Math Skills



#44

I just read an article in our local paper (Louisvile KY Courier Journal) where it stated that 25% of the students tested could not do math at their grade level. I feel this is a function of the way math is taught these days. Kids are taught to rely too much on calculators. Thanks to the marketing blitz of TI in the past where TI would give teachers curiculums to teach from. I feel students today are taught how to perform functions instead of why they work, which if you understand why they work there usually are several possibilities as to how to solve the problem.

Just my $.02

delete 123


#45

Hi, gs;

I'm a teacher (university) and I include the use of calculators as for checking and comparing results from "hand made" calculations. I agree with you when you mention this:

Quote:
I feel students today are taught how to perform functions instead of why they work, which if you understand why they work there usually are several possibilities as to how to solve the problem.
To be honest, I'd not give calculators ALL of the responsability for this fact, although I agree with you that missusing a calculator reinforces this. I'd dare telling you I believe that teaching how to fair use a calculator while studying math would enhance their skills. I see that when their attention is slightly deviated to reason about how to use the calculator to solve a problem they are actually trying to solve the problem twice, ONCE they have already master calculator operation. In all subjects I teach where calculator usage is somehow an improvement, I take one or two classes in order to teach students how to use their calculators. It happens that, in some cases, there are lots of different models that operate a bit different of each other. I don't care showing how to use each of them, provided their manuals are in hands. BTW, I take also the chance to show them that having their manuals/user's guide in hands is always handy.

I don't actually tell them what calculator to buy unless they ask me to. It is too bad for a student having a calculator and the teacher telling him to buy another. I preffer teaching him how to use it the best way possible, because it both enhances his productivity as a student and gives him confidence. If the student knows how to use his calculator, he is not gonna spend time pressing the buttons in order to see what happens: this is distractive and not forthcomming. In other hand, if he knows how to use it, he'll not waste time pressing buttons, he will pay attention to your class and, as I mentioned already, he'll think twice on the solution if he wants to use the calculator.

Well, why didn't we, the guys about and more than 35 YO, need this sort of "appealing" in order to use our calculators at the time we had them in our hands? Because at that time they were the appealing tools, they were fresh and neat, and mastering a calculator in the 70's and 80's was "cool" (although cool was not the term , I guess it was "heavy", right?). What is cool to the youngers today? Mastering a mobile, a palm, hacking, "cracking", and other appealing "freak" activities. Oops, maybe we were the freaky younger people at our time...

You see, I'm in Brazil and things here a bit different, our reality is not exactly the same as the one seen in USA. Yes, each country and continent has its own, no way to straight comparisons, but today we have "instant access" to information, and changes spread too fast around all of us. So, I guess that given some particular adjustments for each case/country/reality, what I wrote here applies to many cases.

I gladly accept additional comments, suggestions, corrections, the like.

Cheers.

Luiz (Brazil)

Edited: 23 Jan 2005, 9:44 p.m.


#46

Hi Luiz,

Isaac Asimov wrote a short story about this matter (The Feeling of Power). The year was 1957 and he was over concerned about this problem. I don't agree with him, but it is worth giving a look:

http://www.themathlab.com/writings/short%20stories/feeling.htm

Or if you prefer to read it in Portuguese:

http://sobral.tripod.com/poder/poder.html

This can also be found in print:

ASIMOV, ISAAC
Sonhos de Robô. Círculo do Livro Editora. Pgs 320-331.

Best regards,

Gerson.


#47

Hi, Gerson;

thanks for the tip!

Novo na área? Não me lembro de ler mensagens suas aqui... (New in the block? I don't recall reading your posts here...)

BTW, do you have the Círculo do Livro e-address? I'd like contacting them again.

Hope to see you (read your posts, of course) more often. If you want to, drop me a line... lcvieira at quantica dot com dot br.

Cheers.

Luiz (Brazil)

Edited: 23 Jan 2005, 11:14 p.m.

#48

Could it be the way math is presented in the classroom (I'm not an expert on this subject)? In society, having math smarts is not looked at as a highly desirable trait (which is sad). While I believe using a calculator is beneficial, I also believe a cirriculum solely on calculators is a bad thing and it misses out on critical thinking (or even creative) thinking skills.

Would it be a good a idea to force the students to do the calculations without a calculator and use it only to check results (and that goes for higher math as well - we all can use highly tuned algebra skills)? I am about to go back to school this year and complete my degree in math and what I do is try the problem on paper first, then use the calculator to verify results.


#49

Hi, Eddie;

just to point out one excellent example: Mathews/Eidswick 'An HP48G Calculus Companion', Harper Collins College Publishers, 1994. An excerpt from 'To The Student' section tittled 'Calculus and your two HPs':

Quote:
(...)You might even wonder why bother studying calculus if a machine can do it all. Of course, the answer is that you need to know the ideas of calculus before you can effectively use the HP48. At the same time, these caclulators provide you an interesting context in which you can learn the ideas of calculus. (...) By combining your two HPs - your HP48 and your Head Power - with what you read in this book, you will find that learning and using calculus is both rewarding and enjoyable.

I consider 'An HP48 Calculus Companion' as the best publication I know (are there others like this?) that actualy "fuses" both learning calculus and using a calculator as a tool to achieve the learning. The 'what if' possibilities are actualy on line.

Cheers.

Luiz (Brazil)

Edited: 23 Jan 2005, 6:24 p.m.


#50

I'll have to see if I can get the book at my library (I work, and soon hope to be a student again of Cal Poly Pomona). Thank you for the reference.

One of the best calculator books that I got to learn from are Math On Keys (TI's TI-30 companion book). Published circa 1978, it shows the different applications one can do with the calculator beyond what most calculator companion books do today. I am also impressed with the HP49G+'s User Guide (the one on the CD), the authors go in to calculus and linear algebra theory that compliments the corresponding functions.


#51

That is math professor Urroz speaking there in the 49g+ User Guide. The extra steps on matrix manipulations have never been published on a cacluclator vendors manual before.


[VPN]

#52

Calculators are very effective for numerical calculations, but not very good for real math problems.

Example : solve y^3=x^2+2.
Anyone can find a solution with a calculator.
I don't know any calculator which solves this for integer x and y.

There are lots of such examples everywhere. So you can teach using "calc-proof" problems - but, really, that may require too much effort.


#53

Quote:
Example : solve y^3=x^2+2. Anyone can find a solution with a calculator. I don't know any calculator which solves this for integer x and y.

Well, that's because there's an infinite number of solutions. But rearrange it into y^3-x^2-2 = 0 and the 48GX will happily solve y for any x or x for any y. And it will graph the function, etc.

I think there's a role for such a calc in the hands of the more inquisitive student, as part of an exploratory learning style. But they have to have the discipline not to bypass the thinking that leads to learning.

Best,

--- Les

[http://www.lesbell.com.au]


#54

But they have to have the discipline not to bypass the thinking that leads to learning.

How about: they have to have the curiosity that leads to learning? Textbooks with dumbed-down problems that bear no relationship to real-world applications of algebra and calculus do nothing to fuel that curiosity, they just make you want the quickest and easiest solution (so you can finish the damned test already and leave!).

My maths teachers in high school used examples like position/velocity/acceleration vs. time of a skier going down a slope. That kind of stuff makes immediate sense, and I never saw anyone's eyes glaze over during those classes! (And the fact that I was playing with my HP-41C and figuring out analytical solutions to geometrical problems didn't cause any problems, either -- the teachers didn't mind if I solved problems using my own methods, as long as they were correct methods.)

I think a lot of kids fail to see how cool science is because it's never made clear to them what it is all about, both in terms of our deeper understanding of the universe and in terms of us being able to build planes, predict hurricanes, and whatnot. If all they have to go on are the stereotypical "scientists" in Hollywood movies, who can blame them.

#55

(Math zealots only)

Just a small note : there is ONE SINGLE solution to y^3=x^2+2 with x and y BOTH integers. Again, no calculator I know can find it.


#56

Hi, GE:

GE posted: "Just a small note : there is ONE SINGLE solution to y^3=x^2+2 with x and y BOTH integers."

Wrong. Without actually bothering to find solutions, it's easily seen that since variable x appears squared, any solution with a integer, positive x will also result in another solution with the same y but integer, negative x.

If you had stated your sentence as "... with x and y BOTH integers AND x positive" you would be right in affirming it would have just "ONE SINGLE" solution. As you omit the "positive x" part, there is actually an even number of solutions, namely TWO in this particular case.

This kind of equations, asking for solutions in integers, are called Diophantine equations. Your specific equation is a variant (with x and y exchanged) of a particular case of the Diophantine equation y^2 = x^3 + k, which is known as Mordell's equation, and has extensive literature about it and copious on-line info. See for example:


Helmut Richter: Solutions of Mordell's equation y^2 = x^3 + k

"Again, no calculator I know can find it"

I'm not sure what you mean with this statement. Any programmable calculator, however humble, would easily find the
solution pair:

    x=-5 and y=3 
x=+5 and y=3
If you mean finding them without a program, that's unfeasible in the general case. In fact, it can be demonstrated that there's no general algorithm to solve arbitrary Diophantine equations, or even determining whether they have one solution, several, infinite, or none.

Take
for instance Fermat's Last Theorem:

 "The Diophantine equation  x^n + y^n = z^n  has no integer solutions for n > 2 and x,y,z not zero"
Though this hard nut has been proved (recently), you can't realistically expect any calculator or symbolic math package to automatically solve or demonstrate this, nowadays. It's taken several centuries of the combined efforts of the very greatest mathematicians in the world to crack it.

On the other hand, the "converse" theorem:

 "The Diophantine equation n^x + n^y = n^z has no integer solutions for n > 2 and positive x,y,z"

is more affordable. Matter of fact, I have discovered
a truly remarkable proof but this message is already too large to include it ! :-)

Best regards from V.


#57

Damn, you're right : two solutions.

>> "Again, no calculator I know can find it"
> I'm not sure what you mean with this statement.

I meant : 'no calculator can prove that there are exactly two solutions', specifically "no more than 2".

> Matter of fact, I have discovered a truly remarkable proof
> but this message is already too large to include it ! :-)
Hmm I read something like this somewhere...
and that last equation looks really easy. Isn't it ?


#58

Hi, GE:

GE posted:

"I meant : 'no calculator can prove that there are exactly two solutions', specifically "no more than 2".

As I said in my posting, "that's unfeasible in the general case. In fact, it can be demonstrated that there's no
general algorithm to solve arbitrary Diophantine equations, or even determining whether they have one solution, several, infinite, or none.".

So you see, no calculator can, but also no man and no inhabitant from advanced planet ZYDJ3123, because
for the general case it's simply impossible. That said, though, in the case of your specific, particular equation it might perhaps be proved, but unlikely. In other words, your expectations are way too unrealistic.


"Hmm I read something like this somewhere... and that last equation looks really easy. Isn't it ? "

Well, just try and find a short, simple demonstration, would you ? :-)

Best regards from V.


#59

Hello,
Yes some classes of problems don't have general solutions while some problems of these classes do.

On the equation n^x+n^y=n^z, I still think it is not very complex :
1. you can reduce it to x=0 or y=0 or z=0 by dividing by n
2. then 2 cases :
2.a. n^x+n^y=1 not many solutions if n>1 !
2.b. n^x+1=n^z, rewritten as n^x.(n^(z-x)-1)=1, not many solutions either.

So the equation has not many solutions. Is that correct ?

#60

It happens to have happened only yesterday. A 15 year old had to draw the function Y=X²-2X-8 for X between -5 and 5 to learn about paraboles. Basic approach is computing results for several X values ("use your calculator" it says in the text book) and sketching a curve through the x,y points. His drawing resembled a thirdish powerish plot, so I tried to find out where things went wrong.
It quickly appeared to me that he had typed in -5^2-2*-5-8 for the first value and he hadn't the foggiest that -5^2 came out -25. Of course he should have typed (-5)^2 on the darn thing.
He obviously has to have an RPN calculator (an HP21 will do). One then cannot possibly make a mistake like this.


#61

Hi, Bram:

I'm probably wasting my time but I'll try anyway:

Bram posted: "It quickly appeared to me that he had typed in -5^2-2*-5-8 for the first value and he hadn't the foggiest that -5^2 came out -25."

Well, then an explanation of the rules of procedence would suffice. Something like

   "a power takes precedence over a change of sign, so if
you type -5^2, the power will be executed first (5^2 =
25), then the sign will be changed (-25). If you want
the other way around, simply enclose in parentheses
what you would want to do first (-5)"
Instead of that, which is by no means complicated and that most 15-year old teens will have no problem understanding, specially since it's an universal rule ("whenever you want to force a particular order of operations, simply use parentheses"), you say this:

"Of course he should have typed (-5)^2 on the darn thing."

which immediately begs the question: Why "darn thing" ? The mere fact that it does things differently than what you consider "The Holy Bible of Calculating Expressions" is enough to belittle it and insult it ? Great ! Very rational ! Openmindedness at its best ! But wait, there's more:

"He obviously has to have an RPN calculator (an HP21 will do). One then cannot possibly make a mistake like this. "

Why "obviously" ? Because you like it to be so ? What may be so "obvious" to you, many other people can blatantly disagree with. I take it that you firmly believe that *only* RPN calculators can compute the coordinates of a parabolic curve, and "obviously" non-RPN models are grossly inadequate for such rocket-science applications !? C'mon !

Perhaps your rationale is that "One then cannot possibly make a mistake like this" in an RPN calculator, such as an HP-21. True, operators precedence is not an issue in RPN, the person will not make that particular mistake. But he/she can make many others.

For instance, the typical sequence to evaluate that expression for X=-5 in your HP-21 would be something like this:

   5 [CHS][ENTER][X][LASTX][2][X][-][8][-]
As stated, there are other possibilities, but the above is typical. And, what's obvious about this sequence ? Do you think that having to press [CHS] instead of [-] is obvious ?
Or having to use [LASTX] to retrieve back the X value ? Or the "first enter the values into the stack, then perform the operations upon them" basic RPN rule itself ?

Of course this is obvious to *you* and to any other experienced RPN user, but not to anyone else !! The typical 15-year old, given an algebraic calculator might make the precedence error you mention, but if told the cause, he would understand it right away, and would use parentheses when necessary, or when wanting to make sure.

Same 15-year old, given an RPN calculator would do mistake after mistake, and throw the thing in despair, unable to even add two values. I've seen exactly this happen any number of times. And so do most of you.

Unless told how it all works. But the same applies to your "damned thing" and your "obvious" better way. All things are "obvious" once explained and understood, and algebraic calculators are just as able for computing expression as RPN ones are, even much better in many ways, we must not let RPN addiction to blind us to the objective facts.

Best regards from V.


#62

No, no, no, that's definitely NOT what I meant. I obviously cannot express myself in English properly (enough). Apologies.

1. Of course one has to know the precedences. Of course the calculator comes at the final stage and must be operated the right way after having read the manual(s), but it is too simple to type things from the text book. This was only meant as a mere example of how things can go wrong.

2. With "the darn thing" I meant ANY calculator, whichever brand. In the old days when most calculations still were done on slide rules, you could easily tell the rich students: their results contained (always) 8 or more digits, because their calculator (whichever brand) told them so.
Some teachers almost became desparate sometimes.

3. My experience with RPN is that it _helps_ you with your calculations by showing intermediate results and I think it would have done so in this particular case as well. I really don't care what one uses for computation, for the right answer is the only thing that counts whichever way you choose to use to get it.


Sorry for making you angry.

Thank you for making me clarify.


#63

Hi again, Bram:

Bram posted:

"No, no, no, that's definitely NOT what I meant. I obviously cannot express myself in English properly (enough).
Apologies.

Sorry if I misunderstood what you meant, I'm perhaps a little too "computer-like" in that respect: I'm usually quite good at understanding what people say (write, post, utter), not so good at what people meant. No need to apologize in either case, you're entitled to your own opinion, even if it's derogatory of some object (a person would be quite another matter).

"My experience with RPN is that it _helps_ you with your calculations by showing intermediate results and I think it would have done so in this particular case as well."

Quite possible. But when trying to plot a parabolic curve, such as your Y=X^2-2*X-8, you're really not interested at all in losing time getting to know (or even see) what X^2 amounts to, what 2*X is, what X^2-2*X evaluates to. The only thing you want or need is the final result, Y=X^2-2*X-8, but RPN forces you to see intermediate results, whether you want them or not. And this for every point X you might evaluate. On the other hand, you can use an HP-71B (or any vintage SHARP) like this:

>FOR X=-5 TO 5 STEP 5 @ PRINT X, X^2-2*X-8 @ NEXT X [ENTER]

-5 27
0 -8
5 7

which I personally find much easier, faster, easy to verify for correctness, and easier to understand than arcane juggling acts with RPN, which was best in its prime, back then in the 70's, but it's been obsolete for more than two decades as of now.

"Sorry for making you angry.
Thank you for making me clarify."

I wasn't angry, excuse me if I gave you that impression. And yes, it's been clarifying indeed, hopefully for you as well.

Best regards from V.


#64

Valentin posted:

Quote:
The only thing you want or need is the final result, Y=X^2-2*X-8, but RPN forces you to see intermediate results, whether you want them or not. And this for every point X you might evaluate. On the other hand, you can use an HP-71B (or any vintage SHARP) like this:

>FOR X=-5 TO 5 STEP 5 @ PRINT X, X^2-2*X-8 @ NEXT X [ENTER]
-5 27
0 -8
5 7

which I personally find much easier, faster, easy to verify for correctness, and easier to understand than arcane juggling acts with RPN, which was best in its prime, back then in the 70's, but it's been obsolete for more than two decades as of now.


Naaah! This must be "debate-bait", but I'm game...

RPN will never be obsolete. As we all certainly know, on RPN-based H-P calc's made since the latter 1970's:

  1. The user could easily program and run the simple expression to avoid seeing the intermediate results. On certain models (e.g., 34C, 15C) the program is ready-made for input to built-in rootfinding (SOLVE) and numerical integration functions.

  2. Using a 27S, 17BII, 32SII, or 33S, the user could enter and evaluate the expression nearly as it reads, without seeing intermediate results.

Either is more straightforward than your short 71B BASIC program, which, although certainly not arcane, still incorporates syntactical rules.

One modern non-graphing calc that is ideally suited for this problem is the Casio fx-115MS. Here, the user enters the expression left-to-right exactly as it reads, then evaluate it repeatedly using "CALC". However, this type of basic scholastic problem is about the only thing the fx-115MS is ideally suited for.

Best regards and a belated "welcome back"...

-- KS

Edited: 25 Jan 2005, 7:20 p.m.


#65

ANY programmable will get it right

[VPN]


#66

VPN posted,

Quote:
ANY programmable will get it right

Thanks for the revelation, but my obvious point was that the user needn't always see every intermediate result when using an RPN-based calc; work-arounds are avialable.

-- KS


#67

Actually I should have posted this to Valentin and ask for CALC mode solution

[VPN]

#68

Hi, Karl:

Karl posted: "Naaah! This must be "debate-bait", but I'm game..."

That wasn't my intention but I'm game, too ... :-)

"The user could easily program and run the simple expression to avoid seeing the intermediate results".

Gotcha ! :-) You absolutely *need* to write a program in order not to see the intermediate results, else it's impossible (unless you close your eyes or look aside).

It seems to me you're mistakenly thinking that my HP-71B (or SHARP) example, namely:

>FOR X=-5 TO 5 STEP 5 @ PRINT X, X^2-2*X-8 @ NEXT X [ENTER]
is a program. Well, it is not. Just notice the ">" which is the prompt, and the fact that I simply wrote the expression (without a line number or such) and pressed [ENTER] (not "RUN" or something similar). This is, I'm not writing any program and running it, like you're forced to do in RPN to avoid seeing intermediate results, I simply write a command-line, immediate mode calculator expression and evaluate it on the fly, without ever storing it in program memory or creating a program file for it or running it.
I'm just using its immediate mode expression evaluator. The current program at the time you do this, if any (else the Workfile) is not disturbed or altered in the least by performing this immediate mode calculation.

If you're not familiarized with the HP-71B (or SHARP models) it's understandable that you would think it was a program, but nevertheless it's a mistake on your part.

"Using a 27S, 17BII, 32SII, or 33S, the user could enter and evaluate the expression nearly as it reads, without seeing intermediate results."

But the expression is entered in algebraic or nearly algebraic form, not RPN, right ?

"Either is more straightforward than your short 71B BASIC program"


See ? I repeat once more: it is not a program, never was.
As for being "more straightforward" I fully doubt. What can be more straightforward than:

     "for X beginning at -5 and incrementing in steps of
5 display the value of X and X^2-2*X-8, then repeat
until X exceeds +5."
which is the verbose version of what my more terse, immediate command line (not program) is telling the HP-71B to do ?


"Best regards and a belated "welcome back"..."

Thank you very much, it's always a pleasure reading your postings, even if we eventually disagree.

Best regards from V.


#69

Hi Everyone.

"it seems to me you're mistakenly thinking that my HP-71B (or SHARP) example, namely:

>FOR X=-5 TO 5 STEP 5 @ PRINT X, X^2-2*X-8 @ NEXT X [ENTER]

is a program. Well, it is not. Just notice the ">" which is the prompt, and the fact that I simply wrote the expression (without a line number or such) and pressed [ENTER] (not "RUN" or something similar)"

http://dictionary.reference.com/search?q=program

pro-gram

# A set of coded instructions that enables a machine, especially a computer, to perform a desired sequence of operations.

# To provide (a machine) with a set of coded working instructions.

" "for X beginning at -5 and incrementing in steps of
5 display the value of X and X^2-2*X-8, then repeat
until X exceeds +5.""

sounds like a program to me.

best,

.


#70

As one who has spent a lot of time on my HP-71, I could say that only in a strict theoretical sense could it be considered a program. Many things that would require a formal program status with line numbers, a file, whatever, on other BASICs, can be done on the 71 as a command line, including the FOR-NEXT. The 71 has the CALC mode too which allows you to enter an equation in algebraic notation with the parentheses and all, and still see intermediate results. It's kind of cool that if you backspace after a * or ) for example, the last result is undone and you again see the last previous partially calculated equation. When I want a calculator though, I reach for my RPN HP-41, not the 71.

As for theoretical v. practical, I'm not sold on the value of being able to key in the equation exactly as written. That algebraic notation tells you what you end up with. RPN shows how you get there.

From hiring a few fresh-out-of-engineering-school graduates, I have to say that their heads were full of theory and their math skills were better than mine, but they had a hard time making the connection to the reality of what was wrong with their circuit on the workbench. They were just plain lost. I myself am a little like the man the other poster said just resorted to Simpson's rule for integration. On those few occasions when the math gets too heavy (Laplace transforms, for example), I still understand the circuit well enough to write a program to simulate it, and manipulate the coefficients until I get what I want. Results are confirmed valid when I actually build the circuit up.

Years ago I also hired quite a few graduates of these two-year programs that are advertised so much on TV (ITT Tech, DeVry, etc.), and found that they were given a grand tour of the "land of electronics" but were not taught to be a valuable part of it. These who should have been good technicians and problem solvers basically turned into electronics assembly-line workers with a huge school debt to pay off. They could see that their school had _not_ prepared them nearly as well as they thought. One of them went back to the cabinet-making business to pay off his school loan, since he could make more money there than he could in electronics.

It has been very hard to find the people who really understand the circuits well enought to be creative and come up with solutions to the design challenges that don't have canned answers. I'd be glad to have them even if they occasionally needed math help to solve something. I once had a calculus teacher who would refer to certain things as "very easy-- you just turn the crank," meaning if you just memorize certain forms, you don't have to try to think about or understand what makes them work. That might get you through school but it's not very useful in industry.

Similarly, if all the students have to have the same model of graphing calculator so they can all key things in the same way to be assured of so-called success, I'd say they're not doing much thinking or learning.

#71

GS wrote:

"I feel students today are taught how to perform functions instead of why they work"

Very true. Many years ago when I was in college, I had the opportunity to take two statistics couses that ready drove this home.

The first couse was a Stat corse designed for the business majors. All the equations were given to the students. THe tests would actually have the formulas printed at the top of each page so the student didn't even have to memorize them. The couses stressed the fact that anyone could memorize formulas but the key was to know which one to use for a particular problem. Really simple - read the problem, pick the correct equation, solve the problem with a calculator.

The second couse was for math majors. This couse only dealt with deriving the formulas and proving them. No attempt was made to actually use them in a problem. Here you got to know where the equations came from, but really didn't put them to use. No calculator required.

What a contrast. What made it even more interesting was that the same professor taught both classes and his abilty to adjust the courses to the two quite differnt grounp of students.

Bill


#72

Hi Bill,

Your two examples point out that in college education, there are many conflcting requirements.

On the one hand, true scholarship--to the average person, quixotic in appearance;

On the the other hand, practical, "career-minded" "education"

What I have found is that as far as my university time was concerned, only the first aspect provided me with any lasting benefit. When a university attempts to become a trade school, or apply formulaic "practical" approaches to teaching, you end up with students who, unless they take their own initiative, have no foundation for lasting understanding.

In fact, in the trade school I attended, the same rule applied: the fundamentals are what mattered and were remembered; the "practical" stuff was good, but not foremost--it was the theoretical understanding which mattered.

Daily, in my field of work, I find that it is my fundamental understanding of a problem which gets me to the right answer. As far as the "practical", that is easily self-taught--you can go find a reference document--perhaps even on-line----.But if you don't have the reference document, and you cannot derive the solution, then you are stuck.

Only if you truly understand a subject and can derive the solution, can you be truly prepared in life.....

So perhaps there is a sad trend in U.S. universities (and perhaps others) of creating a seeminly beneficial "practical" education, which ultimately is a waste of time.

Regards,

Bill


#73

Hi Bill,

That's what made my experience unique. I had both worlds - practical & theory. At the time I was majoring in computer science and took the business stat - later I decided to do a dual major in Mathemathics. Hence the theory stat class.

I'm not sure I would be so quick to downplay the usefulness of practical courses. I'm a consulting engineer and spend a lot of time with both theory type persons and practical contruction persons. Quite often, I think a project would never get built if left to theory only. One can get so bogged down in the theory details to a degree that the practically of the project can disappear.

If i have a choice:

Work with someone who can calculate to the n'th degree and still not know what is wrong with the project.

or

Work with someone who can tell me what's wrong, but may not be able to actually calculate it - but just knows from experience..

then I'll take the second person.


But I still like to do the theory :)

Bill


#74

Hi Bill,

You are right about practical project-oriented people being better to work with. But that same person, if he can solve the problem, demonstrates a level of understanding at the "theoretical" level--not neccessarily a computational level--there is a difference. My old professor (not maths) used to say "I don't know how to do integral calculus, I just use simpson's rule." Yet he was being unfair to his abilities. One time, I was having a problem figuring out how to do something, and he steered me to a particular chapter of a particular book--in which the whole explanation was double and triple integrals. I approached him about this and said, "I thought you couldn't do integral calculus." His reply, "I cant. But those squiggly things just amount to a fancy way of saing take the sum of all the little pieces relative to the one with the "d" in front of it...."

My point is that he had perfect theoretical understanding--even without the practical ability to do conventional computations---this was his way of being practical. He could derive the solution--albeit with his own notation--he understood the theoretical underpinnings, and that is the important point.

So to elaborate or clarify my last post, I would say that teaching a method without a theoretical basis is undesirable; yet for a person to develop a practical capacity is a good thing.

#75

I can do both - where do I sign the contract?

(VPN :)

#76

I am quite opinionated on this subject and as much as I love and do use my (HP RPN) calculators, I am also of the opinion that the calculator is prematurely introduced in American middle, junior, and senior high schools.

I blame the way our general culture is going for this- we glorify certain professions, of course our brightest and best will head there. We practically demean or devalue the teaching and research professions at all levels, unless it is biomedical, especially teaching, which we researchers do really want much to do with. But we still have a real need to fill these positions in society. We must get somebody, anybody, a warm body to walk in front of the kids in the classroom. Many of these bodies are not really up to their designated tasks, especially in science and mathematics. And then, we have too much of a fascination with gadgets; why look at ourselves, a bunch of CALCULATOR enthusiasts... of a specific kind, even!

They then embrace the calculator as advanced technology and as such they must teach its use to the kids, yet they have not the foresight to see that this will prevent developement of true mathematical and even scientific understanding and insight, nor the bent for hard work to teach the mathematical nuts and bolts without the calculator first. If the teachers really don't get it, how will the kids?

And then they hit college and face better prepared competitors from overseas...


#77

Calculators are absolutely introduced too early and are overused in education. I come from the school of thought that calculators are not necessary in learning math/science. Emphasis should be placed on theory, problem solving techniques, and application. Test problems can be designed to have relatively clean intermediates and final answers (which eliminates the argument that the numbers are too difficult to work out by hand). Calculators are indispensible real-world tools, but I believe techniques for their use should be mostly delayed until after solid groundwork in algebra and trig has been established.

Here in the US scientific carreers are viewed as the realm of geeks. In schools, the excitement of scientific achievement and discovery is virtually ignored. We glamorize athletes, movie stars, and TV personalities.

Universities dump a large portion of their resources into athletics. Out of all the collegiate athletics programs in the US, only a dozen or so are profitable... Take away broadcasting deals and merchandizing and the number drops to about 6. Many of these kids pass through with full scholarships (and under-the-table cash incentives), top priority to register for classes, and sub-par GPAs. Meanwhile, they are among the last programs to be dropped when universities are hit with budget cutbacks. How do they cope with budget issues? They cut funding for its' component colleges. My opinion is extremely unpopular, but the economic facts exist to back it up. A business would not hesitate to cut a division with a history of consistent monetary losses despite significant investment.

The bottom line: Our education system in the US needs to encourage science as a lucrative and rewarding path instead of creating the impression that science is laborious, difficult, and only for geeks.

That's my rant. No flames, please! I understand that many people enjoy following college sports, but it is my opinion that they are a generally a waste of resources.

Best Regards,
HDE


#78

Hugh,

I agree with you, calculators are introduced in school too early. My nephew last year upon entering the 7th grade was told that he had to get a graphing calculator by his school and it had to be a TI and the minimum type was the TI-83 Plus. That is entirely too early of an age and grade to be using something that complex and expensive of a calculator.

It has been 23 years sense I was in high school and we were discourage from having calculator until we were in our junior or senior year. That was when you took Chemistry and Physics. At that time as well, we could use whatever calculator we had. It was up to us to know how to use what we had.

At this day and time, I can see having a good scientific calculator in your junior and senior years in high school, of your choice, but not a graphing calculator. In you 3rd and 4th year of college I can see moving to a graphing calculator, again, of your choice not a mandatory this or that brand.


#79

Actually, personally, I don't think even college students really need a graphing calculator, not for chemistry, physics, nor even calculus. Part of the skill of a chemist, physicist, mathematician (calculator?, calculist??), engineer, is the ability to recognize by eye or within a very short time whether a set of data, a response, a set of numbers, whatever, is exponential, or quadratic, or- our favorite- linear.

A student can't get that by constantly relying on something, someone else to plot your functions. You'll never really know what they look like, never get a feel for them. What if they hear terms like, sinusoidal, sigmoid, hyperbolic, can they spot them from a whole bunch of curves, relatively quickly?

I have a 48G, 48G+, 49G+, and I have yet to use their graphing functions. Seriously, I still grab a pencil and a piece of paper, draw two axes, and off I go... sometimes even without the axes. And we all go to the computer (PC, mini, mainframe) if it gets tougher than that... especially since we now have USB connectivity to PCs.

I sincerely hope today's kids who have graduated and are currently functioning using graphing calculators will realize that they need to get a deeper feel for what those equations mean.


#80

"A student can't get that by constantly relying on something, someone else to plot your functions. You'll never really know what they look like, never get a feel for them. What if they hear terms like, sinusoidal, sigmoid, hyperbolic, can they spot them from a whole bunch of curves, relatively quickly?"



see: http://www.hp.com/calculators/graphing/49gplus/technical.html look for "plots" and some examples are given

[VPN]

#81

... that, in most cases, students have access to the www and all of the c..p available on it prior to get in contact with calculators. I guess that we should either consider that calculators today are "restricted" tools and computers, "played" by kids that are 8 YO or less, are "open wide" devices, and by nature, all available OS today are graphics. As they are the 3D games... About these I worry must.

Whatever technology we have at the time we analize facts related to education, it should be taken into account. And to the best use. When calculators were available for the first time, slide rules did not even felt their arrival because they were much more powerfull. And slide rules were neither RPN nor algebraic... ;-)

My .2¢

Luiz (Brazil)

Edited: 26 Jan 2005, 4:19 a.m.


#82

And advanced slide rules were RPN, because you could select the function once the numbers were aligned.


#83

That's the beauty of them! In some cases, you needed to align the arguments and then you'd select the operation amongst a few valid ones, sort of polish notation "as is"; in other cases, you'd select the operation prior to align the arguments, that should be aligned accordingly, i.e., pure RPN.

#84

When Donna Shalala, who was later in the Clinton cabinet and is presently at the University of Miami, became president of the University of Wisconsin the athletic programs had been through a period of deemphasis. Alumni giving to the Universiy waa also slow across the board. Ms. Shalala studied giving patterns and found that the supporters of successful athletic programs were also substantial givers to other University programs. Conversely, detractors of the athletic programs gave frugally or not at all to any programs. Her response was exactly what one could expect. She re-emphasized athletics!

(Ms. Shalala's experience was reported in the Hew Yorker magazine back in those years -- the late 1980's or early 1990's as best I can remember.)


#85

Very interesting. I've heard figures going both ways in terms of donations. For most universities donations from former athletes or other supporters would be the only significant chance at turning a net profit. I will certainly look into the articles you mentioned. Thanks for the tip.


#86

The article that I mentioned appeared in the April 26, 1993 issue of The New Yorker. The discussion of athletics stated in part:

"... The political-correctness faction was just one of several groups that made demands on Shalala in Madison. Another one, perhaps even larger, was the football fans. A series of losing seasons had led to a dramatic drop in attendance at Madison's Camp Randall Stadium, from about seventy-four thousand per game in 1984 to fewer than forty-two thousand in 1989. When Shalala took over "the athletic program was a disaster," she told me. "We had no Notre Dame tradition, so we had to build a new tradition." ...

...Shalala's passion for improving the Wisconsin football team was only part of her plan to revive the university, but she never underestimated its importance. ...

... Barry Alvarez, who had been brought in as coach by Shalala, says "I'd been an assistant coach at Notre Dame, which is pretty well known for supporting football, but I had never seen the kind of support Donna gave our football program, ..."

End of quotes. With apologies to Ms. Shalala and to Jeffrey Toobin, the author of the article, I must admit there is no mention in the article of a tie-in between athletic success and giving by alumni as I stated in a previous submission. So far I have been unable to recall another source for that idea but I'll keep searching my files and my memory. At 76 I am starting to have difficulty remembering some things. (I think my read amplifier is going bad!) Hopefully, I have not starting remembering things that didn't happen.


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