Pierre van Hiele and Gerald Goldin (2)

Listening to Andriopoulos & Odysseas Elutis (1984): Prosanatolismoi


Let us discuss Gerald Goldin (2003), Developing complex understandings: On the relation of mathematics education research to mathematics. I presume that the reader has checked earlier discussions on Goldin (1992), on epistemology and on Stellan Ohlsson.

The paper’s abstract is:

Goldin (2003), p171

Goldin (2003), p171

It took me a while to come to grips with this paper. Suddenly it dawned on me that the English speaking world, including Goldin, makes a distinction between science and the humanities. This is what C.P. Snow (1905-1980) called The two cultures.

For Dutch readers these categories are crooked.

  • When Goldin opposes mathematics education research (MER), which in the English world belongs to the humanities, to science including mathematics then this is the distinction between science and the humanities. But for Dutch it sounds very strange to suggest that MER would be non-science.
  • Dutch has the single word wetenschap. How can Goldin oppose things that are the same (learning) ? How can he lump together things that are different (science and mathematics) ?

Dutch convention categorises the humanities as alpha (α), science and mathematics as beta (β) and the mixture as gamma (γ): those deal with alpha subjects but use beta methods. MER would be gamma.

I am not too happy with the Dutch categories since they don’t account for the separate position of philosophy and mathematics. The better distinctions are in the next table.

Categories for general science (science and the humanities)

Categories for general science (science and the humanities)

Above table is intended to categorise whole disciplines, like physics and economics. But we can also look at sub-areas within a discipline. Since both philosophy and mathematics can run astray without some link to the external world, my suggestion is that they both take mathematics education research as some anchor to reality (but they remain what they are when they refuse to do so). They might take an example of history writing, in which most history writing uses non-experimental methods (see the history on Pierre van Hiele) but some historians will rely on the experimental sciences to recover data from the past.

We are now ready to look at the paper.

Below we will see that Goldin opposes α + Φ versus β = δ + μ, forgetting about γ, while my analysis in Elegance with Substance (EWS) (2009, 2015) (pdf online) diagnoses the problem as α + μ versus γ, while δ + Φ have run away and no longer want to take part in cleaning up the mess. Professor Hung-Hsi Wu of Berkeley calls for help by research mathematicians μ to clean up the mess in ME and MER, but in my analysis we need help from engineers and other researchers in the empirical sciences δ + γ, see here.

Goldin 2003 on the decade since 1992: integrity for the disciplines

Goldin’s paper discusses his background, and he seems very well placed to discuss mathematics, ME and MER. Goldin sees a math war and tries to bring calm by increasing complexity. His article is complex itself so that those who pass the test of reading it will understand enough of the various sides of the discussion and be less likely to vilify the other side.

Goldin’s position is that discussants on MER must respect what other discussants on MER are doing and good at. Scientific integrity tends to focus on ethical behaviour of the individual but Goldin widens this to whole disciplines. Scientists must respect the humanities. The humanities must respect science. Otherwise there is no communication and no progress.

Goldin (1992) looked back at the New Math in the 1960s and behaviourism in the 1970s. When those ‘isms’ failed to produce improvement in mathematics education, the educational departments in the humanities grabbed the opportunity to claim their way to success. Goldin would agree partly, since he in 1992 also opposed the New Math and behaviourism. The humanities however created their own ‘isms’. We can now better understand Goldin’s position w.r.t. the decade 1992-2003.

Goldin (2003), p177

Goldin (2003), p177

A key observation is that Pierre van Hiele (1909-2010) is missing in this list and that Hans Freudenthal (1905-1990) committed fraud w.r.t. the work by Van Hiele: so that Goldin has a somewhat rosy view about the “without the far-reaching dismissals, oversimplifications, and ideologies”. The reference to Leen Streefland (1998†) may highlight the ‘ism’.

A Pro Memory point is that David Tall in 2002 apparently misunderstood the Van Hiele theory, as applying only to geometry and not to epistemology in general. This doesn’t seem to be due to ideology on Tall’s part, but there seems to have been some influence of Freudenthal in the misrepresentation of Van Hiele’s work. See my paper on getting the facts right.

An example with Leen Streefland (1998†)

I have not studied Streefland’s work any deeper than the following internet links just now. Those links fit the diagnosis of sectarian behaviour of Freudenthal’s “realistic mathematics education” (RME), and thus I see no reason yet to read more. Streefland belonged to the Freudenthal sect, see this ESM 2003 issue. Pierre van Hiele suggested in 1973 to look into the abolition of fractions, but Streefland (1991) perseveres with a book on “realistic education” on fractions.

See my 2015 book, pdf online, A child wants nice and no mean numbers, also commenting on the US Common Core program and professor Hung-Hsi Wu on fractions. Professor Wu does not belong to the RME sect but his traditional answer on fractions suffers from the intellectual burying of Van Hiele, which the RME sect so effectively achieved. The ‘isms’ are not without cost.

The strategy by Hans Freudenthal and his Utrecht sect – and these are adults who know what they are doing – is to absorb elements of Van Hiele’s work, but misrepresent it to fit their own ideology – which change does not diminish the intellectual theft. They achieve two effects: (i) for an innocent audience they ride the wave of the success by Van Hiele that they are jealous about, and (ii) they exclude Van Hiele himself from the discussion since “they tell it better” – and thus Van Hiele’s protest that his work is abused will not be heard. After all, Freudenthal was a professor in Utrecht with his own Ph.D. students who later became professors, and Van Hiele remained a mere mathematics teacher doing his writings in the weekend.

In this book on fractions, Streefland (1991) p2 states the following. We can excuse authors for the uncreative use of the word “level” that pops up everywhere. The true problem lies in the ideological following of Treffers (1987) and the neglecting in 1991 of Van Hiele’s own work (not only on fractions of 1973).

Streefland (1991), p2

Streefland (1991), p2

This closes the circle: (a) Treffers (1987)’s misrepresentation of Van Hiele’s work is not only in Streefland (1991) but (b) was also copied in the 1993 MORE study, (c) critically discussed by Ben Wilbrink, here, (d) which alerted me to Wilbrink’s misrepresentation of Van Hiele. Wilbrink namely follows the RME abuse, and he also tends to include Van Hiele in the RME sect instead of saving him from it.

Thus we are back into the Dutch math war swamp, with on one side the RME sect and on the other side Jan van de Craats and others who try to save “traditional mathematics education sanity” alongside psychologist Wilbrink with his misapprehension of empirical science and Van Hiele. My position is that of Sherlock Holmes observing it all from the high ground aside.

Traditional mathematics is crooked as well. It e.g. involves torture of kids by fractions. There is every reason to desire change. It doesn’t help when mathematicians, who don’t have empirical training, team up with the humanities who don’t have empirical training either (i.e. α + μ).

Popper’s falsification

One reason why the humanities might be disrespectful of science has to do with Karl Popper’s demarcation theory to use falsification to distinguish science from non-science:

Goldin (2003), p178

Goldin (2003), p178

Goldin reminds us that the humanities are non-science, as seen from science and its experimental method. The humanities should heed the risk of turning this property into a claimed superiority.

The humanities seem to have learned that they should not claim higher wisdom, which they and only they can discover by reading old documents and watching plays by Aristophanes and Shakespeare and have reception parties afterwards to discuss the faculty gossip. But the humanities might still take the Humean skeptic position, and make fun of physics who can put electrodes upon skulls and in that manner likely will never be able to create the insights that a study of the humanities can generate (though they might actually prove some of the gossip).

Goldin’s argument: Physics can be skeptic too. Save those skeptic arguments for your autobiography, for they contribute nothing to the discussion.

My warning: Don’t make too much of falsification. See the discussion on epistemology and the definition & reality methodology.  Above δ and γ sciences rely for the empirical realm upon definitions, and a mathematician μ might well hold that definitions are non-experimental.

David Hume and Ernst von Glasersfeld

Reading a bit more about Ernst von Glasersfeld (1917-2010) was long upon my to-do-list, and Goldin’s article finally caused me to do so. Advisable is his own article Thirty Years Radical Constructivism, Constructivist Foundations 2005, vol. 1, no. 1, p9-12. It is very useful to see Von Glasersfeld’s background in mathematics (not completed because of WW 2), linguistics and cybernetics: γ rather than α. For methodological justification he might be forced to do some philosophy, but he rejects doing that.

Von Glasersfeld (1995) Radical Constructivism at ERIC is too much for now, though. I checked that he indeed discusses Hume, and also mentions the “problem” of induction (see my discussion of epistemology). Von Glasersfeld holds that the issue is not philosophy but finding mechanisms of cognition.

Comment 1: Reuben Hersh (2008) Skeptical Mathematics? Constructivist Foundations 3(2): 72, suggests that “radical constructivism” would be Humean skepticism, and I tend to agree.

Comment 2: Being a Humean skeptic is agreeable too. This (wiki-) quote by Von Glasersfeld seems accurate:

“Once knowing is no longer understood as the search for an iconic representation of ontological reality but, instead, as a search for fitting ways of behaving and thinking, the traditional problem disappears. Knowledge can now be seen as something which the organism builds up in the attempt to order the as such amorphous flow of experience by establishing repeatable experiences and relatively reliable relations between them. The possibilities of constructing such an order are determined and perpetually constrained by the preceding steps in the construction. That means that the “real” world manifests itself exclusively there where our constructions break down. But since we can describe and explain these breakdowns only in the very concepts that we have used to build the failing structures, this process can never yield a picture of a world that we could hold responsible for their failure.”

It is hard to disagree, except when you want to resort to Wigner’s magic again (see Appendix 2). But it doesn’t tell us how to design a course so that Johnny can learn arithmetic. Or how to abolish fractions.

Comment 3: Von Glasersfeld refers to Jean Piaget. Pierre van Hiele developed his theory of levels of insight, starting from Piaget as well, but eventually rejecting Piaget’s age-dependency and choosing for the logical structure that generates a general theory for epistemology.

It is a question what the contacts between Von Glasersfeld and Van Hiele were, and whether Hans Freudenthal was an interfering factor again. We find Von Glasersfeld (ed) (1991), Radical Constructivism in Mathematics Education, Kluwer, that contains a chapter by Jan van den Brink, since 1971 a member of Freudenthal’s sect in Utrecht. Searching the book generate 0 references to “Hiele”. The RME wiki on RME refers to Von Glasersfeld’s book but not to Van Hiele, even though we saw above that Streefland refers to Treffers who considered the Van Hiele levels a “pillar” of RME. Not referring saves the effort of thinking up a lie.

It is a question how the departments on education at the humanities were influenced by RME and Von Glasersfeld and others, and how they got so entangled that Goldin seems to tend to refer to them as one side of the equation (or rather imbalance). It is no key question, but something for historians of MER to be aware of (see the handbook on history of MER).

Comment 4: I started getting lost on what makes “constructivism” so special that it must be mentioned. Originally I knew about constructivism as an approach in the foundations of mathematics, as distinguished from formalism and platonism. My book Foundations of Mathematics. A Neoclassical Approach to Infinity (FMNAI) (2015) creates a ladder of degrees of constructivism (avoiding “levels”), in which the highest degree allows non-constructivist methods. When people use different approaches we should at least describe what they are doing.

But now there are all kinds of “constructivism” in education, psychology and philosophy, without authors taking the time to shortly explain what the non-constructivist opposition would entail. Fortunately, there is wikipedia that might help or contribute to confusion, here with disambiguation. and here with the general denominator in epistemology. The opposite of constructivism would be that people could know objective reality, by magic, and I wonder whether that is so useful an idea. My impression is that there is more to it. Thus authors should still specify. (And then I would not have time to read it.)

Ben Wilbrink is horribly erroneous about Pierre van Hiele and in breach of scientific integrity for not looking into it  to correct his misrepresentation, and Wilbrink can fulminate against constructivism: but at least he referred to this article by Gerald Goldin so that I found it, and he also has this page with all kinds of references on constructivism.

One book mentioned there is by Kieran Egan (2002), Getting it Wrong from the Beginning: Our Progressivist Inheritance from Herbert Spencer, John Dewey, and Jean Piaget. This fits Van Hiele’s rejection of Piaget’s theory of stages of development. But does Egan refer to Van Hiele ? Not likely, since the wikipedia portal speaks about the constructivist “idea that things (especially learning) always go from simple to complex” – and this is not how Van Hiele would phrase it: who discussed going from concrete to abstract, and who used the notion of proof to identify the highest level of abstraction.

Wilbrink also has a quote on Von Glasersfeld:

“The basic idea of The Georgia Center was to establish a community of researchers in mathematics education working on problems of interest to the community, where the experience of the researcher, conceptual analysis, and social interaction replaced the controlled experiment as “normal science.” No longer did it seem necessary to use the controlled experiment with its emphasis on statistical tests of null hypotheses and empirical generalization to claim that one was working scientifically.”

This is complex. Before you do such a costly double blind randomized trial, with the huge numbers required because of the large number of variables, variety in pupils, and sources for measurement error (see John Hattie), it is useful to have clarity on concepts, definitions, operationalisations, methods, controls, and the like. Confronted with annual unpredictable changes from the Ministry of Education, you might want to give up on such statistical ambitions, and settle for the Google “do no evil” approach. It may well be that modern MER only serves for Ph.D. students to defend a thesis, and the relevance for education may be discussed at the reception party afterwards along with the faculty gossip.

The increasingly popular Japanese Lesson Study is one promising method (tested under Japanese conditions) to deal with the data problem.

However, see the suggestion for Academic Schools modeled after the Medical School, also included in A child wants nice and no mean numbers (2015).

Goldin’s crucial blind spot

What I consider Goldin’s blind spot is that he lumps together science and mathematics, while mathematics is no empirical science but deals with abstraction and patterns.

Goldin (2003), p179

Goldin (2003), p179

Education is an empirical issue. Also mathematics education is an empirical issue. Thus the involvement of mathematicians in such education can be disastrous, when they are trained for abstraction μ and not for empirical science γ. The epitome of the abstract mathematician who got lost in this is Hans Freudenthal who invented a whole new ‘reality’ just to make sure that at least he himself understood and was happy how the world works (including the oubliette for Pierre van Hiele).

The only reason that Goldin lumps together β = δ + μ is that he is so much worried by the ‘isms’ by α + Φ that he forgets about the real problem at bottom of the case: the disastrous influence of μ in 5000 years of education of mathematics. (Fractions were already a problem for the pyramids.)

Check out this example of mathematical torture of kids on fractions. This torture is also supported by professor Hung-Hsi Wu of Berkeley for the US Common Core programme, see A child wants nice and no mean numbers.

Goldin (2003) p180 suggests a seemingly good argument for lumping together science and mathematics.

Goldin (2003) p180

Goldin (2003) p180

Thus, the abstract thinking mathematician has a special trick to describe the physical world ? Without lookin ? Like with Wigner’s magic wand ? I don’t think that we should believe this. It is physics that selects the useful model from the mathematical possibilities. Thus:

  • This misconception about the role of mathematics may help explain why Goldin (2003) does not quite see the disastrous influence of abstract thinking mathematicians upon ME and MER. Golding does make some comments that mathematicians should not think that ME is simple and can be tested as in behaviourism, but he misses the fundamental problem as discussed in Elegance with Substance.
  • The remark about mathematics and empirical modeling remains relevant for the definition & reality methodology. It supports the empirical status of say the definition / law of conservation of energy, and it supports the empirical status of Van Hiele’s theory of levels of insight (abstraction) in epistemology (and application in psychology).

(1) We can support Goldin’s conclusion and plea for eclecticism (yes, another ‘ism’).

Goldin (2003), p198

Goldin (2003), p198

(2) Since the Freudenthal Head in the Clouds Realistic Mathematics Institute (FHCRMI) in Utrecht doesn’t do MER but performs sectarian rituals, also based upon Freudenthal’s fraud, it is a disgrace to general science – including the humanities – and thus it should be abolished as soon as possible. Dutch Parliament better investigates how this could have happened and endured for so long.

Appendix 1. Kurt Gödel

W.r.t. the following I can only refer to A Logic of Exceptions (1981, 2007, 2011) (pdf online). For interesting systems the Gödeliar collapses to the Liar paradox, with no sensical conclusions.

Goldin (2003), p187

Goldin (2003), p187

Appendix 2. Philosophy of mathematics

W.r.t. the following I can refer to the discussion on Wigner on the “unreasonable effectiveness of mathematics”. Given the common meaning of “unreasonable” Wigner must refer to magic, or he didn’t know what he was writing about, as a physics professor lost in the English language. It is some kind of magic that his paper got so much attention. This discussion has also been included in Foundations of Mathematics. A Neoclassical Approach to Infinity (2015) (pdf online). Goldin uses the word “extraordinary” rather than “unreasonable”. Given that the effectiveness is ordinary for physics, he seems to take the humanities’ point of view here (whom his article is addressed at).

Goldin (2003), p188

Goldin (2003), p188


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