February 2013 already gave a weblog entry Cause and Cure of the Crisis and November 2013 a YouTube video – though with a rather slow speed of presentation.

Now in September 2014 there is a short paper (MPRA  58592) with the same title. The paper gives a review of the books DRGTPE and CSBH that are at the core of this weblog – apart for our admiration for Art Buchwald and attention to the education of mathematics.

This may be occasion to pay a small tribute to John Maynard Keynes (1883-1946) and Jan Tinbergen (1903-1994). The analysis on the cause and cure of the crisis is based upon their work, and thus the world can only cure the crisis by properly reading up on Keynes and Tinbergen as well. It requires an understanding not only on economic theory but also econometric methods, with a role for mathematics.

There is another storyline that links up to mathematics education.

Tinbergen studied physics and wrote his thesis with Paul Ehrenfest (1880–1933). As a student he walked in Leiden, saw the poverty, decided to study economics, and doing so he created econometrics.

Ehrenfest’s wife Tatyana Afanasyeva (Kiev, 1876 – Leiden, 1964) wrote on math education too. Her work was hotly debated in small circles. One of the discussants was Hans Freudenthal (1905-1990), another German immigrant to Holland. Freudenthal became the promotor of Pierre van Hiele (1909-2010), which makes the circle round, see this entry on this weblog.


Paul Ehrenfest’s students, Leiden 1924. Left to right: Gerhard Heinrich Dieke, Samuel Abraham Goudsmit, Jan Tinbergen, Paul Ehrenfest, Ralph Kronig, and Enrico Fermi. (Source: Wikimedia commons)

Both President Obama of the USA and President Putin of the Russian Federation have somewhat illogical positions. Obama repeats the ritual article 5 “An attack on one is an attack on all” but the Ukraine is not a fraction of NATO. So what is the USA going to do about the Ukraine ? Putin holds that Russia defends all Russians everywhere but claims that Russia is not involved with the combatants in the Ukraine. His proposed 7 point plan contains a buffer zone so that he creates fractions in a country on the other side of the border. Overall, we see the fractional division of the Ukraine starting, as already predicted in an earlier entry in this weblog.

What is it with fractions, that Presidents find so hard, and what they apparently didn’t master in elementary school, like so many other pupils ? There are two positions on this. The first position is that mathematics teachers are right and that kids must learn fractions, with candy or torture, whatever works best. The second position is that kids are right and that fractions may as well be abolished as both useless and an infringement of the Universal Declaration of Human Rights (article 1). Let us see who is right.

An abolition of fractions

Could we get rid of fractions ? We can replace 1 / a or one-per-a by using the exponent of -1, giving a‾¹ that can be pronounced as per-a.  In the earlier weblog entry on subtraction we found the Harremoës operator H = -1. The clearest notation is aH = 1 / a. Before we introduce the negative numbers we might consider to introduce the new notation for fractions. The trick is that we do not say that. We just introduce kids to the operator with the following algebraic properties:

0H =  undefined

a aH =  aH a =  1

( aH ) H = a

Getting rid of fractions in this manner is not my idea, but it was considered by Pierre van Hiele (1909-2010), a teacher of mathematics and a great analyst on didactics, in his book Begrip en Inzicht (1973:196-204), thus more than 40 years ago. His discussion may perhaps also be found in English in Structure and Insight (1986). Note that aH = 1 / a already had been considered before certainly in axiomatics, but the Van Hiele step was to consider it for didactics at elementary school.

From the above we can deduce some other properties.

Theorem 1:

(a b) H = aH bH

Proof. Take x = a b. From xH x =  1 we get  (a b)H (a b) =  1. Multiply both sides with aH bH, giving (a b)H (a b) aH bHaH bH, giving the desired. Q.E.D.

Theorem 2:

HH = H

Proof: From addition and subtraction we already know that H H = 1. Take a a H = 1, substitute a = H, get H HH = 1, multiply both sides with H, get H H HH = H, and thus HH = H. Q.E.D.

It remains to be tested empirically whether kids can follow such proofs. But they ought to be able to do the following.


The expression 10 * 5H or ten per five can be simplified into 10 * 5H = 2 * 5 * 5H  = 2 or two each.

Equivalent fractions

Observing that 6 / 12 is actually 1 / 2 becomes 6 * 12H = 6 * (2 * 6)H = 6 * 2H * 6H = 2H. Alternatively all integers are factorised into the primes first. Note that equivalent fractions are part of the methods of simplification.


a bH * c d H  = (a c) (b d ) H

Comparing fractions

Determining whether a bH  > c dH or conversely: this reduces by multiplication by b d, giving the equivalent question whether a d > c b or conversely.


That (a / b = c)  ⇔  (a / c = b) may be shown in this manner:

a bH  = c

a bH  (b cH) = c  (b cH)

a cH  = b  


Van Hiele’s main worry was that we can calculate 2 / 7 + 3 / 5 = 31 / 35 but without much clarity what we have achieved. Okay, the sum remains smaller than 1, but what else ? Translating to percentages 2 / 7 ≈ 28.5714% and 3 / 5 = 60%, so the sum ≈ 88.5714%, is more informative, certainly for pupils at elementary school. This however requires a new convention that says that 0.6 is an exact number and not an approximate decimal, see Conquest of the Plane (2011). The argument would be that first calculating 31 / 35 and then transferring to decimals would give greater accuracy for the end result. On the other hand it is also informative to see the decimal constituants, e.g. observe where the greatest contribution comes from.

Another argument is that 2 / 7 + 3 / 5 = 31 / 35 would provide practice for algebra. But why practice a particular format if it is unhandy ? The weighted sum can also be written in terms of multiplication. Compare these formats, and check what is less cluttered:

a / b + c / d = (a / b + c / d) (b d) / (b d) = (a d + c b) / (b d)

a bH + c d H  = (a bH + c d H) (b d) (b d) H = (a d + c b) (b d) H


In this case kids would have to see that H can occur at two levels, like any other symbol.

a bH + H c d H  = (a bH + H c d H) (b d) (b d) H = (a d + H c b) (b d) H

Mixed numbers

A number like two-and-a-half should not be written as two-times-a-half or 2 1/2. Elegance with Substance (2009) already considers to leave it at 2 + 1/2. Now we get 2 + 2H .


Part of division we already saw in simplification. The major stumbling block is division by another fraction. Compare:

a / b / {c / d} = (a / b)(d / c) / { (c / d) (d / c) } = (a / b)(d / c)  / { 1 } = (a d) / (b c)

a bH * (c d H)H =   = a bH  * c H d = (a d) (b c ) H

Supposedly, kids get to understand this by e.g. dividing 1/2 by 1/10 so that they can observe that there are 5 pieces of 10H = 1/10 that go into 2H = 1/2. Once the inversion has been established as a rule, it becomes a mere algorithm that can also be applied to arbitrary numbers like 34H (127H)H = 1/34 / (1/127). The statement “divide per-two by per-10″ becomes more general: divide by per-a is multiply by a.

Dynamic division

A crucial contribution of Elegance with Substance (2009:27) and Conquest of the Plane (2011:57) is the notion of dynamic division, that allows an algebraic redefinition of calculus.

With y xH = y / x as normal static division then dynamic division y xD = y // x becomes::

y xD ≡ { y xH, unless x is a variable and then: assume x ≠ 0, simplify the expression y xH, declare the result valid also for the domain extension x = 0 }.

A trick might be to redefine y / x as dynamic division. It would be somewhat inconsistent however to train on xH and then switch back to the y / x format that has not been trained upon. On the other hand, some training on the division slash and bar is useful since it are formats that occur.

Van Hiele 1973

Van Hiele in 1973 includes a discussion of an axiomatic development of addition and subtraction and an axiomatic development of multiplication and division. This means that kids would be introduced to group theory. This axiomatic development for arithmetic is much easier to do than for geometry. Since mathematics is targeted at “definition, theorem, proof” it makes sense to have kids grow aware of the logical structure. He suggested this for junior highschool rather than elementary school, however. It is indeed likely that many kids at that age are already open to such an insight in the structure of arithmetic. This does not mean a training in axiomatics but merely a discussion to kindle the awareness, which would already be a great step forwards.

His 1973 conclusions are:

  1. In the abolition of fractions 1/a a part of mathematics is abolished that contains a technique that stands on its own.
  2. One will express theorems more often in the form of multiplication rather than in the form of division, which will increase exactness. (See the problem of division by zero.)
  3. Group theory becomes a more central notion.
  4. In determining derivatives and integrals, it no longer becomes necessary to transform fractions by means of powers with negative exponents. (They are already there.)
  1. Teachers will have to break with a tradition.
  2. It will take a while before people in practice write 3 4H instead of 3 / 4.
  3. Proponents will have to face up to people who don’t like change.
  4. We haven’t studied yet the consequences for the whole of mathematics (education).

His closing statement: “We do not need to adopt the new notation overnight. It seems to me very useful however to consider the abolition of the algorithms involving fractions.”


Given the widespread use of 1 / a, we cannot avoid explaining that aH = 1 / a. The fraction bar is obviously a good tool for simplication too, check 6 * (2 * 6)H .

Similarly, issues of continuity and limits x →1 for expressions like (1 + x) (1 – ) H would benefit from a bar format too. This would also hold alternatively for (1 + x) (1 – ) D.

But, awareness of this, and the ability to transform, is something else than training in the same format. If training is done in algorithms in terms of aH then this becomes the engine, and the fraction slash and bar merely become input and output formats that are of no significance for the actual algebraic competence.

Hence it indeed seems that fractions as we know them can be abolished without the loss of mathematical insight and competence.

Europe tends to be divided between North and South by Rhine, Alps and Danube. There is also the cultural divide between East and West by the historical split of the Roman Empire. Northern Europe tends to be Protestant, Southern Europe Roman-Catholic and in the East we see the Orthodox Church of Constantinople.

The Dutch minister of Foreign Affairs and now candidate EU Commissioner Frans Timmermans (1961) speaks about troubles in the Ukraine at the “European border“, but he means the EU border, since the continent of Europe extends beyond Moscow. There is a price when one doesn’t study some history.

My warning alert on Frans Timmermans consists of the following points.

EU federalist

He is a EU federalist. Given the differences in Europe it is rather silly to be such a federalist, see this mind map for a better perspective. But Timmermans was member of the 2003 EU Convention to create a EU Constitution, wanting to abolish the sovereignty of the Member States. In the Dutch referendum he lobbied for Yes but fortunately the majority said No. He has been uttering words of understanding afterwards but we better be cautious given his background as a diplomat.

Blind to Russian revanchism

He did not warn us for the last 23 years about Russian revanchism. See my text on the Repeat of Versailles. Timmermans had an early training on Russia and in Russian. When in the army in 1986 Timmermans was trained to interrogate simulated Russian captives, and later worked at the Embassy in Moscow in 1990-1993. His command in Russian was remarkably useful in last year’s diplomacy. But why not warn us over the last 23 years ? In a speech in 2008, Timmermans said about Putin:

“For all his faults, he has brought the necessary stability that will allow Russia to develop into a democracy based on giving people the chance to work for their own prosperity.”

Playing with dynamite on Maidan Square

He was at Maidan Square increasing that revanchism. In this Dutch TV broadcast of December 2013 he visited the Maidan Square opposition to Janukovich, claiming that he was impartial, but the reporter indicates that his visit of course was interpreted as a sign of support. In his September 2014 lecture he is aware of the threat: “The fear of Putin is not that protesters wave blue flags with yellow stars on Maidan Square, the fear of Putin is that they wave those blue flags with yellow stars on the Kremlin Red Square.” So explain: why provoke revanchism ?

Double standards on MH17

His MH17 diplomatic success should not distract us. Timmermans’s major diplomatic success is his performance in the EU in July and in the UN on the 2014 MH17 disaster, in which he succeeded in getting the world to stop and consider the loss of 298 lives. He must be complimented for this, and for his soothing role in Dutch society. This success as a diplomat should not be confused with competence as a policy maker.

But, he does not take responsibility. There are also the 2593 local civilian deaths due to the conflict, see the UN estimate for April to August. Timmermans might have explained that almost 2900 deaths were partly his own responsibility for not warning about Russian revanchism since 1991 and stimulating it in 2013 on Maidan.

Civilian deaths in the Ukraine, the conflict vs MH17

Civilian deaths in the Ukraine, the conflict vs MH17

Blind to the energy trap, blind on sanctions

In that 2008 speech, on energy:

“Another widespread belief, which I disagree with, is the notion that since Russia is a provider of energy and we are its customers, dependency only goes one way. We depend on their energy, and therefore they call the shots. This is simply untrue. Look at the facts. Yes, it is true that they provide the energy; yes, it is true that we are the customers. Yet given the nature of their energy sector and the nature of the energy they provide, customers like the European Union can dictate a lot of the behaviour of the supplier country.”

Now in his September 2 2014 lecture (in Dutch) he simply states that the EU has become too dependent upon Russian energy. Instead of acknowledging that this has been also his own responsibility, for having been so dumb and not seeing the problem of revanchism and dependence.

His answer to the Ukrainean conflict is sanctions for Russia, without explaining that this runs the risk of stimulating revanchism. He now hopes to get natural gas from Iran, when the sanctions there are lifted, but does not mention that Russia would be needed to solve the problems in Iran.

Under the lure of heroism and fighting

His closing statement on September 2 that the borders of Europe are there ‘where there is fighting about them’, sounds like the frustrated child that starts thrashing about.

Deaf to criticism

He apparently neglects crucial criticism. Karel van Wolferen, renowned for his analysis on Japan, wrote on August 15 a remarkable criticism of the US neo-conservatives who meddle with world peace: The Ukraine, Corrupted Journalism, and the Atlanticist Faith (Dutch text here). It is difficult to judge on various points but one would assume that the Dutch minister of Foreign Affairs would have had such sources before and would have a clear reply by now: but we find none of this. Instead, Timmermans September 2 lecture claims that the extension of NATO to the East brings stability, instead of showing some understanding that it increases Russian revanchism. Timmermans also presents the George W. Bush and Tony Blair lies, on WMD as the excuse for the invasion of Iraq, as mere incidents. It is actually a quite confused lecture, also with a ritual calling for “realism” but then criticizing Russia for supporting Assad.

Little understanding of economics and its importance

He has little understanding of economics. Timmermans studied languages and EU law and mostly worked in diplomacy. Apparently he has not extended his studies on the key issues that have to do with economics (in a setting of history, culture, world politics). Beware of politicians who think that they do not need to study economics. Yes, he says that much in diplomacy is guided by economics, but that does not mean that he draws the conclusion that he must study it. It only means that he continues doing the same as before but now calls it economics.

He takes a position that Holland originally preferred for Jeroen Dijsselbloem. This is my weakest objection. I just mention it for completeness. The selection of Timmermans might be presented as a success for Holland and the EU, certainly by himself, but it actually means a defeat. There has been some discussion about Dijsselbloem’s remark on Juncker’s drinking. This strange discussion raises eyebrows. The best way for Juncker to show the contrary is to agree to stop drinking when in office and accept Dijsselbloem rather than Timmermans. NB. As economists argue for a bit less austerity in the present economic stagnation, this better be argued by Dijsselbloem rather than Juncker  & Timmermans.

A minister in an undemocratic coalition government

He is a member of an undemocratic coalition government. In the 2012 elections, the free market conservative party VVD and social democratic party PvdA strongly opposed each other. This attracted voters who wished to prevent that the other party would be strong enough to form a government. VVD and PvdA managed to jointly collect more than 50% of the seats, and actually formed a joint coalition government, in betrayal of the earlier claims of ‘vote for us so that those others will be blocked’ (no actual quote). It is still somewhat a minority government since they lack seats in the Senate. The betrayal and the political mess make the present Dutch government quite unpopular. The VVD-PvdA coalition now has 45 seats in a Parliament of 150 seats (recent poll). Timmermans joined in that voter betrayal, though he now gained much popularity because of his performance on MH17.


FransTimmermans (source: EU put into public domain)

PM. Potentially a victim of revanchism himself

The text above puts much emphasis on Russian revanchism. There is also a source of revanchism in Dutch politics. This weblog reported on the cultural divisions in Holland before, and indicated the inferiority complex and revanchism from the Southern Catholic provinces Brabant and Limburg that had been under Northern Protestant rule. See the entry on the Dutch Taliban, and Geert Wilders from Limburg.

Frans Timmermans comes from a Northern country but is actually from the province of Limburg from below the Rhine too. In an interview, Timmermans says “to miss Wilders as a buddy” (VN December 27 2013), meaning that he disagrees with him politically e.g. on Islamistic Terrorism or the Polish Hotline, but can appreciate him “personally as a very nice guy”.

The latter doesn’t mean that Timmermans himself suffers from an inferiority complex and revanchism or would be at risk of becoming a Dutch Taliban too. There is no need to delve into these deeper psychological questions since the arguments above on his general failure as a policy maker are sufficient to advise to block his appointment as a EU commissioner.

Jeroen Dijsselbloem (1966) isn’t perfect either, of course, see here. He also partook in that voter betrayal, and comes from the other Southern Catholic province of Brabant. His only advantage is having little responsibility w.r.t. policy w.r.t. Russia.

To: Professor I. Daubechies
From: Thomas Cool / Thomas Colignatus
Subject: For IMU / ICMI: Integrity of science in Dutch research in didactics of mathematics
Cc: secretary of the IMU, president of KWG, professor Andre Ran

To the president of the International Mathematical Union (IMU),
that has the International Commission on Mathematical Instruction (ICMI)

Dear professor Daubechies,

My email of July 16 can be updated integrally as follows, and I will put this present email on my weblog.

Let me invite you to read these two weblinks:



Let me invite you to also read this paper: “Pierre van Hiele and David Tall: Getting the facts right” (version 2, 2014-08-30) at http://arxiv.org/abs/1408.1930.

Let me invite you to keep the matter on your desk with priority, involve others in IMU / ICMI who could advise you on this, and aim at board decisions that result into proper resolution.

Since this email concerns research in didactics, your tendency would be to forward it to ICMI. My suggestion is not to give them total freedom but set up an overall IMU committee to monitor the process within ICMI on this. In itself it might be proper to hand the issue to ICMI, since when they succeed in resolving the issue, then it would meet with greater acceptance in their own circles. On the other hand, there will be a tendency to reject criticism. Hence my suggestion to keep the issue on your desk as well.

One of the problems is that ICMI has a “Hans Freudenthal Award / Medal“, which indicates that ICMI has not been able to detect the fraudulent nature of Freudenthal’s “research” and appropriation of ideas of Pierre and Dieke van Hiele. A related problem is that the Dutch representative to ICMI might not have transferred my earlier message on didactics in general.

Since your background is Belgian, I presume some knowledge of Dutch, and then let me also directly include the link to my letter to KNAW-LOWI, which is the Integrity of Research department of the Dutch Royal Academy of Sciences:

I imagine that IMU might not have the resources available at KNAW-LOWI. My suggestion is that IMU supports my suggestion to KNAW-LOWI to look into this, even though they have already declined my original suggestion. In that case I would hope that there is international monitoring of the investigation at LOWI too, since they might be less critical on what went wrong in Holland. There are some issues here, some of which seem quite local but that still would greatly benefit from international monitoring: (1) the habit of abstract thinking mathematicians and such teachers to forget about the real world and empirical methods, (2) Hans Freudenthal and his “work” (much in Dutch), (3) the renaming of the ICMI Award, say to a “Piaget & Van Hieles Award / Medal”, (4) the abolition of the Dutch “Freudenthal Head in the Clouds Realistic Mathematics Institute” (FI = FHCRMI) here in Holland. It would seem that the last would not be in the ballpark of IMU but it is important to be aware that the institutional drive of that institute is to defend Freudenthal’s “legacy”, and thus to oppose criticism on the other points too, at the detriment of IMU / ICMI. It is better to be straightforward on the logic from the outset, and have international monitoring.

I alerted the Presidents of MAA (Bob Devaney) and AMS (David Vogan) and the director of the US Institute of Education Sciences (IES, John Easton) on the two weblog links, but not on my recent paper on Van Hiele and Tall (yet). After putting this letter on my weblog, I will alert the board of NCTM (Diane Briars) to this. I now copy to the IMU secretary and the chair of the Dutch KWG, now professor Geurt Jongbloed but formerly Andre Ran. In my perception KWG has been seriously failing on this issue but if IMU would indicate that there is an issue indeed then they might perhaps be willing to help out, with some international monitoring.

My position in all of this is quite limited, and mainly described in my books “Elegance with Substance” (2009) and “Conquest of the Plane” (2011) and the Dutch “Een kind wil aardige en geen gemene getallen” (2012), see my website, where the PDFs of the first two can be found. I do not claim particular expertise on Freudenthal’s “work” but what I have read didn’t appear so practical, except for what he took from Pierre van Hiele. I am amazed both by RME’s adoption for education and the lack of interest to repeal it now that everyone can see that it doesn’t really give results, except for the part taken from Van Hiele. My main point is no 1 above: the habit of abstract thinking mathematicians and such teachers to forget about the real world and empirical methods. My suggestion is that we need “engineers in education” rather than such mathematicians, and that education requires the “medical school” model in which education and its research are attuned, see this other link:


Sincerely yours,

Thomas Cool / Thomas Colignatus
Econometrician (Groningen 1982) and teacher of mathematics (Leiden 2008)
Scheveningen, Holland


PM. Readers of this email may also be interested in: Elizabeth Green: Why Do Americans Stink at Math?, NY Times July 23 2014

TC: Thank you very much, your highness King Willem-Alexander of the House of Orange, for inviting me to interview you during your Sunday morning breakfast. I hope that your highness has had a good night’s sleep and enjoys the excellent coffee as well as I do ?

Queen Maxima

King: Well, thank you for interviewing me. Yes, I slept well, and my wife, Queen Maxima, too. Please note that I always mention her in interviews, not only to please her, but people tend to know me as “the husband of Queen Maxima”, so that it is useful to make clear that I really am who I am saying who I am.

TC: Yes, I was going to ask about her too. Unfortunately she isn’t here for the interview, but I would like to compliment her with her presentation yesterday on occasion of 200 years of the Kingdom of the Netherlands. She was elegant as always but it struck me also how slender she looked.

King: Elegant and slender. Yes, I can remember that. I will pass on the compliment.


TC: I hate to spoil a good breakfast – did you try those sausages ? Oh, yes, you have this every morning – but, of those 200 years you have been king now for 1.5 years, and there has already been this MH17 disaster killing 283 passengers and 15 crew members, of which 193 Dutch people. One victim was senator Willem Witteveen, son of Johannes Witteveen who we once featured in this weblog with his important analysis on the economic crisis. Also Willem’s wife and daughter were in that plane. The elder Witteveen is devastated and will likely not speak in public anymore, which is sad for the economic discussion too.

King: I suffer with all the families of the victims. I or a member of the Royal Family have been to all receptions of the planes that returned with the remains of the victims. The Dutch government is doing their best to support the families, identify the remains of the victims, and prosecute those who are responsible for this.

Vladimir Putin

TC: Do you hold Vladimir Putin accountable ?

King: I am somewhat disappointed in the President of Russia. Last year November we met at a reception in the Kremlin for the Year of Dutch-Russian Friendship, and then this year in February at the Winter Games 2014 in Sochi we drank beer together. Besides, our vacation villa’s in Greece are at just walking or swimming distance. We try to be good friends and neighbours. If President Putin hadn’t supported those Russian rebels in the East then they would not have shot that plane. Putin could have exerted more influence to stop the fighting, so that our research team would have been better able to investigate the crash area.

TC: But do you hold Vladimir Putin accountable ?

King: I wished thing were that easy. The situation is complex. There is also the warring in the Middle East.

The Middle East

TC: Yes – apart from that really delicious pancake and syrup – what are your thoughts about the Middle East ? All civilized people in the world feel that they are dragged into a medieval horror show. Where is the world leadership ?

King: My impression is that history is repeating itself. The Roman Empire split up into the Eastern and Western parts, and those parts started fighting each other. General Belisarius of the Eastern Empire devasted Italy around 550 AD. When Islam came up around 600 AD, it started as a small sect, but because the Western Roman Empire had collapsed the relatively small Islamic tribes could conquer its domains in a mere 30 years. For us, when Europe and Russia are at loggerheads over the Ukraine, they forget about handling that “Islam State” that is filling the void in the Middle East.

TC: An amazing historical parallel ! What will you do about this, as King of Holland ?

Breaking News

King: I have decided that the House of Orange will claim the throne of Russia. As you know, my great-great-something-grandmother was Anna Pavlovna, so we are directly descendant from Czar Paul I of Russia. We had a family meeting and discussed the Russian problem of Putin. We observed that Russia never had a constitutional monarchy to make the common transition from feudal dictatorship to modern democracy. Hence it is sensible that a member of the House of Orange takes the throne there, and allows Russia to make the change. We will oversee that there are free democratic elections with a free press, and if Putin’s party loses, so be it.

TC: Excuse me for goggling and gurgling, sire – I am somewhat choking on this great Russian Salad. But, recouperating: You are claiming the Romanov throne ?

King: Ah, be careful here. First of all, the Romanov family is just a romantic relic. Prince Nikolai Romanovich is a good chap, but he is not a ruling king. Secondly, crownprince Charles of England has also good papers, but his eyes are on the English throne, just as prince William’s, while Andrew and Harry are too wild. Besides, the House of Hanover is much overrated. The House von Anhalt that produced Catherine the Great is too much of a good thing. History shows that the House of Orange is modest middle of the road, given that people still don’t know about the Dutch Empire. Thirdly, Maxima and I will remain in Holland, since it is much more difficult to manage a mature democracy like Holland. Besides, our vacation villa is close to Putin’s – well, I said that – and we want to remain good neighbours. Fourthly, my brother Constantijn is good in languages. Thus Constantijn and Laurentien will take the throne of Russia.

Laurentien and Constantijn

Laurentien and Constantijn, 2011 (Source: wikipedia commons)

Closing statement

TC: Thank you very much, your highness, for this exquisite breakfast, that was very nourishing for both body and mind. I hope to hear soon about your ideas of how to realize this ambitious plan. Could you also give me Constantijn’s telephone number, for his view on this ?

King: Just dial any number and ask for him. At least, that is how my phone works. You should try to get one like that.

Sharp readers will have observed that Vladimir Putin of Russia closely follows the suggestions in this weblog. After the last weblog discussion of To invade or not to invade ?” we now see the “Alea iacta est” with Russian tanks crossing the Ukrainean border.

Putin’s dilemma reminded of Shakespeare and the Danish prince Hamlet: “To be or not to be ?”  We shouldn’t be surprised that we got a response from Peter Harremoës from Denmark as well.

On the issue of taking a loss, be it the Crimea or now larger parts of the Ukraine, or children losing their fingers in Iraq-Syria or Israel-Gaza, but rather mathematically more general in the form of the subtraction of numbers in arithmetic, and thus the creation of negative numbers, Harremoës has developed a creative new approach that might stop the combatants in amazement. His 2000 article might stop you too, since it still is in Danish, and Google Translate still isn’t perfect. Harremoës mentions that he considers an extension in English at some time, so let us keep our fingers crossed till then – while we still have those.

In the mean time I would like to take advantage of some minor points on subtraction, partly relying on Peter’s article and thanking him for some additional explanation too.

Namely, in the last weblog discussion on confusing math in elementary school I stated that it is important to distinguish the operator minus from the sign min. Peter referred to a – (-b) and commented that problems of subtraction better be transformed into addition, and that subtraction can be seen on an abstract level as much more complex (or mathematically simple) than commonly thought.

One of his proposals is to create a separate symbol for -1 without the explicit showing of the min-sign. He took an example from history in which 1-with-a-dot-on-top already stood for -1. I have wondered about this, and would suggest to take a symbol that is available on the keyboard without much ado, where we e.g. already have i = Sqrt[-1].

A-ha ! Doesn’t the reader hear the penny drop ? Let us take i = quarter turn, H = half turn =   = -1, then i³ = H i = - = 3 / 4 turn =  three per four turn, and H H = full turn = 1. It would appear that H best be pronounced as ‘eta’, both for international exchange, and in sympathy for German teachers who would otherwise have to pronounce H H as ‘haha’, which would form a challenge for the German sense of humour. I considered suggesting small η or h but the nice thing about H is that it has a shade of -1 in it. In elementary school we can use just the Harremoës-operator H = -1 without the complex numbers. Later in highschool when complex numbers would arise we can usefully refer to H as something that would already be known (or forgotten).

Kids can understand that a debt is an opposite from a credit, or that losing the Ukraine is opposite to winning it. Thus if a is an asset then H a is a liability of the same absolute size. Calculation of gains and losses could be done with a + H b for counting down, or H b + a for counting up. If you lose a debt, then you gain. Losing a debt H b then would be introduced as a + H (H b) = a + b.  Actually, I suppose that it would be even better to start with the absolute difference between two numbers, Δ[a, b]. A sum would be to determine that Δ[a, H b] = a + b, presuming that a and b are nonnegative integers.

Thus H would be used in the creation of the negative numbers and the introduction of subtraction, and for later remedial teaching for who didn’t get it or lost it. Peter Harremoës seems to be of the opinion that there would be no need, in principle, to introduce minus and min, but agrees that people would currently want to stick to common notions. Once the basics of H are grasped, it is no use to grind them in, since it is better to switch to minus and min that must be ground in because of that commonality.

First the min sign and the negative integers are introduced by extending the number line: -1 = H 1 , -2 = H 2, … -100 = H 100 and so on. The teacher can show that applying H means making half turns, or moving from the right to the left, or back.

Subsequently the minus operator is introduced as a – b = a + H b.

Hence there arises the exercise a – (-b) = aH b = a + H (H b) = a + 1 b = a + b.

Or the relation between minus and min: -b = 0 – b = 0 + H b = H b.

A pupil who has mastered arithmetic will do a – (-b) = a + b directly. Otherwise return to remedial teaching and practice with H again.

Arithmetic seems simplest in a positional system. Earlier, we already discussed that English better is regarded as a dialect of mathematics. A number like 15 is better pronounced as ten-five than as fifteen. A sum 15 + 36 then fluently (yes!) translates into “ten-five plus three-ten-six equals (one plus three)-ten-(five plus six), equals four-ten plus ten-one, equals five-ten-one” which is 51. Let me introduce the suggestion that children can use balloons in handwriting or brackets in typing to indicate not only the digits but also the values in the positional system.

15 + 36

Adding 15 and 36 using the positional system, with balloons or brackets

In the same manner, the positional system allows us to state -1234 = [-1][-2][-3][-4], where we might rely on H if needed.

For subtraction, the algorithm for a - b is to keep that order if a ≥ b, or otherwise reverse and calculate -(b – a). But, it is useful to show pupils the following method if they forget about reversing the order. For example, 16 – 34 = 16 + [-3][-4] and the rest follows by itself.

16 - 34

Subtract 16 – 34 when forgetting to reverse the order

One might compare the above with other expositions on subtraction. An obvious source is the wikipedia article on subtraction, while google gives some pages e.g. from the UK or the USA. Some texts seem somewhat overly complex.

Originally I thought that the subtraction a - b for a ≥ b would be harmless, but on close consideration there is a snake in the grass. A point is that corrections are made above the subtraction line, so that the original question is altered. In the Wikipedia example of the Austrian method the final sum doesn’t add up any more. The Wikipedia example of the American method is okay, provided that indeed 7 is replaced by 6, and 5 is replaced by 15. But this is not a proper positional notation anymore. The method also assumes that you use pen and paper, which is infeasible in a keyboard world. Below on the right there are two examples that keep the original sum intact, and that only use the working area below that original sum. One approach is to rewrite 753 = [6][15][3] and the other approach is to do the borrowing a bit later, which is faster. These methods rely on the trick of using balloons or brackets to put values and sub-calculations into a positional place. If we allow for adaptation above the minus line, then the use of H = -1 and T = 10 would work as well, without the need to dash out digits. The second column combines the American & Austrian methods with the Harremoës operator H but now treated as a digit, and using [H][T][0] = HT0 = 0.

Subtraction of 753-491, in the American manner (source Wikipedia), with comments

Subtraction of 753-491, in the American manner (source Wikipedia), with comments

Evaluating these methods, my preference is for the last column. It follows the work flow, in which the negative value is discovered by doing the steps. The method accepts negative numbers instead of creating some fear for them. A practiced pupil would not need the 2[10-4]2 line and directly jump to the answer, so that the number of lines is the same as in the first and second column. The American / Dutch method with HT0 = 0 inserted as a help line creates the suggestion as if borrowing is required before one can do the subtraction, which goes against the earlier training to be able to do such a subtraction that results into a negative value. The borrowing is only required to finalize into a final number in standard notation.

Overall, my conclusion is that the emphasis in teaching should be on the positional system. The understanding of this makes arithmetic much easier. Secondly, the Harremoës operator H indeed is useful to first understand the handling of credit and debt, before introducing the number line and the notation a - b. Thirdly, in a combination of the two earlier points, this operator also appears useful into decomposing -1234 = [-1][-2][-3][-4]. I want to thank Peter again for starting all this (apart from the more advanced ideas in his article). For completeness, let me refer to the 2012 paper A child wants nice and not mean numbers, with a discussion of the pronunciation of the numbers and some more exercise on the positional system.

But these mathematical operations don’t explain that Ukraineans first lose the Ukraine but subsequently gain it once they have turned into Russians.

The problems in Russia-Ukraine, Irak-Syria and Israel-Gaza are so large since the combatants are hardly aware of the concept of fair division and sharing. Something must have gone wrong in elementary school with division and fractions. Let us see whether we can improve education, not only for future dictators but for kids in general.

English as a dialect

In 2012 I suggested that English can best be seen as a dialect of mathematics. The case back then was the pronunciation of the integers, e.g. 14 as “fourteen” (English) instead of “ten-four” (math & Chinese). The decimal positional system isn’t merely a system of recording but it contains switches in the unit of account. In this system the step from 9 to 10 means that ten becomes a new unit of account, and the step from 99 to 100 means that hundred (ten-ten) becomes a new unit of account. This relies on the ability to grasp a whole and the notion of cardinality. Having a new unit of account means that it is valid to introduce the new words “ten” and “hundred”, so that 1456 as a number differs from a pin-code with merely mentioning of the digits. When the numbers are pronounced properly then pupils will show greater awareness of these elements and become better in arithmetic – and arithmetic is crucial for division and fractions.

When education is seen as trying to plug mathematics into the mold of English as a natural language, then this is an invitation to trouble. It is better to free mathematics from this mold and teach it in its own structural language. It is a task for the teaching of English to show that it is a somewhat curious dialect.

Rank numbers

After the recent discussion of ordinal or cardinal 0, it can be mentioned that the ordinals are curiously abused in the naming of fractions. Check the pronunciation of 1/2, 1/3, 1/4, 1/5, … With number 4 = four and the rank 4th = fourth, the fraction 3/4 is pronounced as “three-fourths”. What is rank “fourth” doing in the pronunciation of 3/4 ? School kids are excused to grow confused.

Supposedly, when cutting up a cake in four parts, one can rank the pieces into the first, second, third and fourth piece. Assuming equal pieces, or fair division, then one might borrow the name of the last rank number “fourth” to say that all pieces are “a fourth”. This is inverse cardinality. Presumably, this is how natural language developed in tandem with budding mathematics. Such borrowing of terms is conceivable but not so smart to do. It is confusing.

The creation of “a fourth”, as a separate concept in the mind, also takes up attention and energy, but it doesn’t produce anything particularly useful. Malcolm Gladwell alerted us to that the Chinese language pronounces 3/4 as “out of four parts, take three”. Shorter would be “3 out of 4″. This directly mentions the parts, and there is no distracting step in-between.

For a reason discussed below we better avoid the “of” in “out of”. Thus it might be even shorter to use “3 from 4″, but a critical reader alerted me to that his might be seen as substraction. Thus “3 out 4″ seems shortest. However, there is also the issue of ratio versus rate. In a ratio the numerator and the denominator have the same dimension (say apples) while in a rate they are different (say meter per second). Thus the overall best shortest pronunciation would be “3 per 4″, which is neutral on dimensions, and actually can be used in most European languages that are used to “percent”.

This pronunciation thus facilitates direct calculation, like “one per four plus three per four gives four per four, which gives one”.

Dividing and sharing

The Dutch word for “divide” also means “share” (Google Translate). Sharing a cake tends to generate a new unit of account, namely the part. In fair division each participant gets a part of the same size, which becomes: the same part. This process focuses on the denominator and generates a larger number and not a smaller number. It actually relies on multiplication: the denominator times the new unit of account (the part) gives the original cake again. The process of sharing is rather opposite to the notion of division that gives a fraction, that maintains the old unit of account and generates a smaller number on the number line.

A fraction 3 / 4 or “three per four”, when three cakes must be shared by four future dictators, requires the pupil to establish the proportional ratio with “three cakes per four cakes” (virtually giving each a cake even though there are no four cakes but only three), and then rescale from the four hypothetical cakes down to one cake. PM. The pupil must have a good control of active versus passive voice. The relation is that “4 kids share 3 cakes” (active) and “3 cakes are being shared by 4 kids” (passive). Thus “3 per 4″ or “3 out of 4″ is shorthand for “3 units taken out of 4 units” (or “4 (kids) take out of 3 (cakes)”) but not for “3 (kids) take out of 4 (cakes)” (which would give 1 + 1/3 per kid, and would require a discussion of mixed numbers).

Hence it is unfortunate that the Dutch language uses the same word for both sharing and dividing. Fraction 3/4 reads in Dutch as “3 shared by 4 gives three-fourths” (“3 gedeeld door 4 geeft drie-vierde”), which thus combines the two major stumbling blocks: (a) the sharing/dividing switch in the unit of account, (b) the curious use of rank words. When 3/4 = “three per four” would be used, then the stumbling blocks disappear, and teaching could focus on the difference between the process of dividing and the result of the fractional number on the number line.

David Tall (2013) points to a related issue in the language on sharing and dividing: “The notion of a fraction is often introduced as an object, say ‘half an apple’. This works well with addition. (…) What does ‘half an apple multiplied by half an apple’ mean? (…) However, if a fraction is seen flexibly as a process, then we can speak of the process ‘half [halve] an apple’ and then take ‘a third of half an apple’ (…) the idea is often simply introduced as a rule, ‘of means multiply’, which can be totally opaque to a learner meeting the idea for the first time.” (p97) Note that Tall’s book is rather confused so that you better wait for a revised edition. He indeed does not mention above issues (a) and (b). But this latter observation on the process and result of division is correct.

The rank words thus are abused not only as nouns but also as verbs (“take a third of half of an apple”). We better translate into “(one per three) of (one per two)”, which gives “(one times one) per (three times two)”. The mathematical procedure quickly generates the result. The didactic challenge becomes to help kids understand what is involved rather than to master confused language.


Speaking about Tall and multiplication: Apparently the English pronunciation of the tables of multiplication can be wrong too. E.g. ‘two fours are eight’ refers to two groups of four, and thus implies an order, while merely ‘two times four is eight’ gives the symmetric relation in arithmetic. Said book p94 compares a table with 3 rows and 4 columns, and Tall argues: “the idea of three cats with four legs is clearly different from that of four cats with three legs. The consequence is that some educators make a distinction between 4 x 3 and 3 x 4. (…) I question whether it is a good policy to teach the difference. (…) [ reference to Piaget ] (…) So a child who has the concept of number should be able to see that 3 x 4 is the same as 4 x 3.”

An exercise in marbles

An exercise in marbles

Tall doesn’t explain this: Pierre van Hiele focuses on the distinction “concrete versus abstract”, would focus on the table, so that children would master the insight that the order does not matter for arithmetic. Once they have mastered arithmetic, they might consider “reality versus model” cases like on the cats and their legs without becoming confused by arithmetical issues hidden in those cases. Instead, Hans Freudenthal with his “realistic mathematics education” (RME) would present kids with the “reality versus model” cases (e.g. also five cups with saucers and five cups without saucers, a 3D table), and argue that this would inspire kids to re-invent arithmetic, though with some guidance (“guided re-invention”). Earlier, I wondered why Freudenthal blocked empirical research in what method works best (and my bet is on Van Hiele).


Overall, the scope for improvement is huge. It is advisable that the Parliaments of the world investigate failing math education and its research. When kids have improved skills in arithmetic and language, they would have more time and interest to participate in and understand issues of fair division. Hurray for World Peace !

PM 1. Conquest of the Plane pages 77-79 & 207-210 discuss proportions and fractions.

PM 2. See also COTP for the distinction between dynamic division y // x and standard static division y / x.

PM 3. Some say “3 over 4″ for 3/4, hinting at the notation with a horizontal bar.  I wonder about that. The “3 per 4″ is actually shorter for “3 taken from 4″, and this puts emphasis on what is happening rather than on the shape of the notation. An alternative is “3 out of 4″ but my inclination was to avoid the “of” as this is already used for multiplication. Also, my original training has been to reserve “n over k” for the binomial coefficient (that can be taught in elementary school too). However, a reader alerted me to Knuth’s suggestion to use “n choose k” for the binomial coefficient, and that is better indeed. In that case I would tend to avoid the “over”. It was also commented that “3 from 4″ sounded like substraction: but my proposal is to adhere to “3 minus 4″ for “3 – 4″ as opposed to “3 plus 4″ for the addition. It is just a matter to introduce plus and minus into general usage, so that it is always clear what they are. Note that we are speaking about mathematics as a language and not about English as a natural language. Also -1 would be “negative-1″ or “min-1″, with the sign “min” differing from the operator “minus”.


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