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HowIGotHerePart1 23 Jun 2006 - 13:15 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
MathInTheBlood 23 Jun 2006 - 13:16 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
NowThatWereBothHere 23 Jun 2006 - 13:24 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
StrugglesWithLongDivision 07 Jul 2005 - 20:37 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
SwoopAndSwoop 07 Jul 2005 - 20:38 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
SwoopAndSwoopPart2 23 Jun 2006 - 13:24 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.
SwoopAndSwoopPart3 23 Jun 2006 - 13:25 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.
HowIGotHerePart2 23 Jun 2006 - 13:27 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
BeingYourChildsFrontalLobes 23 Jun 2006 - 13:27 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
PracticeAndOverlearningPart1 23 Jun 2006 - 13:29 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
GreatMomentsInWorldHistory 23 Jun 2006 - 14:01 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
MathInTheBloodPart2 08 Jul 2005 - 00:44 CarolynJohnston
Carolyn's side of the story
See also: MathInTheBlood (Part 1)
I should explain that for my son, school has never been an ordinary undertaking. As a young child, he was diagnosed with an autism spectrum disorder (Pervasive Development Disorder, which is a diagnosis that means 'looks like some kind of autism to me'). His preschool years were a nightmare of trying to treat his developmental problems with Applied Behavioral Analysis therapy, while simultaneously searching for a medical treatment that would help him. The tough thing about having a kid with this disorder is that you have to work on him hardest in the earliest years, when you're most clueless about his prognosis: it's utterly crazy-making, and I was pretty crazy.
In his elementary school years, my son has made great progress; but he still has an attention deficit, severe organizational difficulties, and problems with deep reading comprehension and social cognition. So the fact that he was flying independently with Saxon math, and hit a mountainside when we encountered Everyday Math in fourth grade, was a Big Deal.
Besides, he's a smart kid with an autism spectrum disorder. Math is his greatest strength, and a career in math, science, computers or engineering is his most likely future. In those fields, his colleagues will know how to deal with him (given the sheer numbers in which kids are getting autism-like disorders these days, they'll probably be just like him).
At the end of fourth grade, during a conference with his teachers, I floated the possibility of his doing fifth grade math on his own, with me as his tutor, using Saxon math. It's legal in this state to homeschool in one subject like that, but we all had big reservations about it. We've worked so hard to enable Ben to function in a regular classroom with the other kids that the thought of separating him from the other kids at that point, just because we didn't like the math curriculum, seemed unbearable. So I sighed, gave up, and we entered fifth grade with Ben still signed up for Everyday Math.
Somewhere early in fifth grade, Catherine and I struck up an Internet Friendship (we have never actually met in the flesh!). Among her other interests, Catherine is a noted non-fiction author who specializes in autism research and treatment... we encountered each other in the way that people do online, and I figured out who she was.
Catherine is a true Math Revolutionary. While I, with all my math degrees and our successful experiences with Saxon Math, was still dithering about whether or not to pull my son out of school and teach him myself, Catherine was actually doing her ten-year-old son's fuzzy math homework for him every night, so she could get that over with quickly, and move on to teaching him mathematics from what she regarded as a better curriculum.
Completely independently, she had chosen Saxon Math for him.
Catherine and I, in spite of our different paths in life, have a heck of a lot in common.
more to come...
MathInTheBloodPart3 08 Jul 2005 - 00:50 CarolynJohnston
Carolyn's side of the story
Third in a series: Part 1, Part 2
Catherine talked me into doing something about my own misgivings about the Everyday Math program: starting Ben on a course of Saxon math. I didn't pull him out of his Everyday Math classes at school, although I could have, because I wanted him to remain in class with his peers.
So we started doing the two curricula side by side.
Saxon Math homeschool has a very regular format: there are warmup exercises, a short and simple lesson, a targeted practice set consisting of exercises from the lesson, and a much more extensive practice set consisting of problems that may come from any portion of the text leading up to that lesson.
The Saxon problems aren't easy, but the problem sets are very well designed; there are never any huge leaps, never anything that's clearly over a child's head: no 'discovery' problems requiring the child to intuit the meaning of something he hasn't been taught yet.
Saxon may not be inspired, but it's solid, and as Catherine posted here, it does build mathematical intuition. It is an excellent choice for a homeschooling parent who wants a solid foundation in mathematics for their child.
But I didn't stick to Saxon Math as religiously as Catherine did. I'm not as disciplined as she is, and I kept finding things I wanted to skip, and things I thought I could teach better in my own way.
But although I taught mathematics at the college level for a number of years -- and encountered all too often the results of an inadequate preparation for math at that level -- I never taught elementary mathematics until I tried to teach my own son. And that turned out to be very different from anything I've ever done before.
I remember the night I decided to teach my son how to solve a linear equation. A linear equation is any equation of the form ax+b=c, where a, b and c are numbers, and x is the number to be solved for. I just can hardly imagine anything simpler and more straightforward than a linear equation.
But I was wrong. It turns out there are a lot of skills that go into being able to solve a linear equation.
You need to understand that if two things are on the opposite sides of an equals sign, they are the same, even if they don't look the same. You need to know that if you do something to one side of an equation, you have to do the same thing to the other in order for the equation still to hold. You need to know that you can undo the addition of b on the left hand side by subtracting b, and that it's okay to do that, and a whole host of other things, as long as you do it on both sides of the equation.
That was too much understanding to impart in one night. The poor kid's head was swimming, and I quickly realized I'd made a big mistake, but I wasn't going to just drop it completely; one thing I think I know about how my son learns is that he needs to end every lesson with a small bit of success in order to stay motivated.
And so I needed to leave him with a little more understanding about equations than he'd started with. I told him that an equation was like a balancing scale, something that he'd had experience with in primary school science.
"What happens if you have a scale with weights on each side, and it's balancing, and you take one of the weights off one side?" I asked him.
"It goes 'thunk' on the other side," he said.
"Right! And what can you do to balance it again?"
"Put the weight back."
"Uh, yeah. But another thing you can do is to take an equal weight off the other side. What happens then?"
"It balances again," he said.
"Right!" I said. "An equation is just like that. If you subtract a number on one side, and then subtract the same number on the other side, that's like taking the same weight off of both sides."
And then I showed him how to solve one, just one, very simple equation: x+6=10. And then he did one on his own. And then we had high fives and we were done.
And I felt daunted, because for the first time I realized that there was knowing mathematics, and there was teaching mathematics, and they weren't the same. I might have the former down, but not the latter.
And right about then, at Catherine's urging, I read Knowing and Teaching Elementary Mathematics.
ATeachersStory 16 Sep 2006 - 19:56 CatherineJohnson
Carolyn (J) has just alerted me to the fact that there are comments under some of our posts . . . so apparently my Next Action vis a vis KTM is: ask Carolyn how to keep track of comments.
('Next Action' is Getting-Things-Done-speak. Carolyn and I are both fans of David Allen's Getting Things Done, and in fact last week Carolyn tipped me off to a whole Getting-Things-Done blog that I am hoping will change my life.)
Anyway, this is a comment from a teacher who has a fascinating situation with Saxon Math.
(I've inserted extra paragraph breaks to make this easier to read):
I teach in a private Christian School. My 5th graders continue to score above all other grades on SAT's.
I am now the only teacher who teaches Saxon, although when I came 11 years ago, all grades used Saxon.
It was felt that there were gaps in the Saxon program for lower grades, so they changed to another program for K-3. That program didn't work, so they are now trying another curriculum. They also felt there were gaps in Saxon for high school, so that has changed. Then they changed 7-8 grades to Mc Dougal-Littell's Passport to Algebra and Geometry, leaving only 4,5,6 using Saxon. Then, they added Passport to Mathematics in 6th. Now, this year they have changing 4th grade to the K-3 curriculum. After three years of complaints from parents and after losing many families, they realized they were going to have to do something about the problems between 5th and 6th grades.
But because of my success in Saxon, they are allowing me to remain with the curriculum.
I know this is a long story, but I find this incredible: one grade in the school continues to be at the top on SAT's, year after year, no matter the class's Math abilities and strengths -- it's my 5th grade class and I use Saxon.
Now, I do use Saxon as it is designed to be used (students make corrections and corrections until they get it right) and that's very important. And I require all the proof, rather than merely answers. Students who have hated math for years learn to love math. Even if they don't understand the total concept, an algorithm allows them to get the right answer and they feel successful for the first time. Their self esteem jumps because they are successful.
The bottom line is: Saxon, when used properly and as designed, works.
Then, the students go into Passport and good students make F's. I'm trying to determine if Passport is considered to be "constructivist" but can find no informatiion on that. I've read the reports from Mathematically Correct's seventh grade review. Passport to Algebra/Geometry is given an A, Passport to Mathematics is given a C. That's all I have found. I see no reference to its being constructivist.
All I know is this: students fall apart, parents ask me to help tutor them, yet it does little good.
Our new secondary principal describes the two programs (Saxon and Passport) as being very different, so I'm guessing that our students are having to go from a very traditional, incremental approach that is successful to a very non-traditional approach. I'm very glad that I found your blog site. I'm going to refer parents to you. Perhaps, they can get insights that I can't yet offer them because I can only teach the "old fashioned, traditional (and successful) way". Thanks for listening and God bless.
I'm pulling these lines out for emphasis:
Students who have hated math for years learn to love math. Even if they don't understand the total concept, an algorithm allows them to get the right answer and they feel successful for the first time. Their self esteem jumps because they are successful.
This is absolutely my own experience.
When I started teaching Christopher math, in the wake of his two failed Unit exams, I was hearing 'math is for geeks,' 'math is for nerds,' 'I hate math,' 'math stinks,' and 'I'm not from Singapore.'
A few weeks into the program all that went away. He was getting As on his tests, he understood the lessons, and suddenly math wasn't for geeks after all.
Self-esteem comes from being able to do something. If a child can do math, he feels good about math. It's that simple.
The other day Christopher actually said to me, spontaneously, in the midst of doing his Saxon homework when he could have been outside shooting baskets or upstairs playing WWE Here Comes the Pain on his PlayStation, "I like math, I just don't like doing math problems."
I had to stop what I was doing and check this out.
"You like math?"
"I like the idea of math."
He's not ready to Commit, but he sounded happy.
ILikeMathPart2
CompareAndContrast
FromAReader
PracticePracticePractice
BarModelingVsGraphing (interesting comments from a KTM reader)
BeingYourChildsFrontalLobes
GreatMomentsInWorldHistory
ProgressReport
BonusPreTeenPost
SummerSupplementTimePart2
SundaySchool
ILikeMath
TheGoodNewsFromHere
GoodNewsBadNews
ImGoingToPlayland
ImportantQuestionFromJoanneCobaskoOfSocmm
ImportantQuestionPart2
OutsmartingTheTests
ConversationsWithKids
BonusPreTeenPost 07 Jul 2005 - 21:21 CatherineJohnson
I just asked Christopher if he thought this joke was funny:
He said, "No."
Then he said, "I just put down Who cares? for everything."
I love this age.
BeingYourChildsFrontalLobes
GreatMomentsInWorldHistory
ProgressReport
ATeachersStory ("I like the idea of math")
SummerSupplementTimePart2
SundaySchool
ILikeMath
TheGoodNewsFromHere
GoodNewsBadNews
ImGoingToPlayland
ImportantQuestionFromJoanneCobaskoOfSocmm
ImportantQuestionPart2
OutsmartingTheTests
ConversationsWithKids
SummerSupplementTime 07 Jul 2005 - 21:25 CatherineJohnson
Too much going on today!
I'm eager to think about 'teacher boredom' and ed reform . . . plus I have a terrific email from a teacher on the subject of summer regression that needs a few identifying details deleted before I can post --- and I have a life beyond this bliki, too, or at least I used to.
But all that can wait!
summer regression
I've just stumbled across what I think may be a good source of information (pdf file) on summer regression.
Tilley, Cox, and Staybrook47 studied summer regression in achievement for students receiving no educational services for three months. They found that most students experience some regression during the summer recess. Cooper et al.48 reviewed 39 such studies and found that achievement test scores do indeed decline over the summer vacation. Their meta-analysis revealed that the summer loss equaled about one month on a grade-level equivalent scale, or one tenth of a standard deviation relative to spring test scores. The effect of summer break was more detrimental for math than for reading and most detrimental for math computation and spelling. Also, middle-class students appeared to gain on grade-level equivalent reading recognition tests over summer while lower-class students lost on them. Possible explanations for the findings included the differential availability of opportunities to practice different academic material over summer (reading is much more easily practiced than mathematics) and differences in the material’s susceptibility to forgetting (factual knowledge is more easily forgotten than conceptual knowledge).
The critical points bear repeating:
- Summer loss equaled about one month
- The effect of summer break was more detrimental for math than for reading and most detrimental for math computation and spelling
Think about it.
One month's loss, for kids who are already at least a year behind their peers in high-achieving countries.
I think it's important to keep up your child's math skills in the summer!
(Carolyn and I have been brain-storming ways to use KTM to help-----)
TO BE CONTINUED
FreeWorksheets
TreadingWater
SummerSupplement
SummerSupplementTimePart2
SummerSupplementTimePart3
SummerSupplementTimePart4 (resources for kids who have fallen behind)
SummerSupplementTimePart5 (resources for preventing summer regression)
SaxonPlacementTestsAndGuides
SingaporeMathPlacementTest
TeachYourChildToTypeThisSummer
BeingYourChildsFrontalLobes
GreatMomentsInWorldHistory
HowToSpell
HowToSpellPart2
MoreSpelling
TheSaxonMathOfSpelling
Summer Supplement Time
linking decline in high school scores to elementary school
research on summer regression
the time costs of not teaching to mastery
U.S. fourth graders not doing as well as thought
Phase 4 topic list, grade 6 class
comments thread on pre-algebra as algebra
PrenticeHallArrives 07 Jul 2005 - 22:17 CarolynJohnston
See: SummerSupplement
Ben's new 6th-grade math text, Prentice-Hall Mathematics Course 1, arrived in the mail yesterday. I ordered it so that we could work in it over the summer. I've seen some good things about it, and I liked the table of contents; also, it's the text series that Ben's junior high school will use, so I thought I'd get him accustomed to working in it over the summer.
Catherine, though, who has seen a lot of the math texts that are out there, has been telling me that she hates the look of it, and I can certainly understand why. Even in a world of busy textbooks, this one stands out. It's got text in a thousand colors and fonts, it has multicolored inset boxes everywhere with brightly colored graphs and tables, and photos of jumping happy children or athletes on almost every page. Just looking at it puts me back in touch with my inner ADD child.
I think the intention of designing a book this way is to keep the kids awake and stimulated, but I think it backfires. This book overstimulates me, never mind Ben; I'd have to stick index cards all over the inset boxes and jumping kids just in order to focus on the text (I do have a touch of ADD, so normal types might not be so rattled). The contents do look pretty good, when you strip away the excess.
What do you suppose is the ideal balance to strike between monotony and overstimulation in a math text? You don't want to blow the kids away with dry monocolor text and equations (at least not until they get into grad school!), but you don't want to overwhelm them with trimming either. The principles of good graphic design surely apply here as much as they do elsewhere. Is there a related principle of good textbook design waiting to be discovered?
SaxonPlacementTestsAndGuides 07 Jul 2005 - 21:42 CatherineJohnson
Saxon placement tests
(pdf files):
Math K-3 Placement Inventory
middle grades math placement test
Placement Test for Algebra 1
Saxon Placement Test for Algebra 2
upper grades math placement test
Terrifically helpful: short, easy to use, easy to interpret.
Christopher and I had gotten through 10 or so lessons in Saxon 7/6, normally a 6th grade book, when Carolyn sent me this link. I'd been feeling that 7/6 was too easy, but didn't trust my judgment.
The test confirmed my feeling, and Christopher and I are now using Saxon 8/7 'with prealgebra.'
A wonderful resource if you're considering supplementing -- or homeschooling -- using Saxon Math.
ATeachersStory
CompareAndContrast
FromAReader
PracticePracticePractice
BarModelingVsGraphing (interesting comments from a KTM reader)
FreeWorksheets
TreadingWater
SummerSupplement
SummerSupplementTime
SummerSupplementTimePart2
SummerSupplementTimePart3
SummerSupplementTimePart4 (resources for kids who have fallen behind)
SummerSupplementTimePart5 (resources for preventing summer regression)
SaxonPlacementTestsAndGuides
SingaporeMathPlacementTest
TeachYourChildToTypeThisSummer
TwentyFirstCenturySkills 17 Jul 2005 - 21:02 CatherineJohnson
update
I shouldn't be flip about this lesson.
In fact, teaching young children to build the next set of math facts on the math facts they already know is a good idea.
I'm pretty sure Parker & Baldridge recommend this approach (I'll check).
for more on 21st century skills, see MoreSingaporeMath
ILikeMath 07 Jul 2005 - 21:22 CatherineJohnson
Yesterday, after Christopher's 'I like bar models' confession, I decided I needed to hear more about this.
So I asked him, 'Why'd you start liking bar models?'
'I don't know. I got good at them.'*
'Yeah?'
'Yeah . . . when you can do something, then you like it. Like math, I used to hate math. Well at school now I like it.'
'You like math?'
'Yeah.'
'In school?'
'Yeah.'
'Do you like math at home?'
'No.'
EOC [end of conversation]
When I started teaching math at home, I wasn't remotely thinking about creating a kid who would like math. Christopher hated math.
'Math is for nerds.' 'Math is for geeks.' 'I'm not from Singapore.'
The best I was hoping for was to have the math-is-for-nerds language go away, which it did.
Apart from that, my entire focus was on catching him up to the rest of his class, then catching him up to his peers in other countries.
We have had screaming, we have had yelling, we have had hysterical sobbing and crying. Kids really don't like their moms teaching them extra math after school.
But we kept at it.
We've had good moments, too. One night, just before bed, Christopher said, 'I love you, Mommy. I love you because you teach me math, and L.'s mom doesn't help him with his math.'
Then he got all embarrassed.
I can tell Christopher is happy I'm teaching him math; I've even heard him boast to his friends about how hard the math I 'make' him do is.
But it hadn't occurred to me that I might be creating a kid who actually likes math.
Not a bad year's work.**
*
I'd say this is a classic example of the high confidence levels you see in American school children in TIMSS surveys. I wouldn't have said that Christopher is 'good at bar models,' and I was surprised to hear him say so. It's true, though, that just in the past couple of days he's moved from absolute novice to . . . advanced beginner.
**
Christopher had two terrific math teachers this year: Amy Panitz (of whom Christopher once remarked, "Mrs. Panitz is a better teacher than you") and Nancy Woeckner.
ILikeMathPart2
TeacherAppreciationWeek
Number 2 Pencil
Which brings me to a blog I like called Number 2 Pencil, written by Kimberly Swygert, psychometrician.
In a post today, she writes:
Wouldn't it be fun to produce research showing that the students who learn the most in school and do the best on standardized tests are also the ones who are happiest and have the most love of learning? I'm not saying I know that's so; I'm saying it would be fun to poke at the anti-testing folks with those kinds of correlational results.
I hope someone does that study.
I like math
BeingYourChildsFrontalLobes
GreatMomentsInWorldHistory
ProgressReport
ATeachersStory ("I like the idea of math")
BonusPreTeenPost
SummerSupplementTimePart2
SundaySchool
TheGoodNewsFromHere
GoodNewsBadNews
ImGoingToPlayland
ImportantQuestionFromJoanneCobaskoOfSocmm
ImportantQuestionPart2
OutsmartingTheTests
ConversationsWithKids
ILikeMathPart2 07 Jul 2005 - 20:43 CatherineJohnson
from Barry Garelick (I've added paragraphs to increase white space):
When my daughter tells me she hates math, my response is always the same. "Well, I have good news for you. You don't have to like it. You just have to know how to do it."
She's stopped telling me she hates math.
We shouldn't be so concerned with whether kids like or hate something. I hated history and English, but you either toe the line or get bad grades, and I didn't want bad grades.
In terms of math, kids hate it when they can't do it. When my daughter catches on to something, she likes doing it.
Math is not easy sometimes and it takes work, and that message should also be imparted to children. Not that it's impossible; but that it can be difficult, and that we all had to work at it.
When math isn't taught properly, then kids are not able to do it, and then they hate it. I've been talking to various adults lately who fit the description NCTM wrote about in the Jay Mathews' Post column of May 31 in which they talked about adults groaning when they heard the familiar story problem about distance, rate and speed. (A man starts out at 9 AM at 15 mph, etc etc). The adults I talked to said they hated those problems because they couldn't do them. When pressed, they admitted their teachers were not very good.
This is not a definitive sample by any means. I lucked out and had a very good algebra teacher who gave us very good instruction on how to solve story problems. As a result, I liked them. The fact that I ended up majoring in math may or may not be coincidental.
I like math
I like math, part 2
BeingYourChildsFrontalLobes
GreatMomentsInWorldHistory
ProgressReport
ATeachersStory ("I like the idea of math")
BonusPreTeenPost
SummerSupplementTimePart2
SundaySchool
TheGoodNewsFromHere
GoodNewsBadNews
ImGoingToPlayland
ImportantQuestionFromJoanneCobaskoOfSocmm
ImportantQuestionPart2
OutsmartingTheTests
ConversationsWithKids
NewComments 07 Jul 2005 - 20:47 CatherineJohnson
SteveH has a new comment about Base 5 & fuzzy math in the CompareAndContrast thread.
update: More from Steve!
Thank you!
I love this, especially:
when my son was born, I told my mother that I wanted 3 things for him in life: 1. To care about other people. 2. To know the value of hard work. and 3. To be happy. Her response was that if he did numbers 1 and 2, then number 3 will take care of itself.
And this:
If Everyday Math (as an example), thinks that doing things in different ways is helpful, then why do they completely avoid the standard algorithms (the best ways)? While doing Singapore Math with my son at home, he ends up doing a number of things in different ways than his EM at school. This can be helpful, or it can be an overload of the brain.
I think SteveH is also the commenter who pointed out that ed school students are taught constructivist teaching methods via direct instruction.
I say that's not fair.
If our kids have to discover math, ed students should have to discover discovery.
Guess and check, guys!
Lots of sharp observations on math & practice, math & creativity, math & solving problems more than one way here: ILikeMath
TheSaxonMathOfSpelling 20 Jul 2005 - 15:35 CatherineJohnson
Boy.
Blogging (or blikki-ing) takes time.
I've got all kinds of great stuff to post on engineering & discovery & creativity, and it's still sitting around in emails & Stickies.
And now it's 7 pm.
A comment from Susan got me going on Megawords, so anyone interested in the research on how children learn to spell should click on MoreSpelling.
BeingYourChildsFrontalLobes
LiveBloggingTheSpellingBee
GreatMomentsInWorldHistory
SummerSupplementTime
SummerSupplementTimePart2
HowToSpell
HowToSpellPart2
MoreSpelling
AnotherWikiPossibility 19 Sep 2005 - 23:07 CatherineJohnson
Another possibility for communal Wiki pages is to do something like the thread for RussianMathPart3: pose a problem or a lesson everyone can comment on.
I'm interested in comments on the fraction lesson J. D . Fisher has posted at Math and Text.
My immediate reaction to J.D.'s post is that it would be terrific for developing teachers' conceptual understanding of mathematics, and possibly for developing teachers' pedagogical content knowledge (pdf file).
But I wouldn't be able to teach it to Christopher, even though he does know that a fraction is (also) a division problem.
(I'll pull my thoughts together on this later--time for a bike ride now.)
I'd love to get other people's reactions.
KitchenTableMathIsAWiki
WikiPagesForReadersAndCommenters
WikiHowTo
AnneDwyersSingaporeMathClass
FreeWorksheets 07 Jul 2005 - 21:26 CatherineJohnson
from SusanS:
Two more sites with free math worksheets (and other free stuff) are edhelpers.com and superkids.com. I do love the free stuff.
Thank you!
our favorite math supplements
We are slowly but surely pulling together the sidebar pages, so you might want to take a look from time to time.
We also need to get a reader recommendation page going.
I'm adding Susan's recommendations to the 'our favorite supplements' page so they'll be where people can find them easily.
I'll also gather together the grammar, spelling, handwriting, etc. book & curriculum recommendations into one place, with links to the original reader comments. These are invaluable, so keep them coming!
Back to online math resources, also remember Carolyn's recommendation:
... These math worksheet generators can come in very handy.... very configurable; you can set the number of columns and rows of problems, and the difficulty of the problem, and the numbers of significant digits in the solution, and so forth....
We especially found the sheets for fraction and decimal long division useful. That's a skill that just takes a lot of practice.
computer learning versus paper-and-pencil
Susan inspired me finally to track down some of my favorite online resources and get them entered on the Our Favorite Supplements page.
But first I should say that I'm leery of online math practice, for 3 reasons:
- Christopher has never learned well using a computer
- I've seen research showing a slight decline in student achievement in Israeli schools after the introduction of computers in classrooms
Christopher didn't really get his math facts down cold until we started doing the Saxon fast fact paper-and-pencil worksheets.
He didn't make any headway that I could see using a software math facts program, and I don't think he made much progress using standard flash cards, either.
To be fair, we have problems using materials like flash cards, since I'm constantly having to hide them from Andrew, which of course makes it harder to find them when I need them, which, in turn, makes me tend to use them less than I would if they were easy to get to ...
So I don't know whether anyone should be drawing conclusions from my flashcard experience.
But when it comes to computers-versus-paper and pencil, if you've got time to print out the worksheets Carolyn & Susan have pointed you to, that's probably the better choice.
Online 'worksheets' may be to paper worksheets what fast food is to homemade.
That said, I've eaten plenty of fast food in my day, and so have my kids.
So here's one of the main online resources I've liked thus far.
Saxon Math online problems and math activities
- I've seen a number of parents around the web recommend this Saxon Math 'fast facts' generator. The page is clean, simple, and visually compelling. You decide which math-fact problems you want to do, how difficult the problems should be, and how many you want to do. You can also do timed or untimed problem sets. That's great, because kids love seeing their timing get faster.
- Here are the 5th grade activities.
Apparently the site now tells you which activities to do after which lessons in the book; plus you can download the activities for use when you are not online.
- Saxon online equivalent fractions These are great. OK, I'm sold. Forget the Israeli kids; we're doing online equivalent fractions this summer.
TreadingWater
SummerSupplement
SummerSupplementTime
SummerSupplementTimePart2
SummerSupplementTimePart3
SummerSupplementTimePart4
SummerSupplementTimePart5 (resources for preventing summer regression)
SaxonPlacementTestsAndGuides
SingaporeMathPlacementTest
TeachYourChildToTypeThisSummer
And lots more....
SummerSupplementTimePart5 02 Jul 2006 - 17:49 CatherineJohnson
In SummerSupplementTimePart4 I mentioned that I think I have useful advice for 3 groups of kids:
- kids who, for whatever reason, have fallen significantly behind their classmates
- kids who are right on track, doing well, and you want to keep their math skills in shape over the summer
- kids whose parents want to accelerate their math learning -- in particular, to get them in position to take and master algebra in the 8th grade
My own strategy for kids who have falllen behind (Christopher's situation last summer) is in that post.
But please! Everyone! Chime in.
These are the ideas I've come up with working with one child, and talking to a group of 4 people (Carolyn, Ed, my neighbor & friend Laura, and my friend Debbie), with as many on-the-fly advice sessions as I could get with Christopher's teachers thrown into the mix.
One of the main reasons I wanted to do a bliki with Carolyn was to find out what other people are doing!
avoiding summer regression
For kids who are doing fine, here are my thoughts.
Assuming the research I've found (pdf file) is to be trusted (it makes sense to me, for what it's worth) there are two points to bear in mind:
- summer loss equals about one month of a child's learned skills and knowledge from the previous school year
- summer vacation is more detrimental for math than for reading, and most detrimental for math computation and spelling
I find the math-versus-reading factoid ironic given that schools universally hand out summer reading lists, not summer math lists.
So here's my own stab at a summer maths list. (I think the British plural works for this.)
summer maths list
- 'mad minute' worksheets daily (be sure to include fractions, decimals & percents if your child has gotten that far)
- a word problem or two each day, if you feel ambitious (Carolyn is posting problems from the Singapore series)
- a Math Olympiads word problem each day, if you feel really ambitious (I'll probably post some of these)
books (worksheets)
I did a quick scan of the various 'Mad Minute' books on Amazon, and folks seem to like this one best:
The Mad Minute covers Grades 1 through 8, and includes fractions & percents.
If any of our teachers or parents have used this book, let us know.
- Saxon Math Tests and Worksheet Booklets for each grade level. 120 'fast fact' worksheets to be completed in under 5 minutes. These are the worksheets that finally got Christopher up to speed, and we're doing them again this summer. Cost for the Tests & Worksheets book alone is around $20, probably less at the Homeschool Super Center. If you're just going to use the worksheets you don't need to buy the textbook or the solution manual.
books (story problems)
- Singapore Math Challenging Word Problems series. These are terrific books. Almost 300 story problems in each, grouped according to subject area (e.g. measurement, time, multiplication-and-division, etc.) All problems are multi-step, & all answers are in the back. $7.80 plus shipping.
caution: your child almost certainly needs to use a book 1 or 2 grades younger than the one he's in. So you might want to have your child take the placement test before ordering.
- Math Olympiad problems -- you can find Math Olympiad books all over the place. They're expensive, so try to rustle up a used copy.
- Math League Contest Books from Math League. Wayne Wickelgren strongly recommends these books for everything from building your child's math achievement to preparing for SAT's. I love them, too. Filled with the kinds of problems, including logical reasoning, children are going to need throughout their lives & much more 'sensible' than the showy problems from Math Olympiads. Each book spans 3 grades, and all answers are in back. $12.95 a book plus shipping.
worksheets
virtual worksheets & problem-solving
I've mentioned that I'm leery of online learning, but you can't beat it for convenience and speed. I like Saxon's offerings:
- Saxon Math 'fast facts' generator The page is clean, simple, and visually compelling. You decide which math-fact problems you want to do, how difficult the problems should be, and how many to do in one set. You can choose between a timed & untimed option. That's great, because kids love seeing their times get faster.
- Check out 5th grade activities.
Saxon now has online exercises for each grade. They tell you which activities to do after which lessons in the book, and you can download the activities for use when you are not online.
- Saxon has lots, lots more, so take a look
- Batter's Up Baseball Game I can't find the 'addition facts baseball games' the kids at school love so much, so here's another one. Christopher told me just now that he loved playing online 'addition baseball' when he was in 2nd grade.
I found it!
The kids at school were crazy about Funbrain, especially math baseball.
update: reader recommendation
Also check out Singapore math's Intensive Practice books. These books cover all sorts of fun things including word problems, computation, puzzles and patterns etc... They are not joking when they call it intensive. Some problems are extremely difficult (and some are quite easy too) and we cover them orally and together with the view that exposure to these types of problems will only expand abilities!
I agree. I have two of these books, and they're terrific. [Catherine]
FreeWorksheets
TreadingWater
SummerSupplement
SummerSupplementTime
SummerSupplementTimePart2
SummerSupplementTimePart3
SummerSupplementTimePart4 (resources for kids who have fallen behind)
SaxonPlacementTestsAndGuides
SingaporeMathPlacementTest
and:
Summer Supplement Time
linking decline in high school scores to elementary school
research on summer regression
the time costs of not teaching to mastery
U.S. fourth graders not doing as well as thought
Phase 4 topic list, grade 6 class
comments thread on pre-algebra as algebra
WickelgrenOnIntroducingAlgebra 08 Jul 2005 - 17:19 CarolynJohnston
I've been looking again at one of Catherine's favorite books, Math Coach (by Wayne and Ingrid Wickelgren).
Wayne and Ingrid have a lot to say about what they consider the most difficult aspects of elementary math -- long division and fraction manipulation. But it's what comes after that that interests me now: their discussion of the importance of teaching algebra early. Wayne suggests that the most important thing you can show your kid, what should motivate them most to want to continue in math, is the power of algebra to solve hard problems.
Most problems in prealgebra and early algebra start out something like this:
John is 27 years old. If his age is 3 times Pete's age, how old is Pete?
If you have a kid like Christopher or Ben, you know he's going to spit out the answer on the spot and tell you not to waste his time with this stupid letter stuff.
That's why Wayne Wickelgren suggests that, when you're ready to introduce your kid to the notion of algebra, the first thing you should do is sit down with him and let him watch you do a problem like this one:
In two years, Jean will be twice as old as Chris will be. In six years, Jean will be four times as old as Chris was last year. How old is Chris now?
In short, start with a demonstration of how algebra-at-your-fingertips gives you mindblowing powers. I was reading this last night and thinking: if I tell him that this problem is what algebra is all about, Ben will be blown away. Why scare him off? Maybe start with something simpler...
But the hard thing about this sort of problem isn't going to be doing the algebra: it's going to be setting up the equations, given the word problem. And that's going to be hard no matter how I try to teach it. Doing the mindless rote stuff required to crank out the answer, once you have the equations, is the easiest part of the problem. And I know Ben: he'll think that's the cool part.
Given that, I can't see a reason to hold off introducing algebra. Once a kid is at the sixth or seventh grade level in math, the heck with guess-and-check and pan-balance problems; the heck even with bar models. The most general tool that we currently have for solving word problems, and the only one that we have that isn't stymied by some word problem or other, is algebra. He may as well be motivated to go full speed ahead with the letters and symbols. Wickelgren says that algebra is the key to the castle; it's the most effective means for solving tricky math problems that's ever been devised. As such, you want it to be the tool that kids reach for instinctively when they have a tricky math problem to solve.
Here's a quote from a great article by Ethan Akin, "In Defense of Mindless Rote":
On the other hand, mathematics is cumulative and there are a great many skills that you have be unthinkingly familiar with. Every grumpy calculus teacher will tell you that most of the problems his students have come from weaknesses in algebra. For the students who say "I really understand it but...." the but is that for them algebra is not easy background knowledge. They are trying to build on a foundation of dust. A lot of college majors need a bit of calculus or statistics which are simply walled off to students who don't have sufficient skills in algebra. These are basically not hard subjects but they appear unnecessarily terrifying to such students.
Conversely, a practiced facility with algebra can provide its own positive reinforcement. Not only is the mathematics built on the algebra, but facility in algebra gives the student confidence in the face of new mathematical challenges. As the above discussion makes clear, such confidence is entirely justified.
I am motivated now to try to introduce real algebra by the end of the summer. No more pussyfooting around!
Wickelgren on introducing algebra
Wayne Wickelgren on algebra in 7th & 8th grade
Wickelgren on math talent & when to supplement
late bloomers in math & Wickelgren on children's desire to learn math
Wayne Wickelgren on mastery of math & on creativity & domain knowledge
Wickelgren on why math is confusing
CalculusProfessorEmailExchangeWithParent 01 Aug 2006 - 19:53 CatherineJohnson
Number 2 Pencil links to this exchange of emails between a math professor and one of his students, who is flunking calculus.
The professor is using a pedagogy known as Process Guided-Inquiry Learning, or POGIL:
I mentioned in an earlier post that one of the more controversial -- and to me, appealing -- aspects of POGIL instruction is that the instructor is not seen as a source of knowledge but rather as a facilitator of learning. In non-eduspeak, this means that the instructor is there to observe and to guide, rather than to tell students what to do or think.
I also mentioned, and one commenter pointed out, that students HATE this (at least at first). Typically, students -- even upper-level students in the major who have been around the block -- just want to get the darn problem done, and when people like me insist on students asking the right questions rather than just forking over the answers, things can get heated.
In practical terms, POGIL means that when 'Pat' comes to his professor's office for help, the professor refuses all requests for one-on-one demonstration of the problems being taught in class:
…when Pat would ask me a question such as, "Can you tell me how to do problem 7?", I would say: Let's start by asking the right questions. What are you being asked to do in this problem? What information is given to you in the problem statement? And what do you know from the course, your reading, or your work on other exercises that will help get you to the goal? I made it a point to NEVER give Pat explicit help on content unless it was a last resort -- Pat absolutely HAD to cut the apron-strings from me an learn how to approach, analyze, and solve a problem alone, or else Pat's chances for success in a future career or even making it through college didn't look good.
This goes nowhere.
Finally 'Pat' sends an email explaining that he requires direct instruction in order to learn.
The professor tells him he is wrong.
Pat sent me an email just after midterms that said something like: I now understand why I am not doing well in your class. My learning style is such that you have to show me exactly what to do, or else I can't do it. But you always answer my questions with more questions, which isn't showing me exactly what to do. So from now on, please show me exactly what to do first, and then I should be able to do it. My response was something like: Pat, we've been doing this every day in class -- I work a few problems at the board all the way through during lecture, and then I give you exercises that are based on the stuff you've seen. So you are seeing me show you what to do, and yet you're still having difficulty solving problems on your own. So perhaps your assessment wasn't quite right, and we should be working on your problem-solving skills in office hours.
Then Pat's mother gets into the picture.
(via email): I know [Pat] tried to explain to you that when [Pat] asks questions [Pat] needs answers not another question. We had [Pat] tested at [a local university] in January through the suggestion of [an academic counselor at my college].
During this testing we found out [Pat] has a learning disability. [Pat] does better with visual explanations then being asked another question. [Pat] needs to see how to physically work a problem so he can comprehend it. [Pat] is a slow reader which also frustrates [Pat]. If it is a word problem [Pat] has problems figuring out what are the essential parts of the question to find the answer.
This infuriates the professor, and, subsequently, all of his commenters as well, who pretty much stomp mom to death in the comments thread.
Pat fails the class.
The commenters are united in their view that Pat is a lucky guy to have experienced POGIL calculus, and he had no business hosing the course.
Memo to self: the time to begin instilling core take no course from professor who blogs principle in 10-year old son is now.
POGIL
POGIL, POGIL, POGIL
This does not sound good, POGIL.
I should reserve judgment.
I should, but every one of the little-red-light thingies on my Constructivist Nightmare Detector is flashing wildly, and all the sirens are going off—
So I’m not doing a very good job of reserving judgment.
POGIL is a student-centered method of instruction that is based on recent developments in cognitive learning theory and results from classroom research that suggest [sic] most students experience improved learning when they are actively engaged, working together, and given the opportunity to construct their own understanding. POGIL emphasizes that learning is an interactive process of thinking carefully, discussing ideas, refining understanding, practicing skills, reflecting on progress, and assessing performance. In a POGIL classroom or laboratory, students work on specially designed guided-inquiry materials in small self-managed groups. The instructor serves as facilitator of learning rather than as a source of information. The objective is to develop skills as well as mastery of discipline-specific content simultaneously. (Emphasis added)
OK, that does not sound good.
I'll get to the professor’s various posts on POGIL as soon as I can.
I do want to read them.
But in the meantime, there's one homeschooling mom on the thread (son has LD) whose comments make sense to me:
Pat says clearly that your Socratic style doesn't work for [him]. Why do you then believe that it does work? You (rightly) want [him] to learn problem solving, but just because your method of teaching ... works for others doesn't mean it works for [him]. Maybe [he] needs repetition, repetition, repetition of the underlying content before [he's] ready for process. Other students may grasp the process after going through the underlying solutions three times, or six times, but maybe [he] needs thirty times.
You model the problem solving that you want [him] to do-- Where have you seen a problem like this? What rule did we use?-- but in a sense that's no better than modeling the actual rule for Problem 13. You want [him] to intuit that [he] is supposed to be asking [himself] those questions. But what if, as seems to be the case, [he] isn't intuiting that? Then [he's] not learning anything.
The bad news here is that, clearly, constructivists are giving lots of workshops to math professors.
Even worse, math professors are attending them.
inflexible knowledge, flexible knowledge, and expertise
One of the problems with constructivism (and, apparently, with POGILism) is that it tries to teach higher-order problem solving first, instead of second.
That option probably isn't on the menu.
According to Daniel Willingham, knowledge is always inflexible before it's flexible. You can't hopscotch over the inflexible stage by teaching process, or asking students to discover addition.
Problem solving and critical thinking seem to grow out of extensive practice of surface, shallow, inflexible knowledge.
I’d like to know more about how this happens.
At a minimum I’d like to know what cognitive psychologists (psych department cognitive psychologists, I mean) understand about the process at this point.
And I hope that Robert, who writes the brightMystery blog, will join us at KTM once in awhile to think about these things.
update
WelcomeRobertTalbert
MeasurementAdviceFromCarlL 08 Jul 2005 - 21:46 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:
- Metric OK, Inches Not -- While the kids can't (or won't) measure in inches, many (but not all) can measure using a centimeter ruler. Fractions rear their ugly head again.
- Estimation, Schmestimation -- The kids do not know when it is, or is not, appropriate to estimate. The kids have trouble estimating measurements between the lines of the ruler. But the kids are very willing to make bad estimates to avoid having to figure out what the little lines mean. 2 5/16 inevitably becomes 2 1/2.
- What is a protractor? -- The kids REALLY don't know how to use a protractor (except as a frisbee). Most don't even know that its purpose is to measure angles.
A side note related, I believe, to measurement. Each year I do a lesson where we compare the kids height in inches to their shoe size. The majority of the kids do not know how tall they are, let alone how to convert the height in inches.
So by all means get a ruler, protractor, some measuring cups and spoons, and a kitchen scale (or even better a pan balance) and start measuring everything around the house!
I intend to take this advice.
SummerProgramUpdate (measurement skills)
EarthboxDay
EarthboxDay 21 Nov 2005 - 04:14 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 Math
To 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 Investigation
Christopher 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.
update
I 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.”
Mary M. Forestdale, MO.
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, 2005
Green 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
UKFrameworkForAlgebraPreparation 08 Jul 2005 - 17:13 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.'
SuchACuteAngle 07 Jul 2005 - 22:21 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.
HappyJulyFourth 22 Jul 2005 - 18:04 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!
LoneRangerHomeschoolerReportsIncredibleMathProgress 11 Apr 2006 - 20:55 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!
update
This 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
MoreFromLoneRanger 11 Apr 2006 - 20:55 CatherineJohnson
I wanted to make sure everyone saw this follow-up (I've added bullets & formatting because Jakob Nielsen told me to):
- I used Singapore math books 2B, 3A, 3B and half of 4A before having my daughter take the ITBS test Iowa Test of Basic Skills.
- 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. [ed: Saxon at Home School Center may not be more expensive; I'll check.]
- 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, I find 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. [ed: oh! are these what I call 'bar models'? If so, I'm getting incredibly good at them myself.]
- 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. [ed. note: based in my own experience, I think it's a good idea for parents to learn & re-learn elementary maths along with their children.]
- 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.
LoneRangerHomeschoolerReportsIncredibleMathProgress
PriceComparisonSaxonSingapore 13 Nov 2005 - 18:47 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
$8.00 4A workbook
$8.50 4A Intensive Practice
$6.80 gr 4-6 Answer Book
$8.00 4B textbook
$8.00 4B workbook
$8.50 4B Intensive Practice
$55.80 total Singapore Math 4A & 4B
Singapore Math 4A & 4B w/Home Instructor's Guide
$55.80
$14.95 Home Instructor's Guide
$70.75 Singapore Math 4A & 4B & Home Instructor's Guide
Singapore Math 4A & 4B 'the works'
2 textbooks, 2 workbooks, 2 intensive practice books, 1 'Challenging Word Problems' book, answer book, home instructor's guide
$70.75
$7.80 Challenging Word Problems [I love this book!]
$78.55 total, Singapore Math 'the works'
Singapore Math 4A (one semester)
$46.25, roughly
bang for the buck
Singapore publishes its textbooks by the semester, Saxon by the year.
So if you're going to experiment with a curriculum to see how it goes before making a commitment, it's cheaper to start with Primary Mathematics, U.S. Edition.
Once you're committed, however, you'll end up spending about the same for either one.
Unless you get fancy and start ordering all the Singapore Math extras.
Which you will.
update
OK, ktm readers are much more disciplined than I am.
see Comments
ChallengingWordProblems 07 Jul 2005 - 22:16 CatherineJohnson
Here's where to order Singapore Math Challenging Word Problems Book 3 if you're interested.
I love them.
I've done all of Book 3 myself, and will start Book 4 when I'm finished with Russian Math.
UPDATE 10-4-2006: I've only done a handful of the Book 4 problems, but I have begun to create a complete, hand-drawn solution manual. Don't ask me why. I was in Cambridge last spring, cruising Bob Slate Stationer's, when I spotted an expensive spiral-bound acid-free quadrille paper notebook that cried out to become a solution manual for Challenging Word Problems Book 4.
So I'm making a solution manual.
- almost 300 problems per book
- coherent groupings of like problems with like
- each problem set divided into a less difficult & more difficult group
- each problem set opens with 3 worked-out bar models
- all answers (in numbers, not bar models) in back


source:
artstuff.net
HaroldJacobsAlgebra 07 Jul 2005 - 23:59 CatherineJohnson
Someone (Lone Ranger? Susan?) asked about Harold Jacobs' Elementary Algebra text.
I've heard lots of good things about the book, the Amazon readers all rave about it, and it turns out Barry Garelick thinks Jacobs' geometry text is good.
ELEMENTARY ALGEBRA has been sitting in my Amazon cart for awhile, so I'll have it pretty soon.
In the meantime, I found a reader review that sums up the approach to teaching our kids that I've come to believe in:
At the time I started homeschooling my sixth grader last year, I was completely math-phobic. I had forgotten every bit of algebra I ever learned (and any math I did learn in high school, more than 20 years ago, was just barely learned at that). My now seventh grade son and I are learning algebra together with Harold Jacobs's Elementary Algebra book.
This is really an exceptional self-study guide. We will read a chapter, then independently try to solve the problem sets given. We then compare our answers. If our answers don't agree, I will either explain to him how I solved a problem that he got stuck on, or vice versa.
The delightful thing about this book is that I am learning to enjoy a subject I always thought I detested. Harold Jacobs makes everything clear, comprehensible, meaningful and often humorous. I am learning that I am not left-brain impaired, as I've thought I was ever since second grade, and actually look forward to my algebra time with my son! My son, too, has overcome his own math phobia, and become a math lover. I can't recommend it highly enough.
This reminds me of articles I've read about Chinese-American families.
Apparently the whole family sits around the table at night, even the little kids, and does schoolwork together.
If the school doesn't send home enough to do, the parents add more.
That's what we're doing these days.
update
I just looked at the Amazon reviewer's web site.
It's great.
She and her husband are escapees from the city who've taken up farming & homeschooling.
I must say, the farm I grew up on looked nothing like this:
(you can click on the photo)
Although we did have a big red barn.
ChristopherOnSingaporeMath 08 Jul 2005 - 00:02 CatherineJohnson
Christopher managed to bargain me down today.
Instead of doing:
- Megawords 2, Worksheet 10-J
- Saxon Math 8/7 Lesson 11 Mixed Practice
- Saxon Math 8/7 Lesson 12 Warm Up
- Saxon Math 8/7 Lesson 12 Lesson
- Saxon Math 8/7 Lesson 12 Lesson Practice
- Math Olympiads: 1 problem
he's doing:
- Saxon Math 8/7 Lesson 12 Mental Math
- Primary Mathematics 3A Workbook, problems 8, 9, & 10
So maybe he has a future as an agent.
He just looked up from his bar modeling and said, 'I like the problems in Singapore Math.'
I said, 'You do?'
'Yeah.'
'How come?'
'They're not stupid.'
No idea what that means.
update
Christopher got all 3 of his bar model problems right today. (ummm....no, he didn't. He flubbed the arithmetic on the first one, but he got the bar model almost exactly right.)
I checked his answers & models, and when we got to the 3rd problem, he said confidently, 'This one's a two-parter.'
I was happy to hear that.
I think this signals a new category inside his mind.
- one-part problems
- two-part problems
He can tell the difference!
what bar models do for your brain
I'm trying to figure out how to write about bar models and what I think they do for my 'math brain.'
It's incredibly difficult to articulate, and will involve printing out sample bar models, scanning them back into iPhoto, and reducing the image size...so it will be awhile.
But I'll get there.
For the time being, I'll say that I could do the 3-variable problem from Primary 6 that Carolyn posted using algebra.
But I couldn't do it using a bar model.
There's a reason for that, but I'm going to need visuals to express it.
OTOH, once I'd done the problem algebraically, I realized how to interpret the (correct) bar model I'd drawn--thanks to the Math Olympiads problems I did this weekend.
So today's hypothesis is that the perfect 'problem-solving' curriculum for me would be an amalgam of PRIMARY MATHEMATICS & MATH OLYMPIADS.
math-heads & word-heads
Carolyn has mentioned that mathematicians think facility with geometry may be a good indicator of mathematical talent.
I wouldn't be remotely surprised to find out that's true, if only because of the connection between spatial-visual ability & maths. (I've decided I like 'maths' better than 'math.' fyi)
I don't remember having trouble with any of the high school math I took. (Maths!) It may have been an easy curriculum, I don't know.
But I do remember having lots of fun with algebra. The X's and the Y's and all the neatly stacked-up linear equations....it all just felt right.
I could still solve a two-variable equation 30 years later, without even having to think about it.
This has made me wonder if there is something 'word-like' about standard algebra.
Temple, btw, absolutely could not learn algebra.
She's a brilliant person, but algebra was out.
'I couldn't make a mental picture of it,' she told me. 'It was too abstract.'
I have to remember to ask how she did with geometry the next time we talk.
AWonderfulGame 08 Jul 2005 - 21:42 CarolynJohnston
AnneDwyer has a wonderful math game for kids that she wrote about on her wiki page.
The kids pick the number of digits (we usually start with 5). They put 5 dashes on their paper. I turn over 5 cards in a deck one by one. They have to decide where to put the numbers. Then each kid reads their number to me while I put it on the white board. The kids with the highest number wins.
For some reason, they love this game. On the next round, we go up one digit. Today, we went all the way up to 100 million.
It's a great game.
- They gain familiarity with large numbers. They get a lot of practice with reading large numbers out loud and hearing large numbers read out loud while it is being written on the board.
- They have to use strategy. In some games, we have a lot of high numbers at first which every kid puts in the same place. Then, they winner is the determined by the numbers in the ones and tens place. Conversely, sometimes we have a lot of low numbers in the beginning. Then the winner is determined by the highest digits. Much more interesting is when we have medium and low cards. Then, they have to do a lot more thinking about where the cards go.
- There are very concrete results from this game that allow us to explore numbers even further. In one game, 5 out of 8 kids had the same highest number. So we talk about why and when does this happen? In one game, we had one winner that was a lot higher than anyone else. When does this happen?
We have a gang of kids that run semi-wild in our neighborhood in the summer. They are very mixed in age (ranging from 7 through 11). I have thought about corralling the whole lot of them and bringing them in to teach them all some math together; it would do them all some good to work on it over the summer, and Ben would enjoy his math sessions more if he shared them. I'm a little stumped, though, about how to teach a wide range of ages and interest levels simultaneously.
I'd love to collect some more math games that are as simple and elegant as this one is, especially games that might appeal to a broad range of ages, and (like this one) start a math session off on the right foot.
MathBootCamp 11 Jul 2005 - 19:40 CatherineJohnson
Christopher at breakfast this morning:
What if there's a math boot camp?
Then it'd be like, Come on, maggot! Drop and give me 20 multiplication problems!
Come on! Go faster! Go faster!
For that sluggish work you're gonna have to do 20 more division problems and scrub the toilets for a week! While doing mental math!
Do I make that clear?'
Sir! Yes, Sir!
update
Christopher says I have to give him 50 dollars for the licensing of this post.
Way too much WWE wrestling around here.
Time for Boy Scout camp.
VisualLearningKThru2WikiPage 17 Jul 2005 - 16:51 CatherineJohnson
The comments thread of this post will be the page Becky C requested on the use of direct, explicit visual aids in teaching K-2 math.
Everyone can comment, edit & revise, so please share your experience & thoughts.
OneParentsConversion 29 Jun 2006 - 16:25 CatherineJohnson
Susan just pointed me to the most amazing personal story at Illinois LOOP.
I'm bulletting the main points from the introduction for readability:
- author is a mother of 3 in a state that instituted progressivist reforms in the early 90's.
- she and her husband hold doctoral and masters degrees in non-education fields and provided their children an enriched environment
- all 3 children have professionally-assessed aptitude in the superior range.
- their complete reliance on teaching professionals and progressivist methods resulted in learning difficulties resembling those of the 'disadvantaged'.
- after-school remediation of elementary skills has, over the course of about 18 months, made significant improvement to the children's grade-level achievement and attitudes toward learning.
- family will now homeschool 2 of the children next year using the 'classical' method.
read the whole thing
update
I'm halfway through the story—it's incredible.
Read this:
By the end of elementary, we acknowledged to ourselves that something had gone badly wrong, though the causal link from early elementary instruction was not yet clear. It was easier to place blame on ourselves, on an exaggerated sense of homework neglect. Still, we took the precaution of moving the children to a private school billed as 'traditional' - only to eventually discover it to be an upscaled version of the progressivism offered at no extra charge by the public school next door. That discovery, too, was years in coming; I was so consumed with the career that paid the tuition that I barely took note of the continuing deterioration in scholastic achievement, much less delved deeply into the reasons why.
2 themes:
- parents knowing there's a problem, but not knowing what it is
is it your child? (poor aptitude, 'average' ability, 'math-reading-spelling's not his thing,' watches too much TV')
is it me? (didn't supervise the homework, wasn't paying attention, didn't read the stuff in the backpack)
is it the teacher? (It didn't cross my mind that Christopher's problems in math might be related to the textbook until I tried to teach out of the it myself. And not until this year did I begin to perceive problems with the system (no core curriculum, no articulation between grades & schools, etc.)
- private schools just as pervaded by constructivist philosophy as public
update 2
oh boy. this is harrowing:
What was it that finally broke through my unquestioning faith and mindless optimism? A recognition that certain elements of a 7th grade math program were badly askew, some research for purposes of a teacher conference, and finding the Mathematically Correct website. A binge of research ensued which continues to this day.
As full understanding of how progressivism had failed my children finally dawned, I was furious - more with myself than anyone else. But, I can no longer spare the emotional energy which anger consumes. It takes all I've got to stay attuned to three children from 3:00 to 10:30 p.m. sufficiently to correct Kumon math, direct grammar remediation, go over their SRA reading comprehension work, monitor the writing process program, and check assigned homework for the knowledge gaps which have undermined so much prior learning...and somehow attend to the non-tutoring aspects of parenting.
7th grade.
That is horrifying.
My perception—and I hope everyone will chime in on this—is that many parents hit the wall at the end of 4th or 5th grade.
I've heard through the grapevine that there are lots of unhappy 5th grade parents here thanks to the TONYSS tests.
(The TONYSS aren't mandated by the state, and aren't the same test everyone has to take in 4th and 8th. They're created by a private testing company, and purchased by individual school districts.)
The TONYSS are graded on a scale of 1 to 4.
Almost no one earned a 4 on the English language arts half. Only 2 children in Christopher's class of 19 kids got 4s, Christopher being one.
(Poor thing. Christopher's glaring, obvious talent in life is not math. It's history & social science. Not surprising given that his father is a historian.)
Back to the TONYSS. There were 4 or 5 kids in Christopher's class who earned 4s on math.
It sounds like a lot of kids who had been getting good grades all school year suddenly came up with 2s & 3s on the TONYSS.
I could be wrong about this.
But that's what I'm hearing.
For me, Christopher's '39' on Unit 6 at the end of 4th grade was a lucky break.
Even Christopher said the same thing last fall.
He actually said, 'If I hadn't gotten a 39 you wouldn't have started teaching me.'
Up til the moment Christopher came home with that 39 I had no clue there was anything wrong with U.S. education that couldn't be fixed by moving to a super-expensive suburb and paying a small fortune in property taxes to get small class size and high per-pupil spending.
When it came to education, the sum total of my sophistication was 'you get what you pay for.'
update 3
I've felt anger, but there are no easy targets. I knew every teacher and administrator involved. I knew that they had cared about my children and appreciated my work on behalf of the district; many of them are my friends. I saw them as well-intentioned, doing their best to use effectively the pedagogical tools to which they were limited by the progressivist reform vision that had been imposed from a policy level, one in which millions in professional development funds were being invested.
Check, check, & check.
This is what I've come to realize: the problem is at the 'system' level...
You can certainly have a bad teacher; I think we've had one so far. (She was a terrific lady; I feel bad saying anything publicly. But she didn't seem to be able to teach math out of the SRA book, something I couldn't do, either.)
I love this, too:
If I have anger left for anyone, it is the educationalists who control accreditation standards that shape teacher training and professional development, and incidental to such, education policy.
[snip]
...for all their power to effect or impede change at the critical level of teacher training, this is the last group to feel the heat of public accountability. They will never have to confer with the parent of a 4th grader who can't read. They will never see a performance review based on the achievements of their students. They will never face the electorate with their records. And they are, in a practical sense, insulated from legal liability for malfeasance.
I'd like to file a class action suit against Columbia Teacher's College.
CanChildrenMakeUpForLostTime 11 Jul 2005 - 18:06 CatherineJohnson
I'd like to put this question out to readers of ktm:
Can children make up for lost time?
I ask, because I've now read at least 5 personal stories of children or young adults struggling to make up ground they lost to bad curricula.
Some of the most hair-raising stories I heard from Carolyn were about college kids who simply could not learn algebra because they didn't get what they needed in grade school mathematics.
Carolyn made me wonder whether there might be a critical period for learning math the way there is for speaking a foreign language without an accent.
I've come to think there isn't, mainly because I find it possible (and pleasurable) to learn math as an adult, and I don't think I'm unique.
I started thinking about this because last night I did an impromptu interview with my cousin who, it turns out, pulled her daughter from public school because of a wretched experience with Everyday Math. (I'll post it shortly.)
Her daughter used Everyday Math for 3 years, from 2nd to 4th grade.
Then it took her 'several years' to make up the lost ground.
She just finished her freshman year in high school, and is doing great in high school math. (Her private school used Saxon.)
I talked to another woman who pulled her son out of the Tribeca schools because they use TERC.
He's now high school age and still doesn't have rapid fluency with his math facts. (She spent a lot of time working with flash cards, too. Another flash card failure.)
How can we remediate kids who've fallen behind because of constructivist math?
two immediate thoughts
To me, it seems like it has to be possible to make up lost ground more quickly than this.
At least, I hope so.
Here are my first thoughts:
- remediation has to mean doing timed worksheets every day, day in and day out, until the child or young adult has his calculations down cold
- remediation also means doing story problems every day, day in and day out (probably a coherent sequence of story problems, like those in the Singapore Math Challenging Word Problems books) [I have no idea how many story problems to do per day]
- finally, remediation may mean that you need to back up to the beginning of math, or close to: back up to content well before the point where the child became lost--and move quickly through a coherent 1st, 2nd, or 3rd grade curriculum, regardless of the fact that the child or young adult already 'knows' most of the material
I'd love to hear people's thoughts.
WillinghamOnLearningModalities 22 Jul 2005 - 20:14 CarolynJohnston
From Daniel Willingham on learning modality theory, an explanation of why learning modality theory might make sense from a teacher's viewpoint:
There are two ways that a teacher might see what looks like evidence for modality theory in the classroom. First, a teacher who believes the theory may interpret ambiguous situations as support for the theory. For example, a teacher might verbally explain to a student - several times - the idea of borrowing in subtraction without success. Then the teacher draws a diagram that more explicitly represents that the 3 in the tens place really represents 30. Suddenly, the concept clicks for the student. The teacher thinks "Aha. He's a visual learner. Once I drew the diagram, he understood."
But the more likely explanation is that the diagram would have helped any student because it is a good way to represent a difficult concept. The teacher interprets the student's success in terms of modality theory because she has been told the theory is correct and because it seems to explain her experience.
Willingham offers the following suggestion: teach to the best modality for representing the idea, not to the student's best modality.
But what if there are multiple modalities to choose from, for an idea? More generally, what if there are a whole host of different ways to represent an idea, and the kid's not getting any of them?
I ran into that situation recently, when teaching Ben how to do simple problems by adding and subtracting constants on both sides of an equation. Actually, trying to help Ben get the hang of this has taken quite a bit of effort this week, and I don't think it's a hard idea. I've got kinesthetic, visual, and auditory ways of teaching it, too. I could even sing it, though that's getting a bit ridiculous.
For the kinesthetic learner, you could get out a balancing scale or use Bornstein manipulatives. You could draw pictures of pan balances for a visual learner. You can explain verbally, as I did repeatedly, that what you're doing to solve the problem x + 4 = 13 is to 'undo the addition' of the 4 on the left hand side of the equation. If none of this works, what do you do then?
Try each modality over again, I suppose. Round 2: in case he was a kinesthetic learner, I had him copy each step I made in his own handwriting (laugh, if you will, but it works for me when I do it). In case he was visual, I drew pan balances again, next to the equivalent equation: no dice. "Subtracting the 4 is applying the inverse operation to get the x by itself," I said, auditory-like, but that didn't help either.
All this time, of course, he was able to do the problems by repeating the steps I made; he is a fabulous rote learner (is 'rote' a modality? If not, it should be). But I could tell he wasn't really getting the gist of it. Finally, in exasperation, I said, "Look, Ben, what's the opposite of adding 4?
"Subtracting 4."
"Good! And what's the opposite of subtracting 13?"
"Adding 13."
"Good. All you're doing to get the x by itself is doing the opposite of adding or subtracting the number that's with it," I said, but I didn't even get it all out before he said, "OH! I get it!"
And that's the sound I love to hear.
So, knowing Ben's best learning modality didn't help, and wouldn't have helped. I wish teaching, and learning, were so predictable that all you needed to do to teach a whole class reliably was to know what each kid's best learning style was. But I think that learning is inherently unpredictable. The trick is to be able to hit the teaching problem from a bunch of different angles, and you need to know lots of different ways to present the information. The more, the better (by the way, this is a major part of what Liping Ma's Chinese elementary math experts do with their release time; sit around together, thinking up new ways to teach problems to tough cases).
As an aside, I have never been able to figure out Ben's best learning modality (aside from 'rote'. His raw memory is unbelievable). As a person on the autism spectrum, he's supposed to be a visual learner; this is accepted theory to such a degree that teachers will assume he needs to learn visually, but it's not always the right approach.
What Ben really is, is an unpredictable learner. You never know what's going to be easy, where he'll get stuck, and what will unstick him. He's the kind of kid who keeps a teacher on her toes.
BestPractices 15 Jul 2005 - 21:16 CatherineJohnson
Just came home to the WillinghamOnLearningModalities thread—incredible!
The comments reminded me of my favorite teacher-mentor story.
A friend of ours here in Irvington--he's now chair of the University of Iowa College of Medicine*—told us this story about his own mentor at Columbia Med School.
This particular teacher was legendary. Everyone who was anyone—boatloads of brilliant future researchers & clinicians—were taught by this man.
So what was this fellow's main piece of wisdom, which he conveyed to each & every one of his dazzling students?
If what you're doing isn't working, try something else.
*
This has caused Christopher to tell us, frequently, that 'Daniel lives in a mansion in Iowa!'
MathAndTextPrototypeLesson 21 Jul 2005 - 13:56 CatherineJohnson
When I was in graduate school (DID I MENTION THAT I HAVE A PHD IN FILM STUDIES?) one of my professors told me that the definition of a reader is a person who owns more books than he can read before he dies.
I have now updated that definition for the impending ERA OF THE BLOOKI.
The definition of a reader is a person who owns so many books she can't even get her own web site read before she dies.
Now that's out of the way, I have managed to make a circuit of my favorite blogs this afternoon--and have discovered that J.D. has his prototype lesson up at Math and Text!
It looks wonderful.
I'm going to read it now.
update
It is wonderful.
I love clean, lots-of-white-space invitations to maths...and there was something about the final lesson on figuring out which number is larger that made me happy.
I had the 'click' sensation Carolyn Morgan talks about.
That sensation is so reinforcing, that I think it ought to be an item on textbook write's & editor's lists: Does the student feel a click?
I was confused by just one part of the lesson, which was the first visual display. A middle school teacher has left a detailed comment explaining why she stumbled over it, too.
Take a look.
update 2: more on the click
I'm realizing I've had many, many conversations in which people who like math bring up the click--that moment of knowing you've got it.
Either you've got the right answer, or you've got the concept.
That's what my cousin was talking about when she said it's incredibly boring never to know whether you got the right answer or not:
It’s boring when you don’t have the light bulb go off in your mind because, ‘Oh! I got it right!’
The best you could think was, ‘Well, maybe I got it right.
Our friends Fred & Wendy were here a couple of weekends ago, and Fred said exactly the same thing about maths.
He loved maths (I may have to give up on 'maths'....) and he wanted to study it at Yale, as an undergraduate. What he especially loved was the click.
He quickly realized that college-level maths was a different animal, and he shifted to statistics, eventually earning a Ph.D. in experimental psychology (and then a law degree after that).
Fred is a seriously smart guy (clerked for one of the Supremes, etc.).....and what's he talking about when he remembers math?
The click.
FirstPerson (interview with my cousin about Everyday Math)
WhyIsSubtractionHarder 18 Jan 2006 - 14:23 CatherineJohnson
Christopher is sitting here doing his mixed practice, and he just asked me, "Why is subtraction harder than addition?"
He was doing the problem:
$20 - e = $3.47
I have no idea why subraction-with-borrowing is harder than addition-with-borrowing, or even if it is harder.
I'm asking all of you because I've noticed that sometimes the answer to incredibly simple-seeming questions tell you a huge amount that you didn't know before. Can't think of any examples offhand, but I'm going to start keeping track.
update
Oh!
It's probably the left-to-right issue, yes?

TutoringAdvice 24 Jul 2005 - 01:55 CatherineJohnson
I'm probably going to spend some time working with a friend of Christopher's on his math.
They're the same age--both going into 6th grade--and my sense is that math is probably this boy's strong suit.
I just gave him the Saxon placement test, and he placed into Saxon 7/6, which is the 6th grade book.
That would be great, but here's the hitch: he has been taught almost nothing about fractions at all. (He had a good math teacher--he and Christopher were in the same Phase 3 class for the first half of this year--who left to have a baby. So it seems that the subject of fractions & decimals fell through the cracks.)
So.....if anyone has thoughts, I'd like to hear them. I'll probably go ahead with 7/6, but that means I'm starting a 6th grade book with a child who's been taught virtually nothing about fractions and decimals.
update
Here's the fraction worksheets site Carolyn J found.
whose job is it, anyway?
This is the kind of thing that I just don't get.
Why should I be the person figuring out that this boy hasn't been taught fractions & decimals?
Why shouldn't the school be figuring this out? (Yes, the school might say he was taught fractions and decimals, but didn't learn them. However, it's clear to me that there are certain topics he simply hasn't even heard of, because with some topics he'll say, 'I kind of remember that.' In other words, he can tell me which topics he failed to learn, or didn't learn well enough to retain, or whatever it is. With topics like adding fractions, he simply doesn't know anything about them, and has no memory of having been taught.)
So, yes, the school might say, 'He was taught, but he didn't learn.'
But so what?
If he was 'taught' and 'didn't learn,' then he wasn't taught as far as I'm concerned. It's the school's job to perform formative assessment to know what students have and have not learned.
Then it is the school's job to re-teach if a child has not learned.
Then, if the child still isn't learning, it's the school's job to figure out what else he needs.
I don't want to take this too far, of course. Parents & students are responsible, too:
That's one of my major concerns with NCLB. When students don't do their homework or study for exams, or even attempt to do classwork, it's still considered to be the teacher's fault if the students don't achieve their federally-mandated level of proficiency in reading, math, and science.
And yet NCLB doesn't give me, as a teacher, the authority to require student's who aren't even attempting the work to stay after school and complete their assignments. Unless the kid has committed some breach of the school's disciplinary policy, I can't keep them any later than the school regular dismissal time.
The No Child Left Behind Act holds me solely accountable for my students' academic progress but doesn't give me the authority to help make that happen, especially for children that are considered to be "at risk" of failing to meet minimal standards of academic progress.
Sadly, under the law as it is now written, a large number of children are going to be left behind.
He's right. If a student doesn't do his work, and the parents don't require him to do his work, that isn't the teacher's responsiblity.
But that's not the case with Christopher's friend. This boy has done all of his work; he's a serious student; his parents are serious parents.
It's the school's responsibility to know whether this boy has or has not learned how to add, subtract, multiply, and divide fractions, and to teach or re-teach the subject if he hasn't.
QuestionAboutReciprocals 31 Jul 2005 - 18:04 CatherineJohnson
My 'benchmark' for the moment when I understand elementary mathematics well enough to move on is:
reciprocals
I find reciprocals utterly mysterious.
They're not quite in the magic category anymore, which is my benchmark for complete and total lack of comprehension. If a maths concept seems like magic, that means I know nothing.
Of late, inside the expanding math section of my brain, reciprocals have put a toe outside the magic category. But not much more than a toe.
Danica McKellar
Tuesday's SCIENCE TIMES has a profile of Danica McKellar, who played Winnie on Wonder Days. It turns out she's a UCLA math major who published a proof (pdf file) now known as the Chayes-McKellar-Winn theorem.
car wash problem
McKellar's web site has a mathematics section where she answers reader questions. She's a natural born teacher:
Q: Hi Danica, I heard a question from Mr. Feenie on a "Boy Meets World" episode which he claimed to be unanswerable. After hearing that, I decided to figure it out. If it takes Sam 6 minutes to wash a car by himself, and it takes Brian 8 minutes to wash a car by himself, how long will it take them to wash a car together?
Danica Answers: Hm, unanswerable? That's TV for you. :)
Let's do it: This is a "rates" problem. The key is to think about each of their "car washing rates" and not the "time" it takes them. Alot of people would want to say "it takes them 7 minutes together" but that's obviously not right, after you realize that it must take them LESS time to wash the car together than either one of them would take.
So, what is Sam's rate? How much of a car can he wash in one minute? Well, if he can wash one car in six minutes, then he can wash 1/6 of a car in one minute, right? (think about that until it makes sense, then keep reading). Similarly, Brian can wash 1/8 of a car in one minute. So just add their two rates together to find out how much of a car they can do together, in one minute, as they work side by side on the same car: 1/6 + 1/8 = 7/24 of a car in one minute. That's their combined RATE. (Note: that's a little bit less than 1/3 of a car in one minute). From this point, the way you want to think of it depends on your favorite way of dealing with fractions. You now have their rate. It's 7/24 cars per minute. You can either just take its reciprcal and say: 24/7 minutes for one car, and you're done.
Or, equivalently, you can think of the 7/24 cars/minute RATE as 24 minutes for 7 cars. (think about that until it makes sense, too) So just divide 24 by 7 to find out how many minutes it would take to do just one car. You get around 3.42 minutes for one car, just a little less than 3 and 1/2 minutes. Done! Yes, I think they should work together, it gets done much more quickly that way. :)
By the way, you said when you watched the TV show you decided that YOU would figure it out, right? How did you do?
I love this. McKellar is teaching two things here:
- how to solve a rates problem
- how to read, study, & learn maths (that's metacognition)
First of all, she knows that math novices transfer their normal reading habits to maths books. By normal reading habits I mean that most of us, when we read a book of prose, read straight through at a fast clip, pausing only to underline or make notes in the margin.
You can't read a math book that way; in fact, I've come to feel you can't really read a math book at all. You have to do a math book, or work a math book. McKellar explicitly instructs her reader not to read the solution straight through, but to stop at key points and ponder until he or she gets the point, and is ready to go on to the next point.
She doesn't stop there, either. She also knows the precise spots in her explanation where most novices will need to stop and mull, and she tells them where those spots are.
She's giving novices direct instruction in metacognition.
As to the problem itself, she addresses the most common error novices will make confronting this particular rates problem, which is:
- figure that it takes the 2 boys 14 minutes to wash 2 cars
- so logically it must take them 7 minutes to wash 1 car
Amazing! And all in the space of a few short paragraphs.
I think McKellar's teaching skill here is connected to her acting. There's a large element of 'performance' in teaching, at least in my experience, and to be a good performer you have to know where your audience is, what they want to hear & what they need to hear.
She does.
back to reciprocals
Here's my reciprocal question.
From this point, the way you want to think of it depends on your favorite way of dealing with fractions. You now have their rate. It's 7/24 cars per minute. You can either just take its reciprcal and say: 24/7 minutes for one car, and you're done.
Or, equivalently, you can think of the 7/24 cars/minute RATE as 24 minutes for 7 cars. (think about that until it makes sense, too) So just divide 24 by 7 to find out how many minutes it would take to do just one car. You get around 3.42 minutes for one car, just a little less than 3 and 1/2 minutes. Done!
I don't understand why you would use the reciprocal to solve this problem.
I understand perfectly well (let's hope) why you would divide 24 by 7.
I didn't even know you could use the reciprocal to find the answer.
7 fact families
I haven't had time to sit down and think this through, but I suspect the reciprocal answer to a rates problem is the same concept as the 7 fact family I put together after teaching the Primary Mathematics lesson on ratio & proportion (Primary Mathematics 6A Textbook, p. 21-46):
7 fact families

(back to top)
WorkingWithTeachersAndPrincipals 12 Dec 2005 - 16:25 CatherineJohnson
I make no bones that parents whose children are struggling with a poor mathematics curriculum should find a good curriculum and teach that one instead.
But that raises the issue of what happens politically and socially when a parent rejects a school's math curriculum.
Good question.
it doesn't have to be a battle
My own experience this year was terrific.
Of course, I wasn’t rejecting a curriculum the school had embraced; I was rejecting a curriculum the school had rejected (SRA Math, which is being replaced by Trailblazers).
Even so, I was using a different curriculum at home, and everyone knew it. The reason they knew it was that I printed out copies of the Table of Contents for Christopher to take in and show his teacher.
She was great. She admired all the lesson headings, and told Christopher, “All the parents should be doing this.” It was incredibly sweet of her.
At one point I sent an email saying I was having trouble getting Christopher to cooperate (that’s an understatement) and asking if she could tell him he needed to do my homework, too.
She did.

When we told her, in January, that our goal was to move him to Phase 4, the accelerated track, she blanched. There were already a number of kids in Phase 4 who were struggling; the class was oversubscribed. One child had just been moved ‘down’ to Phase 3, and it had been upsetting to all concerned.
She’d never thought of Christopher as ‘a Phase 4 kid,’ she said. She didn’t want to see him try Phase 4 and fail. (Neither did we.)
It took her about 2 minutes to decide she probably could think of Christopher as a Phase 4 kid, and the reason she could think of him as a Phase 4 kid was that ‘you’ll give him the support he needs.’ She saw clearly that Christopher’s dad and I would do whatever we needed to do to help him succeed—and she saw that we would be taking responsibility for the move. If it didn’t work out, we weren’t going to be back in the school yelling at people. (True.)
Once she'd turned her point of view around 180 degrees, she told us that if we were going to move him we needed to do it now. Suddenly it was our turn to blanch; my plan was to move him in the fall, after we'd had another summer to work on his math.
She said, in so many words, that my plan was going to be problematic. For years the middle school has been hammering the elementary school about placing too many kids in the accelerated class, giving inflated grades, etc., etc., or so I gather. ('Hammering' is not the word she used or implied.) The middle school had made crystal clear to teachers & to parents that they would be placing fewer kids in Phase 4 come fall, not more. Which meant they probably weren't going to think Christopher, who'd been in Phase 3 from day one, and who'd done badly in 4th grade math, was an obvious candidate for the accelerated track.
As she put it, 'They aren't going to know him the way we do.' If we wanted to do it, she said, we needed to do it now.
We said, OK, then, we'll do it now.
She got Christopher moved to Phase 4 within the week.
Not only did she support us in doing something she didn’t necessarily think was a good idea, she told us how to work the ropes. Then she worked the ropes for us.
using TIMSS
Christopher's second math teacher, in Phase 4, was just as terrific. She once sent home a formal, hand-written explanation of the compound interest problems in SRA Math. Yes, it’s mortifying to reveal that I needed a hand-written explanation of compound interest, but there you have it. It was a darn good explanation, too. Later on I learned she’d been an accountant for 15 years before changing careers.
The TIMSS data on U.S. students is a big part of the ‘secret’ to working well with your school district when you object to the math curriculum. The first time I mentioned to our principal, Don, that I thought Christopher maybe ought to move to the accelerated track, he got that tight not-now-not-ever look on his face administrators always get when parents start bugging them to do things they don’t think they ought to do.
I backed off, because in fact Christopher wasn’t ready to move to the accelerated track. But I publicly raised the issue of why the accelerated kids were using a book that was a full year ahead of the rest of the kids without any of us parents having been told.
Naturally it turned out Don hadn’t been told, either; he’s an interim principal. He looked into it, and was obviously pretty dismayed at what he found (not worth going into here).
When we sat down and talked about it, I took the tack that I didn’t think Christopher is Secretly Gifted And Talented In Math; I just wanted him to be on the same track kids are on in high-achieving countries. Which is true. Here are the Magic Words to use with principals, teachers, administrators, & school boards:
In high-achieving countries, students take and master algebra in the 8th grade.
Here in America, only the accelerated kids take and master algebra in the 8th grade.
I told Don: if kids in Germany pass Algebra 1 in 8th grade, I want Christopher to pass Algebra 1 in 8th grade, too.
He had exactly zero problems with that, and the minute Christopher was ready to move to the faster class, he moved him up.
The fact is, our problems in math ed are national, not local, and everybody knows it.
Everybody knows it, but nobody knows how to fix it. Ideological constructivists think they know how to fix it, but your basic principal and/or teacher is living in the real world, facing real children and real parents who blame them when math scores are bad. They’re on the firing line. I don’t think too many principals & teachers truly believe ‘reform math’ is going to be the miracle we’ve been looking for for the last 100 years.
So basically, his feeling was: I’d like to see all our kids learning at the same rate as kids in Singapore. So would I. I don't blame him for our school having the same problems every other school has.
giving respect where respect is due
Once I started teaching Christopher my own hand-picked curriculum, I was on the firing line. For awhile there I was actually having him do the homework I assigned instead of the homework he brought home from school…..so exactly whose fault was it going to be if he didn’t succeed?
It was going to be my fault.
Everyone sensed this. I had moved out of the potentially ticked-off parent category and into the junior colleague category.
That’s another thing.
I also developed a healthy new respect for the teachers he’d had thus far. I couldn’t teach Topic One out of SRA Math, but all of them had managed to teach him a huge amount of math from SRA, which he had retained. His math knowledge from 2nd and 3rd grades was solid as a rock.
So I stopped being a critic, and became a teacher. That meant I asked the school’s teachers for help & advice, and made clear I respected their seniority. Christopher felt the same way. When he told me, ‘Mrs. Panitz is a better teacher than you,’ I sent her an email letting her know.
bullet points
For me, in this school district, putting together public school & home teaching worked during the one year I've done it. Would my approach work everywhere? Most places? I don't know. What I do know is that your basic teacher went into the profession because he or she wanted kids to succeed. Teachers are rooting for the kids, not against them. If you're helping your child succeed, their inclination is going to be to root for you, too.
My (tentative) advice thus far:
- tell your teachers what you’re doing, within limits. I didn’t announce the fact that I was substituting my homework for the school's, and I don’t think I should have done so. It would have been nervewracking for Christopher's teacher—it was nervewracking for me—and since I was going to do it anyway, why get her worried?
- respect the teacher's experience and authority. Show respect even if you're a math major working in a mathematics-related career. Your math knowledge greatly exceeds the teacher's, but your pedagogical content knowledge almost certainly does not.
- ask for help (but don’t suck up lots of the teacher's time)
- tell your child his teachers are good, it's the curriculum, or the too-slow American track, that's the problem
- whenever you talk to teachers or principals, keep the focus on international standings, not local failings
keywords: afterschooling politics of math math wars conflicts with teachers conflicts with schools
TrustButVerify 31 Oct 2005 - 21:58 CatherineJohnson
This bears repeating:
don't rely on state tests
In theory, I'm in favor of standardized tests.
In practice, I'm still in favor of them, but I don't rely on them. High-stakes testing is subject to enormous political pressure from all concerned. Years ago Ed worked on the California History Social Science Frameworks. He helped the CA Department of Ed develop assessments for the Frameworks, evaluating off the shelf tests, which were, in his words, 'insanely easy.' 12th graders were evaluated at a 9th grade reading level.
The Dept of Ed developed its own tests, & tried them out. (They didn't test the entire state, and he doesn't remember which groups took them.) Two political groups objected: some conservative Christians objected to the critical thinking portion of the tests, and some minority groups objected that their children's scores would go down (which they probably would have, at first). These two groups put enough pressure on their respective representatives that the new tests were scotched before they were ever rolled out. CA went back to using off-the-shelf tests.
No state test will survive a high failure rate in my opinion. That's why I view the current situation in NYC, where Mayor Bloomberg's campaign is based on a sudden, monster increase in student scores, as being far from ideal. I'm fine with the idea of a mayor campaigning on improving student scores. And now that I've seen what can happen to one child's scores thanks to simple, hard work, I believe that you could have a sudden, monster increase in student scores on a broad scale. It's possible.
But I want to see independent audits of those scores. I want to see the test items, and I want to see an audit. Sunshine laws are a good thing. Let's have sunshine laws for state & local testing.
I once read a Diane Ravitch essay on this issue (if I find it again, I'll drop in the reference). She argued that the solution is to establish different levels of 'Pass,' as they do in British universities. Students could pass exit exams with high honors, honors, no honors, and so on. That would probably allow states to maintain rigorous testing in the face of parent opposition.
You might still have an inflated pass rate, but then again, maybe not. Competition spurs people on to higher achievement, and not just because people are naturally competitive, which I believe we are. Seeing someone you know & like do well implies that you can do well, too.
Given the pressures on state testing, I don't rely on New York state tests to tell me how well Christopher is doing. At the end of 4th grade, when Christopher had flunked fully one-third of his year's math course, he earned a '4' on the state math test. 'Exceeds state standards.'
I'm sorry, but a 68 on Unit 5, a 39 on Unit 6, and a 4 on the state exam don't square.
(This is kind of funny. A couple of months later I called one of the guidance counselors at the Middle School to ask about Christopher's chances of moving to Phase 4 when he entered 6th grade. The counselor said nobody ever moves to Phase 4 from Phase 3, so the chances were slim to none. I said, 'But he got a 4 on the state test!' He said, 'That doesn't matter.' I was outraged at the time, but even in the midst of my outrage I knew exactly what he was saying. He was saying Don't rely on state tests.)
So today I'm reminding everyone about these Practice Problems for the California Mathematics Standards Grades 1-8 for the Los Angeles County Board of Education, which David Klein developed for the Los Angeles County Board of Education.
The state of California has the best math standards in the country, according to the Thomas B. Fordham Foundation assessment of state math standards. David's problems will tell you whether your child meets CA standards--and, if not, which topics he or she needs to work on.
I count 85 questions on the 5th grade test in all, divided into 4 areas:
- number sense
- algebra & functions
- measurement and geometry
- statistics
The test isn't as time-consuming as it sounds, since often there are 4 separate questions in one larger question (such as identifying several points on a graph). Answers are included.
If giving the test seems like a lot to do in the face of Massive Pre-teen Resistance, just divide it up across a few days' time. That's what I did.
related posts:
Assess Your Child for Free Part 2
Assess Your Child for Free
and
David Klein at the AEI
DimensionalDominoesFromDan 26 Jul 2005 - 14:47 Main.guest
DimensionalDominoesPart2 26 Jul 2005 - 17:01 CatherineJohnson
Don K's lesson on dimensional analysis
Don's page includes PowerPoint manipulatives, showing how to teach dimensional analysis! I think this is going to be very important for us--it's our first entry on the 'Math Lessons' page listed on the side bar, isn't it? (Unless I've missed something, in which case, please remind me.)
Thank you so much for taking the time to do this, Dan!
OnlineTIMSSTest 27 Jul 2005 - 23:40 CatherineJohnson
This is a terrific resource. You can give your child 10, 15, or 20 questions from the 1995 & 1999 TIMSS tests. The web site scores them for you.

Explore Your Knowledge
NewStudyOnManipulatives 29 Jul 2005 - 17:27 CatherineJohnson
I want to follow up on Carolyn's post on Congressional math incentives, but before I do that here's a reminder:
Anne Dwyer has posted new notes on her summer math class.
And...quickly checking her page just now, noticed this comment:
So, what have I learned so far?
- they like games where they compete with one another
- they prefer pencil and paper exercises
- they like to figure out puzzles
This is fascinating, because Kevin Killion, of Illinois Loop, just pointed me to a new article in Scientific American showing that manipulatives are less effective than pencil and paper with young children. I'll write a bit more on this later (bike ride time!), but here's the critical passage:
Teachers in preschool and elementary school classrooms around the world use "manipulatives"--blocks, rods and other objects designed to represent numerical quantity. The idea is that these concrete objects help children appreciate abstract mathematical principles. But if children do not understand the relation between the objects and what they represent, the use of manipulatives could be counterproductive. And some research does suggest that children often have problems understanding and using manipulatives.
Meredith Amaya of Northwestern University, Uttal and I are now testing the effect of experience with symbolic objects on young children's learning about letters and numbers. Using blocks designed to help teach math to young children, we taught six- and seven-year-olds to do subtraction problems that require borrowing (a form of problem that often gives young children difficulty). We taught a comparison group to do the same but using pencil and paper. Both groups learned to solve the problems equally well--but the group using the blocks took three times as long to do so. A girl who used the blocks offered us some advice after the study: "Have you ever thought of teaching kids to do these with paper and pencil? It's a lot easier."
Dual representation also comes into play in many books for young children. A very popular style of book contains a variety of manipulative features designed to encourage children to interact directly with the book itself--flaps that can be lifted to reveal pictures, levers that can be pulled to animate images, and so forth.
Graduate student Cynthia Chiong and I reasoned that these manipulative features might distract children from information presented in the book. Accordingly, we recently used different types of books to teach letters to 30-month-old children. One was a simple, old-fashioned alphabet book, with each letter clearly printed in simple black type accompanied by an appropriate picture--the traditional "A is for apple, B is for boy" type of book. Another book had a variety of manipulative features. The children who had been taught with the plain book subsequently recognized more letters than did those taught with the more complicated book. Presumably, the children could more readily focus their attention with the plain 2-D book, whereas with the other one their attention was drawn to the 3-D activities. Less may be more when it comes to educational books for young children.
This perfectly supports the study Carolyn mentioned way back when, showing that fraction manipulatives are good for middle schoolers.
CA state study on manipulatives Fraction Manipulatives
Quick Thought about Fraction Manipulatives
Fraction Manipulatives Part 2
New Study on Manipulatives Part 2
NewStudyOnManipulativesPart2 28 Jul 2005 - 20:15 CatherineJohnson
I'm reading the Scientific American article about manipulatives & symbolic representation now:
About 20 years ago I had one of those wonderful moments when research takes an unexpected but fruitful turn. I had been studying toddler memory and was beginning a new experiment with two-and-a-half- and three-year-olds. For the project, I had built a model of a room that was part of my lab. The real space was furnished like a standard living room, albeit a rather shabby one, with an upholstered couch, an armchair, a cabinet and so on. The miniature items were as similar as possible to their larger counterparts: they were the same shape and material, covered with the same fabric and arranged in the same positions. For the study, a child watched as we hid a miniature toy--a plastic dog we dubbed "Little Snoopy"--in the model, which we referred to as "Little Snoopy's room." We then encouraged the child to find "Big Snoopy," a large version of the toy "hiding in the same place in his big room." We wondered whether children could use their memory of the small room to figure out where to find the toy in the large one.
The three-year-olds were, as we had expected, very successful. After they observed the small toy being placed behind the miniature couch, they ran into the room and found the large toy behind the real couch. But the two-and-a-half-year-olds, much to my and their parents' surprise, failed abysmally. They cheerfully ran into the room to retrieve the large toy, but most of them had no idea where to look, even though they remembered where the tiny toy was hidden in the miniature room and could readily find it there.
Their failure to use what they knew about the model to draw an inference about the room indicated that they did not appreciate the relation between the model and room. I soon realized that my memory study was instead a study of symbolic understanding and that the younger children's failure might be telling us something interesting about how and when youngsters acquire the ability to understand that one object can stand for another.
here's the anti-constructivist moment:
[The] capacity [to] create and manipulate a wide variety of symbolic representations .... enables us to transmit information from one generation to another, making culture possible, and to learn vast amounts without having direct experience--we all know about dinosaurs despite never having met one. Because of the fundamental role of symbolization in almost everything we do, perhaps no aspect of human development is more important than becoming symbol-minded.
symbols aren't 'natural'
The first type of symbolic object infants and young children master is pictures. No symbols seem simpler to adults, but my colleagues and I have discovered that infants initially find pictures perplexing. The problem stems from the duality inherent in all symbolic objects: they are real in and of themselves and, at the same time, representations of something else. To understand them, the viewer must achieve dual representation: he or she must mentally represent the object as well as the relation between it and what it stands for.
A few years ago I became intrigued by anecdotes suggesting that infants do not appreciate the dual nature of pictures.
[snip]
.... the Beng babies, who had almost certainly never seen a picture before, manually explored the depicted objects just as the American babies had.
The confusion seems to be conceptual, not perceptual. Infants can perfectly well perceive the difference between objects and pictures. Given a choice between the two, infants choose the real thing. But they do not yet fully understand what pictures are and how they differ from the things depicted (the "referents") and so they explore: some actually lean over and put their lips on the nipple in a photograph of a bottle, for instance. They only do so, however, when the depicted object is highly similar to the object it represents, as in color photographs....
[snip]
it takes several years for the nature of pictures to be completely understood. John H. Flavell of Stanford University and his colleagues have found, for example, that until the age of four, many children think that turning a picture of a bowl of popcorn upside down will result in the depicted popcorn falling out of the bowl.
Andrew makes Barney fly
A couple of weeks ago Andrew (10, autistic, nonverbal) brought me Christopher's yellow plastic airplane, on top of which he'd mounted one of his Barney's, and handed the whole big package to me with an urgent look on his face. He was on a mission.
Martine came in and said, 'He wants you to make Barney fly.' She'd been sitting in the family room when Andrew had put his Barney on top of the plane, and then flung plane & Barney up into the air, apparently thinking Barney would fly around the room.
Andrew had been very unhappy with the outcome, and was now appealing to me. Clearly he believed that making Barney fly was one of those things, like operating the TIVO, only adults know how to do.
I was flattered, but also dumbfounded. What goes on inside this child's head? was my exact thought.
I was thinking....does he not understand gravity?
Does he not understand toys?
What's with this kid???!!
The Scientific American article makes me think that Andrew, although he can read, hasn't completely figured out the dual nature of symbolic representation.
He probably couldn't understand the plastic airplane as being TWO THINGS:
- an airplane
AND
- a symbolic representation of an airplane
What I'd like to know is: what does he think about Barney?
is this a shoe?
Here's a little guy trying to put his foot inside a picture of a shoe.
lost in translation
I constantly have the experience of reading constructivist texts, noticing that the ideas they're advocating are good ones or at least not obviously bad ones.....and then, five seconds later, finding that they've taken a sound idea and just completely gummed it up in the application.
Assuming this work on manipulatives & symbolic representation is correct, the constructivist obsession with manipulatives looks to be another instance of a good idea lost in translation. Constructivism is majorly obsessed with manipulatives, that's for sure. I understand that the TERC curriculum is basically just a huge box of manipulatives, with no textbook or 'consumables'--workbooks--at all.
Following in Piaget's footsteps, constructivists believe children don't reach the stage of 'formal operations' until age 11; from 7 to 11 they're in the Period of Concrete Operations. (Often you'll see the word 'developmental' used to designate constructivist curricula. Apparently that's a reference to Piaget.)
Wayne Wickelgren says this is nonsense; children can handle abstract concepts long before age 11. But constructivists are the people time forgot, and they're still basing their pedagogy on work done in the 1950s.
That's bad enough in itself, seeing as how the field of cognitive science was just getting started around that time, and Piaget's work hasn't fared so well over the past 60 years.
But the more glaring misstep, it appears, is that they failed to grasp the nature of the concrete.
The reason constructivists think children should spend their grade school years working with manipulatives is that manipulatives are concrete. But they're not. Manipulatives are symbolic objects that require the child to have mastered the concept of dual representation.
Skinnies and bits are not concrete. They are symbolic representations of the Hindu-Arabic numeral system. Worse yet, they are more intellectually demanding, and hence more confusing, symbolic representations than pencil marks on paper.
They're harder to understand, not easier.
Lost in translation.
question
I hope I'll get a chance to talk to these researchers at some point.
My question is: why should pencil and paper be less challenging than manipulatives?
I can see why pencil and paper wouldn't be any more challenging than manipulatives, but why should pencil and paper be easier? Do pencil marks somehow not involve dual representation? That's what the authors seem to imply, but they don't say so directly.
CA state study on manipulatives Fraction Manipulatives
Quick Thought about Fraction Manipulatives
Fraction Manipulatives Part 2
NewStudyOnManipulatives
New Study on Manipulatives Part 2
XtremeBehaviorismTeachingAndScripts 29 Jul 2005 - 18:34 CatherineJohnson
I just found a wonderful comment after the post on bullying:
smart constructivism
I haven't looked at the book, but I find the concept interesting. I believe that it takes a special skill to remember your own child accurately, through the lens of childhood, and if you can remember it, then you can teach children anything.
You can teach them math or history or art or how to be polite or how to handle a bully.
Teaching is a puzzle. It's a puzzle where you must navigate backwards in a maze. A child is at point K, but they are supposed to be at point Z. If you just show them again how to go from A to Z, you are missing the point of how they got to K.
And usually, kids made a rational mistake: they misunderstood something, or misheard something, and this thing is embedded in their minds. It leads them (Rationally) to this bad position K.
Teaching is about figuring out how someone got into that position, so you can FIX that misunderstanding. It's not enough to tell them that K is the wrong place; you have to help them never follow that wrong path in the first place.
The best way to help kids learn is to remember the typical misconceptions YOU had as a child, and ones similar to it, to try and understand why they would think what they think. Then, you can see how they are really very smart--just misguided.
a child must feel like himself
re: the aspergers/high functioning autism stuff: this kind of description is very similar to what behavioral psychologists teach to help children with anxiety and attachment disorders. I personally believe that there is a high correlation between attachment disorders and what's called asperger's, but I caution people to refrain from just teaching these techniques to children.
The problem with just teaching this techniques is that you need your children to feel like themselves. That may sound silly, but it isn't helpful to teach your child how to act. You may want them to learn how to behave, but they need an emotional makeup capable of backing up the behavior.
For a short term case like a bully, maybe it doesn't matter so much, but in terms of making friends, you need your child to have an emotional makeup that feels these behaviors are natural. If not, the other children will recognize that the behavior is still off, and worse, the child can often feel that they are not capable of making friends by being themselves but have to act like someone else. That's a painful experience for a child, and can do a lot of damage in the long run. Be careful at behavioral solutions that make a child feel that their personality isn't acceptable.
joannejacobs comment thread on bullying
Interesting comments on bullying at joannejacobs.com
how to stop a bully
Comments thread on bullying at joannejacobs.com
MathTalkInTheCar 01 Aug 2005 - 16:50 CarolynJohnston
We took the kids to a bar tonight, as it happened. Colin (17) is into playing the bass these days; he has a band that he plays with during the school year. I have a friend at work who is a hot guitar player and who just joined a classic rock band, and he was playing his first gig tonight, and they were letting kids stay through the first set, so we went to see him. It was a long drive for us -- all the way out to Greeley. The place was an authentic roadhouse with motorcycles parked out front, and the food was good -- it was Cajun food, and very authentic given that we were not in Cajun country but in Greeley, Colorado, home of the Feedlot You Can Smell All The Way To Denver.
On the way home, Colin asked us about the difference between the median, the mean, and the mode of a data set, and what each of them is good for. This is, of course, the sort of thing we love to pontificate about. He then told us that he felt he had never really quite gotten the idea of a function, and asked us to explain it.
It's a smart kid who understands what he doesn't understand. Most adults can't do that very well.
Actually, most kids coming into calculus classes are confused by functions. A function is just a black box; you put in an input, and get out an output. What makes it a function is that, when you put in the same inputs, you always get the same outputs. You can't put the same number in the black box and get 2 one time, and 5 the next.
Most texts teach functions using formulas to define the functions; all the functions kids see look like f(x)=3x-5, or g(x)=x/6. But functions don't have to have formulas to go with them; they can defy description by a formula. The only rule is that if you put in the same input multiple times, you get the same output, every time.
The reason kids confuse formulas with functions is that it's hard to define functions that don't use formulas, even though in real life we encounter them all the time. When a function totally defies description with a formula, we often resort to trying to describe it with only a couple of numbers, such as the mean, median, and standard deviation (this is how the whole field of statistics arises).
We played a 'figure-out-the-function' game on the way home from Greeley. Bernie and I would think of a function, and Colin and Ben would give us numbers for inputs, and we would then tell them the output. They'd then try to guess the formula we were using to define the function.
They are both aces at extracting patterns. If anything, Ben would try to generalize from too little data; once he guessed, after one try, that the function was 'add 2'; he'd given me a 2, and I'd come back with 4 (the function I'd thought of was squaring; he got it on the next try). Bernie was giving Colin some functions that are so simple they trip up students with their obviousness, like the function that returns the same number you give it, and the one that returns '3' no matter what you give it. He gave Colin one function that was so bizarre you can't describe it with a pattern.
Ben knew more about functions than I thought, even piping up with "that's the constant function 3" at the appropriate moment. Did they do functions one day for 5 minutes in Everyday Math? Well, he was definitely on the ball that day.
DanKOnMakingMathInteresting 30 Jul 2005 - 16:50 CatherineJohnson
Great comment from Dan K!
- Go bowling. Ignore the automated system, and keep score manually. Then, work through the calculation for some counter-factual cases (“What would my score have been if I hadn’t missed that @#$! spare in the fourth frame?”). Try to figure which one roll would have boosted your score the most if it would have knocked down all the pins.
- Check the standings. Develop the formula for computing “magic numbers” for clinching the division in baseball. Just please don’t tell me how small the Cardinals’ magic number is to eliminate the Cubs.
- Follow the market. Each person picks five stocks to watch. Invest your pretend portfolio in them. Track their performance throughout the month of August. Figure out how to plot their daily performance on a graph, comparing their performance to the Dow, the NASDAQ, and the S&P 500. Trade into other stocks along the way.
- Try Mathmania. Look for interesting problems in the Mathmania booklets put out by Highlights publishing. These periodicals are probably aimed at 4th or 5th graders, but you can upscale some of the problems by trying to describe them using algebra.
- Look at MATHCOUNTS. The MATHCOUNTS web site (www.mathcounts.org). They’ve got a “problem of the week” archive (with solutions!) that you can browse through. These problems are often topically related to current events. They’re designed to interest kids, so maybe some of them will succeed with your kid. MATHCOUNTS is for math-oriented middle schoolers, so it will challenge most high school students, too.
- Graph the logical flow. Develop a flow chart—or pseudo-code, if you’re already into programming—describing scoring in tennis. Nest a loop for point scoring within a loop for set scoring. Sometimes deuce is an infinite loop.
- Play Jeopardy. Write up your own problems and arrange them in categories. This could be a lot of work, depending on how hard you make the problems. Don’t be too strict about answering in the form of a question.
- WARNING: High risk of failure. Plan a math rally around the yard or neighborhood. Students must solve clues in the form of math problems to find out, say, which envelope to open to get the next clue. Then they must determine which direction to walk to find the next clue. If you open the wrong envelope (or box, or whatever), you lose points, but it then tells you what would have been correct, so you can get back on the right track. If this turns out to be fun, that’s great. If, however, the kid thinks it’s bogus, then you’ve invested a lot of time to end up looking pretty foolish.
I think I'm going to start a user page for this subject....I've forgotten how to set it up so anyone can edit it--Carolyn, do you want to do that?
I'm going to start it from the User Page Index.
HotMath 14 Aug 2005 - 15:09 CatherineJohnson
Thanks to Dan K, I've found a fantastic resource:
Hotmath.com
[Hotmath provides] explained solutions to the odd-numbered homework problems from most of the popular secondary math textbooks used in California. Thus, teachers can now assign practice problems for homework where teacher-prepared, explained solutions are instantly available, and can mix in even-numbered problems for challenges. Students who do not need to see the worked solutions needn't bother, and students who might abuse the availability of worked solutions will be tested on the even problems.
Here is a sample worked-out problem: algebra problem
And here are the 2 critical paragraphs from the Hotmath 'white paper'. I've begun to come across these studies elsewhere, and I'm inclined to trust these summaries, in part because this discussion jibes with my own experience re-learning maths:
Providing students with worked out examples of math problems has been found to be more effective than simply assigning the same problems for the students to work out on their own. In one experiment (Carroll, 1994), 40 high school students were instructed in how to solve linear equations. In an “acquisition phase” the students were divided into two groups and their instruction differed in the following way: in the “conventional learning” group, students were assigned 44 unsolved problems to work out (in the classroom and at home homework), and in the “worked examples” group students were provided with the same problems, but half of the problems were accompanied by correct solutions. After completion of the assigned problems, both groups were tested on 12 related problems, 10 of which were very similar to the linear equations presented in the acquisition phase, and 2 of which were word problems, used to test whether students could transfer and extend their knowledge to a new context. No worked out examples were available during the test. The test results revealed that students in the “worked examples” group outperformed students in the “conventional learning” group on both types of the test problems. A second experiment, employed a similar methodology but focused on “low achieving” students (students with a history of failure in mathematics, and students identified as learning disabled). Here, the data revealed that students in the “worked examples” group required less acquisition time, needed less direct instruction, made fewer errors, and made fewer types of errors than students in the “conventional learning” group.
Related research (Pass & Van Merrienboer, 1994) sheds light on the cognitive underpinnings of the effects described above. In this study, 60 college-aged students were instructed in geometry concepts. As in the Carroll experiments, students were assigned un-worked problems to solve or worked out examples to review (unlike the Carroll study, the “worked examples” group was assigned no un-worked problems to solve). In this study, the researchers manipulated the nature of the problems presented to the students: within each group, some students received problems which were all similar to each other, while others received a more varied problem set. Furthermore, the researchers measured the “cognitive load” experienced by the students. This research revealed that while students in the worked examples group completed their work more quickly, they perceived the work as less demanding and displayed better transfer performance at test. The effect was most pronounced for the students given highly-variable problems. The researchers suggest that the reduced cognitive load associated with the worked examples enabled students to “take advantage of” the variability in problems by using the available cognitive resources to process the underlying similarity in the problems (i.e., the mathematical concepts being taught), and to integrate the current problem with existing knowledge (Linn, 2000).
The site covers Prentice Hall Pre-Algebra, the book Christopher will be using in the fall, so I'm going to subscribe. Cost is $49 for 12 months.
I think it's going to be fantastic for Christopher to have an answer source that isn't His Mother.
Especially since it looks like I'm going to have to start some heavy-duty Writing Instruction this year. (That's another story.)
cognitive load
This is going to be an important term for me. It perfectly captures what it is we're trying to do when we push our kids to practice to the point of automaticity.
We're trying to reduce cognitive load.
update
I've just re-read Dan's original post, and I don't see a reference to hotmath. hmmm. Maybe one of the sites he mentioned pointed me to hotmath. In any case, I'm recommending hotmath, not Dan. (He'll let us know what he thinks, I'm sure.)
MathForumArchivedNewsletters 14 Aug 2005 - 01:37 CatherineJohnson
I've just been alerted to a terrific resource, the Math Forum Newsletter.
They have an article about Kitchen Table Math in the latest issue! (Although so far I haven't been able to find it.....I don't think....)
Sigh.
However, I have managed to attach and display the logo they sent me!
GiftForPrincipals 15 Aug 2005 - 12:27 CatherineJohnson
I think it's worth posting the one reader review of Elaine McEwan's The Principal's Guide to Raising Math Achievement:
Having read this book as a parent and a school board member, I am giving it to both the principals in my district. This book explains both many of the things that are done badly in many schools in the country and shows the path for how to do them well. I found the comparisons with the Japanese and Chinese methods of teaching particularly helpful. This book was pleasant to read as well as enlightening in how to promote the effective teaching of mathematics.
That is not a bad idea. I was on the verge of buying a copy for our principal (whose wife is a high school math teacher) all year long. I didn't do it, ultimately, because the book is awfully pricey ($28 for a paperback).
It's still a good idea.
Principal's Guide to Raising Math Achievement
school starts soon
TallDarkAndMysteriousThread 27 Aug 2005 - 10:58 CatherineJohnson
Interesting thread at Tall Dark & Mysterious sparked by Daniel Willingham's article on different learning styles.
One of the commenters there, Meep, has this to say:
I think it’s best, the younger the student and the more important the concept, to teach the content in every possible relevant way….it’s also a matter of students realizing the different ways to relate to a given subject. When there are more handles on a particular subject, you’re more likely to remember it.
Teaching the same content in more than one way has become a guiding principle for me, whether I'm teaching Christopher or me.
TD&M replies by pointing out that students hate being taught the same material more than one way:
My weakest students hate this. Hate, hate, hate. They want to know the ONE TRUE WAY to understand and approach a problem, and they get very very cranky when I don’t oblige them. (One true way => one-size-fits-all formulas, whereas different approaches => need to think about problems.) I’m remembering one student who asked me about intercepts, and I sketched a graph and explained about setting variables to zero. “So which one is it?” she asked when I was finished.
She's right. Kids hate this.
But when you're talking about young kids, at least, all kids hate it, no matter what their skills. I myself find it quite uncomfortable, once I've understood and solved a problem, to go back and start all over again trying to solve the problem a whole different way--or to work through and understand a different solution offered by the book.
My neighbor and I spent some time trying to figure out why this should be so. Why is doing-it-a-different-way so unpleasant? I think this passage from W.W. Sawyer's Prelude to Mathematics explains it:
I think it's unnatural to throw out the pattern you've just discovered & go off to find a whole new one. It goes against the grain.
I think that's the source of the aversion children, 'weak students,' and adult students like me feel to doing it.
it's not the same thing for us
Here's my other thought.
Around the blogosphere I see an awful lot of complaining about weak students and lazy students and recalcitrant students and students who think their learning style matters.
I'm sympathetic, to a point.
That point is here, asking myself why a 'weak student' would not enjoy having his teacher tell him to 'do it another way.'
I strongly suspect that the practice of doing a problem 'another way'--or simply perceiving that two or more different ways of doing a problem exist--is a different thing for the expert than it is for the novice.
I say this because of my own experience. Slowly, I'm turning myself into what is called, I believe, a talented novice. (I'll check the phrase.)
A talented novice is neither fish nor fowl, neither expert nor beginner.
For that very reason, a talented novice can be a terrific teacher.
As I move into talented novice territory, I find that 'doing a problem more than one way' is becoming a whole different thing. It's starting to be fun. It means making connections--extending the pattern--rather than throwing the pattern out and starting again.
Learning math is hard. Children do not spontaneously see a string of beads as elements in a set, or points on a line as numbers. If you give them a bunch of blocks and tell them to do something together, they will exercise their intuitive psychology for all they're worth, but not necessarily their intuitive sense of number. (The better curricula explicitly point out connections across ways of knowing. Children might be told to do every arithmetic problem three different ways: by counting, by drawing diagrams, and by moving segments along a number line.) And without practice that compiles a halting sequence of steps into a mental reflex, a learner will always be building mathematical structures out of the tiniest nuts and bolts, like the watchmaker who never made subassemblies and had to start from scratch every time he put down a watch to answer the phone.
This is my answer to asking a child to do the same problem more than one way.
Yes, he's going to have to do a problem he's already solved all over again, solving it a different way.
BUT I'm asking him to use the same methods he used yesterday & the day before. It's not chaos; there's a predictable pattern to the two or three different ways he's going to solve the problem.
I'm trying to give him a stable structure he can hold onto while he does (and I do) the hard work of learning math.
LipingMa 24 Aug 2005 - 20:09 CatherineJohnson
Here's Liping Ma:
Multiple Approaches to a Computational Procedure: Flexibility Rooted in Conceptual Understanding
Although proofs and explanations should be rigorous, mathematics is not rigid.... Dowker (1992)asked 44 professional mathematicians to estimate mentally the results of products and quotients of 10 multiplication and division problems involving whole numbers and decimals. The most striking result of her investigation "was the number and variety of specific estimation strategies used by the mathematicians." "The mathematicians tended to use strategies involving the understanding of arithmetical properties and relationships" and "rarely the strategy of 'Proceeding algorithmically.'"
"To solve a problem in multiple ways" is also an attitude of Chinese teachers. For all four topics, they discussed alternative as well as standard approaches. For the topic of subtraction, they described at least three ways of regrouping, including the regrouping of subtrahends. [they also talked about which approach worked best in which situation] For the topic of multidigit multiplication, they mentioned at least two explanations of the algorithm. One teacher showed six ways of lining up the partial products. For the division with fractions topic the Chinese teachers demonstrated at least four ways to prove the standard algorithm and three alternative methods of computation.
For all the arithmetic topics, the Chinese teachers indicated that although a standard algorithm may be used in all cases, it may not be the best method for every case. Applying an algorithm flexibly allows one to get the best solution for a given case. For example, the Chinese teachers pointed out that there are several ways to compute 1 3/4 dividedby 1/2. Using decimals, the distributive law, or other mathematical ideas, all the alternatives were faster and easier than the standard algorithm. Being able to calculate in multiple ways means that one has transcended the formality of an algorithm and reached the essence of the numerical operations--the underlying mathematical ideas and principles. The reason that one problem can be solved in multiple ways is that mathematics does not consist of isolated rules, but connected ideas. Being able to and tending to solve a problem in more than one way, therefore, reveals the ability and the predilection to make connections between and among mathematical areas and topics.
AnneDwyerOnSingaporeMath 27 Aug 2005 - 00:06 CatherineJohnson
You might want to check out the discussion on teaching things more than one way.
I started by saying that my principle has become 'teach things more than one way.'
Carolyn, Bernie, Chris & others objected.
I have to say that while 'teaching things more than one way' is a core principle for me at this point, whether rightly or wrongly, I don't really know what I mean by that.
In practice, what I've been doing so far is to teach bar models each and every day, along with, each and every day, the standard American 'symbolic' approach. I had Christopher start with the very first word problem in Primary Mathematics Book 3A, which is the first semester of 3rd grade in Singapore, & do one word problem a day, drawing a bar model to illustrate the problem set-up, and then doing the math using the standard algorithms.
And that's it. Each problem takes him a couple of minutes (a little more when he was starting out).
His 'real' math lesson obviously takes a lot longer.
Another example. A couple of days ago a Saxon 8/7 lesson taught two different ways of prime factoring a number. I threw out one of them, and substituted the RUSSIAN MATH approach, which I insisted he learn, almost entirely because when I learned it I found it incredibly fun to do. Christopher ended up liking it as much as I did.
Then yesterday, after Drew & Marc taught Christopher how to subtract-a-fraction-with-borrowing, I forced him to sit with me and watch while I subtracted the same fraction without borrowing, ending up with a whole number and a negative fraction. Then I subtracted the negative fraction from the positive whole number and voila. Fraction subtracted without borrowing.
I didn't make him do the subtraction-problem-without-borrowing himself, but only because he was in a MOOD. If he hadn't been in a MOOD, I would have insisted he do one or two such problems.
Now, I wouldn't insist he practice this approach to mastery, because it's Clunky, and forcing a child to practice Clunky Subtraction would be Wrong. IMHO. It's wrong because math isn't clunky, or shouldn't be.
The only reason I'd insist he work a couple of Clunky Subtraction problems is to make sure he really saw that the reason we borrow or regroup is that regrouping is an elegant, mathematically powerful way to do things--NOT because we can't subtract a bigger number from a smaller number! I know for a fact that a lot of kids think the reason you borrow-or-regroup is that you can't subtract a larger number from a smaller. Well, I don't want Christopher thinking that.
(I actually vividly remember the day, just this year, when my neighbor showed me that YES YOU CAN subtract 17 from 25 without borrowing. She's a statistician, and yet even she was puzzled for a moment when I asked her, 'Why do you have to borrow?')
The point is that I'm feeling my way, basing a lot of what I do on my own experience of relearning math, and on what I read in Liping Ma or see in the PRIMARY MATH series. I have no idea whether & when what I'm doing is a good idea, and whether or when it's a waste of time.
Here is Anne on PRIMARY MATHEMATICS:
I have been studying the Singapore math textbooks and workbooks. This is what Dr. Ma says the math teachers in China do.
When a new topic is introduced for the first time, there is an illustration which visually explains the topic. It is very simple and straight forward and ties into all the other illustrations that have been used in the book. There is usually a short English explanation and an equation if appropriate. For example, in 1B on the topic of comparing numbers: the illustration is comparing the number of stamps. The first illustration has 3 stamps. There is a cartoon of a child saying the number 3. The second illustration has 4 stamps, but 3 of them are exactly the same as in the first picture. The exact same cartoon child is saying the number 4.
Then, there are more illustrations but with all different types of things. For example, when learning about ten and ones, sometimes the illustration is bundles of sticks, sometimes blocks of ten etc.
Finally, there is a set of problems by themselves with no illustration.
Then, the workbook has all different exercises for the same type of problems. For example, Daniel is working on equivalent fractions in 3B. There are about 5 different exercises on this subject, some with illustrations to help and some without.
Since topics are always introduced in the same way with the same type of illustrations, you can tie back to what was learned before.
Additionally, word problems are very uniform also. For subtraction word problems for one, two and three digit numbers, there will always be one that uses the words more than, one that will use how many left, and one that will be how many of one type of thing.
Also, Singapore math introduces the first multistep problems in 2A, but only in the textbook.
So in Singapore math, the student is introduced to the concept first by visual illustration and then the procedure. And he has learned to do problems in several different ways right from the beginning. No one asks him to do the same problem in a different way but different exercises in the workbook show him how to do different problems in different ways for the same concept.
As for Everyday Math...well, I've been studying that too for comparison. I won't bore you with my rantings here. I have just one example that I think sums things up:
In the Everyday Math journal that students use in class, there are pages of Math Boxes that are review. In the first semester second grade, there are 120 Math Box pages with 6 problems on each page.
In one particular box, there was a problem to count back by 5s starting with 45. And there were spaces to put in the numbers. Then underneath is said, "Can you keep going?" And had this: 0, , .
Well, of course, my daughter had left this blank. Her teacher filled it in for her with -5 and -10.
What possible sense does it make to throw in negative numbers in a problem in second grade?
But that is Everyday Math.
fraction subtraction without borrowing
SusanOnMadMinutes 13 Nov 2005 - 18:34 CatherineJohnson
Susan used Mad Minutes with both kids--
I agree about the "mad minute" approach for all levels of math ability. I used it with both kids and I'm glad I did. It just helps with the speed and proficiency.
Gifted kids are notorious for not wanting to memorize anything. It's too boring. I had to make mine do speed drills all through first and second grade. I'm still doing it with my LD 8th grader.
I remember when math kid was learning multiplication in the first grade. He informed me that it was stupid to memorize them because he could always just count the the groups. I said, "Quick, what's 7 X 8?" His eyes went up looking for those groups, and after a couple of seconds of his looking for them and then trying to skip count by 7, I said, "That's why you memorize them."
I think it helps in the confidence department, as well, to get that speed going earlier than if it just came through practice. And if practice is the key, like "exposure" to lots of words was to the whole language crowd, then what exactly happens to the kids who don't get enough of that practice? Where is the place where enough practice crystalizes into math fact proficiency, especially with these "spiraling" curriculums that keep pushing mastery on down the road?
Trailblazers, as well as other NCTM curriculums, never seem to have a plan for the ones left behind. And I guess you can't ever know who is accountable since mastery was never the goal to begin with.
Interestingly, both kids do love the minute drills where they try to beat their last best score, so I never have any trouble giving them to them.
Well, that's two.
I have to say....I simply can't see any reason on the planet why worksheets would be bad (though TRAILBLAZERS explicitly states that worksheets are destructive!)
Given that there's no (apparent) downside, and given that they've worked for other kids, I'm certainly going to be using them with any child whose math education I'm involved in. (I'm starting to pick up a few! It's incredibly fun. More on that later.)
modifying worksheets
That reminds me.
One of the kids who took my Singapore Math class just could not get through a Saxon work sheet--and he didn't improve any over time, either. All the other kids got fast fast. (It really was remarkable.)
This particular boy is BOUNCY; he is one high-energy kid. He just can't stand the thought of a 5-minute worksheet; he probably takes one look at those sheets and sees what I would see if I were contemplating singlehandedly painting a two-story house.
I told his mom: try having him do ONE LINE of the worksheet as fast as he can.
I don't know if she'll get to it, but when they get back from Italy in January, I think I'll experiment with him, and see how he does. Just click the stop watch and tell him to call 'TIME!' when he hits the end of one line.
I bet that will work.
FromAroundTheEdusphere 06 Sep 2005 - 04:32 CarolynJohnston
Here and there in the "edusphere" I've seen mention of Professor Plum. He's a fellow educational radical (as I've grown to think of people who favor actual instruction in the classroom), and today I checked out his website.
I learned, among other things, that Direct Instruction actually refers to a very specific method of instruction, and to a commercially available set of curricula. It's not just what happens when I Directly Instruct Ben on how to do a math problem, as I had thought. Professor Plum has a lot of material on it here, if you're curious.
But on a quick perusal, I wasn't attracted to Direct Instruction. I couldn't find what I thought was a sufficiently clear description of what Direct Instruction is about. I learned that it is scripted interaction between teachers and children, and that a great deal of teacher training is needed to implement it properly -- all of which statements I've also seen recently in the Connected Math context. I'd like to see more beef, up front and center.
One of Professor Plum's links also took me here, to a site for parents on how to develop contracts for children that help them achieve academic success. I really like this guy's ideas, which are built around a principle I've been using to good effect around here since Ben was a little toddler, namely bribery. It's not really bribery, of course; it's merely setting up a system of targeted incentives intentionally, rather than accidentally setting up the wrong ones haphazardly. There are lots of good suggestions and examples on this website; a lot of detail of the sort that makes you braver about actually implementing his suggestions.
I also did very much like a recent post of Professor Plum's, entitled Basic Features of Effective Instruction. This post is a gem; it summarizes the features of effective teaching very well, I think (I'd love to know whether KTM teachers agree with me on that!). While reading it, it struck me that I hadn't seen teaching methods of any sort described with such clarity since Ben was very young, and I was working with Applied Behavioral Analysts to implement the Lovaas curriculum, which is designed to treat young autism-spectrum children. There is no tougher customer to teach than a very young autistic child; they are extremely disinclined to pay the teacher any attention at all, and they are often not motivated by the things that motivate typical children (like praise and attention). A teacher can't mess around; her message has to be crystal clear, and her incentives have to be right on. Many of the principles he outlines here are typical-kid versions of those one uses in Applied Behavioral Analysis, to decrease confusion and ineffectiveness (and no surprise either, since he has worked with autism spectrum kids in his career). A terrific post.
WallStreetJournalSingaporeMath 12 Sep 2005 - 19:32 CatherineJohnson
I'm teaching my little Singapore Math class again this fall, in the Main Street School after-school program. Last year I had one blinding success, a boy who took to the Singapore bar models like a fish to water and decided, apparently as a direct result, that he liked math and wanted to do well in it.
He was a Phase 3 kid, now boosted to Phase 4!
So I'm looking foward to it.
(The other kids all did great, too; I don't mean to draw negative comparisons. They just didn't experience major life epiphanies as a result of drawing bar models.)
I was revising my course writeup today, and had to go hunting for the WALL STREET JOURNAL article on Singapore math, which I apparently had neglected to post anywhere on the site. So here's the link.
Excerpts:
Singapore's curriculum was developed over the past few decades by
math experts hired by the Ministry of Education, who continually
interviewed math teachers to find out what works and where kids need
help. The elementary textbooks cover only one-third of the topics
typically found in U.S. textbooks, but the material is taught far
more thoroughly. While rote learning plays a part, kids in Singapore
also learn to use visual tools to understand abstract concepts.
Singapore math texts, for example, ask kids to draw bars and other
diagrams to visualize problems -- a technique called "bar modeling."
When this strategy is applied consistently over a number of years,
children tend to be better able to break down complex problems and do
rapid calculations in their head.
[snip]
The National Council of Teachers of Mathematics in
the U.S. suggests that it might not be possible to copy what
Singapore's done simply by importing its books. The success of its
math program may have roots in Singapore's highly disciplined
culture, where the entire community -- particularly parents --
expects kids to buckle down and work hard, argues the NCTM.
There's little doubt, though, that math teaching in America needs to
be overhauled. Tuesday, Boston College will release a four-year
global study that is expected to show the math gap with Asia remains.
The college's last study, the 1999 Trends in International
Mathematics and Science Study (TIMSS), ranked eighth-graders in
Singapore the best in math, while U.S. kids came in 19th, just behind
Latvia. American kids also fall further behind the longer they're in
school; as fourth-graders, American kids ranked 7th on the 1995 study.
That decline has already had an impact on U.S. universities.
today's horror factoids:
- Among U.S. freshmen who plan to major in science or engineering, one in
five requires remedial math courses
- Enrollment by U.S. citizens or permanent residents in graduate science and engineering programs, meantime,
dropped 10 percent between 1994 and 2001. Enrollment of foreign
students grew 35 percent.
another link to the WSJ article: As math skills slip, U.S. schools seek answers from Asia
key words: decline in U.S. engineering math and science enrollment
InnumeracyPart2 13 Nov 2005 - 19:59 CatherineJohnson
A section of the innumeracy article Carolyn linked to caught my eye:
Wieman says getting students comfortable with math as a way of describing the natural world is a nut he has had trouble cracking. He said methods such as those developed by his Physics Education Technology program can give students without science backgrounds a deep understanding of scientific concepts, "yet when something involves a simple arithmetic calculation, their brains click into this totally different mode."
Steven Pollack, a CU physics professor whose research focus is improving the education of physicists, says the problem also happens in reverse. Physics students often can't conceptualize or explain the results of the equations they so breezily manipulate.
This is something I've been wondering about.
This may sound crazy, but, as a kid, I was reasonably good at math. I got straight A's, I had no trouble learning whatever I was supposed to learn (my one bad moment, in 2nd grade, WHICH I REMEMBER TO THIS DAY, involved--guess what?--fractions).
I took my SATs cold, with no practice, a year after I'd looked at my last math book and got a 620, which put me way up in the top percentile of the nation's 17 year olds at the time. (IIRC, I may have been in the top 95th percentile for girls.)
So....I was reasonably good at math.
That's why when Christopher came home with his 39 on Unit 6 it never crossed my mind I couldn't simply sit down and teach him what he'd missed.
no can do
You all know the end of that story. I discovered I knew practically NOTHING about math.....which is an exaggeration, but is sure the way I felt. I've been obsessively re-teaching myself elementary mathematics ever since, and intend to go on to trig & calculus & and a bit beyond.
So what does it mean to say that I was 'reasonably good' at math?
It means I could set up two-variable algebra problems and solve them in a jif. Thirty years later, I could still do it. Easily.
But I had no idea why setting up two equations (or 3 or 4) worked.
This is why I'm such a fan of the Singapore Math bar models (one of the reasons); it was a bar model that first explained to me what subtraction really meant.
the difference between two numbers
I had a Helen-Keller-at-the-water-pump moment the first time I drew this bar model. I had simply never noticed that the 'number' of boys and girls in Mrs. Johnston's class, up to the number 10, is the same number. The 'extra' five boys are the difference.
For my entire life I had heard the word 'difference' used to name the number you end up with when you subtract one number from another, but I had never, ever realized that 'difference' actually did mean difference. It wasn't just some random word that had gotten attached to the operation somewhere back in the mists of time.
I now point this out to any kids I teach--and they all seem to find it extremely cool, too. I say, and then I repeat frequently, Subtraction is finding the DIFFERENCE between two numbers.
Then I point out that, if you're subtracting 3 from 5, 3 and 5 are the same number until you get past the 3-that-is-inside-the-5.
quick question re: number partition theory
The article on Everyday Math that I linked to yesterday, Weighing the Factors says this is number partition theory.
Is it?
odd man out
Teaching the how-many-boys-and-girls problem to kids, I also point out what would happen in Mrs. Johnston's class if you paired each girl with a boy.
You would have five boys left over.
That seems to make enormous sense to grade school kids, perhaps because they spend their grade school years being assigned partners or buddies to walk in lines with, or go to the bathroom with, etc.
This is obviously the way to teach the concept of even and odd, too. If you have an odd number of kids, there's going to be one child left over when you assign them to teams.
AND this image works great for teaching the idea that you always get an even sum if you add two even numbers or two odd numbers, but you get an odd sum if you add an even & an odd. In my experience so far, kids can instantly see that, when you add two odd numbers, you get two 'odd men out'--and now those two finally have a partner!
why don't numbers & concepts connect more easily?
This brings me back to my original point: somehow, you can learn numerical manipulations, including more advanced numerical manipulations that require you to set up equations and solve them, and not have a clue what it all means.
I don't understand this.
I don't understand how I could have so much fun setting up equations & solving them, and never gain the slightest idea why what I was doing worked.
keywords: conceptual understanding & bar model difference between two numbers comparison of numbers subtraction as comparison subtraction has two meanings
partial product division in Everyday Math
fighting innumeracy at CO
subtraction as the difference between 2 numbers
study sheet: subtracting integers & absolute value
notes on integer, subtraction, & absolute value study sheet
WilliamKSmithCalculus 16 Sep 2005 - 12:16 CatherineJohnson
Here's another recommendation from Barry Garelick:
Calculus with Analytic Geometry by William K. Smith (also available at Amazon)
I've already ordered my copy.
Have I mentioned I'm planning to take calculus?
Well, I am. I'm planning to take calculus.
But first I have to 're-take' algebra & geometry. Then trig, which I've never studied.
You folks here at ktm are helping me so much. Even though I'm a writer, I can't locate the words to describe what you've given me. The reason I can't 'locate the words,' of course, is that I don't actually know what I'm learning from ktm. I study & absorb what people say, but then forget the source of my new knowledge once it's been assimilated into my store of old knowledge. I'm left with the hazy feeling that 'I'm learning a huge amount from the Commenters at ktm.'
So I'm going to start taking notes. God is in the details.
thank you!
integers! integers!
So Christopher's math class started integers on Monday—a topic he knows virtually nothing about—and he's having a test tomorrow. He is way not prepared, so I'm busy today writing an Integer Lesson. Probably won't be posting much (though I may have a couple of things from Barry.)
I'm taking a moment to make one more plug for Mathematics 6 by Enn R. Nurk and Aksel E. Telgmaa, though.
I could probably add & subtract integers in my sleep. (Though I did have to do some review last year when I first re-encountered the topic, which I take as a sign that my knowledge was more procedural than conceptual.)
But last night, after working with Christopher for awhile, who was semi-lost (I don't think he could pass a test at this point) the Math Fog rolled in.
This is the good thing about working with people who know less math than you do. Concepts and procedures you thought you understood turn out to be not quite so clear. I assume that's what Bernie meant when he said the other day that he'd realized there were aspects of reciprocals he hadn't thought about (if I've got that wrong, Bernie, I'll change it!) Carolyn has said something similar at times. I'll be asking her about some elementary concept that, for her, is as simple as breathing in and out, and suddenly she'll see why Ben--or anyone else--might get confused.
lost in translation
This is another one of those constructivist insights that's been lost in translation.
For me, and I think for most teachers & writers, teaching or writing about a subject always forces you to understand it far better than you did.
Radical constructivists conclude from this that children should explain all of their answers in words.
I'm pretty sure that's wrong, because math is not language. Math is math. A child who can explain his answer by showing the mathematical steps he took to find it has produced a proper mathematical explanation as far as I'm concerned. (Russian Math & the Chinese teachers in Liping Ma all offer mathematical explanations & demonstrations.)
But what really bothers me about the 'explain your answer in words' business is that it puts the onus on the child to teach himself. The teacher doesn't have to work and fight and struggle to find the right words; the child does. I know that's wrong.
While I'm on the subject, why don't I just go ahead and take umbrage at the suggestion that a child is capable of explaining math in words?
Writing is hard. Writing well is extremely hard. Finding the words to explain any mathematical concept well is a vast and ambitious undertaking in itself, not a toss-off in the middle of a homework assignment or state assessment. (I'm seriously against the extended response (pdf file)requirement that's taken over IL state rubrics. At least, for the time being I am. [update 5-14-06 sorry, link no longer works])
back to Russian Math
I shouldn't be putting words in people's mouths, so if I've misunderstood Bernie or Carolyn I'll issue a CORRECTION.
In the meantime, why don't I just return to quoting myself.
It's true for me that when I work with a child for awhile, I realize I don't understand things as well as I thought (or hoped).
After Christopher went to bed, I got out Mathematics 6 and turned to the section on adding & subtracting integers.
The first thing that struck me was the fact that this topic appears at the very end of the book. Prentice Hall Pre -Algebra* opens with integers, and I question that. I question it not based on any profound grasp of pre-algebra as a coherent whole. I question it on grounds that Nurk & Telgmaa are geniuses, and they put adding & subtracting integers last, not first.
I'm sure they have their reasons. (I intend to figure out what their reasons were.)
Reading through Nurk & Telgmaa's discussion, I realized why I was confused. I think I realized why Christopher was confused, too. I hope so.
We were both, I believe, stumbling over this type of problem:
5 - (-7) = ?
Both Saxon Math 8/7 & Russian Math teach addition & subtraction of integers using the number line. Saxon's lessons were particularly strong, I thought.
But when I tried to untangle myself by resorting to the number line, I got stuck.
Start at zero, move five to the right, then.......then what?
What was my next move? My very next move, without renaming or re-expressing - (- 7) as + 7 ?
I was stuck.
Reading through Mathematics 6 I realized that the problem is something Wayne Wickelgren & his daughter Ingrid have raised: the same letter or sign has been made to stand for two different things.
There are two 'minus signs' in 5 - (-7). One means 'opposite,' and the other means 'subtract.'
One means 'perform an operation' and the other doesn't (I don't think. Is 'taking the opposite of a number' considered an operation? I don't know.)
In any case, for both Christopher and me, 'subtract' and 'take the opposite of' are two different things.
Mathematics 6 has a formal demonstration of the fact that:
5 - 7 = 5 + ( -7 )
This is something I think I figured out on my own many, many years ago. I've been using it ever since to de-confuse myself when dealing with long lines of integers to add & subtract. At some point, if I'm getting confused about whether I can or can't use the commutative or associative properties, I just turn the whole thing into addition.
Reading Mathematics 6 I realized that's what needed to happen with 5 - ( -7):
5 - ( - 7) = 5 + [ - ( - 7) ]
Voila!
Christopher and I both understand that 'the opposite of the opposite' is the number you started with originally; the opposite of the opposite of 7 is 7. (This wasn't an especially hard idea for Christopher, but the number line really nails it down.)
Once you convert '5 minus negative 7' to '5 + the opposite of the opposite of 7' it's in a form Christopher understands, and can do.
AND it's in a form you can perform on the number line, if you like or just want to check.
5 - ( - 7) =
5 + [ - ( - 7) ] =
5 + [ 7 ] =
5 + 7
Once you've converted a 'double negative' subtraction problem into addition, you no longer have an anomaly, The One Subtraction Problem That Cannot Be Done On A Number Line.
We'll see how it goes. This morning I had Christopher quickly rewrite 12 subtraction problems as addition problems. (I haven't explained to him why a subtraction problem can be rewritten as an addition problem, and I don't know whether I'll get to that today. I haven't closely studied Mathematics 6's presentation to see whether I can introduce it 18 hours before the test.
Fortunately, Ed had already introduced the idea that 'subtraction is addition' last night, when he used the addition-of-debt-to-debt (a concept that is not foreign to our household) to show Christopher that:
- 7 - 7 = - 14
I think he had a lesson in Saxon on subtracting a positive from a negative being the same thing as adding a negative to a negative, so he probably had some knowledge to build on before Ed gave him the add-one-debt-to-another example.
It's the minus-minus issue that's throwing him.
I hope.
one last thing
Looking at this, it strikes me I'm also going to have to create some problems that I ask Christopher to 'simplify'—'simplify' defined broadly as 'write it in the simplest possible correct way that will allow you to recognize what the computation is and do it.'
For instance:
-7 + 5
He probably needs some practice rewriting this as 5 - 7.
I'll see.
I'm also going to try to put together an incredibly simple 2 - 1 type problem that he can always solve quickly when he gets jumbled up. Something like this:
1 - ( - 1) = 2
-1 -1 = -2
-1 - ( - 1 ) = 0
He hasn't learned the Polya line about how 'For each complicated problem you can't do, there is a simple problem you also can't do.' I realize it's not clear that you can explicitly teach problem solving, but I'm going to have to try. He's got to learn the strategy of creating a super-simple version of a hard problem in order to see how to deal with the hard problem SOON.
*new title:
Prentice Hall Mathematics: Explorations & Applications
keywords: subtraction negative minus absolute value subtraction is addition integers extended response
TeachingSubtractionAndIntegers 18 Sep 2005 - 02:42 CatherineJohnson
click on Printable Version to print
What is subtraction?
Subtraction is the ______________ of addition.
When you subtract, you __________ ___________ ___________________ of the number you are subtracting.
An absolute value is always _________________.
1 - 2 = _________
1 - ( - 2 ) = _________
-1 - 2 = _________
-1 + -2 = _________
1 - | 2 | = _________
-1 - | 2 | = _________
-1 - | -2 | = _________
answers
study sheet for class quiz on pages 2 - 16, Prentice Hall Mathematics: Explorations & Applications & Prentice Hall Pre -Algebra
outloud sheets: integers & absolute value
answer key
notes on outloud sheets for integers & absolute values
Carolyn on introducing absolute value
keywords: integers subtraction addition absolute value opposite add study sheet outloud out loud
PracticeSheetIntegersSubtractionAbsoluteValue 16 Sep 2005 - 14:38 CatherineJohnson
I wrote up a study sheet for Christopher's test (it's in the next post) & dragged him through it kicking and screaming.
I think it worked, but we'll see.
If you hit 'Printable Version' it prints out great, exactly enough space for answers in big, round middle-school handwriting.
update
Christopher said last night he doesn't like it when I tell people he screams when we do math.
I told him, Stop screaming and I'll be happy to stop telling people.
We are at an impasse.
StudySheetIntegersSubtractionAbsoluteValue 16 Sep 2005 - 14:45 CatherineJohnson
(study sheet is here: subtracting integers & absolute value)
Here is how Christopher does this problem:
-1 - ( - 2 )
He pencils in a vertical line across both of the minus signs in the middle, turning them into plus signs:
- 1 + ( + 2 ) =
That works for him every time, no matter what the numbers, and he isn't thrown off by the same problem written with an absolute value:
-1 - | - 2 | =
This reminds me of Carolyn's belief that you need to get math into a child's hand.
For some reason a problem like:
-1 - 2
makes sense to him. He 'sees' that he's adding two negative numbers.
Here, too, however, he does a swoop and swoop thing: he squeezes in a plus sign between the 1 and the second minus sign, like this:
-1+-2 =
Ed's explanation to Christopher that you can think of -1 - 2 as adding two debts -- first you owed 1 dollar, then you borrowed 2 more dollars and you owed 3 -- seems to have been the ticket.
I tried that explanation on a friend of mine who is severely math phobic, and she instantly got it, too. Adding debt to debt is something everyone can grasp! It's EVERYDAY MATH FOR THE MASSES!
From one of Carolyn's first posts:
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.
swoop and swoop
the craft of math
subtraction as the difference between 2 numbers
outloud study sheet: subtracting integers & absolute value
answer key
notes on integer, subtraction, & absolute value study sheet
Carolyn on introducing absolute value
keywords: integers subtraction addition absolute value opposite add study sheet outloud out loud
ILikeMathPart3 17 Sep 2005 - 02:47 CatherineJohnson
I almost forgot!
Monday or Tuesday night, when Christopher was doing one of his first homework assignments from Prentice Hall Mathematics: Explorations & Applications, he saw an illustration on the side of the page with the caption:
The early Egypticans drew pairs of legs walking in different directions to stand for addition and subtraction.
He looked up at me and said happily, "I like math. I just don't like math when you make me do it."
BeingYourChildsFrontalLobes
GreatMomentsInWorldHistory
ProgressReport
ATeachersStory ("I like the idea of math")
BonusPreTeenPost
fun with Saxon Math in the summer
SundaySchool
I like math
I like math, part 2
TheGoodNewsFromHere
GoodNewsBadNews
ImGoingToPlayland
ImportantQuestionFromJoanneCobaskoOfSocmm
ImportantQuestionPart2
OutsmartingTheTests
ConversationsWithKids
Christopher on his 39
I like math, part 3
PrenticeHallPreAlgebraQuestion 21 Sep 2005 - 10:24 CatherineJohnson
Well, Christopher managed an 85 on his first math quiz.....but we're gonna need to step up the pace around here.
Ed checked his homework tonight, which prompted vast quantities of screaming and yelling (maybe there's something to that brain periodization business after all), and now reports that Christopher has essentially zero comprehension of how to solve a story problem involving negative numbers. He's just looking at the problem and trying to figure out which operations to do. Ed says he's more or less guessing.
The good news is he got points off for mechanics, failing to put in the degree sign and the like. He would have had an 89 if he'd LABELED EVERYTHING CORRECTLY. So from now on he will label everything correctly.
The bad news is that the teacher is doing what she did last year, which is putting problems on the test they've never done before or even seen in class or on homework. He had two story problems like this one:
The boiling point of oxygen is -297 and the boiling point of nitrogen is -320. How much higher is the boiling point of oxygen?
Here's my question.
Distance, I know, is always expressed as an absolute value.
Is 'distance' on a thermometer the same thing?
Say the question had been written as, How much lower is the boiling point of nitrogen than the boiling point of oxygen?
Would the answer still be 23?
anyone know of a good source of story problems?
To get Christopher through this course, I'm going to need two things:
- LOTS of practice computation problems (integers, fractions, the works)
- LOTS of practice word problems
Does anyone know of good supplies of either?
I am also going to need vast supplies of Iron Will.
teach kids good handwriting in school
Christopher's handwriting was so bad in Kindergarten that his teacher told us he was considered 'at risk' for dyslexia. (Kids with learning disabilities often (usually?) have bad handwriting.)
That was one of those four-star fun-with-childbearing moments. Two autistic kids, and this one's gonna be dyslexic!
Ed pooh-poohed the whole thing (he is the pooh-pooher in the family), and in fact Christopher began reading on his own literally 2 weeks later (THANK YOU, GOD).....and that was the last any of his teachers had to say about handwriting until he had Ms. Duque in 5th grade last year.
So tonight he missed at least one problem on his homework because his handwriting is still so bad he can't read it himself. His test is a mess; I don't know how he managed to do as well as he did given what a visual morass it is.
Two summers ago I researched handwriting programs, and we spent one summer working on handwriting....and then Ms. Duque pushed him on it last year, though she didn't teach it. When my parents went to school, handwriting was taught in formal handwriting-practice programs that worked. (Ever noticed that ALL members of the greatest generation have beautiful handwriting?)
Today there are schools on Long Island that don't even teach cursive anymore, or maybe it's the other way around.
Anyway, handwriting is one of those ROTE NON-CONCEPTUAL NON-CRITICAL-THINKING SKILLS that have been drop-kicked right out of the curriculum. Replaced by character education. Last year the school spent 20 minutes each and every morning for six months doing their No Put Downs program. This year the program's even bigger as far as I can tell. The teachers are all being trained, and one of Christopher's teachers told us on back-to-school night that, thanks to all the character education she would be doing during class time, 'your children will be better people.' [update: This teacher was Mrs. R. 3-25-2006]
The point is, if Christopher is going to speed through these tests, he's going to have to develop fluency not just in math facts & computation, but in handwriting, too.
Well, at least he had those couple of months with me. I have a couple of cursive practice books sitting in his homework file, so maybe I'll pull those out and get started again. One more thing to brawl over.
I feel a rant about character education coming on.
That stuff I just wrote?
That wasn't it.
keywords: character education bullying no putdowns
keywords: good handwriting Write Now
MathLessonRepeatingDecimals 22 Sep 2005 - 20:01 CatherineJohnson
My neighbor showed me this yesterday. Naturally no one had ever taught me how to do this, which is par for the course. But she's a statistician & she'd never learned it, either.
I love this. It reminds me of the shenanigans I go through trying to force Microsoft Word to do graphic design.

I've entered this on the Math Lessons page.
other resources
Purple Math
Math Wizz on converting repeating decimal to fraction
update: Saxon meltdown (3-2-06)
Maybe I'm just tired, but I practically had a nervous breakdown tonight trying to convert 0.013333....(repeating decimal) to a fraction.
I just could not get it.
Finally Math Wizz saved me. Of all the websites I looked at, Math Wizz had the simplest, cleanest, & most follow-able explanation.
Math Wizz also has gigantic gifs.


LoneRangerOnHandwriting 21 Sep 2005 - 19:49 CatherineJohnson
I'm bringing Lone Ranger's advice on handwriting up front, because it's important.
For one thing, there's research showing teachers give higher grades to papers with good handwriting, whether they're supposed to or not. That makes sense. Any teacher who's spent more than 5 seconds in the classroom will have noticed that children with learning disabilities aren't the ones with the nice, neat handwriting.
Personal anecdote pause: Ed was giving me some grief about my handwriting lessons with Christopher. He thought they were silly. Everyone thinks they're silly.
So then we had a birthday party for Christopher, and one of the kids we invited was a boy who was in the multisensory class, considered 'at risk' for LD, etc.....the handwriting on his birthday card was so erratic and wild it was almost scary. And.....it didn't look all that majorly different from Christopher's handwriting.
Ed took one look at that birthday card next to all the other kids' birthday cards and got with the program.
(He has horrific handwriting himself, btw. He's a leftie.)
Back on topic: the other reason kids need good handwriting is the same reason they need good-everything: they need automaticity. Again, there's RESEARCH SHOWING that kids with good handwriting produce better content in writing essays. (I'm not going to look it up; you'll just have to trust me.)
Kids need to be able to write legibly and quickly.
Last but not least, this is another one of those curriculum decisions that I'm betting favors girls. Every boy I know has had handwriting woes; Christopher's handwriting was so bad he was being pulled out of Kindergarten for free O.T. services.
My understanding is that girls have better fine motor skills.....I could be wrong, but that's my impression.
Here is Lone Ranger:
To remediate handwriting buy some handwriting paper with a dotted center line. Miller Pads and Paper sells some for upper elementary aged kids and it is great stuff. www.millerpadsandpaper.com Now teach him two things. First, all his letters must basically be the same width (there are some exceptions here). Second, all letters must bump the top and bottom lines. Lower case would bump the dotted center line and the base. Upper case and numbers would bump the top and bottom lines. (For some reason kids like the expression "bump the lines") Have him practice on this paper until this new skill is internalized. He will see instant improvement and that is usually quite encouraging. Good luck!
update
I was Googling around, trying to find the terrific penmanship-paper software I had for my PC, and I found this instead: SpellWrite Books 1-4
It's a spelling and handwriting program from Oxford that sounds terrific. I've never seen it, obviously, but I like the idea, and I generally like Oxford Press. Offhand, and knowing next to nothing about the teaching of reading, writing, & spelling (now there's a caveat for you), I like the approach:
SpellWrite is a series of four workbooks for middle and upper primary that assist students to develop their spelling skills. It focuses on the four forms of spelling knowledge: Phonological, Morphological, Visual, and Etymological.
Features:
• Spelling is presented as an integral part of the writing process
• 25 units of three pages designed to provide a full week’s work for students
• Each book in the series includes: Words to Learn, Assessment for the appropriate year, Rhymes to read, Word Bank, Test pages, Revision units, and teacher ideas and suggestions for each activity
• Each unit begins with a rhyme, tongue twister, or word game designed to introduce the relevant sound and excite the children’s interest in words
• Allows teachers to explicitly and systematically teach students the four forms of spelling knowledge
• Covers a variety of text types with a focus on rhyme and poetry.
The books cost $12 apiece. I'm almost tempted to get one, just to see if I can kill two birds with one hand.
I love Megawords, but the student writes everything on the text pages themselves, which frequently don't have enough space.
Which brings me to another Personal Anecdote moment.
Ed's always been a tad skeptical of the Home Spelling project I've got going. Then, on back-to-school night, we were all handed class schedules written out by our kids. Christopher had spelled cafeteria cafitrea.
Ed has got religion
StartWrite software
I found it.
This is a dandy little program, that apparently works on a Mac.....wonder if I can find the original disk down in the basement?
review of StartWrite Handwriting
The great thing about StartWrite is that you can make the lines as dark or light as you like, and you can space them the way you like. You can also print any words at all for tracing. Extremely easy to use.
free online penmanship sheets
handwriting worksheets at Teachnology. These aren't bad. Upper case, lower case, printing, cursive, and numerals.
MathLessonsPage 21 Sep 2005 - 15:48 CatherineJohnson
I've started to get the Math Lessons page pulled together. I'm sure I've forgotten posts that should be indexed there, so if you know of any, let me know. (Any lessons you especially like from other people's sites, like MathandText, for instance, should also be added.)
There's a link to 'Math Lessons' on the sidebar.
TeachnologyFreeWorksheets 21 Sep 2005 - 20:07 CatherineJohnson
Teachnology seems like a useful site.
Here are free online word problem worksheets.
And here are lots of free math worksheets.
I like this addition and subtract equations worksheet.
BestMadMinutesBook 22 Sep 2005 - 04:06 CatherineJohnson
I keep forgetting to ask.
I'm teaching the Singapore Math after-school class again, and I don't want to use Saxon's 5-minute sheets.
I need a 1-minute sheet (or online source).
Thanks--
FunBrainNumberLine 22 Sep 2005 - 19:26 CatherineJohnson
Line Jumper, an online number-line competition at FunBrain.
The kids in my Singapore Math class LOVED the FunBrain site. They especially liked the Math Baseball game.
Normally I'm skeptical of online activities (because Christopher seemed to learn nothing from software math facts programs) but the kids I've known really did like these math facts games, and could play them for a lot longer than they'd do a worksheet.
You can also use Math Baseball to teach mental math, because the kids have to do the calculations in their heads, unless they pull out a pad of paper & a pencil.
I had Christopher do the integers worksheet from Saxon Math 8/7 last night. I'm going to have him keep doing it until he can finish it in 5 minutes & get everything right. That will help.
OnlineMathResources 22 Sep 2005 - 22:30 CatherineJohnson
I came across all kinds of interesting-looking math web sites last night while looking for:
- integers worksheets
- downloadable number line worksheets
I didn't find either of the things I wanted (and almost spent $29.95 to join some teacher site linked to by FunBrain just to be able to printout their number line sheet...).
But I found all of these:
- AAA Math (resources listed by grade thru gr8)
also has a potentially interesting page called World Education Levels. Unfortunately, I can't tell what 'world education levels' are without spending a lot more time on the site than I want to spend. LOTS of online quizzes that are corrected by the site, and they seem to be selling a software program on arithmetic.
- the aforementioned FunBrain Math Baseball is a classic.
- FunBrain's teacher site, the page that almost sold me a $30 sheet of number lines. Has articles on behaviormanagement in the classroom that look good.
- Harcourt School Publishers' number line express Blecch. But maybe little kids would enjoy it. There's a talking lion railroad engineer.
- Math Cats how-to for teachers Definitely worth looking at.
- math clip art! possibly for autistic kids (I was on a major clip art tear a few years ago, when Andrew was in his PECS genius phase...)
- Mathsurf teacher's site word problems from Pearson Scott Foresman. If you're looking for story problems with multiple answers, this is the spot. Possibly (probably?) a good site to visit for problems your child may encounter in constructivist math courses -- worthwhile problems, as far as I can tell on cursory inspection.
- Mathsurf telling time worksheet (to print)
- Room 108 Looks decent. You can create online Mad Minute pages (must be answered & graded online)
- odd & even numbers possibly good for autistic kids? this site speaks the directions, although I don't think the directions are also written out in words. But any time an autistic child can hear the same words spoken by the same recorded voice it's a good thing, I believe. Site is simple and graphically compelling. Has a HUGE cursor (also great for autistic kids.)
- Primary Games good for autism? I have a feeling this might work with Andrew at some point in the near future. Very simple, has ONE moving image--'Squigly,' a little worm inside one of 10 apples who pops out of his apple and then disappears back inside every couple of seconds. The child has to tell which apple Squigly is in (first, third, fifth, and so on). The only bad part is that there's a lot of advertising crud at the top and the bottom of the page.
- Primary Games fishy counting game good for autism? terrific. Very, very simple counting game (as nice as the counting game they used to have on the Barney web site....
- Primary Games Tetris bubbles Great! I've been meaning to post a TIME MAGAZINE article saying girls improve their spatial-visualization skills when they play Tetris. This is, I think, a somewhat slower version of a Tetris game. (Slower is always good for me....) Stupid music, though.
- Primary Games time clock Terrific! Very simple & cute. You have to be able to use a mouse (Andrew & Jimmy both have huge MOUSE difficulties, unfortunately.)
eureka
I will never, ever speak ill of the NCTM again.
They have FREE NUMBER LINES, 8 to a page!
Unfortunately, all 8 number lines start at 0 and contain only positive numbers....
update
I take it back.
I will carry on saying bad things about the NCTM.
They do not appear to have posted a single number line on their web site that includes negative numbers as well as positive numbers and 0.
keywords: online interactive math resources tools nets manipulatives
BestGrammarBook 15 May 2006 - 02:07 CatherineJohnson
I have appointed Susan grammar diva, because....she knows grammar! (And, more to the point, grammar books!)
Susan, what book should I order RIGHT THIS MINUTE?
Christopher got a 63 on his grammar test, because he 'mixed up subject and predicate.'
I can't take it.
He's ELEVEN.
And he doesn't know subject & predicate.
So.....which one of the books you told me about should I get NOW. I need something with MAXIMUM direct instruction, MAXIMUM coherence (if possible), and PRACTICE EXERCISES.
Sigh.
Another commenter once recommended the Shurley grammar series--how involved is this series?
(Does anyone know?)
Can I fit it in with everything else?
OwlPurdueUniversityOnlineWritingLab 23 Sep 2005 - 20:27 CatherineJohnson
For future reference -
The OWL - Online Writing Lab - at Purdue University is a fantastic resource.
I'm posting the link on the Math Supplements page.
GrammarSchool 14 May 2006 - 15:09 CatherineJohnson
So, yes, I am now in the grammar instruction business, too.
Ed asked Christopher last night what the subject and predicate were in the sentence, I ate too much food, and Christopher didn't have a clue.
He flat out couldn't say what the subject was, and he thought the predicate was 'too much food.' Then, when Ed corrected him, he sobbed for 15 minutes.
Middle school stinks.
We're only....3 weeks in? Already I've got at least 4 crying children stories, 4 that I can remember, anyway; there may have been more. Today Christopher's close friend M. started crying when the math teacher docked him a point on his math test for telling his twin brother, 'It's easy, you can do it.'
M. protested that he had only been telling his brother he could do the test, and the teacher said that didn't matter, he could have been cheating.
So back to grammar, Christopher has no clue what a subject and a predicate are. He rejected outright Ed's claim that 'I' was the subject: How can 'I' be a subject??????' Then collapsed into sobs brought on by the sudden realization that the reason he 'put the line in the wrong place' was that he didn't know where the subject ended and the predicate began. A classic example of a child not knowing what he doesn't know, which Willingham has written about. (Why Students Think They Understand—When They Don’t and How To Help Students See When Their Knowledge is Superficial or Incomplete)
I'm guessing Christopher probably thinks 'subject' means 'topic,' as in the topic of an article or book; and, by extension, 'predicate' means the topic of the second half of the sentence. Which would pretty much rule out pronouns & verbs as subjects & predicates, respectively.
Christopher is 11.
His school has two hours of 'English language arts' a day, TWO. And in two hours a day this teacher--this tenured, health insuranced, pensioned individual--did not manage to teach Christopher what a subject and a predicate are.
Teaching math is hard. I'm not going to be wildly critical of a math teacher who is trying. (A math teacher who docks a twin a point because he might have been cheating is another story.)
But teaching subject and predicate to a bright child with a good attention faculty whose strength is English language arts.......
Rolling off a log.
And I'm the one who's going to be doing the rolling.
I'm not happy.
update
I just thank God I started teaching Christopher spelling when I did.
FunbrainMadLib 23 Sep 2005 - 23:12 CatherineJohnson
Carolyn suggested I try Christopher on some Mad Libs, to teach him the parts of speech.
I'd never heard of Mad Libs!
OK, that's not right.
I'd heard of Mad Libs. Didn't know what they were; especially didn't know they were fun things that could teach parts of speech. fyi, I am aware of the fact that I have just written two incomplete sentences joined by a semi-colon. It's a secret technique of mine.
Turns out Funbrain has extremely fun Mad Libs.
Here's an Eleanor Rigby Mad Lib.
The Grammar Bible
I've mentioned this book before, but I'll do it again: The Grammar Bible: Everything You Always Wanted to Know About Grammar but Didn't Know Whom to Ask by Michael Strumpf is supposed to be the best book out there.
I heard about it from my editor at Holt, who said Strumpf ran a web site on grammar for years; grammar was his life. Copy editors all over New York would call him up on the phone to ask him how to edit sentences. Finally he wrote a book, and this is it.
I read the chapter on subjects & predicates and then taught it to Christopher--it was great.
He has a rhyme: the subject & the predicate are the name and the claim.
I always like rhymes.
I've got grammar stuff posted on the Math Supplements page for safekeeping.
Steps to Good Grammar
OK, I did it. I ordered Steps to Good Grammar. Has raves on Amazon & B&N.
I'm gonna get Rod & Staff, too.
Given that we've managed to miss THE FIRST TWO SUNDAYS OF SUNDAY SCHOOL, I figure we need it.
DougsNumberLines 25 Sep 2005 - 14:03 CatherineJohnson
I can't thank Doug enough for his number lines--and ALL OF YOU SHOULD USE THEM, TOO!
Adding & subtracting positive & negative numbers is one of those areas that can be severely procedural if you have a good memory, which most children do as far as I can tell.
Christopher was already becoming entirely 'procedural' with his integer problems: if he saw two minus signs in a row he automatically penciled in little vertical lines and turned them into two plus signs. Then he added.
ooops--must take Christopher to his playdate
will finish this when I get back
I love these number lines.
back again
Well, I'm back from my Excellent Adventure at Barnes & Noble with Andrew, who is obsessed with pulling Arthur books off of their shelves and lining them up on the floor, while I apologize to clerks & customers.
Now he's shrieking at the top of his lungs & jumping as hard as can on his bedroom floor upstairs, which has shaken loose all the lightbulbs in the kitchen light fixture, which means someone will have to climb up on a ladder, take down the fixture, and screw the bulbs back in.
I'm in a nuts-to-autism mood.
back on topic
as I was saying.....Christopher's memory is good enough that he's reaching the point of procedural fluency with integer computations, and that's got its bad side as well as its good side, because I'm sure he still has no idea how to do the problem I posted a couple of days ago, the one asking what the difference was between the boiling point of oxygen and the boiling point of nitrogen.
[pause]
This is grueling. I've just spent 15 minutes trying to deal with Andrew, who has slapped himself on both sides of the head so hard that he'll be bruised again. We're both trembling. I loathe this disorder.
I'm going to make one more stab at writing about Doug's number lines before I go have my own nervous breakdown.
What I'm trying to say is that it's clear Christopher is reaching the point where he's going to have procedural fluency with virtually no conceptual understanding, and Doug's number lines are the answer.
Number lines are to integer problems what bar models are to story problems. Perfect.
I'm going to have Christopher do a page of Doug's number lines every day for a few weeks, and I'm going to do them myself.
I was telling Ed about all of this, and he said number lines were essential in the GED math course he taught to high school drop-outs in Newark years ago. He was pretty successful with those kids, and number lines were a big part of it. These kids--young adults, actually--were years and years behind in math; they'd never, ever gotten it. Ed had to have visual ways of teaching math to them, he said, or he couldn't have done it.
I haven't got all of the number lines posted yet, but will get to it soon. Right now one page is here, at the top of the Comments thread.
number lines in your head
I'll have to track this down, but one of the neuroscientists who studies math argues that we have number lines--a kind of number line, though not a formal visual image of a number line--in our heads; number lines are essentially there already, along with basic counting & very simple fractions such as 1/2. (I'll have to see whether this particular researcher thinks animals have number lines, too.)
This is all the more reason to use number lines frequently, I think. Any time you can hook a new concept to something a child already has inside his head you've got an advantage.
While we're on the subject of visual models, I've been reading Sawyer's book, which my neighbor gave me for my birthday. It's challenging, but incredibly useful & rich.
Andrew is better
Lately Andrew seems to be having low blood sugar crises. Either that, or dehydration, or both. He has an autistic eating disorder on top of everything else, and won't drink water or milk, etc....and eats only a couple of foods. So he's chronically low on calories, nutrients, and fluids.
I forced grape juice down him 10 minutes ago, and now he's making cheerful noises. His face is bruised, and he's urinated all over his TV stand, videos, and whichever of my books he'd lined up as part of the still life.
BUT, he's OK.
I feel like Annie Sullivan after breakfast.
He folded his napkin.
(So is Kumon Math sounding like just the ticket for Andrew?? I say yes!)
GrammarQuestion 14 May 2006 - 15:10 CatherineJohnson
What is the complete subject of this sentence?
While taking the dog for a walk, she stepped in poop.
Thank you in advance.
WickelgrenOnPrealgebra 16 Jul 2006 - 20:48 CatherineJohnson
Gulp.
A student can learn a year of pre-algebra math in three to six months studying three to ten hours per week, depending on the child's math aptitutde.
I'm gonna have to pick up the pace around here.
I've been working my way through Mathematics 6 since the beginning of June.
It is now the beginning of October.
RUSSIAN MATH has, estimating conservatively, 10,000 problems. At least 10,000. I have now worked 8000. In the process, I've learned a huge amount, although, sadly, even Enn Nurk & Aksel Telgmaa have not been able to dissuade me from the conviction that 7 x 6 = 43. If they can't do it, probably no one can.
I've just begun the last of RM's six chapters, and I was getting excited about starting algebra next. I can't wait.
So last night I took Saxon Math's placement test (pdf file) for algebra 1.
I got a 72.
conclusion number one:
I am going to stop expressing reservations about the Saxon math series until I can actually take and pass a Saxon math test.
conclusion number two:
wow
There are a boatload of topics I still don't know after doing 8000 complicated Russian computation, geometry, & word problems.
They are:
- using four 'unit multipliers' to convert 630 square yards to square inches: I have no idea what a unit multiplier is, or how to use it
- what a decimal part of a number is (I got the answer right, but only because I made a blind guess as to what a decimal part would be)
- negative exponents
- how to find the volume of a cylinder
- 'the method of cut and try' to find the square root of 20: to my knowledge, I have never heard the words 'cut and try' in my lifetime
- how to use a straightedge (what's a straightedge? I still don't know) and a compass to copy an angle
- how to find the area of a triangle (all I remember is: hypotenuse)
- how to find the probability that a die will first roll a 6 and then roll a 2, in that sequence
- base 2
- update 7-16-2006: I know all these things now, and will finish Lesson 81 (of 120) in Saxon Algebra 1 today.
So my first reaction, in Western polarizing fashion, was: I know nothing.
I know nothing, and I need to work through all 857 pages of Saxon Math 8/7 with Pre-Algebra before I can even think about setting foot inside a real algebra textbook.
I was depressed.
But then I calmed down a little and thought, mmmmm....maybe not.
Maybe I can just go through Saxon 8/7 and do every single lesson & every single problem related to these 9 topics.
Is that wrong?
update 7-16-2006: I ended up working through the entire book. Every lesson, every problem, every test. Then I took the Saxon placement test and placed into Algebra 2, but decided to start with Algebra 1. I'm glad I did.
Christopher began teaching himself Saxon Algebra 1/2 this summer (he starts 7th grade in th fall) so I'm reading through those lessons to make sure I didn't skip anything I need to practice - and just for the joy of encountering John Saxon's take on topics I already know.
Algebra 1 integrates algebra and geometry, though without proofs. I'll start Algebra 2 in September.
In one year I will have worked through:
- final chapter of Russian Math
- all of Saxon Math 8/7
- all of Saxon Algebra 1
That pace seems OK to me.
AnyNumberCanBeAFraction 01 Oct 2005 - 05:52 CatherineJohnson

Steve, on the thread need for speed thread, pointed out that any number can be a fraction, and when I said I ought to put together a worksheet on this subject for Christopher, Dan directed me to this frame, DimWksheet010.ppt. of his dimensional dominoes!
It's wonderful.
I'm going to have Christopher do it.
Which reminds me, yet again, I have got to get Doug Sundseth's number lines attached. I've used them two nights in a row, and today I sent a bunch in to Christopher's teacher, in case she wants to use them with the kids.
math ed is a riveting subject
Obviously, I've become obsessed with math education. I'm constantly trying to figure out what it is about math that makes it confusing, and what one can do to make it less confusing.
Liping Ma talks a lot about fragmented knowledge, and cognitive scientists all wrestle with the problem of expertise, which means the ability to generalize what you know to novel problems and solve them.
I've noticed (I may be quoting others without realizing it) that one of the problems with the 'novice' stage of learning is a kind of over-solidity of numbers, a thingness.
Doesn't Freud talk about children first playing with words as if they were things?
Does he say the same of numbers?
I don't remember.
In any case, what I've seen in myself, and in Christopher, is that numbers are too-solid. Both Saxon & Singapore spend a great deal of time conveying the idea that numbers are fluid, in a away, blinking constellations that can be one thing one moment (-10, say) and another in the next (-5 + -5, or -20/2, or any of an infinite number of combinations & expressions).
The Everyday Math article called this 'number partition theory,' and I haven't been able to figure out whether it is or is not number partition theory, but for my purposes, at the moment, it doesn't matter. Just knowing that the number 10 doesn't have the stability of a chair or a tree or a car is a big help.
So I've been trying to convey this to Christopher.
Dolciani's classic algebra text, btw, opens with this idea. '6 + 4' is another expression for '10.' Ten is not the answer to '6 + 4,' but another expression of '6 + 4'. The difference is huge.
Saxon 8/7 constantly uses the word 'Simplify' to mean 'Find the answer,' which I think is excellent. One day Christopher actually said, 'When he says simplify, he means find the answer.' And I thought that was fine. He's getting the idea that simplify and answer are synonyms.
generalizing knowledge
I'm wondering whether making numbers less thing-y for a child might help him or her to generalize a bit more easily, or a bit sooner -- or at least help him to generalize when he's practiced enough that he/she ought to be generalizing.
DougSundsethNumberLines 30 Sep 2005 - 21:37 CatherineJohnson
blank number lines (pdf file)
symmetric number lines (positive numbers, negatives numbers, 0 (pdf file)
number lines: all positive numbers (pdf file)
number lines: all negative numbers (pdf file)
update
If anyone is interested in, or has time to, critique these study sheets, that would great. (There's no pressing need for this; I'm reasonably certain these are accurate, especially since the second document came straight from the pages of Mathematics 6.
addition & subtractions of integers review sheet
integers problems from RUSSIAN MATH
MathmanOnPractice 01 Oct 2005 - 15:03 CatherineJohnson
from mathman:
So how many exercises should I assign? I can't possibly grade them all. This is not an easy question to answer.
It's much easier to say how many exercises the student should do although most students won't care for what I have to say. The student should work as many exercises as it takes to be able to do them correctly most of the time as fast as he can physically write out a complete solution. When informed that he has made a mistake, he should be able to find and correct his error quickly. When it counts, given time to review his work carefully, he should be able come up with the correct solution every time.
This level of mastery opens the door to calculus, differential equations, linear algebra and the quantitative elements of any science.
I'm going to print this out, ask Christopher to read it out loud to me, and then post it above the dining room table. (We're still waiting on delivery of the Ikea desk I ordered a couple of week ago.)
Willingham on overlearning
I re-read Practice Makes Perfect--But Only If You Practice Beyond the Point of Perfection every few months.
NumberBondsVersusFourFactFamilies 13 Nov 2005 - 20:07 CatherineJohnson
From the Comment thread about Lone Ranger's approach to teaching an 8-year old why it's OK to write the number 5 as 5/1: I mentioned that Saxon Math uses four-fact families to teach the operations of arithmetic, while both constructivist curricula and Singapore math seem to use 'number bonds.'
Here's an example of a number bond flash card:
You can download these cards from DonnaYoung.org, a homeschooling resource that looks pretty good, and has a page of mostly terrific paper math manipulatives, including lots of circular fractions, terrific large-print math facts drill sheets, graph paper, play money, scale paper for household furniture arrangements, and some cool-looking empty worksheets with number lines on top.
It also has triangular addition and subtraction flash cards (pdf file).
from the directions:
To use the cards, hide one of the corner numbers with your thumb or finger and let the child tell you what the hidden number is.
Saxon's fact families
Saxon Math does not use triangular flash cards.
Saxon uses four-fact families combined with Extreme Practice. If there is One Thing Christopher & I have overlearned from Saxon 6/5, it is FOUR FACT FAMILIES:
1, 2, 3
1 + 2 = 3
2 + 1 = 3
3 - 2 = 1
3 - 1 = 2
Same deal with multiplication and division.
Here's a typical four-fact family problem from Lesson 2 in Saxon 7/6:
23. Rearrange the numbers in this addition fact to form another addition fact and two subtraction facts.
12 + 24 = 36
Christopher can do that in his sleep.
So can I.
I probably have done it in my sleep.
I've been doing so much grade school math I sometimes dream about it.
four weeks into Saxon 6/5
Quoting from a post I wrote on this subject awhile back:
About a month after Christopher and I began working with Saxon Math 6/5, he told me,
Multiplication and division are the big brothers,
and addition and subtraction are the little brothers.
Then he said,
And multiplication and division are cousins.
This is a 9-year who, just 6 weeks earlier, had been flunking math.
You have to do a lot of four-fact fact families to come up with a thing like that.
I vote for fact families
Triangular flash cards and number bonds are everywhere these days, but I don't like them. Here's why:
- First of all, the potential for confusion is huge. An addition & subtraction number card looks extremely similar to a multiplication & division number card, and separating factors from addends in a child's mind is a challenge under any circumstances.
- Second, triangular number bond cards aren't all that easy to 'read.' Kids don't naturally undestand visual displays of data; far from it. There's too much info on these cards, IMO.
- Third, number bonds are incredibly static, and I don't think math is static. Math is something you do, not something you look at. Four-fact families are action-packed; you get so good at them you can whip one of those babies out in a couple seconds flat. They're fun, and they absolutely (I'd bet money on it) prepare kids for the time when they're going to start solving problems like 2 + a = 5. When Christopher segued to 2 + a = 5 in Saxon 7/6 he didn't have a second's difficulty. He'd been inverse-operationing 2 + 3 for a year at that point, so 2 + a was just obvious.
- Last & certainly not least, I haven't had any luck with flash cards, period.
Not nearly as beautiful as Doug's number lines, but a good idea.
oops
I've just noticed that Donna Young prefers sites not link to her printable forms, and in fact these links won't access the forms. Just go to her homepage, click on math, and then find what you're interested in. The math page is clear & easy to use.
Curricular Game Playing
Curricular Game Playing, part 2
number bonds vs. 4-fact families
Numicom Dominoes
RonAharoniOnTheFifthOperationOfArithmetic 14 Sep 2006 - 14:53 CatherineJohnson
Carolyn has kindly left my two favorite passages in Ron Aharoni's What I Learned in Elementary School for me to blooki.
Here's the first:
What Arithmetic Should Be Covered in Elementary School?
The embarrassingly simple answer is: the four basic operations—addition, subtraction, multiplication, and division.
Yet, this seemingly simple answer is deceptive in two ways. One is that there are actually five operations. In addition to the four classical operations, there is a fifth one that is even more fundamental and important. That is, forming a unit, taking a part of the world and declaring it to be the “whole.” This operation is at the base of much of the mathematics of elementary school. First of all, in counting, when you have another such unit you say you have “two,” and so on. The operation of multiplication is based on taking a set, declaring that this is the unit, and repeating it. The concept of a fraction starts from having a whole, from which parts are taken. The decimal system is based on gathering tens of objects into one unit called a “10,” then recursively repeating it.
The forming of a unit, and the assigning of a name to it, is something that has to be learned and stressed explicitly. I met children who, in fifth grade, knew how to find a quarter of a class of 20, but had difficulty understanding how to find “three-quarters” of the class, having missed the stage of the corresponding process of repeating a unit in multiplication.
I've thought about this observation every day since reading Aharoni's article. I probably can't explain why. At least, I can't at the moment. (Good thing I'm not taking the Regents, I guess.)
But it reminded me of a post Carolyn wrote early on:
Catherine mentioned that she is a fan of tile fraction manipulatives over the more usual 'pie' manipulatives:
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.
I prefer tile manipulatives too, for what I think are solid pedagogical reasons, and here is why: if you want to talk about improper fractions -- fractions greater than one -- with your kid, then the pie-shaped manipulatives add potential for confusion because you can't make a single connected object that represents a quantity greater than one. If you want to represent, for example, 3/2 with pie manipulatives, then you'll have one whole circle and a half circle. You can tell a kid that that represents a single object, the quantity 3/2, all you like; but to him it will look like two objects. Fractions are confusing enough without that.
Conversely, you can make a single line of tiles that is as long as you like.
So unless your child is really off and running with the pie manipulatives, I'd recommend the tile manipulatives.
These are the fraction tiles I like:

You can order extra tiles, too, which I have done. I've used these over and over again, with Christopher, and with at least two of his friends.
Worth their weight in gold.
Aharoni article, part 1
Aharoni article, part 2: America's 'new math' goes to Israel
Aharoni on the fifth operation of arithmetic
Ron Aharoni on teaching fractions & forming units
What I Learned In Elementary School by Ron Aharoni (AMERICAN EDUCATOR)
DistributiveProperty 13 Oct 2005 - 17:57 CatherineJohnson
I keep forgetting to post this story.
A couple of nights ago I was doing my Russian Math on too-little sleep plus a glass of wine, and I found myself drawing a blank when the text asked me to multiply 24 by 7. I was sitting there complaining, '7 x 24, what's 7 x 24, oooohhhhhhh' (More Sleep, More Exercise, Less Wine coming right up) when I heard, from within my fog, Christopher calling out, "Distributive property! Distributive property!"
I was really tired.
So I kept moaning about What is 7 x 24, and Christopher kept calling out Distributive property! until finally I said, 'What are you talking about?"
Christopher said, 'It's 168! Use the distributive property! 7 x 20 is 140, 7 x 4 is 28!'
I've spent practically a whole year trying to teach Christopher the distributive property.
I had no idea he'd learned it.

INeedAPlan 18 Sep 2006 - 17:21 CarolynJohnston
(blank Saxon answer sheets are attached in the Comments thread.)
Ben is not liking his new math plan. He's been doing Saxon math with an aide for around 3 weeks now, and is doing okay, but he complains a lot about wanting to go back to Ms. Math Teacher's class. She's a nice teacher, that's one thing, but also the other kids are there and he doesn't like feeling singled out. Especially not when the singling-out means he's working a lot harder than he used to in the regular class.
The other problem we are having is one we could be having with any math curriculum; Ben is trying to whip through his math homework, as fast as he can, in order to finish as quickly as possible. As a result, he makes a lot of careless mistakes. I really want to get him working more meticulously, because any work habit Ben acquires will be very hard to get rid of. Here is the plan I've implemented in order to get him checking his work more carefully. It's behavioral, a reward system; I've used behavior plans with Ben since he was a young spacey toddler because if I design them well, they work.
With Saxon 8/7, most of the homework on any given evening is review; it's mostly stuff that he could do correctly the first time if he were careful. So I've taken the risk of telling him that I expect him to get it 85% correct the first time through. If he does that, he gets rewarded (the current favorite reward is getting to watch The Simpsons on DVD when he's done with homework). If not, he does without. But I'm not crazy about this plan, and I'm looking for a replacement.
One thing I don't like is that the reward is tied directly to performance. If he ever gets a homework that is on a topic he finds more difficult than usual, the 85% plan isn't going to work very well.
The other thing I don't like about it is that it isn't working very well. He still makes careless errors the first time through his work. When he is done with that first pass through, and I say "you'd better double check your work now before you hand it in," he usually passes up the opportunity to check his work again. Checking Your Work Again is the brussels sprouts of schoolwork; I remember loathing it myself. Apparently it is SO loathsome that he'd rather risk losing his Simpson's reward than Check His Work Again..
I call this an almost-failed intervention. It's not a total failure; his error rate has dropped considerably since we started, but his first-time-through success rate is topping off at just under 85%. He still has a baseline careless error rate of around 10%, I'd say; he usually has one or two problems he's skipped over completely. You really want a behavioral plan, too, that the kid accomplishes successfully most of the time, not one where he's always just barely missing the target.
Can anyone think of a better metric for success?
And, clearly, Checking One's Work Again from the beginning, at the end of an assignment, is too disgusting to contemplate. How do you get a kid to Check His Work Again on a problem-by-problem basis?
OnlineLessonInUnitMultipliers 25 Oct 2005 - 19:36 CatherineJohnson
unit multipliers
AnotherDistributivePropertyQuestion 31 Oct 2005 - 04:21 CatherineJohnson
Quick question.
I mentioned Christopher has a wicked time trying to simplify this type of expression:
5 ( 6 - x )
Even worse is an expression like:
-5 ( 6 - x )
or:
-5 [ 6 - ( -x) ]
I've been trying to figure out how to teach and explain this. (I didn't get to show him either Doug's or J.D.'s graphics last night, because he had to study for his English test, and do math homework....which reminds me, I have a question about last night's homework, too.)
Back to the distributive property.
It's correct to say that we are distributing multiplication over addition, right?
We're distributing an operation?
That's what I had thought we were doing, but when I went Googling around about it, I found some dissenters.
Unfortunately, when I say we are distributing multiplication over addition, Christopher gets even more confused by the minus-minus aspect of 'addition' when what he's looking at is a subtraction.
I've written some 'Out loud' sheets on the concept of addition being subtraction of the opposite....but if anyone has other ideas, please let me know.
I'm having trouble breaking this problem down into its component parts.
Should I have him, at this point, rewrite the question as addition of the opposite?
That's what I'm thinking at the moment, but I'm not at all confident this won't introduce even more confusion and angst.
finding x - 15 = 30
Here's my other question.
Christopher came home last night with a bunch of simple equations to solve.
He knows how to solve all of them using inverse operations, because he practiced that a lot in Saxon Math.
The teacher told them they couldn't do it that way. She wanted them to do it this way:
x - 15 = 30
+15 +15
____________
x + 0 = 45
x = 45
So naturally we had a whole battle royale about that. Christopher didn't understand the teacher's explanation, and forgot to bring his notebook home, so I had no idea what he was supposed to do. When I said he needed to call a friend and find out he exploded; when I called one of his friends to find out what he was supposed to do he triple-exploded.
His plan was to just find the answers the way he always does, write them in the blank, and take the half-credit.
Here's my question.
To me it seems like a good idea for Christopher to do a bunch of problems the way his teacher showed him.
But why is that?
How do I explain that using the inverse operation is different, sometimes, from isolating the variable?
And is that what we still call it?
Isolating the variable?
I just remembered
Last question:
a = b
b = a
Is this an official property?
Or is it just obvious?
I ask, because I have two Out loud sheets based on this principle.
The first one is full of problems like this:
5 + (-2 ) = _____
The answer is:
5 - 2
Then I have a second sheet filled with the opposite problem:
5 - 2 = _____
The answer is:
5 + (-2 )
I plan to ask Christopher why, if 5 + ( -2 ) can be rewritten as 5 - 2, then 5 - 2 can be rewritten as 5 + ( -2 ).
And I intend for him to answer that if a = b, then b = a.
All a & b have done is switch sides.
But is that right?
Or is this an Official Property?
update from J.D.
You know, it strikes me that this is another Out loud sheet.
I probably better write up a lot of simple multiplication problems, and have Christopher tell me how & where the distributive property is used in the algorithm.
keeping track of graphics
I'm trying to get all these incredible graphics stored where people can find them, which I have taken to mean stored in multiple locations:
Book-style index
Math lessons
Our favorite supplements
I also have a page devoted to Carolyn's math explanations (seriously behind, unfortunately, but I'm working on it).
I do need a Kitchen Table Math intern.
I bet I could rustle one up.
I'm not quite as behind on this project as I am on others (I realized today, I should just go ahead and post Dan's fraction-multiplication graphic now, before I've had time to sit down & study it...).
In any case, these contributions should all be findable.
IsSaxonPlusSingaporeTooMuch 07 Nov 2005 - 23:47 CarolynJohnston
We had a request today for some information about supplementing Saxon Math with Singapore Math...
I found this site several weeks ago and I LOVE IT! I started homeschooling my two sons last year after taking them out of public school. I have been using Saxon math. Last year they were in second and third grade and I had them in Saxon 2 and 3. This year, I have them both in Saxon 5/4. I like the Saxon program because it seems to be very thorough and they have plenty of practice. Neither I nor they are very strong in mental math and I have wondered about supplementing Saxon with Singapore Math. I'd like some advice on this. Would it be overkill? To let you know about where they are now: It takes them about an hour a day to do their math lessons. They are at lesson 28 in Saxon. (It's all pretty much review—nothing they haven't had before.) They have had four tests and have done well on all of them. (They both scored 100 percent on the first three.) My older son knows his multiplication facts through 12s pretty well. My younger son is shakier on these and hasn't learned sixes, eights and twelves. I tried giving them the Singapore 3a placement test and they just couldn't do it. I started giving them the Singapore 2a placement test and they are handling that fine (though with a lot of complaints because they have to THINK about what to do in the word problems.) They both like to have me walk them through problems instead of making a stab at it on their own. Thanks in advance for any help anyone can give me. Diane
First responders on the scene (with math tourniquets) were Susan and Dan...
Susan's response:
A homeschooler friend of mine once told me that many homeschoolers use both Singapore and Saxon at the same time. I'm presently using Saxon as the core supplement curriculum for my public school child, but I add Singapore problems to whatever chapter I'm on.
Singapore's word problems are better than any of the other books I've seen because they start with one and two steps and move up to 4+ steps by their level 5.
I don't know if you've seen The Well Trained Mind book, but it has an easy to follow schedule for homeschooling all subjects throughout the years of your child. You might get some ideas of how much to do from there. Since I'm an "after-schooler," as they call me, I haven't ever looked closely at the way they set up the teaching schedule, but it looks fairly thorough.
Dan's response:
I haven't homeschooled, so I feel a little uncomfortable commenting...but only a little.
I just wanted to ask if you were testing the multiplication (and, for that matter, addition) facts with timed tests. I'm pretty sure that timed fact tests are part of the Saxon school curriculum. It seems to be a consensus opinion here at KTM that these facts must be mastered to the point of automaticity. I certainly agree, and have found any lack of automaticity to be a major hindrance as students try to move forward.
And Diane replied..
DanK, Yes, I am using timed tests for addition and subtraction, and I use multiplication fact worksheets for drill, though I don't usually time them. We are just now moving into timed multiplication tests with Saxon.
SusanS, I have read "The Well Trained Mind" and I just revisited her suggestions for scheduling. An hour a day for math seems pretty typical for what most other homeschoolers I know are doing.
I am leaning towards getting Singapore and supplementing with it. Some of my friends who use Saxon with their kids just have the child work every other problem. I've been having my sons do every problem, and, as I commented earlier, it takes them about an hour. I don't want them to get overwhelmed by having an hour and a half of math every day, so I guess I would have to cut out some of the practice problems in Saxon.
So I'll weigh in now with a few thoughts...
I think an amalgam of Saxon and Singapore is a good choice for homeschooling. With Saxon, especially in the early grades, you can be sure that you're not missing out on any essential skills. I think Singapore has a good emphasis on word problems, and I like the way they get kids thinking algebraically very early.
I home-supplemented my son a lot the last two years (we had a constructivist curriculum in 4th and 5th grade—Everyday Math), and even though I'm knowledgeable about math, there were days when I felt up to the task of 'constructing his curriculum' (so to speak) and days when I just didn't. Saxon is a great support for homeschoolers who don't want to be carefully preparing their kids' lessons every day. Singapore takes a greater background knowledge of math, and is much harder for the kids to do independently than Saxon, so to do Singapore, you'll be making a commitment to get really involved with your kids' math. Not every homeschooler wants to do this.
I'd be reluctant to cut out every other Saxon problem on a regular basis, because I think those mixed practice problem sets are the genius of Saxon. They'll revisit a skill intermittently, and if your kids are only doing even problems, they'll miss getting the practice they need if the skill only appears in odd problems (it would be genius indeed if they had enough forethought to put a given skill alternately in even and odd problems!).
You could start by trying to add Singapore word problems to each math session, and see whether that worked; you might find the kids tolerate it pretty easily. If not, you might try switching off days. You wouldn't get through either curriculum as fast, but Saxon has a lot of repetition from one year to the next, so even if you didn't get all the way through a Saxon book you'd have little cause for worry.
Another thing you might consider doing is making Saxon your main text, and supplementing from one of the Singapore books that specializes in word problems, since that's where I think Singapore really has the most to offer. Singapore has a workbook series called Challenging Word Problems Books 1 - 6 ($7.80 plus shipping; 129 pages), in a U.S. (as opposed to British English) edition. You can start at the workbook that's at the level your kids placed into; the problems are marked at a mixture of difficulty levels. This is definitely what I would do if I were constructing a homeschool program.
One more thought—my son, who has Asperger's Syndrome, got balky in second grade about doing math timed tests. He would basically refuse to deal with them, in class; although he knew the facts, he wouldn't do the timed tests because he was reluctant to deal with the time pressure. We ended up doing some heavy bribing to get him to move on those tests (once he did, he was fine). I think adding the time pressure factor is important to nudge the kids toward automaticity. Rewards in the form of treats or outings or privileges are good, I think. Competition can also be good, if it's friendly competition and not cutthroat (and if they're siblings that close in age, it could get ugly).
HowToWriteAlgebraEquations 05 Nov 2005 - 03:08 CatherineJohnson
Christopher's friend Marc just asked me for help writing equations for word problems.
Here's the question.
Is it the case that you can't write something like:
3 + 5 = x
Russian Math says the convention is to have the variable on the lefthand side of the equation.
However, Prentice-Hall seems to want not only the variable on the left-hand side, but also one of the numbers.
To pass muster you'd have to write:
x - 3 = 5
That seems wrong to me, and in fact, Pre-Algebra: An Accelerated Course by Mary Dolciani has one equation in the answer key in which the variable is isolated on the left side of the equation.
What is the convention?
Meanwhile Marc's dad told him, 'Just write the equation in the most complicated way you can think how.'
Marc is good at math, but even he was a little befuddled by that.
I came up with: Write it whatever way makes sense, then flip it.
That worked for Marc, so he is now writing the equation the way it makes sense, then inversing it.
"I can inverse it," were his exact words.
I have a sneaking suspicion this is the kind of homework scene that led people to think Reform Math might be a good idea.....
EmailToMathTeacher 08 Oct 2006 - 22:14 CatherineJohnson
Hi—
I think Christopher probably did poorly on yesterday’s test, which is distressing. When the test comes home I’ll have him re-do all the problems he missed, and I’ll write worksheets containing similar problems for him to do as well.
We are very committed to Christopher learning to mastery every topic you teach.
Christopher says the test included a number of very long equations to simplify.
That’s great; the kids should be able to simplify long equations. But he hasn’t had any long equations to simplify in his homework, and unfortunately it didn’t occur to me to write such problems myself until Sunday night, when it was already too late. (I’ve written several sheets of practice problems for this chapter.)
I’m really hoping you can send homework at the difficulty level of the items that will be on the test. Kids only learn through practice, and a test isn’t practice!
Thanks—
Catherine
P.S. This is funny. I just pulled up my Chapter Two worksheets, and on the very first page I have written:
Distributive property to do list:
Write some long, complicated equations incorporating all the properties
Also—
I’m attaching my Chapter Two worksheets. Feel free to use them if you like, but be sure to check the answer sheets yourself—
Christopher was having a lot of trouble distributing a factor over subtraction, so I focused on the various permutations of distribution over subtraction.
I also used the technique used in MATHEMATICS 6 & in KUMON, which is to create problem sets in which a student does the same thing over and over again before doing any mixed practice:
The first column of problems distributes a positive factor over subtraction.
The second column distributes a negative factor over addition.
The third column distributes a negative factor over subtraction.
The fourth column distributes a negative factor over an expression with two opposites.
Last but not least, I'm sending my ‘Out loud’ subtraction sheets. Those were very helpful, so you might want to give them to the other kids. I’ve started doing ‘Out loud’ sheets, because it’s a technique used by Mathematics 6, the award-winning Russian textbook.
Enjoy!
to send or not to send, that is the question
Ed read this and said, 'Don't you want to wait 'til you see the test, and find out if Christopher is right about the long problems?'
I think normally that would be good advice.
But in this case I'm going to email first & ask questions later.
I've mentioned that there was a lot of parent furor over this course last year. A major part of the problem—perhaps the problem—was that the tests contained material far more difficult than anything the kids had seen or done in or out of class.
That may be fine in college. (I don't see why it's good there, either, but ktm readers will have informed opinions on this, and I don't.)
It's not good teaching in 6th grade.
Christopher is taking a class in pre-algebra, and the school's job is clear.
The school's job is to teach pre-algebra and make sure the kids learn it.
So my thinking is:
- Christopher is most likely to be right, which means the sooner Ms. Kahl hears from me the better.
- If he's wrong, that's important in and of itself, and is information Ms. Kahl should have. Why is a committed student who's clearly working hard perceiving the test incorrectly?
- Christopher's situation aside, the words 'teach to mastery' probably cannot be spoken often enough. Spoken, written, emailed, tattooed to one's forehead: Teach to mastery.
This is The Message.
I'm hitting SEND.
question
Does it make sense to have the kids simplifying very long equations at this stage?
To me, it seems as if maybe we're getting ahead of the game, but I don't know. (I'm thinking the kids need more practice on the component parts of equations....but, as I say, I'm just not sure.)
I'm serious about having Christopher learn to mastery every topic the teacher covers. I don't question her authority to decide content—especially since the course content has been excellent so far, apart from the Extended Problems, that is, and even those are probably coming under control. They did their last extended problem in class, and the kids were able to manage it on their own. That's as it should be.
I'm curious what math-savvy readers & teachers think.
HomeschooledPreTeens 10 Nov 2005 - 15:41 CatherineJohnson
What are homeschooled pre-teens like?
Are they as hostile to their parents as public school pre-teens?
Does anyone know?
HomeschoolersTalkAboutPreTeens 10 Nov 2005 - 21:44 CatherineJohnson
My neighbor told me, back when we first met nearly 7 years ago, that every homeschooled child she'd ever met was a nice kid.
I never forgot that.
Now, having read Steinberg's Beyond the Classroom: Why School Reform Has Failed and What Parents Need to Do at the very moment Christopher is changing from child to adolescent, I'm remembering what she said.
Yesterday I asked what homeschooled pre-teens are like.
Here are some answers.
♥ ♥ ♥ ♥ ♥
My sons are 9 and 8 now; they were 8 and 7 when I started homeschooling them last year. My oldest son was acting a lot like what I remembered from the middle school years: his friends and teachers had more control and authority over him (in his mind) than his parents. What did we know about his life? He spent the best part of his day with other people. Homework was an (almost) daily battle from kindergarten through third grade with him. If I suggested a way of doing something that was different from what he'd learned at school, or what he thought he'd been taught, he would object that my way was not what his teacher told him. The implication was that SHE was in charge, not me.
I figured that if I homeschooled him, at least I'd be fresh and alert for our daily schoolwork battles instead of waiting until the end of the day for them when I was tired. I also figured he couldn't point an authority who wasn't there (the teacher or the textbook that he couldn't bring home) when he disagreed with me.
In fact, this has worked for our family. We attend a small co-op of about 10 families who meet once a week and do projects for science, history and study various fun electives. The other kids are not as sarcastic to their parents as the general public schooled population seems to be. (Of course, I'm looking at a very small sample.)
I think we've been releasing our kids too soon into the wide world. I know I did. Both of my sons were in daycare or with sitters 40+ hours a week from the time they were each about 3 months old until they started kindergarten. I was "outsourcing parenting" but I was working and didn't know what else to do at the time. My kids were spending their days with people who had no real stake in their lives.
I may not always be able to homeschool my kids, and it is absolutely NOT EASY, but, for now, I am grateful that I can do it and I think they will benefit academically, emotionally, and, yes, socially.
Maybe this goes without saying, but, you never know, so I'll say it anyway: I do not think myself or my children superior to anyone who does not homeschool. I am only able to do this by the grace of God. I'm just doing the best that I can do right now.
-- DianeAustin - 10 Nov 2005
I watched a homeschooled 10 year old come up to his mom and give her a big, snuggly, bury your head in mom's neck kinda hug—in public (McD's), AND in front of a dozen other homeschooled kids.
I've seen even older homeschooled kids enjoy playing cards with 6 and 7 year olds.
Being exposed to these types of kids really cememted the idea of homeschooling for me.
BTW, if you bring your child home from public school to homeschool, you need to read up on DE-Schooling.
-- NicksMama - 10 Nov 2005
Homeschooled kids are nowhere near as hostile. Not only are they less hostile in their pre-teens, but also less hostile in their teens. Not that it would be all smooth sailing for homeschooling parents, but having the kind of close relationship that you can really only get when you are together for a lot of the time really eases some of the tension.
Try reading John Taylor Gatto for information on the way school was designed to create disharmony between family members and contributed to the more extreme problems for teens today. The Odysseus Group
-- SamanthaRawson - 10 Nov 2005
Well, from what I've heard from the other moms at the Well-Trained Mind message boards, the kids are still spacey at that age, no matter where they learn. They turn 12 and act like their brains have rolled under the bed with the dust bunnies, not to be found again for 2-3 years. It's not uncommon for them to forget how to do stuff they mastered three years before. They discover the opposite sex. Some of them get moody, etc.
I have a friend IRL whose oldest is Mr. Know-It-All now that he's a teenager. He's very bright, and enjoys letting everyone else know it. :-P
I hope my kids won't be as insufferable as I was at that age. Of course, my folks hope for just the opposite to occur. ;-D
-- BrendaM - 10 Nov 2005
Steinberg on effective parenting
Steinberg is actually an expert on adolescent development, not education (which I think accounts for some of the ways in which he goes wrong, IMO).
I'm hoping to get around to writing about his study of effective parents, but for now the good news is that probably most of your ideas about what makes an effective parent have been born out by lots of research.
Steinberg talks about the 3 basic kinds of parents who turn up in study after study after study:
- permissive
- authoritarian
- authoritative
Judging by the Comments left here, ktm readers are in the 3rd category, and that's a Good Thing, as Martha would say.
Kids with authoritative parents are in lots better shape going into the Teen Peer Disaster Zone that is an American public school.
The horrifying thing, however, is that effective parenting is more effective for some kids than others.
Specifically, authoritative parenting for white kids packs a wallop. Not enough of a wallop, IMO, but a wallop nonetheless. You can actually define the difference in terms of a child's grades. The exact same child, in terms of SES, parent education, etc. will be getting Cs if his parents are either authoritarian or permissive, Bs if his parents are authoritative.
For black parents, the situation is completely different. The same black child with authoritarian or permissive parents earns Cs; the same child with authoritative parents earns a C+.
Asian parents, good, bad, and in between, have an even smaller difference on their children's grades, but in their case it doesn't matter. The Asian child of effective parents earns just one-quarter letter grade higher, overall, than the Asian child of ineffective parents. Asian kids have one peer culture open to them, and that's Brainy Asian Kid Culture.
the kids I know who've turned out well
I know a lot of them. These kids are glaringly from authoritative homes—and, I would add, from strongly authoritative homes.
They've been sat on.
Interestingly, all of Christopher's friends but one comes from an authoritative home (and the one who doesn't is the one we complain about!)
They've got seriously intense moms, that's for sure.
CongratulationsBen 17 Nov 2005 - 18:44 CatherineJohnson

(I'm sorry. I know this is blinking. But I had to do it.)
SusanOnBeingYourChildsSecretary 08 Oct 2006 - 22:14 CatherineJohnson
great comment from Susan
I have no idea what work you "show" for greater than/less that questions.
You might want to write down managable questions like what you put up above for your meeting with her, which I know you'll be having soon. That's perfectly legitimate and it will get you clear and perhaps make her realize that she's not so clear. She has to tell them what she means or the book must have had them doing it that way unless it is some standard way of doing it that everyone knows about.
We were having similar issues with not having enough homework for the work being asked to be done. We've had to use the other books I have for extra practice.
children don't know what they don't know
Again, children don't know what they don't know. They don't know about flexible/inflexible knowledge. They don't know how much is enough. An experienced teacher whose had children bomb on sections would probably anticipate problems with certain chapters. My son's algebra teacher is a veteran. He has stretched and redone some chapters with extra practice. After 25+ years of teaching math he knows exactly what's going to happen and when he can trust the text and when he can't. Even with that, some kids aren't going to make it and I still have the feeling it has more to do with not having enough practice.
the parent as personal assistant
I have had to become his personal secretary because of the school's expectations of him regarding homework and projects and deadlines. He is given all kinds of things to do with all kinds of deadlines and no real guidance on how to manage his time. Many of these things are lacking in specificity. I have to make him pull out his assignments and go through them one by one. If he can't explain something I ask why he didn't write down more so that he would understand it when he got home. We've had much whining and crying over this, but he's starting, finally, to realize that I am going to look at it when he gets home and it must make sense. Just my hammering away at the assignment book and his responsiblity to accurately get his work written down thoroughly has started to make him realize what he has to do to succeed, but that is a gargantuan assignment in and of itself.
I seriously don't remember this kind of juggling of assignments myself much before high school, so it irritates me that I have to take so much time to teach him how to even write it down properly.
I think as a parent you can point out these kinds of murky expectations by the teacher (like the show your work problem) and that they need to be clarified better.
Test-taking has been more difficult for my son, too. There's a stamina and a maturity needed that's a little different than is required for the quizzes. We were doing great on the quizzes, but tanking on the tests. We've talked it through with him and he's improving, but he still isn't as strong on them as he is on the quizzes.
It sounds like you are trying to turn it into a Life Lesson about perseverence and I think you are so smart to do so. Like you said, quitting soccer is no big deal, but he needs to see that some things he can't quit and that it will be alright. They really think it's the end of the world.
With all that blasted "character" stuff they're teaching, you'd think they'd include some of what he's going through.
the veteran
My son's algebra teacher is a veteran. He has stretched and redone some chapters with extra practice. After 25+ years of teaching math he knows exactly what's going to happen and when he can trust the text and when he can't.
This is exactly my concern with Ms. Kahl.
She is, I think, a 2-year veteran, and last year was a trial by fire.
Plus she's up for tenure this year, and while I don't know whether she should have tenure or not, I don't feel that she shouldn't. I know what a tenure year is like; we went through two years of he**. I'd have to feel strongly that she's in the wrong business to want to make Ms. Kahl's tenure year more stressful than it already is.
Christopher has said to me, several times, 'Ms. Kahl is a good teacher,' or 'Ms. Kahl is a pretty good teacher.'
Ms. Kahl isn't a crowd-pleaser; I'd be stunned to learn that she plays to the kids in any way, or grooms fans.
So if Christopher is telling me she's a good teacher, one thing he's not saying is that she's a narcisstic teacher winning love from kids. Plus he doesn't love her. He sees her as a good teacher who wants him to do well.
She's someone who might be a terrific teacher in 5 years' time.
chipperness restored
OK, Christopher just walked in chipper as usual; so far so good.
He's in particularly good spirits because they had another bomb threat today, so they had to walk down the hill to the Main Street School and mill around with their friends until The Danger Had Passed.
That's two bomb threats this fall, both at the middle school, and both, oddly enough, starting in the girl's restroom. "They always come from the girls' restroom," Christopher says.
I know my school didn't have bomb threats in the girls' restroom when I was a kid.
So we finished up with the bomb threat and segued to the subject of, "Do you have my Feature Story?"
"Yes, why?"
"Mr. Fried wants to see it."
HelicopterParentsPart3 10 Jan 2006 - 13:43 |