Beyond Motivation to Self-Identity

Why do people want to do things?

I’ve been asking myself this question a lot.  If you believe that teaching is about more than simply imparting knowledge and skills, that it is also about inspiring students to achieve greatness through academic study, then you have to ask yourself: how do you give students experiences that will make them want to learn your subject and then apply it in a future career?

For many educators, the answer is often  to show students that the subject is beautiful.  For others, it is to show students that the subject is useful.  Educators talk about motivation, and sometimes intrinsic vs. extrinsic motivation.  These are all very useful conversations.  I think they are also insufficient.

Let me give an example from my own life.  I love playwriting.  I wrote some plays that I consider decent while in college, and I would love to write some more plays and perhaps to see them performed.  Yet I “don’t have time” to keep writing plays.  Why not?  Well, for one thing, I have a job doing something I love—education—and that job probably pays better.  But I could still write plays as a hobby!  I don’t, because there are other things I can do for fun, and because most of my friends are math/science-oriented and so I don’t have a community of people to talk to about it and encourage me.  Instead, I go to see a lot of plays each year and content myself with occasionally imagining what plays I would write if I had the time.

My point is that even if someone likes doing something, that is not enough for them to dedicate their time to it.  Just as we recognize this fact in our own lives, we should think about the many factors that play into our students’ decisions, both consciously and unconsciously:

  • Do they see it as a productive/gainful use of their time?
  • Do they have a supportive peer group?  Will they be able to do it with others?
  • Will it give them social status?
  • Can they see themselves doing it in the future?  Is that future self someone they like and admire?  Do they have role models?
  • Is there an established pathway for doing it that they understand?
  • Do they expect success at it?

I believe that focuses on curriculum or study skills or content knowledge are all good steps, but they are insufficient to the task.  Just because students learn that mathematics is useful for building bridges does not mean that they see themselves doing it.  We need to consider the whole context surrounding a child, the whole environment that might encourage them to become a scientist (or not), and how we can make it more likely that they see math and science as a viable pathway.

Ultimately, I think we need to build their self-identity as scholars.  To me, self-identity goes much farther than just motivation, be it intrinsic or extrinsic.  Motivation plays a part, but self-identity is about how they see themselves.  Indeed, I claim that without a resilient self-identity, all of our efforts to teach knowledge and skills are less effective.

Of course, saying that we should accomplish all these things is a far cry from specific proposals to do so.  I hope to explore more about self-identity in future posts over the coming weeks: to try to give a better definition, and to give concrete thoughts for how to help students develop it.

Careers in Mathematics Education

You’re a well-qualified graduate in a STEM field.  You could make lots of money in Silicon Valley or in finance, doing interesting things, but you want to be in education.  What can you do that makes good use of your talents and maybe even lets you feed your family?

Too many people don’t understand that there are good career opportunities available.  There are curriculum development roles; education technology companies; programs that cater to more motivated students; and all kinds of exciting smaller initiatives.

Teaching can also be a tremendously rewarding career that mixes many different kinds of very interesting challenges.  You engage with academic material on a fundamental level, but you also have some very deep engagement with ideas in pedagogy and psychology.  You also are in a very social career, so you get to interact with many interesting people and see the impact that your work has on them. The pay might not be great, especially initially, but there are prestigious fellowships that can supplement your pay and ease your transition.

This post exists to share what I’ve learned about exceptional opportunities in education that can be part of a serious career.  I hope that it will be a resource for those who would want to pursue work that we so desperately need.

You want to…

  • Change the mathematics classroom: Consider Reasoning Mind, a 140+ employee company that develops math software for elementary instruction.  Their work produces great results and is based on serious mathematics.  Math specialists can be “Knowledge Engineers”, and they are very rigorous about who they hire; many of them have PhD’s.
  • Work with talented students: A natural place to go is Art of Problem Solving, which creates outstanding curricula for elementary through advanced high school level and teaches it online.  They have a huge online community of dedicated students and lots of innovative online tools for them to do math.
  • Work with talented, underserved students: My own program, the Summer Program in Mathematical Problem Solving, is now hiring a Director of Programs to take over leadership and expansion of our work.  During the summer, we also hire instructors, and while you’re in college, you can be a residential counselor/TA.  Over the coming years, there may also be year-round curriculum development work.
  • Develop material about exciting mathematics: The National Museum of Mathematics is developing curricula surrounding their exhibits and also gives you the opportunity to teach to students who come in for field trips.
  • Work with math circles: Josh Zucker makes his living as an “itinerant math teacher”, running and teaching at math circles as well as online at Art of Problem Solving.  My friend Japheth Wood, on the other hand, makes his living as Executive Director of the New York Math Circle.
  • Teach: Do you want to understand the subtlety and rich intellectual life that goes into teaching really well?  Take a look at Sameer Shah’s blog, or Dan Meyer’s blog, to see how really smart people approach mathematics teaching seriously.  If you’re worried about the pay, consider either the Math for America Fellowship or the Knowles Science Teaching Fellowship.  Another way that many people try this out (tends to produce a love-it or hate-it result) is through Teach for America.
  • Do technology: I recently had a great conversation with Zach Wissner-Gross who founded School Yourself which makes lovely interactive tools to study mathematics and is a startup you might want to look at.  You could work at edX or Coursera or Khan Academy.  If you want the kitchen sink, here’s Quora’s list of education technology startups, but beware that there may be a mismatch between many startups and what works in practice.
  • Science enrichment: Unfortunately the sequester has put something of a hold on this for now, but NASA has always produced excellent science outreach materials.
  • Do Research: A number of my colleagues, including Yvonne Lai and Nina White, have made the transition from mathematics to research in mathematics education as well.  Math ed research is in serious need of qualified mathematicians right now!

Even more exciting, this is not a comprehensive list of opportunities.  There are many, many other organizations out there, and you can always start your own.  In fact, just as many underserved students don’t know the landscape of colleges, enrichment programs, or selective schools, many of us in the math world don’t know the landscape of education organizations that would love to have more qualified mathematicians working with them.

If you want to pursue education, you should go for it.  There are great careers awaiting you if you’re very good at what you do.

SPMPS is Hiring!

The Summer Program in Mathematical Problem Solving is hiring!  For those who are not familiar: SPMPS is a free three-week residential summer program for underserved NYC middle school students held at Bard College.  We do year-round followup to help students apply to selective schools, summer programs, and math circles; we also bring them to places like Google NYC and the Museum of Mathematics to help keep them engaged in a mathematical world and culture.  Our goal is to pay attention to all aspects of students’ development, helping prepare them academically, socially, and emotionally for advanced study.  It’s also a lot of fun, and everyone leaves thinking of it as a second family.

We’re actively looking for two roles and I would love it if folks could pass these on to interested people:

  • Instructors design and teach their own classes during the summer in addition to becoming part of the social fabric.  Topics could be combinatorics, number theory, voting theory, math and the arts, astronomy, and more.  Instructors get a stipend, free room and board, and travel.  You can see our past instructors here to get a sense of the folks we’re looking for.
  • The Director of Programs is a year-round full-time position.  She or he will take over the management of the program and also lead its expansion in future years.  This needs to be someone entrepreneurial, able to do something of everything, and very competent and aware.

The full job postings are here.  Please pass both on to anyone whom you think would be a good fit!

Minority Students and Private Schools

I just finished up at the Summer Changes Everything conference in Pittsburgh, held by the National Summer Learning Association.  I was a skeptic coming in but it was a great experience, and I got some great ideas for staff training, fund raising, and family and alumni involvement in SPMPS.  Also some crazy ideas about starting year-round math circles in kids’ schools.  We’ll see what comes of that.

There was a lot of talk at the conference about Paul Tough’s new book and non-cognitive skills in general.  It’s fun watching these fads go through the ecosystem!

On my reading list for the way home was this article in the New York Times about minority students and their struggles at integrating into private schools.  Despite schools’ best efforts, minority students often feel left out.  Even though schools provide much more than full scholarships by paying for books, clothes, and so forth (so that scholarship status is not obvious), it’s not enough.  Backgrounds are different, stereotypes are strong, skin color is different, and cliques persist.

This is a well-known problem, and my program, the Summer Program in Mathematical Problem Solving, seeks to counteract this by preparing students socially and emotionally for high-performing math environments so that they can integrate more easily and feel less different.  (For example, we teach them games like Set and Ultimate Frisbee, and we give them camp-like experiences such as games, hiking, and a trip to Six Flags.)  The Times’ article just goes to show what a big challenge this is, and I can only hope that our efforts will be successful.

Still, I have to admit that I was shocked at what the article describes.  Thinking back on my own high school experience, it makes sense that high schools are so “hands-off” when it comes to socialization.  But in contrast, at SPMPS and at Canada/USA Mathcamp, the counselors meet regularly and discuss what students seem isolated and how to help them integrate.  They look at cliques that might be forming and consider how they might subtly help those cliques open up to more students.  They find lonely students, befriend them, and introduce them to their peers.  If either of the programs saw race-based or scholarship-based cliques forming, we’d certainly take action to help students integrate.

Perhaps this is the difference between a summer program (where social/emotional growth is an explicit priority) and schools (a more laissez-faire environment).  Perhaps the summer programs have an advantage because they have undergraduate counselors who are role models and friends to the students, so it is easy for them to integrate into social groups and help students mix.  Regardless, I wish that schools considered taking a more proactive role in helping students integrate and meet the full diversity of peers available to them.  I wonder what such a school might look like!

What should we teach young students?

Recently, Paul Tough released a new book about non-cognitive skills (link to a highly-recommended This American Life episode where Tough discusses the book).  Tough emphasizes the importance of things like grit, the ability to deal with setbacks, the ability to postpone gratification (whose importance can be seen in the remarkable Stanford Marshmallow Experiment), curiosity, self-confidence, and so forth.  The book is meant to present a new view of education: that, like KIPP schools, we must focus on teaching kids these essential skills for success.  It also emphasizes that kids generally have to learn these skills quite early.

There was also recently a New York Times story about admissions for the specialized high schools, emphasizing the word gap: that kids with higher socioeconomic status just plain hear a lot more words growing up.  Like any social science experiment, it’s hard to formally establish a causal link, but it’s intuitively clear that a word gap would impact their vocabulary, comprehension, and communication skills from kindergarten on up.

In E.D. Hirsch’s review of Tough’s Book in Education Next, Hirsch emphasizes that Tough has only part of the story.  He says that vocabulary—and also basic information (such as if a student can locate Africa on the map)—is the major determinant of a student’s outcomes.

Everyone agrees that good early education is essential.  Many kids do start kindergarten well behind their peers.  They’re behind in reading and math.  Even in early childhood, there’s an illusory IQ gap (I say “illusory” because it becomes smaller when adopted students from low-income backgrounds are raised in more affluent homes).  Underserved students do worse in school and often act out more.  In the end, they have just an 8% graduation rate from college.  (Compared to the population at large, an also-depressing 33%.)

So what should we teach kids?  Should we teach them non-cognitive skills, or should we focus on building their vocabulary?  Is the bigger challenge that underserved students have few non-cognitive skills, or less basic knowledge?  What are the raw materials that allow students to learn better later?

Tough’s answer is to focus on non-cognitive skills.  The New York Times editorial seems to suggest just the opposite: focus on building vocabulary.  Hirsch would add that we must focus on cultural literacy, which enables communication and further learning.

A better answer, though, is obvious: the “all of the above” strategy.  There’s evidence that all of these kinds of early learning are important, and so we should design early experiences that develop all of them, either through direct work with students or through parent education, like what Geoffrey Canada does with the Harlem Children’s Zone’s Baby College (another great This American Life episode).

I claim, further, that this whole argument is a red herring.  The best way to teach both non-cognitive skills and basic knowledge is to do both at once.  In fact, I’ve never seen a content-less effort to build skills succeed.  You learn to study hard by studying something hard.  You learn to solve hard problems by tackling a real problem—but real problems require serious background knowledge.  In other words, these different skills are not in conflict at all.  They are best taught together, with conscious awareness of all the elements that are being incorporated into a curriculum or program.  You still need a rich environment to impart simple facts that do not come from traditional studying: the location of Africa or a large vocabulary.  But beyond that, deep material learning should be paired with skills building, and any other way is a major failure of education.

A real education requires learning facts and learning skills to learn more facts.  While it is possible to design a class that teaches facts without teaching skills to learn more facts, it’s dumb and it doesn’t work in the long term except for a select few students who pick up the necessary skills.  Cognitive and non-cognitive skills must come together, and by putting them together in a conscious way we will develop the strongest educational experiences for students.

What I Want From Science Magazines

Middle school is an age where interest in math and science drops, severely.  It’s terrible timing, because students are just getting to the point where they can understand (or be frustrated by) serious scientific study.  Fortunately, middle school is also a big opportunity: becoming excited when you’re young can carry you through many years of potentially tedious science in school.

I’ve been thinking about how to help students from SPMPS become excited and energized about math and science.  At their age, I was reading things like Discover Magazine, which made me yearn for understanding stars and galaxies and black holes, higher dimensions, particle physics, and science in general.  I had a model of the ideal scientist in my head, and that was who I wanted to be.  It was the driving force of my learning in school and beyond.

Discover, it turns out, is not what it once was—as I found out when I bought myself a copy while waiting for a tow truck.  It’s not a bad magazine at all, but it is very much about “hey look this discovery happened!” and very little about the science being done.  (For all I know, it was like that when I was a kid, too, but I’ve forgotten.)  It also has an inconsistent level of language throughout, so that many articles might be above the reading level of the SPMPSers.

I’ve also tried out Science News and Scientific American.  I didn’t know Science News when I was younger, and it is a nice magazine that also focuses on what happened and not understanding the underlying mechanisms.  Scientific American, which I remember being too advanced for me when I was younger, has become much simpler, and does talk about the underlying mechanisms of science.  It’s probably the best of the magazines I’ve looked at, although it doesn’t convey what doing science is like—it just talks about the results, the discoveries, and why they’re true.

Searching for a way to interest young students in science has made me realize how much is lacking.  Why is there no magazine that explains how the experiments (which are often quite lovely!) were set up, that asks students to think about what conclusions they might draw from whatever facts are available?  It could be a mix of long-established science (“why is the sky blue?”) and new discoveries to get people excited.  As far as I know, there’s nothing like that—nothing that aims to develop scientific thinking with professional writing, great production values, and good content.

Another gap in the field is communities for like-minded students to discuss.  At SPMPS, I was showing one of the students Art of Problem Solving.  His question: was there any discussion areas like this for science?  Not that I know of, unfortunately.

Classrooms often try little tricks to get students excited.  They play games, or watch videos, or do overly-structured lab experiments.  That was never where my interest came from, however.  Part of my interest came from the joy of figuring out how different pieces of the scientific puzzle fit together, and most of my drive from admiring the science being done now and my desire to be doing it.  There should be more avenues to encourage this kind of passion.

SPMPS June Update

Yesterday was the student and family lunch for the Summer Program in Mathematical Problem Solving, and we have an absolutely phenomenal group of students this summer.  We carefully set a tone of serious academic study combined with great fun, and the students are very excited about what they’ll be doing.  You can follow the events on the SPMPS summer blog being maintained by Ana Portnoy.

The rest of this post is an e-mail that I send informally to supporters and other people interested in SPMPS.  If you’re interested in receiving these updates, drop me a line.  Future e-mails will be duplicated here as well.

Dear friends of the Summer Program in Mathematical Problem Solving,

The past months have been incredibly exciting as we’ve met about 115 students from across the city who have interviewed for SPMPS.  The selection process is done, and most of the students have confirmed that they’ll be attending this summer.  We’ll have 40 students, more than twice as many as last year, all coming from schools where 75% or more receive free or reduced-price lunch.  I can say from having interviewed many of the students myself that they are really outstanding kids.

Last summer was documented in a fantastic video produced (pro bono!) by Big Green TV.  It will be broadcast on July 8 at 10:30pm on WMHT, public television for New York’s capital region.  Take a look!

Before I get to more updates, I’d like to invite all of you to visit this summer.  We’re very happy to show visitors the program especially during weeks 2 and 3, pending availability.  If you’re interested, just reply to this e-mail.  The dates would be any time July 17-27.  Visitors are welcome to sit in on classes, participate in activities, eat meals with us, and meet some of the staff.  Also, if you’d be interested in giving a one-hour math talk to the kids, let me know.

With that, here’s some of the news on our end:

  • We are very pleased to announce a $100,000 grant from the Jack Kent Cooke Foundation to support our expansion this summer and the future success of our students.  You can read more about this grant and the other wonderful programs that received funding on their website.   Combined with support from the American Mathematical Society’s Epsilon fund and many generous individual donors, we have been able to fully fund our work this summer.
  • We extended our partnerships this year to 18 schools across the city.  The schools are primarily non-charter public schools, and they also include several charter schools (such as KIPP: STAR and Promise Academy II in the Harlem Childrens’ Zone).  Our students are focused in the Bronx and Harlem, with several from other parts of Manhattan, Brooklyn, and Queens.
  • Our summer staff is hired: top instructors and amazing college students who are stepping up to help underserved kids with mathematical talent.
  • Students from our first summer are going on to high school!  We’ve heard from some students who are going on to private schools on scholarship, and others who are continuing to a variety of public schools including specialized schools such as Brooklyn Tech.

Two last notes for you all.  First of all, we’re still looking for a strong Associate Director who can help run the program and direct a site in the future.  If you know of someone who might be interested, please put them in touch with me.  Second, a number of factors prevented the launch our mentoring program to pair program alumni with volunteers to mentor them in studying mathematics and finding additional opportunities, but we are expecting to fully implement the program after this summer.  If you know of mentors who might be interested or could forward along an e-mail to potential mentors, please let me know!

Over the next year, I will continue to send out periodic updates to those that have expressed an interest in our program.  I’m not using any special software to send them out, so if you would rather not get these e-mails, just let me know and I’ll stop cc’ing you.

Have a wonderful week and thank you all for your support!  I hope to see many of you this summer.
Dan

Should We Bother With Online Classes?

Why should we bother with online classes?

It’s a serious question.  Without the sheen of new technology and “innovation,” there are few clearly-articulated reasons to make this model work.  Online classes for their own sake is nonsense.  There are some good reasons: for example, creating accessibility to material through online videos or specialized classes that cannot be offered locally is great.  Allowing students to go at their own pace can potentially be very helpful, if they are pushed to make progress.  But if you want to replace schools with online classes, you’d better have a good reason why doing classes online is better.  After all, if they are disconnected from the instructor and their peers, will students be motivated to work?  Without the human connection, will students just drift away?  While sitting at the computer, will students just get distracted as we all do?  There seem to be more challenges than benefits.

Of course, there are lots of initiatives in online schools right now.  At the college level or similar, Stanford’s experiments have launched Coursera (where professors from numerous top universities offer their courses) and Udacity (“a digital university with the mission to democratize education”); there’s Minerva (an online MBA program for third-world countries, but see this take-down) and Udemy (which enables anyone to create and take courses online); and of course there’s MITx, soon to become edX, which offers MIT and Harvard classes (and credit).  At the high school level, there are numerous online charter schools (although their success is hotly debated) and a number of options for motivated/talented students (such as EPGY, CTY’s online courses, and Art of Problem Solving, aka AoPS, where I have been an online instructor).

But wait a moment.  Look at all those models.  They’re all about access to specialized material.  Indeed, online charter schools, which try to reach “every” student, have very mixed results.  The emotional distance that comes from sitting behind a computer screen detaches you.  If you are not driven to learn, you will not become driven just because it’s online.

When you look at all of the online schooling models that get Silicon Valley types excited, there is a common trend.  Although they are not aimed at motivated students, that is often where they’re likely to find success.  It’s true for all of the university programs above (which are based around the idea that people will seek out university educations); it is true for Khan Academy.  These are wonderful programs that create educational access, but they do not drive people to their doors.

I’m not saying that there aren’t ways around this.  Perhaps we can create an engaging classroom environment that takes advantage of social networking. (Folks have tried, unsuccessfully, to create more of a real classroom environment with Second Life.)  Perhaps we can create an online culture of learning.  Right now, though, we don’t have that solution.  We should use online schooling where it works, for students who are motivated, who want extra challenge, or who want to go at their own pace.  We shouldn’t try to force everyone into it as some magic panacea until we have a good justification for why it helps, and we shouldn’t say that it’s the wave of the future until we have some clue how that future actually improves on the status quo.

Low Cost, High Impact Ideas to Spread Math Opportunity

Too many students are ready to do more mathematics but

  • do not know where to do it, and
  • do not even know that such opportunities exist or that they should be doing it.

The students don’t know; their parents don’t know; their teachers don’t know. They have no way to discover that their peers, successful math students from other communities, do more than just what they see in school.

It’s not just that they need to be told about it; it needs to be part of their culture. It’s not just that they must know that such programs exist and that successful people do them; they should feel it is expected of them, that lots of people they’ve known and admired do math beyond school.

How can we possibly create this culture and community where it does not already exist?

The Summer Program in Mathematical Problem Solving (SPMPS) is one answer to this question.  I believe that it does amazing things for the students who go, and it might provide a new pathway to developing future mathematicians, scientists, and engineers from communities where none might otherwise arise.  Nonetheless, it is not cheap.  How could we create a real systemic solution?

I don’t have an answer, but over the past two weeks I’ve been visiting low-income schools to do interviews for SPMPS, and I’ve had some opportunities to think on it. Here are two half-baked ideas seeking comments, criticisms, ideas for how to implement them, and ideas for how to find funding.

A Math Opportunities Newsletter

[Idea developed jointly with Japheth.]

One double-sided sheet of paper, sent out each month by e-mail, postal mail, and posted on the web. It lists opportunities across NYC, from math circles to competitions to classes to MoMath events, and possibly some online opportunities as well. It includes up-front information about costs and scholarships, and about the mathematical background and maturity required for each program. It also has a feature story each month, about students in different programs and the successes they’re experiencing, setting a cultural expectation that this is what kids should do.

The newsletter would be mailed to every school in NYC and anyone who signs up for it online, increasing knowledge of good math programs. If the newsletter is really successful, it might strain those programs’ capacities—but such success might provide the impetus for more support to scale successful programs.

Costs would be mild: an editor to compile stories (and keep an eye on the overall tone of the newsletter), a graphic artist to lay it out an in appealing way, printing and mailing costs. Although it could be supported by advertisements, it might be difficult to do so in a way that maintains the positive tone of the newsletter and doesn’t get lost in test prep programs.

Math Club Materials

While I was visiting KIPP STAR, I had a discussion with their math coach. She was telling me how she used to run a math club, but she lacked the time to prepare problems, and she suggested that a ready source of good problems would help her school (and possibly many others) run such clubs.

While there do exist materials for running math clubs, they suffer from a number of shortcomings. They are not widely-known. They often rely on an instructor who is already familiar with sometimes esoteric math, or who has time to learn it.  They’re geared towards students who’ve already had substantial enrichment. They’re not completely assembled: you still have to put together the problems you want, decide on difficulty, and so forth.

What I imagine instead is having something teachers can just print out and go with each week: problems prepared, perhaps six questions to be done each week, along with short pointers on how to run an effective math club. Each week would have four different problem compilations: one for “Level 1,” another for “Level 2,” and so forth up through “Level 4.” In this way, it’s easy to try out a set with your students and decide if they are too easy or too hard, and then change levels appropriately. Ideally, there might also be special problem sets designed around specific school topics, so that you can reinforce recently-learned material. We might even sync the problems somewhat with Common Core.  This is not my ideal model for a math club—I would rather have one that does interesting mathematics different from school math in addition to contest-style problems—but this is an easily-replicated and highly-engaging model with good payoffs for the students.  It works well with a teacher’s limited time.

Naturally, the problems would have to be tested in a wide variety of settings and schools, and the key to success is getting teachers to actually use the problems. In addition to mailing schools, there are a number of schools that participate in math competitions but do not prepare for them; we could contact teachers who bring the teams to the events. Through contacts at schools of education, we could reach out to alumni. We could reach out to Math for America alumni. We could make the materials available at NCTM conferences. As use of the materials becomes more widespread, it becomes easier to bring them into new schools.

Costs would be moderate: the problems must be written/compiled (permission must be gotten for existing problems) and then put into PDF format. A webpage must be created and made easy to navigate. Much active work must be done initially to get teachers using the materials, including advertisements, conference presentations, and so forth.  However, after initial adoption, word of mouth might be enough to keep them going.

Closing Notes

These two ideas are clearly geared towards students who already have some success in mathematics.  While I could imagine the newsletter expanding to include opportunities for those who are not already on the bandwagon, the much thornier problem of helping those who are having serious mathematical difficulties is not one to be addressed here.

With that said, dear readers, I invite you to join me in my brainstorming.  Are these ideas feasible?  Worthwhile?  Are there other pathways to creating a culture of excellence in mathematics?

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.