MATHEMATICS
I
seem to learn best by exploring things in my own way, and writing about it as
I go along - an approach called constructivism. What follows is a record of some
of these explorations.
Genetic Algorithms. These are computer programs that model
evolution and natural selection in looking for 'good' solutions to problems that
do not have a prescribed solution procedures or algorithms. This article describes
a demonstration GA program.
Data Mining involves the extraction of useful patterns from
large apparently unrelated data sets using various Artificial Intelligence techniques.
Application software is now available that allows the non-statistician and non-computer
scientist to 'mine data' for 'knowledge'. I dream of the possibility of making
millions by mining stock statistics for patterns that will allow me to predict
the future value of commodities! Anyhow; the articles in this section attempts
to explain and clarify what it is all about.
Ballistics of Falling Bombs. Equations used to calculate
trajectories of projectiles, that take into account air resistance are described.
Pascal code for doing your own calculations. Incidentally, this is a link to a
detailed explanation of How Bullets Fly: http://www.povn.com/~4n6/index.htm
Iteration and Recursion HyperTalk and Pascal procedures
that illustrate recursion.
Phase State Diagrams. Graphs depicting behaviour of dynamic
systems commonly use the independent variable, time, along the horizontal axis.
Phase state diagrams display dependent variable along both axes.
Chaos vs Determinism. 'Chaos' was a popular term. Now 'non-linear
dynamics' is preferred. The universe is supposed to be, in principle, predictable.
But some systems are inherently so unstable that a tiny change in conditions can
lead to completely unpredictable consequences. The 'butterfly effect'.
Part 1: Recursive vs Newtonian Functions.Two ways of calculating.
Part 2: The Real World. How do things work in reality?
Part 3: The Logistic
Function. A study of population growth and extinction.
Fractal Geometry Euclidian Geometry describes triangles
and squares and circles and speaks of one and two and three dimensions. But we
do not live in a Euclidian world. We live in a fractal world in which objects
have fractional dimensions like 2.34 !
Part 0: Introduction
Part 1: Dimension
Part 2: Self
Similarity
Part 3: Coastlines
Part 4: Sierpinsky Carpet
Part 5: The
Mandelbrot Set
Part 6: Applications
Centroids. Recently someone wrote me asking how to find
the centre of gravity of polygons. I didn't know but was intrigued enough to find
out.
An Argument Against Screening for HIV . Medical tests are
not infallible. This can cause problems.
Investigations with Iterative functions. The behaviours
of moving objects can be modelled (perhaps not perfectly- but pretty closely)
by breaking their motion down into very may small steps.
Part 1: Doing iterative functions on a spreadsheet.
Part
2: Period of a pendulum
Part 3: The comet's path.
Introduction to Numerical Calculus. How to do differential
and integral calculus numerically without learning calculus.
Ray Diagram Calculations. The paths that light rays follow
though lenses is explored. Lots of trigonometry here! How to find the focal length
limitations of a single double convex lens.
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SCHOOL MATHEMATICS
Editorial
School
Mathematics is what is taught in schools. It is not the same as what mathematicians
do. Unlike what is the case in other subject areas, most of the curriculum has
remained essentially unchanged since the mid 1800's.
I
believe that much of what is done in the name of (math) education is unsupportable. Also, I know that just about everyone I talk to about this
feel the same way. Yet in all the years that these questions have been posted here, I have had perhaps
three or four people write and say they agree with the implication of what I say
- and not one has taken issue. Why is that?
Most
visitors to this site are referred by reference pages at university libraries
where this site is listed as "School Mathematics Journal" which is what
I used to call it at the beginning - some eight years ago now. Presumable this
means that most visitors are interested in mathematics education. If that describes
you, have a look at the questions I ask, and maybe
tell me what your answers are..
Items
of General Interest
JERRY
UHL has made a collection of quotations about
how Mathematics is taught.
VECTOR is a quarterly publication of the British Columbia
Mathematics Teachers' Association. My article "Why is Mathematics so difficult
to teach?" was included in a recent issue with invitations to comment:.
GRAPHEQ The act of learning how to use a piece of unfamiliar,
but well designed software can be very "educational" - even for teachers!
Recently I sat down with the manual for this different graphing software
(written by a friend, Gary Tupper).... and learned a few things. For example,
this programs graphs relations as opposed to functions, which no other
graphing software that I am aware of can. This is much more difficult to do from
the programming point of view, and the results can be very interesting. It employs
an amazingly easy-to-use equation editor which will allow you do copy/paste math
expressions into for example word processors. You might find this item of interest.
MuPAD & Maple. These are mathematical software of the
same genre as Mathematica. Maple is a product of Maplesoft, which is affiliated
with the University of Waterloo, Ontario. MuPAD was similarly developed at the
University of Paderborn, Germany and is distributed by an outfit called Sciface.
Both of these do symbolic manipulations as well as numeric ones, and both produce
some pretty incredible graphics. This article an ongoing project. Participation
invited.
Precision and Accuracy in measurement An important, but
neglected arithmetic topic.
Phase State Diagrams. Graphs depicting behaviour of dynamic
systems commonly use the independent variable, time, along the horizontal axis.
Phase state diagrams display dependent variable along both axes.
What is Algebra, and why do we teach it? Do we ever ask
"Why are we doing what we do?"
Experientialism vs Objectivism. Two different paradigms
of Mathematics. Implications for curriculum.
Why Teach Mathematics? The answer is not obvious.
The Museum of Mathematics . Is traditional calculus relevant?
Why is Mathematics Hard to Teach? "I was never a good
math student" is something we hear too often. Why is mathematics perceived
to be harder than other school subjects? Some suggestions.
Puzzles or Problems? Few public school students actually
read textbooks. To all intents and purposes textbooks at this level are simply
a collection of "exercices". Some of these are referred to as "problems".
But how do problems differ from puzzles?
You know about trigonometry, right? You know that sin(30)
= 0.5. You can even prove that sin(30) = 0.5. Now, explain to me how, from first
principles, sin(32) is derived. Really, I'd like to know. One not very good answer
is "You use the Taylor Series". A question: Is it possible that you
understand trigonometry if do not know how to find sines from first principles?
If yes, are you saying that it is possible to understand something without "knowing
how to do it"? What does it mean to understand?
Math You Need This is the latest of many ongoing projects
that are still in the beginning stage. The idea is to write some material that
illustrates convincingly real, practical, and useful mathematics... from the high
school students perspective. What do you think?
Symbolic Calculators. Maple and Mathematica are symbolic
calculators. I have a question about their use. Do you know the answer?
Excel in Math education. Free, easy-to-use math software.
Graphs, linear algebra... lots of good stuff.
A Better Mathematics Curriculum. What should the curriculum
look like?
The following are external links to mathsites I found interesting:
This is the
Mother of All Mathematics websites. Sponsored by Wolfram, creators of Mathematica...
well, there isn't much more one can say.
The
Interactive Mathematics Site
has math blogs, but also some really impressive math explanations including
wonderful animations. Very suitable for classroom use. Murray Bourne, the
author/webmaster also has some complimentary things to say about me at here
Dr. Math. These are
real questions with real, very good answers. Search for keywords. Good.
Roger Shank (external link) writes some provocative stuff
on education in general and Mathematics education in particular. He does go a
little overboard in some places, but if you are interested in this topic - and
I assume you must be if you are reading this - I recommend you have a look.
Want to improve mathematics education?
Teach less of it. Report of an experiment that suggests that math instruction
is better left out of elementary schools altogether. I believe this.
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SAILING AND CRUISING
I
am fortunate to live in the Mediterranean of Canada - the Gulf Islands. The wind
in these parts is moderate, and the climate and the scenery is ideal. Those of
us who have sailboats - well, we're pretty lucky.
Improving the glowplugs circuit for better starting.
Installing a digital voltmeter. The second article on modifications
to GYPSY's electrical system. A digital voltmeter is indespensible for diagnosing
electrical problems and monitoring battery charge state.
Discharge characteristics of a deep cycle battery.
Random Notes from cruising - Summer 2002
Cruising Around Vancouver Island 2000. A couple of summers
ago I circumnavigated Vancouver Island with a friend. The distance is in the order
of 700 nautical miles.
Howe Sound. Just north of Vancouver.
Horton Bay is on Mayne Island. San Juan Islands.
A visit across the border.
A week in Puget Sound 2002
You've Come a Long Way. How to treat beginning sailers.
The Smithsonian Institution. Nothing to do with sailing.
Drift. The track of a boat depends on its speed and direction
in still water plus the effect of cross currents.
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MISCELLANEOUS
A
collection of ramblings on various topics. Not to be taken too seriously.
Vacuum Cleaner Specifications. There is a lot of nonsense
and misunderstanding out there.
Topping Trees is not Clever. The destructive practice of
tree topping is alive and well on Saltspring. Another example of misguided nature
lovers at work.
I Have a Hearing Problem. A few comments that might be useful
to people who live with someone with a hearing problem.
Hearing Aids. Hearing aids are expensive, and there is a
lot of non-knowledge out there about how they work and what they can/cannot do..
On Mattresses. Buying a mattress can be a very frustrating
business. The jargon and lack of useful standards confuses the hell out of most
people. Here I suggest one approach to injecting some sense into this field.
Viruses of the Mind . (External link) Richard Dawkins, author
of The Blind Watchmaker and The Selfish Gene, introduced the notion of memes.
In this article he suggests that "mental memes" (opinions, beliefs,
etc.), biological viruses, and computer viruses have essential characteristics
in common
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