Mathematicians and other math junkies: lets talk about the study and practice of math

In My Humble Opinion
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Here, have a seat… I just put a pot of coffee on, and there’s mineral water and soda in the fridge. There may even be a couple of beers left over from when Mike and Gail stayed over on their way to Cupertino… please, help yourself.

I want to talk about mathematics, specifically the study and practice of math in the real world, and get your opinions on how to best to prepare for a career in math.

I’m not going to give my life story, but the condensed version is this: I never took math in highschool past very basic, elementary algebra: converting fractions> percents> decimals (which I’m aware isn’t algebra), 3x + 4 = 13, solve for x, etc. I chose a career path that almost fully removed me from having to practice any sort of math – to the point I couldn’t remember how to do basic multiplication and division.

I returned to school thinking I would pursue a degree in business, but found while taking the needed remedial math courses that I absolutely love mathematics. I’ve now decided that I want to pursue mathematics as a career, but that brings up a couple of issues.

First, while I’ve earned straight A’s in mathematics so far, I haven’t really done any advance work – I’m still studying highschool level algebra. I understand how to do it, but these, naturally, are crash courses in remedial math. Therefore, we are shown how to solve, say, a quadratic equation, but not why it’s important. For instance, we were taught Euler’s substitution, but not given any real-world examples for why this is useful. Later, I found online the much easier ‘addition method’ of solving such equations, and why and how these figures apply in various aspects of our daily lives. However, I still feel that the school is skipping over much-needed instruction.

So, did I miss out on vital information by not taking math in highschool? I guess it’s a moot point, because I still need to have a better understanding of why this stuff is important. Another example: graph & shade: -2x + 3y is greater or equal to 6, and y is less than – 1/2x +8. Ok, I can put that on a Cartesian plane, get lots of points, and pass my math test and make the honour roll. But I don’t know what it means… and of course the math textbook is mum on the subject.

I find this stuff fascinating, but like black magic or certain political beliefs, I just can’t seem to wrap my around the why’s of it. Again, I can solve the equations, but I’m having a hard time understanding the real-world (or, even how it applies to more advance mathematics) applications.

Second issue: I’m 31, and have no desire to throw away big bucks and many years on a college education with a major that will not provide a good return on investment. So, the obvious question: is mathematics a good career choice? There seems to be a high turnover of math instructors at my community college, but I’m also hearing that the US is woefully short of math and “science” teachers.

So my two questions:

  1. How do I become a better math student? Passing test means little if I don’t understand the concept. Manipulating numbers and variables is a worthless endeavor if I don’t understand why I’m doing it. Considering I’m looking down the barrel of trig and calc over the next year, I really want to have a better understanding of why I’m doing what I’m doing.

I’m very much a book learning and hands-on kinda guy. I feel math textbooks are the most useful, closely followed by visual instruction. So pouring over a math text, then watching a how-to video from something like Kahn Academy seems to help. But again, I’m really interested in the theory and practical applications of math, even at my admittedly currently low level. Are their certain textbooks that have appeal because of their ease of understanding and thorough coverage of the material?
2) Let’s talk jobs. I want to be an educator, and have for a long time. The idea of teaching math to high school or college students is quite appealing. Is this a pipe dream? As far as I can tell math jobs are found in teaching, government and the computer industry. Is this an accurate assessment?

When I tell people I’m thinking of studying as a math major I’m met with incredulity or big guffaws and finger pointing. Nobody, but nobody at my little white bread school is a mathematics major as far as I can tell. Those who like math are majoring in other fields: engineering, business, accounting… they have no interest in math for the sake of knowing the beauty of math.

I’m curious what math boffin Dopers went through: their experiences with school and employment, and of course any opinions and thoughts on my chosen future career and opinions on how to maximize my return on an investment that, while I’m good at it compared to the average college student, is not without difficulty.

I hope the coffee was good.

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I relate to you very much. I also love math but dropped out of school young. Today I find myself seeking out hobbies that employ math, for instance I am designing a catapult that needs to throw a 10# object 500 yards. How much potential energy do I need to do that. How many g forces will be in the throwing arm. What demensions will I need for a throwing arm. What velocity will it take to reach 500 yards. What kind of efficiency can I expect and where do I expect to loose it. These are good excersizes for basic math.
I started doing the khan academy a while back and am sailing through it easily but unless I find away to apply the rules to a real life problem it doesn’t stick long after the test.
Last week I built a moonshine still just for fun and enjoyed the math used in the sheet metal layouts. Just solving a problem as written is very boring compared to appraching an actual problem and formulating your own solution.

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It’s sort of a catch-22 for students like the OP. You can’t grasp the concept without real-life examples but in HS you don’t have much in terms of real-world experience. Once you attain enough real world experience, you’re already shaky in the foundation.

Take the shading example.

Most teachers or textbooks at some point probably did try to explain the shading at some point with a real world example. The most common application would be 2 components needed to assemble something made in a factory. Cogs and widgets make a toy and you need 2 cogs to every widget to make toys. This can be graphed and shaded accordingly to show if you have enough/not enough/exactly the right amount of cogs/widgets in your inventory. Your eyes probably glazed over even now in reading the example much less your high school selves.

Cogs and widgets aren’t exciting unless you work in a cog and widget factory. I suppose they could change the wording to include 2 thighpads for every helmet or 2 cute pairs of shoes for every scarf but even then it wouldn’t do much. If you can’t be interested to learn math without it directly and personally speaking to your personal interests, it means your attention span is a bit wanting.

Not a judgment. I applaud **HoneyBadgerDC **and his ‘I’ll use my math however I please and learn it whenever I please’ approach. That is arguably a more productive approach but impractical for a school setting.

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There are other jobs that use mathematics; for instance, I work in the financial industry and insurance companies employ actuaries. (Not surprisingly, many jobs nowadays use computers, so saying “the computer industry” is sort of all-encompassing, depending on what you mean.)

I don’t think it’s a pipe dream to become an elementary or high school mathematics teacher starting at age 31. A lot of other jobs would require graduate school, and I don’t know how keen you are on that. Like you, I loved math in high school, but by the time I finished my master’s degree I had reached my “level of incompetence”, so to speak. :slight_smile: YMMV, of course.

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Algebra is a language. It is almost meaningless without calculus. Algebra is the language of calculus. And it is very beautiful. Calculus rocks!

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The big problem is also that what math is relevant depends immensely on what profession you’re going to be taking. Nobody is ever going to come up with an example of when knowledge of surjective functions is going to be useful while repairing your car, shopping for groceries, or taking care of your kid.

I had to use a hyperbola the other day and I almost fell over. Of course, the context was that I needed to make an XOR function linearly separable for machine learning, but that’s because I’m in a math heavy field (speaking of eyes glazing over).

The fact is, if you’re going to be a manager or an HR rep or any number of fields nobody is ever going to come up with a great reason why conics or finding the orthonormal basis for a Euclidean space is going to be useful to you. Hell, my mom has been an accountant for years and says she basically never uses anything more advanced than arithmetic.

A lot of the real world applications are buried beneath layers of reasons, and those reasons themselves are often based on even more complex math than you’re already doing (at least until you’ve learned calculus and linear algebra). Even if you’re going to be a mathematician, there’s a good chance that you’ll rarely if ever use entire subfields of math. I had a math professor (actual professor, not just a lecturer or grad student) once who said he had to look up trig stuff because in his subfield it almost never came up.

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Could you give us an example of a real life problem we might use calculous to solve. I will be getting into it soon if I stay with Khan.

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HoneyBadger – it really depends on what you mean by “real world”, I could give you plenty examples in statistics, or machine learning. I could give you somewhat silly examples of finding the area enclosed by a fence with a shape that exactly matches some easy-to-integrate mathematical function. But the chances you’ll ever be walking about in the woods or at the grocery store and yell “holy shit, I could use calculus for this!” and whip out your pen and paper are vanishingly small.

I’ll give a fluffy answer and say that rather than the actual math itself, the concept of a derivative is probably the most useful thing I’ve gotten out of calc for non-academic use. The idea of rates of change, and rates of change of rates of change, is a very useful concept to be able to grasp, and I think derivative calculus gives you a cool way of thinking about it critically. Also, maybe limits, the ability to think about how something would behave if it kept going until some point that’s either unreachable (i.e. infinity) or unattainable (i.e. dividing by zero).

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The concepts you learned in arithmetic will be very useful in algebra. The concepts you learned in algebra will be very useful in trigonometry, calculus, and other higher math. These are all interconnected. (ie solving trig identities is basically substitution and manipulation of equations which you learned in algebra)

My students would often ask me about the real life application of mathematics. I for one once asked this same question. You can think of a lot of practical use of arithmetic but how about algebra or trig? Would you be using x and y when you go buying in the market? Would you really be in a situation where you have to measure the angle formed by the ladder as it leans on the wall? It seems like you will only be using these knowledge if you become an engineer, a statistician, or a math teacher. But for the average student whose goal is just to finish school, mathematics is just another subject they need to pass.

A great teacher of mine, one of my inspirations, answered this question for me. She said that we are not going to use everything we learned in math in real life. Most of them we will never get to use. But the thing is, studying mathematics improves our way of thinking. We become logical. We become critical and analytical. These skills will help us in decision making. These skills will help us solve real-life problems. These skills will help us in life. And that is the reason why we study mathematics. And this is the reason I give to my students now.

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Not too very long ago, I wanted to find the center of mass of a pyramid. (Much like the Great Pyramid of Egypt, but, no, that wasn’t the actual object involved.) I’m sure there’s an elegant geometrical way of figuring this out, but, for me, it was easier just to set up the integral and solve it. Sum of a big pile of diminishing squares.

(Jeez…I just described my old workplace…) :wink:

It helps a lot if you can find it in you to love math. Read the old “Mathematical Games” essays by Martin Gardner. Play with it for fun. Solve problems you don’t need to solve. Go beyond your homework. Write simple computer programs to simulate math-like things – predator/prey sims, or the volume of n-dimensional hyperspheres.

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I can share some experiences which may help but may not be what you want to hear. The highest levels of math I’ve taken are calc w/ analytical geometry, matrix algebra and a mixed bag of techniques called operations research which involves things like queuing theory (poisson distributions), shortest route estimation, Bayesian decision analysis and things of that nature.

What I can tell you from my perspective is that while some concepts seemed utterly natural and intuitive others frankly seemed like magic. Beyond that, some of the proofs I worked in calculus, while they made sense, simply blinded me with brilliance of insight that must have been behind them. Only you get to the higher levels, you have a tendency to feel very small and alone. However it is the same type of small aloneness you feel looking at a galaxy - at least that’s how I feel. And over time, you may even start to see pieces fitting together which let me tell you is an absolutely glorious feeling.

But the mental pursuit alone is also something to keep a hold of even if you don’t pursue a career. I’ve been working on a problem for years and although I have yet to close the deal, the mere exercise has been both intensely frustrating and exhilarating.

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Ok. Say you live near a lake and you want to know how many gallons it holds. Plot the curve of the lake, take it to the limit and there is the answer.
A pitcher throws a 98.7 MPH fast ball to the plate. How many seconds does the batter have before it reaches the plate? Stay with Khan!

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I wouldn’t say that second one requires calc to solve. Not unless you want to take into account wind and air resistance at least, which most people wouldn’t (especially since the deceleration due to air resistance would likely be negligible).

And really, can anybody think up a good example where calc is exceptionally useful in Joe Everyman’s life that isn’t about the pursuit of knowledge for knowledge’s sake? I seriously want to know, but most of the examples people have given so far are about figuring out trivia for yourself. Interesting, to be sure, but nothing too useful. Something where knowledge of calc would drastically reduce the time it takes for Joe to do something an average person is likely to have to do, or save him a lot of trouble? I’m not being anti-math here. I just can’t think of anything.

I mean, other than cowching’s “it teaches you how to think logically” which is a good point. I mean a specific mathematical process or technique.

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Thanks for the replies, everyone. Don’t go away yet; I’m posting from my phone right now but will address specific issues when I have access to my laptop in the morning. There’s some things in this thread that I want to discuss further.

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I really don’t think there’s anything even in just basic trigonometry that Average Joe will ever actually need.

But when does Average Joe ever require having read Shakespeare in his every day life? This is the big dichotomy. Average Joe has no problem with the fact that his Shakespeare knowledge has no practical use but nonetheless enriches his life. But Average Joe demands math to have practical value or else it is a waste of time. In grade school, math never seems to be taught just for the sake of knowledge, it always has to justify itself as a tool.

We seem content with the idea of extending basic literacy into something beautiful beyond practical value, but don’t allow the same for math, despite math having all the same potential for beauty and wonder.

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Right, I’m not arguing learning it is useless, I’m just pointing out that the question “when am I ever going to need this in the real world?” isn’t really a question you can answer satisfactorily. You have to give the same fuzzy answer about “teaching you how to think” and “viewing the world through different perspectives” you do about literature analysis and 14th century East Asian History. Of course those are wonderful things to learn, and I’m in no way arguing we shouldn’t teach them. I was just trying to point out that there’s not really a great answer to the question “what’s a real world situation where I’d use this?” Because you’re never going to have to integrate anything, nor is being able to integrate, specifically, going to save you a ton of effort or time. All of the benefits are tangential (heh…) to the material itself.

Edit: Though you can make a case that at a certain level calculus is good to know so you can call BS on a news article or advertisement using calculus incorrectly to mislead you (i.e. to justify some technobabble junk science).

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You want a real-world example? I’ll give you a real-world example – involving a little bit of calculus and a lot of algebra. (Any little bit of calculus ALWAYS involves a whole lot of algebra. Get used to it.)

I was an assistant at a laboratory, where we studied dolphins. And we also sent a research team up to Alaska every year to study humpback whales. The bays and inlets were full of whales, but one year the migrating whales largely abandoned one particular bay. (Forgot which. This was 25+ years ago.) Various grants were given for various research projects to try to find out why. Ours was one of those projects.

Our team sat on a bluff top overlooking the bay, with a theodolite. (Sort of like a surveyor’s transit.) Every time they saw a whale stick its head up, they took a sighting and recorded the time and coordinates. Also they took a sighting of a nearby boat and recorded the time and coordinates. From these data, we could reconstruct the path a whale was following, and the path of the boat.

Our research question: Were the whales disturbed by the boats? Could we find evidence that they altered their paths when boats were about, compared to when there weren’t boats about?

SURELY you can see the importance of calculus for a question like that! :dubious:

More specifically: We wanted to know exactly when each whale came closest to a nearby boat, and how close it came. We wanted to determine if the whales seemed to be altering their paths to avoid boats.

I didn’t know much calculus at the time. Just enough to know that this is exactly the kind of problem that calculus is for. Given some function, determine the value that you can plug in, to get the minimum (or maximum) value out. Here, the problem was to develop an algebraic equation (from all that data) giving the whale’s distance from the vessel at any point in time.

[Insert about six pages of messy algebra here.]

Then, given that, use a little calculus to minimize the function – that is, find the input value (a time, in minutes or seconds or whatever) that produces the smallest output value (distance between whale and boat at that time).

I knew enough calculus to know that this was a problem for calculus. But I didn’t know enough calculus to actually solve it. For that, guess what I had to do.

I enrolled in the local community college (this was in Honolulu, Hawaii, circa 1983) and took Calculus I. Upon completion of the class, I knew exactly how to solve that problem.

My assignment was to write a computer program that would read all this data (for lots of whales and lots of boats, collected over a week’s time) and do all the calculations to accomplish this.

Stay tuned, and I’ll tell you what parabolas had to do with all this!

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Conic sections in real life!

So, in high-school algebra, you learn those formulas for conic sections. Hey, this is classic stuff! The ancient Greeks studied stuff like that!

One of the simpler conics is the parabola. You’ve probably seen problems with them, from first-year high school algebra. The problems typically go something like this:

Somebody tosses a ball to somebody else. You’re given an equation that somehow describes the ball’s trajectory – usually, giving the height (in feet) for any given length of time (in seconds) after it’s thrown. If you graph that, it’s always a parabola that “opens downward”. You’re asked questions like:
(a) How many seconds after it was thrown was the ball at its highest height?
(b) How many feet above the ground was the ball at its highest height?
© How far (horizontally) did the ball travel before hitting the ground?

In beginning algebra, you’re given some straightforward formula to compute the answers to those questions. That’s simple algebra. But where do those formulas come from? That’s where the calculus comes in! With calculus, you learn how to derive those formulas to maximize and minimize functions. Then, if those formulas are simple enough (like they are with parabolas), they can be taught to beginning algebra students, but there they just give you the formulas and tell you “That’s how it is. Now memorize it!”

As it turned out, that big hungus minimization problem with the whales – surprise, surprise – got really simple by the time I was finished with it. It took me six pages of messy algebra to develop the formula for the distance between whale and boat, as a function of time. Then, by the time I got it all simplified down, guess what! It turned out to be just an equation of a parabola! But I had to do all that work, just to discover that!

Once I got that far, then it was easy to do the actual minimization. I could have used various slick techniques to find the minimum value (closest point of approach). But it was just a parabola. And to minimize that, there’s just that simple little formula from beginning algebra. (To be sure, I had to look it up to remind myself of the exact formula.)

By the way, “just a parabola” does NOT refer to the path of the whale, or the path of the vessel. It simply refers to the form of the equation I ended up with, that gave the distance separating the whale and the boat at any given time. It wasn’t something that you would have actually drawn on any graph.

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But wait! There’s more!

Those whales and boats were often several miles away from the guys on the bluff top with their theodolite! (I find it hard to imagine that you could even get a good view of whales from that distance, but somehow they did.)

So the Principal Investigator (a grad student) asked me to compute the distances, taking into account the curvature of the earth! I thought that any error that we made by not doing this would be insignificant. And the P.I. agreed. BUT. Hey, other researchers were doing it, so they were going to point out how our work was flawed if we didn’t. So we had to do the same.

We started (both the P.I. and I) by poking around to see if we could find a formula for it. I went to the library and poked around in various algebra and trigonometry textbooks. The P.I. chose to poke around in various surveying textbooks that he found. We never did find a formula that did what we wanted.

Given to points on the earth’s surface, it’s easy to find the straight-line distance between them. We wanted the distance along the curved surface of the earth.

So I had to get creative and develop a formula myself. (A little trigonometry helped out here.) The raw data was something like:
(a) Azimuth angle of the point. (That is, degrees from north of the point, as seen by the observer with the theodolite.)
(b) Declination angle. (That is, degrees below the horizontal from the observer to the point. Remember, the observer is on a hilltop, looking down at the water.)

So from this, we needed to get the x-y coordinates of the point (whale or boat), AND its distance from the observer. And the distance had to be from the bottom of the hill-top (the sea level point directly below the observer’s feet), along the curved surface, to the whale or boat.

I developed a formula for that, and it was rather messy, and involved trigonometry. I had to look up the earth’s radius, since that was needed as a constant in the formula. The altitude of the hilltop above sea level was another number in there somewhere.

The moral of this story: Mathematics, at one level, means knowing and applying formulas that you learned in school. But you’re just learning formulas that somebody else had to figure out from scratch (like that formula for how high the tossed ball got). But on another level, it’s about knowing the rules well enough that you can develop the formulas for yourself, just from knowing the requirements of the problem. Once you get into calculus in particular, there’s a whole lot more of that.

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This is an especially good reason why everyone should learn some statistics! Statistics is one branch of math that is especially commonly used and abused to confuzle and bamboozle.

How often have you heard TV commercials that say something like:
Clinical trials have proven that Crunchy Brand toothpaste significantly reduces the number of cavities in your children’s teeth!!!

It helps to know some statistics, just to know what bullshit that statement probably is. Here’s the catch: In the world of statistics, the word “significant” has a very special and specific technical meaning, and it’s not what most people think it means. So that statement doesn’t mean what most TV viewers think it means!