Listening to Roy Orbison – Pretty Women

Let me give a clear and unbiased assessment of the qualifications and work of “dr.” Michiel Doorman at “Freudenthal Institute” at Utrecht University.

Let me first present four neutral points and then follow Sherlock Holmes.

(1) Michiel Doorman defended his “thesis” in 2005: “Modelling motion: from trace graphs to instantaneous change” (online), written under the supervision of P.L. Lijnse and Koeno Gravemeijer.

(2) He might best be introduced by his cv on p243 of his “thesis”:

“Michiel Doorman was born on 1 october 1962 in Eindhoven, the Netherlands. He completed his secondary education in 1981 at the Minkema College in Woerden. In1988 the Utrecht University awarded him a masters in Mathematics for his thesis on the extension of a proposition in intuitionistic logic for automated theorem proving. He minored in Computer Science. From 1988 he has been working at the Freudenthal Institute. Until 1992 he was mainly devoted to software development. During the following years he has been involved in curriculum and teacher training projects, mainly concerning the role of information and communication technology in mathematics education. Since 1994, this work concentrated on upper secondary (pre-university) mathematics education in a research project on the integration of the graphing calculator, in a curriculum development project (Profi), and in a project that aimed at guiding the Biology, Chemistry, Physics and Mathematics departments in schools to cooperate. In 1998 he started his PhD research study.” [my emphasis]

(3) There is also his Utrecht University webpage, that states:

“Interests are context-based mathematics education, modeling as a lever for learning mathematics, inquiry based learning and coherency between mathematics and science learning.”

(4) Michiel Doorman was also invited as one of the keynote speakers at the 3rd International Conference on Research, Implementation and Education of Mathematics and Science (ICRIEMS) at Yogyakarta, Indonesia, in May 2016.

  • This is Doorman’s presentation pptx (on “inquiry based learning”) and
  • this is the proceeding pdf: “What Can Mathematics Education Contribute To Preparing Students For Our Future Society?“.
Michiel Doorman at ICRIEMS 2016 (fourth from left) (Sources: ICRIEMS website)

Michiel Doorman fourth from left (Source: ICRIEMS website 2016)

Why does Doorman in 2016 claim success for RME while it failed ?

Around 2005 there was much discussion in Holland – a real math war – about arithmetic in elementary schools. The academy of sciences (KNAW) set up a committee to look into this.

Recall the graphical display of the math war between RME and TME, and the solution approach of NME. These abbreviations are:

  • RME = realistic mathematics education
  • TME = traditional mathematics education
  • NME = neoclassical mathematics education

(i) This KNAW-committee concluded in 2009 (see the English summary on page 10 in the report) that pupil test scores for RME and TME did not really differ. Paraphrased: one cannot claim a special result for RME. Observe that many test questions contained contexts.

“Growing concern about Dutch children’s mathematical proficiency has led in recent years to a public debate about the way mathematics is taught in the Netherlands. There are two opposing camps: those who advocate teaching mathematics in the “traditional” manner, and those who support “realistic” mathematics education. The debate has had a polarizing effect and appears to have little basis in scholarly research.” [This neglects my third position with NME.]

“The public debate exaggerates the differences between the traditional and realistic approaches to mathematics teaching. It also focuses erroneously on a supposed difference in the effect of the two instructional approaches whereas in fact, no convincing difference has been shown to exist.” (KNAW 2009)

(ii) Doorman in Yogyakarta 2016 is unrepentingly for RME. He refers to key authors on RME, and takes a question of TIMMS 2003 with an international score of 38% and a Dutch score of 74% and claims, misleadingly:

“This cannot fully [be] attributed to the implementation of  RME, but it strengths [sic] the feeling that this approach contributes to the quality of mathematics education.” [my emphasis]

(iii) Subsequently, I criticised the KNAW report on these counts, and neither KNAW nor Michiel Doorman have responded to this criticism:

  • Before the report was published by alerting the committee chairman to Elegance with Substance (2009, 2015), that however is not included in the references.
  • In 2014 explicitly for the collective breach of research integrity, for either neglecting or maltreating my books Elegance with Substance (2009, 2015) and Conquest of the Plane (2011) and Dutch Een kind wil aardige en geen gemene getallen (2012) notably on issues of arithmetic (the present subject): the pronunciation of numbers and notation of mixed numbers.
  • In 2015 for neglecting the issue that TME prepares for algebra while RME doesn’t. The KNAW report uses the outcome of test questions and not the intermediate steps. Pupils who can only use RME will be very handicapped for algebra in secondary education.

See my 2016 letter and its supplement to the president of KNAW and director of CPB about the failure of the KNAW report and the neglect of criticism.

A repeat exercise that isn’t quite superfluous

I have explained, to boring repetition, that the Freudenthal Head in the Clouds Realistic Mathematics Institute (FHCRMI) should not be at a university. Please observe that first there was criticism on the failure of “realistic mathematics education” (RME) and only later it was discovered that Hans Freudenthal had actually abused the work by Pierre van Hiele. There also is a sound scientific explanation why it is a failure: namely a confusion of processes of learning with applied mathematics.

Thus it holds:

There is little advantage in repeating this analysis,
neither for each and every individual working at FHCRMI.

For example, stating that Michiel Doorman works at FHCRMI should be sufficient. That he is at FHCRMI does not imply that he can indeed be at university and that his “thesis” and “PhD title” are proper.

However, the following points cause that it isn’t quite superfluous to look into Doorman’s qualifications and work.

  • Michiel Doorman is member of the board of NVvW, the Dutch association of mathematics teachers. See my recent letter with a Red Card for this board. Thus it helps for the next annual meeting of NVvW in November 2016 to be specific.
  • Also, there is my letter of April 15 2016 to NRO, the Dutch organisation for the distribution of funds for research in mathematics education. I advise them to stop subsidising FHCRMI. It so happens that Michiel Doorman did a project ODB08008 for them in 2009-2012 on the “digital mathematics environment” and “efficient exercising mathematics” (DWO). It will be helpful for NRO to see that, for example, Doorman is an ideologue and no scientific researcher. This is related to the following.
  • There is the new impulse for “21st century skills” or in Holland “Onderwijs2032“. Part of the attention is for soft skills, part of the attention is for computer programming, part is elsewhere. ICT brings us to the work of Doorman too. There has already been a major disaster with the neglect of computer algebra since 1990. For example DWO at FHCRMI tends to present many Java applets that lack the flexibility of computer algebra. Don’t think that these issues are easy to resolve, but I do hold that the decisions have been driven by ideology and that the results are a disaster and a great waste of funds: penny wise pound foolish. See for example these two reports by the Inspectorate for Education: In 2002, mathematical topologist Hans Freudenthal is called a “pedagogue” while he had no education or training on this, and they assume that FHCRMI knows about ICT while the report doesn’t mention computer algebra but applets on “wisweb”. In 2006, the “waarderingskader” (inspection standards) doesn’t seem to realise that computer algebra can used in all subjects that use mathematics.

Above, I mentioned four neutral points. Following Sherlock “Google” Holmes I already debunked the event in Yogyakarta. Let us look at the other three points. Beware of confusion.

Ad 1. Doorman’s “thesis” of 2005

Michiel Doorman defended his “thesis” in 2005: “Modelling motion: from trace graphs to instantaneous change” (online), written under the supervision of P.L. Lijnse and Koeno Gravemeijer.

  1. In the “thesis”, Doorman basically refers to Paul Drijvers at FHCRMI for computer algebra, but Drijvers is no light on this either. For the subject of the thesis (“Furthermore, it has been examined what role computer tools could play in learning mathematics and physics.”) it would have made much sense to look deeper into computer algebra.
  2. Also check my analysis that Koeno Gravemeijer is no scientist but an ideologue for “realistic mathematics education” (RME) who misrepresents issues on “21st century skills” (in Holland “Onderwijs2032“), and who doesn’t see the revolution by computer algebra. (Dutch readers can look here too.)
  3. On p58-59 Doorman critically adopts RME, and remember that this was in 2005, while RME was under discussion, see (a) the discussion in 2006 between Robbert Dijkgraaf (who has no qualification for math ed at this level) en Kees Hoogland (a RME ideologue, see below), which report was written by RME supporter Martinus van Hoorn, and (b) while Jan van de Craats (who has no qualification for math ed at this level) was protesting about RME, and published “Waarom Daan en Sanne niet kunnen rekenen” in 2007. See my criticism w.r.t. Jan van de Craats (fighting his math war on the side of TME and neglecting NME since 2008).
  4. Doorman refers to Freudenthal for “guided reinvention”, but this is a wrong reference (he may only have coined the phrase but not the concept), and Doorman’s thesis does not refer to the true inventor of the concept (guide through levels of insight) Pierre van Hiele at all.
  5. I will not look at this “thesis” in detail because there is really no reason to so so now.
Ad 2. Curriculum vitae

Doorman’s cv shows that he got a mathematics degree and continued at Freudenthal Head in the Clouds Realistic Mathematics Institute (FHCRMI), thus without a teaching degree in mathematics and without proper training in research of mathematics education.

  • The KNAW report of 2009 showed that FHCRMI doesn’t do research on arithmetic education, and one should not suppose that this is different for other areas.
  • Thus Michiel Doorman is neither a teacher of mathematics nor a researcher in the education of mathematics.
  • We find no qualification for teaching and research, but immersion into ideology, and while he is involved in programming and the role of ICT for (mathematics) education, there are only perfunctory statements on computer algebra (for all subjects that use mathematics).
Ad 3. Webpage

Observe that “context-based mathematics education” is a rephrasing of “realistic mathematics education” (RME). Also “inquiry based learning” is basically RME, with contexts as the starting point for the “inquiry” (constrained by learning goals).

Someone really interested in didactics and RME would also have been interested in my analysis that shows that RME is a confusion and an ideology.

Observe also that the sciences are easy victims of RME. The sciences do not care much about mathematics education, and when RME people flock in to assist in the learning of the sciences, and when student learning time for mathematics is actually spent on the sciences, then professors of physics or biology might hardly object. For RME the sciences are useful window dressing, since those provide both contexts and an aura of respect and acceptance, and an argument that “students are learning something” (even if it isn’t mathematics). There is a curious historical link-up of mathematics with the beta sciences while there are also the humanities (alpha) and social sciences (gamma), see here. But is it really curious, and isn’t there a method, that the human and social scientists who know the techniques and who also do research by observation are kept out from this association between “mathematics education” and “science education” ?

Possible confusions that are triggered immediately

Stating the above might immediately trigger some confusions.

  • As member of the NVvW board Doorman might argue “not to look into the criticism on FHCRMI because of an interest there”. Instead, he should rather take the initiative and make sure that this criticism was answered in decent manner rather than burked. If his interest is so large that he cannot speak freely on science then he should rather not be in the board.
  • Doorman in a 2015 text in Euclides, the journal of NVvW, referred to (intellectually stealing) Freudenthal and not to (victim) Van Hiele. When asked to correct, he didn’t reply to this very question (see page 8) but talked around it, see my deconstruction of his “reply”. Potentially Doorman just didn’t have the relevant knowledge about didactics, for histhesis” refers to RME but not to Van Hiele. If Doorman had looked into this criticism of mine, he could have been a bridge of understanding for the other members of the NVvW board and readers of Euclides, but he wasn’t.
    PM 1. I don’t understand either why these people didn’t see that he dodged the question.
    PM 2. Doorman in Indonesia sheets 44-49 on RME repeats the reference on anti-didactic inversion to (intellectually abusing) Freudenthal at the cost of (victim) Van Hiele.  Thus I asked a correction, he dodged the question, and proceeds as if there would be no problem (and likely not informing the audience about the criticism).
  • Thus Doorman is neither teacher nor researcher: so what is he doing in the board of NVvW ? From qualifications, actually their lack, and work, actually the lack of answers, it is a small step towards wondering about motivation. A good hypothesis is: he is spreading the gospel of RME and blocking criticism. If Doorman wants to become professor at FHCRMI he must show that he is a true sectarian of RME. He is polishing up his cv and can now claim that he has been involved in the community of teachers, even when it was dodging questions. This is a hypothesis only, and one might also offer other explanations like blindness.
  • I wonder who paid for this trip to Yogyakarta. It is also known that Freudenthal Head in the Clouds Realistic Mathematics Institute (FHCRMI) still is busy in establishing footholds over the world even though RME has failed. See also, who apparently neglect the KNAW report or my criticism.
  • Yes, there is also NRO-project supervisor Frits Beukers, but he is professor of mathematics also without qualification for mathematics education research. Beukers presently is chairman of the Platform Wiskunde Nederland (PWN)-education committee, but would represent the universities since he has no qualification for primary or secondary education or the trade colleges. In that committee we also see Kees Hoogland, who abused the biography by John Allen Paulos for RME, and who has not corrected yet and who refuses to give an English translation of the key section.
Disclaimer: Can I be unbiased ?

I stated that I would give an unbiased assessment. Can I really do so ? Undoubtedly the reader will make up one’s own mind, but my perception is that I have been fair and unbiased.

Doorman’s “thesis” of 2005 is closely related to the education on the derivative. There was ample scope for a meeting of minds. When Doorman sticks with RME and Java applets, instead of NME and computer algebra, it is all of his own choosing. The differences in positions can be mentioned:

  • Check my proposal since 2007 for an algebraic approach to the derivative, see e.g. Conquest of the Plane (2011).
  • COTP was also programmed in Mathematica – a system for doing mathematics on the computer (a.k.a. “digital environment”) .
  • Also, I am a user of computer algebra since 1993, while Doorman tends to use other programs that don’t have the flexibility of computer algebra.
  • Doorman was editor of TD-Beta when I submitted a short note in 2012 on my invention of the algebraic approach to the derivative. This was maltreated. See here anonymised  and see here with full names (it is a scientific discourse and no private exchange).

At the NVvW annual convention of 2015 when Doorman was elected to the board, I had my doubts but opted still mildly optimistically for the benefit of the doubt. I had no experience with this TD-Beta journal and perhaps everything was an unfortunate misunderstanding. It doesn’t happen so very often that someone can propose a new approach to the education on the derivative. The performance of last eight months however gives evidence of the mindset of an ideologue.

Conclusion: Doors of perception

Any link to Doorman’s name is coincidental, and it is also a coincidence that the Dutch family name “Doorman” indeed is related to the English “doorman” (at least according to the Meertens institute).

The phrase “doors of perception” comes to mind when looking at Doorman’s presentation sheet “Aim of Primas” that states:

“A question not asked is a door not opened!”

  1. This implies that an opened door is a result of a question. (This need not be an open door.)
  2. This doesn’t imply that asking a question will open a door. (The question might e.g. be ignored.)

The message of this present weblog is, amongst others, that there are some crucial questions that Michiel Doorman refuses to look into and apparently doesn’t want to answer. He is employed at the Freudenthal Head in the Clouds Realistic Mathematics Institute, that should not be at a university. Apparently Doorman did not inform his hosts in Indonesia about the existing critique either.

I am quite neutral on the questions whether or not there is a Brexit and / or the UK breaks up. It is all up to the people in the UK what they wish to do.

My view is only that what we see isn’t democracy but political abuse. The data of the June 23 outcome are generally known, and summarised in the Appendix below. These data still leave unclear what UK voters really think. It is utterly false when politicians and their lawyers claim that the outcome represents the “views by the voters”.

This weblog entry on the Brexit referendum outcome has two aspects:

  • First there is an advice for young UK on safeguarding their future.
  • Secondly, there is the scientific outrage that voting theory has been neglected. Let me suggest to my fellow scientists to protest en masse to the political world that this Brexit referendum was & is political abuse, with deliberate neglect of scientific results on voting theory. Civilisation should not let such developments be determined by such political savagery.

Let us start by observing that the Brexit referendum question neglected the possible break-up of the UK. This isn’t hindsight but people warned for this early on, e.g. William Hague in December 23 2015 just when the referendum question was published. One might argue that people thus were warned, and if they still voted for Leave then they must have included the risk of a break-up. Thus we would live in the best-possible-world (Candide), and it is up to Scotland and Northern Ireland to decide now. However, science has shown that one cannot really interprete referenda in such manner. For example, foreign secretary of Sweden Margot Wallstrom warned on June 12 that the UK Leave might mean the end of the EU. Perhaps she intended to stimulate the turnout of Remain, but, if she was heard at all, she likely encouraged the turnout of Leave who wanted to destroy the EU too, and helped them forget about the break-up of the UK. Thus, it is too rosy to hold that people in the ballot box are fully informed and can deal well with misleading referendum questions.

For young UK (age < 50 years)

An advice for young UK who face Brexit is to study the basic theory of elections and then organise a new decision, done properly.

Four texts to study are:

  • the former weblog text on Brexit
  • my online book Voting Theory for Democracy
  • a paper that compares district representation (DR) as in the UK with proportional representation (PR) as in Holland. A), with a summary that appeared in Mathematics Teaching 222 in the context of the 2011 UK referendum on “Alternative Vote”.
  • a warning (in Dutch) to be careful with the mathematics of voting schemes.

Let us consider, for the sake of argumentation, a potential new referendum that might combine the options of Remain / Leave with UK / EU, so that voters can express better whether or not they want to leave the EU at the possible cost of a break-up of the UK. The votes on remaining or leaving the UK must be aggregated for the four areas separately, with perhaps the creation of England as a political entity too.

2016-06-29-Brexit-true-optionsWould this scheme work ? The scheme doesn’t quite deal with conditionals, like (England’s ?) “Leave the EU provided that others don’t leave the UK” or (Scotland’s ?) “Remain in the UK provided that the UK remains in the EU”. (1) Opinion polls “help” people to get an idea what others will vote, but such polls can be quite misleading. (2) A scheme is to let voters rank their preferences. For example, a youngster might rank A > C > B > D and an elderly person might rank B > D > A > C, but this ranking doesn’t quite resolve conditionals, e.g. that might cause strategic voting. And what about other issues, like on immigration ?

This hypothetical case fits the general scientific finding that referenda are predominantly no good way for collective decision making. It is remarkable that this isn’t general knowledge or even part of common sense.

Best seems that people vote for parties and that parties have negotiations in Parliament. However, this also depends upon whether Parliament has been chosen by DR or PR.

  • The political tensions in the UK rose likely because the UK has DR instead of PR. The political system in the UK excludes minorities and reduces political competition, whence non-represented minorities grow ever more extreme.
  • The Dutch system with PR includes minorities and enhances competition for votes. PR forces parties to compromise, and bearing responsibility reduces extremes.
  • Referenda are normally PR and used to correct errors of DR, but that doesn’t make referenda true corrections.

Therefor, the best solution would be to have a decision by Parliament selected in PR manner, or, paradoxically, given that this Brexit referendum has been allowed, have a nullifying referendum. One would put four issues on ballot:

  • “The Brexit referendum question was too simple, and the outcome is annulled.”
  • “Adopt for Parliament a system of Proportional Representation (PR) like in Holland, with a Prime Minister and cabinet appointed by Parliament.”
  • “Adopt a constitutional Economic Supreme Court (ESC).” The ESC provides necessary checks on the quality of information for policy making. For this, see DRGTPE, and this memo in the RES Newsletter.
  • “If there is an ESC, then also have annual elections.” When the ESC is in place, then one can have annual elections to enhance voting power on preferences, with less risk of political chaos.

These latter two conditions for a modern democracy are still lacking in Holland. Dutch readers are referred to “Democratie & Staathuishoudkunde” (2012).

The political abuse

The New York Times reminded us on June 21 2016 that David Cameron used the referendum to resolve a political fight in his own party.

“In 2013, besieged by the increasingly assertive anti-European Union wing of his own Conservative Party, Mr. Cameron made a promise intended to keep a short-term peace among the Tories before the 2015 general election: If re-elected, he would hold an in-or-out referendum on continued British membership in the bloc. But what seemed then like a relatively low-risk ploy to deal with a short-term political problem has metastasized into an issue that could badly damage Britain’s economy, influence the country’s direction for generations — and determine Mr. Cameron’s political fate.”

This use is not necessarily an abuse, since, for example, 52% of the vote legitimise the idea, and these were not party members only. Instead, the abuse is the neglect of voting theory: the misrepresentation of a multidimensional issue by a binary choice. It is like asking “Do you still beat your wife ?” and allowing only a Yes or No, so that when the answer is No then the subsequent conclusion is: “Ah, so you admit that you did beat her before !”

Surprisingly, both David Cameron and Nigel Farage got away with the misrepresentation in the question in this plebiscite. (1) There was not enough discussion on this irresponsible simplification of the issues. At least the possibility of the break-up should have been included, see the early warning by William Hague. (2) Whatever the question up for vote, a referendum can be hijacked for another populist cause, as happened in this case with immigration. Angry voters send a vote-of-no-confidence to the government whatever the consequences.

For scientists

The case for scientific outrage is obvious. It is remarkable that we haven’t heard much from UK scientists on this. Stephen Hawking warned about Brexit but didn’t say that the referendum question is silly and dangerous.

There is this open letter of June 14 by a long list of scientists who protest:

“A referendum result is democratically legitimate only if voters can make an informed decision. Yet the level of misinformation in the current campaign is so great that democratic legitimacy is called into question.”

Curiously, however, this letter doesn’t make the point that the referendum neglects voting theory, since the very question is misleading w.r.t. the complexity of the issue under decision. Would these scientists be willing to admit this ex post ?

There is a critical article in the New Scientist of June 1 for example, but the issue of misrepresentation isn’t quite mentioned. The reporters adopt the frame that the question is sound and the voters are irrational, while the truth is rather that the referendum question is a misrepresentation and that voters are upset (albeit unconsciously) by being boxed into a corner, and by being given responsibility but no means of control (the recipe for stress).

“THE EU referendum could be the most irrational yet. Uncertainty over consequences, and contradictory economic and political information, mean that voters will be swung even more than usual by feelings and biases that have nothing to do with the issues at stake.” (Michael Bond, Jacob AronHal Hodson)

It is well-accepted by students of voting theory that referenda can by abused by politicians for their own agenda. Thus the scientific outrage should be felt by many more scientists.

Insert: The Queen: “Why did nobody notice it?”- in 2008 at the financial crisis

After the financial crisis in 2008, the UK Queen is reported to have asked: “Why did nobody notice it?” (Telegraph 2008, Guardian 2012). There is the plain answer that some people warned but were not listened to, and this is the ancient issue of Cassandra or perhaps The Boy Who Cried “Wolf”. See e.g. this discussion. The same question can be asked now w.r.t. the referendum: why did not more people warn that the referendum and / or its question wasn’t sound.

Potentially, organising a new referendum would better show how the people in the UK think about a break-up. There is one catch: it may be impossible to restore the status quo ex ante. Now that Scotland has discovered that it might be dragged along by England into undesirable waters, perhaps Scotland still wishes a new referendum on independence, even when the June 23 referendum is annulled and when a new referendum confirms that the UK would remain in the EU. This is something that someone in the Policy Simulation Room should have seen coming.

Many observers already mentioned that if there would be general elections before a government dares to invoke article 50, then these elections would turn into a repeat referendum too. In that case DR doesn’t quite square with the PR of a referendum, and thus one would rather first have PR and perhaps secondly also split parties in Remain and Leave subparties.

Insert: How the UK Electoral Commission advised on the question

A kind reader informed me that the UK Electoral Commission advised on the referendum question. Looking into this is another mer à boire.

  • The Commission has the task to check that even a misleading question is “intelligible”.
  • There is a useful discussion about the difference between “remain” and “be”, and whether yes / no creates a bias for people who hate to say no. My impression is that the Electoral Commission deserves a compliment w.r.t. the clarity about the question, so that everyone can see that it was a misleading question.
  • However, the Commission entirely overlooks the possibility of including a “None Of The Above” (NOTA), while this inclusion might have tickled people into wondering about the misleading question itself.
  • There is no mention that the UK might break up after a Leave outcome. Is it really so that no-one in the UK was aware of this and that the Electoral Commission could neglect this ?
  • There were general warnings, like in the section on “Contextual understanding of the European Union” points 3.23-3.27 on page 16:

“3.25 Whilst overall awareness of the UK’s membership of the EU was found to be relatively high, many reported that more contextual information would be required regarding the voting outcomes. Particular queries included what a vote to remain a member would mean in terms of membership status: continuation of current terms of membership or something different? A small number of participants thought that a majority vote to stay would result in the UK becoming a member of the Eurozone.  
3.26 There were similar queries about what a vote to leave would mean in terms of membership status: completely leaving the EU or some other form of membership?
3.27 Those who were undecided about how to vote were particularly likely to report a lack of contextual information enabling them to make an informed vote. They reported a lack of clarity regarding what each voting outcome would mean in practice. This is considered in more detail later in this chapter.”

I didn’t find the required details in this chapter. My advice to the Electoral Commission is to refuse to participate in the creation of misleading questions. It is okay to clarify questions, but one should also be aware of voting theory that referenda can be silly and dangerous. At the minimum discuss the inclusion of NOTA.  The reason of the referendum is to recover the views of the electorate, and what happens with the view is not only “context” but key information for developing these views.

Insert: Partial agreement with Martin Wolf in the FT

Martin Wolf stated in his “What a Prime Minister Boris Johnson should do next” (FT June 28 2016), that I only saw after basically completing this text:

“Might it be possible to abort the entire process? Legally, yes. As Brexiters rightly say, the UK is a parliamentary, not a plebiscitary, democracy. The step that must be taken, if the UK is to leave the EU, is for it to issue a declaration under Article 50 of the Lisbon treaty, to trigger the process. In law, a referendum is solely advisory. Only parliament can do this, because only it makes valid law.”

It is a subtle point that many will not have been aware of. The Dutch advisory referendum on the Ukraine had always been communicated as “advisory referendum”, and in the future such clarity would be advisable.

Now, both David Cameron and Jean-Claude Juncker assume that the UK will automatically invoke article 50, and the debate is only about delay (supposedly to the advantage of the UK) or speed (for the EU wishing to have it over with). There are rising tensions. Juncker’s attitude would have been different when Cameron had treated the referendum outcome only as an advice. Today, David Cameron is not present at the informal meeting of the HOSGs, and EU Council website has a statement by the 28 – 1 = 27:

“In their joint statement following the meeting, the 27 leaders announced: “We, the Heads of State or Government of 27 Member States, as well as the Presidents of the European Council and the European Commission, deeply regret the outcome of the referendum in the UK but we respect the will expressed by a majority of the British people. Until the UK leaves the EU, EU law continues to apply to and within the UK, both when it comes to rights and obligations.””

Nigel Farage was jeered and booed at with Juncker’s unkind question “Why are you here ?” when he presented himself in the EU Parliament. It is unkind not to allow Farage his moment of victory and for exporting his message to a wider EU area, and to imply that he was only there for the attendance fee. One should respect that it remains a political view whether one wants more integration or a return to a trade area. With this unkind and non-sportive treatment, one may better understand how Europe got into wars so often in the past. But also Farage did not treat the referendum outcome as an advice only.

Still, I agree with Wolf that the UK still has the option to backtrack. The real question is what would be the convincing argument. The argument must be made convincingly, otherwise tensions in the EU will rise, and businesses will reduce the risk of trading with the UK. For me, the convincing reason lies in the observation that the referendum question is an abuse and neglect of voting theory.  For me, it doesn’t matter whether the UK leaves or breaks up, if only the voters have been offered the true options.

Wolf is more worried about the economy, and subsequently considers some options, including re-negotiating on immigration, and starting with:

“After selection of a new leader by the Conservative party, and perhaps even a general election, Prime Minister Johnson might, to paraphrase Emperor Hirohito’s remarks at the end of the second world war, declare that, given the “unexpected” economic damage and the risk of a break up of the UK, the situation “had developed not necessarily to the UK’s advantage”. He might forget the whole thing or, alternatively, call another referendum, merely to make sure the people remained as determined.”

The argument on the Brexit is rather thin, given that the Brexit had been presented as an issue of sovereignty and the monster of the superstate. It belongs to the possibilities that Johnson doesn’t really care much whether the UK is in or out of the EU, and that the Brexit campaign was only a last resort to get Cameron and Osborne out of the way. It is also possible that Johnson might become Prime Minister now and is appalled by the chaos that he has created, and thus becomes willing to change his position. However, it is less likely that the EU will agree with re-negotiating on immigration to try prevent the UK from invoking article 50. Politics might be blackmail but the UK cannot claim a special position w.r.t. the problems in Syria or Africa.

Thus I regard this line of reasoning as not so convincing. Little stops us from combining the principles on voting with the practice of economics. However, why would we complicate a clear issue of scientific clarity on principles of voting with a messy assessment on economics ? For the experts, as Wolf indicates, the economic assessment is not messy, but we lack an Economic Supreme Court, and thus non-economists are lost in the game of guessing who the experts really are, and experience shows that this process actually is quite messy.

David Cameron might have selected his time window till October for mere party politics, but it would provide time indeed to let these arguments percolate. I would not wait for Boris Johnson but rather look to young UK and the world of science.

Insert: Gideon Rachman’s non-belief in a Brexit

Gideon Rachman doesn’t quite believe the Brexit, given some precedents (FT June 27 2016), that I also only saw after basically completing this text.

“In 1992 the Danes voted to reject the Maastricht treaty. The Irish voted to reject both the Nice treaty in 2001 and the Lisbon treaty in 2008. And what happened in each case? The EU rolled ever onwards. The Danes and the Irish were granted some concessions by their EU partners. They staged a second referendum. And the second time around they voted to accept the treaty.”

“Boris Johnson (…) hinted at his real thinking back in February, when he said: “There is only one way to get the change we need — and that is to vote to go; because all EU history shows that they only really listen to a population when it says No.”

“And what kind of new concession should be offered? That is easy. What Mr Johnson would need to win a second referendum is an emergency brake on free movement of people, allowing the UK to limit the number of EU nationals moving to Britain if it has surged beyond a certain level.”

With the EU refugee crisis (an emergency for Syria and a structural problem for Africa), it is not likely that the UK will get such exception, for the other 27 members will not be able to explain this to their constituencies. There doesn’t have to be a concession from the EU. It suffices for the UK to come to terms with voting theory, apologise to the EU for the confusion, and redo the decision making process to find out what it really wants.

A cocktail of uncertainties and possible sources for confusion

The Brexit referendum has the advantage of illuminating various uncertainties and possible sources for confusion.

  • My correspondent in The Hague argues that the Brexit outcome injects new energy into society, namely the idea of “getting a life”, and being freed from the suffocating bias by the collective hive. I am reminded of 1914, when people were energised to go to war. Sebastian Haffner‘s book of 1964 still needs an English translation: Die sieben Todsünden des deutschen Reiches im Ersten Weltkrieg.
  • One might argue that Scotland voted to remain in the UK in 2014, and thus now has to suffer the cost. The EU might welcome Scotland but might still have greater fear for break-ups like with Catalunya. Overall, my impression is that the nation states of Vienna 1815 still have an important role to play in the immediate future. Eventually a perspective of a “Europe of the Regions” (Heineken map) makes more sense. One could make a plan for the next 25 years for gradual change with both integration and distribution.
  • The 2011 UK referendum on “Alternative Vote” was crooked because the proposed system was too complex, likely by wrong advice from mathematicians. There is the curious Dutch D66, liberal democrats who neglect science and prefer DR.
  • The demographic breakdown shows that younger people turned out less and were more likely to vote for Remain, while the elderly turned out more and for Leave. Issues are: (1) When people don’t turn out, we don’t know their vote. (2) Are non-voters really indifferent to the outcome or merely confused ? (3) Don’t young people know that elections are important for their future ? (4) Was it relevant that younger people are used to a digital world while the referendum is old technology that the elderly are used to ? (5) When they have regrets, should they sit on the blisters ? (6) Will they be able to understand that this referendum was an outrage and neglect of science, and would they be able to explain this to others ?
  • An interesting point is that this Brexit outcome challenges the “one man one vote” principle. In Public Health we oppose “lives saved” to “life years saved”. It matters whether one saves a baby or a 95-year old. See my essay on the Value of Life. In economics we have intergenerational accounting. Potentially one might argue that young people are more affected by a decision like a Brexit than the elderly. The elderly might argue that they know better what is good for their grandchildren. A 95-year old might also argue that the world has a lot of babies but few 95-year olds, and that he or she represents a huge investment in human capital.
  • Remarkably, the City of London had a great interest in remaining in the EU, but was caught in the problem that others might (obviously) think that they put their interests before those of the UK. Somehow Finance lost from the Tabloids. A mediating role might have been played by Education, but Education apparently didn’t explain about the political abuse.
Dutch EU Presidency disaster, with Dutch PM Mark Rutte

The Brexit outcome means that the Dutch EU Presidency 2016 is a disaster, under the leadership of Dutch Prime Minister Mark Rutte.

Indeed, even Dutch PR has its problems. The Dutch general elections 2012 might have allowed a coalition under the leadership of Diederik Samsom, with parties PvdA & SP & CDA & D66 & CU & GL & PvdD = 38 + 15 + 13 + 12 + 5 + 4 + 2 = 89 seats in a house with 150 seats, but Samsom preferred a quick deal with Rutte (VVD, 41 seats). Samsom forgot that VVD and PvdA had made rather conflicting campaign promises, so that a coalition was strange, and both are much lower in the polls now (VVD 24, down from 41, PvdA 8, down from 38). In 2017 there are Dutch elections. VVD and PvdA had been hoping that the Dutch economy would be doing well by then, but the Brexit makes this less likely again. However, my advice is that all (decent) parties are represented in the executive of the government too, since being in government helps realism. The selection of the Prime Minister can be done by the Borda Fixed Point method.

Potentially the EU regarded the UK referendum as an internal affair of the sovereign UK. European heads of state or government (HOSG) have been advising the UK voters, but always at some “respectful distance”, and not in town hall meetings as Barack Obama toured the USA. It is part of European folklore to have a European project but while maintaining the outward rituals of sovereignty. This emphasizes the distance rather than the togetherness like in the European Song Festival. The Dutch Presidency should have been more alert on this. People in the UK might argue that there are ample people from other EU nations in the UK, so that there are ample opportunities for discussion, and that the purpose of the referendum is to let the British make up their minds themselves. But why not have town hall meetings with EU HOSGs at various locations in Nigel Farage’s backyard ?

Another comment (lost the reference) was that it was Margaret Thatcher who pushed for the neoliberal agenda, as opposed to the Continent, with the Christian or Social Democratic agendas. This caused the austerity programs and the financial crisis of 2007+ and the call for more austerity. Thus the UK exported the policies to Brussels, about which the Brexiteers complain that those came from Brussels. To this, we might add that Tony Blair proceeded along Thatcher’s path, and assisted in the false pretentions for the invasion into Iraq that eventually caused ISIL and the Syrian refugees to Europe. How blind can you be about your own crookedness ? Also, Mark Rutte has hero worship for Thatcher, and thus was the last person to explain this boomerang to the UK. See what Rutte neglects: my paper on the Cause and Cure of the Crisis.

(PM. Perhaps, however, Rutte has other objectives. Now that the House of Hanover leaves the EU, perhaps the House of Orange has better cards to get adopted as the Imperial House for the EU empire, later to be united with Russia. My correspondent in Moscow informs me that president Putin is already looking at pictures of young Dutch princess Amalia. As Putin intends to rule for a thousand years, the age difference doesn’t bother him. But the Moscow correspondent  also states: “Then he swears violently, for he realises that he must also find a diplomatic solution for the MH17 problem.”)

A problem in science

There is a related issue that must be mentioned usefully. For the above, science has a positive role. However, we should not paint a picture that is too rosy. Science can also contribute to political confusion. A problem in science is that here is a common and fundamental misunderstanding w.r.t. the Impossibility Theorem of 1951 by Kenneth Arrow, a mathematician who got the Nobel Prize in 1972 for this theorem and other work. The impossibility theorem might seem complex but is rather simple, for: assume some properties and give a counterexample. The impossibility arises by a simple mechanism, namely by assuming that collective decisions can be built up from voting on pairs of issues. Obviously the world is more complex and one cannot neglect the context. However, mathematicians are very fond of the misconceptions that Arrow created, and political scientists are very fond of acting like they understand the mathematics. Thus there is a serious problem of malpractice in science itself. Cases of this malpractice in Holland are here 1, here 2, here 3. A problematic situation exists in the USA w.r.t. Donald Saari, last year the chairman of the Conference Board of the Mathematical Sciences (CBMS), the union of all mathematical associations. See also the relation of CBMS to the US Common Core in mathematics education.

In 1990, one of my papers at the Dutch Central Planning Bureau (CPB) showed that Arrow confused voting and deciding. The mathematics of his theorem is okay but his interpretation is wrong. One can use pairwise voting results but one should not regard these as decisions. The final decision can only be made when all pairwise results are in and related to each other. A cycle in voting scores translates into collective indifference. When there is a tie, then there must be tie-breaking rules. Thus, democracy is not impossible, but a bit more complex than Arrow suggested. Unfortunately, this paper of mine of 1990 got hit by censorship of science, as also my other papers, including the new approach to resolving unemployment and stagflation. The paper on Arrow’s impossibility theorem has been transformed (partially) into the book Voting Theory for Democracy, but the results on unemployment and stagflation are still importantly much under censorship that should be lifted.


Young UK and scientists all over the world are advised to protest against the political abuse of this Brexit referendum, and it would also help when they start boycotting Holland till the censorship of science by the directorate of the Dutch CPB is resolved.

Appendix. The Brexit referendum outcome

The following data can be found at YouGov (exit poll on ca. 5000 people) and there is a nice graphic at wikipedia (a portal and no source). John Burn-Murdoch at the FT presents a graph that turnout rises with age, from a low 65% for age 30 to a high 80% at age 50 (not copied here).

wikipedia brexit

YouGov Brexit


The former weblog entry discussed how a school can “improve” its success rate by ditching weak students. I used a small theoretical model to show this. It is more advantageous for a weak student to try for graduation twice, and by using a corrected success rate the school is not punished for that. Let us now look at some real data from a real school.

A disclaimer is that I am not at home in this field of study. My interest has been in the didactics of mathematics and overall economics of education, and while I have looked at issues of testing, this present application is a new topic for me. The Dutch Inspectorate of Education reports on this since 1817, and I am merely feeling the water and asking questions for my understanding.

Jan Jimkes has been critical of the government policy on arithmetic tests in highschool, see here and here. So let us see – tongue in cheek – how his own school is doing. Jimkes got his mathematics degree in 1966 and has been a math teacher for 36 years and former conrector of St. Bonifatiuscollege in Utrecht. Arithmetic indicates that he must have retired around 2002. Our data are from 2013-2014 and thus have not been affected by Jimkes directly. This discussion might not be impartial because I disagree with Jimkes on some points (see here), and because this is also my own highschool where I graduated in gymnasium in 1972. We might have arrived at Boni around the same time, but perhaps at the different buildings for I have no recollection of him back then, and my math teachers were Van Gils and Andringa.

The following discussion is not about mathematics education but about equality of opportunities in general. In Holland VWO = 6 years preparation for university, HAVO = 5 years preparation for college, VMBO = preparation for trade school. Within these curricula, there again is the distinction between the humanities (A, qualitative and likely not quantitative) and science (B, quantitative and likely also qualitative).

In elementary school at the end of grade 6 for pupils of age 12, teachers advised whether they might do VMBO, HAVO or VWO. At Boni grade 9, students have been allocated to HAVO and VWO, and we can see how the prediction worked out. Boni might get good graduation scores on VWO by sending weaker VWO students to HAVO.

The history of Boni is that it originated in the emancipation of Catholics in a Calvinist country. Originally Boni wanted to make sure that students capable of university got a real good VWO education (HBS, atheneum or gymnasium). The addition of HAVO is an outgrowth and originally no core business. The option of HAVO is agreeable for students who don’t fit VWO but who can remain in the same school. A student may feel better with high grades in HAVO than with low grades in VWO. Having graduated at HAVO at Boni, 17% continues in VWO again.

The data basically come from the school itself. Boni reports to DUO, and there is a visit by the Inspectorate of Education. The results are reported on by the Inspectorate and on the website “Scholen op de kaart” (SodK) where schools compare with each other. For Boni the link is here. I assume that these data will be updated to a new school-year, and thus I copy a graph below.

Report by the Inspectorate of Education

The Inspectorate of Education gives us the Boni report of 2015 about school-year 2013-2014. On page 9 of the pdf we find the following text. This text refers to the “scorecard 2014” while we will look at the scorecard 2015 with data on 2013-2014:

“The success rate for grades 7-9 is satisfactory for the scorecard of 2014 [sic]. However it is unsatisfactory for the years before. Relatively many students with an advice for VWO transfer to HAVO. The success rate for grades 10-12 is good.” (My emphasis and translation (English teacher Boerlage) of: “Het onderbouwrendement is volgens de opbrengstenkaart 2014 [sic] voldoende maar in de jaren daarvoor onvoldoende. Relatief veel leerlingen met een vwo-advies stromen af naar de havo. Het bovenbouwrendement is goed.”)

In the scorecard 2015, Boni had 1459 students in 2013-2014, 33% in the first two formative years (grades 7-8), 20% in HAVO (grades 9-11), 47% in VWO (grades 9-12).

We indeed find that an initially surprising percentage of potential VWO students are actually at HAVO. See also the graphs below.

  • In grade 9 at VWO: 71% had an original VWO advice from elementary school, 27% had HAVO advice and 2% mixed.
  • In grade 9 at HAVO: 52% had HAVO advice, 47% VWO advice and 2% had a mixed advice.
Report by SodK, schools comparing with each other

SodK gives graphs of above data. HAVO is on the left, and VWO is on the right.

  • The red bar gives students with an original advice for VWO. Many are at HAVO indeed. However, HAVO is a smaller fraction of the school, so there is also the effect of the denominator. The “comparison” is awkward.
  • The purple bar gives students with an original advice for HAVO. Surprising for me, still about a quarter of VWO classes are filled with these students. Apparently, prediction at the end of elementary school is difficult.

Original advice for students in grade 9 at HAVO (left) or VWO (right) (Source: SodK)

Comparison and translating the graphs into numbers

The grey bar is a “comparable group”, not necessarily the national average. This “comparison” however is distorted by the mixed HAVO / VWO advice, that is important for the “comparable group” but not relevant for Boni.

For VWO, the 2% mixed advice for Boni can be allocated equally to 27+1 = 28% HAVO advice and 71+1 = 72% VWO advice.

For VWO, the mixed advice in the “comparable group” is about 18%. Allocating this equally, we find a HAVO advice of about 16+9 = 27% and a VWO advice of about 64+9 = 73%.

Thus Boni is not exceptional.

VWO Boni “Comparable”
VWO Advice 72 73
HAVO Advice 28 27

For HAVO, the 2% mixed advice for Boni creates a choice, and let us assume that the Boni split is 53% versus 47%.

For HAVO, the “comparable group” is about 22%. Let 55+11 = 66% have had HAVO advice, and 14+11 = 25% have had VWO advice. Then we still lack 9%. Probably this is VMBO advice, not shown here. The graph is crooked, and creates some uncertainty.

Boni may seem exceptional but when a normal outflow from its large VWO intake enters a smaller HAVO department, then this share must be higher. My suggestion is that the Inspectorate develops a better comparison for the effect of the denominator (I don’t feel like trying).

HAVO Boni “Comparable”
VWO Advice 47 25
HAVO Advice 53 66 + missing 9 = 75
More data from the Inspectorate

The three year data show for the central exam (and not the school exam nor the joint result) for 2013-2014:

  • At HAVO grade 9-11, 73% didn’t retake a class, and the graduation grade point average (GPA) on a scale of 10 was 6.3 (slightly below average). At SodK we find the final success rate: 89.6% or roughly 90% graduated in 2014.
  • At VWO grade 9-12, 79% didn’t retake a class, and the graduation GPA was 6.5 (above average). At SodK we find the final success rate: 95.1% or roughly 95% graduated in 2014.
Is Boni cooking the books ?

Boni’s profile for VWO hardly differs from the comparison group. If someone is cooking the books then everyone is. We have no data for the final selection point at grade 11 that we discussed earlier.

Boni’s profile for HAVO is partly explained by the denominator effect. It is unclear how a correction would look like, and thus we must postpone judgement. We still would expect a graduation GPA that is higher than the national average. However, it is a bit less. Thus it is more likely that VWO students are transferred to HAVO for the mere reason that HAVO actually fits them. Perhaps these students are disappointed and not motivated to work hard for HAVO ? However, 17% of the HAVO graduates continue to VWO (see here). It is not clear to me whether these are only original HAVO students or whether there may also be former VWO students who get a second chance.

Overall, a possible explanation is that (some) elementary schools give their pupils an advice of VWO to try to get them into well respected Boni, after which the true selection happens at Boni.

Apparently, forecasting a career is difficult. Testing on mathematics skills at elementary school is not necessarily difficult but rather deliberately crooked in Holland. There are two approaches: “realistic mathematics education” (RME) and “traditional mathematics education” (TME). The tests created by CITO still allow that both methods score equally, looking only at the outcome of sums. However, only TME prepares for later algebra in highschool. Thus, CITO better develops tests that also attach value to the intermediate steps that are used to find the outcome of sums. There is also my proposal for better “neoclassical mathematics education” (NME), see here. See my letter to KNAW and CPB.

Boni seems to have a useful school model. By selecting pupils who are closely related (core and related non-core), it can concentrate on the core, while still providing well for the non-core. This model benefits from the fact that there is no education higher than VWO. Schools with a core on HAVO must provide for surrounding non-core VWO and VMBO.

A similar question arises when one can create two classes of the same denomination: mix the students, or create a faster and a slower class ? A criterion should rather be on learning styles, to allow teachers to economise their methods.

What is this discussion about ?

This discussion is actually rather on determination. The distinction between VWO and HAVO is close to the distinction between university and college (professional school). Some people argue that medicine at university is actually a professional education and not an education in science. This indeed also has to do with the learning styles, like Kolb’s contested theory of abstract / concrete and active / passive styles. Potentially VWO and HAVO differ in characteristics as in the diagram below, and the determination test would allocate students, depending upon school capacity or prospect for graduation. It is more likely that there are more dimensions however.

Determination of VWO vs HAVO ???

Determination of VWO vs HAVO ?

The Inspectorate of Education now uses a time horizon of three years. Within this time frame, they can already link graduates to an advice three years earlier. Thus for HAVO they can compare graduation at grade 11 to the start in grade 9.

  • With one year extra, they can link graduation at VWO to grade 9, and allow for resits of HAVO. This look at HAVO and VWO would test the performance of the school itself.
  • With a time horizon of seven years they can link graduates (with resits and return from HAVO to VWO) to the advice at elementary school. This comparison looks into the quality of this advice (with teachers differing from official CITO) rather than performance at Boni.

This model uses graduation as the golden standard (always on top), and uses the common nomenclature for the prospective tests (always on the side). Unfortunately, wikipedia (a portal and no source) presents this table in transposed form. Indeed, better look at wikipedia’s “worked example” that has (inconsistently) the proper orientation.

  • When we spoke about the success rate above, we took VWO as the success. The success rate translates here as the positive predictive value, PPV = TP / (TP + FP).
  • For determination, the discussion was too simple, since there is also the success for HAVO students, with the negative predictive value, NPV = TN / (FN + TN).
  • There is a whole machinery on this kind of test analysis, and it would lead too far discussing this.
Golden standard vs test
Graduated at VWO Graduated at HAVO
VWO Advice True Positive (TP) False Positive (FP)
HAVO Advice False Negative (FN) True Negative (TN)
Returning to the original problem w.r.t. the success rate

Let us return to our detective job that we started out with in the earlier discussion. In this case, graduation is not the golden standard but actually only an imperfect test. We now also take account of students who graduated but shouldn’t have and only were lucky. Thus we assume that there is some golden standard that can determine whether a student is a true VWO student or not. The Inspectorate of Education should study on such a golden standard. For example, when a student is not admitted or fails at Boni but succeeds later, perhaps elsewhere, then this would be a true VWO student. This gives the table below.

The statistical success rate at SodK, say the 95.1% success of the VWO graduation at Boni in 2013-2014, only compares the two rows of students who participated, and then passed or failed. Our problem were the students who were excluded from participation who were true VWO students, while Boni erroneously thought that those were false VWO students (who would either fail deservedly or not be lucky enough to pass anyway). This information is not presented by the Inspectorate.

Golden standard vs test
True VWO False VWO
Participated and graduated True Positive (TP) False Positive (FP)
Participated and failed False Negative (FN-Part) True Negative (TN-Part)
Did not participate False Negative (FN-Nonpart) True Negative (TN-Nonpart)

Perhaps the Inspectorate should first clarify how big the problem is for the whole country before we look at schools. Obviously, a student who is not admitted to the senior (graduation) year, is ill prepared, and after a while it makes excruciating sense not to allow this student to participate in the exams. However, why was this student not admitted to the senior year in the first place ? The issue can thus be paraphrased in the familiar discussion about the risk factors for retaking a class. Still, the earlier point that the success rate better be corrected so that schools are not punished for allowing students to resit graduation, remains, and this would percolate down too to lower grades.


Boni is innocent till proven guilty. These data don’t prove that Boni doesn’t cook the books. With these data, the issue is elusive, rather more on determination than on graduation. We would need more intel based upon the individual capabilities. Potentially we need only information about students in the critical range (whose grades cause discussion in the teacher meeting), but with all the selection going on (e.g. also on A and B flows), we would include all students (also for comparison and denominator). The report by the Inspectorate is still oriented at statistics in the style of 1900 looking at the unit of the school, rather than at statistics in the style of 2000 looking at the unit of the student. It reflects the correct sentiment that schools matter, but still. Management requires measurement. When you don’t measure something, then management runs risks which otherwise might be avoided. Apparently it is not clear to the Inspectorate yet what they really want to know about students. Are you able, now, to formulate your suggestions to them ? A disclaimer is that I didn’t read up on their research agenda, but now I know better what to look for.

There is a trade-off between the success rates of students and those of schools. A school can enhance its image by evicting less performing students, so that only good students contribute to the success rate. Less performing students are directed to schools with lower requirements. Nicely said: there can be schools with different levels that better fit their target populations. Is the latter too good to be true ?

De Maatschappij” (“The society”) is an independent network for people active in business and public service. On June 14 they organised a discussion about education (in-) equality. This obviously relates to success rates.

(PM. The location was the beautiful Hodshon House in Haarlem, seat of the “Koninklijke Hollandsche Maatschappij der Wetenschappen” (KHMW) (“Royal Society of the Province of Holland for the Sciences”) founded in 1752. In this case “Holland” stands for the province and not the whole country.)

Arnold Jonk of “Onderwijsinspectie” (Inspectorate of Education) presented results of the annual survey “De staat van het onderwijs” (a series of reports since 1817). He observed that inequality increases, notably by the statistical phenomenon that children of parents with lower education have increasingly less enjoyment of higher education themselves. Jonk argued that the data showed no single cause. He also observed that inequality between schools is rising.

The discussion caused me to think that the success rate for schools should be corrected for the opportunities that are granted to weaker students. Giving an opportunity should not be punished, in the event when the student still fails.

Other people have already thought about this, see this recent discussion about including CITO scores, e.g. on the website of schools comparing each other. Yet, I find it helpful to organise my thoughts on this, so that I know better what to look for in the future. The discussion at KHMW was open, and then I don’t mind mentioning an idea that isn’t fully developed yet, even though other people may not understand what you are getting at, or think that you are reinventing the wheel. Science is open. The following exposition should be helpful, and is still preliminary.

Item Response Theory clarifies that tests say something about students, but also conversely that students say something about the test. Something must be said about the school too. At the Dutch Inspectorate of Education there are measures on the output of schools, e.g. the “Opbrengstenkaart” formerly known as “kwaliteitskaart” for secondary education. Google showed this book “Het oog der natie, scholen op rapport” (2001). An informative discussion in Dutch is by Janssens & Visscher (undated). The PISA methodology aggregates results for countries, and a similar method might be used for schools. In Belgium, Frank Roels (emeritus) has a popular and partly entertaining discussion (in Dutch) about success rates and ways to manipulate those. Yet, I haven’t looked at these methods in detail.

For this weblog entry I only want to get some preliminary clarity about the problem, by using a simple example, so that it should be clear what the problem is.

Must John retake the junior year or will he be admitted to the senior year ?

Consider a highschool with grades 9-12: freshmen, sophomores, juniors and seniors. Each class consists of approximately 25 normal students. The school accepts that 1 student normally fails at graduation. This failure rate of 4% translates into a success rate of 96%.

Consider John at the end of the junior year (grade 11). He wants to enter the senior year (grade 12) and graduate next year. John is a weak student and differs a bit from the 25 normal students that are admitted without problem (including the one who failed last year). The school wonders about what to do with John. If John would be admitted but fails graduation next year, then this will affect the school success rate unfavourably, for this means a failure of (1 + 1) / (25 + 1) = 2 / 26 ≈ 7.69% or a success of 24 / 26 ≈ 92.31%. Perhaps it is better for John to retake the junior year, and be better prepared for the senior year ? The school has a policy that students may only retake one year. If John would fail twice in the junior year, he is removed from the school. Then it is favourable for the school that he no longer shows up in the school graduation success rate. For John it might be better to go to the senior year directly, for then he has the option to try for graduation twice.

In sum: for John it is better that he is admitted to the senior year (path A), for the school it is better to let him retake the junior year (path B). Assuming some transition probabilities, the following table summarizes the cumulated rates of success for John versus the school for the two paths. The Appendix below shows how this table was constructed.

When a school district has a rule that schools should have a two year average success rate of at least 95%, then the school will be inclined to steer towards path B. In this case the school is small and fluctuations can be explained, but still.


John’s success rate

School weighted average success rate

A. Admit



B. Retake




W.r.t. rising inequality, it is unclear to me whether above perverse selection effect occurs. Having different levels of education already mitigates the effect. Schools with low education levels might already have high success rates, because they collect the dropouts of the higher levels. Still the overall effect might be a reduction of the level.

Schools with high levels may also have high success because of the selection, and not be in need for additional money. The better schools might have the market power to select the better students. Higher educated parents may provide for extra stimulus. For lower educated parents this stimulus would have to be provided by the schools, but those may not have the resources. In that case, the government might give more money to the lower level schools. In the middle, the schools with more aspiration who give students more opportunity are punished because their success rates drop, and they are less likely to get government money.

This discussion tends to interprete the school success rate as applying to the school, but it is a student average. Our real focus is on the overall success rate for all students. This includes the probability that the weaker student John also graduates. Thus the school success rate should better be corrected for giving John this opportunity.

In Item Response Theory (IRT), only students with their competences and the tests with their challenges occur, and there is no school indeed. The individual transition probabilities however are affected by the quality of the teachers, whence IRT must be corrected for the school effect. Indeed, it is a general notion that the quality of teachers is one of the main instruments, as argued by Cornet et al. (2006) at CPB. Jochems (2007) however argued that we know relatively little about education of teachers.

A quick fix

Above weighted average punishes John for requiring one more year to graduate. A quick fix is to eliminate John from the calculation when he fails in the first senior year, conditional on that we know that he will succeed the second time. When the school success rate is based upon the remaining normal students, then this suffices for the school image. In the table below we replace the low success rate of .9231 (in red) by the normal .96, as if John didn’t participate. As a result, the weighted average rises from 94.9 to 95.5. The school is rewarded by a higher success score than along path B.

Now, however, the average duration at school enters the discussion. This quick fix is neutral for the delay that already occurs for John. It however assumes that schools and students do not manipulate by turning a failing normal student into a weak student. Allowing students an additional year is a luxury for the school. Potentially though students want to graduate as fast as possible. This quick fix introduces for policy makers a delay in the outcome (with first an estimate only). I have only looked at this numerical example and not looked at the conditions on the parameters. This is just an example case, and we would have to look how it works out, say in IRT. A label for this kind of research might be “opportunity neutral or rewarding success rates”, but perhaps a known or better label already exists. Perhaps this small model then is helpful for discussion.

Why is this important ?

Better control of the transition probabilities and good measures of success are important for the management of the educational system in general. There is also this important question of design: we have differentiation in Secondary Education, so why not in Primary Education ? This gives more application for success rates. Would it not be better to replace the rather uniform system of Primary Education by a differentiated system that is more sensitive to the capabilities of the pupils ? This proposal comes from Henk Boonstra, and I tend to agree, see here, though be warned that I am not qualified for PE.

When you are sick, you go to a hospital, get monitored, and when cured return to society. For prevention, there should be constant monitoring and regular checks. For schools, we would look not only at the dropouts and low achievers but also at students functioning below their capacity in general. If John in the model above fails, he should get another diploma fitting to his results, and the true question is whether  he has reached his capacity (or good basis for a career of choice). We are going to a society that keeps a digital image of your body and mind. The science of testing is with us since phrenology started, but now we need ever better systems to check upon both science and how it is applied.

Appendix on model and numerical example

There are two main paths A and B with each three endpoints, see the diagram. Let us use the following numerical example of John’s individual chances.

  • the probability that John graduates next year: α = 50%
  • the probability that John graduates after retaking the senior year: β = 70%
  • the probability that John becomes a senior after retaking the junior year: γ = 80%
  • the probability that John graduates after retaking the junior year: δ = 60%



Path John’s success rate School success rate, years 1 and 2
(A1) Senior-Graduate α = .5 25 / 26 ≈ .9615,    24 / 25 = .96
(A2) Senior-Senior-Graduate (1 – α) β = .5 × .7 = .35 24 / 26 ≈ .9231,   25 / 26 ≈ .9615
(A3) Senior-Senior-Fail (1 – α) (1 – β) = .5 × .3 = .15 24 / 26 ≈ .9231,   24 / 26 ≈ .9231
(B1) Junior-Senior-Graduate γ δ = .8 × .6 = .48 24 / 25 = .96,   25 / 26 ≈ .9615
(B2) Junior-Senior-Fail γ (1 – δ) = .8 × .4 = .32 24 / 25 = .96,   24 / 26 ≈ .9231
(B3) Junior-Fail (1 – γ) = .2 24 / 25 = .96,   24 / 25 = .96

View on Haarlem, June 14 2016

The Dutch research subsidy allocator NWO had its annual Spinoza Prize event, in which science meets journalism. About this annual event I reported critically in 2012.

The event this year carried the theme of “The scientist as activist”. NWO had invited Alice Dreger as keynote speaker to explain about the advantages and pitfalls of mixing research in the morning with social activism in the afternoon.

Thus, all of a sudden we have sex change on the table. Also, when there is controversy, then one is obliged to look into details. Thus I spent Friday morning listening to Dreger and the discussion, and was forced on Saturday “the morning after” to fact-check it all.

NWO Bessensap in Amsterdam

The invitation at the NWO website was:

“On Friday 10 June 2016 the Netherlands Organisation for Scientific Research (NWO) will organise the sixteenth edition of Bessensap together with the Dutch Association of Science Journalists (VWN). The event will take place at the Rode Hoed in Amsterdam. Bessensap has been revamped this year to be even more in keeping with current developments, both in science and scientific communication.

The goal of Bessensap is and remains to encourage interaction between researchers, science and mainstream journalists, and other communication professionals. The former title ‘science meets the press’ is being replaced by an annual current theme, however. This year it is ‘the scientist as activist’: professors protesting against cut-price meat and climate scientists warning of the present and future disastrous effects of climate change. What role should scientists play in the public debate? And how should science journalism approach activist researchers?

Keynote speaker this year is the American activist researcher Alice Dreger []. As a historian, she studies the history of science and medicine. At Bessensap, Dreger will discuss what happens when science (the search for truth) and activism (the search for justice) collide. After her keynote address, Dreger will continue her discussion with visitors during a debate on this theme.” (NWO website)

Dreger informed us about her personal experience. She had participated in a social controversy, defending a fellow scientist J. Michael Bailey against harrassment, and had become a target of harrassment herself too. Her own university also hit her work with censorship, after which she eventually resigned as professor of Clinical Medical Humanities and Bioethics at Northwestern. She relates her experiences in the bookGalileo’s Middle Finger: Heretics, Activists, and the Search for Justice in Science“.

Dutch journalist Asha ten Broeke was in the audience and praised Dreger’s book, as a thriller that should become a movie. Google shows a twitter exchange between Ten Broeke and Dreger, and an earlier report in a newspaper, Volkskrant June 4, that opens with the issue of prenatal dexamethasone.

Alice Dreger, 2016-06-10, NWO, "The scientist as an activist"

Alice Dreger about “The scientist as activist”

Developing a hypothesis on the controversy

I only want to develop a hypothesis about what is happening. I have spent a major part of the mornings of Friday and Saturday on this issue, with the only objective to have a fair grasp of the situation. It will not be possible to look into all details, which would require e.g. buying and reading Dreger’s book and all commentary. Dreger observes that books are often not read and still rejected, but I don’t intend to read a full book nor to reject or accept it. Once I have my hypothesis, then it is a later option to test it, but I doubt whether I will ever have time to do so.

The situation is complicated by that Dreger may be right on many aspects, like on the matter of prenatal dexamethasone. Dreger seems also to be right in the protest against censorship at Northwestern, but one can doubt whether resignation was the proper response.

Eliminating noise, it appears that the core issue is relatively simple. This is whether Michael Bailey has a sound scientific approach or only a journalistic report on the “Clarke Institute theory of gender crossing”.

Let me invite you to read these two texts, and for readers of Dutch also a third:

Bailey apparently states that there are only two types of crossing and when McCloskey states that her personal experience doesn’t fit those two categories, then Bailey must either call her a liar or revise his theory. Why not respect personal testimony ? There is no need to concentrate on McCloskey, for there are more people for empirical testing. Thus there is no need for controversy but need for more research, and the research question is already clear too.

We find light in the tunnel by the following approach: (1) Common sense. (2) McCloskey is a brilliant economist, and I am an economist who appreciates her work very much. Her statement is to the point. For example, McCloskey is a world authority on ethical theory, and when she observes that Dreger is shallow on ethics, while Dreger’s chair is on bioethics, then this is very relevant observation. McCloskey agrees with Dreger that Andrea James is an activist and no scientist, and this is actually easy to check.

The Huffington Post article has a curious treatment of McCloskey’s position. Using your thumb to invent that two critics of Dreger “talked many times” and still disagree, and implying that both then are wrong, is bad logic.

“Well, which is it? “Proven wrong” by “almost everyone” (McCloskey) or “unfalsifiable” and without “predictive capabilities” and “untestable” (Conway)? McCloskey and Conway must have talked many times. This discrepancy in how they attacked Blanchard’s theory shows how little they cared about its truth — or that they knew it was true.” (Seth Robert)

Robert also argues: “Deidre McCloskey and Lynn Conway are both powerful persons.” This is a misrepresentation. McCloskey has no power and can only use words. People who read her work tend not to take things for granted. I have no information about Conway.

As a scientist, McCloskey is Dreger’s best ally, and it is curious when these two minds don’t meet. When McCloskey invited Dreger to send a draft text so that she could comment to prevent later confusion, then this was proper science.

A background check on potential sources of bias

Bailey’s website informs us that he originally had a BA in mathematics, and after teaching secondary school for a couple of years went to graduate school in clinical psychology. Mathematicians are trained for abstraction, and it is not impossible that Bailey’s attitude still is rather abstract and theoretical rather than focused on empirical observation, even though he has been an intern in psychiatry. An empirical scientist would be much interested in the evidence that causes a rejection of a theory.

Dreger earned her PhD in History and Philosophy of Science. The topic of the PhD study apparently was on the history of “Hermaphrodites and the Medical Invention of Sex“. This background suggests that she has read about methods of science, but has no training by actually doing so. Dreger’s historical research apparently has alerted her to misconceptions by so-called scientists in the past, but dealing with current science today is a different issue. My impression is that Dreger has misread McCloskey’s accurate criticism of Bailey’s approach, and did not properly distinguish this criticism from social activists.

Adding to confusion and reducing it again

You should read the two or three texts above but let me mention that there are more sources, that contribute to information overload. For example there is Julia Serano, who has this criticism. Or there are withdrawn nominations for lammies. Etcetera, etcetera.

The bottom line is: it would be up to professor Bailey to answer to his critics.

It has been kind of Dreger to want to protect a fellow scientist from abuse by social activists. It is better to avoid the risk of becoming the next target. Best is to provide for a climate in the scientific world itself, so that Bailey indeed provides such answers. For example, Dreger might have translated McCloskey’s criticism into words such that Bailey would have understood better that this is criticism that needs a reply. One should not think that management of controversy is simple.

Insert of Tuesday June 14 2016

Though I really didn’t want to spend more time on this, I now located Dreger’s article at PubMed 2008, in which she clarifies that Bailey’s book, published at a scientific publisher, was not purely science but also intended to express personal opinions and speculations.

“From the start, Bailey intended this book to be very different from anything he had published before. Whereas most of his previous work consisted of peer-reviewed articles for scientific journals, this book would be a popularization—based on certain sexological findings of his lab and others, but replete with vivid stories of people the author had met, stories provided to put a human face on those findings. Along with accessible, abbreviated accounts of key scientific studies, the book would also feature the author’s hunches, speculations, and personal opinions. It would include suggestions for further reading, but no other documentation (Bailey, 2006b). Thus, TMWWBQ was never envisioned as a work of science in any traditional sense; instead, Bailey viewed the book as his chance to expose to the masses what he saw as the often politically incorrect truth about “feminine males”: boys diagnosable with “gender identity disorder” (GID); surgically feminized, genetic male children; male homosexuals; drag queens; heterosexual male crossdressers; and MTF transsexuals. Bailey also saw the book as an opportunity to make some money; when he was ready to sell the book, he engaged an agent, Skip Barker, who negotiated in November 2000 a contract and an advance from Joseph Henry Press (p.e.c., Bailey to Dreger, October 2, 2006). Joseph Henry Press is “an imprint of the National Academies Press […] created with the goal of making books on science, technology, and health more widely available to professionals and the public” (Bailey, 2003, copyright page).” (Dreger’s article at PubMed 2008)

Thus, Bailey was an activist himself, and it looks like Dreger may have defended not a fellow scientist but an activist.

Obviously, there is no objection to personal opinions and speculations, and these actually are an important source of information, as these for example might guide future research. However, the issue is to clearly distinguish those from corroborated findings. For example, I use a science name Colignatus. Apparently Bailey nor Dreger nor the editors of the Joseph Henry Press nor the editors of the journal that published Dreger’s article have been careful enough. Both Bailey’s book and Dreger’s article better be retracted. The abstract of Dreger’s article states:

“Dissatisfied with the option of merely criticizing the book, a small number of transwomen (particularly Lynn Conway, Andrea James, and Deirdre McCloskey) worked to try to ruin Bailey.” (In the abstract of Dreger’s article at PubMed 2008)

This fails as a description of what actually happened. Reading McCloskey’s statement on Dreger, referred to above, shows her position on content. This shouldn’t be misrepresented as being targeted deliberately at ruin. Perhaps others have stated such explicitly but McCloskey (p7-8) even explicitly denies this. Thus retract.

Dreger is right that the case causes some questions. When Bailey’s book is published at a science publisher, then McCloskey is right that research may be needed to have been submitted to the Institutional Review Board (IRB). If the book is “science journalism”, then this IRB is not needed, but then it shouldn’t be at that publisher. One cannot use one argument for the other issue. Dreger may also be right that “oral history” is excluded from IRB rules, but if Bailey uses such reports to put a face on statistical results, then he himself creates a mix that still falls under IRB (because one aspect is). Again you cannot use one argument for the other issue. Also Dreger should ask Bailey to retract and restate his views in a manner that avoids confusion.


Given this hypothesis, some tentative conclusions are:

  • The organisers at NWO should have had the same problem as I had, in needing to understand the situation. They should have been able to reason as above. They didn’t do so. They gave Dreger the floor, as if there all of this was entirely new and nobody had time to look into this. This is misleading to the audience, and generates a wider circle of confusion. It is costly to the audience, like I lost time in recovering what they should have done. The better alternative would have been to present the hypothesis as above, and allow both Dreger and others to comment, so that there would have been an informed discussion, leading to more information and reduced confusion.
  • The organisers at NWO left it there, and after Dreger had reported on the censorship, there was no statement by the board of NWO that they were appalled, and would investigate and potentially write a letter of protest to Northwestern. NWO has a department of science communication and they found it useful to give Dreger the floor for their own reasons of selling NWO, but, apparently, there was no commitment to really defend science against censorship.
  • This framing doesn’t help Dreger much. The newsmedia reported on the Spinoza Prize winners but not on the censorship of science at Northwestern.
  • Journalist Asha ten Broeke already reported on Dreger but should look into above hypothesis, in order to prevent misleading people.

After this discussion on controversy and censorship in the NWO lecture hall, various people in the audience went out onto the street, not to protest with banners, but to enjoy the good weather and the view of Amsterdam’s canals. Dutch people aren’t easily shocked about censorship of science.


It is awkward to state the obvious, but let me do it anyway.

  • This weblog advises to boycott Holland till the censorship of science since 1990 by the directorate of the Dutch Central Planning Bureau (CPB) has been lifted (see the About page)
  • the censored analysis concerns unemployment and poverty
  • resolution of unemployment and poverty is crucial for resolution of world hunger (unless you want to distribute food for free, which isn’t likely to happen)
  • when I was a highschool student in 1972 and wanted to resolve world hunger, I decided that I better study econometrics rather than archeology
  • when I studied econometrics I decided that unemployment was the key topic, since this affects the basic needs. At that time I also wrote an article (in Dutch) that the New International Economic Order (NIEO) was no basic needs programme. Jan Tinbergen read the article and responded on the phone that I very likely was right on this. There is some rumour in Holland that Tinbergen was an impractical dreamer who tragically pursued unattainables, but see my In Memoriam (in Dutch) for praise of his wisdom and common sense.

Jan_Tinbergen_1982-smallThus this weblog is of key importance for world hunger too. It is almost impossible to make a dent into world hunger when you don’t get the economies up and running, with systems of social welfare that are supportive of full employment.

Jean Ziegler has called attention to the issue in clear words, though he lacks the economic theory that has been hit by censorship by the Dutch CPB directorate. See this article “We let them starve” in The Guardian. I haven’t read his 2013 book but I am convinced that it is depressing reading, especially when you know that the solution is at hand, and blocked by censorship. It is depressing too, to read this report on Ziegler’s flirt with Gadaffi (and then check out the pictures of Gadaffi’s visit to Paris and president Sarkozy in 2007).

zieglerMartin Caparros also takes issue at world hunger. See this article in the New York Times. See this interview at the University of Barcelona. Remarkably, there appears to be no English edition of his book yet, and the German edition of his book costs close to $100. Readers of German may also check out this article in Die Zeit.


There are various organisations worldwide that try to deal with world hunger. However, their approach is bottom up in the Third World, and not targetted at protecting science in the Western World and debunking fallacies by Western governments. They are fighting symptoms and not tackling the causes. Admittedly, someone who is hungry is helped immediately with food, and thus it is difficult to be critical of micro management. However, the leaderships of such organisations should be aware where the real solution would come from.

It is the same in Holland. The Dutch chapter of The Hunger Project interviewed Martin Caparros (in English). Let me quote two questions and answers:

You argue that hunger is the consequence of a system. Capitalism is the culprit. At the same time you say that of all the major problems, hunger is easily fixed. It seems too simplistic: changing the system is not simple, is it?
“That’s a problem. Because what would it mean to solve hunger? Make sure people take in more calories each day? That is easily done. It requires some technical changes, but you can realize those if you invest enough money on for instance roads or agricultural innovations. But the point I am making is that hunger is a metaphor for poverty. You cannot solve hunger if you do not first solve the problem of poverty. And that requires an overhaul of our system.”

How do we do that?
“Well, who knows? Until the 80s and 90s people thought equal welfare and justice could be reached with socialism or communism. By now we know better. I think politics should help us change to a moral economy.”

(Quoted from an interview of Martin Caparros by Leontine Aarnoudse, OneWorld, 2016-02-19)

Morality would help but, but the track record of morality shows its limits. People will be willing to help out, but the economic process must support this. Thus, there is my economic analysis, that however has been censored since 1990 by the directorate of the Dutch CPB.

Obviously, when these Dutch organisations like The Hunger Project do not question the censorship of science by the directorate of the Dutch CPB, then they convey an image as if there would be freedom of thought in Holland, and that the problem indeed cannot be solved except by micro management and global morality. What can one do, to make these blind and deaf Dutch people grow aware of their blindness and deafness ?

I collaborated on a book in Dutch that discussed unemployment and poverty. When Holland has such difficulty to manage its own unemployment and poverty (with its level of education and technology and natural gas resources) then one should be modest about claims for the Third World, unless more can be said along the lines of this censored economic analysis. Not all is in this booklet yet, since there is this censorship. Economic scientists should look at DRGTPE. I did not yet collaborate on a book on world hunger. Is that the reason why the coin does not drop ?



The Hunger Project Nederland apparently was founded in 1980, and they “celebrated” the 35th anniversary in 2015. I find this difficult to square, for how can you celebrate continuation of something that should have ended in 1981 ? The official text reads that they celebrate the progress towards elimination of world hunger, with a new target for “in the next generation”, but this reads as an official excuse. The subject is a minefield, with this “celebration”, Ziegler & Gadaffi, and other such issues. I suppose that it is quite acceptable to have parties, for man is not only an animal but also a party animal, but please avoid the cognitive dissonance created here.


The last discussion on teaching the quadratic function was fun, but it left me with some questions, and these clarify that there is a long road ahead.

The recipe for re-engineering didactics of mathematics is:

When you see a student struggling with a topic, ask two (introspective, sympathetic) questions:
– What causes that a (and/or this particular) student experiences this struggle ?
– What causes the mathematics community to create this didactic situation ?
Subsequently put your hunches to the test, and recycle till you have a satisfactory answer.
Then proceed with experimental design and hypothesis testing.

Conventional studies in didactics consider only the first question. They assume that the math is sound and that the struggle is caused by external factors like the use of the calculator or perhaps the order of topics in the curriculum. I agree that much can be improved on these external factors, but my diagnosis is that we better first solve the internal problems within “mathematics” itself. A student who struggles with a math question is often quite capable in dealing with complex issues (check what kids can learn) and a struggle indicates that the cards are rigged. The large number of examples of crummy “mathematics” that has already been identified in my book Elegance with Substance (pdf online) shows that “mathematics” education and its conventional research have lost their innocence and even the benefit of the doubt, and are actually the prime suspects.

Didactic divide from counting to addition

I am not qualified for elementary school and the following is only a hypothetical idea, to provide us with a useful example for below.

In elementary school some kids don’t quickly master addition, and get stuck in counting only. For sums like 7 + 2 = 9 they rely on their fingers, which eventually becomes awkward for a sum like 2 + 7 = 9. The didactic question is rather not why these kids get stuck but why others take the leap across this divide from counting to addition. Apparently, a majority of kids have a way of mastering both commutation that 2 + 7 = 7 + 2 and also the tables of addition, so that the outcomes are automated, and are available directly from permanent memory without the need to find the result by actually counting (and check and double check). Didactics helps to identify the points for a natural learning process: (1) awareness of the didactic divide, (2) diagnosis of what is the cause for slow progress (not math itself), (3) treatment, namely create the tables of addition, create awareness of these and the properties, and practice makes perfect. In this manner the teaching environment can help the slow kids to find the learning route they were unfortunate not to find themselves – and later they may be fast themselves again on another subject. (See Henk Boonstra on elementary school reform w.r.t. structural slow and fast.) Learning goals are, even at this age: (a) knowing how, (b) knowing why, (c) knowing about knowing. A child who has mastered addition has every right to be proud of it, and can be invited to explain what it has learned.

PM 1. For numbers above 10 see A child wants nice and not mean numbers.
PM 2. This also clarifies why the tables of multiplication must be known by heart, to cross the divide between addition and multiplication.

Quadratic functions revisited, conventionally

The last discussion on teaching the quadratic function caused a focused search on what others wrote about its didactics. We find a similar issue of exploration versus automation (counting versus addition). Perhaps we may speak about a didactic divide between quadratic equation solving (factoring or completing the square) and automation (quadratic formula). The situation is a bit more complex than with counting and addition (perhaps also because it happens in highschool and not in elementary school). The quadratic formula can be automated only for the standard polynomial form. Handling the quadratic function requires a certain level of algebra overall.

Let us first follow conventional didactics that assumes that the “mathematics” is sound. In conventional didactics, “completing the square” is a solution technique, and not the basic form. The quadratic formula is called an algorithm, a fancy word for recipe. However, the other solution methods follow recipes too, and the key notion is automation.

Colin Foster (2014), ‘Can’t you just tell us the rule?’Teaching procedures relational, distinguishes procedural knowledge (knowing how) from relational knowledge (knowing why), and finds two uses for the quadratic formula: (1)  to learn more about how and why, perhaps prove it, (2) to use it automatically with a focus on a different target. In itself this is sound, but this doesn’t generate a specific didactic strategy yet, except perhaps the order first (1) then (2). Foster laments the current “political” focus on teaching to the test, and thus (perhaps) emphasis on the quadratic formula. Foster usefully indicates that tradition has been strong and research has been weak:

“Vaiyavutjamai and Clements (2006) comment on the lack of research into students’ difficulties with quadratic equations, and since then a number of studies have explored this area (…).”

David Tall, Rosana Nogueira de Lima & Lulu Healy (2014), Evolving a three-world framework for solving algebraic equations in the light of what a student has met before, suffer from Tall’s idiosyncratic verbosity on “embodied” versus “crystaline” notions (see related critique), but it is helpful to see some of the literature and experiments and student confusions. De Lima (2011), summarizes a study in Portuguese:

“In this paper, we present an analysis of 77 14-15 year-old students’ work with a non-familiar situation: the solution of a quadratic equation, written in a factorized form. Data is analysed in the light of a theoretical framework that considers three different worlds of Mathematics and the influence of ‘met-befores’ derived from learning experiences related to them. We show that having the quadratic formula as the only met-before to solve quadratic equations may not help the students to face all kinds of situations involving such equations. In addition, we claim that it is necessary to present to students learning situations that involve at least two worlds, the embodied and the symbolic, but in ways which also allow consideration of characteristics of the formal world, without which students may create their own inappropriate techniques.”

Alwyn Olivier (1989), Handling pupil’s misconceptionsis informative. Students have minds of their own and they may induce themselves to create a recipe like:

(x u)(x – w) = c solves as x u = c or x w = c, which works for c = 0 and hence for any c

One way to approach such frequent misconceptions is to explicitly discuss them, and for example let students prove a theorem that the above is true only for c = 0. Mathematicians create theorems for what they regard as relevant steps. At the level of students, such issues can be considered to be relevant steps that eliminate their confusions. PM. For this particular equation, it is a step further to analyse it more. We can observe that the vertex lies at = (+ w) / 2 and that solutions will be x1,2h ± d for some d. Substitution of generates = (c + (– w)2 / 4). Perhaps this must be mentioned too, perhaps this distracts, and the traditional approach has chosen the latter.

The traditional approach causes the following didactic issue.

  • When a student knows only the quadratic formula, and doesn’t know about factoring or the basic (vertex) form, then teachers will tend to see this as problematic. An equation (x u)(x – w) = 0 must first be expanded to fit the standard polynomial form, with b = -(u + w) and c = u w, to subsequently apply the quadratic formula, and then generate the solution that already could be seen at the start, namely x = u or x = w. Teachers may still opt for the latter level of competence, since the quadratic formula generates a solution for all cases.
  • Teachers may reason that mastery of the quadratic formula still involves some algebraic competence. The latter may also be a delusion, since not-seeing the direct solution to the factored form isn’t quite competent. Lack of algebraic competence can also result in other errors, like when expanding the factored form. A student may however still earn some points for “knowing the quadratic formula” and “making proper substitutions (even from the wrong derivation)”.
  • When a student is mathematically proficient and uses the quadratic formula, few teachers will object. However, such a student would tend to recognise the basic (vertex) and factored forms, and directly state their solution. It would only be an oversight (inattention) when such a student would still adopt the quadratic formula.
A work of art

Source: David Tall, Rosana Nogueira de Lima & Lulu Healy (2014), p14

Quadratic functions revisited, re-engineered

In the re-engineered approach, teaching starts with the basic form. This is not mistaken as a solution method (“completing the square”). It is only a solution method when one starts from the standard polynomial form, but such is a wrong place to start from.

  • The three forms – basic, factored, standard polynomial – arise in structured order.
  • The relationships between parameters and solutions directly follow from this.
  • The relevant questions concern the turning point (vertex) and zeros. Above discussion concerns the quadratic formula and focuses on the zeros, but there is also the turning point.
  • Each form has its own solution method for turning point and zeros.
  • A 3 by 3 table can show how one can transform one form into the other. It can be emphasized that transformation need not be remembered since the questions in the upcoming test concern vertex and zeros. Special attention would be given to special forms like a x^2 = – b x. The latter can be solved directly for x = 0 or assuming x ≠ 0 as x = – b / a. It can be clarifying to also write it as a (x – 0)(x + b / a) = 0.
  • The quadratic formula is derived directly from the basic form, and is usefully remembered for automated application for the standard polynomial form only.

Essential for the handling of the quadratic function are the recognition of the graph and the three algebraic forms and the associated solution approaches. Essential is the attitude to check the solution and correct errors. Essential are the knowing how, knowing why, and knowing about knowing.

After the above has been clarified and assessed in examination, subsequent lessons can be spent on derived questions. The above has a specific learning goal, but the overall goal of competence in algebra and analytic geometry remains. For example, given two points and a particular parabola, state the formula in this form and / or that form. For example, check reactions to solving forms like x^2 h^2 + v = 0 or a x^4 + b x^2 + c = 0. It is fair that students have a stage during which the names of parameters help support memory, but the next step is to recognise forms and infer the role of parameters.

As said, this re-engineered form must be tested in experiments before it can replace current convention. Didactics remains an empirical science, and the students themselves must show what works for them.

There is one issue of design that needs close scrutiny. Consider students who didn’t get adequate arithmetic in elementary school, who didn’t quite learn the tables of addition and multiplication, and thus fail to directly recognise the factors of say x^2 – 10 x + 24 = 0. I always feel sorry for students who have diligently learned the quadratic formula and who lose time on tedious arithmetic, only because of this background, so that they are severely punished on their weak spot.

  • My null hypothesis is: It is only optional to factor a standard form by using the rules on adding and multiplying zeros. This is a different kind of competence. Focusing on this competence is derivative, inessential, and distracting from the real learning goals on the quadratic function. It is proper to show the method of direct factoring and train on this, since it helps algebraic insight and skill, but in another chapter. It should not be a prime element in the discussion of the quadratic function. Having test questions with functions in polynomial form that are easy to factor by students with that level of algebra, puts a bonus on a competence that is irrelevant here, and puts a malus on students that follow the longer route of the algebraic formula.
  • The alternative hypothesis: It might empirically still be possible that students are served by this repeat training on the tables of addition and multiplication and this factoring, perhaps since they better learn what factors are, and perhaps since this is adequate skill in arithmetic so that the quadratic formula can be put on the graphic calculator. (See also Filloy et al..)

There is a curious key remark by David Tall, Rosana Nogueira de Lima & Lulu Healy (2014), p18-19, on the relevance of the didactic divide. It is correct that the form ‘quadratic expression = number’ doesn’t have a simple recipe, see Olivier, but this does not warrant a conclusion that students need not cross a divide. Also, Lima and Healy (2010) mention the basic (vertex) form of the quadratic function and are close to finding the re-engineered approach. Unfortunately, their frame of reference is that “mathematics” is sound and does not need to be questioned itself.

Tall-Lima-HealyPM 1. Relevant is also Jan Block (2016), Flexible algebraic action on quadratic equations, who advises “tasks which focus not on finding solutions of equations by on classifying different types of equations” (p397). Block uses dashed lines and particular shapes to indicate kinds of relations or operations. This might be useful for teachers. For students, it is indeed a reminder that recognition of shapes can by supported by icons. The basic (vertex) form can be identified with an icon of a vertex (U), the factored form by an eye since the zeros can be seen directly (ʘ), and the polynomial can have this (√) icon to remind of the quadratic formula.

PM 2. Here is a long discussion on how to prove the quadratic formula, which doesn’t quite use the easy one of the re-engineered approach (but of course the methods are the same implicitly).


The quadratic function is an example of the usefulness of the recipe for re-engineering. Do not take “mathematics” for granted but accept it as the prime suspect itself for blocking student learning.

Empirical testing of this issue requires the design of integrated lesson plans, with detailed learning goals and exam questions, and worked-out presentations and materials for practice.

Once the re-engineered method has been developed and field tested and shown successful, then publishers can step in and reap the profits from the hard work of the re-engineers.


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