Archive for April, 2012
This morning, my friend AJ called to ask for help in solving a problem from his ten-year-old daughter’s homework. When he explained his dilemma, the first thing I did, of course, was laugh. “Wow,” I said. “You really aren’t as smart as a fifth-grader, are you?”
AJ and his daughter are both intelligent, and his daughter loves math. The problem they were trying to solve was this:
What is the units digit of the product of the first 21 prime numbers?
Once you solve the problem, of course, you realize that the problem would have the same answer if asked as follows:
What is the units digit of the product of the first n prime numbers, for n > 3?
This made me think that this could be a good problem for the classroom. Have all students randomly generate a positive integer, and then have them solve the problem above using their random number to replace n. It would be impactful for students to see that everyone gets the same answer; and those who multiplied things out might be compelled to look for a pattern and figure out why everyone got the same answer.
But then I realized: this problem is gender biased. Well, maybe. The problem asks for the units digit of the product of the first 21 prime numbers. The choice of 21 was very deliberate, I’m sure. It’s small enough that an industrious student might actually try to calculate the product. In my experience, female students are more industrious than males and therefore more likely to do the computation. But the number is large enough that male students, who are lazy like I am, will think, “That’s too much work. There’s got to be a trick!”
I mentioned to AJ that if a larger number were chosen — for instance, if it involved the product of the first 1,000 prime numbers — then it might be more obvious that students ought to look for a pattern. “You haven’t met my daughter,” he said. “She’d still try to compute it.”
You may think my assertion is crazy. There is nothing in the problem that appears inherently biased against females.
A few years ago, the AAUW published a report about gender bias in math questions. One of the selected questions was something like, “What is the value of n if n + 2 = 7?” Despite the neutrality of the content, girls scored significantly lower than boys on this question, so it was deemed to be biased. (Sorry, I wasn’t able to find a reference to the report. If anyone knows the report to which I’m referring, please share in the comments.)
Further, FairTest claims that the gender gap all but disappears on all types of questions except multiple choice when other question types were examined on Advanced Placement tests. What is it about multiple choice questions that makes them implicitly unfair to females? I have no idea.
The 2012 Annual Meeting of the National Council of Teachers of Mathematics (NCTM) is happening next week, April 25‑28, in Philadelphia, PA. As it winds down, the USA Science and Engineering Festival starts in Washington, DC, and will occur April 28‑29. It will be a busy week for me — I am performing twice at each event! If you happen to be attending either event, please stop by and say hello.
At the NCTM Annual Meeting…
- To 10 and Beyond Using Free Illuminations Resources
Friday, April 27, 8:30-10:00 a.m.
Salon A/B (Philadelphia Marriott Downtown)
- Using Free NCTM Resources to Promote an Understanding of Proportion
Friday, April 27, 1:00-2:30 p.m.
Salon A/B (Philadelphia Marriott Downtown)
At the USA Science and Engineering Festival, Washington, DC…
- Puns and Puzzles
Saturday, April 28, 2:00-2:30 p.m.
Franklin Stage (Washington Convention Center)
- Puns and Puzzles
Sunday, April 29, 3:00-3:30 p.m.
Franklin Stage (Washington Convention Center)
I am expecting an engaged crowd at each event, and I am hopeful that my presentations are received better than this…
A mathematician and an engineer attend a physics lecture. The topic is Kulza-Klein theories involving physical processes that occur in 9-dimensional space. The mathematician is enjoying the lecture, but the engineer is confused and frustrated. At the end, the mathematician comments about how wonderful he thought the lecture was. The engineer asks, “How do you understand this stuff?”
The mathematician replies, “I just visualize the process.”
“But how can you possibly visualize something that occurs in 9-dimensional space?”
“Easy,” says the mathematician. “First, I visualize it in n-dimensional space, and then I let n = 9.”
One of the most exciting plays in the history of professional (American) football was the opening play of the second half of Super Bowl XLIV, when the New Orleans Saints recovered an onside kick. They then scored to take a 13–10 lead, and eventually won the game 31–17.
But onside kicks could be a thing of the past. Yesterday, New York Giants’ co-owner John Mara suggested that kickoffs might someday be eliminated from the NFL. This caused a lot of sports pundits to react, saying that it would inherently change the game. On the Mike and Mike Show, analyst Mark Schlereth responded with these rhetorical questions:
What’re you gonna do, flip a coin three times in a row? You gotta get heads three times in a row to get an onside kick?
Once again, probability was placed front-and-center in recent football discussions. While I like Schlereth’s new, less violent, and more mathematical approach to onside kicks, I just wish he had gotten the math right.
If you flip three coins, the probability of getting three heads is 12.5%. That’s not enough. Data shows that onside kicks in the NFL are successful 26% of the time. So the following would be a reasonable modification to Schlereth’s proposal:
Flip two coins. Two heads results in a successful onside kick.
Then the probability would be 25%, closer to the current reality.
Unfortunately, that’s not exactly right, either — it’s based on a misleading statistic. The success rate of onside kicks is highly dependent on whether the team receiving the kickoff is expecting it or not. When teams are expecting it, the success rate hovers around 20%; when teams aren’t expecting it, however, the success rate jumps to 60%. Considering that data, the process might be modified as follows:
- Kicking team indicates to referee that they will try an onside kick.
- Of course, this must be done secretly, so as not to arouse the suspision of the receiving team. I propose that one referee be assigned to each team; the team would encode the message using RSA encryption, and the assigned referee would be given the corresponding RSA numbers. A message can then be passed without fear of interception by the receiving team. To ensure that this procedure does not signficantly delay the game, messages stating “we WILL try an onside kick” and “we WILL NOT try an onside kick” could be prepared in advance, and unemployed math PhD’s could be hired as NFL referees to decode the messages.
- The receiving team must similarly indicate whether or not they suspect an onside kick.
- Again, use RSA encryption.
- If the kicking team chooses an onside kick, and the receiving team suspects an onside kick, then:
- Flip 9 coins. If 9, 8, 3, or 1 of them land heads, the onside kick is successful.
- P(9, 8, 3, or 1 head with 9 coins) = 20.1%
- If the kicking team chooses an onside kick, but the receiving team does not suspect it, then:
- Flip 9 coins. If 9, 8, 5, 4, or 2 of them land heads, the onside kick is successful.
- P(9, 8, 5, 4, or 2 heads with 9 coins) = 60.0%
- If the kicking team does not choose an onside kick, then:
- Flip 9 coins, just so the receiving team is unaware of what the kicking team decided to do, which will allow for the element of surprise with future kicks.
If the NFL decides to accept Mark Schlereth’s suggestion for using coins to determine onside kicks, I am hopeful that they will give my proposal serious consideration. If necessary, I have an Excel spreadsheet that I would be willing to share with them.
We finished a meal at our favorite Mexican restaurant, and my wife said, “I’m not going to finish my margarita. Would you like the rest?” My response was:
Now there’s a question to which I’ll never say, “No.”
That got me to thinking… there are quite a few questions to which my answer would never be, “No.” The following is a partial list:
- Do you want to tell me a math joke?
- Paper or plastic?
- Do you want to play Scrabble®?
- Will the Barbershop Harmony Society’s international convention be a harmonic function?
- Would you like to hear a really great math problem?
- Would you like to give a talk to our math club?
- Isn’t 2 to the power of infinity equal to infinity, and therefore isn’t 2ℵ0 = ℵ0?
- Do good math jokes exist?
- Do you want to go see the Escher exhibit at the art museum?
- Aren’t almost all numbers very, very, very large? (See Frivolous Theorem of Arithmetic.)
- Do you want to learn a new math game?
- Is there a seed number A for which A3n will always be prime, for integer values of n?
- Is math cool?
And all this talk of yes/no questions reminded me of a joke:
Professor: Are you good at math?
Student: Well, yes and no.
Professor: What do you mean?
Student: Yes, I’m no good at math!
As it turns out, a large number of statistics that aren’t made up don’t really provide much help, either.
The Pew Research Center says that men are happiest over age 65 and that we are least happy in our 20’s. Friends Reunited says that people are happiest at age 33. A Gallup poll from 2009 said that men are least happy in their 50’s and late 80’s, but a different Gallup poll from 2008 claimed that people are happiest at 85. This last result agrees with a report from the National Academy of Sciences, which states that people are most depressed at age 44, as shown by the U-bend happiness curve below:
Well, shoot. With all this conflicting information, how will I know when to be happy? Until I get this all sorted out, I’ll just have to keep doing the activities that make people happiest. (Are you really surprised by the first item on that list?)
Jean Jacques Rousseau once defined happiness as follows:
Happiness: a good bank account, a good cook, and a good digestion.
But I would define it thus:
Happiness: a sharp pencil and some paper, a good problem, and a quiet place with some time to think.
It has been said that happiness adds and multiplies, as we divide it with others. But let’s not forget how subtraction can bring happiness, too…
Some people bring happiness wherever they go. But you? You bring happiness whenever you go.
[Update, 4/13/12: When I checked into a Comfort Inn hotel last night, I was given a bag with fresh cookies. On the outside it said:
Happiness is a warm chocolate chip cookie.
That might be the best definition yet!]
“Alex,” I said, “on our walk to the gym tonight, I have a game for you and Eli to play.”
Alex responded, “Daddy, you have a lot of games.”
Yeah, it’s true.
Earlier in the afternoon, I played a game with them that I had created. On a set of index cards I had written animal names, with one catch: All of the vowels were removed. So instead of DOG, the card had DG, and instead of ZEBRA, the card had ZBR. You get the idea.
Before we started playing, I told them an elaborate tale about how I had tried to write animal names on index cards, but the Vowel Thief kept stealing the vowels from me. At the end of my story, Eli asked, “Did he steal all the vowels, or just some of them?” A-ha, the ruse worked! Amazing how easy it is to pull the wool over a four-year-old’s eyes. (As I explained the game, I also mentioned that “it’ll be like reading from the Torah.” Sadly, my best joke of the day, but it fell on the deaf ears of the wrong audience.)
Here’s my list of vowel-less animals, roughly in order from easiest to hardest. Good luck.
Ask a silly question, get a silly answer.
Teacher: If you have $4, and you ask your father for another dollar, how much would you have?
Johnny: Four dollars.
Teacher: Young man, you don’t know your addition facts!
Johnny: Ma’am, you don’t know my father!
Johnny’s father and my dad seem to have a lot in common. But my dad would have been proud of me yesterday. While walking home from the local coffee shop, I noticed a corner of a dollar bill on the ground. Not the whole bill, mind you, just a corner that had been ripped off. I thought not much of it, until two feet later I saw another scrap of the dollar bill… then another… and another…
I know and understand Calculus, and I realized that a lot of little things can add up to a lot, so I spent 15 minutes scouring the area for as many pieces of the dollar bill as I could find. I took them home and asked my sons, “Wanna do a puzzle?” We spent a half-hour reconstructing the bill and taping it together. The pictures below show the before and after:
The bill was not in good enough shape to be accepted by a vending machine (too much tape, I suspect, and the missing piece on the right side surely didn’t help, either), but it was in good enough shape for my bank to give me four shiny quarters in exchange for it.
I know that a penny saved is a penny earned. But what is a dollar found?
And the bigger question: What should I do with my new-found wealth?
I decided to buy a lottery ticket. The state gambling commission organized a raffle that boasted an infinite amout of money as the prize. To my great surprise, I won! When I showed up to claim the prize, they told me it would be disbursed as 1 dollar now, 1/2 dollar next week, 1/3 dollar the thrid week, 1/4 dollar the week after that, and so on.
But the joke’s on them. My winnings for the third week will include a one-third cent piece, and that’s gotta be worth something, right?
(Note: Almost everything above is true. I really did find the pieces of a dollar bill on the ground yesterday. As best I can tell, the bill had been on the lawn when it was cut by the blades of a power mower. And my bank really did give me four quarters in exchange for the taped-up, reconstructed version.)