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1500 questions
21
votes
6 answers
How rigorous should high school calculus be?
In the UK, calculus taught in secondary school focuses mainly on computation of derivatives and integrals and solving simple differential equations. There is a small amount of discussion about limits and the definition of the derivative, but…
A. Goodier
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21
votes
6 answers
Examples and applications of the pigeonhole principle
The Pigeonhole Principle (or Dirichlet's box principle) is a method introduced usually quite early in the mathematical curriculum. The examples where it is usually introduced are (in my humble experience) usually rather boring and not too deep.
It…
András Bátkai
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21
votes
5 answers
Teaching a student who refuses to learn
How to deal with a student who refuses to learn?
I've met a few of those over the years as a a private-class math teacher. They don't want to learn anything about the subject. Some of them are just not motivated, some have serious gaps in their…
Mefitico
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21
votes
8 answers
How can I learn to write better questions to test for conceptual understanding?
I'm worried that I'm bad at realizing when a question I've written requires little or no conceptual understanding to answer. Like, when I'm writing a question for a homework assignment or exam, I'll be thinking of it conceptually because that's how…
Mike Pierce
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21
votes
4 answers
How to help motivate math when tutoring low level algebra (High school)
I was tutoring a student today and we were doing basic factoring of quadratics and expanding terms like $(x+2)(x+5)$. Now he ended up being able to do this by the end of our 2 and a half hour session, but it was clear that he didn't understand why…
RedWings
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21
votes
6 answers
Teaching indefinite integrals that require special-casing
I encountered the following concern when teaching indefinite integrals. I believe that many of us may overlook this. May I be wrong?
Let's consider the following example.
Find the indefinite integral
$$
I=\int\dfrac{dx}{x\sqrt{x^{2}-1}}.
$$
Some of…
Hoa
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21
votes
5 answers
How students write their work, and learning outcomes
While teaching students mathematics, I have noticed that some seem sloppy in the way that they write down their work.
For example, a student is given a question: What is the area of the rectangle?
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ctrl-alt-delor
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21
votes
4 answers
How does one create "good" math problems?
As lifelong students of mathematics, we find problem-solving to be absolutely essential to enhance our understanding of the subject. Teaching others what we know serves to reinforce our existing knowledge and disseminate information to…
user718
21
votes
5 answers
Should I tell my students that math is hard for me?
I have read and heard from some other instructors that they attempt to encourage students who find math hard by saying "math is hard for me too, in fact it's hard for everyone!" I have tried this a few times, but it always rings false to me,…
Mike Shulman
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21
votes
5 answers
What is the required mathematical background of a US elementary school math teacher?
I'm currently a research mathematician, getting involved in more and more outreach activities. One of these involves delivering lessons for elementary school math teachers (K-6). The purpose is two-fold: to motivate them about the coolness of math,…
j0equ1nn
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21
votes
5 answers
Grating mathematical phrases---How to correct?
As mathematics educators, we all have come across students using mathematical notation incorrectly (looking at you, $\frac{d}{dx}$ vs $\frac{dy}{dx}$ or $\frac{\infty^2}{\infty}$). My question focuses on "verbal notation." For example, my hackles go…
erfink
- 1,129
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21
votes
10 answers
What makes cosets hard to understand?
I have been teaching introductory group theory to undergraduates. We reached cosets several weeks ago, but the combination of the textbook, my explanations and various practice questions has left the students still very uncertain about cosets. I've…
Jessica B
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21
votes
2 answers
Is this just a mistake or a more serious misconception?
One of my main research areas is algebraic thinking at different levels. Yet, from time to time, I observe something that I cannot relate to anything else that I know.
This is the story of one of these things that happened today in my calculus…
Amir Asghari
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20
votes
7 answers
Notation Conflict between Teachers and Textbooks
In mathematics notation plays an important role in clarifying the subject. A bad notation could be confusing. Recently I use a logic textbook which has a very nice approach and content but an inappropriate old notation which I never use personally.…
user230
20
votes
3 answers
Good problems that uncover difficult points in a theory
There is a great quote of Yitz Herstein:
The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would-be solver."
A number of such problems can be found in Herstein's classic book…
Jon Bannon
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