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

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

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

Warning 1. The object of study concerns a chaotic situation

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

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

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

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

Warning 2. Results will be useless

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

Validity and reliability (source: wikimedia commons)

Validity and reliability (source: wikimedia commons)

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

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

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

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

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

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

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

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

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

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

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

The relevant points are:

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

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

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

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

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

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

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

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

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

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

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

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

Two relevant books of mine on this issue are:

Warning 4. Mathematics education research has breaches of scientific integrity

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

This warning holds in general

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

Comment w.r.t. the Dutch MathChild project

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

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

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

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

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

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

On p197-198 we find, my emphasis:

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

Criticism 3 thus generates the sub-criticisms:

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

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

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

An earlier weblog discussed that English is a dialect of mathematics. Compare:

Number    Math                  English
14              ten⋅four             fourteen
21              two⋅ten⋅one      twenty-one

Professor Fred Schuh of TU Delft wrote about the different mathematical pronunciation in books in 1943 and 1949, and addressed the Dutch minister of education in 1952 (see here). Many others observed this issue too. There is a gap though between understanding the idea given by these few examples and seeing it developed fully. Thus I decided to write out the alternative.

The issue is discussed in A child wants nice and no mean numbers. For some particular languages, the suggestions are in Marcus learns to count with ten. Select your language (text in English, numbers in the particular language):

EnglishGermanFrench –  Dutch –  Danish

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

The devil hides in the details

When you look at details then subtle problems show up. First of all, it appears better to use the middle dot instead of the hyphen, to prevent confusion with the negative sign.

Secondly, when mathematical pronunciation in German would use zehn for 10, then 90 in math would be neun⋅zehn, which would conflict with the current use of neunzehn for 19. You cannot seriously propose such a change because it would create confusion. Germans would have to ask each other continuously: “Are you speaking math or dialect ?”

There is already a regularity in German for the numbers of ten 20, 30, 40, …, 90: zwanzig, …, neunzig. Hence, German has at least these options: (a) adopt English ten for 10, (b) use zig for 10. The latter would give least change.

Number    Math in English      English           Math in German ?     German         Math in German !
19               ten⋅nine                   nineteen         zehn⋅neun                  neunzehn      zig⋅neun      
90               nine⋅ten                   ninety              neun⋅zehn                  neunzig          neun⋅zig

A compromise would be to accept 10 = zehn zig, and to use zehn up to 20 and zig from 20 onwards. When you are accepting change then rather do it properly though. My suggestion is to use zig. Dutch has the same problem, and here my suggestion is to use tig.

Danish might use their current word ti for 10. However, I have listened in Google Translate for the Danish pronunciation of ti⋅ti for 100, and though it sounds like tea-tea, I find it less convincing. My proposal for Danish is to use ten, which they already use for the numbers 13-19.

English actually has the same issue. The numbers of ten 20-90 use ty (e.g. forty, fifty), so that we might consider using ty instead of ten. This would give least change as in German. Then 90 would be nine⋅ty instead of nine⋅ten. However, English ty⋅ty for 100 is less convincing again. Thus ten for English is best.

German, Dutch and Danish might all adopt English ten. They already adopt Google or computer, and it would be curious when they are prim on 10, while change would benefit the learning of arithmetic by their young children enormously.

French can use dix for 10 without problem. French has some curious twists and turns. It suddenly relies on addition (soixante-dix = 60 + 10 for sept⋅dix = 7 × 10) and then swiches to multiplication with 20 (80 = quatre-vingts for huit⋅dix = 8 × 10). When 20 changes from vingt to deux⋅dix, then it becomes advisable to change the whole system.

Accepting responsibility

Overall, we again see that mathematicians are trained for abstract thought and have insufficient awareness of the empirical realities of education. Mathematicians should explain to both teachers and language managers about the difference between mathematical pronunciation and national dialects. The problem doesn’t necessarily lie in education but rather in mathematical neglect.

Professor Fred Schuh explained much of this already in 1943-1952, and thus one can argue that mathematics did explain it to education, so that it is the responsibility of education that they did not make the change. This is too simple a view.

This simple view does not square with devoted teachers who explain to their beloved pupils that soixante-quinze + seize = quatre-vingt et onze (check the confusing hyphen), in the belief that they are doing perfect arithmetic. These teachers should have had proper mathematics education, so that they know that they are short-changing their pupils.

Mathematics education should accept its reponsibility and free itself from the stranglehold by the abstract thinking mathematicians who have no idea about the empirics of education. (See my earlier letter to IMU / ICMI.)

Fred Schuh painted by Han van Meegeren in the Dutch hunger period during World War II (source: Schuh's book of 1949)

Fred Schuh painted by Han van Meegeren in the Dutch hunger period during World War II (source: Schuh’s book of 1949: “De macht van het getal” (“The power of numbers”))

The discussion of Putin’s proof gave me an email from Alexis Tsipras, who just resigned as prime minister of Greece and is busy with the general elections of September 20 soon. Rather than reporting on it, I might as well fully quote it.

To: Thomas
From: Alexis@formerprimeministerofgreece.org
Subject: My proof of Fermat’s Last Theorem
Date: Fri, 28 Aug 2015 11:58:03 +0100
Google Unique Message Identifier: 23DFGA@671

Dear Thomas,

Thank you very much for your discussion of President Putin’s proof when he was a youngster of Fermat’s Last Theorem. I know his mother Vera Putina very well. The Putin family has a vacation home here in Greece, and she can stay there on the condition that she immediately leaves when Putin himself comes down. She has shown me his proof too. I can only agree with your conclusion that it shows how smart President Putin was when he was young.

Putin’s proof inspired me to find a proof too. I am sometimes exhausted by the tough negotiations with the European Heads of State and Government, if not with members of my own party. Thus I resort often to a sanatorium for recuperation. Thinking about such issues like Fermat’s Last Theorem helps to clear my mind from mundane thoughts. I was very happy last Spring to indeed find a much shorter and more elegant proof. 

For the theorem and notation I refer to your weblog. My proof goes as follows.

Theorem. No positive integers n, a, b and can satisfy an + bn = cn for n > 2.

Proof. (Alexis Tsipras, April 31 2015)

Let us assume that an + bn = cn holds, and derive a contradiction.

There are two possibilities: (1) n is even, or (2) n is uneven.

(1) If n is even, then we can write A = an/2 and B = bn/2 and C = cn/2 such that A, B and C are still integers. Then we get the following equation:

A2 + B2 = C2

This equation satisfies the condition that n = 2, and thus it doesn’t satisfy the condition n > 2.

(2) If n is uneven, then we can write A = a(n-1)/2 and B = b(n-1)/2 and C = c(n-1)/2 such that A, B and C are still integers. Then we get the following equation:

a A2 + b B2 = c C2

This equation does not satisfy the form of an + bn = cn so that it falls outside of Fermat’s Last Theorem.

In both cases the conditions of the theorem are no longer satisfied. We thus reject the hypothesis that an + bn = cn holds. Q.E.D.

This is much shorter that President Putin’s proof. And, I prove it while he only came close. I have been hesitating to tell him, fearing that he might become jealous, and be no longer willing to support Greece as he does in these difficult times for my country. Now that you have confirmed how wonderful his proof at only age 12 was, I feel more assured. Will you please publish this proof of mine too, like you did with President Putin’s proof ? I have put my best efforts in this proof, just like at the negotiations with the European Heads of State and Government. Thus I hope that it will be equally convincing, if not more.

After the next elections I will probably be exhausted again. I would like to work on another problem then. Do you have any suggestions ?

Sincerely yours,

Alexis

Fermat and Tsipras (source: wikimedia commons)

Fermat and Tsipras (source: wikimedia commons)

The door rang. I was surprised to see Vera Putina. It appeared that Putin’s mother was visiting her granddaughter’s fiancé’s family in Holland. “It is not safe for me to go to Moscow,” she explained, expressing the sentiment of many.

When she was settled in the safety in my living room with a good cup of Darjeeling, it also appeared that there was more.

VP: “I am upset. The Western media depict my son only as a sportsman. They show him doing judo, riding horses, fighting bears, and the last week they featured him as a diver in a submarine. Of course he is very athletic, but he is also a smart man. I want you to look at his intellectual side too.”

Me: “It is fine of you to ask, because there indeed is a general lack of awareness about that.”

VP: “Let me tell you ! When my father had his love affair with Grand Duchess Kira Kirillovna of Russia [see the Putin family tree here], one of their secret meeting places was in the royal archives of the Romanovs. Sometimes my father, because he was unemployed, took some of the old documents to sell on the market. You know, my mother Kira had an expensive taste.”

VP said this without blinking an eye. The unperturbedness when taking other people’s possessions and territories must have a family origin.

VP: “My father once found the application letter by Pierre de Fermat for membership of the St. Petersburg mathematical society. It detailed his proof of his Last Theorem.”

Me: “Ah, that might explain why he never published it ! He used it for his application, and this got lost in the archives ?!”

VP: “I don’t know about that. My father sent it in for the Wolfskehl Prize of 100,000 gold marks, but it was rejected for it didn’t satisfy the criterion of having been published in a peer-reviewed journal.”

Me: “This is historically very interesting. If you still have that letter by Fermat, no doubt a historical journal will gladly publish it. It doesn’t matter on content since Andrew Wiles now proved it.”

VP: “No, no, no !” She gestured with passion as an Russian woman can do. “Fermat is not important ! It is what Vladimir did ! When he was twelve, he also looked at Fermat’s letter, and he found an omission ! Moreover, he worked on the problem himself, and almost solved it. Here, I brought along the papers to prove it to you.”

She delved into the bag that she had brought along and produced a stack of papers. I also saw a wire bound notebook such as children use in school.

Me: “Almost solving means not solving. Mathematics is rather strict on this, gospodina Putina. But it is historically interesting that you have Fermat’s original proof and that your son worked on it.”

VP: “For this, he had to learn Latin too !’

She gave me the stack. There was a great deal of difference between her nonchalant and triumphant handing over of the papers and my hesitant and rather reverent accepting of them.

VP: “You look it over, and inform the Western media that my son almost solved Fermat’s Last Theorem when he was only twelve ! If I hadn’t told him that he had to go to his judo lessons, he would have finished it for sure !”

She said the latter as proof that she had been a good mother, but also with a touch of regret.

Confronted with such motherly compassion I could only respond that I would oblige. Hence, below is Vladimir Putin’s proof. First I translate Fermat’s own proof from Latin (also using the Russian transcript that Putin made) and then give Putin’s correction.

Fermat (1601-1665) and Putin (1952+)

Fermat (1601 – 1665) and Putin (1952 – ∞) (Source: wikimedia)

Fermat’s Last Theorem, using middle school algebra

Theorem. No positive integers n, a, b and can satisfy the equation an + bn = cn for n > 2.

Proof. (Pierre de Fermat, April 31 1640, letter to czar Michael I of Russia)

Without loss of generality b. Take k = n – 2 > 0. We consider two cases:

(1) a2 + b2c2

(2) a2 + b2 > c2.

(1) When a2 + b2c2 then a2 + b2 + d = c2 for d 0

Then a < c and b < c. Then also ak < ck and bk < ck for k > 0.

If the theorem doesn’t hold, then there is a k > 0 such that:

ak+2 + bk+2ck+2

ak a2 + bk b2 = ck c2 = ck (a2 + b2 + d)

a2 (akck) + b2 (bkck) = d ck ≥ 0

negative + negative ≥ 0

Impossible. Thus the theorem holds for (1).

(2) If a2 + b2 > c2 then obviously (see the diagram) for higher powers too: an + bn > cn.

Fermat's drawing for his proof (right rewrites left)

Fermat’s drawing for his proof (RHS re-orders LHS)

Since (1) and (2) cover all possibilities, the theorem holds.

Q.E.D.

Putin’s correction, age 12

The comment by schoolboy Vlad on this proof is:

“While (2) is obvious, you cannot rely on diagrams, and you need to fully develop it. At least I must do so, since I find the diagram not so informative. I also have problems reading maps, and seeing where the borders of countries are.”

Hence, young Putin proceeds by developing the missing lemma for (2).

Lemma. For positive integers n, a, b and c: if a2 + b2 > c2 then an + bncn for n > 2.

Proof. (Vladimir Putin, October 7 1964)

Without loss of generality a b. Take k = n – 2 > 0.

If a2 + b2 > c2 then a + b > c. (Assume the contrary: a + b c then a2 + b2 < (a + b)2c2, which contradicts a2 + b2 > c2.)

Expression an + bn > cn is equivalent to (an + bn)1/n > c. The LHS can be written as:

f[n] = a (1 + (b / a)n)1/n  with a b.

This Lemma has the Pythagorean value f[2] = √(a2 + b2) > c. The function has limit f[n → ∞] = a. (See a deduction here.) Thus f[n] is downward sloping from f[2] >  to limit value a. We have two cases, drawn in the diagram below.

Case (A) Diagram LHS: c ≤ a, so that there will never be an intersection f[n] = c.

Case (B) Diagram RHS: a < c < f[2] = √(a2 + b2). There can be an intersection f[n] = c, but possibly not at an integer value of n. Observe that this case also provides a counterexample to Fermat’s claim that “obviously” f[n] > c, for, after the intersection f[n] < c. Young Putin already corrects the great French mathematician ! This is a magnificent result of the future President of the Russian Federation, at such a young age. His grandfather’s Marinus van der Lubbe’s submission to the Wolfskehl Prize would also have failed on this account.

a (1 + (b/a)^n)^(1/n) and parameter cases

f[n] = a (1 + (b / a)^n)^(1/n) and parameter cases

At this point, young Putin declares that Case (A) on the LHS is proven, based upon above considerations. He adds:

“I accept this proof on the LHS, even though I have difficulty understanding that limits or borders should not be transgressed.”

As so often happens with people who are not entirely sure of their case, the schoolboy then develops the following simple case, just to make certain.

Case (≤ b). Use numerical succession from a2 + b2 > c2.

Given an + bn > cn then prove an+1 + bn+1 > cn+1.

a an + b bnb an + b bn = b (an + bn) > b cnc cn

Thus the Lemma holds for this case.

To be really, really, sure, Putin adds an alternative proof that assumes the contrary:

ak a2 + bk b2ck c2

ak a2bk a2 + bk a2 + bk b2bk c2ck c2bk c2

a2  (akbk) + bk (a2 + b2c2) ≤ c2 (ck bk)

nonnegative + positive  ≤  nonpositive

Impossible. Thus the Lemma holds for (≤ b).

It would have been better when he had looked at Case (B) on the RHS, notably by proving that f[n] = c cannot hold for only integers.

At this point in his notebook, young Putin writes:

“I have to go to judo training. Perhaps I will continue tomorrow.”

I have looked in the remainder of the notebook but did not find further deductions on Fermat. Apparently the next day young Putin continued with what was more on his mind. It appears that he had a fantasy land called Dominatia in which he played absolute master, and it took much of his time to determine what was happening there. Something of the unruly nature of the natural numbers however must have stuck in his mind. In a perfect fantasy land everything is already as wished, but in young Putin’s Dominatia land he fantasizes about unruly citizens who must be put under control.

Conclusions

The above supports the following conclusions:

  • The theorem & lemma are not yet proven for Case (B) on the RHS. We must still rely on Andrew Wiles.
  • Nevertheless, Vladimir Putin doesn’t do just sports but also has amazing intellectual powers, at least when he was at age twelve.
  • Fermat’s original own proof of his theorem seems to have had a serious error, but it is not precluded that it was only chance that it did not get published (with or without corrections).
  • Fermat’s Last Theorem has dubious value for education. It seems more important to develop the notion of limits, and in particular the notion that you should not transgress borders. When students do not understand this properly at a younger age then this may cause problems later on.
Appendix 1. Case (c > a ≥ b)

It may be nice to see how f[n] = c is sandwiched, when a + b > f[2] > c > a ≥ b.

Case (c > a ≥ b)  There is a point f[n] = c or an + bn = cn for reals but perhaps not for integers.

(i) At the intersection:

ak a2 + bk b2 = ck c2

Take ak c2 + bk c2 and substract the above on both sides:

(c2 – a2) ak + (c2 – b2) bk = (ak + bkck) c2

positive + positive = ?

The latter must be positive too, and hence: ak + bk > ck

Thus, assuming that the theorem doesn’t hold for n requires that it holds for k = n – 2.

(ii) After the intersection: Since f[n] is downward sloping we have f[n+1] < f[n] = c. Reworking gives:

an+1 + bn+1 < cn+1

Another way to show this is:

(a – can + (b cbn < 0

a  an+ b bn  < c (an + bn)

an+1 + bn+1cn+1

Comparing (i) and (ii) we see the switch from > to <.

Appendix 2. Parameter restrictions in general

Assume that an + bn = cn holds. There are restrictions for this to occur, notably by the remarkable product:

(an bn) (an + bn) = (an bn) cn

a2n b2n = (an bn) cn

an (an cn) = bn (bncn)             (*)

For example: when c = a, then an + bn = cn is only possible in (*) if c = a = b, but this is actually also impossible because it requires that cn + cn = cn. The table collects the findings, with the LHS and RHS now referring to equation (*).

an + bn = cn c <(LHS +) c =(LHS 0)
c > a  (LHS -)
c < (RHS +) (=),
but Case (c b)
impossible opposite signs
c b  (RHS 0) impossible impossible
cn + cn = cn
impossible
c > b  (RHS -) opposite signs impossible (=) the only risk

This table actually also proves Case (A) that Putin took for granted. Only Case (B) remains, and requires proof that f[n] = c cannot hold for only integers.

Available now

Mathematics education is a mess. In primary education the development in arithmetic, geometry and other math topics is greatly hindered.

Mathematicians are trained for abstraction while education is an empirical issue. When mathematicians meet in class with real life pupils then they solve their cognitive dissonance by holding on to tradition. But tradition and its course materials have not been designed for empirical didactics. Pupils suffer the consequences.

The West reads and writes from left to right but the numbers come from India and Arabia where one reads and writes from right to left. In English 14 is pronounced as fourteen but it should rather be ten·four. 21 is pronounced in proper order as twenty·one, but is better pronounced as two·ten·one, so that the decimal positional system is also supported by pronunciation.

This is just one example of a long list. An advice is to re-engineer mathematics education for all age-groups. This book looks at primary education. Parliaments around the world are advised to have each their own parliamentary enquiry to investigate the issue and make funds available for change.

The author is an econometrician and teacher of mathematics.

Cover of "A child wants nice and no mean numbers" (2015)

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

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

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

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

Linking up and down

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

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

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

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

K.P. Hart on mathematics

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

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

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

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

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

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

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

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

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

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

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

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

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

Overall: Hart’s weblog is biased and unscientific.

Religion studies

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

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

Censorship at CPB

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

Economic Supreme Court

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

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

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

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

Democracy and voting theory

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

Other questions

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

Conclusion: An appeal to ARK and BdG

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

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

Now available

Elegance with Substance (2nd edition)

Mathematics and its education designed for Ladies and Gentlemen

What is wrong with mathematics education and how it can be righted

On the political economy of mathematics and its education 

Elegance with Substance (Cover)

Elegance with Substance (Cover)

National parliaments around the world are advised to have their own national parliamentary enquiry into the education in mathematics.

There is a failure in mathematics and its education, that can be traced to a deep rooted culture in mathematics. Mathematicians are trained for abstract theory but when they teach then they meet with real life pupils and students. They solve their cognitive dissonance by embracing tradition for tradition only. Instead, didactics requires a mindset that is sensitive to empirical observation which is not what mathematicians are trained for.

When mathematicians deal with empirical issues then problems arise in general. Other examples are voting theory for elections, models for environmental economics and growth, and the role of ‘rocket scientists’ in causing the stock market crash in 2008 (Mandelbrot & Taleb 2009).

While school mathematics should be clear and didactically effective, a closer look shows that it is cumbersome and illogical. What is called mathematics thus is not really so. Pupils and students are psychologically tortured and withheld from proper mathematical insight and competence.

The mathematics in this book is at highschool level.

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