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25

First of all, I should point out that the standard definition of a normal subgroup is A subgroup $N \subset G$ is normal iff $g n g^{-1} \in N$ for all $n\in N$ and $g\in G$. When I say "the" standard definition, I mean that this is how working group theorists think of normal subgroups, and this is one of two basic ways to prove that a subgroup is normal....


22

I did this once with a high school group. My approach was to use the metaphor of a scientist doing a controlled experiment: You study a complicated system by introducing a single change, while trying to keep everything else constant, and watch how that change affects the system overall. In this case, the system is arithmetic, and the "one change" we are ...


20

In my experience, one of the problems with series is that usually you have two sequences if you investigate the series $\sum(a_n)$: the sequence $(a_n)$, and the sequence of partial sums $S_n=a_1+\ldots + a_n$. I noticed that trying to stress this distinction helps a lot. To the intuition, I like R. Péter: Playing with Infinity, the chocolate bar example on ...


15

The way I like to approach this is as follows. After discussing subgroups, the natural question as to forming the quotient $G/H$ arises. I then proceed to look at the cosets and prove that if $gH\cdot g'H=gg'H$ is a well-defined operation, then the cosets become a group, which we call the quotient group. This is a very easy proof with basically nothing to do....


15

I think it is important to develop mathematical intuition in one's work, but not necessarily of the naive sort. Of course, one's intuition may turn out, ultimately, to be incorrect. An important outcome in this case is not to assume that you have bad intuition, but instead to think about how you can make the end result sync up better with what could be ...


15

To answer the question in the title, I would say that one problem with no-symbols reasonings is that one need to use a lot of pronouns. Problem is, pronouns usually leave too much ambiguity. At some point, mathematical sentences where written without symbols; the introduction of letters to denote mathematical object improved quite a lot the depth and ...


14

Just supplementing Benjamin Dickman's nice answer, here is $x \mapsto x^2 - x$ in $\mathbb{Z}_{18}$ in the same style: For example, the pentagon wheel reflects the fact that $$(5+3k)^2-(5+3k) = 9k^2 + 27k + 20 = 9k(k+3) + 20 = 2\bmod 18 \;.$$


13

Homotopy equivalence v. Homeomorphism. I believe an accessible difference between homotopy equivalence and homeomorphism is that one preserves an intuitive (though hard-to-define) topological invariant, while the other almost never does: the invariant of dimension. For instance, any $\mathbb R^n$ is homotopy equivalent to a point. Similarly, $\mathbb R^n - \...


13

Modular arithmetic is beautiful and simple but tends to be frighteningly clotted with detail and notation at first. Whenever I present it I start off by telling the audience they already know modular arithmetic. I proceed to ask them: if it is now 1 o'clock (and then I write a '1' on the board), what time will it be in 2 hours (and I write a "+ 2" next to ...


13

(I have not done this exact presentation, so I cannot vouch for its efficacy. But I have used the main idea before, and it seems to help some students, and is at least a bit of fun. Also, this is meant to address the intuition aspect of the question, not the motivation.) Demonstrate by walking! Convergence means there's a spot that will be approached to any ...


12

It seems to me that the reason for learning series in calculus is to analyze functions through power series representations. There are other reasons series are important in mathematics, but in the standard calculus course, it seems to me that series are introduced for the sake of explaining power series. The key is then, Can one motivate power series ...


12

I have given this issue a lot of thought over the years -- in fact a large portion of my dissertation is devoted to related issues. I think of this challenge in terms of a "mathematical sensibility" -- a way (more precisely, a cluster of related ways) of appreciating and participating in mathematics -- and frame the question as "What is the role of school ...


12

One of the approaches taken in some areas of mathematics (e.g., in arithmetic dynamics and considerations of preperiodic points, etc) is to create these graphs by drawing discrete points and then using arrows to show which values map to which other ones. Figuring out a "canonical" way to draw these pictures might be a bit tough (this is related in some ...


11

This answer is from my experience running a Maths Learning Centre. I help students learn and use maths, mostly when they are struggling, and I also hear their opinions of their lecturers and other teachers. I am very interested in all sorts of maths, and I will take almost any opportunity to be excited about a mathematical idea. For example, I made t-shirts ...


10

As @BenCrowell mentioned, the transfer principle proves that direct algebraic manipulation of infinitesimals in single variable calculus is allowed. But I stumbled upon this post in theshapeofmath.com, which shows how things can break when switching to multiple variables. It provides the following basic example of a possible error when handling partial ...


10

Most functions that are studied by physicists and other scientists are continuous. However, more and more discontinuous functions are appearing in the various sciences. This is due to: Computers and their digitization of data. Many computer routines produce discontinuous output, even if the data is near-continuous. Quantum theory is a mixture of the ...


10

My favorite video for this is powers of ten from 1977. Though we can get a little smaller today, I think it still does an excellent job with getting the scale of things starting from what we know. They should pick up on that these numbers are far far far bigger than the largest scale in the video.


9

Unfortunately, my example is not a full answer to your question, but I think, it helps students to be at least beware of what can happen. You can explain series as a inifite summation of areas (at least as long as everything is non-negative). And convergence of a series means: Is there a big rectangle such that all the given areas fit in? Intuitively, most ...


9

My feeling is that the $\epsilon$-$\delta$ formulation is already pretty close to what one should think about limits; that is, the language can be hard to grasp at first, but the idea is very literally expressed in that language and can also be expressed in words. I find this more convincing using a physics measurement approach (which is slightly more ...


9

Have a look at the paper written by Nunez et all: EMBODIED COGNITION AS GROUNDING FOR SITUATEDNESS AND CONTEXT IN MATHEMATICS EDUCATION. In essence, they argue that it is better to be causious if you want to "motivate the formal definition of continuity starting from the intuition" you have suggested in your question. In the following passage, "natural ...


8

Another point that oughtn't be neglected was that, historically, numerical computation/approximation used Taylor-Maclaurin expansions (not to mention Newton-Raphson when convenient) to approximate root-taking. Newton was apparently very happy with his discovery of the binomial expansion for general exponents, although surely not only for numerical purposes. ...


8

I am not entirely sure on the best way to convey the difference between homotopy equivalence and isomorphic homology groups (or even isomorphic homotopy groups, though on CW-complexes I guess this isn't as big of a concern), except by way of examples. I remember my algebraic topology exam had an explicit example of spaces with all the same homology and ...


8

This is an excellent question. Some good advice on this can be found in the writing of Bill Thurston, some of which I have posted in an answer to this question on Math Overflow. The opening of the quotation I posted there is particularly telling: "Mathematics is a paradoxical, elusive subject, with the habit of appearing clear and straightforward, then ...


8

We have two cookies. We divide them into pieces of 1/2 cookie each and end up with four pieces. Thus 2 divided by 1/2 equals 4. We have two cookies. We take 1/2 of the collection which is one cookie. Thus 2 multiplied by 1/2 equals 1. Each of those examples can be criticized. In the first example, one could claim that it shows that 2 divided by 4 equals 1/...


7

I would be inclined to choose my words more delicately. Namely, it's not that intuition-per-say is misleading, but that untrained or inexperienced intuition is... untrained and inexperienced. Of _course_ many reasonable, normal things (in this complicated world) will be surprises to a naif. That doesn't mean they're truly counter-intuitive. I would advocate ...


7

The prototypical way for a function to not be continuous is that of a jump discontinuity. Imagine a jump discontinuity on the order of a few micrometers, like the width of a hair. If you are tracing the graph of the function with an everyday pencil, you would slide right across the discontinuity without even noticing its presence. However, if you shrunk ...


7

A somewhat analogous point of view is the one of "continuous extensions". What looks weird in saying $\frac 1 x$ is a continuous function is, of course, what is happening around $0$. What is happening around $0$ can be summarized by saying: there exists no continuous function on the whole $\mathbb R$ extending $\frac 1 x$ The whole idea of removable ...


7

One suggestion would be to take an approach similar to that used for describing the $52!\approx 8.063*10^{67}$ ways to arrange a standard deck of playing cards as outlined in Scott Czepiel's Blog and popularized in this vsauce youTube video @ 14:40. The example given in the blog post is to set a timer for $8.06*10^{67}$ seconds and pass the time by ......


6

The motto I use for teaching is this: Subsets are what you get when you throw elements away. Quotients are what you get when you glue elements together. In the case of $\mathbb{Z} / n\mathbb{Z}$, we glue together any pair of numbers if the number of steps to get from one to the other is a multiple of $n$. I like this interpretation better because you don'...


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