# Calculators in education

1. Mar 12, 2007

### 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!

2. Mar 12, 2007

### 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.

Last edited: Mar 12, 2007
3. Mar 12, 2007

### 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 accomodate this view is a factor.

4. Mar 12, 2007

### 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.

5. Mar 12, 2007

### 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.

6. Mar 12, 2007

### 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!!!???

7. Mar 12, 2007

### arildno

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.

8. Mar 12, 2007

### 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...

9. Mar 12, 2007

### 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.

Last edited: Mar 12, 2007
10. Mar 12, 2007

### ZioX

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.

11. Mar 12, 2007

### ZioX

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.

12. Mar 12, 2007

### 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.

13. Mar 12, 2007

### Integral

Staff Emeritus
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.

14. Mar 12, 2007

### 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.

15. Mar 12, 2007

### 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).

16. Mar 13, 2007

### 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.

17. Mar 13, 2007

### mathwonk

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

18. Mar 13, 2007

### 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.
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.

19. Mar 13, 2007

### arildno

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

20. Mar 13, 2007

### 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 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.