What effect do you think calculators have?

I do some SAT tutoring, and I find that students are helpless at basic arithmetic if they can't use a calculator.

My suspicion is that calculators prevent students from developing mental math skills, and that this hurts conceptual understanding across the board.

But I can only see the results of the system, so I don't know what actually produced the types of students I'm seeing. I may be wrong about calculators - from what you say, it could simply be that the system in general is bad.

I grew up learning math from calculators. I still today am terrible at mental math, but having immediate, accurate graphing capabilities from precalc onward was an incredible boon.

Even before that, mechanical skills such as polynomial factorization or algebraic reorganization ("fiddly symbol manipulation") were actually, in my opinion, aided by having a calculator which could perform them for me. By releasing the burden of actually performing the computations, I was focused on larger patterns which made the computation meaningful and possible.

There are two camps here, of course. As an engineer or mathematician, doing mental arithmetic is a dead skill---all the effort comes from modeling and understanding the structure of math. However, "practical" math such as computing tips or adding up grocery bills---and this was the kind of math application stressed in my public school education---probably benefits much more from rapid mental sums.

So, ironically, I'd support the heavy involvement of calculators (even ones as powerful as Mathematica itself) in education if it's to be aimed to produce engineers, scientists, or mathematicians. Otherwise, they're probably damaging.

The issue is whether you learned the actual skill (graphing, factorization, etc.) or just learned how to get the calculator to do it for you.

In many cases, the end results appear identical. However, I'd rather the upcoming engineers, scientists and mathematicians be banned from using calculators for anything more than basic arithmetic until they prove they do understand the concepts.

Factorization and graphing aren't actual skills, though. They are mechanical computations that basically scream "please use a tool to do this".

We don't pine for the days when people had to hand-calculate logarithms, because we've come up with better ways of using that tool. CASes and graphing calculators are the same thing.

This is not true. Calculating a log by hand would be the same process every time.

Factoring quadratics is mostly the same every time, while factoring in general is not, and graphing definitely is not.

Graphing, in particular, is a crucial math skill, because it forces you to understand what is actually going on with each function. As a rule, students who have no idea what a graph is going to look like before they press "graph" also don't understand many important underlying fundamentals.

There is no reason middle/high school students should be using graphing calculators on a regular basis. Scientific calculators for things like logs are fine, but beyond that the calculator replaces actual learning.

You do realize that the tools have to be created based on some underlying process. If you have an understanding of that process you'll have a better understanding (and appreciation) of that tool. And maybe be able to come up with a better one.

"A computer lets you make more mistakes faster than any invention in human history - with the possible exceptions of handguns and tequila." ~Mitch Ratcliffe

As a physicist, I use math environments like Matlab and Mathematica to do calculations all the time. Some are simulations that are too massive to even contemplate without a fairly powerful computer. Sometimes I just use them to speed up things I could do by hand. In the latter case, I usually do some samples by hand just to check if my program is working as expected. Later, when using the program, I always try to have some estimate of what the results should look like, even if that intuition of what is correct comes from a very different way of thinking. This is something a lot of students I've taught simply do not do, and it really bites them in the ass a lot!

For example, there's one standard junior lab experiment where students work with a very weak sample of a radioactive isotope. It's just barely strong enough to register a small amount of clicks if you practically stuff the sample inside of a Geiger tube. As an exercise, we have students calculate the equivalent dose of radiation they receive over the course of the lab. If the calculation is done correctly it should come out to being barely above background exposure and nowhere near as much as an intercontinental flight or diagnostic X-Ray. However, without fail, every year at least some students will calculate an exposure that is fatal, either by incorrectly converting units or just simple "calculator error". "Calculator error" is what we call it when a student writes out their calculation and everything is correct except the final answer. They got all the input numbers and units right, but they just failed to punch it into their calculator correctly.

Many students probably make this mistake but catch it right away. Others do not catch their mistake and how they deal with it in their lab reports varies quite a bit. Some simply don't notice it. They clearly had no idea what to expect from the calculation and didn't bother to compare their result to what was in the lab materials. Other students are at least smart enough to note that they think their answer is too high, although they don't know why. Occasionally, a student will actually write that they are concerned that the experiment was unsafe, perhaps because a horrible mistake was made and they were given the wrong isotope samples. These last students are easy to laugh at, but at least they are thinking, unlike the first group of students who write down a dose that would have left them dead without any further comment.

That first group of students who trusts their calculation blindly (or simply don't care) is much larger than the other groups, and that's my greatest beef with our education system. It turns children into calculators. i.e. Devices that can mechanically turn input into output without understanding the process they are executing or having any intuition about what the answer should look like. Grade school texts rarely teach students to critically examine their results. They present students with a way to do a calculation, ask them to execute it like a mechanical device, and then knowing whether they're right or wrong boils down to checking the back of the book to see if the book's answer matches their own. Grade school students can get perfect marks without having any understanding at all, provided they practice the methods enough.

Understanding the method and having the ability to recognize what results should look like are skills that should be at the very core of math curriculum's. Learning to be a calculator is boring. Math is infinitely more interesting when you know the why and not just the mechanical how.

"calculator error" is about 30-50% of the mistakes I see on the SAT.

I end up retraining students to do mental math and only use calculators to confirm a calculation already done mentally or on paper.

I believe this helps them spot possibilities, and understand what the less than straightforward questions are asking them to do.

As a member of the last slide rule generation, I agree. The trusty slipstick was a calculation and thinking aid only. By not automatically handling units, for example, it forced you to think through the steps of your calculation. Back then students learned to approximate the result they were expecting.

I wonder how my college students today have already estimated the total of their purchases when they arrive at the cashier. Such thinking doesn't seem to be part of the culture anymore.

The strange thing is, I'm only 26. I think we only started routine use of calculators in high school. Now it's in elementary school.

I've become an old fogey faster than I expected to.

I agree with this. I also think curricula designed with the assumption that a calculator is available can be much more challenging. After all, the student is given a more powerful tool.

The goal is definitely to avoid turning people into calculators. I found that for myself and my peer group, having a calculator was the best panacea to doing that. It meant that we'd use those extra free moments wondering why things worked and what it meant instead of "carrying the ones".

WolframAlpha is a good visual tool also, and I often use it when I tutor math.

I agree with you about calculators. People need an intuition about numbers that comes from actually thinking about them rather than pushing buttons in order to be able to notice when they make serious errors. Calculator trained cashiers take in a $10 bill on a $5 purchase and then return $95 in change because the computer told them to.

Out of curiosity, why is it that you would prioritize differential and integral calculus? I studied calculus very early myself, but outside of later classwork, I've only found it useful on rare occasions. I surely would have used it more if I had gone into a research job or into material sciences, but I estimate that very, very few careers are like that.

What would have been much more useful for me would have been a stronger grounding in statistics.

I didn't mean to prioritize it, I'm saying that level of understanding should the goal of average achievement for high school graduates.

It is also really really far from what is being accomplished on average. High schools have seniors who can't do division and don't understand operations on fractions. Calculus is something anyone can understand, but it is seen by teachers as some super advanced mysterious stuff for rocket scientists while they struggle along to teach subtraction to 18 year olds.

Imagine a physical education fitness curriculum whose goal after 13 years of training was for 18 year olds to be able to turn over by themselves and begin to crawl. That's what we are doing, being satisfied with goals for graduates that shouldn't be challenging even for 6 year olds. It shouldn't be tolerated at all.

Schools that are graduating students with no math skills have had 13 years of failed instruction with these students. Not just one teacher along the way. All of them. The system should be burned to the ground and started over. That will never happen though, it's too corrupt and incompetent. Myself I've given up hope reform of the system is possible.

Speaking as a recent high school graduate, please, please don't give up. I made it out of school knowing calculus and more because I had one physics teacher and one math teacher that hadn't given up. They wrote their homework themselves, used college textbooks and twenty-year-old texbooks they'd collected from eBay, and railed constantly at the district's purchasing people. They weren't able to change the system all by themselves, but they still made - and continue to make - all the difference for dozens or hundreds of students. You can do the same. Please.

To expect all students to take calculus is prioritizing. There has to be some give and take.

In any case, I believe the OP was asking why it's important that every student takes calculus? I'm very curious about that as well. Why again is that so important?

Here as well - please don't just down vote but please explain why you are if you are going to.

I don't see why we can't learn both. I really enjoyed AP Statistics. It was a class I took my senior year of high school because AP Calc scared me, and it ended up being a very rewarding and memorable experience, but I regret not having taken AP Calc alongside of it. Better statistical reasoning is something society might benefit more from as a whole, but I feel like that first (college) year sequence in Calculus is something that everyone should experience, be it in college or high school.

I know a lot of people who struggled in classes like Algebra 2 and Precalculus (which is kind of an abomination in retrospect) who finally learned to appreciate the power of mathematics by going through Calculus. Not everyone will walk away with that perspective, but I feel like society can only benefit from a greater respect for math.

I really do wish there were courses available in discrete math at high school level.

Probability, graph theory, statistics, number theory, game theory, groups, sets, logic , these are topics which can be interesting and basics of them can be taught before college. Similarly, linear algebra is not that scary.

I completed Calculus BC test in my Junior year in HS and went to nearby University for Calculus 3(forget what it was called exactly), got a bad professor with worse English than mine and burned out of math for a long time. I was under the mistaken impression that Calculus was end of all math.

Differentials and integrals are part of life. People need to be able to understand when a politicians is giving them the differential of the figure they asked for, rather than the figure itself.

But yes, statistics is very important too, for similar reasons.

Do you think the Singapore method at one point was a fad (or new math teaching methodology) and if not, why or how not?

Secondly -- You say "Teaching math is easy." What does that mean? Just from personal experience, 10 years or so ago I took Calc 1 and 2 in high school, and calc 2 again in college, and did well in all, and know how to integrate and differentiate and use various formulas due to a recognition of certain inculcated patterns, however would I say I am skilled at actually using calculus as any sort of real life thinking/problem solving tool? Absolutely not and not even close. Is this a success? Absolutely not. One of the most important things was completely overlooked by the "traditional" method I was taught in.

> 100% of high school graduates should have passed differential and integral calculus.

I'd argue that a similar pass rate for introductory statistics is vitally important for our continued viability as a society. Calculus is, frankly, less important than knowing when a politician is lying to you, or whether that test result means you need to draw up a living will now.