Saturday, September 20, 2014

Puzzling your way through the GRE -- logic edition

I was looking through a GRE sample test the other day and I came across a question that looked oddly familiar. I am not going to repeat it verbatim (not quite sure of the copyright status) but the gist was that the question gave some names and some properties associated with those names and some conditions.

After a moment, I realized that this problem reminded me of an old-fashioned logic grid puzzle.






Here's an example from a popular site:
You can eliminate pairs you know aren't true with an X, and pencil in pairs you know are related with an O. If you know, for example, that Lauren wasn't born in 1961, you can add an X in the box where the Lauren column and 1961 row meet. Similarly, if you know that Bryant was born in 1971, you can add an O in the appropriate box. Furthermore, since every option can only be used once in any given puzzle, you can eliminate the four other options for Bryant in that category (1937, 1946, 1961, 1975) and the four other options for 1971 (Anahi, Jayden, Lauren and Nikolas).




You probably won't see a grid like this on the GRE or the SAT but you will run across the underlying concepts so if you're preparing for one of these tests, it might be worth your while to spend a little of your study time as play time with these puzzles.




Tuesday, September 16, 2014

An annecdote from my teaching days -- sometimes it just takes a nudge

A few years ago, I did a stint teaching math at an urban prep in Watts. It wasn't a tough school by Watts standards (and the Watts of today isn't nearly as tough as it was twenty years ago). No school with a self-selecting student body is going to be that tough.

It was, however, academically challenging. The kids came in about a year and a half below grade level and the objective was to get almost all of them into college and by that I mean almost all of the kids who enrolled. We never resorted to the high-attrition model common in 'no-excuses' charters. We also didn't bury the the kids with work. We had to find a way to get virtually all of our students up to speed while keeping their workload reasonable.

It was a daunting assignment and when I started grading their first test, I was momentarily tempted to give up. Probably half the questions were left completely blank. I was shocked. My approach to teaching math requires a lot of one-to-one interaction so I had seen all of these kids do similar problems successfully in class.

To make matters worse, I had been very clear that I would give partial credit if the students were close or even if they showed a good understanding of the question. Even with this added incentive, they were frequently not even attempting the problems.

I wrote up a new test (treating the first as a practice run), reviewed all of the material with the class and announced one change in the grading policy. I told the class that, since a blank was the most wrong answer, a blank would get the lowest score. If the students wrote anything, including "I don't know," they would get at least a point. The closer they got the better they'd score but writing anything would get them one point out of ten.

From that point on, unless the student ran out of time, I never saw a blank, but here's the kicker: I never saw anyone write "I don't know" either. Pretty much every student attempted every problem. Furthermore they were in pretty good shape for college, good enough that a fair number of them went on to some very good outcomes, including engineering degrees at schools like UCLA and USC.

There are lots of potential take-aways here, but one seems particularly relevant to some recent discussions: learned helplessness and what we might call students' static friction. Students who are having trouble, particularly those on the wrong side of the achievement gap, often find themselves in this situation. This has both a bad and a good side. It can be extraordinarily frustrating for students and educators but often, if you can manage to nudge the student while applying the right kind of instruction and motivation, you'll see a dramatic improvement.

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

(2x+5)(x-3)

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.