Are Calculators Hindering Math Education? Share Your Opinion!

[drpizza]

I just received a memo from New York State's department of education:
"Calculators are instrumental as an investigative tool in the teaching and learning of mathematics to enhance students’ conceptual understanding. The graphing calculator should be used for all types of classroom activities and homework..." This is in regards to teaching algebra, geometry, and trig.

I'm interested in knowing your opinions on this. I agree that graphing calculators can be used as an investigative tool in algebra (i.e. examining parabolas in the form y=ax^2 and seeing what happens as the value of a is varied.) However, I've long felt that the reliance on calculators prevents many students from learning some of the important concepts/skills. ('Why bother learning how to do something by hand, if a calculator can do it for you?' seems to be their reasoning.)

I'm experimenting with a calculus class this year. Other than showing that a calculator can find the value of a derivative at a point, and can approximate a definite integral, we're avoiding calculators.

I'm interested in your opinions on the use of calculators in high school math classes. (Please note in your opinion where your opinion is coming from - student? teacher? engineer?)

Thanks!

 

[mathwonk]

after trying several different combinations, i now and for some iime, avoid almost all calculator use in classrooms. occasionally i ask a student to verify my mental calculation by calculator, to illustrate how easily i have matched the calculator with a simple approximation method anyone can use.

The memo you quote is full of erroneous statements in my experience. actually calculator use seems more to inhibit understanding of concepts, and weaken computational ability. this has been borne out in many different classroom settings in my teaching career.

there is nothing sadder than watching a calculus student try to multiply 13 by 64 by hand, by adding a column of thirteen 64's. I have seen this on a test.

trying to teach the fact that exponentiation changes addition into multiplication is lost on students who have never multiplied anything out. conceptual understanding is based on obseving the feautures of examples. calculator use deprives students of close familiarity with the working of computational examples.

just try teaching uniqueness of prime factorization in algebra to students who have not spent time trying to factor integers, or the root factor theorem or the division algorithm, to ones who have not tried to factor polynomials over Z.

on a calculator, they only see the result of the computation, and do not learn either how it is done, or how to generalize or improve it.

sometimes for fun i show them a computer antidifferentiating 1/[1- x^20] or some such silly business.

to demonstrate the difference between speed and intelligence, one could then let the computer attempt to integrate (1+ln(x))sqrt(1+ [xln(x)]^2).

however as Edwards and Penney observe, this yields easily to the substitution u = xln(x).

I am 2006 PF math guru of the year, a profesional mathematician, researcher in algebraic geometry, BA from Harvard, PhD from Utah, NSF postdoc, frequent PhD committee member in mathematics and mathematics education, parent of 2 children, user of personal computers since 1980's, author of notes on foundations of real numbers, calculus, differential topology, algebra, complex analysis, algebraic geometry, sheaves, cohomology, riemann roch theorem, and teacher of mathematics to students from 2nd and third grade, junior high, high school, college, and grad school, lecturer at regional, national, and international conferences, for over 40 years.

 

[mathwonk]

some other people think the value of letting a computer do the visualizing for the student helps very weak students enter a subject like geometry, who never could do so alone.

i am sceptical, and suspect also that the huge amount of grant money available to those who accommodate this view is a factor.

 

[symbolipoint]

They are computational efficiency tools for use on tests and quizes. Their use needs to be restricted to certain portions of work ABOVE the intermediate algebra level (except for the sensible use of a scientific calculator just for computational efficiency).

A more mature mentality not often found in some younger students is to use a graphing calculator just for checking his/her work only; not for performing the exercises.

 

[mathwonk]

one subtelty lost on students is that calculators have finite capacity, i.e. finite degree of accuracy. if a calculator say ahs only 12 place accuracy, then calculator numbers are not even dense in the real line, and do not contain any irrational numbers, or even any powers of 1/3.

hence literally ALL the thoerems in calculus books are FALSE for calculator numbers, intermediate value, mean value, fundamental theorem of calculus, differentiability, continuity of basic functions.

indeed ALL calculator fucntios are step functions, which are constant on intervals of length less than the accuracy of the calculator.

hence none of them are continuous or differentiable.so conceptual understanding is impossible for a stduent who thinks in terms of calculator accuracy.

try this. ask you student to compute sqrt(2) on his calculator. then when he says something like 1.414 ask him if that is correct. instead of observing that it cannot be correct because when you square it the last digit is a 6, h will instead square it no his calculator, and triumphantly declare it is right, because his calcualtor may tell him it is!this kind of blind idiocy is the opposite of conceptual understanding.

thus instead of elarning that the theorems are true only in an ideal sense, for numbers that are limits of the ones on calculators, the stduents "learns" that the theorems are true because he memorized them, and are true in settings where they are not, and will adhere to this even in the face of obvious contradictions.

 

[JasonRox]

I practically never pull out my calculator.

On the other hand, I was tutoring someone and he pulled his calculator to calculate 3/4*(1)^2-3/4(-1)^2 (for a definite integral). I wanted to shoot myself. He even put in the 1's in the calculator. One time he had the expression (1/3)x^3(3) and I told him to break it down, he had no idea how! Where is the gun?

Anyways, he's doing 1st year Calculus, and has like a 45% in the class because he copies assignments off his friends. His calculator keeps him alive the rest of the time. If the calculator helps enhance his mathematical abilities, then please explain WHY THE **** WE HAVE A 1ST YEAR UNIVERSITY STUDENT WITH GRADE 7 OR LOWER MATH SKILLS!?

 

[arildno]

I just received a memo from New York State's department of education:
"Calculators are instrumental as an investigative tool in the teaching and learning of mathematics to enhance students’ conceptual understanding. The graphing calculator should be used for all types of classroom activities and homework..."

This is a patently FALSE statement. If it were true that calculators were INSTRUMENTAL in developing understanding of maths, no one prior to the age of calculators would have understood maths.

I fully agree with mathwonk.

By and large, calculators are utterly useless in developing UNDERSTANDING and SKILL in maths, their only benefit is to shorten the time on intermediate calculations.
Thus, if you have a problem that as a trivial sub-problem contains the performance of several operations, then indeed calculators are handy tools, but rarely ever else.

 

[Cincinnatus]

In some ways I was the ideal result of the sort of education advocated by that memo...

I hated math more than anything all through elementary, middle and early high school. I don't think I ever really understood it until they allowed us to use calculators. Once calculators were allowed, I started to do well in my math classes, still disliking the subject though.

By the time I went to college I had been doing well in math for a while so I took a more theoretical linear algebra class and amazingly actually enjoyed math for the first time. While taking that class I did two things; I decided to major in math, and I put away the calculator.

Now several years later I don't use the calculator really at all, but it's possible that if I never had used the thing, I wouldn't have ever made it far enough in math to find it interesting. I may have stayed just as "mathaphobic" as I was when I was younger.

Or maybe it was just a coincidence and the calculator had nothing to do with why I started to do better in math in the later part of high school...

 

[drpizza]

That's an interesting comment from Cincinnatus. I wonder, how much of it can be attributed to your teacher's skill in how to integrate calculator use into the curriculum? I can come up with many many examples of areas where calculator use replaces a fundamental understanding of "simple" concepts. In NY, we include combinations and permutations in our algebra curriculum. If I were to pose the question "how many different combinations of 3 letters are there from the English alphabet?" I'd estimate that 80% of the students in the state would know that it's a combination, and to enter 26 nPr 3 on their calculators. However, I doubt very many would even understand why it's (26*25*24)/(3*2*1) (the 1, of course, not being necessary)

So, I ask the broader question; does the ability to answer that question (with the use of a calculator that does combinations) indicate a mathematical skill? Or does the ability to answer that question indicate a calculator skill? Certainly, there is some understanding - a student recognizes it as a combination problem (or gets lucky with a 50/50 guess.)

Maybe I should also add, a typical question about combinations might look like this multiple choice question:

10C8 is the same as which of the following?
2! 10C2 10P2 8!
Sadly, all one needs to know is which buttons to press on a calculator to calculate each of those. Of course, someone who has never seen this notation might be able to figure it out by finding ! on the calculator as well as the nPr and nCr functions.

 

[ZioX]

...i.e. examining parabolas in the form y=ax^2 and seeing what happens as the value of a is varied...

All the stretches/translations can be verified algebraically. If the student has trouble understanding the algebra than he can just plot it himself...which gives better insight into what's going on than letting a calcualtor graph it.

Calculators are garbage. The only time I ever use them is to calculate something I wouldn't want to calculate by hand. Calculator dependency is a bad bad thing.

And of course, everytime a class gets these nifty graphing calculators to do an assignment a good majority of them goof off and draw random junk.

 

[ZioX]

Of course, someone who has never seen this notation might be able to figure it out by finding ! on the calculator as well as the nPr and nCr functions.

That's right too! I remember some friend of mine never being able to remember the angle sum formulas for sin and cos, so he would just plug it into his calculator to figure it out.

 

[Eighty]

My calculator sparked my programming interest. Also, I think it helped in developing my intuition for functions by making it quick and easy to graph functions. But I'll admit it did inhibit my mental arithmetic abilities.

 

[Integral]

I self discovered the concept of a fixed point by repeatedly hitting the square root key.

I did some work with some of the first Calculator algebra (HP28c in 1987) and was not impressed, seemed that it was harder to manipulate expressions on the calculator then by hand. Also I found the HP28 to very useful for unit conversion. I used it a lot when I was searching for and comparing furnace insulation since there is no standard units for thermal conductivity, the calculator made it very easy to compare materials from different manufactures.


While I can see some possibilities for teaching fundamental math, it is to easy to turn them into a crutch while permanently crippling the student.

 

[mathwonk]

my first experience with calculators was as a program changer in an insurance firm in 1961 at age 19. I had an adding machine on my desk which i used daily. After that brief experience, I had changed myself from a bright quick mental calculator into a mathematical imbecile who could not make the simplest mental computation without great effort. I remember being somewhat scared bY the ease with which my mental abilities had been eroded.

 

[mathwonk]

in calculus class i regularly give my class illustrations such as the fact that the series for sin(x) gives us the approximation x - x^3/6 +..., and compare the resulting approximation of 5/6 = .833... for sin(1), or 1-1/6 +1/120 = .84166... to whatever their calculators give.

or i do a Newtons approximation which is more fun and easier to iterate. these can be done in ones head, and are usually almost as good as what their $100 calculators give to a few places.

I am trying to teach them how simple it is to do these thigns, insteqd of forever remaining in awe of the simplest computations (and paying through the nose for them).

 

[Crosson]

I am in total favor of technology, I think high school students should be trained in Mathematica or similar.

Seriously, why bother practicing something by hand if you can instead use a machine? It doesn't prevent the learning of anything other then outdated skills. Square roots by hand, anyone?

I am student, but I have also taught using technology in the classroom.

 

[mathwonk]

well, since you decline to learn by the experience of others, continue on your way until you have more data.

 

[AlphaNumeric]

As someone mentioned, there's a few 'pleasant suprises' to be found as a child messing around with a calculator. As a kid of about 8, we were asked to find two numbers which multiply to 20 and add to 10 and I spent many a bored lesson doing this for other pairs of numbers. Obviously at age 8 quadratics were a little advanced. Similarly fixed points in iteractions were something I came across early because I such investigations. However, I only appreciate such "wow, I came across this years ago messing with a calculator" because I now understand the generalisations behind them.

Generally however, they seem to do little more than destroy someone's mental arithmetic abilities. I remember seeing on old teacher (who remembered good old slide rules) scalding (verbally, obviously!) a fellow student for reaching for a calculator to do something like 5*14, we were 17.

Calculators seem to teach a disjointness to maths. Two neat numerical results are often not seen for the general case they illustrate, but just two seperate, unrelated sums.

Square roots by hand, anyone?

As a kid of 12, I used to be proud of the fact I'd do 2 decimal square roots of numbers less than 100 in my head. Not instantly, more a kind of iterative process but still pretty good. I used to have a thing where I'd look at car number plates (which usually have 3 digits here in the UK) and see if I could make 10 using them, made car drives slightly less boring and to keep up to fairly fluid traffic it sped up very basic arithmetic I did.

I've not used a calculator in 5 years and even in high school, it was only for the pointless "Give tan(34) to 3 decimal places" questions or things like Simpsons Rule for integrals.

I managed an entire maths degree without using a calculator and in exams we were banned them anyway (not that they'd have been of much use). 'Proper' maths isn't about numerical calculation, it's about the underlying relationships and structure beneath the surface which doing a specific numerical example totally obscures.

 

[arildno]

Square roots by hand, anyone?
Very simple!
\sqrt{17}\approx4(1+\frac{1}{32})=\frac{33}{8}

 

[Gib Z]

Square roots, Newtons method :) *does a little dance*

In good relation to the topic, i once reiterated Newtons method 5 times, which the starting approximation the one i keep in my head, 19601/13860. It gave hundreds of digits of accuracy, a calculator can't do that :P

The numerator and denominator are tens of digits long, but even the first iteration was enough decimal places to trick the calculator. You enter it, then square it, it said 2, exactly.

Most calculations are very easy to do by hand at the very least, mental Newtons method for sines as mathwonk said seems to induce fear into me >.<

I think calculator use should be limited. There should only be certain section of the test that require calculators, others only mathematical expressions.

In Australia for calculus, no one has or even heard of graphical calculators to numerically integrate or antidifferentiate for us...so at least we have that on us.

 

[JoAuSc]

I took high school calculus with a graphing calculator, and I don't think it hurt me, though 1.) we did not use them much until at least halfway through the year, and then mainly for figuring out integrals that didn't have analytical solutions (problems such as the word problems on the AP calc exam (AB)); 2.) back in elementary school I got this book called Mathemagics which focused on mental calculations, and that coupled with my grade school teachers not allowing calculators meant I had experience in calculating numbers. I was also the kind of kid who would watch Square One.

I think it would've been nice if we were taught basic programming in math class, though. I didn't really understand the significance of the definition of the derivative (f(x+h)-f(x))/h until I learned to use the definition in order to calculate derivatives numerically.

there is nothing sadder than watching a calculus student try to multiply 13 by 64 by hand, by adding a column of thirteen 64's. I have seen this on a test.

Oh my God, that's horrible.

And of course, everytime a class gets these nifty graphing calculators to do an assignment a good majority of them goof off and draw random junk.

Yes, but some of us used them to draw these cool spirals using the polar graphing function, stuff like (5 + sin(10*theta))*theta.

 

[dimensionless]

Square roots by hand, anyone?
Very simple!
\sqrt{17}\approx4(1+\frac{1}{32})=\frac{33}{8}

I was going to say that. Of course I'm slower than I would be (had calculators not been invented) when it comes to deviding 5 digit numbers in my head. On the other hand I'm not convinced that that skill would be all that useful. I still don't know how to solve for square roots.

 

[drpizza]

"Generally however, they seem to do little more than destroy someone's mental arithmetic abilities. I remember seeing on old teacher (who remembered good old slide rules) scalding (verbally, obviously!) a fellow student for reaching for a calculator to do something like 5*14, we were 17."

Back in 1983, my pre-calculus teacher chastised me and 2 others for picking up our calculators to add two 2-digit numbers together. I'll never forget that teacher! And, I'll forever be grateful.

 

[JasonRox]

I am in total favor of technology, I think high school students should be trained in Mathematica or similar.

Seriously, why bother practicing something by hand if you can instead use a machine? It doesn't prevent the learning of anything other then outdated skills. Square roots by hand, anyone?

I am student, but I have also taught using technology in the classroom.

Did you read my post?

 

[JasonRox]

I self discovered the concept of a fixed point by repeatedly hitting the square root key.

But you understood why this happened, right?

Others wouldn't have a clue.

 

[AlphaNumeric]

I'll never forget that teacher! And, I'll forever be grateful.

Same here. He disliked calculators in general, had no qualms about calling you stupid (it was a small class and we all got on very well, so no hurt feelings, it was for our own good) and would often go at a problem from multiple angles to get us to see the underlying structure not just mindlessly compute results. Had a lot more passion in him for the subject, despite being much old than any of my other maths teachers. Oh he was a cynic but that's probably why I liked his lessons so much.

 

[Doodle Bob]

Same here. He disliked calculators in general, had no qualms about calling you stupid (it was a small class and we all got on very well, so no hurt feelings, it was for our own good) ...

Yes, I will never forget that great time when my math teacher locked me in the closet for three days for mistaking a polynomial rate of change for an exponential one. Or the time he shot my dog for forgetting to carry that one. I think he kidnapped my family for a time too, but I can't remember what I did to deserve that one. I'm sure that I deserved it, though. Great times, and I'm a better mathematician for it!

Keeping in mind that the *vast* majority of humanity does not particularly like to be tortured (however benignly) into learning a subject, I would say that most of these anecdotes do not make much of a case for withholding information and tools from the students in order to make them learn arithmetic.

 

[arildno]

But they DONT learn arithmetics by using a calculator!

Being able to perform a mathematical algorithm, has absolutely no connection with whether you understand the algorithm or not.

Even less understanding can be gained from acquiring the ability of hitting the right buttons on the calculator.

 

[Doodle Bob]

But they DONT learn arithmetics by using a calculator!

That would depend on the pedagogy of the teacher using (or not using) the calculator. Don't assume that they all teach the way you've seen them teach. Some of the most inspired mathematics teaching that I've seen was firmly calculator-based; and some of the most godawful-boring, tired, why-hasn't-this-teacher-retired-already lessons that I have been witness to were calculator-free. *And* vice versa: some of the best lessons I've seen have been calculator-free, yada yada.

I was mildly amused to read about all the calculator naysayers who admitted to finding clever and interesting explorations of numeracy using calculators. Yet, somehow, the hoi polloi of K-12 students are not allowed into that club. I guess they have not yet been tortured enough into hating mathematics by their teachers to be allowed to have fun.

In addition, although this may be tangential to the conversation, I would have to say that such software programs such as Sketchpad have vastly improved what had become a rather stale geometry curriculum.

 

[drpizza]

I'd have to agree with Doodle Bob that calculators can be used to create a very stimulating lesson. However, do you think that mandating that all math teachers use them as an integral part of the course will really improve things in general? I believe that the level of skill necessary to pull it off and accomplish two things - that the kids get a better understanding, and that the kids don't become overly calculator dependent - is uncommon.

 

[JonF]

A TI-89 was one of the worst investment I ever made.

 

[AlphaNumeric]

I would say that most of these anecdotes do not make much of a case for withholding information and tools from the students in order to make them learn arithmetic.

When you're 17 and doing 1st order ODEs, you shouldn't be reaching for a calculator to do something like 14*11.

As for my old teacher, comments like "You're being stupid" were not meant as "I think you're stupid for doing that" but more "You're better than that". We were a class of 3 students, the only 3 taking 'Double Maths' in a year of 200, and as such he expected a touch more from each of us. We'd not be scalded for using a calculator to compute the decimal expression for a trig answer, but if someone used a calculator to do a method they were supposed to know the algebraic method, he'd comment, not to put us down but to make us realize it was not worth using a calculator as a crutch in the long run.

I still think of such rationale when I see 1st year students reaching for a calculator to give the decimal expansion for an answer like '2pi', in a maths class. The question (and entire course!) makes no mention of decimals or calculators, but emphasises algebraic expressions but still it's been ground into some students that 'decimals are better' or 'decimals are the proper answer'. Solving x^{2}-2=0 doesn't gives x=\sqrt{2}[/tesx] but &quot;x = 1.414 to 4 sig fig&quot;. Fine if you&#039;re a physicist or engineering (well, sometimes!) but not as a mathematician.

 

[Crosson]

The machine allows us to expand the scope with which we see mathematics.

Consider the falling scenario:

Teacher: Give me a number.

Student: 6


Teacher: Give me a polynomial.

Student: x^3 +3x -1


Teacher: Give me a matrix.

Student: {{1,4,7},{8,7,0},{4,3,5}}


Teacher: Give me a group.

Student: Z_4 x Z_2

Do you guys see the pattern? All too often students walk away with the unconscious misconception that every mathematical object is built out of a handful of small integers. Its not that they would deny large integer or irrational matrices if they saw them, but that their mind is lacking in examples of these and so they do not think of these things.

I agree that "1.414 with 4 sig figs" is a joke compared to \sqrt(2), but this is why I am in favor of systems that do symbolic math along side numerical math, like mathematica. Mathematica will always return \sqrt(2) or \pi when appropriate, and so it becomes a matter of interest when a approximate decimal is returned.

Without the computer we are confined to a small range of integers. A little time spent with the computer gives us perspective on the smallness of human calculation, a perspective that frees us and empowers us.

I think this debate is rather like an ancient debate that must have occurred over notation. The adoption of arabic numerals vastly expanded the scope of western mathematics, although it is true that with them arithmetic becomes a less mental activity.

When you're 17 and doing 1st order ODEs, you shouldn't be reaching for a calculator to do something like 14*11.

Of course you should, you must have long since mastered a basic facility in arithmetic and could better spend that time learning something new.

For that matter, you should see how the computer generates the analytical solutions for the general class of linear ODEs you are learning about, and the computer will show you how even relatively simple ODEs lead to solutions which are totally out of the realm of human possibility.

 

[morphism]

Are you kidding? If someone asked me for an example of a number, I'm not going to give them pi^(17^(sqrt(3)/2)) * gamma(4.5); unless I'm trying to be a smartass, I'll pick the most natural thing that springs to mind: a natural number. Also, what does this have to do with the topic? Are you somehow implying that calculators help you "think outside the integers"?

And you're saying someone should use a calculator to compute something as trivial as 14*11, when in fact it would take more time and effort to use one for such a thing?!

Sure, computers can be useful, but let's not get carried away...

 

[uart]

I think that calculators and numerical mathematics have their place in mathematics teaching, but I also agree that over-reliance on calculators is a bad thing (and it's also extremely wide-spread these days).

Seriously I’ve got students that will use their calculator to do something like adding 2 to a number. One thing that particularly irks me is when a student reaches for the calculator for something trivial and then to add insult actually gets the wrong answer due to operator error. A typical example might be where a student has an expression like \frac{x+2}{7} and they have to substitute x=9. So they dive straight for the calculator and come up with 9.2857 which they happily write down without a second thought that it can’t be the correct answer.

You’d be amazed at how frequently this type of thing happens when students are over-reliant on the calculator. I suppose the moral of the story is that if you are going to be over reliant on a calculator then at least you’d better learn to use it correctly.

 

[mathwonk]

doodlebob and others are stating correctly that some teachers can make a calculator based program fun. nonetheless their students will never learn any ideas until they lay down the calculator and start thinking about what has been displayed there.

the stimulation from the calculator needs complementing by actually thinking. and thinking is not necessarily torture. we are thinking beings after all.

fun is good motivation for work, but it is not necessarily useful work.

being asked to think, should not be equated with torture.

for extremely weak and frightened students, afraid to begin to think or imagine, calculators may help take away some of the fear. but at some point they must be abandoned for real conceptual work, or there is no gain.

for very strong students, already in possession of computational skills and conceptual understanding, a calculator does little harm, but has little value either except for experimenting in cases of great computational complexity.

For the great majority of students who are able to begin to learn concepts, but need practice both in computing and in reasoning, calculators make this process harder, and are actually harmful.

thus calculators may help motivate beginners to learn, and can be used valuably by those who have already learned, but for those who are in process of learning unless used with great care, they are useless and even detrimental.

this is the lesson of a lifetime of teaching.

 

[Trail_Builder]

im only high school level student and always use a calculator if allowed (within reason, like i know most easy powers and stuff), and find i make silly mistakes trying to work without it.

though i do try...

o well, whatever floats you boat

 

[mathwonk]

you are the perfect illustration of someone who is beign harmed by a calculator, since you are not learning to stop making silly mistakes.

there is a reaSON FOR SILLY MISTAKES, LIKE misunderstanding basic ideas.

unless you change this, in a few years you may well be in college wondering why you are failing calculus, or abstract algebra.

 

[Crosson]

there is a reaSON FOR SILLY MISTAKES, LIKE misunderstanding basic ideas.

Silly mistakes come from lack of concentration, it is the fact that they don't come from real misunderstanding that makes them "silly".

If the complaint is that students make errors in the use of the machine, how does this argue against teaching the use of the machine in the classroom?

 

[mathwonk]

i recommend locking this thread as it is potentially endless. everyone, especially those with no experience, has an opinion.

 

[JasonRox]

Silly mistakes come from lack of concentration, it is the fact that they don't come from real misunderstanding that makes them "silly".

I disagree.

You shouldn't have to concentrate much for small things, and I believe that's what he's saying.

 

[Gib Z]

Eventually the thread will die, just you wait mathwonk.

I have little experience compared to that of mathwonks, but I still think calculators disrupt the theory of math. People in my school know Newtons method just fine at my school, but if you told them to square root something by hand they would stare blankly at you.

The sqrt button of the god damn calculator stops them from realizing they can use what they know rather than use a calculator!

 

[arildno]

Silly mistakes come from lack of concentration, it is the fact that they don't come from real misunderstanding that makes them "silly".

Partly true, but wholly misunderstood.
"Concentration" is not an automatic, inherent ability in everyone, the skill to concentrate and not let yourself be distracted must be meticulously, and painstakingly, built up. In particular, close attention to what you write AND HOW YOU GET THERE is a crucial feature about the concentrated, working mind that requires an on-going understanding of intermediate steps that the short-cuts in calculator use inhibits, by way of concealing these steps within the machinery.

Only those students that are good at maths ought to be allowed to use a calculator, everyone else should be denied that "aid", for their own good.

 

[chaoseverlasting]

Calculators inhibit your ability to learn techniques that would enable you to do those calculations quickly. I am a student and I try not to use calculators at all. For one thing, they arent allowed in my exams, where the time I spend on calculations is very valuable. Therefore, I must be quick and accurate when I calculate something. If I am dependent on a calculator, I won't ever be able to do that. Like 14*11 =140+14=154.

 

[mathwonk]

the only time i have found a calculator useful was in teaching graduate algebra once, and trying to carry out examples of some of the proofs of the fundamental theorem of symmetric functions, that all symmetric functions are polynomials in the elementary ones.

the proofs consisted of very impractical algorithms for transforming a symmetric function into a polynomial in the eleme ones, but these transformations took forever and were very hard to complete without error.

thus even finding the basic expressions for the discriminant of a cubic polynomial was essentilly undoable for me. , i.e. if r,s,t, are the roots of a cubic, the discriminant is a formula for the symmetric product
(r-s)^2 (r-t)^2 (s-t)^2 in terms of the coefficients of the polynomial, namely in terms of r+s+t, rst, and rs+rt+st. Try it.

I this case only, I used mathematica to cary out the operations. But then i learned a better way, using the resultant, to do it by hand. Its all in my webnotes.

 

[mathwonk]

by the way, just because i have decades of experience does not mean your experience may not differ from mine, even if i often disallow this possibility. that's what makes it hard to tell just which advice to take from your elders.

 

[D H]

im only high school level student and always use a calculator if allowed (within reason, like i know most easy powers and stuff), and find i make silly mistakes trying to work without it.

A calculator is no guarantee against silly mistakes. If you can't do simple problems in your head or on paper, how are you to know the answer on the calculator is correct? Over-reliance on calculators is lazy and stupid.

Keep in mind that when calculators are not allowed on a test, the test questions are usually designed to yield nice, neat answers. Moreover, the instructor would much rather see an answer like (\sqrt5-1)\pi[/tex] than 3.88322207745093 because the former shows that the student was thinking while the latter shows the student was punching buttons. This is particularly so if the answer was supposed to be (\sqrt5-2)\pi[/tex].&lt;br /&gt; &lt;br /&gt; How many of you have given $20.50 for an item that costs $5.32 and received a stymied look from the cashier followed by $5.18 in change? I asked for the manager when the cashier would not relent (&amp;quot;I gave you a twenty and change. I should get 15 and change back.&amp;quot; &amp;quot;Well, the cash register told me to give you $5.18.&amp;quot;). The manager quickly saw a twenty at the top of the 10 dollar bill bin and fixed the problem.

 

[JasonRox]

How many of you have given $20.50 for an item that costs $5.32 and received a stymied look from the cashier followed by $5.18 in change? I asked for the manager when the cashier would not relent ("I gave you a twenty and change. I should get 15 and change back." "Well, the cash register told me to give you $5.18."). The manager quickly saw a twenty at the top of the 10 dollar bill bin and fixed the problem.

Sounds like the cashier has mistaken your $20 for a $10. Not a mathematical mistake.

 

[D H]

She made several mistakes. First, she put the twenty in with the tens. Second, her fingers by-passed her brain and automatically entered $10 because that is where her fingers put my bill. Third, she refused to believe her cash register could be mistaken. Fourth, she did not double-check the subtraction. She knew I gave her a twenty and still insisted the amount on the register was correct.

 

[JasonRox]

She made several mistakes. First, she put the twenty in with the tens. Second, her fingers by-passed her brain and automatically entered $10 because that is where her fingers put my bill. Third, she refused to believe her cash register could be mistaken. Fourth, she did not double-check the subtraction. She knew I gave her a twenty and still insisted the amount on the register was correct.

The first mistake led to all the others. Not a mathematical error here.

How do you know she knew you gave her a $20? You're not her.