... And Now for Some Math

This was a Challenge Problem last summer at our Bard College site. Challenge Problems encourage students to work collaboratively; once it's solved, the whole program gets a prize. (For this one, a sleepover on the last night!) So now it's your turn... can you solve it?

Malachi takes 42 pieces of candy, each of which is either a Skittles or an M&M, and arranges them in a circle letter side down (so they look identical!). He tells Hodaya that 23 of the candies are next to at least one Skittles, and 35 are next to at least one M&M. If Hodaya can figure out how many Skittles and how many M&Ms there are, she gets all of them!

How can Hodaya figure this out? (The answer appears below the image.)

Solution:

The important thing is to go step-by-step to figure out what you can.

If there are 23 candies next to at least one Skittles, that means 42-23=19 candies are not next to any Skittles — so they must be next to two M&Ms.

If 19 candies are next to two M&Ms and 35 are next to at least one M&M, that means 35-19=16 are next to exactly one M&M.

Now here's the key insight. Suppose there are x M&M's. Each M&M has a neighbor on the left and on the right, so if you add up for all the pieces how many M&M neighbors they have, you should get 2x. Well, we know that 19 candies are next to two M&Ms, so they have a total of 38 M&M neighbors. And 16 are next to one M&M, so they have 16 M&M neighbors. That gives 54 total M&M neighbors.

Divide by two to get 54/2=27 M&Ms in total.

With 42 candies in total, subtract out the M&Ms to get 42-17=15 Skittles.

Sometimes, it's amazing how much you can figure out with what seems like too little information!

Our Favorite Math

This summer, at BEAM Summer Away at Harvey Mudd College, we asked students, What was your favorite math problem of the summer?

Adib chose a problem from the class Solving Big Problems: Can you add or subtract the numbers between 1 through n inclusive to get 0?

Don Laackman, BEAM’s Program Coordinator and the Site Director for Summer Away at Harvey Mudd, said of this problem: “This is a central problem for the class; it requires a proof of impossibility, thinking about divisibility issues, and the solution goes on to be very useful in a problem in graph theory that students tackle later on.”

Check out Adib’s work below.

The students in Solving Big Problems take time to explore this problem, building up the pieces to solve it. This summer, after the students had completed the problem, they got to see a faculty member and a counselor tackle it together, talking through their thought process so students could see how people with lots of advanced math experience approached the very same problem they already knew well.

Check out more student work (and more about what BEAM has been up to this year) in our 2019 Annual Report.

What It's Like to Teach at BEAM

Cecily Santiago, a first-year faculty member at BEAM Summer Away at Bard College, shared the following with her friends after her summer with BEAM (and we wanted to share it with you)!

This summer, I spent three weeks working as a math instructor at BEAM Summer Away. This has been one of the best experiences of my life, personally and pedagogically, and BEAM's mission is so much in line with my own values that it was a perfect fit.

Let me tell you about why I think BEAM is so awesome:

Last winter, I saw a post in a Facebook group about this job, teaching math at a three-week summer math camp. I clicked on it and immediately knew I wanted to apply for this. It didn’t pay as much as some other summer opportunities, and the application was long and daunting—they wanted a sample lesson plan! They wanted paragraphs about my pedagogical philosophy! They wanted to know how I would react in certain situations! But this didn’t deter me—their focus on excellent math education as a vehicle for social justice was right up my alley. My goal in life is to give the world a group hug using mathematics, and this was an opportunity to work on that goal.

BEAM finds their students from the most underfunded school districts in NYC and LA, testing their ability for abstract problem solving regardless of academic preparation. For those who shine, they run a day camp the summer after 6th grade and a sleep away camp the summer after 7th grade. Here, they teach math in a fun and accessible way, helping to strengthen the kids’ skills in school math, like fractions, but also giving them exploratory courses in math that few people who aren’t math majors ever get to see, working on their ability to reason, prove, and generalize. After these camps, students continue with Saturday Classes, with specialized support for getting into other math programs and competitive high schools. Throughout high school and even into college, the program supports these kids on their academic journeys, giving them the tools, resources, and information necessary for them to excel as students and preparing them to enter math-related careers if they choose to. I applied to be Junior Faculty at the summer camp.

BEAM students work on toppling sandpiles in Cecily’s class.

The next step was a TWO HOUR video interview. I emphasize the application and interview process so much because it indicates how much BEAM cares about finding excellent people to staff their camps. Even the counselors, who were not teaching, went through a similar process. It wasn’t just an interview—it was a conversation. This is also typical of BEAM—they are collaborative and appreciate feedback and input from students and staff alike. Nothing is ever a certain way “just because.” There is always a reason, usually a good one, and if things can be done better everyone works together to find a better way.

I was offered the position and enthusiastically accepted! I finally met some of the BEAM staff on a planning retreat in NYC in May, and it was so amazing to be around people who care so deeply about math education. In many ways, I felt like I’d found my people. The staff come from a wide variety of backgrounds, intellectually and socially, but they are all united in how much they care, how much good they want to do, and how humble they are as they do it. Everyone was working incredibly hard, with passion and joy, and no one was patting themselves on the back.

When I arrived at Bard College, the “Summer Away” site that hired me, I was nervous about working with 13 year olds and didn’t know what I should expect in the classroom. I knew that these kids had an affinity for math, but I also knew their math background was lacking. I knew I was supposed to plan fun, engaging, somewhat rigorous, and very inquiry-based courses, but I didn’t know if the students would want to be engaged. I had never planned a course like that—in fact, I had never planned a course about something I wanted to teach, or planned a course completely from scratch, or planned a course that was in line with how I believe math should be taught.

Turns out my fears were unfounded. The kids were AWESOME. They started off shy, but soon they were all shining with their vibrant and unique personalities, befriending each other and the staff alike. In the classroom, they were interested and engaged and insightful. Every single one of them pushed themself to grow and learn and come out stronger than they came in. They all started in different places and they all ended in different places—although these students are all strong mathematical thinkers, some had better preparation coming in and some had more intuition for certain aspects of math than others—but every single one of them ended as a better mathematician, and probably also better future-adult than they came in as.

Cecily works with a BEAM student.

Teaching was amazing. I taught a course on Sandpiles and another on Sets. I learned so much about how to run a classroom. I got to teach at the pace that was right for the students. I got to take the whole class down unplanned avenues that sparked their interest. I got to interact personally with every student. I got to create games and challenges that supported the course material. I got to make learning a truly fun adventure. I walked away from almost every class with a happy glow. I was given so much freedom, but also a lot of support and feedback. This isn’t just a place for students to grow—this is a place where adults grow too.

Discipline here was really interesting, and different in some significant ways than any other model I’ve experienced. While there were systems and consequences, there were only four rules. Honestly, the first three are now my personal rules for life.

1. Be here to grow

2. Be excellent to one another

3. Don’t do stupid stuff

4. Follow staff instructions

Pretty much any behavior that we would want to discourage falls into the first three rules. You called another student a mean name? You’re not being excellent. You slacked off during class? You’re not being here to grow. You jumped down a whole flight of stairs? You’re doing stupid stuff. And then number four is a catch all. You’re only eating fries for dinner? I’ll tell you to go back and get some veggies. You refuse? You’re not following staff instructions.

But moreover, the way we handle the kids when they break the rules is, to me, revolutionary. We don’t say “Hey, Dan, stop that! I’m reporting you!” We treat them as future adults rather than as just kids. We would say “Hey, Dan, do you know why I came over to talk to you? Do you think that you’re being excellent right now? Why or why not? Have we talked about this before? Why did you choose to take this action? What could you do better next time? Do you think that reporting you would be an appropriate punishment? What do you think would be appropriate?” We ask them to reflect and learn from their behaviors. When a kid is reported too many times, we have a serious discussion with them. There are ways to get reports taken off your record, if you behave well enough for long enough after something happened. There are more serious consequences if patterns of behavior don’t change, especially if they make the camp unpleasant for the other students, but we start by acknowledging that kids can reflect and make choices and decide how they want to act, and that kids will one day be adults who will need these skills in life.

There is so much more I could say about why I am completely in love with this program, but I just want to emphasize how utterly rare and amazing this organization is. As far as I know, nothing else like it exists. They get everything right, from selecting students to selecting staff, from pedagogy to discipline, from site leadership to overarching structure, from an individual class to a lifetime of support. The students and staff alike thrive here, and I am certainly planning to come back.

Thank you BEAM supporters!

Thanks to everyone who has contributed to BEAM!

Now that our summer programs are over, our students would like to tell you what they accomplished.

“I solved a lot of challenge problems with my friends.” —Ruth, BEAM Discovery

“I solved 12 problems in the 100 Problem Challenge.” — Vernon, BEAM Discovery

“I learned some coding and got WAY better at it. BEAM made me love math and realize my passion for it!” —Mia, BEAM Discovery

“This is my first ever program and I loved it! I’m glad that you helped support BEAM.” —Hayden, BEAM Discovery

“BEAM helped me see math in a whole different way. THANKS.” —Sanjana, BEAM Discovery

“BEAM is a glowing star.” (left) —Estefani, BEAM Discovery

“BEAM is out of this world.” (right) —Allison, BEAM Discovery

Exponents, Reciprocals, Expected Values, Oh My! Day 19...

There was a lot going on at BEAM Discovery Downtown on Day 19! In Exponents: Super Powered Numbers, students tested their classmates’ laws about exponents. Students explored reciprocals in Fractions: Puzzles, Problems, and Games. And in Big Questions and Big Answers, students learned how to calculate expected values and played a dice game where they tried to beat the odds.

In Open Math Time, students worked on Problem Sets and the 100 Problem Challenge. Downtown students are getting close to claiming the prize for solving all 100 problems!

Relays! BEAM students can choose collaborative or competitive categories for Relays, where teams work to solve problems together.

During activities, students played chess, learned new knots in knitting, worked with Snap Circuits, and more.

“What’s it really like to work at Google?” “How do you know when something is fake news?” BEAM students asked great questions of STEM professionals from Google, Reuters, Harvard, MIT, and more as part of Career Day at BEAM Discovery Downtown.

BEAM Discovery Uptown Day 18!

Students were busy on Day 18 of BEAM Discovery Uptown. After breakfast, some students started the day in the class Playing with Logic, where they explored the Shuttle Puzzle by becoming the puzzle pieces. (Classes at BEAM are unlike anything most students see at school. Other classes this week include Math for Pirates and Learning from The Number Devil.)

During Open Math Time, when students get to choose what math they work on, some students completed problem sets from their classes. Students are often encouraged to work together to find the answers.

In Applied Math, students talked about game strategies and how to win at games, like Nim. They asked the question, “Can strategies help you win even at a game like Rock, Paper, Scissors, that seems all about luck?” Yes, they decided, by doing things like looking for patterns in your opponent’s play. To test their theories, they squared off in a Rock, Paper, Scissors tournament!

Students checked out the 100 Problem Challenge. As of today, only one problem remains! If the students solve all 100, they win a special prize.

A group of students discussed Problem 74 with Xavier, a veteran faculty member.

Are you up to the challenge? Here’s Problem 74:

In Problem 53, you found a way to link three rings so that if you cut just one of the three rings, the other two would come apart.

In this problem, your goal is to do the same thing but with four rings. Together, all four should be linked. But cut any one of the four rings, and the other three should come apart. (Hint: try using pipe cleaners, like the students above, to help visualize the problem.)

BEAM Summer Away at Bard Week 1!

Week 1 at Bard was great!

Tuesday through Saturday, students attended classes of their choice. First week classes included Geometry, Cryptography, Number Theory, Paper Folding, Solving Big Problems, and Math Team Strategies.

Adri and Zhixing work on problems involving triangles in Geometry.

Students also participated in a variety of activities, from basketball, to making a solar oven, to playing Ricochet Robots, to Billie Eilish appreciation. In addition, there was a five-mile hike and a trip to Zoom Flume Water Park!

Stay tuned for week 2 news!

Fun at the waterpark

Students at BEAM Summer Away Union College enjoy a day at the waterpark!

Mathematician Edray Goins Visits BEAM at Harvey Mudd

This weekend, Dr. Edray Goins, mathematics professor at Pomona College, gave a brilliant talk to students and faculty at BEAM Summer Away at Harvey Mudd College. Dr. Goins grew up in Los Angeles just like our students and graduated from two of the best universities in the country, California Institute of Technology and Stanford University. Dr. Goins explored with students how the geometric mean and recursive sequences of numbers can be used to find an algorithm for square roots. Students left not only inspired to continue their journey in mathematics, but also driven to keep learning new techniques in math. BEAM students Maxine and Karla said of Dr. Goin’s presentation: “Many new ideas and theories were presented in a way that was straight forward and precise.”

Thank you, Dr. Goins, for truly inspiring us!

Interested in getting to know Dr. Goins better? Check out this profile of him, featured in The New York Times in February.

BEAM Discovery LA: A fantastic first week

Students did great work the first week of BEAM Discovery LA!

Each week students can choose to work on the Problem of the Week on their own or with others. The first week, 27 students solved the problem, 10 within the first two days!

Are you up to the challenge? Find out:

Is there a 10-digit number where the first digit is equal to how many 0s are in the number, the second digit is equal to how many 1s are in the number, the third digit is equal to how many 2s are in the number, all the way up to the last digit which is equal to how many 9s are in the number?

If yes, can you find all of them? If no, how do you know for sure?

Week 1 featured classes unlike anything most students see in school:

Elementary My Dear — Students explore different techniques to solve complicated riddles and learn how to apply them to tough mathematical problems.

Truth, Lies, and Logic — Welcome to the land of liars and truth-tellers. What can we figure out in such a crazy place, and how can it help us solve really difficult math problems? 

Fractions and Food — Students’ thinking about fractions (and, hmmm, food) is challenged.

Words Meet Numbers: An Algebra Story — Students learn how Algebra can translate words into numbers and vice versa.

When we asked our students what they thought about the classes, one told us (about Elementary My Dear): 

“One of the many topics that are good about this class is that they teach us how to use clues, vocabulary in word problems, and solve mysteries like a CSI. Also, the staff are really nice.”

Another student said (about Truth, Lies, and Logic):

“You get to learn many things and the problems get you frustrated meaning you'll just have to try harder.”

But it’s not all math all the time. Students also participated in fun activities like a Toy Story Marathon to prepare for the field trip to see Toy Story 4. 

Others played board games, challenging their friends and counselors to Uno, Connect 4, Settlers of Catan, and more. Students were also excited by our science activity where Skittles and M&Ms were placed in water and their color slowly seeped out to create beautiful designs.