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Currently, I'm working as an assistant at the mathematics department of my university. My job consists of sitting for 20 hours a week in an office to which students come with doubts regarding all the basic math courses.

Today, for example, I was solving a limit in a class full of engineering freshmen. After I solved it, I checked the answer in Wolfram in front of the students and I recommended the website, as well as other useful sites such as Desmos, or Khan Academy.

My afterthoughts were: "what if they start using it when they're not supposed to? (i.e. exams and such)". Does the act of recommending such a site carry any moral consequence?

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    I remember when schoolteachers used to say the same about calculators.. – CaptainCodeman Apr 15 '15 at 8:42
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    Is it immoral to teach them how to read and write? – Peter - Reinstate Monica Apr 15 '15 at 9:46
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    You may know Conrad Alpha's TED talk ted.com/talks/… (tl;dw: computing is the least interesting step in applying math in the real world and is much better done by computers [read: Wolfram Alpha]. Nonetheless computing is what math education (he probably means pre-college, admittedly) mostly consists of. We need to change that. – Peter - Reinstate Monica Apr 15 '15 at 9:55
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    I suggest it's immoral not to show it. Some are bound to know of the site; and not making it common knowledge grants "unfair(?)" advantage. – user2338816 Apr 15 '15 at 23:39
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    @Davor Me too. I mean does anybody here seriously think any not completely incompetent math student in university hasn't heard of mathematica and wolfram alpha? Should we mention google next time too? Anyhow, as an engineer - sure I learned how to do integrals manually, but now if I actually want to apply my knowledge I certainly use mathematica. If the only thing a course teaches you is to do something a computer can do better in a fraction of the time, it's a waste of time. – Voo Apr 16 '15 at 16:34
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I always like to show the students Wolfram Alpha in freshman math courses. (Many will already know about it, whether you show them or not.)

There are several reasons for this. The first is that it's a useful tool, both for checking solutions to homeworks, and also for later in life.

But the more important reason is that many students are skeptical of why they have to take a course in calculus. I double down on their skepticism by demonstrating to them that a computer can solve most computational questions on a calculus exam in about 0.05 seconds, and can even "show its work". Having gotten their attention, I now have a good opportunity to make my case for the value of a math education.

As far as exams go, I recommend prohibiting the use of all electronic devices. Some students will cheat on their homework, but that will be true no matter what and in the end they are cheating themselves.

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    What you say about Wolfram Alpha being able to solve the problem in about 0.05 seconds, when I took differential equations, I learned to solve problems that Wolfram Alpha couldn't directly solve (I had to make clever substitutions to another form before it could handle it, then substitute back). This was the first time I encountered a computational problem that I could solve and Wolfram Alpha could not. – Justin Apr 15 '15 at 5:47
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    "This website can solve your math problem in 0.05s" seems like a terrible case for the value of a math education. It makes it sound like any other obsolete thing that we used to teach people but which are now done by computers or machines. – David Richerby Apr 15 '15 at 7:11
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    ^ It's not a point for the value of a math education, but against. They said that by using that, they were able to essentially confirm / recognize an opposing argument that students may have in order to (presumably) present a strong counter-argument which supports the program. – Sean Apr 15 '15 at 9:45
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    The point I make is that, Wolfram Alpha or no, just being able to mechanically compute derivatives and integrals for its own sake is of questionable value. Rather, the point of a calculus course lies in the big picture and appreciation for concepts which students subliminally develop, in the problem-solving muscles which get exercised, and in the good habits of precision which get refined. – Anonymous Apr 15 '15 at 11:16
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    So, basically, better computation tools help good teachers and hurt poor teachers. – KSmarts Apr 15 '15 at 15:42
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In addition to the prior good answers, I would say that I feel the existence of tools like Wolfram Alpha doesn't fundamentally change education any more than the existence of calculators does. We still teach people how to add and subtract, we just raise the bar on the expectation of how easily they can deploy those skills with tool assistance. Likewise, Wolfram Alpha means we have to raise the bar in what we expect students to achieve in more complex mathematics: the goal is not to be able to integrate, it's to be able to use integrals in solving mathematical problems, and tools like Wolfram Alpha just expand the range of problems that are feasible for a student to solve.

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    Indeed, like I have a calculator and can say "Why would I ever try and handsolve dividing a 7 digit number by a 5 digit number?", a CAS system lets me say "Why would I ever hand solve a moderately difficult derivative?". Sure I can do it by hand, but the point of the task is to (for example), identify the time delay of this circuit, not to show I can do calculus (except when it is). – Lyndon White Apr 15 '15 at 6:58
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    Answer: So that later you can solve a horribly difficult derivative that Alpha can't. – JeffE Apr 16 '15 at 10:44
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    Second answer: So that you can tell when the answer the CAS system has spat out is nonsense (you typoed the input, it picked the wrong zero or the wrong side of the branch cut, etc). – zwol Apr 17 '15 at 14:41
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I'm a strong proponent of using computational softwares and engines in elementary courses wherein computation is heavy. In particular, I think that the geometric benefits of using Wolfram (Alpha, or Mathematica) are immense. For learning elementary material I think it is an excellent tool for checking homework to, say, solutions that don't have answers reported or to satisfy curiosities about broader behavior of calculus or certain functions.

In general, I think given that in advanced classes I still use these softwares to check limiting cases and perform routine calculations with which I'm confident, that becoming familiar with them early on is benign and even important. In terms of examinations for beginning students however, I tend to agree that at most scientific calculators or similar tools should be allowed. The tools allowed to the students should scale with the familiarity they have with the processes capable of being performed by the tools.

tl;dr I don't see any trouble or moral issue here. Stress the importance of learning and present the software in your own context acceptable to the level of the class, and all is well.

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    IMO satisfying curiosities about broader behavior is far more important than preventing cheating. With powerful enough tools, even a cheating student might get curious and learn something. – ShadSterling Apr 15 '15 at 14:10
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If only a couple of students knew about Wolfram Alpha and used it to do well in your class, they'd have an unfair advantage over the rest of the students, especially if you grade on a curve.

Either all of your students should know about this tool, or none of them should.

Since you have no way of knowing that none of them know about Wolfram Alpha, it would only be fair for you to show it to them.

  • The same can be said of pretty much anything: a nice textbook, a good set of exercises, a friend that is good at explaining the topic... – Davidmh Mar 10 '16 at 12:44
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I think you definitely should show Wolfram|Alpha to them, and you should take the opportunity to explain why such tools will never be a replacement for mathematical thinking.

Personally I found Mathematica extremely useful for learning calculus some 15 years ago (Wolfram|Alpha didn't exist at that time). It makes it easy to plot functions, check results, and encourages good students to experiment and learn. Graphics can be fun and will encourage students to do such things as trying to figure out the parametric equation of a sphere or torus even before they study it.

But it is also important to understand that such tools cannot replace thinking about the problem. I am quite active on Mathematica.SE and I often see people (presumably students) ask questions such as "Why doesn't Mathematica solve this equation?", "Why won't it compute this integral?", "Why won't it simplify this expression?", "Why does it give such a complicated result, I need a simple one!" They seems to treat it as a magic box that just gives solutions, and when it doesn't, they feel stuck. They don't think about such issues as: is the equation still solvable is this parameter is negative or complex? Does it at all make sense to use the (the complicated and expensive to evaluate) closed form solution of this 4th order equation in my code, or should I solve the equation numerically? Does the number of roots to this function depend on the parameter values? Can one reasonable expect a closed form solution at all? Why do I want an analytical solution at all? Should I use approximations when solving this physics problem?

Mathematica can solve quartic equations, but does this really look like a useful result? This is the kind of result a blind "solve this for me, W|A!!" will give to students. Isn't this explanation so much more informative and useful?

I believe that as their teacher, you owe it to your students to explain the proper use of computer algebra systems and explain why they will never replace thinking for yourself. If anything is unethical, it is allowing them to fall into this trap and treat this tool as some sort of magical oracle.

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I'd go further than the other excellent answers here.

Not showing new students how to use Wolfram Alpha is immoral.

Students should be taught how to use all of the tools they can be taught to help prepare them for the real world, and Wolfram Alpha for a math/stats/physics professional is a very valuable tool. Explicitly avoiding teaching them this tool is counter-productive: you fail to teach them a useful and valuable tool, and you get little in exchange. You can avoid cheating by not allowing electronic device use during tests; and ultimately if a few cheat their way through the homework, it doesn't do much harm to the folks who aren't cheating and are getting the most out of their education.

For some, it's a good way to learn as well: if you're stuck and don't understand a concept or why a solution works, instead of having to wait for a TA session, you can ask Wolfram Alpha to show you, then learn on your own.

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It's not immoral to make them aware of a useful tool.

Used effectively it will enhance their learning:

  • It will allow them to check there homework, giving them immediate feedback.
  • Allowing them to follow the steps shown to see exactly where they've gone wrong. Identifying precisely where a mistake was made while the problem is still fresh on the students mind. This is some of the most valuable feedback they can get.
  • When encountering a problem for which they're unfamiliar with the technique necessary to solve the problem, the work shown can give them that insight, so that they may tackle similar problems in the future.

Establish guidelines on how you expect them to use the tool. Nothing will stop them from disregarding those guidelines, but that will be to their own peril.

  • Make it clear whether or not exams will be given in a setting that allows access to the tool. It should almost certainly not,
    • It should most likely not. Allowing use on exams will encourage them to focus on leveraging the tool exclusively, which will exclude important skills that future instructors will expect them to have.
    • One day it might be so standard that this is not an unreasonable expectation. (We don't learn how to do square roots the long way by hand anymore.) However, until such time as tool use is the standard across all of academia, students should have the expectation that they need to be capable of solving without the tool.

Highlight the value of learning the skills, and not depending completely on such a tool.

  • Emphasize that being able to apply the appropriate technique for each problem is not only important to being able to solve the problems, but also key to understanding what the equations represent.
    • When they move into their respective career fields, a lot of what they struggle through now should be starting to become second nature by the time they begin their career. An electrician doesn't stop to reference the basics of housing wiring everytime they work on a light switch. That has to be reflex so they can focus on the task at hand.
    • In your career you may need to be able to identify what problems can be solved directly, and which may need to be solved using a computational technique that estimates the result.
    • Even if tool usage were standard or expected, you may need to manipulate some problems to get them into a form that the solver recognizes as solvable.
    • Having a solid understanding of solving problems reinforces skills that will be useful in careers where one must design/choose a formula that models a certain scenario.
  • Whenever I show it, I always say: its a great tool for checking your answers, but please do not be dependent of it. Great answer. – Miguelgondu Apr 18 '15 at 16:56
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You mentioned that you showed this to class of engineering freshmen, so I will answer this question as a former engineering freshman (now an engineering junior).

Short answer, yes. Wolfram Alpha is very helpful for basic things like solving simple differential equations and complicated integrals. However, like many have said, tests are a different story. As an example, I took differential equations last semester and was allowed to solve the homework however I wanted, but on the exams all I could use was a pencil. We didn't even get a table of Laplace transforms!

Furthermore, in two short years these freshman will be doing things that Alpha can't handle, such as solving systems of four or more nonlinear equations, as a regular part of their homework. I used Alpha extensively in high school and as a college freshman; now I almost never use it. I use MathCAD or something more powerful (ex I recently solved a large homework problem involving heat exchanger design by writing my own C++ code).

Last, as some others have pointed out, math is not the hardest part of engineering. Arguably more important is developing a physical intuition about what the math actually means. For example, in heat transfer, when I check my work with other students or a TA almost all my errors involve the physics or the assumptions behind the equations. Sometimes they're subtle, such as using the wrong temperature of air to get properties to calculate a convection coefficient, or using diameter instead of radius as the critical length to calculate a Biot number. None of them are the sort of thing Wolfram Alpha, or indeed any computational math tool, could help me with.

In summary, the goal of engineering is problem-solving, and math is one of the tools that is used to solve problems but not the only one. Wolfram Alpha is one tool to help students learn math, but it's not the only one. For elementary calculus, I think it's extremely helpful.

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Whats more important, hollow academia or real-world solutions? Engineers are free to use whatever tools they want to solve problems. Only in the fantasy world of academia do real-world tools start to be a threat, because the problems in academia are meaningless. Make the problems in academia real, and the question will resolve itself.

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    Perhaps I'm wrong, but I guess there is some relation between lazy cheating engineers and broken bridges. I'm just talking about basic calculus: of course they could use these tools in other advanced subjects. – Miguelgondu Apr 18 '15 at 16:40
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    Hi, and welcome to Academia.SE! Could you please adjust the tone of your post to be more informative and less of a belligerent rant? – jakebeal Apr 18 '15 at 16:40

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