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Given that the theme for Math Awareness Month this April is Math and Voting, this week’s choice by Murray Bourne seems particularly appropriate:

Friday Math Movie – Uncounted – squareCircleZ

Green Globs & Graphing Equations Home Page

“Holding theorems in their hands” is a blog post about the Hyperbolic Crochet Coral Reef Project. It’s a wonderful story about collaboration on many levels and across many interest groups – and with beautiful images to boot. I saw it via Stephen Downes.

In Do the math – The Boston Globe columnist Jan Freeman dismisses objections to the common usage of “three times less than” to mean equal to one third of.

But the Merriam-Webster editors (per JF) are completely off base if they claim that “times less” has never been misunderstood. …more »

I got this post by Roger Schank via Stephen Downes and share Stephen’s concern with some of its polemical conclusions. The following counterpoint to Schank’s article may be a bit silly, but no more so than the original. …more »

In this post at squareCircleZ, Professor Bruce Armstrong from the Sydney Cancer Centre at the University of Sydney is quoted as saying “The probability the that increase is due simply to chance is about one in a million so we are looking at something that is almost certainly a real increase in risk”. But this is almost certainly a misstatement since the probability that something is due simply to chance is not computable and probably not even meaningful whereas the probability of its happening in a randomly chosen situation from a well defined population of cases is meaningful and often computable. Either concept could be expressed by the ambiguous title of this post but they are definitely NOT the same – as can be seen from the following example. If I win the lottery without cheating then the probability of it having happened by chance (in the sense of having only chance factors involved) is in fact 1 but the probability of it happening by chance (ie of it happening given that only chance factors were involved) was less than one in a million. Of course, if we don’t assume that I didn’t cheat, the probability that my win was due only to chance may be less, but in any event it is not the same as my chances of winning a fair game. For a more practically relevant example consider the case of an experiment which identifies an effect of some sort “at the 95% confidence level”. What this means is that the probability of the observation occurring if only random effects were present is no more than about 1 in 20. But then in a set of many trials it is likely that up to about 5% of them will actually appear to show the effect. Users of statistics need to be aware of this distinction since in an experiment which collects more than six variables (as many in the social sciences do) there are more than 21 pairs to consider and so in an average such experiment at least one such pair will seem to have a significant relationship even when no such relationship actually exists.

All this is actually relevant to the story about cancer clusters since, in a world with several million observed groups of a hundred or so people, if the chance of a cluster happening given only random factors is one in a million then we may expect to see several such clusters occurring just by chance.

squareCircleZ » Zipf Distributions, log-log graphs and Site Statistics

mathschallenge.net
Fundamental Theorem of Arithmetic

squareCircleZ » Another semi-log graph from Alexa – imeem

This from Stephen Downes is, for me, a reminder to consider whether any of my stuff might be useful One Laptop Per Child and/or the Commonwealth of Learning.

BBC NEWS reports British chemists as pointing out the difference between an admission test for Chinese science undergrads and a UK university’s diagnostic test for incoming students. But perhaps they are comparing Chinese apples with British oranges (or vice versa?)
After all, in the Chinese question (assuming that by “square prism” they mean “right prism” – which is what it looks like) part (ii) could be on the senior level of our own BC high school math contest (and so could the rest if our high school students had any exposure to vectors), and the UK one could be a soft pitch from our Langara  Math Diagnostic Test.

In squareCircleZ » Trig graphs – how do you feel today?, Murray Bourne at squareCircleZ has a nice Flash gizmo to show biorhythm graphs and links to a site debunking their validity. Actually I think a nice lesson based on this would be to use a similar gizmo to support a demonstration of how sensitive the timing of coincidences between the different cycles is to small perturbations in their frequencies (most of those who experience biological cycles know that they usually aren’t entirely regular!). This might also lead to discussion of other more real biological cycles and periodicities – including nontrig ones like junebug and locust populations – and to the fact that astronomical regularities do have such unearthly precision that they can in fact be followed for many hundreds of cycles to make predictions about coincidences which are actually observed.

April is Mathematics Awareness Month and this year the American Mathematical Society, the American Statistical Association, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics have announced that the theme for Mathematics Awareness Month 2007 is Mathematics and the Brain.

Thanks to Zac at squareCircleZ for pointing out the dawn and dusk graphs at Gaisma as real-life examples of approximate sine graphs.

In fact the true time of noon appears to oscillate slightly with a 6 month period so that the Tokyo graphs are modelled pretty well by 6+1.5sin(2pix/12)-0.15sin(2pix/6) & -6-1.5sin(2pix/12)-0.15sin(2pix/6).

(Update: thanks to zac also for pointing out in the comment below that I should either have used cosines or have said “where x is the number of months since the spring equinox”)

It will be neat to be able to give Gaisma as a source of reference data for the hours of daylight modelling examples in my precalculus classes.

Phase, Frequency, Amplitude, and all that.. is an example of a university math course adopting the convention that identifies “phase shift” as angular shift as opposed to horizontal displacement or “time shift”

And at the time of this posting, the Wikipedia article on Phase (waves) takes the same point of view.

Murray Bourne at squareCircelZ has taken the time to respond to a comment I made on one of his interactive math pages, so I thought I should make an effort to explain my concern in a bit more detail.

In high school and college precalculus courses, the material on graphing trig functions often includes a definition of “phase shift” which is contrary to the way the term is used by many in applied fields and also by many mathematicians (including me when I have a choice).

The usage demanded by high school examiners corresponds to the horizontal shift of the graph from a purely scaled basic trig function. So, for A*sin(bx+c)+d it would be given by s=-c/b since, with that value for s, we get A*sin(bx+c)+d=A*sin(b(x-s))+d , so the graph of y=A*sin(bx+c)+d comes from y=sin(x) by first scaling to get y=A*sin(bx), and then shifting horizontally by s units along the x-axis (and vertically by d units along the y-axis).
But in fact, the concept of phase arose from a need to identify the part of the cycle being considered (ie rising, peak, falling, mid-point, trough, etc) and is usually identified quantitatively by an angle. So we typically talk of two waves interfering constructively when “in phase” and destructively when “180degrees (or pi radians) out of phase”, and we also speak of a process such as reflection or refraction as introducing a “phase shift” of so many degrees or radians in the propagation of the wave. With this usage, the phase shift of A*sin(bx+c) relative to A*sin(bx) is just c (radians) rather than the math teachers’ -c/a.

Some authors seek to avoid the conflict by identifying “phase shift” as what the high school teachers insist on and “phase angle” for what the other camp prefers. But I think this is a mistake for several reasons. My main objection is that even if it might be a good idea to implement such a change, it should not be taught to students as fact if it has not in fact yet been established as a convention agreed to universally in the professional mathematics community. There is nothing wrong, and much to value, in admitting to students that not all terms have universally agreed definitions and that when they face such terms it is important to *ask* what convention the user intends rather than to blithely assume something that may be wrong (which is just the sort of thing that leads to expensive space probes crashing into Mars and causes international airliners to run out of gas in the middle of the Atlantic).

But if that particular convention were proposed I would argue against it as I believe it serves no purpose other than to “save face” for the math teachers, and does so at the expense of abusing the language. I say this for three reasons.

First, the word “phase” was introduced to refer to the position in a cycle (which is basically an angle), so to speak of a “phase angle” is redundant.

Second, there is a phase (angle) corresponding to every point on a wave and the term “phase angle” does not properly denote a shift.

Thirdly, to use the term “phase shift” for what in any other graph would be called the “time shift” or “horizontal shift” introduces a completely useless extra bit of language by having a special context-dependent term for something which already has a perfectly good name that works in every other context.

And fourthly (I know you weren’t expecting the Spanish Inquisition, but do you know the three kinds of mathematician?) wasting a term where it is not needed makes it unavailable for where it is actually useful.

When two split light waves are brought together again (as in the creation of a hologram) it is not the phase (angle) itself at each point but the angular shift between the two waves that is directly relevant to the outcome rather than the time shift between two signals. We could of course convert the time shift to a phase (angle) shift just by using the known frequency and velocity of propagation, but it would be silly to use both terms to refer to the time shift and leave ourselves without a name for the quantity that is actually most directly relevant.

The convention that makes most sense to me is therefore to use the term “horizontal shift” (or whatever term they’d use for the x-displacement in any other function) for what the math teachers call “phase shift” and keep “phase shift” for its traditional role as what is now being proposed as “phase angle”.

This posting by Roger Shank
(found via Stephen Downes) uses widespread ignorance of the quadratic formula by successful people as evidence that mathematics requirements in our schools are excessive.

But I know that in BC it is quite possible to graduate from high school without knowing the quadratic formula. So, unless this jurisdiction is more unusual than I think it is, Shank doesn’t seem to know his head from a hole in the ground.

Of course it will always be true that “We need more people who can think. We need to teach job skills, people skills, and reasoning skills. And we need to make education exciting and interesting.” But Shank surrounds these observations with so much incoherent and contradictory posturing that I would consider his polemic virtually useless for persuading anyone who actually does know how to think.

For example, his “Here are reasons why” (teaching math and science “better”) “is simply the wrong answer“(to the question of “why American kids aren’t interested” in science and engineering) is followed not by reasons but by a series of rhetorical questions directed not at that issue but rather at the motives of foreign students – which he does not relate at all to the lack of motives for domestic ones.

Then later he says “The right answer would be to make math and science actually interesting” – but isn’t that exactly what teaching them “better” would consist of??? But then again, why *is* this the right answer if, as he asserts a bit further on, “What also makes no sense is the idea that math and science are important subjects.”? Of course they aren’t *essential* to everyone, and he seems in various places to acknowledge the need for at least some people to know these subjects, but someone who can’t correctly say what he means shouldn’t be pontificating about how to teach people to think.

squareCircleZ � When am I gonna use this stuff?

Philosophy of Real Mathematics: Dawid on probabilities

Producing mathematics for the Web is a posting by Zac in Singapore on his ‘squareCircleZ’ blog about the tools he uses to produce his Interactive Mathematics website