How is Physics taught without Calculus?

[haushofer]

I was referring to the "calculus-free ideologists"...

I suppose those are cousins of the "jet-bundle free ideologists" who teach classical field theory without jet bundles.

 

[swampwiz]

So why are so many teachers thinking, they make things easier with hiding from their students this simple hands-on use of calculus. It's of course far from rigorous, but in high school nobody expects rigor at university level but just good propaedeutics!

I am impressed by your vocabulary.

 

[EventHorizon]

The book "5 steps to a 5: AP Physics I" is entirely algebra-based

 

[vanhees71]

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

 

[Vanadium 50]

It teaches how to score well on tests. One test in particular. The first actual physics is around 15% of the way through the book, and the first problem around 20%.

 

[fisicaltibs]

Sounds to me like you just figured out how to teach the necessary calculus well - you didn’t avoid it.

As someone who has taught introductory algebra-based physics (Mechanics, E&M and "Modern" Physics) several times at the university level, I disagree. Admittedly, I had initial doubts whether it could be done properly without calculus. However after doing it, my doubts evaporated and now I have become an apologist for algebra-based physics.

My clientele consisted of undergraduate students at a U.S. university who were pursuing degrees in the health and biosciences: medicine, biology, biochemistry, physical therapy, sports medicine, radiation technology, etc. Their curricula required them to take two semesters of introductory physics taught in a physics department and had no room for calculus. I set my apprehensions aside because It was clear to me from the start that if I did not teach algebra-based physics to these students, I would not be doing my job. Furthermore, if I taught algebra-based physics badly, I would be doing my job badly. Therefore, I had to teach introductory physics without calculus and do it well.

Take any calculus-based introductory physics textbook and carefully examine how much calculus is in it and whether it is really necessary. Yes, the mathematical description is more compact and elegant with calculus. Yes, it is necessary for students to see calculus introduced at the beginning of their study of physics, but only if they are headed towards a career related to physics and/or engineering. Most of the examples and problems in calculus-based introductory textbooks are artificial physical situations in which polynomials with constant coefficients are given as hypothetical models for a dependent variable and one is asked to find related variables using integration or differentiation. There is little physical understanding gained by the calculus formulation in such problems.

<snip>

 

[hutchphd]

Yes, you're right, everything has been said.

Not quite. How many of the non-calculus students will answer the following question correctly: "A ball is thrown upward and reaches the top of its trajectory. What is the acceleration of the ball at this highest point?" More than half will not give the correct answer IMHO. Because, having not been carefully taught, they do not appreciate the subtlety.

 

[jack action]

You cannot express velocity other than by the time derivative of the position vector. Everything else makes this issue unnecessarily more complicated. But we argue in circles. Let's just agree to disagree.

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.

So you ARE differentiating and integrating. I don't get, why you don't make the step from geometric concepts to just name a slope of a function graph derivative and the area under the function graph an integral. You can very easily motivate this. I don't think that you need the university-level $\epsilon$-$\delta$ formalism in physics but plausible arguments. E.g., you define the derivative of a function, $f'(x)$, as the slope of the tangent at the point $(x,f(x))$ as the limit of the slopes of secants, i.e.,
$$f'(x)=\lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}.$$
It's very easy to show the linearity of the derivative, the product rule, and the chain rule from this by plausibility arguments, and with this you can use derivative for all purposes needed at high school. Many derivatives of concrete functions can be easily derived just using the definition, e.g.,
$$f(x)=x^n$$
for $n \in \mathbb{N}$. The students for sure know the binomial formula, from which you get
$$f(x+\Delta x)=x^n + n \Delta x x^{n-1} + \mathcal{O}(\Delta x^2),$$
and plugging this into the definition of the derivative, you immediately get
$$f'(x)=n x^{n-1},$$
etc.

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.

The same with integrals as the area under a function graph.

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.

 

[haushofer]

Not quite. How many of the non-calculus students will answer the following question correctly: "A ball is thrown upward and reaches the top of its trajectory. What is the acceleration of the ball at this highest point?" More than half will not give the correct answer IMHO. Because, having not been carefully taught, they do not appreciate the subtlety.

Well, that would be an interesting experiment: does knowledge of calculus increase the number of students answering this question correctly?

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.

 

[kuruman]

Not quite. How many of the non-calculus students will answer the following question correctly: "A ball is thrown upward and reaches the top of its trajectory. What is the acceleration of the ball at this highest point?" More than half will not give the correct answer IMHO. Because, having not been carefully taught, they do not appreciate the subtlety.

From personal experience, I agree that a lot of students, even more than half will not give the correct answer. Conflating velocity and acceleration is a common occurrence which IMHO is not the result of careless teaching or lack of appreciation of a subtle difference. Students carry to the classroom the Aristotelian preconception that "motion implies force" which persists even after finishing a two-semester introductory physics sequence regardless of whether it was algebra or calculus-based. This was first described in Am. J. Phys. 50(1), Jan. 1982, 66 and conveniently reproduced here.

It is easy to understand the origin of the preconception. I place a small block on the table and I notice that it just sits there at rest. I push it with one finger and I notice that it moves. I remove my finger and I notice that it stops moving. Therefore, as long as I exert a force on the block with my finger, the block will move. Motion implies force. It is a natural conclusion which does not affect is any significant way the everyday life of most people unless they take physics. Someone with this preconception would explain that a block moving straight up in the air has two forces acting on it, one from the hand that pushed it and gravity. The force of the hand is continuously diminishing until the block reaches maximum height at which point gravity takes over and the block returns back down. Thus, maximum height is seen as a point where forces as balanced. If, in addition, someone has heard that force and acceleration are proportional, the confusion becomes worse.

After I became aware of this preconception, I deemed that I had to remove it when teaching my classes before leaving kinematics. I did not want to wait until I got to Newton's laws to clarify why motion does not necessarily imply force but non-zero force necessarily implies change in velocity. In my opinion, the primitive idea of force that everybody has must be sharpened as soon as possible in the physics classroom.

For the benefit of the readers who think that they have sufficiently explained in their classroom why the acceleration is not zero at maximum height, I have a followup survey question based on demo to test the students' understanding.

I place a coin on a book held with its plane horizontal. I move the book up and then down so that the coin is tossed straight up in the air. I then ask "Describe what must be true for the velocity and acceleration of the book so that the two separate in the manner shown."

The two most common wrong answers are "The book must stop moving" and "The book must change its direction of motion". Both can be debunked by demonstrating that the book can be moved according to each with no separation occurring. Eventually, we get to the correct answer.

 

[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. until 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. until 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 Schrödinger 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

 

[PhDeezNutz]

i was brushing up on some thermo the other day because I plan on working for a tutoring company to teach physics for MCAT students and I ran into a problem and wondered how I would teach this without calculus.

We know that Work and Heat for each of the 4 prototypical processes (Isobaric, Isochoric, Isothermal, Adiabatic) can be figured out using the

1) The first law of fhermodynanics
2) Equipartition Theorem
3) Ideal Gas Law
4) Some calculus

One of the first steps is to “differentiate” the Ideal Gas Law.

$PV = NkT$

To

$P \Delta V + V \Delta P = Nk \Delta T$

How do you convince someone of the validity of the second equation without alluding to the “product rule”?


Edit: Another point.

Without using calculus you would get the wrong answer for work done by an stretched ideal spring restoring itself

$W = F \Delta x = (k \Delta x )\Delta x = k (\Delta x)^2$

Instead of the usual $\frac{1}{2} k x^2$

I guess you could say we’re looking for area “under the curve” and “since force is linear we essentially form a triangle and divide by 2”

 

[renormalize]

Without using calculus you would get the wrong answer for work done by an stretched ideal spring restoring itself

$W = F \Delta x = (k \Delta x )\Delta x = k (\Delta x)^2$

Instead of the usual $\frac{1}{2} k x^2$

Where does the the spring force equal zero? A more plausible approximate equation for the work done by the spring restoring itself is: $$W=F\Delta x=(kx)\Delta x\approx k\Delta(x^2/2)$$ which includes the 1/2 factor.

 

[PhDeezNutz]

Where does the the spring force equal zero? A more plausible approximate equation for the work done by the spring restoring itself is: $$W=F\Delta x=(kx)\Delta x\approx k\Delta(x^2/2)$$ which includes the 1/2 factor.

Let me see if I follow, I shouldn’t have used $\Delta x$ twice as you pointed out

$\Delta (x^2) = 2 x \Delta x$

So

$x \Delta x = \Delta (\frac{x^2}{2})$

Isn’t that still using the “product rule” (calculus) in a way?