BEAM Summer Away Faculty Reflections

BEAM is currently hiring summer faculty for Summer 2022. If you are interested in working at one of our programs, please apply here.

BEAM’s summer staff members are what make our summer programs special. Knowing how much we get from our staff, we recently asked some of our long-time Summer Away faculty what they’ve gotten from working with us. The answers we got were resoundingly encouraging.

What we consistently heard was that our faculty learn from our students just as our students learn from them. Our staff learn how to be flexible, how to connect with students in a way that makes them more excited about math, and how to bring the spirit of mathematical exploration and discovery back to their year-round jobs. Here’s what our faculty have to say:

Tanya Finkelstein  

For Tanya, one of the most special things about BEAM students is how much they can figure out on their own. With only a definition, they can really run! Of course, letting students run requires curricular flexibility and a willingness to be confused that most teachers are taught not to have. It’s a learning experience:

“The most important way you’ll grow as a BSA faculty is learning not to have expectations, which so many teachers come in with. Teachers are taught to have expectations of themselves that they’ll always have the answers and know how to guide their students to the solution. The most important thing kids need from you is to guide them along their own path to the answer without an expectation of what that path will look like. Some of the best teachers I’ve had in my life were kids. A teacher should be ready to feel challenged. It’s important to be confused yourself and make sense of things in front of the students so they can see a model of exploration. This is how you grow over time with BEAM.” – Tanya Finkelstein

Siddhi Krishna

Siddhi has taught both Cryptography and Knot Theory every year she’s been with us, but her classes have never looked the same twice. Siddhi attributes this to the fact that the direction the class goes depends on what the students are buzzing about any one summer. Bringing the exploratory culture of BEAM classrooms to her year-round teaching has ultimately translated to better outcomes for her year-round students. In her own words:

“I’ve learned how to listen much more closely to what students are saying. I’ve also learned to ask better questions. When a student says something, you want to draw the best thing out of them without telling them the answers. What BEAM does well is create a culture where students aren’t expected to know the answers. Trying to inject that spirit into my teaching at the university level has been important. Making it clear I don’t expect you to know the answers, but I do expect you to ask questions and try to bridge that gap, has made a difference in what my students get out of my college classes.” – Siddhi Krishna

Andre Mathurin

For Andre, teaching with BEAM was a stark contrast to the formulaic, almost sterile culture of mathematics education that exists in most high schools. Being able to think about big concepts and problems without the overhead of midterms or standardized exams was not only personally motivating for Andre, it also helped him learn to better connect with students in his year-round position. 

“My first summer I had this idea that it would be super structured. I realized that you can have a game plan and that’s fine, but the need to be flexible and meet students where they are is important. That’s one of the big learnings I got from BEAM, how to leverage students by getting them to work together and make it about discovering math versus me being the one to point things out to them. One of the incredible things about BEAM students is that they really can do this. They leave the summer program being better thinkers and better communicators.” – Andre Mathurin

Cory Colbert

Cory has worked with BEAM every summer since 2014, and he’s taught topics ranging from Analytic Number Theory to Aviation Theory. Working with BEAM has not only given Cory the same tools for leading a discovery and exploration-based classroom as other staff, it’s given him a vehicle to reflect on how he thinks about equity in STEM more broadly:

“I used to not think very explicitly about equity, diversity and inclusion, especially in 2014. I think I had gotten so used to being the only Black person in a classroom (or one of the only), and one of only 2 Black graduate students in math at my school (out of over 100), that I think I grew numb to how much STEM struggles with diversity… Recently, with regards to teaching, my focus has been on combating bias and micro-aggressions in the classroom. For example, my women students are less likely to volunteer an answer than their male counterparts. So, I'll make extra efforts to encourage women to share their results with the class, which are usually right, by the way... My hope is that maybe BEAM faculty can learn how to be effective anti-racist teachers both at BEAM and elsewhere.” – Cory Colbert

BEAM is currently hiring summer faculty for Summer 2022. If you are interested in working at one of our programs, please apply here.

Announcing BEAM NYC High School Results!

High school admissions were turned upside down in New York City this year.

The pandemic forced major changes in the admissions process and meant families faced delays and uncertainties. Under-resourced middle schools, still struggling with online learning, were often unable to help students.

That’s where BEAM stepped in, to fill the gaps and provide the support students and their families needed to successfully navigate the process.

BEAM 8th graders, with our help, earned admission at great high schools this spring!

Results to date:*

• 86% of BEAM 8th graders earned spots at high schools BEAM rates at Trusted+. These are schools we think have good course offerings and support.

• 54% of BEAM 8th graders earned spots at high schools BEAM rates as Tier 1. Tier 1 high schools offer Advanced Placement calculus or its equivalent (like the opportunity to take a college-level math course), and more than 85% of graduates are prepared for college. BEAM counts only about 40 high schools citywide, or about 7% of New York City high schools, as Tier 1; all are highly selective for admissions.

• 12 BEAM students were admitted to Specialized High Schools, including Stuyvesant, Brooklyn Tech, and Bronx Science.

In New York City, what high school you attend determines a lot about what opportunities you’ll have in the future. So, we know it’s important to find a strong, good-fit high school. Given all the uncertainties right now, finding a strong school was even more vital this year.

BEAM provides individualized support to our students and their families throughout the admissions process. This year, we also built an online high school admissions portal to connect students and their families to even more resources.

Here’s what Brandon C. said about his admissions experience:

“The high school admissions process was smooth for me. BEAM made this possible. They were able to give me a list of top schools that fit my interests. From there I looked through the schools and made a list and then ranked them based on what they offered and what I liked. My first choice was Bard High School Early College. They required me to do essays on humanities and STEM [to apply]. Elyse [BEAM’s Enrichment Coordinator] was very helpful in this process. She helped me review my essays and gave me feedback on what to change and what to add. I am very pleased to say that I got into my first choice. I am very grateful for BEAM’s support. I felt very happy and delighted when I got my results. When applying I was very nervous but at the same time felt confident because BEAM helped me in the process.
”

Brandon is looking forward to attending Bard High School Early College Queens in the fall, where he hopes to play on the basketball team.

Way to go BEAM 8th graders! We’re incredibly proud of you. <3

Want to learn more? Check out this article in Chalkbeat featuring BEAM 8th grader Nevaeha Giscombe, and BEAM’s own Elyse Mitchell.

Here’s a complete list of high schools admissions for BEAM students to date:*

• A. Philip Randolph Campus High School (2)

• Academy of Software Engineering

• Academy of American Studies (2)

• Art and Design High School

• Aviation Career & Technical Education High School

• Bard High School Early College (7)

• The Beacon School (2)

• Bedford Academy High School

• Benjamin Banneker Academy

• Benjamin N. Cardozo High School

• Bronx Early College Academy

• Brooklyn Secondary School for Collaborative Studies

• Central Park East High School (4)

• Civic Leadership Academy

• Coney Island Prep

• East Side Community High School (2)

• Eleanor Roosevelt High School

• Energy Tech High School

• Fannie Lou Hamer Freedom High School

• Francis Lewis High School

• Frederick Douglas Academy

• Frederick Douglass Academy VI High School

• High School of Economics and Finance (2)

• Hostos-Lincoln Academy of Science

• Leaders High School

• Manhattan Center for Science and Mathematics (6)

• Midwood High School (5)

• Millennium Brooklyn High School (2)

• Millennium High School

• Morris Academy for Collaborative Studies

• NYC Lab School for Collaborative Studies

• NYC Museum School

• Park East High School (3)

• Pathways in Technology Early College High School

• Science, Technology and Research Early College High School

• Thurgood Marshall Academy for Learning and Social Change

• Townsend Harris High School (3)

• University Heights High School (7)

• The Urban Assembly School for Law and Justice

• Urban Assembly Maker Academy

• Washington Heights Expeditionary Learning School

• Williamsburg High School for Architecture and Design

• Young Women's Leadership School of Brooklyn

BEAM students also received admissions offers from the following Specialized High Schools:

• Bronx High School of Science

• Brooklyn Latin

• Brooklyn Technical High School (4)

• Fiorello H. LaGuardia High School of Music & Art and Performing Arts

• High School for Math, Science and Engineering at City College

• High School of American Studies at Lehman College (2)

• Queens High School for the Sciences at York College

• Stuyvesant

Students admitted to Specialized High Schools will choose between these schools and other admissions offers they received.

We are incredibly proud of our students!

Ange was admitted to Beacon High School.

Brandon was admitted to Brooklyn Technical High School.

Estefani was admitted to Midwood High School.

Mansour was admitted to Bard High School Early College.

Yeshua was admitted to Central Park East High School.

Precious was admitted to Manhattan Center for Science and Mathematics.

Yasong was admitted to Townsend Harris High School.

*We say to date because every year a few BEAM students are under-matched in this process. We are currently working with students who were not admitted to high schools that meet our standards to make sure that they can navigate the appeals process and find a good fit for the next four years.

... And Now for Some Math

BEAM Los Angeles has started a math team! The goal is for students to have fun doing challenging math problems, but also to be prepared to participate in math competitions once pandemic closures lift. Some weeks students work on a specific math competition, and other weeks they focus on particular kinds of problems that often appear in math competitions, like the one below, which involves a guessing game. See how you do! (And if you like this sort of challenge, you should also check out this problem from last summer's newsletter.)

You may have played this game before: I am thinking of a number between 1 and 1000; try to guess my number in as few guesses as possible. Each time you guess, I will tell you whether my number is greater than or less than your guess. What is the best strategy so that you are guaranteed to find the correct number with as few guesses as possible?

In this case, the answer is fairly straight forward and not too difficult.

But what if we add a twist: You are allowed to say three numbers in each guess, and I will tell you whether my number is greater than or less than each. Now what is the optimal strategy to find the correct answer in as few guesses as possible? What if instead, I let you ask three yes/no questions each time? Would you use the same strategy?

Let's start with our initial question. If you want to guess a number from 1-1000 in as few guesses as possible and you can make one guess at a time, what's the best way to do it?

One way to think about this is that when you guess a number, call it x, you turn your original problem (searching for a number 1-1000) into a smaller problem: either searching 1-(x-1) or (x+1)-1000. If one of these ranges is too big, and we get unlucky and our number is in that larger range, then we will have more guesses to do! Hence, a good strategy is to always divide the range in half (or as close in half as we can). That way, no matter which half we end up in, the worst case is under control.

Hence, the first time you'd guess 500. If you're over, then you need to check 1-499; if you're under, you need to check 501-1000. (You could also have guessed 501, in which case you'd get ranges that are the same size.) Keep splitting in half. In total, it will take you no more than ten guesses.

In fact, it turns out that 10 guesses is enough to go higher than 1000. One way to look at this is by building up from smaller values. To guess a number 1-3 requires two guesses (your first guess would be 2, and then you might need to guess either 1 or 3 depending on if you were over or under). To guess a number 1-7 thus requires three guesses: start by guessing 4, and then you're left with two ranges of size three (1-3 or 5-7), which you know you can do in two guesses no matter which range it turns out to be. Guessing a number 1-15 requires four guesses (guess 8, and now you've reduced to a range of size 7 which you can do in three more guesses); 1-31 requires five; and so forth.

Notice the nice pattern: with n guesses, you can do a range of size 2ⁿ-1. That makes sense, because each guess basically lets you double the range you can check. So with 10 guesses, you could do 2¹⁰-1 = 1023 numbers. This search strategy is called binary search and comes up all the time in computer science.

Sidebar: Extra credit: This also exposes a neat pattern: if you double one less than a power of two and then add 1, you get one less than the next power of two. Written with an equation, that's 2(2ⁿ-1)+1 = 2ⁿ⁺¹-1.

So what happens when we get to name three numbers at a time? When we only guessed one number, it would split our range into two subranges (less than or greater than the number we guessed). Now, knowing if the target is greater or smaller than each of three numbers splits our original range into four subranges. We might be smaller than all three numbers, between the lowest and the next two, between the lowest two and the highest, or above the highest.

Effectively, this allows us to take two steps at once: instead of guessing just the number in the middle, we can guess the number in the middle and the number in the middle of both of the possible resulting ranges. Thus, instead of needing 10 guesses, we only need 5 guesses.

This might be surprising. After all, you're guessing three times as many numbers; maybe you would have expected to only need a third as many rounds of guessing. However, because you have to pick all three at first (without knowing the answer to your first guess), you can't do quite as well.

Sidebar: Extra credit: We can also build things recursively like before. We can do 1-3 in one round by just guessing 1, 2, and 3; we can do 1-15 in two rounds by guessing 4, 8, 12 — the resulting range is going to be of size 3, so we can finish it off next round. And so on. The next step would be 15x4 (the four ranges) plus 3 (the three numbers you guess) which would be 63. You'd guess 16, 32, and 48, and whatever range the target number was left in would be size 15, which you can now do in two guesses. The relationship here is that 4(4ⁿ - 1) + 3 = 4ⁿ⁺¹-1.

So, you'd expect the same with asking three yes/no questions, right?

Well, this is the neat part, and why we decided to highlight this problem. With yes/no questions, you have more flexibility than just "is the number too high or too low." Now you can ask questions whose answers are actually independent of each other, and truly cut your search space in half with each guess. (As an example of two independent questions, you could ask: "Is the number bigger than 500?" and "Is the number even?" Both answers cut your search space in half no matter what order you ask them in.)

So how can we come up with three questions that are all independent of each other? That's actually not too hard if you know binary numbers. For example, written in binary, the number 953 is 1110111001, because 953 = 2⁹ + 2⁸ + 2⁷ + 2⁵ + 2⁴ + 2³ + 2⁰ — a power of 2 for each place there's a 1. To turn these into questions, just ask:

• Is the units digit in binary a 1?

• Is the 2's digit in binary a 1?

• Is the 4's digit in binary a 1?

• Is the 8's digit in binary a 1?

• ...

• Is the 512's digit in binary a 1?

In total, that's 10 questions you need to ask, one for each binary digit, and then you know the number. That means we have to do four rounds in total. Three questions in each of the first three rounds, and then one more question to finally know the number. But now that we have arbitrary yes/no questions, it really does mean we only need a third as many rounds (rounding up).

What We're Reading

Supermath: The Power of Numbers for Good and Evil

by Anna Weltman

“Math is, at its core, problem solving,” writes Anna Weltman.

But, she admits, if your experience of math all happened in a classroom, this might sound a little hollow. Math problems in school often don’t seem like real problems, at least not problems we really care about answering. (A problem, notes Weltman, is not “an exercise in using a technique that someone taught us.”)

School math can feel pretty disconnected from, say, showing that our electoral system is unfair, or figuring out what symbols on an ancient tablet represent. But math can solve these problems, says Weltman. In fact, math can solve a lot of problems that might seem more cultural than mathematical, because math (like anthropology or sociology or science) is a cultural practice, a practice that can shift and change as cultures change; just ask the Oksapmin of Papua New Guinea.

In Supermath, Weltman takes on incredibly diverse topics, from Incan Khipu (and the Oksapmin of Papua New Guinea) to a perfect-checker-playing computer named Chinook to the Optimal Stopping Algorithm and Twin Primes Conjecture. If this is beginning to sound a bit intimidating to you, don’t worry; Weltman presents sophisticated concepts in clear and understandable language.

[Full disclosure: Another topic that Weltman writes about is our organization, BEAM. We’d recommend the book even if BEAM wasn’t part of it, but we are, so we thought you should know.]

Weltman’s point is not that math can solve everything, or even that math is always used for good (thus the “evil” in her title). Take Weltman’s example of setting bail. Algorithms can (and do) decide how bail is set in some places; but when designing algorithms to make such complex decisions, it’s incredibly difficult to account for all the criteria you (should) care about. Too often the outcome is an algorithm that is biased, even if unintentionally so — this has been proven to be true in the case of bail — because humans write algorithms and humans have biases.

Algorithms can also be biased by design. Think about voting districts. Algorithms are very efficient at drawing gerrymandered voting districts. It turns out, in fact, that drawing districts that everyone thinks of as fair is much harder than drawing districts that are clearly not fair. But proving that an algorithm is unfair turns out to be pretty hard; it takes an algorithm to expose an unfair algorithm.

How can math do better? One way is by providing more opportunities for those who have been excluded from STEM, and math in particular: people of color and women. Weltman notes that this is much more than an academic problem. When people of color and women are shut out of math they miss out on the opportunity to join a prestigious and lucrative field. They are denied opportunities to solve problems that affect their communities. When whole groups, like people of color, aren’t adequately represented among problem solvers, inequities are much more likely to result.

One answer to How can math do better, then, is that math can open its doors to underrepresented groups. But what does it take to become a professional mathematician? A lot more than doing well in school math, at least for a lot of mathematicians.

Many mathematicians grow up with a whole math ecosystem around them. They may have a mathematician in the family; they almost certainly have adults in their lives to help them navigate the world of math camps and teams and competitions. They may be encouraged to learn advanced math concepts starting at a young age. They often see others who look like them in this mathy world, giving them a sense that they belong.

For those who don’t see themselves reflected in this world, math can seem like a pretty foreign and unwelcoming place.

This, writes Weltman, is where organizations like BEAM step in. BEAM both provides opportunities, through summer programs in advanced math, and also connects students to other opportunities, like other summer STEM programs, good-fit high schools and colleges, internships, and ultimately, jobs.

BEAM also helps students build a rich community with their mathy peers. Finally, BEAM helps students develop their own voices to push the culture of mathematics to accept them and to value them for who they are.

Weltman closes her book with the unlikely story of Marjorie Rice, who in 1975 at her kitchen table discovered a new type of pentagon that could tile flat space. She was a 52-year-old “housewife” with one high school math course under her belt. Yet she found something no one had found before. Was it an earth-shaking discovery? No, writes Weltman, but it was a beautiful piece of math. “Even more importantly, this beautiful piece of math had the power to change Rice’s life.” And that, concludes Weltman, is a superpower.

“The power of mathematics,” concludes Weltman, “comes in part from the problems it solves. But the mathematicians, likely and unlikely, are ones who wield it. Math would have no power without them. Let’s distribute that power widely and see what people do with it.”