A rebuttal to Jarrah's latest masterpiece

[nomuse]

No, Tournesal. Tryphon Tournesal (known in the States as Cuthbert Calculus). Had a boat that looked like a fish. (Okay, actually a submarine that looked like a shark), plus a moon rocket that looked like a V2 -- complete with test markings.

[JayUtah]

I remember integrating and differentiating pages of complex contrived formulas that I knew I would never see in the real world.

Yeah, not helpful.  Today we have mathematics engines that can correctly (if not concisely) integrate practically any integrable function you can express to them.

To instill concepts, I prefer graphs to equations.  I'm a visual person anyway, but I've found by experience that the essence of behavior is revealed more readily at first using a picture of some kind.  After that happens, then we can go back and derive the mathematics necessary to quantify the behavior and explain how the math does that.  Too many people expect students to comprehend behavior at first from the equations.  Few beginning students are yet suitably conversant in mathematics.

But times have changed. Numerical integration is a backbone of science and engineering because it's useful in the great majority of real-world situations that cannot be solved analytically, at least without too many simplifications.

Indeed, if you pull a few practical engineering texts off the shelf and see whether the authors jump from foundation concepts to practical solutions, most of the ∫'s are quickly replaced with a Σ.

The advance of terascale computing (Jay takes an elaborate bow) and the techniques that develop to use it practically do not fully eliminate analytical methods, but they tend to force things to a state where you write algorithmically simple programs that can perform a wide variety of integrations.

So I think there may be merit to using a rudimentary form of numerical integration (vs the classical analytic approach) as you teach, e.g., Newtonian mechanics.

That's exactly how I taught it:  with straightforward differentiation in the form of "measure the slope of this graph between these two data points" and "find the area of this rectangle" integrals.  I don't introduce calculus concepts, other than to say that there exists a mathematical technique they'll learn later as an improvement, invented by lazy mathematicians who got tired of adding up rectangles.

[ka9q]

Yeah, not helpful.  Today we have mathematics engines that can correctly (if not concisely) integrate practically any integrable function you can express to them.

Exactly. I found it fairly easy to learn those branches of math for which I could see obvious applications, but I knew very early that I would never become a mathematician. I found high school trigonometry/analytic geometry quite easy. Its utility in electrical engineering was obvious, and I was already doing satellite orbit predictions. But some of the others, including pure calculus, were always a struggle.

Then again, in college I had trouble with differential equations despite their fundamental utility in all of engineering. I think that had to with some poor teachers as well as skipping a semester with advanced placement. That's not always a good idea even when you can qualify.

Part of my problem was that so much emphasis was on manipulating formulas and very little on understanding the underlying concepts and when and how to apply them to solving other problems. I often wondered how many math teachers got into the job precisely because they didn't know how to solve anything in the real world.

Most of my favorite mathematician jokes are based on this image, so I know I'm not the only one in the boat.

[JayUtah]

I found it fairly easy to learn those branches of math for which I could see obvious applications,

Well yes, because the application helps you understand the behavior among quantities that the formulation is trying to express.  You need conceptual understanding and quantitative rigor, and there's only a modest amount of overlap between them in most presentations.

kipping a semester with advanced placement [is] .. not always a good idea even when you can qualify.

One of my assignments as an advanced student in college was mentoring the "prodigies"  -- the high-school and younger students taking college classes in science and engineering.  I saw a lot of burn-out.  I think you need time-in-rank for some of these bodies of knowledge.  It has to gel a bit and become second-nature before you build upon it.

Part of my problem was that so much emphasis was on manipulating formulas and very little on understanding the underlying concepts and when and how to apply them to solving other problems.

They have to go hand in hand.  Knowing how to integrate is important because you'll need that to derive something down the road.  But at the same time, knowing why kinetic energy looks like the integral of something is essential to knowing how our formulation of the natural world works.

I often wondered how many math teachers got into the job precisely because they didn't know how to solve anything in the real world.

All of them, as near as I can tell.  I made a big push to have essential mathematics taught within the engineering department by engineering professors, not farmed out to the mathematics department.  I had seen good results with essentials like differential equations and linear algebra taught from a practical perspective.  But budgets are what they are, so engineers still get the esoteric, abstract presentation.

It's not really a dig against mathematicians.  There's a time and a place for abstract quantitative thought, and many of these advanced concepts like eigenvectors eventually bear much practical fruit even if they seem at first like conceptions for their own sake.  But presenting abstract mathematics to people who need a more down-to-earth understanding is like trying to explain woodcarving to someone by extolling the metallurgical virtues of the chisel.  Just hit it with a hammer already.

[Echnaton]

I think that had to with some poor teachers as well as skipping a semester with advanced placement.

I had a great Calc 2 teacher and an even better TA.  I had taken busines calc before so I skipped the pre reqs and Calc 1 because the math I really needed to learn was taught in Calc 2 and matrix algebra.  Thank god that was one semester because it kicked my but trying to go back and learn the trig.  I still did better than half the class on the first test, less so on the trig centric second test but got a B in the course.   I also identified and made friends with some of the smart hard working students. That was another trick that served me well in school.

[ka9q]

It's not really a dig against mathematicians.  There's a time and a place for abstract quantitative thought, and many of these advanced concepts like eigenvectors eventually bear much practical fruit even if they seem at first like conceptions for their own sake.  But presenting abstract mathematics to people who need a more down-to-earth understanding is like trying to explain woodcarving to someone by extolling the metallurgical virtues of the chisel.  Just hit it with a hammer already.

Funny you should mention eigenvectors, because that's another subject I struggled with. Even today I'm not sure I really understand them.

So I have to tell my favorite mathematician joke. Like all such jokes, it comes in three parts.

An engineer, physicist and mathematican are attending a conference. Unfortunately for them, a pyromaniac is starting fires in their hotel.

In the middle of the night, the engineer  wakes up, smells smoke and sees a fire in his wastepaper basket. He runs to the bathroom, quickly fills the ice bucket with water and rushes it back to put out the fire. Satisfied  that the fire is out, he goes back to bed.

The physicist wakes up, smells the smoke and sees the fire. He pauses for a moment. Running to the bathroom he puts just enough water in the bucket to extinguish the fire given its current size, rate of growth and the oxygen distribution within the room. The fire out, he goes back to bed.

The mathematician wakes up, smells the smoke and sees the fire. He thinks for a few seconds and then goes back to sleep confident that there exists a solution to the problem.

[JayUtah]

Funny you should mention eigenvectors, because that's another subject I struggled with. Even today I'm not sure I really understand them.

I should sneak up behind you sometime and yell "quaternions!"

So I have to tell my favorite mathematician joke.

Yep, that's mathematics.

[Luke Pemberton]

There's a joke based on Zeno's paradox:

A group of boys are lined up on one wall of a dance hall, and an equal number of girls are lined up on the opposite wall. Both groups are then instructed to advance toward each other by one quarter the distance separating them every ten seconds (i.e., if they are distance d apart at time 0, they are d/2 at t=10, d/4 at t=20, d/8 at t=30, and so on.) When do they meet at the center of the dance hall? The mathematician said they would never actually meet because the series is infinite. The physicist said they would meet when time equals infinity. The engineer said that within one minute they would be close enough for all practical purposes.

[ka9q]

Wouldn't that depend on the exact value of 'd'? They'll still be at d/64 at 1 minute.

Oh, it would also depend on just which purposes were considered practical.

[ka9q]

I should sneak up behind you sometime and yell "quaternions!"

Funny you should mention them, I dug into them when I began playing wiith those little accel/gyro/magnetometer sensors for use on a balloon. Everybody says they're more numerically stable.

[Luke Pemberton]

Wouldn't that depend on the exact value of 'd'? They'll still be at d/64 at 1 minute.

Oh, it would also depend on just which purposes were considered practical.

Now you are sounding like a physicist

[ka9q]

Hey, engineers have to be precise too, ya know.

[Luke Pemberton]

Hey, engineers have to be precise too, ya know.

Actually, talking about being precise, another particular bugbear of mine is the title of each of Jarrah's radiation videos, 'radioactive anomaly.'

This irritates me that he invokes expertise but clearly does not understand the difference between the terms 'radiation' and 'radioactive.' His videos should correctly be titled 'radiation anomaly', not 'radioactive anomaly.' This grates me as he is prepared to denigrate the hard working people that made people a reality, but cannot get the basic correct. I don't know, it just seems such a trivial thing, but it annoys me.

[JayUtah]

I don't know, it just seems such a trivial thing, but it annoys me.

It may seem trivial but I think it's the ideological core of the objections to conspiracism.  Whether one's interest in some particular body of knowledge is professional or merely passionate, to see it deliberately and arrogantly misused for some selfish purpose is legitimately offensive.  Science exists as a systematic approach to knowledge in order to make it better serve humankind.  It is ostensibly altruistic.  It should be offensive to see it being perverted for one person's individual benefit.

And science is hard.  It takes a lot of dedication and sacrifice to become proficient at science and its attendant professions.  Roughly 4 out of every 5 people who sit for the professional engineer's license exam do not pass.  And these are people who already have baccalaureate degrees in engineering.  Pretenders to that throne are indeed odious.  Jarrah quite clearly wants to be seen as the "young Australian genius," and wants that perspective to come at the expense of legitimately qualified practitioners.  I see absolutely no moral justification in that.

[nomuse]

Well, in a sort of devil's advocacy here...

I am not so sure conspiracy theorists, free-energy mavericks and other "I have overturned all of physics!" types, and the like think of their process as having shortcut the onerous task of actually learning a subject.

My argument comes in two parts; one, that they fail to grasp the effort it takes to even earn a science or engineering degree, much less become an established professional. Nor do they understand the width, depth, or the specificity and applicability of that knowledge. If they think about it at all, they think of it as memorizing a bunch of essentially meaningless gibberish that can be regurgitated on command to win the sheepskin.

In the second part, they think that the work they have done is hard. Because to them, it was. I run into this all the time; they have managed to notice something in a photograph and they think no-one else has mentioned it because they lack the same keen attention to detail. It never crosses their mind that almost everyone who saw that photograph already saw the same thing -- and has already gone through the next necessary mental steps in attempting to understand what it is they saw.

They think they are the one-eyed man in the land of the blind, and they keep crying "Can't you see that glowing basketball-sized object right over our heads!" Instead they are the one-eyed man in the land of those with normal depth perception, who all reply, "What, the Moon? That's far away, not right overhead!"