What level of calculator use is acceptable for students?

Great Debates
41

Did I miss something? Did the mods move this into IMHO without me noticing? Did I seriously miss the posts where people gave actual evidence or arguments of any of these positions?

Here. I assert that for non-trivial problems, calculators are measurably faster than computation mentally, and there are precious few situations where solving on paper is more certain or quick then solving on a calculator.

I guessed .9 and a bit.
0.90909090909090909090909090909091
You know how I did that? In the course of computing grades where an assignment is 10 points and a test is 100 points, I got used to dividing numbers by 11. My estimation formula for n/11 is .((n - 1) +(x where x decreases as n increases). I worked that out simply by computing lots of n/11.

You say that we should be focused more on what math means than the answer? So, since integrals are defined as infinite Riemann sums, does that mean that we should be forced to evaluate integrals without using (x^(n+1) / n+1)?
(My calculus teacher thinks, in certain instances, yes, we should. Yes, I’m bitter.)
(x^(n+1) / n+1) and nDeriv() are both shortcuts. In both cases, you get students who will merrily plug 1/x into them, and move along. In fact, I would posit that you get more students writing down things like ((x^0)/0) then ERROR.

If you want to fight the attitude of “plug it in, get it out”, then fine (I disagree, but fine). But please distinguish between a formula memorized and a formula in a calculator.

42

No argument about this from me. I think clearly setting up the problem on paper, or on the computer screen, is a vital step in problem solving. However, this doesn’t have anything to do with learning the mathematical theory behind the operations, or proving the validity of theorems, or onerous drill on hand calculations.

At all times and in all places, as far as I can tell, even the greatest of mathematical theorists have looked for and used the best methods they had available in order to reduce the labor of calculation and to lessen the chances making mistakes. Logarithms, the abacus, multiplication tables, tables of integrals, La Place transforms and on and on.

The calculator is only the latest of these and I favor starting students early on its use and making use of the time saved in not doing calculation drill to learn how to set problems up, estimate order-of-magnitude etc.

43

Reminds me of a (probably untrue) story of a graduate student taking an oral exam when he was asked to give the Taylor expansion of a hyper-geometric function (ok, that probably is nonsesne, but he ws asked to do something). The student said, “Well I don’t really know that but if I needed to I could just look it up, it is in any advanced algebra text.” The Professor shook his head, “Sure, except you need it now.”

44

Okay. When outside the artifical world of academia will you need to know the Taylor expansion of a hyper-geometric function and not be able to look it up?

45

[ul]
[li]Jeopardy![/li][li]Interrogated by Terrosits[/li][li]First Date Small Talk[/li][li]In your car and the radio announcer says the “The tenth caller who can give me the Taylor expansion of a hyper-geometric function wins a trip to Paris.”[/li][/ul]

I’m sure i am just scratching the surface.

46

Academia would be referring to situations set up by professors to test whether or not you knew the Taylor expansion of a hyper-geometric function without consulting a book. Sorry.

47

Octopus, that’s all too like the argument that “drugs are harmful because they can get you arrested.”

48

Drop the condescension. Many of us, including you, have provided our arguments. You know as well as the rest of us that there will not be “evidence” to directly answer the question in your subject line.

This in no way contradicts what I said. I said that calculators are necessary in science classes. My objection was to the attitude held by many that “the calculator is always right.” No, I don’t expect my students to be able to evaluate fractions in their heads. I do expect my engineering students at one of the best engineering schools in the country to be able to tell when the answer their calculator gives them to such a simple problem is off by 20% or so.

I most certainly did not say that. Of course the answer is important; this is why students should be able to recognize when it is blatantly wrong, and take steps when solving a problem to catch mistakes along the way.

Let me say it again: My problem is with some students relying too heavily on their calculators, i.e. thinking that the answer that comes out is always right. Obviously calculators (and computers) are useful and necessary tools. But I don’t think it takes much “evidence” to assert that having engineers who can’t tell that an answer to a calculation is way off is a Bad Thing.

49

I’m in the student position here, being in a UK sixth form and, for a rough idea of where we are, we’ve been doing things like integrated lnx and so on recently. Yes, very basic and all that, but you’ve got to start somewhere :slight_smile:

I’m doing a Further Maths course, which means I do it for two hours a day and to be honest, my TI-83 is on for about an hour and a half of that. Personally, I don’t find this a great problem (it certainlly saves a lot of my time…), but I can see why you might.

One thing I do seem to be discovering is that there seems to be two different angles which people come at the subject from. Some (including my teachers, which is where this rant probably comes from to some extent) treat it as a great puzzle, with the next goal always being to discover the next formula. Where it comes from is unimportant, as it long it can be proven quickly with some random algebra.

To give you a few examples of what I mean by this. my class was introduced to logs by the statement : “Definition: log(base a) c is the power which a must be raised to give c” (or whatever, my mathematical notation and definitions suck :)). From here we did some basic rules. There was no attempt to give any historical context, or reason why we did this, or look at what we were actually doing graphically, or anything like that. It was just a meaning less exercise.

With logs, I can sort of see why you do that. On the other hand, it drives me mad when we are just told that the definite integral equals the area under a graph, some algebra is thrown down unto a page which we hurriedly copy, and that is that. Most of the class just accept it. Personally, I’m trying to drown out the teacher in my mind and figure out why that is true - to come up with a reason.

The other position then, mine at least, is to try and look at the subject scientifically and try and understand how numbers work and what exactly is going on. Maybe I’m being unfair and that’s what the other side’s doing too, but it doesn’t seem that way. To me a proof is often useless - well, not quite useless, but I don’t feel particularly inspired by it. The ‘why’ is much more important.

To finally get round to the subject, IMO if you see Maths as a big puzzle, with proof the ultimate goal. then using a calculator probably seems a cheat to you. You’ll much rather do it on your own, in the same way that a person will climb a mountain instead of taking the chair lift. If instead you’re trying to explore the subject, you’re much more happy to play around with a calculator and rely on it, if needed. I personally forumulas you don’t need the reason for as just as bad.

(On the more mundane issue, calculators may hurt basic arithmetic. However this can be cured without completely giving them, and I think a lot more harm is done by the time wasting that will take place if they’re not used.)

50

I wasn’t making an argument, so much as just relating a story that the debate reminded of.

51

If a student has a formula for, say, differentiating x^n stored in a calculator, then that operation is all the student can do. But if the student knows the formula in their head, then at least the possibility exists of teaching the student why that formula is true, and how it in turn relates to other formulas, such as the chain rule for multi-variable functions…Calculators are great for answering “what?” but they’re lousy at answering “why?”.

Your previous posts seem to suggest that your teachers neglected the “why?” somewhat…I would consider that to be less than ideal. You’ve also been strident about not neglecting the answer in favour of the means. But ideally (especially at the college level) teaching should strive to give the student an understanding of both the means and the answer both the “why?” and the “what?”. Over-relying on calculators obscures the means in favour of the answer.

52

Thank you. Arguments beyond “I say so” are good.
I concede that having a too-friendly user interface in calculators would be detrimental to learning math. But, if I can bring up the formulas in the calculator, then is being able to remember (n-1)x^(n-1) still useful? If I have a mental association of derivates —> Prgm>Edit>CALCFORMS - - > (n-1)x^(n-1), and can bring up (n-1)x^(n-1) with my calculator whenever you could go from derivates —> (n-1)x^(n-1), is there any real difference?

Also, I would say that a calculator that couldn’t bring up (n-1)x^(n-1), but could only solve formulas, didn’t “know” (n-1)x^(n-1), in the same way that a student who could evaluate (f(x+c) - f©/ delta-x) would.

The assertion has been made that it is bad that students cannot evalute trivial math without a calculator. I am challenging this assertion. If a student doesn’t know that 2+2 = 4 without a calculator, but has a calculator, then what is the problem?

I am also challenging the assertion that use of a calculator promotes a “write the result down” mentality moreso than a formula-driven math class. In fact, I am positing that the way that mental estimates are made are not by doing lots of problems by hand, but by doing lots of problems, and that calculators make it easier to do lots of problems.

53

In primary school when we were learning how to do multiplications such as 34*568, we worked them out by hand.

In high school when we were learning algebra we did multiplications such as 34568 on our calculators. It would have been a waste of time to work out 34568 by hand when we already knew how to do it and were trying to understand algebra.

To understand maths (for me at least) it helps to do problems. If we weren’t allowed to use calculators, it would have taken us much longer to do calculations, and therefore longer to get through the problems.

As a result of this, understanding the work would have taken much longer. If understanding took longer, we wouldn’t have been able to learn as much in our class because there wouldn’t have been enough time.

So, by using calculators in high school to do calculations we knew how to do by hand in grade 3, we saved time and learnt more.

In my mind, not being able to use a calculator would be a hindrance.

Instead of punching 9^6 in your calculator, you have to write 99999*9 and work it out bit by bit. What a pain!

54

Let’s scale back a bit. Let’s ask, what is the difference between learning the product rule and knowing how to call up the product rule on a calculator?

The difference is, if I can learn the product rule I stand a chance of figuring out the quotient rule. I stand a chance of learning how the system works, and getting to the point where I can teach myself. It may not be a great chance, and I may not get there depending on how good at math I am. But if I just know how to push the “product rule” buttons on a calculator, I’m denied even the opportunity.

The problem is that the student is now dependent on the calculator. The calculator has become a crutch.

Hey, your grocery bill was $22.58, and you payed with $30 and the cashier gave you back $6.42. Didn’t have your calculator in your hand just then? That’s a shame. Hey, you’re driving in Canada and you’re down to a 1/4 of a tank of gas. Why, there’s a station there…and the gas costs (Cdn) $0.83/litre. Should you stop and fill up or wait a while? Put that calculator down, you’re driving! And look, now you’re hosting a business lunch, and need to figure out the tip for a $251 tab, but your briefcase with the calculator in it is back in the office…well, you get the idea.

A student who depends on a calculator even for basic arithmetic is not even going to be able to hazard a guess in the above situations. It’s like the Griswold’s in “European Vacation”, trying to order in a French restaurant with only an electronic French-English dictionary. (Only over-emphasizing calculators raises the spectre of an entire generation of people looking that silly, instead of just the occasional fictional tourist…)

Wha? So students, in your view, should be able to learn to do rough arithmetic by osmosis, just by watching their calculator does without being told how it does it? That might work with language, but I’ve never seen any evidence that that’s possible with mathematics.

55

I think the thing is that you are interested in knowing the “why” of mathematics. I am too, at least to some extent. But there are millions upon millions of people who aren’t.

And why is this bad for someone who has trouble getting the right answer any other way?

I really question whether these hypotheticals cause such massive problems in life that it is worth having millions of students and hundreds of thousands of teachers spend hours and hours trying learn mathematical theory, which is beyond the ability of the majority of them, and drilling on paper and pencil computation.

56

Mathematical theory on the arithmetic level is beyond the ability of the majority of people? Surely you jest.

57

True. But robertliguori asked the difference between learning a formula/algorithm and learning how to use a calculator function. My response was an answer.

I admit that I’m reacting to an extreme (perceived) position here. I’m aware that new tools change the way we do things. I realize after the invention of writing, there were probably Memory Geeks everywhere bemoaning how no one taught advanced mnemonics anymore.

But the idea that basic arithmetic skills are completely useless, now that we have calculators, is to me as extreme and wrong as the idea that basic walking skills are completely useless now that we have wheelchairs. I suppose that places me in a similar position to RTFirefly on the previous page…and in retrospect I suppose I haven’t really answered the OP, except to say that “use calculators for everything” isn’t an acceptable level of use.

With regards to your second question:

It’s not, if such a person honestly cannot do without. But if we don’t at least try to teach pen-and-paper arithmetic, then we don’t know who can do without and who can’t.

Again, I’m responding to the (perceived) idea that teaching pen-and-paper arithmetic isn’t necessary, that we should just go straight to the calculators. Which, to me, is like saying that we should never give anyone physical therapy. True, it won’t benefit everyone, but that’s no reason not to try.

58

Calculators don’t cause the inability to perform arithmetic, in the sense that it gives you some sort of anti-math virus. However, Performing arithmetic in your head is something that typically requires practice to remain adept at. Up through the end of my career as an engineer, I could multiply three-digit numbers in my head fairly easily, because I did that kind of stuff all the time. (Saved a lot of time pushing buttons.) It’s now been three years since I left the profession, and I’m waaaaay out of practice. Not the point where I need a computer to figure out 3+6, but I’m not as quick as I used to be. The students I mentioned were all very smart kids, but they’d grown so accustomed to their little machines that they had lost the ability to perform even the most basic arithmetic in their heads, and it slowed them down. When you have to whip out your calculator before you can add 10 and 12, you’re not going to be as fast.

Further, using a calculator kinda kills your ability to see the relationship between numbers in the problems you’re working out. It may be that the solution to your problem is 3e/2, and that this is an elegant solution to a difficult problem. It helps to see that specific relation, and you don’t get the same insight from “4.0774” as your answer. Especially in problems involving trig, where the answer may be expressed in terms of sines and cosines and pi, getting a symbolic answer can be very useful.

Yet another advantage of symbolic answers: When you screw up, it’s easier to find the problem, once you see the answer. In my classes, we were always assigned problems that had answers in the back of the book, so that we would know when we got the right answer. (Of course, we had to show our work.) If the answer is 4pi, and you got 2pi as an answer, it’s easy to trace back through your work and find where you may have introduced a rogue factor of two. If your answer is just a giant decimal number, you can’t do that.

You may say that this should be the choice of the student, and if they want the advantages that not using a calculator offers, so be it. Well, I can tell you that if you give students the option of using a calculator or not, 9 times out of 10 they will use it. It has the strange allure of technological advantage. Hey, what isn’t made easier with the aid of a computer?

Not that I think calculators are the Tools of the Devil, or anything. I had a wonderful HP all through college, and it was my baby. (I especially liked playing Minesweep during the boring classes.) But I’ve seen those things used as a crutch too often, especially in high-school.
Jeff

59

Which is faster, adding 2 and 2 in your head, or plugging it into a calculator? By relying on it for everything, students may end up working slower, not faster.
Jeff

60

This is true IF your problem has been designed to have ‘nice’ answers.

I don’t know how often the problems you encountered as a professional engineer worked out that way, but my observation is that it’s a lot easier to generate problems that doesn’t work out ‘nicely’, whether in terms of 1 or e or pi or whatever. (For example, what percentage of quadratic polynomials with 1- or 2-digit integer coefficients (e.g. 34x[sup]2[/sup]-83x+5) factor over the rationals? I can’t remember, but it’s pretty small.)

My point is that if you can track down that rogue factor because you’re in an artificially ‘nice’ environment, that won’t help you much when you’re in a more normal situation.