Useless math

[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 seperate and not to have a higher order on top. also, i believe, that some kids cant complete squares easily, so they need to use polynomial long division to get the +d-d part as a -d remainder. but i dont do that, so i cant 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 gotta 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! :rofl:

[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 langauges. 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 langauges. 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 wanna 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. :frown:

[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 wanna 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 thats 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 thats 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 thats 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. :wink:

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 criticise 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?

[matt grime]

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


What other answer than: because mathematics is important, it is used in.... used for...? can we offer? Exactly the same reasons as why we teach French, History, Biology and so on. The difference seems to me to be that students expect some better answer in respect of mathematics because it is perceived to be geeky and dull, and they need to be convinced before they'll study it. It is perhaps the indirect nature of the application of mathematics that is the problem.

However, we should draw a distinction between why we learn mathematics as subject, which we should explain, and why you are taught are particular technique, which shouldn't need an explanation.

[Math Is Hard]

My little sister (in high school) has the same gripe about her English classes. She says, "I don't care about this, I don't want to be a writer, and I have all the English and writing skills I need."
She sees absolutely no pratical reason for her composition classes, and she's stubborn as a mule.
I think math teachers aren't the only ones who have to put up with this attitude.

[franznietzsche]

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?


So they can calculate the life-long expense of their drug habits.

The typical high school student (in this country anyway) not only doesn't care, but won't ever. You can't convince them, its beyond their understanding.

[JohnDubYa]

RE: "My little sister (in high school) has the same gripe about her English classes. She says, "I don't care about this, I don't want to be a writer, and I have all the English and writing skills I need."

Her English instructor has probably not done his or her job very well. I teach an English course right now and every one of my students knows exactly why they need to work hard in my class. I craft my assignments to demonstrate the importance of solid English skills.

And if I can do that in an English course, why shouldn't I be able to do that in a math course?

RE: "She sees absolutely no pratical reason for her composition classes..."

Probably because she has never been shown a reason, yes?

[JohnDubYa]

RE: "Exactly the same reasons as why we teach French, History, Biology and so on. "

And what are those reasons? And are these reasons likely to motivate a student? If not, what do you do to motivate students.

So put yourself in the role of the teacher. The school principal is sitting on your course for an annual review. You have just finished a lecture on (say) polynomial long division, and a student says "So what?"

What do you say in response?

[JohnDubYa]

RE: "The typical high school student (in this country anyway) not only doesn't care, but won't ever. You can't convince them, its beyond their understanding."

I would guess about 5% of the class will study mathematics for its own sake. They have genuine interest in the subject on its own merits, irregardless of practicality.

Roughly 40% of the students are probably unreachable. They will not put out any effort no matter how important they perceive the subject.

What about the other 55%?

"Screw 'em! If they don't see that mathematics is the most wonderful subject in the whole world, then let them drift off while I teach my beloved 5%."

Is that the attitude that a high school teacher should adopt towards his students? Would you hire that teacher?

[Math Is Hard]

Probably because she has never been shown a reason, yes?

Either that or she hasn't been shown any consequences for not doing the work. She managed to squeak by with a D minus, and was satisfied with that.

Maybe I'll pack her up and send her to you! :biggrin:

p.s. after Algebra 1, I never thought I'd see polynomial long division again, but it made a cameo appearance in Calc 2.

[Goalie_Ca]

I see it every now and then, but most of the time i'm using it, i'm using maple anyways.

[Caldus]

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.

But what if I'm just going into a job where I program all day at a cubicle or be part of a software engineer team. Where would all of this math stuff come in? I mean I would just have to know how to write good documentation and good code. And as far as sorting algorithms go, couldn't I just use a built-in sorting function (or choose from different ones) for Java or whatever language I happen to be coding in? I wouldn't even have to know how the sorting algorithm itself works or how efficient it is.

[Hurkyl]

Then don't call yourself a computer scientist, and hope you're never expected to write efficient code.

[krab]

irregardless of practicality.BTW, that should be "regardless". English does have some mathematical rules: e.g. a double negative becomes a positive.

What about the other 55%?
I would say, make sure you have the respect of your students. Teach enthusiastically and if they cannot be made to see the beauty of mathematics, then at least they will come away with an impression that there is something there that some people can appreciate.

[HallsofIvy]

Teach enthusiastically and if they cannot be made to see the beauty of mathematics, then at least they will come away with an impression that there is something there that some people can appreciate.

Very well said. They may think you are a fool but they will remember that you care!

[e(ho0n3]

Speaking of useless maths., whatever happened to Lamba Calculus? There are very few universities offering this course and all the books I know on the subject are getting old now. The only thing I remember from Lamba Calculus is the wierd symbolism.

[JohnDubYa]

RE: "BTW, that should be "regardless". English does have some mathematical rules: e.g. a double negative becomes a positive."

Groan.

RE: "I would say, make sure you have the respect of your students."

Respect must be earned. The best way to earn respect is to show your students that you are acting in their best interests. When I teach English, I show my students why the material we learn can help them. (One of my first assignments is a letter of inquiry. They learn immediately that they would be in real trouble in the real world unless they pick up significant writing skills.)

RE: "Teach enthusiastically and if they cannot be made to see the beauty of mathematics, then at least they will come away with an impression that there is something there that some people can appreciate."

Teaching enthusiastically with no regard for the benefit to the students is called "self-absorption." Have you ever had such a professor? They go on, and on, and on. At some point you wonder if they would continue lecturing if everyone left the room.

Again, how do you respond when a student says "So what?"

[styler]

BTW, that should be "regardless". English does have some mathematical rules: e.g. a double negative becomes a positive.

sorry to interrupt but this is NOT true.
Irregardless means regardless. As inflamable means flammable. There are other cases too. These are somewhat informal but they were never meant to be antonyms.

Not the best authorties but:

irregardless definition (http://dictionary.reference.com/search?q=irregardless)

more (http://www.bartleby.com/64/C003/0184.html)

opinion (http://web.uvic.ca/wguide/Pages/UsIrregardless.html)

scroll down (http://andromeda.rutgers.edu/~jlynch/Writing/i.html)

another def (http://www.wsu.edu:8080/~brians/errors/irregardless.html)

[arildno]

I quote the entry from the dictionary in full:

"ir·re·gard·less ( P ) Pronunciation Key (r-gärdls)
adv. Nonstandard
Regardless.


--------------------------------------------------------------------------------
[Probably blend of irrespective, and regardless.]
Usage Note: Irregardless is a word that many mistakenly believe to be correct usage in formal style, when in fact it is used chiefly in nonstandard speech or casual writing. Coined in the United States in the early 20th century, it has met with a blizzard of condemnation for being an improper yoking of irrespective and regardless and for the logical absurdity of combining the negative ir- prefix and -less suffix in a single term. Although one might reasonably argue that it is no different from words with redundant affixes like debone and unravel, it has been considered a blunder for decades and will probably continue to be so.

[Zorodius]

When a tutee of mine is expressing their dislike for math, if they are not persuaded by whatever example I can come up with where they would use the subject matter in real life, I show them some renderings of the mandelbrot set that I always have handy. Many students in earlier math classes do not understand how conceptually meaningful and elegant math is, but few fail to see the beauty in the graceful curves and swirls of a fractal set. Being shown that math is not all rote and pedantry can soften them up towards the subject.

Of course, the ultimate "reason why I should learn this garbage" is, "because you would like to pass the class." That's a perfectly valid response, if someone has rejected the other possible motivations for studying the material.

[JohnDubYa]

RE: "When a tutee of mine is expressing their dislike for math, if they are not persuaded by whatever example I can come up with where they would use the subject matter in real life, I show them some renderings of the mandelbrot set that I always have handy."

And the tutee then asks "How is this related to polynomial long division?"

No one has answered my question: You are lecturing on (say) factoring of polynomials. A student asks "So what?" (A perfectly legitimate question, I might add.)

What do you say in response? Because it's beautiful? Because I'm interested in it?

RE: "Of course, the ultimate "reason why I should learn this garbage" is, "because you would like to pass the class." That's a perfectly valid response, if someone has rejected the other possible motivations for studying the material."

Well, that will motivate them to reach at least a D.

[JohnDubYa]

BTW, I'm not trying to antagonize anyone. Those who teach will encounter this situation, and I think it behooves future teachers to be prepared. Showing pictures of Mandelbrot sets is not going to motivate someone to learn factoring.

[Zorodius]

RE: "When a tutee of mine is expressing their dislike for math, if they are not persuaded by whatever example I can come up with where they would use the subject matter in real life, I show them some renderings of the mandelbrot set that I always have handy."

And the tutee then asks "How is this related to polynomial long division?"

No one has answered my question: You are lecturing on (say) factoring of polynomials. A student asks "So what?" (A perfectly legitimate question, I might add.)
I was not bringing up the fractals in response to a specific topic, like polynomial long division, but rather as a response to someone who is frustrated with math in general and has not experienced its elegant side. Resenting a subject makes it much more difficult (or impossible) to learn that subject. When someone discovers a subject has an appeal that they were previously unaware of, it can reduce their level of resentment towards it, making it easier for them to proceed.

You are correct that some of what is taught in math classes will not be of direct use to most of the students who take the class, and that polynomial long division fits into this category. However, even if you never actually wind up wanting to divide one polynomial into another, it's still good practice at manipulating algebra, and can be viewed as a "case study" in following an algorithm to arrive at a result.

[master_coda]

No one has answered my question: You are lecturing on (say) factoring of polynomials. A student asks "So what?" (A perfectly legitimate question, I might add.)

What do you say in response? Because it's beautiful? Because I'm interested in it?

What generic answer could possibly be of any use?

A "why should I learn this" question is impossible to answer without knowing more about the student (other than the "to pass this course" answer, which isn't a very good answer; even then, you are assuming that the student cares about passing the course).

Motivating students by telling them why something is important is actually just telling them why you think something is important. This works if the students are like you; but given the wide variety of interests among high school students, it's not very often that you can give a good answer that works for almost everyone.

And what do you do in a situation where there is no widespread, immediate practical use for something? Almost everything you learn in high school has no immediate value; this includes what you learn in core courses like math and english.

[JohnDubYa]

RE: "What generic answer could possibly be of any use?"

Well, as a teacher you better think of something. You can't just stand up there and say "Oh, nothing, I suppose." (Well, you could, but it wouldn't go over well.)

RE: "A "why should I learn this" question is impossible to answer without knowing more about the student..."

I think it is safe to say that the student wants to know why this particular topic will possibly (but not necessarily) affect his future. I also think it is safe to say that the student is not a born mathematician, otherwise he probably wouldn't be asking the question.

RE: "Motivating students by telling them why something is important is actually just telling them why you think something is important."

I think you tell them how it CAN be important for certain people. The student can decide for himself if he falls in the category. (I don't think calling something important because it is beautiful will fly. It doesn't even fly with me.)

RE: "Almost everything you learn in high school has no immediate value; this includes what you learn in core courses like math and english."

The lessons you learn in core English do have immediate value. But I never said anything about IMMEDIATE value. I have no problems with topics that will not become handly until they are late in their college career, as long as you can express that importance.

Keep in mind that we are talking about middle and high school students.

[master_coda]

Most of the things a person learns in high school do not have immediate value to them; if they did, people would tend to remember a lot more of what they learned.

Of course there are things that will be of immediate value to some people, and there are a lot of things that will be of future value to some people. However very few things will be of either immediate or future value to everyone.

But we can't predict what will be of value to who; and even if we could, it isn't really practical to try and ensure that nobody has to learn something that is of no value to them. So we try and teach a wide breadth of material, hoping that everyone will learn at least some things that are useful.

I think it is safe to say that the student wants to know why this particular topic will possibly (but not necessarily) affect his future. I also think it is safe to say that the student is not a born mathematician, otherwise he probably wouldn't be asking the question.


But unless you know more about the student, it's difficult to be sure what kind of impact a subject will have on their life. I can think of reasons why everyone should learn how to read and write, or why they should learn simple arithmetic. It's much harder when the knowledge they are learning becomes more specialized.

And what good is a response that explains how something may possibly affect his future going to be? If a student wants an answer, they probably won't be satsified with you just making up some hypothetical situation where the subject material will benefit them.

I think you tell them how it CAN be important for certain people. The student can decide for himself if he falls in the category. (I don't think calling something important because it is beautiful will fly. It doesn't even fly with me.)

The lessons you learn in core English do have immediate value. But I never said anything about IMMEDIATE value. I have no problems with topics that will not become handly until they are late in their college career, as long as you can express that importance.

In your previous posts you expressed dissatisfaction with other responses because they would not be effective at motivating students. But now your standards seem to have changed - you don't seem to care if the response is worthless as motivational material, as long you provide a "correct" answer.

Of course, a well thought out explaination of why learning Y is useful in life for people in career X might look good to the school principal. That's probably useful for getting good evaluations.

[JohnDubYa]

RE: "And what good is a response that explains how something may possibly affect his future going to be? If a student wants an answer, they probably won't be satsified with you just making up some hypothetical situation where the subject material will benefit them."

Sure, but if you point out that biologists use XXX for doing YYY, a student will appreciate the topic more then if you say "because it's beautiful." The student may not know whether or not he is going to be a biologist, but at least he realizes the mathematics isn't simply being taught for its own sake.

RE: " you don't seem to care if the response is worthless as motivational material, as long you provide a "correct" answer."

When did I ever say anything remotely like that? Let's not put words in my mouth. I have always maintained that the practical importance of material should be stressed as much as possible. I never said that the relevance needed to be immediate. That was a straw man stated by someone else.

[master_coda]

RE: "And what good is a response that explains how something may possibly affect his future going to be? If a student wants an answer, they probably won't be satsified with you just making up some hypothetical situation where the subject material will benefit them."

Sure, but if you point out that biologists use XXX for doing YYY, a student will appreciate the topic more then if you say "because it's beautiful." The student may not know whether or not he is going to be a biologist, but at least he realizes the mathematics isn't simply being taught for its own sake.

RE: " you don't seem to care if the response is worthless as motivational material, as long you provide a "correct" answer."

When did I ever say anything remotely like that? Let's not put words in my mouth. I have always maintained that the practical importance of material should be stressed as much as possible. I never said that the relevance needed to be immediate. That was a straw man stated by someone else.

You said that is doesn't matter if your example of how the material is important shows that it is important to the student...just that it is important to someone.

Your very first post included a remark about theoretical physicists that suggests you want examples from a career you consider to be "practical", and not just an example from any career. So why would the student be any different? What makes you think they care about the fact the a biologist uses something, unless you know they want to be a biologist?

If a student wants to know about the importance of something, they probably want to know why it's important to them. Not why it's important to a career that you think is important, but they care nothing about.

[JohnDubYa]

Theoretical physicists are so rare and study at such a high level that very few students would appreciate math topics that are relegated to their discipline. Biology is unlike theoretical physics in this regard. If I show how solving simultaneous differential equations is used to model wolf populations, students will understand that and appreciate it. Quaternions? Probably not.

[master_coda]

Theoretical physicists are so rare and study at such a high level that very few students would appreciate math topics that are relegated to their discipline. Biology is unlike theoretical physics in this regard. If I show how solving simultaneous differential equations is used to model wolf populations, students will understand that and appreciate it. Quaternions? Probably not.

What do teachers do in an ordinary high school where the majority of students have no appreciation or interest in either physics or biology? Most teachers will not be lucky enough to work at a school where the students are specializing in biology.

Quaternions are sometimes used in computer science to do 3D modelling.

[futb0l]

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.

Wow, I totally agree.. u couldn't explain it better.
:wink:

[JohnDubYa]

RE: "What do teachers do in an ordinary high school where the majority of students have no appreciation or interest in either physics or biology? Most teachers will not be lucky enough to work at a school where the students are specializing in biology."

Students don't specialize at the secondary level. But students, by and large, are interested in biological applications, even if they never plan to study biology.

There is no way around it. Modelling wolf/sheep populations is simply more interesting, and accessible, to high school students than gauge field theory. And those students who don't even find population modelling interesting? Well, they belong to that 55% who are completely unapproachable. They may as well not even be there.

RE: "Quaternions are sometimes used in computer science to do 3D modelling."

Yes, I'm aware of that. Good point. As long as the connection can be explained to students, I have no qualms about teaching subjects like quaternions to high school students (although it is a pretty advanced topic in algebra). Without the connection, forget it. They will just think you are wasting their time.

[master_coda]

Students don't specialize at the secondary level. But students, by and large, are interested in biological applications, even if they never plan to study biology.

There is no way around it. Modelling wolf/sheep populations is simply more interesting, and accessible, to high school students than gauge field theory. And those students who don't even find population modelling interesting? Well, they belong to that 55% who are completely unapproachable. They may as well not even be there.

Do you actually have any evidence to back up these assertions, or are you just talking out of your ***? I'm not just going to believe that students are interesting in biology or biological applications just because some teacher says so.

You aren't even consistent. One moment, you're complaining that students deserve a proper explaination for why they have to learn something, instead of just getting a dictatorial "because we said so". Then you're making sweeping statements saying that students who aren't interested in applications you consider important are unapproachable.

Yes, I'm aware of that. Good point. As long as the connection can be explained to students, I have no qualms about teaching subjects like quaternions to high school students (although it is a pretty advanced topic in algebra). Without the connection, forget it. They will just think you are wasting their time.

I sure hope they don't get taught. Whenever any kind of difficult scientific or mathematical subject gets taught to high school students, they always get a watered down version of the real thing. It might work better if teenagers were taught as if they were intelligent people instead of generic faceless children.

[JohnDubYa]

RE: "Do you actually have any evidence to back up these assertions, or are you just talking out of your ***? "

I think you are taking this a little too personally. So maybe it is best we end the discussion here.

[Hurkyl]

He has a point; you seem particularly biased towards what you find interesting. You call quaternions advanced, though they are just barely beyond complex numbers. You compare solving a ridiculously simplified toy population model with differential equations to gauge field theory, hardly comparable as techniques. I wouldn't have said anything, but you seem to be attempting to offload criticism onto Master Coda rather than address it.

[JohnDubYa]

RE: "He has a point; you seem particularly biased towards what you find interesting."

Actually, I'm a theoretical physicist, not a biologist. You won't find a single post from me the biology section of this forum. If anything, my stance runs counter to my background. Physicists tend to elevate the importance, meaning, and beauty of their work. I do, but I have enough experience with high school students to know when I am doing them a disservice for the sake of my own interests.

RE: "You call quaternions advanced, though they are just barely beyond complex numbers."

Yes, and we are talking about middle school and high school levels here, not college level.

If you want to teach quaternion algebra to high school students, go for it.

RE: "You compare solving a ridiculously simplified toy population model with differential equations to gauge field theory, hardly comparable as techniques."

Which was my point all along --- that some issues are far too advanced to be concerned with at the middle school and high school levels and that we should concentrate on those issues that are more accessible. Obviously I exaggerated for effect.

RE: "I wouldn't have said anything, but you seem to be attempting to offload criticism onto Master Coda rather than address it."

I have not criticized him one iota. And what criticism have I not addressed?

[Math Is Hard]

I find all your comments very interesting, JohnDubYa. I am studying to become a teacher one day, and I know that one of the biggest challenges I'll face is holding students' attention. I am particularly interested in learning what will make a subject or technique meaningful for them.
I think that in the society we live in we are bombarded with so much information that a lot of kids are programmed to instantly filter, filter, filter everything as it comes in and only hang on only to the small amount of information that think will be relevant later on. Anyway, I find your comments on your first-hand experience with teaching very informative.

[techwonder]

The most useless math I have learned so far is fraction calculation.

[master_coda]

And what criticism have I not addressed?

Well, you haven't backup up your assertion that a lot more students will be interested in a biology example than a physics one.

And you haven't addressed the fact that you are very understanding when students aren't interested in a physics example, but if they don't appreciate your biology one then all of a sudden you classify them as unreachable.

To me that makes it sound like you have a much higher opinion of biology compared to physics (despite your specialization in biology). Otherwise why would you consider it to be normal for students to be uninterested in physics, but unacceptable for them to be uninterested in biology?

[JohnDubYa]

RE: "Well, you haven't backup up your assertion that a lot more students will be interested in a biology example than a physics one."

I base it on personal experience (I have taught physics at the high school level) and common sense. Students can relate to biology because they interact with biological examples. They know wolves and sheep. They don't know abstract algebra.

And I referred specifically to THEORETICAL phyiscs in my examples. Practical physics DOES interest them to a significant degree.

RE: "And you haven't addressed the fact that you are very understanding when students aren't interested in a physics example, but if they don't appreciate your biology one then all of a sudden you classify them as unreachable."

The key word is UNDERSTAND. Yes, I do understand when they don't appreciate topics in theoretical physics. I probably wouldn't have at that level either.

RE: "To me that makes it sound like you have a much higher opinion of biology compared to physics (despite your specialization in biology)."

Who cares?? Why are you trying to turn this into a personal issue? It doesn't matter what *I* like, appreciate, understand, or respect. What matters is reality. And when you step into a class filled with 30 high school students who are wondering why they need to take a math class, reality rules.

RE: "Otherwise why would you consider it to be normal for students to be uninterested in physics, but unacceptable for them to be uninterested in biology?"

Again, it's all about reality, not my personal wish. Sure, I would love high school students to become interested in theoretical physics. Not gonna' happen.

In essence, whenever I try to cite examples of the practical applications of mathematical topics, I look for biology, economics, psychology, and so on. If the only application I can surmise appears in theoretical physics at an advanced level, I have to wonder if the topic is worth teaching.

[remcook]

about Quaternions: They are very much used (not sometimes, often!) in Satellite Control and other control problems! Definately not useless!
(excuse the exclamation marks)

[master_coda]

I base it on personal experience (I have taught physics at the high school level) and common sense. Students can relate to biology because they interact with biological examples. They know wolves and sheep. They don't know abstract algebra.

And I referred specifically to THEORETICAL phyiscs in my examples. Practical physics DOES interest them to a significant degree.

The key word is UNDERSTAND. Yes, I do understand when they don't appreciate topics in theoretical physics. I probably wouldn't have at that level either.

Who cares?? Why are you trying to turn this into a personal issue? It doesn't matter what *I* like, appreciate, understand, or respect. What matters is reality. And when you step into a class filled with 30 high school students who are wondering why they need to take a math class, reality rules.

Again, it's all about reality, not my personal wish. Sure, I would love high school students to become interested in theoretical physics. Not gonna' happen.

In essence, whenever I try to cite examples of the practical applications of mathematical topics, I look for biology, economics, psychology, and so on. If the only application I can surmise appears in theoretical physics at an advanced level, I have to wonder if the topic is worth teaching.

Except you're missing the whole point I was making...I'm not saying that you're using biology examples because you love biology and wish that all students would study it. I'm saying that you have this strange idea that certain subjects like biology are "practical" while other subjects are not, and then are assuming that students are reachable if and only if they can relate to your chosen subjects.


Students do not generally specialize into their fields of interest until after high school, although it starts to happen a little bit in the last year or two of high school. That means that in any given class, it is unlikely you will be able to pick a specific field of interest and find more than a handful of students who actually care about it. So examples drawn from a specific field are not going to interest very many students in your class. The fact that biology is straight from the real world doesn't change that; we're utterly dependant upon the state of the economy for our survival, yet even most college students can't be bothered to learn anything about basic economics. Being in the real world does not magically mean that people are going to better relate to a subject, or have an easier time understanding it.

Of course, you don't seem to be drawing examples from just a single field. You probably can reach a lot students by exposing them to uses in a wide variety of fields - almost everybody gets at least one example that interests them. But then why are you presupposing that students are going to be interested in a select group of subjects? Do you actually never have students who are interested in fields you don't consider practical?

[JohnDubYa]

RE: "I'm saying that you have this strange idea that certain subjects like biology are "practical" while other subjects are not, and then are assuming that students are reachable if and only if they can relate to your chosen subjects."

No, I think that there are subjects that STUDENTS think are more practical than others. It isn't about ME, ME, ME. It's about the STUDENTS, STUDENTS, STUDENTS and what THEY value.

High school teachers need to work within that framework. Those teachers that try to turn their high school courses into college courses do the students a disservice, because the students (by and large) have not developed an attitude that appreciates esoteric subjects. If you ignore that reality, you will see nothing but glazed-over eyes in the room and the students won't learn a damn thing.

RE: "Do you actually never have students who are interested in fields you don't consider practical?"

Yes, but you cannot just teach to those students. That is why you must consider whether or not a mathematical topic has interests beyond the esoteric. And if it doesn't, then you should consider whether or not the subject is worth discussing at that level.

That is all I have been saying. And there is no need to parse every last word I write looking for an inconsistency somewhere. This isn't about ME.

[master_coda]

No, I think that there are subjects that STUDENTS think are more practical than others. It isn't about ME, ME, ME. It's about the STUDENTS, STUDENTS, STUDENTS and what THEY value.

Then why don't you teach based on what they are interested in, instead of insisting that we only use examples that you think they should be interested in? I've been saying that you should decide upon examples based on what students actually care about, and that you shouldn't just use examples based on subjects you've predetermined to be "interesting". IF YOU CARE ABOUT STUDENTS THEN STOP INSISTING THAT THEY CAN ONLY BE INTERESTED IN CERTAIN SUBJECTS.

RE: "Do you actually never have students who are interested in fields you don't consider practical?"

Yes, but you cannot just teach to those students. That is why you must consider whether or not a mathematical topic has interests beyond the esoteric. And if it doesn't, then you should consider whether or not the subject is worth discussing at that level.

That is all I have been saying. And there is no need to parse every last word I write looking for an inconsistency somewhere. This isn't about ME.

I don't care about you. I care about the unfortunate people you are going to teach who aren't interested in what you care about. You're telling us it's wrong to use esoteric examples because we're ignoring all those students who want "practical" examples, whatever the hell that means, and then you go on to tell us that we should only use practical example, thereby ignoring all the students who don't care about practical examples.

If you cared about the students, you wouldn't insist they be taught using only certain examples, automatically excluding a whole bunch of students.

[JohnDubYa]

I don't care about biology, okay? I am not a biologist. I have no personal reason to use biological examples. I have no reason to insist that students must be interested in biology.

I resort to biology because in my experience that is one of the topics that students find most interesting. And in my experience, they are not intersted in esoteric theoretical physics. I play with the hand that is dealt me.

I go by my experience teaching the subjects to actual high school students and common sense and I have yet to meet a high school student who wasn't interested in practical examples.

I mentioned earlier that you were starting to take this to the personal level. Now you are claiming that I don't care about my students, and that any student that enters my classroom is unfortunate. I have more important things to do then to read personal attacks on my character, so I suggest you change your tone if you want this conversation to go any further.

[Hurkyl]

I suggest neither continue this conversation further, in this thread.

[e(ho0n3]

I resort to biology because in my experience that is one of the topics that students find most interesting. And in my experience, they are not intersted in esoteric theoretical physics. I play with the hand that is dealt me.

I go by my experience teaching the subjects to actual high school students and common sense and I have yet to meet a high school student who wasn't interested in practical examples.

Since when do they teach theoretical physics in high school!? I think most of what one learns in primary and secondary school is practical knowledge (expect for philosophy, theology, etc.), so when a student asks you "So what?", your answer should immediately be "Because you will most likely need this later in life if you don't want to become an uneducated bum."

I believe that teaching is not only about transfering knowledge. I think a teacher's prime responsibility is to develop a student's problem solving ability regardless of what is being taught. Giving only practical examples will only get you so far.

I believe most highschoolers will prefer practical examples since they are 'easier' to digest than some deep abstract result (which would require them to think abstractly and most highschoolers (as I've experienced) are lazy to do so).

This is what I believe.

[master_coda]

I suggest neither continue this conversation further, in this thread.

Agreed. Clearly my experiences are very different than those of JohnDubYa, so there isn't much progress that we can make going back and forth like this.

And JohnDubYa, I really don't know why you thought I was taking it so personally. I don't hold back from attacking any point of view I disagree with; if someone introduces their personal experience into the argument, it doesn't get any special treatment from me.

[JohnDubYa]

RE: ""Because you will most likely need this later in life if you don't want to become an uneducated bum.""

That's the same argument their parents have been using on them since Day One. If it doesn't work for their parents, it most likely won't work for you.

RE: "I don't hold back from attacking any point of view I disagree with;"

You didn't just attack my point of view, you questioned my motives.

I am finished with the discussion.