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68
votes
11answers
8k 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, "...
71
votes
6answers
6k 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}=...
59
votes
4answers
4k 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 ...
36
votes
25answers
9k 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 ...
85
votes
13answers
10k 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. ...
27
votes
15answers
4k 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 ...
25
votes
9answers
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....
18
votes
3answers
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 ...
17
votes
9answers
3k 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 ...
19
votes
1answer
1k 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 ...
50
votes
14answers
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 ...
24
votes
9answers
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 ...
29
votes
14answers
2k views

Revisiting topics from previous courses

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.) ...
34
votes
26answers
2k 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 ...
18
votes
12answers
1k 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: ...
14
votes
4answers
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 ...
12
votes
3answers
902 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,...
80
votes
17answers
23k 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?
68
votes
17answers
9k 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 ...
29
votes
4answers
3k 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 ...
46
votes
10answers
4k 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 ...
30
votes
5answers
38k 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 ...
28
votes
17answers
6k 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, ...
19
votes
2answers
3k 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 ...
20
votes
5answers
658 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? <...
13
votes
4answers
777 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 ...
12
votes
7answers
2k 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 ...
10
votes
3answers
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 ...
12
votes
1answer
269 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 ...
72
votes
21answers
20k 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 ...
82
votes
15answers
14k 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 ...
54
votes
24answers
55k 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 ...
63
votes
14answers
2k 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 ...
43
votes
17answers
9k 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 ...
40
votes
15answers
2k 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 ...
20
votes
4answers
9k views

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 ...
38
votes
5answers
2k 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 ...
52
votes
14answers
14k 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 ...
31
votes
3answers
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 ...
13
votes
7answers
888 views

Where do you find math tasks?

What the title says. I got my degree in math last year and now I'm working on a master's independent study project through my education department finding math tasks for K-12 curriculum aligned with ...
44
votes
16answers
27k 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, ...
33
votes
4answers
4k 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 or Hodge et al'...
30
votes
20answers
4k 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 ...
28
votes
4answers
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 ...
22
votes
4answers
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 ...
14
votes
4answers
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 tought at this age and I am unfamiliar with the ...
47
votes
3answers
7k 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 ...
31
votes
10answers
1k 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 ...
19
votes
4answers
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 ...
7
votes
10answers
2k views

How to explain fractions to 7 year old kid

I am finding it difficult to convince my kid that 2/4 and 1/2 are same. As per the kid, 2/4 is more than 1/2 since in first case the boy gets 2 candies out of 4 and in second case he gets 1 candy out ...

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