A Look Back at an Outstanding Summer!

Mathematical Wins

At Summer Away (our program for rising 8th graders), students were exceptionally engaged both inside and outside the classroom. In particular, they spent many hours outside of instruction time solving Challenge Problems, weekly problems beyond their daily coursework completed in teams or on their own.

Want to try out a challenge problem for yourself? Here’s Oscar and Yanny’s favorite: “What is the digit sum of the product of the 94-digit number consisting of all 9’s and the 94-digit number consisting of all 4’s?” Let us know what you think!

When we asked students, What’s the longest time you spent working on a single problem this summer, the median response of students at Summer Away, Pepperdine, was 48 hours — hats off to their perseverance! The median student at this site also grew an incredible 22% in national ranking on a national math contest (which we use to measure problem-solving growth) over the course of the summer. We consider 10% growth a “great” result, so this was truly exceptional.

This summer at BEAM Discovery (our program for rising 7th graders) in New York City and Los Angeles, students eagerly tackled the 100 Problem Challenge (100 difficult, puzzle-like math problems that involve teamwork, pattern-finding, and strategy). Students at all three sites completed all the problems, earning an ice cream field trip! Consider this problem from the 100 Challenge: Fill in each of the hexagons (at right) with a positive integer so that the number in each hexagon is equal to the smallest positive integer that does not appear in any of the hexagons that touch it.

Learn more about the puzzle and its solution here.

When asked what she would tell another student about BEAM, Alice T said “I’ve never been this excited to do math, especially the 100 Problem Challenge.” 

A New Course

How do mathematicians know that a particular fact is true? For example, how do we know for certain that there are infinitely many prime numbers? It's because we found a proof of that fact: starting from the basics of math, we built out logical reasons for why there must be infinitely many primes. Proofs can be beautiful and elegant, but they also form the underpinning of math, and for students they're a great step to develop rigorous thinking and to work more like a mathematician.

How, then, can we ensure that students develop the skills to create their own proofs? This past summer, at our Summer Away site at Union College, BEAM piloted a new Introduction to Proofs course developed and taught by BEAM’s Learning and Pedagogy Manager, Javier Ronquillo Rivera.

“One of the main goals of the courses at BEAM Summer Away is getting students to start owning the process of proving things mathematically. This involves not only understanding the content that the class is covering, but also developing some habits of mind (and naming them) that mathematicians use every day. For example, looking at patterns, making conjectures from those patterns, trying to find counterexamples, or trying to find ways of being certain that the pattern always holds,” said Javier. “We decided to run this pilot to ensure that every student has the opportunity to develop the habits of mind and to have a common language around them.”

In the second half of the week-long class, Javier introduced the class to the MU puzzle, created by Douglas Hofstadter (and presented in Godel, Escher, Bach). The puzzle contains the letters M, I, and U, which can be combined to produce strings of letters. The solver is asked to start with the string MI and transform it into the string MU using one of four rules in each step. (Spoiler alert: the puzzle can’t be solved. The rules and more here.)

Javier explained, “By introducing this puzzle before developing the tools to prove its impossibility, there were two things I hoped the students would understand and feel: The first one is understanding the subtle difference between knowing that a result is impossible versus having tried many ways of getting to that result and not being able to. In the MU puzzle — without having completely developed the tools needed to prove the impossibility — we fall in the space of saying: ‘Is it really impossible to get the MU word, or have we just not found a way to get there by using the rules we have?’ I call the students’ attention to that uncertainty to show that in order to avoid that feeling, we need proof. The second thing I hoped students would feel, while using the tools we had at that time, is the need for mathematical proof.”

The puzzle stuck with Zayden, who wrote, “Although ‘MU’ is impossible, I built a connection that taught me to see patterns in these problems that I never saw before.” Andrew also reflected, “[I] worked on this for a while…it was very interesting because there are so many ways to try to come up with a solution.”

Led by Javier, students discussed their own definitions of arithmetic properties like even, odd, and divisible. Students explored how utilizing slightly different definitions led to different results, and collaborated to create strong definitions that required abstraction. For example, taking the definition “An even number is divisible by 2” and strengthening it to “An integer n is even if and only if n is twice some integer.” With these strong definitions, they were able to answer a variety of questions about different integers, included in the video below!

Visit Javier’s class [10 minutes].

BEAM Pathway students mentor a new cohort of mathematicians

BEAM students gave back in a big way this summer — 38 BEAM high school and college students served as near-peer counselors and teaching assistants at Discovery and Summer Away, mentoring BEAM middle schoolers while gaining valuable (paid) work experience.

For the first time, BEAM students also held leadership positions at all of our New York summer sites, serving as the Directors of Student Life (DoSL) across these programs. It’s a big job — DoSLs are responsible for coordinating activities, managing counselors, communicating with parents, and more. At Union College, the Guidance Counselor – who supports students in their social growth while at BEAM – and Associate Site Director were also BEAM alumni!

We’re incredibly proud to see BEAM students step up as role models for our newest cohort of students.

BEAM’s Learning and Evaluation team charts success!

The Learning & Evaluation Team at BEAM not only helps us demonstrate our impact, but also drives improvement across BEAM. Established two years ago, the team has brought new rigor to the evaluation of our work, which is, in turn, shaping how we design and implement our programs.

This summer for the first time, the L&E Team measured student growth in six areas that research has shown to be predictive of students’ future STEM success, and that are central to our work:

• Math appreciation

• Sense of community with BEAM

• Growth mindset

• Math-specific perseverance

• Math self-efficacy

• Math enjoyment

Students at both Summer Away and Discovery showed statistically significant growth in all six areas, with large effect size — which means we can attribute the effect to BEAM’s programs. (For all you nerds, we promised we’d get there, so thanks for hanging in.)

We’re incredibly excited by the strong results — and by everything our L&E Team is doing to make our work more effective.

For more information about BEAM’s 10 year vision for supporting our students’ STEM dreams, please read our strategic plan.

Quotable Quotes

We could write so much more about this summer, but our students say it best:

• “BEAM wasn't just about math, I did learn new things but I was able to make connections with people and learn how some problems [have] a key to solve them.” -Lauren M, Los Angeles

• “I grew a lot in BEAM. I think I became more independent and mature. I also made friends and the activities are fun.” - Desani M, New York City

• “BEAM is a very interactive and rewarding program. It manages to make math feel like a sort of reward.” - Brian R, New York City

Jane Street and Hudson River Trading Summer STEM Scholars

We are immensely proud to announce the recipients of the Jane Street and Hudson River Trading Summer STEM Stipends, awarded to BEAM high school students in partnership with Jane Street Capital and Hudson River Trading.

At BEAM, we are working to make sure our students get access to the same sorts of STEM enrichment opportunities that their affluent peers already have, especially academic summer programs. To that end, BEAM provides individualized support in finding and applying to rigorous summer programs like Mathcamp, PROMYS, and more. This year, thanks to our corporate partners, we have been able to extend that support even further.

The Summer STEM Stipends are designed to address the fact that BEAM students who attend summer programs often cannot take summer jobs, which they and their families may be counting on. The stipends reduce the pressure of having to choose between getting paid for a summer job and participating in an intensive summer program. Attending such programs allows our students to explore diverse topics in STEM and prepares them for college and beyond.

Here is what BEAM students wrote last spring in advance of their summer STEM experiences.

Los Angeles 

Karla G - 11th grade - TeenArch Studios, UClA

I have been a part of BEAM since 2018, doing courses such as cryptography, topology, abstract expressions, and have worked with the Pathway Program as well as the BEAM Math Research Workshop. I enjoy learning more about fields of mathematics and the branches, doing programs on architecture and engineering during my spare time! I’m really interested in such things as the Artin braid group and the Torus knot, which I learned more about during my time at BEAM and reading books provided such as “Abstract Nonsense” and “Love and Math.”

Over the summer I am taking a course on Politics, Law, and Economics, at Yale University as a Yale Young Global Scholar. We’ll be learning about economic theories, government, and legal frameworks. This program will complement my activism centered on giving aid to marginalized communities and underprivileged areas. During the month of July, I will be taking a course at UCLA known as TeenArch Studios, a highly intensive, three-week architecture program, in which students work together to complete a project.

Daniel T - 11th grade - The Summer Science Program

Throughout my math journey up until 10th grade I viewed math as simply a subject I was good at that had very little applications to the real world. Then during my sophomore year I learned calculus and its applications in the field of physics. Then I really realized how interesting math can be. I became so hooked on calculus and physics that I wanted to learn a lot more than what was normally taught at school. I did this by taking higher level math courses such as calculus 3, linear algebra, and ordinary differential equations. BEAM also offered opportunities to learn math that was not normally taught in school which I was always fascinated by as well. After learning a lot of new things in these courses I am excited for what is to come in my math/science journey. This summer I am participating in the Summer Science Program at UNC Chapel Hill and will study astrophysics. I hope to learn about astronomy/space and how math and physics relate to it. Since I enjoy math and its applications in physics I know that I will enjoy everything that I will learn at this program. 

Alexis M - 11th grade - Canada/USA Mathcamp

I have been interested in STEM ever since the summer of 6th grade when I was first introduced to BEAM. The problems that we worked on at BEAM and the tight-knit community that existed was what hooked me. Alone, I probably wouldn't have grown as strong a passion for math. What excites me the most about STEM has to be the depth of many different topics, whether they are all math based or scattered throughout the sciences. For this reason, I continue studying STEM and hope to keep learning. The program I'm attending this summer is MathCamp, which I am returning to [after first attending last summer]. I hope to further advance my math skills and critical thinking.

Ashley V - 10th grade - Pomona College Academy for Youth Success 

"Falling down is a way for you to see another world, a world you have never seen before." This is a quote that stayed on my mind after hearing it, it felt so familiar. Now I realize, it felt so familiar because I had experienced it — with math. 

As I get older, math has become quite intimidating at times. I have fallen and fallen, over and over again. Failing to solve problems and spending hours of my day breaking my head over them, but there is something I cannot deny with every difficult math problem I have encountered: I always see something new. I always get to experience another world of math that expands my knowledge. So, if you ask me what has helped my interest in math grow, I would say I have fallen. I have fallen over and over again, and to be frank I can't wait to fall some more, if it means I get to keep exploring and discovering new worlds of math. This is my second summer attending PAYS. Last summer, the math class we took exposed us to math problems that truly challenged us.

Jacqueline O-V - 10th grade - Pomona College Academy for Youth Success

What most excites me about math is the many ways of coming to a single answer. At BEAM, I attended a Saturday class about stocks, which really piqued my interest in applying math to a job in the real world. This summer, I'll be attending PAYS, Pomona Academy for Youth Success, for the second time. Last summer at PAYS I discovered myself. I leaped out of my shell and made many close friends. I left behind timid me in Los Angeles and brought forward a confident me at Pomona. I made sure to participate every time I had an answer, right or wrong. I carried this confidence into all the other aspects of my life, including school. Going back to Pomona will give me the chance to, once again, discover new things about myself while still pursuing my education with the intense coursework provided.

Maya P - 11th grade - Turner-UCLA Allied Health Internship

I am what most people call a "STEM Nerd." I am proud of the title as my passion for STEM has driven me to discover new things about myself. The uncertainty that comes with learning something new within the STEM field is what really excites me and makes me want to continue pursuing STEM. I've attended BEAM and a club called MESA since 6th grade, and they have fostered and nurtured my love for STEM. 

This summer I will be part of the Turner-UCLA Allied Health Internship program which will further my knowledge of STEM careers I could pursue. I applied to the program because one day my mother was telling a story about a medical professional that worked at her job. Out of curiosity I asked my mother this person's job title and she said some hard to pronounce long technical name. I had never heard of the profession and looked it up. Upon investigation I realized I really knew nothing about what the medical field has to offer except for the typical doctor or nurse route. This internship is set to expose me to over 80+ medical professions that involve STEM and I hope that by the end of the internship I have gained a better understanding of the different career pathways within the medical field. 

Karen G - 10th grade - Scripps College Academy

I believe math and all fields in STEM have always been of my interest because of the career path that I want to pursue. I always knew I wanted my future career to be in the medical field but for a while I wasn't sure what position would best fit me. I decided becoming a delivery nurse would be ideal for me because you get to be part of welcoming life into the world which I think is a special moment to be able to be a part of. This summer I'm attending Scripps College Academy Program. During their summer program, I will be staying on Scripps’ campus for two weeks, analyzing topics in the real world that affect people's day to day lives.

New York City

Helen G - 11th grade - Summer High School Academic Program for Engineers, Columbia Engineering

I have been interested in math/STEM since I was in 3rd grade. My teachers really encouraged me to expand my knowledge. My 7th grade math teacher even took it a step further and helped me apply to BEAM, and I will always be grateful because to me STEM always has new things to learn about. I love problem solving and working with others, and that is what STEM is about.

Previously, I have done summer programs about different branches of engineering. In the future I see myself majoring in computer science but these programs helped me learn other potential fields I can major in. At SHAPE [Columbia Engineering's Summer High School Academic Program for Engineers], I hope to learn a lot more about computer science and also get to work with other students who share the same interests as me.

Yasong F - 10th grade - Lab Internship at Weill Cornell

Since sixth grade, BEAM has provided me with a lot of opportunities, resources, and support and has helped me develop a stronger interest in math. In elementary school and middle school, math class was always the class I looked forward to the most. I loved the way everything made sense and the satisfaction whenever I got a question right. It was during that time when I found out about BEAM. At BEAM Discovery, I met counselors and teachers who encouraged me to ask questions and made my questions feel welcomed. I also learned about math beyond the math taught in the school curriculum for the first time. 

I am interested in studying medicine. To become closer to my goal, I took AP Biology last year and I am taking a multi-year research course this year. Over the summer, I will be interning at a lab at Weill Cornell studying the effects of estrogen on blood pressure; I'm very excited to learn more about a subject I'm so interested in. Along with that, I will also be volunteering at a hospital.

Joelle N - 10th grade - Cooper Union Summer STEM Program

STEM excites me because STEM subjects are constantly evolving, with new theories, techniques, and discoveries emerging all the time. This means that there is always something new to learn, and the opportunity for growth and development are endless. The  logical reasoning and problem-solving abilities needed for mathematics and STEM topics are intellectually engaging and can help us solve complex challenges.

Both with BEAM and at school I was able to study math that addresses real world challenges. This summer I will be attending Cooper Union [Summer STEM Program] and based on my passions and interests I will create my own problems and projects that could have real world impact in the future. I hope to learn how to combine theoretical knowledge with practical skills while learning what the world needs through project management, project application, and teamwork.

Shreeya K - 10th grade - Science Research Mentoring Program, American Museum of Natural History

My STEM journey began as a child when I would always break things and rebuild them in an attempt to learn about how those things worked. Although I was a STEM enthusiast from a young age, I always found myself struggling in math as a child, primarily because I was new to the country and struggled to adapt to the new environment. However, when I entered the BEAM community during the summer of my 7th grade, it opened a whole new realm of math for me. I'll never forget that experience. I saw math with a new mindset and it completely changed the way I approached challenging math problems. Not only did I gain math skills, but I also learned how to apply those skills to other things such as finding the distance from sun to earth. I think seeing math in a new way unlike in school really helped me appreciate math even more and I will always thank BEAM for that. This summer, I'll be researching at the American Museum of Natural History with scientists in the astrophysics field to gain more experience in analyzing astro data. Through this program, I hope to gain hands-on experience working with astro data and improve my data analysis skills.

Lorraine A - 10th grade - Cooper Union Summer STEM Program

I am excited about the impact that STEM has. It makes things better and creates opportunities for a lot of people. I enjoy math because of the challenge; it also helps me improve my problem solving skills. Something that has helped my interest in STEM grow is learning about game development at The School of Interactive Arts. Although I don't want to work in game development in the future, I really enjoyed exploring it and learning coding. The program I am attending this summer is the Cooper Union Summer STEM Program.  I chose this program because I'm interested in learning about engineering and about Cooper Union. I hope to learn about the type of careers connected to engineering and if they fit with what I like to do.

Justin S - 11th grade - All Star Code

STEM subjects have real-world applications that shape the planet around us. They allow us to understand the complicated workings of the universe and develop practical solutions to address complex questions. Whether it's unraveling the questions of the universe, designing cutting-edge technology, or evolving life-saving cures, STEM fields have the ability to transform lives and create a positive impact on society. This potential to contribute to the advancement of information and better the quality of life for society globally is an important prospect that excites me about math and STEM.

Tasneem T - 10th grade - All Star Code

I liked dinosaurs growing up, it’s a weird fascination I've had since I can remember. One of my earliest memories is going to work with my mom and playing with a dinosaur figurine in the play area. I remember reading a lot of books about them too (I even cried when I found out they went extinct.) So since the age of 3, I've wanted to be a paleontologist.

That's what got me into STEM, simple curiosity and fascination led to something I see myself doing for the rest of my life. My interest has only grown since that time. [I chose to go to] All Star Code because it will give me tools that will help me succeed and give me an advantage in the future. There are many personal projects I have in store that require advanced computer skills like the ones I’ll be learning at All Star Code.

Ashfaq S - 10th grade - All Star Code

During 6th grade, I was opened up to the world of mathematics when my math teacher saw potential in some students in my class and introduced us to the math team and competitions and then, BEAM. This allowed me to explore my interests in STEM. Since 2020, participating in STEM programs has been hard, but I’m still interested in subjects like computer science and coding, which is why I chose to attend a coding program this summer. I hope to continue exploring computer science and hope that I can dive back into mathematics.

David L - 11th grade - All Star Code

When I joined BEAM, I got really interested in computer science. I remember going on repl.it and taking a python course. It expanded my creativity since there is no one right answer in python, you code to create your own world. In middle school, I had beginning coding classes and the usage of scratch and code.org. Then as I got to high school, I took a software engineering course where I learned HTML, CSS, and JavaScript. This summer, I'm attending All Star Code where I hope to learn more HTML, CSS, and JavaScript because I feel like there's more to learn.

Javier V - 9th grade - All Star Code

I first got introduced to BEAM by Ms. Perez, my 6th grade teacher. I was really into math, in fact, it was my favorite subject. I loved it even more when I got into 8th grade and took algebra with Mr. Joanis (I even got a 92 on my Regents exam). BEAM has given me the opportunity to express my natural passion for math and other related topics

I am attending a program in the summer called All Star Code. They teach people many different types of computer languages. You can also learn about different careers, such as game development, computer science, and much more. The reason I applied for the All Star Code program is because of coding’s connection to math. I love math so much because most of the time, there's one direct and definitive answer. Math also has many ways to affect the real world. When coding, I either input exactly what is needed and proceed with running the script successfully or unsuccessfully at which point I attempt to figure out the bug in the script. I wish to use all available resources to the full extent of my capabilities.

Kai G - 9th grade - All Star CODE

I first became interested in STEM around 5th grade. I was excited about how things fly, robots and technology, and how it could be applied. I started to explore STEM by going to various STEM camps, for example I went to a space camp during one of my spring breaks. When I went to BEAM, I got really curious about how I could solve different problems and the different ways to go about it! I chose to attend All Star Code this summer so I could learn about coding. I hope to learn how to make websites and how to make games. 

Mamadi K - 10th grade - Biorocket Research Internship Program

In STEM, I’m really excited about finance. I've taken classes in counting and statistics to help my interest in math/STEM grow. I'm attending the 2023 Biorocket Research Internship Program this summer because of the focus on science. I hope to learn about astronomy, human anatomy, microbiology, geology, neurobiology, and botany.

Kristen J - 10th grade - WorkForce 2000 Pre-RN Program

One thing that excites me most about math/STEM is how there are many ways to solve problems. I love how math involves a lot of critical thinking which allows you to explore the subject deeper and deeper so you can fully understand it. During my summer at BEAM Summer Away, I was given the opportunity to learn and explore different kinds of math. I saw how to logically think, different ways to work with fractions, and how to solve problems I would have never thought of. This opportunity led my interest in math to grow way more. The program I am attending this summer is a Pre-RN program. I chose this program because I would like to pursue a career in nursing in the future and I believe this program will help me learn more about it. I will use the math/STEM skills I have learned at BEAM and it will help me throughout my learning and research process as well.

Selina L - 10th grade - Brown Pre-CollegE

Math/STEM excites me because it allows me to use my hands and my brain at the same time. It also lets me have fun in the process. BEAM has helped grow my interest because during the summer of my 6th grade year, I was introduced to new kinds of math that I wouldn't be able to learn in school. This made me more interested in my math and science classes in school, making them my favorite. 

This summer I will be attending the Pre-college Program at Brown University and taking the materials engineering class. I chose this program because I get to explore one of my potential college majors, as well as experience college life and how it would feel being more independent, away from home. I decided to take materials engineering because I have always wanted to do some form of engineering, but I was never sure what. So this will help me learn if I would like to pursue materials engineering as a career. 

Selina Z - 11th Grade - Girls Who Code

Initially, I only liked math because I was good at it. But coming to BEAM, I realized that math isn't all formulas and unreasonable problems; it's actually fun. During BEAM Summer Away, I loved solving challenge problems and especially liked the sense of accomplishment when I finally solved it. BEAM allowed me to take classes outside of what was offered in my school and opened me up to new forms of math. BEAM has helped me foster my interest in STEM fields and made me want to pursue STEM not just because it pays me well but because I like it. This summer I'm attending Girls Who Code because I am interested in coding. I am excited to create fun projects!

Sylvan C - 10th Grade - STEM Institute City College of New York

I first became interested in math when my sister was preparing for the SHSAT [NYC Specialized High School entrance exam] and we reviewed the math questions together. In my first experiences with math, I was just given equations with operations that I had to evaluate to find an answer. I think too much of math is like this, where you are told what to do and how to do it and only asked to compute. I think it is much more interesting when a question is given without a set path to the solution. Finding how to solve something and why a solution works allows you to understand more and come up with new solutions rather than recycling formulas that you are just told to trust. My first experience at a math program was with BEAM Discovery. I took a class about patterns where we made formulas for patterns. This class is memorable to me because it represents the side of math that I love. We were asked to turn one mathematical concept into another. To represent observations using the language of math. 

This summer, I will be attending the STEM Institute at City College of New York. Last year I attended this program and took Precalculus as well as Chemistry. I would like to take these more advanced STEM courses in order to provide me with more scientific and mathematical background. The math that I am learning is fundamental to so many broader mathematical fields and concepts that I would like to understand.

Highlights from Summer Programs

BEAM Summer Away at Pepperdine University, Malibu

BEAM Discovery, Los Angeles

BEAM Summer Away at Union College, New York

Discovery, Uptown New York City

Discovery, Downtown New York City

Congratulations, Jane Street College Scholars!

BEAM, in partnership with Jane Street, is excited to announce the second class of Jane Street College Scholars. Congratulations to scholarship recipients Hawa, Ilana, and Bryan. Their scholarships are made possible by the generous support of our corporate partner, Jane Street.

A core BEAM supporter, Jane Street is a research-driven global trading firm and market maker trading a wide range of financial products. Read more about their work at janestreet.com.

Hawa D.

BEAM opened the door to many opportunities for me. In 2017, I was part of BEAM Summer Away, which truly transformed my life. I learned independence, determination, and how passionate I am about the field of STEM. I took courses like logic, graph theory, and combinatorics that made me value perseverance, patience, and teamwork. The abstract thinking intrigued me. I loved how math outside the classroom pushed me to think outside the box and explore different possibilities. 

BEAM has also been the support system I could count on. BEAM helped me apply to high school and navigate my high school years. I remember walking through the snow with Sarah, a BEAM staff member, to make sure I mailed my portfolio to Beacon High School on time. BEAM Saturday classes enriched my learning. Algebra 1 and Geometry helped me get a head start in my school courses. Game Theory helped me learn how I could apply math to the real world. Even today I use game theory to help my probabilities at rock, paper, scissors, or to make sure I never lose at tic tac toe. The courses I took at BEAM also helped me complete my non-routine problems at school for extra credit. Biostatistics was a course I took online during the pandemic that I loved. I love biology and computer science and the course taught me there are ways to combine both passions.

In STEM, I always approach every question with a determined and unwavering attitude, committed to seeing it through to the end. It brings me great joy to learn new things and then be able to apply that knowledge to real-world situations. I am deeply appreciative of the ongoing support and resources provided by BEAM, as they have truly been instrumental in helping me achieve success.

BEAM also introduced me to other opportunities that helped me grow, such as my internship at BioBus. For 3 years, I was able to create my own experimental questions, conduct research, collect data, and present my results. During this experience, I learned more about research and further developed my interest in STEM. I was also able to do a summer internship at Memorial Sloan Kettering that introduced me to various jobs in medicine. As a child, I dreamed of becoming a pediatrician. But as I explored various STEM career options, I realized that there are many paths within STEM that also pique my interest. So now, no matter what the future holds, I am excited about the possibilities.

I am truly grateful that BEAM will continue to support me in college, and I am deeply grateful for the scholarship that BEAM has awarded me. Thanks to this generous support, I can pursue my passion for STEM without worrying about the financial strain it may put on my family. This scholarship has truly lightened my load and allowed me to fully concentrate on my academic pursuits. I am certain that I want to major in a STEM field and I am considering majoring in biology and minoring in computer science while pursuing the pre-med track. I also want to take advantage of college opportunities to continue research and explore more interests in the field of STEM. Although I know college will be difficult, I am confident I will succeed because I have the support of BEAM and the skills I have developed through all my experiences.

Ilana O.

If I told my eight-year-old self all the things we have accomplished, I know she would never believe me.  

Sunday mornings in Bogota, Colombia, where I was born, consisted of binge watching Disney movies and American shows. I secretly dreamed of having a spacious house with an open kitchen, a walkable distance to school, bike riding on the weekends, and the freedom to not wear uniforms. Life seemed so easy because I didn’t see my character inside the classroom, doing homework, working to make ends meet, or even taking over my younger siblings' care. 

When I moved to the States I barely spoke any English, and what I knew, I learned from an Australian CD from the 90s. Naturally, I struggled in many ways, from making friends, to my education. What hit me the hardest was the fact that my American dream had to remain a dream. After I finished 4th grade, my mom searched up “free summer camps nyc” in Google. She made it a tradition each summer, until in 6th grade, she came across BEAM. Naturally, my 6th grade self was not excited about spending a whole summer doing math. At that point in my life, I was finally fitting in, and thus, I just wanted to spend my summer with my newly made friends an hour away from home in Manhattan, eating overpriced yogurt at 16 Handles.

After the BEAM application process and the admissions challenge, the acceptance came as such a surprise as I thought there were smarter kids, who definitely had the right answers. But looking back, I know it was my determination that counted most then and will drive me forward now.

At BEAM, math was challenging and fun at the same time, something I thought was not possible. I made wonderful friends, and I adopted a new patience for math. There was a certain beauty about taking hours, days, and even weeks to solve a problem. I loved it so much that I reapplied for BEAM Summer Away. Once on campus at Bard College, I saw college as a very real possibility for me. They showed me the vision of being at school, the beauty of living in a learning environment, and complicated problem solving in a way applicable to real life. 

BEAM changed my priorities as to what I wanted my summer to look like. I desired knowing and learning. BEAM connected me to STEM programs that slowly but surely started reshaping my vision of the American dream.

To this day, BEAM has continued to be a central part of my life, an incredible support system helping me apply to other programs, high school, internships, and now, college. They guided me, and connected me with other kids who attended my high school of choice, Manhattan Hunter Science High School. 

Four years later, BEAM was with me every step of the way as I applied to college, double checking every essay, checking in after every email sent, or meetings done, and most of all helping me prepare for one of my biggest accomplishments, the Posse Foundation Scholarship. On December 6, I got the acceptance call to one of their partner schools, Brandeis University. I truly believe that without BEAM’s help I would have never made it this far. Their trust, investment, and energy spent not only on me, but also on my fellow BEAM students, have pushed us to greater heights than I would have envisioned. BEAM’s support will allow me to not put a financial strain on my parents, and focus my energy at school.

My eight year old would have never believed that my American dream now consists of majoring in Biochemistry at Brandeis University, and doing great things in my field.

Bryan A.

My math and STEM journey started when I was in elementary school. Every day I would watch Wild Kratts on PBS. Watching the Kratts brothers save exotic animals from poachers and evil designers trying to steal their fur inspired me to save the animals around me how ever I could, and that’s how I started to love animals.

As I continued to go to school, I always looked forward to learning more about science. The more science I learned, the more my love for nature grew. The fact that there was always more to learn fascinated me and drew me in. But it wasn’t until my summer at Bard College with BEAM that I began to understand how math and engineering work in harmony alongside science. Spending the summer learning about number theory and probability made mathematics fun, and the time spent living as a “college” student gave me a unique environment where I was able to cultivate a new way to learn. Even after that summer, BEAM continued to give me new topics and new ways to learn during weekend classes held at NYU. BEAM gave me a new way to love learning and a new community that was just as curious as I was.

Being a part of a program like BEAM encouraged me to find more programs that allowed me to explore my interests. I spent my junior and senior years of high school interning at the Kenny Lab of Neuroscience at Mount Sinai Medical School, where I learned how to conduct and present research. I matured as a student and came to understand what it was like to follow science as a career. Seeing the inner workings of a medical school and taking on responsibilities that most high schoolers don’t have allowed me to get a glimpse of what a medical career would be like. My time interning at the Kenny Lab further strengthened my love for science and inspired me to pursue a medical career.

It has been curiosity and ambition that have constantly pushed me to search for and become more. My childhood love for animals has turned into a desire to help others and build a compassionate world with STEM at the forefront of the tools used to do so. I will forever be grateful for the experiences and community BEAM has given me. I am confident in my ability to effect positive change and eventually influence a new generation of STEM students who aim to do great things just as I plan to do in my next four years of college.

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Congratulations also to BEAM students Emmanuela and Cindy, who are receiving tuition assistance scholarships from BEAM.

Emmanuela B.

BEAM has been extremely integral throughout all the vital stages of my academic career. I started BEAM in 7th grade going to Union College upstate. I fell in love with the encouraging and caring environment that I remained committed to till now as a graduating senior. Growing up I was always really good in my math classes, so I knew BEAM  would be the place for me to face a real challenge when it comes to my academics. BEAM was the first place I really saw how vast and complex math could be, yet also extremely rewarding because of the amount of effort and time I learned to put into it.

As a first generation, low income student who has continuously been a high achiever, BEAM guided me during the high school process so I would be at an institution where my knowledge can be fed and I can delve more into STEM. They helped me study for the SHSAT and prepared me for high school math level courses like Algebra 1,which helped me be successful in the class freshman year. I ended up at the highly competitive Central Park East High School. 

 Throughout the past 4 years, BEAM has given me the opportunity to explore math outside of my high school curriculum with classes like Infinity and Finance, which helped me better understand my future career. This past summer, BEAM supported me through their summer stipend program where I studied mathematical theories and computer science with New York University professors at NYU Center for Mathematical Talent. 

Attending BEAM mas made me fall in love with math. There are many things to learn about math, as it's a never ending journey of discovery and growth revealing how powerful math can be at advancing our society. BEAM has fostered my love of math and encouraged me to delve deeper, helping me conclude that I am interested in majoring in economics at  Brown University. Similarly to how BEAM is dedicated to helping low income, first generation students achieve success in STEM, I hope to utilize my economics degree to help immigrants financially integrate into American society.  As I work towards that goal, this scholarship will provide book assistance to succeed in my classes and keep up with the course work. Additionally, I will be able to take educational trips to learn from Providence city. 

Cindy T.

With BEAM’s support, I can happily say that I am proud of the things I have accomplished and the person I have become. I remember joining BEAM in 6th grade, unaware of the massive opportunities awaiting me outside my school and neighborhood. It was the first time I learned to embrace stepping out of my comfort zone and was able to connect with other bright-minded students throughout the city. Throughout middle school and high school, I regularly attended BEAM’s Saturday enrichment classes and field trips.

From those opportunities, I gained new perspectives and explored the many ways math could be applied in the real world. I made long-lasting friendships at BEAM and being able to attend high school with some of the BEAM students made the experience less nerve-wracking and more supportive. During the summer, BEAM encouraged me to apply to summer programs and even helped us throughout the whole process. With their guidance, I’ve been able to intern at the Intrepid Museum, take biomedical engineering classes at Columbia University, and subsequently conduct tissue engineering research at one of their labs. This has ultimately shaped what I want to pursue in the future and fueled my love for STEM. 

In college, I plan to major in biomedical engineering at Johns Hopkins University. It feels like a dream come true and I can’t thank BEAM enough for their support throughout the long and stressful college process. I decided to pursue BME because my experiences at Columbia University helped me realize the impact of novel research on improving quality of life, such as regenerating instead of reducing the progression of diseases. I watched Dr. Warren Grayson's TED talk on tissue engineering and I was deeply motivated by his explanation of how facial appearance is linked to one's sense of self, and that as biomedical engineers, we have the ability to help people regain their confidence through regenerative medicine. As I look ahead, I hope to be at the forefront of rekindling people's joys and smiles via research.

This scholarship will help me achieve my goals because it will give me the opportunity to attend Johns Hopkins University’s renowned BME program and access their research opportunities on the campus and at Johns Hopkins Hospital. I am extremely grateful for the scholarship as it will ease the stress of affording college on me and my family as a first-generation student. 

BEAM's Statement on Affirmative Action in College Admissions and Equity in STEM

In the wake of the Supreme Court’s decision severely limiting affirmative action in college admissions, BEAM’s work is more vital than ever.

Creating equity—true equity—is long and arduous work that has been taken up by many before us and is carried on by many around us. We join them in this work, and reaffirm our commitment to our students and their journeys.

We know that race deeply impacts our students' journeys and that racism restricts their access to opportunities in STEM and beyond. We've seen first hand the way that our students have been subject to both systemic and, yes, very explicit racism at their schools and colleges.

Being blind to that racism does not vanquish it.

We must all step up following this court decision, which will make it harder for our students to achieve their dreams. Supporting our students' aspirations is more important and more urgent than ever. We remain deeply committed to creating equity in educational and STEM spaces, and we believe that the work we do is essential to creating a more just and equitable society.

Our students deserve nothing less than our best, and that is what we pledge to give them.

Graduation Letter: Congratulations Zeñia!

Graduation season is an exciting time of year at BEAM, as the students we’ve known since middle school graduate and continue their incredible STEM journeys. This year, we introduce to you Zeñia Alarcon, who graduated from Worcester Polytechnic Institute this spring with a bachelor’s degree in mechanical engineering and a master’s degree in management. This month, she is beginning her career as an assistant superintendent with a construction management company. Congratulations, Zeñia!

Zeñia has been a passionate, energetic member of BEAM’s community since we met her in 2014. Lynn Cartwright-Punnett, Chief Programs Officer, said, “I know that if I’m hanging out with Zeñia, someone’s laughing.”

See photos throughout her STEM journey with BEAM in the slideshow below! Hover over the photos for additional details.

Zeñia attended BEAM Summer Away in 2014. She is pictured second from the left, wearing sunglasses on her head and working on a paper geometry project.

At Summer Away, Zeñia took classes in combinatorics, problem solving, proofs, and Hamming codes. In the photo, she receives support from a faculty member.

Zeñia participated in many summer enrichment opportunities, including Cooper Union Summer STEM in 2018. Click the photo to learn more!

Zeñia attended college preparation workshops with BEAM to create her application list and work on her personal essay in 2018.

Zeñia visited Caltech with fellow pathway students Maria and Teo in 2018.

Teo, Zeñia, and Maria also visited Harvey Mudd College with BEAM’s CEO and Founder, Dan Zaharopol.

Zeñia was a junior counselor at BEAM Discovery in 2019. She stands in the back row on the far right.

As a counselor at BEAM Discovery, Zeñia also TA’ed and provided support during Open Math Time (OMT), when students work on math of their own choosing.

BEAM Discovery students took a field trip to the movie theatre! Zeñia sits on the far left.

Zeñia plays cards with a student during lunchtime at BEAM Discovery.

Each summer, counselors and students participate in a camp-wide talent show! In 2019, Zeñia and a few other counselors performed a choreographed dance routine.

Congratulations to Zeñia as she graduates with her bachelor’s degree in mechanical engineering and a master’s degree in management (after just four years!).

Student Spotlight: Congratulations, Lennin!

Lennin’s STEM Journey

Lennin has always been deeply curious.

His curiosity led him to one of his first loves, chess, which he began playing in first grade. Lennin was all in, attending tournaments every weekend, taking classes before and after school, winning trophies, even flying to Dallas (his first time on a plane) to play in his first (of many) national tournament.

Eventually, Lennin, who grew up in Brooklyn, landed at I.S. 318, a public middle school famous for its chess program (they have been national champions 58 times). It was there he learned about the Summer Program in Mathematical Problem Solving (SPMPS for short, later renamed BEAM — Bridge to Enter Advanced Mathematics) from his chess teacher. “At the time, I didn’t know these programs existed,” he said. “The idea of going to a college campus to study advanced mathematics and have some of my friends who also play chess be a part of the program, I remember just thinking ‘Oh, wow, this sounds like a really good, dope opportunity.’”

When Dan Zaharopol, BEAM’s founder and CEO, came to I.S. 318 to recruit students for SPMPS, Lennin remembers telling him about all the “really nerdy things” he was into, like math, chess and coding. There was an immediate connection. Lennin says he was really excited to learn there were other students and adults just like him — passionate about learning and sharing the mathematical world.

That July at SPMPS, Lennin’s view of what the future held and what math could be was transformed.

Lennin at Bard, 2014

“It was the most impactful summer I’ve had because that was the first time college became a reality for me — going to the campus, seeing what it’s like being away from the city, staying at a dorm, and being with people who were also nerds.” He took classes in logic, cryptography, and more. “I never even knew you could think of life or the world through these lenses,” he said. He was exposed to ideas he’d never seen in school. "Seeing these incredible professors passionately teaching us mathematics — and not only sharing, but sparking joy with me — radically changed how I viewed education and what I was capable of. I felt cared for, appreciated, and seen. It's moments like these that transform your life direction. I'm eternally grateful."

At the end of the summer in response to the question, “What does math mean to you?” Lennin wrote: “Exploring an infinite realm and deciding which course is yours.”

Lennin sits third from the left in a blue t-shirt.

“It was a really transformative summer for me,” said Lennin. He returned to school feeling less shy, a little more open, and far more prepared academically.

What happened next was equally important. “Not only did BEAM give me this opportunity, but they’re like, ‘Hey, we really want to stick around in your life.’ Having that encouragement and support allowed me to achieve and set my ambitions higher than ever before. And for my parents, who sacrificed so much for me, it meant the world to them to have BEAM take care of me in ways they didn’t know how.”

BEAM’s support and belief in Lennin were pivotal. BEAM introduced him to Bard High School Early College (BHSEC), where he graduated with his associates degree at 17 years old. “Going to Bard was one of the most important decisions in my life,” he said. “I feel like that’s such a crazy thing about BEAM. They help you make what appear to be small decisions at that age, but they’re the decisions that set you up for life — and they have your best interests at heart.”

Lennin spoke at a High School Admissions panel in 2016…

…worked as a summer counselor/Teaching Assistant at BEAM Discovery, and attended field trips with BEAM while in high school

At BHSEC, Lennin explored many passions. “I feel like I’ve definitely had a huge journey in my academic interest,” he said. He became an activist, working on behalf of undocumented students with the Immigration Task Force under Queens District Attorney Melinda Katz, and wrote short stories and poetry. When he was accepted to college, he intended to pursue his interest in government, politics, and public policy. “I have a strong voice [and] I really care about people,” he noted. But when he got to Franklin & Marshall, his interest in STEM reemerged. His thinking was: “The landscape of the world today is heavily shifting towards a digital future, and I don’t want to be left behind. Let me switch and go study computer science and mathematics and use these skills to pursue my humanitarian goals.”

During college, in addition to his studies, Lennin was driven to give back to his community. He completed an internship in product management at Coded by Kids, a nonprofit dedicated to supporting underrepresented students in Philadelphia to become future innovators; there he found a mentor in Sylvester Mobley, a Philadelphia social impact and tech leader. He went on to become the project manager at Draft Studios, a web development agency (part of Coded by Kids) that built websites for nonprofits and startups. Fast forward to today, and Lennin is working within a venture studio, where, among other things, he’s developing an internship program aimed at equipping high school and undergraduate students in Philadelphia with the skills, knowledge, resources, and space to build their own startups.

In the midst of all of this, Lennin also graduated from Franklin & Marshall in May with a double major in mathematics and computer science.

So, what’s next? For Lennin, it’s developing his skills to tackle the issues he cares deeply about, including education for marginalized communities and climate change. He’s drawn to blockchain as a potentially game-changing technology, and to the startup world. “Everyone is trying to figure out what to do and I’m no exception. But I’m only 22, with an intense desire to keep learning and provide genuine value to the communities I love and care for. With this in mind, I’ll always move forward.”

Lennin has his sights set high, hoping to one day build his own startup, while continuing to mentor others. “I know it’s a big thing, but I genuinely just want to build something that helps the people I care about a lot and provides value in the world. I know at the heart of it is this idea of educating youth. There’s so much more that you are capable of, there’s so much more that you can do, so much more that we can build for our communities.”

...And Now for Some Math on Pi Day!

Curious about more math this Pi Day? Here’s the problem we included in this year’s Pi Day Card:

In this game, x starts at 0, and your goal is to make x as big as possible. You can apply the four rules below, but each rule costs you points, and you only have 100 points to spend. What's the biggest you can make x, and how do you do it?

• Rule A: Add 0.1 to x. [1 point]
• Rule B: Replace x with 2x. [5 points]
• Rule C: Replace x with x². [10 points]
• Rule D: Replace x with 2^x. [50 points]

You do not need to prove your answer is the highest possible. The answer may be big enough you can't write it out; writing it in symbolic form is fine.

Are You ready for the solution? Here it is…

Let's ask a big question first: when should we use Rule D? It costs 50 points, and we only have 100 points in total.

We could use Rule D twice, but that only gets us to 2, and as you'll see we can get much further than that. Hence, we should use it at most once.

It feels like we should save it for last. And indeed, that's right. If you consider using it before or after each of the other rules, it always does better to use it after. So if you imagine writing out all the rules you want to use, you always do better if you swap Rule D with the rule after it until it's last. (For the details, see the note at the end.)

OK, we want to use Rule D at most once, and we want to use it last. Then what? Our goal is to get the biggest increase per point spent. Let's compare.

Rule A gives us an increase of .1 per point. Whereas Rule B gives us an increase of .2x per point. This means that applying Rule B can only match applying Rule A in increase once we have gotten x to .5. It also implies that once x exceeds .5, Rule B will be a better option than Rule A.

What about Rule C? It doesn't actually increase x until we're above 1, so for now we only need to trade off Rule A and Rule B.

Now we're ready to begin. To make x as big as possible we have to start by applying Rule A five times, and x is now .5. 

From here we can either apply Rule A five more times or apply Rule B once (because as we mentioned above when x=.5 these rules have the same average increase). It doesn’t really matter which we do, so for the sake of the game let’s just say we have applied Rule A 5 more times. x is now 1, and we have used 10 points. 

Now Rule C still can’t make x any bigger, so we need to apply one of the other two rules and we know that the average increase of Rule B is .2 at this moment, clearly beating Rule A. So we will go ahead and apply Rule B once. That brings x to 2, having used 15 points.

Rule B and Rule C get us to the same value of x, but Rule B is cheaper, so apply it again and we are at x=4, having used 20 points.

From here we know that we can use Rule B twice more or Rule C once with the same average increase over the next points. In both cases we will arrive at the same point, x=16 having used 30 points.

At this point, the average growth per point of Rule C is much higher: it's 24 per point when x=16, so we should use Rule C until our one use of Rule D. We apply Rule C twice, so x goes from 16 to 16⋅16 = 256 to 256⋅256=65536. We've used 50 points.

Should we use Rule C more, or Rule D? Rule C does keep increasing x, up to 2²⁵⁶. Use Rule D, though, and we get x=2^65,536 having used all 100 points.

In case you're wondering, this number is huge. It is vastly larger than the number of atoms in the entire observable universe (i.e., not just Earth but Earth plus every other solar system and galaxy we could possibly see with a telescope). It has over 20,000 digits. This is the biggest number we can get.

Note: 

Here is one way to verify that you always do better if you swap Rule D with the rule after it until it's last. 

Rule D first then Rule A 2^x +.1 vs Rule B first then Rule D 2^(x+.1) = 2^x (2^(.1))=2^x(1.07…) =2^x + 2^x(.07)

Rule D first then Rule B 2(2^x)=2^(x+1)= 2(2^x) vs Rule B first then Rule D 2^(2x) =(2^x)(2^x)

Rule D first then Rule C 2^x⋅2^x = 2^(x+x) = 2^(2x) vs Rule C first then Rule D 2^(x⋅x)

By comparing these outcomes we see that if x is at least 2, Rule D will always yield the greatest increase when used after another rule not before.  If we were to use Rule D when x is less than or equal to 2, we would be using at least 50 points to get x less than or equal to 4, for an average increase per point of less than .08, which we certainly know that we can beat with rule A. 

This shows that in all cases we should save D for last.

Want to Learn about the Digit Distribution of pi?

Check out our Pi Day blog post!

Happy Pi Day from BEAM!

What do you know about pi?

Maybe you would say it’s a handy constant, helpful when trying to figure out a circumference or area. Maybe you’d say it’s irrational and transcendental. Maybe you’d say you know the first five digits, 3.1415 (or maybe you even know the first 10 or 15 or more). But whether you know the first 5 or the first 5000 digits, there are a lot of questions we can ask about the digits that make up pi. 

We could wonder about whether a specific string of digits (such as your birthday) exists somewhere in pi or even if all possible finite strings of digits appear somewhere in pi. 

The first 100 digits of pi have plenty of birthdays!

Happy birthday to everyone celebrating their birthday on pi day! Your birthday is underlined. We also found Leonhard Euler’s birthday, April 15, and highlighted it in green.

Mathematician Emmy Noether’s birthday, March 23, is also in pi! We highlighted it in purple.

Looking deeper into pi’s digits, we noticed mathematician Alberto Calderón’s birthday (September 14, 1920) in the 295-7th digits of pi, highlighted in yellow:

Even later in the sequence of pi, we even found mathematician Benjamin Banneker’s birthday (November 9, 1731) in the 3254-7th digits of pi, highlighted in blue:

The first 1000 digits of pi are here if you want to look for your birthday!

You could also ask what seems like a simple question: does the digit 7 appear infinitely often in pi? Probably yes, right? Would you expect 5's to show up more or less often than 7's, or equally often? Do you expect each digit to show up 1/10th of the time?

Amazingly, we don't know the answer to those questions, not for sure, anyway.

All these questions have to do with the sequence of digits that makes up this famous constant, and it turns out that, despite its fame, there’s still a lot to learn about the digit distribution of pi. The answers are wrapped up in mathematical concepts called normality and simple normality. In mathematics, simply normal means every digit appears equally often as we go out to infinity, and normal means that every possible finite string of digits appears equally often as we go out to infinity. To make this more formal a number is simply normal in an integer base b if its infinite sequence of digits is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b. A number is said to be normal in base b if, for every positive integer n, all possible strings n digits long have density 1/bⁿ.

All these questions have to do with the sequence of digits that makes up this famous constant, and it turns out that, despite its fame, there’s still a lot to learn about the digit distribution of pi. The answers are wrapped up in mathematical concepts called normality and simple normality. In mathematics, simply normal means every digit appears equally often as we go out to infinity, and normal means that every possible finite string of digits appears equally often as we go out to infinity. To make this more formal, a number is simply normal in an integer base b if its infinite sequence of digits is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b. A number is said to be normal in base b if, for every positive integer n, all possible strings n digits long have density 1/b^n.

As an example, the number 0.12345678901234567890123… (where we just repeat all ten digits in consecutive order) is simply normal, but not normal. Each digit appears equally often as we go out to infinity, but the string 11 never appears! 

To get a better picture of how the digit distribution looks for pi in particular, we have color coded the digits and plotted the digit distribution for the first 10, then 100, 1000, and finally 10,000 digits of 𝛑. We see as the number of digits we are looking at increases, the slices start to look more similar in size. If this holds true as we head out to infinity then that would mean that pi is simply normal in base 10. 

When thinking about math, it is often helpful to think about statements in terms of stronger or weaker. Stronger statements give us more information, but they are also harder to prove, while weaker statements are easier to prove but don’t tell us as much about the subject or topic we are interested in. You will notice that saying a number is normal gives us a lot more information about its digit distribution than simply normal does. Normality tells us something about how often all possible finite strings appear, instead of just how often each digit occurs. That’s a lot more information! 

If every possible finite string occurs at least once, well then we can certainly find our birthday. We can find everyone’s birthday. We could find the whole works of Shakespeare encoded in pi — or any book ever written for that matter. Normality even implies that all finite information is encoded in a normal number. What’s more, if a number is normal, then it will automatically be simply normal, and if a number is simply normal then it has to have infinitely many of each digit. 

However this is where our wild wondering about what can be found encoded in pi comes to an abrupt disruption. It turns out that it has yet to be shown that there’s even infinitely many 7’s in pi, much less whether it is simply normal or, beyond that, normal. 

So why do we think it’s normal anyway? 

When we zoom in on a specific number like pi, it can be hard to show anything about its normality. However, zooming out and looking at the whole set of normal numbers and the whole set of non-normal numbers, mathematicians CAN prove things about these sets and we can even try to measure these sets. The mathematical machinery that we use to try to measure sets is called (maybe you guessed it) measure theory. While we don’t have time to go into how we measure a set like normal or non-normal numbers, it turns out that the set of numbers that are NOT-normal has “measure 0.” (Itching to see an actual proof of this fact? Don’t worry, just jump to the further reading section at the end of this post.) Intuitively this means that almost every real number is normal, because when we measure the set of numbers that are not normal it’s vanishingly small (measure 0). Since almost all numbers are normal, and we have no special reason to think pi would not be normal, we can guess that it probably is. 

All that’s to say, as special and unique as pi is, it’s likely that it’s just a normal number like (almost all of) the rest of them. 

While it would be nice to settle the question more definitively, the unknown is part of the joy and frustration that comes with thinking about open problems, problems without any known solution. Sometimes the lines between the known and unknown appear in unexpected places. The frontier of mathematics is not a smooth, well defined arc, but rather a wild, craggy expanse. Even when it comes to pi, there’s still much more to explore. 

Further Reading: 

If you still have room for more musings about numbers here is a great Numberphile video that goes over different types of numbers and includes a nice discussion on normal numbers (starts right about the 8:15 mark). It’s also helpful to see where normal fits in with other properties we may be better acquainted with such as rational/irrational, or transcendental: 

And finally for those wanting to check out an actual proof of the fact that non-normal numbers have measure 0 here are a couple options: 

• If you have a working knowledge of measure theory under your belt here is a proof for the fact that non-normal numbers have measure. 

• Or if you are a math undergrad student and haven’t seen measure theory yet, but have an understanding of calculus, here is a neat approach that tackles proving the same fact without measure theory; instead you’ll just need some knowledge of sequences and series, as well as how to integrate step functions on an interval. 

Looking for the solution to the Pi Day Math Problem?

Check out this blog post!

It's a BEAM First!

In January, Tanasia Gordon accomplished a BEAM first by becoming the first BEAM alumnus to join our full-time staff; she is BEAM’s new Enrichment Coordinator. A recent graduate of University of Rochester, Tanasia will be advising 8th-12th graders on high school admissions, applications to other summer STEM programs, and more.

In 2013, Tanasia joined the other students at Bryant Park to travel to Bard College to attend BEAM Summer Away. “It was my first time away from home,” she remembered, “and I was learning how to be an adult.” She spent the next three weeks immersed in challenging math problems and making new friends who also loved math. At the end of the summer, we asked her, “How have you changed this summer?” She wrote “I learned a lot more math than I ever thought I would.”

BEAM Summer Away, formerly called Summer Program in Mathematical Problem Solving (SPMPS), 2013.

Tanasia is seated in the front row (left center) wearing a green t-shirt.

Since that summer, Tanasia has worn many different hats at BEAM. “BEAM has aimed to employ staff who come from backgrounds similar to our students, but it was a real game changer when we finally had roles, like math team coaches and summer counselors, that allowed us to employ our older students as near peer mentors for current BEAM middle schoolers,” said Lynn Cartwright-Punnett, BEAM’s Chief Programs Officer.  In 2017, Tanasia returned to BEAM to help prepare middle school students for the AMC 8 and other math competitions as a math coach. Since then, Tanasia has worked as a counselor and teaching assistant almost every summer at BEAM Discovery including in 2020 as programs moved online, and again last summer as we returned in person. 

Tanasia (far right) with her fellow coach, Andy (far left), and her math team students.

Last summer was a big one for Tanasia, when she was BEAM’s first-ever student to become a faculty member at one of our summer programs. Tanasia stepped in to teach “Exponents: Superpower Numbers” at BEAM Discovery in New York City, after having served as the teaching assistant for the class. (Due to unexpected circumstances, the teacher had to step down mid-program.) She was excited about the opportunity. “I’ve always wanted to teach and have my own classroom,” she said, reflecting on the summer. “But as soon as I got home, the responsibility set in. Can I handle being responsible for [18] students’ learning?”

Tanasia working with her students in “Exponents: Superpower Numbers” at BEAM Discovery.

On her first day, she was nervous, but quickly grew comfortable leading the class. “It really lifted my spirits post-pandemic and woke me up again,” she said, remembering the energy of the classroom. In her class, students explored exponents and modular arithmetic – a system integral for computer science and cryptography. Since this is often BEAM students’ first encounter with modular arithmetic, the curriculum leans into the feeling of the unknown by introducing students to a far away planet called Z11, where there are only 11 numbers from 0 to 10. Students got the opportunity to work in small groups to explore the properties of the planet, and learn how those rules may apply to other planets, like Z13. 

Students loved the activity so much, they asked for more problems to do! It was like they were in a candy store, said Tanasia, thinking about her students’ eagerness for more problems. The students spent a whole week of Open Math Time (where students get to do math of their choosing) solving the additional problems, even pulling in their friends from other classes to teach them about Z Planets!

What is Tanasia looking forward to in her new role at BEAM? The constant connection with the students. After a summer of math and mentorship, she wasn’t sure she’d get to work with her students again. “I’m excited to show students that they’re capable of so much and to not let their surroundings limit them.” 

As NYC Programs’ Enrichment Coordinator, Tanasia is working with students year round to help connect them to opportunities that match their goals. This means helping BEAM high schoolers apply to Summer STEM programs, and guiding 8th graders through New York City’s high school admissions process. With the growth of BEAM’s programs, current Pathway students have access to programs that didn’t exist when Tanasia was younger, such as Summer STEM stipends. Students no longer have to choose between attending a summer enrichment program like PROMYS or Canada/USA Mathcamp or holding down a job. Students can receive direct aid from BEAM, made possible by our corporate partners, Jane Street and Hudson River Trading, to attend competitive STEM programs.

This summer, Tanasia will be part of the leadership team at BEAM Summer Away in upstate New York, serving as the Associate Site Director. “It’s amazing to know my career has grown so much in a year,” said Tanasia. 

About Tanasia coming into her new role, Lynn Cartwright-Punnett said, “Kids just listen differently when the mentor in front of them can say, quite literally, I have been in your shoes and I know you can handle this challenging math problem or this application to a summer camp. Having Tanasia on our staff year-round?  It's absolutely a dream come true!”