Do formulas prevent students from understanding concepts?

[Ja4Coltrane]

I recently made several comments about the sole use of formulas when a logical approach would be as effective.

https://www.physicsforums.com/showthread.php?t=150999

I am a high school senior taking AP physics C. Last year I took a more basic course, and in both of these classes (more so in the more basic course) I couldn't help but feel that students abuse formulas and feel that the key to succeeding in physics is memorizing these formulas.
Earlier this year I watched a girl move through a huge pile of flash cards with every formula that she had learned all year. Formulas like the acceleration of an atwood machine and the acceleration of a box car with a pendulum inside making an angle with the verticle. I know that if she instead understood the concepts well enough to realize the reason that the formula is what it is, she would not only feel like there is much less to memorize, but she would also be able to manipulate the formula in order to solve different types of problems.

Physics is not about memorization--it's not a language, history, or geography class. A physics course should be about thought and the pure mathematics should always follow that thought.
I'd like any feed back. What do the other members of PF think on this issue?
Do you all think that formulas prevent students from understanding and enjoying physics?

 

[sunbunny]

I completely agree with you! I'm a university student taking introductory physics and I've never been really strong with the conceptionalization and the thinking process that goes along with this course. I find memorizing formulas as comforting because I think that I've leaned something when in reality I haven't learned much.

 

[Kurdt]

I agree that you need to know the concepts behind a formula but my point from the earlier thread was that mathematics is the embodiment of logical thought. With a little mathematical knowledge you can manioulate almost any basic equation presented without knowing too much about it and I agree this is counter-productive. But so is learning concepts and not modelling them mathematically. If you want to learn physics the both go hand in hand.

Another point is that sometimes mathematical methods can glean valuable insights that are otherwise obscured if one is limited to solely thinking about problems with no maths. This is exceptionally evident in the thread where this started which was about the equivalence principal which is something counter-intuitive and has puzzled many of the greatest minds throughout history.

 

[Claude Bile]

The New South Wales (in Australia) high school Physics syllabus was changed six years ago to remove the emphasis on the equation/calculation aspect of Physics. The current syllabus as it stands requires NO mathematical prerequisities (other than the compulsory maths taught at junior level).

Is this the right balance though? From a personal point of view, I do not think so. (I think this change was implemented more due to the lack of science teachers in high schools that have studied physics at a tertiary level). Students ought to have some exposure to equations and the role they play, and not every concept in physics can be tackled using qualitative logic alone.

Do you all think that formulas prevent students from understanding and enjoying physics?

So on that note I would have to answer this question with a no (I'm assuming that understanding and enjoyment go hand in hand ). It is imperative though that students see past the symbols that comprise an equation and understand what they represent, and under what conditions the equation is valid, because without this understanding, the equation is means nothing more than squiggles on a page.

The best approach to teaching I feel is a synergistic approach using both qualitative logic and equations. You can't do good physics without one or the other. Equations are important because they - in a sense - encapsualte the logic of the problem (though it may sometimes be difficult to decipher), while qualitative logic is important as what is commonly reffered to as 'sanity checks' because no equation is guaranteed to give a sensible result.

I'll make one final comment as to why equations are important - when it comes to not simply learning physics, but actually USING this knowledge - whether it be in engineering, applied physics or theory - does require the use of equations, if for nothing more than being able to calculate a quantitative result.

Claude.

 

[Gokul43201]

A physics course should be about thought and the pure mathematics should always follow that thought.

I disagree. Any thought that you apply to physics that lacks a mathematical structure is simply intuition. You can only take intuition so far, and it is not the right way to learn physics.

This doesn't mean you merely memorize formulas either - that approach is even more flawed, for it results in no learning.

 

[Ja4Coltrane]

Thanks for all of your replies--I have always found this to be an interesting argument in physics. Personally I feel that this goes down to the fact that every person thinks in a slightly different way--thus, the different types of IQ. Just a little note, Claude Bile mentioned that the New South Wales has this zero math idea. Last year, when I took my first physics class, I started with Paul G. Hewitt's Conceptual Physics--that book, although it has some very basic mathematics (like calculating the magnitude of the acceleration of a 1 kg block when a 1 Newton net force acts on it) the book made physics very interesting and very enjoyable.
Thank you for your response Kurdt. I like your point--I do feel that by using mathematics one can often find very interesting results that he would never actually notice just by thinking. Of course, once that result is seen in the equation, one can understand the conceptual aspect of it. I think this proves one thing that I said wrong--that thought should always come before math. Perhaps math often stimulates thought and thus has a place before it. (not always--hehe)

 

[Kurdt]

Well I think we can safely say that the two go hand in hand and to replace one exclusively with the other would be a mistake.

 

[tylerp]

I recently made several comments about the sole use of formulas when a logical approach would be as effective.

https://www.physicsforums.com/showthread.php?t=150999

I am a high school senior taking AP physics C. Last year I took a more basic course, and in both of these classes (more so in the more basic course) I couldn't help but feel that students abuse formulas and feel that the key to succeeding in physics is memorizing these formulas.
Earlier this year I watched a girl move through a huge pile of flash cards with every formula that she had learned all year. Formulas like the acceleration of an atwood machine and the acceleration of a box car with a pendulum inside making an angle with the verticle. I know that if she instead understood the concepts well enough to realize the reason that the formula is what it is, she would not only feel like there is much less to memorize, but she would also be able to manipulate the formula in order to solve different types of problems.

Physics is not about memorization--it's not a language, history, or geography class. A physics course should be about thought and the pure mathematics should always follow that thought.
I'd like any feed back. What do the other members of PF think on this issue?
Do you all think that formulas prevent students from understanding and enjoying physics?

I don't think history is about memorization either, I think it's similar to Physics in the aspect that you have to know the whole story to understand the most important question, why?

 

[Ja4Coltrane]

absolutely! Sorry about that impulsive and not-very-well thought out claim.

 

[berkeman]

From my perspective (which is a mix of real-world EE R&D work and a love for general physics), you should memorize formulas and equations that you use often, so that you don't have to slow down to look them up, and beyond that memorize enough fundamentals so that you can derive other formulas and equations within a reasonable time when you need them (instead of looking them up). Like, I absolutely need to have memorized and ready the values for $$\mu_o$$ and $$\epsilon_o$$ and equations for capacitance and characteristic impedance and many other daily things. And I need to have several E&M fundamental equations readily memorized (like the Biot-Savar Law (sp?) ), and I prefer to be able to re-derive things like the s-parameter equations instead of looking them up again.

But especially with the easy access to the Internet and wiki and google, there are lots of formulas and equations that I don't bother memorizing, like Snell's Law (which I had to wiki a few minutes ago to provide some PF homework question help).

My advice would be to be sure that you can re-derive important relations when necessary (because you understand how to, and what underlying relations to use), and then decide what you want to have memorized for immediate daily/weekly use, what you want to be able to derive with a few minutes work, and what you use rarely and can wiki for.

 

[Ja4Coltrane]

berkeman--I'm sure that in your 3600 posts you have at some point felt that the sometimes people look at formulas without really thinking about them and just struggle to plug any numbers in that sort of work (aka have the right units or something) take a look at post number five in this thread: https://www.physicsforums.com/showthread.php?t=152028

that sort of shows what I mean.

 

[samh]

I'm taking an introductory physics course at my university right now.

Let me say something about myself: I *hate* memorizing things. I do anything I can to avoid it. I've held on strong to this viewpoint for years but this physics class is starting to change my mind.

I've spent the semester so far trying very hard to understand everything. I can derive the formulas from thin air now and haven't memorized a thing. But it took me so long to get this understanding that I'm falling way behind in this class and very far behind in all my other classes. At this rate I can't keep up. Everyone else is memorizing and they're doing better than I am. That's BS. These super crunched, 10-week semesters aren't meant for people like me.

You might say, "well it doesn't take that long to acquire a real understanding." But I would say that it doesn't take YOU that long. I'm not good at physics and neither are most people (i.e. 98% of the population). For us physics takes a long time. Memorizing a few formulas saves a lot of effort. As of right now I do nothing but homework constantly and I'm sick of it. Next chapter I'm just going to memorize the formulas so I have some free time for once and maybe I'll be able to enjoy my life a little bit.

Also, don't forget that most people taking physics courses are not aspiring to be physicists. For a lot of people, physics is just another class they're forced to take and they just want to get it out of the way.

 

[DougD720]

i agree 100% with the poster.

My teacher emphasizes this and makes sure we understand the concepts and then uses a General formula that works for All cases instead of the many special ones that work for each. We don't use a formula in class unless we can derive it somehow.

My teacher is excellent i must say.

 

[dontdisturbmycircles]

Do formulas prevent students from understanding concepts?

No. There are two ways to learn high school physics, in my mind. One is to understand the formulas and what they mean and do, and where they came from. There is another way that I see some people using to try to learn physics, they write out the equation and say "Ok, well I need to get v so I need to find an equation with v in it. Ok I see one, but what is t? temperature?" In other words they just learn how to use equations, but aren't really learning physics.

In order to be successful in high school physics (which is the only level of physics I am qualified to speak about) I think it is neccesary to understand the underlying concepts. But without the formulas, physics is just a conversational course, and I don't think that is physics.

 

[dontdisturbmycircles]

i agree 100% with the poster.

My teacher emphasizes this and makes sure we understand the concepts and then uses a General formula that works for All cases instead of the many special ones that work for each. We don't use a formula in class unless we can derive it somehow.

My teacher is excellent i must say.

This is a good point. If the student can use one or two equations to derive the 6 or so kinematics equations (for example) then I think it is safe to say that they understand the equations.

 

[Kurdt]

This is a good point. If the student can use one or two equations to derive the 6 or so kinematics equations (for example) then I think it is safe to say that they understand the equations.

The problem with this sometimes is that at the level where the kinematic equations are introduced, the students don't yet have the skill to derive them from first principles. I'm not sure if this is the case in all countries though but I've certainly encountered it.

 

[berkeman]

berkeman--I'm sure that in your 3600 posts you have at some point felt that the sometimes people look at formulas without really thinking about them and just struggle to plug any numbers in that sort of work (aka have the right units or something)

Absolutely. I even fell into that trap a couple times myself way back in undergrad. Like in my first solid state physics class, I didn't really have a good intuitive feel for what was going on in the early parts of the class, and spent time "looking for equations" that might fit the problem. But I managed to start figuring stuff out after not too long, and then I got a lot better at deriving what I needed from the fundamental equations, and then applying the derived equation to a particular problem.

 

[3trQN]

Another point is that sometimes mathematical methods can glean valuable insights that are otherwise obscured if one is limited to solely thinking about problems with no maths. This is exceptionally evident in the thread where this started which was about the equivalence principal which is something counter-intuitive and has puzzled many of the greatest minds throughout history.

What about thinking geometrically, i often find myself understanding a concept and abstract concepts relations to each other geometrically (as pictures and geometrical objects in my mind, intersections, tangents etc) yet not being able to fully express those algebraically or in a symbolic form very easily (or rapidly).

I can say to myself "I understand this, and i know it is related to this other concept because of the way these pieces fit together", yet i can't show that very easily.

This comes from early development of conceptualising things without mathematical equations (through a late development of mathematical skill), symbolic, and developing a picture/model approach (which basically achieves largely the same thing in terms of understanding and abstract relation).

The problem is then one of communication, learning the language to express these concepts unambiguously, not one of understanding. If you see my dilemma.

 

[CPL.Luke]

if your talking about an intro college physics class, then everybody should have the skill to derive the kinematic equations from first principles, heck even more than that those concepts should be so intuitive to a student that they don't even need to derive it. It should come naturally.

 

[Kurdt]

Yeah I see what you're getting at. I think this thread has firmly established the value of both conceptual and mathematical understanding but has raised the uglier question of when and how to introduce both successfully to physics lessons. Its a tough question because the tutors and lecturers I've encountered have always preferred either one way or the other.

 

[Kurdt]

if your talking about an intro college physics class, then everybody should have the skill to derive the kinematic equations from first principles, heck even more than that those concepts should be so intuitive to a student that they don't even need to derive it. It should come naturally.

I was talking about high school and pre calculus for the kinematic equations.

 

[complexPHILOSOPHY]

What the hell is pre-calculus exactly, I never took the class. I taught myself trigonometry after high school (because my highest math was algebra) and then my college let me take Calculus 1.

What do you learn in Pre calc? Is there anything I need to go back over and review? I hate missing math courses.

 

[dontdisturbmycircles]

In pre-calc the teacher basically just wants to make sure that you are ready for calculus, and if you are not, attempt to get you ready. It is mainly trigonometry. But also introduces the concept of an open and closed interval and set notation. Composition of functions and stuff is also covered. They also make sure you know all of your algebra rules like adding fractions since it is very important for the study of derivatives, but that's about it. Oh, and exponent laws and log laws.

Edit: Here are the contents from my old precalculus book

1. Numbers and their disguises (introduction to the set of real numbers, what is a rational number, what makes a number irrational, etc.)
2.Completing the Square
3.Solving equations
4.Functions and their graphs (graphical addition, subtraction, inverses, compositions, all that fun stuff)
5.Cyclic phenomenon : The Six basic trig ratios
6.Exponential functions
7.Logarithmic Functions
8.Inverse trigonometric functions
9.Changing the form of a function (factoring, canceling, long division, synthetic division, rationalizing)
10.Simplifying Algebraic Expressions (Difference quotients, cancelling common factors, more rationalizing of expressions)
11.Decomposition of functions (sort of an intro to the Chain Rule)
12.Trig identities

Hope that helps you.

 

[complexPHILOSOPHY]

In pre-calc the teacher basically just wants to make sure that you are ready for calculus, and if you are not, attempt to get you ready. It is mainly trigonometry. But also introduces the concept of an open and closed interval and set notation. Composition of functions and stuff is also covered. They also make sure you know all of your algebra rules like adding fractions since it is very important for the study of derivatives, but that's about it. Oh, and exponent laws and log laws.

I haven't run into any problems studying Calculus so far, but do you think it would benefit me to take pre-calc anyways? I enjoy doing maths, I still go back and work through algebra problems to stay fresh (is this a waste of time? I have a fear I will forget my algebra and then all of a sudden need to relearn it in some higher math class). I just feel behind because I hated math until a year ago (which was a year after I graduated high school) so I don't have a formal math education prior to calculus.

Sorry for all of the questions, I just despise not having a precise, informed perspective about something.

 

[dontdisturbmycircles]

I haven't run into any problems studying Calculus so far, but do you think it would benefit me to take pre-calc anyways? I enjoy doing maths, I still go back and work through algebra problems to stay fresh (is this a waste of time? I have a fear I will forget my algebra and then all of a sudden need to relearn it in some higher math class). I just feel behind because I hated math until a year ago (which was a year after I graduated high school) so I don't have a formal math education prior to calculus.

Sorry for all of the questions, I just despise not having a precise, informed perspective about something.

I am basically in the exact same boat. I sluffed off until just last year. It took me quite a while to build up my base, about 1-2 months of studying algebra very hard. But I really truly did start from scratch. I didn't even know how to add fractions :yuck:. But once that was over Ihad a fairly easy time of basic diff and integral calculus. Pick up http://www.amazon.com/dp/0321320506/?tag=pfamazon01-20 if you have any worries about your precalculus skills. And no it is not a waste of time to practice algebra skills. A big mistake some people make is to skip over the algebra and head on over to the harder stuff. I really do believe that when my old teachers used to say that getting good at math is like building a house, you need a good foundation, they were right. My personnal oppinion is that you probably don't need to take a formal pre-calculus course. Just pick up that book, it is actually designed to be a supplement for your calculus course and help you along throughout it.

 

[complexPHILOSOPHY]

About a couple of months ago I started with rational expressions and basic functions and moved on from there, so I essentially started from scratch as well. The last math class I had was in 11th grade (discrete mathematics) and Algebra in 10th grade, so I was pretty retarded. I will probably pick up that book when I have some funds.

How are you doing in comparison to the other kids in your classes? I feel like even though I started late, I have already developed a better intuitive feel for the maths we are doing (then again, I doubt the other kids chill on physicsforums and obsessively create notes).

I drive my girlfriend crazy, lol. The life of a physics major. <3

 

[Kurdt]

If you want to do pre-calc then do it. It certainly won't hurt because maths is all about practise and the more the better.

 

[complexPHILOSOPHY]

If you want to do pre-calc then do it. It certainly won't hurt because maths is all about practise and the more the better.

Yes sir, sexy Mr. Kurdt. :P

 

[dontdisturbmycircles]

About a couple of months ago I started with rational expressions and basic functions and moved on from there, so I essentially started from scratch as well. The last math class I had was in 11th grade (discrete mathematics) and Algebra in 10th grade, so I was pretty retarded. I will probably pick up that book when I have some funds.

How are you doing in comparison to the other kids in your classes? I feel like even though I started late, I have already developed a better intuitive feel for the maths we are doing (then again, I doubt the other kids chill on physicsforums and obsessively create notes).

I drive my girlfriend crazy, lol. The life of a physics major. <3

Well I am definitely doing better than most people in my class, but only because I really need the math because I want to become an engineer. I don't blame them for not liking math and putting the time into it. I don't think that many students have the motivation to do math, they just don't see the point of it. I sure didn't. I am 20 now and sitting in a class room with people 2 years younger than me :P.

 

[Kurdt]

Yes sir, sexy Mr. Kurdt. :P

Another life saved!

 

[Ja4Coltrane]

What about thinking geometrically, i often find myself understanding a concept and abstract concepts relations to each other geometrically (as pictures and geometrical objects in my mind, intersections, tangents etc) yet not being able to fully express those algebraically or in a symbolic form very easily (or rapidly).

I can say to myself "I understand this, and i know it is related to this other concept because of the way these pieces fit together", yet i can't show that very easily.

This comes from early development of conceptualising things without mathematical equations (through a late development of mathematical skill), symbolic, and developing a picture/model approach (which basically achieves largely the same thing in terms of understanding and abstract relation).

The problem is then one of communication, learning the language to express these concepts unambiguously, not one of understanding. If you see my dilemma.

I don't think that is a bad thing though--those intuitive abillities that you gain from just living are so important for physics. I also think that a development of mathematical ability can happen more easily if you can already see these things in your mind. After all, math is nothing but a flawless way of showing logic.

 

[Knavish]

Let me say something about myself: I *hate* memorizing things. I do anything I can to avoid it. I've held on strong to this viewpoint for years but this physics class is starting to change my mind.

I've spent the semester so far trying very hard to understand everything. I can derive the formulas from thin air now and haven't memorized a thing. But it took me so long to get this understanding that I'm falling way behind in this class and very far behind in all my other classes. At this rate I can't keep up. Everyone else is memorizing and they're doing better than I am. That's BS. These super crunched, 10-week semesters aren't meant for people like me.

I have had these exact thoughts and experiences myself. As a fellow student, I can commiserate. You will have to find the balance for yourself.

Ironically, coming to the university has caused to me to crave for free time to escape and study.

Probably the best way to expedite your studying is to search for better books. More "advanced" books generally give greater focus to theoretical underpinnings and historical development; that is, they are meant for students who care to understand. For instance, if you're currently learning mechanics, try An Introduction to Mechanics by Kleppner and Kolenkow, if you aren't already using it.

 

[Moonbear]

I've spent the semester so far trying very hard to understand everything. I can derive the formulas from thin air now and haven't memorized a thing. But it took me so long to get this understanding that I'm falling way behind in this class and very far behind in all my other classes. At this rate I can't keep up. Everyone else is memorizing and they're doing better than I am. That's BS. These super crunched, 10-week semesters aren't meant for people like me.

I think the overall message here might help you out. You need to find a balance. You should understand how each formula is derived so that you know how to apply it properly rather than simply hunting for formulae that have variables matching those in your problems. However, once you have that understanding, there's no need to rederive formulae over and over again if you use them frequently.

You simply cannot learn any subject well without a combination of memorizing the basic formulae or definitions and developing an understanding of the concepts that allow you to use those properly.

And, some people need more time to learn than others. This can be very frustrating if you're one of those people, and are capable of learning the material, but not at the pace the class is being taught. But, this is also why students are told that college is not just the 13th - 16th grade of high school. The pace is much faster and more demanding. Usually, the only way to deal with this is to lighten your courseload so you have more time to work on each course, even if it means taking an extra semester or extra year to complete your degree.

I think it's also worth pointing out that it is not just physics to which this problem applies. I see this in biology classes as well. Students think they can just memorize definitions and terms and don't actually understand what that definition is saying, or understand how concepts relate. When they actually have to apply the concepts, or relate concepts from different chapters, they can't.

The same is even true in subjects like history (as named in the OP). I despised history classes in high school, because that is how they were taught...memorize dates, places, names. I couldn't fathom a more boring and pointless subject. It wasn't until I was out of high school that I gained an appreciation of history when I developed an understanding of the human motivations and relatedness of events, it was much more enjoyable, and much easier to remember.

And that's a critical point. If you understand something thoroughly, it will be easier to remember. You won't need to spend hours poring over flash cards to memorize things because it will begin to feel intuitive.

 

[arildno]

In a nutshell, you should have as many formulas in your memory or bookshelf so that the derivations depending on them do not get inordinately long.

Depending on your work/knowledge situation, the optimal balance will change.

This is why an engineer, for example, should have, say a book of integral formulas in his bookshelf, whereas a first-year calculus student should not.

 

[healey.cj]

I'm in year 12 this year and the one thing i have disliked most about studying maths is really the lack of good, understandable explanations as to WHY something works. And we always seem so pressed for time at school that i don't even bother asking, and when i did people used to get annoyed so i now just make a note and go and find out for myself. I'm doing tutoring this year so it should give me a chance.
I think the biggest failing of school mathematics is that it is all about memorising formulas. Realistically we should be able to write a formula to suite the question. Apparently though, if you don't use taught formule then you fail. A teacher told me at another school they had to fail a mathematical Genius because he wrote his own formulas for the majority of questions... He got around 98% if you just look at the =____.
It's quiet pathetic the education system. We reserve the Best teachers/educators/minds for the best students when could you imagine the success of school students if they had one of them teaching them who could clearly articulate in an appropriate style?

Chris :-)