# Calculators in education

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
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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.

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mathwonk
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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 accomodate this view is a factor.

symbolipoint
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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
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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
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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
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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.

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...

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.

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...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.

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 in to his calculator to figure it out.

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
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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
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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
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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).

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

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

Gib Z
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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 cant 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.

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.

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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.

"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
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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.

JasonRox
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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.

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.

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
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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.

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.

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.