Rethinking Physics Education

[Sudarshan_Hebbar]

Honestly, I'm still under the impression that the main problem here is teaching students why they should care rather than spending ages trying to physically ground every single equation and mathematical operation they learn - once they learn to have a passion for the subject, their curiosity will lead them to try and develop an understanding for the equations they learn in a way that suits them.

My reasons:

1. At least from my understanding, often some physicists don't do physics because it holds practical value or even a physical meaning that is easily reflected in our everyday world. And that's completely fine.

2. Not every equation has an easily viewable physical interpretation, and many equations are dealing with an abstracted version of what is actually happening, because that offers deeper, richer insight into the system at hand. Like what was said previously:


3. Physics isn't intuitive. To learn physics means you need to leave your intuition behind completely (from my experience).

Trying to give every equation a concrete physical grounding might lead to problems later on when the physics becomes more and more abstract, and when giving a physical interpretation via simulations etc is either difficult or misleading. You shouldn't give your students the idea that physics is an intuitive discipline where everything mathematical can be easily reflected through the real world, or that there isn't lots of abstraction/abstraction is vague and not a tool to gain further insight into a system (a nice simple example of where abstracting a system has lead to valuable insight: 3Blue1Brown's video on colliding blocks computing pi) .

If you make kids want to understand everything, if you make them want to know what is going on and learn more physics, then if they feel as if they are manipulating equations with no idea as to what they mean, they will pursue further understanding themselves, asking questions or looking through the internet to gain insight until they are satisfied. Usually, a mathematical derivation that the students can follow and interpret well is good enough to satisfy this curiosity. But if the students don't care about anything except whether what you're teaching will come up in the exam (which, in my experience, is what usually ends up happening), then they probably won't even try to fully understand or get to grips with what they're being taught, in fact they might even prefer a teacher who just teaches them to solve questions and not what any of what they are doing actually means. Even if you do try and teach them the physical intuition behind things... the chances that they will listen to, retain, or care about any of the insight you have provided them could be quite slim...

Disclaimer: these are all my opinions, not facts in any way and not based on anything but my own (limited) experience of physics and physics education :)

When I look at this derivation, I notice that some steps are purely mathematical manipulations without any clear physical meaning. But does it have to be that way? Wouldn’t it be valuable to explore those steps more deeply to gain a better understanding of the underlying process? I do agree that this depends on the student's interest. However, there are students who genuinely seek deeper insight. So why not offer that level of explanation? Those who are interested will appreciate it and learn from it, while those who aren't can simply skip it—no harm done.

Note: The tone of this reply is one of Curiosity.

 

[weirdoguy]

When I look at this derivation, I notice that some steps are purely mathematical manipulations without any clear physical meaning.

Which one? Besides, I don't think that every single mathematic manipulation has physical meaning. Again, what's the physical meaning of squaring an equation?

 

[Sudarshan_Hebbar]

Which one?

This one

 

[weirdoguy]

This one

I see a lot of steps here. Which one do you want to consider? And if you treat all of it as one step, then I don't really think that you will find a "physical meaning" to all of this. Derivations are derivations.

 

[Sudarshan_Hebbar]

Which one? Besides, I don't think that every single mathematic manipulation has physical meaning. Again, what's the physical meaning of squaring an equation?

Okay, Maybe you are right. Maybe not all steps have Physical meaning or some kind of logical meaning. But, Unless we analyse each and every step how do we know which steps have a Physical meaning and which don’t. If you are interested, Maybe we can decipher this derivation together.

 

[TensorCalculus]

When I look at this derivation, I notice that some steps are purely mathematical manipulations without any clear physical meaning. But does it have to be that way? Wouldn’t it be valuable to explore those steps more deeply to gain a better understanding of the underlying process? I do agree that this depends on the student's interest. However, there are students who genuinely seek deeper insight. So why not offer that level of explanation? Those who are interested will appreciate it and learn from it, while those who aren't can simply skip it—no harm done.

Note: The tone of this reply is one of Curiosity.

View attachment 363214

Have you ever come across a student who has told you they would really appreciate this sort of insight though? I feel that there would not be need to spend time physically explaining everything, and as I said sometimes abstraction can provide greater, more satisfying insight for the curious.

exactly as @weirdoguy said, not every mathematical operation needs to have a physical interpretation, and in my eyes that's completely fine. I would like to think of myself as a curious student who pursues physics in their own time out of curiosity yet I'm struggling to see why I would gain much deeper insight into this by physically grounding every single step. I really like the intention but I don't see how physically explaining mathematical operations is going to help satisfy students' curiosity. (But of course this is again just my opinion so maybe there are other students out there who would find it valuable, from my personal perspective, it makes no sense).

(I'm not trying to be confrontational/aggressive, incase the post came across like that, I'm genuinely just struggling to understand it and as I've said I do love the intent of this all, improving physics education is something I am passionate about)

 

[kuruman]

The only thing that I don't understand is why this "I don't want to learn mindset" is more common in physics than any other subject (which, at least for my peers, is the case.

I had a similar question when I started teaching and an older colleague said this: "Physics requires people to think unnaturally. By this I mean that, for most people, it is unnatural to look at events happening around them and be able to translate said events in terms of variables and equations, turn the mathematical crank to obtain a relation linking one variable to the others and then translate the result back into an event happening around them."

That clicked with me because it is relevant to how I chose to study physics. My first physics class was in 10th grade or sophomore in high school. At that time I had three years of algebra, three semesters of geometry and one semester of trigonometry under my belt. One of the first things my physics teacher showed me was the kinematic equation for a rock thrown straight up in the air, $$y=y_0+v_0t-\frac{1}{2}gt^2.$$ I stared at him and the equation in wonder mixed with awe and thought to myself, "Is he telling me that all this math I had so far can be used to predict where the rock would be at any time $t$ that it spends in the air?" This revelation made me realize that the math I learned up to that point empowered me to understand the world around me in a way that I hadn't before. I decided to continue my study of physics after high school. Once I tasted this empowerment, I couldn't see myself doing anything else.

In terms of my colleague's statement, it turned out that I am one of the people to whom thinking unnaturally comes naturally. I am not alone. Physics Forums is full of such people.

 

[Sudarshan_Hebbar]

Have you ever come across a student who has told you they would really appreciate this sort of insight though? I feel that there would not be need to spend time physically explaining everything, and as I said sometimes abstraction can provide greater, more satisfying insight for the curious.

exactly as @weirdoguy said, not every mathematical operation needs to have a physical interpretation, and in my eyes that's completely fine. I would like to think of myself as a curious student who pursues physics in their own time out of curiosity yet I'm struggling to see why I would gain much deeper insight into this by physically grounding every single step. I really like the intention but I don't see how physically explaining mathematical operations is going to help satisfy students' curiosity. (But of course this is again just my opinion so maybe there are other students out there who would find it valuable, from my personal perspective, it makes no sense).

(I'm not trying to be confrontational/aggressive, incase the post came across like that, I'm genuinely just struggling to understand it and as I've said I do love the intent of this all, improving physics education is something I am passionate about)

@TensorCalculus I think you make a fair point. I might be wrong or I might be biased. So, I would like to invite you to try analysing a derivation with me. Then, If you get something out of it you may conclude that it makes sense. If not it doesn’t make sense. Does that sound fair?

 

[TensorCalculus]

I had a similar question when I started teaching and an older colleague said this: "Physics requires people to think unnaturally. By this I mean that, for most people, it is unnatural to look at events happening around them and be able to translate said events in terms of variables and equations, turn the mathematical crank to obtain a relation linking one variable to the others and then translate the result back into an event happening around them."

That clicked with me because it is relevant to how I chose to study physics. My first physics class was in 10th grade or sophomore in high school. At that time I had three years of algebra, three semesters of geometry and one semester of trigonometry under my belt. One of the first things my physics teacher showed me was the kinematic equation for a rock thrown straight up in the air, $$y=y_0+v_0t-\frac{1}{2}gt^2.$$ I stared at him and the equation in wonder mixed with awe and thought to myself, "Is he telling me that all this math I had so far can be used to predict where the rock would be at any time $t$ that it spends in the air?" This revelation made me realize that the math I learned up to that point empowered me to understand the world around me in a way that I hadn't before. I decided to continue my study of physics after high school. Once I tasted this empowerment, I couldn't see myself doing anything else.

In terms of my colleague's statement, it turned out that I am one of the people to whom thinking unnaturally comes naturally. I am not alone. Physics Forums is full of such people.

Interesting. I had always thought the same as you - when I learnt that every projectile, no matter it's mass or shape or the velocity it is thrown at or how it is thrown or the angle (etc, etc) always travels in a parabola (assuming an idealised scenario with no air resistance etc of course), I was stupefied. I went outside and tried this with a ton of objects in my garden and was amazed at the fact that the shapes they made all seemed parabola-like (I don't know why I expected the textbook to be wrong but oh well). The fact that it could all be described under mathematics for me felt not unnatural but wonderful (if a bit surprising). But I guess I do understand how it could seem quite confusing/tricky/unnatural. Thank you for the insight, I have learnt a lot!

@TensorCalculus I think you make a fair point. I might be wrong or I might be biased. So, I would like to invite you to try analysing a derivation with me. Then, If you get something out of it you may conclude that it makes sense. If not it doesn’t make sense. Does that sound fair?

Of course! I may also be wrong (as I have hopefully made that clear), and I am genuinely curious as to how/if it would bring insight. So yes, this seems like a good idea. Whether it works or not, it's a win for me :D.

 

[vela]

If the desire to learn ain't already within a student, it ain't happening. It is, perhaps, a jaundiced view but I was backed into it.

Yeah, nothing more annoying than the student who can't be bothered to even try. I do, however, generally find that most of my STEM students genuinely want to learn and understand the material. Some just don't know how to, though.

 

[vela]

Which one? Besides, I don't think that every single mathematic manipulation has physical meaning. Again, what's the physical meaning of squaring an equation?

I too think focusing on the individual steps is misguided. What would be helpful to many students is learning how to read an equation and interpret individual terms physically, so they don't see it as just a bunch of symbols.

 

[TensorCalculus]

Yeah, nothing more annoying than the student who can't be bothered to even try. I do, however, generally find that most of my STEM students genuinely want to learn and understand the material. Some just don't know how to, though.

I mean... in the case of my peers... it is not that they don't want to understand what is going on at all... but their only motivation to understand what is going on in the lesson is so that they don't fail their exams. And don't even try to get them to learn anything extension - "if it's not on the test, I don't care" is what I have been told numerous times when trying to talk about a "cool physics thing" that I have come across. Obviously, this desire to understand isn't fuelled by passion for the subject and doesn't prompt them to seek any deeper understanding than the bare minimum needed to answer the questions.

 

[WWGD]

What does dividing both sides by a number represent physically? What does squaring both side represent physically?

A way of checking this which I found interesting was with systems of linear equations and trying to understand why, how, adding a multiple of a , say, plane , to another plane, preserved the solution set. Deals with both Physics/graphic as well as Mathematical ones.

 

[TensorCalculus]

A way of checking this which I found interesting was with systems of linear equations and trying to understand why, how, adding a multiple of a , say, plane , to another plane, preserved the solution set. Deals with both Physics/graphic as well as Mathematical ones.

Very cool, but surely if they are learning something like v=u+at (or some other simple linear equations) then their maths might not be good enough to understand that without more confusion... of course it depends on the circumstances though

 

[WWGD]

I mean... in the case of my peers... it is not that they don't want to understand what is going on at all... but their only motivation to understand what is going on in the lesson is so that they don't fail their exams. And don't even try to get them to learn anything extension - "if it's not on the test, I don't care" is what I have been told numerous times when trying to talk about a "cool physics thing" that I have come across. Obviously, this desire to understand isn't fuelled by passion for the subject and doesn't prompt them to seek any deeper understanding than the bare minimum needed to answer the questions.

Maybe profs could reward such risk-taking /Exploring. I sought to do this as s T.A/Adjunct through extra-credit questions.

 

[Dale]

Step 1: Develop a visual definition for every term in the equation.
Step 2: Connect the mathematical operations between terms to their corresponding physical logic.

I would replace step 1 with laboratory exercises. You can learn more about voltage and current with a multimeter than with pictures.

I believe what your are asking for in concept is laboratory activities. Data treatment is also part of the post-laboratory analysis of those lab activities.

This ^

 

[Averagesupernova]

Is he telling me that all this math I had so far can be used to predict where the rock would be at any time t that it spends in the air?" This revelation made me realize that the math I learned up to that point empowered me to understand the world around me in a way that I hadn't before.

I find this very strange. It's difficult for me to believe that no one had ever explained this to you. Not calling you a liar. I'd say anyone prior to this dropped the ball concerning teaching you math.
-
I don't remember not realizing math has a use when I was a little kid. As I got older I realized that math is involved in more things every day.

 

[kuruman]

I find this very strange. It's difficult for me to believe that no one had ever explained this to you. Not calling you a liar. I'd say anyone prior to this dropped the ball concerning teaching you math.
-
I don't remember not realizing math has a use when I was a little kid. As I got older I realized that math is involved in more things every day.

Explained what to me? That math has its use in every day life? I had seen "word" problems involving ages of people, interest paid compounded yearly, distances traveled by cars passing each other, and so on. These involved formulas that one could apply to get answers. Although I didn't realize it at the time, the revelation I had was grasping the difference between an equation in mathematics and an equation in physics. Up to that point in my education I viewed $y=ax^2+bx+c$ as a quadratic where the symbols could be anything; I could just as well have written $z=bw^2+ex+f.$ A quadratic is a quadratic. When you set it equal to zero, it has generally two roots which I knew how to find.

The revelation was that if one takes this very familiar quadratic and writes it in terms of predefined symbols that are dimensioned quantities, out pops an equation in physics. Mathematically, it is still a quadratic but has the added value that it's a description of a physical event couched in terms of measurable quantities.

Mathematics is not concerned with dimensioned quantities, but physics is. If I write $$y=h_0+v_0 t−\frac{1}{2}gt^2$$ where all symbols have their usual meaning, I am specifying the height of an object thrown straight up in the air at any time $t$. Furthermore, the equation as written is shorthand notation for complete English sentences requiring no symbols. Correctly transcribed, the shorthand says

The height above ground of an object thrown straight up in the air at any given time is the same as the sum of three terms.

1. The first term is the height above ground where the object is when it starts moving.
2. The second term is the additional distance the object would travel in the given time if gravity were not acting on it.
3. The third term is the distance it would travel in the given time if released from rest with gravity acting on it; this distance must be subtracted because it reduces the gain of the second term.

Although mathematics is rightly seen as the language of physics, there is often a disconnect between the mathematical expressions used and their precise physical meaning.

So why not ask students to reconnect the mathematical expressions they use to their physical meaning by interpreting and transcribing the shorthand mathematical notation to complete sentences using no symbols as shown above?

I have found that a common sticking point with equation interpretation is the, deceptively simple, Newton's second law $$\mathbf F_{\text{net}}=m~\mathbf a.$$ Students read it, memorize the shorthand as "eff equals em ay" and, if asked to tell you what it is saying in English, they will say "Force is equal to mass times acceleration." Nope.

It says, add up all the external forces acting on the system; you will get a vector which is the net force, i.e. the vector sum of all the forces, acting on the system. Now form a second vector by multiplying the mass of the object, a scalar, by the object's observed acceleration. Newton's second law says that these two vectors obtained in two entirely different ways are the same. And remember, what is true when two vectors are the same? Answer: They have the same magnitude and point in the same direction.

 

[Paul Colby]

I learned as much math as I could and knew how to apply it to problems, but in the back of my mind, I wondered how it worked.

Isn’t it reasonable to say mathematics, in it’s beginning, began by fitting nature? People built logical systems around things about the world they observed?

 

[Averagesupernova]

@kuruman concerning my post #47, I think I'm going to quit while I'm ahead. You replied to my post in a quite lengthy manner. A detailed list of the types of things you were able to do math-wise isn't necessary. My point was that I find it odd you didn't realize early on that math described virtually everything. I had always assumed I was not unique in realizing this. While I was not able to do some of this math (and still can't) I did realize it was possible to use math to describe the way the world works.
To answer your question, explained the following to you:

....that the math I learned up to that point empowered me to understand the world around me in a way that I hadn't before.

 

[Beyond3D]

Chuck gets a charitable 1 point only to distinguish his answer from no answer at all. Chuck has failed to establish a relation between the numbers and physical reality and it doesn't matter that the answer is "20" as opposed to "42".

Was that one on purpose?

 

[weirdoguy]

I had always assumed I was not unique in realizing this.

To the contrary, you were quite unique A lot of students "know" that, yes, but really knowing and realizing, well that's a different thing.

 

[kuruman]

Was that one on purpose?

This is a hypothetical example

If I were grading this hypothetical exam, this is what I would do and why.

It never happened. The example illustrates how I would evaluate the extent to which each hypothetical student has understood the connection of their mathematical manipulation to the underlying physics.

 

[Beyond3D]

This is a hypothetical example

It never happened. The example illustrates how I would evaluate the extent to which each hypothetical student has understood the connection of their mathematical manipulation to the underlying physics.

I meant was the reference on purpose (42)

 

[Beyond3D]

"Is he telling me that all this math I had so far can be used to predict where the rock would be at any time $t$ that it spends in the air?" This revelation made me realize that the math I learned up to that point empowered me to understand the world around me in a way that I hadn't before. I decided to continue my study of physics after high school. Once I tasted this empowerment, I couldn't see myself doing anything else.

In terms of my colleague's statement, it turned out that I am one of the people to whom thinking unnaturally comes naturally. I am not alone. Physics Forums is full of such people.

Oh man, I remember those good old days. Physics started out just okay at first, but once it clicked, I couldn't see myself doing anything else. I remember when I first understood projectile motion. I'm getting goosebumps just thinking about it. Doing physics was the first time I could focus for hours on end. It (physics) just "thinks" like I do.

 

[Mark44]

From post #13:

Visual Definitions:​

• V: The displacement an object would cover in 1 second after acceleration is complete — that is, its final velocity.
• U: The displacement the object would cover in 1 second before acceleration begins — its initial velocity.
• a: The change in displacement per second between two consecutive seconds — in other words, how the velocity changes each second.

This has become a fairly long thread. I've quickly read most of it, so I might have missed some comments. What I've quoted above has a heading of "Visual Definitions." How are these definitions visual?
For another thing, the definitions for V and U seem to conflate the concepts of average velocity and instantaneous velocity. In the video, the definitions are given as average velocity over an interval (i.e. $\frac{\Delta s}{\Delta t}$ rather than as instantaneous velocities that involve the time derivative of displacement.
Also, the definitions for V and U give incorrect units; i.e. distance vs. distance/time. There is a similar problem with how acceleration is defined, especially the part "between two consecutive seconds." This sounds more like an average acceleration rather than the acceleration at a particular moment, which is not necessarily constant.

 

[Sudarshan_Hebbar]

From post #13:

This has become a fairly long thread. I've quickly read most of it, so I might have missed some comments. What I've quoted above has a heading of "Visual Definitions." How are these definitions visual?
For another thing, the definitions for V and U seem to conflate the concepts of average velocity and instantaneous velocity. In the video, the definitions are given as average velocity over an interval (i.e. $\frac{\Delta s}{\Delta t}$ rather than as instantaneous velocities that involve the time derivative of displacement.
Also, the definitions for V and U give incorrect units; i.e. distance vs. distance/time. There is a similar problem with how acceleration is defined, especially the part "between two consecutive seconds." This sounds more like an average acceleration rather than the acceleration at a particular moment, which is not necessarily constant.

Well, The idea at least was that since both velocity and Acceleration are defined with the help of displacement and displacement is visual quantity. It should be possible to define Velocity and Acceleration in a Visual way.

 

[Mark44]

The idea at least was that since both velocity and Acceleration are defined with the help of displacement and displacement is visual quantity. It should be possible to define Velocity and Acceleration in a Visual way.

Velocity and acceleration are defined in terms of positional displacement and time (a temporal displacement). If you want to define velocity (i.e., instantaneous velocity) in a visual manner, it's the slope of the tangent line to the curve s = f(t) at a specific point on this graph. Your video should clarify that what you're calling "velocity" is actually the average velocity.

 

[gleem]

I see a lot of steps here. Which one do you want to consider? And if you treat all of it as one step, then I don't really think that you will find a "physical meaning" to all of this. Derivations are derivations.

It's all about the Unreasonable Effectiveness of Mathematics, as noted by E. Wigner. Math can take us from one physical realization to another as a guide might take us from one location to another through unfamiliar territory. Without that guide, we would be lost.

 

[vela]

Well, The idea at least was that since both velocity and Acceleration are defined with the help of displacement and displacement is visual quantity. It should be possible to define Velocity and Acceleration in a Visual way.

I don't know about defining velocity and acceleration, but Knight, for instance, uses what he calls motion diagrams to illustrate uniform and uniformly accelerated motion. Try taking a look at Hewitt's Conceptual Physics book. He uses a lot of drawings to try get ideas across.