Listening to Andriopoulos & Odysseas Elutis (1984): Prosanatolismoi


Let us discuss Gerald Goldin (2003), Developing complex understandings: On the relation of mathematics education research to mathematics. I presume that the reader has checked earlier discussions on Goldin (1992), on epistemology and on Stellan Ohlsson.

The paper’s abstract is:

Goldin (2003), p171

Goldin (2003), p171

It took me a while to come to grips with this paper. Suddenly it dawned on me that the English speaking world, including Goldin, makes a distinction between science and the humanities. This is what C.P. Snow (1905-1980) called The two cultures.

For Dutch readers these categories are crooked.

  • When Goldin opposes mathematics education research (MER), which in the English world belongs to the humanities, to science including mathematics then this is the distinction between science and the humanities. But for Dutch it sounds very strange to suggest that MER would be non-science.
  • Dutch has the single word wetenschap. How can Goldin oppose things that are the same (learning) ? How can he lump together things that are different (science and mathematics) ?

Dutch convention categorises the humanities as alpha (α), science and mathematics as beta (β) and the mixture as gamma (γ): those deal with alpha subjects but use beta methods. MER would be gamma.

I am not too happy with the Dutch categories since they don’t account for the separate position of philosophy and mathematics. The better distinctions are in the next table.

Categories for general science (science and the humanities)

Categories for general science (science and the humanities)

Above table is intended to categorise whole disciplines, like physics and economics. But we can also look at sub-areas within a discipline. Since both philosophy and mathematics can run astray without some link to the external world, my suggestion is that they both take mathematics education research as some anchor to reality (but they remain what they are when they refuse to do so). They might take an example of history writing, in which most history writing uses non-experimental methods (see the history on Pierre van Hiele) but some historians will rely on the experimental sciences to recover data from the past.

We are now ready to look at the paper.

Below we will see that Goldin opposes α + Φ versus β = δ + μ, forgetting about γ, while my analysis in Elegance with Substance (EWS) (2009, 2015) (pdf online) diagnoses the problem as α + μ versus γ, while δ + Φ have run away and no longer want to take part in cleaning up the mess. Professor Hung-Hsi Wu of Berkeley calls for help by research mathematicians μ to clean up the mess in ME and MER, but in my analysis we need help from engineers and other researchers in the empirical sciences δ + γ, see here.

Goldin 2003 on the decade since 1992: integrity for the disciplines

Goldin’s paper discusses his background, and he seems very well placed to discuss mathematics, ME and MER. Goldin sees a math war and tries to bring calm by increasing complexity. His article is complex itself so that those who pass the test of reading it will understand enough of the various sides of the discussion and be less likely to vilify the other side.

Goldin’s position is that discussants on MER must respect what other discussants on MER are doing and good at. Scientific integrity tends to focus on ethical behaviour of the individual but Goldin widens this to whole disciplines. Scientists must respect the humanities. The humanities must respect science. Otherwise there is no communication and no progress.

Goldin (1992) looked back at the New Math in the 1960s and behaviourism in the 1970s. When those ‘isms’ failed to produce improvement in mathematics education, the educational departments in the humanities grabbed the opportunity to claim their way to success. Goldin would agree partly, since he in 1992 also opposed the New Math and behaviourism. The humanities however created their own ‘isms’. We can now better understand Goldin’s position w.r.t. the decade 1992-2003.

Goldin (2003), p177

Goldin (2003), p177

A key observation is that Pierre van Hiele (1909-2010) is missing in this list and that Hans Freudenthal (1905-1990) committed fraud w.r.t. the work by Van Hiele: so that Goldin has a somewhat rosy view about the “without the far-reaching dismissals, oversimplifications, and ideologies”. The reference to Leen Streefland (1998†) may highlight the ‘ism’.

A Pro Memory point is that David Tall in 2002 apparently misunderstood the Van Hiele theory, as applying only to geometry and not to epistemology in general. This doesn’t seem to be due to ideology on Tall’s part, but there seems to have been some influence of Freudenthal in the misrepresentation of Van Hiele’s work. See my paper on getting the facts right.

An example with Leen Streefland (1998†)

I have not studied Streefland’s work any deeper than the following internet links just now. Those links fit the diagnosis of sectarian behaviour of Freudenthal’s “realistic mathematics education” (RME), and thus I see no reason yet to read more. Streefland belonged to the Freudenthal sect, see this ESM 2003 issue. Pierre van Hiele suggested in 1973 to look into the abolition of fractions, but Streefland (1991) perseveres with a book on “realistic education” on fractions.

See my 2015 book, pdf online, A child wants nice and no mean numbers, also commenting on the US Common Core program and professor Hung-Hsi Wu on fractions. Professor Wu does not belong to the RME sect but his traditional answer on fractions suffers from the intellectual burying of Van Hiele, which the RME sect so effectively achieved. The ‘isms’ are not without cost.

The strategy by Hans Freudenthal and his Utrecht sect – and these are adults who know what they are doing – is to absorb elements of Van Hiele’s work, but misrepresent it to fit their own ideology – which change does not diminish the intellectual theft. They achieve two effects: (i) for an innocent audience they ride the wave of the success by Van Hiele that they are jealous about, and (ii) they exclude Van Hiele himself from the discussion since “they tell it better” – and thus Van Hiele’s protest that his work is abused will not be heard. After all, Freudenthal was a professor in Utrecht with his own Ph.D. students who later became professors, and Van Hiele remained a mere mathematics teacher doing his writings in the weekend.

In this book on fractions, Streefland (1991) p2 states the following. We can excuse authors for the uncreative use of the word “level” that pops up everywhere. The true problem lies in the ideological following of Treffers (1987) and the neglecting in 1991 of Van Hiele’s own work (not only on fractions of 1973).

Streefland (1991), p2

Streefland (1991), p2

This closes the circle: (a) Treffers (1987)’s misrepresentation of Van Hiele’s work is not only in Streefland (1991) but (b) was also copied in the 1993 MORE study, (c) critically discussed by Ben Wilbrink, here, (d) which alerted me to Wilbrink’s misrepresentation of Van Hiele. Wilbrink namely follows the RME abuse, and he also tends to include Van Hiele in the RME sect instead of saving him from it.

Thus we are back into the Dutch math war swamp, with on one side the RME sect and on the other side Jan van de Craats and others who try to save “traditional mathematics education sanity” alongside psychologist Wilbrink with his misapprehension of empirical science and Van Hiele. My position is that of Sherlock Holmes observing it all from the high ground aside.

Traditional mathematics is crooked as well. It e.g. involves torture of kids by fractions. There is every reason to desire change. It doesn’t help when mathematicians, who don’t have empirical training, team up with the humanities who don’t have empirical training either (i.e. α + μ).

Popper’s falsification

One reason why the humanities might be disrespectful of science has to do with Karl Popper’s demarcation theory to use falsification to distinguish science from non-science:

Goldin (2003), p178

Goldin (2003), p178

Goldin reminds us that the humanities are non-science, as seen from science and its experimental method. The humanities should heed the risk of turning this property into a claimed superiority.

The humanities seem to have learned that they should not claim higher wisdom, which they and only they can discover by reading old documents and watching plays by Aristophanes and Shakespeare and have reception parties afterwards to discuss the faculty gossip. But the humanities might still take the Humean skeptic position, and make fun of physics who can put electrodes upon skulls and in that manner likely will never be able to create the insights that a study of the humanities can generate (though they might actually prove some of the gossip).

Goldin’s argument: Physics can be skeptic too. Save those skeptic arguments for your autobiography, for they contribute nothing to the discussion.

My warning: Don’t make too much of falsification. See the discussion on epistemology and the definition & reality methodology.  Above δ and γ sciences rely for the empirical realm upon definitions, and a mathematician μ might well hold that definitions are non-experimental.

David Hume and Ernst von Glasersfeld

Reading a bit more about Ernst von Glasersfeld (1917-2010) was long upon my to-do-list, and Goldin’s article finally caused me to do so. Advisable is his own article Thirty Years Radical Constructivism, Constructivist Foundations 2005, vol. 1, no. 1, p9-12. It is very useful to see Von Glasersfeld’s background in mathematics (not completed because of WW 2), linguistics and cybernetics: γ rather than α. For methodological justification he might be forced to do some philosophy, but he rejects doing that.

Von Glasersfeld (1995) Radical Constructivism at ERIC is too much for now, though. I checked that he indeed discusses Hume, and also mentions the “problem” of induction (see my discussion of epistemology). Von Glasersfeld holds that the issue is not philosophy but finding mechanisms of cognition.

Comment 1: Reuben Hersh (2008) Skeptical Mathematics? Constructivist Foundations 3(2): 72, suggests that “radical constructivism” would be Humean skepticism, and I tend to agree.

Comment 2: Being a Humean skeptic is agreeable too. This (wiki-) quote by Von Glasersfeld seems accurate:

“Once knowing is no longer understood as the search for an iconic representation of ontological reality but, instead, as a search for fitting ways of behaving and thinking, the traditional problem disappears. Knowledge can now be seen as something which the organism builds up in the attempt to order the as such amorphous flow of experience by establishing repeatable experiences and relatively reliable relations between them. The possibilities of constructing such an order are determined and perpetually constrained by the preceding steps in the construction. That means that the “real” world manifests itself exclusively there where our constructions break down. But since we can describe and explain these breakdowns only in the very concepts that we have used to build the failing structures, this process can never yield a picture of a world that we could hold responsible for their failure.”

It is hard to disagree, except when you want to resort to Wigner’s magic again (see Appendix 2). But it doesn’t tell us how to design a course so that Johnny can learn arithmetic. Or how to abolish fractions.

Comment 3: Von Glasersfeld refers to Jean Piaget. Pierre van Hiele developed his theory of levels of insight, starting from Piaget as well, but eventually rejecting Piaget’s age-dependency and choosing for the logical structure that generates a general theory for epistemology.

It is a question what the contacts between Von Glasersfeld and Van Hiele were, and whether Hans Freudenthal was an interfering factor again. We find Von Glasersfeld (ed) (1991), Radical Constructivism in Mathematics Education, Kluwer, that contains a chapter by Jan van den Brink, since 1971 a member of Freudenthal’s sect in Utrecht. Searching the book generate 0 references to “Hiele”. The RME wiki on RME refers to Von Glasersfeld’s book but not to Van Hiele, even though we saw above that Streefland refers to Treffers who considered the Van Hiele levels a “pillar” of RME. Not referring saves the effort of thinking up a lie.

It is a question how the departments on education at the humanities were influenced by RME and Von Glasersfeld and others, and how they got so entangled that Goldin seems to tend to refer to them as one side of the equation (or rather imbalance). It is no key question, but something for historians of MER to be aware of (see the handbook on history of MER).

Comment 4: I started getting lost on what makes “constructivism” so special that it must be mentioned. Originally I knew about constructivism as an approach in the foundations of mathematics, as distinguished from formalism and platonism. My book Foundations of Mathematics. A Neoclassical Approach to Infinity (FMNAI) (2015) creates a ladder of degrees of constructivism (avoiding “levels”), in which the highest degree allows non-constructivist methods. When people use different approaches we should at least describe what they are doing.

But now there are all kinds of “constructivism” in education, psychology and philosophy, without authors taking the time to shortly explain what the non-constructivist opposition would entail. Fortunately, there is wikipedia that might help or contribute to confusion, here with disambiguation. and here with the general denominator in epistemology. The opposite of constructivism would be that people could know objective reality, by magic, and I wonder whether that is so useful an idea. My impression is that there is more to it. Thus authors should still specify. (And then I would not have time to read it.)

Ben Wilbrink is horribly erroneous about Pierre van Hiele and in breach of scientific integrity for not looking into it  to correct his misrepresentation, and Wilbrink can fulminate against constructivism: but at least he referred to this article by Gerald Goldin so that I found it, and he also has this page with all kinds of references on constructivism.

One book mentioned there is by Kieran Egan (2002), Getting it Wrong from the Beginning: Our Progressivist Inheritance from Herbert Spencer, John Dewey, and Jean Piaget. This fits Van Hiele’s rejection of Piaget’s theory of stages of development. But does Egan refer to Van Hiele ? Not likely, since the wikipedia portal speaks about the constructivist “idea that things (especially learning) always go from simple to complex” – and this is not how Van Hiele would phrase it: who discussed going from concrete to abstract, and who used the notion of proof to identify the highest level of abstraction.

Wilbrink also has a quote on Von Glasersfeld:

“The basic idea of The Georgia Center was to establish a community of researchers in mathematics education working on problems of interest to the community, where the experience of the researcher, conceptual analysis, and social interaction replaced the controlled experiment as “normal science.” No longer did it seem necessary to use the controlled experiment with its emphasis on statistical tests of null hypotheses and empirical generalization to claim that one was working scientifically.”

This is complex. Before you do such a costly double blind randomized trial, with the huge numbers required because of the large number of variables, variety in pupils, and sources for measurement error (see John Hattie), it is useful to have clarity on concepts, definitions, operationalisations, methods, controls, and the like. Confronted with annual unpredictable changes from the Ministry of Education, you might want to give up on such statistical ambitions, and settle for the Google “do no evil” approach. It may well be that modern MER only serves for Ph.D. students to defend a thesis, and the relevance for education may be discussed at the reception party afterwards along with the faculty gossip.

The increasingly popular Japanese Lesson Study is one promising method (tested under Japanese conditions) to deal with the data problem.

However, see the suggestion for Academic Schools modeled after the Medical School, also included in A child wants nice and no mean numbers (2015).

Goldin’s crucial blind spot

What I consider Goldin’s blind spot is that he lumps together science and mathematics, while mathematics is no empirical science but deals with abstraction and patterns.

Goldin (2003), p179

Goldin (2003), p179

Education is an empirical issue. Also mathematics education is an empirical issue. Thus the involvement of mathematicians in such education can be disastrous, when they are trained for abstraction μ and not for empirical science γ. The epitome of the abstract mathematician who got lost in this is Hans Freudenthal who invented a whole new ‘reality’ just to make sure that at least he himself understood and was happy how the world works (including the oubliette for Pierre van Hiele).

The only reason that Goldin lumps together β = δ + μ is that he is so much worried by the ‘isms’ by α + Φ that he forgets about the real problem at bottom of the case: the disastrous influence of μ in 5000 years of education of mathematics. (Fractions were already a problem for the pyramids.)

Check out this example of mathematical torture of kids on fractions. This torture is also supported by professor Hung-Hsi Wu of Berkeley for the US Common Core programme, see A child wants nice and no mean numbers.

Goldin (2003) p180 suggests a seemingly good argument for lumping together science and mathematics.

Goldin (2003) p180

Goldin (2003) p180

Thus, the abstract thinking mathematician has a special trick to describe the physical world ? Without lookin ? Like with Wigner’s magic wand ? I don’t think that we should believe this. It is physics that selects the useful model from the mathematical possibilities. Thus:

  • This misconception about the role of mathematics may help explain why Goldin (2003) does not quite see the disastrous influence of abstract thinking mathematicians upon ME and MER. Golding does make some comments that mathematicians should not think that ME is simple and can be tested as in behaviourism, but he misses the fundamental problem as discussed in Elegance with Substance.
  • The remark about mathematics and empirical modeling remains relevant for the definition & reality methodology. It supports the empirical status of say the definition / law of conservation of energy, and it supports the empirical status of Van Hiele’s theory of levels of insight (abstraction) in epistemology (and application in psychology).

(1) We can support Goldin’s conclusion and plea for eclecticism (yes, another ‘ism’).

Goldin (2003), p198

Goldin (2003), p198

(2) Since the Freudenthal Head in the Clouds Realistic Mathematics Institute (FHCRMI) in Utrecht doesn’t do MER but performs sectarian rituals, also based upon Freudenthal’s fraud, it is a disgrace to general science – including the humanities – and thus it should be abolished as soon as possible. Dutch Parliament better investigates how this could have happened and endured for so long.

Appendix 1. Kurt Gödel

W.r.t. the following I can only refer to A Logic of Exceptions (1981, 2007, 2011) (pdf online). For interesting systems the Gödeliar collapses to the Liar paradox, with no sensical conclusions.

Goldin (2003), p187

Goldin (2003), p187

Appendix 2. Philosophy of mathematics

W.r.t. the following I can refer to the discussion on Wigner on the “unreasonable effectiveness of mathematics”. Given the common meaning of “unreasonable” Wigner must refer to magic, or he didn’t know what he was writing about, as a physics professor lost in the English language. It is some kind of magic that his paper got so much attention. This discussion has also been included in Foundations of Mathematics. A Neoclassical Approach to Infinity (2015) (pdf online). Goldin uses the word “extraordinary” rather than “unreasonable”. Given that the effectiveness is ordinary for physics, he seems to take the humanities’ point of view here (whom his article is addressed at).

Goldin (2003), p188

Goldin (2003), p188

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).

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 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.


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