Wednesday, August 6, 2014

Externalizing what you worked so hard to internalize

I've been writing quite a bit on the work of George Pólya, particularly focusing on what strike me as some common misconceptions and overlooked aspects. I started with his emphasis on building self-confidence and making the experience pleasant since these things are so much at odds with the tough-talk rhetoric that has become so popular in education circles over the last few years.

Pólya also spends a great deal of time talking about the importance of "getting inside the student's head," but compared to the parts about self-confidence, I think the point here is less emotional and more cognitive. One of the main ideas of the book How to Solve It is that people who work with mathematics professionally have almost invariably mastered and internalized a number of useful problem-solving tools. Unfortunately, by internalizing these tools we have also in effect hidden them from our students.

When faced with a problem, we quickly and in many cases unconsciously run through a number of techniques that we have found over the years to be helpful. We examine the problem, determine the unknown, compare the problem to those we've encountered in the past, perhaps draw a mental picture and run through any number of similar steps before deciding on the proper strategy. To the student this gives an unrealistically linear appearance to the process, as if "let u equal X cubed minus 8 and then factor by substitution" was the first thing that popped into our mind .

Polya's point, and I think it's a profound one, is that to explain a process to someone who is unfamiliar with it, you have to be self-aware enough to explain the whole process, not just the parts you are still conscious of.

Here's a great bad example from one of our recent whipping boys.

For the math people out there, this is not a bad explanation, but we aren't the target audience. For the target audience, this just terrible, particularly for a video. If you're working live, you can read the room. With a recorded medium, you have to anticipate the room.

Think of the world as being divided into two groups: people who can do this problem without help; and people who can't. The person doing this video is obviously in the first group, For the first group, setting up equations is second nature. It's obvious me, having been through this from every vantage, that the instructor took a few seconds to understand the problem and mentally outline the steps for solving problem, then he hit record and started filling that outline in.

The trouble is, for the overwhelming majority of the people in the second group, getting that outline is the part they were having trouble with. Very seldom do you see students who can effortlessly set up the equations for a word problem but who then get stuck on the basic algebra.

The students who need help look at this problem and see a lot of possible variables and equations. Maybe X should be the amount spent on the expensive paper. Maybe we should set up an equation to show the difference between the two amounts, cheap and expensive. You and I know these are dead ends because we've seen this show before. The key to explaining this type of problem is to imagine what it would look like if you were seeing it for the first time.

George Pólya was, of course, big on word problems. Here's a relevant passage from How to Solve It:

Monday, August 4, 2014

Pedagogical MacGuffins

What's a MacGuffin? A MacGuffin is the key or stolen diamonds or secret code or NOC list that the characters  desperately pursue. Audiences, pretty much by definition don't care about MacGuffins, but they do enjoy watching characters pursue them. Sometimes the audience isn't even clear on what the MacGuffin is.

Do you know what a NOC list is?

A pedagogical MacGuffin is a type of problem we pretend to care about even though we really don't. Like its fictional counterpart, what's important with a mathematical MacGuffin is not the thing but the pursuit.

The classic example is factoring polynomials. A standard part of most algebra classes is to learn how to take a trinomial like  

2x^2 - x - 15

and find two binomials you can multiply together to get it


Every once in a great while, you'll get a trinomial that won't factor but the rest of the time you'll get a nice clean answer where each binomial consists of an integer times x plus another integer. At least, that's how it works with the assignments. You may even be told that polynomial factoring is useful because it can help you solve equations. That part is a lie.

With a couple of notable exceptions (differences between two squares and perfect square trinomials), you will probably never even try to solve a problem by factoring a quadratic for the simple reason that most don't factor.

Not only does solving by factoring usually not work; we already have a simpler method that always works, the quadratic formula.

The truth is, we don't care whether or not you know how to factor a trinomial; we care about what you learned in the pursuit, things like problem solving skills and insights into how numbers work.

(2x+5)(x-3) is just something to keep the plot moving.

If you're interested, try a few randomly generated trinomials and see how many you can get to factor.

Thursday, July 31, 2014

Slow starters -- when the race is not to the swift

In his book  Mastery, George Leonard has some interesting thoughts about slow learners. 
My experience as an instructor has shown me, for one thing, that the most talented students don't necessarily make the best martial artists. Sometimes, strangely enough, those with exceptional talent have trouble staying on the path of mastery. In 1987, my colleagues at Esquire and I conducted a series of interviews with athletes known as masters of their sports, which tended to confirm this paradoxical finding. Most of the athletes we interviewed stressed hard work and experience over raw talent. "I have seen so many baseball players with God-given ability who just didn't want to work," Rod Carew said. "They were soon gone. I've seen others with no ability to speak of who stayed in the big leagues for fourteen or fifteen years."

Good Horse, Bad Horse

In his book Zen Mind, Beginner's Mind, Zen master Shunryu Suzuki approaches the question of fast and slow learners in terms of horses. "In our scriptures, it is said that there are four kinds of horses: excellent ones, good ones, poor ones, and bad ones. The best horse will run slow and fast, right and left, at the driver's will, before it sees the shadow of the whip; the second best will run as well as the first one, just before the whip reaches its skin; the third one will run when it feels pain on its body; the fourth will run after the pain penetrates to the marrow of its bones. You can imagine how difficult it is for the fourth one to learn to run.

"When we hear this story, almost all of us want to be the best horse. If it is impossible to be the best one, we want to be the second best." But this is a mistake, Master Suzuki says. When you learn too easily, you're tempted not to work hard, not to penetrate to the marrow of a practice.

"If you study calligraphy, you will find that those who are not so clever usually become the best calligraphers. Those who are very clever with their hands often encounter great difficulty after they have reached a certain stage. This is also true in art, and in life." The best horse, according to Suzuki, may be the worst horse. And the worst horse can be the best, for if it perseveres, it will have learned whatever it is practicing all the way to the marrow of its bones.

Suzuki's parable of the four horses has haunted me ever since I first heard it. For one thing, it poses a clear challenge for the person with exceptional talent: to achieve his or her full potential, this person will have to work just as diligently as those with less innate ability. The parable has made me realize that ifl'm the first or second horse as an instructor of fast

Wednesday, July 30, 2014

A Tale of Three Students

Most normal people (a.k.a. non-statisticians) tend to think in linear terms. The trouble is most normal behavior doesn't tend to be very linear. As a rule, you're better off thinking in terms of of U-curves (things go either up or down then come back) and S-curves (things are level, they move either up or down, then they level off again). These are still approximations, but they are usually more reasonable approximations. 

This is particularly true in education. Arguably the best way to model learning is with a series of S-curves. We work and study with little progress, then we have a period of improvement, then we hit another plateau.

If we think in terms of straight lines, ranking students is relatively easy.

But if we think in terms of S-curves, which is a great deal more realistic, things get more complicated.

The x-axis doesn't actually mean anything (this is made-up data), but let's say it represents months studying a language and the lines represent daily test scores. Now, who's the best student depends on when you ask, and that raises some troubling points. 

We tend to put too much faith both in the metrics we use to evaluate students and in the linearity of human behavior. We are not straight -line animals but we have a bad habit of making straight line decisions. In this case, think of what would happen if we made a decision on who to drop from a program after two months. 

Monday, July 28, 2014

The Power and Peril of Positive Thinking

Having hammered away at the importance of student self-confidence and positive attitude as a condition for success in math (part of the larger discussion of applying Pólya's teaching principles), it's important to step back and point out that a lot of people have made horrible, costly mistakes thanks to positive thinking and the influence of motivational speakers (for example).

With fantastic successes on one side and horror stories on the other, it is tempting to call this a wash, but if you think like a statistician (and you should always think like a statistician) and start breaking things down, you'll find that a few common sense rules can tell you when to assume the best and when to prepare for the worst.

Being pragmatic about being positive

For the purposes of this discussion, let's decide on a fairly precise definition of what we mean by positive thinking:

To apply positive thinking to a task, you act under the assumption that, given  reasonable and intelligently applied effort, the probability of success is close to one;

Furthermore, this assumption will not be reassessed unless there is a major change in the situation.

The advantages to this approach are: we can waste a great deal of time and energy worrying; overestimating risk can cause us to prematurely abandon projects; thoughts of failure can cause us to "flinch," to hold back and not give the task our best effort. Avoiding these things can allow positive thinking to create self-fulfilling prophecies.

The disadvantages are that underestimating the probability of failure can cause us to waste resources on projects with negative expected value and, more importantly, failing to pay attention to warning signs can leave us vulnerable to otherwise avoidable disasters.

We could have a general discussion at this point about the relative weight of these advantages and disadvantages but it wouldn't be very productive because neither risk nor reward are evenly distributed. In many if not most situations, a fairly clear case can be made for either positive or cautious thinking. To determine which approach is best for a given situation, think about these rules of thumb:

Before you commit yourself, try to think realistically about the expected value in terms of other people's success rates;

Never bet more than you're willing to lose;

Consider collateral damage (are you putting your spouse and children at risk of hardship?);

Is there incremental payoff? This last one is extremely important. If you decide to start a restaurant or move to NYC to make it on Broadway, and you fail, then you will probably have very little to show for the effort. If, on the other hand, you decide to lose forty pounds through diet and exercise or to go from being a C student to an A student, then there is incremental pay off for your hard work even if you fail to achieve your goal.

All of this leads us back to the original point. We often associate positive thinking with business and entrepreneurship where it is, more often than not, a bad idea, while in education, where we have every reason to encourage positive thinking, we are constantly hearing people like Michele Rhee complain that we spend too much time building up kids' self-esteem.

Don't let the posturing and tough talk fool you. Self-esteem is good for kids and you should do everything you can to convince them that they are capable of doing every problem their teacher gives them, as long as they put in the effort.

Thursday, July 24, 2014

Recess was always my best subject

And according to this article from Tim Walker, I might have been onto something.
Like a zombie, Sami—one of my fifth graders—lumbered over to me and hissed, “I think I’m going to explode! I’m not used to this schedule.” And I believed him. An angry red rash was starting to form on his forehead.

Yikes, I thought. What a way to begin my first year of teaching in Finland. It was only the third day of school and I was already pushing a student to the breaking point. When I took him aside, I quickly discovered why he was so upset.

Throughout this first week of school, I had gotten creative with my fifth grade timetable. Normally, students and teachers in Finland take a 15-minute break after every 45 minutes of instruction. During a typical break, students head outside to play and socialize with friends while teachers disappear to the lounge to chat over coffee.

I didn’t see the point of these frequent pit stops. As a teacher in the United States, I’d spent several consecutive hours with my students in the classroom. And I was trying to replicate this model in Finland. The Finnish way seemed soft and I was convinced that kids learned better with longer stretches of instructional time. So I decided to hold my students back from their regularly scheduled break and teach two 45-minute lessons in a row, followed by a double break of 30 minutes. Now I knew why the red dots had appeared on Sami’s forehead.

Come to think of it, I wasn’t sure if the American approach had ever worked very well. My students in the States had always seemed to drag their feet after about 45 minutes in the classroom. But they’d never thought of revolting like this shrimpy Finnish fifth grader, who was digging in his heels on the third day of school. At that moment, I decided to embrace the Finnish model of taking breaks.

Once I incorporated these short recesses into our timetable, I no longer saw feet-dragging, zombie-like kids in my classroom. Throughout the school year, my Finnish students would—without fail—enter the classroom with a bounce in their steps after a 15-minute break. And most importantly, they were more focused during lessons.

At first, I was convinced that I had made a groundbreaking discovery: frequent breaks kept students fresh throughout the day. But then I remembered that Finns have known this for years; they’ve been providing breaks to their students since the 1960s. 

Monday, July 21, 2014

My preferred approach to teaching high school math -- the last twenty minutes

[Dictated to my phone so beware of homonyms]

Note: Though the connection may not be immediately obvious, the following thoughts on teaching will eventually tie in with a larger piece on George Pólya.

Though it varied someone from class to class and situation to situation, my preferred method was to reserve the last part of the class for students to work individually while I went around the room and checked each student's work. Generally, I would give the students a couple of worksheets to be handed in at the end of class. After completing those worksheets, they were instructed to spend the rest of the hour working on their homework. I wasn't always able to get to every student every day, but I came close, and I never let more than a couple of days go by without making sure that I had personally observed a student doing problems in my class.

If a lots of the students were having trouble doing the assignment, I would sometimes interrupt the routine, go back up to the board, and reteach some of the material. That was fairly rare. Most of the time, two or three students would need real help and the rest only needed either a couple of quick suggestions or simply confirmation that they were doing the problems correctly.

The personal help was important, as was the knowledge on the students' part that if they needed help in the future I would be there. This approach also let me make sure that neither the class or any of the students got into a death spiral where confusion and failure started causing a cascading effect. By personally watching students successfully completing assigned problems, I could make sure that everyone was keeping up. Grading was also an important part of that process but for assessment there is no substitute for actually watching how a kid going through a problem.

In some cases, particularly with advanced classes, I might stray from this approach, but if we are talking about at-risk kids in tough environments who need to make up ground academically, I believed then and believed now this is the best way to teach high school math.

If I sound a little over emphatic with that last sentence and perhaps even a little bitter it's because I am more than a little bitter about the direction our schools have headed. I enjoyed that kind of teaching and I got excellent results with it, but if I were to go back into the profession now, there is almost no way I could give that kind of personal attention nor could I take the same level of accountability for students' success. Class sizes have simply gotten too large.

On a completely unrelated note...

Detroit Public Schools EM shifts funds from classroom
By Dr. Thomas C. Pedroni

Many of us are shocked to learn that DPS plans to cut costs in the coming year by further increasing class sizes. Already at an unmanageable target of 38 per classroom in grades 6 through 12, Emergency Manager Jack Martin’s fiscal year 2015 budget allows class sizes in those grades to expand to 43.