I received a sobering email today about the UK “supposed serial killer nurse” Ben Geen, of which a description is given in this weblog page.  Naturally, I don’t know whether Ben Geen is innocent or not, I just was informed about that weblink, and pass it on.

The good news though is that Holland succeeded in getting a release and acquittal for Lucia de Berk. She had also been accused of being a serial killing nurse, and somewhat close to being a witch with her Tarot card reading. Key roles for her eventual release were for Metta de Noo – Derksen MD for analysing the medical data, her brother Ton Derksen, professor of methodology of science, who wrote the eye-opening book on the case, and professors of econometrics Aart de Vos and, crucially, mathematical statistics Richard Gill who broke the statistics of the case and who showed that the courts had been misinformed by their “statistical experts” (who still haven’t offered apologies). In this case I joined in the statistical effort to clarify the issue, see here at arXiv. Whatever the statistics, it are the medical data that are crucial here, since it is the expertise of MDs to decide about issues of life or death. Thus, the greatest heroin in this issue remains Metta de Noo MD. People in the sector should be aware of the legal settings though, see for the USA e.g. the LNC Educational Center with another heroin, Donna Rooney.

Thus, while I overall advise the world to boycott Holland, just now: Three cheers for Holland !

Overall, I hope you agree that it is better to aspire for improvement of the country that you boycott, to ever greater perfection. It is no use to boycott a country that is rather worthless to boycott anyhow. It isn’t quite clear what the EU intends to achieve by boycotting Russia over the Ukraine for example. If the goal is to insult people and a nation, yes, go ahead, and boycott all shops and nations of the world, and be happy with all the insult and misunderstanding that you spread around. But if your objective is some noble goal, like the freedom of economic science, then you should focus on Holland, notwithstanding that wonderful great result with respect to Lucia, that hopefully inspires confused residents in the UK.

PM. A point remains that while Lucia now is free and acquitted, none of the originally responsible MDs, directors, lawyers, experts and judges have apologized or met with some disciplinary measure. If I am correctly informed some continue to believe that Lucia was guilty or at least that they themselves didn’t make any error. See this tv broadcast how director Paul Smits argues that he would do the same: when such suspicion arises, call the police. His defence is that it is bizarre that other agencies simply adopt that suspicion instead of looking into it critically. The critique however is that he didn’t treat Lucia in the manner of “innocent until proven guilty” but rather conversely, while it is difficult for the police to be critical about “medical evidence” (that isn’t really such). Smits later moved on to become director of Maasstad Ziekenhuis, where he mismanaged a klebsiella infection that caused the death of some patients. After that he was given a “second chance” (in which Lucia was forgotten, well, she was acquitted) for the Medical Center Leeuwarden. At least we can say that as science progresses and society becomes increasingly complex, there are ever more ways to make statistical errors of type I and type II.

In Medical Schools, doctors are trained while doing both research and treating patients. Theory and practice go hand in hand. We should have the same for education. Teachers should get their training while doing theory and learning to teach, without having to leave the building. When graduated, teachers might teach at plain schools, but keep in contact with their alma mater, and return on occasion for refresher updates.

Some speak about a new education crisis in the USA. The above seems the best solution approach. It is also a model to reach all existing teachers who need retraining. Let us now look at the example of mathematics education.

Professor Hung-Hsi Wu of UC at Berkeley is involved in improving K12 math education since the early 1990’s. He explains how hard this is, see two enlightening short articles, one in the AMS Notices 2011 and one interview in the Mathematical Medley 2012. These articles are in fact remarkably short for what he has to tell. Wu started out rather naively, he confesses, but his education on education makes for a good read. It is amazing that one can be so busy for 30 years with so little success while around you Apple and Google develop into multi-billion dollar companies.

Always follow the money, in math education too. A key lesson is that much is determined by textbook publishers. Math teachers are held on a leash by the answers books that the publishers provide, as an episode of The Simpsons shows when Bart hijacks his teacher’s answers book. As a math teacher myself I tend to team up with my colleagues since some questions are such that you need the answers book to fathom what the question actually might be (and then rephrase it properly).

At one point, the publishers apparently even ask Wu whether he has an example textbook that they might use as a reference or standard that he wants to support. The situation in US math education appears to have become so bad that Wu discovers that he cannot point to any such book. Apparently he doesn’t think about looking for a UK book or translating some from Germany or France or even Holland or Russia. In the interview, Wu explains that he only writes a teacher’s education book now, and leaves it to the publishers to develop the derived books for students, with the different grade levels, teacher guides and answers books. One can imagine that this is a wise choice for what a single person can manage. It doesn’t look like an encouraging situation for a nation of 317 million people. One can only hope that the publishers would indeed use quality judgement and would not be tempted to dumb things down to become acceptable to both teachers and students. In a world of free competition perhaps an English publisher would be willing to replace “rigour” by “rigor” and impose the A-levels also in the US of A.

In my book Elegance with Substance (2009) I advise the parliaments of democratic nations to investigate their national systems of education in mathematics. Reading the experience by Wu suggests that this still is a good advice, certainly for the US.

About the subject of logic, professor Wu in the interview p14 suggests that training math teachers in mathematical logic would not be so useful. He thinks that they better experience logic in a hands-on manner, doing actual proofs. I disagree. My book A Logic of Exceptions (1981, 2007) would be quite accessible for math teachers, shows how important a grasp of formal logic is, and supports the teaching of math in fundamental manner. The distinction between necessary and sufficient conditions, for example, can be understood from doing proofs in geometry or algebra, but is grasped even better when the formal reasons for that distinction are seen. I can imagine that you want to skip some parts of ALOE but it depends upon the reader what parts those are. Some might be less interested in history and philosophy and others might be less interested in proof theory. Overall I feel that I can defend ALOE as a good composition, with some new critical results too.

Thus, apart from what parliaments do, I move that the world can use more logic, even in elementary school.

I am sorry to report that Holland also fails on the integrity of science in the research on the didactics & education of mathematics. This is my letter (in Dutch) to the Scientific Integrity body LOWI of the Dutch Royal Academy of Sciences KNAW.

Earlier, in my book Elegance with Substance (2009), I made the empirical observation that mathematicians are trained for abstraction while education is an empirical issue. The training of mathematicians to become teachers of mathematics apparently can often not undo what has been trained for before. This basically means that many have lost the ability to observe. Math teachers tend to solve their cognitive dissonance by adhering to “mathematical tradition” that however is not very didactic, and that in fact collects the didactic debris of past centuries.

A key example here in Holland is the difference between Hans Freudenthal as the abstract topologist and Pierre van Hiele who as a mathematician and actual teacher however kept his ability to observe. We need only look at the debris in math textbooks to observe that the majority of math teachers aren’t like Van Hiele. See Elegance with Substance if you cannot identify the debris yourself.

An international example on statistics is the difference between Fisher and Gosset on “statistical significance”. Mathematicians tend to consider mathematical statistics only, and are little aware of empirical significance. Math educators who nowadays use statistics might fall victim to ‘garbage in, garbage out’ but nevertheless be praised as ‘empirical’.

Now in 2014 that empirical observation comes with a sting. When abstract thinking mathematicians make statements about the empirical reality of didactics & education, they actually make statements out of their province, about something they haven’t studied: which is a breach of research integrity. This especially holds when they have been warned for this, say by my 2009 book (listed in the AMS Book List, Notices Vol 58, No 11, p1474), or perhaps even directly by me.

In Holland there now is the case of internationally known Jan Karel Lenstra, who did work in operations research, (linear) programming, scheduling and the traveling sales person, who was selected in 2009-2010 by theoretical physicist and KNAW President Robbert Dijkgraaf to chair a KNAW “Committee on Mathematics in Primary Education”. The Dutch complaint is that children don’t learn arithmetic so well anymore, e.g. aren’t trained on long division as a sure method. It often happens that a committee is chaired by a person who doesn’t know much about the subject beforehand, but then that person tends to be aware of this, and is willing to learn. In the case of Lenstra, he apparently thought that he knew enough about “Mathematics in Primary Education” so that he also understood the didactics & education itself.

Hans Freudenthal had a huge impact on arithmetic in Dutch primary education. Lenstra observes about the Freudenthal madness:

“The core is that we must get more evidence-based research [education ?]. The ‘realistic arithmetic’ has been adopted without the empirical evidence to make it obvious. And also the PABO [training of elementary school teachers] has been constituted on the base of beliefs instead of scientific research.” (my translation, comments in brackets)(Akademie Nieuws July 2011 p5) 

But Freudenthal and his followers did claim that ‘realistic mathematics’ was scientifically warranted and based upon evidence. Thus professor in mathematics Lenstra observes a fraud with respect to empirical research, but doesn’t do anything about it. He doesn’t call for a repeal and annulment of earlier “research” that claimed empirical relevance but without such base. He is quite happy that his fellow abstract thinking mathematician Freudenthal invented a theory and let others suffer the consequences. Lenstra is another abstract thinking mathematician who now thinks that he has solved a problem and then lets others suffer the consequence – like the international community that still considers Freudenthal’s work “research”.

Lenstra’s solution to the arithmetic problem in Dutch primary education is not to retrain the 150,000 elementary school teachers, but shift the problem to the 4,000 math teachers in secondary education. The formula is that “arithmetic skills must be maintained” in the highschool curriculum. Lenstra suggests that it must be tested, but doesn’t quite specify how. The Dutch state secretary on education, Sander Dekker, wants mandatory arithmetic tests for highschool graduation. If you cannot calculate with pen and paper then you can’t get your highschool diploma. Lenstra thinks that this is too strict (see here) but doesn’t provide a practical alternative how to test whether arithmetic skills have been “maintained”. The state secretary apparently is quite happy that he doesn’t have to retrain the 150,000 elementary school teachers, many of whom are likely to fail too, and that it suffices to increase the burden for the 4,000 secondary school teachers, and of course the burden for the kids who turn 16 or 18 and discover that the educational system has given them a raw deal. (It is a bit too easy to blame them that they should have worked harder.)

One might say that Lenstra’s 2009 KNAW Report and recent June 30 2014 KNAW conference presentation (my report in Dutch) aren’t quite research themselves but rather evaluations on educational policy. It may well be that Lenstra’s texts here don’t register under scientific integrity in a strict legal sense, even though Parliament regards it as scientifically warranted. In another respect, Lenstra’s case is just an example, and it is a collective problem that abstract thinking mathematicians expound about empirical issues that they haven’t studied. Hence, my letter to KNAW-LOWI suggests a general exploration into the issue, so that the scientific community grows aware of the issue. Hopefully the specific issues on Freudenthal and his Institute are taken along, as explained here.

PM 1.

Above mentioning of Abstraction vs Empirics might cause the idea that those would be opposite, but these are rather separate axes. We might score the different professional groups on the study hours in each category, with mean and dispersion. I mention two example individuals for lack of an accepted term like “empirical mathematical statistics”. Since teachers teach they aren’t in research like other professions.

Abstraction vs Empirics

Example scores in the Abstraction vs Empirics space (might be tested)

PM 2.

This weblog concentrates on failure on integrity within Dutch economic science, with the case at the Dutch Central Planning Bureau (CPB) concerning economic co-ordination and the example of unemployment. That censored analysis is relevant for the current crisis in the European Union, and for economic recovery in the United States, and for economic policy in the “emerging markets” too. My advice is to boycott Holland till the censorship of economic science here is resolved. KNAW-LOWI cannot officially tackle the case since its mandate concerns universities while the CPB falls directly under the national government.

But now there is a breach in integrity in research in math education as well. I have two academic degrees, one in econometrics and one in teaching mathematics, and it is disappointing to observe that my degrees open up to vista’s of non-integrity. It might soon become a personal thing. But, as Art Buchwald would advise us: while there is a whole country to blame we might as well take a look at the facts. And boycott that country till they get their act together.

The world chooses to use English rather than Dutch or Latin, and hence we fall in the Dutch language sink again. As my letter is in Dutch, we might need to look to whom in Belgium, South Africa or Suriname still understands the gibberish. I presume that the people in New York (“New Amsterdam”, if they only knew) would need a certified translator. There might be some expats living in Holland who have learned some Dutch but I am afraid that some issues are getting complex, and then even many Dutch people would not be able to follow things. Foreigners would have even more difficulty in understanding local conditions. Hence I am quite hesitant to translate that letter.

For comparison on KNAW-LOWI, we may look at the US bureau for research in education, IES, and the office of research integrity ORI of public health, and then also the ethical codes of AMS and MAA.  There are a lot of ethics to look into.

But let us not forget about education itself, and thus let me also alert you to this issue of CF with Forty Years of Radical Constructivism in Educational Research and hope that they put students before method.

Given the last weblog on radians, I noticed that Wikipedia had a nice gif animation created by Lucas V. Barbosa. The article even mentions: “This is a featured picture on the English language Wikipedia and is considered one of the finest images.” Barbosa even made a version with tau = 2 pi. The latter is less appealing since it does not mention pi, while, of course, tau reads like radius r, and then can cause confusion (indeed, run that gif too).

It appears that Barbosa put his gif into the public domain. Thus I adapted it for Archi = Θ = 2 π, including a note of reference that he did most of the creative work.

1 Archi = 2 pi

A radian is an angle measured by an arc of a circle with the same length as the radius of that circle. A full circle corresponds to an angle of 1 Archi = 2π radians. Use 1 Turn ⇔ Θ radians, so 1 radian ⇔ 1 / Θ Turn ≈ 16% Turn.

Interestingly, Barbosa’s original gif has a small shaded disc in the center. If we take the radius of the larger circle as r = 1 then we get the smaller Angular Circle in the center with r = 1 / Θ and circumference 1. My proposal is to speak about “angles” on the Angular Circle (use α and β), and to use “arc” for the radians on the Unit Circle (use φ and ψ). Of course, angles as measured on the Angular Circle are arcs too, but it helps being able to say that angles add up to 1 Turn and Unit Circle arcs to 1 Archi rad.

PM. The Wikipedia article I referred to has a wrong statement on dimensions (today, July 2014). For a discussion of this, see the earlier weblog entry on radians.

Blogger Zendmailer 2012 deserves huge compliments for also thinking about a circle with circumference 1, that I baptised the Angular Circle. See also the figure with both the Angular Circle and the Unit Circle (radius 1) on page 36 of Conquest of the Plane (COTP, 2011).

Zendmailer ponders the question “Why are radians more natural than any other angle unit?” In his words:

“I’m convinced that radians are, at the very least, the most convenient unit for angles in mathematics and physics. In addition to this I suspect that they are the most fundamentally natural unit for angles. What I want to know is why this is so (or why not). (…) Why not define 1 Angle as a full turn, then measure angles as a fraction of this full turn (in a similar way to measuring velocities as a fraction of the speed of light (c = 1). Sure, you would have messy factors of 2π in calculus but what’s wrong with this mathematically? I think part of what I’m looking for is an explanation why the radius is the most important part of a circle.” (Physics Stack Exchange, August 6 2012)

The main thing wrong with this is that “angle” already has been defined, so that it cannot be taken as a unit of measurement. It would have been better when he had chosen 1 Turn as the unit. It is not really very wrong because if he had focussed on this longer he might well have corrected it. It is a pity that he uses 2π instead of Θ = 2π, the unit that I call “Archi” (after Archimedes). (Others want to use tau (τ) for this, see the American Scientist, but this looks too much like r for the radius.)

By chance, if that exists, I applied the Angular Circle recently on Euclid’s fifth postulate. Check the idea in action. It is great to see that more people come up with the same kind of questions and solutions.

It is also great to see that there is room for debate. Zendmailer is convinced that radians are most convenient but there is no need for this conviction. My suggestion is to keep both circles and see which is handier on occasion. For teaching, I would start with the Angular Circle, since it would seem to be easier to calculate in 1 than in Θ. This, of course, needs testing for evidence based education.

Zendmailer rightly refers to sine and cosine functions. If we use radians, the derivative of the sine is the cosine function, so that the slope of the sine at 0 equals 1. When we use dynamic division (I refer to COTP again) then we can write Sin[φ] // φ = 1 at φ = 0, for φ measured in radians, using the Unit Circle. I already knew this, but Bob Palais alerted me to the phenomenon that many graphs do not show the proper slope 1 at 0.

These points arise:

  1. Radians are often called dimensionless, since they arise from dividing arc by radius, thus length / length, but the arc is two-dimensional with the aspect of a turn, whence the dimension is Turn. (Addition July 29 2014: It occurred to me that this shift in focus might also be regarded in terms of the procept-theory of Gray & Tall: as an object we have length / length but as a process we have (length in one direction) / (length around). This may explain the difficulty for some people to “get it”.)
  2. Zendmailer uses a limit expression for Sin[φ] // φ = 1 at φ = 0 but skip the need for limits here.
  3. Zendmailer writes Sin[x] but sine and cosine represent y and x values of an angle φ.
  4. For α on the Angular Circle we can find x and y values on the Unit Circle via Xur[α] = Cos[Θ α] = Cos[φ] for φ = Θ α, and similarly Yur[α] = Sin[Θ α] = Sin[φ], where the “ur” means that those x and y values are relevant for the Unit Circle. See COTP for pictures.
  5. The derivatives (slopes) of Xur and Yur have a proportionality factor since these angles are measured on the Angular Circle and not on the Unit Circle. E.g. Yur’[α] = Sin’[Θ α] = Θ Cos[Θ α] = Θ Xur[α].
  6. Such a proportionality factor also exists for the sine of angles measured on 360 degrees. Try to figure out whether its slope at 0 is higher or lower than 1. Hint: your unit of measurement will be 1 degree.
Angular and Unit Circles

Angular Circle (c = 1) and Unit Circle (r = 1), Conquest of the Plane p36

While trigonometry is less cluttered in using Turns and Xur and Yur, for derivatives it becomes less cluttered from using radians. Note that you can still define what the unit of measurement is, e.g. 1 cm or 1 inch, so there is no real limitation on that choice. The only limitation is the issue of consistency, that once you choose, say 1 m, then the used sine and cosine show such and such slopes.

With this established, the reading of Zendmailer’s questions and the reactions should be easier.

Perhaps these critical comments are still useful:

  1. There is an expression “1 rad = 1″. My impression is that you should not write expressions like this, since this creates confusion. When the rad measurement is transformed from the circle arc to a straight axis in another space (where you plot the sine) then this best be indicated by a functional relationship. Subsequently, keep track of the “turn” in the rad: 1 rad ⇔ 1 / Θ Turn. I also propose Turn = Unit (Measure / Meter) Around = UMA to link up to standard measures.
  2. Note Philip Oakley: “The difficulty in point 2 is that the two lengths are in independent dimensions (as in 3d space). One has just cancelled Lx/Ly and lost information for one’s dimensional analysis (this is a Physics question;-). If one did the same with Charge/Temperature it would be a gross error, but we tolerate it for length. Dimensional analysis is newer than the cubit, so the old inconsistency remains. –  Philip Oakley May 11 ’13 at 20:52″ and “Anybody working in optics definitely cares. There are many measurements that have Angle(radians) as an integral part of their value, and it is a very common error, not spotted by dimension checking, for the angle part to be omitted, double counted, or wrongly applied. –  Philip Oakley May 14 ’13 at 7:30″ I would like to agree but don’t know optics. Also, my impression is that Lx/Ly would cancel as straight lines though this might be different in optics; but then the better format is 2D / 1D = 1D.
  3. There is also mention of Euler’s equation, but this can also be created for Xur and Yur, and thus doesn’t carry weight for the choice between the Angular Circle and the Unit Circle.

Overall, I find that there is no “natural” choice of either Angular Circle or Unit Circle as the “natural” unit of reference. The Angular Circle seems to be best to understand how an angle is measured, the Unit Circle might reduce the clutter for who works a lot with derivatives. Dimensions however tend to arise from the field of application. Having more bodies circling a Sun at various radii destroys all simplicity anyway, especially when those appear to be no circles at all.

PM.

This Wikipedia article has a wrong statement on dimensions (today, July 2014): “Although the radian is a unit of measure, it is a dimensionless quantity. This can be seen from the definition given earlier: the angle subtended at the centre of a circle, measured in radians, is equal to the ratio of the length of the enclosed arc to the length of the circle’s radius. Since the units of measurement cancel, this ratio is dimensionless.” The arc is in 2D space while the radius is in 1D, and 2D / D still leaves a dimension. The proper dimension is Turn. Use 1 Turn ⇔ Θ radians, so that 1 radian ⇔ 1 / Θ Turn ≈ 16% Turn. Turns are measured on the Angular Circle and radians on the Unit Circle. See the earlier weblog entry. I suppose that mathematicians enjoy taking the ratio arc / radius, and then create a bit of mystery, while engineers directly use the Unit Circle, with r = 1 in the standard unit of measurement (meter, foot), with the magic of being practical without the mystery.

Dimensional analysis generally concerns the units such as meters and seconds and dollars, e.g. see here or on wikipedia again. I have been using it to good effect since early university, in particular since F.J. de Jong at Rijksuniversiteit Groningen had increased awareness there. In this case we apply dimensional analysis to our 2D space. I met a mathematician who thought that I thought that dimensional analysis applied only to 1D, 2D, 3D, … and who started lecturing me, and it continues to amaze me how easy it is that misunderstandings arise.

Another possible misunderstanding is this. If you take a circumference of an object in 2D, say an equilateral triangle with sides 1 meter giving a circumference of 3 meter, and divide that circumference by a side, then it is conventionally (3 meter) / (1 meter) = 3 dimensionless, but rather be aware of (3 meter around) / (1 meter straight) = 3 around / straight. Again it is 2D / 1D = 1D. Just like the circle, you can make a turn going around that triangle. As it stands, it is little use to make an issue of this for circumferences in general, and the conventional view has its advantages. But for the circle it is useful to bring it to the fore in the definition of angle and turn. Indeed, here we need it to get the polar co-ordinates {radius, angle}, which uses that Turn is a separate dimension indeed.

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 Langeveld and Dian Geldof under Freudenthal
Hans Freudenthal’s earlier work on mathematics (topology, assistant to L.E.J. Brouwer) Hans Freudenthal on education after about 1965, 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 Dina Geldof on the education of mathematics, and Pierre van Hiele studied with him. 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 the latter.

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. 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, but in civilized manner, 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: http://www.mobygames.com, 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 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 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. 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.

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:

http://homepages.warwick.ac.uk/staff/David.Tall/pdfs/dot1977b-review-freudenthal.pdf

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.

Judith Grabiner has a fine book A historian looks back. The Calculus as Algebra and Selected Writings, MAA 2010, in which she explains how Joseph-Louis Lagrange (actually Italian Giuseppe Lodovico Lagrangia, 1736-1813) developed the derivative as algebra rather than with infinitesimals and limits. His method is more complex than my proposal in Conquest of the Plane (COTP) but his intuition is great.

Grabiner also explains how Lagrange wanted to get rid of Euclid’s fifth postulate. This is equivalent to the property that a point can have only one line parallel to a given other line. The postulate is used to show that the sum of angles in a triangle add up to 180 degrees. With respect to the following diagram: take a line through B that is parallel to AC, and see how the angles α and γ are mirrored, so that α + β + γ = 180 degrees.

Euclid: The sum of angles of a triangle is a half plane.

Euclid: The sum of angles in a triangle is a half plane.

This latter proof is very elegant but also creates a perpetual wonder: Howcome do the angles in a triangle cover a half plane ? And why does this depend upon the fifth postulate that caused so much discussion ?

I always wondered whether it might be explained a bit clearer. Perhaps not as elegant but with faster acceptance and better retention. Let us try to see whether the fifth postulate can be replaced by another one with seemingly less dramatic portent.

Deduction

Three points not on a line define a circle. Alternatively any triangle can be enclosed by a circle.

Circle and triangle

A circle can be defined by three arbitrary points not on one line. Any triangle can be enclosed by a circle.

We can scale triangle and circle up or down to the angular circle with its center at the origin O = {0, 0} and circumference 1. The unit of measurement of angles now is the plane itself. For example a value of a 1/2 means a half plane or a half turn. The angular circle thus has radius r = 1 / Θ, where Θ = 2 π and is pronounced “Archi” from Archimedes. (Check that its circumference Θ r  = 1.)

The angles of a triangle seem to completely exhaust the angular circle. However, angles are measured from the center of the angular circle. Let us draw the diameters from the corners through the center, which gives AA’, BB’ and CC’. We use the letters α, β and γ now for different angles.

Scaled to the angular circle. Diameters drawn through the corners.

Scaled to the angular circle. Diameters drawn through the corners.

There arise three inner isosceles triangles that use the same radius. The corner at A has angle ∠BAC = α + γ. This angle on the circumference associates with ∠BOC at the center (indicated by a tiny arc-sign) with the proper value ∠BOC = Arc[B, C]. Similarly for the other corners B and C.

For the angles at the center we find ∠AOB + ∠BOC + ∠AOC = Arc[A, B] + Arc[B, C] + Arc[C, A] = 1.

Since this can be done for any triangle, we arrive at the following postulate:

(*) For arbitrary corner A there is a proportion f so that ∠BAC =  α + γ = *∠BOC = f * Arc[B, C]

(1) It follows that the sum of the angles in a triangle equals that proportionality factor f too.

∠BAC + ∠ABC + ∠ACB = f (∠BOC + ∠AOC + ∠AOB) = f.

This also gives ∠BAC + ∠ABC + ∠ACB = 2 (α + β+ γ) = f.

(2) Secondly, we can apply the newly found sum rule to the inner triangles too.

For ΔAOB we find ∠AOB + 2 α = f
For ΔBOC we find ∠BOC + 2 β = f
For ΔAOC we find ∠AOC + 2 γ = f

Adding these we find 1 + 2 (α + β+ γ) = 3 f.
With the above: 1 + f = 3 f, or  f = 1/2.

Combining (1) and (2):

(3) The sum of angles in a triangle is 1/2.

(4) An angle that lies on the circumference of a circle is 1/2 of the associated angle at the center of the circle.

For example: You may check for ΔAOB that we find

∠AOB+ 2 α = Arc[A, B] + f (Arc[B, A’ ] + Arc[A, B’ ]).

Using that Arc[B, A’ ] = Arc[A, B’ ]) we find that

∠AOB+ 2 α = Arc[A, A’ ] = Arc[B, B’ ] = 1/2 (a half plane indeed).

Discussion

This proof strategy has these advantages:

(i) It emphasizes the measurement of angles, originally by plane sections but replaced by equivalent arcs. It shows that the angular circle is a natural way to measure angles. The 360 degrees came about historically because of the 365 days in the year but the plane itself makes more sense as a unit. (While 360 allows easy calculation: now use percentages: 1/2 = 50%.)

(ii) It shows clearly where the factor f = 1/2 comes from. There is a neat distinction between angles on the circumference and the actual measurement at the center.

(iii) Reversing the equivalences, we now have an elegant proof that a point has only one line parallel to another given line. Euclid’s fifth postulate has become dependent upon (*).

(iv) Non-Euclidean geometry arises from adapting (*). When the axioms are applied to a sphere then a constant f = 1/2 doesn’t make sense. It depends upon the kind of non-Euclidean geometry what the replacement postulate would be.

(v) Lagrange attached value to this discussion because scientists up to Einstein took Euclidean space also as a model for space itself. Grabiner op. cit. p261 suggests that ancient geometry was “the study of geometric figures: triangles, circles, parallelograms, and the like, but by the eighteenth century it had become the study of space (…ref…).” Her reference is to Rosenfeld 1988 ch 5. The implication would be that The Elements wouldn’t be a study of space as well ? I find this hard to believe – though I didn’t read that reference. It would seem to me that Euclid already thought that he axiomised a theory of space. A theory of geometric figures and their properties would not make sense if they were not conceptualised as being in space. That the theory implied ideas about space (like: a finite line might have any length) would be so obvious that it wouldn’t need mention. A fish in water would not speak about it. The Elements clearly isn’t a kosmology like the Timaeus. Given the importance of Plato, Euclid et al. likely regarded their findings limited to known space below the higher spheres, and they didn’t need to speculate like Plato on what lay beyond. This mental set-up still implies a theory of (known) space. The shift in the 18th century likely would be that Plato’s speculative kosmology fell away, so that Euclid’s known space started to apply to the whole universe. Be that as it may be, nevertheless, let me refer to COTP p195-197 for a discussion that Einstein might have been a bit ‘off’ on the issue of measurement error. It may well be that Euclid’s axioms actually define our very notion of space. At least, I find it impossible to think in terms of a “curved space” – i.e. I can imagine a sphere only as an object within Euclidean space.

(vi) Let us return to the selection of postulates. Euclid’s approach might well be better than the alternative given here. The postulate of a single parallel line feels rather natural. The proof on the triangle is so elegant that it may well have highest impact. However, Euclid’s set-up is that the master selects the postulates and that we pupils follow his results. Nowadays we might adopt more daring didactics. (a) Indeed, start with the fifth postulate and use the elegant proof that the angles of a triangle add up to a half plane. Allow for the sense of wonder. (b) Discuss the alternative approach of assuming a constant proportional factor f, as above. (c) Discuss advantages and disadvantages. Then allow pupils their own choice. Indeed, this didactic structure has been used in this weblog entry. (d) Finally, clean up the mess. (1) In the first triangle, the corners were labelled clockwise to get the Greek letters in alphabetical order. The subsequent triangles have been properly labelled counterclockwise. (2) It would have been stricter if the isosceles triangles used indexed labels α1 and α2 but I opted for legibility. (3) Discuss the actual fifth postulate that Euclid used. Perhaps the original discussion about it was caused merely because of its needless complex format.

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