How is Physics taught without Calculus?

[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.

 

[haushofer]

At least the topicstarter got his question answered.

 

[vanhees71]

They miss a clear way to express the content of the fundamental laws of physics. It starts from the very beginning with kinematics in Newtonian mechanics: You cannot even formulate what's velocity and acceleration etc. This is not even taking into account the use of calculus to make predictions (like, e.g., Kepler's Laws from Newtonian mechanics and theory of gravitation), which is nearly impossible without calculus (though Newton did hide his calculus also in his Principia, but it's very much more complicated and even more out of reach of high-school students than the use of calculus itself!).

Of course, as teacher at a high school you cannot choose your method of teaching freely but you have to adjust to what's given in the curricula some officials have created. In Germany there are 16 such curriculas with pretty different levels. None is really good in comparison to other European countries, as several studies (like PISA) show. For me the main obstacle is the lack of systematics and the low level of math (but still including calculus!).

 

[haushofer]

You cannot even formulate what's velocity and acceleration etc.

You keep repeating it, but it's obviously wrong. Describing change is not applying calculus. You obviously use a different definition of the term than the usual one. I guess e.g. economics then also can't be taught without calculus, according to your standards/definition.

 

[malawi_glenn]

It's like watching a boring tennis match :(

 

[vanhees71]

You keep repeating it, but it's obviously wrong. Describing change is not applying calculus. You obviously use a different definition of the term than the usual one. I guess e.g. economics then also can't be taught without calculus, according to your standards/definition.

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.

 

[haushofer]

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.

So what is it? Is a definition of velocity without calculus impossible or unnecassarily more complicated (but possible)?

You're contradicting yourself in one single sentence. But sure, whatever you want.

 

[haushofer]

It's like watching a boring tennis match :(

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

 

[vanhees71]

So what is it? Is a definition of velocity without calculus impossible or unnecassarily more complicated (but possible)?

You're contradicting yourself in one single sentence. But sure, whatever you want.

In my opinion it's impossible, because after all it's defined by a derivative. I was referring to the "calculus-free ideologists", who claim not to use calculus but define velocity as a limit (of course also without clearly defining what a limit is).

 

[gleem]

Students learn the concept of rates in middle school. Speed is not a new concept, Flow is not new. Other less physical rates such as inflation etc are attainable. But the rate of change of direction with no change in speed needs work but not with calculus to demonstrate the concept and produce understanding.

Perceptible physical phenomena do not need calculus to be appreciated. Transcendent phenomena, on the other hand, do require a mathematical framework to be appreciated.

To do physics requires all manner of mathematics and only a few attain the facility to do this.

 

[Vanadium 50]

It's like watching a boring tennis match :(

Wrestling. Steel cage death match!

 

[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.