What's left to teach?

In his review of the new Wolfram Physics App, Rhett Allain writes:

I think apps like this are going to be more common in the future. So, here is the question: is this a good thing or a bad thing? Should these types of apps be banned from the classroom? For me, I will not ban them. It is essentially just a calculator. If I ask interesting that thoughtful questions, I will be able to determine what a student understands even if he or she is using this app. If you ask questions like: what is the acceleration on an inclined plane? Then this app will not be a good thing.

The app itself solves physics problems, although as Allain points out, it doesn't give insight into the solution process, and it makes some funny decisions about what kinds of problems to solve and how to organize them.  But Allain's question is the crucial one for all of us who are actually confronting the issues presented by new technology, rather than just ostriching them.  (At a conference last Fall, I heard a presenter say that a disadvantage of the new Wolfram Alpha is that we can't just grade answers to math problems--a practice that I thought had been invalidated oh, at least two decades ago.)

Some questions on "good" questions:

• I think part of the distinction Allain is making is between setting up problems and computing--working out--their solutions.  This app is essentially a worker-outer: once you know which kind of problem you're trying to type in, it's only one or two short steps to setting it up properly, because for the kinds of problems handled by the app, the setups are standard.  Interesting problems don't have standard setups.
• This further distinction -- between interesting/nonstandard problems and standard problems--suggests that we should really teach students to set up new problems, not just standard problems.  Is that a reasonable goal for all students?  (I'm not saying it isn't, just pointing out that this is a reasonable question.)
• Conversely, is it worth teaching students to memorize or recognize standard setups? What is the benefit?  This spring, John and I were independently asked to adjudicate a dispute about the interpretation of a contest problem, and the person who asked us commented that both of us started our responses with the same sentence: "This is a standard problem."  I expect my math team students to have a wide repertoire of such standard setups, not because most contest problems are like those, but because by knowing these standards, my students have more intermediate places to take problems that are nonstandard.  For example, a standard problem is "How many routes are there from (0,0) to (5,5) traveling only right or up along lattice lines?"  If you know to set up the solution as permutations of the letters RRRRRUUUUU, it's easy to solve the "hard" problem "How many routes are there from (0,0) to (5,5), traveling only along lattice lines, that take a total of 12 steps?"  In general, I would argue that having a large repertoire of solved problems is an important component of being a strong problem-solver, though hardly the only such component.
• What counts as "setting up" rather than "computing"?  For example, in my upper level classes, I have told my students that they shouldn't bother showing work in solving a straightforward quadratic that results from a particular approach to a problem:  by Precalculus BC, I assume they can do that, and I'd rather they use their solver instead of doing it by hand anyway.  But in an Algebra I or even Algebra II class, solving the quadratic is an important part of the solution.  In Algebra I and Geometry, I tell my students not to show me arithmetic:  simply state what you're adding or multiplying, and give me the answer.
• So a conclusion: what counts as "setup" versus "computation steps" depends on the level of the student and the class.  And that means we as teachers have to be explicit about what our standards are, and in turn, what ultimate goals and outcomes those standards are serving.
• Finally, I think this kind of app reminds us that there are other ways to assess student knowledge--and other things we would want students to be able to do--besides problem-solving.  Producing a clear explanation of a standard solution shows knowledge, and even creativity; being able to address the implications of changing one component of a situation (without necessarily working out every possible solution) is another.  And these are important "skills". 

I'm curious what my physics buddies have to say about the app.  And what do you think?  Are you worried about impending geometry apps?

== pjk