Should One Memorize Definitions?

[Heisenberg7]

Hello,

I've been doing Calculus for a few days now and I'm beginning to wonder whether or not I should be memorizing definitions. By memorizing definitions I mean memorizing them by heart (word for word). This type of memorization hasn't worked very well for me in the past. At this very moment, I don't have the ability to recall any definition I've ever tried to memorize in my entire life (by heart).

I believe that this type of memorization is pretty much useless. Of course, one must understand what they are talking about in the first place. I've always preferred memorizing things by understanding them first and then knowing how to explain the concepts. But memorizing stuff word for word just seems like a loss of time. Unless you recall the material, you will surely lose the ability to recall the definition by heart. Although there are some cases where religious people memorize verses (my case) at a young age and they can still recall them after a decade (by heart). But that of course requires a ton of repetition and I am wondering whether or not that is worth it.

Thanks in advance

 

[Heisenberg7]

I should also add that I am here for the understanding mostly. This is not for school by the way. Only self-teaching.

 

[PeroK]

Whether to memorize definitions or not is not an issue for me, given that I have little hope of memorizing anything these days!

 

[berkeman]

I've been doing Calculus for a few days now and I'm beginning to wonder whether or not I should be memorizing definitions. By memorizing definitions I mean memorizing them by heart (word for word).

In Anatomy/Physiology/Biology there is a *lot* of memorization of words. In Calculus, there are formulas and such that you need to remember, but it's not like memorizing words, IMO. It's more like remembering relationships and being able to visualize mathematical relations.

I'd recommend just keeping written "crib sheets" for now as you do your studying. You can refer to them as you work problems, and start to see which things on your crib sheets you use a lot. Those will be the things that you will probably want to memorize, since you will not always have your crib sheets available.

For example, you should memorize this figure, IMO, as you will use it a lot in your math adventures involving trig functions (and obviously memorize the definitions of sin, cos, tangent based on a right triangle):

https://astarmathsandphysics.com/a-...ple-solutions-of-trigonometric-equations.html

 

[WWGD]

Repeated practice should help you memorize relevant material.

 

[Vanadium 50]

Remembering is more useful than memorization.

 

[WWGD]

This is likely controversial, but there's something to " fake it till you make it". By all means, do your best to understand, remember. If not possible, memorize it, and it will very likely make sense over time, as it's processed by your subconscious. And that's pretty certain, Heisenberg(7).

 

[gmax137]

I was always bad at memorizing things. But like @Vanadium 50 said, there are things that you need to remember. A few examples for your calculus courses:
quadratic formula
trig derivatives ($d sin =cos, d cos = -sin$, etc.)
product rule
quotient rule
etc.

I remember these things because I used them in the solutions to hundreds of homework problems. I do recall other students writing them out on paper and reading them over & over, i.e., "memorizing" them. That just never appealed to me.

 

[symbolipoint]

Memorizing a definition not so easy. Memorizing a figure or drawing with labeled parts from which something can be derived, easier.

 

[mathwonk]

in my opinion, the answer to your question is simply "yes". I believe memorizing is necessary, but not sufficient, for understanding. here is what spivak says after giving a lengthy introduction to the precise definition of a limit:
"This definition is so important (everything we do from now on depends on it) that proceeding any further without knowing it is hopeless. If necessary memorize it, like a poem! That, at least, is better than stating it incorrectly; if you do this you are doomed to give incorrect proofs."

e.g. if you do not know the precise definition of continuity, you cannot possibly even know how to begin proving that a given function is continuous.

if you are not interested in learning deeply, including giving proofs, you may not need definitions so much, but definitions are essential for proofs.

 

[PeroK]

in my opinion, the answer to your question is simply "yes". I believe memorizing is necessary, but not sufficient, for understanding. here is what spivak says after giving a lengthy introduction to the precise definition of a limit:
"This definition is so important (everything we do from now on depends on it) that proceeding any further without knowing it is hopeless. If necessary memorize it, like a poem! That, at least, is better than stating it incorrectly; if you do this you are doomed to give incorrect proofs."

e.g. if you do not know the precise definition of continuity, you cannot possibly even know how to begin proving that a given function is continuous.

if you are not interested in learning deeply, including giving proofs, you may not need definitions so much, but definitions are essential for proofs.

The difference between memorizing and understanding could be exemplified as follows. Should it be:
$$\forall \epsilon > 0, \exists \delta > 0: |x - x_0| < \delta \implies |f(x) - f(x_0)| < \epsilon$$or
$$\forall \epsilon > 0, \exists \delta > 0: |x - x_0| \le \delta \implies |f(x) - f(x_0)| \le \epsilon$$Which is correct and why?

Real understanding is when you know why a definition is precisely what it is and not something else.

Spoiler

The first is the standard definition. But, the second is equivalent and would be equally valid.

 

[symbolipoint]

In #11, most of us may know what you mean in general, but the very very few of us (I am not one of them here) know exactly the answer to that question. It all makes me think of the lyrics to Pretty Vacant.

I did not appreciate enough how important definitions can be. Easier to know and use formulas and understandable relationships.

 

[gmax137]

It all makes me think of the lyrics to Pretty Vacant.

That's pretty funny!

 

[mathwonk]

PeroK has schooled me again. I mentally (mis)understood post #11 as saying:
either 0 < |x-x0| < d implies |f(x)-f(x0)| < e,

or 0 ≤ |x-x0| ≤ d implies |f(x)-f(x0)| ≤ e.

are these the same or different?

 

[PeroK]

PeroK has schooled me again. I mentally (mis)understood post #11 as saying:
either 0 < |x-x0| < d implies |f(x)-f(x0)| < e,

or 0 ≤ |x-x0| ≤ d implies |f(x)-f(x0)| ≤ e.

are these the same or different?

Note that in the definition of $f$ being continuous at $x_0$, we must have that $f$ is defined at $x_0$. That means that we don't need to avoid $x =x_0$ in the definition. You can, of course, include the condition $x \ne x_0$, as you would in the definition of the limit at $x_0$.

Moreover, we can mix and match $<$ and $\le$ in the definition. All four possibilities are equivalent.

 

[billtodd]

Hello,

I've been doing Calculus for a few days now and I'm beginning to wonder whether or not I should be memorizing definitions. By memorizing definitions I mean memorizing them by heart (word for word). This type of memorization hasn't worked very well for me in the past. At this very moment, I don't have the ability to recall any definition I've ever tried to memorize in my entire life (by heart).

I believe that this type of memorization is pretty much useless. Of course, one must understand what they are talking about in the first place. I've always preferred memorizing things by understanding them first and then knowing how to explain the concepts. But memorizing stuff word for word just seems like a loss of time. Unless you recall the material, you will surely lose the ability to recall the definition by heart. Although there are some cases where religious people memorize verses (my case) at a young age and they can still recall them after a decade (by heart). But that of course requires a ton of repetition and I am wondering whether or not that is worth it.

Thanks in advance

Not by heart, but by exercise... (remembering definitions in maths and physics is through doing the exercises).

 

[mathwonk]

oops, I slipped up yet again; here is what I thought I was offering:

(for some L),

either for all e>0, there exists d>0, such that

0 < |x-x0| < d implies |f(x)-L| < e,

or for all e>0, there exists d>0, such that

0 ≤ |x-x0| ≤ d implies |f(x)-L| ≤ e.

are these the same or different?