Here's Doug Sundseth:

Would it be helpful to you to have a sheet of number lines (with whatever arbitrary endpoints you wish) as a .jpg file? It might be as much as a 10-minute task for me to make one and send it to you (though I doubt it would take that long), and I'd be happy to be of assistance.

If so, let me know what you'd like (number of lines per page, end points, title, name line, file resolution, whatever), and I should be able to get to it in a day or two.

This is great!

I'm thrilled!

Thank you!

OK, what do you guys think?

How many number lines should be on the page, and with what kinds of distances between the ticks?

Should there be any numbers associated with the ticks, or should everything be left open so parents, teachers, and students can write in whichever number scale they need? (I'm thinking no numbers except for maybe a 0 in the middle....)

Should some of the ticks be longer than others?

Do I sound like a complete nut?

Don't answer that!


Doug's downloadable number lines

comments...

• list of math sites divided by grade level

• Calculus Help

• Karl's Calculus Tutor

• Purplemath algebra (lists zillions more links by category)

• Regents Prep for New Yorkers (this is the city of Oswego's math prep web site, and I'm very glad to have found it again - test prep for all the various state tests, as I recall)

more later...

the mother ship

Way over my head at the moment, but not for long, I hope.

comments...

I asked them if they would have a teacher's aide work with Ben on his math, one-on-one, using the Saxon Math curriculum. The special ed teacher, bless him, said that he could make it work; that he thought he could spare a teacher's aide during that last period of the school day, and it would just be an (easier) matter of finding them a quiet place to work. I was so relieved I could have hugged him.

It's been two years of struggle for me and Ben, supplementing from Saxon and trying to work around the vagaries and inconsistencies of Everyday Math; and here we were, once again, facing another year of it, after having worked so hard last year to find a school that offered a traditional math class, and then fighting the open enrollment system to get him into it, and then committing to the 45-minute-per-morning commute that it entails. I wanted so much for this year to be the end of it. I never really wanted Ben to have to do two math curricula, especially when one of them seemed to be a total waste of time for him.

And then I found on the first day of school that Ben's math class would be using Connected Math after all. I just about despaired. I've had to give up my dream of having Ben mainstreamed in math -- I always thought it was the one class in which he could hope to really hold his own and have a Typical Kid Experience. But I don't care any more -- math education is a mess in this country, and we're perversely fortunate to be able to opt out.

I got some insight into why Ben's new middle school had chosen to go 50-50 with Prentice Hall and Connected Math this year, following many years during which they had a reputation for doing solid traditional math classes (and for having the best math department in the city). It's not ideology; it's fear.

The special ed teacher told me that if I wanted Ben to be taught from a traditional math class, that I would have to just 'ignore the CSAP' (the CSAP is Colorado's assessment test for students, given in compliance with NCLB). "He'll do badly on it," he told me. "The test is very applications-oriented. You can't hold us responsible for that."

"If he does poorly on the CSAP," I told him, "I'll hold myself entirely responsible." No way will he do poorly on the CSAP. He didn't this last year -- except in those sections, data representation and probability, that I chose not to supplement.

Apparently, on the CSAP, kids are frequently asked to give verbal explanations for what they did on a problem. Math CSAP scores for students at Ben's school have been getting worse and worse over the last few years, and the teachers and principal don't know why, and don't know what to do about it. This adoption of Connected Math is therefore, I conclude, their attempt to grasp at straws. There is no way for them to know in advance whether Connected Math is going to solve their problem; I doubt they even know what the cause of the problem is.

An even deeper question is whether the CSAP itself -- or any other state assessment -- is worth a hoot. Who's vetting the CSAP to check whether kids who do well on it in 5th grade have the skills, on average, to go into calculus in college?

I believe in the value of assessment -- it provides a minimal benchmark of proficiency and keeps people accountable. But the assessment has to be good, and we have to know what to do about the weaknesses it reveals. If it leads good schools astray, I call that backfiring in a big way. I've been assuming that the metrics, at least, are good; now I wonder. The more deeply I look at the problem of math education in our country, the more I realize that there are "unknown unknowns" all the way down to its foundations.

comments...

I'm relieved, I have to say. I've been semi-sanguine about the possibility of having two math curricula in your child's life, a fuzzy one at school & a non-fuzzy one at home.....but the fact is, I haven't (really) had to face that situation.

Last year, in 5th grade, Christopher had SRA Math at school, and Saxon Math 6/5 at home. SRA Math is a very tough textbook to teach from (impossible for me, and experienced teachers have told me the same). But it's not hardcore fuzzy.

David Klein points out that most U.S. textbooks are fuzzy to some degree. That was certainly the case with SRA. Time and again I'd read a passage--this was when I was just setting out to reacquaint myself with math--and not have a clue what it meant. Invariably this was because the text would lay out a couple of observations and then pose a question to the student, who was supposed to draw the appropriate conclusion.

I remember one day I was trying to figure out how to find the equation for the slope of a line, and there was just no way. Finally my neighbor came over, read the passage, and said, 'You'd have to know how to do it to understand this explanation.' Then she showed me how.

Still and all, SRA Math wasn't a b*s book. Not at all. The math was real, and Christopher had two good teachers who'd had plenty of experience getting math into kids' heads in spite of the problems.

I'm pretty sure that in Christopher's case it was a net plus that he had two separate math curricula. He had far more time-on-task, and he had the benefit of seeing the same subjects from slightly different vantage points (which always helps me, and is probably good for everyone).

But I wasn't having that feeling about Ben at all. SRA & Saxon, OK. Connected Math & Saxon? Blech.

So, long story short, I was getting worried about Ben. I'm glad Connected Math is gone.

Saxon into the breach

I've now read quite a few negative assessments of Saxon, by people whose judgment I respect. These are folks on the web--a couple of obviously intelligent homeschoolers, as well as Robert, who writes the brightMystery blog. Robert told me he wants to like Saxon, but just does not--and that students who come to his college courses having been homeschooled in Saxon aren't ready. (That's a paraphrase, so take it with a grain of salt.)

I have misgivings myself. Sometimes I worry Saxon is TOO 'structured'; I worry about pattern training--that Christopher is going to be a Saxon Boy who can only do Saxon Problems typed in Saxon Font.

Thus far that has not been the case. As far as I can tell, all of Christopher's Saxon knowledge has transferred to SRA (and, now, to Prentice Hall).

Other times I've felt the Saxon books are too scattered & fragmented. The fragmentation of topics is a deliberate strategy on Saxon's part, the intention being to use the principles of spaced repetition and distributed practice. That makes sense, but when I taught the Primary Mathematics Grade 3 chapter on fractions to Christopher and his friend Greg it was so much more satisfying and rich, or seemed so.

So.....I've been a heavy-duty Saxon user; I owe Christopher's move to Phase 4 math to Saxon 6/5. And I know the knowledge he's gained from Saxon is conceptual as well as procedural.

But in spite of all these good things, I have Nagging Doubts.

Usually I pay attention to Nagging Doubts. But in this case I think my doubts are either wrong or, more likely, misdirected. Because I keep coming back to Saxon every time I'm in trouble, and Saxon keeps bailing me out.

Saxon vs Dolciani

Christopher has another quiz today, on algebraic expressions.

I was reading along in Prentice Hall, which said that in an expression like x + 7 the x and the 7 are terms.

In an expression like 2x + 7, 2x and 7 are terms, and 2 is the coefficient.

Well, right away I was confused.

Does a term mean you're either adding or subtracting?

Does multiplication mean you don't have a term, you have a coefficient?

That seemed wrong.

So I got out my copy of Mary Dolciani's Pre-Algebra: An Accelerated Course.

I'd been thinking, OK, I'm done with Saxon. There are just too many negative opinions out there, Mary Dolciani's a genius, my neighbor's son liked Dolciani's book, it's shorter than Saxon & we're pressed for time......this year I'm going with Dolciani.

She was no help at all:

In the expression 9 + a, 9 and a are called the terms of the expression because they are the parts that are separated by the +. In an expression such as 3ab, the number 3 is called the numerical coefficient of ab.

Saxon on coefficients

Saxon 8/7 has an entire lesson on algebraic terms. Lesson 84, page 571. I haven't read it yet--I've skimmed--but it's obvious that when I do, my question will be answered.

Here's how he opens:

We have used the word term in arithmetic to refer to the numerator or denominator of a fraction.

Right off the bat, he's made the smart metacognitive move. We have used the word 'term' to refer to numerators and denominators, and it's a good thing to point this out to the student, because otherwise, at some point (probably not now, but later on, when it will really ball things up) the student is going to think, Wait! Doesn't TERM mean DIVISION? Does it mean FRACTION? Does it mean NUMERATOR & DENOMINATOR?

OR WHATTTTTTT?????????

I'm going to go out on a limb and say that Saxon Math is the most metacognitively aware textbook I've encountered to date. Constantly, the books remind you of what you have learned, and point out to you that you are now learning an extension of that concept or you are learning a new and possibly quite different meaning of the same word.

Back to Lesson 84.

Next the book has a table of monomial, binomial, and trinomial algebraic expressions. Wonderful.

THEN the text says:

Terms are separated from one another in an expression by plus or minus signs that are not within symbols of inclusion.

Thank you, John Saxon. I needed that.

More examples follow, and eventually we get to this:

Each term contains a signed number and may contain one or more variables (letters). Sometimes the signed-number part is understood and not written. For instance, the understood signed-number part of a^2 is +1 since a^2 = +1a^2. When a term is written without a number, it is understood that the number is 1. When a term is written without a sign, it is understood that the sign if positive.

Perfect.

At least, perfect for me. What do you think?

deer in the grass

So I looked out, too, and sure enough: the young deer grazing in our lawn is darker than the young deer who was living here a month ago.

But Martine thinks it's the same one. She thinks they get dark in the fall.

It's probably time to give him a name.


(a^2 means a squared - right?)

update

The story problems!

Barry reminded me: they're dreadful. They're just wildly too-easy.

I had meant to put that in the original post, and forgot.

However, the story problems aren't the reason for my 'nagging doubts'.....the story problems are an obvious problem you can remediate easily through supplementation.

It's the other stuff.....




comments...

Doug sent me the number lines he made up--they are beautiful!!!!!

Thank you!!!

waiting on the printer...

......now I'm getting paranoid.....

ah hah

There's no paper in the printer.

I hope everyone's impressed that I managed to check the paper tray before having a nervous breakdown.

pause

I want my number lines!

Right now!

still waiting

I hate that.

success

they're beautiful

Graphic design is my other love in life.

These number lines are so beautiful I'm going to put them on the wall.

I might frame one.

(I'm serious about that. When I first started drawing bar models I really wanted to paint a big, blown-up black-and-white bar model and have it framed. I love the Japanese character paintings, and it struck me that we ought to have number paintings, too.)

number line attachments

thank you, Doug!

First I have to post a screen grab, just to show you what these look like:

hypothetical

Nobody seems to learn math very well in this country, but it does seem to be true that girls are even less likely to learn math well than boys, or to want to learn it, or to think that they could learn it, etc.

Looking at Doug's number lines, it struck me: if math books were visually beautiful, and lovely to look at,......would more girls find themselves drawn in?

Given the way I remember feeling about my 2nd grade math book, I don't think that's necessarily crazy.

update

The OWL - Online Writing Lab - at Purdue University is a fantastic resource.

I'm posting the link on the Math Supplements page.

comments...

Well, my kid spent his pre-school year in the corner playing with the bin of Mr. Potato Heads wearing a red fire engine hat. He had no part of singing, drawing, or learning about anything. Several of the other children knew their numbers and letters and were quite precocious. One was already starting to read a bit. My son read around age 6, no big whup.

I knew he wasn't LD because he seemed so self-posessed and, well, I just "knew," and although he didn't talk much he clearly understood what people were saying.

We didn't read to him or practice counting or take him to museums. I was too depressed about what was going on with my other son whom we had done all of those things with.

One day he walked into the kitchen at the age of 4 or 5. He was in Pre-K at this point. He looked a bit miffed at me. I asked him what was up and he looked at me sternly and said, "I know there are negative numbers," like I had betrayed him by not telling him this. I laughed and said something about maybe learning about a few positive ones first. I rationalized that he must have heard something on TV.

I think these things reveal themselves in their own good time and they often don't look like what we expect. This is one reason why they don't like to identify too early. Same with LDs really.

Ed asked Christopher last night what the subject and predicate were in the sentence, I ate too much food, and Christopher didn't have a clue.

He flat out couldn't say what the subject was, and he thought the predicate was 'too much food.' Then, when Ed corrected him, he sobbed for 15 minutes.

Middle school stinks.

We're only....3 weeks in? Already I've got at least 4 crying children stories, 4 that I can remember, anyway; there may have been more. Today Christopher's close friend M. started crying when the math teacher docked him a point on his math test for telling his twin brother, 'It's easy, you can do it.'

M. protested that he had only been telling his brother he could do the test, and the teacher said that didn't matter, he could have been cheating.

So back to grammar, Christopher has no clue what a subject and a predicate are. He rejected outright Ed's claim that 'I' was the subject: How can 'I' be a subject??????' Then collapsed into sobs brought on by the sudden realization that the reason he 'put the line in the wrong place' was that he didn't know where the subject ended and the predicate began. A classic example of a child not knowing what he doesn't know, which Willingham has written about. (Why Students Think They Understand—When They Don’t and How To Help Students See When Their Knowledge is Superficial or Incomplete)

I'm guessing Christopher probably thinks 'subject' means 'topic,' as in the topic of an article or book; and, by extension, 'predicate' means the topic of the second half of the sentence. Which would pretty much rule out pronouns & verbs as subjects & predicates, respectively.

Christopher is 11.

His school has two hours of 'English language arts' a day, TWO. And in two hours a day this teacher--this tenured, health insuranced, pensioned individual--did not manage to teach Christopher what a subject and a predicate are.

Teaching math is hard. I'm not going to be wildly critical of a math teacher who is trying. (A math teacher who docks a twin a point because he might have been cheating is another story.)

But teaching subject and predicate to a bright child with a good attention faculty whose strength is English language arts.......

Rolling off a log.

And I'm the one who's going to be doing the rolling.

I'm not happy.

update

comments...

I'd never heard of Mad Libs!

OK, that's not right.

I'd heard of Mad Libs. Didn't know what they were; especially didn't know they were fun things that could teach parts of speech. fyi, I am aware of the fact that I have just written two incomplete sentences joined by a semi-colon. It's a secret technique of mine.

Turns out Funbrain has extremely fun Mad Libs.








Here's an Eleanor Rigby Mad Lib.

The Grammar Bible

I heard about it from my editor at Holt, who said Strumpf ran a web site on grammar for years; grammar was his life. Copy editors all over New York would call him up on the phone to ask him how to edit sentences. Finally he wrote a book, and this is it.

I read the chapter on subjects & predicates and then taught it to Christopher--it was great.

He has a rhyme: the subject & the predicate are the name and the claim.

I always like rhymes.


I've got grammar stuff posted on the Math Supplements page for safekeeping.

Steps to Good Grammar

I'm gonna get Rod & Staff, too.

Given that we've managed to miss THE FIRST TWO SUNDAYS OF SUNDAY SCHOOL, I figure we need it.

comments...

When education leaders in New York state decided to require all students to pass five Regents' examinations to get a high school diploma, many observers worried about the consequences. After all, the Regents' exams were generally considered the most rigorous graduation tests in the nation. Logic suggested that—without some remarkable improvements—either the exams would become far easier or there would be a massive failure rate, especially in the big cities, where only one out of four students earned a Regents' diploma.

The early results suggest that the Regents' tests have become easier. A remarkable 96 percent of the state's high school seniors passed the English Regents' exam last January (compared to 65 percent in 1997). High school teachers in New York City complained to the New York Times that the new Regents' exams were a snap. One social studies teacher said of the global history exam that, "students who did no work all semester, who failed tests all year, passed this exam handily." An English teacher complained that teachers were required to give credit for essays that were "barely comprehensible." The state, she feared, was "legitimizing illiteracy."

Sadly, the Regents' exams now resemble the tests that are given to most American students in history and English, in which students are seldom expected to have any background knowledge.

Any time you put through rigorous tests everyone is required to do, either they get watered down or they get abolishes. There is no political support from anywhere for rigorous tests, not the parents, not the politicians, not the schools. Rigorous tests make everyone look bad.

There is a de facto collusion among essentially everybody.

don't trust the tests

the solution

The New York example demonstrates that the political system will not tolerate a denial of diplomas to large numbers of students. That is why it makes little sense to set a single bar for all students. A single standard will inevitably be a low standard. What is needed is a credential system with multiple gradations, for example, a local diploma, a state diploma, and a diploma with honors, each representing different levels of academic achievement. Standards for each level should be both challenging and realistic, making it possible to set goals that inspire all students to increase their efforts without turning the tests into a snap for most students.

If you set the bar too high, performance declines.

If you set standards reasonably high, people will meet and exceed the standard.

People start competing with each other to see who gets the best score, which is exactly what would happen if you had different levels of high school diploma.

update

My real position, in actual practice, is 'trust, but verify.' Here in NY I take our state tests seriously, but I also ask Christopher to do the CA tests David Klein wrote for grades K-8.




comments...

Horray for you!!! Mainstreaming is not always the right answer. Many parents are afraid that if their child does not stay in a regular class, then they will never get back in. In this case, you are doing the right thing so that Ben will be mainstreamed when it really counts: in algebra and above.

Apropos of this, I have a confession to make; I'm glad to be getting my kid out of the mainstream math class he was in. He'll be getting one-on-one work in Saxon -- how is this not an improvement over Connected Math projects like My Special Number?

The first is just that people have fought so long and hard to get special needs kids like mine, who can function in a regular classroom provided they have the support they need, to be educated in mainstream classes. I remember well how isolated and ostracized special needs kids were when I was in school myself; things have improved enormously since the IDEA was passed. And now, here I am, begging to have my kid withdrawn from the mainstream (but only after a long struggle to keep him there).

The second reason I feel guilty is that, in discussions I've had with teaching professionals, the chance has frequently come up to talk about the constructivist math curricula they are using. I've frequently had to say mealy-mouthed things like, "I don't even want to get into my opinion about Connected Math in general; it doesn't matter. I'm talking about what's right for Ben." I play it this way because people are willing to hear that my special needs son can't make do with their clever new curriculum; that makes him the defective element in the equation. They are much less willing to hear from me that their curriculum just plain stinks and they should find another one. It allows them to save face. I allow them to save face.

I could have fought for all kids who are stuck with bad math curricula, but probably not effectively; not when the wall of self-defense is up so firmly. We fight the most winnable battle we can for our own kids; that's really what any parent who tutors his kid on the side, or signs him up for Kumon or other tutoring service is doing. We know that the improved scores their kids get as a result of the special help will be credited to the reformers. And we go ahead and do it anyway, of course.

In the meeting we had the other day, it was mentioned that, while Saxon might be okay for this year, we might want to reconsider mainstreaming Ben when he gets into 7th grade. It was suggested that maybe he'll be ready, by then, to go into Connected Math. Frankly, I'll go to the mat to prevent Ben from ever having to do any such thing. The heck with the mainstream.

As for algebra, my stepson Colin informs us that constructivist methods are creeping into his pre-calculus (is precalculus the same as post-algebra?) class in his junior year of high school; so we may not even be mainstreaming Ben at that point. If Ben's autism ever has had any silver lining, it's this: it's enabling him to opt out of the current foolish educational fad, and it may go on doing that right until he goes to college.

Anyway, there are no requirements here to be fighting for anyone other than your own kid; if there were, then I couldn't be here - at least not right now. This website is about saving your kid first and foremost -- through homeschooling, supplementation, by hook or by crook.

comments...

For me this blog isn't only about saving my own kid or Carolyn's kid or ktm readers' kids....politics takes all kinds of forms, and there's a distinction between power & influence (though I'm not the one to theorize what it is).

One thing I've learned about politics is that effective politicians, inside any organization, don't usually attack something head-on (though this is my inclination). They....form alliances, make horse trades, frame issues in ways that work for them, set agendas, and sell, sell, sell.

I think that's what we have to do. Because we have kids in the school system, we are, ourselves, inside the organization.

For most of us, our most effective tack will be to engage in organizational politics, if that's the term.

This is why I do a lot of 'visual' politicking. I carry my Russian or Singapore math books with me to every meeting; they are major conversation starters. I continually press the issue of Singapore's kids being best, and/or of KIPP's 8th graders having a higher percent passage rate on the Regents A.

Spaced repetition works.

At NAAR I used spaced repetition all the time.

I remember back when I first joined the organization, I was reading a book called BRAIN REPAIR.

BRAIN REPAIR, at that time, was far too radical an idea for the people who had founded NAAR. For a variety of reasons, all realistic and many having to do with the politics of autism science & NIH funding, they were willing to speak at most of treatment and prevention. The word 'cure' wasn't even included in the original NAAR literature.

So I was out there on my own, freelancing the message 'research for a cure.' People used to look at me like I was mad.

Early on, I suggested NAAR sponsor a conference on brain repair.

Here's how those suggestions went.

I'd say, 'Why don't we sponsor a conference on brain repair.'

Whoever I was talking to would look at me blankly, then return to whatever it was he/she had been talking about before I'd said, 'Why don't we sponsor a conference on brain repair.'

I kept inserting the words 'brain repair' into conversation anyway.

About 4 or 5 years into my stint at NAAR, I discovered that NAAR was sponsoring a conference in FL on....guess what?

Brain repair.

Nobody even remembered I'd spent 2 years hawking the idea.

(End thought from Carolyn: I think this also demonstrates the value of a good, catchy, repeatable marketing hook such as brain repair).

comments...

Any ideas for questions and/or categories of questions? Please let us know.

One category will definitely be finding the information you need on KTM.

comments...

I did it! I posted a picture!


Christopher says he and his friend M. are '2 for 2' on crying in Middle School. M. has cried twice, and Christopher has cried twice.

This is probably what the middle schoolists mean by brain periodization.




comments...

Adding & subtracting positive & negative numbers is one of those areas that can be severely procedural if you have a good memory, which most children do as far as I can tell.

Christopher was already becoming entirely 'procedural' with his integer problems: if he saw two minus signs in a row he automatically penciled in little vertical lines and turned them into two plus signs. Then he added.

ooops--must take Christopher to his playdate

will finish this when I get back

I love these number lines.

back again

Now he's shrieking at the top of his lungs & jumping as hard as can on his bedroom floor upstairs, which has shaken loose all the lightbulbs in the kitchen light fixture, which means someone will have to climb up on a ladder, take down the fixture, and screw the bulbs back in.

I'm in a nuts-to-autism mood.

back on topic

What I'm trying to say is that it's clear Christopher is reaching the point where he's going to have procedural fluency with virtually no conceptual understanding, and Doug's number lines are the answer.

Number lines are to integer problems what bar models are to story problems. Perfect.

I'm going to have Christopher do a page of Doug's number lines every day for a few weeks, and I'm going to do them myself.

I was telling Ed about all of this, and he said number lines were essential in the GED math course he taught to high school drop-outs in Newark years ago. He was pretty successful with those kids, and number lines were a big part of it. These kids--young adults, actually--were years and years behind in math; they'd never, ever gotten it. Ed had to have visual ways of teaching math to them, he said, or he couldn't have done it.

I haven't got all of the number lines posted yet, but will get to it soon. Right now one page is here, at the top of the Comments thread.

number lines in your head

This is all the more reason to use number lines frequently, I think. Any time you can hook a new concept to something a child already has inside his head you've got an advantage.

Vision in Elementary Mathematics by W. W. Sawyer

Andrew is better

I forced grape juice down him 10 minutes ago, and now he's making cheerful noises. His face is bruised, and he's urinated all over his TV stand, videos, and whichever of my books he'd lined up as part of the still life.

BUT, he's OK.

I feel like Annie Sullivan after breakfast.

He folded his napkin.


(So is Kumon Math sounding like just the ticket for Andrew?? I say yes!)




comments...

comments...

I still don't know what the point of the project was, but now at least I know what the result looks like.

Ben's exactly right; at the time he wrote the essay, it was exactly 48 months to his 16th birthday and the first legal opportunity he'll have to get his driver's license. I'm a bit alarmed that he's tracking it so precisely.


My Special Number page - hand in assignments here
complete list of Connected Math projects
My Special Number, part 2
All Vlorbik All the Time
Brenda's special number





comments...

BibleLiteracyProject 25 Sep 2005 - 13:45 CatherineJohnson

The Wall Street Journal has a rave review (subscription probably required) of The Bible and Its Influence, a new high school textbook created by The Bible Literacy Project:

"The Bible and Its Influence" is an exceptionally well-executed introduction to the books of the Bible and the shaping effect that it had on the writers and artists of Western civilization. It is a scholarly, clear and richly illustrated amplification of the stories of the Old and New Testaments. And where else will a high-school student find out that the Eucharist was the inspiration for Beethoven's "Missa Solemnis"? Or that when Hamlet calls Polonius "Jepthah," he is pointing to the willingness of Ophelia's father to sacrifice his daughter for his own advantage?

The textbook's intention is to provide precisely the kind of biblical understanding that has drained out of the culture in the past century. (This sort of book itself has a long tradition: family-accessible biblical exegeses began, in English anyway, with the Geneva Bible, brought to this continent by the first settlers.) But once such understanding is on the slide, is there anything to be done about it?

The Bible Literacy Project, which published the textbook, aims to provide a way for students to read the Bible in public schools without trampling on the rights of religious or secular families.

As with every other aspect of my so-called education, it seems, I KNOW VIRTUALLY NOTHING ABOUT THE BIBLE, and I am a person who attended Sunday School each and every Sunday of her entire childhood.

sigh

I'm ordering a copy today.

btw, given the fact that I don't seem quite up to the task of reading the King James Bible (which is the version I want to read), I've made some headway with Walter Wangerin's The Book of God, a re-telling of the Bible in novel form that's supposed to be good.

I was reading it to Christopher, who protested so mightily each time I brought out the book that he finally wore me down.

One of these days I'll get back to it....

OK, I'm off to Sunday School!


Bible Literacy Project Curriculum blurb

comments...

---

GrammarQuestion 25 Sep 2005 - 17:02 CatherineJohnson

What is the complete subject of this sentence?

While taking the dog for a walk, she stepped in poop.

Thank you in advance.




comments...

---

SchmidtCoherentCurriculum 25 Sep 2005 - 18:08 CatherineJohnson

It had been awhile since I'd last read William Schmidt's American Educator article, Coherent Curriculum.

I'd forgotten this section:

Some people might ask, “What difference does it make if we can’t do fancy math problems?” It does make a difference. A typical item on the TIMSS 12th-grade math test shows a rectangular wrapped present, provides its height, width, and length, as well as the amount of ribbon needed to tie a bow, and asks how much total ribbon would be needed to wrap the present and include a bow. Students simply need to trace logically around the package, adding the separate lengths so as to go around in two directions and then add the length needed for the bow. Only one-third of U.S. graduating seniors can do this problem, however. This is serious.

Lately I've been seeing the claim that our seniors blow off the TIMSS test, while Asian kids spend weeks in grueling preparation.

Color me not impressed.

A 17 year old should be able to do this problem in his sleep.




comments...

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EconomistArticleOnHigherEd 26 Sep 2005 - 02:17 CarolynJohnston

There's a large pull-out series in a recent issue of the Economist on higher education in America. I wanted to post a link to the overview article, but it's subscription only, so I'll pull out some tidbits instead.

The article claims that the U.S. still has the best higher education system in the world.

It is all too easy to mock American academia. Every week produces a mind-boggling example of intolerance or wackiness. Consider the twin stories of Lawrence Summers, one of the world's most distinguished economists, and Ward Churchill, an obscure professor of ethnic studies, which unfolded in parallel earlier this year. Mr Summers was almost forced to resign as president of Harvard University because he had dared to engage in intellectual speculation by arguing, in an informal seminar, that discrimination might not be the only reason why women are under-represented in the higher reaches of science and mathematics. Mr Churchill managed to keep his job at the University of Boulder, Colorado, despite a charge sheet including plagiarism, physical intimidation and lying about his ethnicity.

With such colourful headlines, it is easy to lose sight of the real story: that America has the best system of higher education in the world.

Actually, what the above stories really reveals is how difficult it is to get rid of a professor with tenure (but I have heard rumors that CU is still trying to get rid of Churchill; these things take time).

The main reason for America's success, however, lies in organisation. This is something other countries can copy. But they will not find it easy, particularly if they are developing countries that are bent on state-driven modernisation.

The first principle is that the federal government plays a limited part. America does not have a central plan for its universities. It does not treat its academics as civil servants, as do France and Germany. Instead, universities have a wide range of patrons, from state governments to religious bodies, from fee-paying students to generous philanthropists. The academic landscape has been shaped by rich benefactors such as Ezra Cornell, Cornelius Vanderbilt, Johns Hopkins and John D. Rockefeller. And the tradition of philanthropy survives to this day: in fiscal 2004, private donors gave $24.4 billion to universities.

That last figure is not surprising. In my experience, your university will track you down like a dog in order to try to extract money from you. Mine has followed me to numerous towns and through 2 name changes, and I've never even donated. And now, the college my stepson is currently attending is piling on. They are already getting 40K a year from us, but they want more. They can go whistle Dixie at least until he graduates, and probably after that, too.

Limited government does not mean indifferent government. The federal government has repeatedly stepped in to turbocharge higher education. The Morrill Land Grant Act of 1862 created land-grant universities across the country. The states poured money into community colleges. The GI Bill of 1946 brought universities within the reach of everyone. The federal government continues to pour billions of dollars into science and research.

The second principle is competition. Universities compete for everything, from students to professors to basketball stars. Professors compete for federal research grants. Students compete for college bursaries or research fellowships. This means that successful institutions cannot rest on their laurels.

The third principle is that it is all right to be useful. Bertrand Russell once expressed astonishment at the worldly concerns he encountered at the University of Wisconsin: "When any farmer's turnips go wrong, they send a professor to investigate the failure scientifically." America has always regarded universities as more than ivory towers. Henry Steele Commager, a 20th-century American historian, noted of the average 19th-century American that "education was his religion, provided that it be practical and pay dividends".

This emphasis on "paying dividends" remains a prominent feature of academic culture. America has pioneered the art of forging links between academia and industry. American universities earn more than $1 billion a year in royalties and licence fees. More than 170 universities have "business incubators" of some sort, and dozens operate their own venture funds.

The article claims that not all is rosy in American higher education, though. As my son would say, DUH.

The biggest risk to American higher education is the erosion of the competitive principle. The man often cited as the architect of American academia's current success is Vannevar Bush, who was director of the office of scientific research and development during the second world war. After the war he insisted that research grants be allocated to universities on the basis of open competition and peer review. But in the 1980s universities began undermining this principle by lobbying their local congressmen for direct appropriations. In 2003, the amount of money from the federal research budget awarded on a non-competitive basis topped $2 billion, up from $1 billion in 2000.

American academia's merits still outweigh its faults. Many American undergraduates are savvy enough to get a first-class education. Many academics resist the temptation to censor ideological minorities. The vast bulk of research grants are allocated on the basis of merit. Yet American universities are acquiring a growing catalogue of bad habits that could one day leave them vulnerable to competitors from other parts of the world -- though probably not from Europe, which has overwhelming academic problems of its own.

I'll finish with an obvious question. How long can America's higher education possibly remain the best in the world, when its K-12 educational system is admittedly failing?

And as far as competition is concerned, how about starting with some competition in research and development grants for mathematics education? Constructivist programs received a major boost from the NSF when, in the early 1990s, it anointed grant programs for education departments and school districts that adopted the NCTM's constructivist principles, and funded the development and marketing of commercial texts designed along those lines (I was blown away when I first discovered how that worked). How about some funding for competing ideologies, at least until we figure out which methods get better results?


Alan Greenspan on rising inequality
rising inequality, part 2
rising inequality, part 3
median income families UCSC students
another statistics question
channeling the Wall Street Journal
Financial Times on US college costs
Economist on US higher ed
The Economist on rising inequality in universities





comments...

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RisingInequalityPart4 26 Sep 2005 - 16:33 CatherineJohnson

A few weeks back we were talking about 'rising inequality'--whether it's real, and, if so, whether bad schools are a major cause.

For the time being, I believe both propositions: I do believe we're seeing rising income inequality, and I also believe that poor schools are a major cause. (I believe this because I'm taking Alan Greenspan's word for it. I have zero Special Knowledge on this score. )

In one of our exchanges we talked about what it meant that elite universities have a huge percentage of students from the top income quartile. I think it may have been Steve who pointed out that parents with college age kids are in their high-earning years, so you would expect to see colleges mostly populated with kids from top-quartile families.

The Economist article on higher ed has some further statistics on this:

William Bowen of Princeton University and two colleagues, in a study of admissions to elite universities, found that in the 11 universities for which they had the best data, students from the top income quartile increased their share of places from 39% in 1976 to 50% in 1995. Students from the bottom income quartile also increased their share very slightly: the squeeze came in the middle.

Is rising inequality the correct interpretation?

Or do demographics explain this shift as well? Do the delayed childbearing of the baby boom generation, smaller families, and employed mothers account for college students at elite universities today having parents with higher income?

Or are we seeing a 'real' rise in inequality?

those durn Americans

The real threat to meritocracy, however, comes not from within the universities but from society at large. One consequence of the squeeze on funding for public universities, created by Americans' reluctance to pay taxes, has been an academic brain drain to the more socially exclusive private universities. In 1987, seven of the 26 top-rated universities in the US News & World Report rankings were public institutions; by 2002, the number had fallen to just four.

It's always fun reading THE ECONOMIST, because of the little asides they slip in about the shocking woe caused by Americans' reluctance to pay taxes & the like. Every time I see one of those I have to discount the claim being made, because they never offer the slightest evidence that the character foible being cited has anything to do with the subject at hand.

It's interesting to know that there's a brain drain from public universities to private (Ed is part of it, as a matter of fact).

But I wouldn't assume it has anything to with rising inequality in higher education one way or the other. When private universities recruit academic stars, typically they promise them they won't have to teach undergraduates. (Not the case for Ed. He teaches graduates and undergraduates.)

It should be obvious (and it is obvious to people like Ed who teach in elite institutions) that an expensive college filled with world-renowned professors who don't teach undergraduates isn't a good school for undergraduates.

Or isn't obviously a good school for undergraduates, at any rate.

rising costs of college

Between 1971-72 and 2002-03, annual tuition costs, in constant 2002 dollars, rose from $840 to $1,735 at public two-year colleges and from $7,966 to $18,273 at private four-year colleges. True, the federal government spends over $100 billion a year on student aid, and elite universities make every effort to subsidise poorer students. One study of admissions to selective colleges shows that, in 2001-02, students with a median family income paid only 34% of the “sticker” price.

Still, the sheer relentlessness of academic inflation is worrisome. Elite colleges have little incentive to compete on price; indeed, they tend to compete by adding expensive accoutrements, such as star professors or state-of-the-art gyms, thus pushing up the cost of education still further. And the public universities that played such a valiant role in providing opportunities to underprivileged students are being forced to raise their prices, thanks to the continual squeeze on public funding. The average cost of tuition at public universities rose by 10.5% last year, four times the rate of inflation.

The dramatic rise in the price of American higher education puts a heavy burden on middle-class families who are too rich to qualify for special treatment. It also sends negative signals to poorer parents who may be unaware of all the subsidies available. Deborah Wadsworth, an opinion pollster, points out that universities may be courting a popular backlash. Americans increasingly regard universities as the gatekeepers to good jobs, but they also see them as prohibitively expensive. The result is a steady erosion of public admiration for these formerly much-esteemed institutions.

I wonder if this is true. Are universities losing legitimacy because their prices are rising?

Sounds wrong to me.

I love reading THE ECONOMIST. It's like having a mom forever, A British mom. You're always getting little REMINDERS that your behavior is not passing muster, only the behavior in question is economic, not social. Message to America, as Tom Friedman might say, and so frequently does: Behave yourself. Pay your taxes, and stop charging exorbitant fees for services.

my favorite sayings about America

The Americans will always do the right thing...after they’ve exhausted all the alternatives.

- Winston Churchill

What was really amazing was the speed with which the Americans adapted themselves....They were assisted in this by their tremendous practical and material sense and by their lack of all understanding for tradition and useless theories.

- George Rommel

Animals studied by Americans rush about frantically, with an incredible display of hustle and pep, and at last achieve the desired result by chance.

- Bertrand Russell on American laboratory rats 1927



I don't quite know what that last one means, but I like it anyway.

update

Years ago, when I went to England for the first time, I could barely stand it. Everyone sounded like my mom. I love my mom; that wasn't the problem. Our mom is so great, we four kids practically hero-worship her to this day.

The problem was that everywhere I went some complete stranger would step forward to correct my manners. At one point Ed and I were actually scolded by the people seated behind us at a play about Elvis's last night of life. It was a very silly play, featuring a fat Elvis sitting around lecturing his entourage about the Third World in a bad southern accent.

At intermission, when the people behind us asked us how we liked the play so far, and we pointed out that Elvis was unlikely ever to have used the words 'Third World' in the lucid moments of his life, let alone on the night he was overdosing on drugs, they got huffy. Then they made comments about our manners, as I recall, though I no longer remember what they said.

Drove me nuts.

I went back to England a little while ago--just after the Madrid bombings, as a matter of fact--and this time I loved meeting my mom everywhere I went.

It was a lovely trip, and when a 22-year old waiter collecting my plate in a karaoke bar said to me, 'Aren't you going to eat your peas?' I could have kissed him.

No!

I am NOT going to eat my peas!

But THANK YOU FOR ASKING!

That's what ageing will do to you.


I don't know how many of you saw this blog reaction to the London bombings, but it expresses my feelings about England, and about any and all attacks on England. (LOTS of four-letter words, so not for kids.)


Alan Greenspan on rising inequality
rising inequality, part 2
rising inequality, part 3
median income families UCSC students
another statistics question
channeling the Wall Street Journal
Financial Times on US college costs
Economist on US higher ed
The Economist on rising inequality in universities





comments...

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GlobalWarmingBlogs 26 Sep 2005 - 18:04 CatherineJohnson

Global warming has permanent membership in my personal 'unknown unknowns' category.

However, I've found a couple of blogs I think are probably worth reading, and reliable as to what issues serious climatologists themselves are mulling over:

• Climate Science, written by Roger A. Pielke Sr., a 'dissenter' who resigned from the CCSP Committee “Temperature Trends in the Lower Atmosphere-Steps for Understanding and Reconciling Differences," and whose letter of resignation can be read here.

• Real Climate, "a commentary site on climate science by working climate scientists for the interested public and journalists. We aim to provide a quick response to developing stories and provide the context sometimes missing in mainstream commentary. The discussion here is restricted to scientific topics and will not get involved in any political or economic implications of the science."

"Gavin," at Real Climate, characterizes Peilke's position thusly:

Roger Pielke Sr. (Colorado State) has a blog (Climate Science) that gives his personal perspective on climate change issues. In it, he has made clear that he feels that apart from greenhouse gases, other climate forcings (the changes that affect the energy balance of the planet) are being neglected in the scientific discussion. Specifically, he feels that many of these other forcings have sufficient 'first-order' effects to prevent a clear attribution of recent climate change to greenhouse gases.

Another post at Real Climate, on the futures market in global warming, looks like fun (defining fun broadly, that is): Betting on Climate Change



Both of these blogs are tough sledding.

But they're almost certainly worthwhile, and I'll probably check in from time to time.




comments...

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VlorbikOnBibleLiteracy 26 Sep 2005 - 18:57 CatherineJohnson

Seeing as how I'm already wildly off-topic today, here's Vlorbik's advice on acquiring Bible literacy:

who_wrote_the_bible, by richard elliot friedman
is a document i can't recommend highly enough.
asimov's guide_to_the_bible (2 vols) is also
very valuable. copies of those and a _one_year_bible_
(king james if you must, but i only tackled that
on my fourth or fifth go-round), plus a very small
amount of determination, will turn a bible-illiterate
into ... well, a bible reader i guess
(and on many very interesting issues, a better
informed one than churches generally produce
[when they produce bible readers at all]) in,
as advertised, a year.

also BR (the ex-"_bible_review") is a stunning magazine.
i've let most of my hardcopy subscriptions lapse
over the last few years but still get BR.
vlorbik sez check it out.

Thank you!

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MySpecialNumberPart3 26 Sep 2005 - 19:01 CatherineJohnson

My special number
by VlorbikDotCom

6 = 1 + 2 + 3.
6 = 1 * 2 * 3.
6 connects me to my world.
i like clipper ships.

Carolyn and I are waiting for the rest of you to hand in your assignment.


My Special Number page - hand in assignments here
complete list of Connected Math projects
My Special Number, part 2
All Vlorbik All the Time
Brenda's special number





comments...

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AntiFuzzyCatchPhrases 26 Sep 2005 - 19:37 CatherineJohnson

domain knowledge

Remember those words, and stand ready to deploy them whenever you hear superficial knowledge or he/she is good at regurgitating facts.


Charles, Steve, & Carolyn have been complaining that the fuzzies are hogging all the good words.

Hey! No fair!

It's time for us to come up with our own catch-phrases and slogans.

the list so far

'Anything But Knowledge'

Fuzzy Math ('still puts them on the defensive')

When the math teacher at the open house asks if the parents have any questions, you can ask: "Quick, what is 7 times 8?"

"Do you really call this algebra?"

'Two education professors are 100 miles apart and start driving towards each other at noon. One is traveling at 100 miles per hour and the other is traveling at 125 miles per hour. At what time should they put on the brakes to avoid hitting each other? They have to discover the answer while they are driving.'

'empathetic, interactive expository instruction' (giving a lecture)

and don't forget -

domain knowledge

comments...

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MathsurfProblemPart2 26 Sep 2005 - 19:54 CatherineJohnson

A couple of days ago I posted a problem from Mathsurf, the Pearson Scott Foreman & Pearson Prentice Hall web site that supports Middle School Mathematics, a constructivist curriculum.

I didn't get around to working the problem myself, but I posted it because the problem felt wrong.....it felt like the opposite of a challenging problem in Mathematics 6 by Enn R. Nurk and Aksel E. Telgmaa.

All of the challenging word problems in RUSSIAN MATH can be solved with effort. Story problems in Russian Math have only one possible answer.

Doug & Steve don't seem to be too impressed with the Mathsurf problem.

a story problem from Russian Math

Here's a problem from Russian Math I'd been meaning to post:


After competing in the high jump at a high school track and field meet, Mike, Ben Josh, Sam, and Chad compared their results. It turned out that:
a) Mike jumped higher than Ben but not as high as Josh;
b) two boys jumped the same height;
c) Sam, who jumped 1.3 m, did not jump as high as Ben;
d) Mike jumped 20 cm higher than Sam;
e) Chad jumped 5 cm less than Josh but 10 cm higher than Ben.
Question: How high did each boy jump?
page 205, #811


For me, this was quite a challenging problem....but it was clearly solveable, and I did solve it. (Thank God for bar models.) Working on it was fun, challenging, exciting, taxing (I hope I'm not sounding incredibly....delayed), and I learned from doing it.

One thing this problem teaches is to keep your eyes open. I've done enough of the book now that I instantly knew that 'two boys jumped the same height' was going to be the Deciding Clue. But if I hadn't known that, the fact that this 'condition' (is that the right word?) was so different from the others would have helped direct my attention to it.

I also sharpened up my visual modeling skills. I think there's some disagreement on ktm about the value of visual modeling, but for me it's important.

And I 'deepened' my abilty to 'form a unit' (Caroline & I will both be writing posts about that shortly....)

My sense is that the Mathsurf target problem probably wouldn't do any of this for me, or for any student.....what do you think?


(I'll post my answer to the Russian problem after I find out from you guys whether I got the answer right!)

update

Now that Carolyn has graded my answer, I'm posting a scan of the bar models I used to solve this. Carolyn says she used number lines, and I'm not sure these bar models are any different from what she did...

This is pretty messy (because of my distressing handwriting deficit) & faint to boot, so I may copy it over neatly later on...

bar model solution

scratch that

The scan posts HUGE, so it'll have to be reduced in size before I can post it.....

key words: bar model solution Russian math problem scan of bar model solution

comments...

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EngineeringSchool 26 Sep 2005 - 20:50 CatherineJohnson

Via joannejacobs, Confessions of an Engineering Washout:

Interesting.

The United States contains a finite number of smart people, most of whom have options in life besides engineering. You will not produce thronging bevies of pocket-protector-wearing number-jockeys simply by handing out spiffy Space Shuttle patches at the local Science Fair. If you want more engineers in the United States, you must find a way for America's engineering programs to retain students like, well, me: people smart enough to do the math and motivated enough to at least take a bite at the engineering apple, but turned off by the overwhelming coursework, low grades, and abysmal teaching. Find a way to teach engineering to verbally oriented students who can't learn math by sense of smell. Demand from (and give to) students an actual mastery of the material, rather than relying on bogus on-the-curve pseudo-grades that hinge upon the amount of partial credit that bored T.A.s choose to dole out. Write textbooks that are more than just glorified problem set manuals. Give grades that will make engineering majors competitive in a grade-inflated environment. Don't let T.A.s teach unless they can actually teach.

None of these things will happen, of course. Engineering professors are perfectly happy weeding out undesirables with absurd boot-camp courses that conceal the inability of said professors to communicate with words. Fewer students will pursue science and engineering majors, and the United States will grow ever more reliant upon foreign brainpower to design its scientific and manufacturing endeavors. I did my part to fight this problem, and for my trouble I got four months of humiliation and a semester's worth of shabby grades that I had to explain to law schools and employers for years. Thousands of college students will have a similar experience this fall.

So engineering is suffering in this country? It deserves no better.

I have to say, I've given this some thought myself.

I love math, and I'd like to learn more of it.

But I'm not sure there are teachers out there who can teach me.

I'm a self-taught kind of person; I'm constantly diving into new subjects & figuring them out.

But I'm finding math is harder to self-teach than the other subjects I've tackled thus far (and the list includes autism & neuroscientific research).

I think I've mentioned that Mathematics 6 by Enn R. Nurk and Aksel E. Telgmaa makes the Singapore Math series look like a remedial text. A good remedial text, but remedial nonetheless.

If I could use Russian textbooks like Nurk & Telgmaa's, I could learn college math.

I don't know whether I can learn college math from our own texts. Or from our own teachers.

back to Stevenson & Stigler

I suspect that the 'teaching gap' in engineering departments--and I suspect there is a teaching gap--goes back to Stevenson and Stigler, who found that Americans universally see math achievement as being (largely) a matter of innate ability, not effort.

I've never met anyone, apart from Bernie, who sees math as first and foremost a matter of hard work. (And I may be misstating Bernie's position, too.)

People--including mathematically talented people, I'd say--see math as a matter of native ability, talent, genius. I see it exactly the same way, or I did. I had to wrench myself away from this view in order to teach Christopher & me.

When you see math talent as something a person is either born with or not--and in fact math professors are going to be people who were born with math talent & plenty of it--how is that going to affect your teaching?

It's going to tell you teaching isn't what makes the difference.

Overachievement U

I am a firm believer in overachievement.

In fact, AND THIS IS A NONPARTISAN BLOG, LET ME REMIND YOU, overachievement is a quality I vastly admire in Hillary Clinton, who is the hardest working, most overachieving public figure I know. (I saw her give a speech 6 years ago, and it was something. The distance she'd come from the Clintons' first campaign was remarkable. You could see the hours and hours of hard work, on the stage.)

When Ed and I were gearing up to request the Big Switch for Christopher, from Phase 3 to Phase 4, I was a nervous wreck. I had been flatly told, by one of the two Middle School guidance counselors, 'He's a three.'

Our school--everyone in it--thinks kids are ones or twos or threes or fours, and, truth be known, I thought the same. I felt like a delusional over-reacher asking that my child, an Obvious Three, be Crowned a Four.

When we raised the issue with his Phase 3 teacher, she blanched. She'd been singing Christopher's praises, telling us he was the best student in her class, but when we said, "We'd like to move him to Phase 4" she was shocked. She had no idea we were going to raise this possibility.

She had no Mental Construct saying the top student in a Phase 3 class maybe ought to move to Phase 4.

"I've never thought of Christopher as a 4," she said.

I should stop and add that she was (and is) a terrific teacher. I don't tell this story to complain about Christopher's math teachers last year; that's not the point. The point is that Stevenson & Stigler are right; Americans think of math talent as a strange, unique, built-in form of genius.

After that meeting, which had gone terrifically well, since the teacher had rapidly & correctly worked through the logic of moving Christopher and had then advised us to do it sooner rather than later, I was still nervewracked. I couldn't stop thinking about how hard we'd had to work to get him to the top of his Phase 3 class. I couldn't stop thinking that Christopher's math progress was the product of work, not nature.

I couldn't stop thinking he was really a 3.

Ed said, 'We want him to be an overachiever in math. That's our position.'

That was a help.

I'm pretty sure we need to start thinking of math ability as a Spectrum Talent.....some people have lots of it, other people also have lots of it, too, but not at the 'learn it by smell' level of the whiz kids. This second group, the 80 to 90 percentilers, need teachers. Good ones.

The big bulk of people in the middle have whatever level of natural math ability the big bulk of people in the middle do. Singapore's students probably tell us what level of math achievement the big bulk of people in the middle have when they've got a good curriculum & good teachers.

I guess what I'm saying is: Confessions of an Engineering Washout tells me that we have a math teaching problem at the professional level as well as the elementary, middle, & high school level.

I think we need to think of math the way we think of athletics.

Yes, a brilliant athlete is born with something the rest of us aren't.

But none of the greats get there on their own.

They all have coaches--good ones--showing them how to do what they do.


Confessions of an engineering school wash-out
more confessions of an engineering school washout
the Terminator, or 'the magical number 7, plus or minus 2'
On Having a Math Brain (by Carolyn)
Wayne Wickelgren on mastery of math & on creativity & domain knowledge
late bloomers in math & Wickelgren on children's desire to learn math
math brain debunked (by Carolyn)
math professors versus computer science professors
Wayne Wickelgren on math talent





comments...

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HurricaneKatrina 26 Sep 2005 - 21:35 CatherineJohnson





Science News (subscription required)



comments...

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EngineeringSchoolPart2 27 Sep 2005 - 02:18 CarolynJohnston

I read the whole thread about "Confessions of an engineering school washout" here at KTM before reading the original article and all the comments at Joanne Jacobs' site.

I find that I agree with everybody, pretty much.

1. The author is a whiner.
2. Engineering schools do overload their students.
3. Professors at research universities typically do not care as much as they ought to about whether their students are learning, or pay as much attention to their teaching as they ought.
4. Professors and TAs often do not speak English as well as one would wish.

I think item 1 needs little explanation. I think Kern is having a hard time with his first experience of failure; he thought he was the cat's pajamas, and got smacked down hard. I sympathize, but by now I consider the occasional failure routine. It's a natural consequence of overreaching one's limitations -- whether innate or circumstantial -- in life. I hope I don't sound as whiny as he does when I talk about my failures.

As for item 2, it is a fact that engineering schools overload their students, and some are growing concerned about it. At the school Bernie and I taught at (Florida Atlantic University), there was consternation because the number of courses needed for an engineering major had grown to the point where it was impossible to complete them in 4 years (in the end, however, noone was willing to identify any courses that could be sacrificed to keep the program within its boundaries). Engineering school not only requires a lot of courses, the courses tend to be tough.

I did a math major as an undergrad, and engineering school was famously tougher, no doubt about it.

As for professors, they are incentivized to excel at research rather than at teaching. In my experience, this is as true at teaching-centered colleges (such as Bernie and I taught at) as it is at research universities. When professors lose tenure, they do so because their research was poor, not because their teaching was inadequate. Does anyone know of a single counterexample -- someone who did bangup research but was fired anyway for poor teaching (unless the person was so disliked that any excuse to oust them was seized upon)? I'd like to know. It is well worth researching for the good teachers at a school. All schools pay lip service to the importance of teaching; very few really hire and fire on that basis.

At a teaching-focused college, you can be fairly sure that you'll get a genuine (and perhaps even interested) professor for most of your classes. But at a research university, you'll generally find that the grad students that do the teaching have good domain knowledge. The grad students that I went through school with were often considered by the students they taught to be better teachers than their professors; they were less arrogant, more available than the professors, had adequate domain knowledge, and were very conscientious.

I feel that it is a student's job to take responsibility for his own education, and I wonder whether Kern fully did that. But, sometimes, hours of useless head-banging with a math text in the library can be circumvented by one good explanation from a teacher. I think students have the responsibility to utilize every tool available for learning, and one of those is their teacher. Students have the right to expect a good explanation when one is needed.

To that end, weak english is a problem that is very difficult to work around. It's best to avoid such teachers as much as possible, but it's not always possible. Departments generally try not to load themselves up with professors and TAs who can't speak english.

The question of whether a college education is actually worth $40,000 a year tuition is one best not asked around here.

Catherine worries about not being able to find a teacher who can teach her math. I worry more that she will find teachers who can't keep up with her quick mind. I think the Johnston family motto applies here: life is full of vain hopes and groundless fears. This one is a groundless fear.


Confessions of an engineering school wash-out
more confessions of an engineering school washout
the Terminator, or 'the magical number 7, plus or minus 2'
On Having a Math Brain (by Carolyn)
Wayne Wickelgren on mastery of math & on creativity & domain knowledge
late bloomers in math & Wickelgren on children's desire to learn math
math brain debunked (by Carolyn)
math professors versus computer science professors
Wayne Wickelgren on math talent





comments...

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MySpecialNumberPart4 27 Sep 2005 - 16:13 CatherineJohnson


Brenda has handed in her project:

Well, it's not my special number, but I hear that "three... is a magic number." Sorry, couldn't resist -- I was raised on Schoolhouse Rock.

yay Brenda!

Although Brenda did not fulfill the assignment, I am awarding full credit for her demonstrated ability to communicate and make connections.


My Special Number page - hand in assignments here
complete list of Connected Math projects
My Special Number, part 2
All Vlorbik All the Time
Brenda's special number





comments...

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AlgebraicSymbolsHardForStudents 27 Sep 2005 - 16:42 CatherineJohnson

Another interesting comment from a joannejacobs thread on new research about children's abstract understanding of math:

Imagine what a man like Archimedes could have accomplished if he had had the benefits of Saxon math. It is true that we all have some mathematical aptitude and that certain simple skills develop naturally, but this is far from enough mathematics to function at even the minimum wage level in our world.

I have never met a student who could flawlessly manipulate symbols according to the rules of algebra but had trouble with the deeper concepts of mathematics. Most of my students find poor algebra skills to be an almost insurmountable barrier to deep understanding.

Of course the foundations for success in algebra are those tedious skill sheets we "abuse" our children with in primary school.

Posted by: CRW at September 27, 2005 03:42 AM

hmm.

Now that I re-read this, I'm not sure what he or she is saying....is the point that a student who excels at writing & interpreting algebraic expressions can always also understand algebra?




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MultipleAnswerMath 27 Sep 2005 - 17:02 CatherineJohnson

There are all kinds of good comments I want to get pulled up front, but FIRST I HAVE to get something done besides write for ktm!

I'm allowing myself this One Last Post, before I dive into Filing Duty.

Here's Steve on MathSURF's 'no one-answer' math:

No One-Answer Math

M can NEVER equal N

This is classic, modern, ed school math. No problems can have only one answer. Why? Because they say so. Because they don't like the focus on getting only the one right answer. Because they think that in the Real World, there are no (or very few) one right answers. They think that having one right answer means that the kids will focus only on the answer and not the process.

Second, they don't like mathematical techniques, rules, or algorithms. That is why you see them go WAY out of their way to find problems that can't be solved by traditional methods, viz. equations. They don't have a clue about M is less than N, M equals N, and M is greater than N type problems. They don't understand that in the Real World, the goal is one best answer. This could invlove creating a merit function or using some statistical least squares technique. This doesn't include Guess and Check where any old solution is good enough.

Guess and Check Math

This is the holy grail of modern, reform math. I should add that this is Guess and Check without any prior knowledge. This is their linchpin of Discovery Math and Conceptual Understanding. You can't just teach kids stuff and have them practice. To them, this is the definition of rote.

No Knowledge and Skills Math

All Knowledge is Superficial

All Skills are Rote

The idea behind the MathSURF? problem is to apply Guess and Check. It is designed not to have one answer, even though they lead the student to believe that there is one answer. If a student came up with a "creative" solution, like the negative numbers in Doug's example, they would probably be very happy. However, would they be happy if the student wrote down an equation like Doug's and said that there are an infinite number of solutions? This would indicate that the student knew something about math and his/her discovery ability would be suspect.

Knowledge and Skills Preclude Understanding

Steve's right. Constructivist curricula reject right answers. I think that's one of the reasons why elementary mathematics has stopped being elementary mathematics, and has become statistics, probability, and chronic estimation.

While there are better and worse estimations, it's much harder to have One Right Estimation.

Everybody's answer can be right, or at least on a spectrum of rightness.


It's the exact opposite of the problems in Russian Math (see here and here, which are highly challenging, and have one right answer.

update

Oh, heck.

I'm just gonna pull up the whole thread:

I really hope the picture at the top of this post isn't a real problem from a math book (though I suspect it actually is.)

Either it relies on knowledge of an arbitrary convention of target scoring (score increases by one point per ring) that is far from universal, or there are an infinite number of correct answers (Aleph-1 I suspect, but my understanding of transfinites is a bit weak). I'd hate to be the poor teacher who had to determine whether a given solution to w + 2x + 2y + z = 42 is correct.

BTW, my preferred answer is probably something like:

The bullseye is worth 20, the white rings are worth 11 each, and the red rings* are worth -5.5.

* Note that the bullseye is a disk, not a ring, and thus topologically distinct.
-- DougSundseth - 22 Sep 2005


I was going to say that the center ring(?) was worth 42 and the rest zero, but I like your negative scoring better.
-- SteveH - 23 Sep 2005


It's perfect modern mathematics. It's a beautifully nebulous problem, designed to steal whatever pleasure you might have taken in getting the right answer.
-- CarolynJohnston - 27 Sep 2005

update update

Just read Steve's answer closely for the first time, and realized that while this is a lousy math problem, it's potentially an excellent personality quiz.

If we can just figure out what the character difference is between the Doug-answer & the Steve-answer. (I'm on board for Steve's answer, which is probably a Clue, though I greatly admire Doug's answer.)


keywords: no one-answer math multiple answer math multiple-answer math poor word problems bad word problems bad story problems Russian math word problems Russian math story problems

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AdhdIqClassroomPerformanceScatter 27 Sep 2005 - 19:22 CatherineJohnson


I wanted to bring this up front for a couple of reasons:

• first, most people (I find) aren't aware that a normal IQ does not show a large gap between 'Verbal' and 'nonverbal' or 'Performance.' I'm still hazy on the distinction between verbal and performance myself, but if you think of it as verbal & spatial you're on solid ground, I think.
  
  This is important to understand, because we're accustomed to thinking talents are separate & opposed; if you're good at 'A' you're bad at 'B.' I have no idea how this idea plays out later in life (I'm guessing it starts to be significantly true in high school.) But, certainly, a normal child's IQ scores on Verbal and Performance will be close. When the scores are not close, that's often a problem - and it's a hallmark of autism, which is the only reason I know about it. (Autism isn't the only cause of a large gap between verbal and performance.)

• IQ scores, Susan says, predict classroom success quite well

fyi: I used the word 'scatter' in the title of this post, which also comes from autism. Autistic kids will show 'scatter,' or 'islands of ability.' It can be incredibly strange. Donna Williams once wrote that her scores on the various subscales of IQ tests ranged all the way from mental retardation to genius.

That's scatter.


Here's Susan:

Bright children with ADD can get through grade school fairly easily. It's down the road that the problems of attention and/or hyperactivity start to become a serious problem.

I don't know the percentage of states that do this, but in IL some kind of an IQ test is done by third grade. (Here, I believe they did the Otis-Lennin one, which usually corresponds prettly close to the standard WISC III (or whatever WISC they're on.) When the parents are meeting with everyone at the end of 3rd grade/beginning of 4th, they may or may not know that the teachers and administrators have a clear view of what some intelligence test showed on their child. They often won't explain its meaning unless specifically asked for by the parent.

I mention this because in the case of ADD, an IQ score is usually going to be an underestimate of the child's true ability. At the very least, children with ADD or LDs often have wider spreads in their numbers than other children.

In other words, in most IQ type tests the numbers should cluster together. When there is a spread, or when it is below what parents or teachers think it should be, it indicates a possible problem.

Remember that IQ is not an actual measure of intelligence. There is no true measure of that. What it really does is tell you how well a child will do in a standard American classroom setting, and in that regard it is quite reliable. So, the outer edges of the bell curve are really just "falling out of the curriculum" by up to 2 years or more.

So, I'll give you an example of a kid I know who showed ADHD symptoms and had been frustrated quite a bit during his early grade school career. The kid takes the required Otis-Lennin in the third grade and bombs, showing a very low score. The parents are shocked because the kid reads years ahead of his classmates. They demand another test, so they give him the latest WISC (III or IV) and he comes out with a Performance IQ of 146, yet a Verbal of 106, with the overall IQ being around 130. Up to this point, teachers had been treating him as a disorganized, but above-average kid, nothing special. But what they realized they had was a kid with a superior intellect who had a deficiency. While a verbal score of 108 is still considered "above average," the fact that it was in such contrast to the other score told his teachers that he was in need of a completely different approach. They quietly provided the parents with an IEP and made him a part of the gifted program.

So, intelligence testing can often be a useful tool to a parent when dealing with schools. Although, I still don't think many of them really understand ADD, either, or its impact.

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TourDeForce 27 Sep 2005 - 20:01 CatherineJohnson

Engineering school is a rude awakening for most college freshmen. Many students are surprised to learn that their previous thirteen years of formal schooling have not adequately prepared them for the rigors of engineering school. Sadly, about 2/3rds of them, some very bright motivated students, won't make it through the program. This is what you learn by the end of freshman year:

1. You had been coddled the past thirteen years by your K-12 teachers. You were mostly spoon fed the material, at a slow pace, and then tested on how well you could regurgitate the exact same material back to the teacher in the exam. Rarely, if ever, were you required to apply the knowledge you had learned to solving new problems you hadn’t seen before. As a result, you could, and probably did get by, without mastering the concepts as well as you should have. You are finding out the hard way that most of your knowledge is still at the inflexible stage. This would be most apparent in...

2. Algebra: A course you took four years ago and didn’t learn well enough is coming back to haunt you now in calculus. Calculus seems much more difficult than it did when you took it last year in high school. This is because the pace is twice as fast and the exams require more than a regurgitation of what was taught (or rather won't be, see below). You see, mathematics is brutally cumulative. Calculus is really 10% calculus and 90% algebra (which includes a healthy does of trigonometry and geometry); and, the calculus step isn’t all that difficult usually. Most of the difficulty lies in either setting up the calculus step or finishing the problem after the calculus step. Calculus isn't all that difficult provided you've mastered algebra.

In high school, they allowed you over the algebra bridge without paying the full toll and you’re paying the price now, especially if you hobbled over on your graphing calculator. Anyway, you’ll need to know calculus and algebra cold if you expect to pass Physics I next semester. But this is going to be close to impossible because...

3. Your professors don’t teach and you can barely understand your TA’s poor English. This is more of an expectation problem; you’re still expecting to be coddled like you were in high school. Now you are expected to read the new material on your own and attempt to solve the problems before coming to class. This is a feature, not a bug.

By teaching yourself, you will be forced to understand and master the material, assuming you are doing the homework problems beforehand. Which you haven’t been doing because there just isn’t enough hours in the day to teach yourself and then do every problem assigned in every class. So you dutifully copy down the answers that the TA gives you during the class review all the while thinking “hey, that wasn’t so hard, now that someone’s showed me.” But, “understanding when explained by others” is not the same thing as the “ability to explain to others” which will become brutally apparent...

4. When you fail your first exam. The first test you’ve ever failed in thirteen years. You crammed the whole night before, but the test was too hard and too long. Goodbye unearned self-esteem; hello magic number 7. Seven is the number of things you can hold in working memory at one time. Partially learned knowledge uses more of these seven slots and takes longer to process than fully mastered knowledge. Your brain is being tested to its capacity for the first time and it's not prepared. You’ll become casual acquaintances with magic number 7 this semester and good friends next semester in Physics I because...

5. All those damn physics equations. Your brain is full. It feels like every time you learn something new it’s pushing something else out – like your name and your address. Spring semester brings with it Chemistry II (which requires you to remember everything you learned in Chem I), Calculus II (also brutally cumulative with Calc I), Computer Programming (learning new languages isn’t easy, especially when that language is C++); English Composition (your only easy class, too bad you have to do a term paper that’s twice as long as anything you’ve ever written before); and lastly Physics I, which will be...

6. The course that you’ll blame when you transfer to business school. Physics I – the rock upon which many engineering education ships have foundered. Two reasons – word problems from hell and the magic number seven. Physics is your first real test in your education career. It tests how well you are learning not only physics (under a withering course load of other difficult courses), but also how well you previously learned algebra and calculus. It is the latter two that will be your demise because you need every brain cell you can muster to learn physics today.

If you’re expending too many brain cycles recalling how to do the necessary calculus (most likely because you don’t sufficiently know the underlying algebra) sooner or later you’re going to meet the magic number seven. Meeting the magic number seven is like running out of active memory. You become overwhelmed and inefficient. Eventually, it all ends in tears (or an extra year of college after you’ve transferred to a nice soft major like human resources, communications, women studies, etc). So you lash out and look for someone to blame...

7. Like your college engineering department. Wrong. The train was slipping off the tracks well before they came into the picture, most likely sometime in elementary school. Don’t blame them because the train finally derailed at their station. Don’t be like the drunk who’s looking for his lost keys under the streetlights because that’s where the most light is. A career in engineering or in one of the hard sciences was effectively foreclosed to you by the 8th grade,. Most likely, you would have been none the wiser had you stayed in the soft fuzzy land of almost every other undergraduate field of study. Everyone would have been happier too because, well, you don’t know what you don’t know. Anyway, you can at least find solace in the words of Homer Simpson when he said to Lisa and Bart after they failed: “Kids, you tried your best and you failed miserably. The lesson is, never try.” But why blame yourself when you can blame the real culprit...

8. Your rotten K-12 education. Oh sure, they meant well; but look what happened. You see, you’re not part of the lower half of the bell curve who probably shouldn’t be pursuing a career in engineering or the hard sciences anyway. Nor, are you part of the two standard deviations and above gang that have the ability to succeed and compensate for a rotten education. No, you’re part of the curve that needed a good education to succeed and you didn’t get it.

And, it wasn’t a single chop that lopped your head off; rather it was death by a thousand tiny paper cuts. The accumulation of thirteen years of inefficiencies and unsound practices that prevented you from mastering and over-learning the material you needed to succeed in a rigorous college curriculum. Instead of teaching you content and facts and making you practice until automaticity, your well-meaning teachers were feed a bunch of scientifically and cognitively unsound educational fads -- constructivism, discovery learning, child-centered education, and social promotion to name a few. They all sounded so lovely in theory, yet in practice have consistently failed to adequately teach students as you have just found out. The hard way.

This advice may have arrived too late to help you; but it is not too late for that kid who just started kindergarten who lives down the street. This article is really for his or her parents, but they probably need to hear your story first before they begin to take it seriously. After all, you believed everything your K-12 educators told you and your parents, and look what happened.

- contributed by Kenneth DeRosa, October, 2005
(Note from Carolyn: this essay has been rewritten slightly, by its original author, with links added -- Carolyn).

That's going straight into the Math Writing Hall of Fame.

update

the magical number 7, plus or minus 2


Confessions of an engineering school wash-out
more confessions of an engineering school washout
the Terminator, or 'the magical number 7, plus or minus 2'
On Having a Math Brain (by Carolyn)
Wayne Wickelgren on mastery of math & on creativity & domain knowledge
late bloomers in math & Wickelgren on children's desire to learn math
math brain debunked (by Carolyn)
math professors versus computer science professors
Wayne Wickelgren on math talent
grandmasters and the magical number 7


Wickelgren on introducing algebra
Wayne Wickelgren on algebra in 7th & 8th grade
Wickelgren on math talent & when to supplement
late bloomers in math & Wickelgren on children's desire to learn math
Wayne Wickelgren on mastery of math & on creativity & domain knowledge
Wickelgren on why math is confusing





comments...

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OnHavingAMathBrain 28 Sep 2005 - 04:01 CarolynJohnston

(Note -- I've modified this post slightly from its original form. It's surprisingly one of the toughest posts I've tried to write -- Carolyn).

During our recent discussion about whether there are "math kids" -- the consensus seems to be that there are -- you might have noticed that I was quiet. There are definitely kids who find it easier than others -- but that's not the only condition for success in math.

I don't think math is outright easy for anyone. Sooner or later, you're going to hit the wall if you keep up with it. I don't think having a "math brain" that makes math easy for you is what's necessary for success in math.

I do think that some of us, for various reasons, are better equipped to enjoy the work that goes with math than others, because of a combination of their personalities, and the circumstances in which they learn and do math.

For some people math problems are like puzzles (I think that Catherine is discovering this trait within herself these days); for others, they are just a hard slog, a source of pain, and no fun at all. The people who enjoy math problems as puzzles -- as outlets for their obsessiveness, shall we say -- might not find them easy, but they aren't exactly suffering, either.

When I work a problem and get a solution, I check it over and over, obsessively, because I like to be sure that I'm right. I like being absorbed into a problem or a derivation; it beats obsessing over the worldly problems in my head that I absolutely can't solve. It feels as absorbing to me as a crossword puzzle might to you, and it also feels meaningful to be doing it. So it's innately reinforcing to me.

I also, perhaps, am more reinforced than most people by the right answer that comes at the end of all that work. And I feel empowered when I've learned something new and cool. And I've also been reinforced and rewarded by other people for my interest in math. It all adds up to a lot of positive reinforcement.

But there's a wall for everybody if you push it far enough. I hit my own personal math wall in differential geometry -- although I passed it, I took it twice, by choice, and I still don't really get it in my bones (although I do have occasional moments where some missing piece falls into place, presumably because I spent all that time obsessing about it). Differential geometry is abstract, arcane, and mixes badly with my spatial disability (I'm one of those people who can't tell left from right without thinking, and whose intuition about where she is is wrong more often than chance). If a good, tough puzzle didn't suit me, I'd have run from it.

So I don't think you have to have a 'math brain' to go far as an economist or an engineer or a scientist; you can successfully utilize math at a high level in your life without it. You do have to be unafraid of work, because it's just a question of when (not if) you'll first encounter something that's hard to understand. The important thing is that you find it rewarding, not punishing, to do the work most of the time.

That's something we have, as parents, some (though not total) control over. It feels good for a kid to be in advanced math; just being there is reinforcement for doing math. It feels good to see that look in your parent's eyes that says they're proud of you for doing well in math.

And it feels so darn good to get the one right answer.


Confessions of an engineering school wash-out
more confessions of an engineering school washout
the Terminator, or 'the magical number 7, plus or minus 2'
On Having a Math Brain (by Carolyn)
Wayne Wickelgren on mastery of math & on creativity & domain knowledge
late bloomers in math & Wickelgren on children's desire to learn math
math brain debunked (by Carolyn)
math professors versus computer science professors
Wayne Wickelgren on math talent





comments...

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WickelgrenOnCreativity 28 Sep 2005 - 15:32 CatherineJohnson

Ken's & Carolyn's posts, Tour de force and On having a math brain, are companion pieces, and, reading them back-to-back, I see that it's time for me to get Wayne Wickelgren's work posted.

I'll try to get that done today, and am re-reading Math Coach now. But I'm stopping to post this observation:

Creativity is an outgrowth of learning, and a lot of it. The past twenty-five years of cognitive psychology research has shown that the more a person knows about a subject, the more creative he or she can be in it. No question an adult poses is considered creative if someone else has already asked it. Thus, an adult must know what has come before to ask creative questions.

There's more:

This is true more generally as well. A student's ability to be creative in any area of knowledge increases with his or her knowledge of that area. Knowledge forms the fodder for creative new ideas.

sterling advice

Oh, gosh.

I can feel my day getting Sucked Up in pursuit of Wickelgrenisms. Here's another, from his section on how to evaluate your school's math curriculum:

To check that your child's teacher is focusing on math fundamentals, look at your child's homework. It should include lots of math problems to which there are single, correct answers, and no artwork or writing.

Amen to that.

all Wickelgren, all the time!

If your child is now struggling with math or scoring only in the average range in his or her classes, focus on helping your child master the math taught in each grade. Mastery means getting As or the equivalent in math.

Wickelgren on introducing algebra
Wayne Wickelgren on algebra in 7th & 8th grade
Wickelgren on math talent & when to supplement
late bloomers in math & Wickelgren on children's desire to learn math
Wayne Wickelgren on mastery of math & on creativity & domain knowledge
Wickelgren on why math is confusing


Confessions of an engineering school wash-out
more confessions of an engineering school washout
the Terminator, or 'the magical number 7, plus or minus 2'
On Having a Math Brain (by Carolyn)
math brain debunked (by Carolyn)
math professors versus computer science professors





comments...

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WickelgrenOnYoungChildrenAndMath 28 Sep 2005 - 16:22 CatherineJohnson

back story:

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

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

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

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

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

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

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

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

I was bowled over.

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

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

That was the book he wanted to hear.

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

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

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

late bloomers

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

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

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

We were going to keep the doors open.

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

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

more late bloomers

Two more stories.

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

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

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

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

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

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

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

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

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

Go figure.

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

Barbara Oakley's bio

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

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

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

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

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

all English Language Arts all the time

from The Learning Gap by Harold Stevenson and James Stigler:

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

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

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

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

English Language Arts in Irvington

I've seen this here in Irvington.

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

Christopher's 6th grade schedule includes:

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

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

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

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

another story

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

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

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

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

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

Christopher had taught himself to read in Kindergarten, had tested two years above grade level in reading back in the 2nd grade, and had just received 4s on both the ELA & the math sections of the NY state tests. He'd been in the advanced reading group all year long as far as we knew.

So when was he a 3?

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

Wow.

Think about the implications.

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

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

Perfect.

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


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

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

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

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


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

We need the same kind of teachers, with the same kind of know-how and confidence, in elementary mathematics.


Wickelgren on introducing algebra
Wayne Wickelgren on algebra in 7th & 8th grade
Wickelgren on math talent & when to supplement
late bloomers in math & Wickelgren on children's desire to learn math
Wayne Wickelgren on mastery of math & on creativity & domain knowledge
Wickelgren on why math is confusing


Confessions of an engineering school wash-out
more confessions of an engineering school washout
the Terminator, or 'the magical number 7, plus or minus 2'
On Having a Math Brain (by Carolyn)
math brain debunked (by Carolyn)
math professors versus computer science professors

comments...

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MSimonOnEngineeringTeaching 28 Sep 2005 - 19:20 CatherineJohnson

I'm pulling this to the front, because while I understand engineering is hard work, requires lots of tough courses, etc., I'm completely unsympathetic to the idea that engineering professors can't be expected to teach. By teach I mean actually put material across to students in such a fashion that they come away from class knowing & understanding more than they did when they came in.

(I'm also unsympathetic to the idea that an engineering degree should require more courses than can be completed in 4 years, and I'm going to carry on being unsympathetic in spite of the fact that I know nothing about engineering or engineering degrees. Sometimes I get like that. I have my reasons; maybe I'll get to them later.)

I skipped all that college stuff (it did go too fast in some subjects, too slow in others) by teaching myself engineering. I have worked for some large aerospace companies and hardware/software I have designed is protecting you in flight.

In fact some math routines I designed were flying on the F-16. May still be.

BTW I went to one of the top science and math high schools in the country. Omaha Central.

I might add that the US Navy knows how to teach. They cram about 2 to 3 years of engineering training into 6 months of theory and 6 months of practical application. Once you have your specialty down. Mine was electronics. However, I knew that so well that I was often teaching the course and helping the slower students pass.

Being a radio amateur at age 13 helped a lot.

In any case the Navy went faster but for me was easier. Why? The instructors knew their subjects backwards and forwards. If asked for an explanation they could give one. They worked hard to get inside the minds of the students to figure out what the student's problem was. They cared.

Why? Because they were graded on how well they taught the material. They lost their jobs if they didn't do well. No tenure.

++++

I got P-Chem in my first year of college.

I found it rather easy. I hit the wall in multi-variable differentials. (which I now get)

Heat transfer and fluid flow (which I got in the Navy) some find very hard. I sat in the back of the class reading motorcycle magazines and occasionally correcting the UC Berkely Physics Professor's mistakes. Now there was a hoot. The prof rarely called on me. I made him look bad. Still, he was quite good.

The #1 problem in our teaching corps is tenure.

++++

And yet. College was not for me.

So what if it takes 6 or 7 years to learn engineering. Shouldn't desire and tenacity count?

Such desire worked for me. But I had to do it outside of school.

Being outside of school did help me.

When microprocessors were new and there were not enough teachers to go around I taught myself. School can teach you how to learn with help. Learning on your own teaches you how to learn with no help. It ought to be valued more.

In fact learning with no help is exactly what you want on the frontiers.

The US Navy knows how to teach.

If the US Navy can have instructors who teach, universities can, too.

I don't care who teachers are, whether they're full professors or T.A.s. They need to be able to teach. We can talk all we want about the purpose of research universities being research, not teaching. What pays the bills--what keeps research universities running & funded by taxpayers & tuition paying students & parents--is the fact that they award degrees to college & grad school students.

That means teaching.

comments...

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WhatIsConstructivism 29 Sep 2005 - 01:11 CarolynJohnston

AndyJoy asked on this thread: Can someone explain extreme constructivism to me? Is the problem that proponents never want to introduce the standard algorithm for a problem or make children memorize facts?

The short answer is yes, but for the record, here is a fuller explanation. I think the best quick introduction to constructivism and its recent history in U.S. educational practice is Barry Garelick's An A-maze-ing Approach To Math, which appeared in Education Next this year. I'll excerpt a little piece of it to answer Andy's question, entirely without Barry's permission (but hopefully with his blessing).

Discovery learning has always been a powerful teaching tool. But constructivists take it a step beyond mere tool, believing that only knowledge that one discovers for oneself is truly learned. There is little argument that learning is ultimately a discovery. Traditionalists also believe that information transfer via direct instruction is necessary, so constructivism taken to extremes can result in students' not knowing what they have discovered, not knowing how to apply it, or, in the worst case, discovering (and taking ownership of) the wrong answer. Additionally, by working in groups and talking with other students (which is promoted by the educationists), one student may indeed discover something, while the others come along for the ride.

Texts that are based on NCTM's standards focus on concepts and problem solving, but provide a minimum of exercises to build the skills necessary to understand concepts or solve the problems. Thus students are presented with real-life problems in the belief that they will learn what is needed to solve them. While adherents believe that such an approach teaches "mathematical thinking" rather than dull routine skills, some mathematicians have likened it to teaching someone to play water polo without first teaching him to swim.

The Standards were revised in 2000, due in large part to the complaints and criticisms expressed about them. Mathematicians felt that the revised standards, called The Principles and Standards for School Mathematics (PSSM 2000), were an improvement over the 1989 version, but they had reservations. The revised standards still emphasize learning strategies over mathematical facts, for example, and discovery over drill and kill.

So how does this fine-sounding idea play out in the classroom? Kids tend to spend too much deriving everything from first principles. What gets sacrificed is time spent learning advanced skills, as Barry shows:

Concept still trumps memorization. Textbooks often make sure students understand what multiplication means rather than offering exercises for learning multiplication facts. Some texts ask students to write down the addition that a problem like 4 x 3 represents. Most students do not have a difficult time understanding what multiplication means. But the necessity of memorizing the facts is still there. Rather than drill the facts, the texts have the students drill the concepts, and the student misses out on the basics of what she must ultimately know in order to do the problems. I've seen 4th and 5th graders, when stumped by a multiplication fact such as 8 x 7, actually sum up 8, 7 times. Constructivists would likely point to a student's going back to first principles as an indication that the student truly understood the concept. Mathematicians tend to see that as a waste of time.

Another case in point was illustrated in an article that appeared last fall in the New York Times. It described a 4th-grade class in Ossining, New York, that used a constructivist approach to teaching math and spent one entire class period circling the even numbers on a sheet containing the numbers 1 to 100. When a boy who had transferred from a Catholic school told the teacher that he knew his multiplication tables, she quizzed him by asking him what 23 x 16 equaled. Using the old-fashioned method (one that is held in disdain because it uses rote memorization and is not discovered by the student) the boy delivered the correct answer. He knew how to multiply while the rest of the class was still discovering what multiples of 2 were.

Now, consider the constructivists' argument for allowing this lack of 'domain knowledge' to persist -- kids develop deeper understanding, 21st century skills, bla bla bla -- after having read KDeRosa's "Terminator essay" on math education.

That essay just puts this nonsense to death, don't you think?

p.s. from Catherine

I found the smart constructivism post.

Here are the 2 best passages.

Smart constructivism says:

A common misconception regarding 'constructivist' theories of knowing (that existing knowledge is used to build new knowledge) is that teachers should never tell students anything directly but, instead, should always allow them to construct knowledge for themselves. This perspective confuses a theory of pedagogy (teaching) with a theory of knowing. Constructivists assume that all knowledge is constructed from previous knowledge, irrespective of how one is taught (e.g., Cobb, 1940)--even listening to a lecture involves active attempts to construct new knowledge.**

Radical constructivism says:

It is possible for students to construct for themselves the mathematical practices that, historically, took several thousand years to evolve.

comments...

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MoreOnMathBrains 29 Sep 2005 - 01:39 CarolynJohnston

Now that we've got the "math brain" notion on the run, let's catch it and pound it mercilessly, shall we?

Here's some input from Bernie, author of a real math book and therefore someone who might lay claim to having a Math Brain if one existed.

He's reminded me of my own first math wall, which also occurred in 4th grade -- I could not, could not understand the multidigit multiplication algorithm for quite a long while during that year. I struggled massively with it and, in the end, achieved nothing better than procedural knowledge of it. Imagine if even that had been taken from me, as the constructivists would now have it? Conceptual knowledge of that algorithm only crept in over the next few years of use.

From Bernie:

I agree that math has walls all the way down. The first one I remember clearly was in 4th grade. I was required to memorize the multiplication tables. I hate memorization. I don't think I even got the concept of memorization. In that particular class we had to take our seat based on the test score we received on the last test. The "A" students sat at the front. It was a linear ranking. I had to move from the front to the very back because I didn't know that answers to the multiplication questions. They threatened to kick me out of class; my parents got involved and forced me to memorize the multiplication tables using flash cards; I moved back to the front. First wall hit and overcome, first trauma endured.

I had many more. Maybe an advantage I had over Catherine is that I never supposed there wouldn't be walls.

I agree completely with Doug that there are walls in all fields. But I think math is different because for the most part it is nothing but walls.

It's a large mistake to believe that there's a group of people over there who get it all. There isn't. Math is really like a set of mountain peaks, and I think this applies all the way down to grade school. You climb one--it takes a lot of work, but you do it--but that doesn't do anything for all the other peaks out there. They all have to be climbed one by one. Some people have climbed several, a few people have climbed a hundred, but nobody could possibly climb them all. And not just because they're lacking time, but because they're lacking talent. There are all sorts of different kinds of mathematics which require different talents of various sorts. No one has all the talents.

The whole concept that there are "math people" who can get it on the one side, and then the rest of us on the other who can't, is incredibly debilitating. It lets kids off the hook for being lazy when they should have continued on and persevered. It's a horrible concept and completely wrong.

And it lets the more mathematically talented off the hook because they think that just because they have some mathematical talent they don't have to work anymore. I've seen a lot of those, and they were all lying by the wayside. Everybody who does math or physics or engineering seriously has to work just as hard as Ed. Grothendieck was probably the best mathematician of the latter Twentieth Century and he was famous for working all the time.

(Grothendieck invented modern algebraic geometry pretty much singlehandedly. He came from nowhere, and ultimately vanished. Think Good Will Hunting when you think of Grothendieck; he was the rarest sort of bird).


Confessions of an engineering school wash-out
more confessions of an engineering school washout
the Terminator, or 'the magical number 7, plus or minus 2'
On Having a Math Brain (by Carolyn)
Wayne Wickelgren on mastery of math & on creativity & domain knowledge
late bloomers in math & Wickelgren on children's desire to learn math
math brain debunked (by Carolyn)
math professors versus computer science professors
Wayne Wickelgren on math talent





comments...

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BadTeacherStudy 29 Sep 2005 - 14:49 CatherineJohnson

I still have lots more Wayne-Wickelgren blooki-ing to do, plus some tentative thoughts about whether parents have power over their schools, and if so, how much & what kind.

But this thumbnail account of a famous ed study popped up in today's Wall Street Journal & I want to get it posted:

... inept, unkind or unfair teachers can have a huge impact on a child, causing emotional, social and academic setbacks. In a 1996 study that is still widely cited, William Sanders and June Rivers at the University of Tennessee tracked thousands of elementary students' test scores year-to-year and used them to rate teachers as "effective" or "ineffective." Then, they tracked two random groups of similar students who happened to be assigned to either three good or three ineffective teachers in a row between third and fifth grade. The result: a 50-percentage-point difference over three years in the average test-score changes of the two groups, with kids who had the effective teachers progressing more, says Dr. Sanders, now senior research manager at SAS, a Cary, N.C., software concern.

from:
What to Do When You Are Worried That Your Child Has a Bad Teacher
WALL STREET JOURNAL
September 29, 2005; Page D1

I don't know how to put all the different factors together & come up with an understanding of how and why our schools work & don't work.

But I believe this study. I've mentioned this before: after trying to teach Christopher math using the school's textbook, SRA Math, I had higher regard for our district's teachers not lower.

He had learned very little in fourth grade math. He said his teacher couldn't explain things, and I think that's true. (She's not at the school this year, so I hope she won't see this. She was a terrific person; we all liked her very much.)

But what struck me, in my struggle to teach Christopher concepts I was realizing I didn't understand myself, was how thorough his mastery was of the math concepts he'd been taught in K-3. He instantly knew, without thinking about it, that the larger the denominator the smaller the 'piece.' Instantly. And his conceptual understanding was as good as it could be at that age. He could show me, easily, on a drawing, that 1/4 of a pie is less than 1/3. He could generalize this to 1/1005 being smaller than 1/1004, too.

He learned this exclusively from his teachers. His dad and I weren't even paying attention in those early grades, I'm sorry to say. We were leaving things up to the school.

What dawned on me in those first months working with Christopher was the perception that our teachers were so good they could teach math "no matter what you threw at them," as Carolyn would say. They could teach around a book, if the book was bad. And they did.

(SRA Math may not be dreadful, btw; I don't want to be in the SRA Math bashing business. The books have serious content, and are challenging, & I would have opted to keep them rather than change to TRAILBLAZERS though I can certainly understand why the teachers were happy to see it go. BUT I couldn't teach out of SRA Math myself, and I've had several teachers tell me it was murder for them, too.)

So.....there you have it.

Until I know more, I'm sticking with the conviction that A Good Teacher Makes All The Difference.

which means that....

...which means that a lot of good teachers are probably going to de-fuzzify fuzzy math. They're just going to do it. I think Steve has observed that this is what tends to happen; the new fuzzy math comes in, the school works with the curriculum a couple of years, then they start supplementing big-time.

Another commenter, Katherine Prouty had this to say:

I can tell you that my daughter had NO IDEA how to do division at the end of 5th grade. I thought that she didn't get it...

She also wasn't strong in the lattice method of multiplication. Honestly, there were so many addition steps that she was bound to make a mistake -- especially since drilling of any type of math fact was out of the question, although, with my son, now three years later, they are drilling those math facts in the Everyday curriculum like there is no tomorrow. I'm sure the math tests forced them to it (against their better teaching judgement, of course.)

The Schaumberg teacher I met at the airport, the one who was a keen & enthusiastic fan of Everyday Math, told me, 'Well, you have to give them worksheets. Otherwise they're doing this--" and she performed a delightful imitation of a little kid waggling his fingers against his chin trying to add & subtract.

Here was a lifelong teacher who'd spent 15 years doing nothing but fuzzy math, and the idea that kids have to drill & memorize just seemed obvious to her.

I don't understand politics and the nature of social stability and change (I find the subject riveting).

But while my family motto is It's always worse than you think, it could equally be, It's never as bad as you think. Or maybe just, it's never what you think.

That last is true for sure.

rtfm - NOT

This reminds me of the old rtfm line, which I will not spell out, on account of this being a family website and all.

That means no f-words. (No f-words with a few notable exceptions, that is.)

Suffice it to say that the letters r, t, and m stand for read the manual.

Liping Ma & others have pointed out that, in America, teachers' manuals are written with the express & conscious awareness that no one will ever read them.

In the case of constructivist curricula, that's one thing we've got going for us.



^* It's always worse than you think and no common sense-y

worsethanyouthink

comments...

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GagMeWithASpoon 29 Sep 2005 - 19:46 CatherineJohnson







I just about fell off my chair when I clicked on 'bad teacher' Comments and found this--




comments...

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MathProfessorsVsComputerScienceProfessors 29 Sep 2005 - 19:56 CatherineJohnson

Very interesting comment from Lesley Stevens:

Tangential to the "math brain" discussion, my husband has made a very interesting observation.

A smidge of background here: He has always been one who has no fear of questioning or correcting his instructors, something that many of his primary school teachers didn't much care for, as you can imagine. He has a double major in mathematics and computer science and he'll graduate with his B.S. this spring. (He is 31, finishing his degree after a 10 year hiatus.)

What he has noticed is that while his CompSci and gen ed instructors often resent being corrected, his mathematics instructors do not.

His theory is that people who do math are accustomed to being wrong. They make mistakes all the time, and it's easy to do when working a complex problem on a blackboard. He thinks that you pretty much can't do math all the time and still maintain an infallibility complex, or superior attitude towards students. Especially since math is a young person's game, and most math professors are already past their "peak" in math ability, and know it.

In addition, in "soft" liberal arts areas, or conversely, extremely complex areas like programming, mistakes may not be obvious, or may be open to some debate. In math, an instructor can't wiggle around a mistake. If he has added 6 to 7 and gotten 14, that's just wrong, end of story.

What I think I'm getting at here is that making math easy for students through "no one answer", etc. is not helpful because it delays an understanding that math is hard for everybody including people like my husband, and that the best mathematicians in the world make mistakes all the time. This understanding actually makes me feel a lot better about my own anxieties about math.

Oh, and as for "math brains", my husband's major the first time around, before the 10 year break, was Philosophy.

This discussion has been a revelation to me. I'm going to keep all the URLs handy so I can print out these comments out and/or send the links to friends, teachers, & administrators as needed.

The vast majority of people simply assume, without even realizing they are assuming, that doing math comes naturally to the select few AND that those select few are the ones who ought to be doing math, and who deserved to be put in Phase 4.

I was just this afternoon talking to a mom whose son was moved from Phase 4 to Phase 3; according to figures I was given, 35% of Irvington's Phase 4 5th graders failed the Phase 4 placement test at the end of 5th grade, something most parents don't know. Most of these children switched to Phase 3, though some parents refused the move. I know of two; there may be others.

All of this gatekeeping activity is based on the explicitly stated judgment that 'he/she doesn't belong in Phase 4.'

It's an essentialist argument.

I was already off the boat for the whole 'He's a three' business, thanks to Wayne Wickelgren, and to Ed ("We want Christopher to be an overachiever.")

Now I'm seriously off the boat. And I'm armed.


Confessions of an engineering school wash-out
more confessions of an engineering school washout
the Terminator, or 'the magical number 7, plus or minus 2'
On Having a Math Brain (by Carolyn)
Wayne Wickelgren on mastery of math & on creativity & domain knowledge
late bloomers in math & Wickelgren on children's desire to learn math
math brain debunked (by Carolyn)
math professors versus computer science professors
Wayne Wickelgren on math talent

comments...

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WickelgrenOnMathTalent 29 Sep 2005 - 20:34 CatherineJohnson

Differentiating children by their abilities and skills is a controversial subject, but math aptitude can vary greatly among children, just as children differ in their ability to run, jump, give speeches, draw, sing, comfort others, tell jokes, or lead a group. And though it's generally impolite to speak of such differences, it is important to recognize that they exist--and for parents to have a sense of where their children rank among others.

Having a sense of your child's math ability can help you set realistic goals for your child in math. It can help you decide whether your child is progressing in math as fast as he or she can or whether you need to push a little harder or do something different--such as provide supplementary math education. For example, you probably want to supplement our child's math education if

• Your child has a very high ability in math--that is, appears to be among the best 10 percent of students in his or her class

• Your child is at least average in math and has career ambitions in math or a math related field

• Your child scores in the range of college-bound kids, but his or her school math training is inadequate for admission to the college-prep math track or for scoring at the child's ability on the math portion of the Scholastic Aptitude Test (SAT)

• Your child is of normal intelligence, but isn't mastering the basics of arithmetic in school--a necessity for independent living

One way to assess your child's math ability is to have him or her take an IQ test, but a simpler and probably more accurate method is to observe his or her learning rate relative to that of other children. If your child catches on to math concepts quickly compared to others the same age, he or she is probably a fast learner. Similarly if lots of other kids seem to catch on to math concepts more quickly than your child does, he or she may be of only average ability.

My daughter Ingrid says she was well aware of the differences in math ability between herself and her brother Abe, from the time I began teaching them at home at ages eight (Ingrid) and six (Abe). Ingrid remembers times when I would pose a problem to the two of them, and Abe would soon begin scribbling on his paper to come up with the correct answer. Ingrid, meanwhile, would look puzzled, wondering how Abe had figured it out so quickly. She knew she wasn't stupid, but it was obvious to her that her younger brother had a gift for numbers she could not claim.

Still, Ingrid took algebra in the seventh grade (when she was just eleven) without much difficulty. She learned plenty of math to complete a fast-paced college physics sequence and the curriculum required for a biology major. She is now a successful writer specializing in scientific and medical topics.

Meanwhile, Abe wanted to become a professional basketball player as a child and for years worked incredibly hard at improving his basketball skills. After first grade he played in a league every winter and went to as many as three basketball camps in the summer. Year round, he practiced basketball for an hour a day or more. As a result he was named the starting point guard for his seventh-grade team. But after that, his tiny stature and limited natural ability made him less attractive to coaches than kids who were bigger and learned new skills much faster. While his accomplishments in basketball diminished--he did not make the high school basketball team--his success in math continued unabated. He excelled in his advanced classes and remained five years ahead of grade-level in math.

It was clear from watching this drama unfold that Abe's natural talents lay more in math (and other school subjects) than in playing basketball. Similarly, it should quickly become apparent to you from your child's experiences whether your child is very talented at math or has lesser abilities in this area. But remember: Even if your child's natural talents in math do not suggest he or she should become a mathematician, your child could still use his or her math skills to become very successful-perhaps as an engineer [ed: yikes] or a financial manager. My son did, after all, develop a helluva basketball game, making him a top player in the adult recreational leagues he later joined.

[snip]

Your child's talents may or may not be in math. So do all you can to motivate your child to learn math and provide the best teaching possible, but as only one part of a well-rounded life. When you've done that, you've done your best and should accept your child's progress in math at school.

Despite what my dad told me, it is not true that you can do anything if you work hard enough. But if you temper your ambition with realism, you can derive enormous satisfaction from the truly spectacular results of hard work coupled with excellent instruction.

I love this man.



Math Coach



How to Solve Mathematical Problems


Wickelgren on introducing algebra
Wayne Wickelgren on algebra in 7th & 8th grade
Wickelgren on math talent & when to supplement
late bloomers in math & Wickelgren on children's desire to learn math
Wayne Wickelgren on mastery of math & on creativity & domain knowledge
Wickelgren on why math is confusing


Confessions of an engineering school wash-out
more confessions of an engineering school washout
the Terminator, or 'the magical number 7, plus or minus 2'
On Having a Math Brain (by Carolyn)
math brain debunked (by Carolyn)
math professors versus computer science professors





comments...

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ColinPromisesToBlog 30 Sep 2005 - 03:51 CarolynJohnston

Colin, my 17-year-old stepson, is finding his math class pretty frustrating this year for a number of reasons. One is that, having heard me kvetching about constructivist math, he's pretty sure that his precalculus text ("Precalculus: Graphical, Numerical, Algebraic", by Demana, Watts, Foley, and Kennedy -- Pearson/Addison Wesley publishers) is constructivist.

From the preface:

Our text combines appropriate use of technology with standard "paper and pencil" analytic techniques to provide a balanced approach to the study and implementation of precalculus. Technology is fully integrated, rather than just added. The text encourages graphical, numerical, and algebraic modeling of functions as well as problem solving, conceptual understanding, and facility with technology.

It could be worse. From my brief skimming of it, its chief failing is that it's incoherent. It's trying to do too much at once, and the emphasis on learning to use your graphing calculator is stealing time better spent learning algebra.

Also from the preface (speaks for itself):

In writing this edition, we have followed the guidelines and recommendations published by AMATYC, MAA, NCTM, and state guidelines.

I hope to get Colin to weigh in tomorrow.

comments...

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SaxonAlgebraPlacementProblem 30 Sep 2005 - 14:16 CatherineJohnson








I absolutely can't get the answer Saxon gives in the answer key.

Thanks!


keywords: Saxon algebra placement test problem





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CourantConference 30 Sep 2005 - 14:38 CatherineJohnson



I should have posted this two weeks ago--(where is my brain?)


Math and Reading: Delivery on the Promise of Mayoral Control
Sunday, October 2, 2005 2:00 pm - 5:30 pm
Main Auditorium – Ground Floor
Courant Institute of Mathematical Sciences, New York University
251 Mercer Street
New York, NY 10012

In 2002, Mayor Michael Bloomberg, as part of his efforts to fulfill his campaign pledge to turn around the chronically failing public school system, persuaded the State Legislature to abolish the independent Board of Education and to grant him complete control of city schools. He appointed as school chancellor an administrator from outside the education world, Joel Klein, who had directed the Federal Government's anti-trust efforts. In January, 2003 the Mayor announced "Children First," a new agenda for New York City schools. The plan launched a radical re-organization and system-wide instructional reforms, the most far-reaching changes to city schools in a generation.

This forum will examine key aspects of the changes in New York City public education under Mayoral control, with a special focus on the “standard curriculum” for reading and math The analysis and discussions will raise important questions and provide practical guidance for parents, educators, policy makers, elected officials and education philanthropies, for everyone concerned about the future of NYC schools. Audience members will have several opportunities to ask questions and to give comments.

Presentations:

THE PROBLEM WITH MAYORAL CONTROL, DIANE RAVITCH, Research Professor of Education, New York University
NYC'S PEDAGOGICAL TYRANNY, SOL STERN, Senior Fellow, Manhattan Institute; Contributing Editor, City Journal
READING INSTRUCTION - THEORY, RESEARCH, AND REFORM, SANDRA STOTSKY, Research Scholar at Northeastern University
NYC CHILDREN FIRST MATHEMATICS REFORMS, FRED GREENLEAF, Professor of Mathematics, Courant Institute of Mathematical Sciences, New York University
LESSONS FROM JAPAN - THE TIMSS VIDEOTAPE STUDY AND WHY IT MATTERS ALAN SIEGEL, Professor of Computer Science, Courant Institute of Mathematical Sciences, New York University
MATHEMATICS EDUCATION REFORM: TOWARD A COHERENT K-16 CURRICULUM, STANLEY OCKEN, Professor of Mathematics, City College, City University of New York

DISCUSSION MODERATION: ANDREW WOLF, Education Columnist, New York Sun; Editor & Publisher, Riverdale Review and Bronx Press


The Courant Institute at NYU is nearest the following Subway Lines: A, B, C, D, E, F, V (West 4th Street) N, R, W (8th Street)

To RSVP and for additional information, or to request arrangements for translation services, please contact: Elizabeth Carson at CIMSENYU@yahoo.com or call 917.208.7153.

This event is sponsored by the Courant Initiative for the Mathematical Sciences in Education (CIMSE) housed at the Courant Institute of Mathematical Sciences at New York University. CIMSE is a unique collaboration between NYU and area college and university faculty, and New York City parents, teachers and administrators. CIMSE supports activities to educate the parent, educational, business, and philanthropic communities, locally and nationally, on topics and issues in K-12 mathematics education.


I am desperate to attend this event.....and meanwhile Ed's off to Manhattan for an all-afternoon bike ride on Sunday.

I'm thinking I may need to get some babysitting.

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CommunityCollegeAndTheOlderFemale 30 Sep 2005 - 15:44 CatherineJohnson



A year of community-college schooling can raise an older female's income by 10%, according to a Chicago Federal Reserve Board study.

source:
What Color Is Your Collar?
By TODD G. BUCHHOLZ
September 30, 2005; Page W11
WSJ


Just in case there are any older females amongst us.

Which I'm sure there are not.




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ReadingRecommendationsFromKtmReaders 30 Sep 2005 - 16:09 CatherineJohnson

It suddenly occurred to me I'd love to know what books you guys like. Novels, nonfiction, (math, of course)--anything.

Children's books, too.

If you're around, and feel like leaving some favorites in the Comments section, please do.

Thanks!


Also, you might want to check the Recommended Reading and Our Favorite Math Supplements pages every once in awhile. (Both are on the sidebar.) I've been keeping them at least semi-up to date, including recommendations from ktm readers (I think I've got all of the calculus & geometry recommendations), and I also drop in books & web sites I've come across that look good. There's quite a lot of off-topic stuff there, too, now: English language arts including reading, writing, spelling, & grammar; handwriting; even a book or two on how to draw.

If you have other books or resources I should add, let me know those, too when you have a chance.

useability improvements t/k

Carolyn and I are also working on useability improvements, slowly but surely. We're getting together a FAQ page (can't believe we didn't do this from the beginning) and ASAP we'll finally take David Klein's advice and have a dedicated page, with a link on the sidebar, for constructivist curricula. Parents (and teachers) will be able to go directly to that page to find a list of all constructivist curricula, reviews, web sites listing the projecst & assignments for these curricula, as well as our collection of ideas for supplementing and remediating the gaps they leave in children's math education.

We're always open to suggestion (help!) so send your thoughts.




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WickelgrenOnPrealgebra 30 Sep 2005 - 16:16 CatherineJohnson



Gulp.

A student can learn a year of pre-algebra math in three to six months studying three to ten hours per week, depending on the child's math aptitutde.

I'm gonna have to pick up the pace around here.

I've been working my way through Mathematics 6 since the beginning of June.

It is now the beginning of October.

RUSSIAN MATH has, estimating conservatively, 10,000 problems. At least 10,000. I have now worked 8000. In the process, I've learned a huge amount, although, sadly, even Enn Nurk & Aksel Telgmaa have not been able to dissuade me from the conviction that 7 x 6 = 43. If they can't do it, probably no one can.

I've just begun the last of RM's six chapters, and I was getting excited about starting algebra next. I can't wait.

So last night I took Saxon Math's placement test (pdf file) for algebra 1.

I got a 72.

conclusion number one:

I am going to stop expressing reservations about the Saxon math series until I can actually take and pass a Saxon math test.

conclusion number two:

wow

There are a boatload of topics I still don't know after doing 8000 complicated Russian computation, geometry, & word problems.

They are:

• using four 'unit multipliers' to convert 630 square yards to square inches: I have no idea what a unit multiplier is, or how to use it
• what a decimal part of a number is (I got the answer right, but only because I made a blind guess as to what a decimal part would be)
• negative exponents
• how to find the volume of a cylinder
• 'the method of cut and try' to find the square root of 20: to my knowledge, I have never heard the words 'cut and try' in my lifetime
• how to use a straightedge (what's a straightedge? I still don't know) and a compass to copy an angle
• how to find the area of a triangle (all I remember is: hypotenuse)
• how to find the probability that a die will first roll a 6 and then roll a 2, in that sequence
• base 2
• update 7-16-2006: I know all these things now, and will finish Lesson 81 (of 120) in Saxon Algebra 1 today.

So my first reaction, in Western polarizing fashion, was: I know nothing.

I know nothing, and I need to work through all 857 pages of Saxon Math 8/7 with Pre-Algebra before I can even think about setting foot inside a real algebra textbook.

I was depressed.

But then I calmed down a little and thought, mmmmm....maybe not.

Maybe I can just go through Saxon 8/7 and do every single lesson & every single problem related to these 9 topics.

Is that wrong?

update 7-16-2006: I ended up working through the entire book. Every lesson, every problem, every test. Then I took the Saxon placement test and placed into Algebra 2, but decided to start with Algebra 1. I'm glad I did.

Christopher began teaching himself Saxon Algebra 1/2 this summer (he starts 7th grade in th fall) so I'm reading through those lessons to make sure I didn't skip anything I need to practice - and just for the joy of encountering John Saxon's take on topics I already know.

Algebra 1 integrates algebra and geometry, though without proofs. I'll start Algebra 2 in September.

In one year I will have worked through:

• final chapter of Russian Math
• all of Saxon Math 8/7
• all of Saxon Algebra 1

That pace seems OK to me.




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AnyNumberCanBeAFraction 30 Sep 2005 - 20:39 CatherineJohnson








Steve, on the thread need for speed thread, pointed out that any number can be a fraction, and when I said I ought to put together a worksheet on this subject for Christopher, Dan directed me to this frame, DimWksheet010.ppt. of his dimensional dominoes!

It's wonderful.

I'm going to have Christopher do it.

Which reminds me, yet again, I have got to get Doug Sundseth's number lines attached. I've used them two nights in a row, and today I sent a bunch in to Christopher's teacher, in case she wants to use them with the kids.

math ed is a riveting subject

Obviously, I've become obsessed with math education. I'm constantly trying to figure out what it is about math that makes it confusing, and what one can do to make it less confusing.

Liping Ma talks a lot about fragmented knowledge, and cognitive scientists all wrestle with the problem of expertise, which means the ability to generalize what you know to novel problems and solve them.

I've noticed (I may be quoting others without realizing it) that one of the problems with the 'novice' stage of learning is a kind of over-solidity of numbers, a thingness.

Doesn't Freud talk about children first playing with words as if they were things?

Does he say the same of numbers?

I don't remember.

In any case, what I've seen in myself, and in Christopher, is that numbers are too-solid. Both Saxon & Singapore spend a great deal of time conveying the idea that numbers are fluid, in a away, blinking constellations that can be one thing one moment (-10, say) and another in the next (-5 + -5, or -20/2, or any of an infinite number of combinations & expressions).

The Everyday Math article called this 'number partition theory,' and I haven't been able to figure out whether it is or is not number partition theory, but for my purposes, at the moment, it doesn't matter. Just knowing that the number 10 doesn't have the stability of a chair or a tree or a car is a big help.

So I've been trying to convey this to Christopher.

Dolciani's classic algebra text, btw, opens with this idea. '6 + 4' is another expression for '10.' Ten is not the answer to '6 + 4,' but another expression of '6 + 4'. The difference is huge.

Saxon 8/7 constantly uses the word 'Simplify' to mean 'Find the answer,' which I think is excellent. One day Christopher actually said, 'When he says simplify, he means find the answer.' And I thought that was fine. He's getting the idea that simplify and answer are synonyms.

generalizing knowledge

I'm wondering whether making numbers less thing-y for a child might help him or her to generalize a bit more easily, or a bit sooner -- or at least help him to generalize when he's practiced enough that he/she ought to be generalizing.

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DougSundsethNumberLines 30 Sep 2005 - 21:22 CatherineJohnson




blank number lines (pdf file)

symmetric number lines (positive numbers, negatives numbers, 0 (pdf file)

number lines: all positive numbers (pdf file)

number lines: all negative numbers (pdf file)

update

If anyone is interested in, or has time to, critique these study sheets, that would great. (There's no pressing need for this; I'm reasonably certain these are accurate, especially since the second document came straight from the pages of Mathematics 6.

addition & subtractions of integers review sheet

integers problems from RUSSIAN MATH





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-- CarolynJohnston - 04 Oct 2005