Let's start with a simple syllogism:
Modern society requires cheap energy and lots of technology to function.
Technology relies on basic science.
Mathematics is the language of science.
Civilization requires large numbers of technicians to maintain our technology, engineers to solve problems and develop new technology, and scientists to do both basic and applied research to develop new ideas.
Most of the higher-paying jobs now require higher levels of both Mathematics and applied technology.
Therefore, it is good for society as a whole, and the individuals concerned, if we improve Mathematics education.
Our politicians have gotten the gist of this logic several times in my life, beginning in 1957 with the Sputnik scare. Most of the American public saw the resultant Space Race as a matter of US pride. Those able to think knew that the USSR had the one-sided capacity to launch missiles at the US. What was presented as a bold exploration venture was an exercise in self-preservation.
Today, the fear is on the vulnerability of our many computer-based systems. It is just as real a danger as nuclear war, but not quite as obvious. The attitude seems to be that, “if it breaks, somebody else will fix it.”
There is also the small matter of repeated studies showing that success in any higher education is directly correlated with performance in College-level Mathematics.
This awareness led to several rounds of attempts at Mathematics teaching reform, at least four of which happened in my career as a Mathematics professor. I was even involved in a couple of them, trying to do the right thing.
We had the 'New Math' of the late 1960's, which attempted to put the subject on a firm theoretical footing. Next was the 'lean and lively Calculus' movement in the early 1990's, to have students learn 'deeper', using more graphical methods and less Algebra. Then came project-based teaching, which had students working in groups to 'learn' Science, Mathematics and English. Most recently, we have had the push for 'flipped' classrooms, in which students watch videos to teach the lessons, then sit in the classroom while the teacher helps them solve problems.
All of these reforms shared a few features:
The instigators were energetic, enthusiastic, honest and delightful, and were absolutely convinced that they held 'the answer'.
Every such approach wanted to use technology, starting in the 90's with graphing calculators, followed by laptops. Later it was computer Algebra systems.
Every approach ignored human nature.
Each approach showed initial gains, in what is known by Psychologists as 'The Novelty Effect'
( https://en.wikipedia.org/wiki/Novelty_effect ).
Each reform had an obvious flaw built into it.
The New Math failed in large part because the teachers didn't understand what they were teaching.
The graphical/Algebra-light Calculus left students able to generate (sometimes) correct answers, with no understanding of where they came from, and no ability to interpret them. In extreme cases that I experienced with some top students, they could no longer tell the difference between subtraction and division.
In the project-based models, students rarely learn anything significant, but get the impression that they have learned everything. That makes them ready for management, not so much for productive work.
The flipped classroom model ignores human nature. We learn by mimicry, for the most part. In teaching martial arts, I have noticed that, if you give verbal instructions, and do something else with your body to demonstrate, most students will attempt to do what you did, not what you said. Except for the most talented, people learn Mathematics in that way, by watching a teacher solve a problem, then solving several just like it, with numbers changed, then having more of the concept explained.
Every single reform ended up with worse results than when we began. Even more depressing is the fact that, instead of returning to the original ideas, the system just tried the next new model. When I started teaching in 1978, students either mastered their Algebra in Calculus I, or they failed. Now, the bad Arithmetic and Algebra has penetrated as far as Differential Equations (Calculus IV in some systems), because the bad habits are just so ingrained. I once had 2/3 of a class of Honors Calculus III end up with the equation '2x=3', then write 'x=3-2', so 'x=1'. And these were the select of their year.
Meanwhile, the Asian systems are churning out Engineers, while we busily criticize their education systems as not 'inspiring creativity'. There is some justification for this, and many in Asia agree. Students there are taught by rote, and not allowed to stand out. On the other hand, the successful have actually learned something. By contrast, I have met many students who have been labeled as 'creative' who consistently generate ideas and solutions which are as practical as oars on a spaceship.
There is a way to generate more people with talent in Mathematics and the subjects which rely on it: Select them at say age 12, and put them together in special schools. It reduces bullying, and their natural competition will drive them to succeed.
4 comments:
Seems to me teaching is a job one could grow weary of. Yes of course there is the noble art of encouraging young minds but after years of trying to encourage the reluctant young to look beyond Love Island and their Iphone I should think most teachers get a bit sick of it.
Then there is the politico commercial axis. Education is 'a good thing' and results matter - politically. The development of new whizzy ways of teaching boring old subjects is a good way to capture education budget. So sell sell sell those teaching materials. Textbooks are so last year, materials that have to be bought and handed out every year are a much better selling proposition.
So get cynical, invent and package new teaching methods backed up by commercial backers and lobby groups. Much more interesting than getting sore feet in front of a whiteboard and better paid.
Then there is mathematics. Well, Donald Trump does not look much of a maths student. But he is a big shot surrounded by lawyers for whom truth and logic are tools of manipulation. His and their job is well paid and does not need calculus. An individual does not need much maths to do pretty well. But of course a nation does need maths and a whole lot of other hard subjects to succeed. It just need to persuade students that the hard labour is worth the bother - or import people from systems that do encourage students to take the trouble.
Indeed there does seem little point in trying to educate at High School the inner structure of maths. Better to stick to some rote learning, a few may grasp the inner structure later on, that's a bonus. Start with rote and examples, understanding will follow and sometime after that some sense of inner structure. But there is no money in dusty old textbooks and chalk and the politicians want 'whizzy' even if it is shallow and meaningless.
The old Russian education system sought to identify and encourage mathematical talent early on but that seems to be declining. We can probably expect the Chinese to identify and encourage talent as part of a forcible push to overtake the USA. For the USA (and Europe) there is a more vexed choice, what would we actually do with thousands of mathematicians?
Based on some of the models which my colleagues and I constructed, there is a lot of 'low-hanging fruit' to be had. We were able to construct simple and viable models in choice of plays in American football, pollution control, and the economics of state subsidy of higher education.
It is because so few leaders have Mathematics training that a great many bad business and political decisions are made.
Hi Paddington. Yes, pilot teaching methods show great promise because the teachers understand and are enthusiastic about the revolutionary new method. Then the roll-out goes out to teachers who don't understand the philosophy of the new method and are weary of the changes and it flops. Which leads me to think that methods don't matter, but motivation does. Motivation of the teachers that they manage to convey to the motivation to the kids. I went to a scumbag school with a hopeless maths teacher and was consistently close to last in maths. Then about grade 8, the results posted on the school noticeboard showed I had come first in maths!!! I reveled in the glory until the marked papers came back and I realised that the teacher was so bad at maths that he had added the results from two papers up wrongly to promote me to the top. My ears burned with shame but I told nobody until today 50 years later. I came top in maths and hence all science subjects after that without blundering assistance.
Paul - for years I did my best, including working with future teachers, and my colleagues. Overall, there were two things which seemed to matter the most, in terms of results: a) knowledge and enthusiasm for the material and b) a serious desire to communicate that to students.
I had colleagues who were brilliant but hated, and couldn't get through to the students. This included some who had earlier in their careers been brilliant and effective. Perhaps the modern group of students left them cold.
I had other students and colleagues who desperately wanted to teach, but couldn't understand the material that they were supposed to teach. That is a sad disaster, as you can't even explain the issue to them.
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