All Questions
698
questions
74
votes
11
answers
11k
views
Whence the "everything is linear" phenomenon, and what can we do about it?
$$ \color{red}{(a+b)^2 = a^2+b^2}$$
$$ \color{red}{\sqrt{x^4+y^4} = x^2+y^2} $$
$$ \color{red}{e^{t^2+C} = e^{t^2}+e^C}$$
I've observed this phenomenon -- wherein, implicitly, students say, "...
- 10.6k
77
votes
6
answers
8k
views
Issues with "equals", where does this come from and how do I combat it?
An issue I see with students a lot is abuse of the equals sign. For example, one problem asked "what is the degree of the polynomial: $\text{polynomial}$?", and I got answers like "$\text{polynomial}=...
user5108
86
votes
13
answers
11k
views
How to assign homework when answers are freely available or attainable online?
I find that making homework meaningful is becoming increasingly challenging. Let us suppose that I am planning for next semester's first-semester or second-semester calculus course at my university.
...
- 2,115
60
votes
4
answers
6k
views
Is it worth grading calculus homework?
I am a young math educator. I've TAed four semesters of calculus for various instructors. Some instructors have required me to grade selected problems in homework sets. Another required me simply to ...
- 842
43
votes
28
answers
11k
views
Good, simple examples of induction?
Many examples of induction are silly, in that there are more natural methods available. Could you please post examples of induction, where it is required, and which are simple enough as examples in a ...
- 12.2k
28
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
13
votes
3
answers
1k
views
In what curricula are "rectangles" defined so as to exclude squares?
Most contemporary curricula define the word "rectangle" inclusively, so that all squares are automatically rectangles. Are there curricula in which this convention is not followed? That is,...
- 574
26
votes
9
answers
2k
views
Teaching students to find and correct their own errors
Many students have a fairly good grasp of the topics they are learning but fall down because they miss fatal errors in their work. Some don't check for errors at all, while many simply can't find them....
- 8,953
19
votes
3
answers
1k
views
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 ...
- 5,955
18
votes
1
answer
2k
views
How much time to spend on a single question?
When I was self-studying as an undergraduate, I would spend up to two weeks working on a single problem or trying to understand a proof in Rudin's Principles of Mathematical Analysis. I realize now ...
- 283
17
votes
9
answers
4k
views
Was there an SMSG (New Math) "Algebra 2" text?
This question has been kicking around in the back of my head for a couple of years, but the impetus to post it now came from reading the related question at When did the American school system's ...
- 17.2k
51
votes
15
answers
12k
views
How can we help students learn how to read their textbook?
In most secondary and early undergraduate courses, students purchase expensive and carefully-written textbooks. These textbooks contain, roughly, three things:
Exercises and Answers
Reference ...
- 20.8k
32
votes
14
answers
2k
views
Revisiting topics from previous courses [closed]
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.)
...
- 11.5k
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
18
votes
12
answers
2k
views
Mathematical problems for preschoolers
What are some mathematical problems that are feasible for preschool children to stimulate their intellectual development?
There are multiple natural laws that are not apparent to them, for example:
...
- 8,817
82
votes
21
answers
25k
views
Why are induction proofs so challenging for students?
This forum already has many
good, simple examples of induction proofs, a great resource.
As I am soon to teach induction for the $n^\textrm{th}$ time—this time to some perhaps under-prepared ...
- 29k
71
votes
17
answers
10k
views
How shall we teach math online?
Many universities, including mine, are now requiring we teach our courses online because corona. How shall we do this? Let’s brainstorm here.
Some challenges:
My school provides limited online ...
49
votes
13
answers
5k
views
Should we avoid indefinite integrals?
I am very uncomfortable with indefinite integrals, as I have a hard time giving them a precise sense that matches the way they are written and the usual meaning of other symbols.
For example, when ...
- 8,989
36
votes
26
answers
3k
views
What are some great books for exploring mathematics? (not kids' books and not textbooks)
People often think of math as facts and procedure - dry stuff. But it is so much more, even at basic levels. What books about mathematics have you been inspired by? There are some real treasures out ...
- 19.4k
30
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 ...
- 1,307
20
votes
5
answers
748
views
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?
<...
- 321
15
votes
4
answers
1k
views
When should I say "nothing is as it seems"?
"Intuition" is the best friend and worse enemy of mathematicians! Sometimes using intuitive arguments could be very helpful to understand the nature of a phenomenon. Many of the deepest true ...
user230
13
votes
7
answers
3k
views
How can you be perfect at maths (highschool)?
I'm in my last year of highschool. And I'm aiming for a perfect grade in maths. The problem is that this year is the hardest year of maths I have ever faced in my entire life. Especially derivation ...
- 139
84
votes
19
answers
29k
views
What is a good handwriting font for mathematics?
My students frequently mix up my $t$'s with my $+$'s and my $y$'s with my $4$'s.
What is a good handwriting font for distinguishing these and other easily confused symbols?
- 11.3k
63
votes
14
answers
3k
views
Encouraging class participation
I teach calculus to college students, and find it very difficult to get them to speak up in class when I ask questions, or when I'm trying to get a pulse for how much they understand. I think ...
- 2,113
44
votes
18
answers
3k
views
How to teach logical implication?
One of the challenges of undergraduate teaching is logical implication. The case by case definition, in particular, is quite disturbing for most students, that have trouble accepting "false implies ...
- 8,989
38
votes
4
answers
5k
views
Rings before groups in abstract algebra?
The default approach to teaching abstract algebra seems to be groups first, then rings. However, occasionally a textbook pops up (e.g. Childs' A Concrete Introduction to Higher Algebra, Hodge et al's ...
- 4,586
35
votes
24
answers
6k
views
Imbuing a six year old with a sense of mathematical wonder
My six year old started school a few months back and he's loving it. This first year is more about social skills than anything academic and I like that approach. But we're spending some time at home ...
- 570
29
votes
7
answers
45k
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 ...
- 301
28
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, ...
- 3,825
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
21
votes
4
answers
1k
views
How can I choose a free calculus textbook?
As I have been recently informed, it is a good idea to consider free calculus textbooks for college and university courses.
However, this feels risky to me, because:
I don't know anyone who is using ...
- 20.8k
20
votes
2
answers
4k
views
Example "bad proofs"?
As a sidetrack in this question it came up that it is important to have students read texts (in particular proofs) critically. As examples it is nice to have correct proofs at hand (presumably in the ...
- 12.2k
16
votes
4
answers
2k
views
Teaching a very enthusiastic and bright 5 year old
I was asked to give extra lessons to a 5 year old boy who loves math (he says he likes 3 sports: football, swimming and math). However, I have never taught at this age and I am unfamiliar with the ...
- 899
13
votes
6
answers
3k
views
Why do some students struggle so much with fractions?
I read on multiple web pages something that implies that that some students really struggle with fractions but I could never find a detailed explanation of why. This question is different from Are ...
- 501
13
votes
4
answers
920
views
Wiggins' question #12
There's an interesting read: Conceptual Understanding in Mathematics by Grant Wiggins. In that text the author proposes "a test for conceptual understanding" which should be given "to 10th, 11th, and ...
- 8,817
12
votes
1
answer
395
views
Where can I find primary sources from the New Math movement in the 60s?
I'm interested in learning about the New Math movement from a historical perspective. I've located some secondary sources about the topic, mainly parodies, highly critical restrospective articles, or ...
- 1,693
10
votes
3
answers
3k
views
Learning counting and addition: fingers or in their head?
My kids Cindy (5 1/2, in Kindergarden) and Jamie (4, in pre-K) are taking some math enrichment classes. I have been telling and teaching them to count items and do single-digit arithmetic in their ...
- 363
84
votes
15
answers
16k
views
Should I design my exams to have time-pressure or not?
Is it better to design an exam with fewer questions and relaxed timing or with more questions and a resulting time-pressure?
One the one hand, it seems that students who really know the stuff will ...
- 4,046
56
votes
15
answers
16k
views
Student: Why not use a calculator?
The kid I am teaching math (subtraction for large numbers right now) just said this is all too easily done by a calculator, why don't we use it?
Well, I did tell him that you can only learn more ...
- 1,145
56
votes
24
answers
69k
views
Optimization problems that today's students might actually encounter?
Our students are not fencing in farm fields, cutting wires and folding them, or designing windows, so they are often uninspired by the optimization problems we give them. They seem like something that ...
- 20.8k
50
votes
3
answers
10k
views
How do blind people learn mathematics?
I am interested in how blind people learn mathematics at any level, but particularly before college. Math is often taught using a lot of visualization; how does this work with blind people?
My ...
- 777
45
votes
18
answers
12k
views
How to explain the flipping of division by a fraction?
This question is inspired by @DavidButlerUofA's discussion of
"$\div \frac{2}{3}$ as $\times \frac{3}{2}$" in
"Are fractions hard because they are like algebra?"
Q. How can one best convey to ...
- 29k
45
votes
16
answers
32k
views
How is calculus helpful for biology majors?
It's common for majors in biology to take calculus courses, and many calculus textbooks (and calculus professors) try to cater to these students by including applications to biology.
My question is, ...
- 8,119
40
votes
5
answers
3k
views
Effects of early study of advanced books
Context:
There was recently a question on Math.SE: Inferior to Other Younger and Brighter Kids which starts as follows:
I'm a high school student (Junior/Grade 11) and I'm currently studying ...
- 8,817
38
votes
10
answers
2k
views
Reasons for (not) distinguishing $f$ from $f(x)$
Formally, if $f$ is a function, $f(x)$ is a value. So for instance, $f$ can be continuous, but not $f(x)$.
In teaching at school and university, notation is quite often mixed up, e.g. the function is ...
- 4,741
34
votes
5
answers
6k
views
What fraction of the population is incapable of learning algebra?
In the comment thread of this academia.SE question, the following generated some strong reactions:
My very different (community-college) perspective is that the math discipline will end up as a ...
user507
34
votes
3
answers
3k
views
How to cure students from the idea that root and squaring are identity operators?
I tutor high school algebra and I’ve noticed that a lot of my students don’t seem to understand what they’re doing when they “convert” between different ways of writing numbers involving perfect ...
- 451
31
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 ...
- 1,127
31
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 ...
- 11.3k