"Spherical Cow"

Andy Resnick

The title will make sense shortly...

I gave 'reading quizzes' (algebra based Physics I and II) that are due before each lecture- these are 3 short-answer questions designed to identify student misconceptions prior to the lecture so that I can directly address any conceptual difficulties as they arise. Recently, when asked "why does a wire get thinner as it is stretched?", nearly every student gave a highly complicated response in terms of forces and stresses (when all I was looking for was conservation of mass/volume). I was genuinely perplexed, and when I asked the class why they gave such a complicated response, one student replied "well, I didn't think that the simple answer could possibly be correct, since nothing in this class is simple", which got a lot of knowing giggles from the other students.

So I trudged into Jearl's office and we started talking about the 'intellectual baggage' students bring into class, and he mentioned that many students see the teacher as a magician who regularly turns one object into another- for example, turning a cow into a sphere. When he said that, I realized he was absolutely right- the next day I started the class by asking if I acted like a magician and got a lot of nods. I spent a recitation period trying to explain why I do that (abstraction of essential concepts, etc), but didn't get much of a response. The students have a lot of trouble relating real things to the idealized/abstracted models I create. Even when I have them step through the process: for example relating the free-body diagram of a car going around a turn to what happens when they are driving around a turn and hit a patch of ice, they can't make that leap. There's a persistent disconnect between what they know happens from experience and what they think happens "in physics world".

I was just wondering if any of you all are aware of this (teacher as magician), if you have consciously tried to address this, or any other relevant ideas.

FlexGunship

In college, I was the only tutor for Kinematics and Electricity & Magnetism (the two freshman physics courses) on campus, and the best thing I learned from the experience is that people internalize concepts in different ways. And any other attempted method of internalization seems like magic or wizardry.

I turned to a professor that everyone seemed to understand. He was a funny soft spoken guy with a very dry sense of humor. I noticed a pattern with him, he would always dispense the same information in four different ways!

The first was always an analogy: "a transistor is like the floodgate in a dam." This got the concept of what you were talking about into your head. Forget the physics behind it, ignore the symbol for it, and don't even worry about the mathematics used to describe it.

The second way was very expressive drawings and sometimes pantomimes. He would draw a river with a damn and show us the control valve. He would ask the class to explore the analogy he had set up before. "What if the floodgate was controlled with another floodgate?" "What if the valve took no effort to turn?"

The third way was when he dropped the shroud of the analogy and gave us the mathematics. He would ask us to mentally discover the relationship that the math shows us. "As V_gs rises, so does the current through... now imagine that we put infinity volts there, would we get an infinite amount of current? What about zero volts? What about a negative voltage?" And he'd ask us what practical limits we might run in to. He'd ask us to call into mind other things we've learned about that might have some bearing. "Could this be used like a diode? What about a resistor?" And he'd ask us to think about absurdities. "What if I connected all of these together? Any ideas?"

And finally, the fourth way would be an example. Done from start to finish with references to the original analogy. Usually accompanied by the phrase: "Okay, so it's not exactly like a floodgate, but you get the idea."

And we all would!

In my role as TA and a tutor, I've always used this. Now as an engineer I use analogies routinely to help explain complex concepts. But you have to avoid two pitfalls: (1) don't carry the analogy too far, and (2) avoid bad analogies at all costs!

EDIT: When asking the wire question. A follow up question could have been: "What is the fundamental problem if the wire didn't get thinner as it stretched?" The answer should've been: "Well, then you have an infinite amount of wire and that's impossible. You can't create mass out of nothing." Always explore absurdities, especially in the face of an overly complicated answer.

Sankaku

What a great description of your observations.

In my limited experience, the problems grow with class size. The spread of ability levels and learning styles becomes too much to compensate for past about 25 students. Speed is an issue as well; what is perfect for the prepared students will be flatout breakneck impossible for 50% of the class. What is prefect for the slower (but still serious) students will bore the hell out of many of the quicker ones. Finding the best compromise can be a trick.

I like the description of teaching 4 different ways. Also, sometimes seeing the same material again in a different light later in the course can help. Spiralling back to basics (that people should know) using the newer material can help clear up misconceptions.

At the end of the day, learning takes work and many people (especially in the age of instant info) don't want to put much effort into it. Unfortunately, inspiring people to work hard is more difficult than just teaching well.

Andy Resnick

That's an excellent suggestion!

That's somewhat true, but it's also something out of my control.

Maybe I wasn't really being clear originally- here's an analogy.

I often skip mathematical steps when working through a problem on the board- out of necessity. I just don't have enough time in class to grind through "ok, now we subtract 'g*t' from both sides, now we divide both sides by 3..." when solving an equation. Most of the students don't need that level of detail, anyway.

The magical 'spherical cow' business seems to be a skipping of *conceptual* steps. That is, I subconsciously step through a series of concepts and abstractions without explicitly presenting that to the class. And it seems that many more students need to see those conceptual steps than need to see the explicit math steps.

FlexGunship

As a side note, my favorite meal is grilled filet of spherical cow of uniform density.

EDIT: ... with a side of mashed potatoes of indeterminate volume, peas arranged in an optimal packing configuration, and a glass of ideal fluid.

Dickfore

But, although mass is conserved, volume need not be, since the density of a stressed solid is not the same as the density of a solid in a state out of stress, since the inter-atomic distance changes in one direction.

So, they gave a complicated answer because you asked a very complicated question!

EDIT:

You actually asked them about the following Poisson's ratio.

The sentences of relevance are:
"The Poisson's ratio of a stable, isotropic, linear elastic material cannot be less than −1.0 nor greater than 0.5 due to the requirement that Young's modulus, the shear modulus and bulk modulus have positive values. Most materials have Poisson's ratio values ranging between 0.0 and 0.5. A perfectly incompressible material deformed elastically at small strains would have a Poisson's ratio of exactly 0.5."

Andy Resnick

I agree, the question is more complicated than it may appear- but nobody went there. If they did, I would have certainly brought up Poisson's ratio, and how there are engineered materials that get thicker when stretched. I don't want to repeat actual answers here, but they were *way* off target- unusually off target.

Again, this is algebra-based physics I for science majors. The reading quizzes are a simple probe of concepts designed to spark thought and discussion- for example, "why does holding a heavy weight make you tired, even though there is no work performed?" Most students will agree that W = 0, but are then forced into an apparent contradiction. Most students will simply write "work is not exertion", which is fine. That said, I wonder if they are simply separating "physics world" from "real life".

atyy

I think the helpful analogies depend on each person. I personally find electricity more intuitive than mechanics, so when someone tries to explain to me how an electrical model of a neuron is like water, pressure etc - I always have to think "how is water like electricity?" to understand what they are saying.

In the case of why W=0, I don't find "work is not exertion" an explanation, since it only says what exertion isn't, not what exertion is in "physics world".

Andy Resnick

That's sort of what I meant by 'intellectual baggage'. Nearly all of my students have heard of 'quantum mechanics' and 'relativity'. Most students have been fed a steady diet of "Physics tells us that the world is not what it appears" or its variant of "Newtonian mechanics is not valid" from grade school onwards, that "Physics is a lot of really hard math", and that "Physicists are really smart". None of these encourages a biology student to enroll in Physics class.

As a consequence, many of my students come to class fearful that all they are going to do is a lot of hard math, and most don't understand (or even resent) why they have to take Physics I and II.

Now they have a teacher who is making an effort to relate the material to them, on their terms. This causes initial confusion- where is all the math? what happened to the plug-n-chug? Why is he making me think? Most students come through that period intact, and start to appreciate my approach. For my part, I've had to re-learn a lot of material to carefully examine the choices that are made when solving a problem, and I've come to understand that the initial process of abstraction- representing real objects by dots, squares, wiggly lines, etc., representing mathematical constructs (forces, momentum, etc) as real things (arrows, for example) with measurable effects, making choices about what to ignore and what is important- all the things I have been trained to do for years until I don't even think about it- is very confusing for students.

To be fair, most physicists have no idea what 'exertion' is, either.

atyy

Seems to be a biochemistry question? Placing a weight on a table prevents it from falling to the ground without exertion, yet keeping a muscle contracted to hold up a weight requires exertion.

DaveC426913

Yep. A good teacher knows that students all learn in different ways. Providing a variety of modalities ensures the widest array of learning styles is addressed.

I learn visually. Try to describe something to me, like reading a story, and you've lost my attention before you can...Hey a red ball!

Andy Resnick

Not at all- it's thermodynamics. How does a muscle perform work? A muscle, at a molecular level, converts chemical energy (from the nonequilibrium concentration of ATP relative to ADP) into mechanical energy- a myosin molecule physically moves a step. When you hold up a weight, your muscles are dissipating energy- ATP is hydrolyzed, producing both waste products and driving yourself toward equilibrium. Conservation of energy still applies.

The point is, my students have a very good understanding of the process already. What I offer them is a rational basis for understanding the phenomenon.

atyy

Ah, no wonder it's such a mystery. Thermodynamics requires postulates in addition to those of Newtonian mechanics, so "exertion" truly doesn't exist in that particular "physics world". (I'm assuming that microscopic initial conditions cannot be freely specified in thermodynamics, whereas they can be in Newtonian mechanics.) So if questions continue along the lines of why is the table is at equilibrium, but not contracted muscles, etc, etc, eventually the Nernst potential or Gibbs free energy would be called upon, both of which are inherently thermodynamical.

Terminology question: in what sense is a muscle doing "work" when no "work" is done? I presume the former is microscopic work, while the latter is macroscopic, thermodynamic work?

Andy Resnick

That's a good question- I don't understand what happens when a muscle is 'working' (generating a contractile force) but the muscle itself is not contracting: I think that is called 'isometric contraction'. Maybe it's as simple as the motors stalling, but I don't really know. Muscles also work in (opposing) pairs- something that is invariably left out of physics problems. Also note that smooth muscle is very different from skeletal muscle- smooth muscle can remain in a contracted state without expending energy.

Sankaku

I am rusty on my muscular physiology but (crudely) I think there is work being done at a microscopic scale, just no net work at a macroscopic scale.

LaurieAG

I can relate to that Andy.

In high school I was sick for 2 weeks and missed out on the entire introduction of Calculus I. I was totally bewildered and I felt the pressure really build up as a result. On the second day back it just clicked and I have never had a problem since, with pure calculus anyway, maths of finance is an entirely different story though.

Back in high school I would sit next to my best friend in maths and he (he passed his 2 year specialist radiologist course in the top 10% after he became a doctor) would sit next to me and say 'what did you do!!' whenever we were doing differentiation or integration because I would only write down every second step. I had already worked out that I had to do the last 3 steps individually or I could get wrong answers, no marks for wrong answers and half steps. My friend was slow and steady while I never stayed in a calculus exam for much more than half the time he took but we both got the same results.

Its important to try to cover both evenly or you will just turn off the talented ones.

In 1978 I walked out of my last calculus tutorial (on calculus of imaginary units) before I terminated my enrolment (for 12 years I worked in a Geotechnical and Materials testing laboratory (we developed the Australian Standard for Tri-axial core testing) and then worked in communications (Engineering/Design Office, we did Australia's first cable TV network), then went back to uni). I did each of 3 questions in 20 seconds and spent about 40 seconds checking each one. I went up to the tutor and showed him my answers. I asked if there were any more and he said no so I showed my friends how to do the problems and walked out in 7 mins.

That tutorial was a major contributor to my leaving uni. I use my head as a bit of a barometer as a consequence of my experiences. I suppose I matured a bit before I went back and completed my degree. If I'm studying maths now and it gives me a headache just like maths of finance then it is most likely not pure maths that I am analysing.

Now back to this 'Spherical Cow'.

Andy, you are aware of my 'Feedback loop photos' thread. The last Image I posted has the question 'which part of this sphere is imaginary?'. Considering that the setup is in the first post of the thread and we can conceive what we actually see on the screen, what sphere do you think the sphere referred to is if you apply a circular mask and discard the outside?

Dave might be able to help too, considering that both Stephen Hawking and Roger Penrose discuss turning Lorentzian metrics into Euclidian metrics via a Wick rotation (by multiplying t by the imaginary unit i), surely the mental concept of the Lorentzian metric could be considered within 90 degrees of reality?

I have been trying to build a pure bridge across a gaping chasm only to get to the other side and find people who say the bridge is imaginary.

I hope this helps Andy.