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1500 questions
22
votes
6 answers
What theorems from single-variable calculus break down in the multi-variable context?"
In teaching multi-variable calculus, it's insightful to discuss with students not only how certain concepts from single-variable calculus extend to multiple variables but also where these extensions fail. For example, while in single-variable…
Humberto José Bortolossi
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22
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19 answers
Math Proofs - why are they important and how are they useful?
My 13yr old has leapt forward in math during the pandemic. He's taking discrete math right now but is running into a bit of a wall with proofs. I have a feeling he needs to find reasons why they can be important and useful for him, and doing so will…
Agent Zebra
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votes
14 answers
Student asked me if it is necessary to simplify fractions at the end of answering a question. I'm not sure how to respond
I was just tutoring someone and we went through some sort of diagnostic test thing, when the following question came up.
Question:
Here is some information about $50$ people who took the driving test:
$18$ of the $50$ people are teenagers. One…
Adam Rubinson
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22
votes
2 answers
What does research indicate about how one should treat units in elementary school?
Background:
My friend told me that when she was in elementary school, the teacher would ask questions like "If you have $6$ apples and eat $2$ of them, how many apples do you have left?" A kid in the class answered $4$, to which the teacher replied…
Improve
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22
votes
11 answers
What interesting properties of the Fibonacci sequence can I share when introducing sequences?
The Fibonacci numbers are one of the first sequences given as examples of sequences in many calculus textbooks as they have a definition that does not obviously have a closed form and they have many real-world applications.
But that last part has…
Brian Rushton
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22
votes
2 answers
Exam philosophy
I'm curious if anyone knows of any books, studies, or other resources on the philosophy of creating and grading mathematics exams. After working as a graduate TA for 4 years and dealing with a wide variety of issues creating and grading precalculus…
icurays1
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22
votes
16 answers
Why is it possible to teach real numbers before even rigorously defining them?
In mathematics, one can hardly study any mathematical concept unless it is clearly and rigorously defined. For example, without the definition the fundamental group, it is almost impossible to teach anything serious about it.
But as far as real…
Zuriel
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22
votes
15 answers
Explaining why (or whether) zero and one are prime, composite or neither to younger children
There are lots of discussions out there about whether $1$ is a prime number (such as this one) and even about zero (such as this question, though note zero does generate a prime ideal in $\mathbb{Z}$ by the standard abuse of terminology ever since…
kcrisman
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22
votes
16 answers
Tricks for computing things in your head
There are quite a lot of tricks/shortcuts enabling doing calculations efficiently in your head. (One of them that came to me today is the "squaring a number ending with a 5" trick I wrote about in this answer). In a comment, user John Golden…
mbork
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22
votes
3 answers
Polymorphic functions in vector calculus
While teaching multi-variable calculus for the first time in a while, I came across a tricky notational point in our textbook (Thomas' calculus - I'm not sure how widespread this notation is).
When $\mathbf{r}(t)$ is a vector-valued function, our…
Henry Towsner
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22
votes
7 answers
Why are percentages part of the curriculum?
I was reviewing percentages with my son (US equivalent of 7th grade) and the more I dug into explanations, the less I could understand why they are taught.
I understand how they technically work but using them introduces, I think, a complexity…
WoJ
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22
votes
12 answers
Explaining the order of negative integers
Today I happen to have an interesting discussion with a primary school kid.
I asked him "Which is the smallest one - digit integer?" He instantly replied $-1$. I told him that he's wrong and the answer is $-9$. He then replied then "why"? He…
Heisenberg
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22
votes
3 answers
How to advise students who want to do a "Bourbaki"-style study?
There are some good students who understand a lot and are very critical. Such students tend to think that they will only understand abstract algebra if they have followed a course about logic; or they think they can only understand an analysis…
Markus Klein
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22
votes
7 answers
Good examples of functions defined as definite integrals of elementary functions?
I am writing some Calculus content, and I would like a "big list" of useful functions which are defined by definite integrals, but are not elementary functions.
Two examples of such functions are
$$
\mathrm{Erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x…
Steven Gubkin
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21
votes
3 answers
Students problems with reasoning, not exactly math
Consider the following problem:
Maria always buys ice-cream when she goes to the beach. She bought ice-cream today. So, she must have gone to the beach.
Obviously this statement is wrong. Maria could have gone to other place and bought an…
Mark Messa
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