How do I show students the Beauty of Mathematics?

You can't. Mathematics is hard, and its beauty must be earned by hard work.

Think about how you might have realized the beauty of maths. You most probably have thought very hard about a problem, and when you finally put the pieces together, your reward was a feeling of beauty. This reward then made you more likely to put in effort and get more rewards, which became a positive feedback loop.

The point is, you can't just show the beauty of maths. Maths is not a nice painting or a piece of music where beauty can be enjoyed effortlessly. Without the effort put in to understand the concepts, your students won't recognize it.

The best you can do is to present the material in a straightforward, mathematically most honest way, without try-hard ways to "make it interesting". Students recognize immediately if the teacher is trying too hard to "sell" the curriculum, or tries to trick them into being interested, and they resist instinctively. Get them put in the work, by being strict, if necessary, be honest to the mathematics you want to present, and earn their respect. Then some of them, who put in the work, might get hooked by the positive feedback loop.

I can relate to your students, as the education system in most countries is very much like a straight-jacket, especially when it comes to math.

Some tips I can think of that might help

1. When introducing a topic, begin by motivating it with real world examples of where you might apply it.

2. Don't just do the "drill and kill". Instead, mix things up by having group assignments and projects. It needs a little creativity on your part to understand when it is appropriate to use these methods.

3. Try and use heuristics and intuitive explanation where possible, as this help with bridging the gap between everyday thinking and the more formal stuff.

4. When introducing a problem, have an introduction where you work out what you expect using only intuition, and at the end of the problem, do a debrief where you check if your intuitions matched up.

5. Have extra credit assignments that are more open-ended by nature, and encourage exploratory skills.

6. Have a good amount of healthy competition between your students by dividing them up into teams, when possible.

I know these are not all directly related to showing them the beauty of math. The reason I think they are still important is that the beauty of math is multifaceted in nature. Sometimes, math is beautiful for its simplicity and elegance, and at other times, it is the sheer usefulness that draws one to it. Some students may be drawn to its practical power, while others may be drawn to its aesthetic elegance. Let them decide which parts they like most and for what reasons, but by following the above points, you will at least be giving them a chance to get proficient in the subject.

REMEMBER: Sometimes, the math bug can bite you later in life. You want to give them the best chance possible. The rest is up to them and Lady Luck.

Students are hard customers to satisfy, especially after having less than optimal math instruction for so many years, and probably having negative self-talk about their math ability. Make it real. Make it hands-on. Make it immediate and "in their face" as much as possible. What do high school kids do in their daily lives? They play video games. They ride skateboards. They play musical instruments. They play sports.

The Elder Scrolls V: Skyrim, Fortnite, Clash of Clans: how these programs use triangles and other formulas to calculate what is happening.

Skateboarding: Bring a skateboard in to class, or ask one of your students to demonstrate tricks, and then break down the math behind the physics. https://www.tested.com/science/physics/43749-the-physics-of-landing-a-1080-skateboarding-trick/

Musical instruments: play a popular song. Isolate one note. Show the frequency response on a graph. Extract a piece of mathematical information relating to the appropriate math level. Have a student who plays an instrument give a demonstration.

Sports: Have a someone who plays baseball throw a ball as hard as he/she can, and see if the class can determine the mathematical formula.

It's almost reverse from "normal" teaching--observe the physical phenomena first, then determine the mathematical approximation.

I just read a nice piece, Suber's "A crash course in the mathematics of infinite sets". No "higher" math required, just sharp thinking. Suprising results, with simple, clear path to them. I believe this is mathematics at it's best.

As traditional museums try to show the beauty of paintings and sculptures, museums of mathematics also try to do the same with Mathematics. Here are two of them:

Erlebnisland Mathematik | Technische Sammlungen Dresden

MoMath | National Museum of Mathematics

There also big math festivals for the general public:

National Math Festival

Festival da Matemática in Brazil

It is possible to organize such mini museums in schools at low cost:

Cabinet of Mathematics 1

Cabinet of Mathematics 2

I also would like to mention the posters provided by AMS:

Mathematical Moments | American Mathematical Society

As different people like different artistic genres, different students may like different aspects of mathematics: puzzles, generative art, math olympic problems, math circles etc. It is necessary to know your students first to present them aspects of Mathematics that engage them.

There is a very simple game that I've used to illustrate what Mathematics is (the study of patterns). Here is the article:

What is The Name of This Game? Article by John E. Mahoney published in Mathematics Teaching in The Middle School (NCTM)

The three books Mathematical Gems I, II and III by Ross Honsberger have nice examples showing the beauty (gems!) in Mathematics.

Mathematics is a very beautiful subject that is being taught in a totally wrong way, causing it to be perceived as a very boring and ugly subject. One of the biggest problems with math education is the totally flawed idea that one has to focus almost exclusively on the elementary basics and make sure these are mastered 100% before one can move in to the next level. No other subject is taught in this way, although physics comes close and we see the same sort of problems with that subject too.

Imagine that we would teach language in the same way as math. Then children would not read books, write essays etc. in school. They would instead learn lots and lots of grammar, exceptions to the grammar rules etc., and after several years still not have progressed far enough to apply what they've learned to simple conversations.

Music taught in this way would degenerate to practicing playing simple notes on musical instruments over and over again, never to move toward playing an actual piece of music. That would be considered to be "advanced university level music", and you need to first have mastered the basics good enough to qualify for that. There would be no better way to make children hate music class!

So, how can we best remedy the problem with math teaching? I.m.o., the best way is to design a new curriculum that is based on using the tools children like to play with. Computers should be used a lot more. The best way for children to learn mathematics with pleasure is by teaching them computer programming. In trials it has been shown that primary school children have no problems learning to code in languages like C++.

Instead of subjecting children to boring math problems, one can give them assignments where they have to write code to tackle a problem. They'll then be exposed to the rigorous mathematical logic at a far younger age, and yet they'll have a lot of fun. Most of the the feedback they get when they make a mistake will come quite promptly from their own computers, when the compiler complains about errors. The children working on a project will then both practice their logical reasoning skills and yet not get bored by having to do that, as their goal isn't to get to just any code that the compiler will accept, but to finish the project they are working on.

Then, mathematics as we traditionally learn it does need to be presented differently so that it can be presented in this context. For example, instead of introducing calculus via smooth curves, integration as area under a curve etc., one can just as well introduce this topic via coarse graining. Unlike the concepts use in traditional calculus teaching, children encounter coarse graining all the time. Every time they look at a picture on their computer screens, they are looking at a coarse grained representation of the data that makes up the picture.

Limits are then about computing properties of data in a scaling limit, and calculus is about doing commutations directly with the quantities defined in the scaling limit. This way of introducing calculus is i.m.o. not just easier to master, it's also more consistent with what the continuum really is. It's not fundamental, it's an artifact of having taken a prior scaling limit.

This is something that can be obvious to five-year-olds zooming into a smooth pictures and then seeing pixels. In contrast, we are indoctrinated by the wrong idea about a fundamental continuum, and we then need to unlearn this and do computations properly when Nature tells us we're making a mistake, see e.g. here on page 12:

Often, authors forget to mention the first, very important, step in this logical procedure: replace the classical field theory one wishes to quantize by a strictly finite theory.

If you mean literal beauty, audio visualizers like MilkDrop, Friktal, and Particles can be interesting. As of course game engines like Unreal and the demoscene that tends to combine them into incredibly small programs like 64 KiB and even 8 KiB or a single tweet.

One of those demo guys wrote an article about automating boring math. (In JSFiddle you may need to change the JavaScript load type to "No wrap - bottom of <body>".)

Also great is Carl Sagan's explanation of how the ancient Greeks knew the Earth was round and calculated its circumference over 2,000 years ago, which can segue to rocket science via Kerbal Space Program and machine learning for less collateral damage.

It would be nice to draw possible career paths now and it appears that modeling is at the frontier of the landscape/math tree:

I conceive of mathematics as a fantastic citrus tree: the three main branches of oranges, limes, and lemons representing the major fields of algebra, geometry, and analysis. Each part of the upper canopy takes advantage of what is below it. Thus, number theory has two great approaches, one algebraic and the other analytic. Topology makes use of all three fields, as does the very different field of mathematical modeling.

All trees have woody parts, foliage, and fruit. The trunk and branches absorb water and nutrients from the soil. The foliage absorbs energy from the sun, using it to convert the nutrients to usable food, and allowing the tree to grow. The fruit allows the tree to reproduce. I have represented the foundations of mathematics as the trunk of the tree, the support and food supply. The foliage and fruit could represent the two aspects of pure and applied mathematics: which is which, do you think? - Margie Hale, Professor of Mathematics, Emeritus

Even if people decide math isn't for them, their computers can still use it to save the world, save money, or make ever-changing art.

An infinite universe arising from a simple formula is sure to inspire, such as this "Red Roots" Triangle Inequality Average Mandelbrot fractal:

Likewise cellular automata like Conway's Game of Life which can compute anything, even itself, and other articifial life like Boids in 3D.

In short, make math relatable and useful to them. As this excellent presentation quotes:

"Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." - Benoit Mandelbrot

Yet:

"The cosmos is full beyond measure of elegant truths; of exquisite interrelationships; of the awesome machinery of nature." - Carl Sagan

But most importantly:

"Subjects should not be introduced without explaining the purpose or reason, or without giving any way in which the material could be really used to discover something interesting. I don't think it is worthwhile teaching such material." - Richard Feynman

I, for one, would like to learn how to simply scratch my own holograms. and if i were mechanically inclined, build a robot to automate that. And/or do some weird science for real men instead of despairing over strange symbols and formulas that still fail to explain how a bicycle works.

A person doesnt really need to know all of the why in math only what relates to them. Teach people how it relates to them like providing real world examples (for example determining the height of a tree). Show them that you can do cool stuff with math. As far as understanding, be open to tutor a student through something. Generally too though, you never appreciate that you have learned something till after you have taken the course. During the course, it can feel like slave labor. And one last thing, show your enthusiasm and the students will be encouraged to be positive as well. Unfortunately, math is not a well viewed subject with high school students.

I agree with anjama's comment: give them applied examples. Use something like the free Godot engine to create simple "games" (read: interactive apps) that demonstrate concepts of trigonometry, geometry, and linear algebra. It's really quite easy to get started with Godot if you know some basic programming, and once the kids see a game engine, and see how math can be directly used to make video games, you can bet at least some of them will be excited.

If any of my teachers had shown me anything like that, I would have shat myself with excitement. "I can use this math to make games on my computer at home?!" Instead we went outside and used a clinometer to measure the height of trees ....

Simple: You do not.

Some people like maths and will see beauty in it.

Others will not consider it beautiful no matter how useful it is or no matter how much you show them maths that you personally consider 'beautiful'.

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A personal anecdote:

Personally speaking I very much disliked maths at school, despite being told I was good at it.
Like fellow user @SteveSmith, one of my primary reasons for disliking it was that I saw no use for it.
Later in life I took up programming and suddenly found that I actually had a use for all those formerly useless things I despised.

Over time my hatred of maths lessened as I found more uses for it, to the point that I now find most of it tolerable or am willing to attempt things if I have a particular use for them, and in some cases I've even gone out of my way to learn things that I don't need to know (e.g. group theory) purely because I've found it interesting.

However, despite there being a number of things about maths that I find useful or interesting,
I have never found any part of maths that I'd honestly consider 'beautiful'.

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So my advice to you is this:

Don't even bother trying to show your class that maths is 'beautiful'.
The people you convince will only be the ones who would naturally like maths.
For the rest, if you truly want to convince them to take an interest then don't try to convince them that maths is beautiful, try to demonstrate to them that maths is relevant to their lives.

If you cannot manage that, then try to find ways to make maths less painful for them - demonstrate ways to break problems down into simpler problems, demonstrate rules like additive and multiplicative identity and inverse elements and how those things can be used to reduce complex problems to simpler forms.

Failing that, just give up on trying to convince them to like maths.
Let them reluctantly get through the class and achieve whatever grade they achieve.
For some people, it's simply not possible to convince them to like maths.

Péter Rózsa's classic book, Playing with Infinity is an excellent intro of maths to non-mathematicians. Perhaps it would help. MAA review: https://www.maa.org/press/maa-reviews/playing-with-infinity

Many of the students say they don’t like mathematics because they only learn for examinations. How do I show them the true beauty of mathematics?

How about testing them on that beautiful content on the examinations?

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EDIT: And/Or give a wider variety of math assessments that count for grades. Projects, presentations, performances, role playing, games, discussions, etc., all of which can more easily incorporate the true beauty of mathematics.

I'm having a lot of success right now in that area with a class I'm teaching, in which I have several assignments over the course of the quarter that center around self-directed learning. It takes some scaffolding to make it work well (how do you ask a good mathematical question? What counts as a good answer?) but once I got that done, a lot of the students really caught fire - last quarter, they turned in two five-page papers discussing a question they developed and the way they answered it. A few even taught me something I'd never known before!

Of course, doing that came at the cost of a fair bit of class time - if you're teaching in a high-school environment, or a traditional college course, you probably have enough curriculum to get through that you don't have time for a major assignment that isn't directly covering the curriculum. And grading something like this is a bit of a nightmare. But you can do miniature versions of it - a day or two of class where students have a choice of which problems to work on might be successful, especially if you set aside some time after to reflect with them ("which problems were easier? Which were more interesting? Look at this weird little detail, isn't that funny?").

There's a prevailing theme here, and it's the same as what a couple of answers have suggested before: there's no beauty or fun without choice. The more choice you can put into the hands of the students, the better they'll understand the material and the more they'll appreciate it. I make a habit of building this into everyday activities, whenever possible; for example, when demonstrating a problem for the students, I aim to show not just the best way to solve a problem, but a few alternatives that are a little less efficient but might "feel" better for them.

As a child I found maths extremely boring, the biggest problem was that what I was being taught had no relevance to my life as a child, after all apart from counting and spending pocket money what possible use was it to me?

Until I discovered the equation for calculating the optimal size of loud speakers for my bedroom. Finally maths that I could relate to and use, suddenly maths wasn't boring.

So my advice to you would be to try and find some way to make the maths relevant and something your pupils can relate to in their lives.

“The Mechanical Universe,” is a critically-acclaimed series of 52 thirty-minute videos covering the basic topics of an introductory university physics course.

Each program in the series opens and closes with Caltech Professor David Goodstein providing philosophical, historical and often humorous insight into the subject at hand while lecturing to his freshman physics class. The series contains hundreds of computer animation segments, created by Dr. James F. Blinn, as the primary tool of instruction. Dynamic location footage and historical re-creations are also used to stress the fact that science is a human endeavor.

YouTube

I used to watch this as a kid, and although I (still) understand none of the math, I learned that the entire cosmos can be explained mathematically. The professor is as engaging as the 70s animations are dated, but the synopsis above explains why someone who could probably care less can become enraptured.

Imagine you are put in jail. You are forced to paint a painting every day for 10 years. You have no choice in the subject: one month you paint dogs, another month you paint horses, another month you paint lampposts. The prison guard verbally chastises you when your painting is not up to their standard. If you doodle something on your own, outside of the directed painting, you are similarly chastised.

Failure to paint a sufficient number of paintings of sufficient "quality" will result in your sentence being extended by a year or two.

In this context, how can we inspire you to see the beauty in painting?

So my answer to this question is: Dismantle the oppressive system which is causing the damage.

I'd look at the books by Dunham ("Journey through genius", "The mathematical universe" are more or less general, he wrote several others) or at the outstanding "Proofs from THE BOOK" by Aigner/Ziegler. On Quora there is a section on "Beautiful mathematics", mostly elementary stuff.

Probably a search for "beautiful mathematics" will net a selection of blogs by enthusiastic mathematicians.