Freudenthal’s “realistic mathematics education” appears to be a fraud

Sitting on the bench in the park where he died, I hope that Hans Freudenthal whispered: “Forgive me, Pierre, for what I have done to you.”

Here we follow the Amir Alexander method of first selecting the storyline and then fill in the data. In this story Hans Freudenthal (1905-1990) (another link) is the crook and Pierre van Hiele (1909-2010) is the hero.  It might be that Freudenthal is no real crook but never waste a good story. However, Pierre van Hiele remains the hero for a fact. The storyline is given by this table.

Useful for mathematicians, likely higher up in the autistic spectrum
Useful for nobody Useful for education of scientists and the general public
Euclides 300 BC
New Math, after Sputnik 1957 1957 theses Pierre van Hiele under Freudenthal and Dian Geldof under Langeveld (Utrecht archive, math genealogy)
Hans Freudenthal’s earlier work on mathematics (topology, assistant to L.E.J. Brouwer) Hans Freudenthal on education after 1957. His “realistic mathematics education” that isn’t realistic

The education in mathematics had been dominated by Euclid’s Elements. Admittedly, Newton and Leibniz added some aspects on the derivative, but that is small beer in the shadow of the great Greek. In 1957 Russia launched its Sputnik and America woke up to the reality that their system of education didn’t produce enough rocket scientists. Teachers of mathematics rushed in to assist with the New Math curriculum. However, in 1973 Morris Kline wrote Why Johnny Can’t Add: the Failure of the New Math. The New Math was much too abstract and actually quite silly. Hans Freudenthal rushed in with his “realistic mathematics education”. His “realism” looked like the proper answer to the earlier abstractions, and his “guided re-invention” sounded like that every child would reinvent mathematics if guided properly by its math teachers. Nowadays, the International Commission on Mathematics Instruction (ICMI) has the Freudenthal Medal, as if this should be something to be proud of, and not a disgrace to mankind.

The point namely is that Freudenthal was an abstract thinking mathematician too. His “realism” is an abstract kind of “reality”. His invention of this “realism” hadn’t had guidance by real teachers with practical experience. When Freudenthal spoke about statistics he meant mathematical statistics, and he didn’t like it anyway. In his own teaching, he tended to bully his students and when they skipped his course he seemed to regard it as an admission of stupidity – see this article in the Dutch journal of mathematics NAW. David Tall has the story that when he hadn’t met Freudenthal yet but criticised him then he became as nice as can be, which flip-flop behaviour is rather telling, while that story doesn’t tell whether Freudenthal actually corrected his mistakes.

David Tall (2006, top p2): “When Richard [Skemp] was asked to review Freudenthal’s book Mathematics as an Educational Task [1973], having already bought his own copy and not wanting another, he passed the invitation to me. To review a work of the great Freudenthal was a huge task for a young mathematics lecturer and I sought advice from a senior colleague, James Eels, who knew him well. He confirmed that I should say exactly what I felt and, emboldened by his advice, I wrote a welcoming but critical essay. I received a post-card from Freudenthal after the review [1975] appeared: “thank you for the review which I enjoyed, especially the critical parts.”” (Quote added July 11 2014) (July 8 2014: See the newly included Appendix below with an email by David Tall who thinks that he corrects me but he doesn’t.)

Apart from his lunch, Freudenthal had two contact points with reality. He did some work on the history of mathematics, and it is up to the historians of mathematics to check whether it is realistic, rather than to assume that it can be safely referred to. The real historian is B.L. van der Waerden (1903-1996), see this fine interview at AMS. Also, Freudenthal supervised the thesis by Pierre van Hiele on the education of mathematics, and read Dina Geldof too. In his memoirs Freudenthal tells that when he found his mathematical abilities waning, he had the choice to continue with history or education of mathematics. Perhaps it was good for the history of mathematics that he chose education.

In itself it is remarkable that he didn’t resign from the chair of professor of mathematics and switched to the education in mathematics but could proceed as professor. What should have been proper too is that, if he really wanted better math education, then he should have helped Pierre van Hiele to be come professor in the education of mathematics, so that Van Hiele with his empirical ways could teach new students.

In the theses, Van Hiele & Geldof had identified levels of understanding of mathematics. The lowest level is in the realm of the senses. Seeing, tasting, touching, weighing, pushing and so on. In need of a theory of his own, Freudenthal referred to this as “realistic mathematics education”.

Freudenthal did refer to Van Hiele on occasion. Once he had his own publications, it was easier to refer to those, so that Van Hiele passed into oblivion. At conferences people were surprised that he still was alive. Van Hiele remained a highschool teacher for the rest of his life but used his experience to write about the education in mathematics. In sordid manner the University of Utrecht now has a “Freudenthal Institute” for the education in mathematics and science, and Van Hiele has a page in their wiki. (NB: August 18: An error there says that Pierre wrote his thesis under Langeveld.) Here is an interview with Van Hiele in the Dutch journal of mathematics, that shows his disgust of Freudenthal with his partly stolen good and otherwise crooked ideas on education. Van Hiele remains civilized though, so typical of the good math teacher.

My book Conquest of the Plane is based upon Pierre van Hiele’s full approach. My book also explains why the view by Freudental was wrong, and how Freudenthal mistook his own mathematical abstractions for realism.

Holland now has the problem that whoever in Holland proves that Freudenthal was a fraud, is slaughtered. The proof doesn’t count. What counts is the indoctrination from Utrecht. What counts is the need to treat each other politely and not delve skeletons from the closet. What counts is the international great reputation of Freudenthal. What counts is that Dutch mathematicians are afraid of having to admit that they never really look at the education of mathematics but were bullied by Freudenthal.

On June 30, the Dutch Akademy of Sciences (KNAW) in Amsterdam had a conference on the education of arithmetic. Here is my report of the event (unfortunately in Dutch). The conference was an intellectual embarrassment and failure, even though it was led and fed by internationally reputed mathematician Jan Karel Lenstra. The key problem is that Lenstra is another abstract minded mathematician who has no clue about the empirical nature of education. There are also issues of policy. There are 150,000 teachers at Dutch elementary schools that require re-education on teaching mathematics including arithmetic, but the state secretary chooses politically to leave them be, and let the problems be solved by the 4,000 teachers of mathematics in secondary education. Admittedly new teachers should be trained better during their education, but the annual batch of new teachers will have not much effect. Professor Lenstra supports the state secretary on this policy, instead of reporting accurately that this policy will wreak havoc.

It is an international problem. Hans Magnus Enzensberger wrote the book The Number Devil to show children how arithmetic can be great fun. It is telling that teachers at elementary school apparently kill the fun. The latter of course cannot be blamed all on Hans Freudenthal but you will be surprised how much.

The Number Devil

The Number Devil (source:, screenshots) (Guess what the lesson is about.)

I move:

  • that ICMI rebaptises the “Freudenthal award and medal” into the “Jean Piaget and Pierre and Dieke van Hiele-Geldof award and medal”
  • that the Freudenthal Institute is abolished and that a new education research institute is founded that used empirical methods, say the Simon Stevin Institute
  • that everyone calls the present Freudenthal Institute by its proper name the Freudenthal Head in the Clouds Realistic Mathematics Institute to properly describe that their “realism” has no base in empirics, while also to get rid of the psychological connotations that a person who has an institute named after him ought to have had some great results
  • that KNAW supports my proposal that Dutch parliament does an enquiry in the education of mathematics, to determine what went wrong and what funds need to be made available for improvement
  • that the world boycotts Holland till the censorship of economic science by the Dutch government is resolved, see above About page.

PM 1. Let me add that mathematics is an essential part of your life. It may be that the old education has spoiled this for you, but you could understand that you should try that this doesn’t happen to your children and grandchildren.

PM 2. Let me also add that Jeroen Dijsselbloem made his name in Dutch politics when he chaired a parliamentary enquiry into the educational failure of the “studiehuis“. The conclusion of his committee was that policy makers should determine what is taught, and teachers should determine how. See however the proper problem: Policy makers have determined that pupils should have mathematics, but the educators provide something that they call “mathematics” that it is not. While arithmetic scores of Dutch pupils might remain acceptable, there is much to say about testing, and, schools might put more time into language and arithmetic and less into history and music. Thus Dijsselbloem did not really get to the core of the issues. As chair of the Eurozone he is now doing the same with the European economy.

Appendix included on July 8 2014: David Tall’s statement from his own memory

Since I referred above to a text that I only remembered from David Tall, I alerted him to this weblog entry and invited him to correct me if needed. I received the email below and on his request gladly include it. I consider it very important that the witnesses of the history of “realistic mathematics education” are heard on this. Professor Tall complains that this weblog doesn’t allow comments. It is for the simple reasons of both legibility and my limited time to monitor responses. If people want to respond then it better be by reasoned exposition on their own websites. Tall also states that I misrepresent his position. I do not. It appears that I remembered his text correctly. The Dutch journal of mathematics article by Nellie Verhoef and Ferdinand Verhulst quote Kalmijn (p55) stating that Freudenthal quickly divided students in dumb or smart, and did this even with government ministers. This may be exaggerated but gives an indication. Strooker reports: Classes were too hard for many but “if you passed you became a good mathematician”. That particular interviewed person does not reflect whether more students might have passed with a less intimidating approach. I find this telling for a theorist of education. This should not be read as a caricature. Of course there are also interviews with a positive load, but the point is the news of the intimidation in education. Apparently Freudenthal did not intimidate David Tall, and Tall’s direct criticism provoked a kind attitude from Freudenthal. This is telling for what I call flip-flop behaviour of treating some as idiots but becoming kind when being treated with criticism yourself.  It is useful to have that link to Tall’s text and historians may check whether Freudenthal corrected his texts.

My point remains, which is that Tall can only report a positive view on Freudenthal since he escaped the intimidation. Tall still does not seem to be aware of how much of intimidation he escaped. Having success in his own work might make it more difficult to see the students who needed a different approach in education and research. I move that Nellie Verhoef has the article translated into English so that professor Tall and others can read the reports themselves. Of course I reject professor Tall’s qualifications ” vindictive” and “diatribe”. His evidence confirms the story, and he may not have the other evidence to reject his rosy view.

I agree and already decided myself that the Pierre and Rian van Hiele theory applies to all area’s of mathematical development, or indeed any developed discipline. This seems like a matter of logic and definition when finding the right terms to describe what (mathematical) learning is. Professor Tall presents “Realistic Mathematics” as “part of the long-term evolution of theories”. This is complex. With Van Hiele, there is no need for “Realistic Mathematics Education”. The good elements derive from Van Hiele, and the name is taken from the basic Van Hiele level. Thus, take a horse, have it breed with a donkey, call the result “hooves”, and then try to argue that the infertile “hooves” is part of evolution. Of course I ought to return to the issue when I have read professor Tall’s book. (August 27: see here.)

I suppose that professor Tall writes “Thomas C” because of my use of “Colignatus” in science, but this is not how I am normally called.

Tue, 08 Jul 2014

Dear Thomas C,

Thank you for alerting me to your publication on the web. It is in the form of a blog which does not seem to admit a response, allowing you to put forward a discourteous and personalised attack on Freudenthal, including a misrepresentation of something I said to you, without the right to reply.

Your summary of my comments about Freudenthal are quoted out of context to serve your own purpose. The facts are these: In the seventies, as a young mathematician turned math educator with few publications to my name, I wrote a strong critical review of Freudenthal’s work which is still available as a download my website:

Looking at this review nearly forty years on, I am surprised at my own driven directness which clearly appealed to Freudenthal, who sent me a postcard which simply said ‘Thank you for your review which I enjoyed very much, especially the critical bits’. Subsequently he was critical of my work in a very supportive way which I greatly respected. His driven personality was responsible for the introduction of mathematics education to the International Congress of Mathematicians and subsequently to the formation of ICME and then PME. He played a central role in the development of mathematics education as a discipline.

In my book on How Humans Learn to Think Mathematically, which you say you have yet to read, you will see Freudenthal mentioned only in a footnote and Realistic Mathematics considered in the final chapter as part of the long-term evolution of theories. My own perspective builds on the underlying commonalities of many theoretical frameworks from diverse disciplines. You may be surprised to know that the work of your heroes, the van Hieles, in a simpler but broader format, applies not only to geometry but to all areas of mathematical development that even Pierre van Hiele himself did not realise, and denied in print.

Theories evolve, and in that evolution, Freudenthal played a central role. It does you no credit to attack him in a personalised vindictive manner.  Your diatribe does more damage to yourself and your cause than it does to Freudenthal.

If you have a sense of fairness, I suggest that you place this e-mail as a response on your blog page.

I have sent a copy of this e-mail to Nellie Verhoef who, as you know, is closely acquainted with both Freudenthal and van Hiele, and is herself referred to indirectly in your blog.

David Tall

Addendum to this Appendix, July 15 2014: On Pierre van Hiele & Richard Skemp

In above email, professor Tall states: “You may be surprised to know that the work of your heroes, the van Hieles, in a simpler but broader format, applies not only to geometry but to all areas of mathematical development that even Pierre van Hiele himself did not realise, and denied in print.”

Please do not mistake critical admiration for hero worship. But I found this surprising indeed, and as proof, Tall sent me a copy of Van Hiele’s chapter in the book that he edited, the Tribute to Richard Skemp (2002). (Amazon and I have difficulty locating its publisher, but a review is here.)

However, the proof that Tall suggests isn’t there. In that chapter, Van Hiele warns: p28: “The problems in algebra that cause instrumental thinking have nothing to do with level elevation since the Van Hiele levels do not apply to that part of algebra. People applied terms such as ‘abstraction’ and ‘reflection’ to the stages leading from one level to the next. This resulted in a confusion of tongues: we  were talking  about completely different things.” Thus “part of algebra” should not be mistaken for all algebra.

p39: “The transition from arithmetic to algebra can not be considered the transition to a new level. Letters can be used to indicate variables, but with variables children are acquainted already. Letters can be used to indicate an unknown quantity, but this too is not new. ”

p43: “The examples Skemp mentions in his article about I2, R2 and L2 do not have any relations with a level transition. They are part of algebra in which topic, as I have emphasised before, normally level transitions do not occur.” Again “part of algebra” should not be read as “algebra” as a whole.

In his final conclusion, p 46: “In most disciplines there are different levels of thinking: the visual level, the descriptive level and the theoretical level.” Indeed, Van Hiele gives such various examples in my copy of “Begip en Inzicht” (1973) which is more extended in English in “Structure and  Insight” (1986).

Thus, Van Hiele was aware of the portent of his theory, contrary to what David Tall suggests. So much more of a pity that Freudenthal sabotaged and appropriated it. I will return to the issue when I have read professor Tall’s book. If you can’t wait, start reading Conquest of the Plane, section 15.2, p201-206.

Update July 28 2014: There is now my PDF article Pierre van Hiele and David Tall: getting the facts right (or arXiv).


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