Wikipedia (a portal and no source) gives an overview of the Dutch general elections of March 15 2017. For the interpretation of the vote, there is this paper: “The performance of four possible rules for selecting the Prime Minister after the Dutch Parliamentary elections of March 2017“.

The abstract of the paper is:

“Economic policy depends not only on national elections but also on coalition bargaining strategies. In coalition government, minority parties bargain on policy and form a majority coalition, and select a Prime Minister from their mids. In Holland the latter is done conventionally with Plurality, so that the largest party provides the chair of the cabinet. Alternative methods are Condorcet, Borda or Borda Fixed Point. Since the role of the Prime Minister is to be above all parties, to represent the nation and to be there for all citizens, it would enhance democracy and likely be optimal if the potential Prime Minister is selected from all parties and at the start of the bargaining process. The performance of the four selection rules is evaluated using the results of the 2017 Dutch Parliamentary elections. Plurality gives VVD. VVD is almost a Condorcet winner except for a tie with 50Plus. Borda and BordaFP give CU as the prime minister. The impossibility theorem by Kenneth Arrow (Nobel memorial prize in economics 1972) finds a crucially different interpretation.” (Paper)

The paper uses the estimate of March 16, and the official allocation of seats presented on March 21 was the same. Here is a letter (in Dutch) to the Speaker of the House with these results and a summary statement.

Relevance for the world

In addition to that paper, let me mention some other points.

  • The Dutch system of proportional representation (PR) with a threshold of 1 seat is most democratic, and is much better than district representation (DR) or the use of high thresholds. (See this other paper.) The low threshold allows the flexible entry and exit of contestants. For example, in Germany, economics professor Bernd Lucke started the originally decent AfD, didn’t get their 5% threshold, and was ousted by extremist members in his party. For the upcoming elections, France and Germany best adopt the Dutch election model, but likely they will not have time to do so.
  • Within the Dutch system, there still is room for even more democracy. Coalitions can be inclusive or exclusive. Politicians tend to think that a minimal majority is most stable, but in all likelihood voters are better served by a larger majority.
  • The news media of the world tended to focus on the Dutch outcome that Geert Wilders didn’t succeed in getting most seats. Incumbent prime minister Mark Rutte got 33 seats and Wilders only 20. This was interpreted as that the threat of populism in Europe might have a turning point. However, Rutte dropped from 41 to 33 and Wilders rose from 15 to 20 seats, so the gap of 26 seats was halved in favour for Wilders. There is also the new right wing lunatic FvD with 2 seats, and the move to the right by other parties feeling the hot breath by Wilders. Overall, the picture is more mixed than the world news media seem to have reported. A bit more background w.r.t. the Dutch reputation of tolerance is in this earlier weblog text.
Some additional findings on turnout

The official results of March 21 2017 allow an additional statement on turnout. The key data are in the following table.

The Dutch House of Commons has 150 seats. With the turnout of 81.9% actually only 120 seats were fully taken. 27 Seats were lost to no-shows, 2 seats were lost to the dispersion of small parties and 1 seat was lost on blank or invalid votes. One might argue that 30 seats should remain unused, so that the parties that were elected in the House would find it tougher to create a coalition of 76 seats or 50%+1. Alternatively, when the 30 seats are still allocated to the elected parties, then one might raise the majority criterion to 94 seats. Instead, however, the elected parties take the 30 seats anyway and still apply the 76 seats majority rule. See this paper for a discussion w.r.t. an earlier election.

A Dutch – Turkish clash

The vote took place while there was a clash between Holland and Turkey – see the scene on Haberturk TV reported on by Euronews. Much has been said about this elsewhere, but here we continue testing the quality of Google Translate: “They protested the Netherlands by squeezing oranges”.

The Turks should however beware that the House of Orange claims Russia, and you wouldn’t want an orange bear on your doorsteps.

Euronews relaying Haberturk TV. “Hollanda’yı portakal sıkarak protesto ettiler…”

Some Dutch had been prepared for this

In the months before, visionary artist Inez Lenders had already created the artistic reply to maltreatment of oranges. In the match on creativity, the score is 1 – 1.

Art and Photography by Inez Lenders, Nijmegen 2017

The Dutch Official News with a false suggestion

The site Joop.nl calculated that the elections generated 5 MP’s with Turkish roots and 8 MP’s with Moroccan roots, and 0 with roots in Suriname. We may include one Turkish-Kurdish MP, so a total of 14 or 9.3% of relatively new immigrants. There are 4 German names, 2 French and 1 Jewish. Thus a total of 21 MP’s or 14% immigrant names.

Notwithstanding such a composition in the new House of Commons, president Tayyip Erdogan fulminated about descendents of nazi’s, though he is right that the Dutch record in World War II is not so good.

When Angela Merkel and other Europeans supported Dutch premier Mark Rutte, then Erdogan presented a statement for which it is important to provide the right translation. Reuters seems to be okay:

“Erdogan warns Europeans ‘will not walk safely’ if attitude persists” (March 22 2017)

This is a fairly decent warning. The age of European imperialism till 1945 is over. In the world population the European share is dwindling. If the world wants to maintain the idea of safe international travel then we need rules and regulations and consistent implementation.

  • Reuters gives a fair representation that Erdogan warns about the effect of arrogance.
  • Dutch national television turned this into a report that Erdogan threatened Europeans. On this NOS website, the official heading and weblink contain the phrase “Erdogan warns” but the picture on that page has the phrase “Erdogan threatens” (Dutch “bedreigt”) (wayback machine).

I have informed NPO Ombudsman Margo Smit about the difference between warning and threatening, but they haven’t changed it yet.

Official Dutch television NOS falsely states that president Erdogan issues a threat that no European in any part of the world can safely walk on the street. In truth he only warns.

[ This is the same text as the former weblog (here), but now we follow Van Hiele’s argument for the abolition of fractions. The key property is that there are numbers xH such that x xH = 1 when x ≠ 0, and the rest follows from there. Thus we replace (y / x) with y xH with H = -1. ]

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 bH “and” c dH kids are supposed to find (a d + b c) (b d)H instead of (a + c) (b + d)H.

Obviously this is a matter of definition. For “plus” we define: a bH + c dH = (a d + b c) (b d)H.

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

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

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

There are cases where (a + c) (b + d)H makes eminent sense. For example, when a bH is the batting average in the Fall-Winter season and c dH the batting average in the Spring-Summer season, then the annual (weighted) batting average is exactly (a + c) (b + d)H. 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)H 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.

[ Some readers may not be at home in statistics. Let the weight of b be w = b (b + d)H. Then the weight of d is 1 – w. The weighted average is (a bH) w + (c dH) (1 – w) = (a + c) (b + d)H. ]

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 animal 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 female pets and 16 male pets (perhaps “helped”).

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

  1. For the female pets, the proportion of cats in the house is larger than the proportion for dogs.
  2. For the male 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 7H = 0.86, 2 10H = 0.20 and (6 + 2) (7 + 10)H = 0.47.
  • For the dogs: 8 10H = 0.80, 1 6H = 0.17 and (8 + 1) (10 + 6)H = 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 F and M are distributed in unbalanced manner.

The correction

It can be an argument that there should be equal numbers of F and M. 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) H = 0.55.
  • For the dogs: 0.17 * 10 ∼ 2, and (8 + 2) (10 + 10) H = 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).

Conclusions

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: 2017-03-03-data-from-kleinbaum-2003-adapted

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.

2017-02-05-wilders-photoshop-pechtold

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.

seats-d66-pvv

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 female pets and 16 male pets (perhaps “helped”).

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

  1. For the female pets, the proportion of cats in the house is larger than the proportion for dogs.
  2. For the male 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 F and M are distributed in unbalanced manner.

The correction

It can be an argument that there should be equal numbers of F and M. 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).

Conclusions

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-adapted.

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.

pearl-analysis-1

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.

pearl-analysis-2

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.)

Conclusion

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).

Deductions

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 ?

Descartes

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.
  • §12.1.8.1 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.
  • §12.1.8.2 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.)
  • §12.1.8.3 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.
Conclusion

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 §12.1.8.3 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 §12.1.8.2 on the actual calculation.

The basic conclusion is that this “general formula” enhances the consistency of §12.1.8.3. The deduction however is not needed, since we have §12.1.8.1, 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.