Listening to Poulopoulos, O Dromos
While Europe is busy with refugee immigrants, I take a leave, since I already discussed this two years ago: Europe’s bloody border. Some people accuse this weblog of not looking at the real issues but these people don’t boycott Holland and thus are in serious need of a reality check.
Instead, my thoughts and warm feelings go out to Adriaan de Groot (1914-2006), because his work has always been relevant to me, and comes out top again. He is five years younger than Pierre van Hiele (1909-2010) and like him also studied mathematics with Gerrit Mannoury (1867-1956). De Groot got a bachelor in mathematics, switched to psychology with an MA in 1941, and got his 1946 PhD in mathematics & physics, with supervisor psychologist Geza Revesz who remarkably worked at that department. This study is: Thought and choice in chess (online). Original Dutch: Het denken van den schaker (online).
NB. Mannoury was early into semiotics, and found that “the meaning of a word is its use” quite early before Ludwig Wittgenstein (1889-1951) did. For example, immigrant has a different use than future compatriot. Hopefully one day I can say more on Mannoury.
The spelling checker alerts us to problems in this Abstract by Amsterdam University Press. There is also a distinction between a chess master and the computer game chessmaster. But it is great that the English translation is online.
“What does a chessmaster think when he prepartes [sic] his next move? How are his thoughts organized? Which methods and strategies does he use by solving his problem of choice? To answer these questions, the author did an experimental study in 1938, to which famous chessmasters participated (Alekhine, Max Euwe and Flohr). This book is still usefull [sic] for everybody who studies cognition and artificial intelligence.” (Abstract)
A.D. de Groot (1914-2006) (Wikimedia)
Chess and its levels of competence
I will copy much from the obituary, by Fernand Gobet, ICGA journal 240 (?), 2006, p236-243 (?).
“De Groot’s thesis did not, however, attract the interest of chess players only; it was a harbinger of the cognitive revolution in psychology that would occur in the early sixties.Because of its strong impact on cognitive psychology, and because of the breadth of the study, de Groot’s thesis can safely be considered a classic in the field.” (Gobet p36)
Indeed, Gerald Goldin reminded us of the strong influence of behaviorism in the USA, for us: on testing students – see here. The book by De Groot helped the turn to cognitive psychology.
Fernand Gobet p239-240 summarizes the findings by De Groot (1946) and it has been input for Van Hiele (1957).
“De Groot proposed also that a player’s thinking process may be divided into four main phases: orientation, exploration, investigation, and proof. In the orientation phase, players collect relevant information and try to form a first (tentative) judgment of the position. During the exploration phase, sample variations are analysed, and, typically, the number of critical moves or plans is reduced to two. The two candidate moves are analysed in great detail during the investigation phase, which is characterised by a more in-depth search than during the exploration phase. Players strive to validate their favourite move (or plan). Note that most of the argumentation used by chess players consists of convincing themselves that one of the two variations is better than the other. Finally, the proof phase is used to recapitulate the information obtained in the analysis and to check the correctness of the argumentation. De Groot described also several chess methods used by players to reach their solution. These methods include strategic and tactical plans, ideas, and goals. Note that while they differentiate well between strong and weak players, all these methods are tied to the domain of chess. The higher-level thought and choice methods, which organise the structure of the protocol, did not differ between players of various skills.”
I don’t want to go into chess now. Some meta-comments are relevant:
- Personally, I found it quite surprising that the Georg Rasch model (on competence in reading by Danish children) was mathematically the same as the Arpad Elo rating for chess players: Item Response Theory. It is included in my book Voting Theory for Democracy.
- Main idea: When you are not challenged enough then you get bored, when the challenge is larger than your competence then you get stressed, and when challenge and competence match then you get flow.
- This is also a model for competition between theories or academic papers. Currently editors select the articles of a journal – or manage the selection of those. Alternatively, researchers just put their article online. Subsequently the process of Elo rating starts. The main question is how to define the rules of the game. Facebook does it with “likes”. But this is too coarse.
One of the sickest comments in science is to say about an article: “Get it published in a peer-reviewed journal, and then I will look at it.” It is illogical and perverse, and a frequently abused lame excuse for not looking into criticism. When a scientists looks at the article directly, then this is peer-review on the spot. Not looking at it, is shifting the effort to others. For example, when you contact a scientist to report an inconsistency in his or her thesis: then this scientist should look into it, and the reply should not be: “Get …. it.”
There are also other players who discovered that it pays not to stick to the rules. See my game of chess with Vladimir Putin, and the performance of Garry Kasparov on Dutch TV. Surprisingly, in Gobet’s Obituary on page 241 we find the game of chess of 3000 BC that we discussed yesterday, and in this publication Kasparov apparently succeeded in replacing the Egyptian pieces by Staunton pieces. It still isn’t clear yet who must move. The ancient Egyptian or Putin rule is that there are no moves: just proceed at will.
De Groot Obituary, by Fernand Gobet 2006, p241
In retirement: Forum theory
From onset to retirement is a great leap, but it shows De Groot’s most important contribution. This is the Forum theory. Gobet summarizes:
“During his retirement, de Groot spent much of his energy on philosophical questions, most of them related to psychology. A first theme is related to the notion of truth in science. The Forum Theory , which he had been developing over thirty years, insists on the idea that science is a communal activity directed towards rational consensus. As there is no absolute truth in science, all that scientists can do is to strive for truth, that is, to strive for theories having the highest possible level of certainty. This criterion is met in the case of statements that are unanimously endorsed by all pertinent scientific experts. Such statements then are scientifically true to the best of our present knowledge. Neither are the rules for the correct way of conducting science unchangeable or indisputable. These, too, are to be discussed, and agreed upon in what de Groot calls the forum of expert opinion. A second important theme in de Groot’s reflections was a conception of unifying psychology, a field that is now split into innumerable schools. His approach to this gigantic task was to strive for agreements on the definitions of basic concepts in scientific psychology. De Groot conceded that the task of bridging methodological and terminological differences between schools will not promise any early success. However, connectibility of terminology and method is a necessary requirement for any mature scientific discipline, he argued. Working on it is a must.” (Fernand Gobet, p238)
The key book is De Groot (1982), Academie en Forum, as far as I know not translated into English. The book is great and has for example these elements:
- Forum continues where his other book Methodologie (1961) ends (see below). Methodologie already contains the Forum (capitalized) but still ends somewhat depressing: exact results like in mathematics cannot really be gotten. The further development of Forum Theory is a positive idea, and uplifting.
- It indeed also contains the suggestion to look into Elo-rating of research (-ers).
- Academia discusses education research, design of structure and curriculum, selection processes for higher education (equal input of time versus equal output of quality), innovation (at that time). Design of new topics for education should be accompanied by description how those are going to be tested.
- It rejects the triad knowledge, skill and attitude, with the argument that it is rather difficult to operationalise and test attitude; and replaces this with another scheme.
- It highlights the sectarian character of Holland. Two persons are a church; a third causes a schism.
My book Trias Politica & Centraal Planbureau (1994) page 81 quoted from Academie en Forum p 9. Appendix A below contains that quote and a remarkably fair Google Translation of it. Please observe that one objective of this weblog is to contribute to the unification, by showing how findings are related. The suggestion of an Economic Supreme Court is also based upon a Definition & Reality Methodology that supplements De Groot’s Methodologie, and that is also required to qualify the Van Hiele theory of levels.
A bit on significance 1956
Psychologists Eric-Jan Wagenmakers and others found it useful to translate an article by De Groot (1956) on (statistical) significance. I agree that the translation is helpful. Let me immediately refer also to Ziliak & McCloskey (2006) Cult of Statistical Significance. In addition: large sample sizes may easily create statistically significant differences: but with little relevance for meaningful significance (how you want to use the results).
“Adrianus Dingeman de Groot (1914–2006) was one of the most influential Dutch psychologists. He became famous for his work “Thought and Choice in Chess”, but his main contribution was methodological — De Groot cofounded the Department of Psychological Methods at the University of Amsterdam (together with R. F. van Naerssen), founded one of the leading testing and assessment companies (CITO), and wrote the monograph “Methodology” that centers on the empirical-scientific cycle: observation–induction– deduction–testing–evaluation. Here we translate one of De Groot’s early articles, published in 1956 in the Dutch journal Nederlands Tijdschrift voor de Psychologie en Haar Grensgebieden. This article is more topical now than it was almost 60 years ago. De Groot stresses the difference between exploratory and confirmatory (“hypothesis testing”) research and argues that statistical inference is only sensible for the latter: “One ‘is allowed’ to apply statistical tests in exploratory research, just as long as one realizes that they do not have evidential impact”. De Groot may have also been one of the first psychologists to argue explicitly for preregistration of experiments and the associated plan of statistical analysis. The appendix provides annotations that connect De Groot’s arguments to the current-day debate on transparency and reproducibility in psychological science.” (Abstract by E-J. Wagenmans et al.)
De Groot wrote a classic Methodologie, (1961, 1994 online). The recommendation in Dutch by G.J. Mellenbergh on pages v-vi is well-deserved. Apparently De Groot’s empirical cycle is appreciated by the English speaking wikipedians, but the Dutch version looks deserted. Amazon states that the English translation Methodology (1969) is out of print.
Logic, philosophy of science and methodology were an early interest of mine, see A Logic of Exceptions (1981, 2007, 2011). It also caused the Definition & Reality Methodology used in DRGTPE.
Methodology (the study, not necessarily this book) appears to very relevant – see here – when you want to understand the Van Hiele theory of levels of insight.
Psychologist Ben Wilbrink rejects Van Hiele’s theory, referring to Popper: the theory wouldn’t be falsifiable.
- Interestingly, Popper’s approach in the philosophy of science is based upon the approach in psychology by Otto Selz (1881-1943).
- The Dutch wikipedia text states that De Groot was inspired to his thesis on chess by work by Selz.
- However, see this discussion that explains where Popper’s criterion of falsifiability doesn’t work.
- A major problem with the rejection by Wilbrink is that he apparently is not interested in mathematics education research: but it is strange to do “psychology” and not look at the relevant field of application.
One reader at Amazon gave it 100% appreciation, with wonderful words like paedagogocal (kids enjoy a go-go approach) and crimonology (monologues like this weblog are a crime), while the expansion to Ayurveda comes as a surprise out of the blue.
“This book is complete, superb and perfect, therefore handy for research and practical work. It has a high abstract level. Exact for all social sciences; psychology, clinical psychology, social psychology, sociology, paedagogocal [sic] and political sciences, journalism, etc. Regarding the human aspects also for biology, medical science; neurology, psychiatry, law, crimonology [sic], general economics, languages and history. A.D. de Groot became a doctor cum laude in Math and Physics in 1946. He had a Fellowship on the Center for Advanced Study in Behavioral Sciences, Stanford, California, 1959-1960. With this book one can expand too, f.i. like Ayurveda methodology, where cures by placebo effects or self-curings are studied too.” (By JPR Petersen on December 2, 2011 on Amazon)
Gobet summarizes the conundrum of psychology that cannot observe “thinking”. I take two quotes:
“De Groot’s approach to psychology is complex. It is a subtle mixture of “hard” techniques, mathematical and statistical – do not forget that he was a professor of methodology for over 20 years – and of “softer” approaches, such as interpretative analysis of verbal protocols. In this case, the psychologist tries to understand the subjects’ behaviour at several levels, some of which are not accessible through sheer quantitative techniques. His work on chess, starting with his thesis, offers a good example of the concomitant application of these two approaches.” (p 241)
“History may prove de Groot correct after all. In the last two decades, there has been a renewal of interest in more qualitative ways of analysing chess data. There has been a revival of interest, too, in “higher descriptions” and of “global descriptions” of positions, besides the more detailed descriptions at the chunk level. Obviously, this is the level you get when you ask experts to speak about their field of expertise. And this is the level of analysis de Groot emphasised in Thought and Choice in Chess.” (p242)
De Groot 1961 overlooked the 1957 Van Hiele levels of insight
Van Hiele’s thesis’s list of literature mentions De Groot (1946) on chess and De Groot (1955) Cognitive psychology and education in geometry in the introduction to geometry. (This is my translation of the title.) Apparently the Dutch term “aanvankelijk meetkunde-onderwijs” means geometry in junior highschool. (Van Hiele also refers to Emma Castelnuovo Intuitive Geometry. Lehrer Rundbrief VII, 11. See my earlier question on history of math (ed).) (This link generates a different person, teacher A.J. de Groot.)
Given that Van Hiele’s 1957 thesis concerns levels of insight, De Groot could have taken an interest, also with the common background in math and Mannoury. Indeed, Van Hiele’s thesis is in the literature of Methodologie (my copy 1961), and it is mentioned on page 184.
The great disappointment is that:
- De Groot doesn’t include the subtitle in the reference. The full title of Van Hiele’s 1957 thesis is, my translation: “The Issue of Insight, Demonstrated with the Insight of School Children in the Subject Matter of Geometry.” Thus he indicates that geometry is only used for an existence proof for the levels of insight. Observe the pun: geometry itself uses demonstration as a method.
- De Groot puts Van Hiele into the box of “geometry only”, instead of seeing that Van Hiele presented a general theory of insight, cognition, epistemology.
- We can only suppose that De Groot was so busy with Methodologie that he wasn’t really interested in new insights coming from mathematics education.
Never teach generalities by giving only one example. Give strings of examples, so that people see the general portent. Don’t give people any opportunity to put you into the box of a single example.
This is the relevant section of De Groot (1964), Ch 6, p 184. Let me cheat in translation for a first time now, and edit the output of Google Translate. The originals are in Appendix B.
“But what are ‘educational objectives that are considered important ‘ in the case of plane geometry in the first grade of junior highschool ?
One can have rather different views on the aim of teaching geometry. One can even argue whether it is a necessary part, for example of the advanced (HBS) program. As is known it has indeed been proposed to replace plane geometry entirely by something else, for example symbolic logic or set theory. One can see the aim as limited, strictly tied to the program itself: learn to solve certain types of problems. Or one can see the aim as wide in scope, for example: learn to think, analyse a problem, learn to apply systematic methods of analysis and general methods of solution, also for other objectives (cf. e.g. BOS 1955). Or one can put emphasis on the spatial aspect: geometry as a means to develop a structured ‘spatial insight’ (e.g. VAN HIELE 1957). Or, something else again, as a way for a first encounter with a scientific, partly formalised deductive system. There are many sagacious and profound reflections on this – and few conclusions in agreement.”
St Nicholas 1965
Our hero De Groot was no saint but a mere human, and one day became ill. However, he turned his illness into a wonderful opportunity to research the issue of St Nicholas. De Groot (1965), Saint Nicholas: A Psychoanalytic Study of History and Myth, discusses:
“In this book, the fascinating St. Nicholas story is examined from a specific angle. Intrigued by the abundance of fertility and birth symbolism in folklore and legends, the author has tried his hand at a “psychoanalysis of St. Nicholas.” To this end, present-day (Dutch) folklore is traced back to medieval customs and legends and, through a partly historical, partly psychoanalytic interpretation, to pre-Christian beliefs, to Germanic and Greek gods– and in particular to the birthplace of the legend, the city of Myra. The result is an absorbing and often surprising perspective of sixteen centuries of Christian culture.” (cover text)
This is essentially the same analysis as the later one by Tony van Renterghem (1995), When Santa was a Shaman: Ancient Origins of Santa Claus & the Christmas Tree. De Groot has a tougher task as a psychologist, also thinking about semiotics, while Van Renterghem has more liberty to wonder at questions and cultural relevance.
My own proposal since 1992 is to get rid of the religious burden and male chauvism here, and speak about Kidda Claus or Claudia, with helper Jester Peter or Petra. In Dutch: Kinderklaas & Narren-Piet.
Perhaps I am overdoing this. “Santa Claus” is already different from “St. Nicholas”. The fellow in his reindeer sled is not the common religious saint. The situation in the USA may already be neutralised, except for the male chauvism.
Santa in Holland has a helper: not an elf, but a character with the traditional name of Black Peter and the appearance of the black-up Morris Dancers.
- Protests about racism on one side and pleas to keep the tradition on the other side are at boiling point.
- Last year the mayor of Gouda arrested 90 people in the mayhem, see my report. Of these 90, 89 have been acquitted so this indeed smells of an abuse of power, and the remaining 1 likely is a case to cover up police brutality.
- My proposal is to use the name Jester Peter or Petra, and allow all kinds of colours (of which black might be one, but only as one of many). Keep most of the tradition but don’t use a name that plays into racist misunderstandings.
Holland has close connections with Russia, – by chess, – by the Romanovs, – by MH17, – by the current Russian boycott of Dutch agricultural products, – by the Crimea Scythes gold that was on exhibition in Amsterdam when Russia took the Crimea: with now the tantalising question whether it should be “returned” to the Ukraine or Russia. And thus also with St Nicholas, the patron saint of Great Russia. We can only hope that the boycott of Holland is successful for the right reason.
Russian icon (Wikimedia commons)
In Holland, mathematician Hans Freudenthal (1905-1990) rejected empirical methods, like statistics, because he was trained to be an abstract thinking mathematician and he knew little about empirical science anyway. He created “realistic mathematics education” (RME) based upon his own insights on what would work. He also succeeded in establishing this as the norm in Dutch math education. Well, there are plenty of reasons to be wary of abuse of statistics, but the only answer is to become a better statistician. This however was lost to the mathematics education research (MER) community.
Given the void thus created by the absence of MER, psychologists stepped in. At the academia they for example study number sense: in kindergarten and the first years of elementary school, the importance of fingers. As applied science, producing tests as paid-for products, there is CITO. At CITO, (psychological) testers design, process and analyse annual tests of pupils leaving elementary school, for language, math and other subjects, and those support advice on the subsequent school system. It is a huge achievement of De Groot to help set up such a system, that creates some standards, and that doesn’t leave kids in the jungle of well-meaning but perhaps incompetent teachers.
CITO must be praised for this too, since it is by this kind of testing that the failure of RME came to full attention. Teachers at higher and middle education were complaining that students could no longer count, but CITO turned the anecdotal complaints into facts – also eliminating the sneer to similar complaints already in antiquity.
It is however curious that:
- These psychologists rely on their own understanding of arithmetic, and they don’t feel as if they should study MER.
- The psychologists apparently took RME as the authorative standard of MER, and devised tests to measure RME, contrary to Freudenthal’s view.
- Thus CITO created tests with “context sums”, in which text is used to describe situations, such that pupils must detect the underlying mathematical model, and then solve the issue and calculate some answer. See our discussion on Gerald Goldin for similar test issues in the USA.
- CITO allows an uncontrolled experiment on children, while there would be scientific and medical rules for chimps and rabbits. There are now two competing methods in Dutch primary education: RME and Van de Craats’s “traditional method”. A scientific experiment would be stopped once it is clear what method is best, after which the best method is given to all guinea pigs. CITO just allows the mess. See my letter to CITO, in Dutch, October 18, that didn’t get a decent reply yet. CITO refers to the Inspection of Education, as the formal authority for testing elementary school kids. This is a false referral. The Inspection isn’t a scientific institute. CITO does the actual testing and claims to maintain standards of science. My question to CITO is one of scientific ethics, and they simply dodge it. In psychology it might be called cognitive dissonance.
Problematic psychological research on mathematics education
I am no psychologist, and only give my response from teaching practice and MER and econometric technique (including Jöreskog’s LISREL and latent variables).
Overall: Van Hiele opposed concrete versus abstract while Freudenthal misrepresented this as model versus reality (applied mathematics).
To test mathematical competence according to Van Hiele, you would have to test at the various levels of insight. Students would have to know mathematics before they can apply it – by level – and it is not proper to equalise understanding to applied mathematics. Mathematical competence is not a collection of fields of application. Testing Freudenthal is easier, because you can resort to situations of applied mathematics. It reduces to behaviorism again. When the chimps push the right buttons on the calculators then by definition they have mastered some skill.
These so-called cognitive psychologists have reduced to behaviorism again.
Let me mention seven problematic experiences, and correct me if I am wrong, because I am no psychologist, and they do a lot of testing about issues that I am not aware of:
- Research on number sense can be invalid because of inadequate handling of pronunciation of numbers, see here. It is very curious that there is no movement amongst psychologists to reform collective pronunciation of numbers. (Norway had a reform in 1950, the exception.) See my booklet A child wants nice and no mean numbers (2015) (online).
- Stellan Ohlsson inverts the process of learning, saying that it would go “from abstract to concrete”, but he means to say “from vague to precise”, see here. How is it possible to confuse these terms ? It is not just Ohlsson as a single person, because he is member of a community that would have discussed these issues.
- Psychologist Ben Wilbrink shows inadequate grasp of methodology, and doesn’t want to look into this, see here.
- CITO tests mixed fractions in the traditional manner, but those are didactically cumbersome. If psychologists – and especially mathematically capable psychometricians, with the journal Psychometrika founded in 1935 – had been aware of MER then they could have protested early on that this isn’t mathematics but “mathematics” – see the discussion on the torture by Jan van de Craats, see here. (Let me refer to Van Hiele (1973) with a proposal to abolish fractions – here.)
- CITO tends use outcomes of sums as the indicator of achievement, and neglects the methods how the outcomes have been achieved, apart from legal rules that allow or don’t allow a calculator. (Generally, when there is a descriptive text, then it is called a “context sum” and then calculators are allowed.). Thus, kids who use traditional algorithms (e.g. long division) and kids who use RME algorithms (e.g. partial quotients), would be judged equally competent when they have the same outcomes. This is not only the equalisation of “good method but small error” and “hopelessly lost”. Such is a common feature of computerised testing and probably cannot be avoided (except by creative chunking): except by concluding that some tests shouldn’t be computerised. The true problem is that you require the traditional algorithms in arithmetic to do algebra at a later stage. Thus RME might seem to generate “competence in simplifying 165 / 7” but in fact it maims your brain for higher schooling. Hickendorff (2011), a cum laude thesis using CITO data, falls into that trap. This thesis played and plays an important role in the discussion in Holland, with its conclusion as if the traditional and RME methods would be equally effective at the end of elementary school. Such a conclusion only derives from a disregard of MER. Hickendorff explicitly states not to be competent in that field. In itself it is a sign of integrity to emphasize what you are not competent in. This clarity is much appreciated and helps us to identify the problem. For, there still is a problem. Apparently she worked in an environment that was cocooned from the notion that a researcher must develop expertise in the area of application.
- I presume that there are psychologists who supported RME by Freudenthal, that wasn’t empirically tested – but I am new to this world and cannot give references.
- I presume that there are psychologists who supported traditional mathematics education, like from Hung-Hsi Wu in the USA, that doesn’t seem to be tested empirically either. But I am new to this world and cannot quite give references. For example, John Hattie has been educated on education, and I don’t know how psychology features in that, in his part of the world.
I wouldn’t want to be caught in front of a blackboard like that (Screenshot UChicago)
Loose ends we haven’t looked into:
- Wouldn’t Van Hiele have been interested in De Groot’s Methodologie, and have contacted him on that ?
- What about the role of Hans Freudenthal (1905-1990) in all of this ?
- Who are the psychologists supporting Jan van de Craats anno 2015 ?
- When will this misery ever end ?
Appendix A: De Groot Academie en Forum (1982:9) quoted by Trias Politica & Centraal Planbureau (1994:81)
Google Translate actually does a reasonably fair job, for a computer programme.
“(…) is a democratic polity necessary, but not sufficient. For a fruitful development of the politically sensitive social sciences in particular is also needed: a government that understands well its main task of science policy; namely the duty, even in those subject areas, the tradition of critical inquiry, rational discussion and strive for objective judgment – in short – to encourage support, protect the culture Forum. Perhaps a modern democracy was not a triad but with tetras [tessera? / TC] politician must be equipped with the fourth independent power of science. One might think of a corresponding Supreme Court, which is common in severe cases, the government can condemn political prostitution of research, for abuse of expressions such as ‘scientifically proven that …’ and scientifically irresponsible applications. It seems in principle a good idea, at least – in the current situation – a beautiful pipe dream. Realized or not, the idea is dictated by a certainly legitimate concerns about the socio-political climate is such that the last decades has developed in the Netherlands. It seems that the public respect for rationality and integrity, for (better) understanding, and (better) intellectual performance in general, there is no greater on has become. The common aim leveling work a short-sighted anti-intellectualism in the hand; the continued politicization of scientific research standard and rationally decidable problems, not only undermines the (gamma) science but also adversely affects the quality of our entire culture.” (Google Translate)
“(…) is een democratisch staatsbestel nodig, maar niet voldoende. Voor een vruchtbare ontwikkeling van de politiek zo gevoelige gamma-wetenschappen in het bijzonder is tevens nodig: een overheid die haar voornaamste taak van wetenschapsbeleid goed verstaat; namelijk de taak om, ook op die wetenschapsgebieden, de traditie van kritisch onderzoek, rationele discussie en streven naar objectieve oordeelsvorming – kortom: de Forum-cultuur – te steunen, te bevorderen, te beschermen. Misschien zou een moderne democratie niet met een trias maar met een tetras [tessera ? / TC] politica toegerust moeten worden, met als vierde onafhankelijke macht die van de wetenschap. Men zou kunnen denken aan een bijbehorende Hoge Raad, die in voorkomende ernstige gevallen de overheid kan veroordelen voor politieke prostitutie van onderzoek, voor misbruik van uitdrukkingen als ‘wetenschappelijk is aangetoond dat …’ en voor wetenschappelijk onverantwoorde toepassingen. Het lijkt in principe een goed idee, althans – in de huidige situatie – een mooi luchtkasteel. Realiseerbaar of niet, de gedachte wordt ingegeven door een wel degelijk gegronde bezorgdheid over het sociaal-politieke klimaat zoals zich dat de laatste decaden in Nederland heeft ontwikkeld. Het ziet ernaar uit dat het publieke respect voor rationaliteit en integriteit, voor (beter) inzicht, en voor (betere) intellectuele prestaties in het algemeen, er niet groter op is geworden. Het gangbare nivelleringsstreven werkt een kortzichtig anti-intellectualisme in de hand; de voortdurende politisering ook van wetenschappelijk onderzoekbare en rationeel beslisbare problemen, ondermijnt niet alleen de (gamma-) wetenschap maar schaadt ook de kwaliteit van onze hele cultuur.” (A.D. de Groot, “Academie en Forum”, Boom, 1982, p9)
Appendix B: De Groot (1961) on Van Hiele (1957)
De Groot (1961, 1964, Ch 6 paragraph 2.2, page 184) (Source DBNL):
“But what are” considered important educational objectives’ in the case of plane geometry in first class?
One can aim of geometry teaching in general look very different. One can even argue about whether it is a necessary part, for example of the HBS program; as is known has been proposed to replace the plane geometry entirely by something else, such as symbolic logic and set theory. One can see the target limited, strictly tied to the program itself: certain types learn to solve problems; or one can see the large, for example, learn to think, analyze a problem, systematic thinking methods and general solution methods learn to apply also for other purposes (cf. eg forest in 1955.). Or one can focus on the spatial aspect geometry as a means to develop a structured ‘spatial awareness’ (eg from hiele 1957); or different, as a means of an introduction to a science, partly formalized deductive system. There are about many sagacious and profound reflections – and few corresponding conclusions.”
“Maar wat zijn de ‘belangrijk geachte onderwijs-doelstellingen’ in het geval van de vlakke meetkunde in de eerste klasse?
Men kan het doel van meetkunde-onderwijs in het algemeen zeer verschillend zien. Men kan zelfs twisten over de vraag of het een noodzakelijk onderdeel is, bijvoorbeeld van het H.B.S.-programma; zoals bekend is wel voorgesteld de vlakke meetkunde geheel te vervangen door iets anders, bijvoorbeeld symbolische logica of verzamelingsleer. Men kan het doel beperkt zien, strikt gebonden aan het programma zelf: bepaalde typen vraagstukken leren oplossen; of men kan het ruim zien, bijvoorbeeld: leren denken, een probleem analyseren, systematisch denkmethoden en algemene oplossingsmethoden leren toepassen, òòk voor andere doeleinden (vgl. b.v. bos 1955). Of men kan het accent leggen op het ruimtelijke aspect: meetkunde als middel tot ontwikkeling van een gestructureerd ‘ruimtelijk inzicht’ (b.v. van hiele 1957); of, weer anders, als middel tot een eerste kennismaking met een wetenschappelijk, gedeeltelijk geformaliseerd deductief systeem. Er bestaan hierover veel schrandere en diepe beschouwingen – en maar weinig overeenstemmende conclusies.”