Hans Rosling (1948-2017) was a professor of public health and at the Swedish Academy of Sciences. I hadn’t heard about him but his death caused newsmedia to report about his mission to better inform people by the innovative presentation of statistics. I looked at some of his presentations, and found them both informative and innovative indeed.

I applaud this chart in which he tabulates not only causes and effects but rather means and goals. (Clicking on the picture will bring you to the TED talk 2007, and at the end the audience may applaud for another reason, namely when he swallows a sword to illustrate that the “impossible is possible”.)

Hans Rosling 1948-2017

Hans Rosling 1948-2017

Continue the discussion

My impression is that we best honour Rosling by continuing the discussion about his work. Thus, my comments are as follows.

First of all, my book Definition & Reality in the General Theory of Political Economy shows that the Trias Politica model of democracy fails, because it allows politicians still too much room to manipulate information and to meddle in scientific advice on policy making. Thus, governance is much more important than Rosling suggested. Because of his analysis, Rosling in some of his simulations only used economic growth as the decisive causal factor to explain the development of countries. However, the key causal factor is governance. The statistical reporting on this is not well developed yet. Thus, I move one + from economic growth to governance.

Secondly, my draft book The Tinbergen & Hueting Approach in the Economics of Ecological Survival discusses that the environment has become a dominant risk for the world as we know it. It is not a mathematical certainty that there will be ecological collapse, but the very nature of ecological collapse is that it comes suddenly, when you don’t expect it. The ecology is so complex and we simply don’t have enough information to manage it properly. It is like standing at the edge of a ravine. With superb control you might risk to edge one millimeter closer, but if you are not certain that the ground will hold and that there will not be a sudden rush of wind, then you better back up. The table given by Rosling doesn’t reflect this key point. Thus, I move one + from economic growth to the environment.

In sum, we get the following adapted table.

Adapted from Hans Rosling

I have contemplated for the means whether I would want to shift another + from economic growth to either human rights (property rights) or education (I am also a teacher). However, my current objective is to highlight the main analytical difference only.

In the continued discussion we should take care of proper definitions.

What does “economic growth” mean ?

The term “economic growth” is confusing. There is a distinction between level and annual growth of income, and there is a distinction w.r.t. categories within. Economic welfare consists of both material products (production and services) and immaterial elements (conditions and services). If the term “economic growth” includes both then this would be okay. In that case, however, the whole table would already be included in the notion of welfare and economic growth. Apparently, Hans Rosling intended the term “economic growth” for the material products. I would suggest to replace his “economic growth” by “income level”, and thus focus on both income and level rather than annual change of a confusingly named statistic. Obviously, it is a policy target that all people would have a decent standard of living, but it is useful to remain aware that income is only a means to a higher purpose, namely to live a good life.

PM. This causes a discussion about the income distribution, and how the poor and the rich refer to each other, so that the notion of poverty is relative to the general standard of society. In the 1980s the computer was a luxury item and nowadays a cell-phone with larger capacity is a necessity. These are relevant aspects but a discussion would lead too far here now.

What does “environment” mean ?

In the adapted table, the environment gets ++ as both means and goal. There is slight change of meaning for these separate angles.

  • The environment as a goal means that we want to preserve nature for our descendants. Our kids and grandchildren should also have tigers and whales in their natural habitat, and not as photographs only.
  • The environment as means causes some flip-flop thinking.
    (1) In economic thought, everything that exists either already existed or mankind has crafted it from what was given. Thus we only have (i) the environment, (ii) human labour. There are no other means available. From this perspective the environment deserves +++.
    (2) For most of its existence (some 60,000 years), mankind took the environment for granted. Clear air and water where available, and if some got polluted it was easy to move to a next clean spot. The economic price of the environment was zero. (Or close to it: the cost of moving was not quite a burden or seen as an economic cost.) Thus, as a means, the environment didn’t figure, and from this viewpoint it deserves a 0. There are still many people who think in this manner. It might be an engrained cultural habit, but a rather dangerous one.
    (3) Perhaps around the middle of the past century, the 1950s, the environment has become scarce. As Lionel Robbins explained: the environment has become an economic good. The environment provides functions for human existence and survival, and those functions now get a price. Even more, the Tinbergen & Hueting approach acknowledges that the ecology has become risky for human survival. The USA and Europe might think that they can outsource most environmental pollution to the poorer regions of the world, but when the rain forests turn into deserts and when the CO2 turns the oceans into an acid soup that eats away the bones of fish, then the USA and Europe will suffer the consequences too. In that perspective, the environment deserves +++.
    (4) How can we make sure that the environment gets proper place in the framework of all issues ? Eventually, nature is stronger than mankind, and there might arise some natural correction. However, there is also governance. If we get our stuff together, then mankind might manage the world economy, save the environment at some cost, but still achieve the other goals. Thus governance is +++ and the environment is relative at ++. Thus we arrive at above adapted table.
Dynamic simulation

As a teacher of mathematics I emphasize the combined presentation of text, formula, numeric table, and graph. By looking at these different angles, there is greater scope for integrated understanding. Some students are better at single aspects, but by presenting the four angles you cover the various types of students, and all students get an opportunity to develop the aspects that they are weaker in.

Obviously, dynamic simulation is a fifth aspect. See for example the Wolfram Demonstrations project. Many have been making applets in Java and embedding this in html5, yet the use of Mathematica would allow for more exchangeable and editable code and embedding within educational contexts in which the manipulation of text, formula, numeric table, and graph would also be standard.

Obviously, role playing and simulation games are a sixth aspect. This adds human interaction and social psychology to the learning experience. Dennis Meadows has been using this to allow people to grow aware of the risk on the environment, see e.g. “Stratagem” or MIT-Sloan.

The economic crisis of 2007+

What I particularly like about Rosling’s table is his emphasis on culture as a goal. Artists and other people in the world of culture will already be convinced of this – see also Roefie Hueting on the jazz stage – yet others may not be aware that mankind exists by culture.

There is also an important economic angle on culture as a means. In recessions and depressions, the government can stimulate cultural activity, such that money starts flowing again with much less risk for competitive conditions. That is, if the government would support the automobile industry or steel and do specific investments, then this might favour some industries or services at the cost of others, and it might affect competitive conditions overall, and even insert imbalances into the economy in some structural manner. Yet stimulating cultural activity might be much more neutral and still generate an economic stimulus.

For example, Germany around 1920 got into economic problems and the government responded by printing more money, and this caused the hyperinflation. This experience got ingrained in the German attitude towards monetary issues. In the Eurozone Germany follows the hard line that inflation should be prevented at all costs. Thus the eurozone now has fiat money that still functions as a gold standard because of the strict rules. (See my paper on this.) By comparison, when the USA around 1930 got into economic problems and the central bank was hesitant to print money (no doubt looking at the German example), this eventually caused the Great Depression. Thus monetary policy has the Scylla and Charybdis character, with the risks of either too little or too much. Potentially, the option to organise cultural activity would be a welcome addition to the instruments to avoid such risks and smooth the path towards recovery.

I am not quite suggesting that the ECB should print money to pay the unemployed in Greece, Italy, Spain and Portugal to make music and dance in the streets, yet, when the EU would invest in musea and restorations and other cultural services so that Northern Europe can better enjoy their vacations in Southern Europe, then this likely would be more acceptable than when such funds would be invested directly in factories that start to compete with the North. The current situation that Southern Europe has both unemployment and less funds to maintain the cultural heritage is obviously less optimal.

The point is also made in my book Common Sense: Boycott Holland. Just to be sure: this notion w.r.t. culture is not the main point of CSBH. It is just a notion that is worthy of mentioning.

PM. Imagine a dynamic simulation of restoring the Colosseum. Or is it culturally more valuable as a ruin than fully restored ?

By Jaakko Luttinen - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=22495158

By Jaakko Luttinen – Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=22495158

Geert Wilders used a tweet with a photoshopped picture of Alexander Pechtold. The picture displays Pechtold as demonstrating for the introduction of Sharia in Holland. The political message is that Pechtold would be a fellow-traveller and part of the 5th column for political islam, intending to destroy freedom and democracy. Normally Wilders merely says this but a picture tells more than a thousand words.

This falsely portraying of a political opponent is a new low in the Low Countries.

The photoshopped picture would exist since 2009 but there are general elections for the Dutch House of Commons on March 15 which may be the reason why Wilders uses it now. Wilders might have limited campaign funds and the abuse of this picture is politically cunning, since hords of people, including me, are discussing it now. Attention is half of the job, and Wilders knows how to get attention. And when there is a terrorist attack, then he can claim that he has been warning all along.

Yet, the downside of this is, that there are feeble minds on the radical right, like Anders Breivik, who worship Wilders, and who might take this portrayal as an invitation to target Pechtold. The UK saw the assassination of Jo Cox in 2016. Holland already saw a smear campaign against Pim Fortuyn in 2002 who then got assassinated by an activist on the left. Yet a gunman in 2011 who killed six people was a sympathiser of Wilders. Journalist Peter Breedveld has been reporting consistently that the political climate in Holland is getting heated, repressive and threatening of violence. Pechtold is alarmed. He warned that Wilders is deliberately rousing up his followers. One sympathiser of Wilders already threatened Pechtold to kill him, and Pechtold informed reporters that he had to testify in court to get the man convicted. A close political friend of Pechtold, Els Borst, has been murdered by a lunatic in 2014, apparently without political motivation, but it still has an impact.


Wilders and Pechtold have a history of feeding on each other

Geert Wilders and Alexander Pechtold have a history of feeding on each other. They are each other’s best enemies. While Wilders finds great profit in demonising Pechtold as the fellow-traveller of political islam, Pechtold finds great profit in portraying Wilders as indecent and “over the top”. Their political clash was the motor for their rise to public attention in 2006-2010. In the elections of 2010, Pechtold jumped from 3 to 10 seats, and Wilders from 9 to 24 seats.

The following graph shows the number of seats of Wilders (PVV, red) and Pechtold (D66, blue) in the Dutch House of Commons, with a total of 150 seats. (Source: Wikipedia, here adapted.)

  • Wilders started in 2004 as a one-man separation of the Dutch conservative party VVD. The official line of VVD was that Turkey might eventually join the European Union, but Wilders disagreed, and wished to have the freedom to say so. The letters VVD stand for the People’s Party for Freedom and Democracy, but party leader Gerrit Zalm denied Wilders his freedom of expression. In 2006 Wilders got 9 seats, in 2010 he jumped to 24, and in 2012 got 15. (Incidently: Gerrit Zalm had also participated in the smear campaign against Pim Fortuyn, labeling him as a “dangerous man”. Zalm also was the director of the CPB who in 1990 censored my work at CPB and who dismissed me there with falsehoods, the very issue that this weblog is about.)
  • In 2006, D66 had been reduced from 24 seats to 3, and Pechtold began as the new leader. There was talk about ending the party, yet Pechtold managed to get the party back to 10 seats. His strategy was to oppose Wilders.
  • As said, in the elections of 2010, Pechtold jumped from 3 to 10 seats, and Wilders from 9 to 24 seats.
  • In 2010-2012 there was the 1st Rutte Cabinet, a minority government with support by Wilders. This cabinet failed and collapsed, and at the subsequent elections in 2012 Wilders got 15 seats.


There is a major problem with D66

The major problem with D66 is that its party elite and its voters cannot think straight. The name D66 is an abbreviation of “Democrats 1966”, and the idea of founder Hans van Mierlo (1931-2010) was to improve democracy. Van Mierlo was from the Catholic south of Holland, and he was inspired by JFK in the USA. (See my weblog text on the Dutch Taliban.) Thus he suggested that Holland copied democratic conventions from the USA, like district voting, direct elected president and mayors, and referenda. Unfortunately, Van Mierlo had a degree in law and worked as a journalist, and he never really studied democracy. The membership of D66 are mostly lawyers too. They are mostly concerned about the “rule of law”, and less about what the law is about. By now, it should be obvious that Van Mierlo’s ideas about democracy have always been perverse, and actually reduce democracy. Yet, D66 doesn’t openly say so, and they still claim that they and their proposals would improve democracy. Thus D66 is a fossilised lie about democracy.

  • Direct elections with districts causes that in the Bush, Gore and Nader elections, Bush got elected (and we got the lie on Iraq), and that with the Clinton & Trump election, that Trump got elected, while in terms of percentages Gore would have beaten Bush, and Clinton would have beaten Trump.
  • For referenda, see this discussion about Brexit.

See my book Voting theory for democracy and this article about multiple seats elections.

Thus, when D66 collapsed to 3 seats, I hoped that D66 would be abolished, and that there would be room for a new political initiative, to combine sound ideas about democracy with sound ideas about economics and sound ideas about social compassion. Yet, there was Pechtold. He has a degree in art history and a working background as auctioneer, and developed further as a career politician. D66 apparently allows it, and eventually is grateful to him for “saving the party”, as if that would be so useful.

From disaster into greater catastrophy

D66 has been applying its great logical capacities, that they already showed on democracy, also on the issue of Wilders and immigration. Supposedly Pechtold attacked Wilders, but he actually made him bigger. D66 and Pechtold cannot see this fact and this logic, since Pechtold “saved D66” by that jump from 3 to 10 seats. Clearly the attack by Pechtold on Wilders was a great success, namely see the growth of D66 ! Thus they keep themselves deliberately blind about that jump of Wilders from 9 to 24 seats.

The best answer to Wilders would be a party that combines sound ideas about democracy with sound ideas about economics and sound ideas about social compassion. Yet, Pechtold and D66 block this, because of their perverse ideas about democracy and their perverse claim that they have success in attacking Wilders.

Well, it is Holland. Boycott this country till it develops a respect for science so that it lifts the censorship of science since 1990 by the directorate of the Dutch Central Planning Bureau (CPB).

Robert Siegler participates in the “Center for Improved Learning of Fractions” (CILF) and was chair of the IES 2010 research group “Developing Effective Fractions Instruction for Kindergarten Through 8th Grade” (report) (video).

IES 2010 key advice number 3 is:

“Help students understand why procedures for computations with fractions make sense.”

The first example of this helping to understand is:

“A common mistake students make when faced with fractions that have unlike denominators is to add both numerators and denominators. [ref 88] Certain representa­tions can provide visual cues to help students see the need for common denominators.” (Siegler et al. (2010:32), refering to Cramer, K., & Wyberg, T. (2009))

For a / b “and” c / d kids are supposed to find (ad + bc) / (bd) instead of (a + c) / (b + d).

Obviously this is a matter of definition. For “plus” we define: a / b + c / d = (ad + bc) / (bd).

But we can also define “superplus”: a / c / d =  (a + c) / (b + d).

The crux lies in “and” that might not always be “plus”.

When (a + c) / (b + d) makes sense

There are cases where (a + c) / (b + d) makes eminent sense. For example, when a / b is the batting average in the Fall-Winter season and c / d the batting average in the Spring-Summer season, then the annual (weighted) batting average is exactly (a + c) / (b + d). Kids would calculate correctly, and Siegler et al. (2010) are suggesting that the kids would make a wrong calculation ?

The “superplus” outcome is called the “mediant“. See a Wolfram Demonstrations project case with batting scores.

Adding up fractions of the same pizza thus differs from averaging over more pizzas.

We thus observe:

  • Kids live in a world in which (a + c) / (b + d) makes eminent sense.
  • Telling them that this is “a mistaken calculation” is actually quite confusing for them.
  • Thus it is better teaching practice to explain to them when it makes sense.

There is no alternative but to explain Simpson’s paradox also in elementary school. See the discussion about the paradox in the former weblog entry. The issue for today is how to translate this to elementary school.

Cats and Dogs

Many examples of Simpson’s paradox have larger numbers, but the Kleinbaum et al. (2003:277) “ActivEpi” example has small numbers (see also here). I add one more to make the case less symmetrical. Kady Schneiter rightly remarked that an example with cats and dogs will be more appealing to students. She uses size (small or large pets) as a factor, but let me stick to the idea of gender as a confounder. Thus the kids in class can be presented with the following case.

  • There are 17 cats and 16 dogs.
  • There are 17 pets kept in the house and 16 kept outside.
  • There are 17 male pets and 16 female pets (perhaps “helped”).

There is the phenomenon – though kids might be oblivious why this might be “paradoxical”:

  1. For the male pets, the proportion of cats in the house is larger than the proportion for dogs.
  2. For the female pets, the proportion of cats in the house is larger than the proportion for dogs.
  3. For all pets combined, the proportion of cats in the house is smaller than the proportion for dogs.
The paradoxical data

The paradoxical data are given as follows. Observe that kids must calculate:

  • For the cats: 6 / 7 = 0.86, 2 / 10 = 0.20 and (6 + 2) / (7 + 10) = 0.47.
  • For the dogs: 8 / 10 = 0.80, 1 / 6 = 0.17 and (8 + 1) / (10 + 6) = 0.56.


A discussion about what this means

Perhaps the major didactic challenge is to explain to kids that the outcome must be seen as “paradoxical”. When kids might not have developed “quantitative intuitions” then those might not be challenged. It might be wise to keep it that way. When data are seen as statistics only, then there might be less scope for false interpretations.

Obviously, though, one would discuss the various views that kids generate, so that they are actively engaged in trying to understand the situation.

The next step is to call attention to the sum totals that haven’t been shown above.


It is straightforward to observe that the M and F are distributed in unbalanced manner.

The correction

It can be an argument that there should be equal numbers of M and F. This causes the following calculations about what pets would be kept at the house. We keep the observed proportions intact and raise the numbers proportionally.

  • For the cats: 0.86 * 10 = 9, and (9 + 2) / (10 + 10) = 0.55.
  • For the dogs: 0.17 * 10 = 2, and (8 + 2) / (10 + 10) = 0.50.

And now we find: Also for all pets combined, the proportion of cats in the house is larger than the proportion for dogs. Adding up the subtables into the grand total doesn’t generate a different conclusion on the proportions.


Closure on causality

Perhaps kids at elementary school should not bothered with discussions on causality, certainly not on a flimsy case as this. But perhaps some kids require closure on this, or perhaps the teacher does. In that case the story might be that the kind of pet is the cause, and that the location where the pet is kept is the effect. When people have a cat then they tend to keep it at home. When people have a dog then are a bit more inclined to keep it outside. The location has no effect on gender. The gender of the pet doesn’t change by keeping it inside or outside of the house.

Vectors in elementary school

Pierre van Hiele (1909-2010) explained for most of his professional life that kids at elementary school can understand vectors. Thus, they should be able to enjoy this vector graphic by Alexander Bogomolny.

Van Hiele also proposed to abolish fractions as we know them, by replacing y / x by y x^(-1). The latter might be confusing because kids might think that they have to subtract something. But the mathematical constant H = -1 makes perfect sense, namely, check the unit circle and the complex number i. Thus we get y / x = y xH. The latter would be the better format. See A child wants nice and no mean numbers(2015).


Some conclusions are:

  • What Siegler & IES 2010 call a “common mistake” is the proper approach in serious statistics.
  • Teaching can improve by explaining to kids what method applies when. Adding fractions of the same pizza is different from calculating a statistical average. (PM. Don’t use round pizza’s. This makes for less insightful parts.)
  • Kids live in a world in which statistics are relevant too.
  • Simpson’s paradox can be adapted such that it may be tested whether it can be discussed in elementary school too.
  • The discussion corroborates Van Hiele’s arguments for vectors in elementary school and the abolition of fractions as we know them (y / x) and the use of y xH with H = -1. The key thing to learn is that there are numbers xH such that x xH = 1 when x ≠ 0, and the rest follows from there.

PM. The excel sheet for this case is: 2017-01-30-data-from-kleinbaum-2003

Econometrics researches the economy, using mathematical models and statistical data. For me as an econometrician the important relations are given by the causality in economics. The observed causality is put into the model. The model explains what we think that the causal chains are. Statistics can only give correlation. Thus, there is a tension between what is required for economic analysis and what statistics can provide. Different models may meet with the same data, which means that they would be observationally equivalent, yet, they would still be different models with different assumptions on causality.

Judea Pearl in his wonderful book “Causality” (1ste edition 2000, my copy 2007) of which there now is a 2nd edition, took issue with statistics, and looked for a way to get from correlation to causality. His suggestion is the “do”-statement. I am still pondering about this. For now I tend to regard it as manipulating in models with endogeneity and exogeneity of variables. Please allow me my pondering: some issues require time. See here for an earlier suggestion on causality, one on the counterfactual, and one on confounding. Some earlier papers on the 2 x 2 x 2 case are here. Today I want to look a bit at Simpson’s paradox with an eye on education.

The order of presentation in tables

In graphs, the horizontal x axis gives the cause and the vertical y axis gives the effect. For the derivative we look at dy / dx. Thus in numerical tables we better put the y in the top row and the x in the bottom row.

For 2 x 2 tables the lowest row is the sum of the rows above. Since this lowest row better be the cause, we thus better put the cause in vertical columns and the effect in horizontal rows. This seems a bit of a paradox, but see the presentation below.

(This is similar to when we have the true state (disease) (gold standard) vertically and the test statistic (test) in the rows, when we determine the sensitivity and specificity of the test. Check the wikipedia “worked example“, since the main theory is transposed.)

Pearl (2013) “Understanding Simpson’s Paradox” (technical report R-414) has a transposed table. It is better to transpose back. He also mentions the combined group first but it seems better to put this at the end. (PM. A recent discussion by Pearl on Simpson’s paradox is here.)

Pearl’s data example (transposed)

The following are the data from Pearl (2013), the appendix, figure 4, page 10. The data are the count of the individuals involved. Both men and women are treated (cause) or not, and they recover (effect) or not. Since this is a controlled trial, we do not need to look at prevalence and such.

When we divide the effect (row 1) by the total (row 3) then we get the recovery rates (row 4). We do this for the men, women and joint (combined, pooled) data. We find the paradoxical situation:

  • For the men, the treatment causes reduced recovery (0.6 < 0.7).
  • For the women, the treatment causes reduced recovery (0.2 < 0.3).
  • For all combined, the treatment causes improved recovery (0.5 > 0. 4).
Judea Pearl (2013) figure 4

Judea Pearl (2013) figure 4

More models that are statistically equivalent

We may arrange issues in “cause” and “effect”, but the real relations are determined by reality. Data like these might be available for various models. Pearl (2013) figure 1 mentions more models, but let us consider cases (a) and (b). In the above we have been assuming model (a) on the left, with a path from cause to effect Y, in which variable Z (gender) is causally independent. Above data table however would also fit the format of model (b), in which variable Z (blood pressure) would not be independent, and might be confounding issues.

Perhaps the gender is actually confounding the situation in above table too ? The result of the table is so strange that we perhaps must revise our ideas about the causal relations that we have been assuming.

Pearl (2013), part of figure 1

Pearl (2013), part of figure 1

Pearl’s condition for causality

Pearl’s condition for causality is that “the drug has no effect on gender”, see p10 and his formula (7) (with there F rather than here Z). The above data show that there is an effect, or, when we e.g. look at the women, that Pr[Female | Cause] and Pr[Female | No cause] are different, and thus differ from the marginal probability Pr[Female].

In the table above, we compare line (7) of all women with line (11) of all patients. The women are only 25% of all treated patients and 75% of all untreated ones. Perhaps the treatment has no effect on gender, but the data would suggest otherwise.


It would be sufficient (not necessary) to adjust the subgroup sizes, such that there is “equal representation”. NB. Pearl refers here to the “sure thing principle” apparently formulated by Savage 1954, which condition doesn’t modify the distribution. For us, the condition and proof of equal representation has another relevance now.

Application of the condition gives a correction

Since this is a controlled trial, we can adapt by including more patients, such that the numbers in the different subgroups (rows (3) and (7), below in red) are equal. This involves 40 more patients, namely 20 men in the non-treatment group and 20 women in the treatment group. This generates the following table.

For ease, it is assumed that the conditional probabilities of the subgroups – thus rows (4) and (8) – remain the same, and that the new patients are distributed accordingly. Of course, they might deviate from this, but then we have better data anyway.


The consequence of including adequate numbers of patients in the subgroups is:

  • Row (13) now shows that Pr[Z | C] = Pr[Z | Not-C ] = Pr[Z], for Z = M or F.
  • As the treatment is harmful in both subgroups, it also is harmful for the pooled group.
Intermediate conclusion

Obviously, when the original data already allow an estimate of the harmful effect, it would not be ethical to subject 20 more women to the treatment – while it might be easy to find 20 more men who don’t have the treatment. Thus, it suffices to use the above as a statistical correction only. If we assume the same conditional probabilities w.r.t. the cause-effect relation in the subgroups, then the second table gives the counterfactual as if the subgroups had the same number of patients. There would be no occurrence of the Simpson paradox.

This counterfactual would also hold in cases when we cannot simply adjust the group sizes, like the classic case of admissions of students to Berkeley.

While the causality that the drug has no effect on gender is quite clear, the situation is less obvious w.r.t. the issue on blood pressure. In this case it might not be possible to get equal numbers in the subgroups. Not for ethical reasons but because people react differently on the treatment. This case would require a separate discussion, for the causality clearly is different.

Educational software on Simpson’s paradox

There are some sites for a first encounter with Simpson’s paradox.

A common plot is labelled Baker – Kramer 2001 but earlier were Jeon – Chung – Bae 1987. This plot keeps the number of men and women and the conditional probabilities the same, and allows only variation over the enrollments in the subgroups. This nicely shows the composition effect. The condition of equal percentages per subgroup works, but there are also other combinations that avoid Simpson’s paradox. But of course, Pearl is interested in causality, and not the mere statistical effect of composition.

The most insightful plot seems to be from vudlabIt has upward sloping lines rather than downward sloping ones, which somewhat seems easier to follow. There is a (seemingly) continuous slider, it rounds the person counts, and it has a graphic for the percentages that makes it easier to focus on those.

Kady Schneiter has various applets on statistics, of which this one on Simpson’s paradox. I agree with her discussion (Journal of Statistics Education 2013) that an example with pets (cats and dogs) lowers the barrier for understanding. Perhaps we should not use the size of the pet (small or large) but still gender. The plot uses downward sloping lines and has an unfortunate lag in the display of the light blue dot. (This might be dogs, but we can also compare with the Berkeley case in vudlab.)

The Wolfram Demonstrations by (1) Heiner & Wagon and (2) Brodie provide different formats that may come into use too. The advantage of the latter is that you can put in your own numbers.

This discussion by Andrew Gelman caused me to google on these displays.

Alexander Bogomolny has a fine vector display but there is no link to causality (yet).

Robert Banis has some data from the original Berkeley study, and excel sheets using them.

Some ten years ago there would have been more references to excel sheets indeed, with the need for students to do some editing themselves. The educational attention apparently shifts to applets with sliders. For those with still an interest in excel, the sheet with above tables is here: 2017-01-28-data-from-pearl-2000.

And of course there is wikipedia (a portal, no source). (Students from MIT are copying their textbooks into wikipedia, whence the portal becomes unreadable for the common reader. It definitely cannot be used as an educational source.)


This sets the stage for another kind of discussion in the next weblog entry.

Exponential functions have the form bx, where b > 0 is the base and x the exponent.

Exponential functions are easily introduced as growth processes. The comparison of x² and 2^x is an eye-opener, with the stories of duckweed or the grain on the chess board. The introduction of the exponential number e is a next step. What intuitions can we use for smooth didactics on e ?

The “discover-e” plot

There is the following “intuitive graph” for the exponential number e = 2,71828…. The line y = e is found by requiring that the inclines (tangents) to bx all run through the origin at {0, 0}. The (dashed) value at x = 1 helps to identify the function ex itself. (Check that the red curve indicates 2^x).

Functions 2^x, e^x and 4^x, and tangents through {0, 0}

2^x, e^x and 4^x, and inclines through {0, 0}

Remarkably, Michael Range (2016:xxix) also looks at such an outcome = 2^(1 / c), where is the derivative of = 2^x at x = 0, or c = ln[2]. NB. Instead of the opaque term “logarithm” let us use “recovered exponent”, denoted as rex[y].

Perhaps above plot captures a good intuition of the exponential number ? I am not convinced yet but find that it deserves a fair chance.

NB. Dutch mathematics didactician Hessel Pot, in an email to me of April 7 2013, suggested above plot. There appears to be a Wolfram Demonstrations Project item on this too. Their reference is to Helen Skala, “A discover-e,” The College Mathematics Journal, 28(2), 1997 pp. 128–129 (Jstor), and it has been included in the “Calculus Collection” (2010).


The point-slope version of the incline (tangent) of function f[x] at x = a is:

y – f[a] = s (x a)

The function b^x has derivative rex[b] b^x. Thus at arbitrary a:

y – b^a = rex[b] b^a (x a)

This line runs through the origin {xy} = {0, 0} iff

0 – b^a = rex[b] b^a (0 – a)

1 = rex[ba

Thus with H = -1, a = rex[b]H = 1 / rex[b]. Then also:

yf[a] = b^a = b^rex[b]H = e^(rex[b]  rex[b]H) = e^1 = e

The inclines running through {0, 0} also run through {rex[b]H, e}. Alternatively put, inclines can thus run through the origin and then cut y = e .

For example, in above plot, with 2^x as the red curve, rex[2] ≈ 0.70 and ≈ 1.44, and there we find the intersection with the line y = e.

Subsequently also at a = 1, the point of tangency is {1, e}, and we find with e that rex[e] = 1,

The drawback of this exposition is that it presupposes some algebra on e and the recovered exponents. Without this deduction, it is not guaranteed that above plot is correct. It might be a delusion. Yet since the plot is correct, we may present it to students, and it generates a sense of wonder what this special number e is. Thus it still is possible to make the plot and then begin to develop the required math.

Another drawback of this plot is that it compares different exponential functions and doesn’t focus on the key property of e^x, namely that it is its own derivative. A comparison of different exponential functions is useful, yet for what purpose exactly ?


Our recent weblog text discussed how Cartesius used Euclid’s criterion of tangency of circle and line to determine inclines to curves. The following plots use this idea for e^x at point x = a, for a = 0 and a = 1.

Incline to e^x at x = 0 (left) and x = 1 (right)

Incline to e^x at x = 0 (left) and x = 1 (right)

Let us now define the number e such that the derivative of e^x is given by e^x itself. At point x = a we have s = e^a. Using the point-slope equation for the incline:

y – f[a] = s (x a)

y – e^ae^a (x a)

y e^a (x – (a – 1))

Thus the inclines cut the horizontal axis at {x, y} = {a – 1, 0}, and the slope indeed is given by the tangent s = (f[a] – 0) / (a – (a – 1)) = f[a] / 1 = e^a.

The center {u, 0} and radius r of the circle can be found from the formulas of the mentioned weblog entry (or Pythagoras), and check e.g. a = 0:

u = a + s f[a] = a + (e^a

r = f[a] √ (1 + s²) = e^a √ (1 + (e^a)²)

A key problem with this approach is that the notion of “derivative” is not defined yet. We might plug in any number, say e^2 = 10 and e^3 = 11. For any location the Pythagorean Theorem allows us to create a circle. The notion of a circle is not essential here (yet). But it is nice to see how Cartesius might have done it, if he had had e = 2.71828….

Conquest of the Plane (COTP) (2011)

Conquest of the Plane (2011:167+), pdf online, has the following approach:

  • §12.1.1 has the intuition of the “fixed point” that the derivative of e^x is given by e^x itself. For didactics it is important to have this property firmly established in the minds of the students, since they tend to forget this. This might be achieved perhaps in other ways too, but COTP has opted for the notion of a fixed point. The discussion is “hand waiving” and not intended as a real development of fixed points or theory of function spaces.
  • §12.1.2 defines e with some key properties. It holds by definition that the derivative of e^x is given by e^x itself, but there are also some direct implications, like the slope of 1 at x = 0. Observe that COTP handles integral and derivative consistently as interdependent notions. (Shen & Lin (2014) use this approach too.)
  • §12.1.3 gives the existence proof. With the mentioned properties, such a number and function appears to exist. This compares e^x with other exponential functions b^x and the recovered exponents rex[y] – i.e. logarithm ln[y].
  • §12.1.4 uses the chain rule to find the derivatives of b^x in general. The plot suggested by Hessel Pot above would be a welcome addition to confirm this deduction and extension of the existence proof.
  • §12.1.5-7 have some relevant aspects that need not concern us here.
  • § shows that the definition is consistent with the earlier formal definition of a derivative. Application of that definition doesn’t generate an inconsistency. No limits are required.
  • § gives the numerical development of = 2.71828… There is a clear distinction between deduction that such a number exists and the calculation of its value. (The approach with limits might confuse these aspects.)
  • § shows that also the notion of the dynamic quotient (COTP p57)  is consistent with above approach to e. Thus, the above hasn’t used the dynamic quotient. Using it, we can derive that 1 = {(e^h – 1) // h, set h = 0}. Thus the latter expression cannot be simplified further but we don’t need to do so since we can determine that its value is 1. If we would wish so, we could use this (deduced) property to define e as well (“the formal approach”).

The key difference between COTP and above “approach of Cartesius” is that COTP shows how the (common) numerical development of e can be found. This method relies on the formula of the derivative, which Cartesius didn’t have (or didn’t want to adopt from Fermat).

Difference of COTP and a textbook introduction of e

In my email of March 27 2013 to Hessel Pot I explained how COTP differed from a particular Dutch textbook on the introduction of e.

  • The textbook suggests that f ‘[0] = 1 would be an intuitive criterion. This is only partly true.
  • It proceeds in reworking f ‘[0] = 1 into a more general formula. (I didn’t mention unstated assumptions in 2013.)
  • It eventually boils down to indeed positing that e^x has itself as its derivative, but this definition thus is not explicitly presented as a definition. The clarity of positing this is obscured by the path leading there. Thus, I feel that the approach in COTP is a small but actually key innovation to explicitly define e^x as being equal to its derivative.
  • It presents e only with three decimals.

There are more ways to address the intuition for the exponential number, like the growth process or the surface area under 1 / x. Yet the above approaches are more fitting for the algebraic approach. Of these, COTP has a development that is strong and appealing. The plots by Cartesius and Pot are useful and supportive but no alternatives.

The Appendix contains a deduction that was done in the course of writing this weblog entry. It seems useful to include it, but it is not key to above argument.

Appendix. Using the general formula on factor x a

The earlier weblog entry on Cartesius and Fermat used a circle and generated a “general formula” on a factor x a. This is not really factoring, since the factor only holds when the curve lies on a circle.

Using the two relations:

f[x] – f[a]  = (x a)  (2u – x – a) / (f[x] + f[a])    … (* general)

u = a + s f[a]       … (for a tangent to a circle)

we can restate the earlier theorem that s defined in this manner generates the slope that is tangent to a circle. 

f[x] – f[a]  = (x a)  (2 s f[a](x – a)) / (f[x] + f[a]) 

It will be useful to switch to x a = h:

f[a + h] – f[a]  = h (2 s f[a] – h) / (f[a + h] + f[a]) 

Thus with the definition of the derivative via the dynamic quotient we have:

df / dx = {Δf // Δx, set Δx = 0}

= {(f[a + h] – f[a]) // h, set h = 0}

= { (2 s f[a] – h) / (f[a + h] + f[a]), set h = 0}

= s

This merely shows that the dynamic quotient restates the earlier theorem on the tangency of a line and circle for a curve.

This holds for any function and thus also for the exponential function. Now we have s = e^a by definition. For e^x this gives:

ea + hea  = h (2 s eah) / (ea + h + ea)

For COTP § we get, with Δx = h:

df / dx = {Δf // Δx, set Δx = 0}

= {(ea + hea  ) // h, set h = 0}

= {(2 s eah) / (ea + h + ea) , set h = 0}

= s

This replaces Δf // Δx by the expression from the general formula, while the general formula was found by assuming a tangent circle, with s as the slope of the incline. There is the tricky aspect that we might choose any value of s as long as it satisfies u = a + s f[a]. However, we can refer to the earlier discussion in § on the actual calculation.

The basic conclusion is that this “general formula” enhances the consistency of § The deduction however is not needed, since we have §, but it is useful to see that this new elaboration doesn’t generate an inconsistency. In a way this new elaboration is distractive, since the conclusion that 1 = {(e^h – 1) // h, set h = 0} is much stronger.

A number is what satisfies the axioms of its number system. For elementary and secondary education we use the real numbers R. It suffices to take their standard form as: sign, a finite sequence of digits (not starting with zero unless there is a single zero and no other digits), a decimal point, and a finite or infinite sequence of digits. We also use the isomorphism with the number line.

Thus a limited role for group theory

Group theory creates different number systems, from natural numbers N, to integers Z, to rationals Q, to reals R, and complex plane C, and on to higher dimensions. For elementary and secondary education it is obviously useful to have the different subsets of R. But we don’t do group theory, for the notion of number is given by R.

It should be possible to agree on this (*):

  1. that N ⊂ Z ⊂ Q R,
  2. that the elements in R are called numbers,
  3. whence the elements in the subsets are called numbers too.

Timothy Gowers has an exposition, though with some group theory , and thus we would do as much group theory as Gowers needs. There is also my book Foundations of mathematics. A neoclassical approach to infinity (FMNAI) (2015) (pdf online) so that highschool students need not be overly bothered by complexities of infinity. FMNAI namely distinguishes:

  • potential infinity with the notion of a limit to infinity
  • actual infinity created by abstraction, with the notion of “bijection by abstraction”.

There arises a conceptual knot. When A is a subset of B, or A ⊂ B, then saying that x is in A implies that it is in B, but not necessarily conversely. Who focuses on A, and forgets about B, may protest against a person who discusses B. When we say that the rational numbers are “numbers” because they are in R, then group theorists might protest that the rationals are “only” numbers because (1) Q is an extension of Z by including division, and (2) then we decide that these can be called “number” too. Group theorists who reason like this are advised to consider the dictum that “after climbing one can throw the ladder away”. In the real world there are points of view. When Putin took the Crimea, then his argument was that it already belonged to Russia, while others called it an annexation. In mathematics, it may be that mathematicians are people and have their own personal views. Yet above (*) should be acceptable.

It should suffice to adopt this approach for primary and secondary education. Research mathematicians are free to do what they want at the academia, but let they not meddle in this education.

Division as a procept

The expression 1 / 2 represents both the operation of division and the resulting number. This is an example of the “procept“, the combination of process and concept.

The procept property of y / x is the cause of a lot of confusion. The issue has some complexity of itself and we need even more words to resolve the confusion. Wikipedia (a portal and no source) has separate entries for “division“, “quotient“, “fraction“, “ratio“, “proportionality“.

In my book Conquest of the Plane (COTP) (2011), p47-58, I gave a consistent nomenclature (pdf online):

“Ratio is the input of division. Number is the result of division, if it succeeds.” (COTP p51)

This is not a definition of number but a distinction between input and output of division. My suggestion is to use the word (static) quotient also for the form with numerator y divided by denominator x.

(static) quotient[y, x] = y / x

This fits the use in calculus of “difference and differential quotients”. The form doesn’t have to use a bar. Also a computer statement Div[numerator y, denominator x] would be a quotient.

This suggestion differs a bit from another usage in which the quotient would be the outcome of the division process, potentially with a remainder. We saw this usage for the polynomials. This convention is not universal, see the use of “difference quotient”. However, if there would be confusion between outcome and form, then use “static quotient” for the form. This is in opposition to the dynamic quotient that is relevant for the derivative, as Conquest of the Plane shows.

Proportionality and number

Check also the notion of proportionality in COTP, page 77-78 with the notion of proportion space: {denominator x, numerator y}. Division as a process is a multidimensional notion. The wikipedia article (of today) on proportionality fits this exposition, remarkably with also a diagram of proportion space, with the denominator (cause) on the horizontal axis and the numerator (effect) on the vertical axis (instead of reversed), as it should be because of the difference quotient in calculus. In Conquest of the Plane there is also a vertical line at x = 1, where the numerators give our numbers (a.k.a. slope or tangent).

Conquest of the Plane, p78

Conquest of the Plane, p78

Avoiding the word “fraction”

My nomenclature uses the quotient and the distinction in subsets of numbers, and I tend to avoid the word fraction because of apparent confusions that people have. When someone gives a potential confusing definition of fractions, my criticism doesn’t consist of providing a proper definition for fractions, but I point out the confusion, and then refer to the above.

Below, I will also refer to the suggestion by Pierre van Hiele (1973) to abolish fractions (i.e. what people call these), and I will mention a neat trick that provides a much better alternative.

Number means also satisfying a standard form

Number means also satisfying a standard form. Thus “number” is not something mysterious but is a form, like the other forms, yet standardised.

For example, we have 2 / 4 = 1 / 2, yet 1 / 2 has the standard form of the rationals so that 2 / 4 needs to be simplified by eliminating common prime factors. The algebra of 2 / (2 2) = 1 / 2 can be seen as “rewriting the form”.

What the standard is, depends upon the context. We can do sums on natural numbers, integers, rationals, reals. In education students have to learn how to rewrite particular forms into a particular standard. Student need to know the standard forms, not the group theory about the subset of numbers they are working in.

The equality sign in a is ambiguous. Computer algebra tends to avoid ambiguity. For example in Mathematica: Set (=) vs Equal (==) vs (identically) SameQ (===). Doing computer algebra would help students to become more precise, compared to current textbooks. Learning is going from vague to precise.

The equality sign in highschool tends to mean “of equal value”, which is above “==”. But two expressions can only be of equal value when they represent the identically same value. Thus x == a would amount to Num[x] === Num[a]. The standard mathematical phrase is “equivalence class” for a number in whichever format, e.g. with the numerical value at the vertical position at line at x = 1 (also for the denominator 1).

The standard form takes one element of an “equivalence class” (depending upon the context of what numbers are on the table, e.g. 1 / 2 for the rationals and 0.5 for the reals). (See COTP p45-48 for issues of “approximation”.)

Multiplication is no procept

Multiplication is no procept. For multiplication there is a clear distinction between the operation 2 * 3 and the resulting number 6. When your teacher asks you to calculate 2 * 3 then the answer of 2 * 3 is correct but likely not accepted. The smart-aleck answer 2 * 3 = 3 * 2 is also correct, but then the context better be group theory.

It is a pity that group theory adopted the name “group theory”. My proposal for elementary school is to replace the complicated word “multiplication” by “group, grouping”. With 12 identical elements, you can make 4 groups of 3. (With identical elements this isn’t combinatorics.) See A child wants nice and no mean numbers (CWNN) (2015). If this use of “group, grouping” is confusing for group theory, then they better change to something like “generalised arithmetic”.

The hijack of number by group theory

The world originally had the notion of number, like counting fingers or measuring distance, but then group theory hijacked the word, and assigned it with a generalised meaning, whence communication has become complicated. Their use of language might cause the need for the term numerical value. I would like to say that 2 is identically the same number in N, Z, Q and R, but group theorists tend to pedantically assert that the notion of number is relative to the set of axioms. In the Middle Ages, people didn’t know negative numbers, and they couldn’t even think about -2. Only by defining -2 as a number too, it could be included as a number. This sounds like Baron von Muenchhausen lifting himself from the swamp. The answer to this is rather that -2 is still a number even though it wasn’t recognised as this. I would like to insist that we use the term “number” for the numerical value in R, so that we can use the word “number” in elementary school in this safe sense. Group theorists then must invent a word of their own, e.g. “generalised number” or “gnumber”, for their systems.

Changing the meaning of words is like that your car is stolen, given another colour, and parked in front of your house as if it isn’t your car. Group theorists tend to focus on group theory. They tend not to look at didactics and teaching. When group theorists hear teachers speaking about numbers, and how 2 is the same number in N and R, then group theorists might smile arrogantly, for they “know better” that N and R are different number systems. This would be misplaced behaviour, for it are the group theorists themselves who hijacked the notion of number and changed its meaning. When research mathematicians have the idea that teachers of mathematics have no training about group theory, then they better read Richard Skemp (1971, 1975), The psychology of learning mathematics, first. This was written with an eye on teaching mathematics (and training teachers) and contains an extensive discussion of group theory. (Though I don’t need to agree with all that Skemp writes.)

Quote on human folly

Peter van ‘t Riet edited Vredenduin (1991) “De geschiedenis van positief en negatief“, Wolters-Noordhoff, on the history of positive and negative numbers. Van ‘t Riet allows himself a concluding observation:

“Kijken wij er achteraf op terug, dan kan een gevoel van verwondering opkomen, dat begrippen die ons zo vanzelfsprekend en helder lijken, zo’n lange ontwikkelingsgeschiedenis hebben gehad waarin vooruitgang, terugval en nieuwe vooruitgang elkaar afwisselden. Opmerkelijk is dat begrippen zich soms pas echt ontwikkelen als zij bevrijd worden van een dominerende idee die eeuwenlang hun ontwikkeling in de weg stond. Dat is bij de negatieve getallen het geval geweest met de geometrisering van de algebra: de gedachte dat getallen representanten waren van meetkundige grootheden is eeuwen achtereen een obstakel geweest teneinde tot een helder begrip van negatieve getallen te komen. Achteraf vraag men zich af: hoe was het mogelijk dat eeuwenlang deze idee de algebra bleef domineren?” (p121)

Since we sometimes check Google Translate for the fun ways of its expressions, it is nice to let the machine speak again:

If we look afterwards back, then bring up a sense of wonder that concepts which seem to us so obvious and clear, have had such a long history in which progress, relapse and further progress alternating. Remarkably concepts sometimes only really develop as they freed from a dominant idea that for centuries had their development path that is in the negative numbers was the case with the geometrization of algebra:. the idea that numbers representatives were of geometric quantities is centuries successively been an obstacle in order to achieve a clear understanding of negative numbers retrospect one question himself:. how was it possible that for centuries the idea continued to dominate the algebra?” (Google Translate)

Just to be sure: analytic geometry has the number line with negative numbers too. Van ‘t Riet means the line section, that always has a nonnegative length.

A step to answering his question is that mathematicians focus on abstraction, whence they are more guided by their own concepts rather than by empirical applications or the observations in didactics. I included this quote in the hope that group theorists reading this will again grow aware of human folly, and realise that they should support empirical didactics and not block it.

More sources for confusion on formats

More noise is generated by the different “number formats” that have been developed over the course of history. We have forms 2 + ½ = 2½ = 5 / 2 = 25 / 10 = 2.5 = 2 + 2-1 (neglecting the Egyptians and such). We should not forget that the decimals are actually also a form or result of division. Another example is 0.365 = 3 / 10 + 6 / 100 + 5 / 1000. Only the infinite decimals present a problem, since then we need an infinite series of divisions, yet this can be solved. The various formats have their uses, and thus education must teach students what these are.

An approach might be to only use numbers in decimal notation. However, the expression 1 / 3 is often easier than 0.33333…. Students must learn algebra. Compare 1 / 2 + 1 / 3 with 1 / a + 1 / b.

“But to understand algebra without ever really understood arithmetic is an impossibility, for much of the algebra we learn at school is a generalized arithmetic. Since many pupils learn to do the manipulations of arithmetic with a very imperfect understanding of the underlying principles, it is small wonder that mathematics remain a closed book to them.” (Skemp, p35)

The KNAW 2009 study on arithmetic education and its evidence and research is invalid. It forgot that pupils in elementary school have to learn particular algorithms in arithmetic in preparation for algebra in secondary education. It scored answers to sums as true / false and didn’t assign points to the intermediate steps, so that pupils who used trial and error also had the option to score well. In a 2011 thesis on the psychometrics of arithmetic, the word “algebra” isn’t mentioned, and various of its research results are invalid. There is a rather big Dutch drama on failure of education on arithmetic, failure of supervision, and breaches of integrity of science.

Irrational numbers started as a ratio. Consider a triangle with perpendicular sides 1 and then consider the ratio of the hypothenuse to one of those sides. The input √2 : 1 reduces to number √2.

Standard form for the rationals

There are students who do 2 + ½ = 2½ = 2 ½ = 1, because in handwriting there might appear to be a space that indicates multiplication, compare 2a or 2√2 or 2 km where such a space can be inserted without problem. See the earlier weblog text how Jan van de Craats tortures students. A proposal of mine since 2008 is to use 2 + ½ and stop using 2½.

Yesterday I discovered Poisard & Barton (2007) who compare the teaching of fractions in France and New Zealand, and who also advise 2 + ½. The German wikipedia has also a comment on the confusing notation of 2½. I haven’t looked at the thesis by Rollnik yet.

For a standard form for the rationals, the rules are targeted at facilitating the location on the number line, while we distinguish the operation minus from the sign of a negative number (as -2 = negative 2).

  1. If a rational number is equal to an integer, it is written as this integer, and otherwise:
  2. The rational number is written as an integer plus or minus a quotient of natural numbers.
  3. The integer part is not written when it is 0, unless the quotient part is 0 too (and then the whole is the integer 0).
  4. The quotient part has a denominator that isn’t 0 or 1.
  5. The quotient part is not written when the numerator is 0 (and then the whole is an integer).
  6. The quotient part consists of a quotient (form) with an (absolute) value smaller than 1.
  7. The quotient part is simplified by elimination of common primes.
  8. When the integer part is 0 then plus is not written and minus is transformed into the negative sign written before the quotient part.
  9. When the integer part is nonzero then there is plus or minus for the quotient part in the same direction as the sign of the integer part (reasoning in the same direction).

Thus (- 2 – ½) = (-3 + ½) but only the first is the standard form.

PM 1. Mathematica has the standard form 5 / 2. Conquest of the Plane p54 provides the routine RationalHold[expr] that puts all Rational[x, y] in expr into HoldForm[IntegerPart[expr] + FractionalPart[expr]].

PM 2. Digits are combined into numbers, so that we don’t have 28 = 2 * 8 = 16 = 6. Nice is:

“For example, 7 (4 + a) is equal to 28 + 7a and no 74 + 7a.” (Skemp, p230)

H = -1

A new suggestion is to use = -1. Then we get 2 + ½ = 2 + 2H= 5 2H. Pierre van Hiele (1973) suggested to abolish fractions as we know them. He observed that y / x is a tedious notation, and students have to learn powers anyhow. I agree that the notation y / x generates so-called “mathematics” which is no real mathematics but only is forced by the notation. Using the power of -1 can be confusing because students might think of subtraction, but the use of (abstract) H for the inverse clinches it. See here and my sheets for a workshop of NVvW November 2016.

Above quotient form then becomes (y xH) and the dynamic quotient (y xD), in which the brackets may be required in the dynamic case to indicate the scope of the simplification process.

There are students who struggle with a – (-b) = a – (-1) b, perhaps because subtraction actually is a form of multiplication. Curiously, this is another issue of inversion that is made easier by using H, with a – (-b) = a H b = a + H H b = a + b. See the last weblog entry that division is repeated subtraction. The only requirement is that each number has also an inverse, zero excluded, so that these inverses can be subtracted too. For example 4 3H = (3 + 1) 3H = 1 + 3H translates as repeated subtraction (not for the classroom but for reasons of current exposition):

4 – (1 + 3H) – (1 + 3H) – (1 + 3H) = 4 – 3 (1 + 3H) = 4 – 3 – 3 3H = 4 – 3 – 1 = 0

Group theory is for numbers. It is not for education on number formats

The last weblog entry on group theory showed that group theory concentrates on numbers, whence it (cowardly) avoids the perils of education on the various number formats.

Group theory mathematicians will tend to say that 1 / 2 = 2 / 4 = 50 / 100 = .. .are all member of the same “equivalence class” of the number 1 / 2, whence their formats are no longer interesting and can be neglected.

In itself it is a laudable achievement that mathematics has developed a framework that starts with the natural numbers, extends with negative integers, develops the rationals, and finally creates the reals (and then more dimensions). This construction comes along with algorithms, so that we know what works and what doesn’t work for what kind of number. For example, there are useful prime numbers, that help for simplifying rationals. For example 3 * (1 / 3) = 1 whence 3 * 0.3333… = 0.9999… = 1.000… = 1. (Thus the decimal representation is not quite unique, and this is another reason to keep on using rational formats (when possible).)

When these group theory research mathematicians design a training course for aspiring teachers of mathematics, they tend to put most emphasis on group theory, and forget about the various number formats. This has the consequences:

  • Teachers from their training become deficient in knowledge about number formats (e.g. Timothy Gowers’s article), even though those are more relevant to teachers because these are relevant for their students.
  • There is also conditioning for a future lack of knowledge. The aspiring teachers are trained on abstraction and they will tend to grow blind on the problems that students have when dealing with the various formats.
  • All this supports the delusion:

“We should teach group theory so that the students will have less problems with the algebra w.r.t. the various number formats. (For, they can neglect much algebra, like we do, since most forms are all in the same equivalence classes.)” (No quote)

Bas Edixhoven chairs the delusion

Bas Edixhoven (Leiden) is chair of the executive board of Mastermath, a joint Dutch universities effort for the academic education of mathematicians. They also do remedial teaching for students who want to enroll into the regular training for teacher of mathematics but who have deficiencies in terms of mathematics. Think about a biologist who wants to become a teacher of mathematics. For those students the background in empirical science is important, because didactics is an empirical science too. Such students are an asset to education, and they should not be scared away by treating them as if they want to become research mathematicians. Obviously there are high standards of mathematical competence, but this standard is not the same as for doing research in mathematics.

  • The “Foundations” syllabus for remedial teaching 2015 written by Edixhoven indeed looks at group theory with the neglect of number formats. The term “fraction” (Dutch “breuk”) is used without definition, while there is also the expression “fraction form” (Dutch “breukvorm”). I get the impression that Edixhoven uses fraction and fraction format as identical. Perhaps he means the procept ? The fractions are not the rationals since apparently π / 2 has a fractional form too.
  • At a KNAW conference in 2014 on the education of arithmetic Edixhoven presented standard group theory, presumably thinking that his audience had never heard about it and hadn’t already decided that its role for non-university education is limited. Edixhoven insulted his audience (including me) by not first studying what didacticians like Skemp had already said before about group theory in education.

I find it quite bizarre that mathematics courses at university for training aspiring teachers would neglect the number formats and treat these (remedial) student-teachers as if they want to become research mathematicians. Obviously I cannot really judge on this since I am no research mathematician so that I don’t know what it takes to become one. I only know that I have a serious dislike of it. Yet, the group theory taught is out of focus for what would be helpful for mathematics for teaching mathematics.

PM 1. The Edixhoven 2014 approach at KNAW fits Van Hiele (1973) who also suggests to have a bit of group theory in highschool. Yet, there is the drawback of confusion about the power -1 that students might read as subtraction. I would agree on this idea of having some group theory, but with the use of H = -1 and not without it. Let us first introduce the universal constant H = -1, thus also in elementary school where pupils should learn about division, and then proceed with some group theory in junior highschool.

PM 2. Edixhoven wrote this “Foundations” syllabus together with Theo van den Bogaard who wrote his thesis with Edixhoven. Van den Bogaard has only a few years of experience as teacher of mathematics. Van den Bogaard was secretary of a commission cTWO that redesigned mathematics education in Holland, with a curious idea about “mathematical think activities” (MTA). Van den Bogaard has an official position as trainer of teachers of mathematics but failed to see the error by the psychometrians in the KNAW 2009 study on education on arithmetic. I informed him about my comments on cTWO, MTA and KNAW 2009 but he didn’t respond. Now there is the additional issue of this curious “Foundations” syllabus. Four counts down on didactics and still training aspiring teachers.

Letter to Mastermath

These and other considerations caused me to write this letter to Mastermath.

The following indicates that research mathematicians can have their own subgroups or individuals who meddle with education. None is qualified for education, and one wonders whether they can keep each other in check.

Research mathematicians are at a distance from didactics

Research mathematicians may develop a passion for education and interfere in education, and then start to invent their own interpretations, and then teach those to elementary schools and their aspiring teachers. These mathematicians are not qualified for primary education and apparently think that elementary school allows loose standards (since they can observe errors indeed). Then we get the blind (research mathematicians) helping the deaf (elementary school teachers), but the blind can also be arrogant, and lead the two of them into the abyss.

A September 2015 protest concerned Jan van de Craats, now emeritus at UvA. For the topic of division, his name pops up again. In this lecture on fractions for a workshop of 2010 for primary education Van de Craats for example argues as follows (my translation). It is unfair to have criticism on this since these are only sheets. Yet, even sheets should have a consistent set of definitions behind them. These sheets contribute to confusion. Remember that I didn’t give a definition of “fraction”, and that I propose an abolition of what many people apparently call “fraction”.

  • Sheet 3: “Three sorts of numbers: integers, decimals, fractions”.
    (a) The main problem is the word “sort”. If he merely means “form” (with the decimals as the standard form that gives “the” number) then this is okay, but if he means that there are really differences (as in group theory) then this is problematic. A professor of mathematics should try to be accurate, and I don’t see why Van de Craats regards “sorts of” as accurate.
    (b) If he identifies fractions with the rationals (but see sheet 26) then we might agree that Z Q ⊂ R, though there are group theorists who argue that these are different number systems, and it is not clear whether Van de Craats would ask the group theorists not to meddle in education as he himself is doing.
    (c) My answer: for education it seems best to stick to “various forms, one number (for standard form)”.
  • Sheet 30: “A fraction is the outcome of a division.”
    (a) As fraction is a number (Sheet 3), presumable 8 : 4 → 4 / 2 might be acceptable: (i) It is an outcome, (ii) the answer is numerically correct (as it belongs to the equivalence class), (iii) there is no requirement on a standard form (here).
    (b)This doesn’t imply the converse, that the outcome of a division is always a fraction. Then it is either an integer (but then also a fraction (Sheet 25)) or decimal (but then also fraction (Sheet 26)). Thus fraction iff outcome from division.
    (c) PM. My definition was: “Ratio is the input of division. Number is the result of division, if it succeeds.” (COTP p51), which doesn’t define number but distinguishes input and output.
  • Sheet 8: “Cito doesn’t test (mixed) fractions anymore in the primary school final examination.” As an observation this might be correct, but if Van de Craats had had proper background in didactics, then he should have been able to spot the error by the psychometricians in the KNAW 2009 report, which should have been sufficient to effect change, instead of setting up this “course in fractions” (that he isn’t qualified for).
  • Sheet 18: Pizza model. Didactics shows that students find this difficult. Use a rectangle.
  • Sheet 25: “Integers are also fractions (with denominator 1).” On form, students must know the difference between integers and fractions (whatever those might be, see Sheet 30). The answer of (3 – 1) / (2 – 1) = ? better be 2 and not 2 / 1 because the latter can be simplified.
  • Sheet 26: “Decimals are also fractions.” Thus fractions are not the rational numbers. The example is that √2 is irrational, also in decimal expansion (a “fraction”). Van de Craats apparently holds fractions and the decimals as identical, only written in different form. Thus also an infinite sum of fractions still is a fraction. A fraction is not just the form of the quotient as defined in Conquest of the Plane and above (though perhaps it can be written like this ?).
  • Sheet 27: “However, not all fractions are also decimals.” This is a mystery. There are only three “sorts of” numbers, and w.r.t. Sheet 30 we found that fraction iff division, and all numbers should be divisible by 1. Also, the real numbers contain all numbers we have seen till now (not the complex numbers). Thus there would be phenomena called “fractions” (but still numbers, not algebra) not in the reals ? It cannot be 0 / 0 since the latter would be a result that cannot be accepted. Division 0 : 0 might be a proper question with the answer that the result is undefined. Perhaps he means to say that “1 / 2” doesn’t have the form of “0.5”, and that the expressions differ ? But then we are speaking about form again, and Van de Craats spoke about “sorts of numbers” and not about “same numbers with different forms”.
  • Sheet 28: “This course doesn’t offer an one-to-one-model for discussion at school.” It sounds modest but I don’t know what this means. Perhaps he means that the sheets aren’t a textbook.
  • Sheet 30: “A fraction is the outcome of a division.”  (I moved this up.)
  • Sheet 33: “4 : 7 = 4 / 7”. Apparently the ” : ” stands for the operation of division and “4 / 7” for the result. Apparently Van de Craats wants to get rid of the procept. The equality sign cannot mean identically the same, because otherwise there would be no difference between input and output. Is only 4 / 7 the right answer or is 8 / 14 allowed too ? Perhaps one can teach students that 4 : 7 is a proper question and that 8 / 14 is unacceptable since this must be 4 / 7. However, 4 : 1 would be a proper question too, and then Van de Craats also argues that 4 / 1 would be a fraction (and result of division).
  • Sheet 65: “Actually 2 4/5 means 2 + 4/5.” (Van de Craats read an article of mine.) It would have been better if he stated that the first is a horrible convention, and that he proceeded with the second. He calls the form a “mixed fraction” while the English has “mixed number“. Lawyers might have to decide whether “fractions are numbers” implies that a “mixed fraction” is also a “mixed number”.

If a professor of mathematics becomes confused on such an “elementary (school)” issue of fractions (I still don’t know that is meant by this), why would the student believe that anyone can master this apparently superhumanly difficult subject ?

Will the ivory tower stop the blind ?

Would research mathematicians who do group theory be able to correct Van de Craats ?

Let us consider Bas Edixhoven again, see again his sheets.

Or would Edixhoven argue that he himself looks at natural numbers, integers, rationals and reals, so that he has no view on “fractions”, as apparently defined by Van de Craats ? Though the “Foundations” syllabus refers to the word without definition and Edixhoven might presume that aspiring teachers of mathematics know what those fractions are.

Edixhoven in the 2014 lecture only suggests that there better be more proofs and axiomatics in the highschool programme, and he gives the example of a bit of group theory for arithmetic. He also explains  modestly that he speaks “from his own ivory tower” (quote). Thus we can only infer that Edixhoven will remain in this ivory tower and will not stop the blind (but also arrogant) Van de Craats from leading (or at least trying to lead) the deaf (elementary school teachers) into the abyss.

However, professor Edixhoven also left the ivory tower and and joined the real world. At Mastermath he is involved in training aspiring teachers. Since February 2015 he is member of the Scientific Advisory Board of the mathematics department of the University of Amsterdam, where professor Van de Craats still has his homepage with this confusing “course on fractions”. I informed this board in Autumn 2015 about the problematic situation that Van de Craats propounds on primary and secondary education but is not qualified for this. I have seen no correction yet. Apparently Edixhoven doesn’t care or is too busy scaring aspiring teachers away. Apparently, when a teacher of mathematics criticises him, then this teacher obviously must be deficient in mathematics, and should follow a course for due indoctrination in the neglect of didactics of mathematics.

Jan van de Craats, Workshop 2010, page 28

Jan van de Craats, Workshop 2010, page 28

When we take a ring and include division then we get a field For example, the integers Z = { … -3, -2, -1, 0, 1, 2, 3, … } form a ring, and with division we get the rational numbers Q and also (with completion) the real numbers R. These are concepts from “group theory“. I have always wondered what the use of this group theory actually is.

The change from ring Z to field R is not quite the inclusion of division – since the ring already has implied division namely as repeated subtraction – but the change consists of extending the set with “accepted numbers” by inverse elements xH for H = -1. In that case the results of division are also included in the same set. In terms of Z the expression 2H is not a number, but for Q and R we accept this.

If the ring has variables and expressions, then we can form the expression 1 = 2 z, and we effectively have z = 2H, and then we might wonder whether it actually matters much whether this z belongs to Z or not.

Part of the confusion in this discussion is caused by that we might regard 2H as the operation 1 / 2, while we might also regard it as the number. Thus when some people say that the difference between the ring and the field concerns the operation of division, another perspective is that the field already has an implied notion of division but merely lacks the numbers to fit all answers.

The discussion within group theory might be a victim of the phenomenon of the procept. When the discussion is confused, perhaps group theory itself is confused. We should get enhanced clarity by removing the ambiguity of operation and result, but perhaps textbooks then become thicker.

Subsequently, we get a distinction between:

  • Mathematics for which group theory isn’t so relevant – such that there is a logical sequence from natural numbers to integers, to rationals, to reals, to multidimensional reals, for, all is implied by logic and algebra, and only the end result matters,
  • Mathematics for models for which group theory is relevant – i.e. for models for which it is crucial that e.g. Z has no z such that 1 = 2 z. The crux lies in the elements of the sets, as the operations themselves are actually implied.

A model might be the number of people. Take an empty building. A biologist, physicist and mathematician watch the events. Two people enter the building, and some time later three people leave the building. The biologist says: “They have reproduced.” The physicist says: “There was a quantum fluctuation.” The mathematician says: “There is -1 person in the bulding.”

The following develops the example of implied division. This discussion has been inspired by both the recent discussion of the “ring of polynomials” (thus without division but still with divisor and remainder) and the observation that “realistic mathematics education” (RME) allows students to avoid long division and allows “partial quotients” (repeated subtraction).

An example from Z, the integers

Z rewrites repeated addition 3 + 3 + 3 + 3 = 12 as multiplication 4 * 3 = 12.

Z allows the converse 12 – 3 – 3 – 3 – 3 = 0 and also the expression 12 – 4 * 3 = 0.

Z doesn’t allow the rewrite of the latter into 12 / 4 = 3.

Yet 12 – 4 * 3 = 0 gives the notion of “implied division”, namely, find the z such that 12 – 4 * z = 0.

This notion of “implied division” is well defined, but the only problem is that we cannot find a number in that satisfies 1 – 2z = 0.

If we extend Z with basic elements nH for n ≠ 0 then we can find a z that satisfies 1 = 2z but the extension generates a new set of elements that we call Q, the rational numbers. Since we cannot list all these numbers, it is not irrational of mathematicians to say that they actually include the operation itself.

The following discusses this with formulas.

A ring has implied division

Multiplication is repeated addition. The ring of integers has the notion of subtraction. Define “implied division” of y by x as the repeated subtraction from y of some quantity z, for x times with remainder 0. For x ≠ 0:

y – x z = 0                   (* definition)

To refer to this property, we use abstract symbol H, though we later use H = -1.

xH y =  z    ⇔    y = x z          (** notation)

For x itself:

xH x = x xH = 1

For zero

We have 0 z = 0 for all z in the ring. Then for implied division by zero we have:

y – 0 z = 0    ⇒   y = 0

 As above, for y = 0:

00 z = 0   for any z

0H 0 = z    for any z

Thus the rule is: For implied division within the ring, the denominator cannot be 0, unless the numerator is 0 too, in which case any number would satisfy the equation.

This is not necessarily “infinity” or “undefined” but rather “any z in Z“. The solution set is equal to Z itself. There is a difference between functions (only one answer) and correspondences (more answers).

Compare to the common definition

A ring is commonly turned into a field by including the normal definition of division:

  x ≠ 0     ⇒     xH x x xH = 1 

With this definition we get (multiplying left or right):

xH y  ⇔     x xH y x z     ⇔    y = x z

The curious observation is that a definition of division seems superfluous, since we already have implied division. The operation (*) already exists within the ring. We included a special notation for it, but this should not distract from this basic observation. If you have a left foot then it doesn’t matter whether you call it George or Harry.

An aspect is the algorithm

The natural numbers can be factored into prime numbers. When we solve 6 / 3 = 2, then we mean that 6 can be factored as 2 times 3, and that we can eliminate the common factor.

6 / 3 = z    ⇔    6 = 3    ⇔   2 3 = 3 z    ⇔   3 (2 – z) = 0     ⇔   

3 = 0   or    (2 – z) = 0         

But, again, this algorithm doesn’t work for a case like 1 = 2 z.

The “problem” are the elements

Let us consider the implied division of 1 by 2. This generates:

2H 1 = z

2H = z

1 = 2 z

Thus we don’t actually need to know what this z is, since we have the relevant expressions to deal with it.

The point is: when we run through all elements in Z = { … -3, -2, -1, 0, 1, 2, 3, … }, then we can prove that none of these satisfies 1 = 2 z.

Thus the core of group theory are the elements of the sets, and less the operations, since these are implied.

The basic notion is that 0 has successor 1 = s[0], and so on, and this gives us N. That 0 is a predecessor of s[0] generates the idea of inversion that s[H] = 0. This gives us Z. Addition leads to subtraction, to multiplication, to division. The core of addition doesn’t change, only the “numbers”.

Thus, group theory might have a confusing language that focuses on the operations, while the actual discussion is about the numbers (since the operations are already available and implied).

The fundamental impact of algebra

Thus, once we accept algebra, then the real numbers can be developed logically, and it is a bit silly to speak about “group theory”, since there are only steps, and all is implied. It only makes sense for applications to models, such as the notion that there aren’t half people and such.

It remains relevant that some algorithms may only apply to some domains and not others. Factoring natural numbers into prime numbers still works for the natural numbers embedded in the reals, yet, it is not clear whether such a notion of factoring would be relevant for other real numbers.

Appendix. Potential extension with an inverse for zero ?

We might consider to include the element 0H in the ring, to create 〈ring, 0H〉.

(1) If we maintain that 0 z = 0 for all z in 〈ring, 0H〉 then:

0H 0 = 0   with 0H in 〈ring, 0H

Observe that this is not a deduction, but a definition that 0 z = 0 for all z.

One viewpoint is that there is a conflict between “any z” and “only z = 0″ so that we cannot adopt this definition. Another viewpoint is that the latter uses the freedom of the former.

(2) When we write 0H as ∞ then it might be clearer that 0H 0 remains a problematic form.

If we create the 〈ring, 0H〉, then we might also hold: 0 z = 0 for all numbers except 0H. In that case, the result is maintained that

0H 0 = z    for any z

(3) An option is to slightly revise the definition as repeated subtraction by z until the remainder equals that very quantity z again. Thus:

y – (x – 1) z = z                   (*** definition 2)

xH y = y – (x – 1) z = z                  (**** definition and notation 2)

For = 0 we would now use z – z = 0 which might be less controversial.

0H y = y – (0 – 1) z = z

yz – z = 0

0H y = 0H 0 z

However, the more common approach is that 0H isand that is undefined too, while we cannot exclude that the answer would be z∞.

PM. Partial quotients

PM. See also the earlier discussion on this weblog.

I wouldn't want to be caught before a blackboard like that (Screenshot UChicago)

I wouldn’t want to be caught before a blackboard like that (Screenshot UChicago)