When should calculators be introduced to the curriculum?

lurflurf

Using a calculator when you do not know what you are doing can be quite helpful, one can explore a problem and make sample calculations that provide insight. People should be able to make simple calculations (like my example), but the time needed to make more complicated calculations, to do them quickly, and accurately are better used for something else.

If they cannot make such calculations they need a calculator more. Some basic arithmetic is helpful. It is like bicycles and automobiles, learn to use both. If you need to travel at 100 kph no amount of bicycles practice will help. If you need to travel 17 kph hours use the one best for the situation. If you think you should improve your cycling then do so.

Mentallic

Actually I'd prefer to think of doing arithmetic on a calculator as opposed to doing it by hand as turning on the cruise control while driving down the highway.

And I haven't heard of any Learners that've been encouraged to use cruise control.

Studiot

Unfortunately that is what leads to basic drug errors and the administration of 10 times too much drug.

Mark44

This is a very bad idea, IMO. They should be competent at arithmetic operations first.
Faster and cheaper, sure, but not always more accurate.

Back about 20 years ago, when Intel produced the first of the Pentium chips, some entries in the lookup table for division microcode were omitted, leading to incorrect results. This caused a subtle error in some division problems, with errors in the fifth and subsequent decimal places. These chips were unable to get the correct answer for 4195835 divided by 3145727. Anyone who understands the basic long division algorithm can get the correct answer to any desired accuracy. Intel spent more than $1,000,000,000 recalling the flawed chips.


I'm reasonably sure most of the regulars here at PF can do this calculation, but it will take more than a millisecond. However, we might be able to do the calculation more accurately. Suppose that instead of using a calculator for this product, I write a computer program to do it for me, say in C. Suppose also that the computer I'm working on has an old (pre 1997 or so) compiler that stores int values in 16 bits. The compiler would be unable even to store the 79135 value, let alone be able to do the multiplication. Admittedly this is a contrived example, but I can think of examples that are not contrived.

When it comes to arithmetic that involves real numbers, the fact that computers and calculators are unable to perform exact calculations leads to some surprising problems, such as the inability to add 0.1 and 0.1 and get the correct result. Or if I add a large number and a small, but nonzero, number, and end up with the same large number, such as 253,123 + 0.0000004527. Your calculator might do this calculation correctly, but I guarantee you that I can come up with an example that your calculator gets wrong.

lurflurf, you said in another post in this thread, that if a student can't do arithmetic, then he are she needs a calculator. This, to me, seems to be treating the symptom, not the problem. A better solution, IMO, is to teach the student how to do arithmetic, at least the basic addition facts (addition of single digits), multiplcation at least up to 10 x 10, and basic division algorithm. If we can get this student up to speed with fraction arithmetic, so much the better.

The thing about totally entrusting a calculator to do your thinking for you is, what happens if you drop the calculator and it breaks, or you forget it, or the batteries die?

Studiot

Here are some examples from real life. Back in the old days Mark was talking about, before most calculators had square root buttons, I needed to take some square roots accurately.

Our contract required us to supply the Supervising Engineer with a calculator that had a square root button and this had not arrived.

He spent all one afternoon trying to remember/develop a formula and extracted one root by the end of the afternnon. Meanwhile I had to get the job done so I used the brute force and ignorance method and calculated the required dozen or so at a couple of minutes apiece.

I recently talked to a primary school teacher who encourages her class to learn their tables by offering £1 to anyone who can get the answer on their calculator before she can write it on the board.

Stephen Tashi

I think calculators or computers shoudl be introduced early in the elementary school curriculum. They certainly should be used in any situation where performing arithmetic begins to distract from the mathematical concepts. (Do students nowdays still do manual interpolations of logarithms and trigonometric functions? If you are teaching the concept of interpolation that's fine, but it detracts from teaching trigonometry and logarithms. )

I think programming should also be introduced in the elementary school curriculum. If it's done on a calculator, the machine should use something resembling a traditional programming language - not too many "special function" keys.

chiro

I would love that, but unfortunately many students have problems even with basic algebra let alone grasping a language for programming and linguistic construction to give a more complete and abstract treatment of computation.

I have tutored people personally, and I'm sure this is a common thing, where I get year 10 students that struggle with calculating the other side of a right angled triangle given an angle and a side (not the right angle). I also had to explain a year 12 student how to calculate tax for a few given incomes given a simple tax table (i.e. ranges and cents on the dollar for each region of the income).

These people were over 15 years old and had problems grasping this kind of thing.

Although I think the curriculum in high school is rather pointless, wasteful, and underchallenges many students, your proposal would be something for more gifted students and not for the norm.

Having said the above, a pilot study of the above would be a great thing just to see what the results were because it would probably surprise a lot of people including myself.

lurflurf

Sure, because it is impossible to make a decimal place error in hand calculation.

That is like saying before you can use a hammer you should be competent at nailing by hand. If one wants to improve at hand calculation they can practice and use a calculator when helpful. Most people will not continue to practice past a certain point as it is not a worthwhile activity. If hand calculation is such a valuable skill as its enthusiasts claim they should be able to demonstrate it, not treat calculators like spindles in sleeping beauty.
I had a teacher once whose lessons were even less useful than hand calculations. Once he poured a bucket of bearings into a metal can while flipping the lights on and off and said "that is what is was like to be in world war one". Another time he required each student to report a current event, I reported the Pentium chip error and was given a diatribe that the error was not of any importance to anyone ever. The error makes one question intel's ethics and serves as a reminder to check results. The error has been estimated to occur once per several million or billion calculations. How many errors would a human make in a million divisions done by hand? How long would they take? Hand calculations failed William Shanks and he was a better calculator than most people. I would take my chances with a flawed Pentium chip and outdated compiler.

Computers can perform exact and high precision approximate calculations. There is a speed trade off. Some interesting work involves computers that are faster and less accurate than usual. If a calculation can be done by a human in reasonable time a computer can perform numerous checks in the same time. Frequent use of calculators allows one to predict and deal with problems.

I don't entrust a calculator to do my thinking, I use a thinkulator for that. When my hammer breaks I get a new one, same for a calculator. If it happens often enough carry spares. Other skills of decreased value presently include tracking game, starting fires by rubbing sticks together, and refilling a fountain pen. Technology is often used poorly in education, but the answer is to use it better not eliminate it. Properly used calculators lead to more (and somewhat different) learning and reduction in tedious tasks.

Stephen Tashi

I think that the students who really understand trigonometry after taking a trigonometry course are only going to be the gifted ones. It's just a fact of the distribution of human talents -the same for algebra, chemistry etc. The less gifted ones pick up isolated facts, memorize simple patterns etc. I also think students are more likely to grasp and enjoy simple computer programming than algebra.

If you want to teach people practical manual arithmetic , it's true that you can teach it as arithmetic in a given context, like figuring out interest on a loan. But I don't buy the argument that "everybody" must learn these practical contexts and I don't think that the students who don't remember the trigonometric identities will remember how to compute interest on a loan unless they do it regularly. (If you want to teach people how to figure interest on loans, you could start by forcing them to go into debt.)

Angry Citizen

I don't think kids need to be taught things like long division algorithms at all. What exactly does it teach them? To this day I don't know how to do long division using that ridiculous algorithm. Calculators should be introduced the minute they get beyond the times-tables.

I haven't used a dedicated graphing calculator (MATLAB excluded) since my pre-calculus days. I don't need them or want them. I have a solid knowledge of what a graph of a given elementary function ought to look like. Kids need to learn how to graph functions. It teaches you a LOT about how functions behave, and I think it helps make the transition to concepts like continuity much easier. Thus I think the only calculators kids should have are elementary arithmetic calculators, through all stages of their development.

Vorde

I think thats unfair. You seem far above average in this regard. I consider myself exceptional at math and when I'm given a nasty equation I always graph it just to back up my own mental assumptions.

I agree that things like the long-division algorithm are useless, but basic things should still be taught. The way I see it, as long as the student knows how to do something the manual way, they should be able to use a calculator to their hearts content.

Number Nine

I've encountered too many people who can't reality test even the most basic of computations done with calculators because they've never actually done any calculating themselves, and have no idea what a reasonable answer looks like. The ability to perform basic calculations is such a useful and easily acquired skill that I can't imagine why anyone would forego learning it.

chiro

Thinking about what Stephen Tashi said, I have to say that the best way I have learned something involving math is by having to program it in a language like C++.

I don't know if forcing C++ on to elementary school is wise, but the idea of introducing some kind of programming with a syntax suitable for that age group does sound like a good idea to re-inforce understanding if the student is able to get to this goal independently (not just copying other people's or the teachers code).

Studiot

Forcing programming on schoolchildren was an experiment that was tried twice and failed miserably twice in the UK during the 80s and 90s.

Languages and fashions change in programming such that anything learned at school will be hopelessly out of date often before the child has left, let alone later in life.

That is not to say that programming study not be available as an option for those who want or need it.

chiro

If they are emphasizing language-specific training as opposed to teaching general constructs (typically procedural-programming ones), then the course was badly structured and designed itself.

This is though not a feature for this course, but for mathematics, science, and even languages.

The content emphasizes things that do not teach understanding: syntax is not programming just like numbers, right angled triangles, and trigonometry is not mathematics.

I have a feeling that people that did not have the required experience and the ability to really relate this in terms for the young students. You need the former for the latter, but the latter is a rare skill that the best of teachers possess and unfortunately is in short supply.

Angry Citizen

Because it's pointless. Yes, knowing your times tables to 10 and being able to add/subtract on the fly are all very, very useful and very easily acquired. Long division is where I draw the line. Like I said, I don't know how to do long division, and I likely never will. Hell, I can't even use the kiddie algorithm to multiply numbers I haven't got memorized - if I had to do a large-scale multiplication, I'd have to use algebra to break it up into easily-multiplied pieces and sum the results. But so what? Have I lost anything useful? If anything, I've learned how to use a higher form of mathematics (algebra) to invent my own bloody algorithm, the origins of which I understand.

Studiot

Chiro, I really don't have a clue what you mean.

I would also venture that you don't have much idea of what I was talking about, given your response.

Do you have first hand experience of the events I described?

Please all let's remember the original question was

When should calculators be introduced?

not

should calculators be introduced?