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1500 questions
41
votes
5 answers
Effects of early study of advanced books
Context:
There was recently a question on Math.SE: Inferior to Other Younger and Brighter Kids which starts as follows:
I'm a high school student (Junior/Grade 11) and I'm currently studying Spivak's Calculus on Manifolds. I've finished…
dtldarek
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41
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23 answers
How should I answer questions about the purpose of learning math?
What are some good answers to questions e.g. "why do we need to study square roots"?
Of course the answer depends highly on who is asking. For the scope of my question, I have a student in mind, who is - given their interests for further education -…
BKE
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41
votes
6 answers
How can I estimate the length of an exam?
Background: I am fairly new at teaching, and in every subject I have taught, I have had difficulty estimating the length and difficulty of an exam. I need to write an exam for a university senior-level knot theory course. The exam must last one hour…
Brian Rushton
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41
votes
4 answers
Rings before groups in abstract algebra?
The default approach to teaching abstract algebra seems to be groups first, then rings. However, occasionally a textbook pops up (e.g. Childs' A Concrete Introduction to Higher Algebra, Hodge et al's Abstract Algebra: An Inquiry Based Approach or…
J W
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41
votes
6 answers
What am I supposed to be learning with long proofs of the main theorems in class?
It seems like this is exclusively how (most) people teach graduate/upper div math classes. They go through the proof of some big theorem, sometimes it might take two lectures. It's obviously important. But I honestly have no idea what I am supposed…
user5108
40
votes
14 answers
Why is learning mathematics compulsory?
In most education systems, Mathematics is a compulsory subject from primary school all the way to the start of university. A common reason given is that essential concepts like addition and multiplication are taught to the children.
But for many…
Axel Tong
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40
votes
13 answers
Beautiful planar geometry theorems not encountered in high school
I would like to impress college students (undergraduates in the U.S.)
that there is more to planar geometry beyond what they learned in high school. I would like to show them beautiful theorems they likely never encountered. They surely know that…
Joseph O'Rourke
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40
votes
14 answers
Why do we teach complex numbers?
In algebra II, USA, we teach our students complex numbers. However, after algebra II, they never use complex numbers until pretty much complex analysis. The whole point of teaching them complex numbers is to find the roots of polynomials... but,…
Simply Beautiful Art
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39
votes
11 answers
Reasons for (not) distinguishing $f$ from $f(x)$
Formally, if $f$ is a function, $f(x)$ is a value. So for instance, $f$ can be continuous, but not $f(x)$.
In teaching at school and university, notation is quite often mixed up, e.g. the function is called $f(x)$. This may be problematic but can…
Anschewski
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39
votes
14 answers
How to make Calculus II seem motivated, interesting, and useful?
I am due to teach Calculus II in the fall at an American state university. Our calculus sequence is somewhat slow, due to the fact that many of our students come with limited backgrounds. Most of our typical syllabus consists of the…
Frank Thorne
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38
votes
26 answers
What are some great books for exploring mathematics? (not kids' books and not textbooks)
People often think of math as facts and procedure - dry stuff. But it is so much more, even at basic levels. What books about mathematics have you been inspired by? There are some real treasures out there - what are they?
If your interest is in…
Sue VanHattum
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38
votes
6 answers
How can I give feedback that is not demotivating?
Background: To cope with online education, I taught linear algebra using a variant of the flipped classroom. I recorded videos and put them up on YouTube and students presented the content in these videos during class - sometimes in pairs,…
Divakaran Divakaran
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38
votes
18 answers
How do I show students the Beauty of Mathematics?
I teach many high school students, and all of them complain about being unable to fully understand mathematical concepts. I try to show them the joy of learning and deepen their understanding through games and quizzes. Yet they still feel it is too…
Axel Tong
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37
votes
5 answers
Are there any benefits to having an entire course's homework problems available from day one?
I am designing a course for the upcoming fall semester, and I am tossing around an idea in my head. While planning which topics to cover each week and how to set the pacing of the course, I figured I might as well plan ahead for all of the homework…
Brendan W. Sullivan
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37
votes
14 answers
How to give exercises when students can use ChatGPT
I tried some math exercises we will give to students and ChatGPT does really well answering these. It excels at proofs and often gives details that were not our the example solution, and makes some mistakes when it would need to do real…
allo
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