Meet Po-Shen Loh, the professor at Carnegie Mellon University and head coach for Team USA, who paved the way for the U.S. to win the International Mathematical Olympiad for the second consecutive year. Now he's adding more value to the world with his new product, Expii Solve.
Po, can you tell us about your background? As a young child, did you find math to be your favorite subject in school?
I was born in California, but my family moved to Madison, Wisconsin 6 months later, and I grew up there. I attended public school. Math as my favorite subject because it was the one where if I did something wrong, I could learn the source of my error, and work towards improvement. I always chased after challenge, and I enjoyed a domain in which I could keep setting higher challenges, and work my way toward them. In middle school, I first encountered the MATHCOUNTS competition, which provided a pyramid of challenges leading all the way to the National Finals. That inspired me to keep driving upward until I ultimately finished 3rd in the country in 1996. I stayed in the competition circuit in high school, playing in the Mathematical Association of America's Competitions, which ultimately landed a spot on the 1999 International Math Olympiad team for USA. All of this was competition though. It was later in life (in college and graduate school) where I developed a fuller appreciation for mathematical theory, which brought me to my strongest interest in the subject now.
This is your second year leading a winning team in the International Mathematical Olympiad. Can you tell us about the competition and what this means to the world on an international scale?
The International Mathematical Olympiad is the most prestigious math contest for pre-college students in the world. Each country sends a team of 6 students to compete as their national team. The questions are then of course extremely difficult, far beyond what one typically sees in high school, because they need to challenge and distinguish between the best in the world. They generally ask for contestants to write "proofs", or explanations of why certain facts are true, rather than to apply standard formulas to calculate answers. "Nonstandard" is the key word here, as the IMO searches for the ability to create new mathematical arguments, rather than the ability to remember and carry out methods that others have previously discovered. The competition itself consisted of a total of 9 hours, during which contestants each were presented with 6 questions to work on individually. The team score is the sum of the individual scores.
The USA won in 2016 for the second year in a row, definitively breaking a 21-year period of no wins. The United States takes a distinctive approach to Math Olympiad training, in which the national-level training camp focuses not on IMO problems, but on math in general. Instructors are mostly between 20-30 years old, and are former Math Olympiad contestants from many different countries who are now studying or working in the United States, in mathematics or related disciplines. They put high school Olympiad mathematics in context with topics which will be encountered after high school. The entire program also has a strongly non-competitive atmosphere, as all IMO team selection is conducted prior to the camp, and guest students from many other countries are also invited to the USA training camp, free of charge. This year, 10 out of the 44 gold medalists at IMO 2016, had been at the USA program (from USA, Singapore, and Canada), as well as a silver medalist and 3 bronze medalists. Next year, the USA program will invite 30 international guests, creating the largest scale internationalized training program for the Math Olympiad in the world.
It is very interesting that this strategy wins the International Mathematical Olympiad. At first, it may appear counter-intuitive, but ultimately the wins demonstrate that the long-term focus and collaborative atmosphere contribute great value.
Rather than adding and subtracting numbers, your definition of math relates to the world around us. Can you provide us with deeper insight?
Math is the pure art of creative thinking and problem solving in objective contexts. (This distinguishes it from the creativity for subjective contexts.) Math provides a framework for logical deduction that takes one from point A of basic understanding to point B of fuller understanding. This is particularly useful when faced with counter-intuitive facts. One example I often show is this puzzle, which asks which martini glass is half full: https://www.expii.com/solve/22/1
The most common answer (67% of height) is incorrect. The value of math is therefore in providing the analytical framework which one can use to fully understand and believe why the truth is correct. In this way, it plays a role analogous to legal proceedings in the courtroom. If human intuition were 100% accurate, we wouldn't need math (or lawyers). Since human intuition is often completely wrong (as in that example above), we need both. In school, students learn the formulas which are the great triumphs of historical mathematics. The goal should be to learn to appreciate why those facts are true, as opposed to memorizing them. By understanding why, people will be better equipped to make their own creative and original new deductions, to solve the new challenges that face the future generations.
Please tell us about Expii Solve. Why did you create it, how does it work and who should use it?
When I became the national coach for the USA Math Olympiad team, I decided I wanted to focus not only on the very top students, but on raising the baseline across the country (and world, as collateral damage). Since I also have a technological background, I decided to launch a web/smartphone platform (expii.com) to achieve that goal. It now provides free personalized learning to anyone in the world, where it dynamically customizes an educational experience to achieve any desired learning goal. That is the full Expii itself.
Expii Solve is another branch of the site, where I release a set of 5 interesting challenges each week, which I curate myself. They show a face of math which is often unseen, scaling up from approachable-but-surprising to seriously difficult. Each week, I create them by meditating on a subject of common interest (for example this week is on the Olympics), and discovering what interesting math may be embedded deeply in the fabric of reality. I then share these observations by sorting the resulting challenges in order of difficulty, and posting them on Expii Solve. The first problem in each set is approachable by an advanced middle school student, but its answer often surprises adults. The last problem in each set challenges math contestants. Every middle school or high school student in the county would benefit from the puzzles, as they share with a general audience why math matters.
This was inspired by my frequent travels across the country, where I would give math talks to student audiences, showing how math was interesting, fun, and very relevant. Each talk, unfortunately, only reached about a 100-person audience, and so I decided to scale up the endeavor by sharing the interesting topics over the Internet.
We have since also partnered with the recently released film The Man Who Knew Infinity to search the world for undiscovered talent through the puzzles, through the Spirit of Ramanujan Talent Search: expii.com/ramanujan. Since the puzzles work on every smartphone, this permits us to potentially reach the far corners of the Earth.
The following are questions from my teaching colleagues.
What areas of the math curriculum do you believe can be pulled out because they are no longer effective? For example, do you think we still need long division in our basic math curriculum?
I think that the math curriculum should be augmented, not reduced. I think that the reason why math feels overwhelming is because right now, instruction is not personalized. Cost is the major barrier, as we can only afford to have 1 teacher for every 30 students, not 1-to-1. That is why I created Expii, to deliver 1-to-1 personalization for free, right now.
The reason why math requires 1-to-1 is because the concepts in math are linked in long chains of pre-requisites. Before you can add fractions, you need to be able to both add and multiply. Before you can do certain parts of algebra, you need to know how to add fractions, and so on. I think math actually has fewer concepts than, say, history, but the concepts in history are less deeply ordered. For example, even if you do not know the details of World War II, you will generally not be lost when the teacher is explaining the Vietnam War.
In classrooms, the pace must then be set so that it captures a large fraction of the student body, or else everyone will start getting lost, and then be permanently lost because they have holes in their knowledge, upon which future concepts cannot build. This all results in the pace we have today. With free personalized learning technology, learning can go at one's own pace, and I predict that this will cause everyone to be able to learn far more. The question will then not be what should be pulled out, but what should be added.
For your students on the team: How do you think these students developed their "math" identity? Too often, we hear students say, "We hate math because it's too difficult!" When did they know they were "mathematicians?"
I think that everyone has a different answer to this personal question. I believe that most of the students on our team love math because the creative insights are really beautiful, elegant, and thrilling to come up with. Many of them have known they were mathematicians for many years.
This does not mean, however, that if people aren't interested in math at an early age, they will never be good at math. Quite the contrary: it shows that from the moment one decides to pursue math, it only takes a few years to become amazing. I think that it is never too late to take up math (even as an adult), and indeed, one of the best ways to develop mathematical interest in kids is for the parents to explore challenging math puzzles alongside them!
Math has changed so much; many parents say they cannot help their kids with homework. It seems math is reinvented over and over; it was much easier to learn the way we did years ago. Frustration builds all around for teachers, parents and students. What are your thoughts about this concern, and do you have any suggestions on how to best support these ongoing changes?
Many homework problems are routine, because their purpose is to build standard skills. The most interesting math puzzles are those which actually will build creativity, spark conversation, and ultimately develop problem solving skills that confer great advantage later in life. I think that the best math puzzles to work on are those from brainteaser books, from math competitions like those organized by the Mathematical Association of America (which ultimately feed into the International Math Olympiad), and from online sources of CREATIVE math puzzles, like what we are trying to produce with expii.com/solve.
These harder challenges are akin to learning science through projects, and this provides a mathematical analogue to the "Project Based Learning" that is being advocated in science education. While tackling a non-standard mathematical puzzle, learners will develop the standard skills along the way, and also pick up additional creativity. I think that if one tries to work on challenges, and learns from the solutions, one will soon find that math homework problems are much easier than they appeared before.
Math is difficult when one tries to rote-memorize too many techniques, because then the conceptual fabric is lost. More interesting challenges illuminate the fabric itself. I have heard that the Common Core seeks to promote conceptual understanding, which would be excellent!
What suggestions would you give to an 8th grader struggling to retain and remember mathematical concepts?
I'd recommend trying some interesting mathematical challenges in a no-stress environment. They're fun, and the stakes are low. Nobody is grading you on the results, and they're just for you to learn through something that makes you feel good (for yourself) when you crack a challenge. Ideally, these will look non-routine and of natural interest. It may take longer to build up a base of mathematical concepts this way at first, but you can do this at your own pace, without having to chase after a class. And at the end, you'll build up a mathematical background which is so strong that the other subjects will look harder to master (because they are).
For more information, visit Po-Shen Loh's information at Remake Learning.
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