I will get back to more substantive posts soon, but in the meanwhile, I do believe this video captures my entire philosophy of education.

## 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:

http://www.bard.edu/news/news.php?id=76

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

Dan

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.

## Teaching Log: 0.9999… = 1

It worries me that as I move into leadership roles, I get fewer and fewer opportunities to teach. Hence, I was excited to give a talk at the Sonya Kovalevsky Day at Barnard College. The entire 9th grade of the Urban Assembly Institute of Math and Science for Young Women was at the event, and about a quarter were in my class. It was a challenging talk for a number of reasons. First, the students had not chosen to be there, and had no particular interest in math. Moreover, I felt that it was important to do more than give a fun math talk. I wanted the students to really learn something that would help them in their mathematics in school.

The topic I chose was “Does 0.9999… = 1?” I did this for a few reasons:

- I could pose an interesting question at the beginning and let the students think about it and come up with ideas.
- I could vary the level of the discussion based on student background. At one end I could always retreat to 1/3 = 0.3333…, multiply both sides by 3, and get 1 = 0.9999… (ignoring issues of how you multiply an infinite decimal); at the other end I could talk about infinite geometric series and convergence (although that was very unlikely). In the middle, I could talk about how you prove things in general, give the overall idea of a series, and talk about what convergence means.
- I would be able to emphasize ideas of “math makes sense”—that different pieces of math fit together and work in the same way—and I would also be able to reinforce basic mathematics skills. This is as opposed to a topic like, say, combinatorial game theory, where I could talk about logical reasoning but wouldn’t be able to tie it into what they’ve already seen or build their ability to understand numbers.

I began by introducing myself and having all of the girls tell me their name. I asked about their subway ride to Barnard and generally tried to bond a bit and be friendly. Then I put up the question “Does 0.9999… = 1?” I took some questions (“what does the bar over the 9 mean?”) and took a poll. A bunch thought no, a bunch thought yes, several said they were unsure. Great!

I next asked students to explain why they were or were not equal. I got a few ideas but nothing very deep. I knew that underneath, students were struggling with what this question even means, so I asked them straight up, what does this question mean? I got some not terribly enlightening answers.

Now it was time to actually help them understand the question. “What does = mean?” I asked (and wrote on the board). We discussed the notion of equality for a while, and gave examples of things that were equal (1 = 1, 1/4 = 0.25, 2 + 3 = 1 + 4); part of my goal was to emphasize that = means “are the same as”, not “is the outcome of an operation”. I asked them if x + 3 = 2x, and we had some debate before deciding that it is only true if x = 3 (I’m still not sure if they got this); then we had some more debate about if x + x = 2x before deciding yes. Although I had hoped to doubly address the meaning of = and the hidden quantifiers in most algebra problems, I realize in retrospect that this probably muddled the picture more than it helped.

Once I felt like they understood =, I asked what 1 means (briefly, although I wish I’d had time to talk about it more) and then I asked what 0.9999… means. I ended up getting drawn off-topic in the discussion, which is OK, but I wish I’d gotten more of a bead on “it’s 9/10 + 9/100 + 9/1000 + …”.

Instead, we ended up talking about what 0.3333… is. None of them recognized it as 1/3, which took me rather by surprise. OK, time to talk about 0.3333…

We spent some time discussing how to turn decimals into fractions in general, but quickly concluded that you couldn’t straight-up turn 0.3333… into a fraction; 333333…/100000… makes no sense. So instead I suggested looking at 0.3, 0.33, 0.333, 0.3333, and so forth. I was again a bit surprised: they seemed to have no intuition that these numbers are actually very close together. So it was time to take out a number line and draw them.

Where, I asked, is 0.3 on the number line? Only one girl knew how to put it up; she knew that 0.1, 0.2, …, 0.9 were equally spaced and so she found 0.3. But when asked about 0.33, she didn’t know. So we spent some time talking about how 0.3 is 3/10, and how to find 3/10 (divide the number line into 10ths; there’s 1/10, now we want three of them, so we go over here). Then we decided that 0.33 is 33/100, and we talked about dividing the number line into 100 pieces. Then we saw that 0.333 is 333/1000 and we talked about dividing the number line into 1000 pieces. (I actually liked this part a lot, because it required a certain amount of abstraction to visualize dividing the number line into so many pieces!)

Then I asked, “OK, where is 0.3333333333 on the number line?” The students really had no idea that it would be right by the other numbers. Some thought it would be very far out indeed! Others thought it would be close to 0; an original hypothesis had been that as you add 3s, the number gets closer and closer to 0. Eventually I got them to write it out as a fraction: 3,333,333,333/10,000,000,000. I asked them where this fraction was on the number line, and again, no idea. (Although they were amused, perhaps even impressed, at my ability to rattle off “three billion, three hundred and thirty-three million, three hundred and thirty-three thousand, three hundred and thirty-three.”) Eventually we realized that this fraction was close to the others, although I’m still not sure all of them were convinced. I finally explained that if I offered to give you that many dollars, I’d give you 34 cents and then you’d owe me some money! This seemed to help put it in perspective for them, and we briefly discussed how you might scam someone with this “trick”. Note to my future self: give the example of 5,000,000,000/10,000,000,000; it should be much easier to see that it is just 1/2 despite the large numbers.

At some point we noted that these were getting close to 1/3, although many of them didn’t have an intuition for where 1/3 is on the number line. We talked then about how to convert 1/3 into a decimal and I divided 3 into 1, at which point some of them realized that they *had* seen this before. That said, I realize in retrospect that I didn’t emphasize that fractions *are* division and so something of a learning opportunity was lost.

We’d also gotten that 1/9 = 0.1111… in this discussion, but still no insight on 0.9999…. I had them discover the relationship between 1/9 = 0.1111… and 1/3 = 0.3333…, that in both cases you can multiply by 3 to go from one to the other. Then we saw that 1/3 times 3 is 1, and that 0.3333… times 3 is 0.9999…. At this point I tried not to hint further, and despite having everything right there, the class went off on a wild tangent, thinking they could “add” something to 0.9999… to get 1. (A perfectly good theory, if they were different, and I wish I’d emphasized that more. We spent a while talking about what happens if you add 1, if you add 1/2, if you add 0.1, and 0.01, and 0.001, and so forth; I wish I’d had more time to allow them to do this themselves.) Somewhat flummoxed, I did a poll. Nearly everyone was now convinced that 0.9999… is not equal to 1!

Eventually I brought us back on track, and someone made the magical connection, and a student realized that since 1/3 times 3 is both 1 and 0.9999…, they must be equal. This part could have used more discussion, relating it back to equality, but we were really quite short on time. I ended the class by hinting at a graphical argument, and tantalizing them with the sum 1 + 1/2 + 1/4 + 1/8 + 1/16 + …, and then we were done.

In the end, the class was engaged and had very high energy, but had such a deficit of basic facts that they really struggled. I don’t think I managed to bring everyone with me as we went forward, nor was the class nearly as student-led as I’d hoped because it was hard to get them moving in this context. I’m actually happy with the presentation itself, and I hope to continue to refine it and find ways to make it accessible to everyone.

I am reminded just how much students in schools lack, however. The insistence on moving students forward to algebra and beyond when they don’t fully have the number sense to understand how things go on the number line seems to be doing them a great disservice, and I wish we could find a way to enable teachers to address these basic challenges with them. Otherwise, they’ll be continually forced to memorize procedures and they’ll never understand why they do what they do.

## 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!

Dan

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.