All Questions
3,686
questions
33
votes
5
answers
2k
views
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 ...
- 10.7k
32
votes
13
answers
13k
views
What is the mathematical value of children learning and being tested on Roman numerals?
My 11 year old child recently took an important numeracy test. One of the questions required her to know that M = 1000 in Roman numerals. This made me very angry: I could not see how this relatively ...
- 1,317
32
votes
11
answers
4k
views
What is the current school of thought concerning accuracy of numeric conversions of measurements?
I posted this question earlier today on the Mathematics site (https://math.stackexchange.com/q/3988907/96384), but was advised it would be better here.
I had a heated argument with someone online who ...
- 925
32
votes
7
answers
10k
views
Is it harmful to use the word "Cancel"?
Elsewhere, among a group of high school math teachers, I encountered a discussion of the term 'cancel'. Most (>20) people in the discussion had very strong feelings about why the term should be ...
32
votes
11
answers
31k
views
Why should kids learn how to use a compass and straightedge, and not rely on a drawing program?
I am curious why it is necessary for people to learn how to use compasses and straightedges in geometry, and not just rely on a drawing program.
I have a couple ideas, but I might be missing ...
- 421
32
votes
13
answers
7k
views
Is Euclid dead? or Should Euclidean geometry be taught to high school students?
Apparently Euclid died about 2,300 years ago (actually 2,288 to be more precise), but the title of the question refers to the rallying cry of Dieudonné, "A bas Euclide! Mort aux triangles!" (...
32
votes
8
answers
2k
views
How to solve the problem of Wolfram Alpha?
I teach to predominantly non-majors in college algebra, precalculus, and calculus.
How can one possibly incentivize or rationalize assigning practice problems outside of class when this software is ...
- 423
32
votes
20
answers
6k
views
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 ...
- 1,147
32
votes
10
answers
2k
views
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 ...
- 11.7k
32
votes
5
answers
2k
views
The best way to introduce trigonometric functions in a rigorous analysis course
This is something I have always had issues with. Generally, three approaches are used:
The geometric path: this follows the standard way how you would introduce these functions in school. The problem ...
- 4,229
32
votes
6
answers
2k
views
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 ...
- 4,980
32
votes
6
answers
894
views
Alternatives to University Lectures: Non-lecture Mathematics Classes
I am looking for resources for designing undergraduate mathematics classes that are not lecture-based. (Bonus points if the design is for an introduction to proof course).
For example, Robert ...
- 3,800
32
votes
6
answers
3k
views
Allowing nonstandard mathematical language and/or notation
How important is enforcing standard mathematical language and/or notation?
Today, I was questioned by a writing instructor as to how vital it is to correct students when they explain something using ...
- 9,184
32
votes
3
answers
1k
views
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 ...
user507
31
votes
20
answers
8k
views
‘Lies to children’ in mathematics and statistics education
In teaching, we sometimes necessarily oversimplify concepts. Terry Pratchett famously referred to this as Lies to children:
A lie-to-children is a statement that is false, but which nevertheless ...
- 419
31
votes
10
answers
10k
views
Is this homework problem on counting triangles within a 4x4 grid too vague?
My six-year old daughter was given this maths problem for her homework:
Given a regular square grid of 4 × 4 dots, how many different triangles with one dot in the middle can you draw?
We were ...
- 1,317
31
votes
11
answers
5k
views
What are the arguments for and against learning the multiplication table by heart?
I think, a lot of students are bothered by learning multiplication tables by heart, in particular when it comes to numbers greater than 10.
Why should one learn (or not learn) these things by heart?
- 9,388
31
votes
10
answers
11k
views
Why do we teach even and odd functions?
I've been either a student or an instructor in Precalculus or Calculus 1 at about 6 institutions now, and teaching the definition of even functions (where $f(-x) = f(x)$) and odd functions (where $f(-...
- 499
31
votes
9
answers
7k
views
What to do with students who think they "already know it," but actually don't?
Many students take calculus or algebra courses in high school, then later take college courses of the same name. There are various reasons for this, but in most cases the students in a college ...
- 21.2k
31
votes
8
answers
3k
views
How to react to students saying that they are allergic to applied mathematics?
I'm working in the field of applied mathematics (optimization and numerics) and I meet a lot of students saying that they are allergic to applied mathematics or that they hate it or some quotes like "...
- 9,388
31
votes
7
answers
2k
views
Mathematical education by country
Depending on the university, there are always slight differences in the syllabus and the structure of the standard material undergraduate students learn.
But I also noticed that undergraduate ...
- 419
31
votes
4
answers
7k
views
What websites allow students to purchase solutions to problems?
I am a college instructor who's just had an outbreak of academic dishonesty connected to students posting take-home exam problems on a platform called Chegg. Chegg collects a membership fee from ...
- 651
31
votes
7
answers
16k
views
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 ...
- 4,086
31
votes
1
answer
5k
views
Which product of single digits do children usually get wrong?
(I was inspired by the comments in this answer to ask this question.)
I have some multiplication table cards from Kumon that have a list of commonly mistaken multiplications: $7\times 8, 4\times 8, 11\...
- 10.8k
31
votes
7
answers
46k
views
Software to create video tutorial of mathematics topics
I came across many video tutorials on youtube regarding mathematics.
I found this video amazingly simple to understand for students.
I want to know about the tool/software used for it or similar ...
- 321
31
votes
4
answers
4k
views
Books about elementary mathematics written like a good undergraduate textbook
I've never seen any really good expositions of elementary mathematics (middle school or earlier). A good college-level textbook, written for people with an interest in mathematics, reads like a novel ...
- 1,337
31
votes
4
answers
990
views
How to tactfully discourage casual, implicit disparagement of mathematics
I volunteer with a group that provides tutoring to kids from grades nine through twelve. The included kids have been determined to be 'at risk of not graduating high school'. Of course, many of the ...
- 4,980
30
votes
13
answers
8k
views
What do you say to students who want to apply Banach-Tarski theorem in practice?
Once when I was talking about Banach-Traski theorem (paradox) I said:
OK! This is Banach-Tarski's theorem which is against our intuition but provable from our intuitive axioms! It says you can ...
user230
30
votes
9
answers
5k
views
Teaching by Slides, Yes or No?
Mathematicians use (Powerpoint/Beamer) slides for their lectures. My question is about using slides for teaching math. There are several positive and negative arguments about teaching by slide show. e....
user230
30
votes
9
answers
9k
views
Can mathematics be learned by ONLY solving problems?
Here is the concept:
Student is presented with a problem. He/she may not even understand what is being asked, or may attempt.
Students reads a solution to the problem. In it there may be ...
- 625
30
votes
8
answers
4k
views
Is there a good age/level to start learning mathematical proofs?
I know from my experience I learnt proofs myself way before I learnt them in school and I felt it gave me a far better understanding of math.
What is a good point to start learning proofs? what are ...
- 401
30
votes
6
answers
3k
views
f(x+h) in the difference quotient
When teaching students how to compute the difference quotient in a precalculus or calculus class, we need them to evaluate the expression
$$\frac{f(x+h) - f(x)}{h}$$
for various simple functions, like ...
- 21.2k
30
votes
8
answers
2k
views
Good motivation for the introduction of Lebesgue integral?
When students take a course on real analysis, they have likely learned about Riemann integrals.
What is a good motivation why they have to learn a new way to integrate?
A student don't want to hear ...
- 9,388
30
votes
10
answers
5k
views
Getting students to actually read definitions
I'm teaching a second year "Introduction to Theoretical Computer Science" course, and one of the skills/habits I've tried to instill in the students is to actually read definitions, take ...
- 871
30
votes
11
answers
5k
views
Are the words "easy," "basic," "clearly," "obviously," etc., ever helpful?
This is a very basic fact from...
It then clearly follows that...
Obviously, we have...
The proof is trivial...
I could add plenty of other phrases to this list that mathematicians are prone to use ...
- 2,163
30
votes
7
answers
2k
views
Good definition for introducing real numbers?
In the first lectures about calculus/analysis, you should introduce real numbers. Let's assume students know that rational numbers are.
What are the advantages or disadvantages in the different "...
- 9,388
30
votes
12
answers
8k
views
How to give my students a straightedge instead of a ruler
I'm having a "challenge" in my geometry classes getting students to avoid using rulers as measuring devices in constructions. As natural as that usage is, they're only supposed to use them to connect ...
- 5,599
30
votes
7
answers
1k
views
When $-x$ is positive
This recent question reminded me of a question: this year several students expressed concern about the expression $\sqrt{-x}$, on the grounds that it must be undefined because $-x$ is a negative ...
- 11.6k
30
votes
6
answers
3k
views
What are non-math majors supposed to get out of an undergraduate calculus class?
When I teach a course for math majors (an analysis course out of Rudin, say), I have a more or less clear idea of what the students should take away from the course, having been in their shoes some 15 ...
- 301
30
votes
4
answers
4k
views
Open-Source Math Textbooks
It seems to me that an open-source model could work quite well for textbooks, with issues being raised by the users of the book and different forks of the project being created for different ...
- 21.2k
30
votes
3
answers
553
views
What Math(s) Ed literature is accessible to the working math(s) educator?
This site is - as far as I'm aware - for what I would term working maths educators. That is, on the whole the users of this site are not researchers in mathematics education. Rather, we are the ...
- 3,626
30
votes
6
answers
27k
views
Early vs. late transcendentals
There seem to be two approaches to calculus education:
Early transcendentals: introduce polynomials, rational functions, exponentials, logarithms, and trigonometric functions at the beginning of the ...
- 646
30
votes
6
answers
3k
views
How to motivate an adolescent who has fallen behind in conceptual development?
I tutor a 16 year old girl. As far as I can tell, she has average talent and interest in math.
However, her knowledge of math is that of a 10 year old or even below. She knows the basic operations on ...
- 1,242
29
votes
17
answers
7k
views
Examples of Innumeracy
I read Innumeracy by John Allen Paulos and would like to share more up-to-date and relevant examples of innumeracy to motivate my grade 8, 9 & 10 students. Are there any websites, blogs, books, ...
- 3,875
29
votes
15
answers
5k
views
How do I teach algebra?
I find that soon I'll be working with high school students that are struggling with math. In particular, we'll be talking a lot about algebra and some basic trigonometry. The latter I have experience ...
user37
29
votes
7
answers
6k
views
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 ...
- 945
29
votes
10
answers
2k
views
What are argument one can give to students on the definition $0^0$?
From high school to introduction courses in university, the expression $0^0$ is some (psychological) problems. High school students just apply it to their calculator and either the result is $1$ or ...
- 9,388
29
votes
6
answers
7k
views
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 ...
- 884
29
votes
17
answers
28k
views
How does one explain that transformations 'inside' a function operate in the opposite direction than intuition suggests?
Consider a real function $f(x)$ and imagine its graph in the plane. Then the graph of $f(x+2)$ is simply the graph of $f$ shifted to the left 2 units while the graph of $f(x-2)$ is that of $f$ shifted ...
user1916
29
votes
6
answers
2k
views
What holds your students back in Calculus?
I teach Precalculus to high school kids, and I know a lot of you all teach Calculus.
What are some issues that your students have in Calculus classes that
you wish had been addressed in a ...
- 1,721