Teaching to the Limits of Wolfram Alpha

Maria H. Andersen has posted about a new article about Wolfram Alpha’s ramifications for education in The Chronicle of Higher Education. It’s fairly general about creating a new “math war”, but I’m starting to worry about the specifics: just how do we teach with this thing?

With graphing calculators I tried to convince students they needed to understand their graphs rather than have the calculators do all the work by giving cases where the calculator falls down, hard. (On a Texas Instruments machine, try graphing a logarithm or something with a lot of asymptotes.)

Here are three cases Wolfram Alpha can’t handle (as of now).

1. The alternate form of the quadratic formula

When I recently wrote about the alternate form of the quadratic formula I found Wolfram Alpha was able to derive the standard formula from scratch. I was then curious if it could get the alternate version, but (as far as I know) there is no way, even though the solution simply requires rationalizing the numerator. For example,

solve a=(-b+sqrt(b^2-4*x*c))/(2x) for x

gives Wolfram Alpha a blank.

Now, one might claim this instance isn’t useful, but for some computer calculations the alternate form ends up being more accurate. Thus somewhat ironically a calculation used to aid designing a technology algorithm has to be calculated by hand.

2. Graphing 3-D graphs that are not functions of z

A Twitter conversation came up recently about how to graph equations with x, y, and z that aren’t functions of z, that is like


Now, this particular special case of a sphere works, but even simply switching to


causes graphing to fail. If the z is isolated, however, one obtains the proper graph:

graph (1-x^3-y^3)^(1/3)

This is one case where Wolfram Alpha can isolate for z in the original case, but in general this case requires a conceptual leap to get from the start to the finish to enough of an extent that it would be hard to accomplish without knowing what’s going on with hand manipulation.

3. Assorted solve for x problems that Alpha can’t show the steps for

I’ve run across these more or less randomly, and while I think some cases will eventually be patched up, not all of them will.

solve 5=3/(2+x) for x

The issue

I’m especially worried about problems like case #1, where algebraic manipulation may be required no matter what system is in use. If we take a standpoint of letting CAS take over in general, how will students handle such cases? From my experience with Wolfram Alpha and other CAS systems, manipulation by hand is still sometimes required to get the computer to handle things correctly. However, this manipulation can’t be taught piecemeal, because each case is unique.

This creates a huge dilemma of allowing a CAS system yet convincing students that they need to learn minor calculations. A stockpile of problems like case #3 might be helpful, but to get to such examples one needs to practice simpler problems like 3x+5=2x-1 (which they can just have Wolfram Alpha do for them).

6 Responses

  1. I think you have good points here – and ultimately, it’s nice to know that “human” intervention is still required sometimes. 🙂

    I suspect the system will get better and better with time.

    The convenience of W|A just can’t be beat though, I’m finding myself using it more and more each day. I don’t even know where my graphing calculator IS … I haven’t used it once in the last 6 months.

  2. Another by-hand issue is the cube root of x, which can be done solely in the reals if desired but Wolfram Alpha will give the first complex answer as the principal cube root. (Example.)

    This one could really sneak up on an unsuspecting student because Algebra II books only discuss the real root.

  3. And graph solutions in the complex plane along with solutions in the real plane … that might even sneak up on unsuspecting algebra instructors with rusty complex analysis skills!

  4. x^2+2x+1=(x+1)^2=>x=-1

    solve a=(-b+sqrt(b^2-4*x*c))/(2x) for x

    1?=(-2+sqrt(4(1-x))/2x=(-1+sqrt(1-x))/x=[assume x=-1]=(-1+(sqrt(2))/-1=sqrt(2)-1!=1

    W can’t solve a=sqrt(x) while 7=sqrt(x) is no problem. I guess you see why? Don’t know how to tell W to interpretet the variables as Reals but should not be to cumbersome?

    Don’t mean to be rude but I’m a student and I think that if I can spot a mistake like that shouldn’t a teacher also be able to? Especially before blogging that there’s something wrong w advanced math-tools…

    • As far as I know Wolfram Alpha (as opposed to Mathematica) cannot be forced into using only reals. I would be happy to be proven wrong with this.

      Wolfram Alpha can handle a = sqrt(x) just fine:


      as well as give the solutions. (It can’t give the *steps*, so that is an interesting point.)

      I was simply illustrating the inability to force Wolfram Alpha to get the alternate quadratic formula. The point is educational, that some hand-manipulation is required, not whether or not the software is behaving properly.

  5. Today I tried WA for the alternate quadratic, and it worked.

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