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

My 8 yr old son is in grade 4 in a combined 4th/5th class, having skipped a grade earlier. The teacher does a great job of ability grouping within the class, and he is with the 5th graders for most subjects including math. This is the first time these classes have been combined, and it's really working out beautifully for our son.

Usually the teacher moves the 5th graders with math averages >95% at the end of the first trimester to the next math book up (algebra 1/2 I think). She is changing the cut off this year to make it higher because she only used to give 1/2 credit back for corrections on homework but this year she's giving full credit. She'll either raise the average necessary or make the cut-off on tests only (where kids can't do corrections).

Our son really wants to be in the group that moves up. His current average is 96%, with a 94% on tests. So we're not sure at this point whether he'll qualify --could go either way. Walking into parent-teacher conferences my husband and I were split on whether or not it is a good idea. I am concerned about workload, but my husband thought he should be able to go for it if he wants to.
In conferences, the teacher expressed a desire not to have him move up even if he does make the cut. Her rationale was that while he'd be fine THIS year, next year (when he'd be in algebra) the concepts start getting very abstract, and since his chronological age will be so young that his brain won't be ready to handle it.

So, two questions I'd love some thoughts on:
1) can parents of kids who've been through this validate her logic? Do young kids do fine accelerated in math until the concepts get abstract, and then hit a wall until their brains mature a bit more?
2) how should we handle this with our son? He wants this very badly (he talked about it again yesterday) and he's done everything right in order to earn it. If he doesn't make the cut off that'd be an easy conversation. But if he does...

Thanks for your help!

Our son really wants to be in the group that moves up.

DS7 is in a pre-algebra class. He has run into some issues with the "abstract" in terms of solving word problems. He doesn't have any trouble with the math itself, but there are times when working word problems that he has a hard time setting up the problem.

I was told that this is because of his age and non developed abstract thinking. I would ask your son to do a few word problems and see if he has an issue with setting up the problem. The most important part of all this is my opinion is your son. If e really wants to do this, and shows you that he has some abstract thinking, then go for it! It seems as if we spend half our time chasing down challenges for our kids, this seems like a great opportunity.

My 8 year old, like yours, is in a 5th grade class for math. It is pretty apparent he is ready for pre-algebra and he already is pretty comfortable with basic algebra equations and word problems- his teachers have not expressed any concerns about him hitting algebra too early (although I think it would end up being a year behind your DS). Anyway, there is an 8 year old in our district currently doing Algebra and he is doing great. Like any developmental milestone some kids will get there before others - I do agree with Shari that is would be good to see where he is with the material - there are assessment devices that measure if a child is ready for the "concept" part of algebra I believe.

I think CTY or EPGY or something like that says that kids should generally not start algebra before age 10. But some kids are just ready.

Kids hit develop abstract reasoning skills at different ages. The age that many children think abstractly may have nothing to do with your son - he may already be there. We radically accelerated in math. Well into college math I can report that for our child there were no downsides at all. There was never a wall or if there was he gingerly hopped right over it and nobody even noticed.

The decision to accelerated should be made on an individual basis looking at your son's abilities and performance. Age should have nothing to do with it.

If he makes the cutoff, he should move up--that's what cutoffs are about!

As for brains and abstract thinking, well, the whole point of "gifted" is that your brain is ahead of where brains normally are at your chronological age. If he's worked through all the stuff leading up to algebra, it's not a huge leap to algebra itself. Algebra is not a foreign language, and kids have already learned the idea of variables along the way, whether they called it that or not. He'll be fine. I have an 8-yr-old doing algebra, too.

I think that sounds like a teacher who doesn't really understand a true gifted child. Based on chronological age? If the child is showing mastery, he's ready to move on. If nothing else, rather than holding him back if he is ready, why not do it on a trial basis and see how it goes. My DD was grade skipped and all of the reserach we did before hand stated it's not uncommon for knowledge gaps to exist but the difference is that a gifted child can learn those things within a short amount of time and it shouldn't be an issue. I think he should go for it, if he qualifies, and if something becomes a challenge, you can work with him on it at home. I wouldn't hold him back because of a WHAT IF. We haven't had any knowledge gaps so far and I think we'll be in a boat again where we'll have to decide if we move forward or what we do next. You can't make this assumption based on ALL children not being able to think abstractly at a certain age, but rather, what is YOUR child ready for.

I agree, and have seen no sign whatever of an "algebra wall" with my DS. The only time I remember him having trouble with setting up problems was when he was working with percentages and the problems involved discounts, tax etc., of which he had no experience IRL - I had to explain what the words meant (and he still made slips, having no feel for them).

My 8 yo finds the abstract thinking easier than the computational stuff.

Thanks to all for the thoughtful replies. What a wonderful resource this discussion board is!

Does anyone know any specifics about the assessment devices that Catalana mentions?

Even though we're talking about a hypothetical risk one year from now and it's not like we're going to plop him into full algebra tomorrow, perhaps using such a tool might give us and the teacher that little bit of extra confidence...

I think it depends on the kid. I know for myself and my 4th grader doing algebra, long written out computations are the bottle neck. Understanding abstract concepts has been no problem at all.

I think there are levels of abstraction. In this case, the first level is, "Can he understand that x is a variable, and what that idea means?" A higher level of abstraction requires solving word problems in algebra, which often aren't straightforward in the way that other math problems are (BTW, I see this as a fault of lower-grade math curricula; there should be a few harder problems for the stronger students).

My then DS9 did a formal algebra class last year (more this year, plus geometry soon). I wouldn't say he struggles -- he worked harder last year, but also says his teacher this year is much better.

Second thing: I think your son's teacher has some flaws in her reasoning. Sure, the vast majority of nine-year-olds (or ten-year-olds, for that matter) can't handle that level of abstraction. But if your son is HG+, he isn't in the vast majority and she shouldn't judge him by the standards she uses for other kids.

If he skipped a grade and is doing well, he obviously has the mental function of the kids a year older than him (at least), so why is she suddenly worried that this isn't the case?

If he wants to do it, and you and your husband are also on board, I'd say let him try. He can always step back if he's miserable or over-challenged. You could even do some after school work with him to introduce him to concepts. I did this with my son starting when he was 8-ish.

Val

I would agree for algebra, geometry, statistics, logic, and probably other stuff we haven't tried, like calculus . I'm saving the Calculus for Young People books for a snow day this winter.

DD is 8 and finishing up 5th grade math supposedly in the next week or so. (She is in school, so I've heard she is most probably moving up in Nov, but don't know exactly what that means.)

I have seen a big jump with her computation speed and organization in the past 6 months. (This is more of a typical development area for her.) Last spring, she just couldn't do long division or multiple digit multiplication in any sort of timely or accurate fashion and we dropped it for a few months. This fall she's doing the same stuff, but also with decimals no problem. Not speed demon fast, but solid. So I do think there is something to the math and brain development thing, but not in the way I hear educators talk about it.

There are some Abstract Reasoning and Algebra readiness test out there - I know dd took some at school - but I don't know exactly what they are called.

I think this is Piaget raising his influential head again. There was a thread on here about his developmental theory, so I did some reading on it.

He claimed there are four stages of thinking,

Sensori-motor
(Birth-2 yrs)

Pre-operational
(2-7 years)


Concrete operational
(7-11 years) Recent research suggests that children in Western cultures tend to achieve conservation of number by age 7, conservation of mass and length by age 7 or 8, and conservation of area by age 8 or 9.

Formal operational
(11 years and up) Can think logically about abstract propositions and test hypotheses systemtically

And the last stage, formal operational is where they're supposed to be able to do entirely abstract things like calculus.

I found a couple of articles talking about precocious math children, and it seems that they might be as much as two years early with entering the concrete operational stage, they can get everything through to conservation of area within months (which puts them 3 or 4 years ahead). Then they spend years at that stage before the next switch to formal operational.

I haven't read up on that yet, but I think it's where the teacher's coming from, and I'm interested to hear if it's true or not. Is this the time to delve into other interesting math?

DD12 is doing geometry. She struggles with the logic parts of the proofs. She is getting an A so I think it is a good challenge. She did not have trouble with the abtract parts of algebra I. It has been a good experience for her to be accelerated and I would do it again. Her school used saxon math and they had tests that decided what levels the student was ready for. I think u can get some of those materials from ebay or other used books on line. sent from my very small phone....

I may be way off on this, but I think the schools (Not All) get the Algebra thing wrong. They give very little time in 1st through 7th grade in problem solving skills. Most of what they teach is operational in nature. So you take a gifted math student who has it easy (Mostly just follow the operations to get the answer). They cruse along with little to no effort. Then Algebra hits and they have to use effort, and try and solve a problem, and they hit a wall. They think I am not smart enough, or I am not good at math. When it was the schools who never taught them how to problem solve, or how to work on a problem and give effort towards a math problem or puzzle. I hear of this Algebra wall, and what I see it is self created by a short sighted belief in operation over computation. That�s my 2 cents it, and I've been paid well.

Cookiemom,

One more thought:

Have you read Developing Math Talent? It is a book that comes up on this site frequently. It might have some ideas for you.

Chrys

I don't agree with OP's teacher. Gifted kids are all over the place, and as the comments on this page show, not all gifted kids even walk the same path, let alone with NT kids. I would try to informally assess him at home and if the results are good, letting the teacher in on them to convince her. Collinsmum posted an excellent readiness test from the art of problem solving website. Her post, along with the readiness test website is here:

http://giftedissues.davidsongifted....ion_on_demonstrating_gift.html#Post87936


Here's our experience:
DS7 is much better at problem solving than at arithmatic. In his case, he needs practice and drill for arithmatic skills, but he hates it to the core. So that's why he finshed the entire Primary Grade Challenge book by Ed Zacarro last year when he was 6 - every question provides a different take (luckily he has zero problems with the basics, like multiplication tables). He started on EZ's Real World Algebra before the questions on percentages floored him (yes that's his bugbear too). I'm not a teacher so I never realized children could have problems with the abstract "X".

According to the Gifted Math Specialist from the Ministry, math should be lock step and he should work with number puzzles first. But my son has vision issues and (I now realize) can't see straight lines, so try telling him to do a Magic Square. Apparently, kids who are 6/7yo should not understand probablity, but my son loves!

What he finds challenging at this stage - Singapore Math word problems for Primary 5 and above, so we are leaving them off till he's more develpmentally ready. Instead, he seems to really love the Math Olympiad style questions because they are short, sharp, and usually bring some imagery to him. These are not algebra per se, but they are bursting with abstraction.

So I don't think all young kids are not developmentally ready for abstraction. It's finding the appropriate level to offer them that's the key. Your son's scores are terrific - beginning algebra may not require such a great leap at all for him.

Good luck!

I forgot to mention earlier, you might want to try the ALEKS online math curriculum (www.ALEKS.com) either at home or with the school. DS8 does ALEKS at school during his regular class' math time and in the 5th grade math class that he goes to. It is great because it checks to see what you know and then you just keep moving ahead--you can start anywhere from basic addition and subtraction to graduate level Calculus and stuff. It fills in the gaps as it goes, and when you demonstrate that you know something, it goes on. You can register at home for $20 a month, or if the school has an account or is willing to get one, it's only $40 for a whole year that way.

Thanks again to all for the helpful posts. I don't think we had considered all of our options before this. As a couple of people have pointed out, why worry now about next year? None of us think he'll have any trouble with this year. If it turns out he struggles next year for whatever reason, there are tons of options, including EPGY. We don't have any experience with ALEKS, but our older son did EPGY for a year and it sounds similar.

I can't imagine why we hadn't thought of some of this before. My husband and I talked and agreed we'd reopen the conversation with the teacher. We're pretty sure she'll go along if we feel strongly (she's proven that she really wants what's best for him). IF he makes the cut-off, that is. He has a few more weeks to go.

Thanks thanks thanks for the help.

Whew, yes, does that sound familiar. Add to this the fact that the gifted kids coast through computational math without studying at all, ever, and you end up with students who have no concept of "studying."

I struggled to get Bs in Algebra in school, and at the end of the year, I finally sat down and really studied for the first time the weekend before the final. I got the highest grade in the class on it (I was shocked).

I've noticed that the word problems in math books I've looked at over the years are generally quite trivial and straightforward. I struggled to find books for my kids that have problems requiring thought.

Val

Tallulah I was just talking with the hubby over breakfast this morning about the discussion in that thread about the conservation of area. �The first Singapore math book has it right near the beginning and we're stuck on that page. �It said to place blocks in different patterns to see if the child recognizes equal amounts in different arrangements. �I threw down 2 sets of 5 blocks a week ago and my boy thought the spread out one had more. �Yesterday I threw down 6 pieces of cereal and he still thinks the scattered one's have more. �I said to the hubby that's why we go so slow, when he gets stuck I just ask him again later until he can answer right. �There's a reason they put that there so we shouldn't just skip it. �We'll go for several pages then get stuck again and it could be two weeks or two months before he answers right. �The hubby said, just tell him to count it because he's kind of right in a way. �The scattered one is more, it's more area. �I told him about the water experiment (it's not in the math book, but along the same lines) I know he wouldn't know it's the same if I poured 1/2 c water in a tall glass and 1/2 c water in a short glass because the tall glass would look bigger. �The hubby works in the oilfield and measures pressures and volumes for a living and knows too much about hot rod motors. �Practical daily math applications only. �He explained that the static pressure is more at the bottom of the smaller glass because the wider glass has room to spread out. �So, the hubby's no kind of math genuis but he makes a good point about math having so many applications the answer your looking for may change depending on what you want to use it for. �But you can't quote dh against Piaget and some math teachers and expect to win.

To each her own... But, I can only say for our family the idea that "they" put something in a book so the child should do it makes absolutely no sense. "They" don't know my child. "They" may have not been thinking of an asynchronous learner. "They" may have been assuming that the child would be working with a flexible teacher who would be able to adapt or let go assignments that don't make sense.

Forcing a child to march lock step through a curriculum produces a kid good at worksheets.

Thanks for pointing that out. We use what works until it's not useful anymore or we need something else. I'm never too attached to anything that I can't change my mind within 15 minutes. (cross-eyed emoticon:)
We have so many other things to learn and do I just set that book aside for awhile whenever he's stumped and we pick it up later.
Eta: he's just going through that book slower than some of our other stuff.

Yeah, that's good that you aren't too married to any curriculum. There really are these moments in child development when it is like a light switch. Yesterday they couldn't do it; today they can (or in gifted kid style... today it is too easy). It can be frustrating in the early homeschool years because so much is too easy or too hard... that "just right" category can be hard to find and when you do by tomorrow it is too easy!

La Texican, I would just skip the conservation stuff and have him learn what he is ready to learn. I don't think that conservation of area, as a concept, could possibly be a precursor to so many other math concepts that it can or should hold your son back.

From what I read before, it's just an indicator or milestone if you buy into the Piaget developmental levels-- and so a strict follower of Piaget would probably say that even a math supergenius is perfectly normal if he doesn't get conservation of area at a young age. Do you think someone with Einstein Syndrome would necessarily be able to answer correctly about conservation of area at your son's age?

FWIW I don't think my five year old would possibly have gotten such things right at three, which I ascribe largely to the fact that I hadn't started teaching him about math yet at all, so he knew only a bit of counting and simple math that he'd picked up in passing. I would bet that if you teach him about area and he does some more work with it, conservation will just dawn on him.

I agree with passthepotatoes about doing a curriculum in lockstep. As long as you hit everything he needs to know in time, and he understands fully what he learns, you're doing a great job if you just keep his learning time full of fun and productive.

My goal for my daughter is success in college and post-college. She wants to be a surgeon.

Learning should not always be "fun and productive" for our kids. They will grow up and will need to work according to a boring teacher. I teach in college and I really tire of seeing these kids fail.

They need to work problems that take time - many, many steps and respect the inabilities of some teachers. This is a really tough part of being gifted.

Not that I think they should be in classes they should not be in, but they must learn subjects that they don't care for.

My daughter is now in 8th grade and has never liked Social Studies - but she knows the subject. I teach students who know nothing about social studies. It is so hard to explain the feudal system 10 minutes before an exam.

La Texican - patterns are the most important part of math and you are right about that. Most of upper math is looking for the pattern based on many types of principles. I help college students with their math. If they cannot decipher a pattern, they really never get the entire solution - maybe they can just answer the homework. but the big picture - never.

So many gifted kids are procrastinators (if they ever turn their work). They have been taught to only work the interesting work and carry that idea through their lives. If this is their "rule", they cannot be successful in a job.

I generally believe that parents should make educational choices for their children. Most parents (in the US) believe that a child may choose to take the courses or clubs that the child chooses. A child - no matter how intelligent - does not have the overall picture. They get used to making their choices and are unbearable when they hit puberty and basically think that they should make their own decisions.

I don't agree with your logic, and I certainly don't think that if your idea were valid for some, that it would ever be valid to apply it to a three-year-old. One of your mistakes is assuming that a child can't learn to be thorough in the course of a fun lesson.

I liked this quote from one of the links on the inspiration thread, "Too often we give our children answers to remember rather than problems to solve." � Roger Lewin
But you do have to give enough answers to keep them going. �And you have to give them enough problems to keep them involved. �I want to help give him all the tools, but it's gotta come mostly from what's inside him. �I'll admit it's more like training than teaching. �I've only taught people how to do crafts because I love to do crafts. �I taught them what I know, then they went on to do their own thing with it. �
I think in the case of this workbook page I'll do like my husband suggested and tell ds to count the pieces so it's clear what I'm asking. �I also think I'll hand him a calculator in the grocery store because he wants to learn big numbers. �He wrote down 1000 and told me it was 10,000. �He's not that advanced to add them, but he knows you can add more than 2#'s together. �Boy, was he shocked when he saw me add up a few prices to make a catalog order. �("Mamma! �Equals", he tried to remind me, his eyes wide as he saw me writing in a third addend. �Lol)
I guess I'm trying to guide him from the edge of his proximal development. �I'm afraid if I tell him too many of the answers it will skew what that looks like. �So I give him something and if he doesn't get it we put it up for later. �(overthink things much?).
Maybe I'm gauging it from too close. �Maybe that's supposed to interpret from a broad range birds eye view. �Maybe that's what the teachers are doing; they're trying to expertly judge a child's edge of proximal development from the vantage point of a single subject and a single year. �They need to look from a more wholistic angle.

La Texican, I totally agree. For example, I was pretty upset when I found out that my wife had "helped" my son understand how to do a certain kind of problem (making the largest or smallest positive number using a collection of digits). I felt like she'd robbed him of the chance to figure it out on his own-- after that actually doing the problems is essentially make-work. I was quite despondent over something that probably seemed really trivial at the time!

I also won't let his grandmother do anything learning-wise with him when one of us is not around, because she has a strong tendency to "help" him, even when he doesn't ask for it.

With learning math and problem-solving skills, basically my main approach is to find something that's a little beyond his easy grasp, but which I think he can figure out on his own with a little intense focus. My hope is that this will result in lasting superb self-confidence in addition to increasing his general problem-solving skills, and so far it seems to be going that way.

I dunno what I'd do with the conservation / counting idea. It seems to me that telling him to count up the units is really giving him the answer. I think I would tend to just teach him (or learn him) how to do area, and wait for him to realize on his own that the little regions can all be added up, or something, leading to the general concept of conservation. I think it would happen naturally then, and he'd be allowed to discover it. I just know that after DS5 had learned a fair bit about area, conservation was a no-brainer.

Yeah. �You gotta let it go. �I try not to censor how ds interacts with other people in the world. �Glad you can't hear my brain screeching and grinding as I tell myself that sometimes (over religion).�As you know after they turn five you gotta let them go and trust them with their own mind around other people's questionable teachings. �That's why I just focus on the basic tools so he will hopefully be equipped to take charge of his own education despite all of our adult help. I don't know if this is the right thread to ask but you've mentioned how you teach your son logics and stuff and I'm interested in hearing more of your methods and resources.

This is just my two cents:

For algebra, DC15 took it early and still struggles with some of the abstract thinking. DC20 asked me a lot of questions about that earlier (not his area of strength or interest) and sort of figured it out. I think it really depends on the child.

As for "helping," my father helped me learn a lot of math before I started school ("Daddy, how to I calculate what shake is most likely in Yahtzee?" "If dice had 8 sides, would it be 1/8?"), and I loved math and anticipated the day that I would start school. However, my teachers believed in optimum times to learn things, and I soon started to hate mathematics. I didn't start to enjoy it again until I worked with it on my own after undergraduate studies. I don't think that being early would have been damaging or too difficult; however, learning it too late led me to believe that it wasn't interesting and that I wasn't too good at it...

When DS7 asks for help, I give it to him. I do this because I know my son and know that he has put in the time and effort himself prior to asking. He doesn't like to ask for help, he believes that he should be able to do everything himself.
He is 6 years ahead to grade level in math and I can't even fathom being in his shoes.

So maybe it depends on the child...........

I would answer my son if he asked for help or if he asked a question. �Sheesh, i cant imagine (most) anyone wouldn't. �I would take the time to show him something he wanted to learn (at the moment it's erector sets). �But in preschool level homeschool I feel quite free to put up one of the workbooks until I see him apply a skill related to the problem before we go on in that book instead of just telling him what the book says and moving on quickly. �Since there's no reason to at this time. �

The last time he didn't know the answer and I put the book up it was over bar graphs. �I could have told him "count this and mark this many boxes". �And he could have finished a worksheet and learned bar charts. �Instead I put it up until I thought he could understand better. �After a while we had a reason to mark off a box on the calendar every day for three weeks. �After he saw that one box meant one day I felt comfortable pulling out the book and telling him how to count the stuff and do a bar chart. �I also feel free to explain and teach him stuff. �I really don't have any restrictive philosophy on this. �
I apologize if there was any confusion as to what I meant. �Speaking of confusion. �That was dh's point, that the Piaget's thing was a trick question to ds the way I worded it. �Ds does know more, less, bigger, smaller. �Well, the set did have more space. �Maybe I should ask him which group has more pieces. �That's the thing. �The early Singapore math is big on sets. �So if it says at the bottom of the page this would help I choose to pause there until he gets it. �

Oops and all that mess about cringing over how some people "help" our kids was simply commiseration. Here even strangers in the grocery store feel free, almost privileged and obligated to talk to you if you're walking around with a baby. Which is lovely. But it's a catholic town so sometimes they feel obligated during the conversation to stop and recite a blessing to your baby. Which offends me enough to sympathize with Lucouno feeling quite despondent deeply inside over something so trivial and silly.

Likewise! And while the offensive-to-me comments tended to die down once DD was old enough to understand them, they didn't go away completely. A cashier once told DD-then-6 that she ought to be on "Jon and Kate Plus Eight," for instance.

La Texican, we use Singapore math and skip from chapter to chapter depending on interest. We may not have touched n the arithmetic in a book, but have completed all the geometry and measuring in that book and the next one as well. They also have great activity ideas in the home educator guides.

I'm going to argue with your husband. Both containers and the fluid in them are at room temperature, and we're measuring volume only. He didn't read the question right.

That's a different question. (and an earlier developmental stage according to Piaget)

The way I worded the question (using 2x2 duplo blocks) was "see, this block covers one square of the paper."
I then laid out four blocks tight together.
"this is one way to arrange them"
move the blocks apart
"this is another way. Which arrangement covers more of the paper?"

ETA: oh man, I just reread your post, sorry, I was thinking you were talking about conservation of area, not conservation of number. I also didn't realize he was three. I think he's not ready for math yet.

Yeah that's why we move slowly at a pace he can understand each lesson and not just tell me what I said. � We're working on regular kindy maths too. �He knows which coin is which, not their amounts though. �He can read the hour on an analog clock. �He did tons of pre-k maths already.
Well thanks for helping me with the math, I didn't realize I was looking at it wrong. �I'll look at the page again and show him that. �Somebody don't pay good attention to details (embarrased cheesy smile). I just saw that if you have this many and you arrange them in this pattern or that pattern you still have this many. And. If you have this many and you squish them close togeather or you scatter them far apart, you still have this many. I thought both sentences were the same concept.
I think the hubby was trying to prove to me that the question changes depending on what you're doing with it once you get it off of paper and start using it in the real world. �He was trying to say I should not hold back ds because he didn't answer the question one way when ds' answer could have been correct in a different situation. � And then tried to show me a situation where that answer would be right. �And my answer was like yours that's not the question he was asked to answer.�

should we send your husband and mine off to math geek island together?

I just saw that if you have this many and you arrange them in this pattern or that pattern you still have this many. �And. �If you have this many and you squish them close togeather or you scatter them far apart, you still have this many. �I thought both sentences were the same concept.

Oooh, thanks for the wording! My 7yo does not have conservation of area.

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

Colinsmum I'm still reading that link. I like this:

"�Role of teacher (intellectual midwife)

��������� adapt lessons to suit the needs of the individual child.

��������� be aware of the child�s stage of development (testing).

��������� provide stimulation through a variety of tasks.

��������� produce/provide resources,

��������� produce disequilibrium, i.e. a scenario that is outside the child�s current understanding.� E.g. density.

��������� use concrete examples when describing abstract concepts, e.g. ships floating for density, pumping water around��� houses for flow of current in a circuit."


I'm dubious about the other claims regarding ages and stages (which appeared to be debatable anyway), but I like this outline of what a good teacher's job should look like. Nice!