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August 27, 2003 ARITHMETIC....Here's something that happened to me over a decade ago. It happened so long ago that I've halfway convinced myself it didn't actually happen, but I'm curious to see if it rings any bells with anyone else. I was chatting with the 3-year-old daughter of some friends, and we got to talking about numbers. Could she count? Oh yes: she got all the way up to 20, which I thought was pretty good for a 3-year-old. Could she add? It turned out that she could. 2+2: no problem. 3+5: sure. 8+6: you betcha. This didn't qualify her as the next Isaac Newton, to be sure, but it still seemed startlingly good for such a young girl. Then a couple of years later I was talking to her again. Could she count? Sure, she could still count. How about addition? No dice. Oh, she could do 2+2, but that was about it. On the higher numbers she couldn't do a thing. She was a typical pre-kindergartner. This has puzzled me ever since. Did I imagine it? Or was she genuinely able to do more arithmetic at age 3 than at age 5? Has anyone else ever run into something like this? Posted by Kevin Drum at August 27, 2003 07:26 PM | TrackBackComments
my daughter could recite the alphabet at age one, but cannot do so now with ease -- she just turned three. Why? We drilled the frickin alphabet into her as soon as she could talk, and she memorized it. She really didn't get the concept. Couldn't identify most letters. Had no real idea what the sounds that she recited really meant or related to, but she could do it. Over time, it became less important to us to drill the alphabet into her, and she forgot. She didn't forget the concept of letters -- she actually understands the idea of letters and words much better now, she just hasn't memorized the alphabet. Elvis songs, that's different. For some reason, she's got about 5 of those down by heart (more than me), but the alphabet has gotten shaky. My guess is your friends kid never really got addition -- she just memorized a response, and since people haven't drilled her about it in a while, has forgetten it. Posted by: pj at August 27, 2003 07:32 PM | PERMALINKWhen I was a kid I was a whiz at multiplication tables...could shoot them out as fast as the flash cards could be flipped. Same with division, square and cube roots, etc. But now that I'm 43 sometimes it don't come as fast. Same problem with spelling, although even as a kid I was only slightly better then I am now. Just my .02. Posted by: jdw at August 27, 2003 07:40 PM | PERMALINKI had a similar experience with my daughter. At One and a half she knew the alphabet and could identify them in both large and small case. Now at age four, small case letters and basic memorization have pretty much gone out the window but socialization, imaginative play and asking " Why " questions seem to be her intellectual priorities right now. In her brother's case ( age 2) I'm pleased if he makes his needs known without excessive grunting. I would guess this has a lot to do with physical growth of the brain and the periodic reorganization of neuronal connections. A good book on this topic Kevin if you are interested is _What's going on in There ? _ by Lise Eliot, a Northwestern U. neuroscientist, on pediatric brain development and cognition. Posted by: mark safranski at August 27, 2003 07:42 PM | PERMALINKIn the 1980s there was a fad for giving your kids an academic head start at toddler age through lots of drill-and-practice with flash cards and such. (For all I know it's still going on, but I saw my neighbors doing it at the time.) That may well have led to lots of this conceptually empty parroted-response learning among little kids, who later lost it all. Posted by: Matt McIrvin at August 27, 2003 07:46 PM | PERMALINKMust be a slow news day. Or someone just needed a break from the news. Posted by: Thumb at August 27, 2003 07:58 PM | PERMALINKWell, if she didn't practice, she could easily forget. Personal history: I could do that level of arithmetic, as well as counting out change and so forth, when I was three. I used to do "math in the bath" with my dad sitting on the sink giving me arithmetic questions--I thought it was fun. (It was the early 70s, btw, long before the in-utero flashcard trend.) When I got to school, I could still do it, but I was told repeatedly that of course that was impossible, as I was only almost five. I might well have bought it and temporarily ended up like your neighbor, except my parents escaped me out of standard school (where they don't like square pegs of any sort, especially smart kids) and into a better program. I think there is a lot of BS around about "not pushing kids" that causes people not to help their kids capitalize on early learning for fear of impeding social development or being considered abusive. My bet would be that little girl probably liked discovering her math competence, and would have learned more if she'd been encouraged to do so. Posted by: Charisse at August 27, 2003 07:59 PM | PERMALINKWhen you talked to her at age five: was she drunk? I bet that would have made that whole addition thing a lot harder - especially for such a young drinker. Think back: was she slurring her words and telling you to kiss her ass...? Posted by: Greg at August 27, 2003 08:30 PM | PERMALINKI had a very similar experience in a much shorter time frame in the last few days. On Sunday I subtracted 22 from 35 and got +13. Today, I subtracted 29 from 45, hoping to get that same +13, but somehow kept coming up with 16. Even worse, this 16, was actually -16 because it was for Arnold. I'm real confused, I'd ask the LA Times to explain it, but no doubt they'd include "phantom" flashcards in the explanation which would only confuse me further. Posted by: Mr. Olympia at August 27, 2003 08:38 PM | PERMALINKMost of my knowledge of development comes from developmental linguistics, but a lot of the same stuff probably applies to math. Rote memorization that's drilled in very young and goes unapplied is largely lost. Development at that age is focused on exploration and imitation (which includes memorization), but very few three-year-olds are going to 'get' addition in the sense that they understand the abstraction. My advice to parents is always to relax. I know parents who try to drill in the alphabet and color names and counting and so on for an hour a day. Meanwhile, if they just wait 6 months, the kid will pick it up on his/her own without the parents even noticing. Posted by: sidereal at August 27, 2003 08:42 PM | PERMALINKYou can be drilled at something but you won't retain it if you don't like it. For instance, my parents had me memorizing short books when I was very young and I could read by age 3 and have loved reading ever since. On the other hand, throughtout high school (I went to a magnet school for math) I took tons of math all the way up to Beginning Differential Equations. Although I was good at it, I never really liked it all that much though (except for maybe voting theory). Now afte ra year of not having any math in college and not wanting to, I'm losing my knowlegbe of higher level math quite quickly. Posted by: Lavoisier1794 at August 27, 2003 09:01 PM | PERMALINKSometimes new learning in children can interfere with old knowledge. The classic example linguists discuss has to do with regular versus irregular verbs. It's common for a toddler who correctly says "ate" to suddenly start saying "eated", about the time she first figures out that there's a rule about the past tense and she doesn't simply have to memorize each word separately. Another thing that happens goes back to Socrates, who famously said that reading was bad because it would interfere with memory. My kindergartener has an astounding (compared with me) memory; she knows about a hundred children's books almost by heart if she sees the pictures as cues. When she learns to read in another year (she can already sound out words), all that is going away, because she won't need it any more. So, it's possible that if your little friend is starting to get what is really behind the concept of addition, it will temporarily interfere with what someone taught her to memorize when she didn't really get the concept. Eventually this stuff all gets integrated. Posted by: Joe Buck at August 27, 2003 09:02 PM | PERMALINKIt's hard to know what was going on without knowing more about her situation. But there is a real phenomenon that school, and parents who treat their children as if they're in school, teaches children not to think for themselves. This could have been what happened. Young children are very perceptive and creative in coming up with intuitive reasoning to answer mathematical questions, but U.S. schools usually stifle the spontaneous reasoning in favor of mimicing set pattern. I've read education articles documenting experiments in which kids are interviewed in successive grades --- and they lose the ability say to do 2-digit subtraction precisely at the point when it is taught in school. They're told that their spontaneous reasoning was wrong (even though it gave the right answer, and was logically correct). Kids are a lot smarter than the curriculum. But as they're forcefed banalities in the very common didactic manner used by american parents and teachers, they lose interest and also lose their grip on the material. Specifically, for adding 8 + 6 --- one common intuitive strategy is either to use fingers, or to use some mental substitute for fingers, like rhythm (you can think of 6 as d-d-d--d-d-d or as d-d--d-d--d-d to efficiently keep track of the count from 8 to 14.) Most people continue doing something like this when they're figuring out dates. Today is Wednesday the 27th --- what date will Friday be? Anyway, this and other strategies are discouraged in school Posted by: Bill Thurston at August 27, 2003 09:04 PM | PERMALINKMatt McIrvin writes: "In the 1980s there was a fad for giving your kids an academic head start at toddler age through lots of drill-and-practice with flash cards and such. (For all I know it's still going on, but I saw my neighbors doing it at the time.) " It's probably even worse now. Posted by: Jon H at August 27, 2003 09:08 PM | PERMALINKI believe in mental "use it or lose it". When I was a cashier, I could make change for any amount from any other random amount, all in my head, instantly. Now? I regularly embarrass myself. Um... Twenty five and thirteen ... minus... I totally believe that a 3-year old, with little else to entertain herself and her parents, can get pretty good at arithmetic. And lose that skill when your day consists of driving, talking to people, paying with an ATM card, and never having to do any kind of mental calculation. Hell, there are still kindergarten-age things I never mastered the first time. "Quick! Which is your *right* hand?" Um... Posted by: squiddy at August 27, 2003 09:11 PM | PERMALINKI guess I should have pointed out that I'm pretty sure her parents weren't drilling her in anything at either age. In fact, I think I even asked at the time and her father was surprised that she could add numbers. Her arithmetic at age 3 seemed to be entirely intuitive. (At least that's what I remember. But since I barely trust my memory of the whole incident to begin with, I suppose you should take this with a grain of salt.) She's starting her senior year in high school next week. Bright kid, but not Einstein. Posted by: Kevin Drum at August 27, 2003 09:23 PM | PERMALINKI think verbal and arithmetical concepts cloud early simplicity. I used to play a board game with my two daughters, a sort of "great race" combined with "memory" -- turning cards over and seeing a picture of an animal. The youngest was the best and at times could race around half the board. She had no idea what the animals were. I think that for the oldest and me, the concepts of the animals, the idea of using numbers and colors to identify the overturned cards... all that got in the way of the visual remembering. Posted by: Eric M at August 27, 2003 09:24 PM | PERMALINKAnother thing to remember is that even grown-ups will forget how to do math when they don't do any. I've forgotten a lot of calculus since grad school, which was the last time I used it. Sure, I know I'll pick it back up again fairly quickly if I ever need it, but right now I can't tell you whether 1/x is the derivative of ln(x) or vice versa (off to figure some limits.) Posted by: Phillip J. Birmingham at August 27, 2003 09:43 PM | PERMALINKShe has since realized that doing addition in your head is not cool. If the president can't do it, why should she? Posted by: Hack at August 27, 2003 09:51 PM | PERMALINKVery interesting and non-partisian comments. Lavoisier1794, I crushed Diffy-Q and O-Chem when I was 17, but 23 years later I don't remember any Diffy-Q (I still retain a bit of O-chem). My present job does not require it. Kevin, although Einstein was never a dunce in grade school, not even Einstein was Einstein during high school. If anything, the anti-semitism of the Germans made him disrespect authority- a useful trait for developing new and original concepts. Regarding young kids, even if they are just memorizing addition and multiplication tables, the mental training and discipline puts them several steps ahead for problem solving required for academia and white collared professions. Posted by: gyp at August 27, 2003 09:56 PM | PERMALINKDid her parents hire Donald Luskin as a tutor? Posted by: ItAintEazy at August 27, 2003 11:45 PM | PERMALINKIt probably is a smattering of rote memory vs structural knowledge, old memory vs new, used infortion vs unused. Another effect may be peer or societal pressures. I was reminded recently of how many times as a toy store clerk I'd be asked if something was more appropriate for a boy or girl... And at that age, there really is no difference... But people and society does enforce and reward gender-appropriate behaviors. And one of our societal stereotypes is that girls aren't good with, or don't need to know, math. This may have encouraged her to discard, hide, or just not use the rote learning she had aquired earlier. I hate gender stereotyping, but it does occur. Posted by: Crissa at August 27, 2003 11:57 PM | PERMALINKI suspect that the phenomenon Kevin describes is real and wasn't the result of drill. Consider autistic savants and their calendrical skills (Steven Jay Gould had one of his own). They weren't taught how to do this. Somehow these otherwise defective kids figured it out on their own. Some news source (NYTimes likeliest) recently reported that turning off some brain activity could actually enhance drawing skills, implying that some of the astonishing performance of autists might result from -- what? -- a defect? -- not going on to the socializing phase? At an early age kids can learn multiple languages almost effortlessly. We don't exploit this, and we should. What Kevin reports suggests that we ought to explore exploiting this mathematical potential as well. Posted by: bad Jim at August 28, 2003 12:14 AM | PERMALINKMy late brother was a bit autistic, but functioned at almost normal levels a lot of the time. He was very good at remembering stuff about his favorite basketball team. He could remember a lot of stats for a given game given the date it was played on. Anyway, between the ages of 8 and 10(I'm a bit vague on when this took place) my dad taught him how to backdate a couple of weeks in his head without looking at a calender. He was able to take this technique and use it to give you the day of the week a given date fell on. He could never explain how he did it, but the process that took place in his head did work. Given some of the other things we know about the human mind, the question isn't how he did it, but why we can't learn to do it. Posted by: Dark Avenger at August 28, 2003 12:58 AM | PERMALINKalthough Einstein was never a dunce in grade school, not even Einstein was Einstein during high school. Can't say anything about precocious arithmetic, since there are no pre-schoolers in the neighborhood, I don't have kids, and my only nephew is only 8 days old (if he could add 8+6 and tell me the answer, I'd be flabbergasted). I could tell time on an analog clock when I was four and never lost that skill -- my parents loved to show that one off -- but learning to tie my shoes took considerably longer. The notion of losing some innate skills as new ones are developed, pathways rearranging and what-not, makes sense to me. So does the "use it or lose it" phenomenon. My grandparents lived with us when I was little and I could babble on in their native language when I was four or five, but my vocabulary is probably in the low three digits now and I know next to no grammar. Posted by: Keith at August 28, 2003 04:50 AM | PERMALINKIn the books for children's development for the first 2-3 years they say that the kids develop (in talking, walking, etc) uneven, with big leaps in 1-2 weeks and after that stops and even some regress is possible. The story for Robinson Crusoe is a fraud: It was inspired from the true story of Alexander Sterlick (or Sterling??). BUT: Crusoe lived 28 years on the island without any mental regress. Alexander Sterlick lived for 4 years and had almost completely forgoten to speak English. Posted by: GB at August 28, 2003 07:11 AM | PERMALINKMaybe I was wrong for "Crusoe", there are different versions: "So, Alexander Selkirk was finally rescued. He had not spoken in so long, he had forgotten some vocabulary and the crew of ‘Duke’ had a difficult time understanding him. But his appearance and speech became of no consequence when his outstanding seamanship..." "Alexander Selkirk never adjusted to civilization."
"On meeting his rescuers, Selkirk was so overjoyed that for a while he was unable to speak sensibly. " Posted by: gb at August 28, 2003 07:37 AM | PERMALINKI think there is a phase where kids are great sponges, absorbing (memorizing) phrases and such, but not really getting the concepts. My daughter learned some frat songs from me at age 3 (even though they were cleaned up, my wife was furious!). She could also recite the alphabet, and 'spell' very difficult words such as 'Mississippi', 'antidisestablishmentarianism' and 'supercalligragilisticexcpiallidocious' (sp?). Alas, it was all rote learning, and later lost. That is one explanation for early ability. Also, I remember hearing of a horse once (Clever Hans) that could do simple arithmetic. Posted by: Tripp at August 28, 2003 08:20 AM | PERMALINKI attended a Montessori school from age 2-4...one of the instructors was Spanish and taught me many Spanish words. I even remember being able to read simple sentences. By the time I reached first, second grade I couldn't remember any of it. Posted by: theperegrine at August 28, 2003 09:25 AM | PERMALINKtangent: Einstein may have had Asperger's Syndrome. It's a subtle form of autism, makes people very introspective and analytical, but relate poorly in social contexts. http://news.bbc.co.uk/2/hi/health/2988647.stm Posted by: squiddy at August 28, 2003 09:27 AM | PERMALINKHmmmph, in my day we didn't need no fancy pants PC labels such as 'Asperger's Syndrome.' 'Nerd' suited us just fine! Posted by: Tripp at August 28, 2003 09:33 AM | PERMALINKEinstein wasn't Einstein in High School (in fact he may never have
been Einstein), but he was Albert Einstein. Differently but related, a
friend of mine once walked up to Bob Dylan in an LA music store and said
"You're Bob Dylan", and Dylan answered "No, I'm not." And of course he
isn't, he's Robert Zimmerman. Bob Dylan is an entity which releases
albums and does tours. Would everyone please stop diagnosing every great genius of the past 500 years with autism, or Aspberger's, or ADD, or the hip neurological syndrome of the moment? Brilliant people are distractable in class because they're bored, and socially ill-at-ease because they never fit in at grade school. There's a lot of grey area between "eccentric brainy type" and "Aspberger's". Besides, doesn't that BBC article note an objection to the Einstein-Aspberger's theory, namely that Einstein had a great sense of humor? Posted by: Maureen at August 28, 2003 10:20 AM | PERMALINKMaureen and Trip: Yup. Anyone reading Dilbert lately? Posted by: squiddy at August 28, 2003 10:56 AM | PERMALINKCouple of things: it's spelled "Asperger's." And it's a myth that people w/Asperger's have no sense of humor (I think a more compelling reason for thinking Einstein didn't have AS is that he was a very late talker: people w/Asperger's typically don't have language delays. That wouldn't rule out high-functioning autism, though). Myth # 2: people w/Asperger's are not necessarily introspective, shy types: they may be pretty social--possibly clueless, but social. As for the example Kevin gave: it doesn't sound like rote learning to me, because it sounded like she was thinking through the question and coming up with answers, not spitting them out from memory. I'm not a fan of armchair diagnoses, but I'm sick unto death of people saying that Asperger's is just a way of saying "nerd." It's a real syndrome, and there's an honest to God autism epidemic going on. Yeah, I'm a parent. A cranky one. Which is why I also didn't like one poster referring to autistic people as "defectives." Posted by: emily at August 28, 2003 01:33 PM | PERMALINKOh yeah--the Clever Hans horse? That wasn't actually counting--IIRC, the animal's owner was using some sort of system to get him to tap his hoof a certain number of times. Or so I remember from Intro to Psych, taken about 20 years ago. Posted by: emily at August 28, 2003 01:36 PM | PERMALINKClever Hans' owner was believed to be sending him unconscious signals. He would tense up when Clever Hans was given a problem to "solve" and would subtly relax when the horse had pawed his hoof the right number of times. C.H. would notice that he had relaxed and stop pawing. Posted by: obeah at August 28, 2003 04:09 PM | PERMALINKWell, let me clarify. There was no indication that Al Einstein, high schooler, would become the greatest physicist of all time, as well as an ardent pacifist (until Hitler came around), and also a foremost Zionist. Did you know he was offered the presidency of Israel? In regards to social interactions, the only people with whom he did not interact well were German nationalists and Nazis. Finally, (and this is bascially a non-sequiter), he loved fine cigars. Posted by: gyp at August 28, 2003 09:48 PM | PERMALINKLOng division, early grade. Did the work extremely fast. So? It turns out that WAS the problem! Speed got in my way when I was challeneged. And, if I couldn't do the math fast, the mistakes compounded. Since I went to college late in life, I had to go back and do the math. In a world where I had written off fractions and percentages ... I had to go back and rebuild this skill. Good teacher pointed out that I was operating this vehicle with much too much speed; and if I only slowed down I'd get answers that matched those in the back of the book. You see, I could take a power number, and use it as a multiplier. I'd be amazed the teachers would find my mistakes, when I couldn't. Till I realized my numbers way way off the scale. Had a similiar problem with the parenthesis. Since I could distribute a negative sign (but not against the left paren, certainly not all the time) ... and sometimes I'd continue distributing this into the next room. Sometimes our eyes don't see the whole picture. 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