Kitchen > PrivateWebHome > SubjectArea > TrailBlazers

select another subject area

# Entries from TrailBlazers

CompareAndContrast 10 Oct 2006 - 01:52 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?

CompareAndContrastPart2
CompareAndContrastPart3
CompareAndContrastPart4
CompareAndContrastPart5
CompareAndContrastPart6
CompareAndContrastPart7
MathInSalinaKansas

PracticePracticePractice
ATeacherUsingTrailblazers
BigNumbers

TeacherGuideEverydayMath 07 Oct 2006 - 13:19 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!

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?

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 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
StrugglesWithLongDivision
MathInTheBlood
WhoSaysLongDivisionIsHard
Everyday Math alternate division algorithm

keywords: Sponge Bob Bitter Single Guy

HowToGetParentBuyIn 10 Oct 2006 - 02:01 CatherineJohnson

The TRAILBLAZERS teachers' guide devotes a number of sections to strategies for neutralizing incensed parents.

I had planned to quote some of these passages, and then, tonight, found an online TRAILBLAZERS document (PDF file) that's chock-ful of them:

Be pro-active with parents. Don’t wait until complaints hit. People have done a lot of things to involve parents, from math nights to big math carnivals, where the kids teach the activities to the parents. There are letters in the program that go home to parents.

When this teacher says 'there are letters in the program that go home to parents,' she doesn't mean that her school writes letters to parents once a month.

She means that her school has purchased, as part of the TRAILBLAZERS 'package' (which is enormous, I've seen it; worse yet, I've lifted it) a set of special TRAILBLAZERS Dear-Parent letters to be photocopied and sent home in the backpack at regular, designated intervals.

What the parent sees is a friendly letter from the school about her child’s math program.

What the school sees is a professionally-developed public relations campaign targeted to dissenting moms & dads.

The TRAILBLAZERS Dear Parent letters are not intended to serve an educational purpose. At least, no educational purpose is mentioned in any of the supporting materials I've seen as yet.

The explicit and openly stated purpose of the TRAILBLAZERS Dear Parent letters is to promote parent buy-in.

All of which means that not only am I paying for a program I don't like and don't want, I am paying for the press kit to persuade me I'm wrong. Maybe this isn't exactly the kind of thing the Boston Tea Party was about, but it's getting there.

+ + +

And here is another strategy for dealing with parents!

This strategy was developed by one Barbara Martin, principal of the Holmes Elementary School in Chicago:

[For parents] we do also have a math day, and on that math day, we invite parents to be in the room. The kids do math all day. In order to get the parents in the room, I offer them a little stipend. I only offer the stipend to the parents who can stay in the room all day—they’re helping the teacher, because they’re doing math all day, with Trailblazers and all the manipulatives. At the same time, they’re also getting to see what kids do. There are other parents that visit math day and leave because they can’t stay all day. We have a good turnout.

Ms. Martin has had fantastic success with TRAILBLAZERS ---

"For some of my children, our feeder schools are saying, “Please, please send us more like these.”

+ + +

So let's see how Holmes Elementary School children are faring in the high-stakes world of standardized testing.

+ + +

Oh dear.

Third grade: 30% of the kids meet state standards.
Fifth grade: down to maybe 27%.
Eighth grade: down to 5% meeting state standards.

This is an all-black, poor school, so they've got a lot to contend with. Maybe they'd have a 95% fail rate in 8th grade no matter what curriculum you gave them.

But look at their reading scores.

3rd grade: maybe 17 or 18% meet standards.
5th grade: up to 36 or 37%meeting standards.
8th grade: they're up to around 44% meeting standards.

Math goes down, reading goes up. Same kids, same school, same period of time.

EverydayMathDoesItToo
ILoveTheWorldWideWeb
ATeacherUsingTrailblazers
NoCommentPart2
CarolynFisksBook
AnotherGemFromMathForum
BigNumbers

ATeacherUsingTrailblazers 10 Oct 2006 - 02:01 CatherineJohnson

One of the things that I’ve learned is what homeworks are good homeworks to send home and what homeworks we really need to do in class because of parent frustration. Last year, not yet knowing this, I sent a homework home and got back such venomous mail: “What is this? Why are you asking my 3rd grader to do this? If you ever send another magic square home, I am pulling my child out of the school. I can’t do this, and neither can he.” So now I’m just making better choices on what to send home.

I think we can all agree that it's important for teachers to make good choices (pdf file).

But why any parent would object to an 8-year old child being asked to construct a magic square for homework is beyond me.

After all, think how much conceptual knowledge that child will have after his mom has looked up Magic Squares on the internet and helped him draw one.

EverydayMathDoesItToo
ILoveTheWorldWideWeb
CarolynFisksBook
AnotherGemFromMathForum
BigNumbers

ILoveTheWorldWideWeb 18 Nov 2005 - 16:40 CatherineJohnson

I knew if I just kept looking I'd find them.

Somebody would have made helpful pdf files of all the TRAILBLAZERS PARENT LETTERS and posted them on the web.

NoCommentPart2 23 Nov 2005 - 15:59 CatherineJohnson

Getting Your Message Out to Parents

EverydayMathDoesItToo
ILoveTheWorldWideWeb
ATeacherUsingTrailblazers
AnotherGemFromMathForum

CompareAndContrastPart6 10 Oct 2006 - 01:53 CatherineJohnson

## math facts in Singapore, grade 3:

Studying Exhibit 3 in the big Singapore Math Report (pdf file), we learn that:

Singapore students master multiplication tables up to 10 x 10 in grade 3

## math facts in Math Trailblazers, grade 5:

To be honest, it's difficult to say what, precisely, the MATH TRAILBLAZERS schedule actually is. It seems to vary from one document to another.

I did find this TRAILBLAZERS playlet on page 260 of the 5th grade TIMS Tutor: Math Facts (pdf file).

Suzanne: But the facts with nines are harder. I have to think about them, but I use the tens to make them easier.

Teacher: How, Suzanne?

Suzanne: Well, when I see 15 – 9, I think, “What do I need to get from 9 to 15?” I use counting up: from 9 to 10 is 1 and from 10 to 15 is five more. So, I get 6.

update 11-2005

I talked to a friend whose son is in second grade. He's a brainy kid who loves math, but he can't use the addition algorithm. Apparently, the algorithm hasn't been taught. If he's adding numbers smaller than 20, he counts on his fingers and toes. If the numbers are larger than 20, say 12 + 19, he draws 12 circles, then 19 circles, and finally counts them. Same process for subtraction, only in reverse. 63 - 19 means drawing 63 circles, then crossing out 19 of them.

The kids have the triangular flash cards that portray number families, and her son is working on flashcards with numbers 1 - 10. A friend of hers whose child is in 3rd grade told her the children in her child's class are working on the exact same cards.

CompareAndContrast
CompareAndContrastPart2
CompareAndContrastPart3
CompareAndContrastPart4
CompareAndContrastPart5
CompareAndContrastPart7
MathInSalinaKansas

TitlesOfConstructivistMathCurricula 19 Jul 2005 - 01:46 CatherineJohnson

Jo Anne Cobasko has taken the time to construct a complete list of NCTM standards based math programs.

### update: Department of Corrections

This list is David Klein's handiwork, not Jo Anne's.

Thank you, David! (For everything you do.)

All of us should keep this handy, because none of these programs ever calls itself constructivist, and schools don't seem to advertise this piece of information, either.

When I first raised the issue of TRAILBLAZERS being a constructivist curriculum with a teacher on the textbook selection committee, she looked at me blankly. I got a number of those blank looks before I discovered that everyone in the school knows what the word constructivism means, and knows what a constructivist curriculum is.

The reason I know this is that I finally read the original committee report, which states explicitly that the new curricula must have a constructivist approach with modeling. I was a little behind the curve there.

### Elementary school

Everyday Mathematics (K-6)
TERC's Investigations in Number, Data, and Space (K-5)
Math Trailblazers (TIMS) (K-5)

### Middle school

Connected Mathematics (6-8)
Mathematics in Context (5-8)
MathScape: Seeing and Thinking Mathematically (6-8)
MATHThematics (STEM) (6-8)
Pathways to Algebra and Geometry (MMAP) (6-7, or 7-8)

### High school

Contemporary Mathematics in Context (Core-Plus Mathematics Project) (9-12)
Interactive Mathematics Program (9-12)
MATH Connections: A Secondary Mathematics Core Curriculum (9-11)
Mathematics: Modeling Our World (ARISE) (9-12)
SIMMS Integrated Mathematics: A Modeling Approach Using Technology (9-12)

### Programs explicitly denounced by over 220 Mathematicians and Scientists:

Cognitive Tutor Algebra
College Preparatory Mathematics (CPM)
Connected Mathematics Program (CMP)
Core-Plus Mathematics Project
Interactive Mathematics Program (IMP)
Everyday Mathematics
MathLand
Middle-school Mathematics through Applications Project (MMAP)
Number Power
The University of Chicago School Mathematics Project (UCSMP)

printable page

Thanks, Jo Anne, for taking the time to do this!

key words:
DavidKlein
listofconstructivisttextbooks
constructivist textbooktitles
NSFfundedcurricula

MathInIrvington 10 Oct 2006 - 01:51 CatherineJohnson

Just got back from picking up Christopher's other school supplies from the store at the Middle School.

While there I debriefed a high school girl about the math track at Irvington High School.

The Irvington math track is something parents know essentially nothing about unless they do things like debrief high school kids at the school store. There's no brochure; there's nothing on the web site. It's a secret.

OK, it's not a secret. My problem is I don't see why I have to work to find out what the math track is in my own school district.

I've mentioned more than once that for a variety of reasons Irvington grade school ended up with 4 math tracks starting in 3rd grade, a situation no one inside the school liked or ever intended to create. They started with the idea of an enrichment program for the best math kids, then one thing led to another, and they ended up with four math tracks.

At the beginning of 3rd grade Christopher was placed in 'Phase 3,' one step down from Phase 4, the most advanced track. He was 8.

We had no idea what Phase 3 meant, and we were never told. We just thought.....well, I don't know what we thought. At some point I realized they were hitting the Phase 4 kids with a lot of Math Olympiad problems the kids couldn't do. Often the parents couldn't do them, either. Apart from that, both phases were using the same textbook (SRA Math) and moving through it at basically the same rate.

Giving kids a lot of Math Olympiad problems they couldn't do seemed like a waste of time (and in fact is a waste of time), so I didn't worry about it.

At the end of 4th grade we were told, directly, by Christopher's 4th grade teacher: 'Don't worry about the phases. They don't make any difference. All the kids have the same ability.'

Because of the funky way the Phases evolved in the first place, she was probably right that there wasn't a significant difference in ability level, so we took her word for it that there was 'no difference' between Phase 3 and Phase 4.

Then, at the beginning of 5th grade, we showed up for school and discovered that, lo and behold, the Phase 4 kids were using the 6th grade book. Phase 3 kids were using the 5th grade book.

All of a sudden this difference that was not a difference was a difference of one year.

That's the back story.

The point is: none of us parents knew, back in 3rd grade, that all but the Phase 4 kids had just been tracked out of calculus in high school.

We had no idea. Zero. Christopher was 8; we were one year out from 9/11 and 10 months out from the anthrax attacks. (We lost our TONYSS tests that year because they went through one of the anthrax post offices. So we didn't know how he'd done on the state tests.) We weren't thinking about high school calculus.

This is not the way a school district should work.

Track a kid out of high school calculus in the 3rd grade and not tell the parents?

That's not the social contract I thought I was signing when we moved here.

So today I debriefed this girl.

Like Christopher, she was placed in Phase 3. Then, at some point, she 'turned out to be good at math.'

This was not discovered until her freshman year in high school, it seems. A week from now, when school starts, she'll be joining the honors track.

To jump tracks, she had to spend her entire summer taking math at Rye Country Day School, which I'm sure cost an arm and a leg. She also had to get permission from the high school; she had to petition them to move her to the honors track this fall.

When I got home and figured out exactly how much ground she had to make up in one summer, I was stunned. The advanced kids are about a year and a half ahead of everyone else, which means she had to take and master all the math those kids have been taking and mastering for the last 2 years. And she had to do it in 8 weeks.

She said it was torture. She was up at 7 am every day doing math 'til she went to bed. I'm impressed as heck that she did it, but in my view it's pedagogically unsound, and she should not have been put in this position in the first place.

Worse yet, my own experience is that you can't cram math. You need time for math to sink in. Unless you're a natural born whiz, you need to be doing math every day, and living with it.

And, of course, we know from years of research on learning & memory that crammed knowledge disappears rapidly. (See Practice Makes Perfect But Only If Your Practice Beyond the Point of Perfection.)

I think it's extremely unlikely that her parents knew, when she was put in Phase 3 math, what kind of heroic effort it would take for their daughter to get back out of Phase 3 math.

I know for a fact that none of the parents around me have any idea Phase 3 means no calculus in high school.

The incredible thing is, they still don't know.

I made noises about this all last year, to anyone who would listen, which apparently did some good, because some 5th-grade parents raised the question in get-together meetings with the Middle School principal. By the time Ed & I went to our own get-together, on the last available date, the principal told us that parents at the other meetings had been asking whether their Phase 3 kids would be able to take calculus in high school. He acted surprised anyone would ask such a thing.

Then he said Phase 3 kids wouldn't be able to take calculus in high school, at which point the vice principal jumped in and said, Yes, they would be able to take calculus if they wished.

And there we left it.

That is not what I call Information. The principal says no & the vice principal says yes.....and that's an answer? That's it? They've had 3 weeks since the first get-together to figure it out and they still don't know?

And if the principal & vice principal of the middle school don't know whether a Phase 3 kid is on track to take calculus in high school, how am I supposed to know?

After the meeting, I was thinking the vice principal was more likely to be right, because she's been here awhile and the principal is new.

But no.

The principal was right.

Phase 3 kids are not going to be taking calculus in high school unless their parents sign them up for a brutal summer of 12-hour a day algebra & geometry catch-up 4 years from now.

Of course, now that Trailblazers is coming in and tracks are going out.....it'll be interesting.

I own the 5th grade Trailblazers book, which is the final book in the series. I've read it.

I don't see anyone coming out of Trailblazers on track to take calculus in high school.

UPDATE 10-9-2006: Based on what I hear from other parents, the tracks seem to have been preserved. It's possible the administration finally looked at the calculus track and realized they'd abolished it. I surmise this because two years ago parents of mathematically gifted children were pressing Raph Napolitano, the Assistant Superintendent in charge of curriculum, for an answer to the question of whether their children would be able to take calculus in high school. He didn't know. That was his answer. He didn't know whether mathematically gifted 3rd graders taking Math Trailblazers would be able to take calculus in high school. That's typical of this district. Parents are given no syllabi, no scope and sequence, no topic matrix. Unless we debrief other parents and their children we have no idea what lies ahead, or what our children need to know today to be prepared for advanced high school courses tomorrow. It takes many weeks and many emails and telephone calls to get a simple answer to a simple question. So I could be wrong about the tracks. Maybe we have them; maybe we don't.

UPDATE 10-24-2006: A friend whose child is in 4th grade says the tracks are gone. I have no idea what's going on.

### question about calculus and college

The girl I was talking to says her brother has the impression that colleges want to see 'BC calculus' on kids' high school transcripts.

Is that true? (He's applying to the Ivies.)

My close friend in CA says that all colleges now require kids to take calculus....(her son is a freshman at Occidental). So either you need to have taken it in high school, or you'll have to take it in college.

James Milgram on long division & time lag in math learning
James Milgram statement to Congress

key words: summer school cram cramming math cram math sophomore Irvington High School freshman

SusanOnMadMinutes 13 Nov 2005 - 18:34 CatherineJohnson

Susan used Mad Minutes with both kids--

I agree about the "mad minute" approach for all levels of math ability. I used it with both kids and I'm glad I did. It just helps with the speed and proficiency.

Gifted kids are notorious for not wanting to memorize anything. It's too boring. I had to make mine do speed drills all through first and second grade. I'm still doing it with my LD 8th grader.

I remember when math kid was learning multiplication in the first grade. He informed me that it was stupid to memorize them because he could always just count the the groups. I said, "Quick, what's 7 X 8?" His eyes went up looking for those groups, and after a couple of seconds of his looking for them and then trying to skip count by 7, I said, "That's why you memorize them."

I think it helps in the confidence department, as well, to get that speed going earlier than if it just came through practice. And if practice is the key, like "exposure" to lots of words was to the whole language crowd, then what exactly happens to the kids who don't get enough of that practice? Where is the place where enough practice crystalizes into math fact proficiency, especially with these "spiraling" curriculums that keep pushing mastery on down the road?

Trailblazers, as well as other NCTM curriculums, never seem to have a plan for the ones left behind. And I guess you can't ever know who is accountable since mastery was never the goal to begin with.

Interestingly, both kids do love the minute drills where they try to beat their last best score, so I never have any trouble giving them to them.

Well, that's two.

I have to say....I simply can't see any reason on the planet why worksheets would be bad (though TRAILBLAZERS explicitly states that worksheets are destructive!)

Given that there's no (apparent) downside, and given that they've worked for other kids, I'm certainly going to be using them with any child whose math education I'm involved in. (I'm starting to pick up a few! It's incredibly fun. More on that later.)

### modifying worksheets

That reminds me.

One of the kids who took my Singapore Math class just could not get through a Saxon work sheet--and he didn't improve any over time, either. All the other kids got fast fast. (It really was remarkable.)

This particular boy is BOUNCY; he is one high-energy kid. He just can't stand the thought of a 5-minute worksheet; he probably takes one look at those sheets and sees what I would see if I were contemplating singlehandedly painting a two-story house.

I told his mom: try having him do ONE LINE of the worksheet as fast as he can.

I don't know if she'll get to it, but when they get back from Italy in January, I think I'll experiment with him, and see how he does. Just click the stop watch and tell him to call 'TIME!' when he hits the end of one line.

I bet that will work.

EverydayMathLongDivision 13 Sep 2005 - 15:06 CatherineJohnson

Thanks to NYC HOLD I have a graphic of Everyday Math's substitute division algorithm. TRAILBLAZERS teaches the same approach, which it calls 'forgiving division.'

...instead of teaching long division, students are taught to divide numbers using the partial products method, a technique where children guess how many times a number goes into another and keep subtracting the guesses until they come up with the answer (see box). This method works, but it takes more time and doesn't allow the student to divide past the decimal point.

[snip]

Isaacs and others defend the alternative algorithms by explaining that they teach students how math works. The partial product method of division, for example, is a lot more transparent to students than the long division method.

I'm sure he's wrong about this. I found partial product division quite confusing myself when I used it.

otoh, I think partial product division might work as a teaching tool when used on simple demonstration problems. (I tried it on a complicated division problem and got completely lost mid-stream.) I might use a problem like 16 divided by 2 to show that division is repeated subtraction, analogous to multiplication being repeated addition.

I haven't tried it with any children just learning long division, but if I ever get a chance to, I'll take notes.

### the honeymoon

Some parents like the program as well. "It's sort of incredible," said Susan Pottinger, whose son Theo attends kindergarten at P.S. 261 in the Cobble Hill section of Brooklyn. "For him it's great fun. He's fascinated by numbers. He sees patterns everywhere," she said. "He'll put shoes away and alternate shoes with sneakers and say, 'See I'm making a pattern with my shoes.' "

We parents (well, some of us) spend those early elementary school years in a wonderland. Then the you-know-what hits the fan in 5th grade.

source:
Weighing the Factors Does the City's Standardized Math Curriculum Measure Up? By Amy Sara Clark

### update

Lone Ranger supplies this link to lattice multiplication, the method Everyday Math teaches children when they cover multiplication. Carolyn points out that lattice multiplication is distinctly opaque; it obscures rather than reveals the fact that multiplication depends on the distributive property.

Here's another link to lattice multiplication at Math Forum Carolyn posted awhile back.

why long division?

Everyday Math's alternative division algorithm
forgiving division
forgiving division, part 2
try this with forgiving division
who says long division is hard?
Everyday Math division algorithm fighting innumeracy at CO
conceptual understanding vs numbers

keywords: Columbiajournalismstudent EverdayMatharticle

WickelgrenOnYoungChildrenAndMath 17 Sep 2006 - 01:14 CatherineJohnson

back story:

My neighbor, the statistician, showed me her copy of Math Coach: A Parent's Guide to Helping Children Succeed in Math quite awhile back, before either of our kids had had any trouble in math class. I ordered a copy just because I order lots of copies of books I'd like to read but then don't.

So the book was sitting there on my shelf when Christopher came home with his 39 on the Unit 6 test & I subsequently failed to teach him fractions using SRA Math. I needed help.

It was the right book at the right time. A page-turner.

Most of what I believed to be true of math ed & math achievement, I discovered, was wrong. Severely wrong. I had been operating on the basis of sheer ignorance, naivete, and boneheaded cliche.

This is the observation that probably shocked me the most. It appears in Wickelgren's chapter on finding a school for your child:

There are schools with even less structure than Eastside. Take the Sudbury Valley School, a private K-12 school in a Boston suburb. This school gives each child complete freedom to choose how they spend their time at school. There are no classes except those specifically requested by a group of students. Children learn largely on their own, reading books, talking to each other and to teachers or outside experts, solving problems, playing games and sports, practicing musical instruments, doing arts and crafts, and anything else that can be done on the school grounds.

While you can read at length about the school's strengths on its web site, one of its biggest potential benefits is that every child can proceed at his or her own pace, in math and in other subjects as well.

There are also potential drawbacks. Since young children are not generally highly motivated to learn math, they may choose not to study much of it.

I was bowled over.

I had always thought kids want to learn things they're good at. Christopher is good at social studies, and he wants to learn it. At night he'll bug his dad to 'give me trivia questions.' (Give me superficial facts, Daddy!) Ed finally refused to do it anymore, because he ran out of trivia.

Christopher also has a collection of geography trivia books that he reads, and when he was 7 I read all of the first volume in the History of US series out loud to him as his bedtime story.

That was the book he wanted to hear.

So...I assumed kids wanted to learn subjects they had a talent for.

According to Wayne Wickelgren, this is not the case with math.

Or, at least, not generally. Math talent doesn't (necessarily) manifest itself in an obvious desire to learn the multiplication tables. (Or to write essays on My Special Number.)

### late bloomers

That one observation pretty much changed my life. I decided, then and there, that I didn't know whether Christopher had any talent for math or not, or what his eventual level of interest in the subject might be--or, more importantly--could be, given a decent education K-12.

I also knew he had good general intelligence, which meant he had the ability to learn a whole lot of math whether he was going to end up in a math-related career or not.

I decided right then and there that that was what was going to happen. Christopher was going to learn math, lots of it, and learn it well.

We were going to keep the doors open.

When Christopher reached college, he would be in a position to decide to pursue a math-related career or not. That decision would not have been made for him in 3rd grade, when he got sorted into Phase 3.

It wasn't too long after this that I met Carolyn and heard her story: flunked algebra in high school (right?), didn't decide to major in math until senior year in college, then got a Ph.D. In math. Another wake up call.

### more late bloomers

Two more stories.

One comes from Christopher's 4th grade teacher. Her daughter was reaching the end of high school, and it was time to do SAT prep.

So her mom hired a tutor, and within a couple of weeks the guy was reporting that her daughter had strong talent in math.

She had no idea. Neither she nor her daughter had the first clue that this kid had a knack for math. Now, working one-on-one with a tutor who, IIRC, had a Ph.D. in math (or engineering, possibly) she was flying.

I have no idea where that girl will end up, what she'll major in, or which job or career she'll pursue.

It doesn't matter. The point is: she's good at math, and she went through 11 years of formal education thinking she wasn't.

### you can't predict the future, or even the past

Story number two comes from a friend of ours. As a boy he had two or three chums who sat by each other in class & were bright kids. They were the kind of kids who could learn whatever you threw at them, and they got As in all their subjects & went to good colleges & universities. They got As in math, too, of course, but none of them was a whiz. Our friend became a lawyer.

One of the gang shocked everyone by growing up to become a world-famous econometrician.

No one can understand how this happened. This kid never showed any special talent for or interest in math. He was just a smart kid, like the rest of them. Our friend said that to this day, whenever any of them get together, they always ask each other how that friend could turn out to be not only an econometrician, but a world-famous one.

Go figure.

What I like about this story is the fact that not only could this boy's future as World Famous Econometrician not be predicted when he was 8, it can't be back-predicted now, when he's 40.

### Barbara Oakley's bio

I just remembered: Barbara Oakley is in the same category. Here's her bio:

I started studying engineering much later than many engineering students, because my original intention had been to become a linguist. I enlisted in the U.S. Army right after high school and spent a year studying Russian at the Defense Language Institute in Monterey California. The Army eventually sent me to the University of Washington, where I received my first degree–a B.A. in Slavic Languages and Literature. Eventually, I served four years in Germany as a Signal Officer, and rose to become a Captain. After my commitment ended, I decided to leave the Army and study engineering so that I could better understand the communications equipment I had been working with.

Barbara sent me an email that I won't quote without her permission (I'm WAY behind on email). But her story inside an email is more dramatic than her story here, though no different in outline. Barbara is a person who earned an entire B.A. degree in a humanties field and served a full stint in the Army before figuring out she wanted to major in engineering.

And the reason she decided to study engineering is pretty similar to the reason I've suddenly decided to study math; she got tired of not understanding the stuff she was working on. In her case, that was communications equipment; in my case it's K-12 math.

Obviously, Steve H is right, we simply cannoy be assigning grade school kids to our two Standing Committees: math whiz & math's not his thing.

### all English Language Arts all the time

from The Learning Gap by Harold Stevenson and James Stigler:

....American teachers like to teach reading; Asian teachers like to teach mathematics. When we asked teachers in Beijing, nearly all of whom were women, the subject they most liked to teach, 62 percent said mathematics, 29 percent said language arts. The reverse was found in Chicago: 33 percent mentioned mathematics and 47 percent mentioned language arts. There is more to the story than preference, however. Americans simply emphasize reading more than mathematics. Despite the large amount of time already spent in reading instruction, more than 40 percent of the suggestions made by Minneapolis mothers who wanted an increased emphasis on academic subjects said they thought that the subject should be reading. Fewer than 20 percent mentioned mathematics.

These data lead to the obvious conclusion that American children do less well in mathematics than do Chinese and japanese children partly because they spend less time studying mathematics....Conversely, American children may fare better in reading, relatively speaking, because they spend more time on this sujbect.

I mentioned yesterday: it's a commonplace for people to say, 'I was never any good at math.'

No one says, 'I was never any good at reading.'

### English Language Arts in Irvington

I've seen this here in Irvington.

My sense is that Irvington does a good job teaching reading. Not that I know what I'm talking about, but that's my sense. (fyi, after trying to teach out of the SRA Math book myself, I also think our grade school teachers are near-geniuses at teaching math, too.....& I'm not kidding about that. It was tough.)

• 2 periods of English language arts, one for reading & one for writing
• 1 period of social studies, taught by a teacher who told us, on back to school night, "I am an English language arts teacher at heart"
• 1 period of drama

That's 4 periods out of 8, half his day devoted to English language arts. He has 1 period for math, 1 period for science, and that's it. The other 2 periods are specials: study skills, music, art, drama, P.E., technology. Technology will mean creating an online 'portfolio' of his best work in 6th grade, not learning how to program. Study skills is about reading & taking notes, not doing problem sets.

And, on back to school night, the math teacher told us the kids would be keeping a math journal, because a lot of kids in accelerated math probably aren't as strong in ELA, so 'we try to help them with English language arts.'

Thus far she has done nothing of the sort, thank heavens, and she's stopped grading the kids' math tests on spelling, which she did last year. I gather she had a lot of complaints about it, and I made a point of asking her, in front of the other parents, whether she would be grading spelling this year, too. (This is what we call a warning shot.) So she told the kids she wouldn't, and she hasn't. otoh, Christopher is now spelling parenthesis parenthies, so be careful what you wish for.

### another story

This last story pretty much sums it up, I think.

I know I've mentioned the fact that we were clueless back when Christopher was in his early elementary years.

So, unbeknownst to us, he was placed in Phase 3 ELA as well as Phase 3 math. Actually, we're still clueless; I have no idea what kind of sorting & phasing they do with ELA. All I know is that in K-5 they divide the kids up into ability groups within the classroom, rather than separating them into different classes taught by different teachers, as they do with math.

In the hall outside Christopher's 4th grade class, after the year was over, I happened to run into his teacher and we fell into conversation, which led to the subject of Christopher's progress that year. I remember I was expressing gratitude for some especially good teaching she'd done, but I don't remember the details. It was probably about English language arts, since she taught him every subject but math.

One thing led to another, and suddenly I heard her saying, "Oh, I could see when he came into my class he wasn't a 3. He was much better than that. Sometimes you just have to ignore the tests."

So when was he a 3?

It took me a moment to recover, but I managed to keep her talking. "I pushed him," she said. "I knew he could do it." And, again: "You can't believe the tests."

Wow.

Here we have your dufus mom, completely out of the loop about tests, 3s, & 4s. And it doesn't matter; it doesn't hurt the kid. The teacher steps up to the plate, checks out the kid, decides for herself 'he's not a 3,' then sees to it he stops being a 3, and becomes a 4.

No extra reward, no extra praise, no extra payment or promotion. She just does it, because it's her job, and because she's good at it.

Perfect.

(And yes, I know; I'm tired of 3s and 4s, too. But 3s and 4s are a kind of shorthand, and a useful one.)

The point is: I have never heard this story told about a Phase 3 kid in math. Never.

Until this fall (that's another story), only a tiny handful of kids had ever moved from Phase 3 to 4. Maybe one 1 per year.

I've talked to the Chair of the middle school program about this issue, to one of the guidance counselors, to our 4-5 principal, and to numerous other teachers & parents.

Not one of them has mentioned the school or a teacher pushing a kid out of 3 and into 4. Whenever a move is made, the impetus has come from the parent, not the school. And the school resents it. (I've mentioned this before. We have a meta-narrative about pushy parents pressuring the school to put their kids in Phase 4 math when they don't belong there. Everyone subscribes to this narrative, including aides & other parents.)

The lesson I take away from this is that we really do have some major talent in some schools in this country, in the teaching of English Language Arts. I'm lucky to have my own kids in one such school district.

NsfVersusNrc 24 May 2006 - 00:08 CatherineJohnson

I've just become aware of a massive bibliography of
studies on NSF-funded K-12 curricula provided to
school districts by the National Science Foundation.

##### The Changing Mathematics Curriculum:An Annotated Bibliography Third Edition April 2005

The document's 60 pages include 3 studies of
Math Trailblazers:

and:

and:

### round up the usual suspects

• Isaacs, A. = Andy Isaacs, TIMS Senior Curriculum Developer
• Wagreich, P. = Philip Wagreich, TIMS Director and Co-Principal Investigator
• Gartzman, M. = Marty Gartzman, TIMS Senior Curriculum Developer

• Carter, A. = Andy Carter, TIMS Curriculum Developer
• Beissinger, J. S., = Janet Simpson Beissinger
• Cirulis, A. = Astrida E. Cirulis, TIMS Senior Curriculum Developer
• Gartzman, M. = Marty Gartzman, TIMS Senior Curriculum Developer
• Kelso, C. = Catherine Randall Kelson, TIMS Senior Curriculum Developer
• Wagreich, P. = Philip Wagreich, TIMS Director and Co-Principal Investigator

• Sconiers, S. = Sheila Sconiers, The ARC Center
• McBride, J. = James A. McBride, Everyday Mathematics (?)
• Isaacs, A., Andy Isaacs, TIMS Senior Curriculum Developer
• Kelso, C., Catherine Randall Kelson, TIMS Senior Curriculum Developer
• Higgins, T. = Traci L. Higgins, Investigations in Number, Data, and Space (?)

### how to write a letter of recommendation

from Susan:
Boy, if I ever send any resumes out I think I'll also send some fabulous letters of recommendations written by me. That should convince them.

TrailblazerConfusedBoy 13 Nov 2005 - 14:45 CatherineJohnson

from a ktm commenter:
Jim Milgram, the mathematician from Stanford, told me that in writing his math textbook for middle school, the publishers put in cartoons depicting boys acting lost and dumb and asking questions, with girls knowing what was what, and providing the correct answers. Milgram objected to the publisher about this and they were extremely firm in wanting to keep it that way.

### are boys in Trailblazers smart?

Reading the 5th grade 'Student Guide,' I had noticed more than one illustration of a baffled child not understanding math. These images struck me as strange; the 'student helpers' in Primary Mathematics always know math, and are never, ever bewildered by a math lesson.

Now I'm wondering whether the befuddled students in Trailblazers may have been mostly boys.

I'm going to check.

### I've got some scanning to do

The good news is that boys in Trailblazers don't appear to be any more stunned by math than girls, which is a relief. In a couple of cases, interestingly, a boy appears to be considerably smarter than the girl he is shown with.....I wonder how that happened. (I'll scan in the image & post later.)

### meet Professor Peabody

Professor Peabody appears throughout the Trailblazers Grade 5 Student Guide.

He is a professor, not a teacher.

He is stupid.

Teachers in the Student Guide, by way of contrast, are smart. They know what the answers are, and they understand math. Which is something to be grateful for, but still.

Professor Peabody appears for the first time on page 10, "Analyzing Data":

Which Graph Is Which?

Professor Peabody conducted three surveys at Bessie Coleman School. He collected three sets of data:

1. The number of pockets on the clothes of 15 students in a classroom.

2. The number of pockets on the clothes of the same 15 students as they played on the playground. (Each student wore a jacket outside.)

3. The number of pockets on the clothes of 31 students in a classroom.

For each survey he drew a picture, recorded the data in a table, and made a graph. Just as he finished graphing the data, he remembered that he had to get back to his lab to chck on another experiment. When he got back, he discovered that he had left his pictures and data tables at the school. All he had at his lab were his graphs. When he looked at the graphs, he saw that he had forgotten to write titles on them.

He is a real dufus, that Professor Peabody.

meanwhile, back in Singapore

4 boys shared 2/3 of a pie equally.
What fraction of the pie did each boy receive?

source:
Primary Mathematics 5A Textbook, p. 53

TrailblazersProfessorPeabody 15 Oct 2005 - 02:21 CatherineJohnson

TrailblazersProfessorPeabodyPart2 15 Oct 2005 - 02:23 CatherineJohnson

source:
MATH TRAILBLAZERS Student Guide, p. 458

ProfessorPeabodyPart3 15 Oct 2005 - 02:25 CatherineJohnson

source:
MATH TRAILBLAZERS Student Guide, grade 5, page 496

ProfessorPeabodyCorrectCaption 14 Jun 2006 - 21:16 CatherineJohnson

Professor Peabody was having fun exploring different numbers for the circumference and the diameter of circles on his calculator.

SmartBoysInTrailblazers 18 Oct 2005 - 13:46 CatherineJohnson

Checking to see whether Math Trailblazers, like some other contemporary textbooks, observes a girls-are-smart/boys-are-stupid principle in selecting and creating graphic elements, I found these three illustrations, on three successive pages (434 - 436) in the grade 5 Student Guide.

Nila and David again.

The cartoon character in the yellow shirt is a girl, Alexis. The boy is Manny.

Leaving aside Professor Peabody, I didn't see a boys-are-stupid theme in TRAILBLAZERS illustrations & playlets.

But I was surprised to see these three scenes, in each of which the girl is confidently reading or recalling known answers, while the boy is asking the right question.

I'm curious as to how this came about. Every illustration in a contemporary textbook is analyzed and vetted to the nth degree, so how does it happen that we have, on 3 pages in a row, the boy character clearly behaving in a more mathematically sophisticated manner than the girl character?

I should add that this is not simply my interpretation. In each case, the text reinforces the boy character's question, using it as a jumping-off point into further exploration of circumference.

The boy character is providing the narrative force and drive; the girl character is bringing the narrative to a full-stop.*

showing boys as curious

I was especially struck by the TRAILBLAZERS boy characters in light of this chart specifically forbidding images depicting boys as curious:

source:

Banned Words, Images, and Topics: A Glossary that Runs from the Offensive to the Trivial

return of the repressed

I no longer recall what the phrase return of the repressed actually means, if I ever knew in the first place.

But that's what springs to mind.

A large part of me thinks what happened here is that:

a) the intention was to show a confident girl and a non-confident boy

but

b) the intention ran afoul of the editorial staff's unconscious belief that.....boys are better at math than girls. Nobody is supposed to believe boys are better at math than girls any more, and I'm sure nobody at TRAILBLAZERS believes any such thing. Boys are better at math has been repressed.

But that's the problem with the repressed.

It keeps coming back.

forget about marketing math to children

This is Exhibit A in why we should forget about marketing math to children.

TRAILBLAZERS has TRAILBLAZERS Characters & TRAILBLAZERS Playlets on at least 30% of its pages, and my impression is that every last one is emotionally & cognitively false.

Beaming children doing math, page after page.

No one beams when he does math; if you're doing math, you're concentrating, which means you're probably frowning a bit as you think.

Your face is wearing a Thought Frown.

But beyond that, when grown-ups try to fool children into thinking MATH IS FUN!!!!!! (and yes, I know, math can be fun, and often these days is fun for me) the truth slips out.

One way or another.

I would bet real money that the people responsible for these illustrations, if you injected them with truth serum, would tell you: girls can't do math.

That's why the girls are always beaming, and always confidently reading pre-packaged answers out of books and always reminding the doubting boy of what they've discovered in class so far. (Every single black child in the book is beaming, too. I rest my case.)

The boy gets to express the doubt and the skepticism and the non-knowing, because he's going to be smart enough to figure it out.

Unless, of course, it's worse than I think, and the plan is to cut down on the number of boys making it to college.

*This was a major preoccupation of feminists who studied film back in the day. Males were associated with narrative movement; females were spectacle that stopped narrative. Females were 'theorized' as almost a counter-narrative force. This probably sounds ludicrous in the present context, but it's probably true, at least of 1940s film noir...The classic essay on this subject was Laura Mulvey's Visual Pleasure and the Narrative Cinema, which I haven't read in years.

LatticeMultiplicationWithCarrying 22 Oct 2005 - 16:31 CatherineJohnson

This explanation at Math Forum is the clearest I've seen. Perfect.

I'm too tired to think it through right now, but I'm not so sure the final point is right.....

'Doctor Mason,' at Math Forum, says:

Multiplication really takes three steps: multiply, carry, add. The method we typically use does the multiply and carry steps together. The lattice method does all three steps separately, so it's really easier! Centuries ago, the Germans had a method for doing all three steps at once. That method takes a lot of concentration!

But I think the method I left in one of the Comments threads also separates the 3 steps.

Doesn't it?

I'm going to keep an eye out for Dr. Mason.

FourthGradeMathEnrichment 17 Nov 2005 - 01:15 CatherineJohnson

The fourth grade math enrichment program is meant to complement and enhance the distict's new mathematics program, Math Trailblazers. It is a flexible, "push in" model that allows small groups of children the opportunity to explore the previously learned concepts in greater depth.

Most recently, all of the fourth graders constructed runways for various types of airplanes. The focus of the activity was to find the perimeter of different sized runways built with square tiles, noting patterns found during the process. As an enrichment activity, students built runways with shapes other than squares, such as triangles, pentagons and hexagons. Then, we discussed patterns and "function rules" that were devised from shape. With the discovery of each shape's function rule, we could calculate the perimeter of any number of blocks arranged in a line, without having to build the runway.

The math enrichment program can also extend any topic being taught in class. As fourth graders continue to explore our Hindu-Arabic number system with base ten blocks, enrichment activities will focus on other number systems such as the Roman numeral system. By studying the rules and symbols of other number systems, the students can compare and contrast the similarities, differences, pros and cons of our current number system.

As with Math Trailblazers, all enrichment activities align with NCTM and New York state mathematical standards appropriate for grade four. In addition, the children are often encouraged to show their work and explain their thought processes in complete sentences as they will be expected to do on the upcoming state test in March.

think and discuss

Our Hindu-Arabic number system, pros and cons

Hindu-Arabic number system
pros             cons

Roman numeral system
pros             cons

meanwhile, back in Singapore

11.
David spent 2/5 of his money on a story book.
The storybook cost \$20.
How much money did he have at first?

source:
Primary Mathematics 4A Textbook, p. 67

update
A lot of the Regulars (see Comments thread) think these activities are worthwhile.

This tells me I've done a poor job writing this post—and, in fact, thinking it through in the first place.

Thanks to all of you, I'm clearer now.

Here's where I am:

• First of all, it pains me that, apparently, it's only the 'enrichment kids' who are learning about the Roman numeral system and about irregularly-shaped figures. Saxon Math teaches these subjects to all children, as does Singapore Math.

• Second, having now spoken to the mother of a mathematically gifted child struggling with TRAILBLAZERS in 2nd grade, I'm finding myself drawn to the issue of what exactly my school district is doing for these kids. These are very brainy children; I know, because I have them in my Singapore Math class. I do not see or hear evidence of a commitment, in our district, to serve the needs of gifted children as we serve the needs of other children. The reason these children are having 'push-in' enrichment is that our district is philosophically opposed to allowing high-achieving and gifted children to move ahead in the curriculum. So now we're spending money on 'Math Differentiation' specialists to help teachers teach math to below-average, averge, high-achieving, and mathematically gifted and talented kids all in the same time and place. The decision to end tracking was made in a hierarchical, top-down fashion, and imposed upon protesting parents.

• Third, I'm highly skeptical of the use of manipulatives at this age. First of all, we do have some reasonably good research on the subject of manipulatives, which shows—counterintuitively—that manipulatives work for middle school kids, but not for elementary school children. Manipulatives may impede learning in earlier years. (see posts here and here) But leaving aside the handful of studies we have, what bothers me is that all of these kids are bright. Their math skills are strong; they're good readers; they have smart, dedicated parents to help with homework. These children, I believe, can be using the abstract symbols of mathmatics to reason and learn.

• Finally, I simply loathe the tone of this document. I know that's harsh, but that's the writer in me. The fourth grade math enrichment program is meant to complement and enhance the distict's new mathematics program, Math Trailblazers. This is spin, and, reading it, I feel my blood rise. When a document like this comes home in the backpack, I am not being treated with respect. I am being public relationed, in Carolyn's memorable phrase.

ForgivingDivisionIsEasier 10 Oct 2006 - 02:33 CatherineJohnson

TRAILBLAZERS' rationale for replacing the long division algorithm with forgiving division:

Given the vast amount of time and the frustration involved in learning the long division algorithm traditionally taught in the United States, we instead use what we call the “forgiving method.” Sometimes it is referred to as the “subtraction method.” While this method may seem new, written record of it appears in a book published in 1729 while the first record of the traditional method appears in a publication dating from 1491 (Hazekamp, 1978). As with the traditional method, the forgiving method requires students to estimate quotients. The forgiving method is different in two ways. First, the student starts by estimating the entire quotient instead of the first digit. Secondly, if the estimate is too small, the student can continue with the procedure. This greatly alleviates the frustration of having to erase, and to some extent, allows one to get around a forgotten multiplication fact. (page 145, grade 4)

[snip]

Research has shown that low-ability students show better retention and understanding when taught division with this method and become better estimators of quotients. Students who were taught the forgiving method were better at solving unfamiliar problems and were better able to explain the meaning of the steps (van Engen and Gibb, 1956). Another study found that students who were taught both the forgiving and traditional methods did not confuse the methods and that the total amount of time needed to learn both was the same as the amount of time needed to learn one of the methods (Scott, 1963). Understanding rote procedures enables students to perform mathematical tasks with confidence and meaning. When children understand the mathematics they do, they come to believe that mathematics makes sense, and they are better able to think and reason flexibly. (page 146, grade 5)

[snip]

In this unit, an alternative division method is presented, rather than the one traditionally used in the United States. This method, which we call the forgiving division method, does not require that the greatest quotient be found at each step, eliminating the frequent erasing encountered with the standard algorithm. Research shows that students who are taught the forgiving division method are better at solving unfamiliar problems and are better able to explain the meaning of the steps in the method than those taught the traditional method (van Engen and Gibb, 1956). The forgiving division method also gives students the opportunity to practice mental math. (page 166, grade 5)

source:
TRAILBLAZERS background, grades K - 5 (pdf file)
page 167

We have quite a lot going on here.

First of all, we have an explicit statement that TRAILBLAZERS content is geared toward low-ability students. Not high-ability, not average-ability. Low-ability.

Do parents know this?

Second, we have an explicit statement that the authors of TRAILBLAZERS have opted to replace the standard algorithm with the forgiving version because the standard algorithm takes too long too teach ("a vast amount of time") and is too hard ("the frustration involved").

These observations strike me as correct. From what I gather, it does take quite a lot of time & frustration to teach the standard algorithm, although I question how much frustration would be involved using Singapore Math, Saxon Math, or Direct Instruction.

The problem with this line of reasoning is that the standard of diminishing returns has not been applied to activities like Antopolis.

Thirdly, and mystifyingly, we have the inevitable Research Shows passage in which we are assured that in fact it takes no more time to teach forgiving division and long division than to teach either one on its own. That strikes me as unlikely, regardless of what 'research' does or does not show. Under normal circumstances, learning two things takes more time than learning just one. But, supposing the research is right, the obvious question is: Then why aren't you doing it? If it takes no more time to learn both algorithms, and if it's a good idea to learn both algorithms, then—hey! Teach both algorithms!

(For me it almost certainly would have been helpful to have studied both algorithms, though it would not have been helpful to practice both to mastery.)

question

I could probably think my way through this one, but in the interests of efficiency I'll ask you.

Can you do decimal division using forgiving division?

I'm not instantly seeing how that would work....

update

Which means you can't use forgiving division to convert a fraction to a decimal. The Trailblazers grade 5 Student Guide tells children to use their calculators to accomplish this task.

wit and wisdom

This is funny.

TRAILBLAZERS grade 4 has a lesson called, "Oh, No! My Calculator is Broken."

This is Lesson 3 in Unit 7, Patterns in Multiplication.

The Key Content in "Oh, No! My Calculator is Broken" is:

• Recognizing that there are many strategies for doing simple multiplication problems
• Using efficient strategies to do multiplication problems involving the last six facts
• Using the calculator efficiently in problem solving
• Communication problem-solving strategies

I'm wondering how you use a broken calculator efficiently in problem solving.

why long division?

TrailblazersAndTheAlgorithms 29 Nov 2005 - 14:17 CatherineJohnson

I talked to a friend whose son is in second grade. He's a brainy kid who loves math, but he can't use the addition algorithm—because it hasn't been taught, apparently. If he's adding numbers smaller than 20, he counts on his fingers and toes. If the numbers are larger than 20, say 12 + 19, he draws 12 circles, then 19 circles, and finally counts them. Same process for subtraction, only in reverse. 63 - 19 means drawing 63 circles, then crossing out 19 of them.

The kids have the triangular flash cards that portray number families, and her son is working on flashcards with numbers 1 - 10. A friend of hers whose child is in 3rd grade told her those children are working on the exact same cards.

Our approach is based on educational research and is supported by the National Council of Teachers of Mathematics’ Principles and Standards for School Mathematics (2000). It is characterized by the following:

• Emphasis on problem solving. Students will learn the basic facts if they are encouraged to use a problem-solving approach. Students can invent their own strategies, learn from their peers, or learn from the teacher through class discussions. Students will discover the need to learn the facts as they encounter them in labs, activities, and games.

• De-emphasis of rote work. Students learn their math facts, but we de-emphasize the use of rote memorization and in first grade we do not administer timed tests. These can produce an undesirable result—students perceive that doing math is memorizing facts and rules which “you either get or you don’t.” Instead, students should feel confident that they can find answers through the use of strategies they understand.

Throughout first grade the focus remains on the use of strategies that are meaningful to students. Beginning in Unit 11, a systematic organization of the math facts is introduced via the Daily Practice and Problems. Facts are grouped together in ways that may help students think of them in a more efficient manner. However, students are still free to solve the problems using whatever strategies they wish. By the end of second grade, students in Math Trailblazers are expected to demonstrate fluency with the addition and subtraction facts. The first grade curriculum enables them to build to this fluency through experience with and understanding of the concepts of addition and subtraction.

This passage illustrates the difference between 'research' and 'field-testing.'

TRAILBLAZERS is now being used in the field, here in Irvington.

Do we see children demonstrating fluency with the addition and subtraction facts by the end of 2nd grade?

I don't know the answer to that question. From what I hear, it doesn't sound like it.

What I'd like to know is: does anyone have an answer to this question?

the kindness of strangers

My least-favorite reform-math slogan:

• Facts will not act as gatekeepers. Students are not prevented from learning more complex mathematics based on their fluency with the math facts.

All children of normal intelligence can learn math facts. No doubt many children with mild mental retardation can learn math facts as well.

A responsible educator figures out how to teach math facts at the earliest possible age, and makes sure all children have learned.

source: TIMS Tutor grades 1 - 5 (pdf file)

TrailblazersParentLetterOnFractions 27 Nov 2005 - 15:30 CatherineJohnson

I've mentioned before that when a school district purchases the TRAILBLAZERS curriculum, it also purchases a complete package of parent letters to be sent home in the backpack at the beginning of each unit.

Here's how the fifth grade letter on fractions introduces the subject:

Children first learn about fractions from listening to adult conversation: “I’ll be there in a half hour,” or “The recipe calls for 2/3 cup of sugar.” As children’s experiences with fractions grow, they begin to use the language of fractions in their lives. “You can have half of my sandwich,” or “I need half a dollar.” However, many times a child’s understanding of fractions is not complete.
I'll say.

Math TRAILBLAZERS draws to a close

The TRAILBLAZERS curriculum ends with 5th grade. Here is the Parent Letter for Unit 12, the last unit in the series to teach—excuse me, to explore—fractions.

And with that, children are packed off to middle school.

They are now 10 years old. They have 'explored' equivalent fractions, mixed numbers, fraction addition and fraction multiplication. Whether they've practiced these operations to mastery is an open question. TRAILBLAZERS makes no mention of it one way or another. Unless the teacher has supplemented the curriculum with fraction worksheets and word problems, my guess is TRAILBLAZERS students have explored fractions and then moved on.

They don't know how to divide fractions; they don't know how to do decimal division. The only method presented to them for translating a fraction into a decimal is the calculator.

That's a lot of cans to be kicking down the road.

fractions, decimals and percents

SingaporeMathTopicMatrix 30 Nov 2005 - 16:47 CatherineJohnson

from the AIR report (pdf file)

I'm going to try to rustle up the equivalent chart from TRAILBLAZERS so we have a direct comparison.

This seems to say that in Singapore children are using all 4 algorithms in the first grade.

the TRAILBLAZERS track

Christopher told me yesterday that he learned how to subtract with regrouping in 2nd grade. 'It was hard,' he said. He learned the multiplication algorithm for the first time at the end of 2nd grade.

From what I can see, the TRAILBLAZERS track could be as much as a year slower than Irvington's old track, though it's hard to tell:

This unit extends students’ work with place value to four-digit numbers and helps them build an understanding of our number system, the base-ten place-value system. The activities in this unit lay the conceptual groundwork for performing multidigit addition and subtraction. Two-digit addition is reviewed. Three- and four-digit addition and subtraction algorithms are developed in Unit 6.

This is from Unit 3, Grade 4 (there are 20 units altogether).

Here's a passage from the Teacher Implementation Guide (pdf file):

In kindergarten and grade 1, students using MATH TRAILBLAZERS practice their counting skills. They learn to count past 100 by 1s, 2s, 5s, and 10s. They count forward and backward from any given number. They group objects for counting. Students use counting to solve addition and subtraction problems. They learn to write numbers up to and beyond 100. The 100 chart is introduced and used for a variety of purposes, including solving problems and studying patterns. Students partition, or break apart, numbers in several ways (25 = 20 + 5, 25 = 10 + 10 + 5, and so on). These activities help children become familiar with the structure of the number system. Beginning in kindergarten, a ten frame is frequently used as a visual organizer.

The first passage is intended for parents.

The second passage comes from the Teacher Implementation Guide.

subtraction algorithm mastered by the end of 4th grade

It's probably worth skimming through pages 5 through 7 in the Teacher Implementation Guide. This passage lays out the TRAILBLAZERS content, sequence, and timing for teaching the subtraction algorithm. Once again, it's difficult to nail down the exact 'scopt and sequence' TRAILBLAZERS follows:

Later in grade 2, systematic work begins on paper-and-pencil methods for subtracting two digit numbers. Students are asked to solve two-digit subtraction problems using their own methods and to record their solutions on paper. The class examines and discusses the various procedures that students devise. At this time, if no student introduces a standard subtraction algorithm, then the teacher does so, explaining that it is a subtraction method that many people use. The standard method is examined and discussed, just as the invented methods were. Students who do not have an effective method of their own are urged to adopt the standard method.

Problems that require borrowing are included from the beginning. Though this differs markedly from traditional approaches, we view it as important in developing a sound conception of subtraction algorithms. Giving children only multidigit problems that do not involve borrowing encourages the development of a rote and faulty algorithm that may not carry over into problems that require borrowing.

By the beginning of grade 3, students have a strong conceptual understanding of subtraction and significant experience devising procedures to solve subtraction problems with numbers up to 1000. They also have some experience with standard and invented paper-and-pencil algorithms for solving two-digit subtraction problems. In grade 3, this prior knowledge is extended in a systematic examination of paper-and-pencil methods for multidigit subtraction. This work begins with a series of multidigit subtraction problems that students solve in various ways. Many of these problems are set in a whimsical context, the TIMS Candy Company, a business that uses base-ten pieces to keep track of its production and sales. Other problems are based on student-collected data, such as a reading survey.

As in grade 2, the class discusses and compares the several methods students use to solve these problems. Again, any method that yields correct results is acceptable, but now a greater emphasis is given to methods that are efficient and compact. This work leads to a close examination of one particular subtraction algorithm. (See Figure 3.) Students solve several problems with base ten pieces and with this standard algorithm, making connections between actions with the manipulatives and steps in the algorithm. After a thorough analysis of the algorithm, including a comparison of the standard algorithm and other methods, students are given opportunities to practice the algorithm. Practice in paper-and-pencil methods for multidigit subtraction is distributed throughout grades 3 and 4.

TRAILBLAZERS delays mastery of the subtraction algorithm until the end of 4th grade.

This is certainly consistent with the constructivist belief that premature teaching of the algorithms closes off conceptual understanding.

TRAILBLAZERS whole number operations scope and sequence

What they've done here is to use the idea of a math curriculum based in problem-solving to justify not teaching the algorithms.

All of the problems being solved in these first years are of the type:

"How do I add, subtract, multiply, and divide without knowing any algorithms?"

This is not remotely the case in the Singapore series.

Like TRAILBLAZERS, PRIMARY MATHEMATICS is a problem-based curriculum.

But in PRIMARY MATHEMATICS children use the standard algorithms to solve problems. That's why, in Singapore, children can begin using bar models to solve simple algebra problems in the 3rd grade. The bar models help them perceive which algorithms to use in what sequence.

If you don't know the algorithms, a bar model's not going to do you much good.

Introduction to Math TRAILBLAZERS
TRAILBLAZERS (TIMS) Teacher Implementation Guide: Math Facts
TIMS Teacher Implementation Guide Laboratory Method

key words: scope and sequence Singapore Math Primary Mathematics Trailblazers

LatticeMultiplicationAtIllinoisLoop 08 Dec 2005 - 15:06 CatherineJohnson

Becky C reminded me that Illinois Loop has a scathing review of TRAILBLAZERS posted (I've read it at least twice & am due for another go at it).

When I clicked over to the site I found this:

"Yes, New-Math is multiplying, but I am sorry to report that too many children are not learning to multiply with New-Math. ... Multiplication is not all that difficult if one learns the multiplication tables and the logical, precise algorithm for the process. One day I was teaching traditional multiplication when one of the special education students wanted to show me the process she had been taught. Her problem even shocked me, and luckily I had my camera with me.

source:
New Math Multiples by Linda Schrock Taylor

For some reason I've come to love images of lattice multiplication. I'm forming a collection. Any minute now I'll be bugging J.D., Doug, Dan, and perhaps Carolyn, too, to make me one of my very own!

(Just kidding. I do love looking at them, though.)

WayneBishopReviewOfTrailblazers 01 Dec 2005 - 22:20 CatherineJohnson

review of Math TRAILBLAZERS, under consideration for California state adoption

AB 2519 Content Review Panel
REPORT WRITING TEMPLATE
MATHEMATICS PROGRAMS

Name of Publisher: Kendall/Hunt Publishing Co.
Name of Program: Math Trailblazers
Identification Number: None Indicated in Standards Map

conclusions:

• Based upon evaluation of its content only, this submission is NOT recommended for adoption as either a BASIC or a PARTIAL program.

• This submission is not comprehensive, and it does not contribute clearly and significantly to the course of study.

• This submission is not based upon the fundamental skills required by mathematics, including, but not limited to, basic computational skills.

• This submission does not provide for basic skills instruction that is systematic and explicit.

• This submission does not enable instruction in almost all (if not all) of the individual standards for the intended grade level(s) or discipline(s), either in whole or in one or more of the subject area(s) or discipline(s) listed above, in a cohesive, clear, systematic, and significant fashion.

• This submission is not aligned with the standards for the intended grade level(s) or discipline(s), meaning that (1) it does not enable successful instruction in the individual standards it covers, (2) it includes something fundamentally contrary to the standards, or (3) some content extraneous to instruction in the standards does detract from the ability of teachers to teach readily and students to learn thoroughly the content specified in the standards.

• This submission is not factually accurate and the inaccuracies cannot reasonably be corrected.

• To the extent this submission incorporates principles of instruction that are not reflective of current and confirmed experimental research, the bases of those principles are either not appropriately identified or are misleadingly presented as experimental research.

• This submission is not adequate in its coverage.

This submission falls far short of the standards for the grades at which it was submitted. It should not seriously be entertained as a candidate for these grades.

AB 2519 Content Review Panel
REPORT WRITING TEMPLATE
MATHEMATICS PROGRAMS

notes and citations

take-home message:

Objectively measured student performance did not improve.

ChangingDeckChairsOnTheTitanic 10 Jan 2006 - 16:21 CatherineJohnson

from Illinois Loop comes word that &mdash:

At one point, Oak Park District 97 used the merely mediocre Scott Foresman Addison Wesley "Math" series.

Then the district jumped into that land of fuzzy math with both feet by adopting Math Trailblazers.

In May 2005, a parent reported to us that D97 "is leaving Trailblazers behind in Fall 2005 to go to Everyday Math for grades 1-6."

TheTrailblazersSpiral 10 Oct 2006 - 01:49 CatherineJohnson

hoo boy, that was fun

fun, fun, fun

TRAILBLAZERS is going down

bye-bye

UPDATE 9-19-2006: TRAILBLAZERS isn't going anywhere. don't listen to me.

a TRAILBLAZERS spiral

I'll condense the story and post tomorrow, but I wanted to get this down tonight.

After the meeting, Ed was talking to the Dows Lane (K-3) mom who's been agitating against TRAILBLAZERS. Her kid is a math-brain. Maybe both her kids are.

She told Ed that in 2nd grade TRAILBLAZERS teaches kids how to construct graphs.

Then, in 3rd grade, TRAILBLAZERS teaches kids how to construct graphs again — the exact same lessonexcept that, this time around, they teach the kids TO LABEL THE AXES.*

She didn't say whether they teach labeling the axes to mastery.

it's all becoming clear now

All of it.

The huge books, the grinding overwork, the ever-expanding gap between our kids and every other math student on the planet......

I get it.

I have found the basic principle, as Temple would say.

Start from the premise that nothing will be taught to mastery, and everything else follows. Big books, big gap, big backpacks, 11-year old kids breaking down in tears in the middle of a 'quiz.' It all makes sense.

That big sucking sound you hear? That's the spiral curriculum Hoovering up the kids, the mom, the dad, the KUMON operator, and the kindly folks at ktm into the effort to teach basic algebra to just one boy.

I don't like it.

No one at the meeting knew what the term 'spiralling' meant.

Now they do.

Until one year ago, I had never heard the term 'spiral' applied to a curriculum.

But once you start to really work your way through it....once you start to understand that schools deliberately teach skills and concepts so that children do not master them and then grade them on their 'performance'.....

slow burn

* She thought it was 2nd & 3rd grades, but it might be 1st and 2nd grades.

BarryOnMathAndBoys 16 Sep 2006 - 20:39 CatherineJohnson

Barry Garelick left this comment about boys, math, and Congress last October. It's worth reading again.

Here's the section on Askey & Milgram:

Dick Askey from U. of Wisconsin came to meet with me at one point, and the staffer I was working with mentioned to him the "Women in Science" project. He said that right now the problem is not so much with girls but with boys; there are too few of them on campus. (This is what the USA Today story said). The staffer was fairly disgruntled at this, and later I heard her murmuring to people about the "sexist" comments that Askey had made.

Jim Milgram, the mathematician from Stanford, told me that in writing his math textbook for middle school, the publishers put in cartoons depicting boys acting lost and dumb and asking questions, with girls knowing what was what, and providing the correct answers. Milgram objected to the publisher about this and they were extremely firm in wanting to keep it that way. They reached a compromise: They showed pudgy, balding middle-aged men acting lost and confused, with boys and girls providing the right answers. I really don't know that that was much of a solution. Milgram is a bit pudgy and middle-aged and balding, by the way. He's also one of the top mathematicians in the country.

When I went through TRAILBLAZERS looking for gender bias, I didn't find it (that's good!) — although I did find some interesting differences.

compare and contrast

and while we're on the subject of boys in TRAILBLAZERS

Let us not forget Professor Peabody.

Professor Peabody caption contest

-- CatherineJohnson - 27 Jan 2006

AnneDwyersMathBoosters 10 Oct 2006 - 01:50 CatherineJohnson

Comment left by Anne:

BTW, my Math Booster class has seemed to strike a nerve with parents. One of the parents on the PTO at one of the elementary schools is going to speak to her principal about having the PTO sponser my class. I would love to be a fly on the wall at that meeting!!! Politics being what it is in a school district, I don't see it happening.

Excellent.

On all counts.

Irvington redux

Now that I know we have one person from Irvington reading the site, I figure I'll engage in a bit of spaced repetition on the Singapore Math in Irvington front.

personal narrative:

• I co-chaired the PTSA after-school program at the Main Street School for two years.

• During my second year as co-chair, I taught an after-school course in Singapore Math. The principal approved the course, asked me how it was going, borrowed the books to show his wife (a high school math teacher), and told me NY state was moving toward a 'Singapore' model (fewer topics taught in more depth) in state standards. Christopher's teacher helped me out with advice and materials.

• This fall I taught the course again. One teacher asked me for materials to give to the parent of an especially bright child in his/her class. Another teacher told a parent that he/she was eager to learn more about what I was doing.

• Our assistant superintendent in charge of curriculum contacted the president of the PTSA. He told her that 'teachers' and 'parents' had called to complain about the course. He said, too, that I was using my course to undermine TRAILBLAZERS.

• Our assistant superintendent in charge of curriculum did not contact me at any point. Instead, acting in his professional capacity, he chose to make anonymous charges against one parent to another parent in private.

• The PTSA president contacted me. We talked. I met with the PTSA Executive Committee.

• Last I heard, the Superintendent planned to draft a formal policy, to be submitted to and voted on by the School Board, giving her authority to approve and disapprove all parent-run courses.

micromanagement

I have been told, by a board member, that our Superintendent has a tendency to micromanage.

When Ed heard what was going on — for the uninitiated, Ed is a longtime professor and university administrator — he said, "The superintendent shouldn't even know about your course. This should be way below her level of vision. If this is what she's spending her time on, we're in trouble."

I'm sure he's right.

some questions

• This is a small community. I wonder whether my reputation has been harmed by the assistant superintendent's decision to make anonymous accusations against me, in private, to another parent.

• This is a small community. I wonder whether the assistant superintendent has talked to other parents, administrators, teachers, and community members about me.

• This is a small community. I wonder whether the assistant superintendent maybe ought to pick up the phone and give us a call. We're in the book.

one more question

The administration's thinking, I gather, is that under the new policy the PTSA cannot offer after-school courses that cover the same material taught in Irvington schools.

The PTSA can offer enrichment courses — knitting, cooking, all-sports.

The PTSA can offer academic courses not offered by the district — Chinese, for instance. The PTSA is free to offer after-school courses in Chinese.

This means that I cannot teach a writing course in the after-school program.

Irvington parents are actively distressed by the quality of writing instruction in the middle school, and the district acknowledges the problem.

I taught writing to middle-school students for Johns Hopkins CYO; I have a Distinguished Teaching award from the University of Iowa for my teaching. I am a professional writer, author of a well-reviewed book that spent 6 weeks on the New York Times bestseller list. [3-31-2006: 10 weeks on the paperback list and counting]

I would probably agree to teach writing to middle-schoolers in an after-school PTSA program as a service to the community.

Is the administration acting in the interests of Irvington children?

I can think of a dozen parents from whom I'd want Christopher to be able to take an after-school course, and that's just off the top of my head.

All of these parents would be teaching core academic subjects. Math, English, history, science.

I would like to see our administration foster such opportunities for our kids.

Irvington mission statement: The mission of the Irvington School District is to create a challenging and supportive learning environment in which each student attains his or her highest potential for academic achievement, critical thinking and life-long learning.

-- CatherineJohnson - 05 Feb 2006

SpirallingStories 07 Feb 2006 - 16:40 CatherineJohnson

I'm pulling parents' experiences together into one post.

Math Trailblazers

A parent here told Ed that in 2nd grade TRAILBLAZERS teaches kids how to construct graphs.

Then, in 3rd grade, TRAILBLAZERS teaches kids how to construct graphs again — the exact same lesson — except that, this time around, they teach the kids TO LABEL THE AXES. (fyi: She wasn't sure what the grade span was; it could have been 3rd to 4th.)

Everyday Math

My cousin describes her experience with Everyday Math:

Chicago Math gives you advanced math problems sprinkled in with the elementary math your child is learning. They slip it in.

They would have you guess at the answers for the advanced problems, but then they never gave you the answers so you didn’t know if you guessed right or not. You’re always a work in progress with Chicago Math. So you never get a definite answer. And you never had a sense of completion or success on a day-to-day basis.

But my pet peeve was that it sped you along at a rapid pace and you never mastered the material that you left the page before. When my daughter was in the 2nd grade one work page would be coins; the next day you’d be dealing with weather; the next day you’d be dealing with problem solving. My daughter had no sense of what a quarter or a dime was.

When I was taught math, each day you built on what you knew. When you did the coins you learned a penny, a nickel, a quarter. You kept going. Telling time, same thing. You work on time until you get it. You don’t just have a flash of it one day.

In Chicago Math you had one page on one topic, then you went on to something completely different on the next page. There was no repetition. It was irresponsible, very ungrounded.

Mike Feinberg of KIPP on spiral curricula
Steve and Susan J on spiral curricula
acceleration versus remediation

-- CatherineJohnson - 06 Feb 2006

EverydayMathInNewYork 01 May 2006 - 23:08 CatherineJohnson

To see the defects of Everyday Mathematics, one need only examine its treatment of paper-and-pencil subtraction of two multi-digit numbers.

Most adults will have learned to write the smaller number below the larger one, lined up at the right, and write down the result of the subtraction right to left, doing whatever "borrowing" is needed mentally. This is not taught in Everyday Mathematics. Instead, the Everyday Mathematics pupil is exposed to five different subtraction methods, each of them viewed as suitable for the same task. The gymnastics employed to avoid simple methods is truly breathtaking.

There is in Everyday Mathematics a "trade first" variant of the traditional method: Borrow first in all columns where it is needed, recording the intermediate results, and then do the subtractions. There is a "counting up method": count up from the smaller to the larger number, first by ones, then tens, and so on, and then the odd remainder, and then in a second pass, add up the addends. (Example: If we do 425 - 48 then the second stage involves adding up 2 + 50 + 300 + 25 to obtain 377.) The third standard method is left to right subtraction, the way one might well do the problem mentally, but carried out with paper and pencil. The fourth approach is a "partial differences method": subtract in each column separately, keeping track of the sign if a borrow would be needed, and then combine the results by mental arithmetic. Finally there is the "same change rule": change both numbers by the same amount so that the smaller number ends in one or more zeroes and the problem is easier. Addition, multiplication, and division likewise have multiple standard methods in Everyday Mathematics, and in all of this, true fluency in the basic operations appears not to be an aim of the program.

One can well imagine how a pupil who already has excellent mastery of arithmetic can enjoy seeing and understanding how the multiple methods of Everyday Mathematics all lead to the same correct result. The danger of this profusion of methods for pupils who are not so comfortable with the basics is also easily imagined. These pupils, and some teachers and parents as well, will be hopelessly confused. Combine that with the easy tolerance of calculators in Everyday Mathematics and one can foresee that entire classrooms will throw up their hands and rely on the calculator for arithmetic, never to achieve the facility with number and operation that they'll need to advance beyond the grade school level.

Everyday Mathematics combines the very defective treatment of basic arithmetic with some quite sophisticated content elsewhere, resulting in a strange mixed bag that ought never have been selected for city-wide use in the elementary schools. Mr. Klein would do well to reverse himself and listen to the advice about successful curricula that mathematicians and others have provided to him and his staff. As one example, I would suggest that the chancellor look hard at the documented results of the Saxon Math curriculum. He can start by perusing the input provided by New York City HOLD to the Chair of the Children First Numeracy Working Group and available through the NYC HOLD Web pages.

source:
NY Sun editorial
NYC HOLD

Braams' observation that seeing other methods of subtraction would be illuminating for people who've mastered the standard algorithms is certainly true of me. I found all that stuff fun a year or so ago, when I was seeing it for the first time.

Though I can't say I was amused by lattice multiplication.

false dichotomy? for some kids, but not others?

In theory, MATH TRAILBLAZERS is designed to tilt back towards fluency in the math facts, and, I think, in the standard algorithms.

The curriculum also assumes that all children will gain fluency incidentally, rather than through formal practice or drill and kill:

• Early emphasis on problem solving. We believe that children must indeed learn their math facts, but we de-emphasize rote memorization and the frequent administration of timed tests. Both of these can produce undesirable results. Instead, our primary goal is that students learn that they can find answers using strategies they understand.

• Ongoing practice. Work on the math facts is distributed throughout the curriculum, especially in the Daily Practice and Problems and in the games. This practice for facility, however, takes place only after students have a conceptual understanding of the operations and have achieved proficiency with strategies for solving basic fact problems. Delaying practice in this way means that less practice is required for facility with the number facts.

• Gradual and systematic introduction of facts. Students study the facts in small groups that can be solved by a single strategy. Early on, for example, they study facts that can be solved by counting on 1, 2, or 3. Students first work on simple strategies for easy facts, and then progress to more sophisticated strategies and harder facts.

Conceptual understanding first, procedural knowledge second. This is Founding Law in MATH TRAILBLAZERS. It doesn't make a lot of sense to me, speaking as a Naive Relearner of math. Neither does the idea that one would always teach procedures first, and concepts second, although I think this would be different if I were teaching a high-end autistic child....and I wouldn't be surprised to learn that other children do best learning and practicing procedures first....

I don't know.

Now that I'm well into my third Saxon book, I think it's fair to say that Saxon joins the conceptual with the procedural wherever possible. In Saxon 6/5 (5th grade), Saxon even tries to teach the division of fractions conceptually as well as procedurally. Speaking for myself & Christopher, I'd have to admit that the book fails miserably, but they give it a good go.

[note: this is an animated gif.
To see the motion you have to
release the scroll bar.]

-- CatherineJohnson - 01 May 2006

HowToGetParentBuyInPart2 27 May 2006 - 02:30 CatherineJohnson

source:
Getting Your Math Message Out to Parents

how to get parent buy in, part 1

Getting Your Math Message Out to Parents (pdf file)

-- CatherineJohnson - 26 May 2006

TrailblazersAndMathWars 28 Jun 2006 - 18:25 CatherineJohnson

Lots of Trailblazers stuff here. Also sample activities from each grade here.

The FAQ page has this to say about Trailblazers and the math wars:

Q. I heard that the "math wars" are about the two philosophies of teaching math: Traditional/Classical that emphasizes learning math facts, computation skills and applying those with word problems, and the other is Constructivist math which emphasizes discovery in group activities but is supposedly very weak in basic math facts and computation.

A. Not true. First of all there are not only two philosophies of teaching mathematics. Some people may be able to divide the world into black and white, but reality is more complex. Math Trailblazers was developed by the TIMS (Teaching Integrated Mathematics and Science) Project at the University of Illinois at Chicago. It was founded and is directed by Philip Wagreich, a mathematician, and Howard Goldberg, a particle physicist. Goldberg and Wagreich were motivated by the appalling quality of mathematics and science teaching, and textbook, in their children's schooling. In addition, they were confronted daily with college students who could not do the most basic mathematical and scientific reasoning.

In 1984, they decided to take time away from their demanding research activities to find ways to improve education for all children. Before the NCTM Standards, before Math Wars, before they had even known there was a "philosophy of constructivism", they set to developing the foundations of "the TIMS Philosophy." The hallmarks of the TIMS Philosophy are to make mathematics meaningful to children, to challenge them with a rigorous and mathematically sound curriculum, and to help children learn the reasoning skills that are so important in the workplace of today (and will be absolutely essential in the world they will meet when they graduate -- say, in 2012).

I admire the fact they've taken the bull by the horns here, going so far as to print the words "math wars." In politics the rule is, I think, that you don't speak your opponent's name. The fact that they've and named the opposition takes the Math Trailblazers folks out of the realm of politics, to my mind, at any rate.

Unfortunately, they're still in the realm of marketing, PR, and spin:

Q. What is the opinion of the scientific community regarding the Standards set by the National Council of Teachers of Mathematics?

A. Math Trailblazers meets the Standards set by the National Council of Teachers of Mathematics. These standards have undergone extensive review. The opinion of the scientific community is accurately represented by the The Council of Scientific Society Presidents (CSSP)-the leadership organization for more than 1 million scientists and science educators. It commended NCTM for producing its most recent standards document, Principles and Standards for School Mathematics (2000). On its certificate of commendation, the CSSP noted that Principles and Standards is "a significant and high-quality contribution toward the improvement of mathematics education for all students." The Council also encouraged "prompt, thoughtful, and careful consideration of and thorough review of the recommendations and ideas for implementation by all who share a stake in the effective teaching of mathematics."

The CSSP is a nonprofit organization comprised of the presidents, presidents-elect, and immediate past presidents of more than 60 scientific societies and federations, whose combined membership numbers more than 1 million. CSSP serves as a strong voice in support of science and science education, as the premier national science leadership institute, and as a forum for open, substantive exchanges on current scientific issues.

Its praise continues recognition from the scientific community for NCTM's work. A National Research Council report, released in May 2004, gave NCTM high marks for process of creating Principles and Standards. The report says, "The committee finds the process established by NCTM to solicit comments from the field to be commendable and the process established by them to analyze those comments to be exemplary." The National Research Council is the operating arm of the National Academy of Sciences, the preeminent scientific organization in the United States.

This one is worse:

Q. Has Math Trailblazers been around long enough to demonstrate its effectiveness?

A. Math Trailblazers was the end product of 12 years of research and development, partially supported by grants from the National Science Foundation. It has been pilot tested, revised, field tested and revised once more. The TIMS Project is constantly researching ways to improve student learning as well as ways to help teachers be more effective. This is our passion. Contrast this with the dozens of commercial publishers who put together their textbooks using consultants and development houses that have little deep knowledge of the subject matter and few thoughtful ideas on how to make our children better learners. They just go with the fad of the year.

The TIMS Project and Kendall/Hunt have collected data on student achievement that show that schools adopting Math Trailblazers have made significant improvements on standardized test (See Student Achievement) Moreover, a rigorous scientific study of student achievement on standardized tests comparing students using NSF funded reform curricula to students using traditional curricula showed that the students using reform curricula performed at a statistically significantly increased level (See ARC Center study). In short, Math Trailblazers is not new and it has been proven effective.

This is the publisher's website, and I suppose there's some kind of provision in the Constitution holding that a textbook publisher doesn't have to testify against itself.

But these passages go well beyond misleading by omission. There aren't too many mathematicians out there touting the virtues of Math Trailblazers. And the National Research Council has explicitly characterized Math Trailblazers as an "experiment." The NRC says the results aren't in, and the wording in this passage makes me think the folks at the NRC aren't optimistic:

These 19 curricular projects essentially have been experiments. We owe them a careful reading on their effectiveness. Demands for evaluation may be cast as a sign of failure, but we would rather stress that this examination is a sign of the success of these programs to engage a country in a scholarly debate on the question of curricular effectiveness and the essential underlying question, What is most important for our youth to learn in their studies in mathematics? To summarily blame national decline on a set of curricula whose use has a limited market share lacks credibility. At the same time, to find out if a major investment in an approach is successful and worthwhile is a prime example of responsible policy. In experimentation, success and worthiness are two different measures of experimental value. An experiment can fail and yet be worthy.

An experiment can fail and yet be worthy, you say.

Call me crazy, but I wouldn't write such a line unless I thought there was a distinct possibility failure was the direction things were headed.

-- CatherineJohnson - 25 Jun 2006

NctmReformsAgain 14 Sep 2006 - 16:52 CatherineJohnson

In today's Wall Street Journal (\$):

Arithmetic Problem
Critics of 'Fuzzy' Methods Cheer Educators' Findings;
Drills Without Calculators Taking Cues From Singapore
By JOHN HECHINGER
September 12, 2006; Page A1

The nation's math teachers, on the front lines of a 17-year curriculum war, are getting some new marching orders: Make sure students learn the basics.

In a report to be released today, the National Council of Teachers of Mathematics, which represents 100,000 educators from prekindergarten through college, will give ammunition to traditionalists who believe schools should focus heavily and early on teaching such fundamentals as multiplication tables and long division.

The council's advice is striking because in 1989 it touched off the so-called math wars by promoting open-ended problem solving over drilling. Back then, it recommended that students as young as those in kindergarten use calculators in class.

Those recommendations horrified many educators, especially college math professors alarmed by a rising tide of freshmen needing remediation. The council's 1989 report influenced textbooks and led to what are commonly called "reform math" programs, which are used in school systems across the country.

The new approach puzzled many parents. For example, to solve a basic division problem, 120 divided by 40, students might cross off groups of circles to "discover" that the answer was three.

Infuriated parents dubbed it "fuzzy math" and launched a countermovement. The council says its earlier views had been widely misunderstood and were never intended to excuse students from learning multiplication tables and other fundamentals.

Nevertheless, the council's new guidelines constitute "a remarkable reversal, and it's about time," says Ralph Raimi, a University of Rochester math professor.

Francis Fennell, the council's president, says the latest guidelines move closer to the curriculum of Asian countries such as Singapore, whose students tend to perform better on international tests.

So maybe it wasn't such a great idea after all for IUFSD to ban my Singapore Math course.

new timeline

According to their report, "Curriculum Focal Points," which is subtitled "A Quest for Coherence," students, by second grade, should "develop quick recall of basic addition facts and related subtraction facts." By fourth grade, the report says, students should be fluent with "multiplication and division facts" and should start working with decimals and fractions. By fifth, they should know the "standard algorithm" for division -- in other words, long division -- and should start adding and subtracting decimals and fractions. By sixth grade, students should be moving on to multiplication and division of fractions and decimals. By seventh and eighth grades, they should use algebra to solve linear equations.

Here's the Singapore sequence.

Lutherans turning into Catholics

A recent study by the Thomas B. Fordham Foundation, a Washington nonprofit group, found that only two dozen states specified that students needed to know the multiplication tables. Many allowed calculators in early grades.

Chester E. Finn Jr., the foundation's president and a former top official at the U.S. Department of Education, blamed the earlier math-council guidelines for state standards that neglect the basics. He described the new advice as a "sea change," saying that "it's a little bit like Lutherans deciding to become Catholics after the Reformation."

Understanding math, rather than parroting answers to poorly understood equations, was the goal of the council's controversial 1989 standards. Those guidelines called on teachers to promote estimation, rather than precise answers. For example, an elementary-school student tackling the problem 4,783 divided by 13 should instead divide 4,800 by 12 to arrive at "about 400," the 1989 report said. The council said this approach would enable children using calculators to "decide whether the correct keys were pressed and whether the calculator result is reasonable."

"The calculator renders obsolete much of the complex pencil-and-paper proficiency traditionally emphasized in mathematics courses," the council said then. In 2000, in another report, the council backed away somewhat from that position.

Still, in response to the earlier recommendations, many school systems required children to describe in writing the reasoning behind their answers. Some parents complained that students ended up writing about math, rather than doing it.

As the debate heated up, concern grew about U.S. students' math competence. In 2003, Trends in International Mathematics and Science Study, a test that compares student achievement in many countries, ranked U.S. students just 15th in eighth-grade math skills, behind both Australia and the Slovak Republic. Singapore ranked No. 1, followed by South Korea and Hong Kong. Fueling concern about the quality of elementary and high-school instruction: one in five U.S. college freshmen now need a remedial math course, according to the National Science Board.

low-income students

This is very exciting. The AIR report (pdf file) led me to believe that Singapore Math had been a flop in low-income schools because the student mobility is so high (and see Hirsch on this subject, too):

If school systems adopt the math council's new approach, their classes might resemble those at Garfield Elementary School in Revere, Mass., just north of Boston. Three-quarters of Garfield's students receive free and reduced lunches, and many are the children of recent immigrants from such countries as Brazil, Cambodia and El Salvador.

Three years ago, Garfield started using Singapore Math, a curriculum modeled on that country's official program and now used in about 300 school systems in the U.S. Many school systems and parents regard Singapore Math as an antidote for "reform math" programs that arose from the math council's earlier recommendations.

According to preliminary results, the percentage of Garfield students failing the math portion of the fourth-grade state achievement test last year fell to 7% from 23% in 2005. Those rated advanced or proficient rose to 43% from 40%.

Last week, a fourth-grade class at Garfield opened its lesson with Singapore's "mental math," a 10-minute warm-up requiring students to recall facts and solve computation questions without pencil and paper. "In your heads, take the denominator of the fraction three-quarters, take the next odd number that follows that number. Add to that number, the number of ounces in a cup. What is nine less than that number?" asked teacher Janis Halloran. A sea of hands shot up. (The answer: four.)

Ms. Halloran then moved on to simple pencil-and-paper algebra problems. "The sum of two numbers is 63," one problem reads. "The smaller number is half the bigger number. What is the smaller number? What is the bigger number?" (The answers: 21 and 42.)

In this class, the students didn't use the lettered variables that are so prevalent in standard algebraic equations. Instead, they arrived at answers using Cuisenaire rods, sticks of varying colors and lengths that they manipulate into patterns on the tops of their desks. The children use the rods to learn about the relationship between multiplication and geometry. The goal: a visceral and deep understanding of math concepts.

"It just makes everything easier for you," says fifth-grader Jailene Paz, 10 years old.

Cuisinaire rods for bar models!

That's so cool!

TERC time

The Singapore Math curriculum differs sharply from reform math programs, which often ask students to "discover" on their own the way to perform multiplication and division and other operations, and have come to be known as "constructivist" math.

One reform math program, "Investigations in Number, Data and Space," is used in 800 school systems and has become a lightning rod for critics. TERC, a Cambridge, Mass., nonprofit organization, developed that program, and Pearson Scott Foresman, a unit of Pearson PLC, London, distributes it to schools.

parents don't get it part 1

Ken Mayer, a spokesman for TERC, says many parents have a "misconception" that Investigations doesn't value computation. He says many school systems, such as Boston's, have seen gains in test scores using the program. "Fluency with number facts is critical," he says.

parents don't get it part 2

Polle Zellweger and her husband, Jock Mackinlay, both computer scientists, moved to Bellevue, Wash., from Palo Alto, Calif., two years ago so their two children could attend its highly regarded public schools. She and her husband grew suspicious of the school's Investigations program. This summer, they had both children take a California grade-level achievement test, and both answered only about 70% of the questions correctly. Ms. Zellweger and her husband started tutoring their children an hour a day to catch up.

"It was a really weird feeling," says their daughter, Molly Mackinlay, 15. "I do really well in school. I am getting A-pluses in math classes. Then, I take a math test from a different state, and I'm not able to finish half the questions."

Eric McDowell, who oversees Bellevue's math curriculum, says parents misunderstand Investigations.

If it weren't for the parents, teaching would be a great job.

math wars and war wars

In the Alpine School District in Utah, parent Oak Norton, an accountant, has gathered petitions from 1,000 families to protest the use of Investigations. His complaints began more than two years ago, when he discovered at a parent conference that his oldest child, then in third grade, wasn't being taught the multiplication tables.

Barry Graff, a top Alpine school administrator, says the system has added more traditional computation exercises. Over the next year, Alpine plans to give each school a choice between Investigations or a more conventional approach. Mr. Graff, who says Alpine test scores tend to be at or above state averages, expects critics to keep up the attacks and welcomes the national math council's efforts to provide grade-by-grade guidance on what children should learn.

"Other than the war in Iraq, I don't think there's anything more controversial to bring up than math," he says. "The debate will drive us eventually to be in the right place."

wow

I bet things are hopping over at math-teach & math-learn.

[pause]

hmm

No action thus far.

Once Wayne Bishop posts this baby, we'll be in a shooting war.

update: Bishop's got it!

let the fun begin

what Singapore students can do at the end of 7th grade

-- CatherineJohnson - 12 Sep 2006

MathTrailblazersStudent 20 Sep 2006 - 17:13 CatherineJohnson

from one of our friends:

A friend of mine has a daughter who made A's in 4th and 5th grade Trailblazers. She did not know her times tables, ditto division facts, and her fraction knowledge was very basic to nonexistent.

She did not know that you could divide a fraction.*

This was the A student. I can't imagine what the C student looks like in this class.

Another friend's gifted kid dropped dramatically in the standardized tests knocking him out of the high math track for middle school. I told her that it was probably math facts and fractions. She found out through an online assessment that, in fact, it was.

If a teacher looks at the good stuff that curriculums like Trailblazers offer, supplementing what is missing, then it is probably alright. But when they follow it to the letter, it appears to me to be a disaster waiting to happen.

* Math Trailblazers does not teach the division of a fraction by a fraction.

-- CatherineJohnson - 13 Sep 2006

LindaMoranListserv 11 Dec 2006 - 19:25 CatherineJohnson

I think everyone here knows about Linda Moran's Teens and Tweens blog.

I've recently (re)discovered that she has a listserv attached to the blog.

I joined last week, and I think some of you might like to join as well. There have been some very interesting posts to the listserv that I don't believe have been posted to the blog itself — and that I don't expect to see posted to the blog itself.

-- CatherineJohnson - 09 Dec 2006

WebForm
TopicType: SubjectArea
SubjectArea: