Must Everyone Appreciate Math’s Beauty?

Why mathematics?  What purpose does it serve?  Why should we teach it to students?  There tend to be roughly three broad categories of answers to this question:

  1. It is useful.  Math is needed both in daily life and in work, and without a strong math education you are closing doors to excellent careers.  In this vision, the math curriculum should be designed to teach specific knowledge and skills.
  2. It trains you to think.  Math poses deep and abstract problems.  Facility in solving these problems strengthens your critical-thinking skills across the board.  In this vision, the math curriculum should be designed to build abstraction and pose challenging problems.  (Warning: there may be some research that says that critical thinking in mathematics does not transfer to other disciplines.)
  3. It is pretty.  Math is gorgeous; it should be explored and enjoyed.  It is one of the great achievements of humankind, and every student should have the opportunity to appreciate and understand it.

I believe deeply in all three.  It is dangerous to ignore any of them.  If you design a class that explores the beauty of pure and applied mathematics, builds abstract thinking, and poses deep problems, you can easily meet all three criteria together.  But when you ignore one of them, mathematics suffers.

It’s for this reason that I am not nearly as enthralled by Lockhart’s Lament as many of my fellow mathematicians.  Lockhart’s Lament is a powerful and eloquent description of why mathematics is beautiful.  I couldn’t agree more with all of the points Lockhart makes for the beauty of mathematics.  However, when he proposes to scale all teaching towards the exclusive goal of beauty, I believe the result is both dangerous and unrealistic.  (Lockhart later wrote a clarification that he did not mean to be making a proposal nor to imply a dichotomy between pure and applied mathematics; although he walks back from what the Lament says, it fairly clearly speaks for itself and it is what is being passed around between mathematicians.)

Teaching only for beauty is dangerous because without encountering specific content, students would not learn mathematics that is important for their lives.  Those who will not pursue math-oriented careers would not have the tools to manage their lives or interpret the world around them, while those who will pursue math-oriented careers would find themselves without the necessary tools.  Besides, not everyone finds mathematics beautiful for the same reasons.  Some find it beautiful for its applications and relation to the physical world.  Some might not find mathematics beautiful at all, and I do not consider that a failing.  Just as I do not particularly enjoy opera and do not want its appreciation forced upon me, I would not want to force appreciating mathematics on others.

Teaching only for beauty is unrealistic because we simply do not have the teaching force that is capable of teaching mathematics purely for its beauty (as Lockhart notes).  Moreover, even if teachers were in a position to do this, if each teacher designed their own curriculum with no guidance then classes would be filled with students with very different background knowledge and designing coherent lessons would be impossible.

Paul Lockhart teaches at Saint Ann’s School, a private school in New York City whose tuition ranges to over $30,000/year.  It has a free-form curriculum focused on the arts (and with many children of artists attending).  It is a wonderful school, ideal for Lockhart’s self-described teaching style, and I have known several teachers from there who are quite remarkable people.  However, to take that experience and generalize it to all of K-12 education is neither wise nor truthful.

In the end, I believe that it is possible to construct a school mathematics curriculum that combines all of points (1)-(3) above.  We should not focus on one at the expense of another; we should instead see the wonderful ways that they can all work together.  Lockhart’s essay is a wonderful demonstration of the power of mathematical beauty in teaching.  It is now time for the discussion among mathematicians to move beyond this very satisfying lament and on to incorporating all aspects of mathematics.

In his follow-up/reply to critics, Lockhart writes, “My point is that at present we have neither Romance nor Practicality – nothing but a jumbled, distorted mishmash of pseudo-mathematical vocabulary, symbols, and mindless procedures.”  With this I completely agree.  With his proposed solutions and measures of success—well, I’m not there yet.

Can students think like experts?

Which of the following makes for a good history lesson about the underground railroad?

  1. A traditional lesson in which the background of the underground railroad is presented, major players in the story are introduced, an explanation is given of the politics of the time, and the conditions that escaped slaves went through to get to freedom are presented.
  2. For a fourth-grade classroom, have the students bake biscuits, which were a common staple on the underground railroad, in order to gain an understanding of what runaway slaves did and ate.
  3. Give tenth-grade students original documents from the underground railroad that they use to draw and debate conclusions about the underground railroad (like real historians do).
  4. Run a class discussion in which students are asked questions to elicit their understanding of life within the underground railroad.  How did runaway slaves obtain food?  How were they able to prepare it?  How were they able to pay for it?  What might the consequences have been for someone who housed runaway slaves?  And so forth.

Easy answers: (1) sounds fine if you need to transmit lots of knowledge, but boring.  A good lecture will communicate facts as well as connections between those facts, and it will build important neural connections to help students remember and process data.

Option (2) is nonsense.  The students will spend their time thinking about measuring flour, not the underground railroad.  (Math teachers, take note—this happens in your classes, too!)

Option (4) is great.  It prompts students to reflect on what they have learned and it leads them to make connections between topics.  By giving each of these items real thought and relating it to other facts they might know about the time period, their overall understanding will be greatly strengthened.

In his fascinating book Why Don’t Students Like School?, Daniel Willingham argues that option (3) is a mistake unless your goal is building student excitement.  But… surely not, you say!  Is this not what real historians do?  If history class was about analyzing real documents, would that not more deeply engage students’ minds and show them what the field was really like?

Not so, says Willingham.  This kind of exercise is asking students to think like experts.  However, he says, the cognition of experts is fundamentally different from non-experts.  Here’s how:

  • Experts think in terms of functions or deep structure, while novices think in terms of surface features.  For example, suppose you ask novices and expert chess players to memorize a chess board.  When replacing the pieces on the board, the novices will put them down in clumps organized by location on the board.  Expert players will put them down in clumps organized by function—pieces that are threatening each other will go down all at once, even if on different ends of the board.
  • Experts’ focus on deep structure allows them to ignore unimportant details and immediately see the useful information.
  • Experts have procedures that are essentially automatic from repeated practice, so they can execute those procedures/thought patterns without the mental overhead of novices.

Willingham then notes that becoming an expert takes 10,000(ish) hours of practice.  He reflects on judging science fairs where students churn out lots of basically useless science experiments, with huge flaws in experimental design of which they are blissfully unaware.  Besides, for this particular history exercise, students don’t have nearly the background knowledge to make useful connections; they’ll never get into the deep structure, and will instead be stuck on surface features of the documents.

I’ve been wrestling with the idea that we should not challenge students to think like experts.

On the one hand, when I look at the classes I teach, I don’t generally challenge students to think like experts.  I’m not asking students to discover new mathematics in my math classes, and with good reason—I don’t think most discovery problems encourage reflective practice.

On the other hand, I don’t believe the 10,000 hours claim, because it fails to look at why someone would spend 10,000 hours playing chess.  They will probably only do so because they see some of the deep structure of the game, which inspires them to want to explore it further.  Someone’s decision to pursue a topic is a complicated interplay of factors.  To become an expert, you need to feel like you’re already on that track.  So if we want students to pursue expertise, we must challenge them sometimes to achieve it and give them experience with how an expert thinks.

Also, the science fair conclusion seems totally wrong to me.  Just because students do not do something well does not mean that there isn’t value in doing it!

Which leads me to conclude: challenge students to think like experts, but rarely and judiciously.  Option (3) is appropriate, but only after students have built up enough background knowledge of the underground railroad that they have connections and context to draw on.  If you can build up enough experience with that aspect of history that students can work with the documents, then they might be able to draw some new conclusions and learn from the experience.  But most of the class can’t be structured this way, or the basic connections students require to set documents in context will never have time to develop.

Knowledge, Character, and the Next Mole

Yesterday, the New York Times had an article by Paul Tough about developing character in schools.  Citing the academic success of a middle school KIPP charter in NYC at achieving excellent test scores for its low-income students and getting them into selective private schools, the article then went on to show how still only 33% of the KIPP graduates got a college degree.  (Frightening fact: only 31% of middle school graduates nationally get college degrees, according to the article.)  KIPP founder Dave Levin’s solution: help his students develop character.

The article inadvertently demonstrates how education reform is often like a game of whack-a-mole.  We find a target to aim at and we focus our energies on it, only to see that there is another obstacle that pops up.  For example, the No Child Left Behind Act was instituted to guarantee minimum competency for all students regardless of background, but the nature of the tests used to guarantee this “proficiency” focused learning so tightly on a few specific subjects (and narrowly within those subjects) that schools left out important topics and skills students would later need.  Similarly, in the article, KIPP achieved real learning and proficiency on the part of their students, only to discover that the students didn’t have the grit to stick it through college.

Tough goes on to describe how the NYC KIPP schools have started teaching character.  Not “moral character,” but rather “performance character”: things like effort, diligence, and grit.  They’re whacking the next mole.

Unfortunately, I’ll bet that there will be yet more moles after this one.  For example, some of my own work deals with developing education-dependent adult identities for students to understand why they’re going to college.  Students also need a broader understanding of what opportunities exist for them in order to find the right path in college and afterward.

So how do you build a program that really helps students succeed?  It’s not easy to find effective methods.  Imagine trying to test out the effectiveness of this kind of character education.  There’s no test that will show immediate improvement just because some students have a bit more grit each year, so you need to do a longitudinal study (very hard and expensive!) to see how grit helps them in college and afterward.  It would take at least a decade.  And then, what if you made the mistake of doing your study on an ineffective school?  Students without sufficient knowledge and skills wouldn’t benefit nearly as much from character education, so we might conclude that this intervention doesn’t work at all, when in fact it does work in a school like KIPP that has a good grasp on the basics.

But let’s assume we got lucky and tested character education in the right setting.  We’d no doubt find many different interventions that are successful, and now we’d need to understand how these interventions work together.  A rigorous and controlled academic curriculum like KIPP’s might pair well with this kind of character education, while the effort might be wasted elsewhere.  It would take centuries to test all of these combinations.

Building a good program is hard.  (Kudos to KIPP for tracking its students, discovering their failures, analyzing them, and finding an underlying reason.  That takes incredible leadership.)  We can’t possibly test every combination of in-school methods.  Successful methods are often contradictory, or work against each other when implemented together.  Rigorous studies can often provide some clues as to what works, but we still need to trust ourselves to make good choices.  We still need to look at each program holistically and not let ourselves be so beholden to research that we can’t use our common sense to design a curriculum and a school.  Nor can we measure outcomes with a test at the end of the year and conclude that we’ve understood the impact on students.

In the end, this is a flaw with the slew of programs that are developing with laser-precision focus on specific outcomes.  Without thinking about all aspects of a child’s education, we’ll miss things.  The need to explain a program in a single sentence, to give a “statement of need” that explains what niche you fill, misses the real breadth that a successful program has.

We can and should measure outcomes, but we should also admit that our measurement is incomplete and trust our gut.  I really do wish that there was an easy way to find effective methods and combine them into a coherent program.  But because there isn’t, we will have to instead go back to basics: have good people at all levels of the education system and enable them to do good work that is guided by the research but not beholden to it.

Math Literacy vs. Math Workforce Development

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?