Monthly Archives: September 2015

Listening to Izaline Calister “Mi Pais
“Atardi Korsou ta Bunita”, or Willem Hendrikse,
or Rudy Plaate “Dushi Korsou” or IC & CR “Mi ta stimabu“,
and Frank might also have liked Las Unicas “Ban Gradici Senjor” from Aruba


We need two weblogs to discuss Gerald Goldin (2003), Developing complex understandings: On the relation of mathematics education research to mathematics.

We namely start with Goldin (1992), Towards an assessment framework for school mathematics (pages 63-88 in Lesh & Lamon, Assessment of authentic performance in school mathematics).

The Framework gives a baseline and also allows us to see the long standing involvement of Goldin in mathematics education and its research. We all know about the New Math disaster in the 1960s and about Morris Kline (1908-1992) who in 1973 wrote Why Johnny Can’t Add: the Failure of the New Math. I wasn’t aware however of the heavy involvement in the USA in the 1970s with behaviourism. This US phenomenon wasn’t copied to such extent in Europe. Perhaps the European attention for the New Math partly came from the European (Bourbaki) origins. With these two assaults on mathematics education in the USA, both New Math and behaviourism, we may better understand the flower power reaction movement, partly captured by Hans Freudenthal’s scam of “realistic mathematics education” (RME), in which students are left free to discover mathematics for themselves, with perhaps at most “guided reinvention”.

Recovering from New Math and behaviourism

Thus in 1992, after two decades of struggling to recover from New Math and behaviourism, Goldin writes the following Introduction to this Framework.

Gerald Goldin, "Framework", p63-64, Introduction

Gerald Goldin (1992), “Framework”, p63-64, Introduction

The role of the general public in understanding processes and content is that Goldin wants parents to understand what Johnny is doing in his proposed approach.

Test questions, purposes and methods

Goldin (1992) discusses various possible test questions, and then he questions what they are supposed to test. This is a nice example of a problem that Johnny might be asked to handle mathematically.

Goldin (1992), p68

Goldin (1992), p68

While you are solving this problem, you must also wonder whether this fits the RME philosophy, and what would count as a successful answer. Are you allowed to guess yes/no/maybe or would you have to give a proof ?

Goldin shows how the question can be used for various didactic goals with also various possible interventions, ranging from exploration (my suggestion now too: perhaps just give the egg timers and not pose any questions) to standardized solution technique (that flower power calls boring and behaviourism exciting).

A cognitive model to prevent disasters from new fashions

Goldin cannot avoid Freudenthal’s RME (applied mathematics), and gives an answer that fits Van Hiele’s view that mathematics is directed at higher levels of insight (abstraction).

Goldin (1992), p70-71

Goldin (1992), p70-71

Goldin gives the obvious warning that when we don’t know what we are doing, then we may make all kinds of investments in training and computer programs, that later appear to be useless.

A traditional question

Subsequently, Goldin discusses a traditional question.

Goldin (1992), p72

Goldin (1992), p72

RME would allow students to find their own solution strategies, and students perhaps run afoul when they don’t find any (or arrive at the right answer in mysterious ways). Traditional methods would provide a single algorithm that always generates an outcome, but perhaps students run afoul when the question is slightly changed. I refer to Goldin’s paper for these and more angles.


This discussion is only a rough introduction to Goldin (1992) “Framework”. The idea of this text (1) is to set the stage for weblog text (2). It helps to be aware of what Goldin was already proposing in 1992 when we are going to look at his view in 2003.

My problem with Goldin (1992) is that he does not distinguish the Van Hiele levels of insight. For example, it seems somewhat obvious that traditional Problem 2 requires a different approach for novices than for experts. But it helps to be also aware that the same words may have a different meanings depending upon the level of insight.

For novices Problem 2 may require much creativity and thus can also give a lot of fun. It is only when students become experts that problem 2 becomes boring. The discussion about assessment should not be burdened by the situation that mathematics education can create the paradox that students find Problem 2 boring and still cannot answer it properly. Like in Item-Response Theory, a test does not only say something about students, but students also say something about the test (environment).

Listening to Izaline Calister “Mi Pais
“Atardi Korsou ta Bunita”, or Willem Hendrikse,
or Rudy Plaate “Dushi Korsou” or IC & CR “Mi ta stimabu“,
and Frank might also have liked Las Unicas “Ban Gradici Senjor” from Aruba


Frank Martinus Arion passed away yesterday in Curaçao. The English wikipedia site is a bit short, with his 1973 literary debut Double play. His important scientific work is his thesis: “The kiss of a slave”, that traces Papiamentu to Africa.

Kiss of a Slave, by Martinus Arion, Thesis Univ. of Amsterdam 1996

The Kiss of a Slave, by Efraim Frank Martinus (Arion), Thesis at the Univ. of Amsterdam 1996

Masha Danki !

Frank wouldn’t have wanted us to be sad. The best way to to thank him is to have the biggest party of all.


Carneval 2013 (Source: Screenshot)

I met Frank in the bar of the then hotel Mira Punda in Scharloo. These are old pictures taken by its then-owner Jose Rosales in 2005. Nowadays it is refurbished, and you should check out Hotel Scharloo or see pictures, or see


Hotel Mira Punda 2005 before the refurbishment to Hotel Scharloo (Source: Jose Rosales)

A second time in 2005-2006 Frank came by to discuss the future of the Caribbean, and we sat there on the terras of Mira Punda. I was just getting my driver’s licence so it was impossible to drive up to his place.

Just a year later, in 2006, when I had returned to Holland, his book Double Play was presented as the Dutch liberaries book of the year, and I met him again in The Hague.

Here is my view on the future of the Caribbean, no doubt influenced by these brief but powerful meetings about national independence. Perhaps the Caribbean could develop a sense of nationhood ?


Listening to Roefie Huetng with Jamie’s Blues


Roefie Hueting (1929) is an economist and jazz piano player, or a jazz piano player and an economist, who cannot decide which of the two is most important to him. See this earlier report on his double talent.

Hueting’s first public performance was on stage on liberation day May 5 1945 at the end of World War 2, when he was dragged out of his home to play for the people dancing in the streets. He still performs and thus he has been 55+15=70 years on stage.

With the Down Town Jazzband (DTJB) Hueting recorded 250 songs, played on all major Dutch stages, five times at the North Sea Jazzfestival, while the 50th DTJB anniversity of 1999 was together with the Residence Orchestra in a sold-out The Hague Philips Hall.

Hueting was one of the founders of the Dutch Jazzclub from which sprouted The Hague Jazz Club. This HJC has its current performances at the Crowne Plaza Hotel, formerly known as the “Promenade”. This hotel is at the Scheveningseweg, the first modern road in Holland, created by Constantijn Huygens in 1653, connecting the area of the Peace Palace – the area where also Grand Duchess Anna Paulowna of Russia (1795-1865) had her Summer palace – to the sea. See also these pictures of the German Atlantik Wall – to stay with the WW 2 theme.

At the celebration last Sunday September 27 other performers were Joy Misa (youtube), Machteld Cambridge, Erik Doelman (youtube) and Enno Spaanderman.

The Hague Alderman Joris Wijsmuller (urban development, housing, sustainability and culture) came to present Roefie Hueting with a book containing a picture of Mondriaan‘s Victory Boogie-Woogie – also celebrating the end of WW 2. Wijsmuller observed the erosion of “sustainability” that in the opinion of Hueting rather should be “environmental sustainability”.

Roefie Hueting and alderman Joris Wijsmuller at Crowne Plaza Hotel 2015-09-27

Roefie Hueting and alderman Joris Wijsmuller at Crowne Plaza Hotel 2015-09-27

Roefie Hueting solo at the piano, 2015-09-27

Roefie Hueting solo at the piano, 2015-09-27

Hueting introducing a jam session 2015-09-27

Hueting introducing a jam session 2015-09-27

"Victory Boogie-Woogie" by Piet Mondriaan (Source: Wikimedia Commons)

“Victory Boogie-Woogie” by Piet Mondriaan (Source: Wikimedia Commons)

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.)