Reflections on a First Semester Teaching

Currently Reading: Recountings: Conversations with MIT Mathematicians

It has really been a long time since I have updated this blog. There's been some reasons for this, most of which are sad, but I'm feeling willing to take up this torch again. I have successfully completed my Masters in Applied Mathematics. So, on an academic level, I guess I'm a "mathematician" of sorts. I still feel overwhelmingly clueless about the greater scheme of the subject. I found it hard to approach this blog, since I really wanted to share my joy of the subject, but I was not feeling very joyful towards it at the end. I found my degree very hard to complete, especially when I had to take classes that I found very uninteresting. Towards the end I really felt like I just wanted to quit at times and I was also realizing that, in dismay, I was not skilled enough to go into a PhD program. Watching your peers move onto brighter things is difficult, especially when you realize that you probably shouldn't be joining them. For quite some time I was pretty bitter about this and spent a summer not reading and not doing any kind of math.

Despite all this I did realize that I had a decent knack for teaching. I was immediately hired by my University to take on the role of adjunct. I was given three courses to teach in the Fall semester, which I think went okay for my first time teaching a subject I did not like very much. I tried to make the best of it though and I find working with students can be it's own reward no matter the subject matter.

So here are some things I dealt with...

Student's education level:
The class I was scheduled to teach is something my University calls "Management Pre-Calculus." What does that even mean? Well, basically, it's what you'd expect out of a Pre-Calc course, but without the Trigonometry. ...And at the end they do stuff with matrices out of nowhere. Either way it's the lowest level math course we offer.

I find the structure of this course rather frustrating and a lot of it has nothing to do with the Math department's approach. The things they make students do is mind boggling. I literally had students in that class that had taken Advance Placement Calculus in high school, but because of their chosen major the advisers would not let them test out of the subject. This blew my mind, because we give all incoming Engineering and Science majors assessment tests for this. Turns out my saving grace for an uninteresting subject was one motivated student that had taken AP Calculus in high school. She turned out to be a very good student and hopefully a promising future mathematician. I recommended she change majors immediately, because anyone who can grasp the process of Riemann Sums as a further motivation to the process of integration really has no business wasting their time in such a class/major. Plus she really enjoyed doing the higher level math, so I always say you should try to do what you love... and accounting just wasn't that exciting!

Now let me remind you, Management Pre-Calculus is basically considered the lowest level math class the university offers. So amidst students that already had some notion of integration I had students that flat out didn't understand adding fractions, never mind factoring polynomial functions. I had some students that were repeating this remedial course for the third time! Some students came from very strong school systems, but I would say it was about 50/50 on those who could do much. 50% clearly had the ability to grasp what I taught them, which was probably a higher level than when they walked in the door. The other 50% had to basically start to learn mathematical structure for the first time. And I do mean structure, a lot of my students could easily solve problems like 2x - 1 = 3 and come up with a correct answer. However, writing out every step to show me they understood what it meant to solve the problem mathematically was a massive battle.

Fractions, the typical bane for students, was like pulling teeth. The real reason this was such a massive issue for me is because they didn't know the nine rules that created the structure of the algebraic field. The whole notion that "a x 1 = a" and 1 can equal a lot of things was truly hard for them to see. The real reason it was hard for them to recognize is likely due to the fact that they rarely were made to show this in problem solving. Not having that idea at your fingertips makes solving rational functions a real chore. The whole method of completing the square was very hard and I literally had to go over this twice, because it is a really useful problem solving technique. Then it was never even on the department final... I was so angry. It's such an elementary and beautiful technique, just looking at the geometry of it is wonderful.

In the end I hope students got something out of the course. I tried to break things down into enough detail that anyone could solve them. But that whole problem solving sophistication where all you need to know is the technique and you can solve another problem that looks like it... I think that skill set was very hard for some students to grasp. Many students became baffled once the problem changed slightly. I honestly don't know how to overcome this, I feel like this was something I rarely struggled with, so I have no idea why my brain didn't care about the change and theirs does.

Textbooks and Online Learning:
A good text book is hard to find. Almost all of them are crap. The textbook I had to use was virtually unreadable. Examples weren't completely filled out and in one instance on Power Modelling it literally gave a table and said "clearly x^2 is the correct power model". Let's keep this in context, the students I have are considered being on the lower level, how are they possibly going to see that?! I was so enraged I just wanted to throw the thing out the window. But this book had a nice fancy cover, colored graphs and lots of pictures. It also painstakingly tried to develop "real world problems" for students to solve. The first equation a student sees in this book is literally this awful thing with all kinds of decimal points... way to make someone feel this will be accessible. The best books I've read in math have none of this. There is no "touch" of a marketing department. The covers are often blank and unassuming, but these are well written. They aren't written by a team of mathematicians. They are written by one or two authors at most, so you never have a break in a writers continuity. I really resent the textbooks that are being used today. There are a few publishers I really love and trust like Springer and Wiley, but a lot of others are just trying to outsell their competition by making a flashier product.

Now another thing that is all the craze in higher education is using online programs. We use an online program system that basically just bombards students with problems. Instructors really see this as a wonderful thing, because the program tells students if they are correct or not and keeps track of their scores, so Professors don't even have to grade! My most major concern is a students ability to explain whether their logic is correct or not. I worry constantly if students are just going through the motions and not writing out every problem. If the programs are used correctly they can be quite valuable, but often I don't think students really know how to use them or use them in a lazy fashion, because no one has really told them otherwise. A lot of problem solving can be rather intuitive and you can look at a problem and solve it sometimes, but explaining why a solution is true can be far more difficult with rigorous mathematics. Also it didn't take my students to figure out that you can easily cycle through the problems and based on the number changes guess the answer correctly!

How did I fight against this? I honestly made up my own problems and assigned written homework. In the end my students seemed to really grasp what it meant to explain something and what was and was not well written. In a lot of cases I really felt like I was the first person ever bothering to teach them how to write anything. This is why math is so difficult for students. The simple fact that I collected papers and graded them myself every week went a long way, I realize professors aren't required to do this, but let's be honest, how else are we going to train people now? I heard numerous stories of high school instructors that just went around and checked off whether a student did their homework, never bothering to look at a students logic. And we wonder why mathematics is so misunderstood?

Calculators:
I hate these things. HATE THEM. They shouldn't be allowed. I was never allowed to use a calculator in undergrad or grad school. No cheat sheets half the time either on exams. The most you should ever need is a basic hand calculator for any level of arithmetic that might be needed. I am sympathetic on this end, because I don't really have a calculator brain, but I can certainly do Trigonometry and Calculus in my head. I can even do proofs! Very little of this is dependent on your ability to know what 7 x 43 is.

The purpose of learning mathematics, to me, is to train someone in some semblance of logical problem solving in a very particular way. Not all problems are mathematical, but I think it is valuable to at least have some notion of what it means to solve problems in that way. Just training yourself well enough to look at a problem and see where certain steps may or may not lead is invaluable. None of this is done on a calculator. We have calculators now that can solve our Algebra equations. I have a calculator that will do integration! Pushing buttons on a computer, I say, is not doing mathematics and as soon as you go down this road you are entirely defeating the purpose behind learning the subject. Mathematics is done with a paper and pencil. It's been that way for thousands of years and it really shouldn't be changing any time soon.

I took a very old school approach to the course and the response was incredible. I had students telling me that they though this was the first time they ever learned math. I don't think this has anything to do with my ability to lecture. I think a lot of it has to do with the fact that I took the time to grade their homework myself and write meaningful comments on how to solve the problems. The emphasis was on learning how to logically problem solve, not on getting the correct answer. Often times just solving the problem logically is enough to get you to the correct answer anyway, so we focused on that more than anything else. I also focused on writing a lot, which I think paid off in the end.

I'm scheduled to teach Calculus in the Spring semester... classes start next week. My classes are overwhelmingly filled up. I'm not a soft teacher either. The grade I gave out most last semester was a C. I think students genuinely value learning. Sure some care more about the grades... but the simple fact that they felt like they had bothered to learn something and had an instructor that cared about their ability to learn seems to go even further than a GPA. This is what university work is supposed to be about...

Does math need science? Is mathematics discovery or invention?

These are two questions I have been asked lately and I feel they are sort of related to each other, so I will attempt to address them in a single post.  I didn’t really have a well thought out response when one of these discussions was started, but I feel I have formulated a good idea of what’s actually going on.  My professor says my view of math is a lot like that of G.H. Hardy, so I apologize with anyone that heavily disagrees with this and sees things in a different light.  I will try my best to explain why I think my view on mathematics is quite accurate as it interrelates to other subjects and why it is both invention and discovery at the same time.

First off: Does math need science?

I firmly say no and most mathematicians will heartily agree with this assertion.  Does science need math?  I say yes and again many will agree.  It’s this interplay between the disciplines that generates some confusion on the nature of math.  Non-mathematicians, for the most part, seem to view math as an extremely powerful tool for explaining observed phenomena in our world.  I completely agree with this view and I honestly can’t see things being done any other way.  However, does this necessity work the other way?  Certainly, mathematics’ level of abstraction does not require our world, our universe, or anything else physical in nature to define its world.

Most of the tactics used in this discussion deals with the nature of how science proves things versus how mathematics proves things, but I will go with a different tact, because I think it is far more interesting.

In science, scientists spend time doing experiments and gathering data.  Sometimes this data will lead scientists to generalize the data with a mathematical equation.  We see this in physics all the time with things like F = ma.  But does F=ma have anything to say about any mathematical truths?  No, not really.  Please note, I am not saying that scientists can’t prove things in mathematics, this is patently untrue.  Physicists have developed wonderful things in mathematics, but they were not doing physics, they were doing math and these are different techniques.

Here’s the real crux of the problem, scientists do not get to choose their world.  Reality is what it is whether they want it to be that or not.  They do experiments to see what they can find out about the universe.  Sometimes this requires the creation of a mathematical model that helps explain a phenomenon.  This is surely invention on the part of the scientists and they are using rules of mathematics as a tool to delve deeper into this creation they have made.  However, any truths they claim to find about reality by doing math will obligate them to do an experiment to verify it.  If the experiment does not confirm the mathematics, then their mathematical model is simply not the right one.  That is all, so they move on and choose a better one.  This is how science tends to trudge along.

Second: Is mathematics invention or discovery?

It is both!

Mathematics is entirely different.  In mathematics we choose our world and we have no choice over the results.  Mathematics always begins with laying out definitions; this is the piece that is pure creation.  However, we then see what logical implications the definitions behoove us to make; this is why it feels like discovery when a new Theorem is “discovered”.

For example, if we set up an ordered algebraic field for the Real Numbers, then we further come up with definitions for surpemums and infimums… the Mean Value Theorem will result!  As long as all of the conditions exist this theorem is true and it isn’t just true it is an absolute truth.  You are guaranteed this as a result of the definitions.

Science clearly never works this way.  They always have “here is the universe, here is the earth in the universe” etc, these are the givens they must work with and now it is up to them to construct mathematics to help explain what they observe.  The other wrench in this is that there is also some percent error, so things are never absolutely true.  They may be very close, with 99.995% certainty, but there is always a degree of uncertainty.  Furthermore generally accepted ideas in science can be overturned if some other fact presents itself, or some experiment shows it to be false.  Mathematics does not need to worry about this.  Once a theorem is proven there is nothing that can ever come along to show the theorem is false.

I am not trying to downplay science.  Science is very practical and there really is no other realistic way to study the world around us.  Mathematics is a very powerful concept, but you pay a price for that power.  You are no longer bound by reality.  If mathematics were bound by reality then we would never be able to talk about things like right triangles.  There would always be some atom partially out of place throwing off that proper 90 degree requirement.  Mathematics talks about a purely mental idealization of things.  This idealization lets us talk about things that may never exist in reality and that’s all good fun, but it really is just head games.

Hopefully this has made sense and if it hasn’t, I may spend time revising it later.

First Year of Grad School

Currently Reading: “I Want to be a Mathematician” by Paul Halmos

I wish I had more time to update blogs, but I never seem to once I start one.  This is honestly the third one I’ve attempted to start and I figured a themed blog would make me more motivated, I guess not.  On the other hand Graduate School is incredibly time consuming.  I have also found it to be incredibly disheartening.  In some respects I think the main problem is that I’ve done the undergraduate degree far too quickly so it feels like there is a lot I don’t know.  Becoming good at math is more an experience oriented thing than a “natural skill” oriented thing.  I find that it doesn’t seem to matter how good you are at the subject, the length of time you spend doing it is what’s important.  Mathematics is a hugely time consuming subject and those that put the time into it have a major payoff.

I am just worried that I have done things so fast, that I will never be able to achieve the goals I want to achieve.  I don’t see myself doing very well in a general corporate environment.  The only schedule I think I could keep up with is being something like a research mathematician at a University.  I don’t think I even care if it’s a prestigious university… just a university in general would be nice.  But that job market is so utterly competitive that finding tenure track jobs might be out of reach for someone as mediocre as me.  University’s don’t care much about someone’s ability to teach the future generation, what matters is doing original research.  I am certainly not opposed to this since I think doing research would be quite exhilarating.

I feel that I could be up to the challenge.  I just hope I don’t get crushed under the weight of the academic machine that has been put into place for years.  I don’t always have the strongest grades compared to my peers, such as my friends Brendan and Eric, who I believe have far more ability than I.  However, I hold out hope that this will not hold me back for getting into other programs. As I’m reading “I want to be a Mathematician” by Halmos, I am given some comfort that he also did not have stellar grades in mathematics and also found Analysis quite challenging.  I worry my time-line is too fast for the system, but I am reminded of the story of Leibniz who only studied math for a mere five years before turning to original work.  However, Leibniz wasn’t obligated to have a thesis advisor, take a Math GRE, and things of that nature.

Matters have become more depressing, because I have also recently lost my job at the lab I worked at.  I am going to try and look at the bright side of this.  I am going to buckle down and try to solidify the math I’ve already learned at my professor’s recommendation.  My professor, who we affectionately call Kiwi at his insistence, has pointed out that I should know things more quickly than I do, and I really believe that is true.  Hopefully I will have the diligence to amass more skill with this newfound free time.

Also I am hoping to add much more to the blog.  I have recently finished a course on Partial Differential Equations and whenever I searched for useful examples on the web, it was impossible to find anything that had a lot of detail.  Many steps were frequently skipped or not even explained.  I want to spend some time writing up my own solutions and post them on here.  I am debating if I want to teach myself LaTeX in order to publish these or if I should write them up in Mathematica and then just convert the document to LaTeX as I learn that language.  At least in the latter case the brunt of the document writing will be finished… we’ll see.  A decision for next week maybe?

My Review of Atlantic Oskar 1080 CD, 504 DVD Multimedia Storage Tower in Maple or Espresso

Originally submitted at Cymax.com

The Oskar 1080 Media Tower has a clean and simple traditional design ideal for any contemporary home that needs a little help organizing favorite music and movie selections. With plenty of room for CDs and DVDs, this media storage tower will be a welcome addition to any room in your home. Features:...

Atlantic Oskar 1080 CD, 504 DVD Multimedia Storage Tower in Maple or Espresso

CD shelf for Books??

By Adam the Mathematician from Lowell, MA on 10/6/2011

 

5out of 5

Pros: Adjustable shelf heights, Attractive Design, Easy to build

Cons: Adjustable Shelves Thin

Best Uses: Cds, Small Rooms, Dvds, Video games, Books, Large Rooms

Describe Yourself: CD Collector, Movie Enthusiast, Book Collector, Video Gamer

Was this a gift?: No

I now have three of these shelves in my apartment. I bought one many years ago and I was sad to see that they have changed the design slightly when I decided to buy more. The shelves are much thinner than the original design, so they are not as sturdy as the original one I purchased.

That being said, I've decided to re-purpose these shelves to start handling my collection of paperback books. CD/DVD shelves are simply the perfect width for these things and you can fit hundreds of average sized paperbacks on a shelf! Rather than using a typical wide bookshelf to store these types of books, these shelves are perfect for this purpose.

I am even planning on buying a shelf to put hardcover books on, though I am slightly nervous about the shelves holding the weight. I suspect an average sized fiction hardcover should hold up just fine, but these shelves are simply unsuitable for my text books. In that regard I use a typical larger sized bookshelf. A regular fiction hardcover sticks out only a few inches beyond the shelf, but I find the width is still perfect to manage these books as well. We'll see how my future shelf buying goes over the year... I project I need 5 more of these! One will be for my DVD's/Xbox games though.

(legalese)

Update on Work to be done

I thought I would update since I hadn't posted in quite some time. I've just finished my Undergraduate Degree in Applied Mathematics, and I've just been awarded the Teacher's Assistant position for the next two years. This comes with a bit of a tutoring stint, so I've become rather motivated in getting my tutorials up to par for next semester. Maybe if direct tutoring doesn't help students my tutorials can help them further. Also it will help me review the material quite a bit so that I am well prepared to tutor students for higher level math. I suspect most people will have trouble with Calculus and usually the reason that is true is because people never really do well with Algebra. Despite usually passing an Algebra class at the High School level... it does not mean they've learned enough to succeed in Calculus.

Also, I am thinking about switching programs for my Tutorials. Right now I am writing them in Mathematica, and they look fine, but there's a professional quality that is just sort of missing. After speaking with my professors, they made a good point about how it was designed. If you design something to do this one thing and then you want it to do this other somewhat unrelated thing... then it's not going to perform as well as you'd want. I feel this is true, because even though Mathematica is on version 8 at this time, I would still not consider publishing documents with it. The formatting for written documents is really frustrating to deal with. Instead I'm probably going to switch over to LaTeX, which is very widely used in the mathematics community. I just need to learn all the syntax for writing up the documents and then I will start switching everything over. For the time being I am still going to use Mathematica to write my tutorials, luckily Mathematica can be extracted into a TeX file so I can convert it when I know more about writing in LaTeX.

That's about it for now... I do have some interesting articles brewing in my head, but for now I think I'll be focusing on writing the tutorials for a while. They actually take me a long time to write, so they are rather time consuming. Anyway, I'm off to write another one now.

Added New Pages

Currently Reading: Mathematical Sorcery: Revealing the Secrets of Numbers by Calvin C. Clawson

I've got a real treat in store today. I've taken some time to build more pages into this blog. I'm going to be writing and publishing tutorials that I've written using Mathematica. I'm trying to take a more analytical approach to these topics. Rather than list a bunch of theorems or definitions like a typical text book, I try to work through the topic as if I am discovering it on my own. The reason I'm leaving out the intensive rigor is because that has been done time and time again via textbooks. I highly recommend learning the rigor because that gives and added level of depth you just can't get otherwise. However, I think the tutorials, as they stand, will be helpful for many to garner some insight.

I don't want people to misunderstand my approach, so I've included a "Recommended Reading" tab, so that you can go and find where I am getting a lot of my influences. I've read multiple text books on the same topics so I'm only recommending the ones I think are the best. I'll be including links to Amazon.com where you can usually find textbooks at more reasonable prices. I realize over time this could get outdated with new editions, but the books' quality still stands so it may not matter what edition you decide to get, should you decide to purchase a book.

I am also going to try and list some "fun" books on the topics I've come across. I'm going to try and build these recommendations around people that might not have any math background, so the reading should not be overwhelming. My intention here is that this will hopefully get people interested in the subjects and perhaps get the confidence to delve into the topic even though the readers confidence may have been squashed by years of rote learning in classroom settings that didn't help people learn anything.

In the future I am going to include executable webMathematica programs where users can launch a page that you can actual practice topics discussed in the tutorials. All of this right from the web. I am also considering including video tutorials of me actually giving lessons on a subject. That might be much further down the road though. Like a year or more away.

Anyway, I hope people enjoy and appreciate the work I've put into building these things.

The Public vs. the Language of Mathematics:

I’ve got another excellent passage from Oliver Heaviside’s “Electromagnetic Theory”. In this one he talks about the merits of mathematics and some of the things the public (in my observations) get out of the subject.

I read a rather insightful book about how the education system treats children. It’s written by a PhD Mathematician, so it carries some weight. However, I am not sure how widely read it is outside of the university community. The book is called “A Mathematician’s Lament” by Paul Lockhart and I definitely recommend reading it if you get the chance. It’s not a high level math book by any means, it is just a good read to help get insight into the world. I have read reviews on Amazon insisting that education theory has been applied better than when this book was written. However, I should mention my mother is a high school math teacher and her curriculum has only been changed by standardized tests. So, whatever new “theory” these people are professing… people are still losing out on math in literal droves. As Paul Lockhart put it “In school you learn that math is not something you do, it’s something that is done TO you.” Lockhart’s essay is post year 2000, however, I find it very interesting to read similar commentary from Heaviside in the 1890’s, over a hundred years earlier.

“‘Mathematics is gibberish.’ Little need be said about this statement. It is only worthy of the utterly illiterate.
‘What is the use of it? It is all a waste of time. Better be doing something useful. Why, you might be inventing a new dynamo in the time you waste over all that stuff.’ Now, similar remarks to these I have often heard from fairly intelligent and educated people. They don’t see the use of it, that is plain. That is nothing; what is to the point is that they conclude that it is of no use. For it may be easily observed that the parrot-cry ‘What’s the use of it?’ does not emanate in a humble spirit of inquiry, but on the contrary, quite the reverse. You can see the nose turn up.
“But what is the use of it, then? Well, it is quite certain that if a person has no mathematical talent whatever he had really better be doing something ‘useful,’ that is to say, something else than mathematics, (inventing a dynamo, for instance,) and not be wasting his time in (so to speak) trying to force a crop of wheat on the sands of the sea-shore. This is quite a personal question. Every mind should receive fair development (in good directions) for what it is capable of doing fairly well. People who do not cultivate their minds have no conception of what they lose. They become mere eating and drinking and money-grabbing machines. And yet they seem happy! There is some merciful dispensation at work, no doubt.
“‘Mathematics is a mere machine. You can’t get anything out of it that you don’t put in first. You put it in, and then just grind it out again. You can’t discover anything by mathematics, or invent anything. You can’t get more than a pint out of a pint pot.’ And so forth.
“It is scarcely credible to the initiated that such statements could be made by any person who could be said to have an intellect. But I have heard similar remarks from really talented men, who might have fair mathematical aptitude themselves, though quite undeveloped. The fact is, the statements contain at once a profound truth, and a mischievous fallacy. That the fallacy is not self-evident affords an excuse for its not being perceived even by those who may (perhaps imperfectly) recognize the element of truth. But as regards the truth mentioned, I doubt whether the caviler has generally any distinct idea of it either, or he would not express it so contemptuously along with the fallacy.” (Section 9, Electromagnetic Theory, 1893)

I think the first part of this is great and really illustrates the divide some people have. Mathematics is, literally, its own language. The array of symbols used in mathematics indeed looks like gibberish to the uninitiated. But take someone that only speaks English and send them to Russia and you have the same problem. People are incessantly lazy, and learning new languages later in life is actually quite difficult (well for most people). Not to mention a language that is based on pure logic where certain “slang” is not allowed, but it is not different than any language. Such as in English “I can’t not do that”, grammatically makes no sense. Just as 2 = 1, mathematically makes no sense. If you want to start incorporating “slang” in math, you need to prove it makes sense or define it in such a specific way. Take the concept of a gradient for example? I won’t bore you (yet), but it’s sort of similar.

Like any language it has evolved from many rudimentary levels, but, like any language, it has great descriptive power for what is happening. If you ever attempt to read old school math proofs from something like Euclid’s “Thirteen Elements” you won’t see anything like you see in modern math. Instead you see long droning paragraphs setting up a situation that would essentially say something like “sin (x + y) = sin(x)cos(y) + sin(y)cos(x)”. Isn’t that way easier to look at than a long paragraph explaining the same thing? When you look into math, you have to remember it really is short hand for a giant paragraph discussing a specific situation. While it may seem intimidating with all the crazy symbols, trust me its better. Read Newton’s “The Principia” to find out…

In this treatment I am not sure Heaviside is completely clear towards the end. However, he is attempting to illustrate the power of Mathematical Theory versus Application. This reminds me of an interesting thing concerning Tesla and Edison. In America we tend to venerate Edison as the “great American inventor”. However, very little appears to be known as to how inefficient his methods were (never mind how horrible of a person he actually was, but that’s another topic). Anyway, the Edison method can probably best be described as the “brute force” method, as one of my math teachers would have said. Basically, if you want to make a light bulb then you keep trying every kind of material for a filament until something finally starts to glow. It’s pretty messy and not a very nice way to go about doing things. Tesla on the other hand would do the math; he’d study the physics, the heat transfer of materials and so forth. He’d come to solutions much faster than his contemporaries and on top of that they are elegant solutions. This is why most people now work in this fashion and why the concept of Applied Mathematics is so powerful. I think people sorely miss out on that kind of power.

I don’t want to lead anyone astray; this stuff can get real hard. In fact if you are doing mathematical research sometimes you might delve into a particular aspect of the field that has five other people that MAY know what you’re talking about. But this is not a problem exclusive to mathematics; you can find it in about anything, even something like Economics. I even encountered Economics text books that are using Quantum Mechanics to help explain what is happening for Risk Pricing in the stock market. If that’s not a mind bowing application of Quantum Mechanics… then I’m sure there are many others, but I thought that was an interesting one.