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.

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(*) 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.’
https://www.verywellmind.com/an-overview-of-the-dunning-kruger-effect-4160740