## Calculators vs. PDAs in the Classroom 550

Posted
by
michael

from the crutches-for-the-weak-minded dept.

from the crutches-for-the-weak-minded dept.

TheMatt writes

*"CNN.com is reporting about a new conflict perhaps emerging in classrooms: calculators v. PDAs. The article talks about how TI seems to be making their latest calculator more PDA-like, while PDAs are gaining TI-like functionality. A comment on current math education is this quote from the article: "When you have circles and ellipses, there is no way you'd be able to do this without a calculator," Jarvis said. "It helps us visualize what we're doing." Were the compass and geometry uninvented?"*
## PDA?? (Score:4, Funny)

The downside of being a geek is you don't know whether to lose face admitting your system is down and you can't reach it -or- admit you really didn't do your homework, thus can't download it.

## Re:PDA?? (Score:3, Funny)

Like, why not just go straight cellular and connect to the internet or your home beowulf cluster?With WLAN or Bluetooth networking, you could even build a classroom-wide Beowulf cluster _with_ PDAs...

## Why stop there? (Score:3, Insightful)

Like, why not just go straight cellular and connect to the internet or your home beowulf cluster?Why stop there? Put a webMathematica [wolfram.com] server up, and access it though your PDA.

## Re:Why stop there? (Score:3, Insightful)

## Who Wants to Be a Millionare? (Score:3, Funny)

Like, why not just go straight cellular and connect to the internet or your home beowulf cluster?Can students use their cel phones to call their life-lines during exams?

## TI-86 (Score:3, Funny)

## Re:TI-86 (Score:2)

Your other option is to get Linux on one of these babies and try to get one of the many Simcity clones to run on it.

Shouldn't be too hard.

## Re:TI-86 (Score:2, Informative)

## Raising the bar (Score:3, Interesting)

## Re:Raising the bar (Score:3, Informative)

## Re:Raising the bar - sextants (Score:5, Informative)

Maybe you'll be bad with the cheap sextant, but you should still get within 30 miles which will let you make landfall during daylight.

## Re:Raising the bar (Score:2, Insightful)

Remember...somebody has to make the caluclator, PDA, compass, protracotr, or whatever tool ends up aiding in the job at hand.

## Whatever... (Score:3, Insightful)

The compass and protractor are as obsolete as the sextant. If a kid graduates from school and doesn't know how to work a PDA, he's going to quickly learn how to work a deep fryer.Nice troll...

I suppose the PDA is only a requirement if you want to be a marketdriod. For the rest of us,

thinkingis going to be considered a valuable ability. Right now, a PDA is just an interesting toy, and many people somehow manage to exist and lead productive, organized lives without one.For what it is worth, I am all for banning calculators from the classroom. Far better to be able to demonstrate the process by which the student arrived at an answer than to pull some magic number out of the air and expect full marks.

I just graduated from university a couple of years ago and calculation devices of any type were strictly forbidden in my math, statistics, and CS classes. Sometimes it was a pain, but then the answer was rarely expressed as an integer anyways...

## Re:Raising the bar (Score:2)

The professor attended the meeting where the model was presented to the appropriate body for approval. He took one look at the results, then wrote down three equations that showed the model was fatally flawed.

In fact, his motto as a teacher was, "If you can't solve it in half a sheet of paper, you don't understand the problem". A little bit of an exaggeration in the real world, but not by much. {BTW - one of my classmates had his calculator battery go out during the final exam, which was worth 70% of the grade. He was freaking out, so I handed him my calculator without a word. Didn't need it, and those were some of the hardest problems I solved in engineering school.}

There is a difference between being able to do something by rote, and understanding what you are doing. I use calculators as appropriate but I

don'tuse them whereinappropriate, such as foundation classes.sPh

## Re:Raising the bar (Score:2)

## I'm old :[ (Score:3, Informative)

## Re:I'm old :[ (Score:4, Interesting)

So now, If I tutor someone, I made them leave the calculator at home. Everyone to date ended up actually learning, rather than memorizing.

## Re:I'm old :[ (Score:2)

stronglyrecommended we use one of these graphing calculators in our work. I didn't. I think I'm a better mathematician for it now.## Me too (Drafting) (Score:2)

## Re:I'm old :[ (Score:2, Insightful)

The response (Not an exact quote, but it stuck with me), "One needs a good imagination to study math, not a calculator or computer; paper & pencil are helpful when it comes to proofs."

Of course that was my point, but they assumed that I was like most other people today... thinking that a persons ability to use a computers or a calculators make them smart or able in the sciences/math/computer programming.

Ted Tschopp

## Re:I'm old :[ (Score:2)

## Re:I'm old :[ (Score:2)

but graphing isn't one of them. My primary

use for TI-85 is units conversion. The main

thing I dislike about small calculators is

that they don't display several results and

you can't store variables. I like to write

out a complicated arithmetic statement, review

it, correct it, then press enter and see the

answer. It is also handy to be able to recall

last expression you entered. And last but not

least, few small calculators can handle imaginary

numbers and those that can often use hard to

read syntax.

That said, when I went to school, calculators of

any kind were a rarity. Drafting was done by hand.

I think it was better that way.

## You're old? Hell, sonny, let me tell you a story (Score:2)

As for me, calculators were forbidden in my high school math courses, but allowed in science. At that time, though, calculators were pretty much useless for anything but simple math and elementary trig.

## "It helps us visualize what we're doing." (Score:4, Interesting)

## Re:"It helps us visualize what we're doing." (Score:4, Insightful)

atleast not before doing their master's thesis in a university.Actually, at the early level is when calculators and other graphing aids are *most* useful. In my experience, the further along I got in math, the less I used my calculator (and the smaller the books got). I see calculators as a memory aid, sort of like the periodic table. A long-time mathematician doesn't need to turn to his graphing calulator to see what a sine curve looks like, just like a long-time chemist doesn't need to look up the atomic weight of nitrogen. Those things are a crutch for beginners.

## Re:"It helps us visualize what we're doing." (Score:5, Insightful)

Actually, at the early level is when calculators and other graphing aids are *most* useful.I'm a college level math tutor, and I can't even begin to say how wrong that is. Kids don't learn math by using a calculator any more than they learn to spell by using a spell checker or learn grammar through a grammar checker. I've tutored countless students who's teachers thought as you do, and none of them knew a god damned thing about math, despite the fact that they got 'A's all through high school.

When kids are first learning math is exactly the time when you

absolutelyThey need to learn how to do things by hand first, without having to rely on anything else to do it. Then, when you hand them a calculator, it's just a way to do things faster, to get the busy work out of the way so they can focus on more advanced concepts.don'twant them using calculators!In my opinion, graphing calculators should be allowed only at the calculus level and above. Below that level, they can only be a crutch. Scientific calculators should be allowed for Trigonometry and intermediate Algebra, and absolutely no calculators at all at a lower level than that.

## Re:"It helps us visualize what we're doing." (Score:2)

Try graphing polynomials by hand. Once you have several terms, it gets out of hand very quickly. Now try changing the numbers several times to see what changes. It'll take you a while.

I think the proper solution is to learn how to do basic graphs by hand, and then experiment with a calculator to get a better understanding. If you can take two derivates of a function, and know how to draw a graph those results, it's enough. Beyond that, seeing what happens when you change numbers in a calculator is fine.

## HP's (Score:2, Informative)

## The real question is... (Score:2)

## Re:The real question is... (Score:2, Funny)

## Calculatorama (Score:2, Insightful)

## Remember when... (Score:2, Funny)

sneaka calculator into class for fear that you would get caught cheating?## Geometry on Calculators (Score:2)

Cabri or somesuch? I didn't get mine until I was out of that class, but it was pretty nifty and had many ways to describe geometric situations and to get conclusions from that, much as one would with a compass and a straightedge.

Granted, they are fairly pricey calculators...

## Cheating (Score:3, Insightful)

## Math shouldn't be about rote memorization. (Score:2, Insightful)

## Exactly (Score:5, Interesting)

## Re:Math shouldn't be about rote memorization. (Score:2, Insightful)

Actually a few things in math should be drilled into students by rote. That way they will know them without having to even think about them. The multiplication table is one such thing. Also the differences between all numbers from 0 to 100 (so I can get my change quickly in case the cash register is broken.)

If you don't remember a formula there is little chance of applying it is there? At least not until you have looked it up.

## Re:Math shouldn't be about rote memorization. (Score:3, Informative)

Also the differences between all numbers from 0 to 100 (so I can get my change quickly in case the cash register is broken.)Wrong, WRONG,

WRONG!!!!!Disclaimer: I pulled graveyards at a 7-11 in 1982 and 1983.

Everyone should learn the PROPER way to make change. It pisses me off when some clueless idiot goes... "$7.47 is your change". That's not how to do it. let's say my bill was $2.53 and $7.47 *IS* my change. The correct way would be:

Say $2.53

Give Penny (say 54)

Give Penny (say 55)

Give dime (say 65)

Give dime (say 75)

Give quarter (say $3.00)

give dollar bill (say $4.00)

give dollar bill (say $5.00)

give five dollar bill (say $10.00, thank you).

That way, you know that you didn't screw up counting it, or that you didn't fsck up typeing in the amount given. Also, make damn sure you leave the money I gave you on top of the register until I agree that it's the right amount of change. This prevents "I gave you a $20! No you didn't, you gave me a $10!" arguments.

Alas, making change is a lost art.

## Visualisation (Re:Math shouldn't be about rote me) (Score:2, Insightful)

Also, make damn sure you leave the money I gave you on top of the register until I agree that it's the right amount of change. This prevents "I gave you a $20! No you didn't, you gave me a $10!" arguments.How much of these arguments would have been stopped in advance if people in the US were able to see the difference on a 1, 5, 10, whatever note by checking the colour of it?

Take the next step into evolution, colour your notes, and prevent confusion and unnecessary arguments caused by the fact that all your notes are the same colour.

After that it's only a matter of time before you adopt the metric system and your math will be easy again

## Re:Math shouldn't be about rote memorization. (Score:2)

## Re:Math shouldn't be about rote memorization. (Score:2)

## Re:Cheating (Score:2)

The use of these calculators has really changed things. There are a lot of people who still refuse to acknowledge that they exist. If a calculator can do an integral and take a derivative then it forces us to ask what is really important about what we do in a classroom.

There are a number of profs who still refuse to acknowledge that times have changed since Newton. If I were to write a test that someone could just read some formulas off of their calculator and then get a good grade, then that was a

really bad test. Personally, I think some prof's are afraid of the calculators because it forces them to actually think about what they are doing.## It's not cheating. (Score:5, Interesting)

Frankly, anyone who would regard referencing forumulae as cheating is a poor excuse for a teacher. Who cares? Let the student look up the damn formula, already, like real people do here in the real world.

The best mathematics teacher I ever had was strict as hell, but when she gave tests she let students bring a single 3x5 card filled up with anything they thought they might need. Formulae, tables, reminders, tips--anything you could fit on there.

She also held timed open-book pop quizzes. Her reasoning was simple: the more time you needed to spend looking things up the less time you'd have to actually do the math. That policy encouraged students to remember those things they used most often without forcing them to fixate on memorizing every random thing that might be conceivably needed. Both policies also give students some reassurance that a random oversight or memory glitch won't mean failing a whole test.

## And what when you move to higher dimentions? (Score:2, Interesting)

As one young math professior I had in college said

I hope you sometime get the fun of working in at least 11 dimintions. He was a young guy (first you teaching), and was truely serious about that. Now I can deal with 2d graphics just fine, and 3d graphs are normally not a problem, though optical illusions sometimes are possible so I don't rely on them, but the one 4d graph I saw just threw my mind in a loop, and I decided not to bother with them again.Maybe I'm not a visual person, but I can't deal with 4d graphs. I can deal with math in 11 dimentions if I have to, though I'm not good. The ability to work on 2d and 3d problems without a graph helps when you deal with problems that cannot be easially graphed.

Then again, all my college classes allowed calculators, but the time to enter numbers was longer than the time to calculate things in my head so I rarely used my HP-48 after my freshman year.

## Re:And what when you move to higher dimentions? (Score:2)

You don't hit the really geeky math until you deal with spaces that have

uncountably manydimentions (that is, more dimentions than there are integers; or more accurately, as many dimentions as there are points in a real interval.)Most Physics and EE students hit this sometime during their senior year; most math students, sometime in functional analysis.

## Lets crawl before we walk... (Score:4, Insightful)

## Re:Lets crawl before we walk... (Score:5, Funny)

## Re:Lets crawl before we walk... (Score:2, Funny)

## Re:Lets crawl before we walk... (Score:2, Funny)

as long as the "ole fashioned" ways (i.e. by hand on paper) are taught/learned first.Agreed. All computer science education will now start with punch cards and move forward to more modern tools.

## Man (Score:2, Funny)

## PDAs dont' have buttons! (Score:2, Insightful)

## Re:PDAs dont' have buttons! (Score:3, Funny)

## This is a good thing.. (Score:3, Insightful)

Kids these days get these glorious plotting computers that bypass the tedium and take you straight to the insight. They even have algorithms that do their algebra for them. And I am sure they have a much better high level understanding of what they're doing than I did even in college.

Actually I wouldn't be surprised if their ability to actually solve by hand some of this stuff is as good as ours simply because they understand it better than we did.

## Calculators shouldn't even matter in school. (Score:2)

I used a calc in class, we were required to for AP calculus, but we were also required to memorize everything.

## Anecdotal (Score:2)

I looked at him and said 'You're the math major, cant you do simple division?'

He replied 'No man, I need a calculator for that - now whats 23 divided by 2?!'

## The TI 8x series made a great notecard... (Score:2)

I still remember the rather painstaking process of writing down many derivation and integration formulas into my TI85 graphing calculator. I justified it on the basis that if I was actually deriving or integrating in the real world, I'd have a book next to me anyway, while I still knew I was cheating.

In the process though, I got used to typing words and various macros into the graphing calculator, and over a break was able to make a fun little Might & Magic-style maze walking game using four images and a matrix for the maze layout. It's part of why I'm a programmer now.

So, even though it is cheating to use these tools in several situations- learning to cheat with such tools can be a useful learning experience in itself! As long as you don't get caught.

:^)

Ryan Fenton

## A couple of thoughts (Score:3, Interesting)

I'd always wondered how long it would be before the companies that produce software like Mathematica [wolfram.com] and Maple [maplesoft.com] would port their software to PDAs. When I went to college at Rose-Hulman IT [rose-hulman.edu] we were all issued notebooks which ran Maple and CAD software. We used Maple in all of our Calc classes and were able to use it on tests once we proved our ability to do that particular type of problem by hand first. The CAD software could have easily been on higher power workstations. If Maple had been on our PDAs it would have lowered the cost of going to the college by a few thousand dollars (high end notebooks were really expensive back in '95, and sometimes still are)

The main problem is that PDAs were nearly non-existant at that time, but today I can see PDAs like the iPaq doing a grand job of running some of this higher end math software.

Of course cheating would run pretty rampant with wireless transmitting of email and text, not to mention the ability to store files with crib sheets on them. I'm still not sure how our profs back in the day thought they were ensuring that we didn't cheat on our calc exams back then. I think it was more of a matter of honor than anything.

## Re:A couple of thoughts (Score:3, Informative)

## Re:A couple of thoughts (Score:4, Informative)

So a powerful CAS is absolutely possible to run on PDAs, especially ones with ARM processors. It's just not too easy to write a full-fledged symbolic CAS, so nobody's gotten around to doing it yet. But it's entirely possible.

## Re:A couple of thoughts (Score:2)

I don't know how the TI-89 and up (where the whole symbolic stuff comes in) do their fp stuff, but all below that use a simple z80 and do floating point math in software. (in BCD)

Considering the processors usually run at less than 10MHz (and are all 8- or 16-bit), a PDA would be a fine match.

## Grade Inflation... (Score:2)

Does anyone here know how to use a slide rule?

My point exactly. While we may be able to figure one out given a few minutes, we certainly didn't grow up using them. If, however, the need arose, we could figure one out. Likewise with looking trigonometric values up in a table in the back of a book, just like the rules for differentiation by parts. Even if kids today aren't learning to use the tools that we used (our brains) to graph hyperbolas, that doesn't mean they won't be able to do so manually. It may take them a little longer (it would take us longer to use a slide rule) but they could get it. The important point is that they are learning the mathematics behind the concepts.

## It seems that economics is winning again. (Score:2)

## A tale from elementary school (Score:2)

The teacher said "That's too complicated. You don't need to know that."

25 years later, I would wager most of the kids in that class still don't know what that means and don't care.

Every generation complains the kids are getting dumber, lazier, whatever. There will always be kids who are motivated and want to learn, and while using a PDA in class might slow them down, it won't stop them.

## Does this horrify anyone else? (Score:3, Interesting)

"When you have circles and ellipses, there is no way you'd be able to do this without a calculator," Jarvis said.Ok...I know a lot of people don't need to summon Euclidian geometry from memory in everyday life, but the image of a kid

in geometry classlearning an equation thats been around for over 1000 years, and saying that level of math is impossible without a {graphing calculator, PDA} really saddens me. Especially since geometry is usually taught an at honors level - meaning the kids taking geometry are supposed to be the smart ones, on the fast track to college, etc. It makes me think that with all the technology readily available, kids will stop thinking and imagining and innovating.I remember being in school when the TI's started to become popular. My feeling then was that ok, I've done these equations by hand...I've got a good handle on how to do that, and sometimes its a real PITA, so maybe sometimes its better to use the automated functions here. I still think that way -- I CAN configure SAMBA by hand, but there's a nice graphical tool that automates it, so that's simpler for me now.

I just hope with all the automation tools and short cuts technology can provide, we're not engineering out the human quality of wanting to know how things work.

So how do you tell kids today that yes, you can live without the latest gadget, and that it is important to master the fundamentals before you learn all the shortcuts?

## Both are bad for learning (Score:3, Insightful)

PDAs are currently banned because they are "programmable". But so are all graphing calculators. On SATs, the only things that are banned are devices housing QWERTY keyboards, which most PDAs don't. Also, TIs can be programmed (and come with) more functionality than your average Palm. Even my Zaurus comes with only a 4-function calculator app!

Back on the topic of the CASIO, I left it at home nearly every other day of school, if even that infrequently. Yet I survived through every math and physics class often without it. Because of graphing calculators, most kids don't even know what a parabola looks like, let alone how to draw one. Most people even forget fractions and long division, and rather write the answer the calculator gives them, like "3.999999999" rather than "4".

Both calculators and PDAs are tools, and should

## Re:Both are bad for learning (Score:2)

On SATs, the only things that are banned are devices housing QWERTY keyboards, which most PDAs don't.From collegeboard.com:*You may use almost any scientific or graphing calculator on the tests, however, you are not permitted to use:

As for the debate, I can only add my personal experience. I typically always have a calculator in my backpack or otherwise on my person. In fact, for the past 6 months, I've been carrying two calculators (TI-83+ and TI-89) with me everywhere. My calculators hardly ever come out of my backpack.

They're nice to have to do regressions on data, to manipulate numbers with several signifiant figures, and in the case of the TI-89, to do unit conversions. I would not say that the calculator has "crippled" me, only because I view it simply as a tool and for most things I gain more pleasure out of doing math in my head. On the other hand, for MOST people I would say that calculators are a crutch- I've heard horror stories of people taking out their calculators to do 7-11. I think that attitudes towards math develop independently of calculator accessibility.

I've been lucky to have science and math teachers who love math. My physics teacher is notorious for estimating the values of long and complicated formulas largely in his head. It wows the class, and then he shows people how he did the estimation. People, don't blame the calculators. Blame the teachers who taught you to think on the calculator.

## Cheating (Score:2)

You have to wonder about the possibilities for cheating with these types of devices.

When I was in high school, the TI calculators that were programmable had just started coming out. There were several people who enter equations and other cheats into them.

Some teachers would not allow these types of calculators to be used, others would check before the test that they didn't have any equations or other types of cheats stored in them, and others would actually ask people to clear out all the memory in them.

Glad I don't have to worry about this any more. :)

## Hard to draw? (Score:3, Informative)

Circle: Use a compass. A compass is a simple tool that should be easier to learn than any calculator. (Adjust angle, stick pointy end into paper, draw.) And then all kinds of important tricks of geometry are possible, with just the compass - really only learnable with the compass in hand.

Elipse: put two pegs on paper, the chalk board, etc. Toss a loop of string around pegs. Pull loop of string tight with a pendic, chalk, etc. Draw with string kept tight. Lookie! an elipse! How hard was that?

I used my TI-85 to do all sorts of math, but I learned my math in books and on paper.

## Electronic Aids (Score:2)

However, in most cases, electronic aids foster weak learning. First, it discourages analytical solutions in favor of numerical solutions. Second, it impairs the formation of approximate quantitative judgment. (In this regard, slide rules are likely superior educational tools -- you have to know the differences among logarithmic, exponential, and linear responses.) Third, it inhibits the important skill of hand-drawing graphs. (Ok, on a PDA with a graph paper template, you have an expensive etch-a-sketch, but still...)

The biggest problem is that you cannot easily regulate what a device can do, therefore, students rely on a machine too soon after beginning to master a skill. Fifty years ago, or even thirty, science students were MUCH better mathematicians than they are now. On the balance, I think that reliance on calculators has atrophied the minds of two generations now, and it is time to stop the intellectual carnage.

## It's not them I worry about... (Score:2, Interesting)

## There was an SF short story.... (Score:2)

"There is simply no way he'd ever be able to divide or extract square roots without his calculator!"

Yet another SF author accurately predicting the future.

## No kidding -- calculators stifle thinking... (Score:3, Funny)

How annoying. You'd think they'd just switch to calculating the logarithm of the answer, or divide by 10^75, or something. But, no, "very big" was enough for most. These were Stanford students, too -- supposedly the cream of the (western half of the) nation's crop of students...

## Depends on what they're for... (Score:2)

When one is learning basic arithmetic, no calculators of any sort should be allowed. Note: basic arithmetic includes square roots and percentages.

For more advanced courses, when one is presumed to know arithmetic, allow any NUMERIC calculator. Symbolic and graphing calcs should not be allowed. Yeah, you can use them in the

Real World(tm), but in school you're not just supposed to be learning *HOW* to do this stuff, but *WHY* you do this stuff. The symbolic and graphing functions kill the second part.## To a math major, this is scary... (Score:2, Interesting)

I remember back in high school. One time out of curiousity I asked my (I think it was Algebra II) teacher if he could teach me how to find square roots without a calculator. He didn't know offhand, and so I went to EVERY MATHEMATICS TEACHER and NONE of them knew how to do it. I finally found one person who knew how: the ancient librarian. She taught me, and I'm grateful.

Calculators are a tremendous help for solving things faster and more accurately. But if you don't understand what the calculator's doing, what good does it do you when you have to modify it a bit to fit a given situation?

What kind of an "educational" system is this where so many people are utterly incapable of standing on their own two feet without the support of calculators?

This is a really disturbing trend in math, and education in general. And it's only getting worse thus far.

-eosha

When you don't know what to do, walk fast and look worried.

## Re:To a math major, this is scary... (Score:2)

I remember teaching my 12-year-old cousin to extract cube roots in her head. Smart girl! The next year she hit the pubescent wall and suddenly math wasn't cool anymore. Damn.

In case anyone actually reads this far down:

HOW TO EXTRACT CUBE ROOTS (In your head if you want)

(1) Guess the cube root. As badly as you like -- 1 is a good place to start for most small numbers. If you have something like <foo>x10^exp, then try 1x10^(exp/3).

(2) Square the guess.

(3) Divide the original number by the square.

(4) Your next guess can be any number between the

quotient and your last guess; it is guaranteed to be closer to the answer than your last guess.

(5) Repeat as necessary.

Or, for those in a hurry, you can remember the magic three logarithms ( log 2 = 0.3010, log 3 = 0.4771, log 7 = 0.8451 ); using those three and about 10 seconds you can find the logarithm of any number at all! Then divide the log by three and raise 10 to the quotient.

## MathCAD for Palm OS (Score:2)

Assuming where talking about college or precalc and up. Everyone remebers the old TI-85's Visualizing is the most powerful way to learn. I jsut hope TI doesnt' loose it's foot hold. My old Palm Pilot with 2 megs will draft equations and I can usually find an app to do whatever I want. My question is when do you release MathCAD for Palm OS. no seriously.

## It's a Tool (Score:4, Insightful)

Was the compass and geometry uninvented?Back in the day, my Dad got a degree in civil engineering. He was allowed to use a slide rule for many of his classes, even in high school.

Hisdad thought this was inherently bad because it defeated the idea of learning to do the math by hand. Naturally, geometry, trigonometry and calculus didn't lend themselves (graphically) to a slide rule, but he could perform arithmetic calculations like a maniac.When I went to high school, slide rules were out and calculators were pretty damn expensive, so in high school, everything was done by hand. I can do arithmetic calculations in my head like a maniac.

After about 18 years, I went back to college and got my electrical engineering degree. Not only were calculators cheap, but computers were cheap, too. I took Trig, three semesters of calculus, one of differential equations and one of statistics. I used the calculator and computer in each one.

Did it help? Damn straight! Did it hurt? No.

Here's what I think: the mathematical fundamentals that I learned were aided by the electronic tools. Sure, any monkey can poke the keys on a calculator or type in a Mathematica or Maple function, but, fundamentally, the student must have some degree of knowledge of the basics of what he's doing to know that the answer that comes out of the box is the one he wants. I don't know how many times I poked the buttons and watched the calculator or computer toss out the wrong answer because I typed something wrong. But I knew that the answer was wrong because my knowledge of math was such that I could estimate to a reasonable degree what the answer should be.

I do have to admit, though, that the string and two nail method of drawing an ellipse does drive home the idea of visualizing

howthe ellipse works (major and minor axes), but I'm most definitely a cheerleader for using calculators and computers to overcome the mundane mechanics of math. Not only that, but modern calculators like my TI-92 Plus do a great job of graphically modeling things like surface integrals. Computer programs do it even better. Tools like that allow students to progress many times further in their math "careers" than they might have if they didn't have those resources.Fundamentally, though, and I suppose this is what you meant by the calculators and geometry comment, it's vital that a well developed, solid knowledge base is developed in the basics so that the resources become tools and not crutches.

-h-

## Re:It's a Tool (Score:2)

Here's what I think: the mathematical fundamentals that I learned were aided by the electronic tools. Sure, any monkey can poke the keys on a calculator or type in a Mathematica or Maple function, but, fundamentally, the student must have some degree of knowledge of the basics of what he's doing to know that the answer that comes out of the box is the one he wants.Great comment.

To a user, the tool is first a black box. It can be used but not necessarily be

useful. To make a tool truly useful, the human operator has to understand the fundamentals behind the black box enough to check that the output from the tool is meaningful. This process is really the scientific method in action.Tools can be compounded. Computers are a great example viz. libraries. You don't need to understand how to program in assembly language in order to use linux, but you should have some idea of how the libraries and OS work together with your program (read: dependencies).

## Visualize WHAT? (Score:4, Insightful)

"When you have circles and ellipses, there is no way you'd be able to do this without a calculator," Jarvis said. "It helps us visualize what we're doing."We visualized

landing on the moonbefore calculators. Get a grip, young man, and learn your trade before using crutches.## Calculators aren't the end of math (Score:2, Interesting)

## A compass points northwards (Score:2)

## Techno-obscolescence... (Score:2)

I used to use a protractor and ruler to do geometry in school. Damned fine tools... capable of giving a more precise measurement than any calculator or PDA if they're really nice, and does something more than visually expresses the concepts; it gives you a hands-on feel. This contributes to depth-of-processing, which in turn helps aid memory.

Whatever... we already have cashiers who are incapable of performing basic arithmetic when the register dies, I suppose this sort of thing should come as no shock.

But then again, I have to consider the views of the ancient Greeks, as writing was becoming more popular. Some folks had concerns that it would prevent people from memorizing the old stories, since you could simply look up the stories in a book or something instead of having to recall it from memory.

This sort of thing seems to always happen with certain technologies. As they aid us, we lose some skills, only to gain new ones.

So... ideas as to what new skills we'll gain from these advances? Stronger fact-finding skills perhaps? A facility with logic? Better pattern-matching skills?

## Machines don't do math. (Score:3, Insightful)

Just require that the student show their steps in solving the problem. I don't care if the answer's right in a calculus class... I'm not there to teach arithmetic... were the steps used to solve the problem correct? Just because there was a silly addition error doesn't mean the whole problem get's no credit, and just because the answer's right doesn't mean it get's full credit either. A calculator can't help a student who doesn't know the intermediate steps to solving a complex math problem.

## Compass + protractor (Score:2)

## Re:Compass + protractor (Score:2)

## TI Calcs -- more PDA functionality coming soon (Score:3, Informative)

Unfortunately, TI hasn't officially provided much information, but having been involved in the TI dev scene quite a while, I've had the opportunity to play with beta versions of these apps quite a bit. They're slightly limited when compared to Palm because they don't have touchscreen input, although the 92+/Voyage 200 calculators have a full qwerty keyboard. The software is quite nice, and I've been using it full time since my Clie broke a few weeks ago. I'll have the Clie repaired under warrantee, but for the target demographics of TI's calculators (mostly students), the Organizer software is more than powerful enough to make somebody who purchases one of these calcs reconsider whether they need to carry around a PDA as well. And trust me, consolidating the two devices and freeing up a pocket is definitely something to look forward to.

## Calculators and Geometry (Score:5, Insightful)

That said, this is dependent on the student using the calculator only as an _aid_ to learning, not a replacement for it. After I bought mine, I watched as students in courses as simple as (remedial) Algebra I bought 89s, and the calculators solved the problems for them. Then even students in the honors sequence bought them when first getting to limits -- and I do know quite a few students who didn't know how to do limits by hand, yes passed tests solely by using their calculators.

But for someone like me, who actually learns the concepts before resorting to the calculator, it's a great help. Got a tricky integral for homework that you're having trouble with? Check the calculator's answer, and often the "form" of the answer will hint at how to solve it, and the next time you have a problem like that, you'll know how to solve it. Does your homework have even-numbered problems that don't have answers in the back of the book? Use the calculator to check your answers, and if you know you got one wrong, you can go back and figure out

why.Fast forward a few years, and I've just finished up Multivariable Calculus and Linear Algebra at a well-known US university, and the calculator was still a great help. Test and Quizzes were all done by hand, so a calculator won't get you through the course. But I can now check my homework bit-by-bit as I go through it, so a little mistake in matrix multiplication in the first step of a long problem won't result in a completely wrong answer 20-minutes later. It's saved me a lot of time and a lot of frustration, and of course I learn where I commonly make mistakes and can correct them. And you can extend the geometry comment made by this teacher to higher level math, like graphing quadratic forms -- after solving one, I could graph it and see the eigenvectors/principal axes, the signular values, etc. And I was able to take some of those 3d shapes that I had to integrate to find the volume and use the 3d grapher to see what they look like. And the calculator has quite a bit of differential equation functionality that I don't fully know how to use yet, but no doubt it will come in useful in the future.

So the calculators in and of themselves aren't bad; it's those who abuse and overuse them. Can anything be done about that? Well, having calculators banned on all tests did wonders for my math-by-hand skills. Let students use the calculators when learning the concepts, but when it comes to testing their application of those concepts, make sure you're testing the student and not the calculator.

## Re:Calculators and Geometry (Score:3, Interesting)

## all curricula are not equal (Score:3, Insightful)

I'm in the latter category, where the calculator is pretty much irrelevant for the math classes.

I use the calculator for *arithmetic*, and hardly at all for *mathematics*.

## Yes, I uninvented them (Score:4, Funny)

## What you don't Visualise - You Lose (Score:3, Insightful)

whatever you get the machine to do for you - you pay for in letting your own ability to do it atrophy.

If you never learn it manually and always have a machine do it for you - then you're slave to the machine.

once you've Learned It without the machine, then the machine becomes an aid. but if you never actually learn it yourself, then you're slave to the machine.

once you know how to do it manually, then there's a place for letting the machine take the drudgery out of it for you - that's what computers are for after all.

but how many times have i been to a store, and the cashier didn't even know how to give correct change when the register doesn't tell them the right amount!?

john [earthlink.net]

## Re:other conflicts? (Score:2, Redundant)

## Re:other conflicts? (Score:2)

## Re:other conflicts? (Score:2, Insightful)

## Re:other conflicts? (Score:2)

## Re:Tools.... ? (Score:2)

## Re:Tools.... ? (Score:2)

Seriously, though, I don't think that having calculators or PDAs in the classroom is going to be the deciding factor in the quality of a kid's education. A teacher who doesn't know the subject matter is going to compensate any way they can, technology or no, while a good teacher is going to use whatever tools they have to improve the learning experience for their students.

## Re:Favorite Quote (Score:2)

Every I know does this for their math classes. I know most people put all their notes, all their formulas, sometimes even with examples in their calculators. If teachers want to eliminate cheating they're going to have to get rid of calculators entirely.

## Re:cheaters (Score:2)

Typing with a computer keyboard is so much easier.

## Re:Sets back when they get to college (Score:2)

Now Statics, on the other hand, there was a class where I really needed my calculator. Mostly because the prof assumed you had one, and set up the problems in such a way that it was impossible to finish a test in time without one. I know that from experience. My calculator died the morning of a Statics test, and I only managed to get through the first problem and halfway through the second (out of four) in the aloted hour, and I'm pretty fast at working stuff out by hand.