Looking Outside the US

One of my pastimes for the last few years has been to try and get my head around the financial crisis.  I tend to believe what is called the Efficient Markets Hypothesis.  That is, I think that the market does better than anything else we have of predicting value.  So why was the market so wrong?

A lot of people make arguments that go like this:  mortgage brokers lowered their standards and gave loans to people who were speculating or who just couldn't afford the loans.  They could afford to do this because they got the loans off their books by packaging them in mortgage backed securities and selling them off.  When the bubble burst, all those who bought the mortgages were left holding assets that were worthless all along.

I was always a little uncomfortable with this explanation because it didn't explain why the people buying the mortgage backed securities encouraged the mortgage brokers to lower their standards.  Didn't they know this would increase default risk?

I think the answer is that yes, they did know.  But they thought they could manage the risk.  I think a big part of the problem is explained by a quote from Alan Greenspan in 2005.  While admitting to a congressional committee that there was "froth" in the market, Greenspan explained that there was no risk of a national housing boom because "the housing market in the United States is quite heterogeneous, and it does not have the capacity to move excesses easily from one area to another... the behavior of home prices varies widely across the nation."

Investors thought they could smooth out the risk by combining mortgages from different parts of the country. This would work as long as housing prices across the nation were uncorrelated.  But, as it turns out, we actually did end up with a nationwide housing boom.  And since financial institutions had bet so heavily that it couldn't happen we ended up with a financial crisis.

A lot of people had a lot of money on the line, and it seems like they made disastrous decisions based on the assumption that we couldn't have a national housing boom.  Now take a look at this graph:

It wasn't just a national housing boom.  It was an international bubble, and the situation in the US was actually pretty moderate compared to a lot of other countries.  This seems to indicate that all of the funny business going on in the US mortgage industry (or with US interest rates) wasn't the main cause of the bubble.  It was an international phenomenon whose causes were probably international in nature.  One explanation is that developing countries like China had high savings rates, and their investments ended up flowing into the mortgage market.

In any case, how could we keep assuming that we could not have a national housing boom when we were in the midst of a pretty major international bubble?

Math Wars

Over the weekend I started reading a book called Math Wars about the controversy that followed the release of the National Council of Teachers of Mathematics curriculum standards in 1989.  A lot of states modeled their standards after them, and a lot of textbooks were written with them in mind.

The controversy was that many people felt the standards were "fuzzy" because they focused on problem solving and allowed the use of calculators.  They deemphasized memorization and algorithmic skill.

The author of the book points out that math curriculum has been swinging back and forth between two extremes for a long time.  For example, in the 1960's mathematicians themselves had a lot of influence on the curriculum and they developed what was known as "New Math", which focused on things critical to pure mathematics like set theory.  Then there was a period based on problem solving, then a shift to "traditional" math focused on arithmetic and algebra.

The one quibble I have with the presentation is that it seems there is not just one axis of variation for the various kinds of curricula.  One on axis we have the spectrum between abstract and applied.  On the other we have focus on memorization vs depth of understanding.  Below is an image of how I would visualize the four quadrants that result:

Each of these quadrants represents an important part of mathematics, but different curricula and different teachers tend to favor one area over another.  The 1989 standards focused on the top left quadrant, New Math leaned to the top right, and traditional math often emphasizes the bottom right.

The author of Math Wars likes to emphasize that mathematicians (who developed New Math) and math educators (Phd types who wrote the NCTM standards) don't always see eye to eye.   But I think his analysis of this disagreement is a bit muddled because he only has one dimension of variation.  

Today I asked my students which kinds of math they felt most comfortable with.  Interestingly, a lot of them said they didn't like memorizing arithmetic, although that is clearly important as a foundation.  A number of them preferred heuristics.  A handful enjoyed the pure math.  A lot of them didn't vote at all. I would never base my teaching purely on student preferences, but it was interesting to hear their perspective.

In the end I think that a robust math education needs to include all four quadrants, but how to spend class time is still a big question mark.  The right balance probably depends a lot on the age and capabilities of the students.

Psycho

My wife is worried that if we have a child that it might end up with some mental disorder.  Her latest fear is that our child might be a psychopath.  She told me a story that she read about a mother who was afraid of her own child because when left alone he would torture animals or torment his siblings.

My initial reaction was to questions whether psychopathy should even be considered a disorder.  I am not an expert on the matter, but as far as I know, to be a psychopath means that one has a less than normal level of empathy.  This does not mean that they are always violent, as many assume.

My problem with classifying psychopathy as a disease is that it seems to be a primarily moral defect.  Psychopaths can function very well in a lot of ways, but we are afraid of them because they are not moved by emotions that we expect them to be.  Classifying this as a disorder seems a little bit like classifying homosexuality as a disorder.  They can function perfectly well, they just want something different than what is considered normal.  Wanting something different than the rest of society can get one into a lot of trouble.  There is no doubt about that.  But trouble is not necessarily a disease.

This does not mean that I think we should let criminals off the hook if they have a psychopathic personality.  It just means that if my child were a psychopath I would strive to understand what it is that he  (since they are mostly male) wants, and find the best way to ensure that he can live a fulfilling life.

My wife claims I am biased from watching Dexter.

Reflection on Math Curriculum (Part II)

Yesterday my teaching coach came in and we had an interesting conversation.  Some of my thoughts started to crystalize about how I want to construct my assessments.

The first thing I want to do is break assessments into two parts.  The first part will be my "passport".  It will consist of the absolutely critical basic skills (and will probably include some review skills).  This will be much pared down from the usual kind of assessments I give.  In order to pass a student must get every question right, but they can retake it as many times as necessary.  Any student who passess all of the passports and does nothing else will get a "C".

I have been doing some passports the second half of this year, and for the most part it is working pretty well.  It is a good way to ensure that the students get the basics.  I think the key to their success lies in the grading process.  I have to grade them over and over because they are taken multiple times.  But students start to memorize the answers.  Thus, I have to be careful about requiring them to show their work.  No work, no credit.  Of course, then they start to show BS work...so I don't give them credit if there is a critical step missing.

I think I will also require students to complete all of the homework assignments leading up to a passport before they take it.  For some students this may not be necessary.  Perhaps if a student gets an "A" one quarter I will let them challenge passports without completing all of the assignments.

The second part of my assessment process will be challenge problems.  These will be higher order thinking skills such as proving results, modeling realistic situations, and solving complex problems.  But for these I want to give the students a choice.  After they have mastered the basic skills I want to give them some flexibility in how they proceed.  Perhaps some students would even like to propose their own challenge problems.

The purpose for this is that students are different.  I have been focusing a lot on what the goals of a math class should be, but one thing I haven't thought much about is what the goals of the students are.   The fact is that I don't think math education has to be perfectly uniform for everyone.  As long as everyone has demonstrated some mastery of the basic concepts and is engaging in some form of higher order thinking I think I will be satisfied.  Perhaps giving them a choice will give them a stronger sense of ownership.

Reflection on Math Curriculum (Part I)

One of the big themes for me this year has been the dichotomy of direct instruction and discovery learning.  Should we just tell students what they need to know or should we have them figure it out on their own?

My natural tendency is to give students some basic knowledge and then see if they can work the rest out.  I believe students can do hard problems if you set high expectations and give them time.  But a lot of teachers and researchers believe that the best way to teach is to simply show students how to do a problem, do a few together, and then have them do it on their own.

To understand the conflict I think it is important to distinguish between two different kinds of objectives: teaching specific content and exercising general reasoning ability.

Cognitive Load Theory has had a big impact on how I view the question.  It makes two claims.   First, students learn content more effectively if we show them explicitly what they need to know.  Second, it is not possible to teach general reasoning ability.  They justify the first claim by saying that students who are asked to discover concepts get mentally overloaded and shut down.  On the second point they argue that problem solving and general reasoning are biologically primary.  That is, we develop these skills naturally and there is no evidence that instruction has any impact.

As I understand them, advocates of discovery learning argue that in order to learn something a student must care about it and connect it to what they already know.  The discovery process is a way to help students care about a problem by engaging their curiosity.  Furthermore, most of them also believe that it is possible to improve reasoning ability by developing heuristics and practicing them.

I have made a number of observations this year that support both sides.  The first thing that comes to mind is that students often fail to solve problems when they have all of the necessary tools to do so.  They shut down in class or leave questions on a test blank.  When they do shut down, it is often possible to get them going with a little Socratic questioning:  "Ok, what do you see in this picture? What is our goal?  Do we know anything about hexagons? Can we make it into a simpler problem?"

This observation tells me two things.  Just like the cognitive load theorists predict, students shut down when they can't fit a problem into their "working memory".  But, there are thought processes that can help them break the problem down.  The question is whether it is possible to teach a student to engage in their own Socratic dialogue without my help.

The second observation is that students forget.  If we do some problems together and they demonstrate immediate mastery, it is still likely that they will be unable to solve the same problem in a few weeks.  This may mean that they never understood it well enough.  Perhaps they would do better if they had discovered the concept on their own.  Or it could mean that direct instruction just needs to be repeated in order to stick.

My third observation is that given enough incentive, (at least some) students will demonstrate remarkable persistence.  Consider my experience with the "World's Hardest Easy Geometry Problem".  Out of about 50 students that were offered the opportunity, 10 of them seriously took up the challenge.  Out of these about half actually persisted until they found a solution.  Keep in mind that this is a legitimately difficult problem.  I estimate that the ones who solved it spent at least 12 hours on it.  Three of those who haven't solved it are still working diligently.  Two of them gave it a serious effort and gave up (as far as I know).

On the flip side, without a strong incentive most students will give a half-hearted effort. Even a few of those who solved the problem usually slack off in class, copy homework without thinking about whether they understand, and leave test questions blank.  To me, this tends to support the cognitive load theory in that one of the main determinants of successful problem solving, persistence, seems to depend mostly on motivation, which I believe is a biologically primary factor.  That is, I don't think I can do much to teach students how to evaluate consequences.  I can manipulate consequences, but I can't teach them how to interpret them.

My fourth observation is that (many) students like algorithms.  Sometimes advocates of discovery learning, like me, tend to discount the value of teaching algorithmic skill.  Perhaps it is because it seems like that is all some people do in math class.  But the fact is that human begins have a built in capacity to learn algorithms and there is a certain pleasure in carrying out a set of steps that lead to a correct solution.  There doesn't always need to be context or curiosity.  The same could be said for slightly complex problems.  After students learn an algorithm or a concept, many of them are capable of and even enjoy applying the new concept to a variety of problems, even if those problems have no connection to anything else they have ever done.

My final observation is that students can solve a lot of math problems without abstraction if the problem is connected to some aspect of their intuition.  One problem my students were working on recently involved two equations with three variables each.  The problem asked the students to find the value of "x + y", which could easily be done by subtracting one from the other.  A few students saw this immediately, but a few were stumped.  One time I asked a few students who in my room for help the following question:  "Suppose you and a friend go into a restaurant with no prices listed.  You order a burger, two fries and two drinks.  The cost is $7.  Your friend orders a burger three fries and three drinks.  The cost is $11.  How much would it be for one fry and one drink?"  The answer came pretty quick.  There are a number of studies on this topic, and a famous scene from The Wire is a good illustration of the same concept. A lot of the math we do is pretty intuitive, if we could only harness the power of our intuition.  It is the process of abstraction that gets in the way.

Ok, I have written and rewritten this a few times and I can't seem to get my thoughts organized.  So I am just going to post this much and continue later.

Grading on a Curve

A few months ago I read a blog post about grading on a curve.  Unfortunately I can't find a link.  The post listed a few pros and cons of various grading systems.  One thing I gleaned form it was that if you have a linear curve it is better to base your line off of the maximum and the median rather than just dividing each score by the top score.  The reason is that the method of setting the top score to 100% benefits those at the top more than those at the bottom.

As I was tinkering with the scores from my latest test, I had the thought that there is no reason why the curve has to be linear.  I came up with three variations that gave me a rough approximation of a fair distribution:

The red curve is a linear equation, the green curve is logarithmic, and the blue curve is a logistic function.  Eventually I decided that the logarithmic curve had the most appeal.  But it got me thinking about other situations where people tend to use linear curves.  In particular, I started thinking about situations where we need to combine more than one variable.  For example, most law schools assign each candidate a number based on a linear combination of LSAT score and GPA.

But why should the score be based on a linear combination?  Why not consider adding a logarithmic score for the LSAT to a logistic number based on GPA or something to that effect?  Sure, it would be less transparent, but for a number that isn't made public that should hardly matter.  The point is that using different curves allows us to develop a score that more closely reflects our values.

I suppose I could do the same thing when I combine different components of a students grade.  For example, if I think that homework does little good unless it is done with some regularity I might base the homework part of a students grade on a logistic curve (which starts growing slowly, speeds up, and then slows down again).

Actually, grades are probably not the most important application of this modeling concept.  But in general, we should look beyond linear models when trying to quantify things that we value.

Surpassing Shanghai

I just finished listening to an interview with Marc Tucker, the author of the book Surpassing Shanghai.

He compared our reform efforts in the US to the things that successful countries are doing.  The resulting message is that reformers on the right and the left are both wrong.  We don't need to spend more money and we don't need to let market forces disrupt the system.  What we need are 1) high standards for all children (and a commitment to spend more money on those that are hardest to educate), 2) a quality curriculum with matching materials, assessments and teacher preparation, and 3) high quality teachers.

I want to focus on number 3 for a moment because I think it is a huge challenge.  Tucker said that in successful countries, teacher training programs are highly selective and teachers are paid an amount comparable to other professionals, like lawyers or engineers.

Is this possible in the US?  One of the main problems as I see it is that in the US people can make a lot of money by being an engineer, lawyer, or doctor (more than in most other countries).  Take this chart, for example.

The difference is not quite as stark in every other profession, but the main point remains.  Most other countries don't have as much income inequality as we do in the United States so it is (relatively) easy to make teacher pay competitive.  But income inequality in this country is not going away anytime soon.

McKinsey did a study on what it would take to encourage graduates from the top third of their college class to go into teaching.  The results were not exactly encouraging.  Teachers currently start at an average of $49,000/yr and top out at around $67,000/yr.  The result is that about 23% of teachers are top third graduates.  If we increased the pay range to $65,000/$150,000 we could get a whopping 37% of new hires from the top third.

People simply don't respect teaching as a profession, and the pay is just part of it.  Teachers are seen as babysitters.  It is stressful.  The first few years are brutal.  In some circumstances there is a lot of pressure to spoon feed students material to improve standardized test scores.

Luckily for our students it turns out that there are a significant number of very competent and intelligent people who overlook all of this.  But lets face it.  They are not the majority.  And we have a system designed around the teachers we have.  They create the culture and drive expectations.

That is why I don't think we can model our education system after that of other successful countries.  I think we need a complete overhaul.  Contrary to Mr. Tucker's advice, I think we need to unleash market forces and disrupt the status quo, because otherwise I see no hope of getting bright people into the profession on a massive scale.  The United States really is different than Singapore, Japan, and even Canada.

Solved

That didn't take too long.  As soon as I walked in this morning several students rushed into the room to show me what they came up with over the weekend.  One of them had a legitimate proof of the "triangle of death".

Update: Today I also got one student who proved it by looking online for a solution (he will get some modest extra credit because it is still not that easy to comprehend) and one student who brought in a proof his mother did (but he couldn't explain it).

Update II: Another student came up with a novel proof.  This one involved the use of the fact that a central angle of a circle is twice as much as the angle with the vertex on the edge of the circle (if the other two points remain the same).

World's Hardest Easy Geometry Problem

I was stuck on the following geometry problem for a long time:

It was kind of nice in a way, because it reminded me of what it is like to be persistent on a math problem.  I spent hours thinking about it and couldn't find a proof.  Then I came back days later and spent hours again, to no avail.  I don't even want to admit how long it took me.

On a whim I brought it to my class and told my students that I had solved a really hard geometry problem.  I said that anyone else who could solve it would get an A for the quarter.  Heck, if it took me that long then any high school student who solves it has proven their worth.

Surprisingly, quite a few students have taken up the challenge.  After two days, no one has solved it but hey, it took me more than two days.  A number of students are coming in during lunch, study hall, after school, and during breaks to discuss what they have found.  It has actually been one of the coolest things I have done all year.  I have a whole wall of my classroom dedicated to different attempts at the proof.  Some of the students are engaged in what may be the most intense problem solving experience of their lives thus far.

In a way it is kind of cruel because it is a very hard problem and a lot of students won't ever get it.  But some of them will persist and there are a few that are quite close already.  This may be the kind of thing that they remember for the rest of their life. 

Hint:  if you want to try it, try drawing some additional lines inside the triangle.  Aside from that all you need to know are the old triangle congruence rules (SAS, ASA, SSS, etc) and the properties of an isosceles triangle (two sides and two base angles are congruent.