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Something my students* often struggle with is how to react on being confronted by 'new things', including functions, notation or definitions for which they are given sufficient definition but with which they are unfamiliar. I would like to get them used to the process a mature mathematician goes through on encountering new things (e.g. 'read the definition carefully, try to find concrete examples/counterexamples'), but of course to do this I need a stockpile of new things to present them with.

Examples of the sort of things I am describing include:

  • Floor and ceiling functions (e.g. '$\lfloor x \rfloor$ is the largest integer $n$ such that $n\leq x$. Sketch the graph of $y=\lfloor x \rfloor$.')
  • Properties of relations (e.g. 'A relation $R$ is transitive if whenever $aRb$ and $bRc$ it is also true that $aRc$. If $aRb$ whenever $a-b$ is even, show that $R$ is transitive.')
  • New definitions (e.g, 'A value $x\in \mathbb{R}$ is a limit point of a set $A\subseteq \mathbb{R}$ if for any positive $\varepsilon$ there is an element (distinct from $x$) of $A$ less than $\varepsilon$ away from $x$. What are the limit points of the set $(0,1)$? What about $\mathbb R$, $\mathbb N$, or $\mathbb Q$?

Further suggestions would be very welcome!


*My personal context here is helping UK sixth-formers (16–18-year-olds) prepare for university admissions tests/interviews for mathematics, but the question should be more generally applicable.

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  • $\begingroup$ I don't have time to say much now, and besides I've been posting things like this for some 20 years now anyway (phrase google "Dave L Renfro" with various math terms/topics you're interested in), but these comments from 5 September 1999 (and my next day follow-up of them) might be of interest. $\endgroup$ – Dave L Renfro Jul 6 '18 at 10:12
  • $\begingroup$ I managed to locate a collection of some of my pre-Stack Exchange posts that might be useful. Most of these (but several are lower level) cover topics that could be in a U.S. calculus 1 or calculus 2 course (elementary single-variable calculus). Incidentally, to keep people from having to google, you should give a brief indication of what level "UK sixth-former" is. $\endgroup$ – Dave L Renfro Jul 6 '18 at 10:34
  • $\begingroup$ @DaveLRenfro Thanks; I am sure I will spend some fruitful time reading such a wealth of interesting problems and discussions. However, here I'm not looking for interesting problems as such, but rather new concepts about which very simple (or less simple!) problems can be set. I have glossed 'sixth-former' as you suggested. $\endgroup$ – dbmag9 Jul 6 '18 at 11:28
  • $\begingroup$ After looking at the examples you've given, I think it might be easiest and quickest to simply flip through the beginning review material in a lot of undergraduate level texts at a library. There are also quite a few elementary/expository papers that might be suitable, but they're difficult to locate except by exhaustively looking through journal volumes one-by-one. Some of the examples I've given here (continued) $\endgroup$ – Dave L Renfro Jul 6 '18 at 13:08
  • $\begingroup$ and here and here and here and here could be of use. $\endgroup$ – Dave L Renfro Jul 6 '18 at 13:16
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As is often the case on this site, I slightly disagree with the specific premises of the question... and claim that tweaking those premises does help to solve the problem at hand.

Namely, although the logically linearized version of mathematics often does present definitions first, this is mildly perverse, since the people who conceived those definitions rarely wrote them down and then tried to look for examples, or tried to think what they could prove about the just-defined things.

Rather, a "definition" is nearly an end-product of usually very substantial inquiries, including many, many examples whose nature needs to be considered before the "definition" is made, since the definition needs to be inclusive enough, but not accidentally too inclusive to inadvertently lose the key motivations.

So, seriously, I myself only like "definitions" as sort of "wrap-up" of discussions. When grad students giving talks or writing papers want to tell me their "definition" first, I politely interrupt to ask for the examples that led to their definitions...

This does approximately lead back to the operational point of coming up with illustrative examples! Indeed! But/and to more genuinely mimic or simulate or emulate genuine mathematical thinking I'd claim that the examples should be given first. And then ask the students to formalize/definitionalize those examples.

(This also has the psychological benefit of giving students/people more validation, since they're not having definitions dictated to them by some authority-entity, but get to make their own. :)

EDIT:

So, to let the other shoe drop... Groups of symmetries of geometric objects. E.g., regular polygons... first in 2D, and then considering reflections. Then, regular polyhedra...

Also, Galois theory "for citizens", talking about the algebraic indistinguishability of the various roots...

Irrationality of various numbers. A trope.

Complex numbers as a two-dimensional incarnation of "number".

Then, quaternions...

And, then, ask people what was the goal there... ? :)

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    $\begingroup$ +1 for 3rd paragraph about definitions. This makes concrete something I've always felt, and I might wind up "borrowing this" in the future! $\endgroup$ – Dave L Renfro Jul 24 '18 at 11:00
  • $\begingroup$ Thank you for taking the time to write such a thoughtful answer – but I would draw your attention to the footnote to my post, which is that my interest here is specifically in training 16–18-year-olds to cope well when confronted by new things in an interview context. The situation I'm thinking about specifically precludes motivating a the 'new thing' or allowing it to develop naturally in context – otherwise I absolutely would follow your advice. $\endgroup$ – dbmag9 Jul 25 '18 at 21:57
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    $\begingroup$ @dbmag9 Ah, I understand. Unfortunate, though. Those confrontational, adversarial situations are rarely constructive. $\endgroup$ – paul garrett Jul 25 '18 at 23:34
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I submit the function $f$ with rule: $$f(x) = \begin{cases} 1 & \text{if $x$ is rational} \\ 0 & \text{if $x$ is not rational} \end{cases}. $$ Graph $y=f(x)$. Does there exist a point at which $f$ is continuous?

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    $\begingroup$ Or $f(x) = \frac{1}{q}$ when $x = \frac{p}{q}$ (in lowest terms) is rational, and 0 otherwise. That this thing actually has points of continuity (lots of them) is crazy. $\endgroup$ – Xander Henderson Jul 6 '18 at 4:05
  • $\begingroup$ @XanderHenderson strange. Thanks! $\endgroup$ – James S. Cook Jul 6 '18 at 4:41
  • $\begingroup$ I like this example – but would it be common for high-schoolers in the US to know a formal definition (or even informal definition) of continuity? $\endgroup$ – dbmag9 Jul 6 '18 at 11:37
  • $\begingroup$ @dbmag9 given how many of them take AP "Calculus" one would hope they are at least introduced to informal continuity. If it were up to me I might introduce continuity as I motivate functions such as $f(x)=2^x$. Ideally, all rational inputs for $x$ are explained arithmetically, but what is meant by $2^{\pi}$. I think this needs the idea of continuity. $\endgroup$ – James S. Cook Jul 6 '18 at 14:16
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You specified high school students, so I am pretty sure that the following recommendations aren't going to be all that helpful, but I am going to give them anyway. First, a few books (in what I think is a meaningful order):

  • Sergiy Klymchuk and Susan Staples, Paradoxes and Sophisms in Calculus. MAA Press, 2013. This book is full of surprising results that are true, as well as diabolical results which look true but aren't (my favorite example being DOOM KITTY, a poor kitten/kinetic impactor at the top of a ladder which is sliding down a wall). This would be good to have in an AP classroom, I think.
  • Lynn Arthur Steen, J. Arthur Seebach, Jr., Counterexamples in Topology. Dover, 1978. The basic content of this book is probably a bit too high level for secondary students, but there are some really gnarly examples which, I think, could be interesting in a high school setting. It would take work to make it fit in the spirit of the question, though, so it is (as I prefaced above) not a great answer to the question.
  • Bernard R. Gelbaum, John M. H. Olmsted, Counterexamples in Analysis. Dover, 1962. This is probably way too high level for a high school audience, but is good for undergrads. Same caveats as above.

The Cantor set is another example that I think is rich for mining. Showing that it is non-empty requires some careful thinking, and showing that it is uncountable can be done by a careful reading of the usual diagonalization argument. One could also introduce a notion of measure and ask how "long" the Cantor set is. Or even just give students the two functions $$\varphi_1(x) = \frac{1}{3}x \qquad\text{and}\qquad \varphi_2(x) = \frac{1}{3}x + \frac{2}{3}$$ and ask your students to find $\varphi_j([0,1])$ (while introducing the image of a set rather than a point). Then iterate. And do it again. Ask them what they get if they do this infinitely many times.

Another possible source of material might be an introductory undergradate algebra text. Addition modulo $n$ is interesting; even more interesting is the difference between prime and composite values of $n$. Small finite groups could also be interesting to play with.

Finally, I really like to introduce students to things like sawtooth, square, and triangular waves before getting into trigonometry. These provide interesting examples of periodic functions that are not trig functions. They are really good for parsing both the definition of periodicity, and how periods are changed (or not) by elementary transformations (e.g. if $f$ is $T$ periodic, what is the period of $S_a \circ f$ or $f\circ S_a$, where $S_a(x) = ax$?).

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Telescopy (or telescopic induction) is a nice example. It is a simple, prototypical example of induction that will provide strong intuition for later studies. Telescopic cancellation is usually clearer to visualize in products (vs. sums), so I start teaching the multiplicative form first, e.g. below

$\qquad\qquad\, \displaystyle (x-1)(x+1)(x^{\large 2}\!+1)(x^{\large 4}\!+1)\qquad\! \cdots\qquad (x^{\large 2^{\rm N}}\!+\,1)$

$\qquad\ \ \ = \ \displaystyle \frac{\color{#0a0}{x-1}}{\color{#90f}1} \frac{\color{brown}{x^{\large 2}-1}}{\color{#0a0}{x-1}}\frac{\color{royalblue}{x^{\large 4}-1}}{\color{brown}{x^{\large 2}-1}}\frac{\phantom{f(3)}}{\color{royalblue}{x^{\large 4}-1}}\, \cdots\, \frac{\color{#c00}{\large x^{\large 2^{\rm N}}\!-1}}{\phantom{f(b)}}\frac{x^{\large 2^{\large \rm N+1}}\!-1}{\color{#c00}{x^{\large \rm 2^N}\!-1}} \,=\, \frac{x^{\large 2^{\rm N+1}}-1}{\color{#90f}1} $

For a nice example of additive telescopy, in this answer I show how the "proof by picture" below can be mechanically discovered and rigorized by applying a 2-dimensional form of telescopy - which allows us to view a sequence of rectangles as being built-up layer-by-layer from successive differences of prior rectangles - as if they were built by a 2-D printer in FlatLand.

$$ \large \color{PaleVioletRed}1 + \color{DarkViolet}5 + \color{DodgerBlue}9 + \dots + \color{LightCoral}{(4n-3)}\, =\, n(2n-1) = 2n^2-n$$

enter image description here

Many more examples - both simple and complex - are in my diverse MSE posts on telescopy.

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  • $\begingroup$ beautiful answer, I don't understand why someone would downvote this. Of course the totality of what is meant by "telescopy" is larger than the desired answers here, but there is something lucid and intriguing about this family of ideas. $\endgroup$ – James S. Cook Jul 7 '18 at 3:03

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