How is Physics taught without Calculus?

[Will Flannery]

"Regarding simulation: time should not be wasted in core physics classes on simulations."

If we step back a minute, what is physics? Ans: creating mathematical models of physical systems and analyzing the model to be able to predict how the physical system performs.

In classical physics, the models are sets of differential equations. and ideally the model is analyzed by solving the differential equations. Given the solution, we can predict how the real system will perform.

However, the differential equation models of most physical systems are not analytically solvable. Thus, we read in 'Deep Learning for Teaching Physics to Computers' (satirical but accurate) by former AJP Editor R. Price "At Crenshaw-Mellon University,9 in fact, simple computer programs have been developed to recognize and solve the dry-sliding-friction-block-on-tilted-plane, ballistics, and pendulum problems that constitute almost all of university physics."

What are the goals of simulation? The goal of a simulation of a physical system is predicting how the real system performs. What does a simulation start with? It starts with a differential equation models of the system. So, the goals of the classical analysis of a physical system and the simulation of the system are the same. The difference is that simulation can be used to analyze analytically unsolvable systems. That is why computational calculus, i.e. simulation, has been the norm for the analysis of physical systems since the mid-20th century.

 

[Mister T]

Up until about 1960 or so the introductory college-level sequence for physics majors was taught without the use of calculus in America. To the uninitiated using calculus to introduce physics is rather like using magic. Students, on average, show little comprehension of even the most basic concepts and instead rely on memorization to pass tests. Anyone who has ever taught introductory physics, with or without calculus, will recognize this affliction.

Edit: If you look at the textbooks used prior to about 1960 to teach introductory college-level physics to physics majors you will see that they rather resemble the noncalculus introductory college-level physics textbooks used today.

 

[gleem]

Up until about 1960 or so the introductory college-level sequence for physics majors was taught without the use of calculus in America.

Huh! The text I used in 1960 was by Shortey and Wiliams published in 1955 meant to be used with a concurrent course in calculus. I know this text was used by Wheeler at Princeton. Since the college that I attended was of no particular national importance I find it hard to believe that physics was routinely taught without calculus to physics majors during this time.

From the preface of the Australian edition of Sears and Zemansky University Physics

When the first edition of University Physics by Francis W.Sears and Mark W. Zemansky was published in 1949, it was revolutionary amongcalculus-based physics textbooks in its emphasis on the fundamental principlesof physics and how to apply them

 

[Mordred]

You would be surprised. For example the high school I attended didn't even offer Calculus. I ended up taking it as an option when I went to University. Glad I did as I needed it for Particle Physics.
Though not as much for Cosmology or Astrophysics. (Never took Astrophysics for the record). That was a good 35 to 40 years ago though lol.

Now I meet students that graduate that cannot divide fractions

 

[Mister T]

Huh! The text I used in 1960 was by Shortey and Wiliams published in 1955 meant to be used with a concurrent course in calculus.

I remember reading this in a journal years ago but must have got the year wrong. Perhaps by several decades.

I used the same text in 1973-74. The professor didn't use calculus even though it was a co-req. I also took calculus and a calc-based physics class my senior year in hs.

 

[halalaal_hi]

I would say that physics cannot be truly understood without calculus. I took non-calculus physics and calculus I at the same time. My physics book basically just told us formulas and equations whose derivations were "beyond the scope of this course". These were such things as gravitational potential energy, springs, centripetal acceleration, etc which can only be explained with calculus. With just calculus I, I was able to derive these and some things that weren't even mentioned in the book.

 

[weirdoguy]

which can only be explained with calculus.

Derived. They can be explained without calculus. And I can assure you - a lot of high school students do not grasp the physical meaning of the formula just by looking at the derivation.

My physics book basically just told us formulas and equations

Than it was a bad book.

My polish non-calculus high school students understand physics way better then my english calculus-based high school students, because in the latter case books focus way too much on using derivatives in physical context than on the physics itself.

I would say that physics cannot be truly understood without calculus.

Physics in general - yes. I don't know, maybe I misunderstood the premise of this thread. At some level it can be understood, as years of my (and not only my) teaching shows. Besides, physics olympiad (again, polish one) shows directly that knowing derivatives does not equal understanding physics. 99% of tasks is calculus free and is very hard, even for me. I still learn to be able to prepare people for that. And I know that classical fields are sections of jet bundles But that does not help

 

[Mister T]

I would say that physics cannot be truly understood without calculus.

True, because calculus was in large part invented to develop physical laws. But that process was completed by seasoned researchers not by beginning gen ed students.

I took non-calculus physics and calculus I at the same time. My physics book basically just told us formulas and equations whose derivations were "beyond the scope of this course".

The vast majority of college-level non-calculus books derived all those equations. Of course the derivations that use calculus are not convoluted, but students must understand calculus to appreciate that. And even calculus students are struggling too much to be able to appreciate.

These were such things as gravitational potential energy, springs, centripetal acceleration, etc which can only be explained with calculus.

Not true. See just about any college-level non-calculus books true. See just about any college-level non-calculus textbook.

 

[Astronuc]

Up until about 1960 or so the introductory college-level sequence for physics majors was taught without the use of calculus in America. To the uninitiated using calculus to introduce physics is rather like using magic.

I think that the situation depends on the university, the faculty (in the physics department) and the quality of students (i.e., students' preparation). Certainly, when I studied introductory physics at university, and even high school to some extent, the texts included calculus. However, I was aware of physics courses that at most used algebra and geometry.

Some Landau and Lifshitz texts were available in the 1950s.
https://en.wikipedia.org/wiki/Course_of_Theoretical_Physics

I think it fair to say that by the 1960s, there was an increased emphasis in using calculus in introductory physics programs.

 

[Mister T]

I think that the situation depends on the university, the faculty (in the physics department) and the quality of students (i.e., students' preparation). Certainly, when I studied introductory physics at university, and even high school to some extent, the texts included calculus. However, I was aware of physics courses that at most used algebra and geometry.

Yes, as I explained in Post #110 I got my dates wrong.

I think it fair to say that by the 1960s, there was an increased emphasis in using calculus in introductory physics programs.

That's true. Feynman discusses this in the introduction to his famous "Feynman Lectures in Physics" written sometime in the early 1960's IIRC.

 

[Vanadium 50]

And before Newton, physics was never taught with calculus!

We've been teaching physics with calculus for over 60 years. There must be a reason, other than "to torment students". It's because a) it is easier - fewer equations need to be memorized or appear out of thin air, and b) it allows for the student to solve a wider range of problems.

 

[ohwilleke]

The question is not so much "can physics be taught without calculus" as what reasonable goals are for a teacher teaching physics without calculus.

Without calculus you want to focus on conceptual and qualitative ideas, rather than doing calculations.

For example, you can explain that the movement of perceptible physical objects (as opposed to subatomic particles) can be determined more or less exactly by considering all forces acting on it, and you can explain different kinds of forces, and explain how friction muddies the waters.

You can bring in the equation of Newtonian gravity, so that students understand what factors influence it (the respective masses and their distance from each other) without calculating exactly how much motion is caused by gravity or mathematically proving Kepler's laws from that formula as you might in a calculus based course.

You can explain what induction is in electromagnetism without actually pulling out Maxwell's equations.

You can build intuition about how things like leverage work, by doing hands on experiments that will be remembered, rather than doing calculations on paper that mathematically weak students will forget.

You can explain some of the basic ideas of thermodynamics with drawings and hands on experiments, rather than equations.

Rudimentary physics concepts like these are usually taught to some degree starting in elementary school.

Non-calculus based physics isn't sufficient to be an engineer, but it is useful in developing a scientific worldview that can be refined later.

 

[hotvette]

Wow, does this bring back a memory. My high school had no calculus (in 1972), so physics was non-calculus based. When I got to college I put off taking calculus until second semester, but took a calculus based physics class (by mistake) my first semester. I was totally lost. I went back to my high school physics teacher for help and he told me to just think of $\frac{dx}{dt}$ as just $\frac{\Delta x}{\Delta t}$ like we learned in high school physics. It was enough to get me through the class. I started my calculus sequence the next semester.