How does the value of an inverse cosine function change when the unit of its argument changes?

How does the value of \cos^{-1} \left ( 0.1 \, \textrm{cm} \right ) differ from \cos^{-1} \left ( 0.1 \, \textrm{m} \right )?
This is an interesting question, especially because the question of units in trigonometric and inverse trigonometric functions is rarely discussed in mathematics textbooks.
If you’re interested, think this problem through on your own. Then check Problem of the Day 86 at my other site, on or after 23 April 2013, for a discussion.

Posted in Mathematics | Tagged , , , | 2 Comments

Why do Airplanes Fly?

In many physics textbooks, the explanation for lift on a flying airplane is that the top of a plane’s wing is longer than the bottom, and so air must travel faster across the top than the bottom, and therefore the pressure is lower above the wing than below. (This difference in pressure caused by different speeds of a moving fluid is a consequence of Bernoulli’s principle.)
However, the problem with this explanation is that some planes nevertheless manage to fly upside-down! How could these planes fly both right-side up and upside-down? Could the top of the wing and the bottom of the wing both be shorter than the other when depending on the plane’s orientation? This is absurd, and so there must be something wrong with the explanation.
For the correct explanation (briefly, the angle of attack of the wings provides overwhelmingly more lift than any lift produced because of asymmetry in the cross-section of the wing), check out the following references:

Posted in Physics | Tagged , , , , | 4 Comments

New Online Tutoring Site: QED Infinity

Well, it’s been a long time since I’ve posted here. I’ve been involved in a number of writing and education projects, and one of them has now surfaced: My new online tutoring site, QED Infinity.

The main purpose of QED Infinity is to prepare students to make the difficult transition from high-school mathematics to college/university mathematics. Many students need to take a little bit of mathematics, and it’s often scary and difficult. Others need a lot of mathematics. In my teaching career, I’ve been saddened by the number of students who have been unnecessarily stymied by mathematics issues, sometimes having to drop out of their chosen programs, and other times feeling dumb or being stressed out unnecessarily.

At QED Infinity, I intend to help students master as much mathematics as they need to flourish in the college/university program of their choice. There are daily exercises, problems, and thoughts to encourage students to do consistent daily work. There are other resources on the site, and I’ll be adding resources gradually as I produce them. For example, my online textbook will be posted to the site later this year.

Please visit QED Infinity if you’re interested, and please tell others about it.

If you have feedback, or would like to submit a favourite exercise, problem, or thought, please contact me here.


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“Religious Right’s Rejection of Science is Baffling,” by David Suzuki

Religion at its best helps humans to form loving, supportive communities, and helps each person to connect to something larger than himself. Religion at its worst separates groups into the “us” that have the Truth, and the “them” that live in ignorance, and deserve judgement.

Religion at its best is complementary to science. At its worst, it is stupidly contradictory. It is disheartening that the leadership of the Republican Party in the United States is currently dominated by arrogant, anti-intellectual, religious bigots. The Conservative Party in Canada is also burdened by similar types. David Suzuki laments here. Some quotes from the article follow.

Rick Santorum just seems out of touch on every issue, from rights for women and gays to the environment. He’s referred to climate change as a “hoax” and once said, “We were put on this Earth as creatures of God to have dominion over the Earth, to use it wisely and steward it wisely, but for our benefit not for the Earth’s benefit.”

This amounts to saying that playing with fire is for my fun, not for my house’s fun, and ignoring the fact that if I accidentally burn my house down I will suffer the consequences.

Some of these people put their misguided beliefs above rational thought. Republican senator James Inhofe, one of the more vocal and active climate change deniers in U.S. politics, recently said, “God’s still up there. The arrogance of people to think that we, human beings, would be able to change what He is doing in the climate is to me outrageous.”

What is arrogant is the belief that God will save us no matter what we do. Jumping off a cliff in the belief that God will save you is irresponsible, and so is continuing to destroy our environment without a second thought, expecting that God will protect us from the consequences.

That statement is in keeping with the Cornwall Alliance’s Evangelical Declaration on Global Warming, which has been signed by a range of religious leaders, media people, and even some who work in climate science, such as Roy Spencer, David Legates, and Ross McKitrick.

It says, in part, “We believe Earth and its ecosystems — created by God’s intelligent design and infinite power and sustained by His faithful providence — are robust, resilient, self-regulating, and self-correcting, admirably suited for human flourishing, and displaying His glory. Earth’s climate system is no exception.” It also states that reducing atmospheric carbon dioxide and fossil fuel use will “greatly increase the price of energy and harm economies.”

I certainly agree with part of this statement. I do believe that earth’s ecosystem is robust enough and resilient enough that it will survive no matter what we do. What is at issue is whether human society will survive in its present form, or whether we will be relegated to a few bands of hunter-gatherers again. Contemplate for a moment the magnitude of human suffering that would accompany a collapse of human population and the state of the economy will plunge down the priority list.

And then there was this claim from Arizona Senator Sylvia Allen: “This Earth’s … been here 6,000 years, long before anybody had environmental laws, and somehow it hasn’t been done away with. …”

We all wake up every morning, year after year, until one day we don’t. And that’s the way human society is headed if we don’t get our act together soon. And understanding a bit about science is essential to getting our act together, which underlies how important education, particularly science education, is for our collective survival.

Posted in Environment, Society | Tagged , , , , , , | 8 Comments

The Day Niagara Falls Ran Dry

If you’ve had the pleasure of standing right next to the roaring water falling over the precipice at Niagara Falls (and for me, it’s the sound that’s most impressive), then it’s hard to believe that the flow once naturally stopped for more than 24 hours! How could this have happened?

Very strong winds jammed ice against the mouth of the river near Buffalo, damming the flow for almost two full days, starting on 30 March 1848.

You can read more about the strange events of the day here and here.


Posted in Nature | Tagged , , | 3 Comments

“A Lesson in Teaching to the Test, From E.B. White,” by Anne Stone and Jeff Nichols

Anne Stone and Jeff Nichols beautifully make the case against standardized testing at the New York Times SchoolBook blog (hat-tip to Susan Ohanian). Here is an excerpt:

In light of current controversies around testing and teacher evaluation, let’s do a little thought experiment. How would Miss Snug have handled this lesson if it were occurring just before a round of standardized testing? Would she not have had to interrupt the children’s speculations and instructed them that actual circumstances in word problems must be completely disregarded, because the point is to arrive at the answer the test designers have in mind? After all, how could test designers anticipate the lines of thought that spontaneously erupted in her classroom? Real life, and real thought, are too complicated to be foreseen – and so need to be put aside at testing time.

We … are concerned about education because our adult citizens need to be flexible thinkers, ready to adapt to the ever-changing circumstances of the global marketplace. But making standardized tests the center of our curriculums tells children the most important thing they need to learn in school is how to arrive at predetermined answers on the tests. …

So in fact the test doesn’t reflect at all what kids should learn in school. What they really need to master is the kind of imaginative, adaptive thinking Miss Snug encourages in the passage from “The Trumpet of the Swan” – skills that cannot be assessed in any way other than actually knowing the children.

This little episode captures what volumes of education research have shown: we are born curious, and the best education models do not proceed on the basis of “what we want them to learn,” as Mr. Bloomberg correctly describes the goal of test-oriented education, but on the assumption that our job is to foster children’s ability ultimately to shape a world different from what we leave to them.

Now I agree that basic skills in mathematics are important. To learn mathematics effectively, you need to have your multiplication facts at your fingertips, and also your trig identities (if you’re learning at a higher level). However, the problem is that our assessment systems are skewed towards what is easy to test, especially what is easy to test using standardized tests. When resources are strained, as they are now, what is easy to assess ends up dominating the entire educational enterprise.

The best way to engage students is to ask them what they wish to learn. What is it that makes their hearts sing? (However, we should begin asking them early in life; if you wait until 12 or 16 years of the standard education system has crushed the spirit of our children, then you might not get a very satisfying answer to what makes their hearts sing.) And then create an environment where they are lovingly supported in pursuing their own goals. This is the only way to nurture creative and critical problem-solvers that we need so much to confront the serious problems facing civilization. Assessing their work can never be done using standardized tests.

In pursuing their own goals, many students will have need for some mathematical technique or other. With 21st century technology, it should be possible to create a system, a repository, of lessons that teach all of the basic techniques. (I’m sure that the internet is already chock full of such material, albeit scattered and of uneven quality.) When students need them, they can be guided and supported in learning what they need. Making students sit through 12 years of mathematics through high school, and then even more afterwards, in case they should need it, is a terrible waste of effort. Students would be better off doing something of value.

I cringe every time I think about the excellent teachers out there spending precious class time teaching 500 students how to work the product rule for derivatives and such things. Such things are best learned by doing, with feedback, in a combination of small groups and alone. I would much rather see our excellent lecturers spending precious class time on modelling the kind of critical and creative thinking that can’t easily be transmitted in other ways.

A final note: The inhumane standardized testing that is consuming the education landscape like a plague is infecting teacher evaluation as well. Tim Clifford’s article at the same SchoolBook blog is a powerful argument against this horrible practice.

Posted in Education | Tagged , , , , | 18 Comments

“Student Learning Can Only Be Described, Not Measured,” by Rog Lucido

Rog Lucido has written an interesting article against standardized testing, and suggesting better alternatives. (Hat-tip to Susan Ohanian.) He argues that the numerical aggregation of final test scores is not valid and therefore not meaningful, and that subjective assessments together with verbal descriptions are meaningful and valid.

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