# When should calculators be introduced to the curriculum?

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.)

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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.

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

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).

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 possess and unfortunately is in short supply.

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.

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.

mathwonk
Homework Helper
lurflurf:

"Who here can compute 357*79135=28251195 in less than 1.0 milliseconds?"

the teacher's problem is the kid who reads off 357*79135=2825119, and does not realize it is wrong even after much more than a millisecond.

If the teaching is done correctly, I don't think it matters too much when calculators are introduced. Basic four function calculators were introduced to me around the age of 9 or 10, scientific calculators in junior high, and graphing calculators in high school. If anything, having access to calculators at such a young age actually piqued my interest in mathematics ("How does it do that?!"). Calculators never really presented a hindrance to my learning anyway, due to the fact that none of my teachers, from algebra 1 to calculus 2, allowed the use of calculators of any sort during tests. The arithmetic was kept simple enough and the focus was put on the mathematical manipulations.

Calculators should be allowed in courses after trig to speed things up. By the time students get to calculus they know when to rely on a calculator (arithmetic, trig functions that aren't based off 30 or 45 degrees, similar ideas...). A student shouldn't be held up on a multi-variable integration problem because they can't do longhand multiplication quickly. They still know how to do the multiplication, but it would just take a long time.