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I liked this simple calculus exercise

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Re: I liked this simple calculus exercise

#151

Post by physicles »

nohaydeprobleme wrote: post #95
wholinator2 wrote: post #75

Earlier quoted context omitted.

I think this experience is typical of self teaching from a math textbook. It's extremely difficult to find a good book that leaves no gaps while simultaneously explaining everything that might be difficult to understand. The key thing when encountering this for me is to expand my horizons and begin looking for videos or other supplementary materials. A teacher would show you the proof, or at least help you along the…

This is a great comment. To add on to a point that really aligns with my experiences: > "Review from time to time but don't let a hard first couple chapters prevent you from ever learning the concepts." This is a very good approach, and I wish I started doing this earlier. Even in my university math courses, the professors sometimes skipped ahead to have students focus on a few later chapters before coming back, or t…

All great points.

> it's okay to move ahead if I know at least 80% of the material.

One of the worst feelings is moving on in a textbook and realizing you are indeed totally lost.

Here’s my own list of necessary but not sufficient conditions for deep learning:

- Motivation. Easy to overlook; hard to get if you don’t already have it.

- Frequent experience of “I have no idea how to solve this,” followed by hours or days of playing with the problem, followed by a eureka moment. You can’t be sure you’ve learned the thing unless you’ve constructed the solution yourself. Builds confidence too.

- Seeing the same material in different contexts or presented in different ways. It’s like looking at an object from different angles.

And for bonus points:

- Teach the concept to a curious friend. Their questions will lead you to deeper understanding.

Re: I liked this simple calculus exercise

#152

Post by justeleblanc »

mrguyorama wrote: post #146
justeleblanc wrote: post #145

Earlier quoted context omitted.

I don't quite see how that's relevant to what I said.

>the person who wrote it knows that you have the knowledge to find the answer. This isn't always true. The person who wrote it only knows what they taught, not what you have learned. The point of the test is to find what you have learned. This is just a nitpick I think.

Not a very interesting nitpick then. I guess I should have written "knows that you should have the knowledge to find the answer". If we could teach and be sure that students had learned, we wouldn't need to test them.

Re: I liked this simple calculus exercise

#153

Post by Paul-Craft »

red_admiral wrote: post #46

Here's another exercise (resp. exam question) that tests understanding: given a sketch of a curve in a graph, roughly sketch the derivative (or integral). The number of otherwise good students who go "but I can't do the derivative without the formula?" suggests we need more questions like this.

Don't people do this routinely anymore? I remember routinely doing "sketch the derivative" type exercises in calc class pretty frequently.

Re: I liked this simple calculus exercise

#154

Post by Paul-Craft »

BeFlatXIII wrote: post #148
postsantum wrote: post #138

Earlier quoted context omitted.

Funny enough, I ended up with the opposing view - history doesn't matter, narratives do. During the last year I learned about propaganda, psy ops and myths much more than I knew before. And at the risk of sounding as an edgy teenager, I can vouch for the phrase "History is written by victors" which stopped being a platitude for me and became a part of my worldview

I prefer the existentialist "history is written by those who write history." Often, the victors give themselves a better justification for their conquest than admitting that someone bigger and badder may eventually dethrone them. However, some losers write histories praising their new overlords and their incredible military might, not out to gain favor with the new king but to preserve the dignity of the old king. Th…

In the words of Norm McDonald, "It says here in this history book that luckily, the good guys have won every time. What are the odds?"

Re: I liked this simple calculus exercise

#155

Post by postsantum »

deanCommie wrote: post #149
postsantum wrote: post #138

Earlier quoted context omitted.

Funny enough, I ended up with the opposing view - history doesn't matter, narratives do. During the last year I learned about propaganda, psy ops and myths much more than I knew before. And at the risk of sounding as an edgy teenager, I can vouch for the phrase "History is written by victors" which stopped being a platitude for me and became a part of my worldview

You're not wrong but what's the difference between history and narratives? "History" is the study of different narratives to TRY to come to a semblance of truth, but even for recent events this is almost impossible. Pick an example like "Did the US dropping the nuclear bomb on Japan ultimately save lives, or was it unnecessary" and it's impossible to find the truth between 2 conflicting narratives, each fairly justif…

The real history is often a horrible mess, narratives are built by carefully cherrypicking facts to support a worldview. This is similar to what happens to mass media - they don't need to lie, just present only those facts that support their position

Re: I liked this simple calculus exercise

#156

Post by UncleMeat »

postsantum wrote: post #138
deanCommie wrote: post #131

Earlier quoted context omitted.

It's interesting because I was going to argue with you until I applied the same idea to a subject I couldn't care less about if I tried in school, but now think is deeply fascinating: History. See, I loved math. So all through Calculus I could easily remember the big picture. Every time I practiced an Integral I imagined curves and calculating the areas under them and the visual problem and the relevance of what the…

Funny enough, I ended up with the opposing view - history doesn't matter, narratives do. During the last year I learned about propaganda, psy ops and myths much more than I knew before. And at the risk of sounding as an edgy teenager, I can vouch for the phrase "History is written by victors" which stopped being a platitude for me and became a part of my worldview

Frankly, this is spoken like somebody without a history education.

Not only would "narratives matter" be an uncontroversial statement among historians, they'd tell you that all historical writing is narrative construction. And they'd hand you a book on historiography and teach you about the methods that historians use to understand an honestly present narratives in their writing.

Note that this does not mean that historical writing is bullshit. A lot of engineers seem to come up against these observations in the humanities and then just assume that nothing can be done and that entire fields must be discarded while the people working in those fields have been living with this stuff for their entire careers.

Re: I liked this simple calculus exercise

#157

Post by UncleMeat »

postsantum wrote: post #155
deanCommie wrote: post #149

Earlier quoted context omitted.

You're not wrong but what's the difference between history and narratives? "History" is the study of different narratives to TRY to come to a semblance of truth, but even for recent events this is almost impossible. Pick an example like "Did the US dropping the nuclear bomb on Japan ultimately save lives, or was it unnecessary" and it's impossible to find the truth between 2 conflicting narratives, each fairly justif…

The real history is often a horrible mess, narratives are built by carefully cherrypicking facts to support a worldview. This is similar to what happens to mass media - they don't need to lie, just present only those facts that support their position

I think you are mistaking historical memory for history here.

Re: I liked this simple calculus exercise

#158

Post by deanCommie »

Paul-Craft wrote: post #154
BeFlatXIII wrote: post #148

Earlier quoted context omitted.

I prefer the existentialist "history is written by those who write history." Often, the victors give themselves a better justification for their conquest than admitting that someone bigger and badder may eventually dethrone them. However, some losers write histories praising their new overlords and their incredible military might, not out to gain favor with the new king but to preserve the dignity of the old king. Th…

In the words of Norm McDonald, "It says here in this history book that luckily, the good guys have won every time. What are the odds?"

Funny enough, the very premise of "good guys" existing is proof of the concept we're discussing.

Except it's not funny because then it leads to people both-sidesing genocide and actual nazis because the allies also committed war crimes.

History is a mess because humanity is a mess. But we like to think that we aren't. Probably as a defense mechanism.

Re: I liked this simple calculus exercise

#159

Post by Spivak »

red_admiral wrote: post #46

Here's another exercise (resp. exam question) that tests understanding: given a sketch of a curve in a graph, roughly sketch the derivative (or integral). The number of otherwise good students who go "but I can't do the derivative without the formula?" suggests we need more questions like this.

This style question is actually on the AP calc exams. When I came across them I thought they were cute.

Re: I liked this simple calculus exercise

#160

Post by Spivak »

saghm wrote: post #11
ryan-duve wrote: post #2

This reminds me of an exercise I'll never forget from my Math Methods course: finding the derivative of arcsin(x). It seems almost impossible because, just looking at it, there seems to be nothing you can do to simplify it. Then, out of sheer nothing-else-to-do-ism, you take the sin() of it and realize sin(arcsin(x)) = x. Take the derivative of both sides, apply chain rule and draw a right triangle and you have the a…

In turn, this reminds me a bit of a calculus problem I saw freshman year of college that I ended up sharing with my old high school calculus teacher because it similarly looks intractable until you make a "simplification". The problem was to find the integral of `x/(x + 1)` (I forget the exact bounds, presumably from 0 to x with respect to x). The trick is that this the same as `(x + 1 - 1)/(x + 1)`, which you can th…

I liked these problems because they make you feel clever figuring them out but hated them because they were a mountain of tedious algebra.
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