How is Physics taught without Calculus?

[hutchphd]

Really, toddlers get that concept on their own just by simple observation.

While this probably true for most folks when pondering velocity, it is a rare toddler indeed who was pondering acceleration. The step to a second derivative is not at all natural and is a fundamental Aristotelian stumbling block that bedeviled pre-Newtonian natural science.

I know many of my students didn't (I asked this question annually), but I doubt whether knowledge of calculus translate into more insight to this situation.

I think there is a Recapitulation of this for everyone learning dynamics and kinematics be they toddlers or pre (perhaps sans)-Newtonian scholars. Higher order rates of change are "unnatural" for each group.

 

[haushofer]

Indeed. Despite knowledge of calculus students (we all, to some degree) rely on our intuition when faced with these kinds of questions. The conceptual abstraction which goes beyond our natural intuition is what makes high school physics hard, not just the mathematical part.

 

[PeroK]

While this probably true for most folks when pondering velocity, it is a rare toddler indeed who was pondering acceleration. The step to a second derivative is not at all natural and is a fundamental Aristotelian stumbling block that bedeviled pre-Newtonian natural science.

Both Newton's laws and the theory of calculus were so well hidden that neither the ancient Greeks nor the Romans made any progress with either. Despite their sophisticated philosophy, engineering and architectural expertise.

Perhaps the great minds of the ancient world could have learned from a few toddlers of the modern era!

 

[haushofer]

I know from my own PhD-experience that mathematical sophistication doesn't automatically mean intuitive understanding. At some point I needed central extensions for my research, and when asked about it I often got complex explanations with cohomologies and such. Untill a collegue pointed me to a simple example: the mass of a non-relatvistic classical point particle. In this simplicity lies true understanding, if you ask me.

The acceleration-example is easily answered by looking at the expression for the gravitational force on the ball. It's nonzero during the whole trajectory. This clashes with the intuition when the ball is at maximum height. That's imho when the learning really starts. Not at solving a differential equation.

 

[apostolosdt]

Both Newton's laws and the theory of calculus were so well hidden that neither the ancient Greeks nor the Romans made any progress with either. Despite their sophisticated philosophy, engineering and architectural expertise. [...]

But they did try, didn't they? [Inscribed and circumscribed polygons in a circle, that also implied the concept of a limit.]

[...] Perhaps the great minds of the ancient world could have learned from a few toddlers of the modern era!

Well, Kuhn has already said something about not comparing apples with oranges. [And he didn't need mine or anybody else's opinion, wouldn't you agree?]

 

[martinbn]

I know from my own PhD-experience that mathematical sophistication doesn't automatically mean intuitive understanding. At some point I needed central extensions for my research, and when asked about it I often got complex explanations with cohomologies and such. Untill a collegue pointed me to a simple example: the mass of a non-relatvistic classical point particle. In this simplicity lies true understanding, if you ask me.

I am curious. Can you say more?

 

[Motore]

If I would've needed to learn and use calculus in highschool physics, I would never go study physics. And most of my non technical driven schoolmates would fail the physics class.
Does this mean I had more problems in college? Perhaps, but if I remember correctly in half a year it was not a problem anymore.

 

[gleem]

When I was a little kid I used to like those connect the dots puzzles, drawing a line from number to number until at some point while you are not yet finished you realize what the picture that is developing is of. So also in understanding, I think we collect information and make observations connecting the "dots" until we realize the meaning. We need facts and relationships and in time develop understanding, connecting the dots. The Eureka moment. Sometimes the understanding develops quickly because of the simplicity of the situation. Other times we may have skipped a number in the puzzle preventing the realization of the picture until as @haushofer pointed out a colleague gives an enlightening example (shows the dot you missed) and all becomes clear. Certainly, calculus is some of those dots but maybe the picture is evident without it.

 

[haushofer]

I am curious. Can you say more?

At a certain moment in my research I needed to understand central extensions (and deformations in general) of Lie-algebras. So when I asked people about it, they often answered with "you need them in string theory due to quantization (the Virasoro-algebra)", or started talking about Chevalley-Eilenberg cohomologies. But at the beginning I was very confused about the concrete physical meaning of such an extension.

It took some time before a collegue of mine (and later collaborator) remarked that the mass of a non-relativistic point particle is given by a central extension. If you calculate the Poisson brackets for the action of a non-relativistic point particle, you find that the bracket between spatial translations and boosts is non-zero and proportional to the mass. Because the commutator between boosts and translations is zero, this allows for a central extension in the algebra (which can easily be checked by the Jacobi-identities). There you go: the simplest example of physical interpretation of a central extension is the good old fashioned mass you learn in high school.

Of course, if you go quantum, you'll also see this central extension popping up when you look at the symmetries of the Schrodinger equation.

Later on, people more than once asked me why I needed central extensions if I didn't considered quantum mechanics. Very few knew that central extensions already play a role in classical mechanics. So what struck me was that those string theory people talking about Virasoro algebras and Chevalley Eilenberg cohomologies apparently weren't aware of this.

(This central extension plays a crucial role in constructing non-relativistic theories of gravity by a gauging procedure)

 

[brotherbobby]

I noticed someone mentioned deriving the (well known) equations of kinematics. They can be done without calculus. In fact, in my country, a student is required to derive them using (1) algebra, (2) calculus and (3) graphs. While it is tedious to use algebra, it is instructive. Calculus, perhaps because of its power, also "trivialises" the solution to a problem. I put quotes because it certainly doesn't intend to do so. Consider how Archimedes found the volume of a sphere ($V = 4/3\; \pi r^3$) without calculus. Or simpler still, his two methods of calculating the area of a circle $\left( A = \pi r^2\right )$. While it's hard, it is instructive and am of the firm opinion that it is these methods that the student should be exposed to before using calculus. At the very least, he'd realise the enormous power of the discipline to make difficult problems be solved easily.
Same can be said of trying to general relativity without differential geometry. I have done it and most physicists get introduced to the subject using the coordinate-dependent tensor notation. Or doing quantum mechanics without Dirac's bra-ket notation, but algebra and differential equations.

 

[brotherbobby]

The question is, whether it "teaches physics". I doubt it!

Teaching and learning occur incrementally. Algebra, and occasionally geometry, teach physics to a point. Calculus and later group theory and linear algebra teach further still. You are forgetting that we are talking here with the student in mind, who is yet to be introduced to the subject. If you tell him that acceleration $a = \dfrac{dv}{dt}$, you may not see him again. If you told him that instead that if acceleration was uniform, $a = \dfrac{\Delta v}{\Delta t}$, he'd do better. You might ask why talk of acceleration as uniform when it can be non-uniform in general? Because, we teach with the beginning student in mind. (That many accelerations we know of are constant to a good approximation is not the point here).

Algebra came before calculus did - so it must have been easier. Same holds for the student, not the physicist.

 

[vanhees71]

As I said, in this case you have to teach the necessary math (here taking derivatives) along with the physics. You can NOT adequately explain what Newtonian mechanics is without derivatives and integrals (also not without vectors BTW).

 

[vanhees71]

Not only I understood the concept of speed before I knew calculus, but I understood the concept of speed before I knew math, before I went to school. Most likely before I knew how to talk.

It is very intuitive to understand that the faster of two objects is either the one covering more distance in a given time or covering the same distance in less time. It is also easy to understand without math (not just calculus) that two objects not covering the same distance can both have the same speed.

Really, toddlers get that concept on their own just by simple observation.You have just lost me. And I know what a derivative is! I have to go from what I already know to understand what you did. To me, all this is, is a boring, abstract, math puzzle. Just the notation wants to make me blow my brains out. All I want to know is how to make my car go faster than the ones of my friends. Where's the car in this? This is not physics.That's it! This is how the derivative and integral concepts were explained to me! I had a math class - pure math - where the teacher began with a graph of distance versus time at a constant speed where I was shown that the speed was represented by the slope. Then a second graph with a speed change with 2 or 3 different slopes, and finally one where the speed is constantly changing where I can easily visualize that the speed at any point corresponds to the tangent on that line at that point.

My question would rather be: How can you teach calculus without physics? After all, this is how calculus was born: A guy was doing physics, and then at one point he discovered calculus.

It is the intuitive way for most human beings to learn calculus.

I also stressed that for the purpose to adequately teach physics you don't need rigorous analysis but intuitive calculus is enough (for the beginning).

I don't understand, why you claim to understand what a derivative is and at the same time say you don't understand the elementary derivation of the rule to take the derivative of $x^n$. What is unclear in the following derivation?
$$f(x)=x^n \; \Rightarrow \; f'(x)=\lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x} = \lim_{\Delta x \rightarrow 0} \frac{n x^{n-1} \Delta x+\mathcal{O}(\Delta x^2)}{\Delta x}=n x^{n-1}.$$
If you find this boring, maybe physics is simply not the right choice of a subject to study for you?

 

[brotherbobby]

As I said, in this case you have to teach the necessary math (here taking derivatives) along with the physics. You can NOT adequately explain what Newtonian mechanics is without derivatives and integrals (also not without vectors BTW).

We agree that calculus offers insights that algebra does not. But those insights can wait till the student has learnt calculus.
However, that doesn't mean his learning of kinematics should also wait. In my country (India), we are taught kinematics in grade 9 ($\text{class}\;\mathrm{IX}$ as we call it) using only the tools of algebra. Sure we don't understand the crux of the subject, but we get somewhere.
If I were to take what you are proposing further, I'd say that one shouldn't learn the concepts of Classical Mechanics without a thorough understanding of differential geometry first! Just because differential geometry does the "ultimate justice" to mechanics, in the words of my teacher, doesn't mean a college student has to wait for his final years in university before he can learn classical mechanics.

 

[jack action]

I don't understand, why you claim to understand what a derivative is and at the same time say you don't understand the elementary derivation of the rule to take the derivative of $x^n$. What is unclear in the following derivation?
$$f(x)=x^n \; \Rightarrow \; f'(x)=\lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x} = \lim_{\Delta x \rightarrow 0} \frac{n x^{n-1} \Delta x+\mathcal{O}(\Delta x^2)}{\Delta x}=n x^{n-1}.$$

You lost me on the physics aspect. This: $\mathcal{O}(\Delta x^2)$ is an insane math notation to present in a physics class. First, if should be $\mathcal{O}(\left(\Delta x\right)^2)$, and second, I have to go through this mathematical notion to understand it and what it represents in this equation. Still, nothing to help me visualize any physics in this.

And then I end up with $n x^{n-1}$; neat but how is this helping me understand physics? What is $f(x)$ and why is it equal to $x^n$? (It is a rhetorical question by the way)

If this is supposed to help anyone understand the relationship between force and acceleration or acceleration and velocity, I don't see it.

What I see is that you are using a physics case to justify the usefulness of derivatives - which by itself is just a math puzzle obeying arbitrary rules.

 

[vanhees71]

As I said, there's no way to describe the physics of "velocity" without a minimum of calculus. You just confirmed this. If you don't like this minimum of calculus, you should try to understand physics to begin with.

 

[Paul Gustafson]

Later on, people more than once asked me why I needed central extensions if I didn't considered quantum mechanics. Very few knew that central extensions already play a role in classical mechanics. So what struck me was that those string theory people talking about Virasoro algebras and Chevalley Eilenberg cohomologies apparently weren't aware of this.

You might be interested to know that central extensions actually show up even earlier -- Arabic numerals. Z_100 is a central extension of Z_10 by Z_10 -- carrying is a 2-cocycle.

For more, see Dan Isaksen, A cohomological viewpoint on elementary school arithmetic, The American Mathematical Monthly, Vol. 109, No. 9. (Nov., 2002), pp. 796-805