# Tagged Questions

Mathematics education consists in the practice of teaching and learning mathematics, along with the associated research. Research in mathematics education concerns the tools, methods and approaches that facilitate the practice of mathematics or the study of this practice.

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### Why do we first introduce the open set definition for continuity instead of the neighborhood definition?

After (nearly) completing my course in topology, something weird just stuck out to me which I hadn't considered before. When first discussing continuity, we often use the following definition: Let ...
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### Research Projects for 7th Grade Students

I'm going to define some math research projects for 7th grade students. The projects can be both purely mathematical and interdisciplinary. By the way, the students can write simple Pascal codes. I ...
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### Which universities teach true infinitesimal calculus?

My colleague and I are currently teaching "true infinitesimal calculus" (TIC), in the sense of calculus with infinitesimals, to a class of about 120 freshmen at our university, based on the book by ...
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### Determine the set of points $M$ of affix of $z\in\mathbb{C}$

Determine the set of points $M$ of affix of $z\in\mathbb{C}$ such that there exists at least one real $t$ satisfying $z^2=t(t-i)$ My attempt: We look for the form $z=x+yi$ and we want there is a ...
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### Why is polynomial long division being taught in schools instead of Horner's method? [closed]

The Horner´s method is by a long shot easier than the Polynomial long division and serves the same purpose. Why isnt it being taught in school (in germany at least)?
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### Is there an elementary “proof” that the first degree equation has only one solution?

I was asked from a student why the first degree equation has only one solution (if it has a solution) Let's consider the equation $2x+5-3x=-4x+14$ for example. How can I explain to a 13 year old ...
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### How was real analysis & topology taught in the 70's?

What was the 'gold standard' textbook before Rudin? Furthermore, if anyone has knowledge of what textbooks Princeton or Harvard used back in the 1960's or 70's, I would highly appreciate it if you ...
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### Topic for a lecture intended for High School students [duplicate]

I am not sure if this is the right place to post this, but here is the situation. In about two weeks or so I will be giving a 2-3 hours lecture on some topic in mathematics to freshman and sophomore ...
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### Can you recommend a book to learn to teach math to a child?

I am looking for a book which contains some ideas on introducing a child to mathematics. I am not particularly looking for a textbook to be used as part of the teaching (though feel free to mention ...
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### Basic examples of probabilistic method

I'm looking for a truly basic example of probabilistic method proof which could be presented without a board (i.e. speaking only), that is, even moderately complicated calculations are not allowed. ...
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A student of mine has expressed interest in doing an independent project next quarter with me. This would not be for credit and it is purely for her own educational stimulation. She wants to study ...
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### Cartesian product sets

I'm preparing a lesson on the Cartesian product of two sets and I have run into the following confusion: I understand that the Cartesian product is not a commutative operation. Generally speaking, ...
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### Fun results from modular arithmetic

I'm trying to go through various bits of neat, fun math with some junior-high-school students in my local area, and am thinking of doing a short unit on modular arithmetic/finite groups. I'm looking ...
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### Applications of derivatives outside mathematics and physics

I've been teaching calculus for several years and have some doubts about whether derivatives (and integration techniques) of common functions are useful and important outside mathematics and physics. ...
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### How do you teach the correctness of bijections when dealing with counting in combinatorics?

Consider the problem of awarding prizes to people in a school. Let $A$ be the set of awards and $|A| = m = 3$. Let $P$ be the set of people in the school and $|P| = n$. Then in how many ways can ...
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### Examples where it is easier to prove more than less

Especially (but not only) in the case of induction proofs, it happens that a stronger claim $B$ is easier to prove than the intended claim $A$ (e.g. since the induction hypothesis gives you more ...
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### difficulty of accepting $i^2 = -1$ for first timers [duplicate]

While teaching complex numbers for those who are encounter for the first time (usually 10th grader and 11th grader), I get the question like "Can even squared number give negative results? How ...
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### Very elementary proof of that Euler's totient function is multiplicative

Well, I know two or three proofs of this fact $$\gcd(m,n)=1\implies \varphi(mn)=\varphi(m)\varphi(n)$$ where $\varphi$ is the totient function. My problem is this: I'd like to explain this to some ...
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### Why does induction procedure of Euler characteristic fail for non-convex polyhedra? What am I missing?

Euler characteristic of convex polyhedra is always $V-E+F=2$. Induction procedure reduces edges and vertices until we are down to one vertex whose $V-E+F=2$ and hence you are done. The same ...
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### Can I define the limit of a sequence like this?

It is well-known that a sequence has a limit if and only if it is bounded and has a unique limit point. I think this is a better definition of the limit of a sequnece, comparing with the $\epsilon-N$ ...
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### Using Latex in classroom instead of blackboard?

Does anyone find it doable to use Latex in a classroom setup instead of blackboard? I would think it is tempting except for the input speed one can reach. But considering the messiness of chalks, and ...
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### How to teach Critical Thinking

I am currently tutoring a few students in an entry level physics course and had some trouble recently when it comes to helping them with problem solving. The students I am helping don't have many ...
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### Algebraic topology in high school?

This winter I am planning on teaching a small seminar (20 lectures 45 minutes each) for high school students. I was was given the freedom to choose the topic of the seminar, but it is supposed to be ...
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### Confusion on Eigenvalues of Matrix

I'm a TA with Advanced Algebra in school and teach the Jordan Form now. There are three questions about eigenvalues in this chapter: Given matrix $A$, $B$ and polynome $f$, consider the eigenvalues' ...
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### How to explain to a high school student why a linear differential equation is linear?

My mother is teaching a high school course on multivariable calculus, and they were studying linear differential equations of the form $$y' + P(x) y = Q(x),$$ and the question of why this equation is ...
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### Alternative introduction to tensor products of vector spaces

One of the main obstacles in understanding the tensor product is that, unlike many other algebraic structures, you cannot really get hold of its element structure. This confuses many beginners. The ...
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### Tips on “short” math lecture presentation for compsci students

I just got interview to teach college (of applied science) compsci student math. This is the first time I get this type of interview. The type of interview is called a "hearing" where a dozen people ...
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### Intuition for high school students regarding square roots and logarithms [duplicate]

These are some common mistakes high schoolers make: $$\sqrt{a + b} = \sqrt{a} + \sqrt{b}$$ $$\log(a+b) = \log (a) + \log(b)$$ So I can obviously show numeric examples to say why these are wrong, ...
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### Dynamic Geometry Software for teach construction

I would like to know about a software that will help me show construction steps to the students with using a Compass/Straight Edge/Protractor/Divider. Here is example video below. ...
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### When $n$ is divided by $14$, the remainder is $10$. What is the remainder when $n$ is divided by $7$?

I need to explain this to someone who hasn't taken a math course for 5 years. She is good with her algebra. This was my attempt: Here's how this question works. To motivate what I'll be doing, ...
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### Simple examples of math problems illustrating a basic epistemological point.

Instructor: In fact this expression is equal to that expression. Let us see how we can convince ourselves that that is true. Student: I'm more than willing to take your word for it! You're the ...
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### Questions for first year students at the University. [closed]

I will help teach in a introductory class in mathematics for engineers in applied math at the University. Anyone have any good and cool favorite questions or know where I can find some? Anything is ...
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### Binomial expansion for any n

I teach A-Level maths, and in the second year we do the general binomial expansion, which is even provided for the students in the formula book. For values of $n$ that are not positive integers: (I ...
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### Interesting “real life” applications of serious theorems

As a student one sometimes encounters exercises which ask you to solve a rather funny "real life problem", e.g. I recall an exercise on the Krein-Milman theorem which was something like: "You have a ...
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### How is addition on N formally defined in textbooks on real analysis?

This is a follow-up question to Why does the definition of addition require proofs? In Landau's Foundations of Analysis, his definition of addition on the natural numbers seems a bit strange to me -- ...
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### Using math to help people [duplicate]

So I am currently a graduate student at the University of Colorado. I love math. From calculus to category theory to everything in between, I have tried and, for the most part, loved it. However, I ...
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### A basis in the $k$-th exterior power of a vector field

Definition: Let $\mathbb R^n$ be the $n$-dimensional real vector space. An exterior $k$-form call any skew-symmetric tensor on $\mathbb R^n$ of rank $k$. Denote the set of exterior $k$-forms by $E^k$. ...