Gifted Issues Discussion homepage Forums Parenting and Advocacy Math acceleration -- age & abstract thinking

I've just found this interesting article, with a review of literature
http://findarticles.com/p/articles/mi_hb6470/is_3_21/ai_n28728468/

Was this tongue in cheek? I think it should have been :-) You must know the famous experiment that (my interpretation!) demolishes Piaget's ideas about when conservation of number develops by showing that children can answer correctly very much earlier if it's a "naughty teddybear" rather than the experimenter who stretches out one of the lines? IOW, it isn't that young children can't conserve - it's that they haven't learned the phenomenon of adults asking Really Silly Questions. (Well, maybe this is a personal interpretation of the experiment.) Still, I bet it's the same here. "Cover" is ambiguous in English. If a child has been drawing on the wallpaper, we do not say that the same area has been covered if the same amount of crayon has been used - we consider the bounding box of what has been drawn.

I asked my DS7, who enjoys and is used to being asked trick questions and articulating his responses to them. I used exactly the wording given above. He said, up to memory, "Well, it depends what you mean. If you just mean literally how much paper is underneath the blocks, then obviously they cover the same area. If you mean there are invisible lines joining the blocks and you're interested in the area of the invisible polygon, then this way [blocks far apart] covers more."

Here is a page that gives a story about barns build on grassland and how much grass will each of two cows have to eat, depending on whether barns are built in a cluster or scattered around. I'd expect this to be a much better test of what children actually understand about area, because the "grass for the cow to eat" manages to be unambiguous about what area we're talking about. AlexsMom, I'd bet a small amount that your DD will demonstrate fine conservation of area in that context. [The page has some bugs in the names of characters, though, so read it through and sort it out in your head before trying it on a child!]

ETA Here is a page that summarises Piaget and objections to his work very briefly. Incidentally DS7 has no trouble answering questions about two-headed green parrots, either :-)

I haven't read the whole article, but fwiw, DS7 (just!) is clearly more than 2 years advanced through Piaget's stages; he (unsurprisingly to me) passes the first handful of tests I can find for being in the formal operational stage, which Piaget reckoned started around age 11, and others have argued begins later or for some people never. Specifically he easily passed the two-headed green parrot problem I mentioned before, and this one:
http://tlccvc.org/piaget2.htm
The EK47 problem here
http://webspace.ship.edu/cgboer/piaget.html
took him a couple of minutes, and listening to him thinking it through without saying anything myself was painful, but he got there in the end! IIRR, this one is much-researched and many adults, including university students, still fail.

In his case, the reason is clear: he got bitten by the formal logic bug more than a year ago and has had lots of practice at thinking in formal logical terms. Most 7yos have simply not had that exposure. (And I dare say many would not have wanted it - but I just don't buy this as being a non-cultural cognitive development issue.)

Regarding "helping", I don't mean that I never answer any questions, or even that I never give any information up front that he might be able to figure out on his own. I just mean that I keep a constant focus on him figuring things out, that I spend a lot of time finding challenges for him that are at just the right bit past his current understanding, and that I get annoyed when his growth opportunities are destroyed, whether they arose with or without me.


It's nice of you to ask. I haven't taught him any formal logic yet, but do plan to sometime (I am currently putting together our first real home-learning curriculum with the wife).

One thing we have constantly done is play a lot of games. Off the top of my head, some games he enjoys or has enjoyed that involve various sorts of problem-solving skills: chess, Carcassonne, Risk, Bandu, Abalone, Othello, card solitaire, card games including Texas Hold 'Em and Hearts, Chinese checkers, Mastermind, flavors of Blokus, etc. (these are just a smattering based on trying to visualize the games section of the front hall closet). He has enjoyed solitaire logic games tremendously too at various times, including Rush Hour, Clever Castle, Hot Spot, Brick by Brick, Shape by Shape, and some other similar ThinkFun-type toys. He really likes the "Logic Links" toy, which makes me want to explore further offerings from that publisher. He has also played various sorts of computer games, some which are conquest-strategy types, some simulations, some essentially board games on the computer, etc.

We also have bought him various odds and ends of "brain games"-type books or card sets, the names of which I don't recall off the top of my head. I recently got him some workbooks with some problem-solving stuff in them, but I don't remember what the specific series is off the top of my head. My plan with those is to rip out the sheets, scan them to PDF, and combine them into fun little work sets. I want to get him the big Sam Loyd game/puzzle book, the "Moscow Puzzles", etc. but haven't yet, and have saved a long list of mathematical-recreations type of books for later. The IXL website has some types of problem-solving exercises on it; I would guess that Singapore Math does too.

We also have lots and lots of puzzles that he works when he feels like it. I have also given him mini-challenges at times-- I think I wrote before about asking him to put together a wooden train track, using all his pieces, in a restricted space and meeting some other criteria.

He has picked up some rudimentary programming skills with Karel and Java, but we haven't done that in a few weeks, which I will rectify soon. He is enjoying his Lego Mindstorms a lot; the first two robots were put together according to plans shipped with the set, but you can play with the programs with the stock robot bodies, and he's now building his own robot design. When he gets more advanced with his programming, I am going to give him lots of old programming chestnut problems to solve. I am hoping to use programming to increase his modular thinking, introduce him to more patterns, algorithms, etc.

Neither do I.

Yeah, I don't really buy the Piaget POV, but it is what the argument for no algebra before 11 is based on. You need to know their reasoning before you can counter it (or not).

Like, this week DD's teacher needed to tell me about hothoused kids levelling out in third grade and their parents being disappointed. I think I acquitted myself well because I know the reasoning behind her thinking and therefore when it is and isn't applicable. If the OP knows they're talking Piaget and does some reading she'll be able to answer them in their own language.

PS: the E47 answer in that page is wrong, too. Their wording is wrong.

I should stop being shocked at how often this gets said to the people who participate here. And it is so frustrating in its lack of logic - I commend your ability to actually reason with the teacher! How did the teacher react to it?

DeHe

No, but I've been out of psych for 20+ years, and wasn't a huge fan of Piaget back then, so I don't particularly care.

Yes, I think that she assumed that I was asking the second question, since it was clear from her expression that she was wondering what sort of gibberish I was spouting.

I made all the right noises. How can you answer that without rocking the boat?

Speaking of tricky math I got this great illusion in an e-mail from my uncle today. I had to google to find a copy online since I can't forward e-mail to the forum. Wish I could. I get some crazy ones.
http://forum.xcitefun.net/is-it-twelve-or-thirteen-t6050.html
Count the people. Give it a minute for the people to move. Count them again.
Here's the solution:
http://www.defectiveyeti.com/archives/001258.html