comments...

The sample page below -- from a grade 6 workbook in the Singapore Math curriculum -- first appeared in CompareAndContrastPart3.

I draw your attention to problem h: I was horrified to realize that I was not dead certain about what order to do the operations in.

I knew the following: the 1/5+1/4, in parentheses, had to be figured out first. But what about after that? It seemed that there were two things I could do after that, but they don't give the same answer.

1. I could divide 4 by 1/4+1/5, and then multiply the result by 3/10 (answer: 8/3). Or:

2. I could multiply 1/4+1/5 by 3/10, then divide 4 by that amount (answer: 800/27).

They can't both be right!

What's at issue here is the order of operations in a math expression. Most of us over 40 were (once) taught how to parse arbitrary math expressions that didn't have any parentheses in them: but for most of us it's a very rusty skill, because people typically use parentheses in math expressions to clarify what they mean.

Order of operations really matters in computer programming, for example. Most of us who program computers are in the habit of putting parentheses around parts of the expression that we want computed first, because we don't want anyone -- not other programmers, and definitely not the compiler! -- to misunderstand our intentions.

I realized, when I saw this problem, that I've been so softened by years of reading and writing parentheses that I no longer remembered exactly what the order of operations were supposed to be. Do you do equivalent operations from right to left, or left to right? Judging from the discussion in AnotherWikiPossibility, noone else was totally clear on it either.

But JdFisher eventually straightened us out. Here is what he wrote:

[The mnemonic to remember is] Please excuse my dear Aunt Sally. That is, (1) Parentheses, (2) Exponents, (3) Multiplication and Division [from left to right] (4) Addition and Subtraction [from left to right].

So, first you evaluate what is in parentheses, taking care of that 1/4+1/5 term in problem h. After you've done that, then you compute any powers that appear in the problem. Then you calculate all multiplications and divisions. Multiplications and divisions have equal precedence in numerical calculations. But it matters what order you do them in, so we make an arbitrary choice and say that you do the multiplications and divisions in the order they appear, from left to right.

This says that in problem h, the right thing to do is to first divide 4 by 1/4+1/5, because the division, being leftmost, comes first. Therefore 8/3 is the right answer.

Finally, addition and subtraction come in dead last. If that 1/4+1/5 hadn't been in parentheses, that + sign would have been the last thing evaluated. The rule here is also to calculate additions and subtractions in the order they appear from left to right.

This order of operations looks arbitrary -- but it isn't. The reason that things happen in the order they do -- first taking powers, then multiplication and division, then addition and subtraction -- has to do with the way errors in computation grow when you calculate an expression. (The parentheses-first rule exists so that the expression-writer has ultimate control over the order in which things get computed).

This order is designed so that errors that happen early in the computation will have a big impact on the answer, and are therefore more likely to be caught before you get too much farther. It all boils down to the fact that it stinks to discover that you made an error 25 steps ago; better to find an error one or two steps later. (Here is a deeper discussion of this at Math And Text).

What caught me up short, when I saw problem h, was the left-to-right rule. There isn't a rationale for this --it's arbitrary; either left-to-right or right-to-left had to be chosen -- and it's darned hard for some of us to remember arbitrary rules involving left and right. Frankly, it's hard for not only some kids but for some of us adults to even TELL left from right.

So for kids with trouble telling left from right, you can expect order-of-operations problems to be tricky. But they are still well worth doing, because it's a type of logic that is really needed by people doing computer programming of all sorts -- not only geek-type programming, but also spreadsheet (i.e. accounting and finance) programming.

And you can't assume that your kid won't be the type for that, either. Nothing with kids is set in stone.

comments...

He earned a '4' on both tests, English Language Arts and math.

(1 and 2 mean 'did not meet standards;' 3 is 'met standards' or 'sufficient' or something like that, and 4 is, in theory, 'advanced.')

Whew.

He made it to 4 by the skin of his teeth, but he made it.

Plus I've managed to discover that a number of children in his Phase 4 class ended up with 3s. These are kids who were put in Phase 4 on the basis of natural ability, as opposed to having their mothers march them through Saxon Math for a year and a day.

So I don't feel too guilty being happy Christopher nosed ahead of a couple of them.

I mention natural ability because last fall the middle school guidance counselor told me Christopher wouldn't 'qualify' to move to the Phase 4 class in 6th grade no matter how well he did this year, in 5th.

'He's a 3,' he said. (I think he used those exact words, if memory serves. 'He's a 3.')

That ticked me off royally.

btw, I do know that the difference between Christopher's extremely-low-4 and another kid's extremely-high-3 is ... not a difference. At least, I'm pretty sure it's not a difference. (I need to learn some more math, quick.)

I also know I should knock off the gloating until we clear our last hurdle, which is getting Christopher into Phase 4 for the fall.

I realize I said he's in Phase 4 now. That's a bit of a long story, but the upshot is that he moved from Phase 3 to Phase 4 last February. Unfortunately, the middle school, which he enters in the fall, is looking to cut as many kids from the accelerated class as they can. This is a formal, stated policy.

So I keep thinking, Last hired, first fired.

But now that Christopher has almost all As in Phase 4, and a 4 on the TONYSS ... and some of his Phase 4 classmates don't have 4s & don't (necessarily) have almost-all-As ... maybe he'll make the cut.

If he doesn't, he will once we get through working them over. :- )

(Is that how you make a smiley face?)

I need some champagne.

This all started one year ago, when Christopher came home with his 39 on Unit 6 & I discovered that not only was he horrifyingly behind the rest of his classmates, his classmates were horrifyingly behind the rest of the world.

algebra in 8th grade

Wayne Wickelgren says they should, and when I read his book I took his word for it.

All of my work with Christopher has been focused on the goal of getting him into algebra in the 8th grade.

In the U.S., only the accelerated kids take algebra in the 8th grade.

This meant Christopher had to go from flunking regular-track math, to succeeding in regular-track math, to jumping ahead into accelerated math, to succeeding in accelerated math, and he had to do it in one year.

And he did.

Back to algebra-in-the-8th-grade: as I say, I just took Wayne Wickelgren's word for it.

Plus there was the global horserace aspect of things: if perfectly average kids in Singapore take algebra in the 8th grade, I wanted my own perfectly average kid to take algebra in the 8th grade, too.

But since last summer I've become aware that neither mathematicians nor mathematics educators are of one mind on this question.

I don't know what to think.



GoodNewsBadNews

^*Test of New York State Standards




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

comments...

AnneDwyerSingaporeMathClassPart2 20 Jun 2005 - 23:45 CatherineJohnson

I just noticed that Anne Dwyer has new material on her summer course over on her wiki page.

Go take a look!


AnneDwyerSummerMathClass

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GoodNewsBadNews 20 Jun 2005 - 23:51 CatherineJohnson

So I was driving along the freeway (I know, I know, there are no freeways in New Jersey) with Christopher in the back seat, and we were chatting about his two 4s on the TONYSS, and how good he was getting at math, and how great the whole year went ... when he said, "The only thing I'm bad in is science. I'm terrible in science."

"You're not terrible in science," I said. "You're good in science."

Silence from the back seat.

"Why do you think you're terrible in science?"

"I got a 60 on my last test."

update

• why do I find this stuff out in the last week of school?

• I am not going to be homeschooling science. No matter what.
  

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

comments...

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CountYourBlessings 20 Jun 2005 - 23:59 CatherineJohnson

I just found this post from a TERC teacher on a Math Forum thread:

I have parents who are deciding that their child should be advanced in their math levels. In particular, a child who is 5th grade is working with his dad who has been teaching the child division, ratios, and pushing the child through topics. Now the parent feels the child will be bored and wasting time as we go through different topics. The parent feels that he has already experienced these topics and does not need to hear it again in class. He wants the child to be pushed into the 6th grade curriculum. The child is a good student. I told the parent that the child could be challenged by looking at the different ways that we solve problems and investigate the similiarity to what the parent taught the child compared to the class activities. He feels that this is a waste of time because he already knows the topic. How has anyone else handled this type of situation? I did tell the parent that I would find challenging work on the topics that the class is working but expand on the class topics. How have other handled this situation? I know that this must happend in other schools. Any help would be really appreciated.

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AnotherGemFromMathForum 21 Jun 2005 - 00:11 CatherineJohnson

Another from the TERC teacher:

I'd first like to say that I'm sorry, Bob and Mark, for the fact that you hold such an uninformed view of the Investigations program. It really sounds as though maybe your schools have not done a proper job of introducing the program to parents. It is in fact quite different from traditional textbook math. It looks different from what you and I were doing in elementary math. I strongly feel that if a school makes a switch, they need to share their reasoning and market the idea so that all stakeholders are aware and have the opportunity to make an informed judgement.

She is speaking to a homeschooling father who taught 5th grade.


HowToGetParentBuyIn
EverydayMathDoesItToo
ILoveTheWorldWideWeb
ATeacherUsingTrailblazers
NoCommentPart2
CarolynFisksArticle

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TakingABreakPart8 21 Jun 2005 - 01:06 CatherineJohnson

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SuccessAtAllCosts 21 Jun 2005 - 03:57 CarolynJohnston

I actually learned most of what I know about teaching during the couple of years when my son was in intensive early intervention for autism spectrum disorder.

I had done a lot of teaching before that, too, mostly at the college level, with mixed results. I really wish I'd known then what I know now. And here's the most important thing I think I've learned:

To get a kid turned around in math or in anything else, you make every experience a successful one. You put that ahead of everything else, including what most of us would think of as 'making progress'.

Here's some specific directions to go with the overall theory:

1. Don't give a kid a problem unless you are sure that he can do it if he tries (if he's just starting out and help is necessary for him to succeed, that's okay: the point is that you don't let him flounder and fail).

The dog-training equivalent to this is the idea that if you don't think a dog will come when you call him -- if he is too interested in another dog, or chasing a squirrel, or whatever -- you don't call the dog at all; instead, you silently go over and get him. The reasoning behind this is that every time you call him and he fails to come, you are doing actual damage to his training. Same thing with a kid trying to learn math; you don't set him to do something at which you suspect he will fail. His memory has to be dominated by successful experiences (this, by the way, is what will create a kid who likes math).

2. Here's a weird one, but a good one: if you want to give him something challenging, first give him something easy, even though it will mean more work for him. If you want to give him a tough long division, give him three or four easy ones first.

The best example of this that I can give is from Ben's early training. He was doing sequencing problems -- the sort in which you give a kid three pictures from a story, and have him put them in order. He was quite awful at them, and he had gotten to the point where he would sit and stare into space, utterly unresponsive, if I asked him to do one. It became a battle of wills, with him staring into space, and me demanding his attention.

I finally came up with the following idea for getting him to try harder ones; first, I would give him three problems he had done before. He had a fantastic memory, so he could just knock those off without thinking, and then he'd get praise.

Then I'd give him three more that he hadn't seen before, but that were incredibly easy. I did this so he'd know he wasn't being given anything for free.

Then I'd stick in the tough one, the really hard one, that I really wanted him to do. He'd tuck into that one just like he had the others. Ha!

It took less time to do these seven problems, ending with the last tough one, than just the single tough one.

It's paradoxical and unintuitive, but it works like a charm. If you have a kid struggling with long division and you'd be thrilled if he tried two tough ones a night, give him six easy ones first.

Actually, it goes against the parental grain; it feels as though you're letting him get away with something. And you are: you're letting him get away with doing six extra problems.

Try it. It's one of the best tricks I know.

comments...

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AutisticKidsMeetTerc 21 Jun 2005 - 13:02 CatherineJohnson

from Mary, a question posted to the TERC support thread at Math Forum:

autistic childen
Posted: Apr 8, 2002 1:40 PM

Children afflicted with autism in my fourth grade class seem to be confused by problems with multiple solution paths. This approach appears to cause stress and frustration. When I use a more traditional approach, they seem to be more successful. Has anyone else encountered a similar situation?

Hmm.

No one on the TERC support thread seems to have an answer for Mary.

I wonder why that is.

Could it possibly be because all children, autistic or not, experience stress and frustration when you make them do problems with multiple solution paths?

Could it possibly be because all children, autistic or not, prefer repetition and routine to novelty and change?

Could it possibly be because the folk psychology embedded in TERC flies directly in the face of everything we know about the neuropsychology of elementary school-age children?

You got the answer, Johnny?

Good for you!

Now go get it again.

And find another solution path while you're at it.

Progressive Education: One Parent's Journey

This notion that young kids should not be submitted to repetitive practice seems silly. Young brains are predisposed to repeating everything they hear. Five year olds love to sing the same ditty time and again. Parents know the young mind loves repetition, yet at precisely this time the progressive viewpoint refuses to take advantage of this desire to help kids learn basic math.

That observation is from an article (pdf file) by a father who enrolled his children in a progressive school as he was starting a new job:

At about the same time, I started a job administering a large federal education grant project that sought to advance progressive education principles in the teaching of math and science in the northwest region of the United States....

Progressive education is not a new reform. [ed.: I'll say.] .... Developed by John Dewey, progressivism shuns teacher-directed curriculum standards in favor of open-ended projects and hands-on activities. ...”

In the latter half of the 20th century, elements of progressivism became the dominant pedagogy in public schools. ... [Progressivism] is the philosophical belief that schools should be “child-centered.” That is, schools should not directly teach a pre-defined canon of academic knowledge and skills to students, but rather, students should be taught through projects and activities, carefully guided by a trained teacher in an ongoing process of discovery.

All of the professional teachers’ organizations, which determine curriculum standards, accept this idea as self-evident fact. “Within the educational community,” says the noted critic of progressivism, E.D. Hirsch, “there is no thinkable alternative.”

As the modern form of progressivism grew to dominate the public education landscape, a body of teaching beliefs and practices emerged, and to some extent became codified, in the form of the various “standards” documents put out by the professional teaching organizations. I saw the impact of these beliefs in my own kids’ classrooms.

read the whole thing

thanks to: NYC HOLD

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RequestForWordProblemsFromConstructivistTexts 21 Jun 2005 - 14:13 CatherineJohnson

Remember

Barry Garelick, author of 'An A-Maze-ing Approach to Math' in Education Next is seeking examples of word problems. (I'll get mine added shortly -- )

His wiki page is here: ArticleHelp

You can use the Comments box to add a word problem, or you can hit 'edit this page' and type your word problem into the editing box.

Wiki directions here: WikiHowTo

comments...

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WikisForYourDesktop 21 Jun 2005 - 15:29 CatherineJohnson

Ever since Carolyn introduced me to wikis, I've been obsessed.

Wikis are amazing. Everything you need is in one place, where you can find it.

As a direct result, your searching-frantically-for-lost-file-and/or-document time is cut to practically nothing.

Incredible.

Turns out you can get wiki software for your computer, too. Ed has already downloaded Voodoo Pad, for Mac users, and I'm hoping to get to that today.


+ + +


... um ... it's possible I need a little more coaching on procrastination and time management. OTOH, I have completed Mrs. DArcy's memory book not only on time, but early.

Not only that, but I was able instantly to locate the adhesive spray I bought 2 years ago when I was planning to put together a photo album of the kids for my mom.

Twenty years from now, there will be no unsightly yellowing on the cover of Mrs. Darcy's memory book where I glued on the class picnic photo.


+ + +


The 43 Folders blog has a list of all or most of the wiki software here.

43 Folders has its own wiki, too, where people talk about productivity & time management.

They're also posting Personal Mantras.

I should go add our family's motto:

no common sense-y

Of course, no common sense-y probably won't add much to anyone's productivity or time management skills.

comments...

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TeachYourChildToTypeThisSummer 21 Jun 2005 - 21:33 CatherineJohnson

Carolyn mentioned that she wants Ben to learn to touch-type this summer.

Turns out it's easy to teach touch typing; you don't need a book or a software program.

Just use highliters to color in this chart, show your child where to put his fingers on the keyboard, and have him type the alphabet.

And that's it. He doesn't need to type anything other than the alphabet.





Here's a small version of the color chart you'll make.





A large version of this color chart is here.


I picked up this tip yesterday from Faye Gordon, business teacher for BOCES (Board of Cooperative Educational Services) here in New York state. BOCES handles vocational training, quite a lot of adult ed, and special ed.

Faye runs the Office Skills class, where she teaches her students to type using this method.

She happened onto it when she graduated from college and applied for a job that required typing. She was rusty, and typed only a slow 60 wpm on the test.

So she went home and practiced typing the alphabet for the next two weeks. She didn't practice all day long, just a few times each day. Always the alphabet. No text.

When she went back for a second test she typed 80 wpm.

All of her students learn to touch type using only the alphabet.

A couple of years ago Faye saw a headline on Consumer Reports for an article on how to improve your typing speed.

She bought the magazine, and it turned out their big advice was to type the alphabet.

"I could have written the article myself," she said.

+ + +

update

The whole time I was growing up, my dad kept telling my sisters and me, 'Learn to type so you can support yourself in case anything happens to your husband.'

I probably type about 110 wpm.

update 2

Barry G just left this comment & I had to pull it up front:

My mom told me the same thing. In fact, she said learn to type and I'll buy you a typewriter for graduation (from high school; this was in the days when not every kid automatically got a car). So I learned to type and she bought me an Olympia typewriter which I still have. The first job I got out of college was at the U.S. EPA, as a typist. (Jobs were scarce then). They realized I was good at other things besides typing so I made my way up the ladder. Yes, I work at USEPA now, but I had a stint in the private sector for a while. Not as a typist though. But I do think better at a keyboard than writing longhand. I recommend touch typing for all.

I love it!

update 3

I just showed Ed Barry's post, and he said his mom did the same thing.

She told him if he learned to type, she would get him a typewriter for high school graduation.

He did, and she did.

She got him a Smith Corona portable electric typewriter that lasted all the way until we got our first KayPro back in whatever year that was.

He wrote his dissertation on the Smith Corona.

FreeWorksheets
TreadingWater

SummerSupplement
SummerSupplementTime
SummerSupplementTimePart2
SummerSupplementTimePart3
SummerSupplementTimePart4 (resources for kids who have fallen behind)
SummerSupplementTimePart5 (resources for preventing summer regression)

SaxonPlacementTestsAndGuides
SingaporeMathPlacementTest

comments...

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FirstWikitorial 21 Jun 2005 - 23:24 CatherineJohnson

Oh my goodness.

The first 'Wikitorial' has been published, by the Los Angeles Times.

LA Times readers could edit the editorial.

It was up for 2 days before they had to take it down, for reasons I won't go into.

comments...

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TreadingWater 22 Jun 2005 - 03:50 CarolynJohnston

Sometimes, the kid just isn't up to doing a whole math lesson (or more likely, I'm not up to giving him one, since it's something of a battle).

On those nights, something like these math worksheet generators can come in very handy. There are a lot of these generators around, but this one is very configurable; you can set the number of columns and rows of problems, and the difficulty of the problem, and the numbers of significant digits in the solution, and so forth.

Give the kid a worksheet with a few problems on it, and let him get in a little practice. Resist the urge to give him more than 4 or 5 problems on a sheet; make them easy. The most important thing is to make every learning experience a success -- especially true if this is material he is already supposed to know how to do, and will be doing independently.

We especially found the sheets for fraction and decimal long division useful. That's a skill that just takes a lot of practice.


+ + +


Catherine here

I'm so glad Carolyn brought up Homeschoolmath.net. I've been meaning to write a post about them forever.

While you're at their site, take a look at their e-books.

I haven't ordered one of the books, only because I'm swimming in math books already, but they look terrific to me.

I've learned just from perusing them online.

For instance, I was trying to figure out whether, if multiplication is repeated addition, division is repeated subtraction. (Yes, I know. I'm embarrassed.)

Logically, it seemed to me that division had to be repeated subtraction.

But for some reason I couldn't 'see it.' (I don't think I was making the jump from the factor itself, which I was subtracting repeatedly from the dividend, to the number of those factors I could subtract. I can't explain it, but I still have trouble, conceptually, with factors versus number of factors . . . and how that relates to addition & subtraction.)

The little division e-book at Homeschoolmath.net had a crystal clear example & explanation that I have never forgotten.

The books teach the algorithms and explains why they work, with no opposition between those two goals -- and precious little discovery, as far as I can tell. Conceptual understanding is taught through direct instruction, and the text is structured (they say) to encourage children to ask questions.)

I just noticed that you can download the 'Mental Addition and Subtraction' e-book for free to take a look, so I'm going to do that.

Carolyn--sorry to jump into your post! I love this site!



FreeWorksheets

SummerSupplement
SummerSupplementTime
SummerSupplementTimePart2
SummerSupplementTimePart3
SummerSupplementTimePart4 (resources for kids who have fallen behind)
SummerSupplementTimePart5 (resources for preventing summer regression)

SaxonPlacementTestsAndGuides
SingaporeMathPlacementTest

TeachYourChildToTypeThisSummer

comments...

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SingaporeWordProblemSampler1 22 Jun 2005 - 04:48 CarolynJohnston

This is a sampler of randomly chosen word problems from the "Primary Mathematics Challenging Word Problems" series. I'm just going to open the books, and write what I see.

For tonight's problem, pick one and give it to your kid, or try one yourself.

For bonus points, spot the problem that was not written by a native English speaker.

Grade 3: String X is 34 cm longer than String Y. String Z is 58 cm longer than string Y. If the total length of Strings X, Y, and Z is 233 cm, find the total length of Strings X and Z. (!!! OK, that was from the 'challenging problems' section of a 'challenging problems' book. Excuses, excuses)

Grade 4: Jane has 70 balloons. 1/10th of them are green, and 3/5 of them are orange. How many more orange balloons than green balloons does she have?

Grade 5: A man bought a dozen sacks of rice at $18 per sack. Each sack of rice weighed 20 kg. He packed half of the rice into bags of 5 kg and sold them at 6.50 per bag. He sold the rest of the rice at $1.50 per kg. Find his total profit.

Grade 6: Henry has 3/4 as many paper clips as Joyce. Joyce has 4/5 as many as Claire. If the three girls have 96 paper clips altogether, how many fewer paper clips does Henry have than Claire?

comments...

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WhatChildrenMustKnow 22 Jun 2005 - 20:58 CatherineJohnson

This essay, by Jerry Rosen, Professor of Mathematics at California State University Northridge, is probably the single best statement I've read on the subject of what children must know.

I'm posting it in full.

First, two notes:

• The expression x^2 means 2 squared. (Right?) For some reason, exponents often show up onscreen as 2^2 (2 squared).

• Politics makes strange bedfellows.

• The one point with which I will disagree slightly is the observation that studying is hard work, not fun. In my own life studying has certainly been hard work, but sometimes it has been fun, too. Fun, or satisfying, or stimulating, or riveting, or amazing, or strengthening, or moving; the list is long. However, when study has been only hard work, and nothing more, it has been worth it. I want my own child to know how to work hard whether he is having fun or not.

Really Learning Math Helps Us Learn About Life

In Kindergarten through Eighth Grade, children need to become arithmetic experts. They must:

Memorize the addition and multiplication tables as well as learn to explain things such as why 2 + 3 = 3 + 2 and 2 X 3 = 3 X 2.

Know how to carry in addition and borrow in subtraction. The "whys" of these operations should also be explained, but some children may not automatically understand "why." Nonetheless, the procedures must still be mastered.

Know how to do long division. Multiplication tables are important in long division because the students need to be able to try things out quickly, in their heads or on scratch paper, when doing division.

Gain facility with adding, multiplying, subtracting and dividing fractions. This is extremely important for algebra. A child who can’t add fractions will have no chance of dealing with a typical algebra problem such as: simplify the following expression: (x/y + y/x). A student who can manipulate fractions will immediately know that you need to get a common denominator. One way to get a common denominator is to just take the product. The simple arithmetic procedure is then applied and: (x/y + y/x) = (x^2 + y^2)/xy.

This kind of problem is beyond the grasp of students who don’t know the basics of arithmetic. Furthermore, most "real life" problems often reduce to solving simple algebra problems.

Gain facility with decimals and percents. Just a few weeks ago I asked my college math class—the one for prospective K-8 teachers—what is 20% of $155,000? It came from the "real life" problem of calculating the down payment on a house. Not one person in the class could solve it. Students should be able to do such a problem in their heads. It involves several important arithmetic notions.

Avoid calculators and computers. These are the worst "tools" for learning basic math and have no value. Algebra is arithmetic with letters in place of numbers and students who have "learned" arithmetic on a calculator will have not gained the necessary skill to learn algebra. As I said before, studying is not fun; it should be hard work. Also students should be able to see if an answer makes sense just by reasoning. There have been no studies demonstrating that machines have any values in math education. On the other hand, there is overwhelming evidence of great harm done by machines.

Gain good study habits. This will serve them well in all future endeavors, including becoming hard-working communist organizers. Success in the above points will give young people the necessary tools to succeed in high school algebra, geometry and trigonometry. Furthermore, when students are well-grounded in the fundamentals, they have more time to develop deeper conceptual understanding. There is no better application of quantity into quality than math education.

The large-scale failure of the U.S. educational system to teach millions of children reading and math needs to be exposed. But it should not be used as an excuse not to teach properly. Neither should low pay or poor working conditions or the students’ poverty. To be sure, these things don’t help and the Party does an excellent job fighting for improvements and organizing along these lines. But there is only one proper way to learn any skill—practice in the fundamentals. Furthermore, children are resilient and capable of extraordinary achievements if the adults can keep their own bias out of the learning equation.

I would urge all K-8 teachers to review or master (as the case may be) all basic arithmetic procedures as well as to become fluent in high school algebra. Even though I have a doctorate in Math, I have to review the basics before teaching them to future teachers. Weakness in a subject is no shame. The real shame is using our own weakness to find rationalizations for not teaching properly.

comments...

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SaxonMagic 22 Jun 2005 - 22:21 CatherineJohnson

Christopher and I were studying Lesson 7, 'Lines and Angles,' in Saxon Math 8/7, when we came to this passage:

We can imagine a two-dimensional world called a plane, a flat world having length and width but not depth. Occupants of a two-dimensional world could not pass over or under other objects because, without depth, 'over' and 'under' would not exist. A one-dimensional world, a line, has length but neither width nor depth. Occupants of a one-dimensional world could not pass over, under, or to either side of other objects. They could only move back and forth on their line.

page 38

We were both utterly enchanted by this image. It was magic.

(click on the map for an article about the novel FLATLAND)

comments...

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IntroducingTheRequestPage 23 Jun 2005 - 02:38 CarolynJohnston

I've just added a new page on the sidebar, under 'KTM User Pages': a Requests Page. On this page, you can:

• Suggest topics;

• Request help with a problem (usually but not necessarily a math problem);

• Give feedback.

Please stop in and drop off a suggestion. We'll be waiting to hear from you.

comments...

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ImGoingToPlayland 23 Jun 2005 - 17:56 CatherineJohnson

Christopher graduated from 5th grade this morning.

I wept all through the Star Spangled Banner.

Now we're going to Playland, because Christopher, along with another student, won the school's Distinguished Student Award!

We're still spinning; we had no idea Christopher was going to win an award; we had no idea they even had a Distinguished Student award.

In his short talk about Christopher, which Christopher's teacher wrote, the principal mentioned how hard Christopher had worked to improve his math. (Thank you!)

That put fresh wind in my sails.

We also learned, yesterday, that Christopher will be in Phase 4 math next year, in sixth grade.

One year ago, almost to the day, I was panicked. Christopher had failed his final two unit tests in math—two out of 6—and I knew I had to do something.

And now here we are, major thanks to Saxon Math.

If the folks at Saxon want me to write testimonials for them, I will.

reactive teaching redux

I know Christopher is an 'n of 1.'

Nevertheless, after today I'm more strongly convinced than ever that if something is wrong with Christopher's math education -- either in the teaching or in the curriculum -- I need to teach my own coherent curriculum here at home.


MathInTheBlood
NowThatWereBothHere
ReactiveTeaching
ThingsWeHaveLearned



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

comments...

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FreeWorksheets 23 Jun 2005 - 22:15 CatherineJohnson

from SusanS:

Two more sites with free math worksheets (and other free stuff) are edhelpers.com and superkids.com. I do love the free stuff.

Thank you!

our favorite math supplements

We are slowly but surely pulling together the sidebar pages, so you might want to take a look from time to time.

We also need to get a reader recommendation page going.

I'm adding Susan's recommendations to the 'our favorite supplements' page so they'll be where people can find them easily.

I'll also gather together the grammar, spelling, handwriting, etc. book & curriculum recommendations into one place, with links to the original reader comments. These are invaluable, so keep them coming!

Back to online math resources, also remember Carolyn's recommendation:

... These math worksheet generators can come in very handy.... very configurable; you can set the number of columns and rows of problems, and the difficulty of the problem, and the numbers of significant digits in the solution, and so forth....

We especially found the sheets for fraction and decimal long division useful. That's a skill that just takes a lot of practice.

computer learning versus paper-and-pencil

Susan inspired me finally to track down some of my favorite online resources and get them entered on the Our Favorite Supplements page.

But first I should say that I'm leery of online math practice, for 3 reasons:

• Christopher has never learned well using a computer

• I've seen research showing a slight decline in student achievement in Israeli schools after the introduction of computers in classrooms

• Jakob Nielsen says people read 25% more slowly onscreen than on paper

Christopher didn't really get his math facts down cold until we started doing the Saxon fast fact paper-and-pencil worksheets.

He didn't make any headway that I could see using a software math facts program, and I don't think he made much progress using standard flash cards, either.

To be fair, we have problems using materials like flash cards, since I'm constantly having to hide them from Andrew, which of course makes it harder to find them when I need them, which, in turn, makes me tend to use them less than I would if they were easy to get to ...

So I don't know whether anyone should be drawing conclusions from my flashcard experience.

But when it comes to computers-versus-paper and pencil, if you've got time to print out the worksheets Carolyn & Susan have pointed you to, that's probably the better choice.

Online 'worksheets' may be to paper worksheets what fast food is to homemade.

That said, I've eaten plenty of fast food in my day, and so have my kids.

So here's one of the main online resources I've liked thus far.

Saxon Math online problems and math activities

• I've seen a number of parents around the web recommend this Saxon Math 'fast facts' generator. The page is clean, simple, and visually compelling. You decide which math-fact problems you want to do, how difficult the problems should be, and how many you want to do. You can also do timed or untimed problem sets. That's great, because kids love seeing their timing get faster.

• Saxon online math activities I love these things. Don't know why; I just do. They're fun.

• Here are the 5th grade activities.
  Apparently the site now tells you which activities to do after which lessons in the book; plus you can download the activities for use when you are not online.

• Saxon online equivalent fractions These are great. OK, I'm sold. Forget the Israeli kids; we're doing online equivalent fractions this summer.

• Saxon online rectangular area

TreadingWater

SummerSupplement
SummerSupplementTime
SummerSupplementTimePart2
SummerSupplementTimePart3
SummerSupplementTimePart4
SummerSupplementTimePart5 (resources for preventing summer regression)

SaxonPlacementTestsAndGuides
SingaporeMathPlacementTest

TeachYourChildToTypeThisSummer

And lots more....

comments...

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ImportantQuestionFromJoanneCobaskoOfSocmm 24 Jun 2005 - 00:39 CatherineJohnson

This came today from Jo Anne Cobasko, founder of SOCMM in Thousand Oaks, CA:

Re: your way of teaching your son math.

How do you deal with having to use the fuzzy methods for solving [homework & test] problems?

With EM the teachers hold the kids acountable on exams for having to learn Lattice mutiplication and all the other useless methods.

That is the biggest problem we see; parents don't know how to help. Kids miss it on tests; teachers give everybody C's instead of F's on exams.

Talk about your $64,000 question.

There probably hasn't been a day in the past year that I haven't thought about this.

It has been a constant question of:

How much time can I take away from the school's chosen math curriculum to devote to my chosen math curriculum?

Carolyn's going to start writing about this topic tonight, and I'll follow up in a couple of days (we're off to Washington D.C. tomorrow, and I have to get a letter of recommendation written for Christopher's fantastic teacher, Mrs. D'Arcy, first.)

Carolyn and I can do point-counterpoint on this, because while Ben has been using a constructivist curriculum (Everyday Math), Christopher has not. Until this year our school used SRA Math. The little kids started Trailblazers this year; 4th and 5th graders will switch to Trailblazers next year.

So I had an easier row to hoe. SRA is wildly incoherent and hard as the dickens to teach. But there's no lattice multiplication.

Even so, I was on the edge of my seat. I didn't know if I could do what I was doing.

I didn't know if Christopher could do what I had decided we would do.

This year was a leap of faith.

As to the how-to, I'll say one thing tonight:

buy a copy of your child's textbook

Also buy the teacher's guide or teacher's edition.

This is essential.

You must round up all of the 'official materials' you possibly can.

You should buy a copy of your child's textbook and teacher's edition no matter what curriculum your school is using.

Go to the publisher's web site to find out what they're selling, then look for used copies at Amazon, alibris, eBay, Abe books, etc.

You'll find them; they're all over the place. I've mentioned that I own a used copy of the 5th grade Trailblazers Student Guide. I bought it on Amazon. At the moment a copy of the teacher's guide is in my Amazon basket.


This is the beauty of the internet.

Instead of saving for retirement, I can buy used copies of teacher's guides for all of the inferior constructivist mathematics curricula my child isn't actually using.


TERC doesn't have a textbook or a teacher's answer book. (Speechless.) But it's got to have something, or they couldn't sell it. (OK, yes, I do understand that they sell schools a whopping big box of manipulatives. A whopping big expensive box of manipulatives, I'm guessing. You don't want that.)

On the other hand, if your child is using TERC, you may be in luck (well, not 'in luck.')

I think I may have stumbled onto THE, or one of THE, TERC 'answer sites'.

You may find everything you need there -- or, at least, everything that's available.

If you can't get hold of a textbook, or if your child's curriculum doesn't have one, what you need -- what all parents need -- is the list of topics that are going to be covered in the school year.

Many of those lists are probably available online in posted tables of contents. (I'll be scouting for them, and if others find them, please send links.)

You also need a copy of your state standards. (We'll be getting those posted, probably, but for now you can find them all listed in the Thomas B. Fordham Foundation report, The State of State Math Standards 2005, which can always be found quickly on the Recommended Reading page.)

problem help at KTM

Once you have a copy of your child's constructivist textbook, the next challenge is using it.

If there are lessons you don't understand (one year after my Saxon geo-boards arrived in the mail, I still have no idea what to do with the things) Carolyn knows everything!

(OK, I'm a little punchy. Carolyn does know everything, but I'm not sure that's precisely the way to put it in a post....)

Wait.

Stop.

What I'm saying is: Kitchen Table Math is a bliki. Someone here is going to know how to do whatever problem your child is being told to do.

Carolyn's going to know everything in the books, and other people here are going to know plenty, too ... so help is available.

KTM readers -- any parent who is baffled by his child's math text -- should tell us about the problems (via email, Comment, or wiki page edit) and let us help.







ImportantQuestionPart2
OutsmartingTheTests

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

comments...

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ImportantQuestionPart2 24 Jun 2005 - 04:54 CarolynJohnston

A question from Catherine's most recent post:

How do you deal with your kid's fuzzy math curriculum while simultaneously working with him on math at home?

This is what I was doing these past two years. The school my son goes to switched, at the end of his thrird grade year, from Saxon Math to the Everyday Math curriculum. My son, who had been doing well in Saxon Math, immediately began to struggle.

It was impossible to help him with his homework; all that ever came home was a "Student Math Journal", with an incoherent, constantly churning set of problems (there was also a reference manual, as it turns out, but it bore no relation to the sequence of topics and was generally useless).

At the end of fourth grade, I told his teachers that I was on the verge of taking Ben out of the regular math class to teach him myself, just for math. Noone told me I couldn't do this, by the way. Homeschool laws vary from state to state. Ben also has an IEP (Individualized Education Program), which gives his parents and teachers a lot of latitude to determine and implement a curriculum that is tailored for him.

So I could have taken him out of class for math if I'd wanted to; but I didn't want to. Math is his strongest subject, and I wanted him to have the experience of being in the class with the other kids, and being one of the stronger kids. We decided to keep him in regular 5th grade math, which unfortunately meant Everyday Math.

Anyway, that's one of your options; see if you can take your child out for homeschooling in the one topic. If he only has enough room in his life for one math curriculum, I would do that sooner than use, exclusively, a crummy math curriculum that won't prepare him for higher level math.

In the fall, I began supplementing at home from Saxon Math, while Ben was also doing Everyday Math at school. We did both curricula at the same time, and neither one wholeheartedly; we definitely had one foot in each world. By contrast, when Catherine was first working with Christopher (before he was placed in the most advanced math class, at midyear), she would do the regular math homework for him in order to enable him to focus on his Saxon math. That took more courage than I had; but Christopher was quickly doing much better in his regular SRA Math class than he had been, which was encouraging, and also what you'd expect under normal circumstances when a kid is being supplemented with a good curriculum.

Learning multi-digit multiplication in 4 different ways, ironically, means a child is spending more time learning to multiply in Everyday Math than he would if he were learning to do it in the traditional way. This time is taken out of other topics that are important, but that are largely left by the wayside -- like fraction division (fraction multiplication is only briefly touched on). I taught Ben only the standard algorithms, and he used them exclusively (though he had rather taken to the lattice method, and used that for awhile). We were lucky that his teachers weren't Everyday Math zealots, wedded to the idea that every kid should learn 4 different ways to multiply; they just wanted every kid to know at least one way to do the problems.

I think that it's worth trying to get dispensation from the teacher for the child to learn, and apply, only one algorithm, especially if trying to learn more than one is confusing. You could argue that after all, the notion of multiple intelligences (very beloved in modern education) dictates that kids shouldn't be forced to learn in ways that aren't suited to them; and so shouldn't kids be allowed to pick, and stick with, the multiplication algorithm that best suits them, instead of having to learn all the others as well?

Having to learn only one algorithm for multiplication and division (and making it the standard ones, taught in advance) frees up a lot of time to learn math at home, while the rest of the class is learning the lattice method.

In fifth grade, the Everyday Math curriculum was never so meaty that we couldn't deal with both that and our Saxon supplementation. We had times when we had to work pretty hard on it; for example, those double pan-balance problems were a bit over the top, but that was a short-lived unit.

If we'd had to, we would have punted on Everyday Math. I would have withdrawn him for exclusive homeschooling.

Failing that -- if it's illegal to do that in your state -- my final suggestion is this: focus on the supplementation curriculum, and if necessary take lower grades in the fuzzy math curriculum. First of all, it's unlikely that a kid receiving one-on-one intensive supplementation is going to go belly up in his regular class, no matter how fuzzy it is. Secondly, in elementary school it doesn't matter what grades your kid gets; it doesn't impact his future until he gets to high school, at which point it's too late to go back and supplement all the work he missed when he was younger.

The stakes in this game are really pretty high, so it's okay.


ImportantQuestionFromJoanneCobaskoOfSocmm
OutsmartingTheTests

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

comments...

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TheCraftOfMath 25 Jun 2005 - 05:21 CarolynJohnston

Teaching mathematics as a craft seems to be waning in favor in elementary education. As evidence one might consider a quote from the infamous Steven Leinwand paper from Education Week, "It's time to abandon computational algorithms":

Shouldn't we be as eager to end our obsessive love affair with pencil-and-paper computation as we were to move on from outhouses and sundials? In short, we know and should agree that the long-division "gazinta'' (goes into, as in four "goes into'' 31 seven times ... ) algorithm and its computational cousins are obsolete in light of everyday societal realities.

He claims that the requirement to be able to do math on pencil and paper has been rendered meaningless by the calculator:

Today, real people in real situations regularly put finger to button and make critical decisions about which buttons to press, not where and how to carry threes into hundreds columns. We understand that this change is on the order of magnitude of the outhouse to indoor plumbing in terms of comfort and convenience, and of the sundial to digital timepieces in terms of accuracy and accessibility.

And so, in spite of Leinwand's accusation (in the same paper) that school districts make changes only in geological time, we are currently engaged in a huge cultural experiment testing his theory that kids can gain a knowledge of math without having to put pencil to paper (although, as I mentioned in this post, in some of the constructivist curricula kids are spending much more time learning to multiply than they would have in a classical curriculum).

But putting pencil to paper is part of what I would call the craft of mathematics. I think you just don't get intimate enough with numbers and symbols by just watching them flash by on the computer or calculator. I've seen it over and over in students at the college level; the more they've relied on their calculator, the less of a feel they have for numbers and mathematics, and the less able they are at problem solving. They may feel that they understand you while you lecture, but when it comes to actually doing math, to getting the answers themselves, they can't do it; they're impotent.

I may be misreading the situation. It may be that the same kids who have trouble with problem-solving at the college level had trouble learning the standard algorithms for computation, and therefore rely more heavily on their calculators. But we do have parallels in the employment world -- experienced engineers who find that junior employees rely too heavily on the answers given by their computer models, and designers who find that their juniors who have used CAD software have a weakened sense of design. We don't know whether the new tools are enabling people to enter these fields who wouldn't otherwise have cut the mustard, or whether the tools are actually weakening the skills of able people.

Until we understand why kids who have relied too heavily on calculators for basic computation can't do math, and what is truly essential about the process of teaching kids to do math, we would be wise to continue making huge curricular changes in geological time.

Afternote: in the spirit of knowing your opposition, here is a link to the Leinwand paper. You have to register at the Edweek site to access the article.


swoop and swoop
notes on integer, subtraction, & absolute value study sheet
Wayne Wickelgren on why math is confusing, & Carolyn on procedural memory
KUMON & hands-on math
More Singapore math
Pencil and paper
The craft of math

comments...

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UsingMathInMyWork 25 Jun 2005 - 17:59 CarolynJohnston

When you talk a lot about math and math education, sooner or later someone is bound to ask you what the stuff is good for anyway. This is actually not a hard question to answer if you have lots of examples, so I'd like to gather some.

I'll start with a description of what I do with math at work.

I work for a small remote sensing image processing firm on the Front Range in Colorado. My division of the company specializes in an imaging technology called Synthetic Aperture Radar (shortened, usually, to SAR). SAR is a cool technology that wouldn't even exist without lots of mathematics; when it comes straight out of the can, it doesn't look like an image at all. You have to do a lot of mathematical processing to get it to look like an image; convolution filtering, and Fourier transforms.

Once you have the images, though, the math isn't over by a long shot. A lot of the work that I do is related to developing automated ways to get information derived from satellite images into GIS databases.

GIS stands for Geographic Information Systems. GIS databases store information about what's going on at a particular place in space. Examples: who owns what chunk of land, where the nearest fire department is, where the nearest roads are, whether it's in a flood plain, whether the underlying terrain is steep or not, whether the location is wooded, whether the location is raptor habitat, whether the land in that location is sinking and how fast.

GIS databases not only store all this information, they can be used to answer questions about how different features of a chunk of land might interact. For example, if a piece of land is south-facing in Colorado, and is also wooded, it is likely to be dry and loaded with fuel for a fire in summer. How close is the nearest fire department?

Now, find all locations fitting that description, and ask what the mean time to a fire department response would be for all of them taken together. This is the sort of question that governments need the answers to, and GIS is the tool that does the job.

But it's not my job. I don't actually use GIS databases myself in my work; I figure out ways to get data into them with the least amount of hassle -- that is, as close to fully automatically as possible. A lot of useful information can be extracted automatically from satellite remote sensing imagery, with work, and the work involves mathematics.

GIS databases use map projections, for one thing; you probably learned a little about map projections in earth science. The earth is a sphere and can't be represented accurately on flat paper, so people have come up with different ways of representing maps that trade off accuracy in some areas for undesirable distortions in others. Anyone working in geography needs to know what the map projections and their tradeoffs are.

Understanding map projections is high school math; solid geometry. Just look at these diagrams of spheres inside cones and cylinders.

When I work with remote sensing imagery, I need to be able to associate every pixel in the image with its corresponding place on the earth. That means I need to know how to warp the image from its "spacecraft-centric" coordinate system into a map projection. That warping process is pure math, usually involving polynomials -- the same thing kids do in algebra.

Now I've already done a lot of math, and all I've done is to get the SAR image lined up with the map projection. There's more math to come, and I'll talk about that in my next post.

Continued in: UsingMathInMyWorkPart2.

comments...

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UsingMathInMyWorkPart2 26 Jun 2005 - 05:14 CarolynJohnston

In my last post, I was talking about how math is used in the technical part of my job at my firm.

Now, having warped the image into the map projection I need to use, the pixels in my image may still not be lined up with their true positions on the earth. They are likely to be just a little out of position in many places on the images. Perhaps you've seen Google's new map feature, which uses high-resolution satellite imagery aligned with maps of the area? It's worth a visit. But you'll generally find that the features in the images don't quite line up with the map. The reason is that the earth isn't perfectly flat; there is topographic variation causing foreshortening in the image. If you know the topography of the land under the image, you can fix the problem using a mathematical process called orthorectification.

There are a number of different ways to do orthorectification; one popular way uses rational polynomials. These are fractions with polynomial numerators and denominators. Kids study how to work with these in high school algebra.

Now we have an image that is aligned, pixel to pixel, with the appropriate location on the earth. What could we do with this image?

We might want to extract roads from the image. Your eye does this very easily, even with a SAR image (SAR images don't look like typical images, because we don't see the world at radio wavelengths; but most features are still recognizable. Here is a SAR image of an airport; you can see that the runways are dark). But it is surprisingly difficult to get a computer to extract roads from an image automatically! All automated methods for extracting roads use mathematics extensively -- usually filtering the images to try to find the edges of long, narrow features.

You might want to try to put a number of images together to make a much larger image. This process is called mosaicking, and it's what you need to do if you want to make a huge seamless map like the digital photo maps at the Google maps site. It's trickier than it sounds; you need features on the edges of the all the images to line up at the same time. The standard way to accomplish this is called block adjustment. To do a block adjustment, you need to know what kind of camera or other sensor you took the image with, and how imperfect knowledge of the camera's position affects the image. You use the differences in all the images to correct your information about the positions and pointing directions of the cameras that took the images. When you apply these corrections back to the images, voila; they'll be nicely aligned.

The block adjustment problem is a huge linear system of equations. In high school, you solve smaller systems of linear equations by hand, like the one below.

2x+3y- z =15
x - y =1
8x-2y+4z =13

Block adjustment problems are just like the one above, but they have thousands of lines, and thousands of variables; you program computers to solve them, but the basic idea is the same.

This is just a brief tour of some of the math that is used repeatedly in my job (there is more that I haven't touched on). One might think that this is arcane stuff, that very few people can do or would want to do: you'd be wrong. Geography, which was rather a moribund field when I was in college, has received a huge boost as a field with the advent of GIS. There is a call for people to work in urban planning, forestry, environmentalism, and 911 service delivery; really appealing work, too, of the sort that is both interesting and beneficial to mankind and the environment. Knowing GIS and the mathematics that goes with it is the key to getting them.

comments...

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WhatDoesThisMean 26 Jun 2005 - 18:39 CatherineJohnson

Just back from Washington & am addled (hot there & hot here--)

I'm hot, tired, & cranky enough to feel I'm missing something here:

One second-grade lesson encourages students to work with a partner to find various ways to divide 10 cubes into two groups. This lesson helps students identify sums that equal 10, an essential component of addition that will help them later with more-complicated calculations.

Are there 'various ways' to divide 10 cubes into two groups?

Isn't 10 divided by 2 always 5?

What do you think this activity involves?

Are the cubes different colors?

Does anybody know?




source:
Bitter Single Guy

Duval gives 'new math' good grade
(no longer available online 5-14-06)

update

Ed says obviously the kids are working on addition and subtraction.

I am addled today.

I'm going to shape up before tomorrow.

update 2

The Duval gives 'new math' good grade story is majorly aggravating.

The district has brought in fuzzy math, along with beaucoup teacher training & staff development, and lo and behold --

Scores have risen!

Cut to NCTM president Kathy Seeley who, after issuing the standard NCTM disclaimer, takes her bow. (Standard NCTM disclaimer: NCTM 'does not support any specific programs.')

As Dr. Robert Mandell pointed out in an unfriendly exchange of emails with the folks at Everyday Math, teacher training is what we call a confounding variable.

A person who knew a thing or two about math -- the president of the NCTM, for instance -- would know that the rising scores in Duval tell us nothing about Everyday Math one way or the other.

If you want to find out who or what should take the credit for rising scores in Duval -- the textbook, the teachers, or both -- this isn't the way you do it.

Fortunately, some of the Duval teachers have had the gumption to say so:

Sara Stolkner, a fifth-grade math teacher at Sabal Palm Elementary School, said Math Investigations assumes children will discover the lessons on their own, and there is no backup plan for when they don't. She feels the program is getting too much credit for the district's rising math scores.

"No, it's us," she said. "Anyone who is truly a teacher is going to find ways to make things work."

Angela Peterson, a first-grade teacher at Lone Star Elementary School, likes to use old worksheets to drill her students on math skills. She and other teachers feel Math Investigations has been forced upon them and that they are not welcome to use traditional textbooks and worksheets to supplement their lessons.

"Some of the children really need to just go over and over and over and over the skills," Peterson said.

Most of the time a person has no business predicting the future, but in the case of fuzzy math I'm making an exception.

If events continue on their current course, the Master Plan will be complete in a few short years from now:

• implement fuzzy curricula in public schools along with teacher training, professional developing, and lots more class time for mathemathics in the school day (Trailblazers explicitly says that the program cannot be implemented in the standard 40 minutes a day).

• when scores rise, assume that causality has been demonstrated, collect data, publish in non-peer-reviewed forums, and cite liberally in public documents, professional conferences, and all exchanges with parents

• simultaneously push ahead with revision of all known standardized mathematics tests to conform to NCTM standards

If all goes well, by the time the effects of extra teacher training & extra time-on-task begin to wear off, all of the old tests will be gone and the new, fraction-free, conceptual tests will be in place.

The whole country will be one big Lake Wobegon.


LakeWobegonPart2

comments...

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OffTopicEletelephony 26 Jun 2005 - 19:13 CatherineJohnson

I was just cruising the web, looking for jpgs, wnen I found this poem I remember hearing when I was a child:

Eletelephony

Once there was an elephant,
Who tried to use the telephant--
No! no! I mean an elephone
Who tried to use the telephone--
(Dear me! I am not certain quite
That even now I've got it right.)
Howe'er it was, he got his trunk
Entangled in the telephunk;
The more he tried to get it free,
The louder buzzed the telephee--
(I fear I'd better drop the song
Of elephop and telephong!)

a sampling of poems by Laura E. Richards




While we're on the subject of eletelephonies, why are phones so crummy today?

Just asking.


keywords elephant line drawing



comments...

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GodeyLadysBook 27 Jun 2005 - 01:19 CatherineJohnson

SteveH mentioned the Godey's Lady's book.

Naturally I can't find all the Godey's images I dug up the other day, but I found a couple of other great links:

Godey's Lady's Book online and Godey's list of illustrations





This one is wonderful, too.

comments...

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EnglishLanguageArtsBookRecommendation 27 Jun 2005 - 01:45 CatherineJohnson

Writers Express: A Handbook for Young Writers, Thinkers, and Learners by Dave Kemper, Ruth Nathan, Carol Elsholz, Patrick Sebranek, Chris Krenzke (Illustrator)

Christopher's fantastic 5th grade teacher, Mrs. D'Arcy, strongly recommends this book, and it has nothing but 5 star reviews on Amazon.

I'm getting it.

update

No, I'm not.

I'm getting what looks like the sequel, Write Source 2000: A Guide to Writing, Thinking and Learning by Patrick Sebranek, Dave Kemper, Verne Meyer.

+ + +

These books are from the same folks who did the Math on Call series, which I love.

The reviews for all of their books are so glowing I'm getting suspicious, but I have to say I feel the same way about them.


EnglishLanguageArtsBookRecommendationPart2
RoyalRoadToGeometry
BuyThisBookToo
MathRefs

comments...

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SummerSupplementTimePart5 27 Jun 2005 - 13:47 CatherineJohnson

In SummerSupplementTimePart4 I mentioned that I think I have useful advice for 3 groups of kids:

• kids who, for whatever reason, have fallen significantly behind their classmates

• kids who are right on track, doing well, and you want to keep their math skills in shape over the summer

• kids whose parents want to accelerate their math learning -- in particular, to get them in position to take and master algebra in the 8th grade

My own strategy for kids who have falllen behind (Christopher's situation last summer) is in that post.

But please! Everyone! Chime in.

These are the ideas I've come up with working with one child, and talking to a group of 4 people (Carolyn, Ed, my neighbor & friend Laura, and my friend Debbie), with as many on-the-fly advice sessions as I could get with Christopher's teachers thrown into the mix.

One of the main reasons I wanted to do a bliki with Carolyn was to find out what other people are doing!

avoiding summer regression

For kids who are doing fine, here are my thoughts.

Assuming the research I've found (pdf file) is to be trusted (it makes sense to me, for what it's worth) there are two points to bear in mind:

• summer loss equals about one month of a child's learned skills and knowledge from the previous school year

• summer vacation is more detrimental for math than for reading, and most detrimental for math computation and spelling
  

I find the math-versus-reading factoid ironic given that schools universally hand out summer reading lists, not summer math lists.

So here's my own stab at a summer maths list. (I think the British plural works for this.)

summer maths list

• 'mad minute' worksheets daily (be sure to include fractions, decimals & percents if your child has gotten that far)

• a word problem or two each day, if you feel ambitious (Carolyn is posting problems from the Singapore series)

• a Math Olympiads word problem each day, if you feel really ambitious (I'll probably post some of these)

books (worksheets)

I did a quick scan of the various 'Mad Minute' books on Amazon, and folks seem to like this one best:

• The Mad Minute: A Race to Master the Number Facts by Paul Joseph Shoecraft, Terry James Clukey

The Mad Minute covers Grades 1 through 8, and includes fractions & percents.

If any of our teachers or parents have used this book, let us know.

• Saxon Math Tests and Worksheet Booklets for each grade level. 120 'fast fact' worksheets to be completed in under 5 minutes. These are the worksheets that finally got Christopher up to speed, and we're doing them again this summer. Cost for the Tests & Worksheets book alone is around $20, probably less at the Homeschool Super Center. If you're just going to use the worksheets you don't need to buy the textbook or the solution manual.

books (story problems)

• Singapore Math Challenging Word Problems series. These are terrific books. Almost 300 story problems in each, grouped according to subject area (e.g. measurement, time, multiplication-and-division, etc.) All problems are multi-step, & all answers are in the back. $7.80 plus shipping.

caution: your child almost certainly needs to use a book 1 or 2 grades younger than the one he's in. So you might want to have your child take the placement test before ordering.

• Math Olympiad problems -- you can find Math Olympiad books all over the place. They're expensive, so try to rustle up a used copy.

• Math League Contest Books from Math League. Wayne Wickelgren strongly recommends these books for everything from building your child's math achievement to preparing for SAT's. I love them, too. Filled with the kinds of problems, including logical reasoning, children are going to need throughout their lives & much more 'sensible' than the showy problems from Math Olympiads. Each book spans 3 grades, and all answers are in back. $12.95 a book plus shipping.

worksheets

• Carolyn likes these free online worksheets from homeschoolmath.net, a favorite site of mine. (I think their e-books look good.)

• Homeschoolmath.net has fraction & decimal worksheets, too.

• Susan S likes edHelper.com and SuperKids.com.

• Brenda M likes Basic Facts Worksheet Factory from School House Technologies. (Thank you, Brenda!) They offer free worksheets as well as worksheets for purchase.

virtual worksheets & problem-solving

I've mentioned that I'm leery of online learning, but you can't beat it for convenience and speed. I like Saxon's offerings:

• Saxon Math 'fast facts' generator The page is clean, simple, and visually compelling. You decide which math-fact problems you want to do, how difficult the problems should be, and how many to do in one set. You can choose between a timed & untimed option. That's great, because kids love seeing their times get faster.

• Saxon online math activities I love these things. Don't know why; I just do. They're fun.

• Check out 5th grade activities.
  Saxon now has online exercises for each grade. They tell you which activities to do after which lessons in the book, and you can download the activities for use when you are not online.

• Saxon online equivalent fractions

• Saxon online rectangular area

• Saxon has lots, lots more, so take a look

• Batter's Up Baseball Game I can't find the 'addition facts baseball games' the kids at school love so much, so here's another one. Christopher told me just now that he loved playing online 'addition baseball' when he was in 2nd grade.

I found it!

The kids at school were crazy about Funbrain, especially math baseball.

update: reader recommendation

Also check out Singapore math's Intensive Practice books. These books cover all sorts of fun things including word problems, computation, puzzles and patterns etc... They are not joking when they call it intensive. Some problems are extremely difficult (and some are quite easy too) and we cover them orally and together with the view that exposure to these types of problems will only expand abilities!

I agree. I have two of these books, and they're terrific. [Catherine]

FreeWorksheets
TreadingWater

SummerSupplement
SummerSupplementTime
SummerSupplementTimePart2
SummerSupplementTimePart3
SummerSupplementTimePart4 (resources for kids who have fallen behind)

SaxonPlacementTestsAndGuides
SingaporeMathPlacementTest

and:

Summer Supplement Time
linking decline in high school scores to elementary school
research on summer regression
the time costs of not teaching to mastery
U.S. fourth graders not doing as well as thought
Phase 4 topic list, grade 6 class
comments thread on pre-algebra as algebra

comments...

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EverydayMathInDCPart2 27 Jun 2005 - 17:36 CatherineJohnson

from Barry Garelick:

For those who may not know, the DC Public School Board, apparently with little notice, held a meeting on June 15, 2005 at which they adopted various texts to be used in elementary and middle schools in math, English, and social studies. Among the math texts adopted were Everyday Mathematics, Math Trailblazers, Growing with Math for elementary schools, and Connected Math for middle schools (though they also adopted Pearson Prentice Hall Middle School Math which is not great but not disastrous).

A bunch of us wrote testimony to the hearing to no avail. At the Board meeting, Dr. Janey, superintendent, was reported to have remarked on the receipt of the various emails protesting adoption of EM and other texts, characterizing them as "short on research and long on opinion". These emails included protests from Ralph Raimi, math professor emeritus at U of Rochester and Bas Braams, a physicist and chemist and visiting professor at Emory. I have requested information on the decision in a FOIA letter to DC Public Schools (DCPS).

Links to documents on EverydayMathInDC wiki page.

comments...

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OutsmartingTheTests 27 Jun 2005 - 18:02 CatherineJohnson

Just caught a funny thread at WhatDoesThisMean that reminded me of something I'd been planning to post:

Barry Garelick: One problem in a 2nd grade math text asks the students to compare another classroom to yours and tell if it is bigger. It fails to define what is meant by bigger: more volume, more floor space, more seats? Of course, kids will come up with various answers to which the teachers will be delighted—just what they learned in ed school, there's more than one right answer.

Steve H:The silver lining of Everyday Math, which my son uses in school, is that he gets lots of practice with vague or trick questions. I don't want him to be unprepared later on when his math ability is tested with these stupid questions.

That's been one of my issues: how much of this Trick Question stuff does Christopher have to be able to do to look like he's getting with the program on fuzzy math?

Carolyn has already started to talk about this.

I think it was shortly before the TONYSS, which I was intensely nervous about, that Christopher was telling us about some open-ended, mathematical reasoning-type questions he'd had to do on last year's test.

I asked him how he'd managed one of them, and he said, 'I looked for a pattern.'

That jump-started my brain, and we came up with a Standard Verbal Explanation he was to write down for any fuzzy problem he couldn't actually do:

I looked for a pattern, and then I used a strategy of guess and check to see if my pattern was right.

We had him memorize this line and recite it back to us a few times.

Now that you can get points for wrong answers, we figure a core test-taking skill is going to be getting partial credit for using the lingo.

p.s.

I'm serious.


ImportantQuestionFromJoanneCobaskoOfSocmm
ImportantQuestionPart2
OutsmartingTheTestPart2

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

comments...

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AnneDwyerSingaporeMathClassPart3 27 Jun 2005 - 18:28 CatherineJohnson

Check out Anne Dwyer's new entry about her summer math class.

comments...

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ConversationsWithKids 27 Jun 2005 - 18:35 CatherineJohnson

I just saw this on Anne's page:

Conversation with daughter the day of class:

"Did you have fun?"

"You know I don't like math"

Yes!

I've had that conversation!

I had it this morning, in fact!

Only in my case it was longer-winded:

You ruined my summer last summer. I was so unhappy all summer. It was horrible. I hated last summer.

I can't remember the rest of it, but I don't need to remember it.

I'll be hearing it again.


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

comments...

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LakeWobegonPart2 27 Jun 2005 - 19:06 CatherineJohnson

Apparently the Master Plan is farther along than I thought.


WhatDoesThisMean?

comments...

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OutsmartingTheTestPart2 27 Jun 2005 - 20:52 CatherineJohnson

The new essay test on the SAT appears to be is working out well:

"It appeared to me that regardless of what a student wrote, the longer the essay, the higher the score," Dr. Perelman said. A man on the panel from the College Board disagreed. "He told me I was jumping to conclusions," Dr. Perelman said. "Because M.I.T. is a place where everything is backed by data, I went to my hotel room, counted the words in those essays and put them in an Excel spreadsheet on my laptop."

In the next weeks, Dr. Perelman studied every graded sample SAT essay that the College Board made public. ...

He was stunned by how complete the correlation was between length and score. "I have never found a quantifiable predictor in 25 years of grading that was anywhere near as strong as this one," he said. "If you just graded them based on length without ever reading them, you'd be right over 90 percent of the time." The shortest essays, typically 100 words, got the lowest grade of one. The longest, about 400 words, got the top grade of six. In between, there was virtually a direct match between length and grade.

He was also struck by all the factual errors in even the top essays. An essay on the Civil War, given a perfect six, describes the nation being changed forever by the "firing of two shots at Fort Sumter in late 1862." (Actually, it was in early 1861, and, according to "Battle Cry of Freedom" by James M. McPherson, it was "33 hours of bombardment by 4,000 shot and shells.")

Dr. Perelman contacted the College Board and was surprised to learn that on the new SAT essay, students are not penalized for incorrect facts. The official guide for scorers explains: "Writers may make errors in facts or information that do not affect the quality of their essays. For example, a writer may state 'The American Revolution began in 1842' or ' "Anna Karenina," a play by the French author Joseph Conrad, was a very upbeat literary work.' " (Actually, that's 1775; a novel by the Russian Leo Tolstoy; and poor Anna hurls herself under a train.) No matter. "You are scoring the writing, and not the correctness of facts."

How to prepare for such an essay? "I would advise writing as long as possible," said Dr. Perelman, "and include lots of facts, even if they're made up." This, of course, is not what he teaches his M.I.T. students. "It's exactly what we don't want to teach our kids," he said.

... Dr. Perelman is now adept at rapid-fire SAT grading. This reporter held up a sample essay far enough away so it could not be read, and he was still able to guess the correct grade by its bulk and shape. "That's a 4," he said. "It looks like a 4."

full text here


see also: PleaseExplain
OutsmartingtheTests

comments...

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PleaseExplain 27 Jun 2005 - 21:14 CatherineJohnson

OK, so I was looking for a jpg to illustrate the concept of stupid SAT questions, when I came across a site called Free SAT Prep.

Check out the left hand column:

• Discussion Topics
• Ask Admissions Consultants
• What happens if there is mistake in application
• Cornell to begin accepting the common application
• SAT Test

Then there's the last one.

pls xplain

update

They've changed the link.

Here it is, if you're interested.

Looks like they have one of those new site thingies (word retrieval failure) that automatically slots a commenter's essay into the sidebar every day.

wrong again

It's not a spoof. It's real. This is a real news article. (scroll down)

comments...

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EthnoMathematicsIsBack 27 Jun 2005 - 21:31 CatherineJohnson

I've come to believe that the math wars will never be won, or lost.

Here's Diane Ravitch on the return of ethnomathematics, which I thought had been definitively stomped to bits back in the ... 1980s?

I don't even remember, it's been so long.

And now it's back.

Now mathematics is being nudged into a specifically political direction by educators who call themselves "critical theorists." They advocate using mathematics as a tool to advance social justice. Social justice math relies on political and cultural relevance to guide math instruction. One of its precepts is "ethnomathematics," that is, the belief that different cultures have evolved different ways of using mathematics, and that students will learn best if taught in the ways that relate to their ancestral culture. From this perspective, traditional mathematics--the mathematics taught in universities around the world--is the property of Western civilization and is inexorably linked with the values of the oppressors and conquerors. The culturally attuned teacher will learn about the counting system of the ancient Mayans, ancient Africans, Papua New Guineans and other "nonmainstream" cultures.

Talk about your golden oldie.

update

I think I owe a thank you to Susan, who spotted Ravitch's essay posted free on opinionjournal.com.

update 2

I don't think this new wave's going to get too far.

Though I'm surprised it got as far as having a whole textbook in print.

(you can click on this)

update 3

oh.

good news.

it isn't a textbook.

it's a unique collection of more than 30 articles that shows teachers how to weave social-justice principles throughout the math curriculum, and how to integrate social-justice math into other curricular areas as well.

ok, that could be worse.

comments...

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SummerSupplementTimePart6 27 Jun 2005 - 22:54 CatherineJohnson

Two books Anne Dwyer is using this summer:

BTW, I got most of my games and ideas from two sources. The first was a book I got out of the library. Games for Math by Peggy Kaye. The second source is Primary Mathematics 2A Home Educator Support Guide by Jennifer Hoerst. It turns out that the Singapore Math website sells these separate books by Sonlight Curriculum.

update

Games for Math, I see, was first published in 1988.

I have a rule that any book that's stayed in print longer than 5 years is likely to be worthwhile.

This is so because books are more or less like magazines; they're seasonal. Few stay in print longer than the year they spend in hardback and another couple of years in paper.

The only reason a book lasts longer than that is that people want it, and continue to buy it.

Buying a book that has stayed in print for 17 years is like picking the restaurant with the line out the door instead of the one with two tables filled.

comments...

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SlideRules 28 Jun 2005 - 00:56 CarolynJohnston

Apropos of our discussion of Steven Leinwand's recommendation that we quit using pencil-and-paper computations because they are passe, it occurred to me today that we do have a sort of a precedent.

When my Dad was in school, everybody used slide rules to do logarithms (also multiplications). You needed them to do the computations, but they were also a kind of a math manipulative (as one could argue, I suppose, that pencil-and-paper computations are!). Over time, learning to use slide rules, you learned about how the logarithmic scale worked.

By the time I got to school, slide rules were gone. I don't recall having big troubles learning about logarithms, but judging from my dealings with my students from both remedial and college-level courses, I was exceptional that way: nearly everyone had trouble with them, and even those who could manipulate logs correctly didn't have any feeling for how they behaved or what they were good for.

So here's my question: in olden, pre-calculator days, when people used slide rules to do logarithms, did they understand logs better -- the point of them, and how to do computations with them?

In other words, did we give up an important learning mechanism when we gave up using slide rules, and is Leinwand proposing that we make the same mistake with all the other types of computations?

SwoopAndSwoop

comments...

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PreludeToMathematics 28 Jun 2005 - 01:57 CarolynJohnston

Charlie Martin sent me a link to W. W. Sawyer's book Prelude To Mathematics. There are lots of little Dover books like this out there, but I was surprised to see that not only has this one been around since 1982, it's got no less than 5 Amazon reviews, all 5 stars. Here's one (from a reviewer named Nan Zhang):

This is the book that really got me interested in mathematics. I had never thought that a math book could be so engrossing. I finished reading it in a couple days and i immediately seeked out the author's other books. And the quality of the other book are of the same level as this one. It is a shame that the author's other books are mostly out of print. What i appreciate most about the book is that the math concepts are always are related to where it came from. The part on series is a small gem, and the book is full of ones like that. Without having met the author, he is in my mind certainly one of the best math teachers ever. (George Polya is another). Thank you, Mr Sawyer.

It's cheap, by the way: it's a Dover book. I love Dover math books. This one looks like a good one.

I was also thinking that I had heard of W. W. Sawyer before. Catherine has an uncanny knack for knowing where the best reading is, in any field.

comments...

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PreludeToMathematicsPart2 28 Jun 2005 - 16:34 CatherineJohnson

While we're on the subject of W.W. Sawyer, I found this book intriguing, too [update 5-14-2006 — I've now read the first couple of chapters - the book is fantastic]:

The one article I've skimmed about Sawyer, along with this book review (more skimming) make it sound as if Sawyer may have been my kind of constructivist.

Here's an interesting passage from the article:

It is in some sense ironic that his consistency of view has sometimes placed him on the right wing and sometimes on the left of pedagogic controversy, for any rereading of his published work offers considerable evidence that his recommendations were based basic, perennial issues, carefully analyzed, and that his defense of them cannot properly be considered in the context of, say, the 'modern maths' wars of the 1960s. Indeed, the scope of Sawyer's contribution far exceeds curricular issues, and this is as evident in his first book as it is in his latest.

Hmm. I'm trying to get a fix on where Sawyer was politically, in the 'new math' wars of the 1960s and 1970s.

Here is Marc Alder, the webmaster of a site devoted to Sawyer's writings:

It is only very recently that I came to realize an additional role that Professor Sawyer has had in contemporary Mathematics Education. Everybody these days knows the vital importance that Mathematics has assumed now in Basic Skills, Numeracy and Key Skills and the vaste amount of government money that is being pumped into them both here [Great Britain] and in the U.S.A. There are indications that the present lack of mathematical skills in young people (and even adults!) are due to a failure in mathematical educational policies in the USA and UK as far back as the nineteen sixties. I am finding it of immense interest to learn that both Professor Sawyer and the Late Professor Morris Kline have been prophetic voices "crying in the wilderness" since then.

And here is an interesting passage from an article by Sawyer himself, elsewhere on the site, on the question of what happens when students move too slowly through an elementary mathematics curriculum:

Perhaps the first question about the gifted in the minds of educational administrators is 'Do we need to make any special provision for the gifted? Are they not so clever that they will work out their own salvation?' The life of Darwin indicates clearly the damage that can be done if the curriculum is unduly narrow and inflexible. There can also be damage if the rate of progression is rigidly laid down; I found striking evidence of this in the pre-Sputnik California of 1957.

At that time the lock-step was the prevailing fashion in American schools, except of course where an enlightened teacher made special arrangements. All pupils were expected to work through the same book at the same rate. There were historical reasons why the arithmetic textbooks had a rather strange composition. In the formative period of American education, classes contained immigrants from many countries with many different languages; the primary teachers had the responsibility for teaching them the American language and instilling a sense of their new nationality. These were substantial tasks and it was not unreasonable to allow four years for their completion. In this way it came about that an arithmetic syllabus, that by itself would have fitted comfortably into four years, was extended to eight. What no one seemed to consider was that a time would come when children would have absorbed both the language and the national sentiments of the U.S.A. before they came to school. The result was a 4-year intellectual vacuum in the arithmetic curriculum. The Grade 8 (age 13+) textbook was the worst; it was entitled Making sure of Arithmetic, which I took to mean that the youngsters would not meet any new idea in that year. Bill Glenn, a mathematics supervisor in California took me on a visit to a school. We found a boy working outside the Principal's office. Why was he there? They had to put him outside the class room; he was totally unmanageable. Bill Glenn told me such situations were quite normal. He talked to such boys and had usually found them well above average intelligence. He summed up his experience by saying, 'Grades 5 to (ages 10 + to 13+) are the grades in which the superior student becomes a superior delinquent'.

I have no idea whether his history of mathematics education in America is correct, but he's right about the very slow progression through elementary mathematics, a progression that is slower still in constructivist texts.

This slower progress is conscious and intentional; constructivist teacher guides and other documentation explicitly state that students follow a slower track in constructivist curricula than they do in more traditional mathematics instruction. This is seen as a good thing, because, in theory, students spend that extra time acquiring conceptual understanding.

comments...

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FourthGradeSlump 28 Jun 2005 - 16:40 CatherineJohnson

A slump in math gains begins after fourth grade and extends through high school on both national and state tests. The middle-grade slump also appears on the most prominent international test, TIMSS, as the relative ranking of American students falls precipitously after fourth grade. No other country has a sharper drop in math ranking than the United States. A British publication, The Economist, concluded, "The longer children stay in American schools, the worse they seem to get."9 This overstates the case. Older students have made gains in learning. The slump is not in absolute achievement, but in the pace of improvement. It decelerates after fourth grade.

THE BROWN CENTER REPORT ON AMERICAN EDUCATION 2000 How Well Are American Students Learning?


No other country has a sharper drop in math ranking than the United States.


FourthGradeSlumpPart2

comments...

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FourthGradeSlumpPart2 28 Jun 2005 - 17:04 CatherineJohnson

Percentage comparisons of students who scored in the top 10 percent of fourth-graders among the 26 TIMSS countries also show that the United States is lagging. In math, only 9 percent of U.S. Fourth-graders were among the top 10 percent, compared to Singapore’s 39 percent, Korea’s 26 percent, and Japan’s 23 percent. At the eighth-grade level, only 5 percent of U.S. Students were included in this bracket . . . once again confirming that U.S. students do not fare well in international comparisons and drop in rankings the further along they are in school.(14)

source:
SCHOOL FIGURES: THE DATA BEHIND THE DEBATE BY Hanna Skandera & Richard Sousa


FourthGradeSlump

comments...

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FourthGradeSlumpPart3TimeManagement 28 Jun 2005 - 17:29 CatherineJohnson

The math facts acquisition is on a different timetable in Trailblazers probably than it has been in other programs that just did a lot of memorization. Math Trailblazers basically believes that by 2nd grade, kids should be very familiar and have facility with their addition facts through 18; and that by 3rd grade, they should have facility with addition and subtraction and have an understanding of multiplication, but probably don’t have their facts committed to memory. That’s a slower timetable than we’ve had before, and that’s hard for a lot of teachers to deal with. But also in the past, 3rd graders probably weren’t dealing with irregular area or volume, drawing 3-D, figuring out how to build a city, and those kinds of things.

MATH TRAILBLAZERS: An elementary school curriculum for grades K–5
developed by Teaching Integrated Mathematics and Science (TIMS), p. 13 (pdf file)

When you put it that way, I understand.

Figuring out how to build a city in the 2nd grade has got to be time-consuming.

comments...

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CompareAndContrastPart6 28 Jun 2005 - 18:29 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.

That's 5th grade, folks.


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

comments...

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BestMentalMathBook 28 Jun 2005 - 22:34 CatherineJohnson



Arithmetricks : 50 Easy Ways to Add, Subtract, Multiply, and Divide Without a Calculator
by Edward H. Julius


Last fall I got on a mental math kick.

Singapore Math does a lot of mental math, and Saxon opens each lesson with a mental math warm-up.

The constructivists seem to believe in mental math, too.

[pause]

OK, I just Googled 'Constance Kamii,' and yes indeed the constructivists are HUGE mental mathies.

Here's what Parker & Baldridge have to say about mental math:

'Mental Math' means just that: doing calculations in your head. Solving problems mentally is a remarkably effective way to learn place value skills and the use of the distributive property. As students practice mental math they develop quick and flexible ways of doing simple arithmetic, and their understanding of arithmetic deepens. Mental Math is particularly appropriate with young children because it does not require reading or writing skills. For these reasons, Mental Math problems are incorporated into nearly all elementary school mathematics programs. (p. 43)

All of this struck (and strikes) me as correct, so I got on a mental math quest that resulted in the purchase of at least 3 different books, maybe more. (I don't like to think about it.)

Of those, Arithmetricks is the ONE. It's the clearest, easiest to use, and, IMO, has the most 'educational value'...meaning I used it in my Singapore Math class to try to teach the distributive & commutative properties & place value.

Not just party tricks.

Obviously, all mental math is real math, not tricks. But I wanted 'arithmetricks' my elementary school kids would be able to understand, not just memorize.

The funny thing is, I had to use paperandpencil to make this work.

All of the kids had mastered their math facts and the algorithms (I was VERY impressed with Irvington teachers after that class, let me tell you).

So the only way to find out if they'd used the arithmetrick I'd just taught them to do a calculation, or had visualized a two-column addition or subtraction or multiplication problem in their mind's eye and done it that way, was to make them write down the steps they'd used after they'd used them.

Life is never simple.

update

I'm pretty sure I'm right about Arithmetricks, because it has a blurb on the cover from Jaime Escalante.

update 2

I just noticed that Frog Publications, publishers of the Drops in the Bucket series Carolyn likes, has a mental math series, too:

(click on the image)

I think that's adorable.

comments...

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EnglishLanguageArtsBookRecommendationPart2 28 Jun 2005 - 23:50 CatherineJohnson

Another recommendation from a KTM reader:

Check out Institute for Excellence in Writing. Many homeschoolers use this program and adore it.

Now is probably the time for me to say:

Thank God for homeschoolers.

When homeschooling first started, I had the typical noncomprehending reaction ... 'why?' ... 'no social skills' ... 'unregulated' ... 'blah-blah-blah'

Boy, have I seen the light.

If it weren't for homeschoolers, I'd be in big trouble.

update

I'll get all our reader recommendations pulled together on a sidebar page as soon as I can.


EnglishLanguageArtsBookRecommendation
SusanSWeCanTeachOurChildren

comments...

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HowToFindNewCommentsAndUpdates 29 Jun 2005 - 00:34 CatherineJohnson

I've been thinking about our information architecture....which is a tough subject. (Back to Jakob Nielsen.)

Here's how you can find updates & new comments on posts:

• Click on the 'Recent Changes' link in the sidebar to the left.

• You will see a list of all posts that have had 'recent changes.'

• To the right of each post is the name of the person who made the most recent change: Carolyn, me, or a KTM reader.

• A topic has had a recent change when either Carolyn or I has added an 'update' to the body of the post, or when a reader has added a comment.

• If you made a comment on a thread, and you see your name still listed as the last person to have made a change, that means no one has added anything else.

• If you see someone else's name, that means someone added a comment after you did, so there is something new for you to read if you're interested.

It took me a little while to get used to this system, but now that I know how it works, I like it. I can see quickly if anyone has added a comment--and I can see if people have added comments to posts from a few weeks ago, too.

Let us know if it works for you, or if you find the "Recent Changes" page confusing.

keywords: What's New what's new recent changes Recent Changes

comments...

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AdviceFromCanada 29 Jun 2005 - 01:13 CatherineJohnson




(click on image for article)

Parental Attitudes

Parents should be included in their children’s mathematics education in a meaningful way.... Indeed, many researchers feel that it is the parents who are the single biggest factor in a child’s educational success.....

[snip]

There are effective ways that teachers and schools can forge strong links with parents that benefit all involved....as students share at home the problems that they are working on in school, parents will have their own ways of solving these problems, and these ways can be included in school discussions....

After working on the marbles problem Ms. H had three students report that they had learned another way to solve the problem at home. As one student, confided, "I showed my dad my way and he showed me his way. I didn’t understand his and he didn’t understand mine!" Ms. H capitalized on this confidence to look at what was similar between Leah’s way and her dad’s way of solving a division problem. Some of the students were able to find the link between the two methods (shown below). Other students were unsure. Ms. H pursued the comparison because she wanted to make sure that a connection was made between what was happening in her class and what parents were sharing at home. She did not want a disjunction between math at school and math at home; instead, she wanted one to strengthen the understanding of the other.

Teaching and Learning Mathematics – The Report of the Expert Panel on Mathematics in Grades 4 to 6 in Ontario, 2004

If you are going to teach multiple solution paths, this is the way to do it.

Not:

all your children are belong to us

comments...

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WickelgrenOnIntroducingAlgebra 29 Jun 2005 - 04:09 CarolynJohnston

I've been looking again at one of Catherine's favorite books, Math Coach (by Wayne and Ingrid Wickelgren).

Wayne and Ingrid have a lot to say about what they consider the most difficult aspects of elementary math -- long division and fraction manipulation. But it's what comes after that that interests me now: their discussion of the importance of teaching algebra early. Wayne suggests that the most important thing you can show your kid, what should motivate them most to want to continue in math, is the power of algebra to solve hard problems.

Most problems in prealgebra and early algebra start out something like this:

John is 27 years old. If his age is 3 times Pete's age, how old is Pete?

If you have a kid like Christopher or Ben, you know he's going to spit out the answer on the spot and tell you not to waste his time with this stupid letter stuff.

That's why Wayne Wickelgren suggests that, when you're ready to introduce your kid to the notion of algebra, the first thing you should do is sit down with him and let him watch you do a problem like this one:

In two years, Jean will be twice as old as Chris will be. In six years, Jean will be four times as old as Chris was last year. How old is Chris now?

In short, start with a demonstration of how algebra-at-your-fingertips gives you mindblowing powers. I was reading this last night and thinking: if I tell him that this problem is what algebra is all about, Ben will be blown away. Why scare him off? Maybe start with something simpler...

But the hard thing about this sort of problem isn't going to be doing the algebra: it's going to be setting up the equations, given the word problem. And that's going to be hard no matter how I try to teach it. Doing the mindless rote stuff required to crank out the answer, once you have the equations, is the easiest part of the problem. And I know Ben: he'll think that's the cool part.

Given that, I can't see a reason to hold off introducing algebra. Once a kid is at the sixth or seventh grade level in math, the heck with guess-and-check and pan-balance problems; the heck even with bar models. The most general tool that we currently have for solving word problems, and the only one that we have that isn't stymied by some word problem or other, is algebra. He may as well be motivated to go full speed ahead with the letters and symbols. Wickelgren says that algebra is the key to the castle; it's the most effective means for solving tricky math problems that's ever been devised. As such, you want it to be the tool that kids reach for instinctively when they have a tricky math problem to solve.

Here's a quote from a great article by Ethan Akin, "In Defense of Mindless Rote":

On the other hand, mathematics is cumulative and there are a great many skills that you have be unthinkingly familiar with. Every grumpy calculus teacher will tell you that most of the problems his students have come from weaknesses in algebra. For the students who say "I really understand it but...." the but is that for them algebra is not easy background knowledge. They are trying to build on a foundation of dust. A lot of college majors need a bit of calculus or statistics which are simply walled off to students who don't have sufficient skills in algebra. These are basically not hard subjects but they appear unnecessarily terrifying to such students.

Conversely, a practiced facility with algebra can provide its own positive reinforcement. Not only is the mathematics built on the algebra, but facility in algebra gives the student confidence in the face of new mathematical challenges. As the above discussion makes clear, such confidence is entirely justified.

I am motivated now to try to introduce real algebra by the end of the summer. No more pussyfooting around!


Wickelgren on introducing algebra
Wayne Wickelgren on algebra in 7th & 8th grade
Wickelgren on math talent & when to supplement
late bloomers in math & Wickelgren on children's desire to learn math