Mathematics - the crumbling foundation of US wealth and security, by Paddington

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.