On teaching challenging students

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

Failure is (almost always) overdetermined

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

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

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

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

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

Influencing students

This past weekend I had a rather unique experience, so I am going to write about that instead of trying to connect with P.J,’s last entry. There may be some overlap, but it is purely conicidental.

I intend to reflect on an amazing experience I had yesterday. One of my former students, Elliot Damashek, was honored by Stanford University for being in the top 5% of the engineering students in this year’s graduating class. All such students are awarded the Frederick Terman Award. Each of the twenty-one recipients invited a Stanford Professor and a high school teacher to a luncheon and awards ceremony at Stanford. Yes, that means they flew people from all over the world, including Bombay and Malaysia, to Palo Alto for this award. I was honored to be selected by Elliot as a teacher who influenced him during his high school years.

First, it is rare and remarkable that a University is willing to spend this much time and effort, as well as money, to honor secondary school teachers. All of us were thankful to Stanford for doing this, and we all returned home proud of our work and eager to continue doing what we can to help students reach their potential.

All sixty three participants said a few words. The professors introduced the students, and uniformly praised their work ethic and intelligence. But after those two qualities were extolled, each student emerged as an individual, with strengths and perhaps some weaknesses that were unique to them. And each of them had accomplished amazing things, beyond grades earned.

Then the students introduced the teachers, and it became clear that many of the students believed that they were successful at Stanford because of the influence this particular teacher had on their lives. Rarely was that influence focused on content taught. It was always about the concern the teacher had shown for students, and for the speaker in particular. The teachers were not all math and science teachers, either. There were philosophy, government, history, Latin teachers—and a shop teacher—among those mentioned. One woman was a middle school math teacher. All of the stories made me proud to be a teacher.

I realized that sometimes society thinks that the “gifted” will take care of themselves. Anyone can be a successful teacher to “smart, motivated kids”. I agree it is a different task to reach these students than to reach the under-motivated, recalcitrant students, but it is no less important and not any easier to reach the high-acheivers; it is just different.

These twenty-one students are going to change the world one way or another. They are clever, committed, and motivated. Most of them are planning on working in areas that need attention. Some have already created products that meet the needs of society. They will cure diseases, create healthier food, find ways to protect the environment, work to create peaceful, safe environments.

At least: that is what I heard those twenty-one students talking about yesterday. It was exhilarating to meet these young men and women who are dedicated to using their talents to make the world a better place. And I am glad that I had the chance to be part of the life of one of those students.

The work teachers do with each and every student is important, each in its own way. Our influence is not so much on the specific content we teach, but on the attitudes we pass along to our students, the way we treat them and the way they see us treat everyone else, and the things they know that we value. They know our values because they watch us interact with their classmates over a long period of time. And what we do makes a big difference. Perhaps more than we ever know. These students talked about the character of their teachers, not how well organized they were, or how carefully planned their lessons were.

St. Jude of the Classroom

"O most holy apostle, Saint Jude, faithful servant and friend of Jesus, the Church honoureth and invoketh thee universally, as the patron of hopeless cases, and of things almost despaired of. ... Make use, I implore thee, of that particular privilege accorded to thee, to bring visible and speedy help where help was almost despaired of."

An integral part of my experience as a teacher has been struggling with apparently-hopeless cases, and by "cases" I mean students.  These struggles are before, after, and during school: chasing down kids for homework and time to catch up on math; sitting side-by-side puzzling out some theorem; talking, cajoling, entreating them to think ahead, to set priorities besides "what seems fun right now," and to follow up those priorities with action; listening to--or trying to tease out--the details of why the most recent boneheaded decision seemed reasonable at the time.  Some of these struggles have been successful, by some definitions of success, but--without resorting to a detailed tally--I suspect the successes are outnumbered. But these students matter: in fact, when I think about the students who have made the biggest impression on me over the years, about whom I find myself wondering "where is he now?", those "hopeless" students are in the majority.  I suspect that, among long-term teachers, I am not alone in feeling this way.

Today I'm wondering not about whether, but about why, how, and "so what?"  And, just to be clear, the hopeless cases I'm describing are not just "hopeless by the standards of gifted students" or "hopeless by prep school standards."  I'm talking about actual orphans, victims of acrimonious divorces, children who bounce from one underserved home and neighborhood to another,.... Their behaviors reflect the broken circumstances in which they find themselves:  whether because they're unwilling, unable, or uninformed, they don't work outside of class, don't successfully advocate for themselves, don't set and follow through on short-term or long-term goals, ... They are, in short, hard cases.

Why?  The reasons why I'm drawn to these students are the reasons I'm a teacher:

1. I want to improve the lives of young people.
2. I don't believe that teachers should be in the business of giving up on kids or writing them off.  I think the evidence is that we do a poor job of identifying who can and can't "make it", except for the self-fulfilling nature of those very prophecies.
3. At some level, it's hard to accept that there are things I can't do.  Taking on a challenging student is to take on a challenge.  And yes, this shades into hubris: it's hard to believe there's a student I can't reach.
4. I want to be "that teacher": when that kid gets a Grammy, or wins a Pulitzer, or snags a Fields Medal, or just looks back on his high school years from the perspective of a reasonably well-managed and -adjusted adulthood, I want him or her to say "It was Mr. K who turned me around."

So an important part of "why?" is this: I seek out this role for reasons that have as much or more to do with me as they have to do with actually helping the student.  But accepting my own complicity at a personal level doesn't address broader, social concerns.

1. Our society doesn't believe in giving up on children.
2. Our society shouldn't believe in giving up on children. The vast majority of circumstances that make these students lives' difficult are also totally outside their control.  How can we tell a high school student that, after failing to provide him with a decent home, a decent wage for his parents, a decent neighborhood to live in, and decent elementary schools, we're writing him off as a doomed high school student?
3. Some--many--kids in apparently-hopeless circumstances do in fact turn out okay, which benefits everyone. So increasing that number is an important priority.

Still, these decisions have costs:

1. At the moment, I'm giving as much time to teaching as I have.  So the situation is genuinely zero-sum: if I spend more time with one student, that's less time I have to spend with another, or on planning lessons that will impact the entire class.
2. Similarly, at the societal level, unless we're willing to vastly increase the resources we allocate to public education, increasing the allocation for the weakest students means, at least in the short term, decreasing the allocation for other students.
3. Both as individuals and as a group, many of these students are capable of succeeding.  In fact, part of my unwillingness to write them off is that I don't trust my ability--or anyone else's, really--to make good, non-self-fulfilling predictions about who will succeed. As a society, we can't afford to squander talent; as an individual, I hate to see it go to waste.

This has already been a long post, so I'm going to stop here and ask readers for feedback.  Who are the students to whom you feel drawn?  Why do you feel that way?  And are you mostly successful?

More--the "how" and "so what?"--on this topic later.  Post your thoughts in the comments!

A VIP ( Very Important Problem) ( for me anyway)

Consider a line of lockers, numbered consecutively, starting from 1 and continuing without bound.
(As you might have already noticed, this is not a real world problem)
Person 1 walks through and, starting with locker 1, makes sure all the lockers are closed.
Person 2 walks through, and, starting with locker 2, opens every other locker.
Person 3 walks through, starts at locker 3, and changes every third locker, opening the closed ones and closing the open ones.
Person 4 starts with locker 4 and changes the state of every fourth locker.
This continues in like manner, with person n, starting at locker n, and changing every nth locker.
(You might have noticed that this problem is not intended to practice necessary algebra skills)
This problem changed my life. I guess P.J. would call it a puzzle. I am sure most of the readers of this have heard the problem.
Oh, this is not exactly the puzzle that changed my life. That one also included a question. I think it is a better problem if the student has to decide what would be a good question. When I give this problem to students, they ask what the question is. My response is, "Oh no, did I forget to ask a question again? What do you think would be a good question? "
Someone will say something that is roughly equivalent to "Which lockers are open eventually?"
That is the question I was asked. However sometimes student's questions are better than the one I had in mind. Isn't it true that an important aspect of mathematics is deciding what questions to ask? Do we give our students a chance to do this?
But, as is my lifelong habit, I digress.
How did this problem change my life?
It was given to me in a course named "problem solving". I had been teaching for several years. I thought I was making progress but now know I had a lot to learn about how children learn. I now know I still do have a lot to learn about how children learn. I found the problem challenging and interesting, even though it has no basis in reality. I worked on it for a while, and an amazing thing happened. Once I had an answer, I became curious about why the answer ended up being so nice. I started thinking about it and without much effort I had a proof of a theorem about factors of integers, and I understood and therefore learned some interesting mathematics I did not know before.
I realized that what made this problem good was that I found it interesting, and that it created a situation where solving it helped me to understand something. It was not just an answer, but a revelation, that lead to a new understanding.
Was it important? Well, I have since used it to solve other problems, so I would say, yes.
But most importantly, I learned some significant mathematics by doing a problem. I was proud, and happy , and wanted to find other problems to work on. And I got this crazy idea that perhaps I could arrange for this sort of thing to happen in my classes for my students.
And so, for the next thirty five years, I tried to write problems that would lead my students to figuring out something for themselves.
It took a while. I tried and rejected the method of having them work a few examples in the hope they would notice the pattern I wanted them to discover. For one thing, it seemed phony but for another, they didn't learn any mathematics from working on the problem because they didn't really figure out any mathematics, they simply arrived at what I wanted them to arrive at.
I also tried worksheets. That was a total failure. For one thing, once give a worksheet, the objective of a student is to finish the worksheet. If another student has a question or observation, that interferes with the task at hand, which is to finish the worksheet. I also learned that students did not understand that they were supposed to learn something from finishing the worksheet. They were just supposed to finish it.
So, I eventually fell into a groove that I was happy with. One problem at a time. Everyone works on that problem. If you finish, compare your result with others. If you all agree, ask the next question. What if we changed the problem this way? Is there a generalization? Is there a special case? Meanwhile I walk around, rapidly, listening and looking at what is happening, waiting for the teachable moment, that moment when the entire class is ready to share their thoughts on the problem. I usually also share my thoughts at this point, especially with regard to connections and what might happen later.
Then I give them another problem to work on.
And learning happens, the students are actively engaged, and frequently I learn something I didn't know. I also learn a lot about how my students work.
Teaching by giving students a good problem is different than other ways of teaching. The teacher has to be ready for anything that comes up, and must have a plan if nothing comes up. The teacher must be willing to give students time to think of things , time to make up their mind, and time for ambiguity. The teacher must be prepared for long periods of time where everyone is wrong, and be willing to let that happen. It takes practice and a belief that eventually truth will win and incorrect reasoning will be exposed. It means giving up the role of the person who is the authority about truth. But in the end, it worked better for me than any alternative.

And, the best part is, I have a great time doing it.

And it all started with the famous locker problem, a puzzle that taught me how to become a better teacher.

Words on/Problems with Word Problems

Dan Meyer's and Christopher Danielson's excellent blogs have recently been weaving a thread about "word problems"; Danielson quotes one and writes

The apartments in Vincent’s apartment house are numbered consecutively on each floor. The sum of his number and his next-door neighbor’s number is 2409. What are the two numbers?

This is a classic word problem of the sort that I hope to eradicate from mathematics instruction (at least from my own).

And while generally I agree with Danielson that many students find the most interesting problems to be ones drawn from their own real experience--not these contrived, puzzle-y types of problems--I think he's wrong in saying that problems like Vincent's have no place in the curriculum.

Why might they be important?

Consider the following situation we considered in geometry yesterday:

 

Students quickly identified that all three triangles are similar to each other, but to justify the statement, they needed to "chase angles" around the figure.  So we labeled the measure of angle A as x.  What's the measure of angle ABF?

The algebra here is not hard, but it requires translating an idea--two angle measures add to 90 degrees--into an equation or expression.  There are lots of such situations in geometry:

 

To find the exterior angle sum of the pentagon, one of the easiest ways starts by writing:

      a' + a = b' + b = c' + c = d ' + d = e' + e = 180 degrees.

Or how about this early optimization problem: what's the largest area for a rectangle of a given perimeter, say 60?

So while I think it's true that many students find problems like Vincent's uninteresting, the underlying skill--and here, there's really no issue of transfer--is crucial to setting up and doing interesting math on other problems.  I learned this the hard way this week: we spent twenty minutes chasing angles I had thought we would find in just a few.

So what can we do?  Two ideas:

First, we can separate "problems involving translating words into symbols" from "application" and "real-life" problems.  And when we pose problems in the first category, we can be honest about the fact that we're trying to develop a skill students will use in attacking other math problems, not preparing them to do something most people do everyday.

But second, I think we can do a better job of honoring the category of puzzles.  My experience is that kids like puzzles when they know they're puzzles.  For example:

Here's a puzzle:  a 25-foot ladder is propped against a wall 7 feet away.  The bottom of the latter slips back 8 feet.  How far down does the top slip?

The Jain mathematician Mahāvīra poses the following problem:  There were 63 equal piles of plantain fruit put together and 7 single fruits. They were divided evenly among 23 travelers.  How many fruit were there altogether?

In England, there's a tradition of trying to ring all possible permutations (or "changes")  of a church's bells, without duplicating any permutations.  Unfortunately, because of each bell's mass and angular momentum, the only way to change the order is to switch two successive bells: so ABCDE can go to ABDCE or BACDE but not ACEBD.  Is it possible to ring all the changes on five bells without duplicating any?

Indeed, puzzles are an important mathematical tradition; as teachers of mathematics, it's up to us to help students appreciate this vital mathematical aesthetic.  (Great ideas about our role as teachers of mathematical aesthetics are in Natalie Sinclair's paper here.) We teach that aesthetic by including puzzles, riddles, and other mathematical games in our curriculum--not just what students "need", in some sense, "to use". [Not Danielson's quote or exact idea, to be fair!]

I'll go even further: I think that it would be great if we could develop our students' mathematical eyes so that they can notice and appreciate cool number patterns and symmetries just as much as we'd like them to notice and appreciate symmetries in geometric forms and patterns.  That is: our goal is not just that students can do math or that they know math, but that, at some level, they appreciate it.

A last note on puzzles and aesthetics.  Vincent's problem is truly a terrible problem.  But what makes it bad is not that it's contrived or abstract.  What makes it bad is that it presents a "puzzle" with nothing interesting in it to notice.  Who would possibly care about a sum of 2479?  There's no symmetry or pattern.  It's like a painting that is composed of two discrete forms, but with no balance, interesting details.  Even a tiny tweak makes it better--like using the year, or an interesting number, like this:

The apartments in Vincent’s apartment house are numbered consecutively on each floor. The sum of his number and his next-door neighbor’s number is 2011. What are the two numbers?

or this

Don and Leo are going to visit their friend Vincent, who lives in an apartment building where units are numbered consecutively on each floor.  As a puzzle, Don writes down the sum of Vincent's apartment number and his neighbor's, using a scrap of paper he finds in his pocket.  When Leo gets the paper, he asks Don "Which way should I read this?" but then says "Wait, it doesn't matter!" If Don's number has four digits, what's the smallest possible apartment number Vincent could have? 

 And then you can ask interesting, mathematical questions:

Assuming you don't run out of apartments, is every four-digit number a possible sum of adjacent apartment numbers?  [Is this puzzle possible every year?]

What is the sum of the numbers of the apartments on Vincent's and his neighbor's other sides?

A visitor notices that each apartment is directly across the hall from another apartment, and that each pair of across-the-hall apartments has the same sum.  How would you number the apartments so that this happens?

Challenge to readers: in the comments, add your own "apartment problems"!

Notes on Notes

“When I am talking, the students are either ignoring me or listening attentively and trying to take notes, but in neither case are they actually doing mathematics, which is the one activity that I can guarantee will produce learning gains.”

Point 3 on the previous blog (above) registered something I think is important enough for its own discussion. Several years ago I went to a presentation about emerging technology, specifically the tablet P.C., and how it might influence teaching. I thought I was going to a presentation on how the tablet improved learning, but that was not the case. I am a huge fan of the tablet as a way of helping students learn, but the emphasis of this presentation was how helpful and efficient the tablet is to “give notes.” The demonstration proceeded to show how the teacher could have all of the notes for the class prepared on power point slides and how easy it was to annotate them so the students could copy them into their notebooks. Huh?

I firmly believe that we should strive to use every minute of every class for learning useful things. However, as P.J. pointed out, attentive students who are taking notes are not working on problems. So what are they learning? They are probably not thinking about much other than: did I copy this correctly? It is clear to me: the process of copying notes is not a learning experience.

Perhaps there are other reasons why it is important to take notes?

I think there is a belief that if students have good notes, they will have something to refer to when they forget or when they need to study. I propose that if their textbook does not serve that purpose, it is not a good textbook and never should have been selected in the first place. Get rid of it as soon as possible! Replace it with one that does provide an organized summary of what students are studying, with examples and sample problems.

But perhaps you are stuck with a terrible text. Or perhaps you have topics that you think are important but are not in the text. Possibly you have an alternative approach to the topic that you think is better. (I hope both of the last two situations do occur frequently in your class.) I would suggest that it might even happen that a student has an alternative approach that is not in the book and that you had not thought of, and that you have become aware of this approach by walking around and paying attention to how students are solving the problem you gave them to work. If so, doesn’t it make sense to carefully write this up and either post it on your web page for all to read, or print copies and pass them out for all to have.

This approach will provide students with the much-needed supplement, and that supplement will be carefully written, error-free, and well-organized, attributes uncharacteristic of in-class notes. This approach will also encourage students to read mathematics, an important step towards helping them become independent learners. These real or virtual documents will be legible, something my hand-written notes rarely are. I think some would call these documents, “class notes.” I prefer to call them, “supplements to the text.”

There would not be many of them, as we want students to learn to work with their text. But most importantly, it eliminates all the time that is wasted when students think they should be writing down what the teacher writes.

I take notes at every meeting I attend. They are notes, not transcripts of what the speaker has just said, or written. They are thoughts that occur to me as I am learning, thoughts I don’t want to forget. They are nuggets that someone said that I want to remember. My notes from a class I took from Dan Teague include, “There are three kinds of mathematicians, those who can count and those who can not.” I laughed and realized that I wanted to remember it, so I wrote it down. We were working on an amazing problem, and I knew my mind would immediately forget his quip as it focused attention on the problem. I put it in my notes and have used it hundreds of times since.

I think we need to distinguish between notes and transcripts. I think many do not make that distinction and as a result, their students waste a lot of class time. I understand there are even teachers who waste their own time collecting and grading their student’s notes instead of thinking up interesting problems to ask their students to work on.