tag:blogger.com,1999:blog-3722233.post117639472995905572..comments2017-10-20T10:27:19.559-04:00Comments on Computational Complexity: Getting an 8-year old interested in math:Do's and Don'tsLance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.comBlogger31125tag:blogger.com,1999:blog-3722233.post-42375163645204465462017-04-14T11:13:18.182-04:002017-04-14T11:13:18.182-04:00I was always really good at math and I credit my f...I 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).<br />At 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.<br />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). <br />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.<br />I still play the 9s game to this day and try to teach my son. Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1176847693948453322007-04-17T18:08:00.000-04:002007-04-17T18:08:00.000-04:00anonymous 28, it was not Euler's, but Gauß' teache...anonymous 28, it was not Euler's, but Gauß' teacher!Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1176822121238681932007-04-17T11:02:00.000-04:002007-04-17T11:02:00.000-04:00Thanks for the interesting answer! I appreciate i...Thanks for the interesting answer! I appreciate it, anon 27Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1176812906586427242007-04-17T08:28:00.000-04:002007-04-17T08:28:00.000-04:00Dear Anonymous 27:The pattern to 1+2+3+4+5+6+7+8+9...Dear Anonymous 27:<BR/><BR/>The pattern to 1+2+3+4+5+6+7+8+9 is<BR/>(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 <BR/>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 <BR/>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,<BR/>counting, probability... <BR/>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...Milena Mihailnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1176760827791656462007-04-16T18:00:00.000-04:002007-04-16T18:00:00.000-04:00milena, what pattern did they recognize in first g...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?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1176759379435601412007-04-16T17:36:00.000-04:002007-04-16T17:36:00.000-04:00I have many experiences (my son is 8) but I will s...I have many experiences (my son is 8) but I will share this one: <BR/>When 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 !<BR/>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 ? <BR/>Almost all took their pencils meticulously and started doing the addition algorithm ... How sad...Milena Mihailnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1176752701835260872007-04-16T15:45:00.000-04:002007-04-16T15:45:00.000-04:00I just gave my six year old daughter a calculator,...I just gave my six year old daughter a calculator, and she seems to be enjoying it.<BR/><BR/>When 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. <BR/><BR/>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!)<BR/><BR/>Michael MitzenmacherMichael Mitzenmacherhttps://www.blogger.com/profile/06738274256402616703noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1176705248624997372007-04-16T02:34:00.000-04:002007-04-16T02:34:00.000-04:00There is an implicit contrast here between Mathema...There 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 <A HREF="http://www.maa.org/reviews/mathgene.html" REL="nofollow">The Math Gene</A>.) 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.<BR/><BR/>For 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!<BR/><BR/>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.Paul Beamenoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1176686796975017942007-04-15T21:26:00.000-04:002007-04-15T21:26:00.000-04:00I remember this incident about multiplication tabl...I 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".Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1176685623767323182007-04-15T21:07:00.000-04:002007-04-15T21:07:00.000-04:00Another path to math is through engineering. E.g....Another path to math is through engineering. E.g., kids love flying machines -- get him/her <A HREF="http://www.rc-airplane-world.com/toy-rc-helicopters.html" REL="nofollow">toy helicopter</A> or glider, etc.John Sidleshttp://www.mrfm.orgnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1176675655634717562007-04-15T18:20:00.000-04:002007-04-15T18:20:00.000-04:00Getting an 8-year old interested in Math: - Yahtze...Getting an 8-year old interested in Math: <BR/>- Yahtzee<BR/>- Rush hour<BR/>- Set<BR/>- Monopoly: which properties are the best deal and how many houses should one buy?<BR/>- Watch: Deal or no dealclaire Kenyon-Mathieunoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1176660352904542462007-04-15T14:05:00.000-04:002007-04-15T14:05:00.000-04:00Either you're not a theory grad student at MIT, or...Either 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.Aaron Sterlingnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1176645592498632362007-04-15T09:59:00.000-04:002007-04-15T09:59:00.000-04:00That's a very interesting post. You guessed his nu...That's a very interesting post. You guessed his number in just 10 tries. That seems very difficult indeed. How did you do it?Theory grad student at MITnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1176609380711861552007-04-14T23:56:00.000-04:002007-04-14T23:56:00.000-04:00Actually, I can explain this in terms we use in ou...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! <BR/><BR/>Reduced in this manner, the game is much simpler than Nim as Bill describes.Ken Reganhttps://www.blogger.com/profile/06096073630081306116noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1176608394212575222007-04-14T23:39:00.000-04:002007-04-14T23:39:00.000-04:00Anonymous15, the point I should have made clearer ...Anonymous15, 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.)<BR/><BR/>If 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!<BR/><BR/>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.Ken Reganhttps://www.blogger.com/profile/06096073630081306116noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1176543251514170762007-04-14T05:34:00.000-04:002007-04-14T05:34:00.000-04:00I'll draft Mr. Jeff Kent first!I'll draft Mr. Jeff Kent first!Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1176528048908777942007-04-14T01:20:00.000-04:002007-04-14T01:20:00.000-04:00Ken, can you explain the question to those of us t...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?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1176520144926643602007-04-13T23:09:00.000-04:002007-04-13T23:09:00.000-04:00My son at that age (well, 9 and 10)and I started p...My son at that age (well, 9 and 10)and I started playing "20 questions" but with baseball players.<BR/><BR/>The 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...<BR/><BR/>...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?:<BR/><BR/>LF Barry Bonds 55 homers, or Manny Ramirez 50 homers<BR/>CF Sammy Sosa 45 homers, or Carlos Beltran 40 homers<BR/>RF Ken Griffey Jr. 40 or Garry Sheffield 42<BR/>3B A-Rod 45 or Aramis Ramirez 38<BR/>SS Nomar Garciaparra 30 or Miguel Tejada 35<BR/>2B Jeff Kent 28 or Craig Biggio 16<BR/>1B Albert Pujols 48 or Derrek Lee 45<BR/>C Mike Piazza 30 or Ivan Rodriguez 25<BR/><BR/>If you're not starry-eyed at the big totals, can you make the winning first pick?Ken Reganhttps://www.blogger.com/profile/06096073630081306116noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1176508757502483592007-04-13T19:59:00.000-04:002007-04-13T19:59:00.000-04:00'here are other ways to write "\geq" with your asc...'here are other ways to write "\geq" with your ascii keyboard, like ">=", you know'<BR/><BR/>why would you do such a thing?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1176483998438130982007-04-13T13:06:00.000-04:002007-04-13T13:06:00.000-04:00theory grad student wrote:... To express "Is x \ge...<B><I>theory</I></B> grad student wrote:<BR/><BR/><I>... To express "Is x \geq 56?", did you say: ...</I><BR/><BR/>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 -- $\smiley$ ?]Sivanoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1176450570001849482007-04-13T03:49:00.000-04:002007-04-13T03:49:00.000-04:00"His first three questions were as follows."Mmmm....."His first <B>three</B> questions were as follows."<BR/><BR/>Mmmm...<BR/><BR/>Anyway, age does not seem to be the deciding factor:<BR/><BR/><A HREF="http://www.maths.manchester.ac.uk/~avb/micromathematics/2007/01/vladimir-radzivilovsky-mathemaics-for.html" REL="nofollow">Mathemaics for very young children</A>Stefanhttp://www.yellowhead.nl/noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1176441849497517142007-04-13T01:24:00.000-04:002007-04-13T01:24:00.000-04:00If you're gonna try and teach math to an 8 year ol...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 :).<BR/><BR/>Nevertheless, I'll try this with my 7 y.o nephew. I'm sure he'll get it right.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1176429370168648302007-04-12T21:56:00.000-04:002007-04-12T21:56:00.000-04:00Binary search is extremely intuitive. Knowing abou...Binary 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). <BR/><BR/>Also, 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).Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1176425139345024242007-04-12T20:45:00.000-04:002007-04-12T20:45:00.000-04:00As a kid (older than 8, probably like 10 or 11) I ...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.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1176422714708409432007-04-12T20:05:00.000-04:002007-04-12T20:05:00.000-04:00That's quite a lovely post, Bill! For a second, I ...That's quite a lovely post, Bill! For a second, I had a mental lapse and wondered if Lance had a brother named Bill :)<BR/><BR/>I'm curious how you distinguished between inequality and strict inequality. To express "Is x \geq 56?", did you say:<BR/><BR/>- Is x at least 56?<BR/><BR/>- Is x greater than or equal to 56?<BR/><BR/>- Is x bigger than 55?<BR/><BR/>- Is x strictly bigger than 55?<BR/><BR/>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?theory grad studentnoreply@blogger.com