Monday, June 27, 2011

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

Saturday, June 18, 2011


Summer is here, or almost here, for some. Summer is always a special time for teachers, even though most of us cannot afford to take the summer off and rest on the beach, as many non- teachers seem to think we do. It is a time to look at things from a different point of view, to reflect on the last year and take actions that will make next year even better. It is a time to catch up, to rejuvenate oneself, to reacquaint oneself with one's family, and to figure out how to do it better next time.
I have observed that summer is a time for teachers to travel, often to places that connect with what they teach. Language teachers frequently spend time in countries that speak the language they teach, and while there they soak up the culture of the country so they can make their classes more alive. History and English teachers frequent sites that are relevant to what they teach, go to museums, and read books they have been putting off while they grade papers. Artists create art and visit places where there is great art. Musicians create music and listen to music, and physical education teachers participate in their favorite sports and exercise activities.
So, what do math and science teachers do in the summer? Yes, there are historical sites that enhance what we teach, and many of us do travel for that purpose. But summer also gives us a chance to work on some of those hard problems we haven't yet solved or to learn how to use that new piece of technology we just haven't had a chance to master, or perhaps to take a workshop or two. I would also like to point out that many of us use this time to read and study.
On that note: I just finished a book that I recommend to any math teacher who intends to be involved in math education for the next ten years: Mathematics Education for a New Era: Video Games as a Medium for Learning by Keith Devlin. Dr. Devlin is a mathematics professor at Stanford who has become fascinated with the potential of video games as a tool for helping students learn mathematics. He makes a very good case. Even if you immediately disagree, it is interesting, stimulating reading.
Another book I highly recommend is What's Math Got to Do With It by Jo Boaler. This book is about teaching mathematics and is really a case study of two schools, one that achieved remarkable results and another that muddeled along as it had been doing.
And if you are just looking for a book that is about math, I suggest a couple of old favorites: The Mathematical Experience by Davis and Hersh, Journey Through Genius by William Dunham, and The Man Who Knew Infinity by Robert Kanigel. These are all enjoyable, informative and helpful reads.
If only the public really knew how much time and effort we put into our work.
Have a great summer.

Sunday, June 12, 2011

A New Approach to Practice

One great idea I took away from NCTM was giving tiered practice work, aka Challenge by Choice.  We implemented the idea in our geometry classes' last unit of the year, on coordinate geometry and vectors, and for our final exam review.  The way it worked was this:  each time, we created two or three sets of in-class problems for students to work on in groups, sorted by difficulty.  The lowest-level "A" problems were about as difficult as test problems, and--this is important--were labeled as such.  The second level, "B" problems, were a little harder than test questions, and were labeled as such.  Sometimes we provided a highest level, "C", which were even harder.  For example, our second time out, the problems included:
A1:  The midpoint of segment RS  is (2, -3, 7).  If R = (5, 15, -4) find the coordinates of S.
A2:  Two vertices of an equilateral triangle are (-2, 3) and (7, 3).  Find coordinates for the third vertex.

B:  Regular hexagon HEXAGO has H = (4,0) and A = (10,0).  Find the coordinates of point G.

C:  Consider the points O = (0,0), A = (x1,y1), C = (x2,y2), and B = (x1 + x2, y1 + y2).  What kind of quadrilateral does OABC have to be?  Justify your answer.
Often, B & C level problems previewed material we would eventually encounter; later on in the unit, we  proved that the points O, A, A + B, and B form a parallelogram.  After announcing the activity, students were free to pick their own A, B, or C sheets, and made informal groups with other kids working the same problems.

There are two essential differences between this approach and the traditional homework sets with A,B, and C problems:

Most important, students choose their own level of challenge.  As teachers, we worked hard to make the "A" level problems an honorable choice, and we coached some students up who we felt weren't taking on hard enough problems.  But because students experience their challenges as choices they themselves have made, they work harder:  we saw students struggle for ten or fifteen minutes with problems that they would have given up on as part of regular assignments.

Second, the baseline level is "test difficulty."  So the students practicing "A" problems aren't going to be surprised by what's on the test; rather, they know what they're up against in a very explicit way.  Students seeking more challenge work harder and get stronger, rather than coasting.  The results on my tests were striking:  I had more scores above 100 (our tests include 20 points of extra credit, and challenging problems) than in any previous test, and somewhat fewer failures.

Ordinarily we teach a problem-centered curriculum, where students arrive at ideas by solving and discussing one problem at a time.  But we've found it useful to spend 45-50 minutes every few weeks reviewing, consolidating, and extending ideas;  these tiered practices are far better than anything else we've tried.  Everyone was engaged, and everyone got stronger.  We can't wait to try them again in the Fall.

What about you?  What did you do differently this last month, this semester, or this year?