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Kitchen > PrivateWebHome > WebPreferences > Archives > CompleteArchives > May2005

May2005

NowThatWereBothHere 01 May 2005 - 00:49 CatherineJohnson

Carolyn wrote:

Somewhere during the year, I realized that I was teaching him a lot of basic mathematics, but in a completely reactive way; I was allowing the Everyday Math curriculum to dictate the order and the style in which I taught math.



I like that word reactively.

I’m closing in on my 1 Year Anniversary, formally teaching math to Christopher here at home.

At some point along the way I had the exact same feeling about the home-tutoring going on around me here in my own town, but I didn’t have the word for it.

Now I do. It’s reactive. Reactive teaching.

Everyone is scrambling to keep up with the content being taught at school. If a child comes home from school not understanding the distributive property, then mom or dad or Paid Tutor scrambles to explain it in time for the test. If he comes home not remembering how to change a fraction into a decimal (We learned it last year, but I forgot), then mom or dad or Paid Tutor scrambles to explain it again, hoping this time it will stick.

There’s no rhyme or reason.



MathInTheBlood
ReactiveTeaching
ThingsWeHaveLearned
ImGoingToPlayland



-- CatherineJohnson - 01 May 2005

comments...

StrugglesWithLongDivision 01 May 2005 - 05:31 CarolynJohnston

I remember very clearly the problems I had with certain topics in mathematics. I remember getting confused on the day that my fourth grade teacher taught us how to multiply two-digit numbers by two-digit numbers (I had spaced off during the critical fifteen minutes when she explained the moves to us -- I was permanently spaced out as a kid, actually). That confusion was with me for a long time. So I thought I had a particular rapport with any kid who was struggling to learn math, having once been a kid who couldn't do math to save her life. My then going on to be a math Ph.D., and a math professor and researcher, made me what I thought was a pretty decent role model for struggling kids.

I was pretty good at teaching any topic, in fact, as long as Ben could learn it easily. We hit our first big bottleneck at long division. Multidigit multiplication was actually pretty easy for him; particularly since, in Everyday Math, Ben had learned this slick trick for multiplying multidigit numbers called lattice multiplication and was going to town with it. But long division was a different story. Ben had trouble lining up the columns, remembering to pull down the next digit after every step, and knowing where to finish his calculation and what to do with the remainder. Long after he had demonstrated that he knew what to do at every stage, he still couldn't reliably get the right answer.

I couldn't see that anything would help him master long division but long practice. He had learned all the steps and could apply them, but being methodical about it wasn't part of his nature. So, every night for a couple of months, I would give him several long division problems to do; it would always require several revisions before he would be done for the night. I could be what I needed to be -- a brick wall demanding that he apply care to his computations before he could consider himself done. What was doing me no good at all, just then, was my appreciation of the beauty of higher math.

The long division algorithm we all learned is actually just a repeated application of the Division Algorithm, which in its naked form, once understood, sounds obvious to the point of stupidity. The repeated application of the simple division algorithm with divisors that are decreasing powers of ten is just a thing of beauty, though, something written in The Celestial Great Book of Math. A lot of good it did us, though, in helping Ben to learn to apply long division. It took him a long time to learn to do that reliably, but we stuck with it until he got it.

There is the question of whether we even need to do this -- to torment students by making them practice the tedious long division algorithm -- especially now that computers and calculators are everywhere. It's claimed that such drilling kills the joy of math, and that we can teach children to love math better if we don't force them to do computations. I'm claiming (but not yet from any position of certain knowledge) that we do need to teach computation. I'm going by the fact that, in my association with mathematicians and physicists and engineers and computer scientists and finance people in my schooling and various jobs, I've known many people who could apply the long division algorithm, and some few who could appreciate its beauty; but I've never known a single soul who could appreciate its beauty without being able to apply it.


AboutLongDivision
MathInTheBlood
ForgivingDivision
ForgivingDivisionPart2
TryThisWithForgivingDivision
TeacherGuideEverydayMath
EverydayMathEpilogue
ThirteenQuartersInTerc
HowNotToTeachMath
WhoSaysLongDivisionIsHard




comments...

HappyMathematicsAwarenessMonth 01 May 2005 - 13:08 CatherineJohnson

Belatedly.

From the folks at the American Mathematical Society.



comments...

SwoopAndSwoop 02 May 2005 - 02:13 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":

newswoop-and-swoop.gif

And that was it: he got it: those swooping moves with the pencil and the crossing numbers. That's what the standard algorithms are: they are moves that you learn how to make. Those moves get into your fingers, just like learning the piano or the violin or typing, and eventually you can do them completely mindlessly.

But that doesn't mean that nothing is going on in the kid's head. If a kid really has those moves down, it frees his mind to think about doing the next thing, and he becomes more receptive to learning why the moves need to be what they are, because the anxiety of not being able to handle the calculation is gone.

Learning the piano or the violin involves a lot of repetition, while your eyes and your mind and your fingers make the connections that allow you, eventually, to experience the music you're playing on a higher level, without calculating where your fingers need to go next. Math is just like that. Math is something you learn to do, like playing an instrument or riding a bike, not something you learn about remotely, like Magellan's circumnavigation. It has a huge kinesthetic component.


swoop and swoop
SlideRules
the craft of math
Wayne Wickelgren on why math is confusing, & Carolyn on procedural memory
KUMON & hands-on math





comments...

SwoopAndSwoopPart2 02 May 2005 - 20:06 CatherineJohnson

This is probably the time to mention that I’m re-teaching myself elementary mathematics, start to finish.

I’m doing all of the lessons in Saxon Math Homeschool Edition, beginning with book 6/5, which Christopher and I finished a few weeks ago.

I’m also (in theory) working my way through the entire Singapore Math series, beginning with 1st grade.

UPDATE 10-8-2006: I am not working my way through the entire Singapore Math series. I am working my way through the entire Saxon oeuvre, which is all I can manage at the moment. I am, however, for reasons unknown to me, creating a hand-drawn solution manual for Singapore Math's Challenging Word Problems Book 4.

I was always pretty good in math, though I stopped taking it after Algebra II, then hit the wall when I tried to take calculus freshman year in college. I flunked the first test and dropped the course.

But up til then I was fine, I liked math, scored well on my SATs, etc. I don't have any math anxiety and I love statistics. I took one statistics course in college. Correlation coefficients, standard deviations, regression analysis: to me, these things sound like the key to palace.

So, given my general level of math-friendliness, I didn’t think it would be too hard to teach Christopher the math he'd missed in 4th grade.

However, I pretty quickly had the same experience the teacher quoted in the American Institutes for Research report did: “I never realized that I do not understand math until I had to teach mathematics from the Singapore textbooks.”

This time around I’m trying to acquire conceptual understanding of elementary mathematics, and hook it up to my procedural understanding.

It’s not easy.

UPDATE 10-8-2006: Twenty-three lessons into Saxon Algebra 2 the mystery of my Wellesley calculus failure has been solved.

Algebra 1 & 2 in my high school in Lincoln, IL correspond to Algebra 1 in Saxon.

I went to college thinking I'd taken two years of algebra.

I hadn't.

I'd only taken one.

Apparently Wellesley College wasn't big on placement exams in those days.






comments...

SwoopAndSwoopPart3 02 May 2005 - 20:18 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.

ToPostCompFraction8.jpg



comments...

HowIGotHerePart2 03 May 2005 - 12:35 CatherineJohnson

So there we were, Christopher and I, installed at our picnic table, thrashing our way through SRA Math Unit 6: Fractions and Decimals.

Two weeks later, there was blood on the floor.

HowIGotHerePart1



comments...

BeingYourChildsFrontalLobes 03 May 2005 - 13:03 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


dingbatWSJ2.jpg


update 5-23-06: more frontal lobes

sources:
Teenage Brain: a work in progress (NIH)

frontal lobes, executive function, & IQ
hovering is good (MiddleWeb)
being your child's frontal lobes
organization is overrated
executive function, IQ, & hovering, part 1
the discovery of executive function, part 2
executive function self-test
presidents & criminals & the frontal lobes
ISIS initiate sustain inhibit shift

page splatter
page splatter & the frontal lobes

Dear Abby
Susan on dating
Catherine's brain-based dating rule





comments...

MathInSalinaKansas 07 May 2005 - 03:06 CarolynJohnston

From a forum I sometimes visit, I followed a link today to an urban legends website with a page on an internet claim about an 8th grade final exam supposedly given in Salina, Kansas, in 1895. Here are a few of the test questions in the arithmetic section:

Arithmetic (Time, 1.25 hours)

1. Name and define the Fundamental Rules of Arithmetic.

2. A wagon box is 2 ft. deep, 10 feet long, and 3 ft. wide. How many bushels of wheat will it hold?

3. If a load of wheat weighs 3942 lbs., what is it worth at 50 cts. per bu., deducting 1050 lbs. for tare?

4. District No. 33 has a valuation of $35,000. What is the necessary levy to carry on a school seven months at $50 per month, and have $104 for incidentals?

5. Find cost of 6720 lbs. coal at $6.00 per ton.

6. Find the interest of $512.60 for 8 months and 18 days at 7 percent.

7. What is the cost of 40 boards 12 inches wide and 16 ft. long at $20.00 per in?

8. Find bank discount on $300 for 90 days (no grace) at 10 percent.

9. What is the cost of a square farm at $15 per acre, the distance around which is 640 rods?

10. Write a Bank Check, a Promissory Note, and a Receipt.

When I looked at the Urban Legends page about this 1895 test I found that, contrary to my expectation, they weren't debunking the claim that it was a genuine final test from 1895. They were taking issue with the claim that it showed that educational standards had fallen since 1895:

What nearly all these pundits fail to grasp is "I can't answer these questions" is not the same thing as "These questions demonstrate that students in earlier days were better educated than today's students." Just about any test looks difficult to those who haven't recently been steeped in the material it covers. If a 40-year-old can't score as well on a geography test as a high school student who just spent several weeks memorizing the names of all the rivers in South America in preparation for an exam, that doesn't mean the 40-year-old's education was woefully deficient -- it means he simply didn't retain information for which he had no use, no matter how thoroughly it was drilled into his brain through rote memory some twenty-odd years earlier.

Lame, lame, lame. If you can't prove that this is not an authentic graduate exam from 1895, then complaining about it just makes you sound like a whiner (and notice the dig about 'rote memory' -- memorization is in very bad odor these days).

Besides, it's not about us (and what we retained) anymore: it's about our kids. And I am afraid it does imply that we've dumbed down the junior high curriculum. Only a tiny minority of kids graduating 8th grade these days could handle sophisticated word problems like these, even if we gave them the bushel-conversion formulas for free. Apart from the emphasis on farming applications, which is kind of funny and endearing, the application area of problems 6 and 8 (just for an example) is as alive, or more so, in 2005 as it was in 1895, and we simply do not teach it. In the late 1980s, I taught an elective course at LSU on the material covered in these problems. The entering students were completely ignorant of that material, mastery of which I claim is necessary to living adult life competently (and they were very glad to finally learn it, too). Many students who were stronger mathematically, and didn't take that elective math course, are no doubt still ignorant of it, because it is not taught in public schools anymore.

The second thing that leaps out at me is that these are mostly application problems -- word problems -- not problems testing either basic computation or deep understanding of the beauty of mathematics (with the exception of problem 1). It was just assumed that these kids could do the computations necessary to solve these problems, without calculators. What they needed to do was to solve those problems, and get the right answer, and that hasn't changed a bit. And I'll bet there was no partial credit given for having the right idea, either.


CompareAndContrast
CompareAndContrastPart2
CompareAndContrastPart3
CompareAndContrastPart4
CompareAndContrastPart5
CompareAndContrastPart6
CompareAndContrastPart7



comments...

MathInSalinaKansasPart2 07 May 2005 - 15:52 CatherineJohnson

re: MathInSalinaKansas

Wow.

I spoke yesterday to a mathematics professor at a university here in New York state.

When I asked him what level of mathematical knowledge entering freshmen bring to their course work, he said, "We can't assume that a student knows anything we would want him to know."

Specifically, his students can't do algebra.

They can't set up a two-variable word problem and solve it.

These are college freshmen.

Posted on May 07, 2005 @ 11:21



comments...

MathInSalinaKansasPart3 07 May 2005 - 16:53 CatherineJohnson

re: MathInSalinaKansas

Three sample problems from the PRAXIS 1 Content Assessment college students entering the field of education are frequently required to take:

1. Which of the following is equal to a quarter of a million?
a) 40,000 b) 250,000 c) 2,500,000 d) 1/4,000,000 e) 4/1,000,000


2. Which of the following fractions is least?
a) 11/10 b) 99/100 c) 25/24 d) 3/2 e) 501/500


3. Which of the sales commissions shown below is greatest?
a) 1% of $1,000 b) 10% of $200 c) 12.5% of $100 d) 15% of $100 e) 25% of $40

 The Educational Testing Service (ETS) describes these problems thus:

The Pre-Professional Skills Test in Mathematics measures those mathematical skills and concepts that an educated adult might need. It focuses on the key concepts of mathematics and on the ability to solve problems and to reason in a quantitative context. Many of the problems require the integration of multiple skills to achieve a solution. [snip] Computation is held to a minimum, and few technical words are used. Terms such as area, perimeter, ratio, integer, factor, and prime number are used, because it is assumed that these are commonly encountered in the mathematics all examinees have studied. Figures are drawn as accurately as possible and lie in a plane unless otherwise noted.

see also: MathInSalinaKansasPart2



comments...

PracticeAndOverlearningPart1 08 May 2005 - 22:14 CatherineJohnson

Carolyn and I have both been using Saxon Math Homeschool Edition with our kids.

Here is Saxon's explanation of the curriculum:

Saxon Math . . . systematically distributes instruction and practice and assessment throughout the academic year as opposed to concentrating, or massing, the instruction, practice and assessment of related concepts into a short period of time -- usually within a unit or chapter.

I can vouch for this.

SAXON 6/5 has 120 lessons in all, plus 12 'Investigations' & 3 Appendix lessons, and when you get to Lesson 120 you're still practicing the stuff you learned back in Lesson 1.

There are 100 or more problems and computations in each of the 120 lessons: Fast Facts, Mental Math, Problem Solving, Lesson Practice, and, finally, Mixed Practice.

This is what we call drill and kill.

Cognitive psychologists call it automaticity:

Practice Makes Perfect But Only If You Overlearn Ask the Cognitive Scientist: How We Learn by Daniel T. Willingham

review



comments...

GoodReadingPart1 09 May 2005 - 16:25 CatherineJohnson

Just posted to Our Favorite Math Ed Articles:

Daniel T. Willingham's 'Ask the Cognitive Scientist' columns for AMERICAN EDUCATOR (wonderful)

William Schmidt, et al's phenomenally helpful 'A Coherent Curriculum: The Case of Mathematics' (Schmidt headed the Third International Mathematics and Science Study (TIMSS), and summarizes his findings here.)

Specific Learning Disabilities: Finding Common Ground. A Report Developed by the Ten Organizations Participating in the Learning Disabilities Roundtable. This is the American Institutes of Research 2002 consensus report: what findings, hypotheses, and theories do 10 different organizations and insitutions, including the Department of Education and the Learning Disabilities Association of American, agree to be true of 'specific learning disabilities.' (I haven't read this yet.)

See also: PracticeAndOverlearningPart1



comments...

NotTheWholeStory 12 May 2005 - 02:49 CarolynJohnston

Catherine sent me a link today to an article about the Everyday Math curriculum. A host of well-known mathematicians have given Everyday Math a lot of negative press. A group of mathematics professors led by David Klein at Cal State Northridge wrote an open letter to the Secretary of Education urging the U.S. government to publicly withdraw its 1999 recommendation of Everyday Math (among other new-new math curricula).

I am familiar (very familiar) with Everyday Math, and it has clear weaknesses that we'll discuss at length in time, but I was struck by the following quote in today's article:

Klein said that as a result of whole math programs such as EM, CSUN and other colleges must offer entering freshmen remedial math classes at a level as low as third grade. He said he’s seen, for instance, calculus students who can’t add fractions.

"This is kind of the lost generation, ruined by these liberal-minded policies," Klein said. "The truth of the matter is it’s just a crummy program."

It may be a crummy program -- I have certainly found it hugely frustrating to work with -- but it wouldn't be fair to blame Everyday Math for the existence of vast numbers of calculus students who can't add fractions. The problem has been around a lot longer than Everyday Math has.

I taught at SUNY Binghamton in the early 80s, and we had plenty of calc students who couldn't add fractions. When I was a grad student at Louisiana State University, the remedial math caseload on the mathematics department was so heavy that a whole class of 'instructors' -- essentially the equivalent of high school teachers in schooling and training -- were employed by the math department to teach remedial math classes, and a typical grad student was assigned full responsibility for 2 classes of remedial math every semester. That's more than 60 students per grad student.

And these classes were serving just the students who had been identified as needing remedial math classes; many slipped through the cracks. You bet a lot of the students in LSU's calculus classes couldn't add fractions. Nor is the problem confined to LSU; public universities everywhere, with few exceptions, have large remedial math loads. It's been going on for at least twenty years, long before Everyday Math appeared on the scene.

I don't think there are any simple explanations. But I do think we're floundering, and we need to look to countries with a better track record for guidance.

Furthermore, any math professor can point to plenty of failures in math education within his own experience, but individual failures don't help to explain what we're doing wrong at the policy level. For that, we'll need sound research.

comments...

NotTheWholeStoryPart2 13 May 2005 - 00:24 CatherineJohnson

So just how far back does the U.S. fraction deficiency go, you ask.

Answer: really far.

In 1923, the NEW YORK TIMES reported that fewer than half of seventh grade students could convert the fraction 1/5 into a decimal.

The Columbia Teachers College had a plan.

The new aim of the progressive arithmetricians is to abandon drilling in artificial problems and to bring mathematics close to every-day life.

from: 'New Teaching Puts Life into Dreary Arithmetic', NYTIMES December 9, 1923



Apparently, the plan was working.

The new method is so successful, according to its sponsors, that one school has playfully threatened to abandon it for the reason that the pupils are so enthusiastic over arithmetic that their teachers can scarcely interest them in other subjects.


This was the start of progressive education in America.

So flash forward to 1989, and we find NAEP reporting that 60 percent of seventh grade students can 'express simple fractions' as decimals.

A mere 70 years of progress, and 10% of American seventh graders who wouldn't have known that 1/5 is the same thing as 20% back in 1923 do know in 1989.

That was my first thought.

My second thought was, OK, I'll take it. 10% is 10%.

Then I noticed Chris Correa's second post on the subject.

I browsed through the publicly released NAEP questions and found the most comparable question to be from 1992: Of the following, which is closest in value to 0.52?

A) 1/50
B) 1/5
C) 1/4
D) 1/3
E) 1/2

Only 51% of eighth-graders correctly answered this question. Nearly 30% of students responded that 1/50 was closest in value to 0.52.


This is my beef with constructivism.

It's not like constructivism hasn't been given a fair shake.

Constructivists have had a good hundred years to show us what they can do.

I say it's time to move on.

[Thank you, Chris Correa.]

NotTheWholeStory



comments...

GoodReadingPart2 13 May 2005 - 00:58 CatherineJohnson

Posted to Our Favorite Math Ed Research Articles: The A-Maze-ing Approach to Math by Barry Garelick.



comments...

NotTheWholeStoryPart3 13 May 2005 - 01:35 CatherineJohnson

re: NotTheWholeStory & NotTheWholeStoryPart2

Carolyn's right that Everyday Math can't be blamed for the sorry state of college freshmen's ability to add fractions.

I haven't been able to track down the first printing, but EVERYDAY MATH seems to date back to around 1993 or thereabouts.

Garelick reports that approximately 10% of U.S. schools have now adopted E-Math, and I read just this week that another 10% of U.S. schools have adopted one of the other constructivist math curricula. (I've forgotten the source, or I'd link -- sorry.)

Of kids entering college this year, only a small percentage will have spent much time using the latest crop of constructivist mathematics programs.

Of course, that's leaving aside the fact that constructivism has been part of ed school philosophy for a century.



comments...

CarolynIsGobsmacked 13 May 2005 - 03:18 CarolynJohnston

Did you see this chart in Garelick's article that showed the grants that ed departments were given to come up with new math curricula?
ednext20052_garelickfig1.gif
Man, did I ever go into the wrong branch of academia. Five million dollars for Everyday Math! Six for Trailblazers! Fourteen for Contemporary Math in Context!!!!

The path I should have taken is now clear:

  • get a Ph.D. in education from a prestigious department;
  • become a professor and get a HUGE grant to develop a new math curriculum in whatever flavor the government currently prefers;
  • develop it, then sit back and let the government do your marketing for you;
  • then, develop a lucrative side business as a consultant and speaker on the math curriculum you developed, while still enjoying tenure, a light workload, and the envy of your colleagues.

What the heck was I thinking all those years ago? These aren't the only academic rackets I've heard of, by a long shot -- I know of a number of others too (almost any of which beat pure math, in which you work your fanny off for 50K or so in summer research salary, and are glad to be getting it). But I took the road less traveled by!...

I think this math curriculum thing may actually be the sweetest racket of them all.

It also suggests that the push toward constructivist curricula didn't necessarily come from the ed schools themselves. Any school seeking grant money is obviously going to be responsive to the prevailing political winds, which in this case seem to have been emanating from the National Council of Teachers of Mathematics and the National Science Foundation.


MoreBigNumbers
BigNumbers





comments...

CatherineIsGobsmackedPart2 13 May 2005 - 10:51 CatherineJohnson

re: the chart

Oh, yeah. I saw it.
CarolynIsGobsmacked



comments...

CatherineIsGobsmackedPart3 13 May 2005 - 15:25 CatherineJohnson

re: CarolynIsGobsmacked

No question, Carolyn.

When it came time to choose a response, you blew it.

(Sorry. Inside joke. I am WAY ready for summer vacation.)

Around here, educational consultants make a small fortune.

The two consultants I know deserve every penny. They both started out as brilliant classroom teachers. Now they're free agents.

It's a truism in education that the only way teachers can rise in their careers is to stop teaching and go into administration. (Though I'd like to see Caroline Hoxby do a study of this, if she hasn't already.)

I suspect that in affluent districts there is a second career path available to talented teachers these days, which is to leave teaching and become a consultant.

I also suspect that constructivist math creates more work for consultants. Our own grade school, which is adopting Math Trailblazers, now has a Math Enrichment teacher 'helping to support the implementation of Trailblazers,' as well as a 'math consultant,' who is 'working with teachers at each grade level in small workshops to discuss math content and assessment approximately every six weeks throughout the year.'

(Not coincidentally, we also have a publicist to write articles about Math Trailblazers for the school newsletter and the local newspaper.)*

The math enrichment teacher was previously a regular elementary ed classroom teacher. I don't know the math consultant's background, but I assume she, too, began life as a classroom teacher - probably a good one - and then advanced to math consultant.

I would like to see teaching become a profession like other professions.

I would like to see talented teachers able to advance within the realm of teaching.

I would like to see the very highest salaries go to star teachers, not to administrators or consultants.

Good administrators & consultants should be well-paid. But an administrator or a consultant should not automatically, by virtue of being an administrator or a consultant, make more than a teacher.
* OK, that's not fair. We had a publicist - a free lancer - before we had Trailblazers.


keywords: choose a response no putdowns bullying character education lost instructional time



comments...

NewBook 13 May 2005 - 16:41 CatherineJohnson






140396839X.gif



Eduwonk seems to think Joe Williams' book, coming in fall 05, will be good.



comments...

BooksPart1 14 May 2005 - 00:09 CatherineJohnson





rma.jpg




0805829083.jpg

two fantastic books



Elaine McEwan's website





comments...

CatherineIsGobsmackedPart4 15 May 2005 - 01:07 CatherineJohnson



re: career opportunities for ed consultants

In the fall of 2003, Chancellor Klein introduced the
mandated [contructivist] reading and math programs
. . . . teachers frequently complained of micro-
management, due to the heavy-handed imposition
of lockstep constructivism. In some districts,
supervisors roamed classrooms with stopwatches,
and teachers were penalized if they spent a few too
many or too few minutes on a mandated activity.
The new curriculum has proven to be a bonanza for
the education establishment, particularly schools of
education such as Columbia's Teachers College,
which receives millions of dollars each year to
train teachers in constructivist methods.



CarolynIsGobsmacked
CatherineIsGobsmackedPart2
CatherineIsGobsmackedPart3



comments...

CurricularGamePlaying 15 May 2005 - 01:34 CarolynJohnston

Does it matter what mathematics curriculum your kids are using at school, as long as they are getting good grades in math?

Catherine and I both started tutoring our kids, supplementing their math homework, and looking into mathematics education, because our kids weren't doing well in their regular math classes. Had they gotten good grades all along, we might just be rolling along without asking any questions.

But my son was doing poorly in Everyday Math, a new-new-math curriculum, after having been very successful in Saxon Math, a traditional curriculum which emphasizes the incremental acquisition of new skills, including mastery of all the classic computations. It was clear that it was the new curriculum that had derailed him. But was that just my son, whose special needs make him a special case?

Proponents of Everyday Math claim that it integrates a child's mathematics knowledge, and makes it more useful to him, if the kids spend time working with math in the context of discovering and solving real-world problems; gathering data, measuring things, and so forth, at the expense of computation (if necessary). If so, then after (perhaps) a few years of struggle, we ought to see improvement in kids' understanding of math at the level of applications.

In other words, kids raised on real-world data and applications ought to at least be better at word problems. That's what makes this chart so powerful.

ProblemSolvingScoreChart.gif

The chart shows scores on a subtest of math problem solving of the Comprehensive Test of Basic Skills (CTBS), a nationally-normed standardized test. The scores measure the same group of kids from Anne Arundel County's 14 lowest-performing schools in 2nd grade, and again in 4th grade.

The second graders had been working with either Everyday Math or Mathland, a similar 'discovery-based' curriculum (see the blue bars in the chart). When they took the test in 4th grade, they had been working with the Saxon curriculum for a year (see the white bars).

The kicker is that this subtest measures performance on word problems. This is the supposed weakness in traditional math programs that Everyday Math's approach is intended to remedy.

Check out this link to see how the news went over in Anne Arundel.


Curricular Game Playing
Curricular Game Playing, part 2
number bonds vs. 4-fact families
Numicom Dominoes





comments...

CurricularGamePlayingPart2 15 May 2005 - 02:23 CatherineJohnson


About a month after Christopher and I began working with Saxon Math 6/5, he told me,

Multiplication and division are the big brothers,
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.

Any way you slice it, that's conceptual knowledge. In just a few weeks he'd absorbed the idea that addition & subtraction, multiplication & division, are inverse operations, and that multiplication was repeated addition, while division can be seen as repeated subtraction.

I should add that Christopher doesn't consciously know that division can be described as repeated subtraction (I don't think). He probably couldn't put it into words, though he could tell you that multiplication is repeated addition. But a few weeks into Saxon he had intuited the relationship.

This is exactly the goal constructivist math programs have set for themselves: they are trying to help students connect the dots.

Addition, subtraction, multiplication, & division aren't Four Separate Things, as they were for me until I read and studied Saxon Math!

I haven't worked with a constructivist text.

But I know for a fact that Saxon gives children conceptual understanding.


Curricular Game Playing
Curricular Game Playing, part 2
number bonds vs. 4-fact families
Numicom Dominoes





comments...

GreatMomentsInWorldHistory 15 May 2005 - 03:01 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





comments...

SingaporeMathSummerWorkshop 15 May 2005 - 21:43 CatherineJohnson

Scott Baldridge, coauthor of Elementary Mathematics for Teachers is giving a summer workshop in Singapore Math!

Singapore Mathematics Summer Institute
August 1-5, 2005
Madison Country Day School campus, Madison, Wisconsin
Cost: $500


I may have to go.



comments...

HolyBureaucraticNightmareBatman 16 May 2005 - 00:18 CatherineJohnson



. . . sigh . . .
via Eduwonk and New York Daily News



comments...

CalStateStudyIntro 17 May 2005 - 03:36 CarolynJohnston

Part 1 in a mini-series on a review of quality math ed research articles.

In 1998, the California State Board of Education contracted with a group of education researchers from the University of Oregon to conduct a review of high-quality mathematics education research papers. The resulting 100-page report is available here.

Their task was simply to search out all the mathematics education research that had been performed and published within a specified period, cull out the stuff that was of dubious quality (meaning it had unsound experimental underpinnings, or was performed in a setting that was not like a classroom, or had one of a number of other flaws), and see what the remaining studies had to say about mathematics achievement (that is, they avoided papers that did not measure study outcomes quantitatively, using tests of achievement; so studies measuring the impacts of changes in teaching methodology on students' confidence, for example, weren't included).

The results are surprising to me in places. There were studies on the use of manipulatives, studies on kids working with their peers, studies on the use of computers, calculators and technology, studies on motivational methods, and studies on the design of instruction. The researchers seem to have avoided bias, and to be genuinely searching out high quality research. I thought I would do a 'mini-series' describing and discussing their results, section by section. Stay tuned.



California study intro
California state study of group learning
California Board of Ed study part 2
education research - peer reviewed studies - chart





comments...

CalBoardOfEdStudyPart2 17 May 2005 - 15:17 CatherineJohnson

Carolyn wrote:

I thought I would do a 'mini-series' [on the California Board of Education study]
describing and discussing their results, section by section. Stay tuned.

What a great idea!

I've been wanting to know more about the famous California Board of Ed study.

Here's a terrific factoid about Dixon et al, from The Principal's Guide to Raising Math Achievement by Elaine K. Mc Ewan:

From a total 8,727 published studies of mathematics in
elementary and secondary schools, they identified only
956 articles that satisfied the minimum identification
criteria of being an experimental study of mathematics.
. . . The evaluators then used the evaluative criteria
for experimental research . . . Only 231 of the original
956 studies made it through an initial screening of
construct, internal, and external validity. When the
methodologies of those 231 studies were screened
for internal and external validity, only 110 studies
were deemed to be of high quality.


8,727 "studies."

Of which, 231 were scientifically valid.

231

Parents, teachers, administrators, and Concerned Citizens everywhere should have this figure tattooed to their foreheads.

When textbook publishers and ed school types use the words "research shows," you're looking at maybe a 3% chance they're right about that.

Given the fact that, by law, all research findings have to be replicated before they can be certified as facts, the odds are probably closer to zero.

(OK, I'm kidding. There's no law. Anyone can call anything a fact if they want to. It's a free country.)

CalStateStudyIntro


California study intro
California state study of group learning
California Board of Ed study part 2
education research - peer reviewed studies - chart





comments...

RoyalRoadToGeometry 17 May 2005 - 17:45 CatherineJohnson

I had never read this story before today:

When Ptolemy I, the king of Egypt, said he wanted to learn geometry, Euclid explained that he would have to study long hours and memorize the contents of a fat math book. The pharaoh complained that that would be unseemly and demanded a shortcut. Euclid replied, “There is no royal road to geometry.”

I'm sorry to hear that, because a royal road to geometry is exactly what I need today.

I just checked out the next lesson in Christopher's SRA Level 6 book, which turns out to be about finding the equation for a line that's been plotted on a graph.

I could do the easy, obvious problems, but the graph where 'one step to the right' seemed to be followed by '1/3 step up' stopped me cold.

I don't remember ever being taught how to find a formula from a line on a graph. I also don't remember ever being taught a formula for making a line on a graph in the first place, although I do remember plotting out lots of coordinated pairs.

That's got to be be worth something, right?

Unfortunately, while I remember plotting out lots of coordinated pairs, I have no idea when in my mathematics education this occurred, or why. Exactly what Subject Matter Area does finding-an-equation-from-a-graph fall under?

Since the formula-finding problems in Christopher's book are in the unit on 'Algebra Readiness,' I figured this must be algebra, so I went to get my copy of Algebra to Go (buy this book, you'll need it) from the dining room-cum-math-&-spelling-zone.

This is where I feel God Wants Me To Learn Math, or at least not suffer hideously while I try to make sure Christopher Learns Math, because an Unseen Force led me to pull out Geometry to Go instead (buy this book, too), open it up, and land smack dab in the middle of the page explaining the formula for charting linear functions on a graph — all of this before realizing I had the wrong book, glory Hallelujah.

Leading to my first Math Revelation of the day: it's not algebra!

It's coordinate geometry! *

I had no idea!

Thank you!

Then my neighbor, the statistician, came over and showed me how to do it.


* UPDATE 10-8-2006: It's algebra. Algebra and coordinate geometry, I guess. I don't know. I will press on and report back.

bsgconfusedsmall.jpg

source:
Bitter Single Guy




see also:
BuyThisBookToo
EnglishLanguageArtsBookRecommendation
MathRefs





comments...

BuyThisBookToo 17 May 2005 - 18:52 CatherineJohnson

As long as I've got you overspending on math books, you may as well pick up a copy of Math on Call to complete the set.

The Math On Call series is targeted to the school market, though the books are priced well enough that parents can and do buy them, too.

I'd love to know what the sales rep's pitch is.

Essentially, the books cover every topic your child is going to encounter in every level of math, explaining each one directly, conceptually, and procedurally -- and very likely using the same vocabulary, illustrations, and sequence of subtopics his or her school will use to boot, thus putting a stop to the nightly 'I can't help you with your homework, I didn't learn it that way' exchange.

I'm wondering whether schools that have invested in constructivist math purchase these texts as direct-instruction back-ups, for the parents as well as for the kids. [update: I just noticed that there are Parents' Guides available for all of the books.]

This is a less frequently noted problem with constructivist math. If parents have forgotten their own math (that would be me), they're not going to remember it looking at a discovery text.

Which brings me to one of my favorite reader reviews on AMAZON:

My son's 7th grade math teacher recommended this. I don't know what we would have done without it. The school's math textbook was useless. If there was any problem not understanding a math concept, we would just whip this baby out and it was easy to understand. Math homework couldn't have been any less frustrating.

My younger son now takes it with him to school for doing his math work at school. It is invaluable.



One last thing. If you have younger kids, you might want to start with the earlier books in the series. They're easier to deal with when you have a lot of catching up to do yourself.

Grades 1-2 mathtolearn_thumb.gif

Grades 2-3 mathtoknow_thumb.gif

Grades 5-6 mathathand_thumb.gif


for 8th grade & high school

Algebra to Go
Geometry to Go



see also:
RoyalRoadToGeometry
EnglishLanguageArtsBookRecommendation
MathRefs





comments...

CalStateStudyOfGroupLearning 18 May 2005 - 04:04 CarolynJohnston

Part 2 in a mini-series on a review of quality math ed research articles.

Part 1: CalStateStudyIntro

The most surprising thing, to my mind, in the Cal State Study is its strong endorsement of cooperative group learning. The review included eleven studies of group learning, all with positive findings for the use of structured group learning.

The Cal State Study defines "conventional mathematics instruction" as being characterized by teacher explanation of the new material, followed by independent workbook activity. The study makes the strong claim that the conventional approach has absolutely no theoretical support, and is discredited by the totality of the studies in the review that examine cooperative group learning methods.

Actually, this is the one place I've read in the report so far where I felt I might be encountering just a whiff of reviewer bias, particularly in the glowing interpretation of these findings.

But the evidence might really be there. Carefully constructed group learning opportunities beat conventional teaching methods. Go figure.

The phrase 'carefully constructed' is a point to dwell on. Simply sticking kids together in groups to do their homework conferred no benefit. The group learning environment had to be structured, i.e. the nature of the kids' interactions had to be controlled by external reinforcement systems. In particular, higher performing kids had to be motivated to help lower performers.

To some degree, it didn't even matter what the system was, but it had to be present. One study compared cooperative vs. competitive group reinforcement systems: an example of a cooperative system is one in which each kid in a group gets the average of their individual grades, whereas grouping the kids into competitive teams is a competitive reinforcement system. Both reinforcement systems were shown to confer equal benefits above the conventional teaching style.

The reviewers note that "a particularly interesting aspect of these studies on cooperative work is that all of them included students who were at risk in some sense, either by virtue of being inner city urban students, and/or low SES students, and/or students identified as having special learning needs."

While the reviewers seem rather pleased with this study design, to my mind it weakens the broad applicability of the results. I rather wish they'd done these studies with the most typical bunch of learners they could find, since the presence of special needs can really skew a kid's learning style. Kids who are starved for attention, interaction, or positive reinforcement for learning, might benefit much more from learning environments that include these things than a typical learner would.

Also see: CalBoardOfEdStudyPart2, EdResearch


California study intro
California state study of group learning
California Board of Ed study part 2
education research - peer reviewed studies - chart





comments...

EdResearch 18 May 2005 - 15:15 CatherineJohnson



ednext20023_36fig1.gif

graphic from:
The Virtues of Randomness
by Robert Boruch



California study intro
California state study of group learning
California Board of Ed study part 2
education research - peer reviewed studies - chart





comments...

CalStateStudyOnManipulatives 19 May 2005 - 02:29 CarolynJohnston

Part 3 in a mini-series on a review of quality math ed research articles.

Part 1: CalStateStudyIntro

Part 2: CalStateStudyOfGroupLearning

Another surprising fact -- about math manipulatives -- comes out of the Cal State Study.

There were only four studies of manipulative use that were of high enough quality to make the Cal State cut. That really isn't enough to draw a conclusion from, especially given the studies' haphazard coverage of the range of instructional possibilities. Still, there are enough results that they suggest a pattern. See if you can detect it ('benefit' implies that kids did significantly better on normalized tests of math achievement than control groups did).

Kindergarten kids learning counting: no benefit conferred by including manipulatives.

Third graders learning multiplication: two different studies show no benefit to the use of manipulatives before teaching formal computation.

Fifth and seventh graders learning fractions: kids benefit from a fractions game played with or without other manipulatives and pictorial representations.

Elementary schoolers using fraction/ratio manipulatives with fraction/ratio instruction: no benefit.

Seventh graders using fraction/ratio manipulatives with fraction/ratio instruction: benefit.

I love what these results suggest because it is so unexpected and counterintuitive. Most of us think of manipulatives as a stepping-stone from the concrete to the abstract, as something to be used only by the very young when they are first introduced to a topic.

But these results suggest that older kids get more benefit out of manipulatives. In a way, now that I think about it, it makes sense; their relative maturity means kids have a conceptual 'hook' on which to hang the insights that the manipulatives give them. They already have half a clue, and that helps them get the point of the manipulatives.

Perhaps to a younger kid, less able to generalize from the concrete to the abstract, the manipulatives are simply toys.

This is all the evidence I need to get the fraction manipulatives out for my soon-to-be sixth grader.

For more information on math manipulatives, see our favorite math supplements for kids and FractionManipulatives

Also see:
EdResearch
CalBoardOfEdStudyPart2
FractionManipulatives
QuickThoughtAboutFractionManipulatives
FractionManipulativesPart2
NewStudyOnManipulatives
New Study on Manipulatives Part 2





comments...

FractionManipulatives 19 May 2005 - 20:34 CatherineJohnson

re: CalStateStudyOnManipulatives

Over the past year I've used two kinds of manipulatives with Christopher, who is 10:

fraction manipulatives
play money

I didn't need play money and neither does anyone else. I got it only because I wanted to teach Christopher how to make change without a cash register, a lost art, and because . . . if I stacked up a pile of Real Money big enough to make change with, it was going to get raided for lunch money, bake sale money, field trip money (and that's just for starters).

We are chronically short on ONES around here, let me put it that way. So I decided to make things easy on myself and buy some fake money.


+ + +

I'm a huge fan of fraction manipulatives. Christopher and I have spent quite a lot of time using a set of fraction tiles to illustrate:

equivalent fractions
the addition and subtraction of fractions
the addition and subtraction of equivalent fractions

Nothing makes the idea that 2/12 is equivalent to 1/6 more obvious, IMO, than actually lining up two 2/12 tiles below one 1/6 tile and seeing that, yes indeed, 2/12 = 1/6.

These are the fraction tiles I use. They cost $8.75 plus shipping:

C_0743247698.jpg

The same company, (Rainbow Resource, a homeschooling catalogue), also carries a set of extra fraction tiles without the tray that I wish I'd had when we first started trying to learn fractions. (I have them now, but we may be past the point of needing them. We'll see.)

You need the extras because you really want the ability to demonstrate addition and subtraction of fractions with different denominators.


+ + +

There are lots of other fraction manipulatives out there, but I chose these after reading a comment from a mom on a homeschool forum somewhere. (I wish I'd kept the link.)

She said that her daughter didn't get anywhere using the more-common circular, 'pie chart' fraction manipulatives; she needed to see rectangular fractions. I have no idea why this would be, but it 'felt' right to me, so I searched for rectangular manipulatives and found these.

At the same time, SAXON MATH uses circular manipulatives, so Christopher has been exposed to both, which I think is almost certainly ideal. A core principle in teaching math, from what I gather, is to teach the same material from different angles.


+ + +

Another terrific activity to do with fraction tiles:

Show how different combinations of fractions add up to 'one whole.'

To do this you just have your child keep lining up fraction tiles on top of the bright red 'one whole' tile until he's covered the whole thing without anything hanging over the end.

So, for example, he might put 2 1/12th tiles, 1 1/6 tile, & 2 1/3 tiles on top of the 1-whole tile, illustrating the fact that:

2/12 + 1/6 + 2/3 = 1

After awhile it starts to become obvious that you can put 6ths and 3rds & 12ths together evenly to make one whole, or 8ths & 4ths & halves, or 5ths & 10ths, . . . but you can't put 3rds and halves together, or 4ths and 5ths (not unless you have a bunch of 20ths, which you don't), and so on.

You can see your child start to get a feel for multiples* and divisibility, whether he has explicitly studied multiples and divisibility yet or not.


+ + +

That's a whole other issue: is it useful to 'preview' concepts in this way? I have no idea, so offhand my answer is 'It depends.' That's one of the big gripes with constructivist math; the kids are constantly being exposed to advanced topics -- sometimes very advanced -- and then not taught the topics to mastery, because the book will be 'spiralling back' to the same topic the next year and the next year after that. Parents tend to hate this, but parents could be wrong. It happens.

Let's just say that my perception, working with Christopher and the fraction tiles, was that he was developing an intuitive grasp of numbers that are multiples of each other versus numbers that aren't. This seemed like a good thing at the time, but who knows? I'm new at this.

Come to think of it, I'm going to get the fraction tiles out again when I get back to teaching the Singapore Math lesson on Changing Ratios. (My neighbor and I team-taught this lesson to our kids two weekends ago, but it was over Christopher's head. Her son is a year older.)

Singapore teaches changing ratios in the first half of 6th grade:


sp_pmust6a2.gif



+ + +


Since I never remember definitions of even the simplest terms, I am including the definition of a multiple here:

* multiple - The multiple of a number is the product of the number and any other whole number. (2,4,6,8 are multiples of 8)



Also see:
EdResearch
CalBoardOfEdStudyPart2
CalStateStudyOnManipulatives
QuickThoughtAboutFractionManipulatives
FractionManipulativesPart2
NewStudyOnManipulatives
New Study on Manipulatives Part 2



comments...

RussianMath 20 May 2005 - 13:03 CatherineJohnson

I've just ordered a copy of Mathematics: An Award Winning Textbook from Russia from Perpendicular Press.

The translator's press release is here, and Barnes and Noble has posted this 2004 review from Book News:


The textbook won the national competition for best textbook when it was first published in 1987, and is still in use today by sixth graders throughout the former Soviet Union. Harte (mathematics, George Washington High School, Cedar Rapids, Iowa) ran across a copy and decided it was much better than anything he used: there are (almost) no distracting graphics, misguided explorations, or colorful sidebars about courageous people; only half a dozen carefully sequenced lessons with examples and exercises.





Our Favorite Supplements
RussianMathPart2
RussianMathPart3
WhyILoveCarolyn
ItTakesChops
Mike McKeown comment
IndusAcademy





comments...

QuickThoughtAboutFractionManipulatives 20 May 2005 - 13:44 CarolynJohnston

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.

Also see:
EdResearch
CalBoardOfEdStudyPart2
CalStateStudyOnManipulatives
FractionManipulatives
FractionManipulativesPart2
NewStudyOnManipulatives
New Study on Manipulatives Part 2



comments...

FractionManipulativesPart2 20 May 2005 - 16:47 CatherineJohnson

re: QuickThoughtAboutFractionManipulatives

Wow!

Thank you!

This is why Life Changed when I met Carolyn. She's not just a mathematician herself; she spent years teaching math, and she is actively engaged in acquiring pedagogical content knowledge.

Pedagogical content knowledge is a fancy way of saying that the things really good math teachers know are somewhat different from the things really good mathematicians know, and that the difference is important.

(This is why neither Carolyn nor I feel that simply requiring math teachers to major in math is going to do the trick when it comes to raising math achievement. But that is a subject for another post.)

While I was writing about rectangles being better than circles, I was visualizing circle manipulatives, and I was thinking:


Well, you can put together two 2/12 pie slices with
one 1/6 pie slice with two 1/3 pie slices and still get
one whole pie, the same way you can with the
rectangular manipulatives.

So what's the problem?

But then I was thinking,


I know there's something else . . .


Now, here is Carolyn pointing out that it's going to be 'visually' impossible to tell a child that 3/2 represented as 1 and 1/2 circle is ONE THING, whereas it's going to be (reasonably) easy to tell a child that 3/2 represented as 1 and 1/2 of a bar is ONE THING.

This observation has opened a window for me:

I see that I hadn't progressed to the point of realizing that 3/2 should or even could be considered ONE THING.

I have a ways to go.

Still, this makes me hopeful that I'm beginning to develop some intuitive knowledge of math content and math pedagogy or teaching . . . because I could tell there was a reason why I'd grown more attached to rectangular fraction manipulatives over the year, not less. I just couldn't put my finger on it.

Veering off on a tangent here, one of my very favorite books on the cognitive unconscious (tacit knowledge, or, sometimes, intuition) is Arthur Reber's Implicit Learning and Tacit Knowledge: An Essay on the Cognitive Unconscious.

019510658X.jpg
I remember Reber writing that one of the reasons the field of implicit learning got going in the first place was the question of how to make sure experts in one generation passed their knowledge on to the next generation.

As I recall, the first thought everyone had was simply to ask experts, such as surgeons, how they did what they did. They figured the experts could tell them.

It turned out the experts couldn't tell them. They were experts, not teachers.

That raised the question of what we know that we don't know we know.

I hope I'm developing some intuition about teaching math, and about the content of mathematics itself.

But while intuition about how to teach math may be good enough, intuition about math itself probably is not.

To be a good math teacher, it seems, you have to be able to put what you know about math into words and images.


Table of Contents, Implicit Learning


Also see:
EdResearch
CalBoardOfEdStudyPart2
CalStateStudyOnManipulatives
FractionManipulatives
QuickThoughtAboutFractionManipulatives
NewStudyOnManipulatives
New Study on Manipulatives Part 2





comments...

FractionManipulativesPart3 20 May 2005 - 22:00 CatherineJohnson

On the subject of buying fraction manipulatives, if it doesn't break the bank I would also get an inexpensive labeler.

Use it to label each fraction tile with the equivalent percent.

Add a "10 percent" label to each 1/10 tile, "33 1/3 percent" to each 1/3 tile, "100 percent" label to each 1-whole tile, and so on.

If you can fit the decimal representation of the number (.1, .3333, 1.0) on the tiles, put that on, too.

I got this idea from Saxon Math 6/5. 6/5 includes lots of worksheets with fraction circles printed on them, and always, on every sheet, the fractional parts are labeled with all three representations of the number: fraction, decimal, percent.

Brotherlabelergif.gif
Brother PT-65 Home & Hobby III P-Touch Labeling System, $29.95


Saxon also has the kids answer mental math questions about fractions and percents ("How much is half of 5?" "What is 50% of 50?") in virtually every lesson in the book.

At first I didn't get this.

The concepts hadn't really been taught, and it seemed like pure memorization to me.

But I found that this constant practice of simple 'recognition knowledge' -- visually and verbally recognizing 1/2 as 50% and 50% as 1/2 -- meant that whenever we studied a conceptual lesson on fractions, Christopher was ahead of the game.

At least, that's the way it seemed to me.

He could look at a pie chart divided into 10 pieces and see instantly that 50% = 1/2 = 5/10. He already had, inside his head, "50% of 10 is one-half of 5;" it just came naturally.

[*update*: OK, 50% of 10 is not one-half of 5. This is the kind of thing that drives me nuts; I am constantly popping off with statements like 50% of 10 is one-half of 5; I am starting to think I am dyslexic for numerical expressions, da***it. Thank you, Carol Morgan.]

I also began to find that Christopher was getting faster at fraction problems than I was. Faster, and more accurate.

I would ask him a Saxon fraction problem I myself was slightly confused on, he would come back fast with an answer, I'd say it was wrong, he'd say it was right -- and lo and behold, it was right.

Somehow he'd crossed over from knowing the answer to knowing the answer. He knew that the answer had to be right, because it made sense.

I assume he was passing me by because I hadn't been doing all the 'memorization' he'd been doing. I hadn't been doing it because I didn't think I needed to. I already knew the concept of equivalent fractions, and I could do the calculations (which he couldn't) .

And yet by the end of the book Christopher seemed to be overtaking me on conceptual understanding (that's assuming I know what conceptual understanding of mathematics actually is, which I don't).

Christopher seemed to be developing a quicker and more reliable feeling for numbers, for the fact that a particular answer had to be right, or had to be wrong, or was or was not 'in the ballpark.'

So, for the time being, I'm convinced that we want to do solid memory work with our kids. Memorized material seems to give us the base we need to build up something . . . more.

re:
FractionManipulatives
FractionManipulativesPart2
QuickThoughtAboutFractionManipulatives
CalStateStudyOnManipulatives


+ + +


One last thing: Saxon seems to have extended the concept of math facts to include fraction facts.

Fraction facts, decimal facts, and percent facts.

In books Saxon 7/6 & 8/7 he has kids do timed worksheets converting fractions into decimals, decimals into fractions, and so on.

Given how incredibly difficult fractions are for most all students, I think that's probably a good idea.



comments...

MathInTheBloodPart2 22 May 2005 - 00:17 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...



comments...

ColoradoMathStandards 23 May 2005 - 13:53 CatherineJohnson

I was just perusing the 2005 Thomas B. Fordham Foundation assessment of state math standards, and I found this passage drawn from Colorado's standards:

Second grade students will, using objects and pictures,
represent whole numbers including odds and evens
from 0 to 1,000.

Third grade students will, using objects and pictures,
represent whole numbers including odds and evens
from 0 to 10,000.

Fourth grade students will, using objects and pictures,
represent whole numbers including odds and evens
from 0 to 1,000,000.
The authors take a dim view of this requirement:

Grasping the concept of even and odd numbers does
not require three years of collecting progressively more
objects. The time devoted to collecting and displaying
objects and pictures is better spent on other activities.

I'll say.


The report gives Colorado's standards a D, the same grade they received last year.



comments...

CalStateStudyMasteryLearning 24 May 2005 - 04:49 CarolynJohnston

Part 4 in a mini-series on a review of quality math ed research articles.

Part 1: CalStateStudyIntro

Part 2: CalStateStudyOfGroupLearning

Part 3: CalStateStudyOnManipulatives

I had never heard the term "mastery learning" before reading this report, but I immediately understood it, because I am so familiar with the use of the opposite strategy in the Everyday Math curriculum.

The identifying feature of mastery learning is that students have to demonstrate mastery of the current material before moving on to new material. Since students in a classroom generally move on in a block, true mastery learning can be difficult to implement in the classroom, since you'll always have a few kids in the group who are lagging. Still, mastery learning is what happened to many of us when we were in school, and is what parents assume is going on in our kids' classrooms.

But it's probably not. In the Everyday Math primary curriculum, the same material is taught every year, with the kids expected to grow in achievement every time they encounter a topic, and to attain mastery at some time in the indefinite future. This 'spiraling' approach is backed up by the use of both pre-tests and post-tests, given before and after a unit is studied, collectively measuring the student's progress toward mastery of the material during the unit.

My objection to this approach is that The Day of Mastery is pushed off into the indefinite future. If your kid doesn't get it yet, you needn't worry; because he'll get another chance at it; it's never time to press the panic button until the kid switches to another curriculum (or goes to high school).

But even in more conventional curricula, topics aren't taught to mastery on the spot. In the Saxon curriculum, after a topic is introduced, the student moves on to another topic the next day; but encounters the first type of problem over and over again in the practice problem set. As with mastery learning, there is no escape from a topic until it's mastered. New topics that depend on the old topic aren't introduced until the student is competent with the original topic.

How good is mastery learning? Two of the review studies looked at mastery learning by itself and with combinations of other curricula, and found that mastery learning by itself produces better results than what was termed 'conventional instruction'. However, mastery learning got its best results when used with other teaching techniques. One study got decent results for "mastery learning with corrective feedback" (meaning -- electric shock? The review didn't say), but got its best results from mastery learning with 'enhanced cues' -- extremely detailed instructions to the students on how to do problems.

Another study found that mastery learning and cooperative learning strongly enhanced each other (note: cooperative learning is structured working-together among students, as opposed to simply being stuck in groups to do your homework together: see part two of this series).

I can see why cooperative learning and mastery learning would enhance each other. If you're a fast learner and the kid next to you is holding you back from getting to the next topic, you're going to be highly motivated to help him get it. This is fine for the average kid, but for the faster ones...? Frustration, possibly. And yet the study claims that achievement was enhanced by cooperative learning for both the helpers and the helpees...

To summarize, I found it hard to draw conclusions about mastery learning from this review. I'd like to see a study of mastery learning compared to a curriculum like Saxon math, which doesn't demand mastery on the spot, but rather draws it out of the kids over time.



comments...

MoneyClassSizeMathAchievement 24 May 2005 - 20:32 CatherineJohnson

This item made my day.

After our school board announced that budgetary constraints had left them no choice but to increase class size in the 4th and 5th grades (from 19 or 20 students per room up to 23 or 24) parents voted in our second double-digit tax increase in a row.

Our fourth and fifth grade classes will remain small.

I was skeptical.

For one thing, I was aware that Asian math classes are far larger than our own.

For another, I was aware that comparative education researcher James Stigler* actually recommends increasing class size as a means of improving math achievement in America. Larger class size would allow American teachers to meet with colleagues in the lesson study groups that are standard practice in high-achieving countries.

But while I knew all this, I hadn’t quite allowed myself to draw the obvious conclusion.

I hadn’t grokked the possibility that if you’re living in a school district where everyone’s clamoring for small class size, and no one’s clamoring for teacher release time, . . . that might be a problem.

So this afternoon I found this analysis of TIMSS data in Education Next:


When other factors are taken into account, higher
spending and smaller class sizes seem to correspond
to inferior mathematics and science results, though
the overall effect is relatively small.


Well, all I’ve got to say is, thank heavens there’s only a small correspondence between high spending, small class size, and inferior mathematics and science results.

Because if there were a large correspondence we’d be in trouble.


+ + +


I like this chart, too:

ednext20012_69.gif



soucre:
Why Students in Some Countries Do Better
by LUDGER WOESSMAN
EDUCATION NEXT


* James Stigler was one of the investigators in the 1999 TIMSS study and is coauthor of The Teaching Gap: Best Ideas from the World's Teachers for Improving Education in the Classroom by James W. Stigler, James Hiebert and The Learning Gap: Why Our Schools Are Failing and What We Can Learn from Japanese and Chinese Education by Harold W. Stevenson, James W. Stigler.




comments...
HowToRespond 25 May 2005 - 01:00 CarolynJohnston

Although this ship has already sailed for me and Catherine, here's instructions on what to do when your school district announces a switch to a new-new math curriculum.

I'd love to know if anyone is able to use this information to their advantage. My experience is that this stuff is like the flu... once you've caught it, there's little you can do but let it run its course.



comments...

AssessYourChildForFree 25 May 2005 - 01:50 CatherineJohnson

A terrific resource for parents overseeing their children's math education.

File this under Peace of Mind.

When I first began teaching Christopher seriously here at home, I was flying blind.

I knew he was easily passing all his Saxon Math tests, but I had no idea what that might mean in the larger scheme of things. The tests seemed awfully easy to me.

Was he doing as well as a child passing all his tests in any other curriculum would be doing?

What did other kids his age know about math, anyway?

I had a lot of anxious days.

Math Coach tells parents trying to do what I was trying to do (catch Christopher up to where he was supposed to be, and then teach him enough extra math to bump him onto the accelerated track) to start by hiring an educational psychologist to administer a battery of standardized tests of math knowledge and skill.

That sounded like good advice, and in the best of all possible worlds I would have had a trained professional explaining my son's Math Profile to me: strengths and weaknesses, scales and subscales, percentile standings, the works.

But seeing as how here in Westchester County private assessments run into the countless thousands of dollars, I figured, Let's just put our heads down, run like he**, and pray that sheer doggedness and force of will can get us there.

That turned out to work.

On the other hand, the suspense was killing me.

+ + +

Today life is much calmer, thanks to the folks at Mathematically Correct, who've posted a set of sample problems for the CA tests. The problems were written by the LA County Board of Ed, and the test is a perfect length. Just a few pages long, but everything is there.

Keep these in a safe place.



See also:
DontRelyOnStateTests
PenfieldParents
NewYorkStateMathCurricula
CompareAndContrastPart3
FriendlyFractions
PaperFractions
ADifficultChild
ADifficultChildPart2
WorksheetsForSummer
AssessYourChildForFree
AssessYourChildForFreePart2
BonusOnlineAssessmentQuestions




comments...

UhOh 25 May 2005 - 14:24 CatherineJohnson

I just checked my Cart at Amazon: $311.05-worth of books, mostly math books.

Some of them are going to have to go.



FathersDay
FathersDayPartTwo



comments...

FathersDay 25 May 2005 - 14:38 CatherineJohnson

I wonder if my husband would enjoy a copy of What Is Calculus About? for Father's Day.


+ + +


He ended up really liking the TIVO I got him two summers ago.



8769994.gif



re:
UhOh
FathersDay
FathersDayPartTwo
JuneNineteenth




comments...

FathersDayPartTwo 25 May 2005 - 18:40 CatherineJohnson

OK, Ed is not getting a math book for Father's Day.

Even though he got me a new operating system for our computers for Mother's Day.

Nothing says, "You are the mother of my children and I will love and cherish you until the day I die" like "The Tiger upgrade came today."



FathersDay
UhOh



comments...

ProgressReport 25 May 2005 - 19:42 CatherineJohnson



Success!

Christopher came home with a 93 on his Mid Unit review yesterday.

He was very happy.

Later on I heard him talking to his friend S on the phone.

S had called to report that a girl in their class had been 'demoted' (I think that was the actual word Christopher used) from Phase 3 to Phase 2.


Back story:

Our school district tracks kids in math starting in 3rd grade.

There are 4 tracks: Phase 1, Phase 2, Phase 3, Phase 4.

Phase 1 is children with special needs (I assume); Phase 4 is (in theory) accelerated.

The math tracks have been a bone of contention here in the district. Last year 300 parents tried to prevent the school from de-tracking math.

They lost.

So now we're moving to . . . well, I'm not sure what we're moving to. I can never keep the plans straight. I think the idea is to get down to no tracking at all, but in the meantime we'll have 3 tracks, with the highest track being smaller than it is now, because the Middle School is determined to whittle it down.

Their view is that there are gazillions of kids in Phase 4 who don't belong there, and can't do the work.

I'm sick about the whole thing, because I had no idea that the de-tracking business was part and parcel of a district-wide decision to adopt the constructivist Trailblazers series. If I had had a clue we were talking about bringing in fuzzy math, I would have been marching in the streets.



Anyway, back to Christopher and his friend S.

S was in Christopher's Phase 3 class last year, doing even worse than Christopher was.

Over the summer I ran my own little Caroline-Hoxby-like natural experiment with the two boys.

I began formally teaching Christopher using Saxon Math.

S had a normal summer.

Shortly after school began again in the fall, Christopher & S were back together again in Phase 3. I say 'shortly after,' because the school had placed S in Phase 2, so at the beginning of the year that's where he was. Then his parents complained, and he moved back to 3.

Christopher was now getting As while S was still getting Ds and Fs.

That's the difference 6 weeks studying Saxon Math with your mom makes.

Then, in early February, Christopher moved to Phase 4.

This has made a HUGE impression on everyone, and it was the subtext of the boys' conversation.

Both boys were exulting over the 'demotion' of J, a girl in their class, from Phase 3 to Phase 2. (J is about a foot taller than both of them, and looks like she's passed through at least two stages of puberty, so I'm inclined to give them a pass on this).

But then they got into a discussion of their own comparative Math Standings.

I heard Christopher report his 93 on the Mid Unit Review; then I heard him say something about how 'My mom had to teach me.' He sounded really animated.

Afterwards he reported that S had gotten an 87 on his latest test, and that S 'is doing better in math.'

I thought that was pretty interesting. S is a very bright kid who has no business flunking math, and who, last I heard, hasn't had any home tutoring at all. He seems to come to school without his homework on a regular basis.

I got the sense that Christopher's progress is sparking S to greater effort . . . because S definitely wanted Christopher to know he was no longer in the D & F category. I think that was the point of the call.

I'm wondering whether this may be a side effect of formally teaching math to your child, as opposed to doing the reactive teaching Ed and I were doing in fourth grade.

People are natural-born observational learners, and a boy who sees his friend pulling ahead in math is going to feel like he ought to be pulling ahead in math, too. S has come a long way: from Phase 2 in the fall, to Phase 3-with-Ds-and-Fs, to Phase 3-with-solid-Bs. All of this without much extra help.

Pretty darn good.



MathInTheBlood
ReactiveTeaching
NowThatWereBothHere

BeingYourChildsFrontalLobes
GreatMomentsInWorldHistory
ProgressReport
ATeachersStory ("I like the idea of math")
BonusPreTeenPost
SummerSupplementTimePart2
SundaySchool
ILikeMath
TheGoodNewsFromHere
GoodNewsBadNews
ImGoingToPlayland
ImportantQuestionFromJoanneCobaskoOfSocmm
ImportantQuestionPart2
OutsmartingTheTests
ConversationsWithKids





comments...

FromWichitaBoy 26 May 2005 - 00:03 CatherineJohnson

I just found this comment from 'Wichita Boy,' who, I happen to know, is a Real Mathematician:

Peer pressure is probably the most effective technique one can apply to get kids to learn math. I had 5 or 6 kids in school who were good at math with whom I was in competition. At various points things came along which I didn't grasp immediately. When some of the other kids grasped the concepts before me I become highly motivated to learn them. Kids who lack smart peers are bound to do worse in math in general. -- Wichita Boy - 25 May 2005

I think he's right.


ProgressReport



comments...

MentalMultiplication 26 May 2005 - 04:36 CarolynJohnston

I just got off the phone with an old friend. Gerry, Bernie and I all used to be colleagues in the Florida Atlantic University math department, and we more or less independently left and moved to take up new lives in the greater Denver area. Bernie and I went into industry, and Gerry went into teaching; he now teaches mathematics at a private Catholic girl's school in Denver. We see them occasionally (not often enough!).

Gerry is a great innovator when it comes to math education, and a prolific inventor of new and creative math manipulatives, including one of the largest math manipulatives ever: the Sugar Sand Park Moebius Climber, designed with the aid of Mathematica.

mobius1.jpg

Gerry is an extremely thoughtful individual. We are both fascinated by developmental issues and how they affect math education, and we began a conversation tonight that I hope will continue over a long period of time on this website.

But just for tonight: here is a tip he dropped on me for teaching the essence of multidigit multiplication.

At the core of multidigit multiplication is the distributive property of real numbers: (a+b)c = ac+bc. The standard algorithm utilizes it more or less explicitly. But often, these days, the standard multidigit algorithm is not taught: either it's eschewed completely, or some variant like the lattice algorithm is taught instead. If kids are not explicitly taught the distributive property, it will come back to bite them in algebra, where it is used all the time in algebraic simplification and in factoring polynomials.

Here is Gerry's tip; if you want to be sure your kids understand the distributive property, get them to do problems where they multiply one-digit numbers by two-digit numbers entirely in their heads.

Working memory can't hold too much in storage, but it can do that much. If a kid knows his single-digit multiplication tables cold, then he can multiply a multiple of ten by a single-digit number, and add it to a multiple of two single-digit numbers, all in his head. And in doing so, he'll internalize the distributive property, because he has to use it in order to do this sort of problem.

Because unless you have an incredible visual memory, the lattice method isn't of much use for doing mental math.

Brilliant and simple. Like all of Gerry's other math ed innovations.



comments...

ThankYou 26 May 2005 - 16:08 CatherineJohnson



Carolyn and I owe a major thank you to Jo Anne Cobasko, founder of Save Our Children from Mediocre Math (SOCCM).

A couple of nights ago I sent Ms. Cobasko an email telling her about Kitchen Table Math, and asking her to send folks our way.

Well, she did.

This morning I discovered that Kitchen Table Math is listed on top of the What's New at Our Site? section at New York City HOLD.

HOLD stands for "Honest Open Logical Decisions on Mathematics Education Reform." I think I speak for Carolyn when I say that the two of us fervently support honest open logical decisions when it comes to mathematics education reform!

(I can probably also speak for Carolyn when I say that we will have some posts on this very subject in the days to come . . . )

And thank you to Bas Braams and Elizabeth Carson of NYC HOLD. Carolyn and I both have spent a lot of time pouring over your site, reading everything we could.

My son Christopher's mathematics education has been directly improved by your work.

We are thrilled to see Kitchen Table Math on the homepage of NYC HOLD.


+ + +


One more thing.

I'd been telling Carolyn I was 'going to write a press release.'

Then Jo Anne Cobasko did it for me! This woman is on the ball.

Thank you!

(I'm still going to write that press release.)




comments...

ClandestineTeaching 26 May 2005 - 16:59 CatherineJohnson

Following up on an observation made by Jo Anne Cobasko:

Parents who teach their children math at home help raise the apparent success rates of constructivist math curricula.

If Carolyn's son does better in school because he has a Ph.D. mathematician for a mom who's teaching him Saxon Math, his success doesn't get chalked up to Saxon.

It gets chalked up to Everyday Math.

All I can say is, she's right.

I think I first encountered the term clandestine teaching in something Elizabeth Carson wrote.

Carolyn and I are engaged in clandestine teaching, teaching that goes undocumented and unmeasured.

Teachers do clandestine teaching, too, when they close their doors and teach the way they want to.

I should add that my son's teachers and our principal know all about my home-teaching, and have been terrifically supportive. My efforts are clandestine only in the sense that they don't show up in official statistics.


+ + +


This got me to thinking: why shouldn't we know how many children are being tutored at home?

Do we have survey data on this?

Anecdotally, I can tell you that I'm constantly meeting parents who've hired math tutors for their kids, or who are doing a huge amount of 're-teaching' themselves.

I also have the impression that in my district it's the parents of average and above-average kids who are hiring tutors. (I could certainly be wrong about this, so take it with a grain of salt.)


+ + +


I was actually told by one teacher that she preferred teaching kids with special needs, because they have I.E.P.s (Individualized Education Plans) that the school has to stick to, and does stick to. (That is a feather in my school's cap. There are plenty of schools out there not complying with IEPs, and I'm in a postition to know.)

Regular kids don't have IEPs, and if they're not learning math the school has the option, and probably the temptation, of assuming that the problem lies in the child, not in the curriculum or the teaching.

The teacher who filled me in on all this felt that the IEP was a 'protection' for the teacher, not just the child. The IEP empowers her to do whatever she needs to do to make sure this child learns math.


+ + +


Of course, this is one of the standard criticisms of public schools today: a child can't get quality direct instruction until he's been classified as having special needs. If he's average or above average, forget it. He's gonna be discovering his algorithms.

Nobody expects a child with learning problems to discover long division.



comments...

NYCHOLDMailingList 26 May 2005 - 22:59 CatherineJohnson

NYC HOLD has a mailing list.

I just signed up.




comments...

RedLetterDay 26 May 2005 - 23:18 CatherineJohnson

My used copy of Math Trailblazers, grade 5 has arrived in the mail!



comments...

CompareAndContrast 27 May 2005 - 00:22 CatherineJohnson



problems in three grade 5 textbooks


from the last page of Primary Mathematics 5B (U.S. Edition):

18. A fish tank is 2/5 full after Sara poured 14 gal of water into it. What is the full capacity of the tank in gallons?



final problem in Saxon Homeschool Math 6/5 3rd Edition:

Change each of these base 10 numbers to base 5:
a. 31
b. 51
c. 10
d. 100
e. 38
f.  86



from the last page of Math Trailblazers Grade 5:

4. Write a paragraph comparing two pieces of work in your portfolio that are alike in some way. For example, you can compare two labs or your solutions to two problems you solved. One piece should be new and one should be from the beginning of the year. Use these questions to help you write your paragraph:

Which two pieces did you choose to compare?

How are they alike? How are they different?

Do you see any improvement in the newest piece of work as compared to the older work? Explain.

If you could redo the older piece of work, how would you improve it?

How could you improve the newer piece of work?







home%20alone.gif



CompareAndContrastPart2
CompareAndContrastPart3
CompareAndContrastPart4
CompareAndContrastPart5
CompareAndContrastPart6
CompareAndContrastPart7
MathInSalinaKansas

ATeachersStory
FromAReader
PracticePracticePractice
BarModelingVsGraphing (interesting comments from a KTM reader)
HowToGetParentBuyIn
ATeacherUsingTrailblazers
BigNumbers




comments...

LikePracticingTheViola 27 May 2005 - 03:02 CarolynJohnston

The question of the new-new math decade: how do we resolve the need to develop math fluency in children, without sacrificing their ability to think creatively?

This question presupposes that you believe the two to be in opposition - I don't. Math fluency is developed through practice, of the drill and kill variety; it's harder to say how mathematical creativity is developed (and yes, creativity is of immense value in mathematical research -- we don't just sit around thinking about the Really Big Numbers, as one of my grandmothers thought).

But the two really do coexist -- they have to. Mathematical creativity is hard to express when you have to go back to first principles every time you add fractions. But drilling algorithms can be pretty boring. How does the tedium of drilling algorithms coexist with creativity in solving word problems or engineering problems or Fermat's last theorem?

I think learning math is a lot like practicing the viola, which I could never stand to do.

I personally think the tedium of practicing computations is nothing compared to that of practicing viola, or any other instrument, but that's just me. Still, noone doubts that all violists, even the great ones -- especially the great ones -- have had to put in thousands of hours of practice, and probably noone would argue that they weren't necessary.

And how does the need for practice coexist with creativity and inspiration in playing the viola?

Well, pretty much everyone who practices the viola hard, over a number of years, is going to be a competent violist. The concert violists are going to be some subset of those who practiced their fannies off -- in fact, in terms of hours spent practicing, really inspired instrumentalists beat out their merely awesomely competent competitors. That's how you get to Carnegie Hall, after all, and here's a chart to prove it.

cogsci.jpg

How do we deal with the fact that musical practice is boring for most of us? Well, if we don't like to practice, we don't have to play. We opt out if we don't like the arrangement - as I did long ago, and as Ben did this year (although the instrument he is spurning, after a perfect record of non-practice in fifth grade, is actually the cello).

The problem with math is that nobody can opt out of learning it: we all need to be competent at it. An understanding of quantities and numbers and rates and growth are the basis of a lot of thinking in our society. It would be nice if there were a royal road to mathematical fluency, but there isn't one that we've yet found; it takes years for even the most mathematically able child to pick up all the mathematics they'll need as an adult.

Even a merely competent violist has pushed his knowledge of the mechanics of his instrument down out of his conscious brain and into his fingers. This has to happen before a violist can even dream of being creative, because if it hasn't, then his conscious brain is still working on mechanics.

Here is what I saw in my college algebra and calculus classes: people still struggling with the mechanics of math, years after they ought to have had the basic moves down. They didn't practice long and hard enough, and if they ever had the moves down, they'd lost them by then.

So how do you get your kid to practice? You get him into the habit. You provide carrots in the form of praise, trips to Chuck E Cheese, movies, video game time, whatever turns him on. You also provide a stick if necessary. You do what it takes to ensure that your kid does this thing that he needs to do, even if you have to fight with him (this is what Bernie calls being a brick wall, and what Catherine calls being your kid's frontal lobes). You clear out his schedule, if necessary, to ensure he has the time he needs to practice.

And you try to make sure he is taking a line of study that isn't going to let him down in the end.



comments...

PracticePracticePractice 27 May 2005 - 15:24 CatherineJohnson

I have to do something today besides sit around thinking and writing about math . . .

But all that other stuff can wait!

I'm going to be quick, which means this is off the top of my head:


1. Carolyn's friend Gerry on multiplication

For what it's worth, I think he's dead right about the value of mental multiplication.

I've mentioned that I taught a little after-school class in Singapore Math this winter. In every class I had the kids do mental math.

We did a lot of mental multiplication with the explicit purpose of implanting the distributive property inside everyone's heads.

I'm constantly pushing Christopher to do mental multiplication for this very reason.

He now 'knows' the distributive property; I think he can actually write it out in its 'letter form,' i.e. a(b + c) = ab + ac. (I think.)

He also, I think, knows -- and understands -- that the multiplication algorithm is based on the distributive property.

He knows that when you're doing a problem like:

21
x23

(sorry for the funky alignment; neither Carolyn nor I has been able to figure out how to insert extra spaces in the text thus far . . . )

. . . anyway . . . Christopher knows that when you take the 3 times the 2 you are multiplying 3 x 20; he knows that you are splitting the problem up into smaller multiplication problems and then adding the products together, which you can do because of the distributive property.

But even though he knows all this, I swear he's not as good at mental multiplication as the kids in my Singapore Math class (which Christopher boycotted). Nor does he seem to understand mental multiplication.

He didn't get the practice my Singapore Math kids did, and he's still not really making the connection that the same thing that lets you do the standard multiplication algorithm can be used to multiply numbers in your head or to very quickly multiply numbers horizontally.

His knowledge is still inflexible; he's not generalizing it to other situations and contexts. He's not seeing the connections.

This brings me to --


2. Carolyn's post on practice

This is a HUGE subject, but here are my first thoughts.

I've found that practice per se isn't such a hard thing to get kids to do.

My Singapore Math kids loved the timed worksheets I gave them. (I used the 'Fast Facts' worksheets from Saxon Math.) They used to ask to do more of them, because they made it into a competition. They were revved!

I'd have my timer out, and the kids would call out Done! when they finished the sheet; then I'd call their time & they'd subtract it from the starting time of 5 minutes and write it down on their score sheets.

(I gave each child his own 'Singapore Math' notebook with a Saxon score sheet in the front. So each week they could compare their new score to their previous scores.)

Now, you'd think this could go seriously awry, with the slow kids feeling defeated. I was worried about this myself, since I had kids ranging all the way from a fourth grader who may have been classified with some level of special needs (I have no idea--the parent seemed to indicate this) to a fifth grader whose parents immigrated from China and who's probably one of the best math students in the school.

That's a range.

But nobody's ego got crushed. Exactly the opposite.

Since they all had their own score sheets, they were competing against themselves as well as against the class. They also did different worksheets, depending on whether they'd hit the 5-minute mark on the worksheet from the week before.

As soon as somebody could do the 'Fast Facts' addition sheet, he or she moved on to the 'Fast Facts' subtraction sheet. So the faster kids were doing harder worksheets, and the slower kids were doing easier worksheets.

I guess that's like handicapping in golf, right? (I don't play golf, so I don't know.)

Let's just say that levelled the field considerably, and no one seemed to feel remotely humiliated because they were still doing subtraction when someone else was doing multiplication. They just liked the race.

And they all picked up speed incredibly quickly; I was amazed.

I had one child who, the first time he did a 5-minute addition worksheet, took -- gosh, I don't know -- upwards of 8 or even 10 minutes to get through it.

This child has perfect handwriting and is painstaking when he writes numbers, which was slowing him down, so the second day I actually wrote the answers for him so he wouldn't lose time just on penmanship.

But here's the miracle.

This kid did zero practicing in between classes, and yet by the third class he was coming in under the 5-minute deadline.

I couldn't believe it, and I don't know how he did it. He just . . . got faster. They all did.

They were achieving personal bests every week.

This gets back to Carolyn's post on group learning and Wichita Boy's post about competition: under the right circumstances, practice is fun.

I think the problem for Christopher & Ben is that they're sitting at a table with their mom who is forcing them to do math.

If they were sitting at a table with their friends, and everyone was doing math, it would be different. I happen to know for a fact that this is true, because a couple of times Christopher's friends Drew & Marc, who are fraternal twins, have done a Saxon Math lesson with us. Their mother told them they had to, so they did.

When the three of them are doing Saxon Math together, they peddle.

I've been thinking about group learning ever since Carolyn wrote about it, and I'm turning into a believer.

But more on that later.


+ + +


I see I've gotten off-track.

I meant to talk about Carolyn's observations on practice and expertise.

I'll have to do that later, but in the meantime the single best article I've seen on this subject is here.


+ + +


I wonder if you could get kids to practice the viola if you put 3 of them in a room together and set the timer.


ATeachersStory
CompareAndContrast
FromAReader
PracticePracticePractice
BarModelingVsGraphing (interesting comments from a KTM reader)



comments...

MathInTheBloodPart3 28 May 2005 - 02:08 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.



comments...

FrenchCalculatorForKids 28 May 2005 - 19:03 CatherineJohnson

Naturally, I was thinking, Excellent!

1st graders can Practice Their Math Facts AND Learn French at the same time!

Unfortunately, I have no idea what language this person is speaking.

(Click on 'Magic Maths.')


+ + +


The addition & multiplication tables are cute, though.


SpeakingOfTheFrench
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SpeakingOfTheFrench 28 May 2005 - 20:37 CatherineJohnson

Just a couple of days ago I was talking about clandestine teaching.

Now today I find Instructivist has linked to a raft of anti-constructivist books published in France in the past couple of years, one of them being:


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Je suis une jeune institutrice : ma troisième année d’enseignement vient de se boucler. Je sais, le terme de " clandestine " peut faire sourire. Pourtant, j’insiste. J’efface soigneusement le tableau quand je quitte ma classe pour qu’on ne voie pas trace de mon travail, je fais recouvrir de papier kraft les manuels avec lesquels mes élèves apprennent à lire - et que j’ai achetés sur mes deniers. Je tais soigneusement mes convictions et beaucoup de mes méthodes. Elles n’ont pas l’heur de plaire à certains de mes collègues et, en tout cas, elles répugnent franchement aux membres de l’inspection.

En fait, dès mon entrée à l’Institut universitaire de formation des maîtres (IUFM), j’ai presque aussitôt compris que je n’avais rien à en attendre. Nous avons passé en tout et pour tout six heures sur l’année à l’enseignement de la lecture et de l’écriture ! Le credo des formateurs se résumait à : " Le maître ne doit pas être un reférent pour l’apprenant [l’enfant]."

J’ai donc résolu de me comporter en reporter clandestin. De septembre à janvier j’ai tenu un journal tous les soirs, pour résumer mes journées et mes impressions.


roughly:

I am a young teacher: my 3rd year of teaching is about to end. I know, the term 'clandestine' might make people smile. However, I insist. I carefully hide the blackboard when I leave my class so that no one can see a trace of my work, I cover the handbooks from which my students learn to read - and which I bought witih my own money - with 'kraft' paper. I carefully hide my convictions and above all my methods. They're not the sort of things that will please certain of my colleagues and, in any case, they frankly repel members of the inspection team. [I gather that in France inspection teams visit classrooms to monitor quality.]

In fact, since my entrance in the teacher's college I learned almost at once that I should expect nothing from them. We spent a total of six hours in one year on the teaching of reading and writing! The creed of the trainers could be summarized as: "The teacher shouldn't be a 'referent' [probably a source of knowledge] for the student.

So I resolved to [conduct myself as a clandestine reporter ??]. From September to January I kept a diary every night, to record my days and my impressions.



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SpeakingOfTheFrenchPart2 28 May 2005 - 21:08 CatherineJohnson

Spiked (another Instructivist find) translates 'clandestine' as 'illegal.'

[update: Instructivist thinks 'illegal' is wrong. His translation of 'clandestine' in this context is 'stealth.' Diary of a Stealth Teacher.]

[update 2: My husband, who is fluent in French, says 'illegal' is completely wrong. He says 'underground,' 'hidden,' and 'stealth' all capture the meaning.]

Diary of an Illegal Teacher


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StillSpeakingOfTheFrench 28 May 2005 - 21:22 CatherineJohnson



I love this one:


lebris1.jpg

Spiked translation:
And Your Children Will Not Be Able To Read ... Nor Count!


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FrenchPrincipalSaysWakeUp 28 May 2005 - 23:53 CatherineJohnson



Et vos enfants ne sauront pas lire . . . ni compter!
Editions Stock
Avril 2004
Marc Le Bris

« Pendant vingt ans, l'Éducation nationale m'a empêché de faire mon métier. À ma sortie de l'école normale, en 1977, j'étais un jeune instituteur progressiste et militant, convaincu de la supériorité de la méthode de lecture dite "naturelle".

J'ai tout cru. J'ai tout fait, des groupes, des activités d'éveil, de la grammaire fonctionnelle, de la lecture naturelle, des mathématiques modernes, de l'animation, de l'auto-apprentissage, de l'histoire des objets, du décloisonnement, de la créativité, des études dirigées . . .

Pourtant, les élèves des maîtres plus anciens, qui osaient continuer à faire des dictées ou à apprendre la lecture par syllabage systématique, obtenaient de meilleurs résultats. Les miens, dorlotés par les méthodes modernes, ont subi un handicap scolaire dont j'ai honte aujourd'hui. Honte? Pas tant que ça... Car, comme bon nombre d'entre nous, j'ai corrigé le tir.

J'écris ce livre pour alarmer les parents, pour qu'ils sauvent leurs enfants, pour qu'ils fassent le travail de l'école à la maison. La pédagogie moderne ne sert plus qu'à justifier l'abandon des ambitions que nous avions pour nos enfants. Nous avons devant nous une véritable catastrophe culturelle. »

Marc le Bris, 50 ans, est instituteur et directeur d'école à Médréac, en Ille-et-Vilaine. Il est membre de l'association Sauver les lettres.




roughly:

For twenty years the national education system prevented me from doing my job. When I graduated from education school, in 1977, I was a young instructor, progressive and activist, convinced of the superiority of the method of teaching reading known as ‘natural.’

I believed everything. I did everything, I did groups, I did [icebreaking] activities, functional grammar, natural reading [probably whole language], the new math, student participation [l’animation = interaction], self-teaching [self-directed teaching, probably], ‘history of objects’ [l’histoire des objets], taking down the walls, creativity, directed studies . . .

However, the students of the oldest teachers, who dared to continue to do dictée* or teach reading with phonics, obtained the best results. My students, guilded by modern methods, had endured an academic handicap of which I am ashamed today. Shame? Maybe not completely . . . because, like many of us, I compensated for some of the worst excesses.

I wrote this book in order to wake parents up, so they can save their children and teach their children their school work at home. All modern pedagogy does is justify the abandonment of the ambitions we have had for out children. We have ahead of us a veritable cultural catastrophe.

Marc le Bris, 50 years old, is a teacher and principal of d'école à Médréac, en Ille-et-Vilaine. He is a member of Sauver les letters.

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*Le dictée is the classic exercise in which the teacher dictates a passage of prose and the students write it down. This was traditionally an important part of French language arts, because so many French words sound alike. My husband did it when he was first learning French, and said it was incredibly hard. All the adjectives have to agree in gender & number with the nouns, and you can’t hear any of this in the spoken language.

French is still being taught using le dictee in other countries. Recently there was an international dictee contest judged by Bernard Pivot, the famous moderator of the book review show Apostrophes.



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HowNotToTeachMath 29 May 2005 - 01:42 CatherineJohnson


While we're on the subject of illegal teaching, one of my favorite personal stories of a teacher closing the door and teaching is Matthew Clavel's How Not to Teach Math in City Journal.

Clavel was teaching in a Manhattan School that had phased in Everyday Math four years before he arrived; his fourth graders had used the curriculum since Kindergarten.


The curriculum’s failure was undeniable: not one of my students knew his or her times tables, and few had mastered even the most basic operations; knowledge of multiplication and division was abysmal. Perhaps you think I shouldn’t have rejected a course of learning without giving it a full year (my school had only recently hired me as a 23-year-old Teach for America corps member). But what would you do, if you discovered that none of your fourth graders could correctly tell you the answer to four times eight?

[snip]

Instead of rote learning and memorization, students move haphazardly from one seemingly unconnected topic to another. In Fuzzy Math lingo, it’s called “spiraling.” On this view, teachers shouldn’t use a single method to get addition across to students; they should try lots of approaches—like adding the left-most digits first. That way, the Fuzzy Math approach says, you have a better chance of getting students to understand the concept of addition. In practice, however, trying to teach a host of different methods if students haven’t sufficiently mastered any specific one—as is all but inevitable, since they haven’t spent much time practicing any specific one—can be very confusing.

[snip]

Teachers frustrated by this incoherent approach got little sympathy from school administrators. District officials told us that we should just keep going—even if not a single child in our rooms understood what we were talking about. We were going to spiral back to each topic later in the year, they reassured us.

[snip]

According to a 2000 Brookings Institute study, fourth graders who used calculators every day were likely to do worse in math than other students. But it’s minority kids like those in my class who are turning to calculators the most. The Brookings study reports that half of all black school children used calculators every day, compared with 27 percent of white school kids.

[snip]

Then there is the bizarre recommended homework. According to Everyday Mathematics, I should have assigned my students extra-hard material to struggle with at home. Here’s an example from the updated fourth-grade workbook: “Homer’s is selling roller blades at 25 percent off the regular price of $52.00. Martin’s is selling them for one-third off the regular price of $60. Which store is offering the better buy?”

Now put yourself in the place of kid who hasn’t learned how to multiply quickly, who isn’t sure about what a percentage is, and whose knowledge of fractions is meager.

[snip]

I certainly wasn’t alone in hating it. Indeed, I never heard a good word for it from my fellow teachers. At a grade conference one day, one our most respected fourth-grade teachers, a veteran who worked hard and cared deeply about the achievement of her students, summed up the general frustration with the new program: “I can’t teach it.”

[snip]

A third-grade teacher objected to the intimidating complexity of some of Everyday Mathematics’s word-heavy mandatory activities, mentioning by way of example one of her totally lost students, who could not yet read or write. I had a few students in my class who were in the same boat, so there was nothing unusual about her statement. Yet the district official, smiling, just responded, “I don’t believe you.”


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comments...

InflexibleKnowledge 29 May 2005 - 05:17 CarolynJohnston

In HowNotToTeachMath, Catherine posted an example of a fourth grade Everyday Math homework problem:

Homer's is selling roller blades at 25 percent off the regular price of $52.00. Martin's is selling them for one-third off the regular price of $60. Which store is offering the better buy?

I remember this sort of problem from last year, when Ben was in fourth grade. There were a whole series of such problems, more or less just like this. They were the sort of word problems you'd more typically see in a 7th-grade pre-algebra class; fortunately, they were all more or less the same. There was only one way to teach them, and that was to train the kids to do this sort of problem, step-by-step; what you might call by rote. I'm pretty sure this defeated the intention of the Everyday Math curriculum designers, who were trying to get the kids to think creatively about real world problems.

That's the idea behind many of the new-new math curricula. We can skip the tedium of teaching the standard algorithms, and emphasize estimation instead; we can skip teaching algebraic symbol manipulation independently, and teach algebra in the context of the word problems that adults really have to solve. Adults have to work with data, and so in the Everyday Math curriculum, there is enormous emphasis on statistics; kids start learning the median, mode and range before they are even capable of calculating the average. Calculating statistical landmarks is a topic that my son's classes have 'spiraled back to' any number of times in the two years my son has been doing Everyday Math.

And I don't think Everyday Math is even the most extreme of the new curricula: noone gets out of Everyday Math without at least knowing something about how to do multiplication and long division. I credit my son's teachers with taking the extra time needed to ensure that this was the case.

The intent of Everyday Math is to teach kids how to think flexibly about mathematics from the get-go. It's a laudable goal. But apparently it's a misguided one, because that's simply not how people learn new material.

When we're learning something completely new to us, we go through a phase where we understand the new material only in a very inflexible way; we can't generalize it very well, and we find it difficult to apply to new situations.

And that's okay. It's the way our minds work, apparently; we start out with inflexible knowledge, that we can gradually apply more flexibly as we gain more familiarity with it. That's why beginning violinists play stiffly, and why kids learning to read read small words, slowly. Inflexible knowledge isn't the same as rote knowledge, which leads nowhere; it's a necessary precursor to expertise.

This is something Catherine and I will harp on, over and over, because it's really important to understand this hard fact about how humans learn if you want to teach your little humans how to do math, or anything else.

This article from American Educator on inflexible learning, and its relation to expertise, is a must-read.



comments...

ConcernedParentsOfReading 29 May 2005 - 19:36 CatherineJohnson

Carolyn and I have not yet systematically found and posted all the parents' groups (on the to-do list, obviously).

Just yesterday I discovered that I had somehow missed Concerned Parents of Reading, so I wanted to get this link up at once.

Dr. Robert Mandell and his wife, Jackie, seem to be either founders or mainstays of the group (please correct if I've got this wrong!), and Dr. Mandell has been kind enough to send me material on the goings-on in MA--including news articles on the pilot program of Singapore Math which I've been reading about in the AIR report (this links to the press release; you can download the full report from there. The report is quite long--well over a hundred pages--but well worth reading.)

Links:
analysis of 1997 scores
local news article on math scores & Everyday Math
Suggestions of Concerned Parents of Reading, MA


+ + +


I chuckled when I saw this item on the 'Suggestions' list:

You ask the principal about test scores and are told the tests don't measure skills your children will need in the 21st century.

I had been wondering about that phrase, 21st century skills.

It's everywhere in the Singapore report.

You're reading along, growing more enlightened & inspired with each passing page, and then--BAM--you slam into 21st century skills again.

Now I know where it comes from.

TO BE CONTINUED



comments...

ParentPundit 29 May 2005 - 21:14 CatherineJohnson

Carolyn just spotted an incredibly kind post about KTM at Parent Pundit.

Neat!

Parent Pundit also has a number of posts on Everyday Math, which is her daughter's math curriculum, as well as discussion of an online tutoring program I had never heard of: ALEKS A Better State of Knowledge.

Parent Pundit's daughter has just moved to the advanced math class in her school, so I'm going to check out ALEKS right away (maybe for my own use at some point).

Her story of discovering that her daughter had fallen behind in math knowledge while getting A's in her math classes is here: If your school has Everyday Math.


dingbatWSJ2.jpg


a parent's experience with ALEKS
ALEKS Graphic
formative assessment on wheels
ParentPundit uses ALEKS to fix Everyday Math
ALEKS question
ALEKS assessment coming right up





comments...

BadInternetDay 30 May 2005 - 03:38 CarolynJohnston

Service from Comcast has actually been intermittent for the last couple of days, which is very frustrating. It's Memorial Day: what better time to be on the internet?

Sadly, though, it seems a lot of people share my vision of the perfect holiday weekend.

I do want to sneak this post on, however. Parent Pundit's article on Everyday Math, to which Catherine linked a couple of posts ago, is the best short summary of objections to the Everyday Math curriculum that I've ever seen.

I don't want to rant about Everyday Math indefinitely -- my main goal on KTM is to collect and share useful methods and ideas for teaching math, and there are, incredibly, even crazier math curricula to target. So ParentPundit's post will stand as the absolute last word on the failings of Everyday Math, as far as I am concerned.

And don't fail to check out her list of supporting links at the end of the post, especially if you're looking for ammunition to prevent an Everyday Takeover.



comments...

HappyMemorialDay 30 May 2005 - 11:59 CatherineJohnson




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Memorial Day 2005
Memorial Day 2006
Leigh Ann Hester





comments...

BlameTheTeacher 30 May 2005 - 18:24 CarolynJohnston

Reading over ParentPundit's post about Everyday Math, I encountered the following in the comments section, left by aschoolyardblogger. It's an argument one frequently hears to counter parents' and teachers' complaints about reform curricula.

It is a difficult task for teachers to begin any reform mathematics projects - their own math learning at first is being tested and reformed. One of the key ingredients, in my mind, is support provided through teacher training, but almost and maybe more important is the support of parents. One way to understand a math program like EM is to read through and do the exercises in the curriculum consecutively, openmindedly as a learner, not a an assessor. Play with the manipulatives, perhaps even borrow a teaching guide. These programs are much different, and much more exciting than the way we were taught. They are also very hard to describe. With some study, you might find yourself a great parent contributor to something your children's school is attempting to perfect.

Open your mind, Grasshopper: play with the manipulatives. Wax on, wax off.... I think teachers (and parents) need some sticking up for.

Math itself doesn't change much, and neither do people. Teachers who know how to teach math weren't invented by new curricula (for that matter, reform math curricula aren't a new invention, either). Nor have the rare teachers who take pleasure in humiliating children been stopped by the adoption of new curricula.

The truly exceptional teachers aren't the ones who need a supportive curriculum most; they can always roll their own. The whole purpose of a curriculum is to guide the process of teaching and learning for the majority of people. To argue that a curriculum fails only because of the failings of the teachers who must implement it is specious -- like arguing that Communism fails only because of the fallible people who must implement it.

Not to mention that the argument is insulting. God, teachers must get sick of these insinuations that their understanding needs 'reforming'. I know that parents do.

Learning to be a good teacher of math, like learning math itself, is very challenging. There is a depth of domain knowledge and pedagogical understanding that one can acquire over the course of a career in mathematics education; this pedagogical understanding should be what guides a teacher's explanation of mathematics in the classroom, not a 'Teaching Guide'. Only a teacher with a flexible approach that comes from deep understanding can come up with the fifth explanation that meets the needs of an individual child, when the first four have failed.

I've noticed that there are topics where Everyday Math does not offer cool new teaching methods, and they tend to be the topics that have always been difficult to teach: for example, division by fractions. These things are difficult to teach and understand because, well, they just are, no matter whose method you're using.

A math curriculum should be the foundation of a kid's math education. A teacher who has an exciting activity to try can supplement a curriculum, but the curriculum should provide enough guidance to ensure that the ground that needs to be covered, gets covered. The cool techniques that Everyday Math uses to enhance understanding can then serve as grace notes.

And it may sound absurdly pedestrian, but the second valuable thing that a good math curriculum can provide is a good set of problems for the children to work. A good problem set design is worth its weight in gold. Saxon has one. I'm not always crazy about Saxon math's explanations of methods, but its problem set is awesome.

A teacher who is motivated to try to acquire and pass on what Liping Ma refers to as Profound Understanding of Fundamental Mathematics -- and who is respected for trying -- can and must provide the rest.



comments...

GreatMindsThinkAlike 30 May 2005 - 21:15 CatherineJohnson

Good Grief.

Apparently Carolyn and I were not the first people on the planet to think up 'Kitchen Table Math'.

(Scroll down.)



comments...

ProfoundUnderstandingFundamentalMathematics 30 May 2005 - 21:26 CatherineJohnson



p200047e2g15001.jpg


Carolyn mentioned Liping Ma's concept of 'profound understanding of fundamental mathematics' (PUFM).

This chart is Ma's map of the 'knowledge package' Chinese teachers possess for the topic of subtraction. This is what Chinese mathematics teachers know and understand about subtraction.

I don't happen to have this knowledge package inside my own head, and neither does any other parent I know.

This is why it won't do to say:

One way to understand a math program like EM is to read through and do the exercises in the curriculum consecutively, openmindedly as a learner, not as an assessor. Play with the manipulatives, perhaps even borrow a teaching guide. These programs are much different, and much more exciting than the way we were taught. They are also very hard to describe. With some study, you might find yourself a great parent contributor to something your children's school is attempting to perfect.


+ + +


Chinese math teachers develop pedagogical content knowledge over the course of many years teaching and studying elementary mathematics.

There are no shortcuts.

How long does it take to acquire a profound understanding of fundamental mathematics?

I'm guessing 10:

Some evidence that a great deal of practice, and not just talent, is a prerequisite for expertise is the "ten year rule," which states that individuals must practice intensively for at least 10 years before they are ready to make a substantive contribution to their field. What about prodigies like Mozart, who began composing at the age of six? Prodigies are very advanced for their age, but their contributions to their respective fields as children are widely considered to be ordinary. It is not until they are older (and have practiced more) that they achieve the works for which they are known.


+ + +


No parent is going to pick up a copy of Everyday Math, read through the book, work the exercises, and be ready to teach or tutor the curriculum effectively.

That's not the way it works.

Parents have a fighting chance of teaching or tutoring effectively with a direct-instruction curriculum like Saxon Math. We have that chance because the books are written so that anyone who's been through grade school can understand what the lessons are about.

None of us is going to do a brilliant job teaching math using Saxon. Becoming brilliant at anything takes 10 years.

But we can help our children learn math.

It's not just children who need direct instruction. Parents need it, too. We parents need to be able to pick up our child's mathematics textbook, read the lesson, and know what it's talking about.

That school districts consciously select unproved mathematics curricula they know parents will not understand and will not be able to teach or tutor from is, to me, unconscionable.

It's not up to us to go begging for a peek at the teacher's guide.

It's up to our schools to bring us into the loop.



comments...

TeacherGuideEverydayMath 31 May 2005 - 00:12 CatherineJohnson


Wow.

Speaking of sneaking a peak at the teacher's guide, it just so happens that I have open, on my desktop, a bunch of pdf files from the Everyday Mathematics Teacher's Reference Manual, Grades 4-6, The University of Chicago School Mathematics Project, Everyday Learning Corporation, Chicago, IL, 1999, ISBN 1-57039-515-2, pages 127-139, courtesy of one Tsewei Wang, Ph.D., Associate Professor, Department of Chemical Engineering, University of Tennessee and Concerned Parent.

Have I mentioned how much I love the internet?

Interesting to see that Everyday Math teaches the same Guess-and-Check algorithm for long division that's in Trailblazers.

Only, Trailblazers calls it 'Forgiving Division' (pdf file; search for 'forgiving division'):

Forgiving Division Method
(URG Unit 4 pp. 5, 6, 53; SG p. 113)

A paper-and-pencil method for division in which successive partial quotients are chosen and subtracted from the dividend, until the remainder is less than the divisor. The sum of the partial quotients is the quotient.


+ + +


So say you're dividing 239 by 3.

Instead of using math facts to know that 3 goes into 23 seven times, you start by guessing how many times 3 goes into 239.


+ + +


OK, let's divide 239 by 3 using forgiving division!


spongebob_1.jpg 'I'm ready!'



I'm going to start by guessing the number . . . 7!

I guess 7!

3 x 7 is . . . 21!

I write down 21 underneath 239, then I subtract, and I get . . . 218.

Whoa.

That's a lot.

OK, I'm going to use a strategy.

I'm going to guess . . . 10, because 10 is a friendly number.

10 x 3 is . . . 30!

I write 30 underneath 218, then I subtract----188.

Wow.

188 is big.

OK. 188. I'm down to 188.

. . . I'm going to try 10 again.

10 x 3 is 30, subtract 30 from 188, get . . . 158.

158?

bsg%20confused.jpg

Wait.

Wait.

I'm lost.

What number am I down to?

Oh. 158. I'm at 158.

OK, I'm going to try 20.

20 x 3 is 60, subtract from 158, get . . . 98.

Oh good! 98! That's really good! 98 is below 100!

Maybe I could try 30 this time.

30 x 3 is 90, subtract from 98, get 8!

Fantastic!

8!

8 is a really friendly number!

Now I can use my math facts and find that 8 divided by 3 is 2.

2 x 3 is 6, subtract from 8, get 2; 2 is less than 3, I'm done!

Yay!

Finally!

Now I add up all my partial quotients and the answer is------

7 + 10 + 10 + 20 + 30 + 2 = 79 remainder 2.

79 remainder 2!

That's the answer!

That's it!

All done!

Bye Bye!

The end!




Forgiving Division

see:
The Many Faces of the Bitter Single Guy

and:

BlameTheTeacher
ProfoundUnderstandingFundamentalMathematics
ForgivingDivision
ForgivingDivisionPart2
TryThisWithForgivingDivision
ILoveTheWorldWideWeb
TeacherGuideEverydayMath
EverydayMathEpilogue
ThirteenQuartersInTerc
HowNotToTeachMath
AboutLongDivision
StrugglesWithLongDivision
MathInTheBlood
WhoSaysLongDivisionIsHard
Everyday Math alternate division algorithm

keywords: Sponge Bob Bitter Single Guy




comments...

AssessYourChildForFreePart2 31 May 2005 - 15:39 CatherineJohnson



I've just added this post to ThingsWeHaveLearned:


David Klein developed these Practice Problems for the California Mathematics Standards Grades 1-8 for the Los Angeles County Board of Education.

For me, these problem sets are precious. That is none too strong a word.


And here's Carolyn, in an email to David:

It's wonderful that you put together those assessment questions. Those practice problems are golden. One of the most difficult things for a parent to do is to get a solid idea of what kids ought to know -- what it means for them to be on track. CA's state standards are good, but too dense. There could be nothing more succinct than a set of problems that the kids must know how to do, year by year. Kitchen.Catherine and I want to post links to them front and center, and to continually refer parents to them (because, of course, repetition is key :)).


"Golden" is right. A consultation with an educational psychologist can run into the thousands of dollars.

If you suspect that your child has specific learning problems, wrangling a consult from your school may be a very good idea.

But if your question is simply: where does my child's math achievement stand today? then these grade-by-grade problem sets are all you need (I think) to find your answer.

At least, they've worked for me and Christopher.


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AssessYourChildForFree
DontRelyOnStateTests
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comments...

ATeachersStory 31 May 2005 - 20:55 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.)


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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.




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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.


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comments...

TakingABreak 31 May 2005 - 21:38 CatherineJohnson

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