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

Summer

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.

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?

Simulating Mathematics?

Bret Victor, in his online essay Simulation as a Practical Tool, critiques a classic IMP problem, which I'll paraphrase here:

A skateboarder hangs on to a merry-go-round turning at a constant rate.  At a given instant, he lets go, and travels in a straight line at a constant speed, until he runs into a (straight) wall a given distance away.  How much time does it take him to hit the fence?

As I read Victor's essay, he has three main critiques of the problem:

1. The problem is contrived, and fundamentally uninteresting (who would care how long it takes to hit the wall?), especially with the original data in the problem (all triangles are 30-60-90, etc., etc.). 
2. The solution relies entirely on symbolic mathematics, and so is hard to generalize to other situations, e.g. varying speeds, curved walls, etc.
3. A more interesting--and, for many students, more valuable--approach would be to simulate the situation on a computer, so that students can appreciate how changing the parameters in the problem affects the answer.

While the first of these is valid, I think the second and third miss the point entirely, for reasons I'll explain.

Criticism #1:  The problem is contrived and boring.  I think Victor's right about this, and an analysis that stops at "it takes 4.7 seconds to hit the wall" isn't necessarily worth the time it takes to get there.  But used as a springboard for generalization and extension -- I'm imagining one of Dan Meyer's WCYDWT videos -- this could be really cool.  In fact, Victor's own suggested rephrasing of the problem, "What release point minimizes the time to hit the wall?" is one obvious extension.

Here's another approach that we don't ordinarily take:  start with the general problem in roughly the way that I've stated it above, and have students make it specific enough to solve as a first approximation to a general solution.  The task is now "Can we (always) compute the time it takes for the skateboarder to hit the wall?" and the students--much like mathematicians or engineers in real life--start by filling in the variables with plausible initial conditions to see what a solution might look like, before attempting a solution (or a model) with variables.

Criticisms #2 and #3 seem weird to me, for three reasons:

• One goal of mathematics is the creation (or identification) of general, true statements.  Victor's proposed simulation doesn't really allow for those.  At a philosophical level, it could be argued that the simulator itself is equivalent to a set of true statements that describe all possible behaviors of the system, but without some way of tying those together, we lose what philosophers would call the "propositionality" of mathematics.  Victor is claiming that our goal should not be a single answer or set of answers, but rather a computational model that allows us to find answers given a particular set of initial conditions. I'll concede that in some areas of mathematics--dynamical systems, for example--often such models are the best we can hope for.  But even in those we want to make general statements about a system's behavior, and it's hard to imagine how we might do that without invoking symblic mathematics.
• If our goal is to empower students to do more and more interesting mathematics, we can't just hand them simulators and tell them to go play: we need to teach them how to create those simulators.  Doing that requires a lot of math and a lot of programming.  So Victor's "simulation" model of doing math ultimately requires teaching kids a lot of traditional mathematics.
• Even if we do give students a simulation that they can use to find a particular outcome, I think it's natural for students to ask "Why?" and "How can we be sure?"  Those questions ask for justification and generalization that require ... traditional symbolic mathematics.

I think simulations are terrific ways for students to experiment, to learn, and to generate and answer interesting questions.  But what we need to do is not replace traditional math with simulations, but use the power of models and simulations as an incentive to do, test, and generate exciting mathematics.

== pjk

Pentaminos

I have an activity about polyominos. I begin by reminding my students what a domino is.
Then I ask them what a triomino looks like. We decide that there are only two.
The other triominos are congruent to these two, by rotation, reflection and translation. I then ask my students to find all possible tetrominos. I suggest you try it before reading on.

When I first gave this activity, many years ago, I learned that it took about ten minutes with several interesting and useful discussions about double-counting and that there would be several students who did not discover all five. Eventually, those students did find all five and were ready for the problem I really wanted them to tackle: finding all pentominos. I then asked them if they could construct a square out of the pentominos, and if not, why not? What rectangles could they construct? It is a nice activity. But the nice activity is not the point of this blog.

One year my students immediately solved the tetrominos problem, and I mean immediately. Like in fifteen seconds or as long as it took them to write the answer. This task was not a difficult problem for them at all. This class was not particularly exceptional, and I was bewildered, until I heard one of the students ask another how he did it so fast. His answer: Tetris.

What had happened from one year to the next was the game whose pieces were these pieces. What was once difficult, was now simple.

I have often thought about what makes a problem difficult and another problem simple. I am particularly interested in problems that some students seem to understand with little effort and other students struggle with. I think of this class. I am convinced that what we perceive as talent and insight is often just experience and familiarity with similar ideas.

I ask my honors students how many of them played with Lego when they were little. Many of them giggle and smile. Some admit that they still play with Lego. Often the conversation moves to other toys of the same nature. When I ask the same questions of students who struggle with math, I get blank stares. I think we need to pay careful attention to what sort of experiences our students have had. Even more important, I think we need to build in lessons so that they are previewing topics we know will be difficult later. And they can't just be pencil and paper activities.

Peg Cagle, a middle school teacher from California, gave a fascinating talk about how students are having different experiences growing up than we had. She asked her students, I think she said about 180, how many had climbed a tree. None had. I think that effects their sense of what three dimensional space looks like and feels like.

Have a thoughtful Memorial day.

On teaching challenging students

A theme of this blog for several months has been the importance of challenging problems to help students learn. Last week P.J. wrote about one of the most challenging problems every teacher faces: those students who just don't get it, and whom we can't get to no matter what we try. Now that I am retired and am not faced with that particular problem on a day to day basis, I may be able to provide a different perspective.
Of course I agree with PJ: the most important thing is to NEVER give up on the student, no matter how hopeless it seems. Learning is complicated and unpredictable. Some time ago, I was tutoring a student on a weekly basis, one-on-one, for an hour at a time. She was taking seventh-grade math, and it seemed hopeless. I would give her a problem. She would get it wrong. We would go over it. I would give her another. She'd get it. We'd go on to the next topic. Fifteen minutes later, we'd return to the first topic, and she would get it wrong. This pattern happened every session, every week. In eighth grade, a similar experience presented itself as she studied algebra. When high school arrived, she took algebra again, this time at an honors level. She not only passed, but was disappointed when she only got an A-. She managed a B in Geometry Honors and ended up taking four years of math, the last two at a regular level, and going to a good college.
She had resources, and she had encouraging parents who did not give up on her, and she was determined to succeed--but she did succeed, and that is the thing to remember. Each person has his or her own learning personality. Each person can learn, but some in radically different ways, and many are looking for an excuse to quit trying. A teacher who will not give up on the student fights that student's urge to quit.
In her outstanding book, Overcoming Math Anxiety, Shelia Tobias points out that virtually every adult she interviewed had a vivid memory of a moment when a teacher made the interviewee believe that math was not possible for that person to learn. We must ensure that every student believes that we believe that student can learn important mathematics, somehow, sometime. One of my colleagues, Janet Webb, used to continually remind us that every parent sends us the best child they have and expects us to do our best to educate that child.
Another related thought is that the specific content of the course is less important then the intangible things that the student takes away. The impact of their time in your class has more to do with the way you treated them and the respect, enjoyment and excitement about mathematics you demonstrated than the actual algorithms and theorems they were tested on.

Failure is (almost always) overdetermined

This is a pessimistic entry, but it encapsulates some issues I've been gnawing on for a while.
Every year, I see a few students who just can't seem to pull it together, and eventually sink faster and faster.  But when I try to help them, I find that more often than not, the problems are deeper than I first thought.  These students aren't just unprepared, or unwilling to work.  Most often, the problem is a combination:

1. On a purely mathematical level, their skills are weak, and their conceptual understanding is even weaker.  They pull out a calculator to subtract 72 from 180, can't measure an obtuse angle with a protractor, and believe that when the sides of an angle are extended, the angle's measure increases.
2. More generally, they have trouble understanding complex tasks.  For example, one student, when asked to draw four quadrilaterals, measure their angles, and compute their angle sum, instead freehanded four rectangles.  The student didn't realize that shapes without straight lines have angles that are at best poorly-defined; didn't realize that the experiment is essentially pointless with rectangles anyway; and, when he got an angle sum greater than 360 degrees, didn't try to resolve the discrepancy.
3. On the level of practical reasoning, they have trouble connecting present behavior to future results, especially when positive results require sustained effort.  Because of their poor preparation, on the occasions when they do put in effort, they don't get good grades on tests or quizzes.  They quickly learn that these sporadic efforts don't get them the results they want, and then decide that those efforts were not worthwhile.  A student gets a 15% on a quiz; after lots of studying, the test score is a 60%, and instead of seeing a 300% gain, the student says "I studied and I still got a D, so why bother?"  Who would blame him?
4. They don't have the support of families who can help them with the mathematics, or even help them connect their efforts (or lack of efforts) to results in any concrete way.  These parents may not come to parent-teacher conferences, or if they do come, profess an inability to actually change what their children are doing.  I tend to believe them: as a parent myself, I've come to realize how hard it is to actually make anyone do anything.

While I've spent hours thinking about what causes these different deficiencies, fundamentally that thinking doesn't help those students.  But I haven't had much more success figuring out what to do about them.  They need lots and lots of scaffolding: in math and in academic skills generally.  They need our help connecting the dots from incremental efforts to incremental gains, until their gains become large enough to be visible to the naked eye.  And they need to trust the very teacher who--in their eyes--is asking the unreasonable and punishing them for failing to accomplish the apparently-impossible.  Finally, they need all those things on a sustained basis, for weeks and months, rather than days.  And then we wonder why they fail.

Personally, I'm able to help one or two of these kids per year, to the point where they're actually reasonably successful: passing classes, not eternally frustrated.  But I have at least half a dozen, and that's at a super-selective public high school.  I don't see any way to increase my own capacity, both because success requires so much time and energy per student, and because which students I "connect" with seems, at this point, a matter of luck more than anything else.

Fellow teachers: have I missed anything?  Any thoughts on a way to do this better?