In 2009 I wrote Elegance with Substance (EWS), discussing both better education in mathematics and the political economy of the mathematics industry. See the available PDF. Check also Steven Krantz Through a Glass Darkly at arXiv 2008.
The dismal state of mathematics education is generally acknowledged, essentially since Sputnik 1957. People have tried all kinds of solutions. Why do those solutions not work ?
The answer: because of barking up the wrong tree. The finding in EWS is:
- Mathematicians are trained to think abstractly.
- Education is an empirical issue.
- The courses for becoming a math teacher don’t undo what has gone wrong before.
- When abstract thinking math teachers meet real life students, those math teachers solve their cognitive dissonance by sticking to tradition: “School Mathematics” (SM).
- School mathematics isn’t clear but collects the confusions and wreckages of math history.
- Thus we need to re-engineer math education and reorganise the mathematics industry. One idea is that education would use the form of the Medical School: both practice and research.
EWS contains various examples where traditional math is crooked instead of clear. One example is that “two and a half” means addition and should be denoted as 2 + 1/2, but is denoted as multiplication or “two times a half” or 2½.
2009 + 5 = 2014
Now five years later in 2014, this explanation can be enhanced by including:
- There is a collective failure w.r.t. the integrity of science, in that Research Mathematicians step outside of their field of expertise (RM) and make all kinds of unwarranted claims about Education in Mathematics and its research (EM). This aggravates the observation above that the conventional EM is lopsided to SM.
- It is also a breach of research integrity that the warning in EWS is not responded to. When it is shown that the brakes of some kind of car don’t work properly, it should be recalled – and the same for EM.
- This especially holds in Holland. In Holland there is even explicit fraud in EM.
- For the UK there is some worry, see my 2014 paper Pierre van Hiele and David Tall: Getting the facts right.
- For the USA there is now the worry concerning professor Edward Frenkel.
Pierre van Hiele (1909-2010) was the greatest analyst on mathematics education of the last century, with his main thesis in 1957, coincidentally with Sputnik. However, his analysis was maltreated by Hans Freudenthal (1905-1990), who stole Van Hiele’s ideas but also corrupted those – partly claiming his “own” version but without proper reference. Van Hiele looked at the angle of abstract versus concrete, while Freudenthal turned this into model versus reality, which is didactically rather absurd, but which apparently appealed to policy makers after Sputnik 1957. Holland now has a 95% dominant “Freudenthal Institute” that rather should be called the “Freudenthal Head in the Clouds “Realistic Mathematics” Institute”. Apparently, the Dutch RM and EM community is unable to resolve the issue. Internationally, IMU / ICMI (see my letter) has a “Freudenthal Medal” honoring the fraudster.
A leading analyst in the UK is David Tall (b. 1941) who rediscovered the importance of the Van Hiele analysis, but erroneously thinks that Van Hiele was not aware of what he was doing, so that Tall claims the discovery for himself. Part of Tall’s misunderstanding of the situation is the consequence of Freudenthal’s abuse of Van Hiele. Professor Tall should however quickly bring out a revised 2nd edition of his 2013 book to set the record straight.
From Russia with math and confusion
I have discussed some of Frenkel’s ideas. As he hasn’t studied math education empirically, he is not qualified to judge, but he follows the RM arrogance to think that he is. Well, hasn’t he passed through the educational system himself ? Isn’t he teaching math majors now ? These are hard fallacies to crack.
Numberphile has a 9-minute interview with Frenkel, asking him: “Why do people hate mathematics?” I leave it as an exercise to the viewer to identify the amazing number of delusions and fallacies that Frenkel mentions in this short time. Perhaps shortness invites imprecision. However, check this weblog’s texts of the last week, and see that these delusions and fallacies are systematic. Just to be sure: debunking those delusions and fallacies may not be easy. If it were easy, the state of math education would not be as dismal as it is now.
To help you getting on the way, check some of these delusons or fallacies:
- The beauty of art is abused again. Math education would teach you painting fences but not the appreciation of the great results of mathematics. To some extent one can agree. Math history and some encyclopedia of math are very useful to have. But art education is not intended to get people to make masterpieces. Mathematics education is intended to help students develop their understanding and competence. These are different settings.
- Frenkel claims that everything is based upon the language of mathematics. “In a way one can survive without art. No one can survive without mathematics.” Since abstraction means leaving out aspects, it should not surprise that if you start with the world and then abstract from it, then your results may indeed be relevant for “everything”. But you cannot infer from such an abstract position that people should love their math education.
- He again is in denial of the role of mathematics in causing the economic crisis.
- The problem is often stated in the terms of “people hate mathematics” in a manner that is not linked to mathematics education. As if there are two kinds of people, mathematicians and other – the elite versus the peasants. But the true problem is mathematics education. Math teachers have their students for some 12 years as their captive audience, and manage to turn human innate interest into said hate. By stating the problem in terms of some vague “general audience” it becomes easier to run away from the responsibility staring you in the face, and the destruction of human lives going on in the classrooms around the world.
Taking a blame without any consequence
There is no doubt that Frenkel respects education – though it is from personal experience and without empirical research of a national curriculum:
“Now that I’ve had students of my own, I appreciate even more what (… my teachers …. have …) done for me. It’s hard work being a teacher! I guess in many ways it’s like having children. You have to sacrifice a lot, not asking for anything in return. Of course, the rewards can also be tremendous. But how do you decide in which direction to point students, when to give them a helping hand and when to throw them in deep waters and let them learn to swim on their own? This is art. No one can teach you how to do this.” (“Love & Math p129)
The major point is this: Asked who is to blame for the dismal appreciation for mathematics (minute five) he offers himself as the scape-goat:
“If I really were to assign the blame, … I would assign the blame to myself. And my colleagues, professional mathematicians. We don’t do nearly enough, in exposing these ideas to the public.”
Okay, so, Frenkel takes the blame. But there is no consequence. No reduction in salary. No prison term – with use of the library to start studying mathematics education. Just the burden to go out into the public and become a media star by comparing mathematics to Van Gogh, Picasso, and what other artist that can be abused and intimidated into an admiration for mathematics that they don’t understand but generally hate.
In minute six he says that the math teachers are not to blame. “They are overworked and underpaid” and “products of the same flawed system”. Thus, the idea that grown-ups should take responsibility for what they are doing, and that professional educators have an ethic to live up to, is flushed down the drain. Jesus absolves the sins of those who believe in him. The topic of discussion is reduced to “beauty”. This will generally concern topics that require an advanced university degree to understand – and that conventionally are presented in a confused manner to the general public (see yesterday).
About the improvement of education, Numberphile properly aks (minute seven-and-a-half): “Why has that not happened ? It seems so obvious. What you said is not like a huge conceptual link. Why isn’t it not already happened ?”
Since he has no clue about empirical science, the world turns into a conspiracy:
“Sometimes I am wondering myself why it hasn’t already happened. It is almost like a conspiracy. I mean, honestly. It is almost like there is this system of mirrors that has been created which distorts reality, that does not allow people to see what is out there.”
His closing statement turns failure on scientific integrity, fraud and dismal negligence into “irony”:
“This is the coolest stuff in the world. And yet everyone hates it. Isn’t it ironic ?”
PM. The link of Jesus to a scape-goat is no coincidence. December 25 falls in the sign of Capricorn and Jesus was sacrificed as the Lamb of God. See “The simple mathematics of Jesus“ for a discussion that the Bible is an astrological book – and, if you didn’t know, that astrology isn’t science.