Excellent Approximations and Lying to Children

In which I talk about the common complaint that we teach students physics that "isn't true," and the limits on that statement.


Frequent commenter Ron sent me an email pointing to this post by David Reed on "What we “know” that t’aint so…. and insist on teaching to kids!":

he science we teach is pretty old. Mostly 19th century ideas about the world around us are taught as “facts” with little but anecdotal data to support it. We teach it via an ontology that replays the history of science, thus the newest and most powerful scientific understandings are viewed as “too advanced”. If it weren’t so widely discredited, we would be teaching K-12 kids about phlogiston first, and general chemical oxidation reactions second, just as one example of that.

We don’t teach kids ancient Greek and French cave drawing first, do we?

I’ve heard lots of reasons for teaching “old” science, and old models and definitions taught as facts, over the years, and the reasonsnever really made a lot of sense. They certainly don’t make sense today. If we rethink the whole enterprise of what we teach as science, we have a real opportunity – perhaps a bigger opportunity than the big postwar drive towards science surrounding the Kennedy space program.

This is a complaint that comes up a lot, particularly in physics, which is more prone to a historical approach than some other sciences. It's got a certain appeal-- the reason I'm a professional physicist is quantum mechanics, not blocks sliding on inclined planes. But in the end, I think it's misguided, because it relies on the common misconception that approximations are wrong.

To see what I mean, let's look at his two examples of failures of the historical approach:

1) Is the Universe’s geometry Euclidean? We teach Euclidean geometry under the unadorned name “Geometry” as if it were foundationally correct. And we teach physics (including elementary astrophysics and cosmology) as if the universe were Euclidean. But by the end of the 20th century it’s become pretty clear that the universe’s geometry is non-Euclidean. That’s what Einstein proposed in the first decade of the 20th century, and more than 100 years later, it’s pretty clear that he was right in rejecting Euclid. Many (but not necessarily all) of the theorems taught in High School geometry also can be derived without using the parallel postulate in their proof – and non-Euclidean geometry need *only* be complicated to people who have learned Euclidean geometry first.. That postulate, however, is almost certainly *scientifically* wrong…. do we teach geometry according to Euclid because it’s always been taught that way? If so, we should put a caveat in the front of the book that we are teaching something that kids should forget about when they learn more about physics? [Euclidean geometry is a consistent formal theory, but teaching it first is profoundly misleading, if our universe is not that way.]

First of all, the initial question is too vague to do much with. Is the geometry of the universe Euclidian? On what scale? On very small scales, geometry will always appear Euclidean-- that's one way of phrasing the principles of general relativity. Any observer, no matter where they are, will see space inn their immediate vicinity as flat (granted, the range that qualifies as "immediate vicinity" can get very small near a black hole...). On the very largest scales, to the best of our ability to measure it, Euclidian geometry also appears to work-- the universe as a whole is extremely close to flat, and there are reasons to believe that there is no overall curvature to space.

So the question only makes sense on a sort of intermediate scale, about the size of planets and stars and things. And there, it's true, space is curved by the presence of mass. We have any number of experimental measurements confirming the predictions of general relativity, to exquisite precision.

In a narrow, technical sense, then, it's true that the geometry of the universe is not Euclidian. But the thing is, Euclidian geometry is an outstanding approximation. You need to go to fairly extreme circumstances to see any appreciable effects of the curvature of spacetime. The effect on the perihelion shift of Mercury is a bit less than 10% of the prediction you would get from classical physics (the gravitational influence of all the other planets), and that's pretty much the biggest effect of spacetime curvature you can see within the solar system. Other sorts of things you can measure are the effect of the Sun's gravity on radio signals from the Cassini probe, where general relativity shifts the frequency of the 8,000,000,000 Hz carrier by about 4 Hz. If you don't want to leave Earth, you'll need an ultra-precise atomic clock to measure the gravitational time dilation, or a tall building and a Mössbauer spectrometer.

So, yes, it's true that Euclidian geometry is only a special case of the mroe general geometry of spacetime. But it's an amazingly good approximation to any situation you will ever encounter. Which is why we teach it to children-- because it's vastly simpler, and the cases where it doesn't work are very far from everyday experience.

Reed's other example isn't much better:

2) Is there nothing? One of the most painful parts of physics teaching these days is the conceit used in its teaching that a true “vacuum” exists or that experiments can be perfectly isolated… Teaching kids the idea that they can create conditions of complete vacuum, with no fields, no matter/energy, … except for, say, a couple of billiard balls representing masses has a real downside. The downside is that we have no evidence that such a vacuum exists or can exist.

Again, strictly speaking, it's true that there's no such thing as perfectly empty space-- quantum electrodynamics tells us that the vacuum necessarily contains a vast number of virtual particles fluctuating their way in and out of existence. But the practical consequences of this are minimal for any situation you're likely to ever encounter. The Casimir effect is real, and the electron g-factor isn't exactly 2, and we can measure these things to extremely good precision. But none of these effects will have any measurable impact on the motion of everyday objects.

There's something closer to a good point here, when Reed notes that "It’s very hard to differentiate "nothing" from "I haven't found any way to measure all aspects of reality yet, but maybe there is something there that we haven’t managed to notice."" But again, this is a narrow technical point, and does not change the fact that treating space with no macroscopic objects in it as empty is an excellent approximation.

And this is the point. We teach kids "obsolete" physics because it works. There are important situations where classical physics breaks down, true, but they're limited in number, and not really important for most of the situations people learning physics will ever encounter. Even people who routinely make use of physical science in their work will almost never need to use non-Euclidian geometry or think about QED.

I've talked before about the concept, borrowed from The Science of Discworld of lies-to-children, namely the simplified versions of how the world works that we teach to kids who aren't quite ready to handle the full thing. These provide the basic understanding that they need right away, and leave the greater complexity for a time when they're ready for it. My usual example of this is the "How a bill becomes a law" civics explanation American schoolchildren get in grade school: both houses of Congress pass a bill, the President signs it, and it becomes a law. That's the basic level of knowledge you need when you're in elementary school, and all the stuff about cloture motions and reconciliation rules and conference committees can come later.

Classical physics is like that, only a billion times more so. The "Schoolhouse Rock" model of the legislative process fails pretty dramatically these days, because you can't really understand the news without knowing something about the filibuster. Classical physics, though, will work for the vast majority of situations any student is likely to encounter (really, the only place it breaks down that's at all significant is when you start talking about the interaction of light and matter). And it's vastly easier to work with.

As I said at the start of this post, there is a certain appeal to the idea that we need to reinvent physics instruction to be more "modern" from the beginning. There's definitely some value in reconfiguring college physics to look less like high school physics with a more casual attendance policy, and there are places where you can put in some hooks to let students know they're not getting the whole story. But the idea that we're doing students a disservice by teaching them classical physics is just nonsense. Classical physics may not be a complete theory of reality, but it's an excellent approximation of reality.

And while I probably bear some responsibility for promoting the general idea that classical physics has been superseded, by writing books emphasizing the weird and counterintuitive stuff in relativity and quantum mechanics, I've come to think that it's a mistake to think of modern topics as the only attractive elements of physics. While quantum mechanics and relativity are some of the weirdest and coolest things around, you can do an awful lot of neat stuff with just classical physics. If you don't believe me, go read Rhett Allain's Dot Physics, which is full of awesome stuff that rarely if ever requires physics beyond basic classical mechanics.

More like this

Isaac Asimov pointed out (I am paraphrasing from memory here) that Ptolemy's model of the solar system was wrong, and Copernicus's model of the solar system was wrong, but if you think Ptolemy and Copernicus were equally wrong, you are more wrong than either Ptolemy or Copernicus.

Like it or not, most students who take the introductory physics sequence are not physicists. Many are engineers, some are majoring in other sciences (chemistry, etc.), some are pre-meds who have to take a physics course, and a few (admittedly not many) are none of the above. Sure, there are places outside physics where QM or, less often, special relativity come into play, but most of those non-physics majors will never actually need a practical knowledge of QM or SR. Newtonian mechanics adequately describes the stuff they deal with.

Reed did miss some low-hanging fruit, however. Economics courses often teach basic models of economics which completely fail to explain everyday experience, e.g., the Efficient Markets Hypothesis. A contributing factor to the financial crisis we are going through is the fact that policy makers continue to apply these not-even-approximate economic models despite the overwhelming evidence of their wrongness (e.g., Wall Street in its current form could not exist if EMH were true, because the profits they are observed to make should not be available according to EMH). Reed makes no mention of economics in his post.

By Eric Lund (not verified) on 31 May 2012 #permalink

I think it is useful to distinguish tools from concepts. While you are right about tools, for many purposes the old tools work just fine, the historical approach to teaching physics is terrible. It first gives the students a powerful set of intuitive (albeit wrong) concepts that are ingrained in the way they already organize the world, then spends the rest of their education trying (unsuccessfuly, for the most part) to tear those concepts apart and teach less intuitive but more correct concepts. Much better, I think, to let them know how the world really works, then take the approximate view.

For example, in the historical approach we teach thermodynamics, with all its ancient and misguided concepts, first. So students learn that heat and entropy are akin to physical fluids flowing around. Even when starting to talk about stat mech, most students quickly get the intuitive picture that heat is some form of energy having to do with average kinetic energy of the constituents -- an old view specific to ideal gases, which is not true for general systems. The net effect of all that is that almost no physics graduate will be able to tell you what heat, one of the central concepts of physics, really is. Which is a shame, because statistical thinking is beautiful and powerful, much more so than the concepts of thermodynamics.

I think the heart of Reed's complaint holds; we teach classical physics as if it were certain, not an approximation. But I also agree with you that the answer isn't to modernize the elementary physics.

How then do we incorporate the idea that what they are learning is a very useful approximation of a better theory, and not really the way the world works? We would do a disservice to kids to hide that bit of information. But we would immensely confuse kids if we flub how we teach them that Newton wasn't right, but I'm teaching it to you anyway.

By Thad Harroun (not verified) on 31 May 2012 #permalink

What I as a mathematician find VERY sad is that almost nothing is taught of the last two centuries. Including the fact that Euclid missed a number of assumptions, and hence the existence of non-Euclidean geometries, with easy examples like why airplanes flying Madrid to New York do not follow a parallel.

As another example, we do not teach even an outline of projective geometry, despite the possibility to connect with, say, art/art history courses.

we teach classical physics as if it were certain, not an approximation
I don't think "it's true unless you're getting into sub-micrometer or planetary dimensions" is much of an approximation, I fail to see how you could construct a practical example of a ball rolling down the incline is not governed by classical physics but quantum effects or require relativistic corrections.

One of the issues is that you're not teaching to a blank slate - you're teaching to people who already have a (likely patchy and incorrect) rudimentary understanding of Newtonian physics. And I'm not talking about what they learned in 6th grade Science class.

Our daily lives are suffused with Newtonian-scale physics, and the conception that people get from day-to-day living as a toddler meshes well with Newtonian physics. Sure, general relativity and quantum mechanics are more correct, but they're way less coherent with everyday experiences. Start by teaching modern physics, and it'll be a long time before you build up to the point where you can explain things like the path of a thrown ball, and you'll lose most students before then.

My guess is that pedagogy recapitulates ontogeny because the path of discovery reflects the general naive to knowledgeable path we all take when learning. We discovered Newtonian physics before quantum mechanics because Newtonian physics is a closer match to the naive understanding of the universe that we start with. (This is why the comparison to phlogiston is inappropriate - it was a dead end theory, not a stage one would work through as one progressed from naive to knowledgeable.)

I agree though, that teaching of Newtonian mechanics should be different now than it would have been pre-1900. There should be a lot more foreshadowing and eyebrow wagging in the directions of the places where things break down -- but I think that already occurs. My physics teachers certainly pointed out where Newtonian physics made assumptions that weren't fully accurate (but we used them anyway, as they simplified the analysis).

The historical approach to teaching modern physics is wrong because it preserves the wrong paradigm. You can see this in the very way you present physics, here for example as if the curvature of space is something that needs extreme circumstances and the many related mistakes you thus commit: It is curvature of space-time (!), it is stronger at earth's surface than at the event horizon of a very large galactic center black hole, and you can see it simply by throwing up a ball so that it goes up, slows down and comes back to your hand in about one second (light goes to the moon in one second, so that arc, namely up-down stretched out to the moon, is basically the curvature right there). The world is Euclidian in small space-time (!) hyper-volumes, and a few seconds is not a long time even for highschool kids.

By Sascha Vongehr (not verified) on 31 May 2012 #permalink

It seems a shame that an engineering student could get a BS without ever being told their are four fundamental forces and that reality is probalistic. In fact it seems wrong that we don't expose most, if not all, high school students to physics, chemistry, and calculus including mention of Einstein, Bohr, Heisenberg, and quarks. Why should high school (historical) physics end with Newton and Galilleo? These other giants are also part of science history. We live in a highly technological world and understand how nearly everything works; yet the populace is superstitious, doubting Darwin, the age of the universe, and completely ignorant of the foundational ideas of modern science. Unless the general public appreciates the depths to which we have plumbed the well of knowledge, they cannot and will not raise the standards of political discourse, but will instead remain bamboozled by those who use the word "elite" as a perjorative. The problem isn't that we teach the approximation, the problem is that we don't go a few steps farther as if it weren't all that important.

Chad Orzel wrote (May 31, 2012):
> Is the geometry of the universe Euclidian? On what scale?

These questions don't belong to the field of physics, but to applied physics (phenomenology, engineering, "stamp collecting").

A related physics question is instead:
Given a (suitable, selected) set of observational data, _how_ is it to be characterized in terms of "geometry" and whether (or to which accuracy) it is "Euclidean" (straight, plane, flat) ?,

such that unambiguous comparisons can be drawn with results obtained from data sets of other trials or selections.

And ...

> Any observer, no matter where they are, will see space [in] their immediate vicinity as flat [...]

... No! -- surely this is not accomplished by any solitary participant "plainly", idiosyncratically, irreproducibly "seeing", but by several participants analyzing their mutual observations by shared, comprehensible methods, reaching results in mutual agreement.

> [...] is curved by the presence of mass. We have any number of experimental measurements confirming the predictions of general relativity

General relativity doesn't involve any predictions on the presence or distribution of mass (energy, stress).
Instead, general relativity involves a methodology for determining presence or distribution of mass (energy, stress) in the first place;
while cosmology, astronomy, geology etc. summarize these finds and model them together with various expectations on subsequent results.

> We teach kids “obsolete” physics because it _works_.

Teaching physics isn't about teaching what works, but about teaching how to determine what works and what doesn't.
Looks like there isn't much physics taught these days, even though several people seem to have this goal in their job description ...

By Frank Wappler (not verified) on 31 May 2012 #permalink

There's a sort of implicit belief in a lot of these comments that we pretend that classical physics is the be-all and end-all of physics in the intro courses. That's not the case at all-- the book we use makes a point of explaining that there are four fundamental forces, and pointing out where the limits of the models used in the book lie. I make a point of mentioning quantum and relativistic corrections whenever appropriate.

What we don't do in the intro classes is try to calculate anything with the more fundamental theories (with the exception of some fairly trivial applications of special relativity). The intro mechanics class is about solving problems for quasi-realistic scenarios of macroscopic objects moving slowly compared to the speed of light, and for those situations classical mechanics works, to more decimal places than anything we might reasonably calculate. The intro E&M course deals with quasi-realistic scenarios involving charged objects and moderate fields, and for those situations Maxwell's Equations work, to more decimal places than anything we might reasonably calculate. There's no need to bring up quantum mechanics or relativity, because even a Ph.D. physicist faced with one of these situations is going to use classical methods to do the calculation.

RM's point, a little while back, is also a good one: students come in with what they think is a decent understanding of Newtonian-type physics. Unfortunately, a lot of what they believe is just wrong, and so a good deal of intro physics teaching is breaking down false intuitions about how the world works. That false intuition won't serve them any better in dealing with quantum mechanics and relativity than it does for classical physics, and the only effective way to break that down is to confront them with the classical situations where their intuition goes astray, and show them that it doesn't work.

Speaking as a total amateur who came to this page via trying to teach my dogs physics (and never taken it in high school or college) - I found the fact that there are still vast frontiers of discovery out there to be thrilling. I am a tutor for kids who are in crisis and my only goal is to keep some love of learning alive in them. There are just so many ways you can parse a preposition or even care about it.

By Dog'sMother (not verified) on 01 Jun 2012 #permalink

Newton's approximations of c=infinity and h=zero are functional, but are woefully incorrect for the good stuff. Good ideas (GR, QFT) need only be testable. They are believable after approximations to the contrary are empirically falsified. Know when to put on long pants.

1.2x10^(-10) m/sec^2 Milgrom acceleration demands dark matter to stabilize spiral galaxies. Noether's theorems plus isotropic vacuum conserve angular momentum. The vacuum, tested with massless boson photons, shows no refraction, dispersion, dichroism, or gyrotropy, arxiv:0912.5057, 0905.1929, 0706.2031, 1106.1068. It must be dark matter, for the vacuum is isotropic to 14 significant figures.

Common matter is massed fermions, leptons and quarks. Bosons and fermions are not interchangeable. Approximating photon vacuum mirror-symmetry to matter demands hierarchies of manually inserted symmetry breakings: 1) String/M-theory and quantum gravitation with no empirical meaning; 2) Empirically sterile SUSY and enough Standard Model parity violations to choke Phys. Rev. D; 3) dark matter that evades all quantitative detections.

End the approximation. Trace chiral vacuum background only active toward matter allows trace violation of angular momentum conservation in kind, sourcing Milgrom acceleration. It biases all massed fermion anomalies plus matter-antimatter abundance. Opposite shoes specifically fit into chiral spacetime with non-identical energies. This fuels Equivalence Principle violation without contradicting Einstein's inertial elevator. Somebody should look.

There is no reason to approximate. Crystallography offers pairs of perfect opposite shoes as enantiomorphic space groups since 1900 AD. Load up existing apparatus and measure. Stop approximating and curve fitting,

Two geometric parity Eötvös experiments. Left-handed versus right-handed single crystals of alpha-quartz; left-handed versus right-handed single crystals of gamma-glycine.

Space may be flat on large scales, but spacetime isn't. That's why gravity is relevant to cosmology.

By Ambitwistor (not verified) on 01 Jun 2012 #permalink

There’s a sort of implicit belief in a lot of these comments that we pretend that classical physics is the be-all and end-all of physics in the intro courses. That’s not the case at all...

I think that's true at the college level, but when I was in middle school I knew that quantum mechanics existed and felt betrayed by my science teacher when he told us that electrons move around the nucleus in classical orbits. There's no reason that kids can't be told that stuff beyond classical physics EXISTS, and maybe a little about the modern picture, when they're in grade school. Again let me emphasize that I genuinely felt lied to. There's nothing more discouraging to a student than feeling like the stuff you learned isn't true, no matter how good an approximation the classical stuff is. (Also, explaining the limits of classical physics is a great way to introduce what a model is.)

By Andy Perrin (not verified) on 01 Jun 2012 #permalink

As a mathematician, I'm particularly baffled by his first example, Euclidean geometry; even more so by what he said after the paragraph you quoted: "It would be quite easy to start with a course in the Geometry that actually matches the Universe as we know it." This course is called "advanced differential geometry", and I'd really like to see him try to teach this in an introductory physics class without any use of Euclidean ideas. (How does one discuss a pseudo-Riemannian metric on a manifold, without discussing the dot product?)

We don't teach kids cave drawing first; nor do we teach them alchemy first. We teach kids things that they can comprehend, that are relevant, and that are very good approximations. Relevance and comprehension are well worth sacrificing the eighth decimal place of accuracy.

That is a pretty strange list. I personally put statistical mechanics at the top of my list. There is no reason to teach the dynamics of thermo as if heat was a fluid substance as real as the phlogiston that produces it, and pretend that there is no basic energy and momentum of particles behind it until you get to a section that is designed to be skipped.

And why pick on approximations that are barely observable when the biggest error we make is assuming our terrestrial laboratories are inertial? We know the earth rotates and this has huge effects from weather systems to howitzers. After all, there are probably as many meteorology majors taking intro physics as there are physics majors.

However, my biggest objection is to the premise that it is easy to teach non-Euclidean geometry. I'd like to see this person teach ANY geometry to a classroom of average high-school students, since this is where geometry is taught in our school system. It would be an eye opener to teach math to students who don't know what a proof is.

As a professor at a "somewhat less selective" college, I can tell you that our state system makes no attempt to teach geometry from a proof-based approach like I experienced, nor does a typical student even know how to calculate the volume of can with a given diameter and height. Either they skip lightly over geometry, to save a few semesters for calculus, or they use it to focus exclusively on a few topics on a state-mandated graduation exam to help the average student who can barely do algebra.

By CCPhysicist (not verified) on 02 Jun 2012 #permalink

You teach them there are FOUR fundamental forces?

Electroweak unification is a half-century old! Sheesh.

By CCPhysicist (not verified) on 02 Jun 2012 #permalink

Do "real" physicists worry more about f=ma or lagrangians? Edwin Taylor's beautifully thought out but (thus far) quixotic attempt to shift the focus of elementary physics from forces to action: http://www.eftaylor.com/leastaction.html

By Patrick M. Dennis (not verified) on 02 Jun 2012 #permalink

We teach science from old to new largely because science first explained the world within a limited range of energy, size, duration and precision. That's the way we experience the world. Newtownian mechanics and Dalton's laws of proportion adn Euclidean geometry are still quite relevant and very useful.

As science's range of energies, sizes, durations precisions widened, there came new, more general, more powerful theories and abstractions. These often involve phenomena we do not experience directly in our day to day lives, and they often require much more complex mathematics to describe. It really doesn't make much sense to stress the relativistic effects of walking across the room or the quantum effects of stacking blocks, especially to someone who has just been exposed to algebra for the first time.

Of course we want to teach approximations. If you take a desire for final knowledge too far, you could argue that we shouldn't be teaching QED or general relativity because we know that these theories are incomplete and in need of unification. Surely, we are just misleading our graduate students when we expose them to QCD or string theory, rather than waiting for the next set, or perhaps the next set of theories after that, are properly elucidated. How will they be able to comprehend unified field theory if their minds have been polluted by the wretched approximations of the day.

The educational improvement I've long thought would help kids (with just a bit of essential math knowledge) better understand QM is a much better use of the visual arts.

Initially I was imagining things like simple sketches of Bloch spheres, e.g. but lately I'm impressed by some more creative efforts like this one: http://arxiv.org/abs/1112.3560

Perhaps it would be useful to introduce the idea of a model before any specific topic is covered. For an engineer, the ability to create a good black box model for a system is at least as important as what the ultra-precise physics would predict for the system.

The complaint about teaching Euclidean Geometry first is weak. Should we not teach addition and subtraction, because infinities exist, and with infinities the rules will change?

Anyone who thinks that Newtonian Physics shouldn't be taught because it's wrong, ought to take a look at the great Richard Feynman re-creating Newton's derivation of Kepler's laws, starting with nothing but 1/R^2 gravity.


I do think that when we teach physics, we should emphasize that some things are simplifications or approximations. I remember being taught that no coefficient of friction could exceed 1. That makes explaining how dragsters accelerate at 50 m/s^2 difficult!

I'm baffled by several of the commenters complaining that the standard approach to teaching physics should be ditched for teaching how the world really works. Especially after being exposed to the historical process of scientific discovery, how can anyone be so arrogant as to think that the currently most sophisticated model is the "truth" or "reality"?

I have to wonder if it really matters whether we teach physics that is correct and proveable or not to such precision, given the fact that literacy in general is taking such a downward slide in today's world. Since I find it disappointing that apparently even someone with a high degree of education can't bother to accurately proofread and spell check an article before publishing it online, I would find the topic of whether to bother changing longstanding educational formats worthy of debate, because it appears that once a graduate emerges into the world, he cares little about accuracy when he communicates anyway. So does it really matter if what he is communicating is accurate in itself? Accuracy appears to be an unforgiving condition. There is no '' partially accurate'' condition in science. I would hope that communicating a scientific topic would at least try to share a similar effort to be accurate. But it seems these days everyone is so quick to get their words out into the world that they don't bother to carefully select the words that really convey what they are trying to say. This condition is sadly growing more and more prevalent, appearing even in online textbooks, which could easily be proofread and corrected if the author cared to put forth the effort. So the question becomes, is it even worth bothering to be accurate anymore?

Right, because typos never made it to print before the 90's...