Rethinking Physics Education

[Sudarshan_Hebbar]

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. Many students learn to manipulate equations without understanding what those mathematical steps represent in terms of physical definitions and principles.

I’m looking to collaborate with someone to create a course that bridges this gap—one that focuses on interpreting the exact meaning of mathematical operations through the lens of physics, ensuring that every equation is grounded in a clear physical understanding.

 

[jedishrfu]

It's a noble idea to pursue, but the question at its foundation is why mathematics works so well to describe the world, and the answer is we don't know.

There was a NOVA episode that attempted to answer this question:



Perhaps the video can be a starting point for your project.

I hope your project succeeds, as it was something that bothered me as a student. 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. I asked my professors, and only a few noticed my discomfort, but I never got an answer that convinced me until someone said there is no connection.

Its only through observation, experiment and application of math that we learn what we can and can't do. The math may illuminate something new but until we test it we don't know if it is true or not.

 

[symbolipoint]

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. Many students learn to manipulate equations without understanding what those mathematical steps represent in terms of physical definitions and principles.

I’m looking to collaborate with someone to create a course that bridges this gap—one that focuses on interpreting the exact meaning of mathematical operations through the lens of physics, ensuring that every equation is grounded in a clear physical understanding.

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.

 

[kuruman]

Many students learn to manipulate equations without understanding what those mathematical steps represent in terms of physical definitions and principles.

When that happens, in my opinion, these students were not adequately taught. If the answer is a numerical quantity, then a good teacher would provide commentary why it makes sense in physical terms.

Consider the simple case of a projectile fired from the edge of a cliff of height $h$ with some initial velocity $\mathbf v_0$ and one is asked to find the time of flight. One would normally solve the quadratic equation $$0=h+v_{0y}~t-\frac{1}{2}gt^2$$and get two solutions, one positive and one negative. If the teacher proclaims that the negative solution must be thrown out because "there is no negative time", that is inadequate teaching.

The teaching moment is to say something like, "The two solutions of this equation are the times at which the projectile is at the same vertical level. There are always two such times in projectile motion except the time to maximum height which is unique. So the negative solution is an earlier time, before $t=0$, at which the projectile must be fired so that it reaches height $h$ with velocity $\mathbf v_0.$ If you were asked to solve that problem, you would write the same equation and throw out the positive solution."

If the answer is not numerical but an algebraic expression, then one needs to show to students how to interpret the dependence of the answer on the "given" quantities on the right-hand side. For example, if you double this or that on the right-hand side, what happens to the left-hand side? Are there any limiting cases? Does the value of the dependent variable at those limiting cases make sense? If you have two quantities with a relative negative sign what happens when they are equal and in a denominator or under a radical? And so on.

I don't think that the problem you describe will be addressed by creating a new course. Where this problem exists, a new course is not the answer, a better trained teacher who understands it, is.

 

[weirdoguy]

Many students learn to manipulate equations without understanding what those mathematical steps represent in terms of physical definitions and principles.

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

 

[kuruman]

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

I don't know what you think the answer is, but my answer is not much. What you describe is rescaling. If $a=b$ this is shorthand hand notation for "quantity $a$ is the same as quantity $b$. To maintain the logical relation of "is the same as" between the two, if I rescale $a$ by a factor of $\frac{1}{3}$, I need to rescale $b$ by the same factor. Squaring both sides is based on similar reasoning, if I rescale $a$ by a factor of $a$, to maintain the equality I need to rescale $b$ also by $a$, but since $a$ is the same as $b$, I might as well rescale $b$ by a factor of $b$. I see no correspondence between the mathematical manipulation and physical reality here.

Physical reality enters when units are attached to numbers or algebraic symbols are given names that have implicit dimensions. When I see $10~\text{m/s}$, I know that it describes a velocity which is a word that stands for a quantity with dimensions $\left[LT^{-1}\right]$ attached to it. Let me illustrate my thinking with a specific case of a simple exam problem.

Exam Problem
A car traveling at constant speed in a straight line covers a distance of 100 meters in 5 seconds.
Find the speed of the car. (10 points)

Ideal Solution $$v=\frac{\text{Distance}}{\text{Time}}=\frac{100~\text{m}}{5~\text{s}}=20~\text{m/s}.$$
Alice's Solution $$v=\frac{\text{Distance}}{\text{Time}}=\frac{100}{5}=20~\text{m/s}.$$
Bob's Solution $$v=\frac{100}{5}=20~\text{m/s}.$$
Chuck's Solution $$v=\frac{100}{5}=20.$$If I were grading this hypothetical exam, this is what I would do and why.

Alice gets full credit, 10 points. She did not show the units consistently but she did attach units to her final answer. If I were solving the problem for myself, I would write exactly what Alice wrote. The ideal solution is what I would post for others to see.
Bob gets half credit, 5 points. Bob's answer shows a disconnect from physical reality. Although Bob put down the units, he did not establish the implicit attachment of dimension $\left[LT^{-1}\right]$ to the mathematical symbol $v$ through the governing equation $v=\text{Distance}/\text{Time}.$
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".

Any of the last two students who cares enough to come to me and question where he lost points even though his answer was "the same" as the ideal answer, will be enlightened about what it means to use the language of math in order to do physics and be asked to be more careful next time. Mere mathematical manipulation without questioning whether it makes physical sense can easily lead one astray.

A piece of advice I frequently gave to my students was, "As you do your mathematical manipulations, split yourself into a second self looking over your shoulder. This second self is tasked with questioning every step you take and try to prove you wrong. If, when you reach the end of your manipulation, your second self has not been able to prove you wrong, then chances are that what you wrote down is correct."

 

[Frabjous]

I am not sure if it will be helpful, but you might check out “Mathematical Aspects of Physics” by Bitter (1963).
https://www.amazon.com/Mathematical...-Mathematics/dp/0486435016/?tag=pfamazon01-20

 

[Sudarshan_Hebbar]

I am not sure if it will be helpful, but you might check out “Mathematical Aspects of Physics” by Bitter (1963).
https://www.amazon.com/Mathematical...-Mathematics/dp/0486435016/?tag=pfamazon01-20

Thank you @Frabjous Will surely check it out

 

[TensorCalculus]

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. Many students learn to manipulate equations without understanding what those mathematical steps represent in terms of physical definitions and principles.

I’m looking to collaborate with someone to create a course that bridges this gap—one that focuses on interpreting the exact meaning of mathematical operations through the lens of physics, ensuring that every equation is grounded in a clear physical understanding.

Honestly, I think that the main problem is children being taught the equations, and what they mean, but not why it matters or why they should be excited about it, or what it means in the big picture. Like people think that "physics is so boring, it's just like how to calculate the velocity of something and dumb stuff like that, I'm never going to need it and it's never useful" (genuine quote from a classmate, in grade 8)

The general perception around physics for the past 3 years in my school as people have begun to learn it as a separate subject is that it's "boring", "useless" and "confusing". People's perception of physics is just things like V=IR, or v = u + at, and they absorb and apply these equations, in of course, their idealistic scenarios, and have no idea why it matters, or why they should find it interesting. They show people the equations but never the bigger picture: how this connects to other concepts and how it gives valuable, and often very beautiful, insights into how the world works. For this reason many people come to detest physics (at least from my experience) and it is by far the most unpopular subject of all amongst my peers. Which is a shame because in my eyes, if you just pursue it a little further, you begin to see the beauty and why you get taught everything you did.

This is something I personally am very passionate about. My physics certainly is not very advanced (Maybe at the level of a first or second year: I have read through Young and freedman a couple times and done most if not all of the problems (yes it took me ages, no I do not regret it) and of course done other things, nowadays looking through online lectures, learning from lecture notes, lots of practice problems of course) so depending on who this course would be aimed at I might not be able to help much but I would definitely love to help in any way possible!

 

[jedishrfu]

But its more than that. Highschool physics is certainly guilty of the formula with conditions notion where they never explicitly spell out the conditions.

And of course underneath that facade for students who think deeper is why do these equations work and not others.

I think thats what separates the masterful physics teachers from the experienced physics teachers. The masterful ones can tie the physics to the real world through observation, experiment, measurement and analogy.

I had a few teachers in college like that. One in particular, who listened to the students and masterfully guided them through classical physics.

Often students don't know the right question to ask and teachers don't know what they should have asked.

For me, the twin paradox in relativity stood out. We knew from experiment that it was a real effect but the explanation felt hollow.

The prof said the traveling twin accelerated (switched to a noninertial reference frame) during the turnaround and thats why they don't age but the earthbound twin does. Huh?

We pondered that for quite some remembering to recite the explanation for test purposes even though we still didn’t understand why. It was a very frustrating time.

 

[symbolipoint]

Often students don't know the right question to ask and teachers don't know what they should have asked.

One must be really really intelligent to be able to think like that.

 

[jedishrfu]

One must be really really intelligent to be able to think like that.

yes

 

[Sudarshan_Hebbar]

Thank you all for the insightful responses. I'd like to explore whether there’s a structured methodology that can help students better understand concepts and more effectively connect mathematics with physics. I follow a 2-step process to achieve this:

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.

Let’s take the equation V = U + a·t as an example:

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.
• t: The duration for which this change (acceleration) is applied repeatedly.

Connecting Math with Physics:​

When an object moves with an initial velocity U, it covers U meters in 1 second. The term a·t represents the accumulated change in velocity over t seconds — since the object’s velocity increases (or decreases) by a every second, this change builds up over t seconds. Adding U and a·t gives the final velocity — essentially, the object's updated ability to cover more displacement after acceleration has acted on it.

This simple example illustrates how one can approach physics equations by visually defining each term and grounding the mathematical relationships in physical meaning. In this case, defining velocity and acceleration in terms of displacement helps students see the deeper connection.

Applying this same method to other mathematical tools and operations used in physics can lead to better conceptual understanding and help students develop a more fundamental and intuitive way of thinking.

Please watch the below video which tries to explain V=U+at in the above method.

 

[Sudarshan_Hebbar]

Thank you all for the insightful responses. I'd like to explore whether there’s a structured methodology that can help students better understand concepts and more effectively connect mathematics with physics. I follow a 2-step process to achieve this:

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.

Let’s take the equation V = U + a·t as an example:

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.
• t: The duration for which this change (acceleration) is applied repeatedly.

Connecting Math with Physics:​

When an object moves with an initial velocity U, it covers U meters in 1 second. The term a·t represents the accumulated change in velocity over t seconds — since the object’s velocity increases (or decreases) by a every second, this change builds up over t seconds. Adding U and a·t gives the final velocity — essentially, the object's updated ability to cover more displacement after acceleration has acted on it.

This simple example illustrates how one can approach physics equations by visually defining each term and grounding the mathematical relationships in physical meaning. In this case, defining velocity and acceleration in terms of displacement helps students see the deeper connection.

Applying this same method to other mathematical tools and operations used in physics can lead to better conceptual understanding and help students develop a more fundamental and intuitive way of thinking.

Please watch the below video which tries to explain V=U+at in the above method.

My aim is to apply this process to every single equation, every single derivation and every single problem to high school physics to make the most in-depth course for Students. I am planning to do this full time And I would need help to do that. Please suggest your opinions on how I should proceed with this.

Thanks

 

[Frabjous]

Does defining everything in terms of displacement and time cause problems with learning units?

 

[Steve4Physics]

Please watch the below video which tries to explain V=U+at in the above method.

Sorry to be negative but...

The video has the ground moving left (watch the road-markings and the purple bushes in the background) as if observing from a car moving right. So the distance-scale shown is completely wrong.

To teach velocity, first the idea of displacement (as a vector) is needed. Is there a preceding video which does this? Without it the student will not appreciate that speed and displacement velocity are different.

I found the explanation of acceleration very confusing.

Incidentally, the SI symbol for a second is 's', not 'sec'. So 't = 5 sec' is wrong.

Edit - correction as shown.

 

[Sudarshan_Hebbar]

Does defining everything in terms of displacement and time cause problems with learning units?

Actually, I’d look at it a bit differently. If you consider the units of velocity(m/s) and acceleration(m/s^2)
, they’re already defined in terms of displacement. However, we often overlook the importance of visualizing them that way each time we encounter these terms. We tend to treat them as abstract quantities, rather than consistently grounding them in their physical meaning

 

[Sudarshan_Hebbar]

Sorry to be negative but...

The video has the ground moving left (watch the road-markings and the purple bushes in the background) as if observing from a car moving right. So the distance-scale shown is completely wrong.

To teach velocity, first the idea of displacement (as a vector) is needed. Is there a preceding video which does this? Without it the student will not appreciate that speed and displacement velocity are different.

I found the explanation of acceleration very confusing.

Incidentally, the SI symbol for a second is 's', not 'sec'. So 't = 5 sec' is wrong.

Edit - correction as shown.

I acknowledge that the video may have several errors, and the explanation might not be very strong. However, the core idea I want to communicate is that every physical quantity can be visualized through its underlying fundamental quantity. This process of visualization can serve as a powerful bridge between the mathematics and the physical concepts it represents.

 

[PeroK]

Actually, I’d look at it a bit differently. If you consider the units of velocity(m/s) and acceleration(m/s^2)
, they’re already defined in terms of displacement. However, we often overlook the importance of visualizing them that way each time we encounter these terms. We tend to treat them as abstract quantities, rather than consistently grounding them in their physical meaning

"Being abstract is something profoundly different from being vague … The purpose of abstraction is not to be vague, but to create a new semantic level in which one can be absolutely precise."

Edsger Dijkstra

 

[kuruman]

If you want to rely on visualization to develop intuition, you should not be showing things like the screenshot below. The vertical tick marks indicate equal clock intervals of 1 second. Against the background of the road and bushes, anyone looking at this will understandably get the wrong impression that, under constant acceleration, the car travels equal distances at equal times. If the driver of the car were spray painting lines on the road, they would not be sparated by equal distances. So the visuals defeat the exact point that you want to make. A correct drawing would show the separations between black tick marks drawn to scale against the background. A tiny clock at the bottom of each tick mark could indicate the passage of time in equal time intervals.

You talk about $T$ being the time interval over which the object accelerates. That is problematic. In the kinematic equation $V=U+aT$, $T$ is the clock time. The equation implies that the velocity changes continuously from $T=0$ on.

Also, you are defining velocity as the rate of change of position with respect to time. That is, of course correct. But then you also call "velocity" the ratio $\dfrac{\Delta S}{\Delta t}.$ That is not correct. The ratio is the average velocity over the time interval. You don't make the distinction between average and instantaneous velocity, yet somehow the average velocity become the instantaneous velocity in the equation $V=U+aT.$

Personally, I think that the best way to show the link between the mathematical description and physics in the case of constant acceleration is to use the velocity vs. time approach with displacement being the "area under the curve." The simple geometry of rectangles and triangles, if not already known, can be easily explained and the connection between the math and the physical world is fully transparent.

The instantaneous velocity is a point on the v vs. t line. The average velocity from $t_1$ to $t_2$ is the height of a rectangle that has base $t_2-t_1$ and height such its area $h(t_2-t_1)$ is equal to the area under the v vs. t line from $t_1$ to $t_2.$ It follows immediately that the height of the rectangle is the average velocity $v_{\text{avg.}}=\frac{1}{2}(v_2+v_1).$ Furthermore, the distinction between average and instantaneous velocity is in front of everyone to see.

 

[jedishrfu]

One thing i learned from UX design was to not be so detailed. A UX designer should roughly sketch out a customers idea with a crayon so that when the final product is presented the customer wont have any preconceived expectation and will like the final product.

I think this notion could be applied to what you want to teach and that you have a lot of work to do to get there.

Have you checked out YouTube minute physics or 3brown1blue or veritaseum episodes with his use of cartoons?

You will want to illustate the idea without having so much distracting detail that someone will spot a “mistake” and get fixated on it instead of the central idea.

 

[Sudarshan_Hebbar]

Thank you all for the feedback — I truly appreciate it.

I'm currently working through a derivation to understand the adiabatic condition, and while I have no doubts about the mathematical steps, I would be very grateful if someone could help me with the physical interpretation of each step in the derivation.

I'm visualizing this as an adiabatic process involving a gas enclosed in a cylinder-piston setup. With that setup in mind, I’m trying to understand what exactly happens physically when the gas expands or contracts:

• How is the energy distributed ?
• Which terms in the derivation represent what physical interpretations?

Specifically, I’m trying to build an intuition for why the term

[PV][/γ]=Constant

remains constant throughout the process, and how this emerges from the physical behaviour of the system.

Any insights or intuitive explanations would be highly appreciated!

 

[Sudarshan_Hebbar]

For me, the twin paradox in relativity stood out. We knew from experiment that it was a real effect but the explanation felt hollow.

The prof said the traveling twin accelerated (switched to a noninertial reference frame) during the turnaround and thats why they don't age but the earthbound twin does. Huh?

We pondered that for quite some remembering to recite the explanation for test purposes even though we still didn’t understand why. It was a very frustrating time.

Yes, @jedishrfu, I faced the same issue. Concepts like length contraction and time dilation didn’t make intuitive sense to me at first. So I spent a lot of time reflecting on them, watching videos, and studying the Michelson-Morley experiment. Through this, I began to form my own understanding: since time, length, and mass arise from electromagnetic interactions between particles—and since these forces propagate at the speed of light—any motion at high speeds would distort these interactions, leading to the observed relativistic effects.





Although I haven’t fully grasped all the underlying reasons, this gave me an intuitive foundation to start thought experiments that helped me accept and explore the results of Special Relativity. I want to create a space where students like us can engage deeply with these concepts, experiment with their own ideas, and learn through intuition rather than feeling forced to memorize something that doesn’t yet make sense.

 

[kuruman]

. . . and learn through intuition rather than feeling forced to memorize something that doesn’t yet make sense.

Hold on a moment. Who is "forcing" them? You are in control of how you teach them what and when. If you suspect that the subject doesn't yet make sense to them, tell them that they cannot learn everything all at once, but eventually, as they get more involved in a topic, it will make more sense. That said, it's your job to detect, understand and break down the barriers that prevent them from being receptive to the given topic. That is not easy to do. It's much easier to say "Just do the math using the formulas I gave and let's all pretend that you have learned something" but then you would not be a good teacher.

Take the example of relativity. It is counterintuitive only because we live in the ordinary world where one does not normally experience objects moving at relativistic speeds. Newtonian mechanics and dynamics is what we observe and operate under even though they are only approximations in cases where $v<<c.$ So life goes on in the approximate Newtonian description and nobody notices the difference or cares if there is a difference. However, denying relativity just because we don't need it in our everyday experience would be denying what is known to be true about the world around us.

Now here comes the analogy that will get the students, at least those who care, to be more receptive to relativity. We also live in a world where we go about our daily business pretending that the Earth is flat. Of course we know that the Earth is spherical but this knowledge does not affect what we do and how we move on it unless one is a ship's captain, an airline pilot or an astronaut. However, denying a spherical Earth just because we don't need it in our everyday experience would be denying what is known to be true about the world around us.

At this point they need to get involved. Ask them to research "GPS clock relativistic correction" on the web and perhaps write a short essay, in their own words - no math, about their findings.

 

[weirdoguy]

experiment with their own ideas,

In the context of relativity it's a big waste of time. Your students have to understand that they need to rebuild their intuitions. And the best way to do so is to go through a lot of problems. But you know, think deeply about each, not just do the math.

 

[TensorCalculus]

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:

"Being abstract is something profoundly different from being vague … The purpose of abstraction is not to be vague, but to create a new semantic level in which one can be absolutely precise."

Edsger Dijkstra

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 :)

 

[kuruman]

If you make kids want to understand everything, if you make them want to know what is going on and learn more physics . . .

Well-intentioned but unrealistic. This article encapsulates what I gathered after more than 40 years of teaching university-level physics about how easy it is to "make" students want to learn. 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.

 

[TensorCalculus]

Well-intentioned but unrealistic. This article encapsulates what I gathered after more than 40 years of teaching university-level physics about how easy it is to "make" students want to learn. 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.

Wow - very insightful (you have, whether fortunately or unfortunately, convinced me as well). The booklet thing is very cool.

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 go to a selective school where most children are very happy to learn more and further challenge themselves in lots of subjects, however for physics there is drastically less interest than any other subject). Surely there is something wrong with the way it is being taught if this is the case? Or maybe it is just the inherent nature of the subject as a whole, or I'm paranoid that I don't have anyone to share my passion for my favourite subject and the situation is not as bad as I am making it out to be. But if the desire to learn is inherent within a student, why are people so much more motivated to learn in other subjects in comparison to physics? I was hoping that maybe some form of what the OP is trying to do would help bridge this gap but I am not sure now...

 

[Sudarshan_Hebbar]

Hold on a moment. Who is "forcing" them?

Well, I believe my words were taken out of context. I am not a native English speaker, So maybe the choice of words came out wrong. I sincerely apologise for that. My intent was not to blame teachers or students. By “forcing” I meant circumstances. I was just expressing my opinion that some students might feel dissatisfied by the standard approach and might be interested in exploring their ideas concepts further and it might be an opportunity to create a platform for them to solve this.

 

[TensorCalculus]

Well, I believe my words were taken out of context. I am not a native English speaker, So maybe the choice of words came out wrong. I sincerely apologise for that. My intent was not to blame teachers or students. By “forcing” I meant circumstances. I was just expressing my opinion that some students might feel dissatisfied by the standard approach and might be interested in exploring their ideas concepts further and it might be an opportunity to create a platform for them to solve this.

Confused/dissatisfied -> want to explore physics further -> ask questions and chat on physics forums :)

 

[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.