A discussion of what a good undergrad programme in mathematics should be about can be found here. The book concerns mathematics in the American system, where applied mathematics was rarely taught in maths departments (at the time of writing). It also discusses the content often presented in four-year American liberal arts colleges, and two-year community colleges, and criticises the specialist nature of the topics chosen. The courses offered to students of science and engineering by departments of mathematics are also panned, one point being that the professor doing this job knows no science or engineering at all.
The author, Morris Kline, talks of the research professor, who hates teaching undergraduates of any sort, and also mentions the wide use made of graduate student tutors, who have neither training nor experience. Scarce mention is made of full, associate and assistant professors, who form (in my experience) the bulk of the academic staff in most departments of mathematics in American universities, and who do a very professional job in their teaching, as well as research and admin.
Some of the criticism of present methodology is way off centre. For example, Kline objects to the teaching of applied mathematics by using over-simplified models. He argues that there is no point teaching the laws of falling bodies as if there were no air friction: tell that to a parachutist, he quips. In this he fails to capture the essence of science: we must study models, and compare with experiment, so that we may shoot them down, and revise them. More, we can usually find a range of applicability of the simple model, outside of which it is no longer a good one. But worse: he seems to be saying that only a fully developed, correct model should be taught. This goes against the teaching of Picasso: a teacher should mix a little bit of what we do not know with a lot of what we do know. I taught physics at Virginia Tech., a course in which the technician had prepared the "Galileo Bench". This was an inclined plane on an air cushion, in which Galileo's laws of falling bodies, s = s(0) + vt + 1/2 at2, was true, to within the experimental error. Some of the students found these laws hard to understand, even without friction. When they had achieved that, we went on to the refinements coming from friction. To start with the full theory, would place them in a similar position to Galileo... who had to cope with Aristotle's dictum that a force was needed to keep a body moving; when Galileo had abandoned this, he broke the 2000 year stalemate in science.
Most of the points made about the limited syllabus of a maths degree do not apply to the UK, where applied mathematics traditionally forms half of the degree in mathematics. However, with the possible replacement of A-levels with a bacc., our school programme might become more like the American one. We should then think about whether the university course might also move in that direction. Should we have four-year degree courses in mathematics, with the first year devoted to maths, physics, chemistry and computing? These could cover the four subjects to replace the omitted parts of A-level. The mathematics course could cover analytic geometry with calculus, trig, probability and statistics, complex numbers, and vectors, with a little theory of matrices. Physics could include Newton's laws, with examples from one dimension, such as motion under constant gravity, friction, simple damped oscillator, sinusoidal wave motion, interference of waves, the laws of thermodynamics, Eulerian fluid equations, and electricity and magnetism (before Maxwell). Chemistry might contain the Bohr atom, the Mendeleev table, the inorganic chemistry of acids and bases, salts and metals. Some physical chemistry such as the law of mass-action, and some organic chemistry, should be included. Lab work in physics and chemistry should be at the level now done in schools in the UK. Computing might introduce a useful language such as Java, Maple or Mathematica, and should give a general competence in Windows.
Kline suggests that scholarship in mathematics would serve a purpose, to reduce the number of pointless and empty papers, by critical reviews. This used to be the job of Mathematical Reviews, until it changed its policy, and now bans controversy in the reviews. Kline suggests that a new degree of high status, Doctor in Arts, should be awarded, which would not require original theorems, but would be readable and deep. We have something similar, in the M. Phil., which however has not got the status of the Ph. D.
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A nice article on antimatter can be read here.
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Everything you wanted to know about abstract algebra can be found here.
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A good way into the theory of graphs is to start with Locke's page.
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The site of the University of Kentucky hosts a list of free material on mathematics, of which III. above is just one. I found the course on partial differential equations very useful. The course on Hilbert Space Methods for Partial Differential Equations, by R. E. Showalter, is very pleasant indeed. It is slightly informal in its definition of distributions, but this is all that is needed for partial differential equations at this level.
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Wikipedia is a free internet encyclopaedia, written by its viewers. There is quite a large set of mathematics pages, as well as pages on physics and other sciences, and all other subjects. Some pages are sketchy, and others are literally empty, awaiting the first volunteer. I found a mistake in Wightman's biography: it said he was British. I was able to edit that page and correct it. The statement of the Navier-Stokes equations could not be right, as the terms do not all have the same physical dimension.
The site is worth a browse, and might become more reliable as time passes.
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Lots of links to notes about probability, information and complexity.
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Free internet sites that are useful for students of maths or physics will be listed here from time to time. Please send me your suggestions.
Go to my HOME PAGE for more links.
© by Ray Streater, 17 July 2003.