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24 votes
5 answers
785 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 ...
Jennifer Silverman's user avatar
14 votes
3 answers
323 views

Resources for teaching Riemann integration in higher dimensions and on submanifolds, with view toward Stokes' theorem?

Question I am looking for suggestions of good resources (textbooks or lecture notes preferably) for teaching Riemann integration in $\mathbb{R}^d$ with $d\geq 2$ and also for Riemann integration ...
Willie Wong's user avatar
  • 2,731
15 votes
3 answers
5k views

What is a good physical example of Stokes' Theorem?

I find it useful to give physical examples of theorems, especially in vector calculus - for example $\nabla f$ being the direction of maximum ascent on a surface $f$. What is a good example for ...
mirams's user avatar
  • 253
15 votes
6 answers
1k views

Factoring quadratic polynomials

In Secondary education in Australia, the general outline for introducing techniques to solve the quadratic equation $$ x^2+bx+c=0 $$ is first to introduce the idea to find two numbers $p$ and $q$ such ...
Daryl's user avatar
  • 796
45 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 ...
Benoît Kloeckner's user avatar
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?
Markus Klein's user avatar
  • 9,388
12 votes
4 answers
305 views

It is good to have old exams (with solutions) published?

What are the arguments in publishing or not publishing old exams? If yes, should they contain also a solution? Would you publish them directly after that particular exam or publish your old one's just ...
Markus Klein's user avatar
  • 9,388
17 votes
5 answers
478 views

Amount of concrete calculations on board?

Imagine that you are teaching a high school class in the last years of high school, an undergraduate class in university, or you are a tutor of a small group at university. Should one provide ...
Markus Klein's user avatar
  • 9,388
15 votes
4 answers
2k views

Why is rounding half away from zero the only method taught?

Rounding to the nearest even digit is very practical in a lot of areas (e.g. statistics, accounting), but is never taught anywhere from elementary school to college. Even in R, the go-to statistics ...
jpd527's user avatar
  • 499
9 votes
4 answers
274 views

Using original texts while introducing new concepts in class

I'm still a undergrad math student, and my experience in education in math is very limited, however I've been lucky enough to meet teachers that encourage students who are interested in teaching, like ...
Ana Galois's user avatar
4 votes
1 answer
144 views

Morphism-oriented definitions

For some objects there are alternate definitions, which are "morphism-oriented". To give some examples, there are two definitions of a prime number: $p$ is prime if it is greater than $1$ and has no ...
dtldarek's user avatar
  • 8,907
7 votes
1 answer
284 views

Is there a tag/competence classification for mathematics education?

I am looking for both a course hierarchy of mathematics education (for example, Galois theory is part of abstract algebra) and a representation of all competences involved in learning mathematics (...
Jill-Jênn Vie's user avatar
19 votes
7 answers
2k views

"Correct the following mistake"-style questions?

Does anyone have any experience giving students incorrectly "solved" math problems and asking them to identify this error? Being self-critical is one of the skills that I would like my students to ...
David Steinberg's user avatar
18 votes
4 answers
422 views

Are teaching about finding the missing member(s) of the sequences really appropriate?

I notice that in current mathematics education they always have sections teaching about finding the missing member(s) of the sequences e.g. in this way: $1,2,4,8,16$ , the next term is what? Someone ...
doraemonpaul's user avatar
37 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 ...
Sue VanHattum's user avatar
  • 20.1k
17 votes
11 answers
14k views

Looking for simple "interesting" math problems that cannot be easily solved without algebra

I often find students who dislike algebra. They prefer to work with numbers in solving problems. I believe there are many problems that are hard to solve without algebra. For example: Finding the ...
kiss my armpit's user avatar
8 votes
6 answers
417 views

On the use of calculators in elementary/high school? Computer algebra systems later on?

I know this subject is inflamatory... but I also believe it is something that requires broader discussion (and hopefully some consensus). My own position (from personal experience) is that today you ...
vonbrand's user avatar
  • 12.3k
20 votes
8 answers
1k views

What are some good mathematical applications to present in an abstract algebra course?

One of the main difficulties for a student learning abstract algebra is understanding the motivations behind concepts like groups, normal subgroups, rings , ideals etc. Also, many have difficulty ...
user774025's user avatar
19 votes
6 answers
12k views

Good examples of proof by contradiction?

In later courses on automata theory, many students just seem incapable of getting a proof that a language isn't regular right, be it using the pumping lemma (see also the many questions on the matter ...
vonbrand's user avatar
  • 12.3k
14 votes
3 answers
427 views

Do private lessons really help?

When I was in high school I explained a lot of things to my class mates. Later during the first time of my studies, I was hired as a private tutor. Although I was earning some money, I had very mixed ...
Markus Klein's user avatar
  • 9,388
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 ...
Michael Pershan's user avatar
13 votes
2 answers
156 views

Ideas for term papers in a graduate or advanced undergraduate class

In my graduate courses, I often have my students write term papers on original mathematical topics. I explain the process in this answer over at MathOverflow. It works fairly well for me. But I'd be ...
JDH's user avatar
  • 4,086
14 votes
3 answers
245 views

Is there an evidence that learning mathematics influences our characters?

I am teaching mathematics for elementary school (even playgroup) to university students. Most of my students think of mathematics might not be necessary for them. I have an hypothesis that learning ...
kiss my armpit's user avatar
29 votes
5 answers
2k views

How should normal subgroups be introduced?

One standard definition of a normal subgroup is A subgroup $N \subset G$ is normal iff the set of left cosets $\{gN\}$ and right cosets $\{Ng\}$ coincide. There's a class of similar definitions (...
user avatar
23 votes
3 answers
2k views

What are the differences between graduate and undergraduate classes, relevant to course design and teaching?

I will be a postdoc in the fall and will be teaching my very first classes aimed at graduate students. One will be an intro class, and the other a topics class. There are of course many differences ...
Aru Ray's user avatar
  • 846
21 votes
3 answers
1k views

When did the American school system's progression of math classes take its current form?

In the United States, secondary education students generally progress through pre-algebra courses, then algebra, Euclidean geometry, more algebra/trigonometry, then calculus or statistics. I am ...
Brian Rushton's user avatar
13 votes
7 answers
424 views

What different ways do people use to show students that $\mathbb{R}$ is uncountable?

In particular, if you use Cantor's diagonalization argument, do you ignore the repeating decimal annoyance? Or prove that it's not a problem? Is there another clean way that gives students intuition ...
adamblan's user avatar
  • 1,990
13 votes
10 answers
1k views

What are some great books for inspiring children to explore mathematics?

Starting from a young age, children can explore deep mathematical questions and enjoy thinking about basic math within the context of a story. There are some real treasures out there. Parents often ...
Sue VanHattum's user avatar
  • 20.1k
17 votes
4 answers
702 views

What are some good ways to motivate and introduce reasoning abstractly about abstract algebra?

I've found one of the hardest topics to introduce to students early on is abstract algebra. Even if they've already written proofs, it's hard for them to work directly from axioms. They seem to have ...
adamblan's user avatar
  • 1,990
5 votes
2 answers
424 views

Why does current mathematics education often ignore the analyses of complex number solutions on (systems) of non-polynomial equations? [closed]

I have discovered that current mathematics education often teaches students to emphasize all polynomial equations should have complex solutions. But starting from non-polynomial equations (e.g. $\sin ...
doraemonpaul's user avatar
29 votes
7 answers
2k views

Teaching logic with a proof assistant

I am thinking about teaching a university-level "introduction to proofs" class (mainly for math and CS majors) making use of a computer proof assistant like Coq. I feel like there is a lot of ...
Mike Shulman's user avatar
  • 6,520
44 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 ...
vonbrand's user avatar
  • 12.3k
14 votes
3 answers
651 views

Why teach back substitution with row reduction?

Many linear algebra books include two versions of row reduction for solving systems of linear equations: (1) Reduce to echelon form, and then use back substitution. (2) Reduce to reduced echelon ...
Jim Belk's user avatar
  • 8,189
7 votes
2 answers
158 views

When is it more advantaged for students to stop attending lectures? What to do then?

Under what situations is it more productive for math students to stop attending lectures and learn the material alternatively? What are some remedies? I originally posed this as an Example Question.
user avatar
14 votes
6 answers
231 views

Books/(auto)biographies/references on how mathematicians study/studied (as students)?

As Geoff Pointer commented: [...] As a composer I've learnt a lot from studying famous composers why wouldn't that also apply to studying maths and mathematicians of note as well? [...] Are there ...
user avatar
4 votes
1 answer
80 views

What are ideas and strategies on improving at discovering counterexamples? [closed]

What are ideas and strategies on improving at discovering counterexamples? I originally posed this as an Example Question.
user avatar
2 votes
0 answers
79 views

Where to find resources about study skills and strategies of immediate impact? [closed]

Where can research and resources be found, about study skills and strategies for university math of immediate impact? To wit, how do I efficiently find categorical, pragmatic advice about learning/...
user avatar
29 votes
6 answers
938 views

How can we help students who are very anxious about math?

In many parts of the world, the majority of the population is uncomfortable with math. In a few countries this is not the case. We would do well to change our education systems to promote a healthier ...
Sue VanHattum's user avatar
  • 20.1k
14 votes
2 answers
628 views

Splitting the students by abilities

Disclaimer: This question will use as an example a computer-science university program, because it is much easier for me to describe and give some statistics, but it applies to math as well. The ...
dtldarek's user avatar
  • 8,907
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 "...
Markus Klein's user avatar
  • 9,388
21 votes
2 answers
280 views

Pressure vs. Laissez-faire: Literature dealing with balance in university-classes

I am seeking for some pedagogical literature dealing with the following question: Imaging you have an average class in college/university: What is a good balance between Laissez-faire and pressure? ...
Markus Klein's user avatar
  • 9,388
24 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 ...
Markus Klein's user avatar
  • 9,388
20 votes
5 answers
8k views

Best textbooks to introduce measure theory and Lebesgue integration?

What are the best textbooks to introduce measure theory and Lebesgue integration to undergraduate math majors? Many students in such a class will go on to graduate school and be required to take a ...
Gamma Function's user avatar
46 votes
9 answers
3k views

Knowing mathematics does not translate to knowing to teach mathematics. Why?

Many brilliant mathematicians seem to make average or even poor classroom teachers. Is this an accurate assessment? Has there been any research to explain the phenomena? What is the difference ...
Mara's user avatar
  • 888
19 votes
5 answers
389 views

Is required reading of the text effective, and how can it be assessed?

This will likely depend on the class, of course. But I've asked calculus students in the past if (a) they regularly read the textbook and (b) whether this is helpful for them and (c) whether they like ...
Brendan W. Sullivan's user avatar
27 votes
2 answers
1k views

Students who know high-level math before going to college

There is a high school in the city I live in which has some high-level math courses in their curriculum. It's a special math class mentored by some university lecturers, and the children basically do ...
dtldarek's user avatar
  • 8,907
26 votes
7 answers
3k views

What arguments can I give a high school student why mathematics is important?

In almost all countries all over the world, mathematics is a main subject in school. Maybe the subject bringing trouble to families with kids. It is clear that scientist, engineers, etc. need ...
Markus Klein's user avatar
  • 9,388
41 votes
6 answers
15k views

How can I estimate the length of an exam?

Background: I am fairly new at teaching, and in every subject I have taught, I have had difficulty estimating the length and difficulty of an exam. I need to write an exam for a university senior-...
Brian Rushton's user avatar
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 ...
Markus Klein's user avatar
  • 9,388
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 ...
David Steinberg's user avatar

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