# Very High GPA but no understanding is this possible?

1. Jan 31, 2013

### 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.

2. Jan 31, 2013

### Woopydalan

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

3. Jan 31, 2013

### Hercuflea

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.

4. Jan 31, 2013

### Woopydalan

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.

5. Jan 31, 2013

### 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.

6. Jan 31, 2013

### 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.

7. Jan 31, 2013

### micromass

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.

8. Jan 31, 2013

### AnTiFreeze3

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

9. Jan 31, 2013

### Hercuflea

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).

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.

10. Feb 1, 2013

### clope023

Fallacious.

11. Feb 1, 2013

### clope023

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.

12. Feb 1, 2013

### micromass

13. Feb 1, 2013

### rollingstein

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.

14. Feb 1, 2013

### WannabeNewton

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.

15. Feb 1, 2013

### clope023

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.

16. Feb 1, 2013

### clope023

17. Feb 1, 2013

### micromass

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

18. Feb 1, 2013

### clope023

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 possess from problem solving came from mechanical practice coupled with concepts, not an either or thing IMO.

19. Feb 1, 2013

### micromass

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.

20. Feb 1, 2013

### WannabeNewton

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?