Conjecture. ‘Thinking like a mathematician’ is a concept confusing cognition of the individual (student, academic, mathematician) and the fruits of centuries of mathematical and scientific research.
Teaching students to ‘think like a mathematician’ is a huge waste of effort.
This quote from Whitehead (1911, p. 8) “… mathematics … is necessarily the foundation of exact thought as applied to natural phenomena” refers to science, not to the thinking of an individual (scientist, student).
- [Alfred North Whitehead (1911). ‘An introduction to mathematics’ https://archive.org/details/introductiontoma00whitiala%5D
Whitehead p. 11
“To see what is general in what is particular and what is permanent in what is transitory is the aim of scientific thought.”
‘Scientific thought’ here is a metaphor for scientific research , it is not meant to be the thinking of the individual scientist.
Whitehead, p. 11:
“In the eye of science, the fall of an apple, the motion of a planet round a sun, and the clinging of the atmosphere to the earth are all seen as examples of the law of gravity. This possibility of disentangling the most complex evanescent circumstances into various examples of permanent laws is the controlling idea of modern thought.”
Again, ‘modern thought’ here is not the thinking of individual scientists; it is scientific method itself, a collective achievement.
Therefore: teach math, and math only.
Instead of ‘thinking (like a mathematician)’, the clever thing to do is to learn particular achievements of science so one has not to ‘think’ about them anew.
Whitehead p. 61: It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.”
Just like driving one’s car. Amen.
Alfred North Whitehead was a great mathematician and philosopher (which does in itself nothing to protect such a top influencer from being thoroughly mistaken in matters not directly mathematical/philosophical). A great man, worked with Bertrand Russell. (Bio by Victor Lowe, 1985 ‘Alfred North Whitehead. The man and his work’)
More on ‘thinking like a mathematician’, and the damage that this myth (especially promoted by the brilliant mathematician Hans Freudenthal, in Dutch Realistic Math Education RME) does to education: http://www.benwilbrink.nl/projecten/wiskundig_denken.htm
(literature and annotations, partly in Dutch & English)
I will try to use this material in a position paper on the Dutch curriculum reform proposals #curriculum_nu, exposing the many psychologisms, especially in the math proposals. Dutch Parliament will organize hearings and round tables on the curriculum proposals, early in 2020.
Note (via Evert Beth, 1948 ‘Wijsbegeerte der wiskunde’ [Philosophy of mathematics]). Mathematical thinking: not to be confused with mathematical significa (Brouwer [intuitive math], Van Eeden, Van Ginniken, Mannoury, Van Dantzig). https://en.wikipedia.org/wiki/Gerrit_Mannoury
Can I give examples of ‘thinking like a mathematician’? (question by
@SGStotsky, thanks Sandra).
That is the question, isn’t it? If ‘mathematical thinking’ exists, there must be examples. Hans Freudenthal thought he had examples of ‘mathematical thinking’ of his 6 year (or some) old grandson, so it should be possible to teach primary students to ‘think like a mathematician’.
About ’70 Freudenthal got the money/opportunity to reform Dutch primary math education. He turned it around (RME), yet failed realizing his ‘mathematical thinking’ goal as he recognized at the end of his life (about ’90), also recognizing kids could not do arithmetic any more.
I must give the source for the last statement. It is a letter (in Dutch) to the newspaper NRC by a math friend of Freudenthal, Van Zwet, see http://benwilbrink.nl/projecten/debat.htm#Zwet_2008
Dutch results on national (PPON) and international math tests (TIMSS, PISA) are in decline ever since .
What could ‘mathematical thinking’ be? The first problem is with the ‘thinking’ part: thinking is an autonomous cognitive process that is not available to introspection by the ‘thinker‘ herself. There are some tricks that the researcher can use, though:
Click to access d68f76ba21cedd5806b8e2d5374c1e4c07de.pdf
Already in the 70s research was done on the ‘thinking’ of judges. What I remember: judges were not able to give their exact ‘thinking’ in coming to their verdicts (in line with the observation in the preceding tweet). Of course, they were able to present a verdict motivation.
On the psychology of this kind creative professional work, see Stellan Ohlsson (2011). ‘Deep Learning: How the Mind Overrides Experience’ Cambridge UP. Reviewed by Jared Freeman
Click to access Freeman_DeepLearningReview_CognitiveTechnology_2013.pdf
(building further on work by, a.o., Newell and Simon, Newell 1990).
My attempt at explanation in 1 tweet: The professional brain brims with complex knowledge, organized in very large ‘chunks‘ making it possible for the brain to juggle all that knowledge in pursuit of possible answers to new problems. The ‘chunks‘ are not open to introspection!
This is cognitive psychology as already to be found/foreshadowed in the 1946 Dutch dissertation by Adriaan D. De Groot ‘Het denken van den schaker’ http://www.dbnl.org/tekst/groo004denk01_01/ [translated 1965/2008 as: Thought and choice in chess]
Lawyers, chess masters, mathematicians: same psychology.
Maybe it occurred to you that the judge not knowing exactly how his brain came to a particular verdict is uncannily similar to self-learning AI resulting in algorithms nobody can know.
It is therefore important for professions to be strongly disciplined.
mentions
Note on Beth. For those triggered by the above reference to Evert Beth 1948: Beth is more elaborate on mathematical thinking in: Beth & Piaget (1966). ‘Mathematical Epistemology and Psychology’ , see https://www.springer.com/gp/book/9789027700711 for chapter previews.
Ondertussen in Nederland [Dutch actuality]:
Paul Drijvers en Hanneke Kodde-Buitenhuis (december 2019). Wiskundig denken in de centrale examens wiskunde B van havo en vwo. Nieuw Archief voor Wiskunde, 252-258. http://www.nieuwarchief.nl/serie5/pdf/naw5-2019-20-4-252.pdf
- Mijn bijzondere belangstelling heeft figuur 2, daarin de ‘domeinONafhankelijke denk- en werkwijzen’ van #curriculum_nu rekenen en wiskunde. Deugt de psychologie daarvan? Bestaan ze?
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