The Roots of Math Anxiety

I still remember when I got my smartphone.  I’d used some friends’ smartphones to some degree, and so I knew the basics of moving things with my finger, using two fingers to zoom, etc.  However, I lacked a deep understanding of smartphones and apps, and hence, I was very nervous when using my phone.  I didn’t know what effect installing certain apps would have, nor agreeing to various licenses.  I didn’t know what the preinstalled apps did.  Essentially, I was scared of doing much with my phone because I was worried that I would mess it up, and I didn’t want to ask for help because I’d be demonstrating my ignorance.

I’ve felt math anxiety exactly three times before.  The first time was my first summer at CTY, when I first learned proof by induction.  We had morning quizzes, and I had only a surface understanding of the topic; I was worried that I would demonstrate my ignorance to my teachers, whom I greatly respected.  The second time was my first summer at Mathcamp.  Throughout the entire program I felt less prepared than my peers because I was seeing all this math that I didn’t really understand.  I knew the words behind group theory; I could talk about subgroups and say the words “quotient group,” but I didn’t know what they meant.  I was terrified that this would become apparent and that my friends would lose respect for me.  The final time was taking algebraic topology at MIT; I had failed to do my reading and I was worried again that I would be discovered for not having understood what we were learning.

To me, the experience with the smartphone and my experiences with math anxiety are very similar.  In both cases, I knew some words and some surface features of how to work with something, but I knew nothing about what made it tick.  I did not understand, if I poked in one spot, which other spots would be bumped out.  Without understanding that structure, I couldn’t do anything beyond what had been rigidly prescribed to me, and hence I felt like I was lost in a world that I did not truly understand.

I believe that this is what math anxiety is like.  If you do not understand the mathematics, and you are not comfortable repeating a given procedure to get an answer (perhaps because you sense that there should be something more, or perhaps because you are not very good at memorization), then you’ll feel like you’re in a foreign country where you don’t understand the language — or the laws.  You’re terrified that you’ll make a mistake and get arrested, but you can’t communicate with anyone to ask what you should do.  Thus, you mimic what everyone else is doing and hope that it’s good enough.

If you want to teach someone a foreign language and set of laws, we have an established way of doing so.  Start by giving them basic language tools (reading picture books, if you will) and by communicating the basis of the laws (the morality that the people share).  The corresponding aspects of mathematics are understanding the precision with which mathematics is communicated, and understanding basic arithmetic and why it works.

If we move students to more advanced mathematics without those basics, then it is that much more difficult to keep understanding, and they are forced to resort to mimicking what they’re taught.  They’re lost in a foreign country, scared of what’s happening to them.  They’re terrified someone will see them do something wrong, but they won’t really know why it’s wrong.

That’s the disservice that I believe we do to students when we force them to go to more advanced mathematics before they understand the earlier levels.  It’s the disservice we do when we try to teach how to do something to those who don’t understand the earlier steps.

Is the Gender Gap Cultural?

Why do boys so often outperform girls at the top end of mathematics achievement?  This question perplexes academics, is the source of much consternation in hiring, and has caused no end of trouble for those (like Larry Summers) who have waded incautiously into the debate.  As en educator, I want to understand if we as a society are not doing enough to help girls enter math and science.  But I’m not convinced if someone tells me that girls don’t get as many math PhD’s as boys, because it doesn’t address the root cause.  I need to see something about society’s impact on those numbers.

First, some background.  Yes, in school, boys do test higher in math than girls on average, but the difference is usually very small.  If you don’t like relying on tests, girls tend to have higher marks in school.  The key thing to understand is the score distribution: more boys score at a high level, but more boys also score at a low level, bringing the average back to about the same score as girls. Hence, men are overrepresented as top achievers, but also as bottom achievers.  (There is a provocative argument to be made that there are evolutionary reasons for men to be greater risk-takers and to have greater variation in their performance.)

Nevertheless, there is considerable evidence for strong cultural factors that impact girls’ performance.  For example, here’s some evidence for factors that may affect girls in the aggregate (but which, fortunately, are also overcome by many girls):

  • There is clear evidence of stereotype threat.  For example, if the College Board asked for test takers’ gender after the AP calculus exam, rather than before, some 5000 more girls would pass it nationwide.  Surely this stereotype danger affects not just test-taking in the moment, but also impacts students in each classroom, each day, little-by-little.  (Full disclosure: there are also some doubts about the impact of stereotype threat.)
  • Besides, in general, women don’t do as well on tests, and yet tests define a large part of our success and advancement in such subjects.
  • Even when sexism is not overt, we all have mental biases of which we’re not aware.  Just as double-blind auditions significantly boosted women’s admission to symphony orchestras (i.e. having potential candidates audition behind a screen), the same is true of peer review for journal articles.  Again, this effect must surely play out again and again throughout school, with many little encouragements or discouragements building upon one-another.
  • The clearest evidence of social factors affecting girls’ performance might be this study that shows that top-performing girls tend to consistently come from just a few high schools, while top-performing boys come from many different high schools.  In theory, this means that those high schools have figured out how to more fully develop girls’ abilities.
  • Another strong example is this study in which the math anxiety of elementary school teachers (almost all women) strongly affected girls’ math performance and perception of gender roles, but had no impact on boys.
  • Another, similar study that might find causation is this recent one in which it was found that American parents talk to boys age 20-27 months about numbers much more often than they do to girls.
  • Women who are exposed to romantic cues report less interest in mathematics, perhaps a reflection of what society views as “feminine” or desirable in women.
  • This study demonstrates several countries in which girls and boys have the same variance in performance, and also notes that there are more girls on serious IMO teams when measures of gender equity are higher.
  • Another cute example comes from a study that gave spatial puzzles to two tribes, one patrilineal, one matrilineal.  The gender gap disappeared in the matrilineal society.  (Although one has to ask, why was it merely eliminated and not reversed?)

If we search for a core issue, the popular perception of girls’ math ability seems to be a major factor.  Perception of girls’ math ability can explain many of the above bullet points: it is likely the core cause of stereotype threat, it may affect parents’ conversations with children, and it likely impacts the kind of role model that an elementary teacher might be.  If we could just address this one issue, if we could telepathically make it clear to each person in the world that women can be tremendous achievers in math and science, then we might eliminate the gender gap or significantly close it.

Now, I want to be clear: the jury is still out.  Political correctness dictates that it’s much easier to publish a study that explains the gender gap through social rather than biological factors.  The preponderance of evidence points to clear social factors that influence girls’ performance, and those social factors all seem tied to our view of girls’ ability to achieve, but there’s no way to judge what would change if we could correct for social factors.  Women might choose other careers for very legitimate reasons.  There might, yes, be biological differences.  We don’t truly know.

However, this lack of evidence should make no difference to our policies!  If it turns out that the impact of social factors is small, then not very much is lost by making a concerted effort to change the cultural perception of women’s achievement in math and science.  On the other hand, if it turns out that the impact of social factors is large, then we gain tremendous value from repairing cultural perception of women in math and science.  Moreover, it seems likely that the impact of social factors is large.

We should get rid of the cultural factors that prevent girls from making good on their ability, or we should at least strengthen the paths for girls to succeed independent of cultural factors.  Which begs the question: how can we effectively do these things?

Note: Post has been updated to add the Wisconsin study in which some countries have the same variance in performance for both boys and girls.  Post has also been updated to add the meta analysis that calls stereotype threat into question.

Where Does EdTech Fit In?

Let’s face it: educational technology is overhyped.  There’s nothing out there that gives the same education as a great teacher in a great environment.  But education technology is also underhyped!  Nothing else gives the potential for reaching so many people with the full richness of learning.

Yes, there’s a lot of potential in edtech.  But most startups fail, and frankly they fail because they’re often pretty dumb about what education really needs.  If we’re going to improve, we have to stop using technology “because we can” and instead use technology where it actually makes a positive difference.

I’ve long been an EdTech skeptic who believes that great things are possible.  While I am far from an expert on the field, here are my thoughts on the two big categories of EdTech, the failure points they’ve experienced so far, and the potential for excellence.

Technology for the Classroom

AKA “working with teachers,” this is technology designed to aid teachers.  Two favorite examples include Reasoning Mind which offers a computer-based math curriculum using Russian ideas in math education, and the “inverted classroom” model, in which students get content from videos or other sources outside of class and then do problems in class.  (The idea is that you can learn the basics from anyone, but time with a teacher is precious and should be used where the teacher is most valuable.)  I also like BLOSSOMS, videos made by MIT folks where the in-classroom teacher shows part of the video, does an activity with students, shows the next part, does another activity, and finishes up with the rest of the video.  BLOSSOMS is great because it brings experts into the classroom while working well with the classroom structure.

Why do these fail?  It depends on the tech.  Sometimes, the big innovation is trying to give teachers access to more data so that they can diagnose what their students don’t understand and help them—these fail because entering the data is too clunky, and takes too much time.  Sometimes the innovation is giving the teachers better ways to present material—although the SMART board has succeeded, most other technologies fail when the complexity of creating lesson plans goes up.  I’ve seen tech based on helping teachers decide which student to call on (often not enough benefit for time/cost), tech based on giving all students iPads (not enough educational material to make it worthwhile, and too distracting), and so forth.  Everyone has their idea for what teachers really need to become more effective, and they’re usually wrong.  That’s not entirely true—sometimes the innovators are right, but they haven’t taken the time to align to standards, nor have they made it easy for teachers to find and use their resources.

What are the opportunities?  Some innovations have potential for greater efficiency in classroom time (such as the inverted classroom), and technology should be able to help here.  Technology could also allow for better in-class problems customized to each student (adapting to their prior work) or better assessments.  We’ll eventually find a way to get teachers better data about their students.  Finally, people keep talking about well-designed interactive apps that would allow students to explore material in a non-linear fashion or to do projects where students analyze real-world data.  Although nothing seems to have taken hold just yet, it’s a tremendous opportunity.

Technology for Outside the Classroom

This is designed to skip the in-person component altogether, optimized for self-paced study or for online classrooms.  The motivating forces tend to be either “make education accessible to everyone” or “take the work of the best teachers, distill it, and scale it.”  Favorite examples include Khan Academy (even though I think there’s lots of room for improvement) and Art of Problem Solving.  There are also tons of online schools of various sorts.

Why do these fail?  Often, these technologies don’t account for how much less engaging it is when you’re not in front of a real person.  (That’s why live sporting events and live plays are still popular.)  They also don’t account for our short attention spans.  And, most critically, they don’t account for the importance of being around other learners, part of a community that spurs you to greater learning.

What are the opportunities?  The trend towards adaptive learning is a big plus here, although the give-and-take of adaptive multiple-choice questions remains less compelling than real human interaction.  For very motivated learners or those with someone (e.g. a parent) watching over them, online schools may be compelling.  There’s also an opportunity in short, awesome online content that students can watch, share on Facebook, etc., and then perhaps follow up on in greater depth.  However, I think the real future is in social networking, where students can watch a video and chat live as the video plays, or share questions with a small group of trusted online friends who can help them through the material.  As near as I can tell, that’s the only hope for a real online community of learning that will keep students learning.

The problem with education technology is that there’s a lot of “cool stuff” out there that is genuinely very cool—but it just doesn’t actually educate that well.  I think we’ll get there eventually, as more collaborations evolve between knowledgeable educators and really awesome tech innovators.  I’ll even share some of my own ideas in a future post, places I think there’s space for real innovation.

Meanwhile, as an educator, I’m not holding my breath.  I’m happy to adopt things that come around that will really make a big difference, but until they do, don’t expect me (or anyone else) to jump on your bandwagon!

Building a Better STEM Pipeline

Recently, I attended the Building a Better Commonwealth forum hosted by the Boston Globe on “Building the Talent Pipeline.” In other words, how do you produce more high-level STEM jobs? Here are some of the things that I learned:

  • Massachusetts is unique for having a statewide plan for developing STEM talent.   Like the (national) report from the President’s Council of Advisors on Science & Technology, the MA plan discusses the need for nurturing curiosity, suggests a strong attention to standards, wants to get more people pursuing STEM, and wants to more effectively prepare teachers.  Unfortunately, it seems that both of these reports fail to address allowing students to excel.  (See below.)  That said, the Massachusetts plan is very impressive, with an eye towards creating coherent curriculum, good experiences, role models, and much more, and a plan for implementation.  If MA pulls this off, it could be quite significant.
  • Apparently STEM learning in MA is particularly pushed by Lt. Gov. Timothy P. Murray. It’s interesting to see how this kind of project gets high-level support.
  • There is a dramatic hiring shortage in STEM. (Well, I already knew that.) But the STEM jobs are not just the desk jobs that you might have in the financial industry or at Google. They’re also repairing and servicing power lines, because the job is so dangerous that you need to know what you’re doing. They’re also factory jobs, which are so automated that they require special expertise. And so forth.
  • It’s really weird to explain that you live in Boston but run a summer program for New York City students.

Here are three reflections:

  • Parents must know what their kids need and push them towards it.  It’s great that there are all of these initiatives, but ultimately, it’s very hard to make up for low-knowledge parents or communities.  There is too much that students learn about from parents; too much push that parents need to give to apply for or attend other programs (how many times did your parents wake you up on a Saturday morning to make sure you got to your activity?); too much information where parents must be part of the process.  Usually, it’s not that parents don’t care deeply for their kids (just the opposite), but that they don’t have a model from their own parents of pushing kids in this way and so it’s not natural to them to provide the push.
  • Rigid curriculum can be severely limiting.  Enforced curricula from a state or national level can block hands-on learning, customization of a student’s work to their interests, or customization of a class to a teacher’s skills.  I think there are good ways to design a curriculum that is flexible to this kind of work, although it hasn’t been done yet.
  • The community does not understand the difference between baseline achievement and excellence.  Most people in education do not realize that getting all A’s does not make you a world-class student.  Even taking lots of AP courses does not necessarily do so.  Students today do research, attend summer programs, do various competitions, and more.  There’s a national infrastructure set up by small independent groups designed to help students achieve excellence, but no one in education pays attention to it.  In part, it’s because there’s a perception that all-A students have “already made it.”  In part, it’s because it can be politically daunting to want to help top students succeed more.  In part, it’s because of a lack of knowledge.  But this kind of structure is necessary to create the best-possible STEM workforce, and it doesn’t have to be based around selective admissions.  There are many good opportunities that anyone can participate in and gain the opportunity to excel.  Yet the constant focus on creating new STEM standards and testing is never going to produce a STEM workforce because it will always be based around broad ideas that don’t develop the top students.

As you can tell, I’ve been thinking a great deal about what a good STEM pipeline should be.  Look for some systemic thoughts in the coming weeks.


Apologies for the long absence.  The past few months have featured intense work and family obligations.  I have missed blogging, and now I’m back!  My hope is to spend some of the next few weeks thinking carefully about what a good pipeline looks like for developing a STEM workforce.  We begin with the next post, my reflections on a recent Building a Better Commonwealth event.

Assorted: long-term thinking, accepting the other side, and math/science preparation

It’s Splash season, so my blogging has fallen behind.  To keep you satiated, here are some interesting articles I’ve read recently and some thoughts on each.

  • Harvard Education Letter writes about the Waldorf model of schooling.  A small movement, but one that even has some Gates Foundation support, Waldorf schools have a very long-term outlook to children’s success.  They do a lot more art, and a lot more personal exploration.  “In early grades, strict Waldorf classrooms delay overt academic work in favor of imaginative play and movement centered on myths and fairy tales. Multiplication tables, for example, are not taught until fourth grade, although kindergarteners may gain early math skills as they knit. Even high school students studying science find a narrative focus as a teacher describes how Charles Darwin struggled to conceive his theory of evolution. Students may draw muscle cells to learn about them. There are no textbooks; students create their own “lesson books” to chart their learning.”  Test scores are rising at this small sample of schools, although that might just be because their leaders are particularly passionate.  Regardless, the article provides an important reminder that when you focus in single-mindedly on one goal like test success, you can forget what’s really important to bringing a child into adulthood, and in the process sabotage your own test results with short-term thinking. (*)
  • Washington Post reporter Jay Mathews blogs about improvements at his old high school.  The twist?  He’s strongly reform minded (use tests to evaluate teachers, more charter schools, etc.) and the school’s reforms are precisely the opposite.  I normally hate writing from partisans in this debate because they just talk past each other.  Here is one partisan showing how the other side has done something significant.
  • The Wall Street Journal writes about students who switch to easier majors despite lower pay.  College courses in science and math are a huge step up from high school courses in terms of abstraction and the independence required to succeed, and so students change majors.  Some places seem to want to make the college courses easier, but that leads to insufficiently prepared graduates.  Naturally, I think the solution is for high school students to study more abstract topics in greater depth.  That’s not on the horizon, so a lot of colleges offer remedial work—which is just like a high school course, but maybe sped up!  Most students fail their remedial course, or at least that’s what happened when I was at the University of Illinois.  So instead of offering more of the same, I propose that colleges should develop deep, abstract “remedial” courses that teach math and science the way it should be taught.  These courses will adequately prepare students and the ideas within these courses might trickle down to high schools, becoming the standard of college preparatory work.

(*) A more common example of short-term thinking: teachers cram test-prep into the end of the school year, because it’s so important to them and students that the students pass.  But students forget the test prep and lose learning time from it, so in future years, they have less to build on.

The Serendipity of Learning

I’ve spent much of this week doing interviews for Early Action applicants to MIT.  This volunteer role lets me meet a lot of very cool young people.  Moreover, although I live in Boston, I do interviews in my home town of Vestal, NY to a radius of a couple hours out.  It’s a sparsely-populated region, so I meet a wide mix of folks.

This year, I’ve been struck by how much applicants are shaped by their surroundings.  The difference in resources between schools and communities is stark.  At one extreme, there are famous magnet schools like as Stuyvesant or Thomas Jefferson which have highly-advanced courses, advisers to help students complete Intel research projects, training for national-level competitions, and high awareness of advanced summer programs.  At the other end, rural or inner-city schools that have virtually no AP classes.  But right near Vestal there is also wide variation.  Vestal, my alma matter, has a number of academic clubs, advanced study through both AP courses and an IB program, and, for those students who take initiative, the opportunity to take classes at SUNY Binghamton (one of the top New York public universities) or even to do research there.  In contrast, I’ve interviewed students from 30-60 minutes out whose schools have fewer than 5 AP courses available, and where awareness of out-of-school opportunities are minimal.  There, if the family has sets the right kind of tone, then kids can discover serious resources and potentially make a lot of progress.

It’s easy to forget in all of the debates about testing, teachers, and curriculum that our learning and achievements are strongly defined by our communities, their awareness and emphasis on learning, and our opportunity to learn.  The best students I’ve met from more remote schools had families who provided them with reading suggestions and role models.  Creating opportunity to learn (and the inspiration to pursue it) has been a driving force of my career in education from Learning Unlimited to the Summer Program in Mathematical Problem Solving to Mathcamp.

Unfortunately, you can’t just say “here, kids, have an opportunity!”  There are innumerable influences on students’ time.  To pursue real studies, they need some mix of older peers to be role models, teachers who will be supportive, knowledge of how to pursue those studies, a sense of where serious study will take them, and more.  Yet even those opportunities aren’t quite enough.  Most of what we choose to do comes from serendipity.  We hear about an opportunity right when we have a moment to Google it.  We have an argument with a friend and, feeling isolated, we immerse ourselves in our work—just when an interesting topic comes up—and because we happen to do well on that test, we decide to pursue that topic more seriously.  Our attention catches on random things, and if our world isn’t permeated with topics of interest, we’ll miss those serendipitous moments.

Which is to say that we have a long way to go if we’re going to get more people to pursue learning outside of school.  For my part, now I’m thinking about organizing a “math day” for anyone within 90 miles of Vestal sometime in the spring so they can get a taste of advanced math.  What are you going to do?