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SingaporeWordProblemSampler2 01 Jul 2005 - 03:03 CarolynJohnston Here's another random sampling of word problems from "Primary Mathematics, Challenging Word Problems". A KtmGuest (henceforth known as 'Lone Ranger') left the following useful comment on the SummerProgramUpdate thread: FYI...Singapore Math is organized differently than American elementary math textbooks. The book are arranged in this order 1A, 1B, 2A, 2B...6A,6B. When a student is finished with 6B, that student is ready to begin studying Algebra. Therefore the number on the book does not translate into an American grade level. In fact many people comment that children begin the Singapore program 1 number below their current grade. My child began with level 2B even though she was starting 4th grade.(thanks, kemosabe). So who knows what American grades these problems match up to? Just target the problem that suits your kid, and don't worry about whether they're behind what kids are doing in Singapore. Primary 3: The capacity of a bucket is 9 qt. If 3 qt. 3 c. are added into the bucket, how much more water is needed to make it full? (I like that last problem because it ties in with this recent post.) And here is a rather strange one: Primary 4: 5/9 of a box of chocolates are round, and 2/9 are square. How many more chocolates are round than square? Give your answer as a fraction. Primary 5: Martin and Gary had 80 stickers altogether. After Martin gave away 35 of his stickers and Gary gave away 1/5 of his stickers, they had the same number of stickers left. How many stickers did Martin have at first? Primary 6: Linda and Jane set off from City P to City Q at the same time. When Linda reached City Q, Jane was still 140 km away. 2 hours later, Jane also reached City Q. If Cities P and Q were 630 km apart, at what speed was Linda traveling? comments... HappyBirthdayCatherine 01 Jul 2005 - 04:03 CarolynJohnston comments... MeasurementAdviceFromCarlL 01 Jul 2005 - 14:57 CatherineJohnson Re: Measurement My first year teaching high school freshman (I just finished my 3rd year at a urban neighborhood school) I was completely shocked that none, and I mean none, of the kids could measure using an inches ruler. How can they get out of middle school, or even grade school, not knowing how to measure? I still have no clue. I doubt its the constructivists fault due to their fondess for hands-on, manipulatives, and project, which all lend themselves to measurement. What I have observed:I intend to take this advice. SummerProgramUpdate (measurement skills) EarthboxDay comments... KTMReaderPages 01 Jul 2005 - 16:04 CatherineJohnson reminder: KTM reader pages (created by readers of KTM)and, from Carolyn & Catherine:comments... EarthboxDay 01 Jul 2005 - 19:11 CatherineJohnson Since it's my birthday, and since I get to do what I want on my birthday, more or less, and since I DON'T HAVE A CAT TO BLOG ABOUT, I am choosing to blog about EarthBoxes. EarthBoxes are even better than Russian MathTo prove this to KTM readers, I am going to enlist Christopher in a measuring task. No! Not a task! An investigation! WE ARE GOING TO PERFORM A MEASURING INVESTIGATION! WE ARE GOING TO COLLECT DATA! AND WE ARE GOING TO USE A RULER TO DO IT!OK, now we have resistance and rudeness. 'No!' 'Not today!' 'Then I'm not doing a lesson!' Funny how the kids in the Math TRAILBLAZERS PLAYLETS never seem to react this way when a grownup suggests that they collect data in order to solve a problem. Alright, while the moaning and groaning continues in the background, I will locate:
[pause] Question. Why do we never, ever, ever put rulers away in this house? [pause] Rulers located. Anyone care to lay odds on whether the tape measure is living in its designated spot in the kitchen junk drawer? [pause] Yes. Tape measure in its designated spot, along with, apparently, every other smaller-than-8-inch item we have acquired in the past 12 months or however long it's been since the last time I went on a junk-drawer cleaning jag. Time to start tossing. Now Christopher is eating lunch. At 2:31 pm. So it's looking good for the Bad Mother of the Month Award in July, too! Back shortly. In the meantime, this is an EarthBox.
EarthBox InvestigationChristopher and I used a ruler to measure the basil plant planted in the ground, and a tape measure to measure the basil plant planted in our EarthBox. The two plants came from the same nursery, on the same day, and were the same size when we planted them. The EarthBox is directly next to the patch of earth where the other basil plant is planted, and the two plants get the same amount of sun, rain, etc. The basil plant in the earth is scrawny, not too healthy looking, and stands 10 1/2" tall. The basil plant in the EarthBox is a bush. It is 14 1/2" inches tall, and is so huge and fleshed out that Ed is going to cut it back because he's afraid it's blocking the sun for the green bean plants that are also growing in the same EarthBox. Not that the green bean plants look like they need any help. They're bushes, too. The tomato plants in the tomato EarthBox look like the stalk in Jack and the Beanstalk, and we've got corn stalks barrelling up-up-up out of yet another. I just ordered more EarthBoxes. Here is a web site that tells you how to make a homemade EarthBox. What I want to know now is how to duplicate the EarthBox technology for indoor plants in small pots.updateI was just cruising the EarthBox web site. Here's a line from a satisfied customer:"Quite a new wave of gardening. We are having so much fun with our 'MONSTER' tomato plants.”It's true. Our EarthBox plants look like the kind of thing you see in those Fantastic Island—type movies, where the actors shipwreck on an Island Time Forgot and every living thing they find is 10 times bigger than it's supposed to be. It's only July 1 and I'm already wondering how on earth I'm going to use all the basil I've got. (I'm pretty sure I remember where my gazpacho recipe is, so that's a plus.) Oh wait. Gazpacho takes fresh parsley. Not basil. So I have to find my pizza recipe. It's probably in the same place we left the rulers. Well, thank heavens we didn't grow cucumbers. There's another customer quoted on the site shown standing on a ladder next to a cucumber plant that's about 8 feet tall, maybe taller. He says that from June 20 to August 18 he picked 105 cucumbers. The biggest one was 16" long. That's just gross. update July 24, 2005Green bean plants kaput, basil plants victorious. Green beans & basil don't mix?SummerProgramUpdate (measurement skills) MeasurementAdviceFromCarlL EarthBox investigation with Christopher adjustable reservoir for indoor plants EarthBox reminder self-watering pots and planters from Denmark hydroculture sub-irrigation comments... NoProgramForAndrewAsYet 01 Jul 2005 - 21:15 CatherineJohnson This is way off-topic, but I'm posting it anyway. Andrew, who has autism, is entitled by federal law to a summer program. The summer programs began today. Andrew did not attend, because our district failed to enroll him in a summer program, and failed to provide for him in any other way. Nor does he have a speech therapist, to whose services he is also legally entitled. So Irvington is now officially in noncompliance with the law. This in a district that spends $18,000 per student. The new interim director of special ed (he'll be our third in 4 years) sounds great, and is trying to get the situation fixed. But he just started his job today, and there are other parents in our boat. He's got his hands full. comments... HayBalerProblemFromIMP 01 Jul 2005 - 21:30 CatherineJohnson I've just this moment noticed the 'hay baler problem' Barry posted on his page. Here's a problem that appears in IMP for 9th grade It is known as the "Haybaler Problem" “You have five bales of hay. For some reason, instead of being weighed individually, they were weighed in all possible combinations of two: bales 1 and 2, bales 1 and 3, bales 1 and 4, bales 1 and 5, bales 2 and 3, bales 2 and 4 and so on. The weights of each of these combinations were written down and arranged in numerical order, without keeping track of which weight matched which pair of bales. The weights in kilograms were 80, 82, 83, 84, 85, 86, 87, 88, 90 and 91. Find out how much each bale weighs. In particular, you should determine if there is more than one possible set of weights, and explain how you know.” David Klein, a mathematics professor at California State University at Northridge comments on the problem. “The process of solving this problem made me resentful of the stupidity and pointlessness of it. There is nothing ‘real world’ about it. It is completely inappropriate for kids who likely have not been taught how to solve simultaneous linear equations, or exposed at most to two equations in two unknowns. If I had been given such problems at that age, I think that I would have hated math.” Consistent with much of the philosophy of “real life math”, the goal of the exercise is to explore strategies and to be able to write about it. This is made apparent by the “student guide” that accompanies the problem. It is essentially a scoring sheet, containing categories, with points awarded for each, such as “Restate the problem in your own words” (4 points); describe all the methods you tried before reaching your solution(s) (4 points); describe the process that lead to your solution(s) (4 points); describe all assistance provided and how it helped you (2 points); state the solution (2 points); describe why your solution(s) is correct, include all supporting data (6 points). Out of a total of 50 points, only 2 are given for the solution. In fact more points are given for describing why the solution is correct. It's unbelievable. You really do have to see this stuff in the flesh to know what our kids are up against. On the other hand, I'd bet money there are no more than 5 teachers on the planet willing to use the IMP grading rubric, (pdf file) if that. I've been a teacher myself; I've used grading rubrics (teaching freshman rhetoric at the University of Iowa). The IMP rubric asks the teacher to use 18 separate categories for a total of 50 points to score one problem. Unless the NCTM is now allowed to send federal auditors into the classroom (which is pretty much what we've got in Manhattan at this point) that's not going to happen. Students can earn a grand total of 2 points, out of 50, for the right answer. No teacher's going to go along with that. updateCheck out the IMP web site.
"IMP™ Receives Award from the U.S. Department of Education"Here's the Mathematically Correct review. (pdf file) comments... CarolynMorganUseTheBlackboard 01 Jul 2005 - 22:07 CatherineJohnson Just in case I was wondering why on earth I am suddenly writing a MATH BLIKI, now I have my answer. I've just this moment ordered a One Minute Reader recommended by Anne Dwyer (no time to explain at the moment, but the One Minute Readers jibe perfectly with other research I've been relying on...) And here is this from Carolyn Morgan, in the SlideRules comments thread: Catherine asks, "Are you saying that you prefer the blackboard to pencil and paper?" For one-on-one tutoring, "Yes, yes, yes!!" For group review and drill, "Yes!" First, the tutoring: I've noticed that students do much better, learn much faster, seem to gain understanding much quicker. I never really understood why -- it worked and I kept doing it. Then, our Learning Center Director tells us that new studies and new research show that every time the hand (or foot for that matter) crosses the midline of the body, something important happens in the brain. I need to run up to the school or talk to her to get information to explain this properly. But I'm going to take a stab at it and ask that you give me a chance to reference it and get back to you. Apparently, the brain cells really fire away and brain activity picks up every time the hand crosses the midline. There are special drills that our L.C. teachers have their students do just to be sure that the hand crosses that imaginary center line of the body. The brain becomes actively involved as the student is working. Perhaps this is why board work helps my students so much with concepts they've covered before but have never grasped or been able to reason through. Now, for the group review: I've seen "working at the board" just do wonders to help students nail down procedures or recall. (Those not at the board are working in a spiral notebook at their desks on the same problems.) And sometimes students who have lost their way through a multistep problem can see the missing steps in someone's board work, and it helps them recall. Group board work also helps me "see" 3-4 students at a time and I can then zero in on areas where students are still struggling. It helps me "assess" students' needs (assess is a big word right now in education) both as a class and as individuals. So, "yes" there are times that I definitely prefer board work to paper and pencil. Students love it and beg to get to do it. If they're excited about doing it, all the better.Like Carolyn, I still don't understand the whole 'crossing midline' thing, but I know it's important. It comes up in virtually every CSE meeting I attend, and my own kids have a very hard problem doing it (including, I think, Christopher). During our stint in vision therapy, IIRC, I think we found that Jimmy, Christopher & I all found crossing midline challenging under certain conditions. I'm going to start using the blackboard. We have one in the kitchen. comments... MathAndLanguage 02 Jul 2005 - 04:14 CarolynJohnston I was talking to a friend of mine the other day (okay, okay, it was Catherine). She had been to a party at the French Embassy in Washington, and got talking with a French gentleman there about mathematics. "Mathematics is a language," he told her. "It is?" she said. "Furthermore, it is a dead language," he went on (I bet he says this sort of thing to all the women he meets at parties). Later, Catherine asked me if I think that mathematics is a language. I've been hearing this sort of thing for years, and had never really thought too deeply about it. I don't even know exactly what qualifies something to be a language. But Catherine asked me, and Catherine doesn't ask questions lightly, so I gave it some thought. What is a language, exactly? I, for one, certainly don't know. But whatever a language is, I thought, it ought to be able to stand on its own. The jargon of a specialized field shouldn't count as a language by itself; it's just jargon. If you take away the English or German or French or Chinese words that support the jargon, the jargon doesn't stand on its own. And that's how it is with mathematics. You can write out some simple proofs, of course, without using any words at all. But you can only write out the simplest arguments that way, basically those that follow directly from manipulating expressions and equations. I think a language should be able to support all sorts of complicated ideas from all walks of life; if mathematics is a language, then it's a pretty limited one. "No", I said, "I don't think it is." When I mentioned this later to Bernie, he pointed this article out to me. These research results suggest that mathematics and language are pretty much independent functions, as far as our brain functioning is concerned. I would guess that the intersection of language and math occurs at word problems. Word problems are very hard, perhaps because we do have to integrate totally different functions in our brains in order to understand them. But turning word problems into algebraic expressions isn't translation; it's distillation. A lot of the meaning in the original word problem is left behind in the process; the names of the kids who exchanged marbles, the fact that it was marbles that were exchanged, and so forth. You can't work backward from the algebraic expression and uniquely reconstruct the word problem as it was originally. So I think Catherine's French acquaintance was wrong. If anyone knows any more on the topic of what language is than I do, and if mathematics actually qualifies as a language, I'd like to hear about it. What Counts: How Every Brain is Hardwired for Math, by Brian Butterworth The Number Sense: How the Mind Creates Mathematics by Stanislas Dehaene Children's Mathematical Development: Research and Practical Applications by David C. Geary (fyi: It is possible to buy Geary's book for far less than the $124 Amazon wants for it, or the $55 I paid for a used & extensively highlighted copy...) Carolyn on math and language 7-2-05 Carolyn on math and language again 7-3-05 "the language of numbers is not language" 7-3-05 comments... CompareAndContrastPart7 02 Jul 2005 - 14:32 CatherineJohnson caveatThere are lies, damned lies, and statistics....so perhaps it's impossible to say, precisely, what international comparisons on mathematics examinations mean. I don't know. Nevertheless, care & thought have gone into testing equivalent populations, & everyone takes the same test. Take one look at the problems 6th grade Singaporean or Russian kids are doing, and you don't need advanced statistical theory to tell you who's ahead.US world rankingFrom this morning's NYTimes Book Review:China, India, Japan and Europe all churn out more science and engineering degrees than we do. Worse -- and downright embarrassing -- is the state of American education. Globally, our 12th-graders rank only in the 10th percentile in math (that's 10th percentile, not 10th). Our students also rank first in their assessment of their own performance: we're not only poorly prepared, we have delusions of grandeur. item from SAT math testThere are 20 packages of bagels on a shelf in a store and each package contains the same number of bagels. If 3 of these packages contain a total of 18 bagels, how many bagels are there in 7 of these packages? (A) 21 (B) 36 (C) 40 (D) 42 (E) 49 I just asked Christopher (age 10) to do this problem. He did it in his head, while simultaneously plotting out his eBay bid for an Extreme Worldwide Wrestling cage that normally costs $35, and he muffed it the first time. ('Is it 6/7?' 'NO!') When I told him, Christopher, look at the problem, he got it in a couple of seconds. He's 10. This is ridiculous. CompareAndContrast CompareAndContrastPart2 CompareAndContrastPart3 CompareAndContrastPart4 CompareAndContrastPart5 CompareAndContrastPart6 MathInSalinaKansas comments... TwoReaderWikiPagesWithSimilarNames 02 Jul 2005 - 14:49 CatherineJohnson I've just realized we have 2 KTM reader-user wiki pages with similar names. (I'm pretty sure I dropped in an incorrect link for one of them in a comments thread somewhere, so I'll try to find that and fix it). I think Carolyn is still working on fixing the links over on the side (some problem with Hosting Matters??) When that gets resolved, we'll get a good index up. complete list of KTM reader-user wiki pages I think:If I'm missing something, let me know! comments... SybillaBeckmannArticleBarModeling 02 Jul 2005 - 16:26 CatherineJohnson Terrific article on Singapore Math's bar modeling technique: Solving Algebra and Other Story Problems with Simple Diagrams: a Method Demonstrated in Grade 4-6 Texts Used in Singapore (pdf file) by Sybilla Beckmann. Short and readable. I've found that sometimes only the first page of the article opens, so if you have that problem, let me know. I can attach a copy of the text to KTM. comments... MathOlympiadProblem 02 Jul 2005 - 17:24 CatherineJohnson We just did our first problem from Math Olympiad Contest Problems for Elementary and Middle Schools by George Lenchner problem4 min. answerEvery 7 days from 'today' will be Tuesday. Since 98 is a multiple of 7, the 98th day from today will be Tuesday. Then the 100th day will be Thursday.HappyJulyFourth (Moise & Downs) Math Olympiad 2005 comments... PrecalculusAssessmentTestAndCrossingMidline 02 Jul 2005 - 21:42 CatherineJohnson occupational therapy issuesReally interesting comments on the HayBalerProblemFromIMP thread concerning crossing midline (scroll down for related posts), a phrase I've heard dozens of times in CSE meetings over the years. I don't understand 'OT' (occupational therapy) issues at all well, though I can now at least 'see' them. It's possible that The Out of Synch Child: Recognizing and Coping with Sensory Integration Dysfunction by Carol Stock Kranowitz, part of my Great Unread, has useful info on the topic. If anyone has good resources on occupational therapy, let us know.precalculus assessmentAnd Carolyn has found an online precalculus assessment that I'll also put on the recommended reading page. (We'll be figuring out the titles of these pages & what new pages we need shortly.) (pdf file)comments... UKFrameworkForAlgebraPreparation 02 Jul 2005 - 22:36 CatherineJohnson Liping Ma says that math teachers should know where their pupils are headed. What skills will a child most need in the next stage of his education? Since I had no clue, one year ago, what skills a 5th grader needs for algebra in 8th, I found this UK 'Framework for Teaching Mathematics' document, Laying the foundations for algebra, terrifically helpful. After I read it, I spent a LOT of time pushing the distributive property.... which, as my friend Debbie says, 'is one useful property.'
comments... SingaporeWordProblemSampler3 02 Jul 2005 - 23:22 CarolynJohnston Note: solutions to the problems from SingaporeWordProblemSampler2 have been posted here. So here's a whole new set of problems! Primary 3: Margo has 3 times as many pears as apples. If she has 84 pears and apples altogether, how many pears does she have? Primary 4: A cake was cut into 12 equal pieces. Jim ate two pieces and Tom ate four pieces. What fraction of the cake was left? Primary 5: A bag of potatoes weighs 7/8 kg.. A bag of yams weighs 4/5 as much as the bag of potatoes. Find the total weight of the bag of potatoes and the bag of yams. Primary 6: Eric has 75% as much money as Joshua. Carl has 60% as much money as Eric and Joshua have together. If Eric has 36 dollars less than Carl, how much money does Joshua have? I don't know about you all, but I think I perceive these Singapore math problems becoming markedly harder at level 5. comments... MathAndLanguageAgain 03 Jul 2005 - 02:34 CarolynJohnston Now that my opinion of math-as-language is challenged (I say math is not a language, and JdFisher says it is, but neither of us really knows), I've had to go out on the internet to see what kind of stuff people have said about it. One of the first papers I came across was on this very topic: this one by Tony Brown. Brown is perhaps a student of Jacques Derrida, the well-known and recently deceased deconstructionist philosopher. I include a snippet from this paper for your amusement: Whilst Mason's distinction might offer a valuable rhetorical device in initiating or analysing mathematical performance, such a distinction suppresses the historicity endemic in anything commonly recognised as mathematical performance, or even mathematics itself, and thus obscures the values associated with this (cf. Derrida, 1989). In particular, the linguistic forces driving (and being driven by) mathematical constructing get squeezed out of the picture. Mathematical constructing, I would suggest, is always linguistic to a degree, oscillating in a hermeneutic circle, between more or less sturdy linguistic frames.OK, so there you have the philosopher's perspective (in an entirely new language, I think). Here's a column by Edward Willett, and he perfectly expresses where people are coming from when they claim math is a language. In fact, elements of these arts can sometimes be set out in mathematical terms--no one who has listened to Bach can doubt the mathematical underpinnings of music, and Leonardo da Vinci even wrote a mathematical treatise on the depiction of perspective in paintings. The fact that he could do so demonstrates that mathematics is also a language, which, like other languages, uses agreed-upon symbols and grammar to describe objects and relationships. Using nouns (constants), pronouns (variables) and verbs (operations), you construct sentences (equations), which build upon each other to create whole paragraphs and even books.But I'm not quite convinced, though I agree with each of his items in the second paragraph. I've always thought Bach had a style that was reminiscent of math -- but his music itself isn't math, it's music. They're not the same. The presence of nouns and verbs alone doesn't qualify mathematics as a full-blown language, either. One of the most useful items I came across in my search was this article, "Mathematics as a Language". This guy (and I couldn't figure out who the author was -- I think it might have been Alex Bogomolny, but I'm not sure) has a different perspective on the math-as-language argument: When I think of the development of Mathematics over the last 2500 years, I am less surprised that early mathematicians left lasting results than that, given the tools they possessed, they achieved anything at all that could have lived through centuries.The really great thing about his article is a table that he presents about 4 or 5 paragraphs down, comparing the descriptions of the great mathematical results (expressed by their inventors in their native languages, and translated for the purposes of the table) with their expressions in mathematical notation. For example, contrast this statement by Euclid: If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments. (Euclid, Elements, II.4, 300B.C.) with the same thing in mathematical notation: (a + b)^2 = a^2 + b^2 + 2ab. (Note: ^2 means 'squared'). Several other examples are given, and they're impressive: I suggest you have a look. He also concludes that mathematics is a language, but his examples (it seems to me) can actually be used to disprove it! They suggest that mathematics is actually a shorthand because, after all, the ancients were able to express their ideas in their own language. OK, I'm going out on a limb to try to prove a point that a. I'm not sure about myself, and b. I'm not really equipped to prove. I still don't think math is a language, but I think mathematical notation might be a sort of a simple or pidgin language. But the amazing thing about mathematical notation is that, once you've expressed your idea (that is, once you've translated the word problem), you can take off in a completely different direction and solve the equation by manipulating its components. There's really no comparable way to take the building blocks of language and reorganize them until the answer pops out. If mathematical notation is a language, then the addition of that symbol-manipulation capability makes math a lot more than just a language. And now I would like Steven Pinker or Noam Chomsky to wade in and sort this out for us. What Counts: How Every Brain is Hardwired for Math, by Brian Butterworth The Number Sense: How the Mind Creates Mathematics by Stanislas Dehaene Children's Mathematical Development: Research and Practical Applications by David C. Geary (fyi: It is possible to buy Geary's book for far less than the $124 Amazon wants for it, or the $55 I paid for a used & extensively highlighted copy...) Carolyn on math and language 7-2-05 Carolyn on math and language again 7-3-05 "the language of numbers is not language" 7-3-05 comments... WhereWeAreHeaded 03 Jul 2005 - 12:50 CatherineJohnson I mentioned that Liping Ma says teachers should know where students are headed. So, in the spirit of knowing where we are headed, I am posting this:
(Click on image to arrive at the Yellow Pages for Mathematics Multimedia Learning Objects 5/5/04 Perfect Little Programs PLP. These are all short animated programs--funded by OurFriendsAtTheNSF--that illustrate mathematical concepts. They look cool, but require a Flash Player, which I don't seem to have at the moment.) comments... TheLanguageOfNumbersIsNotLanguage 03 Jul 2005 - 15:06 CatherineJohnson I skimmed my 3 books on the neuropsychology of mathematics this morning. It seems there is strong agreement, amongst neuroscientists and cognitive psychologists, that math is not language and language is not math. The title of this post, 'the language of numbers is not language,' comes from Brian Butterworth's book What Counts: How Every Brain is Hardwired for Math. The idea that math is language comes from Jean Piaget. (Surprise!) Noam Chomsky's in there, too. Chomsky believes, or once believed, that 'number was just a special aspect of language.' 'Special aspect' is the critical modifier here. The question of whether math is a language the way English is a language does not appear to have adherents outside ed schools. Neither Piaget nor Chomsky appears to have argued that math is a language per se. Their idea is that math depends on the same core logical-reasoning capacity language depends upon. (As I say, I've been skimming.) talking versus readingAs I was trying to get a quick and dirty read on the issues, I came across an incredibly useful distinction:
off-topic: Of course, this leaves out hyperlexic kids and kids like me. I 'taught myself to read,' which is probably why I was always confused by the reading-isn't-natural idea. Christopher 'taught himself to read,' too. Two weeks after his Kindergarten teacher told us he was at risk for dyslexia, because of his very poor handwriting, he burst into literacy. He just took off. Then there's Andrew, who, back during the days of 9/11, was spelling out words like 'interpol warning' on the floor with his alphabet blocks. Once he spelled 'Osamy' and 'Somaly' on the refrigerator. Somebody should probably come study my kids.... Back on topic, here is a nice summation of the distinction between primary & secondary skills: Even though many of the neurobiological systems that support language also support reading (Luria, 1980), these systems have not evolved to automatically acquire reading skills. estimation versus arithmeticThe same distinction is true of math. Math is not natural. Children don't pick up mathematics the way they pick up walking and talking. Estimating and approximating quantities, on the other hand, are natural. Animals do it, and all humans do it, too. Babies probably do it. But people do not acquire knowledge of algebra in the same way they acquire knowledge of '2 cookies is better than 1 cookie.' Here is Stanislas Dehaene, one of the major researchers in the field:[The] human capacity for arithmetic finds its ultimate roots in a basic cerebral system for perception and mental manipulation of approximate numbers, very ancient in evolution. According to this theory, we share this system with many animal species, and it appears very early in human development, independently of language. Of course, it is a primitive system, capable only of basic computations such as estimation, comparison, addition and subtraction of approximate numbers. On this shared basis, various human cultures invent increasingly elaborate cultural tools such as Arabic symbols, counting routines, algorithms for exact addition, multiplication etc. Thus, the origins of human arithmetic lie in both a universal core system of approximate quantity, and on various cultural tools for exact arithmetic. does brain research tell us that math is a 'special branch' of language?In a word, no. There is now reasonably extensive research on people who have suffered brain injuries that tells us math and language are separate and distinct. We also seem to have a body of brain scan research showing the same thing. This is what's known as converging lines of evidence, and it's important. Researchers have studied people who, because of brain injury, have lost only their ability to do math. Language is intact, memory is intact, logical reasoning is intact. But math is gone. There are also one or two cases of people who have lost everything but math. Here's one:Mr. Bell's language had almost completely disappeared. He was left being able to utter just a few stereotyped phrases, such as 'I don't know' and, curiously, 'Millionaire bub.' His understanding of speech or of written language was almost nonexistent. Nevertheless he was still pretty good at calculation, and could accurately add and subtract . . . He could also select the larger of two-and three-digit numbers, showing that he still understood about numbers as being ordered by size, and the way the Arabic numeral system worked. Millionaire bub. I'm going to remember that. updateI've just read JdFisher's comment in the math and language again thread. I'm pretty sure that the (apparent) fact that math and language are two different things inside the brain does not mean they are necessarily two different things in philosophical or even linguistic terms. But if you're coming at the question from a neuroscientific or cognitive science point of view, math is not a language. (It's not dead, either!)What Counts: How Every Brain is Hardwired for Math, by Brian Butterworth The Number Sense: How the Mind Creates Mathematics by Stanislas Dehaene Children's Mathematical Development: Research and Practical Applications by David C. Geary (fyi: It is possible to buy Geary's book for far less than the $124 Amazon wants for it, or the $55 I paid for a used & extensively highlighted copy...) StevenPinkerOnLearningMath Dehaene on high quality neuro-gear Carolyn on math and language 7-2-05 Carolyn on math and language again 7-3-05 "the language of numbers is not language" 7-3-05 comments... PassportToMathematics 03 Jul 2005 - 18:37 CatherineJohnson Anyone who hasn't already done so should read InterestedTeachers' discussion of Passport to Mathematics at MoreOrLessPaperAndPencil. Incredible. comments... SuchACuteAngle 04 Jul 2005 - 00:51 CatherineJohnson (click on the image) Every time Christopher has to identify an acute angle he squeals, 'Such a cute angle! Such a cute angle!' in his talking-to-the-dog voice. comments... TheQuoteGarden 04 Jul 2005 - 06:27 CarolynJohnston While looking up the genesis of a quote about statistics this evening (AnneDwyer quoted it: "torture numbers, and they'll confess to anything"; the quote was originally by Gregg Easterbrook), I came across some fun pages: The Quote Garden for statistics and mathematics. Here are some of the quotes I especially liked. I don't know what this collection says about me, other than that I'm possibly a very silly person. I notice an interesting thing about the statistics quotes vs. the math quotes: the math quotes tend to be admiring, and the statistics quotes tend to be wry and distrustful. The practice of lying with statistics seems to go back a long way. I've dealt with numbers all my life, of course, and after a while you begin to feel that each number has a personality of its own. A twelve is very different from a thirteen, for example. Twelve is upright, conscientious, intelligent, whereas thirteen is a loner, a shady character who won't think twice about breaking the law to get what he wants. Eleven is tough, an outdoorsman who likes tramping through woods and scaling mountains; ten is rather simpleminded, a bland figure who always does what he's told; nine is deep and mystical, a Buddha of contemplation.... CharlesBabbage comments... HappyJulyFourth 04 Jul 2005 - 14:34 CatherineJohnson notes from Lone Ranger on homeschooling her daughters using Singapore Math: Just a quick note that I didn't know where to put on this forum. I started homeschooling my daughter in August 2004. She had been in public school since kindergarten and was a rising 4th grader when we started homeschooling. She had suffered through 3 years of "Math Their Way" and then 1 year of "Everyday Math" before I woke up to the fact that she was not learning math well. Her third grade test scores showed her to be working at the 50% in math. Well, after one year of homeschooling using only Singapore Math Levels 2B- half of 4A and supplementing with Singapore Math's Intensive Practice her total math score on the Iowa Test of Basic skills is now at the 99%!! More importantly her confidence, fluency, and ability to work through difficult problems have gone through the ceiling as well. Happy 4th of July ![]() We are taking home educating one year at a time. This coming year we will home educate again using Singapore Math. I am quite impressed with the program. At first glance it looks rather simplistic and lacking in review. However, I have found it to be very systematic in its presentation and its ability to build understanding is amazing. This is not your inch deep mile wide program at all. The review is there but usually disguised in word problems. Our school system is in terrible distress and using constuctivist math and science, whole language, and very little basics. The private schools are full and all but one have selected curricula I cannot tolerate. So for now it's home schooling. I'd love to hear what other people are using for high school level math. I keep hearing about the following titles: Jacobs Algebra and Video Text. What are good programs? Lone Ranger I used Singapore math books 2B, 3A, 3B and half of 4A before having my daughter take the ITBS test. She completed the 2B placement exam but took 3 times as much time to complete it as was recommended. I thought better to start her slightly below her level to build confidence, learn the rod diagrams, and build speed and fluency with her facts and basic procedures. We also used Intensive Practice books 2B, 3A, 3B, and part of 4A (not every problem though) I made the decison to use Singapore because through my research 2 titles kept appearing over and over: Saxon and Singapore. Saxon is expensive and did not seem to be a good fit for my youngest daughter. Singapore seemed to be the best one to try first, since I wouldn't be out a lot of money if it flopped! Not very scientific or glamorous but the truth. Once I worked with the program and saw the children's response to it I was sold. I am average in my math ability and studied through Trig in college. I think at first Singapore can be intimidating, but after working with it, it is fairly straightforward. I used the Instructor Guide for 2B and have not really used it since. I try to work out all the rod diagrams, and boy am I getting good at them. Jenny, at the Singapore Forum board, is a great help if I am hopelessly stuck. All problems at this level can be solved without using algebra and Jenny is very helpful for teaching people how to set up the rod diagrams. (singaporemath.com) I also am learning much along with my daughters. I think Saxon is also a great program and a few of my homeschooling friends' kids are doing very well with it. I am going to look into the Russian Math program too. ![]() Rod diagrams are another term for bar models! Honestly, the only thing I did with the Singapore program was to follow it. This is what a day at our kitchen table looked like: First a warm up. At first this consisted of basic facts practice. Usually a worksheet of facts isolated by family (ie: just 9's in multiplication) until enough families were learned to combine them. The text presented them this way as well. Eventually we did our multiplication and division randomly mixed and often multiplication facts presented as missing factors 9 X ___=72. Sometimes the children practiced on a hand held device called "Math Shark" or used flash cards. After the children mastered their multiplication and division facts the warm up was several problems from the series that were difficult for them. These problems came from prior days' instruction and I often changed the story slightly and always changed the numbers. We would repeat "types" of problems each day until these problems became routine and easy to solve. Also, once they learned to compute equivalent fractions and reduce fractions to lowest terms I would have them do a warm up of these types of problems until I saw mastery of the procedure. This part of our lesson took about 5-10 minutes. The second phase of our Kitchen Table Math consisted of 1 or 2 pages of Intensive Practice from a book one level below the text. For example we are working in book 4A but are working in Intensive Practice book 3B. I found this was a great way to provide extra review and also not overdosing on the topic currently being studied in the text. Also parts of IP are quite challenging and having extra skills did not hurt. This part took about 15 minutes. The third part was the actual lesson in the text. The children worked orally and on white boards. They completed most of the practice exercises. Sometimes if I saw they had mastery, they only completed a few. We also completed every word problem using bar modeling if appropriate. This took 10-20 minutes. The final section of our lesson consisted of the children completing the corresponding workbook page(s) independently usually taking 5-20 minutes. I reviewed their work and had the children correct errors immediately. That's it! comments... FourthOfJulyCreek 04 Jul 2005 - 20:28 CatherineJohnson (click on this for mining history of Fourth of July creek) Unfortunately, this shows how far I have to go to understand even the basics of what people like Carolyn do. When I found this gif, which is labeled 'USGS map,' I thought: Hey! This is what Carolyn does! (OK, I do recall that Carolyn doesn't do the mapping. Carolyn designs the software for downloading--or was it uploading?--data from the satellites. I think.) But now, looking at it, I'm not sure. How was this image made? I don't know. These are the moments where you realize that a vague 'conceptual' knowledge of some math-related field doesn't get you too far. God is in the details.HAPPY FOURTHcomments... LoneRangerHomeschoolerReportsIncredibleMathProgress 04 Jul 2005 - 23:18 CatherineJohnson Lone Ranger just left this report on her daughter's progress using Singapore Math: I started homeschooling my daughter in August 2004. She had been in public school since kindergarten and was a rising 4th grader when we started homeschooling. She had suffered through 3 years of "Math Their Way" and then 1 year of "Everyday Math" before I woke up to the fact that she was not learning math well. Her third grade test scores showed her to be working at the 50% in math. Well, after one year of homeschooling using only Singapore Math Levels 2B- half of 4A and supplementing with Singapore Math's Intensive Practice her total math score on the Iowa Test of Basic skills is now at the 99%!! More importantly her confidence, fluency, and ability to work through difficult problems have gone through the ceiling as well. Happy 4th of July - Lone Ranger Congratulations! That is incredible. Your daughter has moved from the 50 percentile to the 99th in 11 months. Incredible. Good work! updateThis should give those of us who aren't working in math-related fields more confidence about using Singapore Math with our kids. It certainly does me-- Comments thread on what 'Lone Ranger' did with her daughter's math education & why.MoreFromLoneRanger comments... MoreFromLoneRanger 05 Jul 2005 - 12:38 CatherineJohnson I wanted to make sure everyone saw this follow-up (I've added bullets & formatting because Jakob Nielsen told me to):
LoneRangerHomeschoolerReportsIncredibleMathProgress comments... PriceComparisonSaxonSingapore 05 Jul 2005 - 15:02 CatherineJohnson fyi Assuming I've done my arithmetic right, Saxon Math is probably either the same price as Singapore Math, or cheaper. This is not to make a case for Saxon over Singapore. I have no idea which curriculum is better, or whether one curriculum works better for some kids and another works better for others. The Singapore curriculum certainly moves much more quickly, and is more demanding by ... 2nd grade? 1st? If I'd had the nerve I would have gone with Singapore. Saxon has worked great for us, so I'm a fan, & plan to remain a fan. But it hasn't bumped Christopher up to the 99th percentile in math skills, that's for sure. price comparison:Saxon Math 6/5 (5th grade)3 books: textbook, answer book, tests and worksheet book$69.50 at Saxon Math web site $51.48 at Homeschool Super Center Singapore Math 4A & 4B (roughly: 3rd or 4th grade): 'small package'
$8.00 4A textbook |