The article is largely about math beyond what’s typically taught in school. You might call it high-level math, more challenging math, more abstract math, or maybe STEM Pathway math because it helps kids develop the skills to succeed on that pathway. Regardless, it basically doesn’t touch at all on what a school math curriculum should be or what math class should be like. It’s important to keep that in mind; this is math for kids who want more math.

And let me say, it’s wonderful that kids have so much opportunity to challenge themselves with interesting mathematics. However, there are two big downsides to the current arrangement. First, some students may feel pressure to advance themselves for the wrong reasons. Second, those without access are being left ever further behind the top achievers.

I’m honored that BEAM was discussed in the article as an effort to address inequity in access, and that I get to contribute to this conversation, but we’re still quite a ways away from really closing the gap. In this post, I want to dive into what’s really going on so that we can understand how the next generation of mathematicians and scientists is growing up.

Disclaimer: This is what I have personally pieced together; some of it is from before my time.

Once, access to higher-level math in high school was hard to come by, unless your parents were scientists. There were a few summer programs, mostly the Ross Program and its descendants, that presented a wonderfully curated journey through number theory. (HCSSiM also deserves mention as one of the old guard. Also, starting in the mid-80s, RSI, a free science research program open to high school juniors.) If you attended, you could be around math professors, learn about beautiful mathematics, be a part of a wonderful community of peers… and then go home and back to school math for a year. Perhaps you might have a math book or two to work through on your own, but otherwise, the life of a high school math fan was mostly noted for a certain sense of isolation without other young mathematicians nearby.

There were also math contests, although not nearly as many as there are now. Those who achieved at the very top levels might do some proofs (for example, through the USA Math Olympiad), which is the key entry point to mathematics. That was about it.

It started opening up a couple of decades ago, with (as far as I can tell) a few key movements. First of all, several more summer programs opened up, creating more opportunities for students around the country to study advanced math. Summer courses at CTY, TIP, and CTD, which have broader national reach, enabled interested students to learn proof-based mathematics. (I first learned proofs at a computer science course at CTY.) Additionally, math circles and other enrichment programs brought some local access to students. Combined with math teams, this started to provide some semblance of year-round access and community.

All that said, when I was a student in the late ’90s, the vast majority of my peers were still not exposed to much in the way of proof-based mathematics. I’d had some the summer after 8th grade at CTY, then a drought, then much more the summer after 11th grade at Mathcamp. But I was not a very sophisticated mathematician when I first went to Mathcamp.

Nonetheless, a great deal was changing. First of all, a few specific geographic regions strengthened their math programs even further than they had been before. Boston, with its concentration of intellectual capital from its universities, grew even stronger with the founding of the Russian School of Mathematics about 20 years ago, and the Boston Math Circle around the same time. It helped that the area also had other outreach programs, such as MIT’s Educational Studies Program which ran both Splash (a weekend of courses to explore new areas of study) and HSSP (with weekly meetings to go into depth on a topic, including math). Even students who didn’t participate in these programs benefited from math teams with students who did. Similar growth happened in the Bay Area with the influx of tech workers and their interest in strong math educations; math circles grew and prospered, and several schools concentrated students who excelled in mathematics. The same happened around the Thomas Jefferson High School for Science and Technology in northern Virginia; and in New York City with the specialized high schools like Stuyvesant and Bronx Science. However, this only helped students in those areas; students who lived elsewhere (like me) could only watch with jealousy (and we were jealous!) as these kids with geographic access continually beat the pants off of us at contests like ARML. Even worse, underserved and minority communities, even those nearby, rarely were involved in these programs. Although the programs were technically open to everyone, they did not have the capacity for outreach, and underserved communities did not have the knowledge and drive to get involved.

However, what really changed things on a national level was the internet. Sites like Wolfram MathWorld and Wikipedia were already tremendous resources that let students taste abstract math. Then came Art of Problem Solving, and I think things were never the same. All of a sudden, regardless of your geography, you could stay in touch with other students interested in math no matter where you were. Before, hearing about contests or summer programs was a matter of good fortune. Someone in your network had to know about them, or you had to get really lucky reading the right flyer. Now, there was a central repository to find them and discuss which ones were good. You could talk about math problems anytime, day or night. You could take classes that were actually *really good* and which stretched your reasoning skills. You could order books (and figure out which books were good) and learn on your own. The homeschooling community especially adopted both the books and online classes.

This was the point when networked students started regularly seeking out enrichment. When they had contact with other people who were also learning all this mathematics, they pushed themselves to do the same. Suddenly, role models in mathematical achievement were easy to find; math olympiad winners had fans, sometimes quite obsessive fans! People who finally met in person, at contests or summer programs or even in college, had known each other for years online. Eventually, those communities would flow over into Facebook and other social media as well.

The spread and growth of in-person opportunities worked in tandem with online opportunities. It remains the case that a good in-person teacher is preferable to an online teacher, and the community created in-person is simply unbeatable. But the online community strengthened the in-person communities, allowing for more consistent contact even for friends who lived across the country or across the world. It allowed students everywhere to hear about the in-person opportunities, and in turn new summer programs sprung up to meet the demand.

Although the internet can be a huge democratizing force, it acts in very particular ways. Much as we’re seeing now with Khan Academy, Coursera, and edX, internet resources like Art of Problem Solving primarily benefit those who are already motivated and who have internalized the importance of a certain type of education. The same was true of the new in-person learning opportunities. People who were really into math took advantage of them. (It didn’t help that many, but not all, had a high price tag, although they often come with financial aid available.) That left a growing gap between those who partook, and those who did not; not surprisingly, that gap tracked racial and economic lines.

A second factor came up as well. These programs became important for college admissions, because they provide a marker of advanced study. As a result, more people started doing them not for the love of math, but instead for the boost to their future chances. Today, cheating on the admissions tests is a regular issue that the programs must deal with.

It is now common for incoming college students to have studied all kinds of topics before college. The mathiest will have done number theory, group theory, or topology while in high school; sciencey folks will have done all kinds of lab research; programmers will have created complex projects. In 2015, there were 167 students in 8th grade or younger who took AP Calculus (compared with only 33 such students ten years earlier). I have friends who entered into MIT and immediately started taking graduate-level courses. This learning is not essential; I also have mathematician friends who first learned what a proof was in college. However, that has become the exception rather than the rule, and it’s harder and harder for students to not feel like it’s a “race.”

But, on the other hand, it’s *good* that students have this access. It’s a travesty to be denied adequate educational challenge growing up. But while greater access is a huge positive, the inequity of access is very problematic. If some students are coming in so much more prepared, then it is that much harder for less-prepared students when they get to college. Worse, those less-prepared students often have many other hurdles to face. To diversify math-related fields, we must broaden access to this kind of material.

The *Atlantic* article proposes universal testing for gifted-and-talented classes. I support this as an improvement over the status quo, although it is not my first choice. For one thing, I still hate the name. (Gifted, really? As if some powers were “given” to you from above, and hard work does not play into it?) Beyond the name, testing for gifted programs is unreliable. It uses proxies that benefit fast thinking (not deep thinking) and does not account for how students change and grow over time. Once it has selected its students, the difference in classmates and curriculum becomes a self-fulfilling prophecy.

I would love to experiment with another system, one where students have the flexibility to switch between courses at different levels of challenge. With synchronized curricula, it would be possible for students who want to seek out more challenge at a later age to change to more difficult courses without having content to make up. The standard testing-based approach is “identify, then teach”; it inevitably looks only for proxies of ability. A choice-based, student-driven approach — “challenge, then adjust” — would make actual performance and motivation the basis of where students end up.

We must also increase access to opportunities for deep thinking in mathematics. BEAM is a first step to this. I hope that BEAM grows but also that solid opportunities for studying mathematical reasoning become more broadly adopted. This could happen through BEAM’s curriculum or through the development of other materials that could be used elsewhere.

Finally, and this is perhaps the biggest challenge, we must make it common for students from all groups to *want* to pursue these opportunities. Perhaps the most successful example I’ve seen of this is Neil deGrasse Tyson, who shows up with some frequency in the Facebook feeds of BEAM students. I’ve long wanted to create some really shareable discussions by diverse groups of scientists, but that would only be a small piece of this puzzle. I don’t have a real plan for what I really want: an online community among young students from many backgrounds who might become interested in math and science and encouraged to seek out more opportunities. Anyone have any thoughts on how to encourage greater involvement in math?

Right now, despite all of the resources that are out there, many students are living with pretty much the same situation of several decades ago, with little exposure to deep mathematical thinking. It’s not that the opportunities don’t exist; it’s that they don’t know about them, don’t understand their value, have not had the necessary preparation, and might feel uncomfortable if they go. However, if we can address these issues, then these same extraordinary opportunities can be made available to all students and we can really broaden the next generation of scientists.

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I was never exceptional at math contests, but I love them. I think they’re fun challenges and they create a vibrant community; used well they are a good tool in mathematics education. However, most contests also promote bad habits. They lead students to focus on speed and winning over deep thought. They encourage students to learn a few tricks well and memorize useless facts. They further perpetuate the myth that math is about speed, even though real-world problems take days, months, or years to solve. Hence, my talk was about transitioning to solving “big” problems. For my own reference later as well as for anyone who is interested, here’s a log of what I said and how I structured the talk.

I began by asking the room to raise their hands if they had ever spent more than one minute on a math problem, and all the hands went up. “More than five minutes?” “More than an hour?” Now many of the hands were down. “More than a day?” Few hands were still up. “More than a month? More than a year? More than five years?” (That last one was especially unlikely for 12-year-olds!)

The real problems in the world, I explained, take a long time. Proving a theorem, building a bridge, creating Facebook or Google — these take years! So I encouraged them to think about mathematics in a different way. Instead of contest thinking like“how fast can I solve these problems,” or “do I have the sides of a 30-60-90 right triangle memorized,” or “do I know the perfect squares up to 25 squared,” get practice thinking deeply about mathematics.

I told them about Maryam Mirzakhani. She’s one of the best mathematicians of our time, a winner the Fields Medal (the “Nobel Prize” of mathematics). It was important to me to show a woman doing mathematics at the highest levels, and I also wanted to go with someone currently living. I decided to go hard on establishing her credentials and to say “she” several times, but I was very careful *not* to say that she was the first woman to win the Fields Medal.

I debated that choice for a long time before the talk. Ultimately, I wanted a woman winning the Fields Medal to seem normal. There is a time when women in math must reckon with their underrepresentation, but MATHCOUNTS, which has a relatively large number of girls, is not that time. I was also wary of Einstein’s strange praise of Emmy Noether (one of the most brilliant mathematicians of the 20th century) in the obituary he wrote: “the most significant creative mathematical genius,” he wrote, “thus far produced since the higher education of women began.” The latter portion of the sentence was thoroughly unnecessary for Emmy Noether. Similarly, saying that Mirzakhani is the *first* female winner of the Fields Medal makes the gender dynamics front and center instead of her raw mathematical accomplishment. (Meanwhile, for the historical record, Einstein was a huge supporter of Noether and the reason she received a paid university position at all!)

Anyway, I’d brought up Mirzakhani because I wanted to share this quote from a Quanta article:

Mirzakhani likes to describe herself as slow. Unlike some mathematicians who solve problems with quicksilver brilliance, she gravitates toward deep problems that she can chew on for years. “Months or years later, you see very different aspects” of a problem, she said. There are problems she has been thinking about for more than a decade. “And still there’s not much I can do about them,” she said.

In short, if a scholar of this magnitude does not think speed is important, then why should you? Think deeply, not quickly.

With that, it was time to tell a story that reinforced this idea.

There has been much recent progress on the Twin Primes Conjecture and it is a very approachable topic. It was to be my avenue in, the bit of mathematical culture through which I could convey that serious breakthroughs take a long time. I began by talking about how it was open since at least 1860; I wanted to show how mathematical results build upon one another, the partial results that come over years of work and progress from many mathematicians.

One of the early results on the twin primes conjecture is that the sum of the reciprocals of the twin primes converges, but I needed to introduce infinite summations for middle schoolers who might not have seen them. I talked about how the sum of the reciprocals of the powers of two converge, but the sum of the reciprocals of the positive integers do not; I also explained that it had been proven that the sum of the reciprocals of the primes diverges while the sum of the reciprocals of the twin primes converges.

This allowed me to point out that we proved something about the sum of the reciprocals of the twin primes without knowing how many there were or what they converged to, hinting at nonconstructive proof. An analogy helps explain the significance of this result: there are a lot fewer twin primes than primes; they are somehow “sparse” like the powers of 2 are sparse. I also pointed out that we might see the convergence result as a “bad” result, because we were hoping to prove that there are infinitely many twin primes, and the convergence leaves open the possibility they are finite. However, no result should be seen as bad, because everything gives insight.

From there, I outlined some of the results up through 2005 in the effort to prove the conjecture. The idea was to tell a story. Look at this steady advancement of mathematics. I was able to talk about how we build results upon other results, and to demonstrate that these real advances take time, returning to my theme that it is great to think slowly but deeply.

Then I jumped to the present day with Yiting Zhang’s result, and I got a nice laugh from saying that he made great progress by showing that there are infinitely many pairs of prime numbers… that are within 70,000,000 of each other! (Considering that we were aiming for infinitely many “2” apart, this still seems far off — but again, partial results are key!) From there I talked about the wonderful online collaboration between mathematicians to narrow 70,000,000 to 246.

Finally, to close off this topic I gave the students a challenge they could take home: to determine how many “prime triplets” there are, i.e. sets of three primes where each differs by 2 from the previous.

By the end of this, I felt like I’d communicated a strong component of mathematical culture and how progress is made. I felt like I’d done good work explaining to students that they must think deeply and take their time. In short, I felt like I had re-aligned them from the typical contest mindset.

Now, the challenge was to switch to doing mathematics. Even if they acknowledged the value of deep thinking and taking their time, they still were not empowered to do so. The problem was that with a 45-minute talk, half of which was already over, there simply wasn’t time for students to solve a “big” problem themselves. (In fact, I joked early on about how they shouldn’t fear, my talk would not be a year in length!)

Something which I think gives many students difficulty approaching mathematics is the appearance of brilliance. Don’t get me wrong, I don’t doubt the *existence* of brilliance. However, it often seems as if proof methods and ideas “come out of nowhere” and someone simply must be a genius to discover them. I decided that if we couldn’t *actually* solve a problem somewhere, I could at least dispel this myth. Seemingly arbitrary but effective moves actually come from looking at patterns, experience, and building a chain of (often ineffective) ideas.

The problem was to take a wheel divided into 6 sectors. As many times as you want, you may add 1 to any two adjacent sectors, like this:

Is it possible, through these moves, to get from the starting wheel

to a final wheel where all the numbers are equal?

You’re welcome to try it out yourself before moving on!

I gave a warm-up problem (where it was easy to find moves leading to the goal wheel), and then we shifted to answering the problem. I told the students that if they were approaching this question, they would probably experiment for a while and get a feel for it. Then they’d look at all their attempts to see if there were any patterns in how this worked.

To save time for the talk, I gave the students a grid of 15 different states of the wheel that are achievable, perhaps the results of their experimentation. Then I gave them two minutes amongst themselves to look for patterns and see what they could discover. I wasn’t expecting them to solve the question: after all, this is intended to be a problem that they might spend hours upon! Instead, they should look for insights to push their thinking forward.

The students came up with some nice ideas. One student suggested that perhaps every attainable wheel has two equal numbers that appear. Although this was true in all of my examples, further experimentation showed that was not the case. We set for ourselves a goal of finding a wheel with all different numbers (which we did). Another student brought up the maximum difference between the highest and lowest values on the wheels (unbounded). Although these ideas did not work, they gave us nice results about what wheels *were* achievable, and I wanted the students to recognize that these results were valuable even when they did not solve the problem. It’s like proving that the sum of the reciprocals of the twin primes converges.

Then a student made a lovely observation: every wheel had exactly two odd numbers on it. Fascinating! Was that always true, I asked?

We soon discovered a wheel with zero odd numbers. However, it was rather interesting that we found 0 odd numbers, and we found 2 odd numbers. Can you have just one odd number? No, you can’t!

This leads you to realize that the sum of the numbers on all the wheels is even; this is easy to prove because each move adds 2 to the total. This partial result *still* does not prevent finding an attainable wheel with all six numbers equal, since such a wheel would also have an even sum. *Again* we did not solve the problem, but this is still strong progress! We now have a theorem that rules out many other wheels.

Perhaps we should take a step back. Using the sum of all the numbers is only using part of how the game works. It would still be true even if you were allowed to add two 1’s to *any* two sectors. We need to find an idea that takes advantage of the fact that the two 1’s must be next to each other. I challenged them to see if they could think of something.

One student did (although I think he had seen something like this before), and we soon narrowed down on creating an alternating sum of the numbers in the wheel, like this:

Now, adding two adjacent 1’s leaves the total unchanged! A wheel with all numbers equal has an alternating sum of 0, but our starting position has an alternating sum of 2! Hence, we know that we cannot attain all numbers equal.

Imagine, I told the students, that you had not solved this problem and someone came up to them with this proof by taking the alternating sum. They might look and say, “That person is brilliant! I would never have come up with that.” That ignores the way that all of this partial work and ideas built on each other; the “brilliant” person did not start out by taking the alternating sum! (Unless they’d seen similar problems before.) Playing around with it led us to counting odd numbers, which led to taking a sum; then seeking a way to think about adjacency led us to the alternating sum. Each step makes sense, in the context of thinking deeply about this problem.

It’s easy to imagine spending a long time on this problem. Perhaps one day, you come up with the idea of counting odd numbers, and you conjecture that there are always 2 odd numbers in an attainable wheel. However, then you discover a wheel with 0 odd numbers, and you conclude that your conjecture was false. You might decide it’s not worth looking at odd numbers. But perhaps you come back to it a day or two later, and, frustrated, you think about odd numbers again — then you realize that there are no wheels with exactly 1 odd number entry, and you’re making progress again! It takes time and persistence, and that is what I want the students to learn.

I ended by saying how no mathematical proof literally comes out of nowhere. It comes from someone who thought deeply about the problem and found a way through it. I talked about the idea of “following your nose”, and looking for maneuvers that seem appealing even if you are not sure they will work. I also pointed out that the next time a similar problem comes up, one where odd/even plays a role or where an alternating sum is useful, this is now a tool in their toolbox.

In short, it is important to develop strong habits of mind, stamina, and a willingness to *try many things*. That was my concluding message to the students in their next steps to solving big problems.

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The case provoked all the expected outrage, especially when Justice Scalia made a half-informed comment about the effects of affirmative action on minority students’ ultimate achievement. However, while listening to back episodes of the Amicus Podcast, I heard a different comment from Chief Justice John Roberts.

“What unique perspective,” he asked, “does a minority student bring to a physics class?”

The lawyer for the University of Texas, not surprisingly, was tongue-tied. (Not exactly part of prep for the case, eh?) A casual internet search revealed many non-response responses explaining why diversity is important and why physics needs underrepresented students to succeed. (That second link, if you’re curious, is a letter from almost 2500 physicists to the Supreme Court.) The *Atlantic* has a lovely piece about Einstein’s journey to discovering relativity and how it relied on philosophy, but the piece still could only hint at an answer to Roberts. Somehow, none of these responses *actually answered the question!*

That’s where I’m stepping in.

The job of a physicist is centered around two things: making new scientific discoveries and *communicating those discoveries**.* A discovery that is not communicated is useless. Physicists write up their work in academic journals and give talks at conferences. For many of them, the bulk of their academic employment will be based around teaching physics classes. Those who go into industry must communicate with coworkers, management, and the public on a regular basis.

Successful communication requires being able to phrase your work in a way that can be understood by those of many different backgrounds. In lab settings, in group projects, in presentations, it provides a key benefit to learn how to communicate with those who don’t share your background.

Many people taking physics classes are going on to think about applications of their work to the real world. Perhaps they are engineers and will be building bridges. Perhaps they are going to work at NASA or SpaceX or Blue Origin and will lead space exploration. Perhaps they are going to work in nanotech, or semiconductors, or… you get the idea.

In all of these cases, applications to the real world are essential. They must design technologies to be used by other people. They must think about how the bridges they build interact with the communities around those bridges. Diverse perspectives allow students to better understand the applications of their work, how it will be used, and how to design it for maximal benefit to society.

For the sake of argument, let’s suppose that there’s a physics student who gets into UT and she’s be the only Black student in her class. She’s doubly underrepresented: one of few women, and the only Black student. Her learning will be negatively impacted because she has no one to talk to about those struggles. There’s no one who can understand the lack of role models or the biases she faces. If she comes from an environment that is not middle- or upper-class, there is no one with whom she can discuss the culture shock not just of attending the university, but of physics, which has its own cultural norms.

This student, although admitted on her own merits, is getting an inferior education to others because she does not have a supportive peer group. This is preventing her successful education, because her class lacks the perspectives of other students that will help her succeed. Without diversity, the University of Texas cannot do its job for her, cannot give her the service for which she is paying tuition.

These are not the only reasons I support affirmative action in educational settings. However, as someone who has designed numerous educational programs in math and science settings, I have sought diversity of viewpoints and backgrounds not for a social justice purpose, but because that is how I can provide the best educations for my students and create the products that students will want. As the country becomes more diverse and as students enter a globally competitive marketplace, access to diverse viewpoints is an essential part of a good education.

To put it in the starkest terms, denying the University of Texas the tools to create a diverse class will decrease their educational effectiveness and put them at a competitive disadvantage against other educational options that offer greater diversity.

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How can we possibly make everything fit?

Seriously, we have a highly ambitious curriculum planned and we must *also* fit a vibrant social schedule. The community formed at BEAM 6 will carry students through their future studies if we get this right; it will provide a beacon that tells them that they can have good friends and be serious about math at the same time. It will also encourage them to continue on to BEAM 7, and we want them to come! What schedule will let us accomplish these goals?

BEAM 6 will run six days per week. Five days will be class days, which will mix both classes and activities. The sixth day will be a field trip or activity of some kind to further build community and friendships.

We need to fit in as much time during the week as possible, but we have a serious limitation: rush hour! Students will be brought to the program by our undergraduate counselors, who will meet them at subway stations near their homes. However, navigating the subway during rush hour with a bunch of 11-year-olds is not a great plan.

We can’t avoid rush hour completely without terribly shortening our day, but we can avoid the worst of it. If we start at 8:15am, then the farther students will be boarding the subway at 7:15am, which is not too bad, and by avoiding an “on the hour” start we also avoid peak times. Then if we end at 3:40pm, we can get on the subway before the afternoon commute.

That’s our day: 8:15am-3:40pm. I’m not crazy about how that schedule makes us feel like a normal school, but we have few other options.

I debated for a while which day should be our sixth day. BEAM 7 runs on a Tuesday-Saturday academic schedule (adopted from Canada/USA Mathcamp), with field trips on Sunday and Monday. The advantage of Monday trips is that you can visit when places are almost entirely empty. However, such a schedule might cause significant confusion among students and families, and the subway has unpredictable problems on weekends that could interfere with students’ on-time arrivals. So, in another compromise, classes will run Monday-Friday, with activities on Saturday. We might reconsider this in future years when we have more capacity to deal with any unexpected challenges an unusual schedule causes.

We have five courses planned: Logical Reasoning, Applied Math, Math Foundations, Math Team Strategies, and Exploring Math. We cannot possibly offer all four in the same day. With lunch, two activities, and study hall, we simply run out of hours. (Study hall is very important to me, because it gives students time to reflect on their work and instills study habits.)

The first decision I made was that “Exploring Math” can naturally be simply “Afternoon Math Circle”. The last thing in the day, it’s a fun piece of math, different every day, taught by different people. We can also use the time for guest speakers talking about how they use math in their work. Regardless, this will be in the final block, 2:40pm-3:40pm.

For the rest of the courses… well, let’s consider two different options.

The four other courses naturally break up into two groups of two, so we can have 2 two-week sessions. Students would focus on one pair of courses during each two-week session.

**Applied Math with Math Foundations**. I paired these because Applied Math will likely be the most intrinsically exciting course, and Math Foundations the least exciting. Applied Math needs as much time as it can get: if students are to become independent in programming, then they need to do lots of programming. Hence, while Applied Math can expect to give 1-1.5 hours of homework per day (depending on the day – see below), Math Foundations should give no more than 15-30 minutes per day. Math Foundations is not designed to drill students in procedures, but rather to encourage creative solutions to problems, so it is all right to give less homework.

**Logical Reasoning with Math Team Strategies**. Math Team Strategies would get the bulk of the homework time, because we want students to become acclimated to using online resources such as Alcumus and the Art of Problem Solving forums. These classes are both in the middle in terms of intrinsic excitement, so they pair well together. Additionally, by putting Math Foundations and Math Team Strategies in different sessions, we know that students are always getting something related to math they’ve learned in school.

There is another way to divide the courses. Instead of two groups of two, they could alternate days. One day is Applied Math and Math Foundations courses; the next day is Logical Reasoning and Math Team Strategies. In this plan, all courses run the full four weeks, but every-other-day.

With Plan #1 (2 two-week sessions), it is easier to find faculty (who can now teach for just two weeks) and students can focus on specific topics as they go along. Moreover, by studying the same thing each day, teachers don’t have to spend as much time reviewing at the start of class.

On the other hand, with Plan #2 (alternating days), students get more practice balancing competing demands on their time with homework assignments. Moreover, their ultimate recall is stronger because they spend a longer time actively engaged with each topic. Finally, it makes things more uniform. For example, suppose that in Plan #1, someone is teaching Math Team Strategies. For the first two weeks, their students are new to the program and haven’t taken our Math Foundations course yet, so they will struggle. But when the course is repeated for different students in the latter two weeks, all of those students have had Math Foundations. If we use Plan #2, this goes away.

After talking with my colleagues, we settled on Plan #2. We feel that it is a better educational experience for the students. While finding faculty may be harder, it is worth it for a stronger program.

At BEAM 7, the courses provide no homework. Students do all their work in class, with attention from the instructor. This allows for a fast-paced, highly-interactive environment. However, there are disadvantages as well. It doesn’t train students to budget their own time and develop independent work skills. Moreover, it doesn’t fit well with part-time faculty for a day program. At BEAM 6, we’ll have shorter classes and time for students to do work.

My first draft of the schedule came out like this:

**8:15am-8:30am:** Breakfast

**8:30am-9:30am:** Class

**9:35am-10:35am:** Class

**10:40am-11:40am:** Activity

**11:45pm-12:15pm:** Lunch

**12:20pm-1:30pm:** Study Hall

**1:35pm-2:35pm:** Activity

**2:40pm-3:40pm:** Afternoon Math Circle

Lunch can be short, because we will almost certainly get catered boxed lunches that students can grab and eat. Since it is right after activity, it still provides a good break from their classes. If we had just one more hour, I could fit two hours of class/study hall/whatever between lunch and activity, but with avoiding rush hour we just don’t have that time.

However, after reflecting on this schedule, I want more time for study hall. Especially for the programming course, there just isn’t much time for student independent work. Currently students would have a total of 20 hours of work on programming (10 with the instructor and 10 in study hall); more time would be a huge asset. Moreover, having 20 Afternoon Math Circle sessions, while delightful, is not really necessary. Hence, on some days we can replace Afternoon Math Circle with a second Study Hall time. In the end, I decided that Monday, Tuesday, and Friday will have Math Circle (good way to end the week!), while Wednesday and Thursday will have extra Study Hall, allowing students to work on projects or longer assignments later in the week when they are in the thick of things.

This is all very complicated. Now we have odd/even days determine which of the four long-running classes are happening, while days of the week determine if Math Circle is happening. I think these are all the right decisions, but we will need clear messaging to make it work and make sure that students feel comfortable with their schedule.

There are, of course, a whole wealth of other details. For example, at what point do students select their courses? Should they do so on the first day of the program (which eats up class time), or in some earlier orientation? Right now, my plan is to schedule an orientation for students and families before the first day to talk about the program and how it will work, and to include course selection there. Unfortunately, some students will miss that event, and we will have to give them another time for course selection.

We must also schedule Saturday trips. We are thinking about movies, or a trip to the Bronx Zoo, or similar events. These will each have their own schedule based on what we are doing.

We may also want some sort of closing ceremonies with parents. Again that will require separate scheduling. Most likely, we will go for Friday night after the program is done, and provide some sort of food.

Finally, we must have training/setup and wrap-up/take-down with staff. I am planning the Friday before the program for the former (full-day for counselors, half-day for faculty) and the Saturday after the program for the latter.

I’m sure there are other details that we will think of as we go along.

Things fit. They don’t fit as much as I want; the day feels too short to me, making it hard to really bond with everyone as much and get as involved in the classes. I am worried that we won’t be able to instill in students the habits we want them to have for their educations. But this is an iterative development process. We will run a great program, and then make it even better for next year.

Despite any shortcomings in our available time, this will be a tremendous experience for students. It will open up so many educational pathways. Seeing a concrete schedule really gets me excited for the summer!

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At BEAM 7 (a program many BEAM 6 students will attend the summer after 7th grade), we tend to throw students off the deep end when it comes to doing math. I mean it: they come in and we teach them about proofs, and we have them solving MATHCOUNTS problems, and we have them learning number theory and combinatorics and even group theory… and a lot of them are kind-of still weak on fractions, y’know?

It varies by school, of course. Not surprisingly, some schools tend to give us more prepared students, while others don’t. BEAM 7 has had seventh graders (top of their class at school!) who were not comfortable multiplying negative numbers.

I’ve been asking myself what I wish our BEAM 7 students knew. They’re held back constantly by foundational math knowledge. They also need to learn how to look at a problem and focus on what it’s asking, rather than guessing at a solution mechanism. Finally, I want them to have more skills in deductive reasoning and case analysis. It’s a little bit crazy to be thrown into a proofs class without that!

In the end, there are five course tracks that I really want to work into the program.

- Logic
- Math Foundations
- Math Team Training
- Applied Math
- Seminars

I want students to have choice, so each of these topic areas will have different courses within it. During their summer, each student will take on course from each track. The exception will be Seminars, where each will be independent (see below).

It’s a minor nightmare to fit all of these classes into a four-week program. Right now, the best I can do is 10 hours of class time each, plus some homework time depending on the course. So they have to be compact and get to the punch quite quickly. Scheduling will be covered in detail in a future post, but lack of time is a huge concern.

Before I get to describing the course tracks, readers who know about BEAM 7’s courses will see right away that this is *super different*. BEAM 7 is basically a playground for our faculty to develop all kinds of interesting math courses and then teach them. These courses are much more structured and targeted. Why?

The primary reason is simple: BEAM 6 has a very different goal. BEAM 6’s main goal is to remedy specific gaps that students need to succeed both in BEAM 7 and in their future mathematical studies. In contract, BEAM 7’s goal is to transition students to other programs for advanced study where they will have to do more abstract thinking. Hence, BEAM 7 invites faculty to rock out in courses similar to what students will do at future programs. In contrast, BEAM 6 is a laser aimed at skills and knowledge that students need. BEAM 6 courses will be lots of fun, but they’ll also have much more concrete goals.

There are advantages and disadvantages to both approaches. One big advantage of BEAM 6 is that I can develop a strong curriculum for students. A second advantage is that it opens our program up to more potential instructors, because they do not need the same experience designing enrichment classes. However, BEAM 6 is still open to those who want to create their own crazy classes through both the Seminars and Applied Math topic areas.

Great! Let’s figure out what’s actually in the courses.

This one is my favorite! How do we help kids think, and do it in a really fun way that will translate to thinking about math?

Here’s my crazy plan. Logic courses will begin by looking at a particular type of game or puzzle. For example, they might look at Ken-Ken or Sudoku (or other Nikoli puzzles); they might look at liar/truthteller puzzles; or they might look at puzzles like Einstein’s riddle. These are different lenses for gaining the same skills: deductive reasoning; case analysis; careful organization; and, to some extent, proof by contradiction, because these puzzles all encourage trying one path and then, if that path fails, eliminating it and trying something else. All of the classes should also build careful reading skills. This is easy with liar/truthteller puzzles or the riddles. For the Nikoli-style puzzles, introducing a new puzzle type and having students read the rules will work well.

Once students have learned about deductive reasoning and case analysis, we transition to mathematical arguments that use those same skills. There are plenty of math problems where a case analysis can be extremely helpful. (One, which might be a tiny bit too sophisticated, is constructing all 3×3 magic squares.) By drawing explicit analogies, students can learn how to reason about mathematics. We can also help them write their own mathematical arguments, going step-by-step and saying which lines follow from which other lines (but without cumbersome rules of logic).

If time allows, I would like the logic classes to include looking at arguments (from other students, but also provided by the teacher) and deciding if they are correct, or if they are not correct, deciding where they go wrong. This reinforces how to think about deductive reasoning. One example of this is the Missing Dollar Riddle, although I will find more mathematical arguments as well.

All in all, I think this is a very solid ten hours (plus homework time).

The rest of these courses draw on more established areas of math teaching, so they don’t need quite as much detail to flesh out. For example, Math Foundations reviews mathematics from school but in ways that lead students to be more reflective on the mathematics. We’ve already developed modules designed for self-study that draw from problems in the Art of Problem Solving *Prealgebra* book as well as problems of our own design. Modules questions require thinking in new ways about math students know, which triggers self-reflection. In the spirit of Arnold Ross, I want students to “think deeply of simple things” — so simple that they’ve seen them in school! (It feels rather sacrilegious to take this quote from Ross and apply it to a program that is not the Ross Program, but perhaps this is the first step to getting there!)

Modules can form the basis of a solid course that reviews some aspect of school mathematics. Students would choose which course area they want to work on: fractions (probably the most important), uses of the distributive property (which could be pitched as mental math tricks), exponents, basic geometry, etc. Then the course would go through modules based on those ideas, led by the instructor.

However, there are some topics so essential and so frequently misunderstood that I believe *all* students should study them. These could include:

- The meaning of the = sign. We should also cover the different ways that = interacts with variables.
- Different representations of the same number. Fundamentally, 1/2 and 0.5 are actually the same mathematical object. Possibly also touching on infinite decimal expansions, depending on how feisty we feel.
- Division by 0, and
*why*it is not possible. Ideally, we would lead this from the standpoint of deciding as a class what the definition of division is, and then analyzing both 7/0 and 0/7 according to our definition to see which makes sense. As a bonus, we can divide 7/(1/2) and see what happens. - Is 0 even or odd? This can again be approached from the standpoint of definitions.

I don’t think we will have time for all of these. Covering the equals sign seems essential. The others are not critical from a content perspective, but seem very helpful from the perspective of understanding how mathematics is done and how to think about meaning in mathematics. Readers, if you have anything else that would go here, I would be very excited to hear it.

This course would prepare students for competitions like MATHCOUNTS. Most students have team meetings for years. We’d just be getting their feet wet, although it’s early enough in their mathematical careers that they could keep pursuing it.

I’d like each Math Team Strategies course to pick a mathematical focus area and to really develop students in that focus area. They might work on number theory, geometry, or combinatorics. As one example, a number theory course might develop prime factorizations in some detail and show students how to use prime factorizations to solve challenging problems. In this way, students see what it’s like to learn and use a mathematical subject in some depth, and they can feel a deeper learning of the material. Plus, this is pretty stuff!

Math Team Strategies might have some sort of built-in contest within the program. I would also like the course to introduce students to the Art of Problem Solving website by using Alcumus or even posting on a forum. In this way, students get started on a tool that can carry them forward mathematically. Of all the courses, this one is the easiest for students to stay engaged with after the summer, and I would like to encourage that as much as possible. The ease of staying engaged is my biggest reason for including the course in the program!

This course track will require getting faculty who can really develop their own courses. The most popular is likely to be programming, and we will have to recruit heavily to find great programming faculty, perhaps finding people in industry who want to teach for an hour a day for a couple of weeks. Other applied math courses could include biology, astrophysics, engineering, etc.

These courses don’t have explicit mathematical goals. Instead, we want students to gain a stronger idea of what math can do, and to learn new ways of thinking about it by seeing genuine uses of math, rather than artificial short problems from school. The programming course should equip students to continue programming after the summer, because this can be a tremendous source of self-directed learning.

Finally, it wouldn’t be a math program (at least not one that I’m running) if there isn’t something to explicitly show students how big and beautiful and wonderful and fun math is. Math Explorations will be the final period of each day, a math circle where students explore something new each afternoon before going home. We can get a variety of presenters to give students many pictures of what math can be. It will also reinforce the thinking strategies they’ve been learning in the rest of the program.

Sometimes, we might replace the math circle session with guest speakers who use math in their careers. Those role models can be very powerful for the kids.

There’s obviously a long way to go here in developing all of these materials. I’m excited to put together logic materials, and we already have the modules for Math Fundamentals. I also have some basic materials on Math Team Strategies. Applied Math will require people who can put together great courses. In any case, we’ll need teachers who can improvise; we’re not going to be handing anyone a fully-formed curriculum with handouts ready, at least not in our first year.

The other big challenge is time. Especially for programming, but really for all of these courses, there just aren’t that many classroom hours and there’s not a whole lot of problem set/homework time built in to the program. I’m still not sure if we want to assign work for students to do at home, since it’s summer vacation, after all! Can we actually get students far enough in programming that they can continue on their own with so few hours?

It’s a lot to think about and a big challenge for our staff. Fortunately, as a young program, we can iterate as we discover what works well. Sometimes it’s hard to remember that the first year is just the *beginning* of many journeys.

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In about seven months, there will be 100 sixth-grade students all ready to learn math. Almost exclusively, their mathematics educations so far will be designed around memorizing procedures and passing tests. We have four weeks to change their lives. What should we do?

No pressure or anything.

It’s rare in education to get an opportunity to work with motivated, talented students with no outside requirements. We can design the program that is best for them. This is the first post developing BEAM 6, and so we will set down the program goals.

If students leave the program and they have not learned about logical reasoning, I will feel exceptionally disappointed. I want students to grasp ideas of deductive reasoning. This might be my single biggest goal.

I also want to change the way they think about mathematics. For many students, math problems are defined by the *solution method*. “Oh, this is an addition problem.” “Oh, this is a related rates problem.” “Oh, this is a Pythagorean theorem problem.” This thinking leads to oversimplification and memorizing procedures. It makes it more difficult to solve multi-step problems. Students should engage with the question, understand the problem independent of its solution, and accept or reject solution paths because they do or don’t solve the problem.

This leads to the broader question of **mathematical communication**. For example, the equals sign. Students often interpret the equals sign as asking a question. In elementary school, it is always used as “2 + 5 = ?”. By algebra, the question changes — “2x – 3 = 15” means “solve for x” — but the equals sign is still primarily used to express a question. Students don’t realize that “25 + 7 = 32” is a statement that can be true or false; that the purpose of = is not to ask a question but rather to give a statement. The result is a failure of both communication and conceptualization.

These goals are less mathematically sophisticated than BEAM 7’s goals. This is in part because the students are younger. It’s also to build synergy with BEAM 7. Students often come out of BEAM 7 with a strong grounding in abstract mathematics but still well behind peers in school-based math. For example, students often do well taking a number theory course at CTY or going to a program like MathPath, but do relatively poorly in a contest like MATHCOUNTS. BEAM 6 can close that gap and set students on a path to deepening their facility with school-based math.

People love math because it is beautiful; because it is thrilling to challenge yourself with a hard problem that you finally solve; and because it is interesting to see how it applies to the real world. We must show students what math really is. That it is not about memorization or following procedures. That it is beautiful and creative and exciting. A love of math will carry you far, and we should develop it in the students.

In my experience, self-identity drives a lot about a person. More than just thinking something is “cool,” self-identity can push someone to pursue an interest; it can create resilience to failure; it can drive life decisions. If we can develop self-identities in our students as scholars, and furthermore as scientists and mathematicians, they are much more likely to succeed on that path.

What contributes to developing self-identity? Here are some thoughts:

- Interest/passion for a topic.
- A feeling of self-efficacy; confidence in your abilities.
- Membership in a distinctive community.
- Role models.
- A sense of future (where will it take you?).

We should harness all of these within the program. We have special expertise in creating a mathematical community. To drive students’ further engagement, creating a very strong community will be essential.

A summer program cannot alone cover the mathematical education of all these students. If they will be successful, they must continue to pursue learning after the summer is done.

Students should be connected with resources for further study, such as Art of Problem Solving. They should get used to these tools during the summer and be encouraged to continue using them when they’re done so that they continue to get better.

These goals feel right. They cover what I feel is very important to develop in young mathematicians. However, they are not complete. While program elements will be tied into these goals, as the program development continues we will also find new goals that we want to achieve. These will be included below as updates to this post.

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Now comes of the work of *designing* that program for a launch this summer. I’m going to do that design here, on the blog, so that others can follow along with the process of creating a new program and see the ideas get developed and change over time.

To begin, I’ve created an outline of the major topics I plan to think through. Each of these bullet points will become a link to a post. Please note that both this post and all of the other posts in the series are likely to evolve over time. They’re likely to get edited to reflect the final state of thinking as we move to launch.

- Program goals
- How do we measure achievement of our goals?

- How is the program communicated to schools and students?
- How are students selected for the program?
- How will we hire staff?

- What topics should the program cover?
- What guidance/materials will be provided to the faculty?
- How closely must faculty stay to planned curricula?

- How do we create a vibrant community?
- What structures do we need to manage student behavior?

- What, if any, additional support is provided to students?
- How does this connect to the existing BEAM program?

- What space will we use?
- What will the schedule be for students?
- How will we ensure that students attend the program for all four weeks?

The program will be known (for now) as BEAM 6, and all posts about it will be labeled as such.

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The core of this issue seems to be: what is the purpose of teaching algebra? For example:

- If the purpose is to preserve the opportunity for all students to enter science/engineering/math, then the standard
*should*be high. It does no good to a student to barely scrape through algebra if they want to be a scientist. - If the purpose is to give everyone exposure to a beautiful subject, then the standard should be kept relatively low: it is the exposure, not mastery, that is important.
- If the purpose is to give people access to math they need for life, then algebra should be dropped or revamped. Many people do not need algebra in life, and a high barrier to graduation does them no good.

Right now, the grade required to pass is being used as a proxy for this kind of battle. Those whose focus is on high school graduation want the required grade to drop. Those whose focus is on preparing students for STEM careers want it to go up. Without resolving this difference of goals, everyone will just keep shouting at everyone else and we’ll end up with a muddled policy that drags students in multiple directions.

Alas, that is not so unusual.

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Sound complicated? We’re not done yet. In a separate process, there are the specialized high schools (think Stuyvesant, Bronx Science, etc.) which have their own test, the SHSAT, which you take for admission. Admission is based solely on this one test, and many people study for years. (A sad waste of talent.)

No, we’re still not done yet. Based on the SHSAT and your Round 1 form, you get a school assignment in March, which could have 2 schools (if you got into one of each, you get to choose), 1 school (if you got into one “normal” school and one specialized school), or 0 schools (if you were not placed with any of your Round 1 schools nor did you score high enough on the SHSAT). At this point, you can complete the Round 2 form, in which the same matching process is used as Round 1, but with those schools that are left over. (Usually, not the good ones.)

If the Round 2 process doesn’t work out, you are automatically placed into a local school that has space. But it’s still not over. You can appeal your choice if something went wrong. You can also apply separately to charter schools (yes, they’re in *another* totally separate system), which are decided by random lottery, but the more you apply to, the better your chances.

All of which is to say, the process is *truly, incredibly complicated*. (I also simplified it, leaving out LaGuardia School of the Arts, private schools, and many intricacies of appealing your placement.) As the *Post*‘s opinion piece justly points out, this process dramatically favors more affluent students who have much better coaching and access to information. (Let alone that they often speak better English.)

You might think, at this point, that I agree with the author that we should abolish this system. After all, it clearly favors affluent students and takes a lot of time for everyone. But there is one key problem: *nothing better has been proposed*. At least the current system gives students a *chance *to go to a better school. At BEAM, we can coach students on how to get into a great school that will challenge them, and we have tremendous success. Some families find their way on their own. Compare to a system that just places students by geography, in a city that is highly segregated (by race and income) — what chance would low-income students have then for access to these schools?

This is a subtle issue. Maybe a system based only on geography would help, because then there might be some mixed-income schools (although I find that unlikely). Maybe the current system has another downside, “creaming” the best students from low-performing schools, leaving them worse off. These are interesting questions, questions that deserve study. But it’s not worth changing a system that offers some kids incredible opportunities unless you’ve done a very careful examination of the trade-offs, and we frankly have no idea!

So what *should* we do? If more students had real advising on navigating the high school system, they could be vastly more successful at getting into great schools. There are so many big mistakes that students make all the time: not filling in all 12 spots on the Round 1 form, for example, or incorrectly judging what school to apply to. (We had a student who decided by looking at schools’ graduation rates, not realizing he’d ranked a dual-language school specifically for English Language Learners, a decision that he is now stuck with for 9th grade even though he does not speak fluent Spanish!) Just as we need more guidance counselors for college applications, help with the high school application process might create a system that offers tremendous opportunities for all students, regardless of background.

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