The *Atlantic* has a new piece about the more and more advanced work now studied by middle school and high school students. This trend is well recognized among those of us who’ve been working with these students for years, although it was interesting to read about it in a publication intended for a general audience!

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.

## A History of Math Outreach

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.

## Where Do We Go From Here?

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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