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,
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:
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.
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).