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I teach grade 8, 9, and 10 maths students who are blissfully ignorant of great mathematical tools available to them such as WolframAlpha or graphing calculators (which are not used in our school). At present I am aware and grateful that I don't have to worry about any issues related to cheating.

I do feel that these students should be introduced to WolframAlpha, since it is a great (not to mention interesting) tool they should use even if they don't study maths in university. Besides, they're likely to find out about it before grade 12 anyway.

But here's my question: When and how should I introduce my students to WolframAlpha?

I'd be especially grateful if someone could suggest a particular topic or question for which WolframAlpha could provide my students appropriate insight.

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    $\begingroup$ Is "Never" an acceptable answer? $\endgroup$
    – Andrew Stacey
    Commented Apr 13, 2014 at 11:34
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    $\begingroup$ I introduce it early as a method to check answers, but I mostly teach grades 13-16. It would be nice if you could find an example which crashes it, but common sense finds the right answer. However, this will be tough. I tried something similar with Maple a few years back, an DEqn which only could be solved approximately, they just solved outright with Maple... $\endgroup$
    – James S. Cook
    Commented Apr 13, 2014 at 12:51
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    $\begingroup$ @JamesS.Cook It's not really possible to get a correct, to-scale graph of something like y = x^3 - 21x^2 + 20x (has three roots at 0, 1, and 20). Since things like Wolfram|Alpha love making sure the scale is consistent across one graph, no such graph can capture both the tiny things happening near zero and the huge things happening near x=10. And this example is even better on graphing calculators, since it looks like a parabola on the default window! $\endgroup$
    – Chris Cunningham
    Commented Apr 13, 2014 at 14:20
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    $\begingroup$ Somewhat like when to teach your kids about sex and drugs -- in time for them to learn judicious self-control. If you can teach them to use it wisely, then why not do it? (Unfortunately, I do not know how to satisfy that hypothesis with 8th, 9th, or 10th graders. I imagine it could done, and I imagine there are difficulties.) $\endgroup$
    – user1815
    Commented Apr 13, 2014 at 14:25
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    $\begingroup$ @JamesS.Cook Yes, I think one certainly can argue for "never" BUT I think that in the long run my students (1) WILL probably figure out WolframAlpha anyway and (2) SHOULD learn how to use computers well. Perhaps WolframAlpha doesn't really count as programming, but ideally some of my students will learn a lot more about programming, at least in Excel or graphing calculators, if not Python, etc. $\endgroup$
    – David Ebert
    Commented Apr 13, 2014 at 14:47

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Never.

I think that in 2014 we can teach students to use opensource projects: Sage does a very good job: Sage Notebook is couple times better than Wolfram Alpha since you can use it with no subscription, save your worksheets and it's based on much better programming language than Alpha is.

You will probably very likely find some other free (really free, not like Wolfram's stuff that doesn't allow you to copy-paste a number) and good alternatives, but most of them are topic-specific and already included in Sage.

If we generalize the question to: How and when should I introduce my students to online mathematics softwares?

Then my opinion is: As soon as you feel they are able to learn to use it. They should have some basics in programming, since learning both programming and mathematics tools is quite difficult. I agree that for WolframAlpha you don't need to do any programming, but you don't get much interesting stuff out of it.

How to introduce it? Some ideas:

  • Make a lecture on it, and then give a (very simple) homework that is solvable without the software, but is easier to solve with the software. Examples: roots of a polynomial of higher degree that can be guessed, but it takes some effort; computing integrals; performing RSA method; solving linear equations; working with complex numbers; compute the 25th Fibonacci number ... This shows that it can ease our job.

  • Use it yourself to verify results of some exercises you do with the students, use it to plot functions you introduce, generate nice fractals if speak about them, etc. That all shows that it can significantly help us do things that are (close to) impossible otherwise.

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    $\begingroup$ Oh you, with your strong opinions :) Mathematica is a fine language, but free is nearly better than, well, not free. But definitely +1 for easing our job. $\endgroup$
    – Sean Allred
    Commented Apr 14, 2014 at 3:18
  • $\begingroup$ Even though I also like free software, I don't agree with this. Both me and my students did not find Sage extremely easy to use (especially setting it up - and I found the "official" server somewhat unstable). Even though I know a bit of Python, I could find a decent tutorial on Sage (by "decent" I mean explaining things, not just showing how to do them). And the main advantage of WA is that you don't have to memorize (almost) any syntax. Very handy if you use it only once a few months... That said, I always remind my students to read WA's terms of service; after all, it's non-free. $\endgroup$
    – mbork
    Commented Apr 14, 2014 at 21:37
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    $\begingroup$ When you say that one should introduce software when they are able to use it, you gloss over a disparity between W|A and Sage. It is hard to use Sage, even when you do know python. The online system is slow and there's an annoying wait between sheet creation and usage. The documentation is poor and misleading (though I've tried to correct some of that). Wolfram Alpha is free, has excellent documentation, and is extremely forgiving in its inputs. So people can use W|A much before they can use Sage. (That said, I support Sage and encourage others to improve it too). $\endgroup$
    – davidlowryduda
    Commented Apr 15, 2014 at 2:24
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    $\begingroup$ +1, Firstly, I think the support for sagenb.com will probably be dropped. But, there will be a free cloud service: cloud.sagemath.org. $\endgroup$
    – kan
    Commented Apr 26, 2014 at 6:26
  • $\begingroup$ -1 I strongly disagree with the notion that software is better simply because it is open source. Even in the open source world, sage is unusually poor as a tool for elementary mathematics. $\endgroup$
    – Mark McClure
    Commented Apr 12, 2015 at 18:20
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[Long winded response!]

Although I do enjoy Sage, I have to agree with others that it is not quite user-friendly enough for high schoolers to tolerate just yet. I would also echo those that discount W|A due to it not being free. Maxima being on smartphones interests me, but I offer another alternative.

The free dynamic geometry software, Geogebra, is perfect for high schoolers. It has a strong (and intuitive) CAS to be able to do anything they'll want to do and combines the power of geometry with algebra, as should be happening in the classes. They're working on phone/tablet versions (just released). The commands for things are simple, and the software helps guide you for input if you're stuck. Please check it out! [Don't use the browser version.]

The question of whether to use these tools or not... I'm kind of surprised by many answers here. To not introduce mathematics students in this century to technology and have them actively using it is just shocking and a great disservice. We should be embracing these tools to provide more legitimate mathematics to our students. Let's take an example.

A cookbook problem that exists in a class with no calculators/software could be something (for example) like:

Solve $x^2-5x+6 = 0 $. And maybe a few of these are necessary to get a feel. However, what is ventured by spending days doing these things by hand with ever more ridiculous problems like $2415x^2+433x-20=0$? That is not a real problem. It comes from nowhere, and no one does these kinds of things without software -- neither mathematician nor applied scientist.

A real problem might result in the cubic $−0.0080556t^3 + 0.11881t^2 − 0.30671t + 3.36$ (from here, example 1) where we are taking data and fitting a curve. Now I might ask (because this came from data about gasoline prices over time) when the prices were increasing. I can ask that of someone in Algebra 2! Even in Calculus if this is asked, a calculator/software is going to be REQUIRED to find derivatives and arrive at actual numbers. Before Calculus, they can use technology to firstly -- find that cubic(!) with the given data and then discuss what it means.

[Point of honesty: Of course I was taught the cookbook stuff! I thought it was awesome! Let's simplify rational expressions and verify trig identities all day long, in my opinion. But I'm a mathy, nerdy teacher. That just isn't doing (real!) mathematics nor applications justice.]

And to finish my rant, I would argue that the technology can more quickly allow teachers to introduce students to conjecturing about very hard things more readily than if they don't have such access. Please see the journal "Mathematics Teacher" for more cogent and detailed arguments in this vain.

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    $\begingroup$ This is such an excellent first post, that I hope we will see a lot more of you around! Welcome to ME:SE! $\endgroup$
    – Chris Cunningham
    Commented Apr 25, 2014 at 18:53
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You should introduce Wolfram Alpha as soon as you like. Some students will already know about it, and it might be an unfair advantage.

In my experience, students will always find ways to cheat if getting the correct answer is all that is needed. See Joel's answer here: https://matheducators.stackexchange.com/a/53/78 . This doesn't really apply directly to your age group; but one idea to take from it is that you could give quizzes with no calculators allowed, or worksheets without calculators, and emphasize how important it is to know how to do all the steps by yourself.

One fun exploration is to have them graph different polynomials to figure out how many 'bumps' you can have for a polynomial with a specific degree.

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  • $\begingroup$ I'm not entirely sure you should introduce them to Wolfram at all. Not because it provides solutions, but because you have to pay for more than 3 solutions per day. I'd much rather introduce them to OpenSource software, to prevent them from getting attached to a paid service. $\endgroup$
    – Alec
    Commented Apr 14, 2014 at 9:33
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There are several CAS around, including my personal favorite maxima, which incidentally also is available on smartphones... Use the computer to do the routine jobs, save the brain for understanding and creating. Sure, some familiarity with the routine work is needed, but I'd concentrate on the "higher level" tasks.

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    $\begingroup$ cool, I did not know Maxima is on smart phones, I've known about the wolfram alpha app a few years now. Also, I share my brother's criticism of graphing calculators: smart phone plus $2 wolfram app will beat the pants off any off the shelf calculators. I will have to go make my phone smarter now, thanks! $\endgroup$
    – James S. Cook
    Commented Apr 13, 2014 at 18:31
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    $\begingroup$ @JamesS.Cook, also check out SAGE for Android. My biggest gripe is that they need reasonable keyboards, so I'd go for a tablet with bluetooth keyboard... $\endgroup$
    – vonbrand
    Commented Apr 13, 2014 at 19:13
  • $\begingroup$ Sage doesn't work on my Android tablet; I run the app but nothing happens. I'll check out maxima though. $\endgroup$
    – Mike Shulman
    Commented Apr 14, 2014 at 5:04
  • $\begingroup$ @MikeShulman, I believe the first time you run it it does a lengthy initialization, downloading a lot of stuff. $\endgroup$
    – vonbrand
    Commented Apr 14, 2014 at 23:26

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