think that it belongs there. Indoctrinating school students with mathematically proven false information is not the same as privacy and database monopolies. This doesn’t mean that the question should be removed to the 2000 unassigned ones, and further forgotten because there are too many of those.

Other questions

I have some comments on the other questions but a weblog should not be too long.

Conclusion: An appeal to ARK and BdG

Rinnooy Kan has a background in mathematics and De Graaf was in the news with her personal background in Christianity.  Jointly they would be able to recognise the points raised here. My appeal to them is to take my 14 questions as umbrella issues, and work from there.

The breach of scientific integrity by K.P. Hart and his misleading weblog must be dealt with separately.

Philippe Legrain (1973) is a British economist who worked as an advisor for former communist and later EU commission president José Manuel Durão Barroso in 2011-2014. Legrain has now written a book on a European Spring, something that Manny tried hard to achieve, as readers of this weblog know, as I tried to advise Manny too.

The real news from Brussels is that Manny’s successor Jean-Claude Juncker doesn’t want a Spring but a Hot Summer. A showdown with Putin should unite the peoples of the EU, and also force Greece to choose sides, JCJ thinks.  Philippe Legrain’s book is old hat. I am still waiting for approval by JCJ to spill some of the beans of our luncheons, dinner parties and fire-side chats. The big secret in Europe is that Angela, François, Jean-Claude and me were all born in 1954, and that we share an obligation to make it all work, though I am for peace-with-strength and they are hopelessly confused.

This Sunday, Philippe Legrain allowed Dutch television to interview him on his Spring eulogy. Legrain is unaware that Dutch TV journalists are hypocrites by profession, and that he is sucked dry like in those vampire movies.

The interview is in English with Dutch undertitles, and you can check how interviewer Marcia Luyten (1971) takes Legrain for a walk in the woods, guts him, and leaves his body somewhere in a ditch.

About a key moment in the economic crisis:

“There was a panic across the Eurozone, and people thought that governments could not pay their debts. And actually it was a panic that ultimately the ECB was able to solve. In the case of the Netherlands it wasn’t even a panic. This was just a mistake made by this government, under pressure from Brussels, and Berlin.” (Philippe Legrain, minute 35).

It is absurd to portray Holland as a victim of Brussels. It is the other way around. Europe is a victim of Holland:

  • It were rather speculators who saw an opportunity to make a killing. It wasn’t a panic but a real threat, caused by wrong legal rules.
  • The solution provided by the ECB still is an improvisation, and we still need a new treaty, see my piece on the two Mario’s.
  • It were Dutch hawks who joined with Germany and imposed austerity on the rest of Europe, also using Brussels as a front for the Dutch electorate.

Other major errors in the interview are:

  • Legrain apparently never really studied Holland. For him it is a small country that falls under his radar. For hm, Holland is not a perpetrator but a victim. For him, these are nice people, and not hypocrites. Apparently he regards Holland as a free, tolerant, open-minded country, that just happens to have made some policy errors, without kids delving into the trash as now happens in Greece. He can’t make himself see Marcia Luyten as co-responsible for making kids in Greece do so. But she is a hypocrite and co-responsible. She never properly informed the Dutch viewers.
  • Legrain just answers the questions that Marcia asks. He doesn’t expose Holland as a major perpetrator. He does not expose Marcia as belonging to the hypocrites who helped cause the problems. He joins Marcia in her “frame” that it are politicians who do not want to lose face, while a major part of the story is that it are journalists who have been misreporting and misrepresenting.
  • Legrain doesn’t expose Jeroen Dijsselbloem as a major stumbling block: an agricultural economist who got into politics too quickly and who apparently isn’t able to deal with these issues properly. (Dijsselbloem already failed on the policy w.r.t. the education on mathematics.) (See Dijsselbloem on Dutch exports.)
Marcia Luyten reads the English book title "European Spring", by Philippe Legrain

Marcia Luyten reads the English book title “European Spring”, by Philippe Legrain (minutes 22-40)

Thus, Philippe, the next time that you visit Holland, first consider my books DRGTPE and CSBH, and let us discuss those, before you face Dutch journalism.

Incidently, we appear to agree with a lot. For example, that the surplus on the Dutch or German exports account was invested abroad, and basically was squandered when investments failed, is a key feature in my analysis since 1990. See also Johannes Witteveen, a former executive director of the IMF.  But the hypocrites on Dutch television will not allow viewers to hear this from home grown economists. They will welcome people like you, Philippe, who criticize Germany and France, instead of the local Dutch incrowd hawks and perpetrators, and their messenger prime minister Mark Rutte, who has a degree in history and who does not know much about economics but still is zealot about Margaret Thatcher, and who got the Rathenau prize on freedom while he censors economic science in Holland.

But, you also state that you have a proposal how a currency union would work with decentralised decision making. I wonder how you could achieve that. My suggestion has been that each nation adopts an Economic Supreme Court. I wonder whether your ideas are the same. Apparently those are behind a pay wall. This will not work.  Even a crisis should not force people to buy into books that might be scientifically unwarranted.

PM. This hypocritical Buitenhof TV broadcast also contains a discussion with Nikos Koulousios, six minutes before Philippe Legrain. This is somewhat amusing, but not really so.

  • The suggestion is that when Greece and Germany collaborate on jokes, and the Germans admit that they don’t have a sense of humour, then we see a proper collaboration in Europe: and all is fine, and people should feel satisfied that the notion of a unified Europe might work. I am afraid that this only confirms national stereotypes and isn’t real humour.
  • Koulousios calls it typical that a letter signed by economists like Joseph Stiglitz and Thomas Piketty did appear in the Greek paper but not in other papers, except in the Financial Times in which it appeared originally. He seems to suggest some kind of conspiracy, in which the Greek readership is manipulated and not given the right information. However, it may just be that the FT expects royalties. Check why I don’t blog at the FT. Thus the proper diagnosis is that journalists can be quite hypocritical, not only in Holland but even in Greece.

There is one real conclusion from all this. I will discuss it with JCJ the next time I see him. Rather than joining Putin in a hot Summer war on the Ukraine, the EU should pay Russian journalists more money for accurate reporting about the state of the world.