## Saturday, December 24, 2011

### "I think I discovered a theorem, Mr. Karafiol!"

This past week, one of my students stopped me on my first pass around the class with the exciting words "I think I discovered a theorem!  In a right triangle, the angles the median makes with the hypotenuse are twice the other angles!"  We had discussed the median to the hypotenuse in the previous class, but I'd never thought about the angles before.  So I started doing algebra in my head, confirming the relationship he was describing:

Before I could get very far, Vik showed me the elegant diagram below:
With the inscribed angle theorem, the proof is immediate.

Eureka!

## Friday, December 16, 2011

### Low Tech, High Impact

Although my department has a reputation for integrating all kinds of whiz-bang, high-tech devices into our classes, we've become increasingly excited by one decidedly low-tech tool:  large whiteboards.

The model is one I learned from our AP Physics teachers several years ago.  Contrary to my usual one-problem-at-a-time practice, I give students a handout with several problems, and instruct them to work on one that they find tough--so that they can build their mental muscles.  Then, each group gets a large whiteboard, and writes up a good solution.  Groups present their solutions and field questions from the rest of the class.

When I first heard about this technique, I thought it was a poor second-best to having every student work on every problem.  But I tried it, and now I've seen some substantial benefits
1. Students get to see a variety of problems worked out carefully, and ask questions in a nonthreatening setting.  And while they're not grappling with each individual problem, the more questions they ask and answer, the more they think through the problems presented--even ones they didn't work on.
2. Students work on communication skills.  In other instructional models, it's rare to have an entire student response of this length subjected to a full-class critique.
3. Students focus on one problem for enough time that they can think deeply about it, and transfer the ideas to other contexts.
The class above is my AP Calculus AB class, working on optimization problems.  After a brief introduction, groups chose between four:  minimizing the distance from a point on a line to a given point not on the line, minimizing the surface area of a can with fixed volume, maximizing the area of a rectangular pen with dividers, and the traditional "box problem" shown above.  In this case, I had two major goals:
1. Students needed to understand and be able to recreate the major steps in solving any calculus optimization problem: identify the constraints and the objective function, rewrite the objective function using a single variable, and apply calculus to find the maximum or minimum value.
2. Students needed exposure to standard problems, both in case they should encounter those very same problems later, and to provide foundational experiences from which they could build out to other, similar but perhaps more-complicated situations.
Whiteboarding was perfect: it was enough for kids to see and discuss the different setups--some of which they had encountered in precalculus--without having to struggle through creating each one themselves.  (Assessment today: they were pretty comfortable doing another distance minimization with a harder graph.  So it worked!)  And the in-depth thought about the process really paid off: again, in today's followup class, students seemed able to articulate the major steps and follow them with little confusion about sequence or "what comes next?" kinds of questions.  Finally, their writing and communication were much stronger than I would have anticipated:  students presenting today wrote clearly and identified major steps and results along the way, and other students used vocabulary correctly without hesitation.

So there you have it: for less than \$15 per board all-in, a mechanism to facilitate thoughtful collaboration, presentation, and discussion.

Postscript: if you're looking to acquire some, we get ours from whiteboardsusa.com--they ship quickly and have reasonable prices.

## Sunday, November 27, 2011

### Homework and more homework.

A year and a half ago, two other experienced teachers and I gave a workshop about teaching geometry. Many of the participants wanted to know how we handled homework. We wanted to have them do geometry activities but decided to allot some time to homework since that was their concern. The other presenters and I decided that the three of us would each get a minute to explain how we managed homework. We did not consult each other beforehand but the three of us were on the same page about virtually everything regarding teaching and learning geoemtry, so we assumed we would support each other on this issue.

It would be hard to find three more different responses than the three we gave. Our descriptions of how we handle homework make the Tea Party and the current administration look like they are in agreement on almost every issue. Each of us had arrived at very different conclusions based on our years of experience.

Having said that, I think I will contribute my thoughts on the subject. I tried a lot of things and all approaches had flaws. I eventually came to several conclusions about homework:
1. Homework is necessary. Students need time, on their own, to wrestle with mathematical ideas, to put them into perspective, connect them to other ideas and to practice. Some of this can happen in a class, but it is the rare students indeed who is able to make connections and internalize concepts in one class period.
2. It needs to be done on a regular basis, so students come to class knowing more than they did when they left class the day before. Otherswise there in not much progress happening.
3. It needs to involve reading and learning from reading. A students who can not read text, study an example, and determine meaning, is not an independent learner and will be forever limited in his or her ablilty to further advance his or her education.
4. Students need incentives to do homework in mathematics even though they know it is helpful and important. They are human and they are immature. Given the choice, they will usually do something that is more social or more to their own interests than doing the problems that the teacher requires that they do. Even my best students, when homework was not required, spent their time doing the Chem assignment or writing the history paper that was due tomorrow, instead of the math, if they could put it off for another day. "Never put off until tomorrow what you can just as well do the day after tomorrow." --Mark Twain
5. Homework is a learning experience, not an assessment activity. As such, it should not be graded. It should be evaluated in proportion to the effort that went into it, not in proportion to the number of correct answers.
6. Students need feedback with regard to the correctness of their work.
7. My time, both in and out of class, is better spent writing interesting problems, asking good questions, and writing appropriate assessments than it is spent grading homework.
So, after many iterations, I devised a plan that worked tolerably well for me. Here it is:
When students enter the class, they are provided with solutions to the homework that is due. Sometimes this was done by displaying them on the screen in the front of the room. sometimes it was done by placing copies of worked-out solutions in their mailboxes so they could retrieve them as they entered the room. (Every student had a mailbox so that I did not have to waste tome passing out papers.) Were I still teaching, I might resort to putting the solutions on my web page at an appropriate time. At any rate, students always had a chance to see what I thought was an appropriate way to solve the problems. These were more than answers; they were solutions.
I also gavemy students a problem to work on during the first few minutes of class. When class started, I walked around and looked at the homework to see if it looked appropriate. If so, I gave them 2 points. If not, I told them to finish it in a proper manner for tomorrow. If it was turned in the day after it was due, they got 1 point. Otherwise, 0 points.
As I walked around, my students had a chance to ask me about a particular problem or two that they were still confused about. I would make a judgement based on this information about the appropriateness of working a homework problem with the class. Usually we did not.
We were on a quarter system, about nine weeks per quarter. At the end of the quarter, any students who had done all of the homework on time (or at most missed one or had one or two late--that is, 2 points off of the maximum) had earned THE HOMEWORK BONUS! That meant the their lowest quarter test grade would be raised by a grade. I graded by letter grades, not points, so that meant that, for instance, a C became a B.
It worked for me with my students. Most students did their homework daily and were very concerned about not losing that bonus.
Oh, there was a dark side. If a student missed more than a week's work, five assignments, their highest test grade was lowered by a full grade for every one they missed after five.
I had wanted the homework score to be entirely positive: If you do this, it will advance your work by a bit. I found that after students lost the bonus, some decided it was no longer necessary to do any homework, so I had to include the negative part. It worked as well. Students do not like losing something they have already earned.
I am not sure how much of the homework I looked at had been copied. I never did find a good way to combat that problem, other than giving two or three unannounced quizzes every week. All of the quizzes counted as one test. They were short and I thought of them as formative more than summative, but they did count. And I did learn quite a bit about what my students had learned and what they had not yet learned, based on these quizzes.
I think this system worked very well for me. I hope there is a part of this you can use as well.

## Friday, November 11, 2011

### Of Babies and Bathwater

Thanks, John, for those kind words.  And yes, I plan on presenting more proofs.

These last two weeks have been busy with end-of-quarter grades, projects, etc.  (A forthcoming post will be "High-school level projects that involve actual mathematics," but I digress.)  This post is reflecting on the not-entirely-successful first iteration of our "no homework grade" policy: nightly homework assignments don't count towards students' grades, but frequent unannounced quizzes use representative homework problems as an incentive to complete assignments and an assessment of whether students know how to do the math.

The data are in and the following seem clear:
• Students are completing less homework.  How much less is unclear, because in the past, the marks on students' papers didn't always correspond to thoughtful effort expended on mathematical problems, but in my upper-level classes, it's typical for only about half the students to have attempted a significant number of problems, and in previous years, it was more like 80%.  In the past, I doubt that the typical homework assignment was copied/scribbled from answers by 30% of my students.  Lower-level classes are doing better on most days.
• Students perceive the policy as "you don't have to do homework," which seems like a misreading to me.  More accurate would be "homework isn't graded and factored into your overall grade."
• Students are doing less well on in-class tests than they did on last year's tests, although results vary by class.  Classes that are giving a lot of homework quizzes are finding less dropoff, but those same classes have younger students.
In his fabulous book, So Much Reform, So Little Change, Charles Payne dissects the etiology of failed reform efforts, discovering that a pervasive symptom is simply discarding changes that don't work, rather than examining and adjusting them.  In that light, our department is trying to revise our new homework strategy rather than revert to one we found problematic.  Some questions and possible answers:

1.  Why don't students do homework under the new policy, when they can see their grades dropping?  First, they may not see the connection between doing homework thoughtfully and actually getting better at math: it's telling that large numbers of students describe the policy (to parents, counselors, and teachers) as "you don't have to do homework" rather than "homework is important to learning, but you're only graded on what you learn, not what homework you do."  Second, they have a lot of other work--we're suffering from being the "first movers" in responding to Race to Nowhere.  When a student is up at midnight and choosing whether to do math or go to bed, the threat of a possible homework quiz is clearly not enough incentive to do a handful of math problems.

2.  What can we do to improve the situation without throwing everything out?  First of all, we can communicate better about what we think it takes to learn serious mathematics.  Maybe we should take a cue from the advertising folks, and make posters saying "192 minutes is not enough" or comparing time spent doing math to time spent doing other valuable activities?  What if we change the homework quiz policy so that homework quizzes are frequent but expected, for example every Monday and Friday?  What if we specifically identify which homework problem serves as the basis for each quiz question?

3.  What other ways are there to get students to do math outside of class without increasing incentives to cheat or skate by?  Webassign?

This is a tough time for us:  we want to do things differently and better, so we need to figure out how to adjust our course rather than simply u-turning.  Any ideas and suggestions are welcome; we'll take them in the comments.

## Sunday, October 23, 2011

### Presenting proofs 2 etc.

I applaud P.J. for working through the proof of Brahmagupta's formula with his advanced students, even though it was perhaps not in line with what we call "best practice," as many of his students may not have understood.

First and foremost, one of the most important things we can do for our students is to share our enthusiasm for mathematics, perhaps even mathematics they do not at this point completely understand. Knowing P.J. as I do, I'll bet his students were as excited by his excitement as they were by the beautiful proof he shared with them. I am sure he did not just stand at the board with his back to them writing. Probably didn't even use a board.

Secondly, it is critical that every student see some beautiful and significant mathematics. Part of being educated is understanding where knowledge comes from. How is it that we come to know the things we know? How did we come to think this way? How did anyone ever think of that?

Seeing a beautiful--if complex--proof is as important to an education as seeing a complicated, if not completely understood painting, perhaps a Picasso, or a DeKooning. Seeing a beautiful proof is as important as hearing music written by Schoenberg or played by Charlie Parker or Ornette Coleman, or attending a play by Beckett or Shakespeare. The proof or the painting or the music or the play may not seem beautiful the first several times we encounter it, but we are aware that creativity and beauty are present, and sometimes it takes us work and time to take it all in. When the complex, beautiful thing does make sense to us, we are changed. That is a part of education. Not the part that will necessarily make a lot of money, certainly not a part that will improve our test scores on a high-stakes test, but a part that will improve the quality of our lives.

Mathematics is overflowing with creative ideas and contributions from creative people. People like Cantor, Godel, Gauss, and Newton had some amazing ideas and made some remarkable contributions. Mathematics teachers have a responsibility to make our students aware of the inventive nature of mathematics, and it is easiest to do so if we share the problems and proofs we love. I used to teach a unit on non-Euclidean geometry just because I liked the ideas so much. A few of my students grasped what I was telling them and studied it further. I suspect others did much later in their lives, and I just have never heard.

I recall my Modern Algebra professor, Dr. Pilgrim, comparing a proof he had just showed us to a Gail Sayers run he has witnessed the Sunday before during a Bear's victory. His comment stayed with me and made me take a harder look at proofs. As it happens, I have made a list of my favorite mathematics. In the top ten, seven are proofs. I didn't always feel that way, but then I didn't always love mathematics the way I love it now. Mathematics has clearly made my life more interesting and the opportunity to share that with other people has been even better.

I rarely share those proofs with all of my students because I know that the timing has to be right in order to have in impact. Many students shut down as soon as they see a proof coming. It is such a shame that they are missing out on such enjoyment. But then a lot of people don't listen to jazz, classical music, go to art museums or serious plays either. I find all of those interesting and fulfilling. I am sure there are things I am missing that are equally important, but no one ever hooked me on them. Such is the way of the world. All I can do—and I must do it—is to try to share with others those things that bring me such joy and hope some of it will rub off, so we can share it together, and so they can keep it going.

This weekend, I attended the Illinois Council of Mathematics Teacher's annual conference. One session I attended was organized by Doug ORourke, a good friend of PJ's and of mine. Coincidentally, part of what he offered was an opportunity to investigate Hero's formula and Brahmagupta's formula in a new and different way. He proposed several versions of what could have been the formula and challenged us to find a way to explain why each variation of the formula could not be correct. The discussion was stimulating and resembled the discussion that comes before a proof and rarely happens in any math class. He then took Brahmagupta, had us enter it into a CAS calculator and then enter all sorts of numbers, to see what would happen. The overriding theme was Plausibility. By this he meant to look at special cases, impossible cases. What a brilliant way to spend an hour. Thanks Doug and I shall take this with me.

Friday evening was devoted to awards. The outstanding secondary School teacher award went to Natalie Jakucyn, truly a giant among us. In her acceptance speech, she thanked her high school math teacher, a nun who held her students to very high standards. Natalie recalled the day Sister put a long proof on the board. When she was done, the Sister wrote, "QED," went the back of the room, and said "Isn't that beautiful?" It was then that Natalie decided to become a math teacher!

So, thanks again, P.J., for taking the time out from the usual hands-on, engaging, student-discovery type of lesson that is typical of your classes and inspiring at least a few of your students by showing them a proof. Do it again. No one should graduate from high school without seeing Euclid's proof that there are infinitely many primes. It is certainly a proof that is in "The Book." and has inspired many a fledgling mathematician. And an important part of excellent teaching is inspiration.

## Sunday, October 16, 2011

### Presenting Proofs

What is the point of presenting a proof?

I teach an advanced—I like to call it “University-Level”—geometry class for 11th and 12th grade students who have finished Calculus, or who are gluttons for more punishment than a single math class a day affords.  And while much of their work is either a group discussion, or quasi-independent—they spent most of this week working on proving concurrences and collinearities they selected from the book 99 Points of Intersection—every so often I find myself doing what was done unto me in my college math courses:  presenting the proof of a theorem.

I like to think of myself as an engaging presenter, and today’s example—a version of the trigonometric proof of Brahmagupta’s Theorem from Zuming Feng and Titu Andreescu’s book 103 Trigonometry Problems from the Training of the USA IMO Team—was one of my best.  I put the trigonometric steps first to motivate the brutal algebra, and many of my students were able to stay the half-step-ahead that’s necessary to fully understand a proof.  (That is: they understood not only why a particular step was justified, but why it might be desirable.)  Cheers came at the end; one student stood up and shouted “That’s freakin’ awesome!”  A good class, right?

I’m not so sure.  I mean, it was fun, and the result extended the guided proof of Hero’s theorem they worked on yesterday.  And as a capstone to a unit on advanced theorems and proofs, Brahmagupta’s formula is hard to beat.  But what did students actually take away from the activity?  The fact is, I don’t even know.  They saw a new-to-them idea about trigonometry in cyclic quadrilaterals, that you can use the law of cosines on opposing angles to relate sides and angle measures.1 A few saw ahead to the fact that factoring complicated polynomials is almost always more useful than multiplying them out.  Frankly, there was no real assessment, so I can’t even say with confidence what any particular student took away, other than awe.  Really, at base, they watched me produce a proof.

And yet I think the experience was worth it.  The students got to see an argument too elegant and too long to generate on their own.  They saw a proof use four different ideas from Algebra, Geometry, and Trigonometry in combinations ordinary classes would avoid.  And they experienced mathematical beauty.

As teachers, we typically entangle two distinct tasks: “generate X” and “comprehend X”, even when only one of those tasks is actually useful.  For example, we spend as much or more time teaching language students to write correctly in the target language (not very useful) as we do teaching them to read it (absolutely critical if you want to understand the street signs, find out what a newspaper says, or experience literature).   But if one reason for this entanglement is lack of clarity about what’s important; another is that sheer comprehension is difficult to assess.  We teachers don’t have tricorder-like comprehend-o-meters:  the way we figure what students have learned is by getting them to produce something.  And it’s hard to create tasks that assess whether a student understands X without asking her to generate X.  Finally, from a philosophical perspective, educational positivists might ask what “comprehending X” really amounts to besides the capacity to generate X or something like it.

I’m not yet sure how (or even whether) I’ll get my students to show me what they learned today, whether they comprehend the proof or can generate a similar one or…. But I stand by my teaching.  I could trot out the old saws about preparing kids for college, in this case, college-style lecture classes, but I won’t:  the pervasiveness of terrible math teaching in higher education is a poor argument for bringing that same so-called pedagogy into the lives of high school students.

Instead, I would argue that the experience of following a complicated argument and seeing different ideas interlock is a valuable one, if only because it expands one’s conception of what is possible.  Being excited about math, and being in a group of people excited about math all at the same time, is also important, maybe just as important as anything else.  And the proof was really cool.

## Sunday, October 9, 2011

### Homework

Have you ever read any research about the effectivness of homework? I suspect not. I have not. Yet math homework is an integral part of many mathematics classes, and I suspect that the expectations, rewards, rules and circumstances surrounding this part of the learning experience are as diverse as anything we do. Teachers have strong feelings about homework. How many times have you heard it explained that a student "doesn't even do his homework, so how could he possibly succeed?" But I know that there are a lot of students who succeed without doing much, if any, homework. Many of my best students did as little homework as they could get away with. Many of my best students did all of the homework carefully and then did extra problems.
I think it is clear that many students will not do very much homework if there are no immediate consequences. Will this lapse stand in the way of their learning? Sometimes. After forty-two years of teaching, there is nothing I am as confused about as the value of homework. So I would like to bring up a few things about homework. I do not have answers to any of these questions and welcome logical arguments that will convince me, or anyone else, of a logical path.
Why do so many math teachers believe that students need to work on math homework every night?
I suspect the most common answer is that students need practice in order to master something. Do all students need the same amount of practice? I doubt it, but most homework is designed as if every student needed thirty minutes of practice every night, or every student needed twenty problems every night. I don't know how to insure that students work on math for thirty minutes every night. If I assign twenty routine problems, some students will finish those problems in five minutes, while others may take an hour. If I assign three challenging problems, many students will never make progress on any of them, and other students will give up after two minutes and will be angry because I did not show them how to do the problems. If I don't assign challenging problems, then students will never practice problem-solving skills and will never encounter interesting problems that will capture their imaginations and lead them to the excitement of mathematics. And the homework will always be tedious and boring.
How should I assess homework? I could create a list of important problems and tell students that they should do these problems in order to learn the material. I can then make solutions available somehow so they can check their work. Or perhaps I should collect all homework and grade the problems so I can give students feedback on their work. I could check to see the homework has been done and then not collect it. I could collect it and grade some of the problems. I could collect it on random days. I could give a homework quiz where they can use their homework paper to copy a specific problem that I would then grade. I could give a quiz containing a homework problem or two but not let them use their homework. There are lots of variations. I know some master teachers who use each of these systems, and others.
How do I know, in the end, if my method of assigning and assessing homework is valid?
I think if my students know the material, as assessed by some sort of legitimate final exam at the end of the course, that perhaps my method works. I am particularly positive if they do well in subsequent courses, and in subsequent work.
I can tell you that I spent considerable time trying to find a way to make homework work for me. I tried most of the things listed above in some form or another, and found one system that seemed to work for me and for my students. It did not over-burden me and seemed reasonable to my students, and so I used it for most of the classes I taught for the last twenty years of my career. A key aspect of my plan was that doing homework every night gave students a bonus at the end of the quarter. If a student had no more than one missing assignment at the end of a quarter (about 35 assignments), that student's lowest test-score was raised by a grade. I determined grades using letter grades on tasks, not points. If a student missed more than five homework assignmets in a quarter, that student's highest test score dropped by a grade and continued to drop a grade for every subsequent missed assignment. I did not collect homework, but I looked at it every day as they worked on the opener for the day. I gave frequent quizzes to inform myself and my students whether they were making progress. More to the point, I spent almost all of every class walking around listening to them work problems, so I knew whether they were learning or not. This system worked very well for my students and for me.
I think the best advice I have is: devise a plan that you are comfortable with and tweak it until you are happy with it. Then: share what works with others.

## Sunday, October 2, 2011

### Checking In...

When we last left our intrepid hero, he had advocated trying something new this year.  But without reflection, trying something new = taking another shot in the dark.  So here I present an update on two initiatives I'm trying this year.....

1.  In our geometry classes, we've agreed to stop giving traditional "points" grades, and instead give students grades based on our assessment of their proficiency in (for this semester, 19) predefined outcomes:  skills we expect them to master, or concepts we expect them to understand and apply.  Quiz questions, for example, now refer to outcomes ("1a") rather than points ("3pts").  We assess overall proficiency at each outcome based on a student's most recent work, not an average that includes failed attempts.  There have been some logistical glitches, due in part to our district-wide grading software.  And there are some things we won't do again:  give a quiz with five different outcomes on it, for example.  But I've noticed two positive effects:
• After giving a quiz, I'm much more aware of what kids know and don't know than I was in the past.  The simple act of recording, for each student, what his/her performance was on each assessed outcome, has helped me focus in on what I've successfully taught and what needs further teaching.
• My standards have gone up. Before, I'd sometimes give an answer full credit--or mostly-full credit--even when it wasn't exactly what I was hoping for, thinking "Well, is this issue really worth 1/2 of a letter grade?"  Now there's no averaging, and kids are, in principle, free to try again as many times as they need to.  The result is that I hold out for answers and explanations that are well-nigh perfect.
2.  In my AP Calculus class, we're still using traditional points, but we've decided not to count homework towards students' grades; instead, we give many short "homework quizzes" that give problems similar to the ones assigned.  I check in HW to see what students have done, but I don't count poor performance against them.  Again, two positive results:
• After an initial drop in HW effort, it's coming back up.  And students appear to be doing homework more mindfully:  they come in with six or seven problems done, saying "I knew how to do the rest" or "I figured I needed more practice on this."  Though I'm still seeing less homework than I did under the old check-for-completion system, I'm not sure I'm seeing less actual work:  before, many students rushed assignments, or copied answers from the back of the book (or their friends) just to have something to turn in.
• Because I'm quizzing more often, I have a better sense of what kids can actually do.  We're retooling the lessons this year anyway, but now, our conversations usually start with a discussion of the most recent HW quiz.  And grading is fast: I usually find I can grade two classes of two-question quizzes in under 30 minutes.
So that's what's up with me.  What are you trying in your classes?  How is it working out?

## Monday, September 12, 2011

### Give this a try this year, if you haven't already

Here is a proposal for the coming year that I hope you will think seriously about.
Invite someone other than an evaluator to observe one of your classes. Invite yourself to observe someone else's class. Depending on how it goes, do it again, with the same person or a different person. Be creative or not depending on how comfortable you are with this idea.

If you are a new teacher and suspect that you may have a few things to polish, ask a veteran teacher to visit your class and share any observations or thoughts. If you are a veteran teacher, if you think you have it down cold, ask a new teacher to observe your class. Tell the teacher you are looking for new ideas and that you are interested in a fresh look at what you are doing. In both cases, the teacher probably will have some unexpected comments, and in both cases the teacher will probably get a few ideas from the visit. Follow up by asking for a return visit to that teacher's classroom.

The teacher you invite need not be a math teacher. I learned some very interesting things one year when I visited two English teachers, an American history teacher, and a physics teacher. My goal that year was to learn how to get my students more involved in discussing what they were thinking about. I asked students and other teachers who in the school was particularly good at fostering class discussion, and I came up with four names. All four teachers did things I didn't expect; all four classes were thoroughly enjoyable; all four had totally different styles of teaching. One of the English teachers had a chair with wheels, and he scooted around the room and sat directly in front of the student who was speaking, as though it were a private conversation between him and the student. Another teacher had students sitting in rows while he stayed in the front of the room. The students seemed to be enthralled with the class, as was I.
I took ideas from these four teachers and incorporated them into my style. I learned a lot from these visits. I also became closer friends with all four teachers.

Visiting math classes also taught me approaches to students as well as approaches to mathematics. Interesting conversations followed about things like why it was important to distinguish between the function f and the value of the function at a number x, f(x).

Try it. Not only will you learn some things about teaching and learning, but you will share some things you know. Furthermore, you will open a dialogue about teaching and learning that will make your school a better place to be. Perhaps the idea will grow, and soon there will be an open door policy with teachers coming and going into each other's classes on a regular basis. At the very least, people will begin to understand that teaching is a very personal activity, and while there are best practices, there is no single best way to teach. One size does not fit all.

Have a great year.

## Tuesday, September 6, 2011

### Two things for (from) your first day...

Today was the first day back for my school, and for many others.  Leading up to it, and reflecting on the drive home, I had two thoughts about what makes for a good year teaching and learning.

First, micro: do actual math on the first day.  It's easy to get caught up in going over rules, procedures, grading standards, etc.  But why do we implement those systems in the first place?  For the most part, the point is to ensure that our students do, actually, learn some math.  How ironic to then spend the first day not doing math in order to communicate that math is the main thing!  Much better is to communicate that math is the main thing by actually spending a chunk of the period doing math!

(For the record, I doubt that going over lists of rules or procedures is even helpful. I myself have a personal capacity of learning about five rules, procedures, or standards at a time--tops--and maybe ten per day.  I doubt most students are much better.  Even at just two rules per period, the students are effectively done by lunchtime.  So there's no point in going through every detail; instead, provide a handout with the fine print, mention the big picture, and make sure to go over rules as (just before) they come up.)

This math doesn't have to be hard.  In Geometry, we go through a sequence of folds on a circle and talk about angles, symmetry, terminology, etc.  It's a great pre-assessment of what kids know coming in, and it allows us to "get through" a lot of vocabulary in a context where that vocabulary is meaningful.  In Calculus, we watched a video of a speedometer:

and a video of my dog's ability to do calculus while fetching a ball--which wouldn't upload, so here's the original "Dogs know Calculus" video.

Both videos generated lots of mathematical discussion, and allowed me to preview the main ideas of the course.  And the students were engaged--doing math.  How excited can a kid get about the details of a quiz makeup policy?

Second, macro: do something new.  Obviously (?) do NEW math with the class on the first day: spending several weeks "just reviewing", as far as I can tell, only communicates to kids that it doesn't really matter whether they learn it the first time around.  But mostly I'm talking about you, the teacher.  Try something new this year.  Change your grading system.  (My department did: regular nightly homework, as such, is no longer counted towards student grades; instead, we'll quiz more often, and assign more interesting out-of-class work that we can collect and grade thoughtfully.)  Or change your pedagogy:
• use videos (á la Dan Meyer), or
• incorporate more formative data into your lesson planning, or
• plan quizzes and tests collaboratively, or
• create tiered work on which students can select for themselves the appropriate level of challenge (aka Challenge by Choice), or
• design (or borrow) writing prompts that get students to think mathematically, or
• anything else you haven't done very well before, like in that cool presentation you heard at last spring's conference.
It doesn't have to be a complete overhaul.  But if you don't set your sights on growing at the start of the year, you're not going to be able to when you can't even see above the stack of exams you need to grade.  Pick something and try it.  And if it doesn't work so great, well, as we Chicagoans know all too well, there's always next year.

Good luck, and do some great math!  In the comments: what math did you do on the first day?

== pjk

## Sunday, August 21, 2011

### Back to School

It is about time for most active teachers to return to their classroom. While I am retired, the beginning of the school year still holds a special place in my routine. My schedule will change soon as tutoring requests pick up, workshops and math teacher meetings increase in regularity, and I start to get more questions about this and that.

Still, the first day is the most important day of the year, and I miss it. I always worked hard to have an interesting lesson that would surprise my students, excite them about the possibilities for the coming year, and give me a chance to meet them. I did not go over rules nor did I have non math getting to know you activities. We did math. I wanted to set the stage for them. It worked.

I made sure the first lesson and every one thereafter was based on what I consider to be my most important in sight about how mathematics is learned and what I think teachers should be doing in their classroom.

We must cover content. We must teach students how to logically approach problems that they have not seen before. If we can also get students interested in math, then we are successful.
It took about ten years for me to figure this out, but the these tasks end up being connected and the outcome often leads to excitement and interest. The idea is to have students solve problems that lead to an understanding of the concepts and content we want them to master. Here is how I was able to manage this.

I gave students a problem to work on. It was one problem and everyone was expected to work on it. Often it was related to the work they did yesterday and flowed into the concepts I wanted them to learn today. It was one problem, one they could all get started on but perhaps could not solve, hence the phrase, work on instead of solve.

As they worked on it, I walked around and listened and watched. I did not help anyone but I paid careful attention to their progress. I tried to give them a problem they has not seen before. They understood that their task was to figure out what to do.

I encouraged them to work alone at first and then discuss their thoughts with neighbors. Sometimes they did arrive at a solution, sometimes they did discover an important idea, and those were exciting moments for all of us. I knew exactly which students had figured it out because I was watching and listening while they did.

Sometimes they did not see the breakthrough idea and could not solve the problem. When I observed that they were about to go off task, I might give a hint, or I might have a student who was on they way suggest a strategy, or I might even demonstrate a procedure for solving the problem. However it worked, I had their attention because they had worked on the problem and wanted to know what they might have done to solve it. contrast this with the traditional method of showing students how to do a problem, when they haven't worked on it. They probably don't even recognize the key idea because they never had a vested interest in solving the problem in the first place. When I start to explain, they want me to explain because they have exhausted their resources.

I believe one of the worst habits teachers have acquired is the belief that they have to explain something before a students can try to solve it. Another bad habit is to give students a worksheet with several problems on it. This encourages students to write down answers rather than ponder the problem at hand. They lose focus and really don't want to discuss the problem as much as they want to finish the worksheet.

It takes a lot of work to get students used to this style. they would complain that I wasn't explaining things. They were frustrated. They were confused about what they were expected to do. But I am stubborn and persistent and eventually there would come a day when I would tell them it was time to discuss the problem, and some of them would say, "No, wait. We almost have it." Then I knew that I had converted them from passive note takers into active problem solvers. I would just stand there and smile and tell them I guessed they could have a few more minutes.

One good problem, let them work on it, discuss it, consider alternate solutions and common errors, then move on to the next problem.

If this is an advanced class, there might only be two or three good problems in a given class period. In a more ordinary class, there might be fifteen in a class period, each a challenge for the students, each within reach.

Not only did it work, but teaching this way is great fun, because there is another unexpected part. Sometimes students think of an approach that I had not considered and I learn to become a better problem solver from them.

Have a wonderful journey with your students.

## Sunday, August 14, 2011

### The Talent Equation

In the last few months, I've found myself getting into arguments with people about talent; in these arguments, I hear myself saying things like "I don't believe in talent."  Less tendentiously, I should say that I'm skeptical about talent, but even that isn't quite right.  So here's what I've come to believe.

First, a basic question: what is talent?  When someone gets an A on a test with little or no effort, we say either "the test was easy" or "she's really talented."  When someone puts years of effort into lab work -- or proving a theorem -- and comes out the other side with an amazingly mind-bending result, we say "Sure, she worked hard--but she was also brilliant to have seen ... "  In both cases, talent is what bridges the gap between the effort somone has put in and the results that come out.  We can express that thought in something like an equation:

Talent + Hard Work = Great Achievements.

As with any equation, we're tempted to apply algebra--in this case, subtracting the hard work from both sides.

Talent = Great Achievements - Hard Work.

Note what this equation doesn't say: that great things can be achieved without hard work, if only you have talent.  Rather, the equation reminds us that talent is explaining something:  why Jane can work in the lab for ten years and come up with only incremental results, while Jeanine works in a similar lab for ten years and revolutionizes her area of research.  In the classroom, when we say "Jimmy is more talented than Jeff" what we usually mean is something like "Jeff studies for a test and Jimmy doesn't, but they get the same grade."

Notice that the word "talent" (or phrase "more talented than") doesn't necessarily refer to any particular characteristic or set of characteristics, even in a single field such as mathematics.  Some kids have good number sense; some are good at seeing patterns; some are good at recalling any idea or problem they've seen once; some can gnaw on a problem for hours or days without getting overwhelmed; some have great metacognition about what they need to improve and how to improve it.  All of these are talents, and for any combination, I can think of a student I've taught.  I think teachers in particular toss this idea (as well as "smart" or "bright") around without even thinking carefully about which talents or strengths they are trying to refer to.  In my view, simply calling someone talented (or "smart") is intellectually lazy.

It's also useless, to the student--because it doesn't reinforce the actual behaviors that need reinforcing, or point to other areas in which he or she could improve--and to other students--who are led to believe that student X has some "special thing" that nobody else has--and to the teacher, whose job is to strengthen the student, not just praise him or her for what he or she already does well (right?).

Talking about "talent" or "smarts" is actually counterproductive because it reinforces the false notion that great achievements are primarily the result of talent, when in fact virtually every great achievement is the result of years, sometimes decades, of sustained hard work.  Genius/smarts/talent are only invoked to explain why Einstein (or Marie Curie, or Andrew Wiles, or...) achieved the great results they did when other people who worked just as hard got nowhere.  But because we can't point to one or even a few specific things that talent consists in, talk about "talent" doesn't help us identify people who will be successful ahead of time.  And in fact I think that we as math teachers are pretty bad, demonstrably bad, at identifying those people ahead of time, especially when those people are girls, or slightly-rambunctious boys, or nonwhite and nonasian.  (See Lee Stiff's article here.)

So when I say that I'm skeptical about--or I don't believe in--talent, what I'm saying is that talent-talk misses the point rather badly.  The real point is hard work, which for some people will result in great achievements; because hard work doesn't do that for everyone, we explain the disparity as a combination of luck, timing, and talent.  Any task worthy of the name requires concentration, effort, and persistence. Talent isn't the point; hard work is.  Going back to the second equation for a minute, we can rephrase the algebra as an analect: talent is the capacity to achieve great things through hard work.

Now that's something I'd hope we'd try to teach everyone.

## Sunday, August 7, 2011

### Hard work

I have no doubt that productive hard work reaps benefits. Malcolm Gladwell, in his book Outliers, claims ten thousand hours of work are necessary in order to become an expert. His examples range from The Beatles to Bill Gates and even talks about child prodigies like Mozart. I am convinced.

Many of those who put in extra time did so because they enjoyed the work. A friend of mine who is an excellent piano player has told me that when he was a child, he couldn't wait until he got home so he could play the piano for several hours. Over the years, I have noticed that many outstanding math students get very excited when telling me about one of their mathematical discoveries. They didn't do the work that led them to these discoveries because they thought it would make them better at doing the work as much as they did that work because they enjoyed doing the work. I see two consequences to this observation.

The first one is that teachers need to put more emphasis on interesting, challenging, intriguing problems; if they do, the mandated test scores will take care of themselves. Drills and practice just are not going to motivate more practice. They will drive children away from our subject and convince students that they are not good at math, and that they don't even want to be good at it because it is boring. (I just googled "boring math class" and got 46,000 hits.)

The second consequence is related to recent work by Keith Devlin about the potential impact of elctronic games on middle school education. I heard him speak at NCTM and read his book, Mathematics Education for a New Era: Video Games as a Medium for Learning. Among other things, he points out that people willingly play video games for hours on end, and that when they do, they get immediate feedback. They also expect to fail at what they are doing for quite awhile.

I can recall several open ended questions and puzzles I posed to my students that kept them thinking for years. Several students inquired when they were seniors if I would explain to the what was wrong with the "proof" I showed them three years before that all triangles are isosceles. Likewise there are several puzzle-like questions I have asked that students sometimes solved many months after the question was first posed. This sort of persistence only happens when students are surprised and puzzled by something they care about.

Two weeks ago I had the pleasure of being at a talk by Dan Meyer. He raised some of these same questions about getting students involved. In particular, he pointed out that we often break a problem down into so many little parts (scaffolding I think it is called) that the student isn't sure what the real point of the question is and never has to figure out how to solve it. Part of being interested is to figure out what needs to be done. We are so afraid that students will fail that we drag them through every step of the problem in hopes of keeping them from being frustrated. Being frustrated is part of the fun and a valuable part of the experience.

If students work hard because they want a good grade, or because they want to please a parent or teacher, or because they have bought into the belief that hard work will get them a good job, they will stop working as soon as they have made progress toward their goal. If students work hard because they find the work challenging and stimulating, they will keep working no matter what. I think the latter motive for doing work is worth cultivating.

## Thursday, July 28, 2011

### "To study for ten years at a cold window..."

The title of this entry is a line from a 900-year-old poem by the Chinese poet Liu Qi, often quoted by Chinese parents to children whose academic performance has been less-than-ideal.  I heard it at a meeting with officials from East China Normal University, who were explaining Chinese attitudes towards learning and education.  The point of the quote is simply that success doesn't come quickly or easily:  only sustained effort, quite possibly unpleasant, produces results.

How different from the U.S. perception of math education!  A Google search on "no good at math" produces over 724,000 results, and while some are actually echoes of the Chinese perspective (such as this thoughtful piece by my high school friend Jean Marie Linhart), the majority (at least on the first pages) seem to be people saying that they themselves are "no good at math."  Well, of course not!  The last thing most people who believe themselves "no good at math" seem to do is more math, which is pretty much the only way I know to get better at it.

I had a student several years ago who is still legendary for how much of our courses he figured out, on his own, before we could teach it to him.  His math contest results were phenomenal.  But what most people who didn't work closely with Alex didn't realize was that he did math for hours every week:  at least 10 or 15 more than required for class and math team, so about 12-17 more hours each week, at least, than most "ordinary" math students at my school.  Over four years, Alex spent at least 2400 more hours thinking and working about math than the typical math student at my school.  Of course he was good at it!  He'd have had to have--quite literally--some kind of learning defect if that much work hadn't paid off.

I'm not saying that there's no such thing as talent or giftedness in mathematics.  But I don't think focusing on talent or giftedness is useful, for two reasons:
1. We're not very good at discerning talent, unless it comes packaged in very recognizable shapes, sizes, and containers.  Quick at arithmetic, good at spotting patterns, doesn't need to show work--these are all recognizable, and so we spot those relatively easily, especially when the person is someone whom, because of our own biases, we are likely to think of as good at math (e.g. white/asian, male, etc.).  But: other mathematical talents, such as the ability to generalize and synthesize different results, to construct arguments containing multiple ideas, to sift through different approaches for the most fruitful one--these don't show up as early and are harder to see.  In fact, by the time a student is old enough to have encountered a mathematical situation for which these talents are valuable, he or she may already have been "tracked out" of the highest-level mathematics.  And again, we're even less likely to spot a student with these talents who's African-American or Hispanic, or not a boy.
2. Math is too hard for sheer talent to get most people very far.  Alex worked on his ideas and chased them down until they made sense.  Discovering and writing a good proof can take hours even when the problem is just a college-level exercise--much less a Putnam problem or (gasp!) an actual theorem.  The people who make it to the ends of such journeys aren't the ones who start out the fastest or the furthest ahead; they're the ones who don't give up because of their passion for the subject and their determination to keep going.  That's where the ten years comes in--whether it's in a study in Princeton, or by a cold window in Xiangsu province.
Marzano points out that there are four ways to explain success:  either the task was easy, or I was lucky, or I was talented, or I worked hard.  Only the fourth of these leads to behaviors that increase the chances of future success:  if my past successes were all based on ease, or luck, or talent, then when I encounter a genuine challenge, I've got nothing.  The Nurtureshock authors go even further, citing research that telling kids they're smart actually decreases future success.

Chinese parents don't tell their kids that they're smart; they tell them that they're smart enough to be successful if they work hard.  And that's a message I think all of our kids should hear.

== pjk

## Thursday, July 21, 2011

### Something I found interesting

I would like to suggest that teachers will find the current issue of The Atlantic (July-August) particularly interesting, appropriate, and thought-provoking.

The article on the cult of self-esteem makes a lot of sense to me; it ought to be brought to the attention of anyone who is involved in the life of children.

"The World's Schoolmaster" is also noteworthy.

Perhaps the most interesting article for me is "The Brain on Trial." Although the article focuses on criminal behavior, I couldn't help but think about P.J's earlier blog about the students he just doesn't reach. I also thought about many of the students I just couldn't understand.

As well, there is a short piece on the public employee as the new enemy that many of us will find interesting.

There are also lots of other fascinating articles that did not relate directly to mathematics education but were otherwise interesting; I hope you find it as thought provoking as I did.

## Saturday, July 16, 2011

### The impact of data

There is conclusive evidence that smoking causes cancer, yet people still smoke. The evidence from probability is overwhelming that it is foolish to purchase a lottery ticket, yet millions of tickets are sold. There are hundreds of other situations similar to this, so I don't see why P.J. is surprised that teachers do not pay much attention to the data--teachers are people too. (If you did not take the time to read the article indicated in the comment to P.J.'s last blog, I suggest you read it as well.

http://youarenotsosmart.com/2011/06/10/the-backfire-effect/

It is a very interesting take on why and how we resist data that tells us things that run counter to what we want to believe and on what happens when we encounter data that provides evidence for something we do not want to believe.)

First: let us get to the heart of what we do as teachers. We wish to provide our students tools so that they can make intelligent decisions as they live their lives. We want them to learn to think clearly about little things and big things, so they can contribute to society at large and so they can make good personal decisions regarding their well being and safety. I hope everyone agrees with that. As a result, we teach students how to collect, analyze and draw inferences from data. The assumption is that if they understand what is true, then they will make informed decisions. It appears that we are wrong. There is considerable evidence that people will ignore data and continue acting according to old habits. Even intelligent, well-informed mature people will ignore the research if it provides an "inconvenient truth." Why is this, what can be done about it, and what are the consequences for education?

It seems true that people trust their own experiences much more than they trust research, most likely because that--experience--is the way we learned to learn. Children try to walk: they fail; they try again until they get it. They learn that the way to learn is to try and to trust experience. Not only does the child learn to walk, talk, and generally function in the world, but the child learns a good way to learn, namely: try, and then trust personal experience.

Second: most classroom teaching is still approached in one way: there is stuff to be learned; it is the stuff that the curriculum wants students to be learned; we have an obligation to have our students learn it. The students usually do not experience the curriculum. That means they might not believe it, because it is not something they personally experienced. If we want students to believe data and live by it, then students need to experience it. That means that we need lessons, beginning at a very young age, in which students state their beliefs, students collect data, the data contradicts those beliefs, and then the students have a way to test their beliefs against the data. The best example I can think of has to do with physics and falling objects. I think of these because they confronted my beliefs when I was a child, and I still remember being proved wrong. There was an exhibit at the museum where one ball was dropped and another was projected from the same height at the same time, and both balls hit the ground at the same time. I had to see the demonstration several times before I could accept the new information about how gravity worked.

The problem is further complicated by the way our culture celebrates people who just have a "gut feeling" that they should try something, and when it works they are celebrated as heroes. A particular instance of this is found in the book Moneyball, where the baseball culture ridiculed and ignored evidence that ran counter to long standing beliefs about baseball, even though considerable amounts of money were at stake.

Even the stories of great scientists seem to celebrate hard work and good fortune rather than a careful analysis of data. Perhaps we need some more stories, plays, films and TV shows about the importance of careful analysis of data.

There may be no better reason to reform mathematics education than to restructure what we do so that our students learn to look at data and make intelligent decisions accordingly.

## Thursday, July 7, 2011

### Resistant to Data

Many bytes have been spilled (can we say that?) about how Americans in general, and even teachers, have trouble interpreting data and reasoning quantitatively.  But there's another syndrome that I think infects education, and that's data-resistance.  Data-resistance is when people--educators, parents, students, whatever--are so gripped by their preconceptions that they overlook or ignore data that suggest a different general picture from the one they are used to.

Here's an example.  Several years ago, Chicago Public Schools created a new, free, voluntary summer program for entering ninth graders.  The program lasted four weeks, and -- as implemented at my school -- was well thought-out, addressing academic, social, and emotional needs, with support in particular content areas (math and English) as well as general discussion and practice of academic strategies.  I helped work on it, and I was really excited about the program.  Everyone agreed it was a terrific success.

The following academic year, I followed up by gathering data.  For each student, a counselor and I collected first semester GPA and number of courses failed,and we looked for differences between the 100-or-so students who attended the program and the 100-or-so students who didn't.  We couldn't find any. We went back to the database and sorted students by F/R lunch status and by score on the qualifying entrance exam, and in no subgroup did students who attended the summer program do better -- either higher GPA, or fewer failures -- than students who skipped it.  Statistically, there was no difference between students who attended the program and students who didn't.  In face, we couldn't find a single academic measure on which there was a difference.

As we started discussing whether and how to implement a similar program the following summer, I took my data back to the planning team and said "Look, we all thought this program was terrific, but in fact it doesn't seem to have any impact."  The response was uniform:  it's a great program; we should do it again; we shouldn't make major changes.

That's what I call resistance to data.

Now it's possible that there are other ways in which the program was helpful.  Maybe students who attended the program found the first weeks of school easier, or more enjoyable; maybe they got more involved in extracurriculars.  But nobody suggested that there were data supporting any of these claims, and -- one could argue -- if these outcomes don't show any ultimate academic impact, there might be easier and less-expensive ways to attain them.  Here's another hypothesis: maybe having half the class attend inculcated a culture that "infected" the whole class and boosted everyone's performance, sort of like how "herd immunity" can those who don't get vaccinated.  But nobody suggested that, either; in fact, the consensus at the meeting was that it was important to get more kids to attend.  And we have.

To be fair, kids love the program, and for one group of our students, we've made it extremely helpful: kids who haven't finished a full year of Algebra I can do so and start in Geometry--an option we created before Freshman Connection, but incorporated into the program when it began.  Nobody would say that Freshman Connection hurts anyone, and everyone who attends is glad they did.  But I'm still skeptical that it actually improves academic or even socio-emotional outcomes for our students.

I'd say that too often in education we make decisions like the way we made these:  without gathering data, or in the face of data that contradict our intuitions and preconceptions.  The worst is when we teach based on what worked for us as individuals.  Teachers lecture for 45 minutes with a little guided practice, then assign 1-47 odds for homework, despite mountains of research about what kinds of tasks, in what quantities, make for effective independent work.  In many cases, teachers have heard of this research or been told about it, but at some level they don't believe it:  lecture + practice + 1-47 odds worked for them when they were kids.  Doing "what worked for me" is particularly harmful in math, where we tend to forget that those of us who are comfortable with math now are really "math survivors".  Imagine if we taught kids to swim by dropping 100 kids at a time into a shark-infested pool; the two or three who made it out would later replicate that method, saying "it worked for me".

Even when the practices are not demonstrably harmful, using "what worked for me" or "what makes sense" as a metric ignores the fact that today's kids are growing up in environments that visibly and palpably different from the ones we grew up in.  Recently, I posted this article (on Facebook) about how schools in Indiana will be giving up cursive instruction (yay!) but teaching keyboarding instead; I asked why bother teaching keyboarding either.  An astonishing number of my friends commented that typing class had been incredibly useful for them, and how would kids learn to type well otherwise?  My response:  yes, but that was when typing was a specialized activity that you only did when preparing final drafts of papers; kids today type all the time and get immediate feedback about the quality of their typing; my students have learned to text blind quickly and accurately, without any formal instruction.  And ... my friends repeated their arguments: it's important to learn to type quickly and accurately, and they only learned to type quickly and accurately in typing class.

I'm not saying that we should totally jettison personal experience and common sense when we teach.  I'm just saying that we forget that both of those are anecdotal, and what memory and common sense tell us about our own experiences is not necessarily relevant to the population-at-large 30 years later.  It took me a little over a year of teaching to realize that, although math homework was mostly irrelevant to my own learning of high school mathematics, it could really help my students--so I shouldn't make it optional.  And it's that kind of skepticism, and openness to data, that we all could adopt more often.

## Sunday, July 3, 2011

### What do we want students to do?

As I finished up last week's post, I was wondering about different kinds of tasks we could ask students to do besides solve traditional or not-so-traditional problems (numerical, proofs, whatever).   Of course, few math tests consist primarily of "problem solving" in the narrower, Polya-type sense of "attacking novel mathematical situations or questions"; most of the time, tests and quizzes ask students to "solve problems" in a somewhat-more-general sense that includes working out answers to exercises similar to ones done in class or on homework.  But in general, prompts look something like
Find all values of w such that...
Draw the graph of a function with the following properties ...
At what time will ...
If P, Q, R, and S are points on quadrilateral ABCD such that ... prove that ...
Find a polynomial with the following properties, or prove that no such polynomial exists.
Compute the probability that ...
As I was thinking about assessing mathematical knowledge, I began to wonder why virtually all test and quiz questions are of these types.  It's not that hard to dream up different types of prompts.  For example:
Write an explanation of how to determine the degree of a polynomial based on its graph, including any uncertainty in the final answer.
Place isosceles trapezoids in the quadrilateral hierarchy drawn below, and explain your choice.
Explain the following statement: Given enough trials, any event with probability greater than 0, no matter how small, eventually occurs.
An ironing board has two legs of fixed length, one of which is free to slide along the length of the board, but both legs are attached at their midpoints as shown in the diagram below.  Explain why this setup guarantees that the board is parallel to the floor.
What is the derivative of a function?  What is its importance and how is it computed?  What information about a function can you get by examining values of its derivative?  Explain using symbolic, numeric, and graphical examples, and including one example of motion.
All of these prompts demand a fair amount of mathematical knowledge, and the ability to construct coherent arguments.  And--unlike most traditional math prompts--they get at skills and concepts in a way that might actually be useful to someone not involved in mathematics at a professional level.  But -- okay, I'm contradicting myself slightly here -- coming up with these prompts is not trivial: it takes creativity, insight into what students will be able to do, time on the exam for students to work on them (so, not the 29th question on a 30-question hour exam), and time for thoughtful grading and assessment.  Yet they are worthwhile:  the last question is part of the best test I ever gave, the final exam for a two-week calculus course I taught for elementary school teachers (as part of the SESAME program at the University of Chicago), and what made it great was that it simultaneously allowed every participant to demonstrate at least some knowledge, while differentiating between teachers who knew a lot and teachers who knew only a little.

So here's the question.  Why we almost-entirely-confine ourselves to the first kind of "problem"?  Is it because
• We think that the most important thing for students to do is to work out these kinds of questions, or
• We want students to do other kinds of reasoning, but we don't know how to assess it, or
• [my sinking gut feeling] We haven't really thought carefully about what we might want students to be able to do besides "solve problems," or whether this is the only (or even most important) set of skills and outcomes for students?
Thoughts, please, in the comments.  And guys, I know it's summer, but Blogger tells me we have substantially more followers than I can count on one hand.  So please do comment.

== pjk