Tanya Khovanova's Math Blog

I recently got a new job — to coordinate math students at RSI (Research Science Institute). RSI provides a one-month research experience based at MIT for high school juniors. The program is highly competitive and kids from all over the world apply for it.

Before the program started, I asked around among mathematicians for advice on how to do a great job with these talented kids. I was surprised by the conflicting opinions on the value of the program. I thought you’d be interested in hearing those opinions, although I confess that I do not remember who said what, or anyone’s exact words. I will just repeat the gist of it.

Former participants:

• I went there, it was awesome.
• I went there, it was underwhelming.
• Canada/USA math camp is more fun for sure.
• RSI is an absolutely fantastic experience for students, and I think the adults who take part enjoy it very much as well.

Potential participants:

• Cool, if I get there I’ll try to prove the Riemann Hypothesis.
• Last year Eric Larsen won $100,000 as a result of this program. If twenty math students participate, then the expected return is $5,000 per one month of work — not bad for a high-schooler.
• MIT is my dream school; just to be there will be inspiring.
• I will prove the Riemann Hypothesis.
• Yeah, I can become famous.
• Cool, I want to be a mathematician — I should try this.
• I love Canada/USA math camp and I’d rather go there.

Grad students, former and potential mentors:

• My professor doesn’t have a good problem for me. If he gives a nice problem to a high school student, that will be unfair.
• It’s just a job.
• What if I solve the problem first, do I keep silent? — That doesn’t make any sense.
• What if this high school student is better than me? That would be a bummer.
• This job was a lot of fun; I enjoyed it.
• I used to participate in RSI myself, and that was great. Now I would like to be on the giving side.
• RSI teaches students how to get versed in impressing people. For the Meet-Your-Mentor Night the students showed up in suits. How many real mathematicians do you know that own a suit?

Professors on the program in general:

• Usually students study mathematics for many years. RSI allows them to actually do mathematics.
• I studied for many years before I could start to do research. This RSI experiment is degrading to mathematics and disrespectful to mathematicians.
• Most students are wired towards problem solving, and very often they need only one basic idea and 15 minutes to solve a problem. Research has a completely different pace; it is important that kids try it.
• Some students go to this program because they want to win competitions and get to good colleges. These goals should be secondary. We should accept students because they want to try research.
• One month for research? Is this a joke? Do you like fast food?
• These are the best students from around the country. It feels nice when a potential future Fields medalist looks up to you.
• These students might be better than average undergraduate students at MIT. It might be fun to work with them.
• I think that the number of students who might be a good fit for such a program is very small; the number of professors who might be a good fit is very small too. If this program grows it might become completely useless.
• High school students are being mentored by grad students, who themselves have just started their own research. Grad students do not have enough experience to really guide people through research.
• It is such a great opportunity to get a taste of research while you are in high school.
• People usually choose projects for their research. These kids are given projects: this is not research — it’s slave labor.
• One month is not enough for interesting research. It would be good if students use this month to jump-start some research and then continue it after the program.
• It’s a waste of time to learn mathematics for many years and then discover that you do not like research. This program gives an opportunity for students to decide whether they are interested in research very early in their lives. This is tremendously useful.

I asked some math professors to suggest problems for these students:

• I have some problems I can give, but they require deep knowledge of topology. The students would need to take some courses to understand the second paragraph of the paper I would give them, which they can’t succeed in doing in a month. Can we replace this program with my course?
• It wouldn’t be nice to give them a problem that is too difficult. If the problem is easy, then I usually have an idea how to solve it. Instead of wasting two hours describing an easy problem to students, I can use this time to solve it myself.
• Ask Ira Gessel or Pavel Etingof. I have heard that they generate problems faster than their graduate students solve them.
• I have some leftover problems I can give away. However my concern is this: what if they solve it or mostly solve it, but then go back to school without writing their paper. What do I do? Giving the same problem to someone else or writing a paper myself without mentioning the student would not be kosher. Writing a joint paper for them is a burden. I need to think about a leftover problem I do not care about.
• If I have a good project, I will give it to my graduate students. Why would I invest in a high school student who is here for a month and probably is not ready for this anyway?
• That’s great, the online database of integer sequences contains tons of conjectures. They even have an index pointing towards “Conjectured sequences” and towards “Unsolved problems”. Besides, you can search the database for the words “conjecture”, “apparently” or “appears”. There is also an article by Ralf Stephan describing 100 conjectures from the OEIS.
• I have some things I need to calculate, but I do not know programming. If someone can do this for me that would be good.
• They usually want to submit papers for competitions, which means they do not want me to be a coauthor. I do not have problems I just want to throw away.
• Richard Stanley keeps a list of unsolved problems, ask him.
• There is a list of unsolved problems on wiki, but they are too difficult.
• They can always try to find a different proof for something.

The 2009 RSI has just begun. We have awesome students, great mentors and quite interesting problems to solve. I am positive we’ll prove the negativists wrong.

There is interesting data to show that MIT takes math students more seriously than Harvard and Princeton. By Michael Sipser’s suggestion I looked at the Putnam Competition results. Out of the top 74 scorers of 2007, 21 were from MIT, 9 from Harvard and 7 from Princeton. Keep in mind that the total freshman enrollment at MIT is much lower than at Harvard or Princeton. This story repeated itself in 2008: out of top 79 scorers 23 were from MIT, 11 from Harvard and 11 from Princeton.

Ironically, MIT’s team didn’t win Putnam in those years. MIT’s team won the third place after Harvard and Princeton. If you look at the results more closely, you will notice that had MIT arranged teams differently, MIT would have won.

It appears that MIT put their three top scorers from the previous year on their lead team. MIT shouldn’t assume that those three continue to be their strongest competitors. Instead they should probably test their students right before the Putnam competition, because if you look at MIT’s top individual performers, had they been on a team together, they would have won.

Maybe MIT should rethink its algorithm for creating teams, or maybe we should just wait. As it is obvious that MIT is more serious about math, all top math students may want to go to MIT in coming years. If this happens, the mathematics field will be absolutely dominated by MIT.

I have a tiny book written by Vladimir Arnold Problems for Kids from 5 to 15. A free online version of this book is available in Russian. The book contains 79 problems, and problem Number 6 criticizes American math education. Here is the translation:

(From an American standardized test) A hypotenuse of a right triangle is 10 inches, and the altitude having the hypotenuse as its base is 6 inches. Find the area of the triangle.
American students solved this problem successfully for 10 years, by providing the “correct” answer: 30 inches squared. However, when Russian students from Moscow tried to solve it, none of them “succeeded”. Why?

Arnold has inflated expectations for kids. The book presents the problems according to the increasing order of difficulty, and this suggests that he expects kids under 10 to solve Number 6.

Arnold claimed that every student from Moscow would notice what is wrong with this problem. I can forgive his exaggeration, because I’ve met such kids. Anyways, I doubt that Arnold ever stumbled upon an average Russian student.

My own fundamental interest is in the state of American math education, so I decided to check his claim concerning American students. I asked my students to calculate the area of the triangle in the above puzzle.

Here are the results of my experiment. Most of them said that the answer is 30. Some of them said that it is 24. In case you’re wondering where the 24 is coming from, I can explain. They decided that a right triangle with hypotenuse 10 must have two other legs equal to 8 and 6.

Some of the students got confused, not because they realized that there was a trick, but because they thought the way to calculate the area of the right triangle is to take half the product of its legs. As lengths of legs were not given, they didn’t know what to do.

There was one student. Yes, there was one student, who decided that he could calculate the legs of the triangle from the given information and kept wondering why he was getting a negative number under the square root.

You decide for yourself whether there is hope for American math education. Or, if you are a teacher, try running the same experiment yourself. I hope that one day I will hear from you that one of your students, upon reading the problem, immediately said that such a triangle can’t exist because the altitude of the right triangle with the hypotenuse as the base can never be bigger than half of the hypotenuse.

So many people liked the puzzles I posted in Subtraction Problems, Russian Style, that I decided to present a similar collection of multiplication and division puzzles. These two sets of puzzles have one thing in common: kids who go for speed over thinking make mistakes.

Humans have 10 fingers on their hands. How many fingers are there on 10 hands?

This one is from my friend Yulia Elkhimova:

Three horses were galloping at 27 miles per hour. What was the speed of one horse?

Here is a similar invention of mine:

Ten kids from Belmont High School went on a tour of Italy. During the tour they visited 20 museums. How many museums did each kid go to?

Another classic:

How many people are there in two pairs of twins, twice?

Can you add more puzzles to this collection?

My son Sergei brought back the Flip-Flop game from Canada/USA Mathcamp, and now I teach it to my students. This game trains students in the multiplication table for seven and eight. These are the most difficult digits in multiplication. This game is appropriate for small kids who just learned the multiplication table, but it is also fun for older kids and adults.

This is a turn-based game. In its primitive simplification kids stand in a circle and count in turn. But it is more interesting than that. Here’s what to say and do on your turn, and how the game determines who is next.

First I need to tell you what to say. On your turn, say the next number by default. However, there are exceptions when you have to say something else. And this something else consists of flips and/or flops.

So what are flips? Flip is related to seven. If a number is divisible by seven or has a digit seven, instead of saying this number, we have to say “flip” with multiplicities. For example, instead of 17 we say “flip” because it contains one digit seven. Instead of 14 we say “flip”, because it is divisible by seven once. Instead of 7 we say “flip-flip”, as it is both divisible by seven and has a digit seven. Instead of 49, we say “flip-flip” as 49 is divisible by the square of seven. Instead of 77 we say “flip-flip-flip” as it has two digits seven and is divisible by seven once.

Flop relates to eight the same way as flip relates to seven. Thus, instead of 16 we say “flop” as it is divisible by eight; instead of 18 we say “flop” as it contains the digit eight; and for 48 we say “flop-flop” as it is both divisible by eight and contains the digit eight.

A number can relate to seven and eight at the same time. For example 28 is divisible by seven and contains the digit eight. Instead of 28 we say “flip-flop”. The general rule is that all flips are pronounced before all flops. For example, instead of 788 we will say “flip-flop-flop-flop” as it is divisible by eight and contains the digit seven once and the digit eight twice.

The sequence of natural numbers in the flip-flop version starts as the following: 1, 2, 3, 4, 5, 6, flip-flip, flop-flop, 9, 10, 11, 12, 13, flip, 15, flop, flip, flop, 19, 20, flip, 22, 23, flop, 25, 26, flip, flip-flop, 29, 30, 31, flop, 33, 34, flip, 36, flip, flop, 39, flop, 41, flip, 43, 44, 45, 46, flip, flop-flop, flip-flip, 50, 51, 52, 53, 54, 55, flip-flop, flip, flop, 59, 60, 61, 62, flip, flop-flop, 65, 66, flip, flop, 69, flip-flip, flip, flip-flop, flip, flip, flip, flip, flip-flip-flip, flip-flop, flip, flop-flop, flop, flop, flop, flip-flop, flop, flop, flip-flop, flop-flop-flop, flop, 90, flip, 92, 93, 94, 95, flop, flip, flopflip-flip-flop, 99, 100.

So how does the turn change? Everyone stands in a circle and says their number the way explained above. We start clockwise and move to the next number. For every flip we reverse the direction and for every flop we skip a person. That means that if we have two flips, we don’t change the direction, while for two flops we skip two people. If we have flips and flops together, for example 28 corresponds to “flip-flop”, then first we change the direction and then we skip a person.

On top of that, there is an extra rule for what you do on your turn. If you say something other than a default number, you switch your position from standing to sitting and vice versa. Sometimes I skip this extra feature — not because I am too lazy to exercise, but because I usually conduct this game in a classroom, where all the desks prevent us from fully enjoying such physical activity.

There are two ways to play this game: as a competition or as practice. When we are competing, a person who makes a mistake drops out. If we’re just practicing, no one drops out. Sometimes I am particularly generous and allow my kids one mistake before making them drop out after the second mistake. So far we have played up to 100. I am curious to see if we can ever reach 700 and how long we will be able to continue the game after that.

Testing in the US is dominated by multiple-choice questions. Together with the time limit, this encourages students to stop thinking and go for guessing. I recently wrote an essay AMC, AIME, USAMO Contradiction, in which I complained about the lack of proofs in the first two rounds of math competitions.

Is there a way to improve the situation? I grew up in the USSR, where each round of the math competition had the same format: you were given several hours to write proofs for three or four difficult problems. There are two concerns with organizing a competition in this way. First, the Russian system is much more expensive, whereas the US’s multiple choice tests can be inexpensively checked by a computer. Second, the Russian system is prone to unfairness. You need many math teachers to check all these papers on the highest level. Some of these teachers might not be fully qualified, and it is difficult to ensure uniform checking. This system can’t easily be adopted in the US. I am surprised I haven’t heard of lawsuits challenging USAMO results, but if we were to start having proofs at the AMC level with several hundred thousand participants, we would get into lots of trouble.

An interesting compromise was introduced at the Streamline Olympiad. The problems were multiple choice, but students were also requested to write proofs. Students got two points for a correct multiple choice answer, and if the choice was correct the proof was checked. Students could get up to three points for a correct proof. This idea solves two issues. The writing of proofs is rewarded at an early stage and the work of the judges is not as overwhelming as it would have been, had they needed to check every proof. However, there is one problem that I discussed in previous posts that this method doesn’t solve: with multiple choice, minor mistakes cost you the whole problem, even though you might have been very close to a solution. If we want to reward thinking more than accuracy, the proof system allows us to give credit for partial solutions.

I can suggest another approach. If the Russians require proofs for all problems and the Americans don’t require proofs for any problem, why not compromise and require a proof for one problem out of the set.

But I actually have a bigger idea in mind. I think that current development in artificial intelligence may soon help us to check the proofs with the aid of a computer. Artificial intelligence is still far from ready to validate that a mathematical text a human has produced constitutes a proof. But in this particular case, we have two things working for us. First, we can use humans and computers together. Second, we do not need to check the validity of any random proof; we need to check the validity of a specific proof of a simple problem that we know in advance, thus allowing us to prepare the computers.

Let us assume that we already can convert student handwriting into computer-legible text or that students write directly in LaTeX.

Here is the plan. Suppose for every problem, we create a database of some sample right, wrong and partial solutions with corresponding scores. The computer checks the students’ solutions against the given sample. Hopefully, the computer can recognize small typos and deviations that shouldn’t change the point value. If the computer encounters a solution that is significantly different from the ones in the sample, it sends the solution to human judges. Humans decide how to score the solution and the solution and its score is added to the sample database.

For this system to work, computers should be smart enough not to send too many solutions to humans. So how many is too many? My estimate is based on the idea that we wouldn’t want the budget of AMC to go too much higher than the USAMO budget. Since USAMO has 500 participants, judges check just a few hundred solutions to any particular problem. With several hundred thousand participants in AMC, the computer would have to be able to cluster all the solutions into not more than a few hundred groups. The judges only have to check one solution in each group.

As a bonus, we can create a system where for a given solution that is not in the database, the computer finds the closest solution and highlights the difference, thus simplifying the human’s job.

In order to improve math education, we need to add proofs when teaching math. My idea might also work for SATs and for other tests.

Now that there is more money available for education research, would anyone like to explore this?

A stick has two ends. If you cut off one end, how many ends will the stick have left?

This pre-kindergarten math problem was given to me by Maxim Kazarian who lives in Moscow, Russia. That got me thinking about math education in the US. Actually, just about anything can get me thinking about education in our country. One of our math education patterns is to provide simplified templates and to train kids to plug numbers into them without thinking.

Math education should be about thinking. We need to give kids a lot of math problems that do not fit into standard templates, in order to encourage creative thinking. Here is another puzzle from Maxim:

A square has four corners. If we cut one corner off, how many corners will the remaining figure have?

I invite my readers to invent additional problems that sound as if a subtraction by one is needed, when, in fact, it is not. Here is my contribution:

Anna had two sons. One son grew up and moved away. How many sons does Anna have now?

To get to the national swimming championship, you need to win the state running championships.

What? Is that a joke? Perhaps you’re having the same reaction. Because this is exactly what is happening with math competitions. The official USA math competition has three rounds: AMC, AIME and USAMO.

AMC is a multiple-choice competition with 25 problems in 75 minutes. To be good at it, you need speed, accuracy and the ability to guess well.

AIME is 3 hours long and has 15 problems. The problems are a different level of difficulty and guessing will not help you. Though AIME is also multiple-choice, unlike AMC where you choose out of 5, in AIME you choose out of 1,000. But you still need speed and accuracy. A small arithmetic mistake will cost you the whole problem.

USAMO is a competition that runs for 9 hours and has 6 problems. The problems are much harder and you have to write proofs. Proofs? What proofs? Where are the proofs coming from? It is like you got to the national swimming championship because you are a great runner, but you do not know how to swim.

As the result of this system of selection, the USA team at the International Math Olympiad has diverse skills: these kids are good at all three types of the math competitions. It is like taking an Olympic triathlon team to the Olympic swimming event.

However, the US loses by selecting in this way. There are many kids who are great mathematicians: they may be good at difficult problems but not that good at speed-racing problems. An arithmetic mistake costs you only one point at IMO, but a whole problem at AIME. There are kids who are deep mathematicians prone to small arithmetic mistakes. They could get a gold medal at IMO, but they can’t pass AMC or AIME.

Not only that. As many AMC problems are boring and do not require ideas, AMC might discourage kids from all math competitions at an early stage.

I will write later with my ideas about how to change AMC. Right now I would like to offer a solution to a smaller problem. I am sure that the US math team organizers know many cases in which a non-senior kid does great at USAMO and is potentially a team member for the next year’s US IMO team, but, oops, next year he can’t pass AMC.

I suggest the following: USAMO participants are allowed to go to next year’s AIME no matter what their AMC scores are. USAMO winners are allowed to go to the next year’s USAMO no matter what their AIME results are. This way kids who have proved that they are great swimmers do not need to compete in running again.

I bought the book “Math Doesn’t Suck: How to Survive Middle School Math Without Losing Your Mind or Breaking a Nail“ by Danica McKellar because I couldn’t resist the title. Sometimes this book reads like a fashion magazine for girls: celebrities, shopping, diet, love, shoes, boyfriends. At the same time it covers elementary math: fractions, percents and word problems.

You can apply math to anything in life. Certainly you can apply it to fashion and shoes. I liked the parallel between shoes and fractions that Danica used. She compared improper fractions to tennis shoes and mixed numbers to high heels. It is much easier to work with improper fractions, but mixed numbers are far more presentable.

Danica is trying to break the stereotype that girls are not good at math by feeding all the other stereotypes about girls. If you are a typical American girl who hates math and missed some math basics, this book is for you. If you want to discover whether the stars are on your side when you are learning math, the book even includes a math horoscope.

I ran an experiment. I copied multiple choices from the 2007 AMC 8 into a file and asked my son Sergei to try to guess the answers, looking only at the choices. I allowed him to keep several choices. The score I assigned depended on the number of leftover choices. If the leftover choices didn’t contain the right answer, the score for the problem was zero. Otherwise, it was scaled according to the number of choices he left. For example, if he had four choices left and the right answer was among them he got 1/4 of a point. Here are the choices:

1. 9, 10, 11, 12, 13.
2. 2/5, 1/2, 5/4, 5/3, 5/2.
3. 2, 5, 7, 10, 12.
4. 12, 15, 18, 30, 36.
5. 24, 25, 26, 27, 28.
6. 7, 17, 34, 41, 80.
7. 25, 26, 29, 33, 36.
8. 3, 4.5, 6, 9, 18.
9. 1, 2, 3, 4, cannot be determined.
10. 13, 20, 24, 28, 30.
11. Choose picture: I, II, III, IV, cannot be determined.
12. 1:1, 6:5, 3:2, 2:1, 3:1.
13. 503, 1006, 1504, 1507, 1510.
14. 5, 8, 13, 14, 18.
15. a+c < b, ab < c, a+b < c, ac < b, b/c = a.
16. Choose picture: 1, 2, 3, 4, 5.
17. 25, 35, 40, 45, 50.
18. 2, 5, 6, 8, 10.
19. 2, 64, 79, 96, 131.
20. 48, 50, 53, 54, 60.
21. 2/7, 3/8, 1/2, 4/7, 5/8.
22. 2, 4.5, 5, 6.2, 7.
23. 4, 6, 8, 10, 12.
24. 1/4, 1/3, 1/2, 2/3, 3/4.
25. 17/36, 35/72, 1/2, 37/72, 19/36.

It is clear that if you keep all choices, your score will be 5, which is the expected score for AMC if you are randomly guessing the answers. Sergei’s total score was 7.77, which is noticeably better than the expected 5.

There were two questions where Sergei felt that he knew the answer exactly. First was question number two with choices: 2/5, 1/2, 5/4, 5/3, 5/2. All but one of the choices has a 5 in it, so 1/2 must be wrong. Numbers 2/5 and 5/2 are inverses of each other, so if organizers expect you to make a mistake by inverting the right answer, then one of these choices must be the right answer. But 5/4 and 5/3 are better suited as a miscalculation of 5/2, than of 2/5. His choice was 5/2, and it was correct. The second question for which he was sure of the answer was question 19, with his answer 79. I still do not have a clue why.

Sergei’s result wasn’t too much better than just guessing. That means that AMC 8 organizers do a reasonably good job of hiding the real answer. You can try it at home and see if you can improve on Sergei’s result. I will publish the right answers as a comment to this essay in a week or so.