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1500 questions
33
votes
11 answers
How can I teach my students the difference between a sequence and a series?
Sequences and series are related concepts but differ extremely from one another. I feel that students in integral calculus frequently mix them up. Part of the problem is that:
Sequences are usually taught only briefly before moving onto series.
The…
Brian Rushton
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33
votes
10 answers
Simple "real world" l'Hôpital's rule problem?
I am on a team which is writing a set of lecture notes for differential calculus.
I am using a format of "Break ground" which poses a problem, "Dig in" which develops the tools to solve the problem, and "Reinforce" which practices using the tools…
Steven Gubkin
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33
votes
3 answers
What is the evidence about the effectiveness of remediation in math?
At many colleges in the United States, incoming students are required to take placement tests in basic skills such as math and reading. Those who score below a cut-off are required to take remedial coursework. Is there solid evidence about the…
user507
33
votes
14 answers
Revisiting topics from previous courses
I teach calculus to students who have almost all taken calculus before. (Primarily first-year college students who took calculus in high school but didn't perform well enough to skip the course.)
For many topics, my students think they know the…
Henry Towsner
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33
votes
17 answers
Natural origins or learned habit: Why do students skip concepts before applications?
When teaching elementary mathematics, it takes a lot of time and effort to teach students that our goal is not to learn the examples, but to learn the concepts first, and then apply them to specific examples or problems. But students have the…
Mahdi Majidi-Zolbanin
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33
votes
18 answers
How to teach someone that $-3>-4$?
I am trying to teach a teenage person math, but he doesn't seem to be able to grasp the concept of negative numbers and $0$.
Again and again he finds $-4$ greater than $-3$ because he has spent several years seeing $4$ greater than $3$. Similarly…
Rijul Gupta
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33
votes
6 answers
What are the best practices for giving online tests?
Many of us our coming off our first semester of required-online classes; and at some of our institutions we are preparing for what is most likely a required-online semester in the fall. (That is: The first time we know in advance of the all-online…
Daniel R. Collins
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33
votes
5 answers
Inability to work with an arbitrary mathematical object
This question is motivated by student responses to homework and quiz problems I have recently posed in an undergraduate real analysis course. I will share some examples and observations first, to present the main ideas, before posing a question.
A…
Brendan W. Sullivan
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33
votes
8 answers
Math topics that reward going beyond cookbook methods
Students fresh out of high school are often under the impression that mathematics is a discipline based entirely in recognizing the type of problem and applying an algorithm or cookbook method. These students think there are finitely many types of…
Chris Cunningham
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33
votes
12 answers
For calculus students, what should be the intuition or motivation behind series?
I've noticed that series are one of the most difficult portions of calculus for new students to learn.
I think the level of abstraction has to do with this. Limits, derivatives, and integrals, as well as continuous functions, all have helpful visual…
Brian Rushton
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33
votes
3 answers
Near-universal student mistake on $\lim_{x\rightarrow\infty}e^{x+1}/e^x$
On a recent first-semester calculus exam, I gave a bunch of limits. The student was supposed to use L'Hospital's rule if possible, or if not, explain why it didn't work and evaluate it by some other method. One of the problems was…
user507
32
votes
7 answers
What are the comparative advantages of open-book versus closed-book exams?
I would like to know the advantages and disadvantages of open-book exams as compared to closed-book exams, particularly in standard undergraduate courses like calculus or linear algebra.
My practice (about 20 years of teaching) has always been to…
JDH
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32
votes
20 answers
How to explain that a negative number multiplied by a negative number is a positive number, and that $-(-x)=x$?
Actually, there is no algebraic problem to show that $-(-x) = x$. This proof can be build upon the concept of the addition of the opposite like this:
$- x + x = - x + [- ( - x) ]$, and thus by eliminating $- x$ from both sides we obtain $x = - ( - x…
Abdallah Abusharekh
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32
votes
10 answers
Should students be asked to use more than one notation for the derivative in an introductory calculus class?
There are many, many ways of writing the derivative of a function $y=f(x)$:
$$\frac{d}{dx}y, \frac{dy}{dx},\frac{d}{dx}f(x), \frac{df}{dx}, \dot y, D_x f,f',y',f'(x),f_x$$
and so on.
Students often feel uncomfortable switching back and forth between…
Brian Rushton
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32
votes
6 answers
What is the rationale for the absent (+) in mixed fractions?
Why are students taught to represent one and a half as $1 \frac{1}{2}$ rather than $1 + \frac{1}{2}$? This mode of expression seems standard at least throughout North America. I believe that it is bad pedagogy for a couple of distinct…
NiloCK
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