‘The big education for me is that civilisation is fragile and can be destroyed in a heartbeat' - Jeremy Brade, former peacekeeper in Sarajevo.

Tuesday, June 30, 2020

Maths learning: go figure, by Paddington

In over 40 years of teaching and tutoring Mathematics, and reading lots of studies in Mathematics Education, I have become convinced of the following:

1. Almost (*) everyone can learn more Mathematics than they currently know.
2. There is a fairly clear hierarchy of difficulty in the subject: Arithmetic, Algebra, Basic Functions, Calculus, Advanced Calculus, Real Analysis and the higher level material. Almost (*) everyone has a maximum level that they can achieve, long before the top.
3. The top level for 80% of the population appears to be Basic Algebra or lower, with only about 5% able to pass a standard Engineering Calculus I course.

Understandably, these observations have met with a great deal of resistance, especially from politicians and administrators who have read the studies that performance in college-level Mathematics classes is a good predictor of overall academic success (undeniably true). This leads to the insistence that we pass more students without lowering standards.

The people who insist that this is possible tend to fall into two categories: Those who themselves do not perform well in the subject, but blame all of their experiences on a single bad teacher, and those who found the subject relatively easy.

Large scale experiments, such as the mess in the O- and A-level syllabi in England from 1980 to 2000 show that increased pass rates mean lower achievement. In the US, cases such as the impressive improvement at Georgia State a few years ago were a result of lowered standards, but the people in charge blinded themselves to the fact.

Nonetheless, a higher percentage of jobs are now tied to higher education (including many trades), and most of the degrees in demand require levels of Mathematics far higher than Basic Algebra. We have also built an Education system which treats students as consumers, and the failure rates in Mathematics are unacceptably high to the administrators and political overseers. Never mind that those rates are close to the same across countries and decades, if not centuries.

What to do?

Form the perspective of a politician or administrator, especially one trained outside of the STEM areas, the obvious answer is to increase pass rates, and pretend to be maintaining standards.

This has been happening for decades, but it is getting worse. Be prepared for the majority of college graduates to have the paper qualifications, but not the actual abilities.

They will, however, be full of confidence in those missing abilities, thanks to the Dunning-Kruger
(**) effect, which is all that really matters.


To follow on, I would point out that one of the loudest and all-knowing groups to criticize what we did were the Engineering and Science professors.

Some decided that they could do much better, and tried to create courses which took students with the base competence to start Precalculus, and tried to get them do do Differential Equations in a semester. That did not go well.

Another group decided that our placement process was too restrictive, and insisted that they could tutor and nurture the students with weak backgrounds. Those students simply couldn't get through.

Yet another group thought that we were just too harsh, and were not getting students through to their courses. They believed the famous 'Calculus is a weeder course' meme. They encouraged students to take their Math courses at online places, or local institutions who were known to have higher pass rates (i.e. lax standards). Those students got great grades in those courses, and then couldn't pass the higher-level Engineering courses.

In short, we Mathematicians generally know what we are doing.

(*) Excluding mathematical greats such as the late Paul Erdos, Terence Tao and the like.

(**) 'The Dunning-Kruger effect is a type of cognitive bias in which people believe that they are smarter and more capable than they really are. Essentially, low ability people do not possess the skills needed to recognize their own incompetence.'


Sackerson said...

Given the limited numbers of people capable of higher mathematics, is there potentially a ceiling to how far our technology-based civilisation can continue to develop (or indeed, manage to sustain its current level)?

Paddington said...

Possibly, but a great many of those with talent are currently under-utilized, so it will be some time yet.

jim said...

A touch of 'deformation professionelle?'. Why are students encouraged to study maths - to A level say. Largely as a marker for 'subset of the reasonably bright', one tick box for a uni course. At least they can think straight - or remember well. But only some A level maths students go on to a STEM degree subject.

Even then, the STEM students do not all become mathematicians or physicists or chemists - at least not for very long. Many move over into business or consultancy or the regulators or government where the jobs are and the money is better.

Mathematics always was a fairly difficult and sometimes abstract subject. That was its usefulness - as a filter. But perhaps that process is too inefficient and wasteful. We already have equation solvers online and integrators and differentiators too. But today one needs to have a fair background in order to use them. Perhaps it is time to de-mystify mathematics and add some AI. Mathematics for the Million. Stop using mathematics as a barrier. In this way we can still keep the elite students but not lose the talents of those who can get by with a mathematical 'walking stick'.

Sackerson touches on the same problem. Unless we make better use of our human capital we will not advance our high tech society. Our society may plateau and lose out to others more ruthless or more efficient. Thankfully a ruthless society may have forgotten the conversation between David Hilbert and Bernhard Rust.

Paddington said...


Many different educational programs have tried to replace learning Mathematics with software, with very dismal results.

Many disciplines, such as Psychology, Education and Sociology, have tried to eliminate learning Mathematics or the underpinnings of Statistics, with software. The results there are that lots of numbers are generated and deductions made, which do not follow from the data, because the users did not understand enough to verify the necessary conditions for the tests that they were applying.

Your idea takes us back to the Dunning-Kruger effect. The very skills and learning required to correctly use software and understand the results turn out to be the very skills taught in those Mathematics courses. Consider that the software which you espouse was developed by Mathematicians, and used by them to solve the problems that are posed.