- Is it bigger than 20? (YES)
- Is it even? (YES)
- Does it have a 7 in it? (NO)
- Is it 80? (NO)

It took him 20 more questions to get it. I bet him a quarter I could get his number with 10 questions. I succeeded and he had to beg his dad for a quarter. I've been told he has learned not to gamble with Uncle Bill. His father told me that the concept of `try to make every question cut the number of possibilities in half' was over his head since he has not learned fractions yet.

I then tried NIM-games. There are toothpicks on the table and you can remove 1 or 2. The players alternate. The player who removes the last toothpick WINS. He played his sister Jordan (who is nine) with different numbers of toothpicks on the table. They DID catch on that if the number of toothpicks is 3,6,9,12, ... like that, then Player II wins, otherwise Player I wins. They then did NIM with removing 1 or 2 or 3 and also 1 or 2 or 3 or 4, They learned the trick and the pattern. They liked it and learned some math.

I do not know if this is indicative, but it may well be that if a kid has not learned fractions yet, binary search may be over his head, while NIM games is fine and fun.

Warning: I once tried to teach my 6-year old nephew Michael that, when doing multiplication, the order does not matter.

BILL: Say you had two pans of brownies. One is 3 by 5 and the other is 5 by 3. Then---

MICHAEL: Do you! I love brownies!

We didn't get much math done ...

well ... the concept of "half" is pretty simple. i'm sure that a lot of kids can understand halves way before they learn about fractions. you might have tried to do it again using 1000 instead of 100.

ReplyDeleteIs it bigger than 320?

ReplyDeleteSorry about my last comment.

ReplyDeleteHi my name is Janet and I think your idea is awsome I will try this to my little brother!

ReplyDeleteoh yea, my little brother knows the multipacation table when he was only 6! He loved it!

ReplyDeleteI don't think binary search is as intuitive as addition/multiplication

ReplyDeleteThat's quite a lovely post, Bill! For a second, I had a mental lapse and wondered if Lance had a brother named Bill :)

ReplyDeleteI'm curious how you distinguished between inequality and strict inequality. To express "Is x \geq 56?", did you say:

- Is x at least 56?

- Is x greater than or equal to 56?

- Is x bigger than 55?

- Is x strictly bigger than 55?

I remember being preoccupied with the following construction in elementary school: "Alice is as tall as Bob." Does it mean Alice's height = Bob's height, or Alice's height \geq Bob's height?

As a kid (older than 8, probably like 10 or 11) I remember thinking binary search was pretty intuitive. I recall using it for some silly minigame on some old gameboy game. I don't think I understood it by thinking about fractions - It was more like thinking about writing numbers in binary and then each question you learned a digit. Not that I could explain my understanding anywhere near that clearly at that age, but I think that was the gist of it.

ReplyDeleteBinary search is extremely intuitive. Knowing about fractions doesn't have a lot to do with it. "Half" is sort of a visual/geometric notion. With guess-the-number, the operation is more of a comparison than divide (the divide is a forced consequence of the intuition).

ReplyDeleteAlso, you mentioned they learned the NIM patterns. But was that on the first try or was that after repeated trials of the game. I bet if you played guess-the-number that many times, they'd figure out the rule too (and perhaps much quicker).

If you're gonna try and teach math to an 8 year old then its better be your OWN 8 year old. Otherwise, when he doesn't get it you'll have to endure his parents excuses (like not learning fractions yet :).

ReplyDeleteNevertheless, I'll try this with my 7 y.o nephew. I'm sure he'll get it right.

"His first

ReplyDeletethreequestions were as follows."Mmmm...

Anyway, age does not seem to be the deciding factor:

Mathemaics for very young children

ReplyDeletegrad student wrote:theory... To express "Is x \geq 56?", did you say: ...wow, you're a _theory_ grad indeed. there are other ways to write "\geq" with your ascii keyboard, like ">=", you know :-) [wonder how you'd write a smiley though -- $\smiley$ ?]

'here are other ways to write "\geq" with your ascii keyboard, like ">=", you know'

ReplyDeletewhy would you do such a thing?

My son at that age (well, 9 and 10)and I started playing "20 questions" but with baseball players.

ReplyDeleteThe first binary-search questions are easy: current or retired? pitcher or not? AL or NL? But then it gets "chunky". Do you ask "outfield or infield"? Do you lump catcher with "outfield"? And now each baseball league has 3 divisions, not a helpful binary 2. So I'd ask "does he play east or west of the Mississippi?" to help his geography along a bit...

...but this hasn't turned him on to math yet. My best question of that type was: Suppose you're drafting a fantasy team with 1 player at each position, and you want to get the most homers. At each position there are only 2 choices---if you pick one, I will eventually get the other. Here are the choices, and you have first pick---whom do you draft first, and why?:

LF Barry Bonds 55 homers, or Manny Ramirez 50 homers

CF Sammy Sosa 45 homers, or Carlos Beltran 40 homers

RF Ken Griffey Jr. 40 or Garry Sheffield 42

3B A-Rod 45 or Aramis Ramirez 38

SS Nomar Garciaparra 30 or Miguel Tejada 35

2B Jeff Kent 28 or Craig Biggio 16

1B Albert Pujols 48 or Derrek Lee 45

C Mike Piazza 30 or Ivan Rodriguez 25

If you're not starry-eyed at the big totals, can you make the winning first pick?

Ken, can you explain the question to those of us that are baseball challenged? why not just pick the one with more homers for each position?

ReplyDeleteI'll draft Mr. Jeff Kent first!

ReplyDeleteAnonymous15, the point I should have made clearer is that you pick one player, then the other guy picks one player, alternating picks. Like picking sides for a sandlot ballgame. (In a "snake draft", you would pick 1, then I pick 2, then you pick 2, and so on.)

ReplyDeleteIf you could take 2 left-fielders then I'd pick Barry Bonds, but the limitation to 1 player at each position defines the problem---in ways Anon16 recognized!

My son loved doing such drafts with real baseball cards, not counting homers or anything, then we kept the lineups in our pockets and pretended to be the players while pitch-and-hitting in our driveway.

Actually, I can explain this in terms we use in our field. The problem is ISOMORPHIC to a simpler one: You have 8 stones labeled LF,CF,...,C. On each stone is a number representing the difference you gain by picking that stone (first). Players alternate picking a stone---or you do the snake-draft thing, 1-2-2-2-1 here. Alternate or snake, the answer is the same: first picker takes the stone with "12"---even though Barry gives twice as many homers!

ReplyDeleteReduced in this manner, the game is much simpler than Nim as Bill describes.

That's a very interesting post. You guessed his number in just 10 tries. That seems very difficult indeed. How did you do it?

ReplyDeleteEither you're not a theory grad student at MIT, or you're being a total jerk. Either way, your post is inappropriate. Everyone from "big names" like Ken Regan and D. Sivakumar, to girls named Janet with little brothers, are helping ensure Bill succeeds with this blog. Please take the hint. Thank you.

ReplyDeleteGetting an 8-year old interested in Math:

ReplyDelete- Yahtzee

- Rush hour

- Set

- Monopoly: which properties are the best deal and how many houses should one buy?

- Watch: Deal or no deal

Another path to math is through engineering. E.g., kids love flying machines -- get him/her toy helicopter or glider, etc.

ReplyDeleteI remember this incident about multiplication tables from my own childhood. We were taught multiplication tables by rote and were expected to "remember" them. I somehow discovered the we could generate the tables by addition. When I mentioned this to the teacher, instead of appreciating the fact, she admonished me and called me "lazy".

ReplyDeleteThere is an implicit contrast here between Mathematics and Arithmetic, between designing and understanding algorithms and merely memorizing and executing them. (They even activate different parts of the brain as Keith Devlin emphasizes in his book The Math Gene.) Most of school math class at age 8 is about the latter. Number search/team selection/NIM algorithms may be good because there is some chance that the 8-year old will understand/develop them intuitively and concretely but they are not much better than arithmetic if all the 8-year old does is memorize and execute them.

ReplyDeleteFor arithmetic, the previous poster is right at some level that recomputing the tables via addition may imply more understanding than memorizing multiplication tables but the advantage of having a look-up table is too much to ignore. My then 8-year old daughter saw no advantage of memorizing addition tables since she could use her fingers but the advantage for multiplication was clear to her. This led to an odd mixed strategy in which the multiplication of the partial products was the easy part but the final summation was the hard part because she had to use her fingers each time!

P.S. Great list, Claire! Binary Arts also makes a good game in the style of Rush Hour called Lunar Lockout that has a different set of rules but is also mathematical and fun.

I just gave my six year old daughter a calculator, and she seems to be enjoying it.

ReplyDeleteWhen I was a kid, there were a lot of simple electronic games that taught math (and other things) quite nicely. The TI Little Professor, Speak and Spell, Simon, Merlin, and some others I can't now remember the name of. I find current games for my daughters too flashy. They have Leapsters and play games on the PBS Web site, but the graphics and such are too good -- they're paying more attention to Dora and Elmo than the learning.

Maybe they're too spoiled by good graphics to play the simple games I grew up with, but if anyone has any suggestions I'd love to hear them. (Thanks, Claire, for your list!)

Michael Mitzenmacher

I have many experiences (my son is 8) but I will share this one:

ReplyDeleteWhen he was at first grade, so they knew addition as a concept but no formal addition algorithms, carries, etc, I asked a group of his friends how much is 1+2+3+4+5+6+7+8+9 ? More than 50% saw the pattern in a few minutes !

When he was in second grade, and by that time they knew the addition algorithm, putting long numbers under each other and carrying carries etc... so then I asked another group of his friends how much is 1+2+3+4+5+6+7+8+9 ?

Almost all took their pencils meticulously and started doing the addition algorithm ... How sad...

milena, what pattern did they recognize in first grade? the n(n+1)/2 formula? (I find it hard to believe, though I've never had a kid that age). Why is it so intuitive that a 6 year old can get it?

ReplyDeleteDear Anonymous 27:

ReplyDeleteThe pattern to 1+2+3+4+5+6+7+8+9 is

(1+9)=10, (2+8=10), (3+7)=10, (4+6)=10, and an orphan 5, for a total of 45 ! The story has it that Euler's teacher, as punishment to his class, gave the kids to compute

1+2+3+...+100, and Euler answered in 2 minutes ! My point is 7 years olds can do it too... Apparently, people stop being able to see the pattern once the symbol pushing

n*(n+1)/2 starts... Oh, Steven Rudich had wonderful notes for Discrete Math for Freshmen, apperently they are not on the Web anymore (are they?), and there are other great resources too... But its a great challenge and obligation to teach, from grade 1 to graduate school the meaning and the symbols at the same time... And final note, because I teach freshmen Discrete Math almost every year, half of the stuff that Freshmen understand, 8 year olds can understand too (when phrased appropriately). 8 year olds (not just mine, most of his friends, boys and girls alike), can understand recursive design,

counting, probability...

Oh they can certainly understand binary search... I keep being surprised by the immense intuitive potential that kids come with, and the systematic supression of intuition once the symbol pushing starts...

Thanks for the interesting answer! I appreciate it, anon 27

ReplyDeleteanonymous 28, it was not Euler's, but GauĂź' teacher!

ReplyDeleteI was always really good at math and I credit my father for it. When I was a kid he would drive me to school every morning and play math games with the license plates of cars (you can also do this while walking anywhere).

ReplyDeleteAt first we would each choose a car in traffic and add the numbers on the license plate - the winner would be who had the highest number.

Then when I was fast at it and was able to choose a licence plate that I knew would have a high result, he changed to a different game, in which we would have to remove all the 9s from the total number to see would have the lowest result (I learned that removing 9's from each digit gives you the result - say 18 + 8+1=9 so result is 0).

The 9s game got more complex and we would get the result from one licence plate, add to another plate, and so on, until we reached 0.

I still play the 9s game to this day and try to teach my son.