12/22/13 11:14:32 PM
It invites confusion.
Newton's laws of motion are not intuitive. Limits and rates of change are not intuitive. Infinitesimals and differentials are not intuitive.
Imagine you're in high school and are just starting to learn about physics.
"s = vt"
If v = 100 cm/s and it's 3:00 in the afternoon, how far have you traveled? Impossible to say.
If v = 100 cm/s and t = 20 s, how far have you traveled? Impossible to say.
If a = 0 cm/s^2, and v = 100 cm/s, how far will you travel in 20 seconds? You'll travel 20 m.
Sometimes "t" means "duration" and sometimes "t" means "instant in time (a point on a graph)". Without careful teaching, it all gets mixed together. And then when it comes time to discussing differentials and limits, it can all just become a mishmash of mathematics.
I've mentioned an excellent book on physics teaching here before - but I can't find the name of it at the moment. But an on-line copy of an essay that makes similar points is here - http://web.physics.u...oy/challenge.html
A similar confusion arises concerning the work-energy theorem, Eq. (2). When students are asked to explain what kinetic energy means, the most common response is that it is "one-half the mass times the speed squared." By fixating on the mathematical definition, they fail to grasp the essence of the work-energy theorem: that the kinetic energy of a particle is equal to the total work that was done to accelerate it from rest to its present speed, and equal to the total work that the particle can do in the process of being brought to rest.
This tendency to focus on a mathematical definition rather than physical meaning was shown convincingly by Lawson and McDermott.  They presented students with a simple question concerning the work-energy theorem. As depicted in Fig. 2, an object of mass m and another object of mass 2m are initially at rest on a frictionless horizontal surface. The same constant force of magnitude F is then applied to each object. The question to be answered is "Which object crosses the finish line with greater kinetic energy?"
Using the work-energy theorem, and keeping in mind the physical meaning of kinetic energy, it can easily be seen that each object has the same kinetic energy upon reaching the finish line. Yet in interviews with 28 students taken from two classes at the University of Washington, an honors section of calculus-based physics and a regular section of algebra-based physics, Lawson and McDermott found that only a few honors students were able to supply the correct answer and the correct reasoning without coaching. While most of the remaining honors students were able to eventually achieve success with guidance from the interviewer, almost none of the students from the algebra-based course were able to do so. No less disappointing results were obtained with a written version of the question presented to a regular section of calculus-based physics. I have had similar experiences with my own students: Their performance on conventional homework-type problems shows that they can compute quantities such as work and kinetic energy, but their performance on conceptual questions shows that they have much more difficulty explaining or interpreting their results.
This example shows again that emphasis on numerical problem-solving can obscure major conceptual deficiencies in students. It underscores the importance of requiring students to apply the fundamental concepts of physics in a variety of different situations, as well as requiring them to explain the logic that they use in solving physics problems of all kinds.
This confusion and lack of understanding can have real consequences. People become great at memorizing equations and doing "plug and chug" but they have no understanding. For example, I heard a story about someone at their physics PhD defense being asked whether "the wavelength of light was bigger than a breadbox." They couldn't answer correctly.
Teaching the basics carefully and clearly is really important if you want to impart understanding. If someone makes things "easy" in the wrong way, it is damaging.
12/23/13 7:27:00 AM
12/23/13 10:50:12 AM
Yes, and no, and yes again
The difference between time and duration ... yeah, I can see people getting thrown by that. I get it intuitively, so it never occurs to me that someone else wouldn't. It's why I shouldn't be a teacher.
Speed vs.acceleration, though ... that IMO isn't simplifying the concept, it's the problem being asked. In explaining the "confusion" you introduced a new variable which "must be zero". You could just as easily introduce direction and say that *it* must be constant. Also true, but defining all the variables that *could* affect the outcome if they *were* included seems like s different sort of problem.
The "force" example ... yeah, that requires understanding what force *means* as more than just a variable in an equation.
I figured out how to explain my thinking better.
You have to understand the units and what they represent for a calculation to be meaningful. I "get" that time means elapsed time; the idea that "3:00 p.m." could show up in that equation just doesn't occur to me.
But in the "force" example, "force" is actually a tricky concept. Tell a student that you apply "the same force" and you've already lost them. The intuitive concept is that "I pushed just as hard", so you would think it takes longer to get the heavier ball up to speed, and therefore you've put more "work" into it, so the kinetic energy would be higher. But the formal definition of "force" doesn't work that way. You've already abstracted away questions of how that force could be efficiently and completely transferred to two different masses. You've also obscured the fact that you wouldn't necessarily be pushing all the way from start to finish.
The correct form of that example if you wanted to teach the concept of force is:
You have two wagons on a frictionless surface with a different amount of weight in them. Each is pulled by a rope that goes around a pulley, up over a second pulley, and down to a hanging weight. (ie: The falling weight pulls the wagon.) The falling weight provides the force. The two wagons will cross the finish line at different times, at different speeds, but with the same kinetic energy because both had the same force applied.
Edited by drook
Dec. 23, 2013, 10:50:12 AM EST
12/23/13 11:43:32 AM
You've brought up another important issue
Writing good word problems is very hard.
One that drove me nuts, and still sticks with me is:
"Consider two identical stars revolving around each other..."
He meant the orbits have the same center, but if you don't realize that instantly, you'll be going off down a rabbit hole - at least I was. :-/
The work, force, and kinetic energy problem is just one example of how things can be obscured by the way physics is taught (at least in the US). Too much of it is all about finding the "tricks" and knowing which equation to apply and how to simplify it. Too little is about clarity in definition of the terms and getting students to the "ah ha!" moment when it finally clicks.
In '83 or so when I was finishing up my college classes and thinking about what to study in grad school, I had some sessions with a TA who was studying for his physics qualifier exam (one of the hurdles to jump before being allowed to write a dissertation). He was going through one of the standard problems that everyone had to be able to do there: Write the equations of motion for a top spinning on a spinning globe and determine various things given their sizes and the friction between them. It turned me off immediately.
Yeah, being able to do advanced mechanics problems like that was something I'd likely never have to do again (though there are exceptions - someone had to figure out the equations of motion of the planets and the rockets that went to the moon and related problems - http://link.springer...0.1007/BF02715967
), and sometimes things like that are designed to be weeding-out problems rather than something that one really should "know". And someone being able to know enough about it to teach it to the next generation is important if you're interested in teaching. But it didn't appeal to me. (I knew I wanted to do research - not teach.)
I have no idea how things are in graduate schools now. Some areas of physics have exploded (solid-state, biophysics, various cosmological theories) while some have changed very little (mechanics). There's more stuff to know that's being crammed into the same standard time frame. I'd hate to think that the same "weeding-out" problems are taught just for historical reasons.
Anyway, I'm rambling.
Before I close, here's the book I was trying to remember. It really should be required reading for those interested in teaching physics - and for those who are seriously interested in trying to simplify physical topics for a lay audience in the hope that they'll want to learn more later.
Arnold B. Arons - "Teaching Introductory Physics" - http://www.amazon.co...ns/dp/0471137073/
(It's spendy, but can sometimes be found for far less used.)
12/23/13 1:40:30 PM
Re: weeding out
When a department intentionally weeds out students in the intro courses, do they tell themselves they're selecting for interest, or for aptitude? Honest question, I wonder if they even consider the difference when doing it.
Here's the weeding out question from my intro physics final. We had been studying pendulums (pendula?), levers and rotation. Then on the final:
Consider a hockey stick laying on the ice. (To simplify, one-dimensional rod on a frictionless surface.) It is struck by a hockey puck, traveling perpendicular to the stick, transferring 100% of its energy. In terms of length of the stick l, distance from the end of the stick of the impact d, mass of the stick m, mass of the puck m', and velocity of the puck v, describe the resulting motion of the stick as it travels and/or spins on the ice.
Sure, I'd like to have spent some time working with the formulas for two- and three-dimensional problems like that. But this was asking us to derive
those formulas on-the-fly during a timed exam.
12/23/13 8:16:38 PM
I don't know if a lot of thought goes into philosophy of the weeding-out questions. The profs know that they have too many people in the program, so they want to cut the numbers down to those able, by hook or by crook, to do the problems. Whether they would actually make good physicists is a question to be answered later (when they need to choose advisers and be graded on their qualifier and dissertation). Engineering schools seem to go through the same process.
On the hockey stick problem - that's a pretty good one. But not one I'd want to see on an exam! I guess I'd attack it by starting thinking about what happens when the puck hits the very end of the stick (pure rotation about its center of mass), and what happens when it hits the center (pure translation of the center of mass). But if the puck transfers all of its energy in the first case, then it will stop while the stick rotates, so presumably the puck will be hit when the stick rotates around. At that point, will all of the stick's energy transfer to the puck? I'd think not in practice, but on a frictionless surface, maybe so. That case is probably a variation of the balls-hanging-from-threads see-saw "Newton's cradle" toy.
It's more complicated if the puck hits between those two extremes - being a combination of translation and rotation.
It's the type of problem one could get lost in on an exam... :-(
I'm glad I don't have to think about those things any more!! :-)
12/23/13 9:23:37 PM
Why would you expect pure rotation with an edge hit?
I knew the center hit would be pure translation, so I started solving for the end strike. It's definitely not pure rotation.
I think I just figured out the trick! Treat the rod as two point masses at the ends of a massless rod. The mass that is struck is at that instant taking the full force. Solve for that motion, and the other end of the rod is motionless.
The general case formula is still a nightmare, but I can solve for the two extreme cases.
12/23/13 10:20:53 PM
If the rod were pinned at the center, it would rotate if hit at the end. If it were not pinned at the center, but were on a frictionless surface, would it just rotate? Hmm... No, I guess it wouldn't. The end point would want to move in a straight line along the direction of the initial impact. It would get more complicated along the length of the rod, I think. It's been too long for me to be confident of constructing a solution...
I'm thinking back on the video out there of the behavior of a slinky that's hanging vertically and then dropped. What happens to the end closest to the ground when the top is released? One can be mislead if one jumps to conclusions based on intuition...
1/30/14 12:19:30 AM
Belated comment to (this, buried-in 'Hardware'.)
(Had meant to return to this topic earlier but TMI elsewhere.. dilutes everything.)
The Arons book (scanned index and opening material) limns Precisely that which was flat-Missing! at the '50s Institute: the actual Concepts!!
And yes.. everyone Was gaming-the-(Learning!)-System via memorizing Identities, algebraic short-cuts ie. "getting numeric answers."
sans any but ... metaphorical? [substitutes for ~Understanding.]
(I might have taken a different course of action, had I seen a copy of this book, then.) Though doubt that any 'debate'
from my nascent perspective, would have been to the slightest effect. Nor had I the foggiest notions -let alone actual info- about 'pedagogy' elsewhere (well beyond my 'pay-in' grade.)
I add that: indeed "accuracy of numeric answers was of quite less import than proper method"; all fine and good, of course--but without the proper emphasis on Concepts..
The gamers win by default. Oh and; the assumption that Grad-students can, innately, Teach--is so false it's not even wrong [some few: Can.]
Can only shudder at the kinds of cramming facing a student today--given the sheer volume and acceleration-rate of incoming factoids.
(Even Max Jammer's opus--though largely of technical formulation methodology--didn't happen until 1957, but he, at least did address Concepts throughout--it's in the title too!)
Just now it could not be more Unclear (to me) whether or not such as Arons, John de Bono (and several similar addressed in these pages way-back):
Have had/are having any significant effect upon the Teachers-of-Teachers du jour. 'Hope' isn't enough.
And if.. you don't Get That right (inculcating effectively) ... then just at the time when the Planet needs as much disciplined Thinking-power as can be found?
Well, you know.. ...
Other link would be most interesting--except, in keeping with Vulture-copyrigt/everything/everywhere ... TWO bloody pages is cutting it a bit Short.
(There MUST be some Int'l Legislation soon which permits individuals to see vastly more of these ideas-locked-up in wads of greenbacks.)
Feeding-dumbth by-design! is about the worst trend I can imagine, given the entire zeitgeist.
Thanks for troubling to give great examples of your (my!) pet peeves about ... retarded pedagogy (?)
1/30/14 12:08:45 PM
Thanks. I remembered the wrong book.
The actual book of his that I have is "A Guide to Introductory Physics Teaching" - the book above is a later, expanded, version.
has it listed for as little as < $25 used. (No preview is available.)
Yes, imagine how much richer physics would be if more people hadn't been turned off in the earliest classes, and if more teachers had taught things clearly. Of course, one could say the same thing about Economics, also too. :-D
12/23/13 6:04:36 PM
12/23/13 6:08:38 PM
Bingo! Max Jammer's 'Concepts of Force' gives you a qed
--and Max takes a bow; Here are the Intro first few sentences: [Pub 1957, Harvard U. Press] 2nd Ed. (alas)
In our present age of technological progress the frightening discrepancy between our technical "know-how" and our philosophical incomprehension, in general, of basic scientific conceptions seriously endangers the integrity of our intellectual outlook. The cogitative activity of the modern scientist, who is more a technician than a philosopher, is strained to its utmost limits by the necessity for digesting the swiftly accumulating information in his specific field of research. He has little opportunity to indulge in the fundamental problems relating to the very concepts which he applies. ... ...
[. . .]
And.. here we Are--57 years later: only, Piled Higher and deeper (PHd)
Think I came across this gem at Stacy's [Sci bookstore in SF) and it seemed a possible illumination of that mysterious F so readily equated-to m x a.
Did skim same subsequently; realized that--amidst being snowed by accelerated sci-type 'teachings' / tantamount to feeling like fois gras.. I (too) had no time to digest [this treatise On 'digesting'!] the actuality behind the formulae--was Max a First then, also to incorporate ~snide recursion?
Now perhaps I Can take-the-time with Max to clarify that remaining.. which was mainly memorized technical Boolean-musical-chairs .. in their inculcation.
No. *Wonder. that so few attend academe in order to be exposed to a liberal education, as prep for the coming vicissitudes: Most-all need to cram stuff in cranoium at light-speed,
get Out and become immersed in the bizness, Big-sci or related $-games of our impoverished 21st Century cultures. Now there's doom with a capital-D.
* Wonder.. ... ... reserve No Time ever to experience That, in any lesson plan: and you Are in the 21st Century (and apparently--most of the 20th.) :-/
57 Years later .. le plus sâ change.
Nice examples, but Max suggests that we spend even more effort in civilizing All those arcane scribblings we osmosed. (Time stays. We go.)
Which always brings us back to Wordsworth and.. The world is too much with us. Getting and spending we lay waste our .. ...
OOps gotta answer an e-mail--or is this one a Summons? ... from the Ministry of Truth??
Ed: PS--seems there's a .pdf of this book, I just noticed; just G. full title 'reviews' etc.
Edited by Ashton
Dec. 23, 2013, 06:08:38 PM EST
12/23/13 10:25:31 PM
Gotta move it up my reading list! Thanks for the reminder.