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HowIGotHerePart1 09 Oct 2006 - 01:17 CatherineJohnson For me, Kitchen Table Math—Picnic Table Math, in our case—began last June (2005) when our fourth grader, Christopher, came home with a 39 on his Unit 6 test in SRA Math. A 39. How does a person get a 39 in 4th grade math, I kept asking myself. An 80 or a 70, OK. Or, if you really learned nothing, maybe a 68 or a 66. But 39? I'd never even seen a 39 on a test; it's not even listed as a possibility on any of the grading rubrics, all of which stop at 65, or maybe a 60 at worst. A 39 is off the charts, only in the wrong direction. That’s when I bought a used copy of SRA Math Explorations and Applications, Level 4 and set up shop on our picnic table outside the kitchen. I figured, OK, I’ll teach him the stuff he missed. -- CatherineJohnson - 30 Apr 2005 comments... MathInTheBlood 20 Oct 2005 - 16:04 CarolynJohnston Carolyn's side of the story of this website My husband and I have always worked with our kid on his math homework at home. We're both Ph.D. mathematicians, and he never had much of a chance to be anything other than wonderful at math. Every night he would either do his math in front of us, or we would check his work to make sure that he understood what had been covered. In fourth grade, last year, his school switched from the curriculum they had been using, Saxon Math, to a new math curriculum, Everyday Math. I knew the change was coming -- it was announced the previous year, and copies of the new book were left out for parents to review and comment on (and did I review it? ... actually, I didn't, because I was too introverted to Get Involved). Math, formerly my son's strongest subject, became an everyday struggle for him and for us. Our biggest problem was the frequent appearance of problems involving skills he hadn't been introduced to yet. First it was multidigit multiplication, a topic that practically all kids learn in the fourth grade anyway; but its first appearance was in a problem set that came early in the year, before the topic was taught. I don't think the Everyday Math guys intended the kids to approach those problems with the standard algorithms. The problems were always of the sort that you could hope to figure out with common sense. For example, the first multidigit multiplication problems were of the 51 times 3 sort... if you were a bright fourth grader with an adventurous attitude, and some energy left over from the day, you could hack around for a bit and discover for yourself that you could get the right answer by multiplying 50 by 3, and then adding another 3 to your answer. But then, in the next night's homework, there was 23 times 4 to be similarly discovered. Some night soon, I feared, there would be 324 times 5, and then 324 times 54. He would be like Archimedes, rediscovering math from first principles every night. Enough, I thought, and I taught the multidigit multiplication algorithm on the spot. Later that year, I taught my son long division... and drilled him on it every night for a couple of months, since it was a sticking point for him. When problems such as 4 times 1/2 appeared, I sighed and taught him how to do fraction multiplication calculations. Somewhere during the year, I realized that I was teaching him a lot of basic mathematics, but in a completely reactive way; I was allowing the Everyday Math curriculum to dictate the order and the style in which I taught math. If I had to teach my child math myself, I wanted to be doing it on my own terms, in the manner that I thought was best -- and I was sure, at the time, that I knew what that was. MathInTheBlood ReactiveTeaching NowThatWereBothHere AboutLongDivision StrugglesWithLongDivision ForgivingDivision ForgivingDivisionPart2 TryThisWithForgivingDivision TeacherGuideEverydayMath EverydayMathEpilogue ThirteenQuartersInTerc HowNotToTeachMath WhoSaysLongDivisionIsHard comments... NowThatWereBothHere 07 Jul 2005 - 20:35 CatherineJohnson Carolyn wrote: Somewhere during the year, I realized that I was teaching him a lot of basic mathematics, but in a completely reactive way; I was allowing the Everyday Math curriculum to dictate the order and the style in which I taught math. I like that word reactively. I’m closing in on my 1 Year Anniversary, formally teaching math to Christopher here at home. At some point along the way I had the exact same feeling about the home-tutoring going on around me here in my own town, but I didn’t have the word for it. Now I do. It’s reactive. Reactive teaching. Everyone is scrambling to keep up with the content being taught at school. If a child comes home from school not understanding the distributive property, then mom or dad or Paid Tutor scrambles to explain it in time for the test. If he comes home not remembering how to change a fraction into a decimal (We learned it last year, but I forgot), then mom or dad or Paid Tutor scrambles to explain it again, hoping this time it will stick. There’s no rhyme or reason. MathInTheBlood ReactiveTeaching ThingsWeHaveLearned ImGoingToPlayland -- CatherineJohnson - 01 May 2005 comments... ReactiveTeaching 25 May 2005 - 19:34 CatherineJohnson Looking back, 4th grade in our house was pretty much wall-to-wall reactive teaching. Christopher didn't understand anything he was being taught at school, so my husband would re-teach the material every night at home. Then Christopher would do his homework. That's why he got a 39 on the Unit 6 test. The teacher didn't assign any homework for that Unit, so my husband didn't do any re-teaching at night. When it was time to take the test, Christopher was on his own. MathInTheBlood ReactiveTeaching NowThatWereBothHere comments... StrugglesWithLongDivision 14 Jun 2005 - 01:42 CarolynJohnston I remember very clearly the problems I had with certain topics in mathematics. I remember getting confused on the day that my fourth grade teacher taught us how to multiply two-digit numbers by two-digit numbers (I had spaced off during the critical fifteen minutes when she explained the moves to us -- I was permanently spaced out as a kid, actually). That confusion was with me for a long time. So I thought I had a particular rapport with any kid who was struggling to learn math, having once been a kid who couldn't do math to save her life. My then going on to be a math Ph.D., and a math professor and researcher, made me what I thought was a pretty decent role model for struggling kids. I was pretty good at teaching any topic, in fact, as long as Ben could learn it easily. We hit our first big bottleneck at long division. Multidigit multiplication was actually pretty easy for him; particularly since, in Everyday Math, Ben had learned this slick trick for multiplying multidigit numbers called lattice multiplication and was going to town with it. But long division was a different story. Ben had trouble lining up the columns, remembering to pull down the next digit after every step, and knowing where to finish his calculation and what to do with the remainder. Long after he had demonstrated that he knew what to do at every stage, he still couldn't reliably get the right answer. I couldn't see that anything would help him master long division but long practice. He had learned all the steps and could apply them, but being methodical about it wasn't part of his nature. So, every night for a couple of months, I would give him several long division problems to do; it would always require several revisions before he would be done for the night. I could be what I needed to be -- a brick wall demanding that he apply care to his computations before he could consider himself done. What was doing me no good at all, just then, was my appreciation of the beauty of higher math. The long division algorithm we all learned is actually just a repeated application of the Division Algorithm, which in its naked form, once understood, sounds obvious to the point of stupidity. The repeated application of the simple division algorithm with divisors that are decreasing powers of ten is just a thing of beauty, though, something written in The Celestial Great Book of Math. A lot of good it did us, though, in helping Ben to learn to apply long division. It took him a long time to learn to do that reliably, but we stuck with it until he got it. There is the question of whether we even need to do this -- to torment students by making them practice the tedious long division algorithm -- especially now that computers and calculators are everywhere. It's claimed that such drilling kills the joy of math, and that we can teach children to love math better if we don't force them to do computations. I'm claiming (but not yet from any position of certain knowledge) that we do need to teach computation. I'm going by the fact that, in my association with mathematicians and physicists and engineers and computer scientists and finance people in my schooling and various jobs, I've known many people who could apply the long division algorithm, and some few who could appreciate its beauty; but I've never known a single soul who could appreciate its beauty without being able to apply it. AboutLongDivision MathInTheBlood ForgivingDivision ForgivingDivisionPart2 TryThisWithForgivingDivision TeacherGuideEverydayMath EverydayMathEpilogue ThirteenQuartersInTerc HowNotToTeachMath WhoSaysLongDivisionIsHard comments... HappyMathematicsAwarenessMonth 01 May 2005 - 21:22 CatherineJohnson Belatedly. From the folks at the American Mathematical Society. comments... SwoopAndSwoop 03 Nov 2005 - 18:24 CarolynJohnston This evening, we are working on long division with decimal divisors, and comparing the sizes of two fractions. We are working merely on getting these skills down: nothing too deep. When I first showed Ben the cross-multiplication algorithm for comparing two fractions, I showed him why it works the way it works. "It's easy to compare two fractions when they have the same denominator, right?" I said. "Well, it's easy to get two different fractions to be over the same denominator. Just multiply on each side by 1, written as the other fraction's denominator over itself. Then you notice what you get on the left side is the numerator times the right side's denominator, and vice versa on the other side. All you do is compare those numbers. That's called cross-multiplication because it makes a cross. Now you show me." He tried to follow the steps in my first demonstration, and didn't get it right. "It's like this. The numbers move in an x when you do cross-multiplication, like this. They just go swoop, and swoop, like this": 
And that was it: he got it: those swooping moves with the pencil and the crossing numbers. That's what the standard algorithms are: they are moves that you learn how to make. Those moves get into your fingers, just like learning the piano or the violin or typing, and eventually you can do them completely mindlessly.
But that doesn't mean that nothing is going on in the kid's head. If a kid really has those moves down, it frees his mind to think about doing the next thing, and he becomes more receptive to learning why the moves need to be what they are, because the anxiety of not being able to handle the calculation is gone.
Learning the piano or the violin involves a lot of repetition, while your eyes and your mind and your fingers make the connections that allow you, eventually, to experience the music you're playing on a higher level, without calculating where your fingers need to go next. Math is just like that. Math is something you learn to do, like playing an instrument or riding a bike, not something you learn about remotely, like Magellan's circumnavigation. It has a huge kinesthetic component.
swoop and swoop SlideRules the craft of math Wayne Wickelgren on why math is confusing, & Carolyn on procedural memory KUMON & hands-on math comments... SwoopAndSwoopPart2 09 Oct 2006 - 01:27 CatherineJohnson This is probably the time to mention that I’m re-teaching myself elementary mathematics, start to finish. I’m doing all of the lessons in Saxon Math Homeschool Edition, beginning with book 6/5, which Christopher and I finished a few weeks ago. I’m also (in theory) working my way through the entire Singapore Math series, beginning with 1st grade. UPDATE 10-8-2006: I am not working my way through the entire Singapore Math series. I am working my way through the entire Saxon oeuvre, which is all I can manage at the moment. I am, however, for reasons unknown to me, creating a hand-drawn solution manual for Singapore Math's Challenging Word Problems Book 4. I was always pretty good in math, though I stopped taking it after Algebra II, then hit the wall when I tried to take calculus freshman year in college. I flunked the first test and dropped the course. But up til then I was fine, I liked math, scored well on my SATs, etc. I don't have any math anxiety and I love statistics. I took one statistics course in college. Correlation coefficients, standard deviations, regression analysis: to me, these things sound like the key to palace. So, given my general level of math-friendliness, I didn’t think it would be too hard to teach Christopher the math he'd missed in 4th grade. However, I pretty quickly had the same experience the teacher quoted in the American Institutes for Research report did: “I never realized that I do not understand math until I had to teach mathematics from the Singapore textbooks.” This time around I’m trying to acquire conceptual understanding of elementary mathematics, and hook it up to my procedural understanding. It’s not easy. UPDATE 10-8-2006: Twenty-three lessons into Saxon Algebra 2 the mystery of my Wellesley calculus failure has been solved. Algebra 1 & 2 in my high school in Lincoln, IL correspond to Algebra 1 in Saxon. I went to college thinking I'd taken two years of algebra. I hadn't. I'd only taken one. Apparently Wellesley College wasn't big on placement exams in those days. comments... SwoopAndSwoopPart3 16 May 2005 - 01:16 CatherineJohnson As a child, I was never taught the reason why the cross-multiplying ‘trick’ worked when you're comparing fractions. So when I read Carolyn's explanation (SwoopAndSwoop), I didn’t understand what she was talking about until I wrote out her fractions myself, and put in the missing steps. 
comments... HowIGotHerePart2 09 Oct 2006 - 01:28 CatherineJohnson So there we were, Christopher and I, installed at our picnic table, thrashing our way through SRA Math Unit 6: Fractions and Decimals. Two weeks later, there was blood on the floor. HowIGotHerePart1 comments... BeingYourChildsFrontalLobes 24 May 2006 - 11:56 CatherineJohnson This morning I explained to Christopher that, when the bus is late, this is an opportunity to complete another page in your Megawords spelling book. He wasn't buying it. But that's the beauty of being your child's frontal lobes. They don't have to buy it, they just have to do it. LiveBloggingTheSpellingBee GreatMomentsInWorldHistory SummerSupplementTimePart2 BonusPreTeenPost ILikeMath HowToSpell HowToSpellPart2 TheSaxonMathOfSpelling MoreSpelling ConversationsWithKids  update 5-23-06: more frontal lobes sources: Teenage Brain: a work in progress (NIH) frontal lobes, executive function, & IQ hovering is good (MiddleWeb) being your child's frontal lobes organization is overrated executive function, IQ, & hovering, part 1 the discovery of executive function, part 2 executive function self-test presidents & criminals & the frontal lobes ISIS initiate sustain inhibit shift page splatter page splatter & the frontal lobes Dear Abby Susan on dating Catherine's brain-based dating rule comments... MathInSalinaKansas 02 Jul 2005 - 14:41 CarolynJohnston From a forum I sometimes visit, I followed a link today to an urban legends website with a page on an internet claim about an 8th grade final exam supposedly given in Salina, Kansas, in 1895. Here are a few of the test questions in the arithmetic section:
Arithmetic
(Time, 1.25 hours)
When I looked at the Urban Legends page about this 1895 test I found that, contrary to my expectation, they weren't debunking the claim that it was a genuine final test from 1895. They were taking issue with the claim that it showed that educational standards had fallen since 1895:
What nearly all these pundits fail to grasp is "I can't answer
these questions" is not the same thing as "These questions
demonstrate that students in earlier days were better educated
than today's students." Just about any test looks difficult to
those who haven't recently been steeped in the material it covers.
If a 40-year-old can't score as well on a geography test as a high
school student who just spent several weeks memorizing the names of
all the rivers in South America in preparation for an exam, that
doesn't mean the 40-year-old's education was woefully deficient --
it means he simply didn't retain information for which
he had no use, no matter how thoroughly it was drilled into his
brain through rote memory some twenty-odd years earlier.
Lame, lame, lame. If you can't prove that this is not an authentic graduate exam from 1895, then complaining about it just makes you sound like a whiner (and notice the dig about 'rote memory' -- memorization is in very bad odor these days).
Besides, it's not about us (and what we retained) anymore: it's about our kids. And I am afraid it does imply that we've dumbed down the junior high curriculum. Only a tiny minority of kids graduating 8th grade these days could handle sophisticated word problems like these, even if we gave them the bushel-conversion formulas for free. Apart from the emphasis on farming applications, which is kind of funny and endearing, the application area of problems 6 and 8 (just for an example) is as alive, or more so, in 2005 as it was in 1895, and we simply do not teach it. In the late 1980s, I taught an elective course at LSU on the material covered in these problems. The entering students were completely ignorant of that material, mastery of which I claim is necessary to living adult life competently (and they were very glad to finally learn it, too). Many students who were stronger mathematically, and didn't take that elective math course, are no doubt still ignorant of it, because it is not taught in public schools anymore.
The second thing that leaps out at me is that these are mostly application problems -- word problems -- not problems testing either basic computation or deep understanding of the beauty of mathematics (with the exception of problem 1). It was just assumed that these kids could do the computations necessary to solve these problems, without calculators. What they needed to do was to solve those problems, and get the right answer, and that hasn't changed a bit. And I'll bet there was no partial credit given for having the right idea, either. CompareAndContrast CompareAndContrastPart2 CompareAndContrastPart3 CompareAndContrastPart4 CompareAndContrastPart5 CompareAndContrastPart6 CompareAndContrastPart7 comments... MathInSalinaKansasPart2 09 Oct 2006 - 01:29 CatherineJohnson re: MathInSalinaKansas Wow. I spoke yesterday to a mathematics professor at a university here in New York state. When I asked him what level of mathematical knowledge entering freshmen bring to their course work, he said, "We can't assume that a student knows anything we would want him to know." Specifically, his students can't do algebra. They can't set up a two-variable word problem and solve it. These are college freshmen. Posted on May 07, 2005 @ 11:21 comments... MathInSalinaKansasPart3 20 May 2005 - 04:16 CatherineJohnson re: MathInSalinaKansas Three sample problems from the PRAXIS 1 Content Assessment college students entering the field of education are frequently required to take:
1. Which of the following is equal to a quarter of a million?
The Educational Testing Service (ETS) describes these problems thus:
The Pre-Professional Skills Test in Mathematics measures those
mathematical skills and concepts that an educated adult might
need. It focuses on the key concepts of mathematics and on the
ability to solve problems and to reason in a quantitative context.
Many of the problems require the integration of multiple skills to
achieve a solution.
[snip]
Computation is held to a minimum, and few technical words are used.
Terms such as area, perimeter, ratio, integer, factor, and prime number
are used, because it is assumed that these are commonly encountered
in the mathematics all examinees have studied. Figures are drawn as
accurately as possible and lie in a plane unless otherwise noted.
see also:
MathInSalinaKansasPart2
comments... PracticeAndOverlearningPart1 20 May 2005 - 04:18 CatherineJohnson Carolyn and I have both been using Saxon Math Homeschool Edition with our kids. Here is Saxon's explanation of the curriculum:
Saxon Math . . . systematically distributes instruction and
practice and assessment throughout the academic year
as opposed to concentrating, or massing, the instruction,
practice and assessment of related concepts into a short
period of time -- usually within a unit or chapter.
I can vouch for this.
SAXON 6/5 has 120 lessons in all, plus 12 'Investigations' & 3 Appendix lessons, and when you get to Lesson 120 you're still practicing the stuff you learned back in Lesson 1.
There are 100 or more problems and computations in each of the 120 lessons: Fast Facts, Mental Math, Problem Solving, Lesson Practice, and, finally, Mixed Practice.
This is what we call drill and kill.
Cognitive psychologists call it automaticity:
Practice Makes Perfect But Only If You Overlearn Ask the Cognitive Scientist: How We Learn by Daniel T. Willingham
review
comments... GoodReadingPart1 09 May 2005 - 16:25 CatherineJohnson Just posted to Our Favorite Math Ed Articles: Daniel T. Willingham's 'Ask the Cognitive Scientist' columns for AMERICAN EDUCATOR (wonderful) William Schmidt, et al's phenomenally helpful 'A Coherent Curriculum: The Case of Mathematics' (Schmidt headed the Third International Mathematics and Science Study (TIMSS), and summarizes his findings here.) Specific Learning Disabilities: Finding Common Ground. A Report Developed by the Ten Organizations Participating in the Learning Disabilities Roundtable. This is the American Institutes of Research 2002 consensus report: what findings, hypotheses, and theories do 10 different organizations and insitutions, including the Department of Education and the Learning Disabilities Association of American, agree to be true of 'specific learning disabilities.' (I haven't read this yet.) See also: PracticeAndOverlearningPart1 comments... NotTheWholeStory 20 May 2005 - 04:21 CarolynJohnston Catherine sent me a link today to an article about the Everyday Math curriculum. A host of well-known mathematicians have given Everyday Math a lot of negative press. A group of mathematics professors led by David Klein at Cal State Northridge wrote an open letter to the Secretary of Education urging the U.S. government to publicly withdraw its 1999 recommendation of Everyday Math (among other new-new math curricula). I am familiar (very familiar) with Everyday Math, and it has clear weaknesses that we'll discuss at length in time, but I was struck by the following quote in today's article:
Klein said that as a result of whole math programs
such as EM, CSUN and other colleges must offer entering
freshmen remedial math classes at a level as low as third
grade. He said he’s seen, for instance, calculus students who
can’t add fractions.
It may be a crummy program -- I have certainly found it hugely frustrating to work with -- but it wouldn't be fair to blame Everyday Math for the existence of vast numbers of calculus students who can't add fractions. The problem has been around a lot longer than Everyday Math has.
I taught at SUNY Binghamton in the early 80s, and we had plenty of calc students who couldn't add fractions. When I was a grad student at Louisiana State University, the remedial math caseload on the mathematics department was so heavy that a whole class of 'instructors' -- essentially the equivalent of high school teachers in schooling and training -- were employed by the math department to teach remedial math classes, and a typical grad student was assigned full responsibility for 2 classes of remedial math every semester. That's more than 60 students per grad student.
And these classes were serving just the students who had been identified as needing remedial math classes; many slipped through the cracks. You bet a lot of the students in LSU's calculus classes couldn't add fractions. Nor is the problem confined to LSU; public universities everywhere, with few exceptions, have large remedial math loads. It's been going on for at least twenty years, long before Everyday Math appeared on the scene.
I don't think there are any simple explanations. But I do think we're floundering, and we need to look to countries with a better track record for guidance.
Furthermore, any math professor can point to plenty of failures in math education within his own experience, but individual failures don't help to explain what we're doing wrong at the policy level. For that, we'll need sound research.
comments... NotTheWholeStoryPart2 14 Jul 2005 - 23:53 CatherineJohnson So just how far back does the U.S. fraction deficiency go, you ask. Answer: really far. In 1923, the NEW YORK TIMES reported that fewer than half of seventh grade students could convert the fraction 1/5 into a decimal. The Columbia Teachers College had a plan.
The new aim of the progressive arithmetricians is to abandon
drilling in artificial problems and to bring mathematics close
to every-day life.
from: 'New Teaching Puts Life into Dreary Arithmetic',
NYTIMES December 9, 1923
Apparently, the plan was working.
The new method is so successful, according to its sponsors,
that one school has playfully threatened to abandon it for the
reason that the pupils are so enthusiastic over arithmetic that
their teachers can scarcely interest them in other subjects.
This was the start of progressive education in America. So flash forward to 1989, and we find NAEP reporting that 60 percent of seventh grade students can 'express simple fractions' as decimals. A mere 70 years of progress, and 10% of American seventh graders who wouldn't have known that 1/5 is the same thing as 20% back in 1923 do know in 1989. That was my first thought. My second thought was, OK, I'll take it. 10% is 10%. Then I noticed Chris Correa's second post on the subject.
I browsed through the publicly released NAEP questions
and found the most comparable question to be from 1992:
Of the following, which is closest in value to 0.52?
This is my beef with constructivism. It's not like constructivism hasn't been given a fair shake. Constructivists have had a good hundred years to show us what they can do. I say it's time to move on. [Thank you, Chris Correa.] NotTheWholeStory comments... GoodReadingPart2 13 May 2005 - 01:12 CatherineJohnson Posted to Our Favorite Math Ed Research Articles: The A-Maze-ing Approach to Math by Barry Garelick. comments... NotTheWholeStoryPart3 13 May 2005 - 01:37 CatherineJohnson re: NotTheWholeStory & NotTheWholeStoryPart2 Carolyn's right that Everyday Math can't be blamed for the sorry state of college freshmen's ability to add fractions. I haven't been able to track down the first printing, but EVERYDAY MATH seems to date back to around 1993 or thereabouts. Garelick reports that approximately 10% of U.S. schools have now adopted E-Math, and I read just this week that another 10% of U.S. schools have adopted one of the other constructivist math curricula. (I've forgotten the source, or I'd link -- sorry.) Of kids entering college this year, only a small percentage will have spent much time using the latest crop of constructivist mathematics programs. Of course, that's leaving aside the fact that constructivism has been part of ed school philosophy for a century. comments... CarolynIsGobsmacked 09 Oct 2006 - 01:31 CarolynJohnston Did you see this chart in Garelick's article that showed the grants that ed departments were given to come up with new math curricula?  Man, did I ever go into the wrong branch of academia. Five million dollars for Everyday Math! Six for Trailblazers! Fourteen for Contemporary Math in Context!!!! The path I should have taken is now clear:
MoreBigNumbers BigNumbers comments... CatherineIsGobsmackedPart2 13 May 2005 - 22:58 CatherineJohnson re: the chart Oh, yeah. I saw it. CarolynIsGobsmacked comments... CatherineIsGobsmackedPart3 09 Oct 2006 - 01:33 CatherineJohnson re: CarolynIsGobsmacked No question, Carolyn. When it came time to choose a response, you blew it. (Sorry. Inside joke. I am WAY ready for summer vacation.) * OK, that's not fair. We had a publicist - a free lancer - before we had Trailblazers. keywords: choose a response no putdowns bullying character education lost instructional time comments... NewBook 20 May 2005 - 18:44 CatherineJohnson 
Eduwonk seems to think Joe Williams' book, coming in fall 05, will be good. comments... BooksPart1 23 Jun 2006 - 13:59 CatherineJohnson 
two fantastic books Elaine McEwan's website comments... CatherineIsGobsmackedPart4 25 May 2005 - 17:55 CatherineJohnson re: career opportunities for ed consultants
In the fall of 2003, Chancellor Klein introduced the
CarolynIsGobsmacked CatherineIsGobsmackedPart2 CatherineIsGobsmackedPart3 comments... CurricularGamePlaying 09 Oct 2005 - 22:44 CarolynJohnston Does it matter what mathematics curriculum your kids are using at school, as long as they are getting good grades in math? Catherine and I both started tutoring our kids, supplementing their math homework, and looking into mathematics education, because our kids weren't doing well in their regular math classes. Had they gotten good grades all along, we might just be rolling along without asking any questions. But my son was doing poorly in Everyday Math, a new-new-math curriculum, after having been very successful in Saxon Math, a traditional curriculum which emphasizes the incremental acquisition of new skills, including mastery of all the classic computations. It was clear that it was the new curriculum that had derailed him. But was that just my son, whose special needs make him a special case? Proponents of Everyday Math claim that it integrates a child's mathematics knowledge, and makes it more useful to him, if the kids spend time working with math in the context of discovering and solving real-world problems; gathering data, measuring things, and so forth, at the expense of computation (if necessary). If so, then after (perhaps) a few years of struggle, we ought to see improvement in kids' understanding of math at the level of applications. In other words, kids raised on real-world data and applications ought to at least be better at word problems. That's what makes this chart so powerful. 
The chart shows scores on a subtest of math problem solving of the Comprehensive Test of Basic Skills (CTBS), a nationally-normed standardized test. The scores measure the same group of kids from Anne Arundel County's 14 lowest-performing schools in 2nd grade, and again in 4th grade.
The second graders had been working with either Everyday Math or Mathland, a similar 'discovery-based' curriculum (see the blue bars in the chart). When they took the test in 4th grade, they had been working with the Saxon curriculum for a year (see the white bars).
The kicker is that this subtest measures performance on word problems. This is the supposed weakness in traditional math programs that Everyday Math's approach is intended to remedy.
Check out this link to see how the news went over in Anne Arundel.
Curricular Game Playing Curricular Game Playing, part 2 number bonds vs. 4-fact families Numicom Dominoes comments... CurricularGamePlayingPart2 09 Oct 2005 - 22:44 CatherineJohnson About a month after Christopher and I began working with Saxon Math 6/5, he told me,
Multiplication and division are the big brothers,
Then he said,
And multiplication and division are cousins.
+ + + This is a 9-year who, just 6 weeks earlier, had been flunking math. Any way you slice it, that's conceptual knowledge. In just a few weeks he'd absorbed the idea that addition & subtraction, multiplication & division, are inverse operations, and that multiplication was repeated addition, while division can be seen as repeated subtraction. I should add that Christopher doesn't consciously know that division can be described as repeated subtraction (I don't think). He probably couldn't put it into words, though he could tell you that multiplication is repeated addition. But a few weeks into Saxon he had intuited the relationship. This is exactly the goal constructivist math programs have set for themselves: they are trying to help students connect the dots. Addition, subtraction, multiplication, & division aren't Four Separate Things, as they were for me until I read and studied Saxon Math! I haven't worked with a constructivist text. But I know for a fact that Saxon gives children conceptual understanding. Curricular Game Playing Curricular Game Playing, part 2 number bonds vs. 4-fact families Numicom Dominoes comments... GreatMomentsInWorldHistory 14 Jun 2005 - 23:07 CatherineJohnson Christopher and I finally finished Megawords 1 today. Megawords 1 is the 4th grade book, and I've been saying for months now that my goal in life is to finish the 4th grade book before Christopher gets out of 5th grade. My new goal is to finish the 5th grade book (Megawords 2, in case you were wondering) before Christopher gets into 6th grade. I would like to be doing the 6th grade book in the 6th grade. I don't feel that's asking too much. Um . . . just so there's no confusion, this post isn't about math. It's about spelling. BeingYourChildsFrontalLobes SummerSupplementTime HowToSpell HowToSpellPart2 MoreSpelling TheSaxonMathOfSpelling comments... SingaporeMathSummerWorkshop 16 May 2005 - 01:12 CatherineJohnson Scott Baldridge, coauthor of Elementary Mathematics for Teachers is giving a summer workshop in Singapore Math! Singapore Mathematics Summer Institute August 1-5, 2005 Madison Country Day School campus, Madison, Wisconsin Cost: $500 I may have to go. comments... HolyBureaucraticNightmareBatman 16 May 2005 - 01:07 CatherineJohnson . . . sigh . . . via Eduwonk and New York Daily News comments... CalStateStudyIntro 22 Nov 2006 - 16:42 CarolynJohnston Part 1 in a mini-series on a review of quality math ed research articles. In 1998, the California State Board of Education contracted with a group of education researchers from the University of Oregon to conduct a review of high-quality mathematics education research papers. The resulting 100-page report is available here. Their task was simply to search out all the mathematics education research that had been performed and published within a specified period, cull out the stuff that was of dubious quality (meaning it had unsound experimental underpinnings, or was performed in a setting that was not like a classroom, or had one of a number of other flaws), and see what the remaining studies had to say about mathematics achievement (that is, they avoided papers that did not measure study outcomes quantitatively, using tests of achievement; so studies measuring the impacts of changes in teaching methodology on students' confidence, for example, weren't included). The results are surprising to me in places. There were studies on the use of manipulatives, studies on kids working with their peers, studies on the use of computers, calculators and technology, studies on motivational methods, and studies on the design of instruction. The researchers seem to have avoided bias, and to be genuinely searching out high quality research. I thought I would do a 'mini-series' describing and discussing their results, section by section. Stay tuned. California study intro California state study of group learning California Board of Ed study part 2 education research - peer reviewed studies - chart comments... CalBoardOfEdStudyPart2 22 Nov 2006 - 16:45 CatherineJohnson Carolyn wrote:
I thought I would do a 'mini-series' [on the California Board of Education study]
What a great idea! I've been wanting to know more about the famous California Board of Ed study. Here's a terrific factoid about Dixon et al, from The Principal's Guide to Raising Math Achievement by Elaine K. Mc Ewan:
From a total 8,727 published studies of mathematics in
8,727 "studies."
Of which, 231 were scientifically valid.
231
Parents, teachers, administrators, and Concerned Citizens everywhere should have this figure tattooed to their foreheads.
When textbook publishers and ed school types use the words "research shows," you're looking at maybe a 3% chance they're right about that.
Given the fact that, by law, all research findings have to be replicated before they can be certified as facts, the odds are probably closer to zero.
(OK, I'm kidding. There's no law. Anyone can call anything a fact if they want to. It's a free country.)
CalStateStudyIntro
California study intro California state study of group learning California Board of Ed study part 2 education research - peer reviewed studies - chart comments... RoyalRoadToGeometry 20 Nov 2006 - 13:39 CatherineJohnson I had never read this story before today: When Ptolemy I, the king of Egypt, said he wanted to learn geometry, Euclid explained that he would have to study long hours and memorize the contents of a fat math book. The pharaoh complained that that would be unseemly and demanded a shortcut. Euclid replied, “There is no royal road to geometry.”I'm sorry to hear that, because a royal road to geometry is exactly what I need today. I just checked out the next lesson in Christopher's SRA Level 6 book, which turns out to be about finding the equation for a line that's been plotted on a graph. I could do the easy, obvious problems, but the graph where 'one step to the right' seemed to be followed by '1/3 step up' stopped me cold. I don't remember ever being taught how to find a formula from a line on a graph. I also don't remember ever being taught a formula for making a line on a graph in the first place, although I do remember plotting out lots of coordinated pairs. That's got to be be worth something, right? Unfortunately, while I remember plotting out lots of coordinated pairs, I have no idea when in my mathematics education this occurred, or why. Exactly what Subject Matter Area does finding-an-equation-from-a-graph fall under? Since the formula-finding problems in Christopher's book are in the unit on 'Algebra Readiness,' I figured this must be algebra, so I went to get my copy of Algebra to Go (buy this book, you'll need it) from the dining room-cum-math-&-spelling-zone. This is where I feel God Wants Me To Learn Math, or at least not suffer hideously while I try to make sure Christopher Learns Math, because an Unseen Force led me to pull out Geometry to Go instead (buy this book, too), open it up, and land smack dab in the middle of the page explaining the formula for charting linear functions on a graph — all of this before realizing I had the wrong book, glory Hallelujah. Leading to my first Math Revelation of the day: it's not algebra! It's coordinate geometry! * I had no idea! Thank you! Then my neighbor, the statistician, came over and showed me how to do it. * UPDATE 10-8-2006: It's algebra. Algebra and coordinate geometry, I guess. I don't know. I will press on and report back.  source: Bitter Single Guy see also: BuyThisBookToo EnglishLanguageArtsBookRecommendation MathRefs comments... BuyThisBookToo 09 Nov 2005 - 17:18 CatherineJohnson As long as I've got you overspending on math books, you may as well pick up a copy of Math on Call to complete the set. The Math On Call series is targeted to the school market, though the books are priced well enough that parents can and do buy them, too. I'd love to know what the sales rep's pitch is. Essentially, the books cover every topic your child is going to encounter in every level of math, explaining each one directly, conceptually, and procedurally -- and very likely using the same vocabulary, illustrations, and sequence of subtopics his or her school will use to boot, thus putting a stop to the nightly 'I can't help you with your homework, I didn't learn it that way' exchange. I'm wondering whether schools that have invested in constructivist math purchase these texts as direct-instruction back-ups, for the parents as well as for the kids. [update: I just noticed that there are Parents' Guides available for all of the books.] This is a less frequently noted problem with constructivist math. If parents have forgotten their own math (that would be me), they're not going to remember it looking at a discovery text. Which brings me to one of my favorite reader reviews on AMAZON: My son's 7th grade math teacher recommended this. I don't know what we would have done without it. The school's math textbook was useless. If there was any problem not understanding a math concept, we would just whip this baby out and it was easy to understand. Math homework couldn't have been any less frustrating. My younger son now takes it with him to school for doing his math work at school. It is invaluable. One last thing. If you have younger kids, you might want to start with the earlier books in the series. They're easier to deal with when you have a lot of catching up to do yourself. Grades 1-2 
Grades 2-3
Grades 5-6
 for 8th grade & high schoolAlgebra to GoGeometry to Go see also: RoyalRoadToGeometry EnglishLanguageArtsBookRecommendation MathRefs comments... CalStateStudyOfGroupLearning 22 Nov 2006 - 16:45 CarolynJohnston Part 2 in a mini-series on a review of quality math ed research articles. Part 1: CalStateStudyIntro The most surprising thing, to my mind, in the Cal State Study is its strong endorsement of cooperative group learning. The review included eleven studies of group learning, all with positive findings for the use of structured group learning. The Cal State Study defines "conventional mathematics instruction" as being characterized by teacher explanation of the new material, followed by independent workbook activity. The study makes the strong claim that the conventional approach has absolutely no theoretical support, and is discredited by the totality of the studies in the review that examine cooperative group learning methods. Actually, this is the one place I've read in the report so far where I felt I might be encountering just a whiff of reviewer bias, particularly in the glowing interpretation of these findings. But the evidence might really be there. Carefully constructed group learning opportunities beat conventional teaching methods. Go figure. The phrase 'carefully constructed' is a point to dwell on. Simply sticking kids together in groups to do their homework conferred no benefit. The group learning environment had to be structured, i.e. the nature of the kids' interactions had to be controlled by external reinforcement systems. In particular, higher performing kids had to be motivated to help lower performers. To some degree, it didn't even matter what the system was, but it had to be present. One study compared cooperative vs. competitive group reinforcement systems: an example of a cooperative system is one in which each kid in a group gets the average of their individual grades, whereas grouping the kids into competitive teams is a competitive reinforcement system. Both reinforcement systems were shown to confer equal benefits above the conventional teaching style. The reviewers note that "a particularly interesting aspect of these studies on cooperative work is that all of them included students who were at risk in some sense, either by virtue of being inner city urban students, and/or low SES students, and/or students identified as having special learning needs." While the reviewers seem rather pleased with this study design, to my mind it weakens the broad applicability of the results. I rather wish they'd done these studies with the most typical bunch of learners they could find, since the presence of special needs can really skew a kid's learning style. Kids who are starved for attention, interaction, or positive reinforcement for learning, might benefit much more from learning environments that include these things than a typical learner would. Also see: CalBoardOfEdStudyPart2, EdResearch California study intro California state study of group learning California Board of Ed study part 2 education research - peer reviewed studies - chart comments... EdResearch 22 Nov 2006 - 16:44 CatherineJohnson  graphic from: The Virtues of Randomness by Robert Boruch California study intro California state study of group learning California Board of Ed study part 2 education research - peer reviewed studies - chart comments... CalStateStudyOnManipulatives 14 Aug 2005 - 14:49 CarolynJohnston Part 3 in a mini-series on a review of quality math ed research articles. Part 1: CalStateStudyIntro Part 2: CalStateStudyOfGroupLearning Another surprising fact -- about math manipulatives -- comes out of the Cal State Study. There were only four studies of manipulative use that were of high enough quality to make the Cal State cut. That really isn't enough to draw a conclusion from, especially given the studies' haphazard coverage of the range of instructional possibilities. Still, there are enough results that they suggest a pattern. See if you can detect it ('benefit' implies that kids did significantly better on normalized tests of math achievement than control groups did). Kindergarten kids learning counting: no benefit conferred by including manipulatives. Third graders learning multiplication: two different studies show no benefit to the use of manipulatives before teaching formal computation. Fifth and seventh graders learning fractions: kids benefit from a fractions game played with or without other manipulatives and pictorial representations. Elementary schoolers using fraction/ratio manipulatives with fraction/ratio instruction: no benefit. Seventh graders using fraction/ratio manipulatives with fraction/ratio instruction: benefit. I love what these results suggest because it is so unexpected and counterintuitive. Most of us think of manipulatives as a stepping-stone from the concrete to the abstract, as something to be used only by the very young when they are first introduced to a topic. But these results suggest that older kids get more benefit out of manipulatives. In a way, now that I think about it, it makes sense; their relative maturity means kids have a conceptual 'hook' on which to hang the insights that the manipulatives give them. They already have half a clue, and that helps them get the point of the manipulatives. Perhaps to a younger kid, less able to generalize from the concrete to the abstract, the manipulatives are simply toys. This is all the evidence I need to get the fraction manipulatives out for my soon-to-be sixth grader. For more information on math manipulatives, see our favorite math supplements for kids and FractionManipulatives Also see: EdResearch CalBoardOfEdStudyPart2 FractionManipulatives QuickThoughtAboutFractionManipulatives FractionManipulativesPart2 NewStudyOnManipulatives New Study on Manipulatives Part 2 comments... FractionManipulatives 14 Sep 2006 - 14:04 CatherineJohnson re: CalStateStudyOnManipulatives Over the past year I've used two kinds of manipulatives with Christopher, who is 10: fraction manipulatives play money I didn't need play money and neither does anyone else. I got it only because I wanted to teach Christopher how to make change without a cash register, a lost art, and because . . . if I stacked up a pile of Real Money big enough to make change with, it was going to get raided for lunch money, bake sale money, field trip money (and that's just for starters). We are chronically short on ONES around here, let me put it that way. So I decided to make things easy on myself and buy some fake money. + + + I'm a huge fan of fraction manipulatives. Christopher and I have spent quite a lot of time using a set of fraction tiles to illustrate: equivalent fractions the addition and subtraction of fractions the addition and subtraction of equivalent fractions Nothing makes the idea that 2/12 is equivalent to 1/6 more obvious, IMO, than actually lining up two 2/12 tiles below one 1/6 tile and seeing that, yes indeed, 2/12 = 1/6. These are the fraction tiles I use. They cost $8.75 plus shipping: 
The same company, (Rainbow Resource, a homeschooling catalogue), also carries a set of extra fraction tiles without the tray that I wish I'd had when we first started trying to learn fractions. (I have them now, but we may be past the point of needing them. We'll see.)
You need the extras because you really want the ability to demonstrate addition and subtraction of fractions with different denominators.
+ + + There are lots of other fraction manipulatives out there, but I chose these after reading a comment from a mom on a homeschool forum somewhere. (I wish I'd kept the link.) She said that her daughter didn't get anywhere using the more-common circular, 'pie chart' fraction manipulatives; she needed to see rectangular fractions. I have no idea why this would be, but it 'felt' right to me, so I searched for rectangular manipulatives and found these. At the same time, SAXON MATH uses circular manipulatives, so Christopher has been exposed to both, which I think is almost certainly ideal. A core principle in teaching math, from what I gather, is to teach the same material from different angles. + + + Another terrific activity to do with fraction tiles: Show how different combinations of fractions add up to 'one whole.' To do this you just have your child keep lining up fraction tiles on top of the bright red 'one whole' tile until he's covered the whole thing without anything hanging over the end. So, for example, he might put 2 1/12th tiles, 1 1/6 tile, & 2 1/3 tiles on top of the 1-whole tile, illustrating the fact that: 2/12 + 1/6 + 2/3 = 1 After awhile it starts to become obvious that you can put 6ths and 3rds & 12ths together evenly to make one whole, or 8ths & 4ths & halves, or 5ths & 10ths, . . . but you can't put 3rds and halves together, or 4ths and 5ths (not unless you have a bunch of 20ths, which you don't), and so on. You can see your child start to get a feel for multiples* and divisibility, whether he has explicitly studied multiples and divisibility yet or not. + + + That's a whole other issue: is it useful to 'preview' concepts in this way? I have no idea, so offhand my answer is 'It depends.' That's one of the big gripes with constructivist math; the kids are constantly being exposed to advanced topics -- sometimes very advanced -- and then not taught the topics to mastery, because the book will be 'spiralling back' to the same topic the next year and the next year after that. Parents tend to hate this, but parents could be wrong. It happens. Let's just say that my perception, working with Christopher and the fraction tiles, was that he was developing an intuitive grasp of numbers that are multiples of each other versus numbers that aren't. This seemed like a good thing at the time, but who knows? I'm new at this. Come to think of it, I'm going to get the fraction tiles out again when I get back to teaching the Singapore Math lesson on Changing Ratios. (My neighbor and I team-taught this lesson to our kids two weekends ago, but it was over Christopher's head. Her son is a year older.) Singapore teaches changing ratios in the first half of 6th grade:  + + + Since I never remember definitions of even the simplest terms, I am including the definition of a multiple here: * multiple - The multiple of a number is the product of the number and any other whole number. (2,4,6,8 are multiples of 8) Also see: EdResearch CalBoardOfEdStudyPart2 CalStateStudyOnManipulatives QuickThoughtAboutFractionManipulatives FractionManipulativesPart2 NewStudyOnManipulatives New Study on Manipulatives Part 2 comments... RussianMath | |||||||