Very High GPA but no understanding is this possible?

[Hercuflea]

So I have a relatively high GPA (3.8) and constantly get compliments from my friends about my grades/achievements, etc. but I am worried that maybe I really am lacking understanding in my field. It seems like I have spent all of my college career worrying about my grades and not the actual material itself, and I have spent more time figuring out the points systems in each class than I have figuring out the topics at hand. For example, in PDE's the other day we had to use the divergence theorem from Calc 3. I was completely stumped and had forgotten it. In fact, I don't think I ever really understood it in Calculus to begin with. However, I got over a 95% in Calculus 3. How can one explain this? I have also pretty much forgotten most of the vector calculus from the course, but my transcript would tell you a different story.

This also happens in other classes, like today I was working on special relativity with a study group and I got almost every problem wrong, because in the words of the group I was approaching each problem "in the hardest way possible." I have a tendency to see really obscure connections in the formulas, which lead me down a completely different path (usually erroneous) from the rest of the class, and I miss the more obvious method that the teacher expects us to use. Also notation completely throws me off when it isn't well explained.

So do my grades really reflect my understanding. I mean surely I couldn't have made it through college with a 3.8 and understood nothing? Do you think there is a chance that I just continually received professors who were not strict enough? But even in the "easy" professors' classes, there were students who got F/D/C. I just can't figure it out.

 

[member 392791]

You don't need to understand math to do well in it, as long as you can follow the steps to solve the problem.

If you were able to prove all those theorems, then yes you would have a complete understanding, above and beyond what is expected of a typical introduction calculus sequence

 

[Hercuflea]

You don't need to understand math to do well in it, as long as you can follow the steps to solve the problem.

Hmm...this seems to be the case with me, but I get the feeling that it is kind of a false success or faulty education system if it is also the case for most other undergraduates in mathematics.

 

[member 392791]

You know how hard proving theorems is, when compared to solving random integrals? Hard enough that it is given to upper division math students to do. Physical scientists and engineers don't need to have a deep understanding. So as far as your education goes, I found a happy medium.

I don't think you need to be able to prove something in rigorous math language, but if you understand conceptually the idea of whatever it is in math you are doing, then you understand it well enough.

 

[micromass]

Yes, it is entirely possible to do well in calculus without understanding anything. The reason is because most teachers tend to severely dumb down calculus. They basically give steps to solve the problem, and all students have to do is follow the steps. They don't need to develop actual understanding or problem solving skills. You just need to be able to manipulate some symbols.

Many people who are taught this way have troubles later. Because if you get more advanced, then problem solving and understanding WILL become important. Following simple steps will not be possible anymore.

It's not really your fault, but rather the fault of the teachers. Teachers should emphasize problem solving and creativity much more. If people understand the material, then they don't even need any steps to follow, it will just come natural. Of course, making people understand is WAY harder than just having them memorize and follow some easy steps. So the teachers and the textbook take the easy way out. The students is the one who pays the price.

Maybe your physics class will teach you a good understanding of calculus. Seeing how it's used in practice can be really helpful. If it helps you, then that's great.
If your physics class doesn't help you in understanding calculus, then I'm afraid you're going to have to relearn it. Get some books like Spivak and Apostol. Those books really don't do plug-and-chug things. You can only solve the problems in there by really understanding the theory.

 

[Astrum]

In regards to math, can you do proofs? Not difficult ones, but general once. If you can, I would say you understand it well enough.

 

[micromass]

You know how hard proving theorems is, when compared to solving random integrals? Hard enough that it is given to upper division math students to do.

This really isn't true. Even high-school students are capable of proving theorems. It's just that the textbooks and the curriculum emphasize "monkey math" instead of actual understanding.

In my high-school, we started learning proofs from when we were 15 years old. So it certainly is possible to let high school students do proofs.

 

[AnTiFreeze3]

This really isn't true. Even high-school students are capable of proving theorems. It's just that the textbooks and the curriculum emphasize "monkey math" instead of actual understanding.

In my high-school, we started learning proofs from when we were 15 years old. So it certainly is possible to let high school students do proofs.

And now, as a ripe and matured 17 year old, he is ready for the world of mathematics

 

[Hercuflea]

In regards to math, can you do proofs? Not difficult ones, but general once. If you can, I would say you understand it well enough.

Yes I have been able to do some proofs. Actually I still haven't taken the required abstract algebra and real analysis courses. Most of the math courses I have taken have been computational in nature(except complex variables - I could do some easy proofs but others totally stumped me).

Yes, it is entirely possible to do well in calculus without understanding anything. The reason is because most teachers tend to severely dumb down calculus. They basically give steps to solve the problem, and all students have to do is follow the steps. They don't need to develop actual understanding or problem solving skills. You just need to be able to manipulate some symbols.

Many people who are taught this way have troubles later. Because if you get more advanced, then problem solving and understanding WILL become important. Following simple steps will not be possible anymore.

It's not really your fault, but rather the fault of the teachers. Teachers should emphasize problem solving and creativity much more. If people understand the material, then they don't even need any steps to follow, it will just come natural. Of course, making people understand is WAY harder than just having them memorize and follow some easy steps. So the teachers and the textbook take the easy way out. The students is the one who pays the price.

Maybe your physics class will teach you a good understanding of calculus. Seeing how it's used in practice can be really helpful. If it helps you, then that's great.
If your physics class doesn't help you in understanding calculus, then I'm afraid you're going to have to relearn it. Get some books like Spivak and Apostol. Those books really don't do plug-and-chug things. You can only solve the problems in there by really understanding the theory.

Physics/engineering have definitely helped my understanding of calculus. I didn't even know what the point of a differential was until I started manipulating them in E&M and thermodynamics! And as a consequence I have a much more intuitive understanding of what an integral is instead of just memorizing seemingly random formulas.

 

[clope023]

. If people understand the material, then they don't even need any steps to follow, it will just come natural.

Fallacious.

 

[clope023]

You know how hard proving theorems is, when compared to solving random integrals? Hard enough that it is given to upper division math students to do. Physical scientists and engineers don't need to have a deep understanding. So as far as your education goes, I found a happy medium.

Physical scientists and engineers don't need deep understanding of the math; their rigor lies in understanding the physical world and building useful products respectively. In a classroom setting mathematics is more difficult but the work physicists and engineers have to do is more difficult than what mathematicians do and quite a few math professors from my university agree with me.

 

[micromass]

Fallacious.

Great, thanks a lot for your contribution. Can you also add some reasoning behind your one-word reply?

 

[rollingstein]

So I have a relatively high GPA (3.8) and constantly get compliments from my friends about my grades/achievements, etc. but I am worried that maybe I really am lacking understanding in my field.
This also happens in other classes, like today I was working on special relativity with a study group and I got almost every problem wrong, because in the words of the group I was approaching each problem "in the hardest way possible." I have a tendency to see really obscure connections in the formulas, which lead me down a completely different path (usually erroneous) from the rest of the class, and I miss the more obvious method that the teacher expects us to use. Also notation completely throws me off when it isn't well explained.

So do my grades really reflect my understanding. I mean surely I couldn't have made it through college with a 3.8 and understood nothing? Do you think there is a chance that I just continually received professors who were not strict enough? But even in the "easy" professors' classes, there were students who got F/D/C. I just can't figure it out.

No. Don't worry. Often there's something called the "imposter syndrome" that can make you think so. High achievers sometimes get this self doubt as to whether they may have just sailed through the cracks in the system.

Unlikely. You are probably a smart guy and know more than you think you know.

Leave the grading to the professors. If you've had a high GPA you probably deserve it.

 

[WannabeNewton]

Physical scientists and engineers don't need deep understanding of the math; their rigor lies in understanding the physical world and building useful products respectively.

You should probably refrain from giving bad advice like this to a person who is seeking help on how much to learn in order to actually understand the material.

 

[clope023]

You should probably refrain from giving bad advice like this to a person who is seeking help on how much to learn in order to actually understand the material.

It isn't bad advice, not having a mathematician's level of understanding about the math does not mean they don't understand their material.

 

[clope023]

Great, thanks a lot for your contribution. Can you also add some reasoning behind your one-word reply?

My one word was replying to your generic oversimplification.

 

[micromass]

My one word was replying to your generic oversimplification.

I am very interested to hear your opinion and to see where my statement is wrong. So can you please provide some more clarifications?

 

[clope023]

I am very interested to hear your opinion and to see where my statement is wrong. So can you please provide some more clarifications?

It is fallacious for me though most people here tend to say the same things, I view the statement as wrong in so far from experience. I've had plenty of experiences where I understood the material but did not in any way feel problem solving came naturally. The comfort I now possesses from problem solving came from mechanical practice coupled with concepts, not an either or thing IMO.

 

[micromass]

It is fallacious for me though most people here tend to say the same things, I view the statement as wrong in so far from experience. I've had plenty of experiences where I understood the material but did not in any way feel problem solving came naturally. The comfort I now possesses from problem solving came from mechanical practice coupled with concepts, not an either or thing IMO.

Thank you!

I certainly agree that mechanical practice is needed to understand the material. I never claimed anything else. I am certainly not saying that we only need to do difficult proofs in calculus and never practice some mechanical things such as the chain rule. Both are important.

All I'm saying is that the focus right now is on following steps mechanically (at least in my experience), and not so much on concepts and proofs. Like you said, it is not an either or thing. Both are very important.

When I studied calculus, I calculated so many derivatives and integrals. This practice was very necessary. But I did tend to notice that after a while, all the methods became very obvious. I remember that I had many troubles with solving related rates. I learned how to do it mechanically. A year later, I looked at it again and it was obvious. I didn't even need the steps anymore. I could invent the steps on my own. This is what I meant with the statement that "if you understand the material, then you don't need to follow steps, it will come naturally". But I made no statement what the best way is to actually come to understanding the material. It is of course both from mechanical practice and conceptual understanding.

 

[WannabeNewton]

It isn't bad advice, not having a mathematician's level of understanding about the math does not mean they don't understand their material.

But why are you telling him to settle for a lower level of understanding a priori? What if he wants to go beyond the hand waving mathematical arguments and methods presented in the undergraduate physics textbooks?

 

[clope023]

But why are you telling him to settle for a lower level of understanding a priori? What if he wants to go beyond the hand waving mathematical arguments and methods presented in the undergraduate physics textbooks?

I never told him to settle for anything, I want deep mathematical understanding and I'm a physics/EE double, but the fact is you don't need to play with mathematical proofs to do science and engineering.

 

[Aero51]

It's very common - much more so in high school than college. I was the opposite, I never cared about grades but tried my best to understand the material. At times it was very frustrating seeing my peers who I helped do better than I did in a class. Still, I don't regret it because I developed a good reputation among my professors and peers.

My advice to you is screw the grades and learn because you want too.

 

[member 392791]

When I studied calculus, I calculated so many derivatives and integrals. This practice was very necessary. But I did tend to notice that after a while, all the methods became very obvious. I remember that I had many troubles with solving related rates. I learned how to do it mechanically. A year later, I looked at it again and it was obvious. I didn't even need the steps anymore.

I have a similar experience to this. After I finished linear algebra and diff eq, I went back and looked at the stuff from Calculus I, such as related rates, optimization problems, and Newton's method, and I was able to understand it conceptually and it was easier, compared to my first time at it just memorizing whatever I needed to do to get the best grade on the test.

 

[rollingstein]

But why are you telling him to settle for a lower level of understanding a priori? What if he wants to go beyond the hand waving mathematical arguments and methods presented in the undergraduate physics textbooks?

That's a gratuitous slight to several excellent undergraduate Physics texts.

 

[AlephZero]

I suspect "continuous assessment" has something to do with this. You are under constant pressure to get a high test score on the last little section of the course.

When I was at university, there was NO marked homework, and NO mid term and end of course tests. Just end of year exams - 6 hours a day on 3 consecutive days.

Either you LEARNED the material, or you failed. Simples...

 

[micromass]

That's a gratuitous slight to several excellent undergraduate Physics texts.

I don't think so. It's just the truth. If you compare physics texts to mathematics texts, then physics texts really are more handwaving. Mathematics texts really do go further into the theory and are more rigorous. This is just a fact.

This is not meant as an insult though. Physics textbooks are not as rigorous as mathematics texts because they don't need to be. The goal is to make students understand physics. There is no point in making everything mathematically rigorous. Knowing the precise construction of the line integral really doesn't help you understand physics, so I understand why it is not being done.

 

[rollingstein]

I don't think so. It's just the truth. If you compare physics texts to mathematics texts, then physics texts really are more handwaving. Mathematics texts really do go further into the theory and are more rigorous. This is just a fact.

This is not meant as an insult though. Physics textbooks are not as rigorous as mathematics texts because they don't need to be. The goal is to make students understand physics. There is no point in making everything mathematically rigorous. Knowing the precise construction of the line integral really doesn't help you understand physics, so I understand why it is not being done.

To me handwaving arguments is a loaded term. Often used in a pejorative sense. We shouldn't be calling every model / simplification / approximation "handwaving".

Undergrad Physics chooses a level of rigor that suits its goals best.

 

[micromass]

Undergrad Physics chooses a level of rigor that suits its goals best.

Completely agreed! Being mathematically rigorous would be an awful way to write physics books!

But if you compare an undergrad physics book with an undergrad math book, then I think it is fair to speak about handwaving. Handwaving is not good or bad in any case, it's just what it is.

 

[rollingstein]

Quoting Wikipedia:

"Handwaving is a pejorative label applied to the action of displaying the appearance of doing something, when actually doing little, or nothing. For example, it is applied to debate techniques that involve fallacies. It is also used in working situations where productive
work is expected, but no work is actually accomplished."

So, not what I want to be calling Physics.

Ok, fine, maybe I'll turn the tables and call the math texts pedantic.

 

[micromass]

Quoting Wikipedia:

"Handwaving is a pejorative label applied to the action of displaying the appearance of doing something, when actually doing little, or nothing. For example, it is applied to debate techniques that involve fallacies. It is also used in working situations where productive
work is expected, but no work is actually accomplished."

So, not what I want to be calling Physics.

Ok, fine, maybe I'll turn the tables and call the math texts pedantic.

You have all the right to call math texts pedantic. In comparison to physics texts, they certainly are!