Consider Learning

Here are two complaints often heard in any math department.  The first is that people do not understand any mathematics; this complaint is typically inspired by some horrendous misuse of statistics or basic misunderstanding about numbers.  The second complaint is about students who have generally succeeded at basic mathematics but who do not truly understand the advanced tools that they’re using.  These complaints are about very different aspects of math education, and we need to keep them straight.

Ok, secret educator speak decoding time!

• “math literacy” == understanding basic mathematics to get through life and interpret your world
• “workforce development” == preparing people for jobs in STEM fields (Science, Technology, Engineering, and Math)

People often don’t give enough thoughts to their goals.  Say, for example, that you’re debating if calculus should be the pinnacle of high-school learning.  If you think the purpose of math education is math literacy, then it clearly shouldn’t be: statistics would be much more valuable.  If you think the purpose of math education is workforce development, then calculus probably should be the goal: you need calculus for engineering, for math research, for any kind of science, and without it you’re sunk.  (Leaving aside other goals such as exposing students to beauty or to deeper thinking, which might culminate a discrete math course.)

Schools basically push everyone to a workforce development track until they get to college.  Even assuming that this track adequately prepares people to study STEM fields (it doesn’t, judging from most incoming college freshmen), it’s pretty bad for anyone who doesn’t want to pursue a STEM field: they don’t get a lot of the skills they need for life (interpreting mortgages, research studies, and politicians) and they’ve learned to thoroughly hate math.

Yet we shouldn’t put everyone into a math literacy track, either, because it’s quite helpful to build up good abstract thinking skills early-on if you do plan to pursue a STEM career.

Naturally, you think, we should just let students choose or perhaps sort them somehow into different tracks.  But this opens up its own host of problems.  For example, a student might actually be quite good at math but lack the self-motivation to pursue it seriously, so it is a disservice not to push them harder.  There are students who develop mathematical skill later in life, but it is very hard to catch up in the workforce development track if you haven’t been given the opportunity to learn advanced algebra or trigonometry.  Let alone that low-income and minority students will track disproportionately into math literacy, and you’ll also create student self-images of being “not good” at math in the literacy track.

Which brings me to the recent New York Times op-ed, How to Fix Our Math Education.  This is probably the best proposal I’ve seen for addressing the situation, but it’s still not good enough.  The authors come up with credible ideas for courses that would demonstrate the connection between the “real world” and mathematics.  However, I don’t believe that the resulting courses would cover enough for workforce development.  Also, the plan feels too much like a retreat: since we can’t actually get most students to deeply understand calculus, let’s find a way to learn topics that don’t involve calculus.  But we know that calculus is possible from the results of other countries!

I do not know how to resolve this.  I don’t know how to make calculus and advanced, abstract mathematics accessible to those who are ready for it without the collateral damage of convincing other people (many of whom could excel in mathematics) that they are not good at mathematics.  Any ideas?

Mathbabe wrote a very interesting two posts about math competitions and the harm they do.  In summary, her argument gives three negatives to competitions:

• Math competitions discourage most participants because low scorers conclude they are not “good” at math.
• Those who do well at math competitions get an inaccurate picture of success, only to be stymied in (for example) grad school, which requires more sustained attention, gives less instant feedback, and does not always have a “right” answer.
• Competitions particularly discourage girls, who in general are not as “into” competitions and are more susceptible to feeling mathematically inferior.

Her posts are interesting and provocative (just see the comments thread!).  Yet there are also numerous advantages to competition.

Perhaps the biggest advantage is that competitions are an incredibly scalable model: a single competition can involve tens of thousands of participants at low cost.  Good competition problems and preparation books offer a solid curriculum for math team coaches across the country (even those who are not very good at math) and hence create mathematical communities where they could not otherwise be.

There are other advantages as well.  Competition really does appeal to something primal in young boys, and so it can get them into doing math when other techniques won’t work.  (Is this infrastructure why boys are still good at math when they fall behind in many other subjects?)  In fact, being on a math team can help raise your social standing relative to just being a generic math geek—it provides a cover similar to being on a sports team, if somewhat less revered.  Competition math also spurs the creation of lots of good math problems(*) that help students to reflect on deeper mathematics.

(*) Some argue that competition problems are not good problems, but that they are forced and arbitrary exercises that do not arise in nature.  I see the point, but I still find them intellectually stimulating.

I believe that competition math is valuable up to a point.  I think that intense studying for high-level performance is generally not very useful; it’s not learning real math (just competition strategies) and it exposes competition problems as arbitrary.  But for someone who has not spent hours upon hours studying, competition problems are engaging and a good way to challenge yourself.

However, mathbabe’s arguments point to two very important changes that we should bring about in the pathways towards studying mathematics.

1. We must create more alternate ways to become mathematically successful.  Math circles are a start, but they still rely on a strong mathematician/educator in the area.  Some specific, scalable ideas:
• Can we harness the internet to create better opportunities with video lectures and online communities?  Art of Problem Solving does a great job at this (see, e.g., Alcumus) although still with competitions as the main focus.
• Instead of timed competitions, can we distribute deep problems for students to unravel over the course of a month, submit, and receive feedback without grading?  The best work could be published, sufficient reward in itself.
• Can we develop deep mathematical curricula for the formation of “math clubs” that could be used out of the box by motivated students or teachers?
• Can we expand math summer programs and create more national opportunities for students?  Successfully doing this also requires new selection mechanisms, as right now most summer program quizzes are competition-like problems.
2. We must change the culture around math competitions.  We should emphasize the joy of the problems over numerical success and treat competitions for what they are—fun, stimulating, but arbitrary collections of problems.  Students should understand that competition math is an inaccurate reading of their chances at becoming a real mathematician.  In particular, they should understand that studying hard and doing well at the top levels of competition math is like training hard to be a good sprinter—you’re great at running 100 meters, but it doesn’t help for the marathon.  There is nothing wrong with that and the competitive spirit should be encouraged for those who want it.  However, it should not be forced on those who don’t.

I feel that throwing out competitions is throwing out the baby with the bathwater, and mathbabe notes much the same towards the end of her second post.  But building a new infrastructure for mathematical success is not easy, and will require some real initiative on our parts to make it happen.

The long tail has been frequently discussed with respect to online companies.  The idea is simple: yes, there are lots of very popular things (like, say, Lady Gaga or Justin Bieber), but there are actually a lot of people with specialized tastes (for example, computer nerds like Jonathan Coulton).  The internet enables those specialized interests to take root: on the local level, fans are isolated, but combine people nationally or internationally and you can get big business.  Amazon and Netflix are often credited for having taken advantage of this: Amazon because it can sell lots of books that local bookstores can’t carry, and Netflix because it can have lots of special-interest DVDs.  The online program Art of Problem Solving, for students with talent and interest in mathematics, is quite similar: it creates a mathematical community that can’t exist locally.  Other forums, Twitter, Facebook, LinkedIn, etc. do the same for other fields.  And magazines with national distribution fill the same niches, even moreso prior to the internet.

Summer programs that draw statewide or nationally can provide much of the same advantage.  Programs for students with interest in particular academic fields (from mathematics to writing to astronomy) allow communities to form based on these common interests.  Without the inter-school draw, there would not be enough students to create these communities.  (The same argument can be made for programs for students with learning disabilities, for sports, etc.)

My view is that these opportunities are crucial to developing young learners.  There should be systemic support for exploring your interests.  Schools alone cannot offer this support, because they don’t have the staff or the size to take advantage of the long tail.  Hence, we must have stronger support for summer programs that can bring students with common interests together, and also for programs like Splash that draw across schools, where the variety encourages students to find their part of the long tail.  (I also admire Citizen Schools for the same reasons: they encourage students to deeply pursue new interests, although because they are within a single school it’s not always the case that everyone in a project is deeply invested.)

Which brings me to the discussion recently about “expanded learning time” (ELT).  The discussion is here and here, and the participants quite nicely talk right past each other.  In a bid to share some of the contradiction myself, I am both a strong supporter and opponent of ELT.

I am a strong supporter when ELT means exploring and developing new interests.  But I am not a supporter when it means, simply, more time learning the way you do in school.  I don’t believe that additional time addresses the root problem that students do not self-identify as scholars, and it is a tremendous missed opportunity for students who cannot explore their part of the long tail.

While I can appreciate arguments that students should not choose their in-school curricula (because there are basic things that they should know), I firmly believe that ELT should center around choice and following your interests.  Summers, weekends, and after-school programs provide fantastic opportunities to do so.  We should make it expected that students pursue these opportunities.

This talk from Joshua Aronson is simply outstanding:

I recall a graduate school course where a guest speaker was discussing the importance of opportunity to learn.  This self-explanatory idea can easily be forgotten in debates about education.  However, contrary to the speaker’s perspective, opportunity to learn is not enough.  Students must also have the drive to learn.

To understand the importance of having drive to learn, consider your own life as you balance different things that you “want” to do.  For example, I want to learn German.  However, I do not want to learn German more than I want to help Learning Unlimited succeed, more than I want to read the new George R.R. Martin book, or more than I want to maintain a blog.  Hence, I do not learn German.  Similarly, all students want to learn and want to go to college—but perhaps not more than they want to hang out with friends, monitor their social status, or earn some extra money.  That’s why wanting to learn or go to college is not enough.  You have to be driven to learn; your self-identity must be that of someone who learns, a scholar, in order to make it a priority for you.

Although Aronson’s talk is about “stereotype threat”, the impact of students’ self-identity as it is reinforced by stereotypes of their groups (gender, race, etc.), one conclusion to draw is the importance of self-identity to academic success.

In case you didn’t watch his talk, here is just some of the evidence he cites:

• Fear has a big impact on intelligence.  “In the two weeks following a homicide in the South Side of Chicago, the IQ scores of the black children in a pretty large radius around the homicide will suffer a half a standard deviation decrement” (time index 15:30).  [Aronson goes on to talk about “threatened belongingness” having the same effect, showing how important social life is to learning.]
• You probably already know that if you’re asked to indicate your race prior to a test and you’re African American or Latino, then your score goes down.  On math tests, Asians asked about their race see their scores go up (but their English scores go down).  Women asked about their gender on math tests see their scores go down, men see their scores go up.  Older people asked about their age prior to a test of memory see their scores go down.  (Back to the math scores for a moment.  Because the AP calculus exam asks for gender before the exam, some 5000 women do not pass who would otherwise have passed — time index 36:30.)
• What may be more surprising is the same effect for miniature golf.  If, prior to playing miniature golf, the group is told that it is a measure of “kinesthetic intelligence” (emphasis mine), then you get a better score if you’re white and a worse score if you’re black.  In contrast, if you’re told that it’s a measure of “natural athletic ability”, then this reverses and black participants see their performance go up!  (Time index 26:30.)
• White male engineering students from Stanford were given a test.  In one group, they were told that the study was on “math ability — what makes some people smarter at math than others”.  In the other group, they were told that the study was about why “Asians are good at math”.  This second group scored 30% less well.  This is a full standard deviation, the size of the black-white gap.  (Time index 29:40.)
• Let’s move on to solutions.  “Students who are high self-monitors are much less likely to experience stereotype threat.”  “People who are high self-monitors are really good at thinking about the situation they’re in and considering the consequences, and the stereotype threat just sort of bounces right off of them.”  (Time index 35:45.)  Teaching self-monitoring might get around the stereotype threat!
• Asking students questions that make them think about membership in intellectual communities changes their scores.  For example, reminding students that they go to a selective college before the test (via the question “Can you tell us a few things a student might like about being a student at a highly-selective liberal arts college?”) nearly eliminates the gender gap.  (Time index 52:40.)  In other words, building a positive community membership can raise students’ achievement.
• Finally, if you listen to nothing else, I highly encourage you to listen to the description of Crellin Elementary School at time index 56:00.  It is a school that has truly managed to give students an intellectual identity, and to create an intellectual community.  In three years, the principal turned the school around from 0% to 100% proficiency on the state exam, with 50% reaching an advanced level.  Seven years after leaving the school, its students form 75% of the AP courses at the high school, despite the extreme poverty and tough home conditions of the students.  I won’t try to describe what the principal does.  You should hear it yourself.

I have been appreciating more and more the importance of students’ self-identity.  Do they see themselves as scholars?  Do they see themselves as people who enjoy learning?  (This is a critically different question from “do they enjoy learning?”)  I’m sure that it matters not just to immediate scores on tests, but also to learning over time.

How much stronger would students be with better self-identities?  How much would their performance improve if we gave them the ability to self-monitor and thereby get around their stereotypes?  One thing is clear: we must design our programs to build students’ self-identity as scholars, and we must do so as early in their lives as possible.

Update 02/29/12: A recent meta analysis calls stereotype threat into question, although there are claims that the meta analysis is, itself, biased.  The meta analysis still found effects among top performers, and was unclear on effects for average performers.

The most powerful experiences of my life have come from membership in intellectual communities.  We draw our identities as people from community membership, and being a part of a community that prides itself on learning inspires us to strive for learning in our lives.  I have been privileged to be a part of intellectual communities at MIT, Canada/USA Mathcamp, Splash, CTY, and SPMPS, among many others.  I have modeled my communities in life after the communities I grew up in, seeking friends who value intellectual achievement as I do.

When I design a new educational program, I give great thought to shaping the community.  A carefully-designed community will turn students’ focus to academics and make them celebrate their growth.  Community should be a conscious part of designing any place of learning.

But how?  For this inaugural blog post, I would like to share my developing thoughts on how to create a successful intellectual community.  I believe that the key is not to make the community yourself but to drive students towards creating an intellectual community themselves.

• Celebrate learning and knowledge, not achievement.  If the primary focus is on test scores, class rankings, or even going to college, then learning is not valued.  If the focus is on learning, the usefulness of the material, or the beauty of the material, then it will be valued by the community.
• Make students feel the value of being here.  For most of the programs above, this was easy: the programs are selective, and so students feel “special” for making it in.  However, any program can make learning feel like something you are privileged to be able to do in this place, and students will in turn value the experience and create a community around that valuation.
• Design the space to center on learning.  MIT lines its corridors with robots, posters, and labs.  Math departments everywhere have mathematical sculptures and posters.  At the Summer Program in Mathematical Problem Solving, I made sure that books were prominent along the walls.
• Give students choices in what they learn, so that they take ownership over it and want to share it with their peers.
• Show passion over the subject.  Give students role models who love learning.  Do not be afraid to digress from the material (within reason); that digression demonstrates love of the topic.
• Treat each student question as sacred.
• Make teachers accessible for out-of-classroom discussion, and create opportunities for a vibrant intellectual life to take root outside of class.
• Humor and weirdness stemming from academics make those academics a part of the culture.