David Afshartous writes:

I recall you had a post awhile back RE the difficulty kids have excelling in statistics versus mathematics, e.g., there are few statistics prodigies yet many mathematics prodigies. In any event, my 10 year old nephew was on his school math team last year and I helped him with his homework which consisted mainly of previous math competition problems (2xweek via skype video). It seemed like they were developing a bag of tricks and not learning the underlying material behind the problems. As he is on the fence about joining the math team in the fall, I’m thinking about continuing our weekly meetings but teaching him basic statistics/probability instead. As I don’t want to turn him off from the subject at an early age, my guess is that I should focus on fun probability problems that he can relate to (e.g., binomial problems related to basketball, or perhaps mix in some intriguing aspects of the history of probability) and then later introduce additional material. I’d like to come up with a plan for the semester and would appreciate any advice you have on what a 10 yr old should be taught in statistics/probability.

My reply:

First off, I envy your nephew. I had zero math education at age 10. No math team, nothing like that. I just considered myself lucky when the teacher let me sit in the library and read books.

I do remember math team from high school, and I agree that much of it was centered around silly tricks. On the other hand, silly math tricks are still math. I don’t know that he really needs to learn the underlying principles right away. Maybe what it really takes is the proverbial 10,000 hours of practice. If he’s enjoying it, that should be fine.

If you’re doing statistics and probability . . . I really have no idea! I personally like a lot of the games in my Bag of Tricks book, so you could start with some of them. A natural area of applications would be board games, if he likes Monopoly or Scrabble or whatever, there are a lot of probabilities to calculate. You could also try getting a little roulette set, if you’re not worried about turning him into a gambling addict.

Any other ideas out there?

Backgammon!

Poker, Soccer, Baseball, Basketball, Football, Hockey statistics. It's a cheap form of cool which will probably open up social space for him to be nerdy. If you have money, hire a female grad student or math major to tutor -and motivate- him in probability and statistics.

It may be my contrarian tendencies, but I always love little tricks and paradoxes in any area of mathematics – they often lead to clearer thinking and counter-intuitive results are fun. I don't have much concept of where this kid is at mathematically (or any ten year old for that matter), but I am thinking about things like monty-hall and the birthday problem. A couple of foundational lessons and you could probably start in on them.

If he's into sports, then there are the statistics heavy simulation games – I remember playing some of them, a few decades ago.

The best math teaching I've ever seen done with kids is at Math Circle

http://www.themathcircle.org/

There are also some good activities in some of the books by Al Possamentier.

1) make 'em flip a coin 50 times and write down H or t: HHHTHTHTHTTTTTTHTTTT and then ask him to find patterns, then have him test if they hold going forward (make him flip coins some more)

2) nothing beats playing cards

Thanks to all for the suggestions. I've discussed the result of the Monty Hall problem and there is a nice website where one can actually play the game. I also did the card "trick" probability example in Teaching Statistics (guessing a card that is upside down in a rigged deck) for his 5th grade class and this was a big hit. I can't seem to find this now in the table of contents or index; Andrew, do you know the chapter it is in? Some of the other activities in the book look doable so I'll start w/ these and check out the other suggestions as well.

Fred Mosteller, "Fifty Challenging Problems in Probability With Solutions" has some gems.

Peter: Wow, that Math Circle looks great!

Bill: I love Mosteller's book too, but I think it may be a bit much for a 10-year-old. My guess is that backgammon, Monopoly, etc., would work better.

Just seconding some of the suggestions here: while I was never taught probability (or anything, really) in any systematic way growing up, my dad used to give me paradoxes and puzzles at an early age. It's surprising how early kids can enjoy this (maybe they're really just enjoying the attention you give them).

Besides the Monty Hall problems, I remember him explaining countable infinity to me via puzzles about the hotel with infinite rooms and infinite guests, etc. (Actually, what he did mostly was torment me with semantic paradoxes, but I suspect that's more idiosyncratic to us.)

We played a ton of card games and sometimes he'd give me probability problems related to them (mostly, I now suspect, when he wanted a way to occupy me for a time while he did something else); this is how I first learned combinations and permutations. If you teach kids to love card games for their own sake — my parents didn't let me win — they may develop a feel for probability even if they can't articulate the principles.

Finally, my dad used to give me a lot of liar/truthteller puzzles; when I was very young, this was his answer to the bedtime story, because he wasn't very good at making up stories. I was quite startled, when I first took a logic class, to discover that these had a name — knights/knaves problems — and were not his invention.

Raymond Smullyan has a book of knights and knaves puzzles that I think would be appropriate for a 10-year-old. (If he has other puzzle books, I'd check those out too; he's wonderful and fun.) There's also a book called Sherlock Holmes Chess Mysteries that are not the standard chess puzzles ("white to win in so many moves") but more interesting variants requiring only the rules of chess ("A piece has fallen off this board. Which one?") These suggestions are not probability, they're deductive logic, but I think they'd be fun for a kid of this age. Hope that helps.

It might be a chance to teach some data collection skills, which is more science than math skills but it's all related. Some things in the world are Poisson distributed, like the chips in a chocolate chip cookies or bus arrival times or number of people in line at the grocery store. Similarly, other things are normally distributed or binomially distributed. You can collect data about things in your environment and recreate these natural distributions.

Here is what I did to teach maximum likelihood to my daughters when they were six and seven years old: I had two equal urns (paper boxes), each with 10 balls, one had 8 black and two white, the other the numbers switched. Then I mixed the urns, draw one of them randomly, and from the random urn draw one ball. It turned out the ball were white. I asked them, "from which urn did we draw, the one with 2 whiteÅ› or the one with 8 white balls? After about one second thinking they answered: "the one with 8 white balls". So I asked them why they thought so. Answer, also fast: "Because then it is easier to draw white". Easier = more probable, that is.

The MathCircle has an interestingly limited (old school?) nondiscrimination policy:

"ENROLLMENT IS LIMITED. For new students a phone interview may be required. Please call

(617) 519-6397 or e-mail mathcircle@math.harvard.edu for further information.

——————————————————————————–

The Math Circle does not discriminate on the basis of race, creed, color, national or ethnic origin in the administration of its educational and admission policies. We are non-profit and welcome contributions. "

What the heck would a statistics prodigy be? To single out stats would be like looking for a combinatorial prodigy.

I suggest one of the many books about how the great problems in history were solved. Even back to Euclid but certainly including Euler and Cantor. The point would be for him to find what he's most drawn to. Is it infinities? Probabilities? Knots? Who knows?

Andrew – they're amazing. I went up to Boston for the day and watched them teach. It's *stunning*. I know you're in Boston sometimes – they welcome visitors.

I wrote about them on Daily Kos

http://www.dailykos.com/storyonly/2008/7/16/13551…

in a diary I called "The joy of participatory learning".

Jonathan: No! Statistics is not a branch of mathematics.

Andrew: You're right, most of the problems in Mosteller's book wouldn't be suitable for a ten-year-old, but some would. And, simplified versions of some others would (use your imagination!). I was thinking, though, that as the child approached college age, a judicious sprinkling of problems like this could sharpen her or his intuition. I must also think that the particular child in question is precocious to a degree that many of the problems in this book might be approachable earlier than the average child could.

There was no math team when I was in high school (over 50 years ago). But my elder son joined the math team and enjoyed it and got a lot out of it. I'll have to ask him how he thought about the "learning tricks" aspect of it.

Andrew, Could you please elaborate more on your "statistics is not a branch of mathematics" comment, or point me to a link that already discusses it. I am a grad student, currently struggling to understand this distinction. Thanks so much.

Lacey: You could start with my link above to Dick De Veaux's talk. Also see my two books on applied statistics, which have a bit of mathematics but a lot that's not mathematics at all.

My mathematics edjucation as a child largely came from baseball statisics and playing strat-o-matic baseball. Understanding basic probability was key to evaluating the player's cards.

My 7yo daughter took out from the library "Desperate Measures" by Kjartan Poskitt which is part of the "Murderous Maths" series. The bit I read to her was on measuring the width of the book and choosing which units to measure it in and what accuracy to use. It was all done very humouresly. She laughed at all the zeros when measuing in nanometres or kilometers. There are lots of cartoons to break up the text. The humour is slightly evolved "English comic book" so it may not appeal to Americans. The examples also tend to be around ideas more appealing to boys (and tomboys in my case). Seven is a little young for it but your ten year old is probably just the right age.

I haven't read this one but you could try

"Do You Feel Lucky? The Secrets of Probability" by Kjartan Poskitt.

There's only a few of them on http://www.amazon.com but there's more on http://www.amazon.co.uk. And more details about content at http://www.murderousmaths.co.uk/usa/index.htm

(There's also a "Horrible History" and "Horrible Science" series too. )

Surprised nobody has mentioned Larry Gonick's

Cartoon Guide to Statisticsyet. My kid slurped up all of his books around age 8-10.Dude, you are so out of it when it comes to board games.

The idea of board games for learning math-like thinking is good, but if Monopoly is your idea of the standard, you totally missed the revolution.

What I would consider the "standards" now (not necessarily the best): Settlers of Catan, Puerto Rico, Ra, Ticket to Ride, Alhambra.

They're certainly not going to learn a lot about math by playing Buy That Guy!