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1. Am I correct in saying that all college students in the USA have to do a math course? If so, how and why was this rule introduced, and what do the students have to do?
To my knowledge, at one time every US college student had to pass a Calculus course. This was gradually weakened over the years, and Mathematics became known as the 'weed-out' subject. As more universities opened up and so needed more students, the requirements were made easier. It came to a head in 1968, when failure (for male students) meant getting drafted into the Vietnam War, and administrators seriously watered down the coursework.
With the push for the STEM subjects (Science, Technology, Engineering and Mathematics) in the early 1980's, especially in Computer Science, many universities tried to increase the Mathematics requirements, only to find out that failure rates were 'unacceptably high'. When 40 years of college-level remediation efforts were shown to have failed, lots of 'experts' began pushing the idea that one should learn Statistics instead, and a watered-down version of an introductory course began to be accepted instead of an actual Mathematics course. Unfortunately, actual understanding of Statistics requires ability in Algebra, which is the very material that the students can't pass. This is all too often the problem with the Statistics in Sociology, Psychology, Education and related areas.
Some universities accept a very cursory course in Logic, doing less in 15 weeks than I used to teach in 3 weeks in Discrete Mathematics for Computer Science. Others accept a course called something like 'Math Appreciation', 'Excursions in Mathematics' or 'Math for the Liberal Arts'.
Theoretically, the standard Mathematics course requirement at many universities is something called 'College Algebra', which is an Algebra course dealing in functions, matrices, logarithms, exponentials and some minor topics. The material is the same as that usually done in high school Algebra II, at age 16-17. This is the material that I did in 3rd year of Secondary School.
Such a course would be the jumping-off point. Students headed to STEM areas would take Pre-Calculus, or Algebra with Trigonometry, and then on to at least 2 years of Calculus. Students in Business would likely take a watered-down Calculus course, and something in Statistics.
Throughout all of this, students and administrators blame the Mathematics departments for the failure rates, which haven't changed much in over 40 years. For reference, when I started teaching in 1978, the typical state university had a graduation rate of about 33% within 6 years. This was blamed on the Mathematics requirements. However, when I asked for the data on that, it did not exist. On the other hand, when we looked at predictors for college success, it turned out that grades in Mathematics courses and standardized tests were the best ones available.
2(a). The start date still isn't clear. Would it have anything to do with JFK and the response to the realization that Russia was pulling ahead in the Space Race? (Over here in the UK, I recall that at least Oxford and Cambridge made a pass in 'O' level maths an entry requirement; a friend who got a scholarship to Cambridge in History tried and failed in maths four times and the college let him through anyhow.)
I honestly don't know when. Certainly, these was a great 'crisis' in 1957 when the USSR launched Sputnik, and leaders recognized that we were behind in ICBM technology. Then, as now, we temporarily fixed the problem with immigrants, at that time from post-War Europe.
2(b). Do you think a universal maths course is still a good idea?
After years of frustration and watching failure, I would abandon the Mathematics requirement. BUT success in all of the 'good' areas (read entry-level salaries) requires Mathematics, from Accounting and Finance to Nursing to Engineering and Actuarial Science. So, we have the do-gooders saying that it isn't 'fair', and watering down all degrees.
3. You have previously told me that maybe only 15% of students are capable of higher level math. Is that because of natural ability, or failures in high school teaching?
When I started in 1978, about 15% of high school graduates, and 20-25% of entering college students had mastered enough Algebra to pass a placement test, and take College Algebra. When I retired in 2017, after multiple rounds of reform and the inclusion of Technology, those numbers were the same. The only difference was that the top 10% of students had weaker skills than their predecessors. I attribute the latter to the overuse of calculators and related software.
I have argued with my colleagues, administrators and all over the internet for decades that the problem appears to be something in the brain, while others argue that it is defective teaching. My argument is that, while the latter most certainly takes place, it would have to be almost uniform across the US to get such consistent results. This is probabilistically unlikely. My argument is aided by some research in brain development, showing that difficulties in learning Mathematics seem to be connected with either immaturity of the hypothalamus, or of the myelin sheaths in the brain, the latter being connected to the ability to move from concrete thought processes to abstract ones.
It is worth noting that the historical pass rates for the standard first-semester Calculus course are the same in Sweden as in the US.
4. You refer us to a paper on the cross-currents in mathematical education from the eighteenth century on:
http://jwilson.coe.uga.edu/EMAT7050/HistoryWeggener.html
What would be your answers to these questions, which seem to be the core issues:
4(a). What mathematics should all people learn, useful to them in their future work and daily lives?
Before I attempt to answer these questions directly, let me note a further problem. Not only does it appear that every person has a natural level of Mathematics attainment (my experience suggests 95% can learn Arithmetic, 85% can learn Algebra, perhaps 5% can learn Calculus, and much less than 1% can learn higher-level Mathematics), but there appears to be a 'window of opportunity' for that learning, as there is for languages. Hence, if we allow large portions of the population to opt out of the subject, they can never get back on track.
Useful math: in a modern world, every functioning person should have an idea of weights and measures, percentages, and basic probabilities. Most do not, and are not even close. That's why many people make such terrible financial decisions.
4(b). What should be taught to all, for the sake of national military security and economic prosperity?
We need as much of the population as possible ready to learn in the STEM areas. As more jobs are automated, the need for technical repair people goes up exponentially.
4(c). What mathematical learning should be reserved for an elite naturally qualified for the study? How, and how early, can such people be identified?
See the answer above, and add the need for experienced Mathematical modelers in all fields of study and research. The National Academy of Sciences in their report on the year 2025, suggested a scheme of collaborative research including an Applied Mathematician in just about every discipline, including the Social Sciences. Much of the issue with research in the fuzzier subjects is that it is Statistical in nature. That means that it is descriptive of what is (if you are lucky and people aren't lying). It is very rare to take the next step, and model the phenomena. Instead, people express opinions as to why things are the way that they are. In short, it is much easier to explain the past than predict the future. In the current climate, it is also more financially and socially valued, but totally stagnant. The science and SF writer Isaac Asimov noted this in his first ‘Foundation’ novel.
5. I was heading for this one and you have anticipated me.
5(a). When would you say the 'window of opportunity' closes?
Our hypothesis was that the window was around the typical age to move to formal operational thinking, at age 12-14 or so.
5(b). Does this mean there should be a wide-spectrum maths education up to that age?
In short, yes.
5(c). Is there a good way to assess aptitude for higher math?
Sadly, the only way seems to be for the student to try, although you can clearly see the tendencies in very young children - counting and sorting.
5(d). Does this also raise the issue of having sufficiently skilled math teaching in school?
Of course. In the US, most Mathematics teachers have far less than an undergraduate degree in the subject. A lot is taught by people who only had cursory education in the field, due to teacher shortages and seniority rules. Most of the most talented Math Ed students that I taught ended up not going into education at all, since the opportunities in areas like Finance were more lucrative and less stressful.
6(a). Continuing with secondary age math education, you have previously told me that your college freshmen come to you thinking they know material when they don’t. There appears to be more behind this than the school-teachers' lack of expertise - can you tell us what goes on in school to allow their students to maintain that illusion of knowledge? How are students assessed in American schools?
I would say that it's partly the Dunning-Kruger effect (https://en.wikipedia.org/wiki/Dunning%E2%80%93Kruger_effect), and largely the difference in views on schooling between the US and Europe. Here in the US, there is zero respect for most teachers, unless they give good grades to the little darlings. Politicians and parents claim that 'good' teachers can teach anyone to mastery. People have tracked high school graduation rates (and tout them) as they rise over time. College grade averages and graduation rates go up every year. At the place where I worked, the rate went from about 31% over 6 years to 50% or more, with a discernible lowering of standards. It's one of the many reasons for these degrees in Media Studies and the like.
As for testing, we have the ACT and SAT, but lots of parents and administrators don't like them, because they show the actual weaknesses, so they claim that 'tests don't measure students'. We did a study on 7,400 of our students, looking at the ACT Math sub-score versus whether they graduated in 6 years or less, and found almost perfect correlation. When we presented this to administrators, they were less than impressed.
Our problems are compounded by the rules by which the state legislature supports the public universities. It used to be based on total numbers. Then, someone thought that we were wasting money by flunking out so many students, and changed the system to reward grades and graduation rates. Surprise, surprise, both went up immediately. By the way, the same legislature artificially increased the requirements (especially in Mathematics) to graduate high school, and then made it harder for the state universities to refuse students. The private schools and colleges had no such issue. Some remained highly selective, others just pretended, as there is no national exit exam in most disciplines.
Over the years, Ohio generated various competency tests for graduation, to be taken by sophomores (5th year students). The ones that I saw could be passed easily by a decent 6th grade student (age 12 or so), but they had to set the pass bar at 42%, and still many students failed it repeatedly.
Then, an impressively well-meaning and totally inept set of reforms changed the minimum Mathematics to graduate high school from Algebra I plus one more year, including numeracy courses, to 4 years, including Algebra I, Geometry and Algebra II, courses only previously taken by the top 30% or so. Because the less-talented students were thrown into the classes, and failure is failure by and of the teacher, students who would previously have obtained C's in courses were suddenly A students, which reinforces their illusion of mastery (a phrase which I coined when I was chair of our department). One of the reforms consisted of having State-wide end-of-year exams in those three courses. When they piloted the one in Algebra I, only 35% of students passed, even though the bar was set fairly low. The 'experts' at the State Board of Education tried to cover their tracks by changing the threshold, only to be admonished by the Federal officials. That test appears to have vanished into limbo. Interestingly, that 35% rate pretty much coincides with the historical 20-25% of incoming freshmen ready for college-level Mathematics. No-one that I ever talked to wanted to hear that either.
6(b). Leaving aside (for a moment) the teacher's own subject expertise limitations, do schools need better texts to guide the students, and better tests to check their progress?
There is certainly an issue with the quality of teachers, since they need to know the material. As the great Mathematician Polya said, "One cannot teach what one does not know". However, that same 15% of 12th-grade students who know enough to take a college-level Mathematics course becomes the 5% or less that can make it through two full years of Calculus and beyond. From that number comes all of our hard scientists (Computer Science, Geology, Chemistry, Physics, Mathematics, Statistics), and a lot of Biologists, plus all of the Medical Doctors and Engineers, the top Finance and Accounting people, Actuaries, and most technicians. That doesn't leave many people to go and teach. In the UK they offered scholarships and signing bonuses for Mathematics teachers, and got very few takers. We worked with a Foundation to take Math majors and get them the Education credentials that they needed. For obvious reasons, they were weak students, or they would not have taken this route. Getting them to pass Education courses was a doddle. Not so much to pass the Math certification exam. We gave them one such, then coached them for 8 weeks, and administered the identical multiple-choice test. Not one student changed their score substantially.
Textbooks are a whole other ball of wax. They are big business here, and written by professional writers who usually know no Mathematics or Education. What they try to do, very badly, is to give a script to a teacher who does not know what they are doing. The good news is that there are loads of free resources out there, such as Khan Academy and quora.com, which can help students. The bad news is that these sites are used instead of brainpower, so that the skills are decaying even further. It doesn't help that so many rely on calculators (many with Algebra and Calculus features built in), to the point that they might get the right answer by accident, but can't correctly transcribe the results, or understand them.
Personally, I would do what Singapore and many Asian countries still do, and that is to not require Mathematics beyond age 14 or so. This cuts you off from the Sciences, but not the Arts and Humanities. I would go further and have licensure (the old O-levels and A-levels would be fine) at several stages. I would not use the new GCSE stuff, as political pressure has degraded their quality as well. I would use those certificates to limit what people were permitted to do in the Sciences.
7(a). There is also (is there not?) an issue (in the UK as well as the USA) of social pressure on academically-inclined students, ranging from under-trying in order to be tolerated by their 'cool' peers, through to outright bullying of the nerd or 'swot'.
There has actually been a lot less of that in the past 20 or more years, now that computers and computer games have become ubiquitous. However, there was also the movement that 'we need more women in STEM, other than Biology'. That meant open encouragement and nurturing, which is largely a good thing. It contrasted with the common experience of older female friends, who have told me about being told that, "Girls can't do Mathematics" (by contrast, in my years as an undergraduate at Exeter, 60% of the Mathematics students were female). This nurturing also meant that many students got great grades thanks to the miracle of 'extra credit', in spite of failing tests. I believe in tests in Mathematics. I will repeat something that I said years ago: In my 39 years of teaching Mathematics, perhaps 5,000 students, I had exactly two who failed tests repeatedly, yet could pass the equivalent of an oral exam. Both had burned their brains with street drugs as teenagers. One went on to work for NASA.
In the binary mode of thinking which is so common in the US, nurturing female students meant elevating them above the males. My eldest son, no slouch in the brains department, told me that the parade of girls getting the high school awards each year were often carefully manipulating their teachers. We had many such operators (both male and female) arrive as undergraduates, to find out that they actually had to perform, and fold under the pressure.
7(b). You have said how hard it is to recruit able math graduates to school teaching. May I suggest that not everyone is motivated solely by money and that such graduates might be more likely to apply for posts in schools where pupils were selected for their academic talent and commitment to learning?
No, it isn't just money. In fact, there are even public-school systems in rich towns and suburbs where typical teachers earn double what public college professors do. This comes at the price of very 'involved' parents, including those who bring lawyers to parent-teacher conferences. Again, a lot of the US is Lake Woebegone, "Where every child is above average".
8. In conclusion, and looking at what you have said here, please summarise why mathematics matters for the USA. What detailed program of action would you recommend for the reform of mathematical education to meet the nation's needs?
The US has always got a lot of innovation from immigrants. First in the 1800's, with many peasants displaced from Russia and Germany by farm industrialization, then by refugees from Europe after WWII, then by emigres from the Soviet Union, India and China in the 1990's. Government policies and racism have discouraged such immigration, although we still get quite a few from Vietnam, Bhutan and Nepal. Many of the other nations have encouraged the educated to return home. Our officials, of both parties, seem blind to this, as they tout 'American ingenuity' and destroy the quality of the education system.
If they realize in time, and make investment in the STEM areas, it will have to be done against the vacuous idea of 'equality'. If my experience and observations are correct, no amount of coaching will help the untalented. What would help would be to create the equivalent of grammar schools in the STEM areas, perhaps one per county, and move students there in grade 6 or so. Such selection would be brutal by tests, and data suggests that it would be called racist. Not to mention the children of the richer parents, who would cause the real stink.
The cost, if we do not do something correctly, will be to sink into Third World status. The coin of a vibrant economy is innovation and technology, and always has been.
Given the large numbers of very loud groups who insist that the Earth is Young and/or Flat, that vaccines are worse than the diseases that they prevent, that twisting people's necks can cure all ailments, and similar stupidity, I do not hold out a lot of hope!
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