Most Useless Math Topics for Experienced Scientists & Educators

[JohnDubYa]

For the experienced scientists and educators, what are the most useless math topics from a practical standpoint. I would list my favorite:

1. Polynomial long division. If the numerator and denominator don't factor, what is the use of performing the long division?

I know of only one -- to find the leading order behavior of a rational polynomial. But who would care about that other than a theoretical physicist or chemist?

 

[Nexus[Free-DC]]

I dunno. Sometimes if you have a polynomial you can spot one factor real easy, but the others are hard to find. It becomes easier if you divide through by the factor you've found.

 

[abertram28]

i think it applies more readily when the numerator is of signifigantly higher order and a simple factor is hard to remove visually. also, checking work for removing a factor from a larger order polynomial.

younger kids are taught to write out the whole part separate and not to have a higher order on top. also, i believe, that some kids can't complete squares easily, so they need to use polynomial long division to get the +d-d part as a -d remainder. but i don't do that, so i can't remember why they taught it to me specifically.

basically its just used to convert improper rational expressions as a proper rational expression plus a polynomial. id suppose its hand for integration using ln x and u'/u formations, since the top is reduced to an order less than the denominator. i still like trying to find factors and completing the square just by looking, but i can see how some kids dont.

 

[Zorodius]

Rather than a specific subject, I'd like to nominate the generic set of all problem solving techniques where students are taught something incredibly overspecialized simply to solve one certain class of problem.

For instance, "mixture" problems. (John has two containers of punch. One is 5% juice, the other is 10% juice. How much of each should John mix together to get a 10-liter solution of 7% juice?) Rather than using these problems as general examples of how to describe and solve mathematical relationships involving percentages, students are frequently taught to set up specialized grids that include the given information, and then compute the missing information using the pattern of the grids. Essentially, rather than being taught problem solving skills, students are taught yet another step-by-step process to follow, without understanding why it works or how they could develop a similar process on their own for a different sort of problem.

I can understand why this would be appealing to math teachers - anyone can learn a process through repetition and reproduce it on demand, given sufficient practice, but teaching someone a concept is harder. Still, I think it's a disservice to students to do it that way.

Also - unrelated to the previous point - quaternions are a subject that will be of no use to most people, even in the sciences, although they do produce elegant solutions to certain problems.

 

[matt grime]

the geometry of the quaternions (and related divisiona algebras over C) looks like it will be of use in theoretical physics.

 

[jcsd]

Is:

\frac{x^2 +12x +30}{x+7}

analytic at x = -7 ?

can the singularity be removed?

I rest my case.

 

[matt grime]

you probably ought to have chosen a better example where -7 was a root of both top and bottom with unknown multiplicity.

 

[HallsofIvy]

A very practical application: if P(x)/(x-a)= Q(x)+ r (r is the constant remainder) then
P(x)= Q(x)(x-a)+ r so P(a)= r. Any computer scientist knows that calculating r by (synthetic) division is faster than evaluating P(a).

 

[JohnDubYa]

Good catch, Halls. I had forgotten some of these applications (although I think few students will ultimately need to know them).

RE: " Essentially, rather than being taught problem solving skills, students are taught yet another step-by-step process to follow, without understanding why it works or how they could develop a similar process on their own for a different sort of problem."

Here! Here! I noticed these techniques cropped up with the onset of new math, and I find them abominable. Why not just reason it out? Sure it's hard, but math is hard.

 

[fourier jr]

Polynomial division is good for curve sketching too. Try finding a slant asymptote (if necessary) without long division.

 

[Goalie_Ca]

Functions and relations have got to be down there. Calling a function a subset of a relation is all that was about IIRC. Pretty stupid if you ask me. But i guess math people like formalizing things.

I wasn't impressed with at least half of the discrete math I've taken thus far.

edit: ooh, i got to go back to calculus two for this one. Rotating things about an axis to calculate surface area and volumes. It's much faster and easier, and works for all conditions, if you do it via triple integration and everything.

edit2: Now that i remember way back to calculus 2, i remember 1/2 of it was trivial integration techniques. Trivial in the sense that they were very limited in uses. Most of the time nowadays i try to use maple/matlab to calculate things. I seriously doubt anyone still remembers those ways of integrating those nasty partial fractions (or something like that) using triangles and trig.

 

[Math Is Hard]

Sure it's hard, but math is hard.

Hey! Leave me outta this!

 

[matt grime]

Although it pains me to say it Goalie ca, many mathematicians do not think of relations in terms of subsets ("orbits" would be my view of them), but that notion you dislike is useful for computer science, certainly for some languages. You define the relation as ntuples, then evaluate expressions on entries in the elements. So is.father.to( , ) is a list of father child pairs, and one can call functions such as father.of( ).

But then there is the other reason for defining functions in terms of sets. If you don't, what are they? Letting one of the many secrets out of the bag, they way you're taught about functions for most of your life is mathematically unsound. One needs only look at the number of questions I see written by supposedly good mathematicians that ask something like is the function 1/x from R to R continuous everywhere. It isn't even a function from R to R.

 

[Zurtex]

edit2: Now that i remember way back to calculus 2, i remember 1/2 of it was trivial integration techniques. Trivial in the sense that they were very limited in uses. Most of the time nowadays i try to use maple/matlab to calculate things. I seriously doubt anyone still remembers those ways of integrating those nasty partial fractions (or something like that) using triangles and trig.

Surely the writers of such programs as MATLAB do...

I'm only very much at the start of my mathematical life and am at the moment revising such integration techniques. But the way I always see it is rather than having a set problem and a set solution is that you have a logical problem and you need to try and find a logical solution, thus getting in the mind frame for solving any particular problem.

 

[Hurkyl]

I dunno, I can do a lot of those integrals from Calc 2 faster than it would take to fire up a computer and run Mathematica.

And, sometimes, knowing a technique can be important even if you never use it; for instance, I've seen theorems in my Complex Analysis and Algebra courses whose proofs would be totally mystifying if you didn't understand partial fractions.

 

[HallsofIvy]

Although it pains me to say it Goalie ca, many mathematicians do not think of relations in terms of subsets ("orbits" would be my view of them), but that notion you dislike is useful for computer science, certainly for some languages. You define the relation as ntuples, then evaluate expressions on entries in the elements. So is.father.to( , ) is a list of father child pairs, and one can call functions such as father.of( ).

On the contrary, many mathematicians do. Mathematicians prefer to have a variety of ways of thinking about the same concept.

 

[krab]

... But who would care about that other than a theoretical physicist or chemist?

... and theoretical physicists and chemists do what? Useless stuff?

 

[Goalie_Ca]

Chances are its useless math if 5 years from now you'll never use it again.

I mean, all math is usefull, in some way or another, but who the hell cares about most of that stuff. Maybe its the engineer in me, but I mostly just want to learn the math i can need and use and get the idea where it came from so i can derive things and understand it. I'm not one for pointless details, particularly techniques for solving stuff that my calculator can do!

 

[Janitor]

When I was in high school, multi-function hand-held calculators were becoming available for something less than a king's ransom. My math teacher, Mrs. W., must have been well past the age at which a public school teacher can retire, but she hung in there for some reason. She diligently taught us how to take square roots using some process that I have forgotten, other than it was laid out on paper kind of like the way you do long division. I have never had a moment when I cursed myself out for having forgotten that technique.

 

[Hurkyl]

I've had a few occasions where I was very annoyed I couldn't remember how to do a square root by hand.

 

[e(ho0n3]

I mean, all math is usefull, in some way or another, but who the hell cares about most of that stuff. Maybe its the engineer in me, but I mostly just want to learn the math i can need and use and get the idea where it came from so i can derive things and understand it. I'm not one for pointless details, particularly techniques for solving stuff that my calculator can do!

Some details may be pointless to you at this stage of the game, but remember that these details needed to be understood in order to develop the calculator.

The attitude here seems to be: Why do I need to learn how prepare meals from scratch where I can just go to the store, buy a TV dinner and be done with it?

If there is a more efficient technique to solve a problem, then by all means use it. If you can evaluate definite integrals by hand faster than with Mathematica, then do it by hand. If you find that Mathematica is faster, then use Mathematica. Whatever floats your boat.

 

[Janitor]

I've had a few occasions where I was very annoyed I couldn't remember how to do a square root by hand.- Hurkyl

Ah, but if you just remember the Newton-Raphson method, then you can always do it that way. Maybe Mrs. W's way was faster, but if it is not something that is easy to remember how to do...

 

[abertram28]

hurkyl>

you can just use Newtons method

or this iteration form...
i think that's the guess, divide, average, guess again.

like sqrt(12) guess is 3
12/3 = 4
(3+4)/2=3.5
12/3.5=3 3/7 (3.428571428571)
(48/14 + 49/14)/2 = 97/28

you can check by squaring the 97/28 and subtracting the original value, and that's the square of your error.
(97/28)*(97/28) = 12.0013 which is pretty damn close.

the longer way is here
http://www.nist.gov/dads/HTML/squareRoot.html

it uses a little guessing, but can be done very precisely without many repetitions.

 

[Hurkyl]

Anyways, one of the biggest dangers of not learning a technique because your calculator can do it is, quite simply, that you won't know the technique.

This can cause a myriad of problems including:

Being unable to solve a problem that requires multiple techniques because you'd have to apply the technique you never learned.

Being unable to recognize part of a problem can be attacked via the technique you never learned.

Being unable to recognize that a problem can be transformed into one that can be solved with the technique you never learned.

Being unable to apply the technique you never learned to transform into another one that cannot be solved, but is still easier to analyze.

Being unable to learn a more complicated technique that uses, or generalizes, the technique you never learned.

 

[HallsofIvy]

hurkyl>

you can just use Newtons method

or this iteration form...
i think that's the guess, divide, average, guess again.

"guess, divide, average (not "guess again"- the average IS the next value) is exactly the same as Newton's method for the square root.

 

[Caldus]

I am a computer science student and have completed Discrete Mathematics and Calculus I so far as far as math goes. I've never quite figured out why they make you take so much math. I mean I still have to take Calculus 2, Statistics, and Linear Algebra. Why all of these math courses when I could be taking more classes about programming or learning more IT-related topics? It seems like I'm spending more time trying to complete all of the math classes. Ah well...

 

[ahrkron]

You never know, Caldus. You may end up working for a company that develops software for finite element analysis, computer graphics, or simulation of physical processes, all of which may require the math that you still have not taken.

Math not only gives you tools to solve specific problems, but also (and mainly) helps you develop abstract problem-solving skills, plus, the many techniques you learn can often be applied in quite different settings if you are able to establish solid analogies.

 

[JohnDubYa]

RE: "... and theoretical physicists and chemists do what? Useless stuff?"

I hope not, as I'm a theoretical physicist.

But keep in mind that my examples were aimed at the algebra/pre-algebra level. Is it really efficient to focus on techniques that only 0.1% of the students will ever use?

Consider polynomial long division. If a teacher never shows how polynomial long division can be used to help plot a function, or optimize a computer code, should they teach it at all?

What I find in most mathematics books is a complete disconnect between mathematical techniques and their practical use. (And contrived word problems don't count.)

 

[Goalie_Ca]

Calculus 2, Statistics, and Linear Algebra

Actually that's pretty skimpy. Aren't you required to take a few more discrete math, calculus III, and some numerical analysis.

Most of computer science deals with developping data types and algorithms. IT ISN'T about writting a text editor with pink and blue text.

 

[e(ho0n3]

Actually that's pretty skimpy. Aren't you required to take a few more discrete math, calculus III, and some numerical analysis.

Most of computer science deals with developping data types and algorithms. IT ISN'T about writting a text editor with pink and blue text.

I don't know why Calculus is required for CS majors. Most problems in computer science make no use of Calculus at all. I think more Discrete Maths. would be appropriate. Of course, all of what I'm saying applies to undergraduate studies. I mean, if you're going to do research in quantum computing, the more maths. you know the better.

"Writing a text editor with pink and blue text" does require knowledge of data types and algorithms, so I find this argument rather flawed.

 

[matt grime]

What I find in most mathematics books is a complete disconnect between mathematical techniques and their practical use. (And contrived word problems don't count.)

And in when learning the basics of French they don't teach you how to use metaphor and simile using the complexity of the language to enrich your written and oral style. There's no reference to Balzac, and you're not learning to act like Madame Bovary.

Finding a subject uninteresting and worthless because of these reasons seems peculiar to mathematics. You are presumably at University so the motivation should be yours. But I do sympathize as I have taught pointless courses, or rather potentially pointful courses (but the habit of setting partial credit ruined that) to some particularly odd sections of the undergraduate community. If anyone can tell me why an Architecture student was made to do multivariable calc I'd be grateful, it's been puzzling me for a while now.

 

[Hurkyl]

I mean I still have to take Calculus 2, Statistics, and Linear Algebra.

It takes more than basic arithmetic to analyze algorithms.

 

[JohnDubYa]

RE: "Finding a subject uninteresting and worthless because of these reasons seems peculiar to mathematics. You are presumably at University so the motivation should be yours."

No, I am talking about teaching these concepts to middle school and high school students.

Math for math's sake is another matter entirely. I have no problem teaching unpractical topics out of sheer interest to college students.

 

[matt grime]

But my comments are even more valid there. Maths is the following of rules. We don't see the need to explain to high school students *why* it goes amo amas amat, it's just the rules of latin. Similarly it is just the rules of maths that mean log(xy)=logx +logy, nothing more. Follow the rules, get good marks, it's not very hard, and all will be well. I don't understand why we *must* motivate, with some higher reasons, the study of mathematics when we don't do so for any other subject. Maths isn't really about concepts, there's no need to pretend it is, it is only the rules that govern an object in mathematics that are important to learning it. Anyone with half a brain and a slight inclination to do so can learn mathematics, perhaps we ought to examine the mentality of people who say things like: you do maths you must be so clever, I couldn't do mathematics at school. Would that person go up to a journalist and say: oh, you use words, you must be so clever because I'm completely illiterate? No, it's an attitude that people believe maths must be motivated by the real world and bear relevance to it, and that only through application to real life situations can any meaning be taken from it. Utter garbage obviously. Otherwise practically no research into pure mathematics, or applied for that matter if were honest, could ever be undertaken.

 

[Zurtex]

I know what you mean matt grime, I've always been talented at mathematics throughout my life so far and I've always had the automatic assumption from people that I must therefore be really clever and it's just not true lol.

 

[Goalie_Ca]

Well think about it, most people cannot grasp the concept that is math. Look how many morons fail high school math.

 

[cookiemonster]

In those morons' defense, the most common reason a student fails math is because they don't really want to pass. And by the time they do want to pass, they're usually so far behind that they're screwed.

cookiemonster

 

[Caldus]

OK, answer me this...

Why would I have to be able to determine when the function:

x^3 + 2x^2 - 5

Is concave up or concave down as a computer scientist? Or better yet, why would I have to be able to do a lot of this stuff without a calculator when I could just use a calculator at the place I happen to work at? I just don't get that.

 

[cookiemonster]

To impress your boss and freak out your coworkers, duh!

cookiemonster

 

[Goalie_Ca]

actually, its probably a good thing to know when doing optimization problems. Quickly noticing whether something is concave up or down (isn't that how now common) is good when solving for (forget name, i think langrange uses it, or maybe I'm mixed up). Well, anyway, sure you'll probably end up using maple and/or MATLAB to do it but you'll need to know that for more theory.

 

[Gza]

Well think about it, most people cannot grasp the concept that is math. Look how many morons fail high school math.

That seems a bit harsh to generalize all who couldn't pass high school math as morons. When I was in high school, I had other ambitions that were heavily at odds with physics and math. I couldn't stay awake long enough to read the first page of the chapter we were studying, let alone try to grasp the material. This kept me from getting past algebra II, since I couldn't even slide by with a D. This even led me to not being able to graduate high school, since I didn't meet the math requirements. It wasn't until some subsequent soul searching, that I realized physics was my future in some form or another. I enrolled in a local JC and proceeded to get straight A's through all my science and math classes, and got accepted to UCB, UCSD, UCSB and Cal Poly. So I've been on both sides of the fence. I guess my point is, don't be quick to criticize those who aren't fortunate to have the same interests as you.

 

[matt grime]

I certainly don't support the view that all people who fail maths must be morons, though presumably if you're a moron you will fail it, and everything else. I do dislike the culture, prevalent in England certainly, which means that this is found acceptable, or at least notinh to worry about, and often seen as a badge of honour in certain parts.

 

[Hurkyl]

Or better yet, why would I have to be able to do a lot of this stuff without a calculator when I could just use a calculator at the place I happen to work at? I just don't get that.

Because the calculator won't suggest to you that concavity might be something useful to use.

 

[Hurkyl]

Some things that a computer scientist will specifically find useful from calculus are limits (asymptotic analysis), infinite summations, and the general method of estimating functions by finding good upper and lower bounds.

Furthermore, some techniques of discrete math bear strong relation to those of continuous math; for example, differences correspond to derivatives, finite sums correspond to definite integrals. The techniques are often easier to learn in the continuous setting.


Statistics is also generally useful. Many very useful algorithms have abysmal running times; the most prominent example is that quicksort, in the worst case, is a \Theta (n^2) algorithm... absolutely horrible for sorting techniques... but it almost always beats out "better" algorithms like heapsort and mergesort. Why? Because, statistically, quicksort has an average case running time of \Theta (n \ln n).

Also, many problems simply cannot be solved in a reasonable amount of time... but probabilistic algorithms can be effective. Without knowledge of statistics, how could you design or analyze such an algorithm?


As for linear algebra, it's just so pervasive throughout mathematics that you'd be disadvantaged without it.

 

[Goalie_Ca]

Okay, in defense of my statement about morons failing high school, i do realize that not everybody who struggle was a moron. I do realize that others who excel at the arts or at something else may totally suck at math or just not care. But from my own experiences most people who fail math were not that smart to begin with though but they'll likely pass their other courses.

 

[JohnDubYa]

RE: "I don't understand why we *must* motivate, with some higher reasons, the study of mathematics when we don't do so for any other subject."

I have taught physics, math, computer science, and English. I try to relay the importance of each subject I teach.

But you are correct -- we don't have to motivate our students. We don't have to teach in a manner that produces a quality learning environment.

 

[master_coda]

RE: "I don't understand why we *must* motivate, with some higher reasons, the study of mathematics when we don't do so for any other subject."

I have taught physics, math, computer science, and English. I try to relay the importance of each subject I teach.

But you are correct -- we don't have to motivate our students. We don't have to teach in a manner that produces a quality learning environment.

Why is it assumed that teaching students "practical" uses of math is the best way to motivate them?

While some people really are motivated by seeing an example of math being used in another, it's been my experience that most people who complain about a lack of practical uses are never satisfied. "Practical" is usually defined in such a way as to intentionally exclude math.

 

[matt grime]

I didn't say we shouldn't have to motivate, i said we shouldn't have to motivate with some higher reason, writing as a (university level) teacher of pure mathematics. I don't mean without reference to a practical application, but that there often is no high metaphysical/philosophical reason why something is true in mathematics. How Euclid's algorithm works is a simple consequence of the rules of the ring of integers. But at school mathematics isn't taught like that. And I feel that it is because maths is lumped in with science that people treat its results as theories and not theorems. If it were taught as rule following, just like conjugating verbs, then people might be in a better frame of mind when it came to actually having to do some *real* mathematics. (real mathematics of course in my case has nothing to do with reality.) There is then the need to teach the application of these rule following constructs to the real world., of course.

 

[JohnDubYa]

I didn't say we shouldn't have to motivate, i said we shouldn't have to motivate with some higher reason, writing as a (university level) teacher of pure mathematics. I don't mean without reference to a practical application, but that there often is no high metaphysical/philosophical reason why something is true in mathematics.

Again, you are thinking of a college course. I am talking about mathematics as taught to middle school and high school children.

My philosophy has been: If a student asks "So what?" and you cannot respond, then step down from the podium.

After all, if you cannot relate the importance of a topic, then how can the student be convinced the topic is important? And if you cannot convince the student that the topic is important, then how are you going to motivate them to work hard?

 

[JohnDubYa]

RE: "Why is it assumed that teaching students "practical" uses of math is the best way to motivate them?"

Well, what IS the best way to motivate a typical high school student to study math?