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1500 questions
29
votes
6 answers
Misuse of parentheses for multiplication
I'd like to raise the issue of constant misuse of parentheses in the U.S., and I'm wondering if anybody else shares the same feelings, has had the same issues, knows any history behind it, and can offer some thoughts on the subject. Let me explain…
zipirovich
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29
votes
11 answers
When asked to by a religious university, how can an instructor make a mathematics course spiritually uplifting?
Several religious universities, such as Brigham Young University, ask all instructors in every area to try to make their courses spiritually uplifting. This is something included in student ratings.
When asked to by a department, how can one make a…
Brian Rushton
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29
votes
12 answers
Why do students like proof by contradiction?
Every-so-often I come across proofs of the form
Assume $X$ is false.
Prove $X$ is true (without using that it is false).
This contradicts that $X$ is false.
Hence $X$ is true.
I've seen students write such proofs, and I've seen them read this…
Jessica B
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28
votes
10 answers
Unusual applications of integration
I am trying to teach my calculus students to apply integration by thinking about what they are integrating rather than just applying formulas. Calculus books are full of formulas like "to find the volume of a solid of revolution obtained by…
Mike Shulman
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28
votes
5 answers
What is a healthy and effective way for a math educator to evaluate his or her performance?
Educators are constantly evaluated by at least the following methods:
Anonymous evaluations collected from students by the school
Anonymous evaluations posted online by students
Grades or pass-rates of the students in the class
Experienced…
Chris Cunningham
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28
votes
12 answers
Should we teach functions as sets of ordered pairs?
The context of this question is an "introduction to proofs and mathematics" class for freshman/sophomore math majors. Most textbooks for such a class say something about functions between arbitrary sets, which are of course central to modern…
Mike Shulman
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28
votes
11 answers
Impressive examples where a "proof by picture" goes wrong
There are many proofs where the whole idea can be expressed by a picture and often naturally translated into a correct formal proof.
Often one has to argue with students that a picture is not a proof (but a good thing to get a proof started).
What…
Markus Klein
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28
votes
8 answers
Is there a simple explanation for calculus classes of why partial fractions work?
I'd be happy even with an explanation in the simplest case: an explanation of why expressions of the form $\frac{ax + b}{(x - c)(x - d)}$ with $c \neq d$ can always be rewritten in the form $\frac{A}{x - c} + \frac{B}{x - d}$. I'm following…
Frank Thorne
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28
votes
7 answers
What value is there in requiring students to answer word problems in complete sentences?
This is related to my previous question What value is there in requiring students to declare the dimensions of an answer when it is already clear from context? , but with a different focus.
A sizeable minority of my primary and secondary school…
Robert Columbia
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28
votes
5 answers
What are some good examples to motivate the implicit function theorem?
I always had problems teaching the implicit function theorem in advanced analysis courses. This result is motivated by later applications, but it would be great to provide easily accessible examples to motivate the whole thing.
I usually use Example…
András Bátkai
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28
votes
10 answers
Should figures be presented to scale?
I've been working with a teacher, helping her with tech. One of the things I help with is to convert PDF formatted quizzes or tests to DeltaMath for the students to take online. The issue that I face is exemplified by the following image (the…
JTP - Apologise to Monica
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28
votes
10 answers
How to justify teaching students to rationalize denominators?
I'm teaching an "intermediate algebra" college course ($\approx$ junior high school or beginning high school algebra) and we have a bunch of problems on rationalizing denominators. How do I motivate this?
About the best I can think of is that it's a…
Dan Drake
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28
votes
10 answers
What is the best way to intuitively explain the relationship between the derivative and the integral?
This is my first post so bear with me, but something I've been thinking about lately is: Why didn't I ever question the relationship between the derivative and the integral when I was taking calculus?
Let me explain what I mean: In most courses, the…
Brain Gainz
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28
votes
4 answers
The Undergraduate Responsibility Gradient
We tell undergraduate students that they should study two to three hours for every hour they spend in class. We know that many students don't follow through with this nearly to the degree that they should.
In response to this, faculty I know do…
Jon Bannon
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27
votes
18 answers
Given a 3 4 5 triangle, how do you know that it is a right triangle?
Without knowing the Pythagorean theorem, and in presenting reasons why the theorem might be true (without giving a full proof), is there any way to give examples of triangles that are intuitively understandable to be right triangles?
For example,…
Mitch
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