# All Questions

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### What was the problem with New Math? Why did it end?

During the 60s, people in the US (and also in Europe), school curricula introduces New Math where students began with set theory in the first grade before learning to perform addition or ...
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13k views

### Applications of Vector Calculus to Economics/Finance

I will be teaching a course focusing on multivariable integration soon, for the millionth time. The most difficult topic in such a course is certainly Vector Calculus, by which I mean line and surface ...
• 1,293
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 ...
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807 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 ...
335 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 ...
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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 ...
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1k views

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 ...
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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 ...
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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?
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306 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 ...
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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 ...
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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 ...
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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 ...
• 835
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 ...
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327 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 (...
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 ...
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425 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 ...
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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 ...
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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 ...
420 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 ...
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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 ...
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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 ...
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430 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 ...
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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 ...
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156 views

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 ...
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250 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 ...
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 (...
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 ...
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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 ...
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425 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 ...
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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 ...
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718 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 ...
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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 ...
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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 ...
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12k 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 ...
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659 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 ...
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159 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.
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 ...
81 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.
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/...
943 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 ...
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648 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 ...
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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 "...
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281 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? ...
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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 ...
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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 ...
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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 ...
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393 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 ...
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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 ...
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