When should calculators be introduced to the curriculum?

[moonman239]

I'm just wondering. When do you think kids should be introduced to the idea of using a calculator to do math?

 

[Simon Bridge]

We don't have to introduce kids to that idea - they come to us with it already.

 

[AlephZero]

Never, IMO. Mathematics $\ne$ arithmetic.

And ban calculators from all math tests and exams, of course. If you want to give people an educational qualification for knowing how to use a calculator, that fine, so long as the certificate doesn't include have word "mathematics" on it.

 

[Infinitum]

Calculators have pretty much nothing to do with mathematics.

 

[Diffy]

Never, IMO. Mathematics $\ne$ arithmetic.

And ban calculators from all math tests and exams, of course. If you want to give people an educational qualification for knowing how to use a calculator, that fine, so long as the certificate doesn't include have word "mathematics" on it.

I understand where you are coming from.

But calculators do have some place within High School education. I think teachers should show and test students on their ability to use tools to solve complex problems. Hell, we do it every day.

I use calculators, excel, matlab, etc. every day to help solve numerical problems at work. Why shouldn't we devote at least some time developing student's skill to use a calculator to help solve problems?

Most of the time yes, students should not be relying on these instruments. They should be learning the core concepts. I agree with you. But if we do not teach them at some point to properly use tools to solve complex problems, where should they learn it? There simply isn't another mandatory subject where that skill fits into as well as "Math class".

 

[Simon Bridge]

Because mathematics is not arithmetic, you need not ban calculators from exams etc. You can set up your questions so you test the mathematics and not the ability to plug numbers into a machine.

You know, there was a time when slide-rules were not permitted in math classes ... or books of tables.

 

[Studiot]

This question is being asked in another guise in many walks of life.

For example, should nurses (and doctors) be able to read an old fashioned thermometer and be examined on this ability?

Should nurses and doctors be able to take a pulse the old fashioned way

or should they be allowed to use a machine with a digital readout in an exam?

My wife teaches drug calculations and it is suprising how many 'trained' ie exam passed staff get it wrong with a calculator and don't realize they have it wrong when they draw up the medication.

 

[Feodalherren]

Hard to give an exact age... In general the calculator should be introduced as an aid to estimate uneven roots and graph polynomials and that type of stuff. It should never be given as a tool for doing arithmetic.

I'm not sure I'm all for banning the use on exams. A calculator in the hands of an educated user is basically a way to quickly check your answers. And if you can check your answers and understand how and why an answer make sense, it means you understood the problem.
It's all about building habits. Because I never used a calculator for arithmetic when I was younger, I still don't. What I do, however, is if a question involves graphs, for an example, is that I check my answers. It saves me from losing points on problems I know how to solve because I forgot a minus sign or something else somewhere.

 

[Simon Bridge]

In general the calculator should be introduced as an aid to estimate uneven roots and graph polynomials and that type of stuff. It should never be given as a tool for doing arithmetic.

Interesting idea - considering that it's use as an aid suggested is by taking over the tedious parts of the calculation: i.e. the arithmetic.

At HS - calculators are useful for teaching math where you need the students to pay attention to the math and not the numbers. Smart calculators can be useful for displaying quick relationships.

This is not to say that skills such as sketching graphs and mental arithmetic should not be taught. Everyone has experienced punching a calculation into a machine severa times and getting several different results (mostly by mis-hitting keys) and yet people are more willing to trust the machine over their own brain. One of the things that needs to be explicitly taught in this computer-age is how to check your answers. How often do we see a question on PF which is the student asking "here's my calculation: did I get it right?"

 

[pwsnafu]

Never, IMO. Mathematics $\ne$ arithmetic.

And ban calculators from all math tests and exams, of course. If you want to give people an educational qualification for knowing how to use a calculator, that fine, so long as the certificate doesn't include have word "mathematics" on it.

Naive at best. All high school mathematics courses worldwide will do statistics at Year 9~11.

 

[Feodalherren]

Interesting idea - considering that it's use as an aid suggested is by taking over the tedious parts of the calculation: i.e. the arithmetic.

At HS - calculators are useful for teaching math where you need the students to pay attention to the math and not the numbers. Smart calculators can be useful for displaying quick relationships.

This is not to say that skills such as sketching graphs and mental arithmetic should not be taught. Everyone has experienced punching a calculation into a machine severa times and getting several different results (mostly by mis-hitting keys) and yet people are more willing to trust the machine over their own brain. One of the things that needs to be explicitly taught in this computer-age is how to check your answers. How often do we see a question on PF which is the student asking "here's my calculation: did I get it right?"

Good point. I do suppose it can be used for arithmetic. But I think it's good that students do as much arithmetic as possible on their own. I understand where you are coming from. I can't honestly say that I remember much of my high school math (I went into the military for a couple of years before I went to college, where I am now) BUT, if my memory serves me correctly, we used easy numbers when studying new concepts to focus on the math and not the arithmetic and then gradually went on to fractions etc. What I do remember, is that we weren't allowed calculators for tests and people who used the extensively for homework didn't get A's.
I quickly learned to use it for what you just said, checking solutions.

 

[Vorde]

I got introduced to four-function calculators in grade 5, graphing calculators in grade 7 and I was required to have my own graphing calculation by grade 9. I think this is about perfect.

Similar to what Studiot said, I find it silly that we are required to have cold things that are no longer useful. The amount of trig formulas I know and will never use is incredible, simply because I'd just plug it into a calculator. To me saying calculators don't belong in mathematics is like saying I shouldn't be allowed to write my english paper on my computer because I should have to handwrite it.

The realistic approach is to teach kids the 'old way' (a.k.a. the manual way) of doing things, then show them the modern (often times fast) way of doing things, and allow them to show proficiency through the modern way in every case but when checking to see if they can do it the old way.

In trig for my final we had two tests. One was all proofs and derivations showing that we could do things the manual way (no calculators allowed), with very little numerical effort required. And the second was all numerical with the calculator permitted to show that we can present actual answers.

A good mix is required, leaning too far on either side is wrong to me.

 

[chiro]

I'm just wondering. When do you think kids should be introduced to the idea of using a calculator to do math?

When they are able to do arithmetic without a calculator at a competent level, and understand the processes behind the methods.

If you don't have this then you have students pushing buttons without knowing what the hell they are doing, and if you force students to have to do everything manually then you are taking up their time with useless problems while sacrificing time for them to focus on more important higher level tasks like putting the math into focus.

Kids should not be given calculators when they start learning arithmetic and numbers but once they demonstrate a good enough understanding of arithmetic, what it means in context, and how to do it independently at a minimum standard, they should not need to waste any more time on computation when there is a better tool to do the job.

 

[Mark44]

When they are able to do arithmetic without a calculator at a competent level, and understand the processes behind the methods.

If you don't have this then you have students pushing buttons without knowing what the hell they are doing, and if you force students to have to do everything manually then you are taking up their time with useless problems while sacrificing time for them to focus on more important higher level tasks like putting the math into focus.

Kids should not be given calculators when they start learning arithmetic and numbers but once they demonstrate a good enough understanding of arithmetic, what it means in context, and how to do it independently at a minimum standard, they should not need to waste any more time on computation when there is a better tool to do the job.

This sounds completely reasonable to me. The part about "pushing buttons" blindly reminded me of when I used to teach at a community college. A physics instructor who had an office near mine told me about a student who had come to see him during his office hour. She had done a calculation, but her answer was off (too small) by a factor of 10. When he told her that, she immediately grabbed her calculator to multiply her answer by 10. Before she could start punching buttons, he grabbed the calculator, to try to get her to use her brain.

 

[lurflurf]

Kids should be introduced to the idea of using a calculator to do math the same day they are introduced to the idea of doing math. Why are all these technophobes on the internet? Calculators and computers are faster, cheaper and more accurate than humans. When I need to make 57389 or so calculations I do not do them by hand. Who here can compute 357*79135=28251195 in less than 1.0 milliseconds?

 

[chiro]

Kids should be introduced to the idea of using a calculator to do math the same day they are introduced to the idea of doing math. Why are all these technophobes on the internet? Calculators and computers are faster, cheaper and more accurate than humans. When I need to make 57389 or so calculations I do not do them by hand. Who here can compute 357*79135=28251195 in less than 1.0 milliseconds?

I have a feeling that most people nowadays encourage the use of software and other tools to do calculations.

Some of my statistics teachers in the past have told me about how their own coursework was full of doing manual calculations, and because of this, couldn't focus on doing the kinds of projects and work that we do on a regular basis in modern statistics courses like the one I took.

These professors certainly had no problem with the new curriculum, but what was surprising was that even though this was only say 30 years ago, this was the nature of things.

But again, would you want someone who knows how to push buttons without knowing what the hell they are doing? If not, then what is the minimum standard you would want for the new generation of youths and students not only in specialized science, engineering, and math programs, but for people in general?

 

[Mentallic]

Kids should be introduced to the idea of using a calculator to do math the same day they are introduced to the idea of doing math. Why are all these technophobes on the internet? Calculators and computers are faster, cheaper and more accurate than humans. When I need to make 57389 or so calculations I do not do them by hand. Who here can compute 357*79135=28251195 in less than 1.0 milliseconds?

And what kids that're just beginning to do math are required to make such calculations?

 

[lurflurf]

But again, would you want someone who knows how to push buttons without knowing what the hell they are doing? If not, then what is the minimum standard you would want for the new generation of youths and students not only in specialized science, engineering, and math programs, but for people in general?

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.

And what kids that're just beginning to do math are required to make such calculations?

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]

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.

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]

If they cannot make such calculations they need a calculator more. Some basic arithmetic is helpful.

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

 

[Mark44]

Kids should be introduced to the idea of using a calculator to do math the same day they are introduced to the idea of doing math.

This is a very bad idea, IMO. They should be competent at arithmetic operations first.

Why are all these technophobes on the internet? Calculators and computers are faster, cheaper and more accurate than humans.

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.

When I need to make 57389 or so calculations I do not do them by hand. Who here can compute 357*79135=28251195 in less than 1.0 milliseconds?

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 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.

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]

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

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

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

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.

...Pentium chips

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.

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.

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.

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?

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. http://www.obsoleteskills.com/skills/skills 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]

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.

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]

Thus I think the only calculators kids should have are elementary arithmetic calculators, through all stages of their development.

I think that's 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 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.

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]

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.

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 possesses and unfortunately is in short supply.

 

[Angry Citizen]

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.

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?

 

[chiro]

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?

I've already given my own response to that earlier in the thread, but in case you were wondering, my response was give them calculators when they understand the processes used on their calculators.

My later response though was targeted specifically at your response for programming, not for the calculator debate and I thought this would be crystal clear given that your response focused on programming.

All I'm saying is understanding programming, much like understanding mathematics is not about understanding a specific language or a bunch of largely un-connected specific examples like we have in the high school curriculum (right angled triangles, angle classifications for straight lines, etc).

Instead real understanding comes from knowing constructs like for example, doing a loop so many times to calculate a polynomial expression, or using an if statement to decide whether to use option a or optio b.

These kinds of things don't depend on languages in the absolute sense and if its taught this way, then the students will not be learning and we will continue the stupidity that is already happening in the classroom, where the students typically know the answers, but they really don't actually understand that much.

I don't have first hand experience of students being introduced to programming in schools on a classroom or school level, but I have helped a variety of people to learn programming of various backgrounds and ages, and I have seen that the learning difficulty shares similarities in mathematics where people often just do things without knowing until suddenly the lights go on and it makes sense.

In some of the above situations, people just read code (or reading symbols in mathematics) and they don't fully know what the code is even doing, so they fudge the code in some way to try and get what they are aiming for until it magically works.

This situation can be amplified when you use specific implementations and examples and if the course is structured bad enough, students can get away with going through the whole course by using a kind of superficial understanding to know what to fudge even if they have no clue why.

This is the gist of what I was trying to get at.