Our students go on to do great things: MathILy participants have attended MIT, Harvey Mudd, Princeton, Stanford, UC Berkeley, Swarthmore, Oxford, Carleton, Carnegie Mellon, Brown, U Waterloo, UCLA, Haverford, Caltech, Stony Brook, and other excellent colleges. They have participated in PRIMES-USA and in Research Experiences for Undergraduates, published papers, scored well on the Putnam exam, and attended national mathematics conferences.

* Athina Avrantini*: "...over the five weeks, I have learned more stuff than my last few years of school combined! After spending five weeks here, I realized how learning math should be and it became clear to me that this is what I want to do. ... As someone who is used to being the best in my class for all my life, coming here has been a very enlightening, eye-opening experience. I realized how little I know and how much there is to learn!"

* Michael Tang*: "When I first arrived in Root class, I was taken aback by the very free and student-driven style of teaching. Since then, I have been continually surprised at the talent of my fellow students for finding patterns and meaning in problems that they and I have never seen before. ... MathILy, more than anything for me, has been a greatly enriching social and learning environment, where I have made new friends, learned about different cultures, and become better at solving anagrams. MathILy has expanded my horizons in many different directions... The connections I've built to friends and mentors will be invaluable, I think, for finding new opportunities later in life."

* Meghana Vemulapalli*: "My goals in coming to MathILy included engaging with areas of math that I had previously not encountered and improving my proof-writing skills. The first goal was met many times over---WoC and Branch have shown me fascinating areas that I hadn't previously known of... MathILy had a powerful impact on me and I'm so glad to have learned more math and met many other excited mathematicians there."

* Hussain Waris*: "Not only did I learn so much more than I thought I would, I also learned that math is definitely what I want to do in the future."

* Luke M.*: "... I think that loads of doors have been opened, or at least pushed wider than they were before... I'm often amazed at how easy many obscure topics, such as graph theory, can be, if only they don't tell you what you're getting into until you're having so much fun drawing spaghetti monsters that there's no turning back."

* Éva Siladji*: "I didn't know it was possible to learn so many things in such short period of time! I think I understood majority of it, and I won't forget them easily, mostly because of the way we learned them, discovering all the things by ourselves, and not having everything handed to us on a plate."

* Allen Wang*: "Overall, I'm having the best time of my life at MathILy... I've learned more math than I ever knew and learned how to tackle unfamiliar problems and problems that I don't even know how to approach."

* Kye W. Shi*: "Only at MathILy did I feel that I was working *with* my peers rather than teaching/assisting/explaining concepts most of the time. Kudos to all of you at MathILy for running and being part of the most engaging and most exciting summer program ever (from my perspective, at least)! Yay!"

* Kaili Liu*: "This summer is definitely a valuable experience, and it is the first time I have learned so much in one summer (and also had fun!)... I now find attempting extra-hard problems a fun activity... after arriving at MathILy, the "levity" part of MathILy kicked in and attempting such problems became a fun activity where I felt determined to find a clear (and hopefully creative/innovative) answer---I learned how to view a problem from varying perspectives from others at MathILy."

* David Gonzalez*: ...MathILy has been a life altering, transcendental, suburb (I didn't mean 'suburb,' but 'superb'---No. suburb. I meant 'suburb.' I made no mistake.), experience that I will inevitably associate introducing me to proof-based math for the rest of my life. Just experiencing this program has grown my love of math to an unprecedented level.

* Tracy Wan*:
The MathILy community consists of people from various culture and geographical backgrounds. During one of our talks, we noticed that together, we speak more than 10 different languages. I really enjoyed the diversity of the group and learning about different cultures.

*Dennis Yatunin*:
I think the most valuable thing I've gotten from MathILy is the ability to think properly. Rather than believe anything that appears correct, I now look towards proving it using the techniques I've learned here... I've gotten more out of this than I ever dreamed I could, and I loved every single minute of it. I put a lot of effort into participating in class and doing the problem sets, and it's all paid off. I feel like I now have a far better understanding of what mathematics is truly about (and how immensely misleading high school math classes are).

*Erica Nathan*:
I know that my time at MathILy will have a tremendous impact on my future studies, math classes and beyond. I loved hearing about so many applications outside of pure mathematics... MathILy has been an absolutely stellar summer program and I can't wait to use what I've learned.

*Josh G.*:
In less than 2 weeks, my outlook on life has changed already. I hadn't met people so similar to me until now, and it's great!

*Jo B.*: I also wanted to make sure that I did indeed want to do a Math major at college because I felt like I had not had enough experience to really know, but now I am sure! I definitely do.

I don't want to go home, I just don't. The only consolation is that I'll get to go to Europe and see some people. I would like to stay here forever.

*Mo K.*:
Thoroughly learning such concepts literally changed the way I view mathematics, from numbers of equations with different variables to concept and logic. Attending MathILy made me determined to study math further, as it showed me how much more math there is for me to explore.

*Cat O'Donnell*:
The teaching style at MathILy gives me the support I need to be comfortable asking questions... This creates an environment where I can be sure I understand what is going on enough to be able to contribute... The really enjoyable and funny group of students and instructors at MathILy are also great.

*Nathan Smith*:
My goal in coming to MathILy was to learn more mathematics and meet with others who share my passion. I can safely say both of these goals have been achieved.

*Sarah Liebowitz*: I was introduced to topics that I did not even
know could exist in mathematics... I did not expect *everyone* to enjoy
and be good at math! I really liked working in that kind of environment. ...Overall, I am delighted that I found my way to this
program. The new topics, skills, and learning techniques I was introduced to
will benefit me throughout my academic career.

*Bradley Zykowski*: The exploration we did in this class is exactly
what I think is so awesome about math, and the class atmosphere was very
exciting. ...I have learned how to think about math and express myself in a
more efficient and accurate way, which I will be able to use throughout my
studies of mathematics.

*BB*: I learned quite a lot in this class, more than my last few
years of school combined.

*SS*: I am immensely happy that I went to this program this summer. ...I had no
idea whether I would enjoy majoring math once I entered college. However, after
taking this class, I have made up my mind, and I am certain I want to major in
mathematics... In the beginning, I had hoped to learn things to improve my
contest-ing. However, I believe I took home something greater than that. Though
I love contests and I'll continue to work on improving on contests, this class
has enriched me and showed me mathematics in a new light, and I will never
forget that.

*Charlotte Kavaler*: I think that the program overall was fantastic. I really didn't know, when I first came here, if I would like it, but I definitely have, and it has taught me so much about things I had never even thought about... On top of learning new things, I have also learned how to think about subjects I already know about in greater depth and from a different perspective.

*Eugene Shao*: I've experienced graduate-level math to math that I didn't even know existed... MathILy doesn't have the normal school environment one would expect; the classes demand students to discover many of the things by themselves. I love this technique of teaching because, instead of a teacher lecturing, we can discover theorems on our own, learn the topic better, and know when to use it.

*Laurel N.*: I got exposed to areas of math that I'd only ever heard about, discovered some areas that I was really interested in, and got to work with people who were interested in math and learning and delving deeper into the problems... the Week of Chaos classes managed to be surprisingly detailed given the amount of time available, which was awesome.

*Hugh G.*: My goals were to... learn how to be the coolest cool kid and with what I know now, well, no one can say that I'm not the swaggiest person at school.

*Yiting Y.*: I gained a deeper understanding of math and realized what real math learning should be---generating conjectures and creating new things. I am extremely horrified at the thought of returning to the traditional math class at school. ...I learnt a great amount of math, realized that math is way cooler than I thought, and have become a bolder and more energetic person in class... The experience here at MathILy has been inspiring and transformative.

*Jamie Gu*: I had a very good experience... The program is different and a lot better than what I was expecting before coming. I like this format of learning because we have more interactions with others and we all have fun.

*Esther Plotnick*: When I was originally applying, I was a little hesitant to spend five weeks away from home to do math... I am so glad that I made the decision to apply... I definitely learned a lot over the past five weeks---way more than I ever anticipated. I now have a great curiosity about mathematics that I didn't have before, and I am very happy about that.

*Kye Shi*: One of my main aims before coming to MathILy was to learn about all sorts of different subjects in math---throughout the past several weeks, that has definitely been covered... I had a ton of fun developing math skills and learning how cool math topics relate to each other and explain things. I'm sure that, later in life, whatever skills I develop(ed) here I will use again whenever I do math.

*Nadav Kohen*: Over the past five weeks, I've gotten better and more comfortable writing detailed and sometimes long proofs more rigorously. ...I feel much more prepared than I was five weeks ago for this coming year and its proof classes... I can't believe I've been here for a full 5 weeks and I'm really sad that it's almost time to go but I'm certainly even more excited about math now and look forward to doing more!

*Andy Zhang*: I'm so happy that our classes are being taught the way they are being taught. Here, the things that we come up with ourselves matter... It feels great and even motivational to see our own conjectures appearing on the problem sets.

The past five weeks have been so amazing. I feel like I have never had a more productive summer than this one, and I certainly never learned this much stuff and had this much fun at the same time.

*...from a parent*: I can't tell you how much MathILy has meant to [my daughter] - it's like she discovered a whole new world. It was the first time in her life that she was truly academically challenged and she loved being part of a community where people were so excited about math.

*Lydia Wickard*: MathILy will help me in the future in 1000 ways.

*Brandon H.*: One of my main goals when coming into MathILy was to learn new types of math. I have known for a long time that I want to do math when I get a job, but I never knew about the many fields of mathematics. I'm happy to say that this goal was fulfilled... I was especially excited for applications of mathematics in game theory and economics, but I was surprised to find myself extremely intrigued by Markov chains and combinatorial optimization as well. I believe now that I have a wider view of the different fields of mathematics and will be better able to make choices regarding what I want to do later in my life.

*Josh Mundinger*: There's no BS here, no showing off, none of the surrounding clouds and fog that shroud any ordinary attempt to understand math.
... I have been having an amazing time outside of the classroom, whether talking about math in the dining hall, playing Pandemic (we won today!) or throwing a frisbee.

*Rashmika Goswami*: I've met a lot of great people who love math, and been able to see the many interesting variations and applications of math in other fields through the Daily Gathers.

*David Lingenbrink*, Harvey Mudd class of 2014, currently graduate student at ORIE Cornell: Before 2009, my math
background was very elementary. That summer, I attended a math program where
both sarah-marie and Thomas Hull taught. I arrived without a clue how to write
a proof, and left *sure* that I wanted to be a math major. I had both
sarah-marie and Tom as teachers that summer, and both were excellent at helping
me through the standard what-is-math-really? obstacles. The summer was very
difficult for me, but every class was filled with humor and a sense that they
cared that I grew as a mathematician. By the end, my ability had greatly
improved, and I was sure that I could be a mathematician. In fact, not being a
mathematician sounded boring.

*Matt Redmond*, MIT class of 2013, currently Data Scientist: After learning all about counting
problems with polytopes (how many faces are on the 120-cell?, etc) from
sarah-marie, I ended up discovering an interest in enumerative combinatorics
and linear algebra that drove me towards finishing a mathematics major at MIT.
sarah-marie's classes are NOT TO BE MISSED - you'll have a fantastic time, and
will almost surely learn something beautifully elegant in the process.

*Miles Dillon Edwards*, Indiana University class of 2013, currently Instructor at Art of Problem Solving: I took
sarah-marie belcastro's class on topological graph theory in 2007, the summer
after my tenth grade. I am still amazed at how much math she managed to fit
into that course. At the end, I had learned a lot about math I didn't even know
existed---and I had new perspectives on math I already knew. I also got to work
with some friends on an open problem, and two of us even got to see our name in
print, as our work became a small part of a paper that was later published. But
the summer itself was really just the beginning. After I left, I kept turning
things over in my head, figuring out and filling out things that were outlined
or hinted at during the summer, and I continue to do so, even as I prepare to
go to math graduate school next year. Did I mention that I had a great time
while doing this? It is difficult to describe how wonderful it is to be
surrounded by people who are really interested in (and good at!) the same thing
as you (math, in this case), so instead of trying to do that, I'll just say
that some of the people I met there are still close friends of mine. I would
recommend this experience to any student who really likes math and wants to do
more of it over the summer.

*Josh Vekhter*, Cornell class of 2013 and entrepreneur: The three weeks I spent in
sarah-marie's classroom were one of the most chaotic and fulfilling experiences
I've ever had. Unlike a typical classroom, where you are taught to memorize and
regurgitate words and phrases, with sarah-marie you learn how to invent words
and how to ask questions in the face of chaos. In my eyes this is one of the
most important life lessons that one can ever learn, and this program is one of
the few places where someone will help you learn it.

A little more pragmatically, if you are curious about mathematical research,
this program will give you an incredible amount of exposure to open questions
which you can actually understand and play with. After taking the course on
topological graph theory, I became very interested in one of the questions
mentioned, and the work that I did on it gave me a chance to go to the
International Science and Engineering Fair (ISEF). This itself was a great
experience, which this course made possible.

*MathILy
is a project of the nonprofit organization Mathematical Staircase, Inc..*