Here's a math problem done by top students in Singapore

This viral math problem shows what American schools could learn from Singapore
Updated by Libby Nelson
April 15, 2015
Vox

...First, for those struggling with the problem, some good news: this is not from a general textbook. It was designed for the top 40 percent of high school math and science students in the country for use in a math competition, and was a relatively challenging question for that competition, according to the Singapore and Asian School Math Olympiad, the group that wrote the question. So this is a question designed to stretch Singapore's better math and science students.
Here's the problem Singaporean high school students were asked to solve, reworded slightly for clarity:

Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is. Cheryl marks 10 possible dates: May 15, May 16, May 19, June 17, June 18, July 14, July 16, August 14, August 15, or August 17.
Then Cheryl tells Albert the month of her birthday, but not the day. She tells Bernard the day of her birthday, but not the month. Then she asked if they can figure it out.
Albert: I don't know when Cheryl's birthday is, but I know Bernard doesn't know either.
Bernard: At first I didn't know when Cheryl's birthday is, but now I know.
Albert: If you know, then I know too!
When is Cheryl's birthday?

[Hint from Maura Larkins: I had trouble with this problem, and here's why: to solve it, you have to focus on WHAT ALBERT KNOWS.  He knows that Bernard doesn't know the answer.  How does he know this?  He knows that there are a couple of months that have dates that would be dead give-aways to Bernard.]


This problem is meant to test logical and analytical reasoning skills. Students have to work backward from the information they're given to the solution. It's not something that can be solved by a formula; it requires a process of elimination.
Bernard, who knows the date, has more information in this situation than Albert, who knows the month. No matter what Cheryl told Albert, there's no way Albert could figure out her birthday.
Bernard, though, might have been able to figure out Cheryl's birthday on his own, but only on two out of Cheryl's 10 possible dates.
If Cheryl told Bernard she was born on the 18th or 19th, he would know her birthday right away. That's because the 18th and 19th only show up once in Cheryl's list of possibilities. The rest of the dates are duplicates: the 14th could be in July or August, the 15th could be in May or August, the 16th in May or July, and the 17th in June or August. But the 18th could only be in June, and the 19th only in May.
But Bernard didn't know when Cheryl's birthday was at first — so she wasn't born on the 18th or 19th.
How did Albert know that Bernard didn't know? When Cheryl told him the month, she must have said July or August, because every possible date in July and August is also in another month. If she had told Albert she was born in May or June, she might have been born on May 19 or June 18, and Albert wouldn't be certain that Bernard was in the dark.
As soon as Albert says that, though, Bernard figures it out. How? He knows Cheryl must have told Albert her birthday is in July or August, because that's the only way Albert can be certain that Bernard doesn't know her birthday. Narrowing the months down to two possibilities is all it takes for him to find the answer. That means the date Cheryl gave Bernard must not be a possibility in both July and August. It's only a possibility in one of the two months.
We already knew Cheryl's birthday isn't the 18th or 19th. Eliminating the dates that are in both July and August knocks out the 14th. That leaves three possible birthdays out of the original 10: July 16, August 15, or August 17.
But now Albert knows, too. That means Cheryl must not have told him she was born in August, because he'd still be confused; two of the possible dates are in August. The only remaining possibility is July 16.
If this doesn't make sense to you, the New York Times has the another explanation of this solution, and the Guardian has a slightly different way of solving the problem.

The best math and problem-solving students in Singapore are really good

These problems show up in Singapore for a reason: they're meant to strengthen and test students' problem solving skills — and they seem to work.
"This kind of problems trains a person to analyze a problem in order to come to a logical solution," Henry Ong, the director of Singapore and Asian Schools Math Olympiad, wrote in a statement. "We are not saying this problem is for every student (since it involves rather sophisticated reasoning)."
Singapore has a disproportionate share of top problem-solvers. Its 15-year-olds had the highest scores, tied with Korea, on the problem-solving portion of the Programme for International Student Assessment, a standardized test administered to students in developed countries in 2012. Almost 10 percent of students performed well enough to be considered "highly skilled problem-solvers" — the highest share in the world. In the US, just 2.7 percent of students tested that well.
Students might be so good at problem-solving because they're very good at math. Students in Singapore scored second in the world, behind students in Shanghai, on the math portion of the PISA. Forty percent of Singapore's students are considered top performers in math, compared with just 8 percent in the US.
In another international test of math abilities, the Trends in International Mathematics and Science Study, Singapore has been ranked at or near the top every time it's been given since 1995...



Here's another point of view:

Want Kids to Learn Math? Stop Teaching It

124 Apr 6, 2015 8:00 AM EDT

By Susan Engel

Bloomberg

The U.S. has a math problem. Despite all the time, energy and money the country has thrown into finding better ways to teach the subject, American children keep scoring poorly and arriving at college woefully unprepared. Just as bad, if not worse, too many students think they hate math.
I propose a solution: Stop requiring everyone to take math in school.
People typically offer some combination of four reasons children should learn math: for everyday functions such as doing taxes, buying groceries and reading the news; for getting a job in an increasingly technologically advanced market; as a powerful way of thinking and understanding the world; to tackle high school or get into a good college.
Let's consider these one by one. To some degree, children naturally learn basic arithmetic just by spending time with people who use it, and by carrying out such tasks as setting the table, going to the store or sharing toys with friends. Research shows that even illiterate children can compute sums quite quickly and accurately in familiar settings (such as selling produce on the street). Babies are born with an intuitive knowledge of numbers. It wouldn’t take much for schools to teach every child how to add, subtract, multiply and divide.
Those interested in highly quantitative fields such as technology, finance or research are likely to have a natural inclination for math. They can obtain the knowledge they need later, in a much more effective and profound way, in college or beyond. People who invent new industries are rarely using math they learned in school, and often aren’t using any at all. Why drag all elementary school students through a compulsory curriculum that turns off as many as it prepares, on the off chance that a few might need it?
True, learning math can give us intellectual strengths different from the ones we get reading novels, studying history or poking around in a petri dish. However, these kinds of thinking are not necessarily tied to numbers, certainly not at the novice level. Advanced mathematics requires students to reason logically, be patient, methodical and playful in trying out solutions to a problem, imagine various routes to the same end, tolerate uncertainty and search for elegance. They need to know when to trust their quantitative intuitions and when to engage in counterintuitive thinking.
However, such abilities are usually precluded by the typical K-12 curriculum -- a dizzying array of isolated skills and procedures, which many college professors say they spend too much time getting students to "unlearn." Research has shown that many students who do perfectly well on math tests often can't apply a single thing they have learned in any other setting. We end up missing a chance to teach them what they would really need in order to go on to higher-level math or to think well.
Instead of a good score in algebra, children need three things:
1. Time. For the most part, children think concretely when they are young, and become more capable of abstract thought later. A huge industry has grown up around the idea that we can game the human system and teach children to think abstractly before they are ready. Such strategies haven’t been very successful, and they preclude activities that would be much more compelling and useful to young minds.
2. Reading. Research has demonstrated that literacy is crucial to abstract thought. Children who read become capable of specific kinds of conceptual and logical thought not available to others. This opens the door to thinking about things that are not part of one’s immediate tangible experience, a crucial aspect of higher mathematics.
3. Intellectual challenges. Children who are immersed in informal quantitative reasoning come to more formal math tasks, at a later age, with much greater ease. Similarly, children who are asked to give reasons for their thinking, or speculate about the past and future, are well positioned to learn various kinds of logic and argument.
So here’s the plan. Teach young children arithmetic, a task that would probably take 20 minutes a day through the end of third grade. Spend the extra time on reading, and on the kinds of play that involve abstract thinking and problem solving. .