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1500 questions
15
votes
7 answers
An introductory example for Taylor series (12th grade)
I (a student) am doing a presentation on Taylor series in my class (12th grade, in Germany if this is relevant). I am looking for a good example where you can see when Taylor series might be useful. Something like
Consider this problem with this…
jng224
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15
votes
3 answers
What can be done to instill good habits in students?
This question might sound vague, but I'm really just looking for particular examples that worked for you.
From my experience, it seems like a large portion of "weak" students remain weak despite their best efforts. I understand that part of this is…
jon
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15
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6 answers
Are there direct practical applications of differentiating natural logarithms?
The textbook I am using to teach Calculus I includes in the exercises of most chapters a number of interesting real-world applications of the concepts from that chapter. However, the chapter on the derivative of the natural logarithm is remarkably…
Amos Hunt
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15
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5 answers
Best form for slide / beamer presentation: display items in a slide as they are discussed or all at once?
I am preparing beamer slides for an online class, and I am unsure whether I should display different items in a single slide as they are discussed or all at once.
To be more precise: I am teaching Linear Algebra. Most topics discussed during class…
Luiz Cordeiro
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15
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1 answer
Analogies for grad, div, curl, and Laplacian?
I want to try making some calculation-less questions about vector calculus identities that are solely based upon picture diagrams of vector fields, or fields that could be sketched out by hand. The purpose of this is to introduce the ideas so that…
user13544
15
votes
2 answers
Can GAP/Magma be helpful for a first abstract algebra course?
Does anyone have any experience using either GAP or Magma in a first abstract algebra course?
The context here is a course for sophomore/junior math majors that focuses primarily on groups. I'm curious:
What are the advantages (and disadvantages)…
Jim Belk
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15
votes
1 answer
Students using l'Hôpital's Rule on the terms of a series, instead of the Limit Comparison Test
I realize this is a very specific question, but it is something I have noticed over the last few semesters teaching sequences and series in Calculus 2 to undergraduates.
The purpose of the Limit Comparison Test is to take an intuition, such…
Brendan W. Sullivan
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15
votes
7 answers
Should we teach simple content quickly or slowly?
The title might be a bit not specific, so let me give an example.
In China, Japan, Korea, etc, there is a type of problem about chickens (or crane, or anything with two legs) and rabbits (4 legs) in the same cage:
Some chickens and rabbits are in…
Ma Joad
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15
votes
8 answers
How should I introduce the Chain Rule
I'm halfway through my first year of teaching AP Calculus to high school seniors. It's been going generally well, but I'm feeling like I really could have done better getting them into the Chain Rule.
I started with it the same basic way that I did…
Matthew Daly
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15
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2 answers
Teaching 100 x 100 times tables
First, why bother: I teach and play math with my son in the mornings. It's a lot about letting him enjoy learning, so the curriculum (my intentionally vague vision of how we'll proceed) is flexible. If he finds something interesting and wants to…
Hal
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15
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4 answers
When Euclid was used as a textbook, what exercises did students do?
Until fairly recently, it was common for students in school to learn Euclidean geometry from a translation of Euclid. I get the impression that ca. 1700 this would have been in college and only for a tiny fraction of the population, whereas maybe…
user507
15
votes
8 answers
How does knowing more about mathematics help one's teaching of lower level course, such as calculus?
A question has been asked about why great mathematicians are not necessarily great teachers. On the other hand, I am wondering if knowing more mathematics actually helps with one's teaching of lower level courses in mathematics. For example, I…
Zuriel
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15
votes
7 answers
Should theorems be proved to students who are not majoring in mathematics?
My impression to students majoring in mathematics is, whenever we teach them a theorem, a proof should be given in the class, or at least as a reading assignment. However, how about students not majoring in mathematics?
One extreme is, proving…
Zuriel
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15
votes
6 answers
Iconic image to explain the fundamental theorem of calculus?
Is there some single, persuading visualization that can be used to convince students of the intuitive truth of the fundamental
theorem of calculus, in the form
$$
\int_a^b f(t) \, dt = F(b) - F(a) \;?
$$
I'm not seeking a "proof-without-words," but…
Joseph O'Rourke
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15
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2 answers
Why do we state the antiderivative of $\sec x$ as $\ln |\sec x +\tan x|+C$?
One easy integration of $\sec x$ substitutes $u=\sin x$, viz.$$\int\frac{\cos x}{1-\sin^2 x}\,\mathrm{d}x=\frac{1}{2}\ln\left|\frac{1+\sin x}{1-\sin x}\right|+C.$$Multiplying top and bottom by $1+\sin x$ and writing $1-\sin^2 x=\cos^2 x$ gives the…
J.G.
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