A Reminder That Personal Relationships Matter

I wanted to briefly point you all to an article by Atul Gawande in a recent New Yorker.  Gawande is writing about the adoption of medical advances, but his remarks about teaching rural nurses in less developed nations are as relevant as anything to education.

Gawande asks: what drives a nurse to internalize that hand washing or warming the baby are important for safe childbirth?  He has a fascinating description about how one nurse was able to persuade another nurse to change her practices by becoming her friend.  Not because the mentor nurse’s training was impressive, nor because she had the force of law behind her.  Because the two of them sat down to tea.

Gawande says that success at getting nurses to adopt improved methods—especially those (like hand washing or warming the baby) whose effects are only visible after the child has left the hospital—come from personal connections formed by mentor nurses.  Otherwise, no matter the law, no matter what classes or informational videos or glossy handouts you offer, change comes slowly if at all.  To me, this sounds a lot like convincing kids to learn mathematics or to want to go to college.

Indeed, I think that these observations, hardly a surprise to anyone who’s seen the success of individual tutoring and mentoring, have implications across education.  I suspect that a difference between successful charter schools and unsuccessful ones is while both shout “college! college! college!” from the rooftops, only successful schools forge persuasive personal relationships.  While MOOCs make great resources available, they still have to persuade people to invest time in their classes.  How much did you learn from your best teachers because you felt like they knew you personally, or because you admired them and wanted to be like them?

Anyway, it’s a great article.  Read it while thinking about teachers—especially the difference between great teachers and merely good teachers—and it will give you provocative new thoughts about education.

The Latest Updates from SPMPS

I periodically send update e-mails to our supporters.  Here’s the latest:

Dear friends of SPMPS,

What happens when almost 40 middle school students from low-income backgrounds get to learn mathematics topics like Logic, Number Theory, and Combinatorics?  Amazing, amazing things.

In Group Theory, instructor Ben-Blum Smith invented a dance.  As Ben played on his guitar, the students responded to calls like “East-West Switch”, “LAX to JFK”, and “Jam in Place” to learn about symmetry groups.  In Proofs, Shelley taught her students about the pigeon hole principle and how to phrase a formal mathematical proof.  In Numbers, Sets, and Mappings, Marcus helped students prove that the quantity of natural numbers is the same as the quantity of even natural numbers, but that the real numbers are a higher order of infinity.  In Circuit Design, Sage helped her students construct circuits to add two numbers in binary.  In Digital Communications, Taylor’s students designed their own methods of sending images across a room using nothing but sound.

Thirty-nine students came to the program and got their first entry point to advanced mathematics.  Of course, the summer is just the launching-off point: over the coming year we will be connecting students with selective high schools and summer programs, the New York Math Circle, math contests of all kinds, and other opportunities that will open a new world up to them.

We’re still crunching the data, but here are a few outcomes that we’re already seeing:

  • Students took the AMC-8, a contest given to advanced students nationwide.  By the end of the program, the average of our students’ nationwide rankings grew by 21 percentile points!  Six of our students, exposed to serious mathematical study for the first time, saw their scores rise by 40 percentile points or more.
  • Students reported huge changes in their dispositions towards mathematics and challenges.  97% agreed that the program “showed me that I can learn more than I thought I could”, and 82% strongly agreed.  Students also learned the importance of hard work (not just natural brilliance) and of frustration and challenges: every student agreed that “Working can improve one’s ability in mathematics”, and 95% agreed that “time used to investigate why a solution to a math problem works is time well spent”.
  • Students raved about the program in their summer evaluations.  “A life-changing thing,” wrote Edson.  Math is “the best subject in the world” wrote Faith, who also said that she learned to “stop being shy”.  “I find math as a more common language than the language I speak”, said Seth.  Tiffany summed up something that many students said: “I want from liking math to loving it.”

Next year, SPMPS is looking to grow to a second campus.  We’ll be able to serve twice as many NYC-area students.  This expansion is possible thanks to a three-year grant from the Jack Kent Cooke Foundation which expires after next summer.  We’re fundraising now to raise the difference to make the second site possible next year, and also to help us transition smoothly as this grant concludes.  If you are able to contribute or you can make an introduction to someone who might be able to, please do let me know!  We are working to raise an additional $150,000 over the next twelve months to help support summer 2014 and summer 2015.

There are two other ways you can help us out:

  • We’re seeking meeting space in NYC on evenings and weekends.  This year, we will be greatly increasing our year-round programming to help students enter future programs for advanced study, and space costs may become an issue.  If you have access to free or low-cost space, please get in touch.
  • We are also looking for someone to help redesign our website to match our growing profile.  Please drop me a line with any connections!

Finally, you might enjoy this Bard news article about this summer:


Thank you so much for all your help and support with the program!  Until next time,


There are lots of exciting things in the coming year with much stronger year-round programming and connections to other mathematical offerings.  (What this post doesn’t tell you, for example, is that we’ve seen an explosion in SPMPS alumni registering for the NY Math Circle.)  This is a very exciting time for SPMPS!

The Many Uses of =

The equals sign is a sophisticated, subtle tool.  You may not think of it as such, but it is deep.  It is hard to learn and use and understand.

I first understood the challenge while teaching a calculus class.  I had two expressions, something like (x – 2)(x + 3)(x – 7) and (x – 2)^3 – x + 50, and our goal was to show that they were equal.  I took one, and I simplified it to x^3 – 6x^2 – 13x + 42, a chain of equals signs spread across the blackboard.  I took the other, and simplified it to the same thing.  So I proudly declared that they were equal, and thanks to the magic of my boardwork, the two identical expressions had ended up next to each other.   I drew an equals sign between them, creating a giant chain of equals signs connecting the two original expressions.  I thought this was a very clever presentation.

They didn’t get it.

I’d heard before that this might be a challenge.  They’d all been drilled in elementary school on problems like 5 + 2 = ?.  They thought of equals as meaning the outcome of an operation, which is why they felt comfortable using it sloppily.  (You know what this looks like: asked to compute “five squared plus two”, they write 5^2 = 25 + 2 = 27.)

I thought, then, that it was time to correct this misunderstanding.  So I speechified at length about how they had been tragically mis-taught about the equals sign.  It actually means that two things are the same thing!  I pointed to (x – 2)(x + 3)(x – 7) = (x – 2)(x^2 – 4x – 21) and said “these two things are the same thing!”  Then I pointed to (x – 2)(x^2 – 4x – 21) = x(x^2 – 4x – 21) – 2(x^2 – 4x – 21) and proclaimed, “these two things are the same thing too!”  Since the first expression was the same as the second, and the second was the same as the third, the first and third must be the same, and a pointed and said it.  I went on, through all the equals signs, until I had discovered that the two things on opposite sides of equals signs were the same thing.  I discussed why I didn’t like the way they did a problem like “five squares plus two”.  I was proud of my improvisation, but somehow, it didn’t seem like the students had the great moment of realization (and repentance for their mathematical sins!) that I wanted.

I say it again: the equals sign is a sophisticated, subtle tool.  Despite my oratorical skill, I had failed to bring together a deeper understanding of all the possible uses of equals.  There are at least four:

  • As the outcome of an operation (used in school, but mathematically the wrong way to look at it).
  • To declare that two things are equal all the time, as in 4(x + 2) = 4x + 8.  This is an equvalence.  There is a hidden universal quantifier.
  • To declare that two things will be equal for the right value of x, as in 4x + 2 = 3x + 9.  (This is disambiguated from the previous case by explicitly saying, “Find x so that…”  To a mathematically sophisticated reader, it is clear that this is the same = sign, just with an existential quantifier first.  This is not clear to someone just learning mathematics.)
  • As a definition, such as f(x) = 3x^2 – 4x.

Without any explicit guidance, we expect students to recognize these different situations.  To recognize quantifiers and definitions without any discussion of what “equals” really means.  All this while learning how to “solve” these problems.  Also without discussions of the other subtleties of equals: symmetry, transitivity, reflexivity.

Embedded here is also the understanding that two things can look completely different and can be the same thing, can be equal.  30/3 and 10 are different representations of the same number.  7 + 3 and 5 + 5 are all different representations of that same number.  All of these things are equal!  They are also equal to 2x when x = 5.  The mathematical symbols alone are not enough; the words nearby must be used to interpret them.

Worse, uses of the equals sign get conflated in different settings.  For example, suppose that you are trying to find an x so that 4x + 2 = (x + 2)^2.  You might write that (x + 2)^2 = x^2 + 4x + 4.  One of your equals signs is only true for a few values of x; the other equals sign is true for all values of x, all in the same math problem.  Imagine how confusing it would be if you wrote 4x + 2 = (x + 2)^2 = x^2 + 4x + 4!

The same thing can happen when you are defining a function.  You might say “Let f(x) = (x + 2)^2 = x^2 + 4x + 4”.  In one line, two different uses of the equals sign.

To me personally, these uses of = are all the same thing, stemming from the same definition.  To a student just learning algebra, however, there are layers upon layers of subtlety, and they must be addressed explicitly or students will not truly understand the mathematics they are doing.

SPMPS School Admissions, Google Trip, and More!

And now, my latest update e-mail about SPMPS.  If you want to receive SPMPS updates by e-mail, let me know.  Perhaps I should have a formal sign-up system for e-mails, but the truth is that I like the informality of a simple text e-mail that I send to our supporters.

Dear friends of the Summer Program in Mathematical Problem Solving,

I would like to take a quick break from preparations for summer 2013 to celebrate the achievements of our students from summer 2012.

Before I get to that: last Friday, we had a reunion at Google’s NYC headquarters.  Google engineers made some fantastic presentations about their work.  In one presentation, we learned about binary and RGB values to encode images on a computer.  In another, we learned about the math behind apps, and the geometry that underlies a tap, pinch, or drag.  In a third, we heard about one Googler’s path from a challenging background to success at Goldman Sachs and then Google.  Then students could ask their own questions in a panel about careers, and got a tour of the great facilities.  Thanks to Google for providing such a wonderful reunion day!

Now… drumroll please… congratulations to all of our students who were admitted to selective schools:

  • Amy, Ana, and Jahdel for gaining admission to Brooklyn Tech!
  • Jamila for gaining admission to Medgar Evers College Preparatory School!
  • Amy, John, Nicole, Quentin, and Salimatou for gaining admission to Bard High School Early College!
  • Emalee and Kiara J. for gaining admission to the Brooklyn Latin School!
  • Joel for gaining admission to Westminster School!
  • Taylor for gaining admission to Manhattan / Hunter Science High School!
  • Jeremy for gaining admission to NYC iSchool!
  • Nathaniel for gaining admission to Manhattan Village Academy, Cardinal Hayes High School, and Cardinal Spellman High School!

Ana was also admitted to the Center for Talented Youth’s Academic Explorations program.  We still have many more students left to hear from about their school admissions, so I hope to have more great news for you soon.

Thanks as ever for all your support!


I’d also like to add something to this e-mail: all of our students are absolutely incredible.  I’m tremendously proud of their admission to selective schools, but the truth is that all of them have bright futures.  I didn’t mention the students attending the NY math circle, or those applying to other summer programs.  And the tremendous intelligence and insight they will bring to whatever school they attend.

I just saw them a week ago, and I miss them already!

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.