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.

Presenting Proofs

What is the point of presenting a proof?

I teach an advanced—I like to call it “University-Level”—geometry class for 11^th and 12^th 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.

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.

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?

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.

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

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.