All Questions
3,559
questions
25
votes
7
answers
4k
views
Why are we so careful in saying that dy/dx is not a fraction?
Calculus instructors are mostly very careful to explain that $\frac{\mathrm{d}y}{\mathrm{d}x}$ is not a fraction, and multiplying both sides of an equation by $\mathrm{d}x$ is nonsense, wrong, or evil....
- 20.4k
25
votes
6
answers
1k
views
How to present $\Bbb Z/n\Bbb Z$ to highschool level audience
I have the oportunity to talk to a highschool class about mathematics, the topic I got to present are the integers modulo $n$, ie, $\Bbb Z/n\Bbb Z$ , however I don't want to be very heavy and formal, ...
- 815
25
votes
4
answers
3k
views
Common Core, threat or menace? Or maybe ok after all?
I have a background in math but no contact with secondary education or kids. I hear all sorts of stories ... horror stories mostly ... about the Common Core math curriculum in the USA. Then I hear ...
- 351
25
votes
8
answers
2k
views
"We already passed that course!" How to overcome this?
I've heard the phrase "we passed that course already!" too many times when asking for e.g. the derivative of a simple rational function or a simple integral, getting blank stares, and digging deeper. ...
- 12.2k
25
votes
6
answers
3k
views
How do you go about writing your own lecture notes for a new course?
In advanced graduate courses, there are often no textbooks on the material being covered, and sometimes no good introductory material at all. Even when there are textbooks, many professors prefer to ...
- 11.3k
25
votes
8
answers
4k
views
How can I convince someone to use a calculator and not worry about the mechanics too much?
I'm trying to help someone pass their final exam (analysis of functions) so they can graduate high school and move on to college. (Not a teacher, just another student, currently in high level math)
...
- 353
25
votes
8
answers
2k
views
How to write like a mathematician?
I learned to do math proofs in college. But recently I have begun studying more advanced math books and I've noticed some mathematicians frequently make assumptions that I don't. When I write proofs, ...
- 1,046
25
votes
6
answers
4k
views
Students confusing "object types" in introductory proofs class
In my intro to proofs (and discrete mathematics) class, I see a common mistake where students make nonsensical statements because, for lack of a better term, they confuse the types of the mathematical ...
- 361
25
votes
9
answers
2k
views
How can mathematics educators encourage innovation and creativity?
Almost by definition, innovation requires that things be done differently than established custom has it, and comes from the young more often than from the old.
In a field as old and established as ...
- 1,745
25
votes
6
answers
886
views
Too much motivation?
This is something that I felt like was difficult for me in some classes, especially lower division differential equations and linear algebra classes. I know professors want to motivate certain topics ...
user5108
25
votes
4
answers
2k
views
Lesson plan to self-teach real analysis to student with comp-sci background
For my background, I'm a software engineer currently studying for his master's degree in information security. But when that's all done, I plan on going back to mathematics to keep me busy. But with ...
- 393
24
votes
13
answers
19k
views
Ideas for high school pure maths projects
I am thinking of giving my high school students some pure maths projects to do. It is a lot easier to think of some interesting stats projects but not in pure maths. The students' maths background are ...
- 349
24
votes
7
answers
3k
views
Why do we care about multiple proofs of the same theorem?
I am teaching a math appreciation course to high school students who are approximately 17 years old, in their last year of high school, and who do not believe they will choose a STEM major in ...
- 543
24
votes
5
answers
1k
views
A Series of Unfortunate Examples!
All of us know, when teaching, the "right" choice of examples is important. Though, making the "right" choice is one of those things that are easier said than done. Here is the ...
- 4,330
24
votes
2
answers
4k
views
Are there science-backed effective teaching strategies?
As a math teacher, I am always trying to self-assess my teaching methods. I am trying a lot of different methods but I would like to organize my study on the subject without weighing too much on the ...
- 673
24
votes
8
answers
2k
views
What are the historical reasons for the hostility against standardized testing in the US?
NB. Some answers appear to be for a question I did not ask, namely, "Why is standardized testing bad?" Indeed, these answers tend to underscore the premise of my actual question, which can ...
user378
24
votes
6
answers
896
views
How can you explain to students that they should not use the same variable in an integrand and in the limits of integration simultaneously?
When teaching Calculus, one thing that many teachers emphasize is that the variable of integration is a `dummy variable' that is unimportant.
Around the same time, we introduce integrals with ...
- 11.3k
24
votes
5
answers
2k
views
Should we "program" calculus students, like the physicists seem to want us to?
If it is true that we first learn formalism...how to do things that we don't understand, should we regard teaching students mathematics as programming dumb machines with formal rules (to the greatest ...
- 5,955
24
votes
5
answers
4k
views
When did US mathematics programs start failing to prepare incoming students for books like "Baby" Rudin?
I've seen in a lot of questions about "which textbook to use for intro analysis", and inevitably Rudin's Principles of Mathematical Analysis comes up, with the (almost cliche) rejoinder that "today's ...
user5661
24
votes
8
answers
2k
views
Counterintuitive consequences of standard definitions
Let me motivate my question with the following situation. While teaching the concept of continuity, I usually start with motivating the concept. Then, when we see that there is an important and ...
- 4,109
24
votes
3
answers
2k
views
The impact of dyslexia on learning mathematics, and available resources
I have always loved the beauty of mathematics and physics. However I'm severely dyslexic and find it hard to keep numbers in my head, any more than 4 numbers at a time and they melt together and lose ...
- 341
24
votes
2
answers
1k
views
Student Poisoned Experience with Math
I’d like to start off with saying I am not a teacher so I don’t know how much of this is already trying to be addressed in math education throughout the world.
In talking to and tutoring fellow ...
- 520
24
votes
4
answers
6k
views
How is teaching calculus in high school different from teaching calculus in college?
I've taught calculus in college for five years, and it's always interesting to see students coming in who already had calculus in high school. Many of them do very well, and don't even seem like they ...
- 11.3k
24
votes
1
answer
665
views
Tutoring elementary student who reverses left and right
I am a volunteer math tutor. My student is in fifth grade. He was evaluated in August and found to be at the first grade level in math. (His reading is very close to grade level.)
He has epilepsy, ...
- 725
24
votes
1
answer
728
views
Is there a Piagetian age at which proofs can be comprehended?
I am wondering if there is literature on the developmental age
(pre-adolescent?, adolescent?) at which the notion of a "proof"
can be understood? I am less interested in mathematical proofs
and more ...
- 28.9k
23
votes
23
answers
5k
views
How can I explain why we need proofs to someone who has no experience in mathematical thinking?
I know someone I really like, but sadly, that person has absolutely no experience in math or mathematical thinking above third grade mathematics (+, - are fine, but division already makes problems). ...
- 331
23
votes
10
answers
7k
views
Why would you teach Calculus before teaching Real Analysis?
Let's assume our students are actual aspiring mathematicians.
Why would we introduce our students to Calculus rather than Real Analysis?
After all, "Calculus is a subset of Real Analysis". He will ...
23
votes
17
answers
3k
views
Examples of Mathematical Slang
Unless you have taught highschool algebra in Iran, you could not make sense of the phrase: Elephant and Teacup Identity! This is what teachers use to refer to the following identities:
$ (a+b)(a^2-...
- 4,330
23
votes
9
answers
4k
views
Why do we introduce the notion that triangles are "congruent" instead of just saying that they are "the same" or "equal"?
The assumed age of the students is 10-15 years old.
What is the danger in saying that two triangles are "the same" or "equal" instead of saying that they are congruent? It seems to ...
- 1,811
23
votes
13
answers
5k
views
What is a free and simple 3D plot software for students?
I need any plot software on Linux or Windows that my students should use it for plotting 3D functions.
I want introduce any software that be free and useful for bachelor students.
- 633
23
votes
11
answers
3k
views
Redundant zeros
How to convince a middle school student that $0.50=0.5=0.500=\cdots$?
I used the fact that $0.50=\frac{5}{10}+\frac{0}{100}=\frac{5}{10}=0.5$ but that far from intuitive.
Then I tried to explain ...
- 1,468
23
votes
6
answers
2k
views
What is the proper way to ask a "find the domain" question?
A function is not really a function unless it's defined everywhere on its domain. So consider these three questions:
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the square root function $f(x) = \...
- 341
23
votes
10
answers
4k
views
Is 'estimating' still considered a valuable skill?
I was with a 2nd year high school class, preparing for our (US) state's standardized test. I asked the class how they would solve this, and they flipped through the sheets to find
$$V=\frac{1}{3}\pi ...
23
votes
5
answers
5k
views
What is the proper verb for "doing" an integral?
It's time to write exams, and when writing in committee we often discover differences in usage between various instructors. Here's an example I noticed today.
What is the proper verb to use in a ...
- 744
23
votes
10
answers
4k
views
How to encourage young student to think in unusual ways?
I tutor a young girl aged 11 (grade 4).
She is doing OK for her age, but I have observed that she has a tendency for rigid ways of thinking. She is usually more inclined to follow rules and stick to ...
- 1,213
23
votes
6
answers
2k
views
What are some good rules for handling student questions during exams?
For example, I gave an exam earlier today with a problem that ended in the sentence
Use the chain rule to find $(f\circ g)'(3)$.
During the exam, one of the students asked me what the circle ...
- 8,079
23
votes
4
answers
3k
views
How to Teach Adults Elementary Concepts
I've recently taken on the task of helping out in my school's Math Center. The courses I assist in range from Algebra to Calculus. While I'm younger (in my 20's), most of the students at the school ...
- 341
23
votes
7
answers
5k
views
Repeated addition: standard notation?
My daughter showed me the picture below, which came from 9GAG. It shows a question on an exam asking the student to "use the repeated addition strategy to solve: 5 x 3." The student answered "5+5+5" ...
- 10.7k
23
votes
5
answers
1k
views
Why are the contents of contest maths so different from contents of degree-level maths?
I wonder why topics examined in high school math contests are so different from the maths learned by those who are seriously studying a math major at a university. Firstly, contests like IMO, ARML, ...
- 1,635
23
votes
6
answers
1k
views
Practical experience with teaching differentials in freshman calc?
There is a well known essay by Dray and Manogue which argues that differentials should be brought back into freshman calculus, and that we shouldn't worry too much about choosing a specific way of ...
user507
23
votes
4
answers
605
views
Keeping quicker students engaged and interested throughout a course
In a college math course one is bound to find a fairly broad range of students in terms of their quickness in understanding the material. This is due to many reasons, including differing mathematical ...
- 2,093
23
votes
4
answers
2k
views
Non-answerable questions on exam: What to do?
What is a good strategy when you realize (e.g. while grading the exam) that a question on an exam was incomplete/wrong?
More concretely:
If it is decided that additional points should be given:
How ...
- 9,122
23
votes
5
answers
726
views
Do all high school students need the same 3-year sequence of math courses?
I continue to be troubled by the amount of symbolic manipulation in a typical Algebra 2 course. Once a student has completed Algebra 1 and Geometry, shouldn't there be another option for them if a ...
23
votes
6
answers
3k
views
What is a good way to explain the Lebesgue integral to non-math majors?
A few days ago I had my last discussion session on probability theory as a TA. In the end I asked students to ask me questions as this is the last class. One of the student asked me about the (real) ...
- 391
23
votes
3
answers
11k
views
Why aren't logarithms introduced earlier?
I've always been puzzled by the unequal treatments of square roots and logarithms in school mathematics. In the United States, most students know what a square root is before they enter high school (...
- 8,079
23
votes
4
answers
2k
views
How should one tutor a student in undergraduate real analysis?
I am an undergraduate. Other undergraduates sometimes ask me to tutor them in an introductory real analysis course that covers the equivalent of the first half-dozen chapters of Rudin's Principles of ...
- 519
23
votes
3
answers
542
views
How to teach perseverance?
I have found that when I give problems that require multiple steps or ideas to solve, students often give up quickly and come to office hours begging for hints. Sometimes I break up such problems ...
- 6,320
23
votes
5
answers
548
views
Any support for mathematical "learning types?"
Back when I was an undergrad calculus TA, I participated in a general TA training class. We were taught to be mindful of different "learning types," such as visual learners, audio learners, and ...
- 2,115
23
votes
2
answers
1k
views
Should geometric algebra be presented early on in undergraduate education?
The Cambridge University GA Research Group’s website along with the “Geometric Calculus R & D Home Page” should serve as a good introductions to geometric algebra, along with the Wikipedia ...
user89
23
votes
1
answer
735
views
Has Benezet's teaching experiment ever been reproduced?
In the 1930's, Louis Bénézet, a superintendent of several schools in New Hampshire made the interesting experiment of teaching no formal arithmetic until grade 6:
In the fall of 1929 I made up my ...
- 1,640