How is Physics taught without Calculus?

[swampwiz]

I remember taking Physics in high school, so I guess it is possible, but it's been so long ago, I can't remember. It just seems that Calculus is indispensable when teaching Physics topics, except for a few like heat expansion or geometric optics. I would imagine that there is a lot of Δ this & that instead of differentials, which is almost like using Calculus.

 

[jedishrfu]

Recall they had different formulas for different conditions and students would lament that they had too many to remember. Later when Calculus came into a more advanced physics course then you got to see that these formulas were from special cases and were easily derivable meaning there's little need to memorize them.

 

[Vanadium 50]

How is physics taught without calculus?

Badly.

 

[BillTre]

They had lots of formulas.
I took calculus in High School and then took physics w/o calculus (don't ask why).
I just just figured out a lot of the equations on the fly in tests instead of remembering them.
It was kind of fun.

 

[malawi_glenn]

You don't need calculus for motion with constant acceleration.

I teach high school physics in sweden. Students learn how to differentiate and integrate in math class in the first half of the second year. The physics class starts in the second half in the first year. So the physics needs to be calculus free for a while (do not blaim me - this is enforced by the swedish school agency).

However, I actually introduce the concept of slope of tangent and area under a curve, in the case of motion with constant acceleration and work done by a constant force. I also show them how to do this algebraically for these cases. In this way, they have something they can relate to once we get to differentiation and integration later.

Students don't need to remember stuff here, we have a formula book (which are allowed at swedish universities too).

 

[Vanadium 50]

They had lots of formulas.

The problem is that these formulas are seeminly un-related and the student never sees the interconnection or the general principles. Especially in E&M.

The other problem is that this develops a problem-solving methodology of "just pick the right formula". This works...until it doesn't.

 

[mpresic3]

I took physics in my high school, and although it had a calculus class, physics was taught without calculus. The textbook was mostly of the variety of presenting one equation after another and textbook problems were plug-and-chug. No attempt was made to unify concepts.

I knew the study of undergraduate and later physics was going to be very different from this. Otherwise, I would never have selected this as my major. I knew I was going to major in physics in my last year of high school, although looking back, I might have given electrical or aerospace engineering more of a chance. I did think quantum mechanics and relativity (what I could learn about them at this level) was (in the language of this earlier day) cool.

 

[vanhees71]

It is simply impossible to teach physics without calculus. It's even impossible to talk about physics adequately without calculus. It's thus simply a flawed concept to think that one can teach physics in a "calculus free" way.

 

[Andy Resnick]

I remember taking Physics in high school, so I guess it is possible, but it's been so long ago, I can't remember. It just seems that Calculus is indispensable when teaching Physics topics, except for a few like heat expansion or geometric optics. I would imagine that there is a lot of Δ this & that instead of differentials, which is almost like using Calculus.

See for yourself:

https://openstax.org/details/books/college-physics

 

[kuruman]

It is simply impossible to teach physics without calculus. It's even impossible to talk about physics adequately without calculus. It's thus simply a flawed concept to think that one can teach physics in a "calculus free" way.

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.

After the second semester, my students were not going to see any more formal physics instruction. However, they were going to see the application of physical models in their professional courses. The primary goal I defined for them was understanding the physical world in terms of predictive mathematical models that they could manipulate within their mathematical abilities. So I worked around the calculus barrier. Necessity is the mother of invention and here are some examples of what I did.

Considering kinematics, my students had to understand motion and acceleration, so that they could understand $F_{net}=ma$, so that they could understand rotational dynamics and equilibrium, so that they could understand the parameters that come into play when they put patients in traction. No calculus needed for that.

I defined acceleration as "the speed of the speed", that is acceleration is a measure of how fast the speed is changing much like speed is a measure of how fast position is changing. This gave students an intuitive understanding of acceleration as a rate of change by relying on their existing perception of speed as a rate of change. I did not integrate $~~\dfrac{d^2x}{dt^2}=a~~$ to get $v(t)$ and $x(t)$ for motion under constant acceleration. Instead, I used the idea that the change in velocity ($v-v_0$) is the "area under the $a$ vs. $t$ curve" and that the change in position ($x-x_0$) is the "area under the $v$ vs. $t$ curve".

There were few instances in which I taught by writing expressions down rather than deriving them. A notable failure was my attempt at deriving the inverse $r$ potential from the inverse $r^2$ force in a way that would not glaze over the eyes of my audience. However, I was able to bring the essence of Maxwell's equations to the algebra-based class. This is what I did with Gauss's law, which is a no-no in its differential form and still a no-no in its integral form. I carried the integral form one more step over to the average form: $$\begin{align} & \int_S\mathbf{E}\cdot \mathbf{\hat n}~dA=\frac{q_{enc.}}{\epsilon_0} \nonumber \\
& \langle E_n\rangle_S=\frac{\int_S\mathbf{E}\cdot \mathbf{\hat n}~dA}{\int_S dA}= \frac{q_{enc.}}{\epsilon_0 S}. \nonumber \\
\end{align}$$In the expression $~\langle E_n\rangle_S~$ is the normal component of the electric field averaged over a closed surface $S$. It is proportional to the charge enclosed by the surface and inversely proportional to the surface area $S$. It can be written down as an experimental result without justification just like the differential and integral forms. It can be used to find the electric field on a surface in cases of high symmetry when the normal component of electric field anywhere on the surface is the same. In that case, the average electric field is the same as the average an idea for which students understand intuitively: when the entire class receives 100% on a test, the average test grade is 100%. One can then list the three cases of high symmetry and the expression for $S$ that accompanies them: $S=4\pi r^2~$ (sphere); $S=2\pi rL~$ (cylinder); $S=2A~$ (pillbox).

The remaining Maxwell's equations can be cast in similar forms. Ampere's law for example (without the displacement current) would be $$\langle B_t\rangle_C=\frac{\mu_0 I_{enc.}}{C}$$for the tangential component of the magnetic field averaged over a closed contour $C$. This post is already longer than originally planned so I will leave the formulation of the remaining two Maxwell's equations in average form as an exercise to the reader. Yes, the equations in this form cannot be used to derive the wave equation and whole lot of other stuff, but the underpinnings of classical E&M can be conveyed to the audience in an easy to understand form.

Finally, I will mention in passing the derivation of Snell's law using Fermat's principle presented without calculus in an earlier post.

In summary, I hope that I have convinced the reader that meaningful physics can be taught without calculus snd reach people who do not plan to pursue careers that require understanding of how physicists have modeled the physical world. Where there is a will there is a way.

 

[haushofer]

It is simply impossible to teach physics without calculus. It's even impossible to talk about physics adequately without calculus. It's thus simply a flawed concept to think that one can teach physics in a "calculus free" way.

I think that's exaggerated; I teach physics without calculus. Sure, sometimes you have to skip insight, but otherwise it's perfectly possible.

E.g., in mechanics we emphasize motion with constant acceleration if we want to calculate, or diagrams and computer models in the case of more complicated motion. The underlying concepts stay the same.

Sometimes I give some extra math for those students following advanced math courses. And yes, life would be better if that level of math would be the standard. But I don't see the impossibility you mention.

Edit And what Kuruman says

 

[haushofer]

Let me give an example of an exercise I gave my students (4th year, age 15) this week about conservation of energy.

A falling object of which air resistance is negligible falls down to the ground. Show that the height function is given by $h(t) = h_0 + v_0 t - \frac{1}{2}gt^2$, give an expression for the velocity v(t) and show that the mechanical energy of the system is conserved and interpret your result (direction up is positive).

You don't need calculus for this exercise. Constant speeds and accelerations can all be understood by looking at the area in a v,t graph. Of course, indirectly this amounts to a Riemann sum, but you don't need to call it that way. Furthermore, if they insert their h(t) and v(t) in the expression for the total energy, they find that $E_{tot} = mgh_0 + \frac{1}{2}mv_0^2$, i.e. the energy is constant at all times. Of course, students who already know the chain rule can show this without using h(t) by differentiating $E_{tot}$ w.r.t time and using Newton's second law (seeing Noether's theorem without noticing it).

When you want to include air resistance (e.g. with a skydiver), you can include a h-t and v-t diagram of the movement. What physics do students learn when they know how to solve Newton's second law explicity and integrate by using partial fractions? It's mainly a useful exercise in doing algebra or integration, but physics-wise it doesn't add that much. All the physics stuff can be understood without calculus (why doesn't the velocity increase without a bound? why does air resistance quadruples when speed doubles? what happens with the falling time if you double the height, including air resistance? etc.etc.etc.). And, importantly: this calculus-free approach can force you as teacher and the students to think more about the physics!

Another great exercise I once did with my students was to calculate the gravitational collision time between two masses. This expression can be found by doing an integral on a level my students can't understand yet, but it can also roughly be found by just looking at the units and thinking about what kind of expression you expect. Which approach does teach you more physics?

So concerning @vanhees71 his statement "It is simply impossible to teach physics without calculus.", I must say I'm really confused (or we have very different notions of what the subject "physics" means). I would turn it the other way around: leaving out the calculus stuff can provide you an opportunity to deepen your conceptual understanding (think e.g. of Hewitt's Conceptual Physics)! Of course, when it comes to quantum mechanics, it becomes a bit more subtle, but even there you can do a lot without calculus.

 

[vanhees71]

We agree to strongly disagree. For me it's not even possible to built concepts without the adequate language to express them, and that's calculus. It's not by chance that after the discovery of calculus also physics in our modern sense came into being!

 

[Bystander]

https://en.wikipedia.org/wiki/Physical_Science_Study_Committee

 

[haushofer]

We agree to strongly disagree. For me it's not even possible to built concepts without the adequate language to express them, and that's calculus. It's not by chance that after the discovery of calculus also physics in our modern sense came into being!

Can you give an example of such a concept from high school physics?

 

[WernerQH]

For me it's not even possible to built concepts without the adequate language to express them, and that's calculus. It's not by chance that after the discovery of calculus also physics in our modern sense came into being!

What about Galileo? Surely he made great contributions to physics even without the special mathematical concepts that you find indispensable and that have become so familiar to us now. ( "not even possible" ???)

 

[haushofer]

Newton doesn't use calculus in his Principia either. Conclusion...? :P

 

[kuruman]

It's not by chance that after the discovery of calculus also physics in our modern sense came into being!

No, it's not by chance that calculus gave a great impetus to the development of physics. However, this doesn't mean that, having gained the insight to the physical world that calculus has given us, we cannot go back and reformulate the calculus-based mathematical models in terms of non-calculus mathematics.

 

[haushofer]

@vanhees71 Ok, let's take the fundament of mechanics, a big part of (high school) physics: Newton's 3 laws. Would you say "it's not even possible to built these concepts without using calculus"?

Inertia? (Resulting) Force? Force as change in momentum? Interaction and the 3d law? Inertial observers? Fictitious forces? Where do I need calculus to define these concepts? Why would it be impossible to explain these concepts without calculus?

The more I think of it the weirder it sounds, to be honest.

 

[weirdoguy]

It is simply impossible to teach physics without calculus.

Just because you don't know how to do it does not mean it's impossible. It is, and people do this. I have a student from England and in comparison, level of our polish non-calculus physics is way higher then that of english with derivatives and integrals. Just because you know how to do integrals does not mean you understand the physics. And most of the physics olympiads tasks show that there is a lot to understand even without formal knowledge of what derivative is.

 

[malawi_glenn]

I teach physics and math for swedish high school pupils in the natural science programme. They are age 16-19.
Their education have three main purposes:
1) grow an interest towards any of the natural sciences so that they know what to study at a university
2) education in natural science methodoly and language
3) take more and more responsibility in their own learning

Thus, my role as a physics teacher is to make physics as interesting as possible, demonstrate the role of mathematics in physics, the interplay between theory and experiment.

As mentioned earlier, the entire physics curricula is calculus free, the approach is phenomenological and there are lots of demonstrations and laboratory work involved.

But as I teach math too - I try to sneak in calculus whenever I can and leave the derivations for the math classes, as mentioned in a previous post of mine. In that way, the physics becomes a motivation for why they should learn trigonometry, vectors, differentiation and integration and differential equations.

How do the kids know that they want to enroll in the natural science high school programme? Well, they have had nine years elementary school too, in which natural science is taught too - at even more basic level.

I am quite interested in how a pupil in Germany is exposed to physics in school. Say a student who is just about to start bachelor in physics programme at a university in Germany - what does he/she knows already?

 

[vanhees71]

I am very surprised that in Sweden the entire physics highschool curriculum is "calculus-free". For me it was a shock to learn that such attempts of teaching physics without calculus at all. Of course, also in Germany physics gets calculus based only in the final grades (11-13). Of course you can teach physics to a certain extent in a qualitative way and do experiments and nowadays also rely on computer simulations, but a true understanding cannot be gained without calculus. It already starts with the very first subject usually taught at this level, i.e., Newtonian mechanics. How do you explain Newton's Laws without calculus? I once had to substitute for a colleague in a calculus-free-physics lecture at the university, and there was calculus all over the place, but it was not allowed to call it "calculus" or to mention that the limit (you were not allowed to call it a limit of course) of the average velocity of a time interval $\Delta t$ for $\Delta t \rightarrow 0$ is a derivative of the position vector with respect to time (fortunately at least vectors were allowed). I had and still have no clue, what's gained by hiding the efficient thinking of calculus in some nebulous language just to avoid "calculus" althouth it's used all the time in this strange hidden way for the simple reason that it's impossible to discuss even the basic kinematics of Newtonian point-particle mechanics. As I said above: It's not by chance that physics in the modern sense started at and together with the development of modern calculus!

 

[Lord Jestocost]

Of course you can teach physics to a certain extent in a qualitative way and do experiments and nowadays also rely on computer simulations, but a true understanding cannot be gained without calculus.

You put it in a nutshell!

 

[kuruman]

We agree to strongly disagree.

$\dots~$ but a true understanding cannot be gained without calculus.

Here I don't agree to disagree, I strongly agree with you. Yes, a true understanding (of physics) cannot be gained without calculus. However, as I have mentioned above, the choice is not between true understanding and no understanding at all. My point is that partial understanding, enough to ensure functionality in a student's chosen field, is sufficient and must be done without calculus in curricula that are peripheral to physics but require a basic understanding of the physical world. Learning calculus, usually three semesters of coursework, is a significant investment to acquire a tool that they will not use in their chosen profession. A radiation technologist does not need to solve a first order ODE or grasp the intricacies of alpha-particle tunneling to understand radioactive decay. A plot of activity vs. number of half-lives provides sufficient understanding, albeit not "true", to get the job done.

 

[weirdoguy]

How do you explain Newton's Laws without calculus?

Does definition of inertial frame of reference (regarding first law) require knowledge of calculus? Does second law require calculus after thorough and long discussion of uniformly accelerated motion? I know a lot of students who know how to differentiate and know how acceleration is defined, but have problem describing what it physically means. Anyways, does third law require knowledge of calculus?

Besides, someone has to teach calculus to students and that takes time. Understanding all of it also takes time. In the meantime you can do a lot of challenging physical exercises that do not require calculus and are sometimes hard even for physicists.

And I really know that calculus and math is important to physics, I specialised in mathematical physics and I would love to describe everything in terms of fiber bundles But reality of teaching is not that simple.

By the way, in July it will be 15 years since I started my teaching adventure

 

[vanhees71]

I don't say that you should only teach calculus and forgetting about the physics, I only say that you can't teach physics without calculus. In other words it's not sufficient to know calculus to understand physics, but it's necessary. Of course, solving physical problems without the adequate tool is very difficult. It's also clear that you can't use abstract fiber-bundle theory in highschool, although it's perhaps the most adquate language for large parts of physics ;-).

 

[haushofer]

How do you explain Newton's Laws without calculus?

Are you serious?

1st law: if an object doesn't experience a resulting force, it travels in a straight line with constant speed.
2nd law: the resulting force is given by mass times acceleration (i.e. a force causes an acceleration of which the magnitude is inversely proportional to it's mass, hence the name "law of inertia")
3d law: a force is always one side of an interaction; if A exerts a force on B, then B will exert a force on A with the same magnitude and opposite direction.

You can do all kinds of explicit calculations with these laws without calculus as long as you're considering constant accelerations at most, or using diagrams if you want to treat more real-life problems.

Why on earth would you need calculus to explain Newton's laws? I'm sensing a True Scotchmen argument here (what do you call "understanding physics"?) What are we doing at our high schools here in Holland and in Sweden (and God knows where else) then? We don't teach "true understanding of physics" because we don't use integrals and derivatives explicitly? A textbook like Giancoli, which for the most part can be understood without calculus, doesn't teach "true understanding of physics"? Newton's Principia didn't explain Newton's laws truely?

Come on. I sincerely cannot believe you're being serious here.

Edit: to be specific: can you specifically explain how exactly calculus would improve understanding w.r.t. the explanation given above of Newton's laws?

 

[haushofer]

By the way, don't get me wrong: I would love to include calculus into my lessons. But mainly not because it enhances physical understanding, but mainly because it gives students a reason why they want to learn calculus, and it gives very nice physical intuition for math which makes it stick more. One of the many examples is the need for the chain rule in differentiating sine and cosine because otherwise the units (the frequency in the argument) won't work out, and the frequency tells you something about change.

Also, the nice thing about teaching physics without calculus is that you can discover how easy it is to fool yourself that you comprehend something because you can calculate it. Understanding physics conceptually is often much harder than being able to calculate stuff. E.g. I can calculate surface integrals, but that doesn't mean I conceptually understand why the bigger balloon will become even bigger if I connect a small and a big balloon with a tube and let air flow between them. That's the physics part (Laplace's law in this case). A surface integral is just a useful tool to do explicit calculations, but by no means necessary to understand this counter-intuitive phenomen. And that's just one example of many.

I can sincerely say that my teaching at high schools has enhanced my understanding of physics greatly, giving me insights I've never had at university as a student. And one reason is because of the emphasis on doing calculations. I guess that explains my reaction to your claim.

 

[weirdoguy]

I can sincerely say that my teaching at high schools has enhanced my understanding of physics greatly, giving me insights I've never had at university as a student.

Oh yes, so true!

 

[vanhees71]

Are you serious?

1st law: if an object doesn't experience a resulting force, it travels in a straight line with constant speed.
2nd law: the resulting force is given by mass times acceleration (i.e. a force causes an acceleration of which the magnitude is inversely proportional to it's mass, hence the name "law of inertia")
3d law: a force is always one side of an interaction; if A exerts a force on B, then B will exert a force on A with the same magnitude and opposite direction.

You can do all kinds of explicit calculations with these laws without calculus as long as you're considering constant accelerations at most, or using diagrams if you want to treat more real-life problems.

Why on earth would you need calculus to explain Newton's laws? I'm sensing a True Scotchmen argument here (what do you call "understanding physics"?) What are we doing at our high schools here in Holland and in Sweden (and God knows where else) then? We don't teach "true understanding of physics" because we don't use integrals and derivatives explicitly? A textbook like Giancoli, which for the most part can be understood without calculus, doesn't teach "true understanding of physics"? Newton's Principia didn't explain Newton's laws truely?

Come on. I sincerely cannot believe you're being serious here.

Edit: to be specific: can you specifically explain how exactly calculus would improve understanding w.r.t. the explanation given above of Newton's laws?

How do you define acceleration without calculus? You can make many words. If these have no meaning, it's empty gibberish! I know, how these ideas of "calculus-free physics" look like: They do calculus all the time without systemizing it and make thus phyiscs even more difficult to understand than necessary!

 

[haushofer]

How do you define acceleration without calculus? You can make many words.

The acceleration is given by $\frac{\Delta v}{\Delta t}$. For a constant acceleration, this is the acceleration; for a changing acceleration, this is the average acceleration; in a v-t diagram, this is the slope of the graph at a certain time.

No derivative needed. Yes, it takes some words, but then again, you need those same words to interpret the derivative.

I know, how these ideas of "calculus-free physics" look like: They do calculus all the time without systemizing it and make thus phyiscs even more difficult to understand than necessary!

A fraction is not a derivative. A derivative involves a limit. Yes, of course we use fractions in doing physics. We calculate stuff.

 

[vanhees71]

It's the derivative, whether you call it such or not! That's my point. You make your physics "calculus-free" by only hiding the fact that you of course use it. That's why everything becomes more difficult. The effectiveness of the adequate math is that you can derive general things in an abstract way and having it at hand for each concrete application. The "avoid math at any cost" makes physics more complicated instead of simpler!

 

[Klystron]

It's the derivative, whether you call it such or not! That's my point. You make your physics "calculus-free" by only hiding the fact that you of course use it. That's why everything becomes more difficult. The effectiveness of the adequate math is that you can derive general things in an abstract way and having it at hand for each concrete application. The "avoid math at any cost" makes physics more complicated instead of simpler!

We have reached the crux of this discussion and a likely place of agreement: teaching physics (above basic demonstrations) requires mathematics.

Due to exigencies of life, I learned, practiced and eventually taught several advanced concepts in physics from an electromagnetic field (emf) and QM perspective before formally learning university Calculus but definitely employed prodigious algebra, geometry, trigonometry and other transcendental functions with healthy use of set theory.

Seeing pages of frantic algebraic scribbling reduced to simple calculus equations, brought me nearly to tears while working with better educated peers. Calculus saves computation time and greatly deepens understanding of functions such as previously mentioned sine, cosine and tangent. One advantage of learning to operate a slide rule even after calculators became ubiquitous included deep exposure to logarithms and exponents; this exposure replaced by studying these functions via calculus.

 

[WernerQH]

It's the derivative, whether you call it such or not! That's my point. You make your physics "calculus-free" by only hiding the fact that you of course use it. That's why everything becomes more difficult. The effectiveness of the adequate math is that you can derive general things in an abstract way and having it at hand for each concrete application. The "avoid math at any cost" makes physics more complicated instead of simpler!

(emphasis added)

Isn't it disquieting when prospective physics teachers are led to see abstractness as beneficial?

 

[vanhees71]

To the contrary! If physics teachers find out that abstractness is beneficial in making complicated things easier to express and explain, they may be better suited to provide this important experience to their pupils. Of course, you cannot start with the abstract things in high school, but you can start to teach the students the more and more abstract reasoning such that they gain an easier understanding through abstraction.

This is also well established by didactics research: E.g., for a long time mechanics has been introduced by considering 1D motions along a line first. Now a physics-didactics colleague of mine investigated this traditional approach compared to another approach, where one starts with motions in a plane and right from the start introduce vectors. That's of course on the first glance more complicated, but the result was better, i.e., the students didn't develop misconceptions easily induced by the older approach in not recognizing the important difference between vector and scalar quantities, which have to be corrected when turning to the more complicated 2D motions (e.g. circular motion) or even full 3D problems. That's of course not an example of calculus but rather algebra, but using the somewhat "more difficult/abstract" concept of vectors (introduced, of course, as directed straight connections between points or "displacements" identifying all such "arrows" as equal when you can parallel shifting them into each other, which is a pretty abstract concept!) leads to a better understanding of introductory mechanics than the older apparently simpler approach.

I don't say that you shouldn't start teaching physics early on without "higher math". Indeed it's clear that you should get pupils very early on excited about the STEM subjects, but it's good for a teacher to be aware that without the abstracter way you can in fact not really teach the full thing, and that you must be very careful to "elementarize" the material you like to teach them, not to induce miconceptions, which are hard to correct later on.

 

[Vanadium 50]

It's the derivative, whether you call it such or not!

You might take a look at Barbeau's excellent book Polynomials, where he introduces the derivative as a function of functions - I suppose in physics speak we might call it a functional. No generalization to trig or other s[ecial functions, no fundamental theorem of calculus, no nothing. It's introduced simply as a polynomial one gets by following a partcular process on another polynomial.

 

[vanhees71]

Nice, but not so useful for physics, I guess.

 

[haushofer]

It's the derivative, whether you call it such or not! That's my point. You make your physics "calculus-free" by only hiding the fact that you of course use it. That's why everything becomes more difficult. The effectiveness of the adequate math is that you can derive general things in an abstract way and having it at hand for each concrete application. The "avoid math at any cost" makes physics more complicated instead of simpler!

With "using calculus" I assume you mean explicit usage of differentiation, integration etc. Newton's 2nd law is also a tensor equation under the Galilei group, but that doesn't mean I have to explain that to my students because "It's a tensor equation, whether you call it such or not." You don't use jet-theory in a first course in field theory, I suppose. Representation theory of the Poincare algebra in a first course on special relativity. And so on.

I disagree with your last point. I think you mean it becomes more complicated for you. Are you aware of any research that students experience it also that way?

 

[gleem]

Are we talking about HS physics courses or college courses for the non scientist/engineer?

 

[vanhees71]

When I went to high school (graduating 1990) physics was taught using differentiation and integration in the final 2 years. This matched also what was done in math, which consisted of introductory calculus (functions of one real variable), vector algebra (Euclidean vectors in 2D and 3D), and some rudimentary probability theory. Seeing this mathematics applied does not harm. I don't understand, why you want to avoid calculus in introductory physics, since you have to learn it anyway and from the mathematics lessons it's also known.

 

[haushofer]

When I went to high school (graduating 1990) physics was taught using differentiation and integration in the final 2 years. This matched also what was done in math, which consisted of introductory calculus (functions of one real variable), vector algebra (Euclidean vectors in 2D and 3D), and some rudimentary probability theory. Seeing this mathematics applied does not harm. I don't understand, why you want to avoid calculus in introductory physics, since you have to learn it anyway and from the mathematics lessons it's also known.

In Holland they chose to allow students to choose physics without learning the math modules involving calculus, to increase the number of students choosing physics.

The students who do learn calculus only learn it after we've treated mechanics.

 

[haushofer]

But again my question: what essential physical (!) concepts and insights do my students miss if I treat mechanics the way I described (i.e. with my version of Newton's laws)? Could you pinpoint that?

 

[vanhees71]

They miss the conceptual understanding. Physics conists of both experiments/observations and theory/model building. A purely qualitative collection of empirical facts is at best half of the achievements of physics.

 

[haushofer]

They miss the conceptual understanding. Physics conists of both experiments/observations and theory/model building. A purely qualitative collection of empirical facts is at best half of the achievements of physics.

No-one is talking about a "purely qualitative collection of empirical facts", so this is a strawman argument. We're talking about applications of calculus. That means explicit integration and differentiation.

As I said, we calculate a lot of stuff. Again: for up to constant accelerations we use simple algebra, beyond that we use graphs and geometry like slopes and areas. Yes, if you formalize this stuff you get calculus. But that's not the point. You seem to claim that if you don't formalize this stuff, students will lack in conceptual understanding. I don't get that.

 

[haushofer]

Let me give an explicit example. I throw a ball right up in the air, under the influence of gravity and air resistance (Fair ~v^2). My standard questions then to my students are questions like

- what's the acceleration at the maximum height, and does this depend on air resistance?
- which way will take the longest time, up or down?
- when will the resulting force be the greatest, half-way up, at it's maximum height or half-way down?

I let them sketch h-t and v-t diagrams, compare these diagrams in the case without friction (why does the speed at the beginning equals the speed at the end geometrically?) and we investigate whether such sketches help to answer the questions.

If students can answer these questions in a satisfactory way, I don't see what it would add in their understanding of the underlying physics if these students could also solve Newton's second law for this case, solve a first order differential equation by using seperation of variables, perform some nasty algebraic manipulations and an integral and give me the function h(t). In my experience, even students who can perform these calculations don't always give the correct answers to questions like the above. Sure, if they want to study more formal stuff and more complex situations, calculus becomes necessary. But we talked about the understanding of Newton's laws here.

Of course, we also calculate in these situations, with energy conservation, forces and what have you, so of course it's not a matter of "purely qualitative" exposure. Again, that's not the point.

 

[vanhees71]

No-one is talking about a "purely qualitative collection of empirical facts", so this is a strawman argument. We're talking about applications of calculus. That means explicit integration and differentiation.

But why not using explicit differentiation and integration in physics? It's taught in mathematics, so why not using it where it's most easy to use?

As I said, we calculate a lot of stuff. Again: for up to constant accelerations we use simple algebra, beyond that we use graphs and geometry like slopes and areas. Yes, if you formalize this stuff you get calculus. But that's not the point. You seem to claim that if you don't formalize this stuff, students will lack in conceptual understanding. I don't get that.

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.

The same with integrals as the area under a function graph. The fundamental theorem of calculus is a one-liner at the level of rigorosity I have in mind, and you have everything you need for integration used in high-school physics.

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!

 

[gleem]

So what to do if the student does not know calculus or is not taking it: not let them take a physics course because they would not fully appreciate or understand it?

 

[haushofer]

But why not using explicit differentiation and integration in physics? It's taught in mathematics, so why not using it where it's most easy to use?

No, in my case it is not taught in mathematics for all students, and when I cover mechanics in the beginning of their fourth year (age 15), none of them has seen any calculus yet. I also teach two different kinds of levels (pre-university and for the more applied sciences we call "havo" in the Netherlands); the last level students are even less mathematically educated. They will never see an integral or differential equation in their life unless they choose a technical study afterwards.

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.

Well, again, because most students haven't seen it yet, and this is what in teaching we call a cognitive overload. I'm teaching 14-18 year old students with all kinds of different mathematical backgrounds, not university students. It seems like you don't get that.

And no, I am not differentiating and integrating. It is a big step to go from the geometrical visual methods to the usual calculus rules. Not for you, of course, but for the average student. You seem to underestimate that. It's like saying "ah, but your ARE using fibre bundles in your first year university course on electromagnetism! Not rigorously, but you can very easily motivate this!" Yes, very easily for you as the teacher. Not for the students you're teaching.

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.

The same with integrals as the area under a function graph. The fundamental theorem of calculus is a one-liner at the level of rigorosity I have in mind, and you have everything you need for integration used in high-school physics.

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!

Yes, I know how calculus can be taught informally; I do that sometimes with interested students and I have written material for it. And yes, it is "very easily motivated" if you have studied physics at a university and are used to think mathematically. But the whole point of teaching is to be able to understand how (in my case) young people in the age of 14 to 17/18 think, isn't it? My impression is that you aren't able to descend to the level of the average modern high school student who wants to learn physics. Did you ever teach at high schools? Teach to children who are desparately trying to pass the exams? If I mention these "very easily motivated" concepts of yours (and trust me, every now and then I mention some calculus concepts for half of my public who has seen it), half of them simply quit. It may be "very easy" for you, but you know, your way of thinking, your cognitive skills and your years of experience may give you a limited view on the capabilities of the average person. At least, that's my impression if I read your posts about this subject.

And most of all: You haven't answered my question of #45. Tell me how their understanding of the physics (!) improves in learning this calculus stuff for that particular example. The big question is not if it adds anything, because of course it does. The big question is: is it worth the enormous investment at that very moment (and then I mean not for you, but for the average student)? And do I have to wait untill senior year to teach mechanics before my students have learned how to solve (simple!) differential equations and integration?

For a high school teacher, those are the relevant questions. And to be honest, that makes it so hard. I could even say: harder than teaching at a university. If I compare my teaching now with my teaching as a PhD and teaching mathematics at an applied university of mathematics, teaching at university level was peanuts compared to what I do now. I really learned how to teach at high school.

 

[vanhees71]

I give up. We agree to disagree.

 

[haushofer]

I think that's rather easygoing because you only explained to me how I could explain calculus to my students (who you don't know) without explaining me why and which physical understanding they miss without this explicit usage of calculus. I think that's one of the core questions in teaching. But whatever you want.