Do formulas prevent students from understanding concepts?

[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 :-)

 

[verty]

Hi Chris. Education like that only teaches one to regurgitate what the educator expects to see. It then becomes a matter of sussing out what will be asked and how those questions are expected to be answered, rather than understanding the work. I myself have been marked down for attempting to answer questions in novel ways, or sometimes for simply answering the question.

The typically example is that the question "Do you know which mountain is the tallest in the world?" deserves the answer "Yes, I do".

 

[symbolipoint]

Healey.cj, verty:

You'll find that in lower-division college courses that more is required in problem solving than just regurgitating formulas. You will actually be expected to understand formulas, and select a set of formulas which are needed for a solution; and then manage the derivation of an appropriate formula to answer the question. If the students in your current classes are getting upset when you ask questions outloud in class, maybe you would want to enroll in college preparatory or "advanced placement" courses, in which the students will be more motivated to learn, and who will share your interest in understanding.

'symbolipoint'