"I am not now and have never been a constructionist"

(This post also appears at Observational Epidemiology.)

After my last post thought I should run this titular disclaimer. For those of you not up on the subject, here's a definition from the well-written Wikipedia entry on the subject:

Constructivist teaching is based on constructivist learning theory. This theoretical framework holds that learning always builds upon knowledge that a student already knows; this prior knowledge is called a schema. Because all learning is filtered through pre-existing schemata, constructivists suggest that learning is more effective when a student is actively engaged in the learning process rather than attempting to receive knowledge passively. A wide variety of methods claim to be based on constructivist learning theory. Most of these methods rely on some form of guided discovery where the teacher avoids most direct instruction and attempts to lead the student through questions and activities to discover, discuss, appreciate and verbalize the new knowledge.

Don't get me wrong. For the right topic, executed the right way with the right teacher and class, this can be a great, wonderful, spectacular and really good approach to education. Unfortunately, education reformers (particularly the current crop), are not good at conditional problems. They tend instead to fall into the new tool camp (you know the saying, "to a man with a new hammer, the whole world is a nail.").

Worse yet, (and I'm afraid there's no nice way to say this) many of the educational theorists don't have a firm grasp on the subjects they are working with. This is never more plain than in constructionist science classes that almost entirely eschew lectures and traditional reading assignments and instead have the students spend their time conducting paint-by-numbers experiments, recording the results and performing a few simple calculations.

To most laymen, that's what science is: stuff you do while wearing a lab coat. Most people don't associate science with forming hypotheses, designing experiments, analyzing results and writing papers and, based on my limited but first hand experience, many science educators don't give those things much thought either.

The shining exception to the those-who-can't-teach-teach-teachers rule is George Polya. Though best known as an educational theorist, Polya was a major Twentieth Century mathematician (among his other claims to fame, he coined the term "central limit theorem") so he certainly fell in the those-who-can camp.

But it it important to note that Polya advocated guided discovery specifically as a way of teaching the problem solving process. I suspect that when it came to simply acquiring information, he would have told his students to go home and read their textbooks.

Reasons to teach what we teach

[note: this is a math-centric post but most of the concepts can, on some level, be generalized to other subjects]

There's a curiously inverted quality to the education debate. We spend a great deal of time discussing revolutionary changes to the educational system and almost no time talking about what we should be teaching, as if the proper combination of reforms and incentives can somehow overcome the rule of garbage in, garbage out.

I spent a lot of my time as a teacher thinking about which parts of the mathematics curriculum were good and which parts were garbage and I came up with a list of reasons why a topic might be worth the student's time. The list isn't in order (I'm not sure it's even orderable) but it is meant to be comprehensive -- everything that belongs in the curriculum should qualify under one or (generally) more of these criteria.


1. Students are likely to need frequent and immediate access to this for jobs and daily life.

and

2. Students are likely to need to know how to find this (Samuel Johnson level knowledge).

(These are the only two mutually exclusive reasons on the list.)


3. This illustrates an important mathematical concept


4. This helps develop transferable skills in reasoning, pattern-recognition and problem solving skills


5. Students need to know this in order to understand an upcoming lesson


6. A culturally literate person needs to know this

Most topics can be justified under multiple reasons. Some, like the Pythagorean Theorem can be justified under any of the six (though not, of course, under one and two simultaneously).

Where a topic appears on this list affects the way it should be taught and tested. Memorizing algorithms is an entirely appropriate approach to problems that fall primarily under number one. Take long division. We would like it if all our students understood the underlying concepts behind each step but we'll settle for all of them being able to get the right answer.

If, however, a problem falls primarily under four, this same approach is disastrous. One of my favorite examples of this comes from a high school GT text that was supposed to develop logic skills. The lesson was built around those puzzles where you have to reason out which traits go with which person (the man in the red house owns a dog, drives a Lincoln and smokes Camels -- back when people in puzzles smoked). These puzzles require some surprisingly advanced problem solving techniques but they really can be enjoyable, as demonstrated by the millions of people who have done them just for fun. (as an added bonus, problems very similar to this frequently appear on the SAT.)

The trick to doing these puzzles is figuring out an effective way of diagramming the conditions and, of course, this ability (graphically depicting information) is absolutely vital for most high level problem solving. Even though the problem itself was trivial, the skill required to find the right approach to solve it was readily transferable to any number of high value areas. The key to teaching this type of lesson is to provide as little guidance as possible while still keeping the frustration level manageable (one way to do this is to let the students work in groups or do the problem as a class, limiting the teacher's participation to leading questions and vague hints).

What you don't want to do is spell everything out and that was, unfortunately, the exact approach the book took. It presented the students with a step-by-step guide to solving this specific kind of logic problem, even providing out the ready-to-fill-in chart. It was like taking the students to the gym then lifting the weights for them.

Long division and logic puzzles are, of course, extreme cases, but the same issues show up across the curriculum. Take factoring trinomials. A friend and former boss of mine wrote a successful college algebra text book that omitted the topic entirely. I had mixed feelings about the decision but I understood his reasoning: this is one of those things you will almost certainly never have to do outside of a math class (what fraction of trinomials are even factorable?).

You can justify teaching the factoring of trinomials because it illustrates important mathematical concepts and because it gives students practice manipulating algebraic expressions, but the way you teach this concept has got to reflect the reasons for teaching it. Having students memorize a step-by-step algorithm would be the easiest way to teach the students to answer these questions (and improve their standardized test scores) but it completely miss the point of the lesson.

The point about standardized test scores is significant and needs to be revisited a post of its own. By evaluating teachers and schools on standardized test scores, we put pressure on teachers to treat all subjects as if they fell solely under reason one. This is not a good outcome.

Even more important than how we should teach something is the question of what we should be teaching. Current curricula tend to be broad and shallow with a tragic evenhandedness that often grants the same amount of time to trivial techniques as it does to fundamental concepts. This is bad enough when a class on grade level and everything is going well but it's disastrous when a large part of the class is struggling. There is tremendous pressure under those circumstances to leave the stragglers behind (a pressure that actually increases under many proposed reforms).

In addition to being overstuffed, the current curriculum omits subjects that are arguably more important than most of what we cover. The obvious example here is statistics, a topic that everyone actually does need on a daily basis (as informed citizens and consumers if nothing else). Perhaps even more relevant is what we might call spreadsheet math (customized worksheets, recursive functions, graphs, macro programming). You could also make a case for discrete mathematics, particularly graph theory (I might even put this one up there with statistics and spreadsheets but that's a subject for another post).

"Counseling out"

(This post also appears at Observational Epidemiology.)

Paul Tough writing in Slate recounts the following:

In Whatever It Takes, in one of the chapters on the Promise Academy middle school, I describe the impact of the KIPP schools in the Bronx and Harlem on the Promise Academy’s leaders and staff. This was during the first few years of the Harlem Children Zone’s middle school, which were a struggle, and those KIPP schools, which had very good test results, were for the Promise Academy administrators both a standard to be aspired to and a frustrating reminder that their own students weren’t performing at the same high level as KIPP’s students.

Terri Grey, the Promise Academy principal at the time, believed the attrition issue was part of what was holding her school back. As she put it to me in one conversation, “At most charter schools, if the school is not a good fit for their child, the school finds a way to counsel parents out”—to firmly suggest, in other words, that their child might be happier elsewhere. “Whereas Promise Academy is taking the most disengaged families and students and saying, ‘No, we want you, and we’re trying to keep you here, and we don’t want to counsel you out.” That policy made it impossible, she believed, for the Promise Academy to achieve KIPP-like results.

I’m not entirely convinced that that was the real problem at Promise Academy—or that the KIPP schools in New York were actually “counseling out” a significant number of students. But I do think it’s true that Geoffrey Canada’s guiding ethic has always been to go out of his way to attract and retain the most troubled parents and students. And that makes running a school, or any program, more difficult, even if it makes the mission purer and, in the end, more important.

For reasons I'll get to later, I suspect that the number of students you have to "counsel out" to have a significant effect on a school's test scores is lower than Mr. Tough realizes, but there are a couple of more important points.

The first is that selective attrition is recognized as a serious issue not just by critics of the reform movement but by responsible people within the charter school community.

The second is that all charter schools and charter school administrators are not interchangeable. There are some gifted educators with great ideas in that system. We've spent almost two decades overlooking the flaws in charter schools. It would be a serious mistake to try to compensate by overlooking the strengths.

"they are purging nonperforming students at an alarming rate"

(This post also appears at Education and Statistics.)

Mike at ScienceBlogs has some thoughts about selection by attrition:

A letter to Diane Ravitch from a Los Angeles school prinicipal documents just how dishonest and harmful this practice is (italics [Mike's]):

I received an email from Dr. DeWayne Davis, the principal of Audubon Middle School in Los Angeles, which was sent to several public officials. Dr. Davis said that local charter schools were sending their low-performing students to his school in the middle of the year. He wrote:

"Since school began, we enrolled 159 new students (grades 7 and 8). Of the 159 new students, 147 of them are far below basic (FBB)!!! Of the 147 students who are FBB, 142 are from charter schools. It is ridiculous that they can pick and choose kids and pretend that they are raising scores when, in fact, they are purging nonperforming students at an alarming rate--that is how they are raising their scores, not by improving the performance of students. Such a large number of FBB students will handicap the growth that the Audubon staff initiated this year, and further, will negatively impact the school's overall scores as we continue to receive a recurring tide of low-performing students."

Ravitch concludes:

Doing better than an under-resourced neighborhood school is not the same as getting "amazing results." Very few charters do. Probably less than 5 percent. Charters are not a silver bullet. They are a lead bullet. Their target is American public education.

This is just par for the course for modern conservatism: have private systems skim the cream, and leave the public sector to clean up an impossible mess. When they can't, this supposedly shows the inability of government to solve problems.

I have a few points to add:

1. This is a brutal way to treat these kids. You build their hopes up, then crush them, then dump the kids in a new school in the middle of the year;

2. We are talking about getting an influx of students who are badly behind and who are ready to give up and/or act out. This will disrupt classes slightly less than having a nearby car alarm go off at random times once or twice an hour;

3. But I think Ravitch overstates the case against charter schools. I've dealt with some small, independent schools that have impressed the hell out of me and I can see them playing an important role in our system, though a radically different role than Arne Duncan sees.

Topologists at play -- the game of Sprouts

It's important to have students think deeply about math in both structured and unstructured ways (I have a guilty feeling that I ought to say more about this, but that will have to wait for a future post). It's the unstructured part that tends to cause problems. That's one of the reasons I liked to make games part of my lessons when I was a teacher.

Games (at least the kind I recommend) require a great deal of focus -- you have to think about what you're doing or you won't do well -- and they encourage exploration and a playful attitude to the material. All of these things help build mathematical intuition.

On the subject of topology, my game of choice is Sprouts, invented by mathematicians John Horton Conway and Michael S. Paterson at Cambridge University in 1967 (as a general rule, you can't go wrong with a game if Conway had anything to do with it).

The rules are simple:

1. Start with some dots on the paper. The more dots you have the longer the game takes so you will probably just want to start with two or three.

2. Players take turns either connecting two of the dots with lines or drawing a line that loops back and connects a dot with itself.

3. The lines can be straight or curved but they can’t cross themselves or any other lines.

4. Each dot can have at most three lines connecting it.

5. When you draw a line put a new dot in the middle.

6. The first player who can’t draw a line loses.

You can find a couple of sample games here.

Benoît Mandelbrot (1924 - 2010) and Education Reform

From A maverick's apprenticeship:

It was then and there that a gift was revealed. During high school and the wandering year and a half that followed, I became intimately familiar with a myriad of geometric shapes that I could instantly identify when even a hint of their presence occurred in a problem. “Le Père Coissard,” our marvelous mathematics professor, would read a list of questions in algebra and analytic geometry. I was not only listening to him but also to another voice. Having made a drawing, I nearly always felt that it missed something, was aesthetically incomplete. For example, it would improve by some projection or inversion with respect to some circle. After a few transformations of this sort, almost every shape became more harmonious. The Ancient Greeks would have called the new shape “symmetric” and in due time symmetry was to become central to my work. Completing this playful activity made impossibly difficult problems become obvious by inspection. The needed algebra could always be filled in later. I could also evaluate complicated integrals by relating them to familiar shapes.

I was cheating but my strange performance never broke any written rule. Everyone else was training towards speed and accuracy in algebra and reduction of complicated integrals; I managed to be examined on the basis of speed and good taste in translating algebra back into geometry and then thinking in terms of geometrical shapes.

Where did my gift come from? One cannot unscramble nature from nurture but there are clues. My uncle lived a double life as weekday mathematician and Sunday painter. My gift for shapes might have been destroyed, were it not for the unplanned complication of my life during childhood and the War. Becoming more fluent at manipulating formulas might have harmed this gift. And the absence of regular schooling influenced many life choices, but ended up not as a handicap but as a boon.

I realize we can't have an educational system focused entirely on the occasional Mandelbrot, but I can't help but wonder how the great man would have fared in the rigid, metric-driven system we're headed toward.

(also posted at Observational Epidemiology)

Wolfmeat -- games edition

(As I mentioned before, games and puzzles have always been a big part of my approach to teaching, partially because they helped make school interesting but also because they often conceal some extraordinarily sophisticated mathematical concepts. The following is a starter set of classroom games I put together a few years ago.)

Czarist-era Russians who had to take long trips by sled, particularly at night, would often gather up several large chunks of meat in a sack before starting out. If the sled happened across a pack of wolves, the driver would throw out the meat a piece at a time in the hope that the wolves would stop for a few moments to fight over the food.

Even the best teachers will have a Russian sled moment now and then, when the wolves are circling your desk and searching diligently for your last nerve, so it's always a good idea to keep a few sacks of wolfmeat on hand just in case.

Hex – Probably my first choice for a "here, do this" moment. A fast, simple strategy game with a great pedigree. You can easily fit two hex boards on one side of a sheet of letter paper.

Checkers – Don't disrespect the lowly checker. Players who have mastered both chess and checkers often argue that checkers is the more challenging game, particularly the Spanish version with long jumps. A few cheap chess/checkers sets are a great classroom investment.

Chess – There are two contenders for the world's most popular game, Chess and Go, but only one is available for five dollars at your local discount store.

Chess Board Games – A chessboard is probably the most versatile playing surface ever invented. There are countless games that can be played on all or part of a chessboard. Here are a few good ones to start with:
Dodgem
Nine Hole
Chomp

Pencil and Paper Games – The only thing wrong with pencil and paper games is that people usually play the wrong one. Tic Tac Toe is the least interesting member of a distinguished family of row games (click here, here, and here for a few examples) many of which can be played with pencil and paper. In addition to row games there are Nim, Tac Tix, Dots and Boxes, Sprouts, Hangman and the very entertaining Racetrack.

Teaching across the Curriculum -- Yes, it's a buzzword, but as buzzwords go it's not bad. Here's a table to help you get started.