That's not what the National Research Council says:

Executive Summary

Under the auspices of the National Research Council, this committee’s charge was to evaluate the quality of the evaluations of the 13 mathematics curriculum materials supported by the National Science Foundation (NSF) (an estimated $93 million) and 6 of the commercially generated mathematics curriculum materials (listing in Chapter 2).

The committee was charged to determine whether the currently available data are sufficient for evaluating the effectiveness of these materials and, if these data are not sufficiently robust, the committee was asked to develop recommendations about the design of a subsequent project that could result in the generation of more reliable and valid data for evaluating these materials.

[snip]

The Quality of the Evaluations

These 19 curricular projects essentially have been experiments. We owe them a careful reading on their effectiveness. Demands for evaluation may be cast as a sign of failure, but we would rather stress that this examination is a sign of the success of these programs to engage a country in a scholarly debate on the question of curricular effectiveness and the essential underlying question, What is most important for our youth to learn in their studies in mathematics? To summarily blame national decline on a set of curricula whose use has a limited market share lacks credibility. At the same time, to find out if a major investment in an approach is successful and worthwhile is a prime example of responsible policy. In experimentation, success and worthiness are two different measures of experimental value. An experiment can fail and yet be worthy. The experiments that probably should not be run are those in which it is either impossible to determine if the experiment has failed or it is ensured at the start, by design, that the experiment will succeed. The contribution of the committee is intended to help us ascertain these distinctive outcomes.

[snip]

The charge to the committee was “to assess the quality of studies about the effectiveness of 13 sets of mathematics curriculum materials developed through NSF support and six sets of commercially generated curriculum materials.”

[snip]

In response to our charge, the committee finds that:

The corpus of evaluation studies as a whole across the 19 programs studied does not permit one to determine the effectiveness of individual programs with high degree of certainty, due to the restricted number of studies for any particular curriculum, limitations in the array of methods used, and the uneven quality of the studies.

source: On Evaluating Curricular Effectiveness: Judging the Quality of K-12 Mathematics Evaluations (2004)
National Academies Press
Mathematical Sciences Education Board (MSEB)
Center for Education (CFE)
available online or purchase, pages 3 & 188

And I seem to recall something in NCLB about....evidence-based instruction?

Evidence-based instruction and receipt of federal dollars?

Yes?

I'm pretty sure.



nationalresearchcouncil

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PartitiveAndQuotitivePedagogy 11 Jul 2005 - 03:01 CarolynJohnston

Catherine mentioned in one of her comments that she always finds it amusing when a mathematician encounters the notion of partitive vs. quotitive division:

I absolutely think there's all kinds of elementary math knowledge real mathematicians don't have, or did have but forgot, etc.

I always crack up when i see or read real mathematicians reacting to the 'partitive'-'quotitive' distinction in division.

They think it's ridiculous!

(And btw, I STILL can't explain the difference, so I'm not even going to bother to try....)

She's absolutely right. When I first encountered the notion of partitive vs. quotitive division (Liping Ma goes into a lot of detail about it in her book) I thought it was unnecessary obfuscation.

I know I never learned it myself. I don't know if my teachers knew it, but I know they never taught it to me (although Liping Ma says they didn't need to). And I don't know whether I need to know it in order to teach young children the full meaning of division, although Liping Ma says I do.

But as it happens, I do know what the difference is: my husband explained it to me in brilliantly simple terms (having learned it at the same time I did, and distilled its meaning more efficiently than I did). Here it is:

Partitive problems ask you to divide number of objects by number of groups, and get number of objects as an answer.

the partitive type of word problem asks this question: if I have x objects, and I want to split them into y groups, how many objects will be in each group?

Examples of partitive problems:

I have a board of length 16 inches, and I need to make 10 shorter boards of equal length out of it. How long can each board be? (16 objects, 10 groups)

I have a batch of 128 cookies. I need to split it into 8 equal bags of cookies. How many cookies will there be in each bag? (128 objects, 8 groups)

I have 12 cans of pears, and I need to serve 24 kids at lunch. How many cans of pears will each kid get? (12 objects, 24 groups)

It is somewhat difficult to frame word problems involving division by fractions as partitive problems, because you are dividing by the number of groups you want. Generally, you don't want a fractional number of groups. Note that in the problems I gave as examples of partitive division, the divisors are always whole numbers.

But here is a partitive word problem that uses a fractional divisor:

I have two cans of dog food that I need to split into 1-1/2 servings for my big and small dog. How many cans will be in a single serving? (2 objects, 1-1/2 groups -- awkward!)

Quotitive problems ask you to divide number of objects by number of objects, and get number of groups as an answer.

the quotitive word problem asks: If I have x objects, how many groups of y objects can I make from them?

Examples of quotitive problems:

I have a board of length 16 inches, and I need boards of length 1-3/4 inches. How many short boards can I cut from the longer board?(16 objects, 1-3/4 objects)

I have a batch of 128 cookies. I need to split it into bags of 12 cookies to give to children at school. How many such bags can I give away? (128 objects, 12 objects)

I have 12 cans of pears, and I need to serve a half can of pears to every kid at lunch. How many kids can I serve? (12 objects, 1/2 objects)

Problems involving division by fractions are easier to frame as quotitive word problems. Note that in the first and third sample problem, the divisor is a fraction; I didn't have to gin up an awkward problem involving big and small dogs in order to give you an example of quotitive division by a fraction.

Liping Ma's only point vis a vis quotitive and partitive division is that teachers should know the difference. It doesn't have to be explicitly laid out for the kids. But teachers need to know about it because they need to give a mix of types of word problems. She says that it may be obvious to us that numerically they are the same problems (in fact it is SO obvious that we miss the distinction!), but to the kids it may not be.

I'm not sure that's true, but I'm willing to give her the benefit of the doubt.

Liping Ma actually gave a set of US and a set of Chinese elementary school teachers the following problem: frame a word problem for 1-3/4 divided by 1/2.

The best of the Chinese teachers gave examples of both partitive and quotitive word problems; they were all able to give at least one word problem for the division. But some of the US teachers couldn't do the calculation.

The difference: in China, elementary math teachers are respected for what they do, and given time to consult with each other in order to improve their pedagogical knowledge. Elementary Chinese math teachers are specialists in math education.

Catherine has studied the Liping Ma book very carefully. I think Catherine concluded that the fundamental problem in the US is that teachers need release time to consult with each other and improve their knowledge.

I believe that the fundamental problem is that teaching is not a respected profession in the U.S., and that the other problems -- lack of release time, and mathematical weakness in the teachers themselves -- all follow from this.

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NewGoogleFeature 11 Jul 2005 - 03:12 CarolynJohnston

I just came across a new and cool Google feature that's being tested: Google Print. (It's actually called Google Print Beta, which means it's passed its first series of tests and is now ready for use but not fully bug-free).

I encountered it in a search for a link to Liping Ma's book. It seems to function very much like Amazon's 'search inside this book' function.

Here's what Google has to say about it:

What is Google Print? Google's mission is to organize the world's information, but much of that information isn't yet online. Google Print aims to get it there by putting book content where you can find it most easily – right in your Google search results.

How does Google Print work? Just do a search on the Google Print homepage. When we find a book whose content contains a match for your search terms, we'll link to it in your search results. Click a book title and you'll see the page of the book that has your search terms, along with other information about the book and "Buy this Book" links to online bookstores (you can view the entirety of public domain books or, for books under copyright, just a few pages or in some cases, only the title’s bibliographic data and brief snippets). You can also search for more information within that specific book and find nearby libraries that have it.

You can view the first three pages of Liping Ma's book through Google Print, and get a feeling for it.

I personally have the feeling that Catherine's and my book budgets are doomed, doomed, doomed.

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MozillaFirefoxVersusSafari 11 Jul 2005 - 15:40 CatherineJohnson

Quick piece of advice.

I've been using Safari to read & edit KTM.

I've just used Mozilla Firefox (that's the name, right?) and it's WAY better.

On Safari the print inside the edit window is basically invisible.

On Firefox I can see the words I'm typing.

What a relief.

update

Forget it.

I just found out why teeny-tiny font size inside an edit window might have its advantages.

what I need in life

A full-capacity word processor-slash-wiki

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WidgetCountdown 11 Jul 2005 - 16:29 CatherineJohnson

At the moment, this household has only the F in FAPE, minus the APE (IMO).

That's not good.

Still, we have to count our blessings, because we do have the Battlestar Galactica Countdown widget.

four days, nine hours, 32 minutes, and 59 seconds until Season 2

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HecksLibrary 11 Jul 2005 - 16:57 CatherineJohnson






Call me crazy, but I don't think this would be a whole lot more fun if every book said Math Puzzles Galore on the spine.....

Of course, I could be wrong.

Either one would be better than a wall-full of books on long division.


MarketingMathProblems

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CanChildrenMakeUpForLostTime 11 Jul 2005 - 18:39 CatherineJohnson

I'd like to put this question out to readers of ktm:

Can children make up for lost time?

I ask, because I've now read at least 5 personal stories of children or young adults struggling to make up ground they lost to bad curricula.

Some of the most hair-raising stories I heard from Carolyn were about college kids who simply could not learn algebra because they didn't get what they needed in grade school mathematics.

Carolyn made me wonder whether there might be a critical period for learning math the way there is for speaking a foreign language without an accent.

I've come to think there isn't, mainly because I find it possible (and pleasurable) to learn math as an adult, and I don't think I'm unique.


I started thinking about this because last night I did an impromptu interview with my cousin who, it turns out, pulled her daughter from public school because of a wretched experience with Everyday Math. (I'll post it shortly.)

Her daughter used Everyday Math for 3 years, from 2nd to 4th grade.

Then it took her 'several years' to make up the lost ground.

She just finished her freshman year in high school, and is doing great in high school math. (Her private school used Saxon.)


I talked to another woman who pulled her son out of the Tribeca schools because they use TERC.

He's now high school age and still doesn't have rapid fluency with his math facts. (She spent a lot of time working with flash cards, too. Another flash card failure.)

How can we remediate kids who've fallen behind because of constructivist math?

two immediate thoughts

To me, it seems like it has to be possible to make up lost ground more quickly than this.

At least, I hope so.

Here are my first thoughts:

• remediation has to mean doing timed worksheets every day, day in and day out, until the child or young adult has his calculations down cold

• remediation also means doing story problems every day, day in and day out (probably a coherent sequence of story problems, like those in the Singapore Math Challenging Word Problems books) [I have no idea how many story problems to do per day]

• finally, remediation may mean that you need to back up to the beginning of math, or close to: back up to content well before the point where the child became lost--and move quickly through a coherent 1st, 2nd, or 3rd grade curriculum, regardless of the fact that the child or young adult already 'knows' most of the material

I'd love to hear people's thoughts.


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FirstPerson 11 Jul 2005 - 22:24 CatherineJohnson

I mentioned earlier that I talked to my cousin last night, discovering in the middle of our conversation that her daughter's school adopted Chicago Math 10 years ago.

Here's the first part of my impromptu interview with her, which she said I could post:

how Everyday Math came to my cousin’s town

The 2nd grade teachers had a grant and were very excited. I think the teachers were turned on by the program. So they started introducing it in the 1st grade.

Nobody else liked it. I hated it, and many parents complained.

Teachers in the upper grades didn’t like it, either. The district was always having these huge teacher-board meetings to convince the other teachers that they had to adopt it, too.

Eventually, when the grade school kids got to high school, the high school teachers were in horror because the kids coming in couldn’t calculate. They complained that the Chicago Math students had to spend all this time guesstimating and figuring out what the answer was to one small step inside a complex problem. [Everyday Math was developed by the University of Chicago. Everyone in my cousin’s town in MA called it ‘Chicago Math.’] The students were too slow; they were hung up on the basics.

This war went on for a decade. I don’t know how it came out. I do know that for at least the first couple of years after Chicago Math came in they were not getting lots of kids proficient on the state tests. I’ll ask my friend who teaches at the high school whether they’re still using the books. She had 3 kids who went through the system, and she hated Chicago Math.



part 2: easier for mathematically talented kids?

One of my daughter’s friends had a very easy time with it, and was successful at it. She really soaked it up. Someone told me that kids who are chronologically older and have math talent, maybe they respond to it better. My daughter was the youngest in the class.

My older daughter, though, had a babysitter who had Chicago Math at New Trier when we were living on the North Shore. She said it was a failure. The New Trier students were the first guinea pigs, because it was Chicago Math. She said Chicago Math came from a bunch of ivory tower people figuring the whole thing out and then trying to disseminate it to all these little children.



part 3: developmentally inappropriate

I once told the assistant principal that in the Saxon book, when you’ve done something wrong you go back. You can’t advance until you get it right. I said that’s what I like about the Saxon program.

He said, “Well children can do that with Chicago Math, too.’ He was suggesting that my daughter had the ability to assess herself in Chicago Math, and that’s what she should have done. She was a little adult who could self-assess.

But she couldn’t. She was too young, and she didn’t know enough about math to be able to assess how much she knew about math.

It’s like driving. When you know how to drive, driving is built into your thinking process.

If you don’t know how to drive, you’re not going to have the confidence to figure out what your problem is. If you can’t get from one corner to the next, you’re not in a position to assess why not.



part 4: spiralling

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

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

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

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

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



part 5: frustrating

They would want my daughter to guesstimate whether something was 50 or not, or 100 or not. And they wanted her to do that before she knew 25 and 25 was 50, before she knew what the building blocks that made a number were. It’s hard to estimate something before you know that numbers are created.

To guesstimate is so frustrating. Math has a yes or no answer. And with math, when you go 5 x 7, it’s 35. That’s the answer. Children at a young age want to have something concrete. They learn from ‘This is wrong’ and ‘This is right.’ They like getting the right answer.

In Chicago Math, children don’t get that reward.



demoralizing

First they give you an intuitive flash that of material that is above your level, that you aren’t successful at. It’s like a prelude.

The thinking is that when you get to the material for real, you’ve had a prelude. But on a day-to-day basis if you’re always getting preludes, the child never has a sense of completion or success.

There was never a sense of mastery; there was never a sense of completing a task successfully before moving on to the new material that you were supposed to pick up intuitively.

Chicago Math was like trying to learn a foreign language by hearing tapes every day and intuiting what the words mean. Then 3 months later you’re supposed to know what the tapes are saying.



boring

It was too abstract and theoretical and boring. It’s boring when you don’t have the light bulb go off in your mind because, ‘Oh! I got it right!’

The best you could think was, ‘Well, maybe I got it right.

I think it’s crippling.



Saxon Math

I moved my daughter to private school after 4th grade. She’s worked with the Saxon Math books ever since.

It took her awhile to get to a stable place in math because she had gaps in her knowledge, and because she didn’t have confidence in the basics. She learned new concepts; she could understand them. But under testing she would crumble, because she didn’t have confidence.

In Chicago Math, computation doesn’t become second nature. I guess in new math they teach you all these steps you have to take. They make multiplication into 5 steps. Chicago Math makes learning to multiply real slow, and so damn confusing.

So she was bogged down in trying to do it in the new math way. It took her several years to overcome that, to get solid in the basics.

She improved greatly with the Saxon book. She’s doing fine at the high school level. She just finished 9th grade, and she does well in math now.

why do kids like math?

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MathProfCrochetsHyperbolicSpace 11 Jul 2005 - 23:03 CatherineJohnson

Fantastic story in the TIMES today:

Dr. Taimina, a math researcher at Cornell University, started crocheting the objects so her students could visualize something called hyperbolic space, which is an advanced geometric shape with constant negative curvature....

...balls and oranges, for example, have constant positive curvature. A flat table has zero curvature. And some things, like ruffled lettuce leaves, sea slugs and cancer cells, have negative curvatures.

This is not some abstract - or frightening - math lesson. Hyperbolic space is useful to many professionals - engineers, architects and landscapers, among others.

[snip]

Math professors have been teaching about hyperbolic space for decades, but did not think it was possible to create an exact physical model.

[snip]

(This is my favorite part)
Dr. Taimina was a good candidate to create a better model. As a precocious child in her native Latvia, she tried her elementary school teacher's patience. When her fellow second graders did not understand a math lesson, she recalled, she would jump up and yell, "I can't stand these idiots," prompting her teacher to send notes home.

By high school she had settled down, and was most impressed by a teacher who was known to keep his advanced students laughing and engaged. When she became an educator, she decided that no student, regardless of aptitude level, would feel out of place in her classroom. One way she assured that was by using everyday objects to explain theories. (She was known for peering so intently at the oranges at her local grocery to see if she could find perfectly round ones to use in her geometry class that she scared the clerks.)

[snip]

In 1997, while on a camping trip with her husband, she started crocheting a simple chain, believing that it might yield a hyperbolic model that could be handled without losing its original shape. She added stitches in a precise formula, keeping the yarn tight and the stitches small. After many flicks of her crocheting needle, out came a model.

One professor who had taught hyperbolic space for years saw one and said, "Oh, so that's how they look"....

[snip]

A year after she created the models, she and her husband gave a talk about them to mathematicians at a workshop at Cornell. "The second day, everyone had gone to Jo-Ann fabrics, and had yarn and crochet hooks," said Dr. Taimina. "And these are math professors."

[snip]

As an adult, when terrified artists started showing up in her math classes to fulfill their degree requirements, she signed up for a watercolor class, thinking, "Then I will know how they feel."

Now when students tell her they simply cannot understand math, she pulls out one of her paintings and says, "I learned that in three months." Then she might pull out one of her crochet models.

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WillinghamOnQuestionOfDifferentLearners 12 Jul 2005 - 00:14 CatherineJohnson

What a day!

First the crocheting mathematician, now a brand-new column from Daniel Willingham!

Do Visual, Auditory, and Kinesthetic Learners Need Visual, Auditory, and Kinesthetic Instruction?


short answer:

no


And see Willingham's deconstruction of Howard Gardner in Education Next.


And remember, Daniel Willingham, like our own Barry Garelick, is prominently featured in the ktm Pantheon!

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KleinBottleHat 12 Jul 2005 - 00:21 CarolynJohnston

If Catherine can blog about crocheting hyperbolic space, then I can blog about some unique geek knitwear: Klein Bottle Hats.

A Klein Bottle is a two-dimensional surface that is something like a twisted-up torus (a torus is the surface of a doughnut). It can't exist in 3 dimensions without intersecting itself, but it can be imbedded in 4 dimensions. If you slice open a Klein Bottle, you get a Moebius Strip.

Since this hat does exist in 3 dimensions, it's not a genuine Klein Bottle Hat, but it comes pretty close.

Here is a picture of Cliff Stoll (an astronomer and computer scientist) wearing a Klein Bottle Hat:

You can get a matching Moebius Strip scarf to go with it.

---

Catherine here:

As it happens, I am an Experienced Knitter.

Hey look!

They have the directions for Clifford's Knitted Klein hat over at Wolfram's Mathworld!

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AMathematiciansApology 12 Jul 2005 - 00:52 CarolynJohnston

I should explain first that "A Mathematician's Apology" is an in-joke -- it's the title of the memoirs of G.H Hardy, a mathematician who was at Cambridge in the last century, and who for a time was (according to himself) the "fifth best pure mathematician in the world". His Apology in the title is for the absolute inapplicability of the highest level of pure mathematics to real life problems.

The current Apology (by an anonymous pure mathematician) is not so much an apology as an explanation of why we really can't look to pure mathematicians as a whole for effective help in the political games surrounding the Math Wars. He's right; mowing over your average pure mathematician, politically, is like shooting fish in a barrel. In addition, the realities of the mathematics research and research funding game are exactly as he describes them; they do not reward political savvy at all; quite the contrary.

Lest I sound too jaded, this is a good time to recognize the efforts of those many pure mathematicians who have involved themselves in the effort to improve mathematics education at the K-12 level. David Klein, Ralph Raimi, Bas Braams, James Milgram, Hung-Hsi Wu, Fred Greenleaf, and many others have spent lots of perfectly good political capital fighting the good fight. As David says, thank goodness for tenure.

A bit of background: the AMS is the American Mathematical Society, the main professional society for research (pure) mathematicians. The MAA is the Mathematical Association of America, which as a group focuses on college-level mathematics education. The Notices are the newsletter of the AMS.
-- CarolynJohnston

Mathematicians are a diverse group of human beings and don't deserve to be stereotyped anymore than any other stereotyped groups deserve. However, society has already done a good job stereotyping mathematicians. There is usually a grain of truth in stereotypes and the mathematician stereotype might well be more accurate than most.

As a group, politics is not our strong point. I doubt that we have the normal spectrum of political smarts within our ranks but the whole spectrum has probably slid down to one side quite a bit.

There is not much in our daily work lives that develops political skills. Better an engineer or physicist who is used to politicking for zillion dollar grants and who cannot do their work without these grants. In math, if you lose your grants you can still plod along and get your work done.

It is worse than that. If a mathematician goes to Washington and raises a hundred million dollars for math in general, their chair won't give them a raise because they didn't do anything. If a physicist or a biologist does that, their lab is cranking out papers with their name on them all the time while they are in Washington.

Our "opposition" in mathematics education works in an environment where political skills are necessary to advance. They are a tough bunch.

A math Ph.D. in academia has two fundamental jobs after helping the institution run itself. One is to do research and one is to teach. Only a handful of academic mathematicians avoid teaching and only do research. On the other hand, probably the majority, far and away, are not doing research but only teaching. If all you do is teach mathematics, then it might be reasonable to be labeled a math educator as opposed to a "mathematician."

The MAA is not really a research mathematician organization. The AMS is a research organization and those in the AMS who gravitate towards the education committee are not your normal mathematicians (by definition).

I am at something of a loss as to why the Notices is so open to the rantings of the education folk. Perhaps it is Andy's way of trying to get mathematicians moving. I don't know.

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FirstPersonPart2 12 Jul 2005 - 14:36 CatherineJohnson

I'm posting the second part of my impromptu interview with my cousin in FirstPerson.

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FirstPersonPart3 12 Jul 2005 - 16:48 CatherineJohnson

Part 3 of my interview with my cousin is up.

Amazing how much work it takes to pull an interview into shape--

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CompareAndContrastTopicThread 12 Jul 2005 - 17:40 CatherineJohnson

When you get a chance, take a look at the Archives organized by thread box Carolyn has been working on, above and to the right.

If you click on CompareAndContrastPosts you'll get a page containing every one of the posts that compare a constructivist text to a non-constructivist text.

A lot of us seem to agree that these posts are the single most effective argument against fuzzy math.

That's why they're all here, in one place.


ways to use the compare and contrast thread:

• send the link to friends, teachers, & interested parties
  http://www.kitchentablemath.net/twiki/bin/view/Kitchen/CompareAndContrastPosts

• pull up the thread on people's computers if you're in the midst of a conversation about math ed (I've pulled up Kitchen Table Math now on a couple of teacher's computers to show them what we're doing)

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HighTechHeretic 12 Jul 2005 - 18:20 CatherineJohnson

Jeff Boulier just pointed me to High Tech Heretic: Why Computers Don't Belong In the Classroom and Other Reflections by a Computer Contrarian.

This reminds me that I never got around to reading The Cuckoo's Egg: Tracking a Spy Through the Maze of Computer Espionage, so I'm ordering that, too!

I think Clifford is right about computers in classrooms.

The research I've seen makes me think that Computers are Calculators writ large, with many of the same negative effects on learning.

Even if I hadn't seen the research, the fact that we have Mystery NGOs actively promoting the use of computers in classrooms--and being cited as authorities by Steve Leinwand--would make me leery.

I'll get around to posting the studies I've found on this question sooner rather than later, I hope.

update

Oops.

I already did post the Israeli study of computer use in the classroom.

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MoneyWellSpent 12 Jul 2005 - 21:02 CatherineJohnson

Bastiaan Braams has just posted the June 15 D.C. Board of Ed resolution, which includes these items:

Based on the evaluation of the submitted materials, the following recommendations are being made to the Superintendent of Schools for immediate adoption to insure delivery for SY 2005 - 2006:

Elementary Mathematics

Mathematics (Grades PK - 5) - It is recommended that the Board of Education for the District of Columbia Public Schools approve the adoption of Wright Group/McGraw-Hill: Everyday Mathematics. Cost: $1,207,875.

Mathematics (Supplemental) - It is recommended that the Board of Education for the District of Columbia Public Schools approve the adoption of Pearson Scott Foresman: Investigations in Number, Data, and Space. Cost: $470,000.

Middle School Mathematics

Middle School Mathematics (Grades 6 - 8) - It is recommended that the Board of Education for the District of Columbia Public Schools approve the adoption of Pearson Prentice Hall: Connected Mathematics. Cost: $875,567.

Puts me in mind of the Boston tea party.

I don't know why.


EverydayMathInDC

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FirstPersonPart4 12 Jul 2005 - 22:03 CatherineJohnson

FirstPerson

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LookingForPrealgebraResources 13 Jul 2005 - 02:23 CarolynJohnston

I've just started introducing Ben to some algebra concepts.. variables, equations, translating story problems into equations with unknowns.

I've found that it really is a conceptual hurdle, totally different from what came before it. It's a Big Discontinuity in one's math education.

Take a sample word problem like this one:

John weighed 78 pounds in 5th grade. When he was weighed in 6th grade, he weighed 86 pounds. How much weight did he gain between 5th and 6th grade?

"You want to figure out how much weight he gained, that's what you don't know," I say. "So you give it a letter name. Let's call it w for weight gain. Write w = weight."

w = weight, he writes.

"So what is that word problem saying about the weight gain?"

He sits there silently for a couple of minutes, so quietly you think he's zoned out; that's his way. And then maybe he'll say "Oh, I get it," and write down 78 + w = 86. Very slowly. And maybe I'll need to give him another hint or two before it happens.

The very idea that he can give a number that he doesn't know a letter name -- that he is even allowed to do such a thing -- is totally new and revolutionary for him. The idea that he can put that letter name into an equation that he translates from a word problem is just as revolutionary, and it all came at him in just one section in Prentice-Hall Course 1. And the section was labeled just like all the others: it didn't say "Huge New Concept" in flashing letters.

Prentice Hall just doesn't quite have enough practice at this new skill, of giving some quantity a letter name that you pick out. The next section is on solving problems like the example above. I just want to take an extra day for Ben to work on the daring act of naming an unknown and putting it into an equation.

So I went looking for some online resources, and I wasn't too excited by what I found. There is a lot of software that purports to give help in algebra; most of it costs money, and being cheap, I was looking for free stuff. There are a lot of sites that give explanations and assistance in algebra, and one or two that have online quizzes and the like. Nothing that fit what I was looking for, though; I think I'll have to go to the textbooks for that.

Here's a brief rundown of what I found that might be worth having a look at:

Algebra section of library.thinkquest.org. I absolutely DETEST the name of this site, which appears to be Math for Morons Like Us. If there were a larger number of good sites out there, I wouldn't recommend it at all, just on principle. However, the site is fairly well organized and the explanations are pretty good, and there are little popup self-quizzes at the end of the sections. All of this puts it ahead of the other sites I looked at. It could be good for a teenager reviewing for SATs, or for a parent trying to brush up before teaching algebra to a child, but they need to lose the awful name.

Algebra worksheet generator. This looks pretty good; it's very configurable, and it's free.

Word problem worksheets. These are algebra and prealgebra word problem worksheets. There are many of them, and all the ones I looked at looked good, like they would stretch a kid without actually breaking him.

However, I've still not found what I was looking for. Any suggestions would be welcome.


PreAlgebraFastFactsFromSaxonMath

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SailorMoonForJennifer 13 Jul 2005 - 16:27 CatherineJohnson



This is for my niece, Jennifer Caslin.





Hi, Jennifer!

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FirstPersonPart5 13 Jul 2005 - 17:26 CatherineJohnson

Whew.

I did it.

The last installment of my cousin's experience with Chicago Math (aka Everyday Math) is up at FirstPerson.

I was thinking, Why is this taking so long?

Then I did a Word Count.

The complete interview is the equivalent of a 5-page document.

That's a lot of work.

the Kitchen Table Math interview feature

I've been planning all along to do some original reporting for KTM in the form of interviews.

After this first foray with my cousin, I'm thinking: now I definitely need a clone.

Oh, well.

Next up: David Klein!

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AnneDwyerOnAssessment 13 Jul 2005 - 21:51 CatherineJohnson

I just noticed this comment from Anne Dwyer on the hay baler thead:

When I give my tutoring clients an assessment test, I give them mostly calculation problems.

I usually give four word problems:

• one problem is slightly below what they should be able to do. It is easy to read and very straight forward.

• One problem is at their grade level. It uses straight foward numbers but is multistep.

• If they are above 3rd grade but still in elementary school, the third word problem involves fractions. It is usually a problem from Singapore math that has several steps clearly deliniated with an a, b, and c. They should be able to get at least part of the problem.

• The fourth problem is a multistep problem that requires that the student have some logical way (ie bar model or equations) of keeping everything straight.

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FunWithVertices 14 Jul 2005 - 00:11 CatherineJohnson

This is fun!

I think it may have come from Carolyn's friend, Charles Martin.....

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AnnouncingSolutionsToSingaporeSampler4 14 Jul 2005 - 04:46 CarolynJohnston

This post is just a pointer to the solutions page for the SingaporeWordProblemSampler4, which consisted entirely of problems considered by the Singaporeans to be challenging.

Didn't want to leave anybody hanging if they're still paying attention! But a session of basic algebra with Young Ben has done me in for the evening. I'll be along soon with more Challenging Word Problems from Singapore -- but maybe not quite so challenging as these were.

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JapaneseMiddleSchoolEntranceExam 14 Jul 2005 - 13:02 CatherineJohnson

Anne just asked about a bliki post or an article comparing a Japanese to an American assessment test showing a 3-year gap between there & here.

I don't think we've had a post on this exact topic, but I do have the URL for a set of sample problems on the Japanese middle school entrance exam.

You can also download or purchase a CD of these problems:

The story problems provided in the software "World Math Challenge Volume 1" are translated from Japan's Junior High School math placement test. This test is given to 12 year olds and each section of the full test consists of 225 story problems. Students are given a time limit for each problem ranging from 1 to 5 minutes. If completed within the time provided, the 225 story problems require over 8 hours to complete.

The problems are logic-based and consist of about 20 different types of story problems. The point of this site is to begin providing quality math content based on Japanese (maybe a world) standards. The Japanese continue to place among the top 3 countries world-wide in terms of their students' math abilities. The US was recently ranked #14 in international math placement among the industrial nations. We think that US students should be exposed to international level math content and this site may represents the first step.

Constructivists have claimed that TIMSS video studies of Japanese math classes show them using constructivist pedagogy.

This claim has been rebutted by Alan Siegel of the Courant Institute of Mathematical Science at NYU in Telling Lessons from the TIMSS Videotape: remarkable teaching practices as recorded from eighth-grade mathematics classes in Japan, Germany and the US (pdf file)

The fact that Japanese 12-year olds are given timed math tests tells me that Japanese schools do not subscribe to constructivist doctrine.

Japanese-online
Free registration required to view assessment problems.

sample problems from Japanese middle school assessment test

Q1 How many 'C' balls does it take to balance one 'A' ball? (2 minutes)

Q2 Jenny wanted to purchase 2 dozen pencils and a pen. Those items cost $8.45 and she did not have enough money. So she decided to purchase 8 fewer pencils and paid $6.05. How much was a pen? (2 minutes)

Q3 Hose A takes 45 minutes to fill the bucket with water. Hose B can do the same in 30 minutes. If you use both hoses, how long will it take to fill the bucket? (1 minute)





Q4 A job takes 30 days to complete by 8 people. How long will the job take when it is done by 20 people? 2 minutes





Look at these time limits.

A 1-minute limit doesn't give you a lot of time to guess and check.

International Red Cross Symbol for Guess and Check

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NAEP's "hard" 8th grade problems are Singapore's 5th grade problems

....my own school district – Montgomery County, Maryland – is one of the most affluent, highly educated counties in America, yet our gifted students scored at the level of Singapore’s average student. NAEP classifies its problems as “easy,” “medium,” or “hard.” I benchmarked the “hard” 8th grade problems, examining NAEP’s highest level of expectation for 8th grade math. Most of these “hard” 8th grade problems are at the level of Singapore’s grade 5 – or lower.

[snip]

8th grade problem, NAEP

Consider: In one problem, for example, the student is shown a “Lunch Menu” with items like Onion Soup for $.80 and Ice Cream for $1.10. The question asks: “What is the total cost of Soup of the Day, Beefburger with Fries, and Cola?”

This is considered a “hard” eighth grade problem.

3rd grade problems, Singapore

But Singapore has harder problems than this in grade 3....


1 ) 5 oranges cost $2.25. What is the cost of 12 oranges? ________

2 ) I want to buy a calculator for $29.70 and a watch for $32.00. I have $28.50. How much more money do I need?

(1) $26.20
(2) $30.80
(3) $33.20
(4) $32.70


Both of these are two-step math problems. They illustrate Singapore’s expectation that all children should acquire mastery of the math skills needed for algebra and beyond. NAEP’s expectation is that children need to be able to order take-out from McDonald’s.

Testimony of John Hoven On Behalf of The Center for Education Reform At the National Public Forum on the Draft 2004 Mathematics Framework
(pdf file)

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NewWikiPageMidChapterAssessment 14 Jul 2005 - 14:17 CatherineJohnson

A new page from Interested Teacher!


MidChapterAssessment




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WillinghamOnLearningModalities 15 Jul 2005 - 04:33 CarolynJohnston

From Daniel Willingham on learning modality theory, an explanation of why learning modality theory might make sense from a teacher's viewpoint:

There are two ways that a teacher might see what looks like evidence for modality theory in the classroom. First, a teacher who believes the theory may interpret ambiguous situations as support for the theory. For example, a teacher might verbally explain to a student - several times - the idea of borrowing in subtraction without success. Then the teacher draws a diagram that more explicitly represents that the 3 in the tens place really represents 30. Suddenly, the concept clicks for the student. The teacher thinks "Aha. He's a visual learner. Once I drew the diagram, he understood."

But the more likely explanation is that the diagram would have helped any student because it is a good way to represent a difficult concept. The teacher interprets the student's success in terms of modality theory because she has been told the theory is correct and because it seems to explain her experience.

Willingham offers the following suggestion: teach to the best modality for representing the idea, not to the student's best modality.

But what if there are multiple modalities to choose from, for an idea? More generally, what if there are a whole host of different ways to represent an idea, and the kid's not getting any of them?

I ran into that situation recently, when teaching Ben how to do simple problems by adding and subtracting constants on both sides of an equation. Actually, trying to help Ben get the hang of this has taken quite a bit of effort this week, and I don't think it's a hard idea. I've got kinesthetic, visual, and auditory ways of teaching it, too. I could even sing it, though that's getting a bit ridiculous.

For the kinesthetic learner, you could get out a balancing scale or use Bornstein manipulatives. You could draw pictures of pan balances for a visual learner. You can explain verbally, as I did repeatedly, that what you're doing to solve the problem x + 4 = 13 is to 'undo the addition' of the 4 on the left hand side of the equation. If none of this works, what do you do then?

Try each modality over again, I suppose. Round 2: in case he was a kinesthetic learner, I had him copy each step I made in his own handwriting (laugh, if you will, but it works for me when I do it). In case he was visual, I drew pan balances again, next to the equivalent equation: no dice. "Subtracting the 4 is applying the inverse operation to get the x by itself," I said, auditory-like, but that didn't help either.

All this time, of course, he was able to do the problems by repeating the steps I made; he is a fabulous rote learner (is 'rote' a modality? If not, it should be). But I could tell he wasn't really getting the gist of it. Finally, in exasperation, I said, "Look, Ben, what's the opposite of adding 4?

"Subtracting 4."

"Good! And what's the opposite of subtracting 13?"

"Adding 13."

"Good. All you're doing to get the x by itself is doing the opposite of adding or subtracting the number that's with it," I said, but I didn't even get it all out before he said, "OH! I get it!"

And that's the sound I love to hear.

So, knowing Ben's best learning modality didn't help, and wouldn't have helped. I wish teaching, and learning, were so predictable that all you needed to do to teach a whole class reliably was to know what each kid's best learning style was. But I think that learning is inherently unpredictable. The trick is to be able to hit the teaching problem from a bunch of different angles, and you need to know lots of different ways to present the information. The more, the better (by the way, this is a major part of what Liping Ma's Chinese elementary math experts do with their release time; sit around together, thinking up new ways to teach problems to tough cases).

As an aside, I have never been able to figure out Ben's best learning modality (aside from 'rote'. His raw memory is unbelievable). As a person on the autism spectrum, he's supposed to be a visual learner; this is accepted theory to such a degree that teachers will assume he needs to learn visually, but it's not always the right approach.

What Ben really is, is an unpredictable learner. You never know what's going to be easy, where he'll get stuck, and what will unstick him. He's the kind of kid who keeps a teacher on her toes.

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MorganOnLearningModalities 15 Jul 2005 - 18:16 CarolynJohnston

This is a comment by Carolyn Morgan on the WillinghamOnLearningModalities thread: it's a beauty, so I'm posting it.

Things I note about her teaching approach:

• It's actually multimodal! She's talking, she's getting the kid to write AND draw his own pictures, she's doing whatever it takes to get the kid on track.
• It's feedback-driven: she switches gears in midstream if the kid isn't getting it.

Carolyn's post:

SteveH is correct. Whatever you do, you must bring all students to UNDERSTANDING. And this is what students really do want. "Understanding" or "not understanding" are the reasons students "hate" or "love" math.

A good math teacher learns how to "approach" a student having difficulty. The teacher has all of these ideas (hopefully) stored away back there some place, ready to be pulled out when needed. But a teacher's most important job is determining where the student's understanding fell apart, identifying where there might be gaps in reasoning, and knowing how to bridge those gaps. This is where choosing the right approach comes in. It might involve reteaching, reviewing a step that is being omitted, or helping a student reason through a difficult story problem.

So a hand goes up, and a student says, "I need help."

(Those are my favorite times of the math hour because it means I get to find the puzzle piece that is needed to make this all fit together in his/her head and give understanding to what I've just taught or to what's needed to solve this problem.)

So I have some choices, but I always look to see what the student has already done or tried. That tells me what to do next.

I then start by having the student read the problem to me (if it is a word problem).

Then I make a choice:

I might say, "OK, draw Bill's house. Now write 'B' on it for Bill. Now, draw the schoolhouse; now write 'S' on it. Now, draw the road from the house to the school. Now, look at the problem again to see how far it is to the school (and the student answers outloud 4 1/2 miles). OK, write that number on the map you've just drawn."

I could have drawn that little map for the student, and might do it under certain conditions, but having a student draw the map involves his sight and his movement (and mouth from speaking and ears from hearing his own voice) and it involves more importantly his BRAIN. (I've got to make sure his BRAIN is working and focused on the problem so he can "understand".)

So I would continue, "Start at Bill's house with your pencil; now walk to school. OK how far did Bill just walk? ('4 1/2 miles') OK, write that down. Now, he's at school, but he wants to come home, so have Bill walk back home. How far did Bill walk to get home? OK, write that down under the first number. How would you know how much he walked to school and back on that one day? ('add the two numbers together.') Good, do that. OK, but that is just one day. Now, let's read the problem one more time and let's see what the questiion was. (How far does Bill walk in a week going to school and back?) OK, now how could we figure that out?"

Many times the problem just works itself out in the student's brain as they begin to draw out a picture of the problem.

Or, if I've checked over his work and seen that he's added 4 1/2 miles 5 times for the 5 days of the week, I can see that he's overlooked a part of the question. So I have the student reread the question. Many times, the student will catch his own mistake when he hears his voice repeat the words "to school AND BACK". If not, I have him read just that part again.

Something really important: for some students it's just a matter of not knowing "how or where to get started". There are gaps in processing the information and gaps in understanding.

Not only does he need to know where to start, but he needs to know that where he is starting will get him going in the right direction and will help him get the right answer. This is very important to a student's confidence. If a student doesn't know "where to start" or isn't sure "if he's going about solving it properly", a teacher's trying to find the right modality isn't necessarily the answer.

This is where an "constructivist" approach is so devastating to the student. That student wants to be able to KNOW what to do to get the right answer.

It's terribly upsetting and deflating to a student not to know "where to start" and "if they're taking proper steps to solve the problem correctly." To leave this student to come up with his own idea isn't helping him.

Hopefully, though, a teacher will NOT just repeat the instructions that were given initially (if any were given). If a student didn't get it the first time, at least try a different approach.

One of my former students said of another teacher, "Why should I ask her for help? She always just repeats the same instructions that didn't make sense the first time." This student, smart as a cookie, just wanted to understand the entire process and to know how to work to get the right answer.

Carolyn Morgan On Conceptual Gaps
Congratulations Carolyn Morgan

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BestPractices 15 Jul 2005 - 21:26 CatherineJohnson

Just came home to the WillinghamOnLearningModalities thread—incredible!

The comments reminded me of my favorite teacher-mentor story.

A friend of ours here in Irvington--he's now chair of the University of Iowa College of Medicine^*—told us this story about his own mentor at Columbia Med School.

This particular teacher was legendary. Everyone who was anyone—boatloads of brilliant future researchers & clinicians—were taught by this man.

So what was this fellow's main piece of wisdom, which he conveyed to each & every one of his dazzling students?

If what you're doing isn't working, try something else.




^* This has caused Christopher to tell us, frequently, that 'Daniel lives in a mansion in Iowa!'

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HowToFindNewCommentsAndUpdatesPart2 15 Jul 2005 - 22:01 CatherineJohnson

a reminder

4 kinds of changes get made to content on ktm:

• Carolyn or I add an update to one of our posts

• ktm commenters & guests add a new comment

• ktm commenters & guests revise their own pages or a group page

• ktm commenters & guests create a new page

update: as of mid-July 2005, the next section is no longer true (click on 'What's New' on sidebar instead)

To find any or all of these changes:

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

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

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

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

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

We need to make some USABILITY changes around here, but until we all figure out what those should be (and Carolyn stays up all night doing them) the Recent Changes link works pretty well once you get used to it.

One nice thing: Recent Changes allows any of us to revive a post from weeks or months ago simply by adding a new update or comment.

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CallingAllKTMReaders 15 Jul 2005 - 22:38 CatherineJohnson

Good news.

Jo Anne Cobasko, of SOCMM, will be interviewed in the next few weeks:

Thank for the info... If you'd like, I'd like to have you on the air for an interview and we can talk about the program and it's shortcomings. As I mentioned to Mike, I saw some of the materials and found much of it to be very odd. My talk show schedule gets dictated by when we broadcast Angels baseball and other interviews, but I should have an opening in the next few weeks, we can do an interview over the phone that would run probably about 20 minutes.

Jo Anne could use advice on talking points.

I've tidied up our two talking points pages a bit, but anyone who has anything further to add should put your 2 cents in now.

Thanks!

TalkingPointsDiscussion
TalkingPointsDiscussionPage

bringing the math wars to the public

Is it just me, or does it seem as if the 'math wars' are far less well-known than the phonics wars of the 80s and 90s?

I don't know the answer to that, but the fact that Jo Anne could line up a radio interview with one letter tells me that we can all be thinking about ways to bring our story to the public.

single best book about writing press releases:

Writing Effective NewsReleases... by Catherine V. McIntyre

I wrote my first and only press release about 5 years ago, for a church program my pastor and I had put together for autistic kids and their families.

I followed every single rule in Writing Effective NewsReleases....

We got a story and photos in the NEW YORK TIMES.

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MathAndTextPrototypeLesson 15 Jul 2005 - 22:57 CatherineJohnson

When I was in graduate school (DID I MENTION THAT I HAVE A PHD IN FILM STUDIES?) one of my professors told me that the definition of a reader is a person who owns more books than he can read before he dies.

I have now updated that definition for the impending ERA OF THE BLOOKI.

The definition of a reader is a person who owns so many books she can't even get her own web site read before she dies.

Now that's out of the way, I have managed to make a circuit of my favorite blogs this afternoon--and have discovered that J.D. has his prototype lesson up at Math and Text!

It looks wonderful.

I'm going to read it now.

update

It is wonderful.

I love clean, lots-of-white-space invitations to maths...and there was something about the final lesson on figuring out which number is larger that made me happy.

I had the 'click' sensation Carolyn Morgan talks about.

That sensation is so reinforcing, that I think it ought to be an item on textbook write's & editor's lists: Does the student feel a click?

I was confused by just one part of the lesson, which was the first visual display. A middle school teacher has left a detailed comment explaining why she stumbled over it, too.

Take a look.

update 2: more on the click

I'm realizing I've had many, many conversations in which people who like math bring up the click--that moment of knowing you've got it.

Either you've got the right answer, or you've got the concept.

That's what my cousin was talking about when she said it's incredibly boring never to know whether you got the right answer or not:

It’s boring when you don’t have the light bulb go off in your mind because, ‘Oh! I got it right!’

The best you could think was, ‘Well, maybe I got it right.

Our friends Fred & Wendy were here a couple of weekends ago, and Fred said exactly the same thing about maths.

He loved maths (I may have to give up on 'maths'....) and he wanted to study it at Yale, as an undergraduate. What he especially loved was the click.

He quickly realized that college-level maths was a different animal, and he shifted to statistics, eventually earning a Ph.D. in experimental psychology (and then a law degree after that).

Fred is a seriously smart guy (clerked for one of the Supremes, etc.).....and what's he talking about when he remembers math?

The click.


FirstPerson (interview with my cousin about Everyday Math)





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RequestForReaderInput 16 Jul 2005 - 04:37 CarolynJohnston

I've been collecting reader recommendations for the last few weeks, from people who are finding Kitchen Table Math a useful site, but one that can be difficult to use. Kitchen Table Math is being challenged by its own success -- by the rate at which the information on it is growing, which is greater than Catherine and I ever expected in May when we launched it.

I like to think of Kitchen Table Math as a successful prototype of a Blooki (part blog, part wiki, and part book, for those of you who just came in) -- like any good prototype, it's lost no time in illustrating the strengths and shortcomings of its own design.

But now it's time to make it easier to use. I would like to enlist suggestions and input from readers in several areas. I'll ask some questions about each feature -- hopefully some of the questions will spark thoughts (and annoyances!) you might have forgotten you had. Please put answers and suggestions into the comments on this topic.

main questions

Is there one feature of KTM that you use all the time and wouldn't want to be without? What about KTM would you most like us to fix or change?

the sidebar

1. Did you know that clicking on the fish picture brings you back to the main blog page? Do you ever use that feature?
2. Have a quick look at the pages under the Kitchen Table Math part of the sidebar. Are there pages there that you look at all the time, or any that you never use?
3. Have you ever used the search feature to try to find a topic that interested you? Were you able to find it?
4. Have you ever used the archives-by-date? (These are month-by-month archives, accessed by clicking on 'archives' under the search part of the sidebar)
5. Do you have suggestions for organizing any of the material on the sidebar? 6. Have you ever used the recent changes feature (under 'special lists') to find posts with new comments?

the index, threading by categories

7. Have you ever used the index (under KTM index and contents, at the top of the page) to search for a post?
8. Have you ever used the "archives organized by thread" feature (at the upper right hand side of the main page) to search for a post?
9. Have you ever created a user page? Was it hard to figure out how?

Help features

10. WikiHowTo is our page with instructions for editing and creating new pages. How could these instructions be improved?
11. Have you been able to find information about posting syntax when you needed it?
12. Was it hard to figure out how to register?

User pages

KTM is part wiki -- this means that users can easily create pages of their own. We have a number of user-created pages, such as AnneDwyerSummerMathClass2005, EverydayMathInDC, and MoreOrLessPenAndPaper. I've been trying to put user-created pages on the sidebar, but it's clear that soon there won't be enough room, and I won't be able to keep up with new and deleted pages.

We're going to demo one possible solution this weekend. It will use a single page, a User Page Index. It will be a single location from which all users can create new user pages and look for new user pages. Like the regular KTM index, there will be a prominent link to it at the top of the page. The page itself will contain instructions for creating new pages.

If you have other ideas for keeping track of user pages from a single location, please let us know.

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CarolynMorganOnConceptualGaps 16 Jul 2005 - 05:03 CarolynJohnston

CarolynMorgan, who wrote the material in MorganOnLearningModalities, has written some more on conceptual gaps in students. She asked me to include it in her earlier post -- but that one was just perfect; just the right message and length. So I'm going to post the new piece here.

This highlights a teaching strategy that we used to use a lot in teaching at the college level, and on ourselves when learning new and difficult research material -- if a kid is stuck, have him work through a much simpler but still analogous example. Then work your way back up to the original problem.

Conceptual gaps

Often no matter what modality you try, a student just won't get it because there are conceptual gaps in his learning.

A precious student comes to mind. He had trouble with fractions. He just didn't understand the concept. He could recognize fractions, write fractions, read fractions properly. He could even (through tough perseverance) add, subtract, multiply, divide fractions. But the concept of a fraction was fuzzy. Very fuzzy.

His learning center teacher and I and other previous teachers, I'm sure, had him cut pies in half, divide groups into equal parts, but he continued to have those gaps. Something was just not clicking and none of us knew what it was, or wasn't.

I say all of this because that "conceptual gap" showed itself, not in computation of fractions, which he became very good at. It showed it's ugly head in the middle of a word problem, that most 5th graders could handle with very little trouble. A specific example comes to mind, which I will share now.

There is a problem in Saxon 6/5 something like this one:

Joe walked 288 feet, to the end of the pier and back. How long was the pier?

This student had no idea how to go about solving this problem. When he asked for help, I realized that it was because he really didn't understand what a fraction was, and how finding a half of 288 feet would solve his problem. I think he knew that there were two distances, but he didn't see them as halves.

To begin with the number was too huge for his mind to grasp. So I picked up his pencil and drew a line on his paper and said, "OK, here is another pier, but it's a very short pier. And when Joe walked on this pier to the end and back, he had walked 10 feet. How long was the pier?

He immediately, said, "Five feet."

I said, "Good for you. How did you know that?"

His answer was, "because 5 + 5 = 10". Notice how his mind was working. He still didn't see it as half of 10. That was why he couldn't solve the 288 feet problem. He didn't know two numbers that equaled 288. But he did know 2 numbers that equaled 10. And he picked 5 because he knew the 2 numbers had to be equal. But he didn't see it as halves.

So I knew we were only a part of the way there.

So I said to him, "OK, now, let's think about how we could work that problem so we could start with the 10 feet and know that he had walked 5 feet one way? Let's see if you can do another one and maybe that will help. Let's make another pier and make it shorter. (I'm drawing the pier.) OK, Joe walked to the end of the pier and back and he had walked 8 feet. How long is the pier?"

He immediately said "4 feet".

And so I said something like, "OK, when Joe walked 10 feet to the end and back, the pier was 5 feet long. (And we wrote that information down by the pier I had drawn.) And when Joe walked 8 feet to the end and back, the pier was 4 feet long (and we labeled that pier also)."

Now, my question: "OK, how could we work that problem to figure out that answer?"

And bless his heart. He said, "2 divided into 10". (Now I would have preferred that he say, "ten divided by two" but I was not going to quibble at this juncture.)

"Good for you," I said. "Now let's try that on another one. (drawing another pier) If Joe walked 20 feet going to the end of this pier and back, how long is the pier? How could we work that problem?"

And he understood the answer, and he smiled and wrote it.

"Now," I said. Let's look at our problem in the book. Joe walked 288 feet to the end of a LOOONNGG pier and back. How can we figure out the length of this long pier?

A HUGE, HUGE GRIN burst all over his face, as he said, "2 divided into 288".

It took getting down into smaller numbers of which he had some concept. He knew 8's and 10's. He could grasp numbers that size. And from there, he was able to know "how" to do the problem. He didn't understand fractions any better. He was just so happy that he knew how to get the right answer. And he felt successful.

That story happened two years ago. I don't know when he will fully grasp fractions. In a new story problem, in a new setting, he may have to be led all over again, step by step, from something small and "graspable" to something larger. And he may need that help for many years. I just hope he has teachers who understand him.

MorganOnLearningModalities
Congratulations Carolyn Morgan

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CognitiveHoles 17 Jul 2005 - 04:53 CarolynJohnston

Bernie and I were talking tonight, and he told me a story that worried me a bit.

Ben came to visit us at work the other day, and wanted to get a snack from the vending machine. So he went into his dad's office and asked for some money. Bernie gave him a few coins, and Ben went into the snack room, picked out what he wanted, and put his money into the machine; but he didn't have enough. So he came in and asked for more; but he couldn't tell Bernie how much more he needed. He didn't seem to have much sense of how much more he needed, either.

Well, it wouldn't be the first time we came across this sort of gap in his understanding. We have a sort of a family byword for these things, very much like Catherine and Ed's no-common-sense-y; we call Ben's gaps his Cognitive Holes. They are located in unexpected places -- they're generally about something, like handling coins, that you think is very easy by comparison with other things he can do, like long division. And they tend to be very big gaping holes in his knowledge, and at first they were very frightening. But we come across them less often now than we used to, and we've found that once we know they are there, we can remediate them pretty quickly.

So I thought this was another run-of-the-mill Cognitive Hole.

Well, you tackle these by filling them in. Ben and I were ready for a change from what we've been doing lately, anyway (introductory equations, solved by adding and subtracting). We've been doing them all week, and struggling, and we finally got a 'click' a couple of nights ago (those babies are practically audible, aren't they?), and last night when he took his section test he got a 100. So tonight, when it was math time, instead of doing algebra, I got out some coins.

I had 3 quarters, and a dime. "OK, you're at our work, and you want a snack, and these are the coins I have", I told him. "The snack you want costs 60 cents. Which coins do you take?"

He went for two of the three quarters, and the dime. Good. "How much do I get back from the machine?" I asked. Nothing: good.

"OK, your snack costs 40 cents". He goes for the two quarters: he tells me the machine returns a dime.

"The snack costs 80 cents." He takes all the coins, and tells me the machine returns 5 cents.

In short, he passed my common sense test with flying colors, and Math Time was fun and a breeze for once. So what the heck was happening the other day? In short, what part of this Cognitive Hole we think we've uncovered am I not mapping correctly?

Tomorrow, we try it a little differently; we'll simulate the precise problem we had the other day with the snack machine at work. I'll give him too little money, tell him the snack costs a certain amount, and get him to tell me how much more he needs.

There may in fact be no Cognitive Hole, this time, just some situational rigidity. This is the deal with smart people on the autism spectrum; sometimes they know what they need to know, they just stiffen up when it comes time to apply it in the real world.

comments...

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MartinGrossColumn 17 Jul 2005 - 16:15 CarolynJohnston

Yesterday the Washington Times ran an article by Martin Gross, Weak U.S. Education Link, that is a broad-band indictment of public education.

He begins by quoting Greenspan's testimony from a recent Senate Finance Committee hearing.

In the long run, he accurately pointed out, our economic strength in the world market eventually rests mainly on one factor -- brainpower, measured by the quality of our education system. In that race, he emphasized, we are failing badly.

Why is it, Mr. Greenspan asked, that our fourth-grade students are superior in international competition, while our eighth-grade students have proven inferior? Also, why are 12th graders hopeless in the key disciplines of math and science? In the Third International Mathematics and Science Study, our high schoolers scored 19th out of 21 countries, beating out only Cyprus and South Africa. They scored 20 percent lower than the Netherlands, a nation that lives on its brainpower -- as America might one day have to do.

Asked why our students become more ignorant the longer they stay in our public schools, Mr. Greenspan's response was typical of America's uninformed leaders: "I have no idea."

Gross has an idea, though.

But for those of us who have studied public education, the answer is clear: Our educators, from teachers through superintendents of schools, are academically and intellectually so inferior that the fourth grade is apparently the outer limit of their teaching abilities. They are so poorly selected, poorly trained and lacking in general intelligence, that failure by our middle- and high-school students is foreordained.

How can we support such a potent indictment? Easily. All standardized exams confirm their shocking inferiority.

He also has some solutions to offer.

(1) Close all undergraduate schools of education, and eliminate the generally ignorant and gullible 18-year-olds from the system. Instead, adopt the system used by most European and Asian nations. They require teacher candidates be graduates of a liberal arts college, and have at least a B average. They spend one year in practical teacher training, not in studying outdated educational theories.

(2) High school teachers of calculus receive the same pay as kindergarten teachers, which is ludicrous. To get satisfactory high school teachers, we must select better and pay more. To save money, we should fire 50 percent of administrators and support personnel and bring the student-bureaucrat ratio back to where it was 40 years ago.

(3) Education is not now a free market, but a closed shop. Scholarly college graduates who might be more independent are purposely kept out. A Yale summa cum laude in math is prohibited by law from teaching math in most states because he or she doesn't have an "education" degree. But the Yalie can teach -- in private schools.

The answer? Change the law so teachers need no education credits at all. Superintendents should be able to hire better college graduates trained in a true academic field. Then mathematicians will teach math, scientists teach science, and historians teach history. For the extra money needed, see (2).

(4) The middle school and high school should, by state law, be separated from elementary school and headed by a separate scholarly superintendent with a Ph.D. in a subject other than "education."

In short, sweeping political reforms are needed; the beneficiaries are generally either vulnerable (children) or clueless (parents), and the opponents (teacher's unions and schools of education) are motivated and politically savvy. It's enough to make even me wonder about homeschooling...

Think about this. For every underqualified person teaching math in public middle and high schools, there is probably an overqualified person teaching math for $3000 per course as adjuncts to university or college faculty. The only thing keeping him out of the public schools is his lack of an education degree.

comments...

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BattlestarGalacticaRIP 18 Jul 2005 - 01:05 CatherineJohnson

So how bad was the LONG AWAITED season premiere of Battlestar Galactica?

Three words.

THE MATRIX RELOADED.

Ed and I were so disappointed we got in a fight about it.

Not a big fight.

More of a since I can't jump through the screen and throttle Ron Moore I guess I will have to argue with you about WHY the second season is so wretched kind of fight.

My position was: They've decided to turn it into WEST WING.

Ed's position was: That's ridiculous.

So I just now clicked on my Battlestar Galactica Countdown widget, which took me to the Galactica Panel at Comic-Con (Comic-Con?) where I read that:

Jamie Bamber (Apollo) describes BSG as “West Wing meets 24 meets Little House on the Prarie.”

Obviously my PHD IN FILM STUDIES was good for something. And may I just add that if I had to sit down and figure out exactly what kind of show I would pay money not to watch, West Wing meets 24 meets Little House on the Prairie would be it.





And from today's NYTIMES MAGAZINE: Ron Moore's Deep Space Journey

Just in case you want to read the last article that will ever be written about BATTLESTAR GALACTICA when it was still BATTLESTAR GALACTICA and not WEST WING.

There was a line towards the end that I liked:

He said he had gained some perspective on the cause that had taken up so much of his time. ''Looking back,'' he said, ''it's hard for me to believe I did what I did.'' He never intended to become so emotionally involved, he said. ''I just felt there's something powerful here. And I just found myself taking a series of small steps that turned into big steps.''

For some reason, this made me think of KITCHEN TABLE MATH.

I don't know why.

comments...

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NavigationSidebar 18 Jul 2005 - 04:52 CarolynJohnston

I've tried to make some usability improvements to Kitchen Table Math over the weekend. The top part of the sidebar at left is now devoted to navigating the website.

The 'What's New' feature displays recent changes by posters and commenters.

You should be able to search by date or keyword or via book-style indices for any content you want to find.

I removed the Archives organized by thread drop-down menu from the upper right hand corner, since it apparently does not work on Internet Explorer. The same function can be found on the sidebar, under Search by Category.

Feedback is welcome!


Catherine here: A round of applause to Carolyn for countless hours of heavy lifting!

(Gee, I wonder if there's a Google image out there somewhere for heavy lifting? I bet there is! I bet I could spend the next 5 hours finding one! OK, then! Back later!)

[pause]

I found one!

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UsabilityFeedbackPart2 18 Jul 2005 - 15:40 CatherineJohnson

1.
Our challenge is the fact that Kitchen Table Math is a New, New Thing.

Standard usability practice says follow the conventions already established by other web sites. Sidebar goes on the side, search engine goes on top, and so on. That way people arrive at your site already knowing how to use it.

The problem for ktm is that there are no conventions for blookis (blog-wiki-book), because there are no blookis.

There aren't even any blikis (blog-wiki) to speak of.

We're starting from scratch.

So, how do we telegraph to new arrivals that:

• Kitchen Table Math isn't a blog, but a blooki
• Kitchen Table Math needs contributors
• Kitchen Table Math is super-easy to use so please give it a try
  (I realize ktm is not super-easy to use at the moment, but it will be)

2.
How do we telegraph all this to readers without gunking up the homepage so that every time you log on you see the same heap of old instructions you're sick of looking at.

3.
How do we do these things while continuing to make the site easier for all of us early adopters to use?

4.
Do we need a FAQ page? (my vote: yes)

5.
Will 'newbies' understand the word 'user'? (I might not...) Should we use the word 'reader'? 'Contributor'? 'Volunteer'? (I kind of like 'volunteer.' It expresses the fact that readers who have created pages are helping others, not just using a site. 'Volunteer' also conveys the idea that ktm is a new, new thing....AND 'volunteer' might be a good recruiting tool. Probably most people would rather think of themselves as volunteers than users.)

update

Well, at least we don't have horizontal scrolling.

update 2

OK, this is fun.

Early adopters, according to Wikipedia:

social leaders, popular, educated

I'm guessing this definition was written by early adopters.

do not attempt to adjust your television screen

I write one-sentence paragraphs because Jakob Nielsen told me to.

here's an opportunity you won't want to miss

from the Nielsen Norman Group Workshop:

Writing Workshop: Writing for the Web (Content Usability)

a full-day workshop

cost: $8000 (plus travel expenses)

question: Does this tell us that writing for the web is far more lucrative than any of us has heretofore suspected?

and: What do you want to bet the folks at Nielsen Norman advise against using words like HERETOFORE when Writing for the Web?

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TeachYourKidsToWrite 18 Jul 2005 - 17:02 CatherineJohnson

This sounds just great:

"If you write for a living," says Jefferson D. Bates in Writing with Precision, "this book is probably not for you." But if what you do for a living involves writing, then this book can help you do so "clearly, concisely, and PRECISELY." Bates is fond of italics, boldface, CAPS, exclamations!, quirky footnotes, and the word crotchet. He's over 80. He's been editorial director of the U.S. Air Force's Effective Writing Program and a chief speechwriter for NASA. The cornerstone of his campaign is the elimination of bureaucratese and jargon. Writing with Precision, originally published in 1978, is divided into four parts: writing (mainly letters, memos, instructions, regulations, and reports), editing (mostly copyediting), usage, and exercises. There is a definite personality behind this readable, conversational book. It's mostly updated, though a little checking by Bates could have prevented the reference to some books as being "probably out of print now."

Talk about a book that's withstood the test of time. 1978. Wow.

I may have to order a copy.

Especially since it has EXERCISES.

Writing with Precision: How to Write So that You Cannot Possibly Be Misunderstood
by Jefferson D. Bates

comments...

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BookPriceComparisonSite 18 Jul 2005 - 20:20 CatherineJohnson

I was trying to remember this book-price comparison site just the other day:

Best Book Buys

I'm putting this in the 'Book-style Index' (Carolyn--good name!) now, so I can find it again.

update: another book price comparison web site

BookFinder.com

Carolyn just added BookFinder.com in the Comments section.

Thanks to Kitchen Table Math, I am never, ever going to be without a book price comparison site again.

comments...

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BelievingInMiracles 18 Jul 2005 - 23:52 CatherineJohnson

As usual, I was looking for something else when I came across something much better:


Anyone who does not believe in miracles is not a realist.
David Ben-Gurion



That is my whole philosophy of life, practically.

That and No common sense-y.

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TitlesOfConstructivistMathCurricula 19 Jul 2005 - 00:38 CatherineJohnson

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

update: Department of Corrections

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

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



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

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

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

Elementary school

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

Middle school

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

High school

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

Programs explicitly denounced by over 220 Mathematicians and Scientists:

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

printable page


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



key words:
DavidKlein
listofconstructivisttextbooks
constructivist textbooktitles
NSFfundedcurricula

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TeacherTrainingInChina 19 Jul 2005 - 00:54 CarolynJohnston

SusanS wrote the following post about teacher training and education on the Martin Gross thread, and it's got me so intensely curious now about the Chinese school system that I've decided to break it off and give it its own thread.

The Chinese system of teacher training and development has garnered a lot of interest because of Liping Ma's incredible book, Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States. The Chinese teachers whup us in understanding and pedagogy, of course, but it's the details of how they whup us, and what they do differently, that are fascinating.

I've long thought that what lies at the base of the difference in "teacher culture" is the difference in our cultures themselves. It appears to me that Chinese elementary math teachers are respected specialists, for one thing. Compare that with teaching in the United States, where teachers are anything but respected (as the Martin Gross column proves).

Here' s Susan's post:

Okay, back from the web... I found an interesting straightforward article concerning China.

"There are three main educational aims for elementary schools in China today. The first is to develop the students moral character by teaching them to love the motherland, the Chinese people, manual labor, socialism and the Chinese Communist Party, and public property. The second goal is to enable students to obtain a fundamental education, develop skills in reading, writing and science, possess social knowledge and cultivate good study habits. The third goal is to enable students to develop physically. At least one hour a day students are required to perform some type of physical exercise."

"Originally, the duration of general middle schools was five years, but now in many places this has been changed to six years. The six years are divided into two levels: junior middle school and senior middle school. There are over 162,000 middle schools in China with over 65,400,000 students, more than sixty times higher than the number in 1949."

(This is kind of interesting....)

"Professional middle schools train middle level personnel for various vocations. Students entering these schools are required to have graduated from junior middle school and have some professional knowledge in a special area. The duration of these schools is from three to four years. Because all professional schools were closed during the Cultural Revolution some students entering these schools now have already graduated from senior middle schools and are completing the professional program in two years."

"There are seven types of professional middle schools: technical, agricultural, forestry, medical, financial and economic, physical education, and art. There are more than 1700 professional schools in China with more than 500,000 students enrolled."

"Teacher training schools are included in these schools. Students are drawn from senior middle school graduates and complete their training in three years. Tuition is free. Elementary school tuition is five yuan a year, middle school is 10 yuan a year. There are over 1,046 teacher training schools in China with an enrollment of more than 360,000 students (29% women)."

http://www.yale.edu/ynhti/curriculum/units/1982/4/82.04.02.x.html

I'm not certain of the validity of the source, but it's a place to start. I do believe that we can learn a lot from the Chinese and the Liping Ma book is absolutely a great book for any parent or teacher. But there are obviously some things that we probably can't do on any official level.

For me, this view into what's going on behind the scenes in the Liping Ma book (and I always had the feeling it was something very different from what goes on here) raises more questions than it answers.

I have no clue what 'middle school' would be the equivalent of in the US or Europe. It sounds like it might be a technical college, or what would once have been called 'Normal School'? (My grandmother, who like many intellectual women of her generation taught school, was educated at a Normal School, and it was a good education).

And are public schools for children called 'elementary schools', all the way through what we would call high school?

And why is there tuition for elementary school and not professional training schools?

And now I wonder whether people are tracked into professions by the government, or are free to choose what they want to do?

Susan raises an excellent point here. What exactly can we do -- what would we really be willing to do -- to have a teaching culture that is more like China's?


Susan S on teacher training in China
how Chinese teachers learn math
teacher release time & Liping Ma & Elaine McEwan's Princepal's Guide





comments...

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ReportOfTheCurriculumCommittee 19 Jul 2005 - 01:06 CatherineJohnson



Just found the Curriculum Committee Report to the Board of Education, June 9, 2005.

Trailblazers: most teachers have positive reaction, as do parents and students. Some say it’s too early to judge its success. Student survey positive. Anecdotal parent reports positive.

Character Education: students positive about 4/5 program (“No put downs”).

question: Is there a formal mechanism for submitting a minority report?

Because I've got one.


In case you're wondering.....'No put downs' is an anti-bullying program, which, for 6 long months, eats up 20 minutes of instructional time each and every morning, when kids are at their freshest.

Among the kids, it is an object of sport. They make ruthless and relentless fun of No Put Downs, the 'Choose a Response' injunction being the favored target of parody, and see the whole thing as One Big Joke.

Ed says that in his view it's never good to put a program in place that undermines adult authority in this way. I agree.


Ah.

I see the Curriculum Committee further reports that:

Parents would like to see it continued at Middle School.

Parents.

Hmm.

That's strange. Because I don't remember anyone taking a vote.

Actually, if we're talking 'parents' as in mothers, they're probably right. I'm the only mother I've met who can't stand the thing. We moms are in charge of the Civilizing Mission, & we'll take all the help we can get.

I'm off the boat only because I started reading about 'loss of instructional time,' and because we successfully dealt with a bullying situation ourselves a few years ago, when Christopher was in 2nd grade.

Needless to say, when Christopher was being bullied I dived into The Research. The No Put Downs program is in one crucial way actually at odds with an effective anti-bullying strategy; if we had taught Christopher to handle things the way No Put Downs tells kids to deal with bullies, he would have been bullied more, not less.

At the very end of this school year, in fact, one of Christopher's friends was being bullied. I told his mom what I'd learned from a fantastic book called Good Friends Are Hard to Find: Help Your Child Find, Make, and Keep Friends by Fred Frankel, and she told her son. Two weeks later his bullying problem was over the same way Christopher's was over.

Compare and contrast: 6 months of No Put Downs versus one parent-son talk about Fred Frankel.

I'd be happy to see the school bring in an anti-bullying program if it worked--and if we were collecting data to see if it worked. But it doesn't (IMO) and we're not.

Getting back to moms & dads, probably most mothers do like the No Put Downs program, and do want to see it repeated in the Middle School, too. 'No Put Downs' tells kids, every day, most of the same things we tell them at home. Taken at face value, it sounds like a good thing.

But if we're talking about dads.....

Let me put it this way.

We ran into our friend R. on the train a couple of days ago, and he was pretty hilarious on the subject of No Put Downs.

Afterwards Ed said there are probably about 2 dads in the entire town who think No Put Downs has any effect whatsoever on normal boy behavior.


keywords: character education bullying no putdowns lost instructional time

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ChineseTeachersPart2 19 Jul 2005 - 13:52 CatherineJohnson

From Liping Ma, on the education of Chinese teachers:

…the U.S. teachers behaved more like laypeople, while the Chinese teachers behaved more like mathematicians…

Obviously, these teachers are not mathematicians. Most of them have not even been exposed to any branch of mathematics other than elementary algebra and elementary geometry. However, they tend to think rigorously, tend to use mathematical terms to discuss a topic, and tend to justify their opinions with mathematical arguments. All these features contributed to the mathematical eloquence of the Chinese teachers.

We're talking pedagogical content knowledge, folks.


Chinese teachers acquire pedagogical content knowledge on the job:

• they engage in intensive study of their students' textbooks

• they study the textbooks together in teaching research groups, or jiaoyanzu

Susan S on teacher training in China
how Chinese teachers learn math
teacher release time & Liping Ma & Elaine McEwan's Princepal's Guide





comments...

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TheIndefatigableCarolyn 19 Jul 2005 - 16:03 CatherineJohnson

Wow!

I just looked at a Comments box and it is huge.

Carolyn is indefatigable.

comments...

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UserPageUsability 19 Jul 2005 - 22:19 CatherineJohnson

1.
Carolyn has set up the new User Page as a central location from which everyone can start his or her own pages.

In the directions, we asked people to insert their page title under the appropriate letter, so that the pages are ‘automatically’ alphabetized as they are created.

However, you don’t have to do this in order to create a page.

You can type your new page title anywhere inside the edit window, and the link will work.

So, if you can’t find the letter (they’re hard to see), don’t worry about it.

Start your page somewhere near the top of the User Page, and someone else can move it to its appropriate letter later on.


2.
Don’t be nervous!

TWiki saves all changes made to all pages, so if you accidentally erase or write over something, we can always get it back.

Same goes for the pre-existing code inside the edit window.

Most of it is bulletproof; even if you erase it, nothing will happen.

But even if something does happen (somebody accidentally erased the Comments window on one of the threads a while ago, for instance) it doesn’t matter.

comments...

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HowToStopABully 19 Jul 2005 - 23:52 CatherineJohnson


Carolyn and I were just chatting about Fred Frankel’s book Good Friends Are Hard to Find: Help Your Child Find, Make, and Keep Friends on one of the Comments threads.

I mentioned that we solved a fairly serious bullying problem Christopher had in 2nd grade in just two weeks, using Frankel’s book.

It struck me that the subject of bullying is so universal I should pull this comment up front in spite of the fact that it has nothing to do with maths.

Carolyn asked, specifically, whether Frankel’s book can be used with very high-functioning autism & Asperger kids.

Xtreme behaviorism in action

Fred Frankel says his book is not intended for kids with autism or Asperger syndrome.

But if my autistic kids were high-functioning, I'd sure give it a shot.

In his book, Frankel precisely breaks down exactly what kids do to make friends.

Exactly, down to the finest detail. It's Xtreme behaviorism.

For instance, he says that when kids approach other kids to play, they are rejected 30% of the time!

I don't know about you, but I find that observation incredibly useful.

Most adults think it's Bad When Kids Reject Each Other--and, from an adult perspective, it is. I certainly wouldn't reject 30% of the people who tried to talk to me at a party, and I would leave any party where 30% of the other guests refused to talk to me.

But Frankel says 30% is what kids do; it’s normal.

(caveat: I haven't fact-checked this figure, but I will.)

Then Frankel tells you what a kid should do when he is rejected, which is: he should accept his rejection and move on!

And that’s it!

There’s no You Can’t Say You Can’t Play!

I had just assumed you’re supposed to teach your child surefire social strategies to change the nasty rejecting child’s mind, but no.

That kid doesn’t want to play with you, and he’s not gonna want to play with you any time soon! So you're outta there!

I don't see how this observation wouldn't be helpful to the parent of a high-functioning child. If regular kids are getting rejected 30% of the time, and your kid is getting rejected 35% of the time....maybe he's not doing so bad.

[Hey! This does have to do with maths!]

Frankel also tells you almost word for word what your child should say and do in order to join a group of kids playing a game. (Hint: always join the losing side.) He scripts it out, and you can rehearse your child before he makes an attempt.

Frankel (and others whose work I’ve read) makes the point that we adults can't see children's social skills; we see their behavior through our adult filter. We don't perceive what it is socially skilled kids are doing, because children's social skills are different from grown-ups'. (I may be grafting something I read in another book onto Frankel....but if he didn't actually say this, he could have.)

Xtreme behaviorism & conceptual understanding

After I read his chapter on bullying, I had all the conceptual understanding I needed to solve the problem.

I knew that children who are bullied share two characteristics:

1. they cry easily, giving the bully bang for the buck
2. they are compliant to other children

Both of these things were true of Christopher.

We didn’t end up using Frankel’s script for anti-bullying, because our neighbor had a better idea. He taught Christopher ‘how to fight,’ which in Christopher’s case meant how to defend himself in a very loud voice accompanied by an equally loud glare & the all-important step forward.

There was also a whole dramatic Second Act Christopher was supposed to launch into if the bully dared to mouth off after he’d been Warned. It was basically Robert DeNiro for the 2nd grade. Christopher spent the afternoon running through the whole thing with the neighbor and his son, and then we rehearsed him at home.

So I didn’t use Frankel’s script, but I based everything I did do and had Christopher do on Frankel’s concepts.

They worked.

How to stop someone else's bully (aka transfer of learning)

When Christopher's friend was being bullied, I was stumped.

I knew he didn't cry easily, and I'd never seen him be compliant to other kids.

Then it hit me.

When other kids bullied him he ran.

Talk about bang for your buck. Number one, motion triggers everyone's 'prey chase drive;' and number two, chasing a running target is fun whether you're planning to kill and eat your prey when you catch him or not.

I told his mother: Tell him not to run.

I also told her that not only should he not run, he should make direct eye contact with the lead bully, and take a step forward.

His message: There are 5 of you and 1 of me, so you can stuff me in a garbage can if you want to.

But I'm not the only one coming out of this with bruises.

I don't know how much of that she told her son, but I know she gave him the basic thrust.

The bullying stopped so fast I almost had to jog her memory when I asked her how things were going two weeks later.

I haven’t read too many books in my life that let me solve a major problem in two weeks’ time, and then follow that up by solving someone else’s problem in 2 weeks’ time, too.

I’m a fan.

update

from Amazon.com:

As the mom of an Asperger child who desperately wants to have friends, I found this book more helpful than any other. It describes -- step by step -- the powerful social dynamics needed to "infiltrate" the mysterious world of friendship. I would recommend this book to the parents of ANY child who had social issues, be they autism, LDA, or just a bit shy or a bit aggressive. A must have for every resource library as well.

update 2

Frankel is now part of UCLA's Center for Autism Research and Treatment, which was established after we left. (fyi, Ed used to be a history professor at UCLA, and I taught in the film department as an adjunct years and years ago. That's how we met.)

Dr. Frankel is the Principal Investigator on the current CART project, “Parent-Assisted Friendship Training in Autism,” which focuses on the friendships of high-functioning children with autism who are included in typical elementary school classrooms from grades. This study is based upon the Dr. Frankel’s published treatment manual Children’s Friendship Training (2002).

update 3

Interesting comments thread on bullying at joannejacobs.com



Xtreme behaviorism, teaching & scripts
comments thread on bullying at joannejacobs.com

comments...

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NerdReport 20 Jul 2005 - 01:14 CatherineJohnson

Through my usual circuitous route (ktm to brightMystery to nerdtests.com) I stumbled onto a web site with a test for nerds.

My score: 50%

This is a Francis Galton moment (more on which later, or see BlookiHelpWanted & scroll down.)

I am always, in every single quiz, poll, or test I take, dead center.

And I mean…..DEAD……CENTER.

It simply never fails.

A couple of years ago I took a famous Are You A Republican Or a Democrat? test and found out I was Colin Powell.

Yes, I know Colin Powell works for the Republicans, but in this particular test he was DEAD CENTER.

I always tell Ed, and this is something he really enjoys hearing 5, 6, 10, or 20 times a month, Forget it, don’t even bother arguing with me about who's going to win the election, or whether BATTLESTAR GALACTICA just turned into WEST WING, for I Am Everywoman.

I am, too.

If I think or like or am keenly interested in X, that means everyone else is thinking or liking or keenly interested in X, too, or at least enough folks are thinking, liking or keenly interested in X that X is going to be everywhere you look until I stop thinking, liking, and/or being keenly interested in X and move on.

Still, even though I have an unbroken string of Dead Center scores on all manner of pop psych quizzes and tests, I did not expect to score Dead Center on a test for nerds.

But I did.

I am a nerd bellwether.

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NerdReportPart2 20 Jul 2005 - 11:42 CatherineJohnson



Are you a nerd?



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UsabilityFeedbackPart3 20 Jul 2005 - 12:38 CatherineJohnson



I've just read this Comment from Brenda M and my hair is on fire:

Hate to say it but I hate having to scroll sideways to read the comments. Just my pair o' pennies.

Is everyone else having this problem?

Anyone else?

Horizontal scrolling is one of the Top Ten Usability Mistakes (see mistake no. 3) in the known universe.

So we have to fix it.

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TheNerdCorner 20 Jul 2005 - 13:30 CatherineJohnson



is here



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CongratulationsCarolynMorgan 20 Jul 2005 - 17:22 CatherineJohnson






I'm a regular reader of The Carnival of Education, a weekly collection of "interesting and informative posts from around the EduSphere that have been submitted by various writers," hosted by The Education Wonks.

From the beginning of Kitchen Table Math, I've been on the look-out for a post or comment to send to the Carnival.

The minute I read Carolyn M on conceptual gaps, I knew that was the one.

The folks at Education Wonks agreed.

Here is their write-up today (a model of good writing for the web, by the way):

Kitchen Table Math is not a blog, but a blooki. (part blog, part wiki, part book) We find the concept to be fascinating. Check out this two-parter about how students learn math, as well as the conceptual gap that many kids experience when learning about abstract operations. See Part I here, Part II over there, and a description of what is a "blooki" (and an invitation to contribute) right here.

Yay, Carolyn M!



Carolyn Morgan On Conceptual Gaps
Morgan On Learning Modalities

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WhyIsSubtractionHarder 20 Jul 2005 - 20:11 CatherineJohnson

Christopher is sitting here doing his mixed practice, and he just asked me, "Why is subtraction harder than addition?"

He was doing the problem:

$20 - e = $3.47


I have no idea why subraction-with-borrowing is harder than addition-with-borrowing, or even if it is harder.

I'm asking all of you because I've noticed that sometimes the answer to incredibly simple-seeming questions tell you a huge amount that you didn't know before. Can't think of any examples offhand, but I'm going to start keeping track.

update

Oh!

It's probably the left-to-right issue, yes?










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LovelessOnInconclusiveFindings 21 Jul 2005 - 01:07 CatherineJohnson

Tom Loveless is one of my favorite ed writers & researchers.

Here he is on the question of inconclusive findings:

The research on tracking and ability grouping is frequently summarized in one word: inconclusive. This pronouncement is accurate in that nearly a century’s worth of study has failed to quantify the impact of tracking and ability grouping on children’s education. It doesn’t necessarily mean, however, that the gallons of ink spilled on these issues have been much ado about nothing. A non-effect in educational research is quite common. It can mean that the practice under study is truly neutral vis-a-vis a particular outcome. But it can also mean that the practice has off-setting negative and positive effects, that positive effects are produced under some conditions and negative effects under others, or that effects occur that researchers either don’t measure, because they’re measuring something else, or can’t measure, because of inadequate methods or expertise.

Non-findings must be interpreted with great care, especially when looking for policy guidance. In 1966, a federal report was released that many scholars consider the single most famous study in the history of education, Equality of Educational Opportunity, otherwise known as the Coleman Report for its primary author, the famed sociologist James Coleman. The Coleman Report was widely interpreted as finding that schools themselves have no significant effect on student learning. Fortunately, policymakers did not rush out to close schools and turn them into car washes or something else more useful.

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LovelessOnTracking 21 Jul 2005 - 01:22 CatherineJohnson

Loveless' survey of the research on tracking is interesting, especially given the philosophical opposition to all tracking that seems to be part of constructivist pedagogy:

Slavin’s support largely resting on the benefits uncovered for grouping in mathematics in the upper grades of elementary school.

I'm confused by the phrase upper grades of elementary school.

Does this mean 4th and 5th grade?

Or is middle school considered technically part of elementary school?


Here in Irvington, de-tracking students was part and parcel of bringing in TRAILBLAZERS.

Differentiated instruction is the buzz word.

tracking good for talented students?

Kulik finds that tailoring course content to ability level yields a consistently positive effect on the achievement of high ability students. Academic enrichment programs produce significant gains. Accelerated programs, where students tackle the curriculum of later grades, produce the largest gains of all. Accelerated gifted students dramatically outperform similar students in non-accelerated classes. Slavin omits studies of these programs from his analysis. He argues that the gains, though large, may be an artifact of the programs’ selection procedures, that schools admit the best students into these programs and reject the rest, thereby biasing the results.38

Three things are striking about the Slavin- Kulik debate. First, the disagreement hinges on whether tracking is neutral or beneficial. Neither researcher claims to have evidence that tracking harms achievement, of students generally or of students in any single track. Second, accepting Slavin or Kulik’s position on between-class grouping depends on whether one accepts as legitimate the studies of academically enriched and accelerated programs. Including these studies leads Kulik to the conclusion that tracking promotes achievement. Omitting them leads Slavin to the conclusion that tracking is a non-factor.

Third, in terms of policy, Slavin and Kulik are more sharply opposed on the tracking issue than their other points of agreement would imply. Slavin states that he is philosophically opposed to tracking, regarding it as inegalitarian and anti-democratic. Unless schools can demonstrate that tracking helps someone, Slavin reasons, they should quit using it. Kulik’s position is that since tracking benefits high achieving students and harms no one, its abolition would be a mistake.

So....just a few short paragraphs ago, Loveless has told us that inconclusive findings have to be interpreted with caution.

Is this finding of 'no harm done' a positive finding?

Or is it an inconclusive finding?

And why aren't we told?

we need editors!

Now that I'm reading think-tank & NRC pubications, I have a Firm View on the question of book editors.

Every book needs one.

I don't care how smart the author is.

talented kids need accelerated classes

High School and Beyond (HSB) is a study that began with tenth graders in 1980. The National Education Longitudinal Study (NELS) started with eighth graders in 1988. These two studies followed tens of thousands of students through school, recording their academic achievement, courses taken, and attitudes toward school. The students’ transcripts were analyzed, and their teachers and parents were interviewed. The two massive databases have sustained a steady stream of research on tracking.

Three findings stand out. High track students in HSB learn more than low track students, even with prior achievement and other pertinent influences on achievement statistically controlled. Not surprising, perhaps, but what’s staggering is the magnitude of the difference. On average, the high track advantage outweighs even the achievement difference between the student who stays in school until the senior year and the student who drops out.40

I say again: think tanks need editors.

I believe what he is saying here is that the gap between the high & low track student in the HSB study was larger than the track between high school graduates and high school drop-outs.

it figures

African-American students enjoy a 10% advantage over white students in being assigned to the high track. This contradicts the charge that tracking is racist. Considered in tandem with the high track advantage just described, it also suggests that abolishing high tracks would disproportionately penalize African-American students, especially high achieving African-American students.

A worthy mission for the fuzzies, de-tracking the whole entire country.

Thanks, guys.

tracking & the achievement gap

Moreover, NELS shows that achievement differences between African-American and white students are fully formed by the end of eighth grade. The race gap reaches its widest point right after elmentary and middle school, when students have experienced ability grouping in its mildest forms. The gap remains unchanged in high school, when tracking between classes is most pronounced.41

Sophie's choice

Third, NELS identifies apparent risks in detracking. Low-achieving students seem to learn more in heterogeneous math classes, while high and average achieving students suffer achievement losses—and their combined losses outweigh the low achievers’ gains. In terms of specific courses, eighth graders of all ability levels learn more when they take algebra in tracked classes rather than heterogeneously grouped classes. For survey courses in eighth grade math, heterogeneous classes are better for low achieving students than tracked classes.42

These last findings are important because we don’t know very much about academic achievement in heterogeneous classes. When the campaign against tracking picked up steam in the late 1980s, tracking was essentially universal. Untracked schools didn’t exist in sufficient numbers to evaluate whether abandoning tracking for a full regimen of mixed ability classes actually works. The NELS studies that attempt to evaluate detracked classes, which thus far have been restricted to mathematics, point toward a possible gain for low achieving students and a possible loss for average and above average students, but these findings should be regarded as tentative.43

grouping versus tracking?

The elementary school practices of both within-class and cross-grade ability grouping are supported by research. The tracking research is more ambiguous but not without a few concrete findings.

Will somebody please get the Fordham Foundation an editorial staff?

What is grouping?

What is tracking?

Why aren't these terms defined?

OK, I'm assuming 'grouping' means grouping kids according to ability within the same class, as Christopher's school does for reading.

I'm assuming 'tracking' means creating separate classrooms with separate teachers for kids of differing ability.

Singapore vs U.S.

Assigning students to separate classes by ability and providing them with the same curriculum has no effect on achievement, positive or negative, and the neutral effect holds for high ,middle, and low achievers. When the curriculum is altered, tracking appears to benefit high ability students.

This is exactly what happens in Singpoare--separate classes, same curriculum--but in Singapore this practice has a large positive effect.

race & income

When it comes to race, the disparities are real, but, as just noted, they vanish when students’ prior achievement is considered. A small class effect remains, however. Students from poor families are more likely to be assigned to low tracks than wealthier students with identical achievement scores. This could be due to class discrimination, different amounts of parental influence on track assignments, or other unmeasured factors.44

what do black parents say?

A study conducted by the Public Agenda Foundation found that "opposition to heterogeneous grouping is as strong among African-American parents as among white parents, and support for it is generally weak."45 If tracking harmed African-American students, one would not expect these sentiments.

choose your poison

The public labeling of low track students may cause embarrassment, but the public display of academic deficiencies undoubtedly has a similar effect in heterogeneous classrooms. There, a low ability student’s performance is compared daily to that of high-achieving classmates.46

At our school the tracking-obsession among the kids is brutal. There's a huge amount of taunting; at least, there has been when I've been around. 'I'm a 4!' 'You're a 2!'

Yuck.

jumped the track

A study of transcripts from five Maryland high schools showed 59.9% of students changed math levels during their high school careers, 65.4% in science. A national survey of high school principals reports substantial movement among tracks, especially upward (see Table 7). But an analysis of NELS data found that only 16.5% of students who were in low-ability classes in 8th grade went on to take either geometry or Algebra II by 10th grade (in comparison to 81.0% of 8th graders in high-ability classes).

Which reminds me, I've been meaning to post the strange goings-on with Phase 3 & Phase 4 this spring.....

you don't say

Without a push, a lot of students remain in low tracks who are capable of moving up.

Singapore vs. U.S. redux

It appears that high tracks are taught by better qualified teachers, however, in the sense of having teachers more schooled in content know-ledge.48 High school principals are inclined to assign teachers who know advanced subject matter to teach advanced subjects.

Another glaring difference.... (more on this another day).

Catholic schools

Reba Page’s 1991 study, Lower Track Classrooms, painstakingly reports on the daily activities of eight low track classes, documenting how they often function as caricatures of high tracks, how teachers and students in low tracks make deals to not push each other too hard so that they can cope with their environment. Low tracks may be used as holding tanks for a school’s most severe behavior problems. Even under the best of conditions, low tracks are difficult classrooms.

Intellectually stimulating low track classrooms do exist, however, and researchers have found the most productive of them in Catholic schools. Margaret Camarena and Adam Gamoran have described low track classrooms where good teaching, lively discussions, and ample learning take place. In 1990, Linda Valli published her study of a heavily tracked Catholic high school in an urban community. The school’s course designations publicly proclaimed each student’s track level. Textbooks and instruction were adapted for each track. Yet Valli discovered that "a curriculum of effort" permeated the entire school, even the lowest tracks. The school culture centered around academic progress, and the tracking system was but another facet of the school that served this aim. Students of all abilities were aggressively pushed to learn as much as they could. Every year, low track students were boosted up a level. By the senior year, the lowest track no longer existed. A judicious tracking system teaches low track students what they need to know and moves them out of the low track as quickly as possible.51

I hate like the dickens seeing Catholic schools go out of business.



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WhitherAmericanTalent 21 Jul 2005 - 03:27 CarolynJohnston

In my line of work, we're already seeing the effects of the dwindling American-born high-tech workforce. It's not hard for us to find mathematically and technically literate people, so long as we are willing to take in people who are foreign-born, and we are. In our commercial business sector, we have Chinese, Korean, Austrian and Canadian employees.

But a certain segment of our business is done in the classified world, and there, we are hurting for skilled employees who are 'clearable'. It's not just us, either -- it's everywhere in this business -- and the problem is getting worse. I don't think the shortage of educated American-born high-tech workers is entirely due to dwindling educational standards, either, though they contribute. I think it's cultural, too, a result of our increasing wealth.

I noticed when I was a graduate student that people were pouring in to study mathematics disproportionately from the struggling, up-and-coming (or trying to up-and-come) countries with good educational standards. We saw a big influx of Chinese, and later Russians, and they all opted to stay (following both Tiananhmenh Square and the end of the Soviet Union, and who could blame them?). They caused the glut of talented academicians in the job market that was discussed in this recent thread. Notable by their absence were any Europeans. The mathematics talent was mostly coming from the 'second world' countries.

Bernie and I also noticed the same phenomenon on a smaller scale in American students. In the generation prior to mine, a lot of the technical talent came from Brooklyn and New York City and other big eastern cities with a lot of bright first-generation American kids. My dad came from Brooklyn, went to good schools and got to go to Brooklyn College for free in those years, and so the son of a bus driver was able to become a pharmaceutical researcher with a much better standard of living than he'd had while growing up. In that generation there were a lot of men like him.

In my generation, there were no longer as many kids from New York and the big eastern cities coming into the graduate schools. Other than Chinese and Russians, we had quite a few Americans from parts of America you never heard of; midwestern towns, and smaller towns in New York and New England. What happened? I'm not sure, but I think the kids from New York felt themselves to have options for getting ahead in life that didn't involve quite so much hard work. Probably the kids in Europe did too.

Anyway, I've gotten far afield from what I wanted to post, an article about some recent testimony about America's critical need for homegrown talent.

Current shortcomings in U.S. education could leave the next generation of Americans ill-equipped to combat terrorism, according to testimony given before the National Infrastructure Advisory Council (NIAC).

"The country's long-term security is tied to the quality of the workforce," Alfred Berkeley, a trustee of the Mathematical Sciences Research Institute said.

Berkeley's testimony before NIAC cited mathematics and science as key areas that need to be addressed at all educational levels. He stressed the importance of young adults being qualified to enter fields such as cyber security. However, Berkeley, who also serves as an NIAC member, said that current elementary education provides a poor foundation for the subsequent pursuit of these fields of study.

"The public has not embraced education as a priority. We must find a way to engage the public with a sense of urgency," Berkeley said.

Besides the problem of education quality, the United States is facing a shortage of students willing to study areas such as engineering.

According to a National Science Board (NSB) report released in 2004, "bachelor's degrees in engineering have declined by 8 percent and degrees in mathematics have dropped by about 20 percent" since 1990.

Check out the rest of it.

As a final aside -- where could the next great influx of American technical talent possibly come from, with birth rates in America falling and people so wealthy that a future in a technical job appears harsh by comparison with their other options?

Here in Colorado, we have a lot of Mexican and other immigrant Hispanic families. I understand that what we're seeing here isn't just local, but part of a larger trend in the U.S.. I'm thinking that their children, born in the U.S., would probably really appreciate the opportunity to make a good salary in a technical field.

If the schools don't let them down. Those families don't have a lot of money to burn on tutors and Kumon.


Whither American talent?
Congressional incentives for study of math
Paul Samuelson on the 'science gap'

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ImALittleTeapot 21 Jul 2005 - 18:00 CatherineJohnson





I'm a little slow off the dime today (I almost typed 'this morning,' which should give you some idea...) in spite of the fact that I put up the 'I feel determined' mood widget at 8:30.

In theory I am going to email the Chinese mathematician I mentioned in a comment on Carolyn's thread, AND post something from THE GEOGRAPHY OF THOUGHT, but first, how does this software work?

And what kind of mathematics is used to do what it does?

Artis image gallery




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PreAlgebraFastFactsFromSaxonMath 21 Jul 2005 - 18:33 CatherineJohnson

Carolyn mentioned that she's looking for prealgebra resources.

The 5-minute 'Fast Fact' worksheets from Saxon Math 8/7 are terrific. There are 21 different worksheets in the 8/7 Tests and Worksheets book ($24.50), with multiple copies of each sheet. Around 150 worksheets in all.

The Solutions Manual is another $29.50, but you don't need it if you're using only the Fast Fact sheets. (I would probably buy the solutions manual if I were ordering the entire 3-book package, but in that case I would purchase from a discount site like Homeschool Supercenter.)

$24.50 plus shipping is a lot to pay for worksheets (the book includes 23 20-item tests as well, which you can use as mixed practice). You could pull together 21 sheets with equivalent problems yourself. On the other hand, it would take you awhile, and you'd be doing a lot of formatting & printing-out & whatnot.

So my feeling is that if you're not on an incredibly tight budget (I've been on tight and not-so-tight) they're worth the money.

Here are the titles of the sheets, and the number of problems on each:


A 64 Multiplication Facts

B 30 Equations - these are simple equations like:
a + 12 =20
a = _____

C 30 Improper Fractions and Mixed Numbers

D 40 Fractions to Reduce

E Circles
sample question: A segment between two points on a circle is a _____

F Lines, Angles, Polygons 15 questions

G Fractions add, subtract, multiply divide 24 problems

H Measurement Facts 33 problems
sample questions:
Water freezes at _____ Fahrenheit & _____ Celsius
1 meter ^2 = _____ centimeters^2

I Proportions 24 problems

J Decimals (add, subtract, multiply, divide) 21 problems

K Powers and Roots 24 problems

L Fraction-Decimal-Percent Equivalents 25 problems
sample question: express 5/6 in decimal & percent

M Metric Conversions 24 problems
sample problem: 50 centimeters = _ mm

N Mixed Numbers add, subtract multiply, divide 20 problems

O Classifying Quadrilaterals and Triangles 9 problems (includes kite, rectangle, isosceles triangtle, right triangle, trapezoid, rhombus, scalene triangle, acute triangle, parallelogram, square, equilateral triangle, obtuse triangle)

P Integers add, subtract, multiply, divide 32 problems
sample problem: (-5) + (-6) + (-2) = _____

Q Percent-Decimal-Fraction Equivalents 25 problems
sample problem: express 83 1/3% as a decimal & a fraction

R Area 12 problems (includes non-square figures)

S Scientific Notation 20 problems

T Order of Operations 16 problems

U Two-Step Equations 15 problems

The book also includes 24 20-item tests that can be used as mixed practice.

SaxonMath Homeschool 8/7 with prealgbra

Third Edition
Tests and Worksheets
Fast Facts (5 minute)



LookingForPrealgebraResources

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NerdWannabe 21 Jul 2005 - 20:41 CatherineJohnson

That's what NerdTests.com has to say about Ed, who took the nerd test this morning.

I thought he was going to score way higher than me, seeing as how he's the one who does all the computer stuff around here. (I had a friend who worked in....graphic design, I think. Her husband was a computer guy at IBM. Every time she had a problem she called him, a practice her office mate called 1-800-DIAL-HONEY.)

So I thought Ed was the nerd. Boy, was I wrong.

This is making me think maybe I am a geek manque....

The funny thing is, I put down math as my favorite subject in high school.

Thinking back, I realized I didn't have a favorite subject. My classes weren't great, and as a matter of fact, the best classes, and the ones I enjoyed the most, were math.

Oh!

No!

That's wrong!

I just remembered: my favorite classes were language classes. Spanish, and then French.

That's a little geeky, too.

Actually, this gets into the whole issue of good teaching & a good curriculum. The best teaching & the best curricula I had through all 4 years of high school were in math & languages (and, come to think of it, in science). History & English were a joke, and nobody ever assigned any writing. When I got to college I didn't know what a 'paper' was.

My favorite subjects were the ones with the best teaching & textbooks.





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NoTVsInChildrensBedrooms 21 Jul 2005 - 22:00 CatherineJohnson

“In this study, we found that the household media environment was related to a child’s academic achievement,” said Dina Borzekowski, EdD, lead author of the study and assistant professor in the Department of Population and Family Health Sciences at the Bloomberg School of Public Health. “Among these third graders, we saw that even when controlling for the parent’s education level, the child’s gender and the amount of media used per week, those who had bedroom TV sets scored around 8 points lower on math and language arts tests and 7 points lower on reading tests. A home computer showed the opposite relationship—children with access to a home computer had scores that were around 6 points higher on the math and the language arts test and 4 points higher on the reading test, controlling for the same variables.”

[snip]

The researchers did not find a consistent negative association between test scores and the amount of television watched per week.

Apparently, a child who spends 20 hours a week watching TV in his bedroom scores worse than a child who spends 20 hours a week watching television in his parents' bedroom.

I believe it.

Christopher had a TV in his bedroom for awhile, because Andrew uses it to watch his Barney videos. Christopher had his Playstation hooked up to it.

Christopher and I stopped having any relationship at all. It was as if he'd moved out.

Finally I complained to my neighbor about how I never saw Christopher any more, I needed to limit the Playstation hours, etc., etc.

She said, 'Take the Playstation out of his bedroom.'

Duh.

I did, and problem solved.

Of course now it's practically impossible for us to watch a DVD, since the Playstation is hooked up to our TV, and playing DVDs on the Playstation, while possible, is a chore.

But we have lots more togetherness.

It's possible that more 'face-time' with your child translates directly to higher scores because of the higher levels of interaction.

Or it's possible that parents who insist on lots of face time are also more involved in homework....

Or it's possible that this study belongs to the 33% of highly cited studies that turn out to be wrong.

That's the great thing about studies.

After you've read a new one, you don't know if you've actually learned something new or not.

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ProgrammingForTheMasses 22 Jul 2005 - 14:29 CatherineJohnson

Just came across a potentially interesting online book from MIT Press: How to Design Programs: An Introduction to Computing and Programming.

The central argument is that everyone should learn how to program:

Many professions require some form of computer programming. Accountants program spreadsheets and word processors; photographers program photo editors; musicians program synthesizers; and professional programmers instruct plain computers. Programming has become a required skill.

Yet programming is more than just a vocational skill. Indeed, good programming is a fun activity, a creative outlet, and a way to express abstract ideas in a tangible form. And designing programs teaches a variety of skills that are important in all kinds of professions: critical reading, analytical thinking, creative synthesis, and attention to detail.

We therefore believe that the study of program design deserves the same central role in general education as mathematics and English. Or, put more succinctly,

everyone should learn how to design programs.

On one hand, program design teaches the same analytical skills as mathematics. But, unlike mathematics, working with programs is an active approach to learning. Interacting with software provides immediate feedback and thus leads to exploration, experimentation, and self-evaluation. Furthermore, designing programs produces useful and fun things, which vastly increases the sense of accomplishment when compared to drill exercises in mathematics. On the other hand, program design teaches the same analytical reading and writing skills as English. Even the smallest programming tasks are formulated as word problems. Without critical reading skills, a student cannot design programs that match the specification. Conversely, good program design methods force a student to articulate thoughts about programs in proper English.

I don't know what to make of this.

For a long time I've felt that I 'need' to know computer programming....and I've come to trust such feelings.

I also feel, quite strongly, that the constructivist emphasis on 'real world applications' would be radically better served by doing real world things other than collecting data. At this point, I find the obsession with data-collection, stem-and-leaf charts, and probability incomprehensible and even bizarre. (I'm sure that as I carry on reading the history of progressive education I won't find it bizarre, because I'll find out where this preoccupation came from.)

If you want to do real-world things that involve math, let's reinstate shop & home ec classes, as a thread on the NY Math Forum list recently recommended. I'm all for that; I think we've lost huge amounts of real-world skills in cooking, housekeeping, carpentry, home maintenance in general--you name it, we've forgotten it.

I don't know a soul my age or younger who can cook conceptually, by which I mean that he or she understands the underlying principles of food and food chemistry, and can apply them usefully.

Everyone I know cooks procedurally; we follow recipes. Our skill in cooking is skill in picking out good recipes and doing what they tell us to do.

I call this missing knowledge.

But if you don't want to teach kids how to use math in real life (data collection is not real life for most of us), then how about teaching them programming?

Isn't LOGO a computer programming software designed for kids? (I'll have to look it up.)

Here's the line that worries me, though:

This book is the first book on programming as the core subject of a liberal arts education. Its main focus is the design process that leads from problem statements to well-organized solutions; it deemphasizes the study of programming language details, algorithmic minutiae, and specific application domains.

That sounds suspiciously like de-emphasizing basic skills.

Does it make sense to teach programming without teaching 'algorithmic minutiae'?

Is that possible?

I don't know enough about programming to venture an opinion.

I do know that I have yet to encounter the discipline in which God is not in the details.

It goes against the grain of modern education to teach children to program. What fun is there in making plans, acquiring discipline in organizing thoughts, devoting attention to detail and learning to be self-critical?
-- Alan Perlis, Epigrams in Programming

I can state as a fact that 'learning to be self-critical' is a huge component of constructivist intention.

It's called metacognition.

update

unlike mathematics, working with programs is an active approach to learning

Wrong.

update 2

Back from the web.

Yup, LOGO is used to teach kids how to program.

I like this guy (see his review of The Guide to Web Publishing), which you can buy from Amazon or read online.

He needs to come visit ktm.

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MathOlympiad2005 22 Jul 2005 - 16:11 CarolynJohnston

Check out the problems from the 46th Math Olympiad competition in Mexico (held last week).

The competition is held over two days. Each day, the kids are given 3 problems, and 4-1/2 hours to work them.

I haven't even dared look at them yet -- I'm afraid I'll be lost for days.

(ht: Charlie Martin).


MathOlympiadProblem
HappyJulyFourth (Moise & Downs)





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TutoringAdvice 22 Jul 2005 - 21:38 CatherineJohnson

I'm probably going to spend some time working with a friend of Christopher's on his math.

They're the same age--both going into 6th grade--and my sense is that math is probably this boy's strong suit.

I just gave him the Saxon placement test, and he placed into Saxon 7/6, which is the 6th grade book.

That would be great, but here's the hitch: he has been taught almost nothing about fractions at all. (He had a good math teacher--he and Christopher were in the same Phase 3 class for the first half of this year--who left to have a baby. So it seems that the subject of fractions & decimals fell through the cracks.)

So.....if anyone has thoughts, I'd like to hear them. I'll probably go ahead with 7/6, but that means I'm starting a 6th grade book with a child who's been taught virtually nothing about fractions and decimals.

update

Here's the fraction worksheets site Carolyn J found.

whose job is it, anyway?

This is the kind of thing that I just don't get.

Why should I be the person figuring out that this boy hasn't been taught fractions & decimals?

Why shouldn't the school be figuring this out? (Yes, the school might say he was taught fractions and decimals, but didn't learn them. However, it's clear to me that there are certain topics he simply hasn't even heard of, because with some topics he'll say, 'I kind of remember that.' In other words, he can tell me which topics he failed to learn, or didn't learn well enough to retain, or whatever it is. With topics like adding fractions, he simply doesn't know anything about them, and has no memory of having been taught.)

So, yes, the school might say, 'He was taught, but he didn't learn.'

But so what?

If he was 'taught' and 'didn't learn,' then he wasn't taught as far as I'm concerned. It's the school's job to perform formative assessment to know what students have and have not learned.

Then it is the school's job to re-teach if a child has not learned.

Then, if the child still isn't learning, it's the school's job to figure out what else he needs.

common sense from The Education Wonks

I don't want to take this too far, of course. Parents & students are responsible, too:

That's one of my major concerns with NCLB. When students don't do their homework or study for exams, or even attempt to do classwork, it's still considered to be the teacher's fault if the students don't achieve their federally-mandated level of proficiency in reading, math, and science.

And yet NCLB doesn't give me, as a teacher, the authority to require student's who aren't even attempting the work to stay after school and complete their assignments. Unless the kid has committed some breach of the school's disciplinary policy, I can't keep them any later than the school regular dismissal time.

The No Child Left Behind Act holds me solely accountable for my students' academic progress but doesn't give me the authority to help make that happen, especially for children that are considered to be "at risk" of failing to meet minimal standards of academic progress.

Sadly, under the law as it is now written, a large number of children are going to be left behind.

He's right. If a student doesn't do his work, and the parents don't require him to do his work, that isn't the teacher's responsiblity.

But that's not the case with Christopher's friend. This boy has done all of his work; he's a serious student; his parents are serious parents.

It's the school's responsibility to know whether this boy has or has not learned how to add, subtract, multiply, and divide fractions, and to teach or re-teach the subject if he hasn't.

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ANerdInTheHouse 23 Jul 2005 - 01:15 CatherineJohnson

This is unbelievable.

Christian, who works with Jimmy & Andrew (and is a surrogate big brother to Christopher) just took the nerd test.

He got a 77:

23% scored higher (more nerdy), and
77% scored lower (less nerdy).

What does this mean? Your nerdiness is:

Mid-Level Nerd. Wow, it takes a lot of hard nerdy practice to reach this level.

My nerd stereotypes are now undergoing massive, rapid, cascading revision.

Christian is a 6'4" black guy from Yonkers who doesn't like math. A 6'4" good looking black guy from Yonkers, I might add, with dreadlocks. I think I speak for many when I say that, prior to 15 minutes ago, if you had asked me to close my eyes and Think of a Nerd, I would have seen a person who looked something more like this.

So now I'm stricken with guilt. Just why, exactly, was I thinking Christian would be less nerdy than I am? (nerd score: 50)

Apart from the fact that he doesn't like math, that is.

I should have seen it coming when he knew what 'cracked software' was. He explained it to Christopher & me, and we still don't know.

Of course, this provides a golden opportunity to climb back up on my Schools Aren't Doing Their Job and While We're on the Subject, What's With The Achievement Gap, Anyway? soapbox. Christian's schools sound dreadful, and I'm sure they were dreadful.

He also said his dad was great at math, and I'm sure Christian would have been great (or good), too, if he'd had half a chance. I've been threatening to sit him down and teach him math along with Christopher. What he's really interested in, though, are the Greek and Latin roots of words. So I should probably give him my copy of Vocabulary from Classical Roots by Norma Fifer & Nancy Flowers, since it's going to be a long time before I get back to the classical roots of vocabulary, if ever.

Gauss Shmauss

Tonight I showed Christian the Gauss-in-3rd-grade problem, which we both thought was pretty cool. But then we got all balled up because when we added the natural numbers 1 to 50, the answer we got (1275) was way less than the answer we got adding 1 thru 100 (5050). Apparently we both were thinking the sum should be half as big since we had half as many natural numbers to add....which is A) wrong and B) demoralizing.

I think number sense is a real thing (I say that because I get the sense some non-fuzzy types may feel number sense is yet another constructivist distraction, like collecting data & laying odds on getting a 6 when you roll a number cube). I sometimes wonder if my own number sense has budged an inch since I started all this.

update

I just Googled 'nerd' again and looked at the 1st two pages.

40 nerd thumbnails:

• 1 black guy (nerd-bg2.gif)
• 1 or 2 women ('Gina the nerd' is Asian)
• 3 Labrador retrievers
• 1 cow wearing a beaner with a twirler on top ('nerd of the herd').

Now I don't feel so bad.

update 2

sample pages (pdf file) from Vocabulary from Classical Roots




Gauss Story
Carl Friedrich Gauss (painting)

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CarlFriedrichGauss 23 Jul 2005 - 01:55 CatherineJohnson






A Nerd In The House
Gauss Story





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QuestionAboutReciprocals 23 Jul 2005 - 12:29 CatherineJohnson

My 'benchmark' for the moment when I understand elementary mathematics well enough to move on is:

reciprocals

I find reciprocals utterly mysterious.

They're not quite in the magic category anymore, which is my benchmark for complete and total lack of comprehension. If a maths concept seems like magic, that means I know nothing.

Of late, inside the expanding math section of my brain, reciprocals have put a toe outside the magic category. But not much more than a toe.

Danica McKellar

Tuesday's SCIENCE TIMES has a profile of Danica McKellar, who played Winnie on Wonder Days. It turns out she's a UCLA math major who published a proof (pdf file) now known as the Chayes-McKellar-Winn theorem.

car wash problem

McKellar's web site has a mathematics section where she answers reader questions. She's a natural born teacher:

Q: Hi Danica, I heard a question from Mr. Feenie on a "Boy Meets World" episode which he claimed to be unanswerable. After hearing that, I decided to figure it out. If it takes Sam 6 minutes to wash a car by himself, and it takes Brian 8 minutes to wash a car by himself, how long will it take them to wash a car together?

Danica Answers: Hm, unanswerable? That's TV for you. :)

Let's do it: This is a "rates" problem. The key is to think about each of their "car washing rates" and not the "time" it takes them. Alot of people would want to say "it takes them 7 minutes together" but that's obviously not right, after you realize that it must take them LESS time to wash the car together than either one of them would take.

So, what is Sam's rate? How much of a car can he wash in one minute? Well, if he can wash one car in six minutes, then he can wash 1/6 of a car in one minute, right? (think about that until it makes sense, then keep reading). Similarly, Brian can wash 1/8 of a car in one minute. So just add their two rates together to find out how much of a car they can do together, in one minute, as they work side by side on the same car: 1/6 + 1/8 = 7/24 of a car in one minute. That's their combined RATE. (Note: that's a little bit less than 1/3 of a car in one minute). From this point, the way you want to think of it depends on your favorite way of dealing with fractions. You now have their rate. It's 7/24 cars per minute. You can either just take its reciprcal and say: 24/7 minutes for one car, and you're done.

Or, equivalently, you can think of the 7/24 cars/minute RATE as 24 minutes for 7 cars. (think about that until it makes sense, too) So just divide 24 by 7 to find out how many minutes it would take to do just one car. You get around 3.42 minutes for one car, just a little less than 3 and 1/2 minutes. Done! Yes, I think they should work together, it gets done much more quickly that way. :)

By the way, you said when you watched the TV show you decided that YOU would figure it out, right? How did you do?

I love this. McKellar is teaching two things here:

• how to solve a rates problem
• how to read, study, & learn maths (that's metacognition)

First of all, she knows that math novices transfer their normal reading habits to maths books. By normal reading habits I mean that most of us, when we read a book of prose, read straight through at a fast clip, pausing only to underline or make notes in the margin.

You can't read a math book that way; in fact, I've come to feel you can't really read a math book at all. You have to do a math book, or work a math book. McKellar explicitly instructs her reader not to read the solution straight through, but to stop at key points and ponder until he or she gets the point, and is ready to go on to the next point.

She doesn't stop there, either. She also knows the precise spots in her explanation where most novices will need to stop and mull, and she tells them where those spots are.

She's giving novices direct instruction in metacognition.

As to the problem itself, she addresses the most common error novices will make confronting this particular rates problem, which is:

• figure that it takes the 2 boys 14 minutes to wash 2 cars
• so logically it must take them 7 minutes to wash 1 car

Amazing! And all in the space of a few short paragraphs.

I think McKellar's teaching skill here is connected to her acting. There's a large element of 'performance' in teaching, at least in my experience, and to be a good performer you have to know where your audience is, what they want to hear & what they need to hear.

She does.

back to reciprocals

Here's my reciprocal question.

From this point, the way you want to think of it depends on your favorite way of dealing with fractions. You now have their rate. It's 7/24 cars per minute. You can either just take its reciprcal and say: 24/7 minutes for one car, and you're done.

Or, equivalently, you can think of the 7/24 cars/minute RATE as 24 minutes for 7 cars. (think about that until it makes sense, too) So just divide 24 by 7 to find out how many minutes it would take to do just one car. You get around 3.42 minutes for one car, just a little less than 3 and 1/2 minutes. Done!

I don't understand why you would use the reciprocal to solve this problem.

I understand perfectly well (let's hope) why you would divide 24 by 7.

I didn't even know you could use the reciprocal to find the answer.

7 fact families

I haven't had time to sit down and think this through, but I suspect the reciprocal answer to a rates problem is the same concept as the 7 fact family I put together after teaching the Primary Mathematics lesson on ratio & proportion (Primary Mathematics 6A Textbook, p. 21-46):

7 fact families




(back to top)



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DrewAndMarcTakeTheNerdTest 23 Jul 2005 - 17:03 CatherineJohnson



Drew got:

94% scored higher (more nerdy), and
6% scored lower (less nerdy).

What does this mean? Your nerdiness is:

Definitely not nerdy, you are probably cool.

Marc got:

97% scored higher (more nerdy), and
3% scored lower (less nerdy).

What does this mean? Your nerdiness is:

Definitely not nerdy, you are probably cool.

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TerrificallyHelpfulAdviceFromDanKAndCarolynM 23 Jul 2005 - 18:25 CatherineJohnson



Dan K and Carolyn Morgan have given me some incredibly helpful advice on teaching fractions 'in a hurry.' I'll get it pulled up front as soon as I can, but in the meantime take a look at the Comments thread.



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WitAndWisdomOfKitchenTableMath 23 Jul 2005 - 20:59 CatherineJohnson


it's here!



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DimensionalAnalysis 23 Jul 2005 - 22:23 CarolynJohnston

DanK brought up dimensional analysis in this thread, and it's such a useful idea that I thought we should have a thread to explain what it is, and talk about it and its possible uses in math education.

Here's a very simple example, where dimensional analysis can help you get the right answer.

Suppose a man drives 60 miles in 50 minutes. How fast is he driving?

There are two answers a kid is likely to come up with: the first (and correct) one is 60/50, but a kid might very well come up with 50/60 and not notice he's made a mistake.

Here's how dimensional analysis could help this student get the right answer: he knows he wants a rate for an answer; distance per unit of time. If he thinks of the 60 as '60 miles', and the 50 as '50 minutes', then his two choices are:

(60 miles)/(50 minutes) = 60/50 miles/minute

or

(50 minutes)/(60 miles) = 50/60 minutes/mile.

This gives him more context to help him choose the right answer. Miles per minute are units that make sense for this answer: minutes per mile don't.

In addition, dimensional analysis is the tool to use to make unit changes. If the question requires the answer to be given in miles per hour, then 60/50 is not the right answer, because the units are miles per minute. How to do the conversion to miles per hour?

As with converting fractions to have common denominators, the trick is to multiply the answer by 'one'. In this case, the conversion factor will be (60 minutes)/(1 hour). (You see why this is really 'one'?)

Thus the answer in miles per hour is:

(60 miles)/(50 minutes) x (60 minutes)/(1 hour).

Notice that (60 minutes/1 hour) is actually 1, expressed in different units in the numerator and denominator!

Now for the trick. Move the units around a little, just as though they were numbers in fractions being multiplied, and you get

(60 miles/1 hour) x (60 minutes/50 minutes).

Now the minutes cancel in that second term, and you are left with 60/50 (otherwise known as 6/5) as a dimensionless number. (A dimensionless number is a number without any units attached. For example, all ratios are dimensionless).

So the answer is: 60 miles/hour x 6/5, or 72 miles/hour.

There's even more that you can do with dimensional analysis. As Dan points out, it's a very handy concept, but hardly any math text uses it to the fullest extent they could.

At the undergrad level, it's something engineers and scientists learn explicitly. They have to know it in order to make unit conversions. I was a graduate student when I learned it in a geochemistry (i.e., thermodynamics) class; I had already had a complete undergraduate math education. I taught that whole class of geochemists how to do differential calculus; in return, they taught me dimensional analysis, and I think I got the better end of the deal.

So: when are kids ready to learn, and to start using, dimensional analysis?

Manipulating dimensions is a lot like manipulating fractions, and largely uses the same skills. You can't add dimensioned quantities, for example, unless the dimensions are the same: for example:

x miles/hour + y meters/minute = x+y miles/hour

doesn't make any sense unless you first convert the y term to miles/hour. Identical units can cancel (as the first example showed, when I canceled minutes in the numerator and denominator). So right about the age Ben and Christopher are now -- tennish or elevenish -- is about the earliest kids could really start using it, and it's also about the time that math texts stop emphasizing units (as DanK pointed out).

Plus, if the parents don't know it, how can they teach it?

Once again, it's the internet to the rescue.

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LindaSeebachColumnOnKtmAndWikis 24 Jul 2005 - 20:42 CatherineJohnson


I'm having a great weekend!

I hope you are, too!

How to get some help with math- wiki-wiki

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BarryGarelickAtEducationNews 25 Jul 2005 - 00:23 CatherineJohnson

Wow!

I just stumbled across Barry's op-ed, "Doing the Fox Trot with Cathy Seely," at EducationNews!

Go read it right now!

That reminds me: we have got to get a link up to Education News.

Also, for anyone who has tried to contact me via my KTM email address, it doesn't work. My 'home' email address works only sporadically; as a matter of fact I have just now discovered that I have been thrown off the NYC Math Forum mailing list for the 2nd time in as many months....

So, if you've emailed and I haven't answered, that's why.

update

It's great!

Cathy: Great! I think what you have in the U.S. is too much “Here’s the rule, now do the problem”; too much teacher instruction. The teacher should refrain from stepping in too early to provide students with answers or tell them exactly what steps they should use.

Me: I think I get it. I was thinking that students actually learn things when you teach them what they need to know. But you’re saying, first throw out the text books. Then instead of “Here’s the rule, now do the problem” you say “Here’s the problem, you figure out what the rules are”. How am I doing?

Cathy: Ummm; I think I probably confused you. The point I want to make is that there’s more than one way to teach.

Me: Ah! So sometimes “Here’s the rule, now do the problem” is OK and Singapore Math meets the NCTM standards? Or are you still looking beyond the textbook?

Cathy: Wow. Good questions. In Singapore and other countries they teach math differently than we do here. They teach it according to the NCTM standards.

Me: Uh, I wouldn’t say that. Singapore actually teaches content, and the content they teach actually matters.

Cathy: I don’t know why you think NCTM standards don’t emphasize content. The vision of Principles and Standards for School Mathematics paints a picture of the depth that we can achieve with all students.

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MissingKnowledge 25 Jul 2005 - 00:40 CatherineJohnson

More good stuff from Education News:

Today's math lessons, Armbrecht said, focus much more on "inquiry-based learning" than the math of yore. Students are given a problem, then asked to use their understanding of number structure, logic and math concepts to solve it. In Armbrecht's generation, most students were told to memorize facts instead of being challenged to understand the underlying concepts, he said.

Furthermore, today's math students use calculators, computers and hands-on objects more often than their parents did. So, like Wilmington resident LaMere Henderson, even well-educated parents aren't equipped to help their children with math.

[snip]

But math teacher Dawn Olmstead, recently retired from Alexis I. du Pont High School, said so many reach high school unprepared that remediation can't be avoided.

"What we're seeing is the kids don't know how to add fractions," she said. "Some don't even know what fractions are.

"When they come into ninth grade, they're supposed to be prepared for algebra, and they're not."

There are so many topics to cover, she said, it's a burden to teach them all by the time of the test, which is given in March.

"How about probability?" she said. "Why would I teach that in an algebra class? Because it's on the test. I have to do both: algebra and what's on the test."

For many kids, math is a low priority

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HighlyQualified 25 Jul 2005 - 00:51 CatherineJohnson

Some of you may be aware that a second provision of NCLB kicks in next year. Teachers must be 'highly qualified.'

I would be in favor of this provision if ed schools weren't in charge of definining who is and who is not highly qualified.

Case in point: One candidate certified in math submitted his application this month for a job in Howard County - less than two months before classes begin.

"He wasn't worried," Mascaro recalled. "He'll have six to seven job offers wherever he goes. There's a lot of competition."

She added, "For the critical-needs areas, it's absolutely a teacher's market."

Adding urgency to recruitment this year is a requirement under the federal No Child Left Behind Act that all teachers in core subjects - English, reading, math, science, social studies, foreign language, economics, geography and arts - be "highly qualified" by the end of the next school year. Otherwise, schools risk losing federal funds.

In Maryland, recent data show that the percentage of classes not taught by "highly qualified" teachers has declined to 24.7 percent this year, from 33.1 percent in 2004. Suburban school systems tend to fare better than urban systems.

Hiring is tough task for schools

(Another thank you to Education News.)

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WelcomeCornerAndRockyMountainNews 25 Jul 2005 - 18:10 CatherineJohnson

Good grief.

I just checked the site meter and we are in the midst of a huge Linda Seebach-lanche (thank you!) AND a Warren-lanche (thank you! thank you!), and I haven't even had time to straighten up the place.

While I'm doing that, newcomers might want to read or bookmark the single best article on the math wars, Barry Garelick's An A-Maze-ing Approach to Math in Education Next.

You can find Barry's ktm comments here & user pages here.

Or, you might want to get started with the running feature we call Compare and Contrast.

Carolyn & I hope some of you will enlist in the cause of better math ed for our kids. Leave comments & advice, edit group pages, or create your own page here (scroll down for instructions; more detailed instructions here).

We need volunteers!

Mathematicians, applied-math people, teachers, parents, students, occupational therapists, any & all interested parties: we need you! When you're trying to tap the WISDOM OF THE CROWD, the first thing you need is a crowd.

Our model is WHO WANTS TO BE A MILLIONAIRE. The audience always knows more than the contestant.

strictly nonpartisan! and we mean it!

The fun thing about the math wars is that they cross party lines.

If you don't believe me, read this. (Hint: the politics-makes-strange-bedfellows moment occurs in paragraph 9).

update

While we're on the subject of Warren-lanches, I'm planning on becoming an avid fan of According to Jim this fall.

Especially now that Battlestar Galactica has become West Wing.

update 2

Some of you may be aware that, once upon a time, I wanted to be Nora Ephron.

Warren Bell actually is Nora Ephron, or he would be if he were 20 years older and female.

Here's my favorite Warren Bell piece so far: Condumb?

Trust, but verify!

For parents who aren't as obsessed with math ed as we are, the one thing to take away from the site is this set of Practice Problems for the California Mathematics Standards Grades 1-8 for the Los Angeles County Board of Education, which David Klein developed for the Los Angeles County Board of Education. For a variety of reasons, we think it's a bad idea to rely exclusively on state tests to tell you how your children are doing in math. True for private schools as well as public.

The state of California has the best math standards in the country, according to the Thomas B. Fordham Foundation assessment of state math standards. David's problems will tell you whether your child meets CA standards--and, if not, which topics he or she needs to work on.


related posts:
Assess Your Child for Free Part 2
Assess Your Child for Free
and
David Klein at the AEI

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ABetsyLancheToo 25 Jul 2005 - 19:36 CatherineJohnson

I just checked the referrals page, and discovered a Betsy-lanche, too!

Thank you!

Thankyouthankyouthankyou!!!!!!

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TheCourantInitiativeForTheMathematicalSciencesInEducation 25 Jul 2005 - 20:24 CatherineJohnson

from Elizabeth Carson, co-founder, with Bas Braams of New York City HOLD:

The Courant Initiative for the Mathematical Sciences in Education (CIMSE) is an activity in K-12 mathematics education, that has been informally in progress since 2000, involving a number of faculty members: Charles Newman, Director of the Courant Institute, Sylvain Cappell, Fred Greenle af, Jonathan Goodman, Alan Siegel, Arthur Goldberg, Al Novikoff, Mel Hausner and Edmond Schonberg.

The CIMSE mission is to help foster excellence in school mathematics education.

CIMSE will support activities to educate college and university Mathematics, Science, and Education faculty, K-12 educators and administrators,

parents, business leaders, education philanthropies and members of the community at large on a range of topics and issues in mathematics education, including instructional programs, curricula, standards and assessments, teacher training, research and development, and education policy at the local, state and federal levels, and internationally.

CIMSE is guided by the belief that an educated and informed community, and innovative partnerships between key constituencies of education stakeholders, can help transform the education enterprise to one where educational excellence in the mathematical sciences is part of the customs, practices, relationships and behavioral patterns of importance in the life of our schools, communities and society.

The Courant Initiative for the Mathematical Sciences in Education

EXECUTIVE BOARD

Chuck Newman

Sylvain Cappell

Fred Greenleaf

Elizabeth Carson

PLANNING AND ADVISORY COMMITTEE

To Be Announced

ADVISORY BOARD

To Be Announced


CIMSE year one plans include support and development for a number of NYC HOLD associated activities.

NYC HOLD Honest Open Logical Decisions on Mathematics Education Reform is a national grassroots mathematics education advocacy association of parents, K-12 educators, mathematicians and scientists working to improve mathematics education. NYC HOLD has established a partnership between Courant faculty and parents, teachers and administrators in the NYC education community, faculty at CUNY schools, and at NYU's Steinhardt School of Education. The partnership has grown to extend beyond New York City, to include parents and teachers in school districts across the nation and faculty at a number of universities including Harvard, Stanford, CalTech^?, Johns Hopkins, Emory, Brown, California State Universities, the University of Texas, and Rochester University.

NYC HOLD was co-founded in 2000 by Elizabeth Carson, a NYC parent advocate who currently serves as executive director. Founding members and advisors are listed at http://www.nychold.com/who-we.html

NYC HOLD activities include:

* NYCMATHFORUM and NYC HOLD news distribution and discussion lists * National e-newsletter * Mathematics education resources on the Web * Information and consultancy services to parents, teachers, university math and science faculty, education policy makers and the media

* National advocacy network * Education Forums and Conferences

Please show your appreciation and support for the work of NYC HOLD by making a contribution today.

Your donation may be made through CIMSE and is tax-deductible.

Suggested levels for Individual Support:

Associate $50 - $499 Advocate $500 - $999 Partner $1,000 - $2,499 Sponsor $2,500 - $4,999 Patron $5,000 - $9,999 Benefactor $10,000 - $25,000

Checks may be made out to:

New York University /Courant Initiative for the Mathematical Sciences in Education

and mailed to:

Courant Institute of Mathematical Sciences at NYU Office of the Director 251 Mercer Street New York, NY 10012

ATTENTION: CIMSE, Elizabeth Carson or Charles Newman

Please contact me with specific questions or comment, or for additional information.

Thank you !

Elizabeth Carson Email: ecarson@nyc.rr.com Tel/Fax: 212.529.1302 Cel: 917.208.7153

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TheBigsAndTheBoutiques 25 Jul 2005 - 21:19 CatherineJohnson

KITCHEN TABLE MATH is still so new we haven't got our sidebar pulled together.

the bigs

These are the web sites I go to every week. (OK, I probably go to these sites a tad more often than that, but I don't like to display the depths of my obsession....)

Mathematically Correct
New York City HOLD
Illinois Loop

(I think we've got most of the smaller parent organizations on the sidebar.....and if not, they'll be there eventually.)

the boutiques

There are a number of terrific ed blogs on the web, and I'm still finding more. I'm especially fond of eduwonk & joannejacobs.com, also The Education Wonks & EducationNews.org, & Number 2 Pencil & Jenny D.

But I wanted to mention especially 2 blogs I visit all the time:

Instructivist
and
MathandText

I always stop by Instructivist because HE'S THE INSTRUCTIVIST!

I go to Math and Text because J.D. Fisher is a writer & producer of mathematics textbooks who's currently writing sample math lessons & posting them online. It was from J.D.'s blog that I learned the reason why U.S. math textbooks call dice 'number cubes' instead of 'dice.' I always thought it was because 'number cubes' was the proper mathematical terminology.

But no. The reason U.S. textbooks call dice number cubes is that you can't say 'dice' in a U.S. textbook.

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WayneWickelgren 25 Jul 2005 - 22:10 CatherineJohnson

I'm finally getting around to posting excerpts from Wayne Wickelgren's Math Coach:

The Third International Mathematics and Science Study, conducted in 1996, found that the material taught in U.S. eighth-grade math classes was taught in the seventh grade in many other developed countries and even earlier in Japan and Germany.
Math Coach, by Wayne Wickelgren

Elizabeth Duffrin explains why:

Researchers blame this pattern on the heavy repetition of basic skills that begins in 5th grade and persists through grade 8. Students fall so far behind in those years, Schmidt [U.S. research coordinator for The Third International Mathematics and Science Study, or TIMSS] explains, that they never have a chance to catch up...
Math teaching in U.S. ‘inch deep, mile wide by Elizabeth Duffrin, Community Renewal Society

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

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WorkingWithTeachersAndPrincipals 25 Jul 2005 - 23:01 CatherineJohnson

I make no bones that parents whose children are struggling with a poor mathematics curriculum should find a good curriculum and teach that one instead.

But that raises the issue of what happens politically and socially when a parent rejects a school's math curriculum.

Good question.

it doesn't have to be a battle

My own experience this year was terrific.

Of course, I wasn’t rejecting a curriculum the school had embraced; I was rejecting a curriculum the school had rejected (SRA Math, which is being replaced by Trailblazers).

Even so, I was using a different curriculum at home, and everyone knew it. The reason they knew it was that I printed out copies of the Table of Contents for Christopher to take in and show his teacher.

She was great. She admired all the lesson headings, and told Christopher, “All the parents should be doing this.” It was incredibly sweet of her.

At one point I sent an email saying I was having trouble getting Christopher to cooperate (that’s an understatement) and asking if she could tell him he needed to do my homework, too.

She did.





When we told her, in January, that our goal was to move him to Phase 4, the accelerated track, she blanched. There were already a number of kids in Phase 4 who were struggling; the class was oversubscribed. One child had just been moved ‘down’ to Phase 3, and it had been upsetting to all concerned.

She’d never thought of Christopher as ‘a Phase 4 kid,’ she said. She didn’t want to see him try Phase 4 and fail. (Neither did we.)

It took her about 2 minutes to decide she probably could think of Christopher as a Phase 4 kid, and the reason she could think of him as a Phase 4 kid was that ‘you’ll give him the support he needs.’ She saw clearly that Christopher’s dad and I would do whatever we needed to do to help him succeed—and she saw that we would be taking responsibility for the move. If it didn’t work out, we weren’t going to be back in the school yelling at people. (True.)

Once she'd turned her point of view around 180 degrees, she told us that if we were going to move him we needed to do it now. Suddenly it was our turn to blanch; my plan was to move him in the fall, after we'd had another summer to work on his math.

She said, in so many words, that my plan was going to be problematic. For years the middle school has been hammering the elementary school about placing too many kids in the accelerated class, giving inflated grades, etc., etc., or so I gather. ('Hammering' is not the word she used or implied.) The middle school had made crystal clear to teachers & to parents that they would be placing fewer kids in Phase 4 come fall, not more. Which meant they probably weren't going to think Christopher, who'd been in Phase 3 from day one, and who'd done badly in 4th grade math, was an obvious candidate for the accelerated track.

As she put it, 'They aren't going to know him the way we do.' If we wanted to do it, she said, we needed to do it now.

We said, OK, then, we'll do it now.

She got Christopher moved to Phase 4 within the week.

Not only did she support us in doing something she didn’t necessarily think was a good idea, she told us how to work the ropes. Then she worked the ropes for us.

using TIMSS

Christopher's second math teacher, in Phase 4, was just as terrific. She once sent home a formal, hand-written explanation of the compound interest problems in SRA Math. Yes, it’s mortifying to reveal that I needed a hand-written explanation of compound interest, but there you have it. It was a darn good explanation, too. Later on I learned she’d been an accountant for 15 years before changing careers.

The TIMSS data on U.S. students is a big part of the ‘secret’ to working well with your school district when you object to the math curriculum. The first time I mentioned to our principal, Don, that I thought Christopher maybe ought to move to the accelerated track, he got that tight not-now-not-ever look on his face administrators always get when parents start bugging them to do things they don’t think they ought to do.

I backed off, because in fact Christopher wasn’t ready to move to the accelerated track. But I publicly raised the issue of why the accelerated kids were using a book that was a full year ahead of the rest of the kids without any of us parents having been told.

Naturally it turned out Don hadn’t been told, either; he’s an interim principal. He looked into it, and was obviously pretty dismayed at what he found (not worth going into here).

When we sat down and talked about it, I took the tack that I didn’t think Christopher is Secretly Gifted And Talented In Math; I just wanted him to be on the same track kids are on in high-achieving countries. Which is true. Here are the Magic Words to use with principals, teachers, administrators, & school boards:


In high-achieving countries, students take and master algebra in the 8th grade.


Here in America, only the accelerated kids take and master algebra in the 8th grade.

I told Don: if kids in Germany pass Algebra 1 in 8th grade, I want Christopher to pass Algebra 1 in 8th grade, too.

He had exactly zero problems with that, and the minute Christopher was ready to move to the faster class, he moved him up.

The fact is, our problems in math ed are national, not local, and everybody knows it.

Everybody knows it, but nobody knows how to fix it. Ideological constructivists think they know how to fix it, but your basic principal and/or teacher is living in the real world, facing real children and real parents who blame them when math scores are bad. They’re on the firing line. I don’t think too many principals & teachers truly believe ‘reform math’ is going to be the miracle we’ve been looking for for the last 100 years.

So basically, his feeling was: I’d like to see all our kids learning at the same rate as kids in Singapore. So would I. I don't blame him for our school having the same problems every other school has.

giving respect where respect is due

Once I started teaching Christopher my own hand-picked curriculum, I was on the firing line. For awhile there I was actually having him do the homework I assigned instead of the homework he brought home from school…..so exactly whose fault was it going to be if he didn’t succeed?

It was going to be my fault.

Everyone sensed this. I had moved out of the potentially ticked-off parent category and into the junior colleague category.

That’s another thing.

I also developed a healthy new respect for the teachers he’d had thus far. I couldn’t teach Topic One out of SRA Math, but all of them had managed to teach him a huge amount of math from SRA, which he had retained. His math knowledge from 2nd and 3rd grades was solid as a rock.

So I stopped being a critic, and became a teacher. That meant I asked the school’s teachers for help & advice, and made clear I respected their seniority. Christopher felt the same way. When he told me, ‘Mrs. Panitz is a better teacher than you,’ I sent her an email letting her know.

bullet points

For me, in this school district, putting together public school & home teaching worked during the one year I've done it. Would my approach work everywhere? Most places? I don't know. What I do know is that your basic teacher went into the profession because he or she wanted kids to succeed. Teachers are rooting for the kids, not against them. If you're helping your child succeed, their inclination is going to be to root for you, too.

My (tentative) advice thus far:

• tell your teachers what you’re doing, within limits. I didn’t announce the fact that I was substituting my homework for the school's, and I don’t think I should have done so. It would have been nervewracking for Christopher's teacher—it was nervewracking for me—and since I was going to do it anyway, why get her worried?

• respect the teacher's experience and authority. Show respect even if you're a math major working in a mathematics-related career. Your math knowledge greatly exceeds the teacher's, but your pedagogical content knowledge almost certainly does not.

• ask for help (but don’t suck up lots of the teacher's time)

• tell your child his teachers are good, it's the curriculum, or the too-slow American track, that's the problem

• whenever you talk to teachers or principals, keep the focus on international standings, not local failings

keywords: afterschooling politics of math math wars conflicts with teachers conflicts with schools

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TrustButVerify 26 Jul 2005 - 13:59 CatherineJohnson

This bears repeating:

don't rely on state tests

In theory, I'm in favor of standardized tests.

In practice, I'm still in favor of them, but I don't rely on them. High-stakes testing is subject to enormous political pressure from all concerned. Years ago Ed worked on the California History Social Science Frameworks. He helped the CA Department of Ed develop assessments for the Frameworks, evaluating off the shelf tests, which were, in his words, 'insanely easy.' 12th graders were evaluated at a 9th grade reading level.

The Dept of Ed developed its own tests, & tried them out. (They didn't test the entire state, and he doesn't remember which groups took them.) Two political groups objected: some conservative Christians objected to the critical thinking portion of the tests, and some minority groups objected that their children's scores would go down (which they probably would have, at first). These two groups put enough pressure on their respective representatives that the new tests were scotched before they were ever rolled out. CA went back to using off-the-shelf tests.

No state test will survive a high failure rate in my opinion. That's why I view the current situation in NYC, where Mayor Bloomberg's campaign is based on a sudden, monster increase in student scores, as being far from ideal. I'm fine with the idea of a mayor campaigning on improving student scores. And now that I've seen what can happen to one child's scores thanks to simple, hard work, I believe that you could have a sudden, monster increase in student scores on a broad scale. It's possible.

But I want to see independent audits of those scores. I want to see the test items, and I want to see an audit. Sunshine laws are a good thing. Let's have sunshine laws for state & local testing.

I once read a Diane Ravitch essay on this issue (if I find it again, I'll drop in the reference). She argued that the solution is to establish different levels of 'Pass,' as they do in British universities. Students could pass exit exams with high honors, honors, no honors, and so on. That would probably allow states to maintain rigorous testing in the face of parent opposition.

You might still have an inflated pass rate, but then again, maybe not. Competition spurs people on to higher achievement, and not just because people are naturally competitive, which I believe we are. Seeing someone you know & like do well implies that you can do well, too.

Given the pressures on state testing, I don't rely on New York state tests to tell me how well Christopher is doing. At the end of 4th grade, when Christopher had flunked fully one-third of his year's math course, he earned a '4' on the state math test. 'Exceeds state standards.'

I'm sorry, but a 68 on Unit 5, a 39 on Unit 6, and a 4 on the state exam don't square.

(This is kind of funny. A couple of months later I called one of the guidance counselors at the Middle School to ask about Christopher's chances of moving to Phase 4 when he entered 6th grade. The counselor said nobody ever moves to Phase 4 from Phase 3, so the chances were slim to none. I said, 'But he got a 4 on the state test!' He said, 'That doesn't matter.' I was outraged at the time, but even in the midst of my outrage I knew exactly what he was saying. He was saying Don't rely on state tests.)

So today I'm reminding everyone about these Practice Problems for the California Mathematics Standards Grades 1-8 for the Los Angeles County Board of Education, which David Klein developed for the Los Angeles County Board of Education.

The state of California has the best math standards in the country, according to the Thomas B. Fordham Foundation assessment of state math standards. David's problems will tell you whether your child meets CA standards--and, if not, which topics he or she needs to work on.

I count 85 questions on the 5th grade test in all, divided into 4 areas:

• number sense
• algebra & functions
• measurement and geometry
• statistics

The test isn't as time-consuming as it sounds, since often there are 4 separate questions in one larger question (such as identifying several points on a graph). Answers are included.

If giving the test seems like a lot to do in the face of Massive Pre-teen Resistance, just divide it up across a few days' time. That's what I did.


related posts:
Assess Your Child for Free Part 2
Assess Your Child for Free
and
David Klein at the AEI

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DiskSpaceProblemSolved 26 Jul 2005 - 16:36 CarolynJohnston

Some of you may have had trouble with posting or commenting in the last few hours -- the problem is solved.

We were out of disk space on our server!

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DimensionalDominoesPart2 26 Jul 2005 - 17:06 CatherineJohnson

Don K's lesson on dimensional analysis


Don's page includes PowerPoint manipulatives, showing how to teach dimensional analysis! I think this is going to be very important for us--it's our first entry on the 'Math Lessons' page listed on the side bar, isn't it? (Unless I've missed something, in which case, please remind me.)

Thank you so much for taking the time to do this, Dan!

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MathsInEngland 26 Jul 2005 - 17:26 CatherineJohnson

I have no idea how I got to this link, so can't give credit....

Maths in England sounds even worse than here, if that's possible, which I suppose it isn't:

Lost count of gloomy reports about the state of maths in schools and universities? For more than a decade mathematicians have been moaning and the government has responded with inquiries, changes in the curriculum, numeracy hours in primary schools, golden hellos for maths teachers and a plethora of other initiatives in England.

Golden hellos, you say. Sounds good to me. Think I'll knock off here and go learn some more Russian Math. Which is an especially good idea given the paragraphs that follow:

Where will the next generation of UK mathematicians come from, asks the group, drawn from university maths departments around the country, learned societies and the government's curriculum watchdog.

At the moment the answer seems to be "from Russia and Hungary". In many university maths departments nine out of 10 of appointments go to candidates from abroad, while the shortage of maths teachers in schools has got so bad that the Department for Education and Skills has stopped collecting the figures.

Oh, boy. This next part jibes unpleasantly with Loveless's report on the importance of ability tracking for the most talented students:

There is also agreement on the need - outlined by Adrian Smith's report Making Mathematics Count - to boost the numbers of pupils taking A-level maths, the pool from which science graduates (and future maths teachers) will come. Maths has gone from the largest A-level entry to third place as numbers have dropped by nearly half from 80,000 in 1989 to 49,000 in 2002.

A curriculum for the most able 25% of pupils is needed to encourage them to progress to A-level, says the report, which also suggests awarding more university admissions points for a maths A-level than other subjects.

Dr Gardiner wants a national debate. He argues that in the last 15 years or so, "much of our mathematics teaching, and most of our assessment at all levels, have become fragmented - with multistep tasks being routinely reduced to (and assessed as) a collection of unrelated 'one-step routines'".

The upshot, he says, is that maths undergraduates cannot solve the kind of problems that 13-year-olds used to be expected to do.

He adds: "Students in general are no longer required to combine simple techniques in the most basic ways - so they no longer understand that the power of elementary mathematics lies in the integration of simple techniques into larger wholes.

This is an interesting assessment of the problem, in terms of Saxon Math versus Singapore Math.

From the get-go--and I mean from the 1st or 2nd grade--the Singapore curriculum (the old one, at any rate) asks children to do multi-step problems. That strikes me as the right way to go, but of course I can't base such judgments on anything more than what I think I see in Christopher & me as we learn math.

Nevertheless, the one aspect of Saxon Math that makes me feel chronically nervous is the one-stepness of the word problems. Christopher and I are now working through Saxon 8/7, which is in theory a 7th grade book, and the word problems are either one-step, or they're two-step problems that we're told upfront are two steps. That can't be right.

otoh, I had a fun moment the other day when Christopher, who is, after all, still only 10 years old, solved a problem (probably in the Primary Mathematics 3A Workbook) and then tossed off the comment, 'It's a two-parter,' like some guy in a bar casually mentioning he just wrestled a bear. He thought he was hot stuff, doing a two-parter.

I loved it. Macho in a 10 year old boy--especially macho about a story problem--is awfully sweet. (OK, maybe that's a mother's perspective.)

Still, if he gets manly I-wrestled-a-bear-feelings from doing maths, I say that's a good thing.

update

I just realized: I am supplementing the Saxon 7th grade book with a first semester 3rd grade book for Singapore.

I should say that the 3A problems are now far too easy for Christopher, thank heavens (although the bar model solutions are not too easy. He still can't fully do them. He'll get the bar model wrong for a problem he can do in a second just setting up the problem and doing the computation.)

However, I've worked all the problems in the Challenging Word Problems Grade 3 book, and I know there are problems in there he's not going to be able to do.


maths in England
maths in England, part 2
more maths in England, part 2
top students in England, US, & Singapore
why do kids like math?
another brilliant person who liked getting right answers (scroll down)
Catherine's cousin talks about Everyday Math

Call for national debate on maths teaching GUARDIAN
Where will the next generation of UK mathematicians come from? (GOVT REPORT: pdf file)





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TwoMathEdBlogs 26 Jul 2005 - 17:53 CatherineJohnson

Stephanie just sent me a link to a fascinating list of prerequisites for college math, which includes a terrific Comments thread, at Tall Dark and Mysterious, a blog written by "Twentysomething curmudgeon seeking employment teaching college math in BC."

And btw, these are not prerequisites for a serious college math course:

A year ago, I would have posted that list under a heading more along the lines of “Things Students Should Know By Grade Nine”, but alas, experience as extinguished such optimism on my part.

This is long, but it's so valuable I'm quoting the entire list, which I'll probably 'archive' over on....the 'math lessons' page? Another Content Question for the folks at Information Architecture, Inc. (Definitely read the Comments section as well):

Based on my experiences, students graduating from high school should, in order to succeed in even the most basic college math classes:

1.Be able to add, subtract, multiply, and divide fractions. Moreover, they should understand that the horizontal bar in a fraction denotes division. (Seem obvious? I thought so, too, until I had a student tell me that she couldn’t give me a decimal approximation of (3/5)^8, because “my calculator doesn’t have a fraction button”.)

2.Have the times tables (single digit numbers) memorized. At minimum, they should understand what the basic operations mean. For instance, know that “times” means “groups of”, which will enable them to multiply, for instance, any number by 1 or 0 without a calculator, and without putting much thought into the matter. This would also enable those students who have not memorized their times tables to figure out what 3 times 8 was if they didn’t know it by heart.

3.Understand how to solve a linear (or reduces-to-linear) equation in a single variable. Recognize that the goal is to isolate the unknown quantity, and that doing so requires “undoing” the equation by reversing the order of operations. Know that that the equals sign means that both sides of the equation are the same, and that one can’t change the value of one side without changing the value of the other. (Aside: shortcuts such as “cross-multiplication” should be stricken from the high school algebra curriculum entirely - or at least until students understand where they come from. If I had a dollar for every student I ever tutored who was familiar with that phantom operation, and if I had to pay ten bucks for every student who actually got that cross-multiplication was just shorthand for multiplying both sides of an equation by the two denominators - I’d still be in the black.)

4.Be able to set up an equation, or set of equations, from a few sentences of text. (For instance, students should be able to translate simple geometric statements about perimeter and area into equations. ) Students should understand that (all together now!) an equation is a relationship among quantities, and that the goal in solving a word problem is to find the numerical value for one or more unknown quantities; and that the method for doing so involves analyzing how the given quantities are related. In order to measure whether students understand this, students must be presented, in a test setting, with word problems that differ more than superficially from the ones presented in class or in the textbook; requiring them only to parrot solutions to questions they have encountered exactly before, measures only their memorization skills.

5.Be able to interpret graphs, and to make transitions between algebraic and geometric presentations of data. For instance, students should know what an x- [y-]intercept means both geometrically (”the place where the graph crosses the x- [y-]axis”) and algebraically (”the value of x (y) when y [x] is set to zero in the function”).

6.Understand basic logic, such as the meaning of the “if…then” syllogism. They should know that if given a definition or rule of the form “if A, then B”, they need to check that the conditions of A are satisfied before they apply B. (Sound like a no-brainer? It should be. This is one of those things I completely took for granted when I started teaching at the college level. My illusions were shattered when I found that a simple statement such as “if A and B are disjoint sets, then the number of elements in (A union B) equals the number of elements in A plus the number of elements in B” caused confusion of epic proportions among a majority of my students. Many wouldn’t even check if A and B were disjoint before finding the cardinality of their union; others seemed to understand that they needed to see if A and B were disjoint, and they needed to find their cardinality - but they didn’t know how those things fit together. (They’d see that A and B were not disjoint, claim as much, and then apply the formula anyway.) It is a testament to the ridiculous extent to which mathematics is divorced from reality in students’ minds that three year olds can understand the implications of “If it’s raining, then you need an umbrella”, but that students graduating from high school are bewildered when the most elementary of mathematical concepts are juxtaposed in such a manner.)

7.More generally: students should know the basics of what it means to justify something mathematically. They should know that it is not enough to plug in a few values for x; you need to show that an identity, for instance, is true for all x. Conversely, they should understand that a single counterexample suffices to show that a claim is false. (Despite the affinity on the part of the high school text I am working for true/false questions, the students I am working with do not understand this.) Among the educational devices to be expunged from the classroom: textbooks that suggest that eyeballing the output of a graphing calculator is a legitimate method of showing, for instance, that a function has three zeroes or two asymptotes or what have you.


also added to the list by commenters:

I would add estimation and verification to that list. Students should know the difference between a sensible and nonsense answer.

Another blog by a college calculus professor: Learning Curves

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MathsInEnglandPart2 26 Jul 2005 - 19:28 CatherineJohnson

I just glanced at the British maths report, Where will the next generation of UK mathematicians come from?, (pdf file) and I think I'm going to read the whole document. It reminds me very much of observations both Carolyn and Bernie have made to me, as well as Carolyn's post, Whither American Talent?.

Still, I'd never quite thought of the issue as a 'failure to reproduce,' as the report sees it. I'm not surprised Britain would be thinking of it this way, because of Europe's declining, or soon-to-decline population, which seems to me to have been covered fairly extensively in the European press.

They're right in framing matters this way. For countries and civilizations to grow and thrive, they must reproduce themselves biologically and culturally--which means, I think, that it's not a great idea to allow math talent to dwindle away, as it seems to be doing.

(fyi, I'm having a metacognitive moment here: I'm asking myself, Do I know, for a fact, that any of these statements are true? Answer: no.)

So I'm assuming these things are true, until I learn otherwise. Excerpts from the report:

• The UK mathematics community now falls far short of “reproducing itself” – as evidenced by the dramatic fall in the number of students taking A level Mathematics and Further Mathematics; the declining number and quality of students entering highly numerate university courses; the lack of qualified mathematics teachers; the shortage of high quality IT specialists; the narrowness of the UK mathematics PhD; and the apparent need to import large numbers of research mathematicians.

• The most urgent short-term action was identified in the Smith report – namely to increase markedly the number of students taking, and enjoying, a serious A level in mathematics.

• However, this goal cannot be achieved by simply easing the apparent demands of A level mathematics. In any effective strategy for recovery two key elements must be
  
  (i) to strengthen the foundations laid at KS3 and KS4 in a way that better nurtures the interest, and raises the aspirations, of more able students;
  
  (ii) to devise a concerted programme of professional development to ensure that current mathematics teachers appreciate why these stronger foundations matter.

• The present situation is far more serious than is generally admitted and needs to be addressed as a whole – since many of the most serious weaknesses arise from a failure to recognise, and to deal with, the interplay between the actions of different agencies.

update

Still reading....

The domestic UK supply of mathematically competent manpower is in such decline that in many areas (including teaching, commercial specialist requirements, post-doctoral fellows and appointments to academic positions) we are now dependent on trawling recruits from other countries for “bread-and-butter” appointments (not just for “key” personnel).

I love it!

Nobody can write like the Brits, nobody. They're unbelievable. (I have GOT to go TRAWLING on the UK ed web sites to find out exactly how they do what they do.)

Have you ever in your life seen a government report in the U.S. produce language like this?

The answer is no.

ok, problem spotted

There are serious shortcomings at the level of individual government departments and agencies. But our failure to nurture the home-grown talent we need has been exacerbated by a consistent failure to coordinate policy between different agencies.

They may write better than we do, but thus far the content is just as stupid.

Sorry.

That was harsh.

it gets worse

(i) We have failed to recognise that the effectiveness of curriculum and assessment change (which is the responsibility of QCA) depends on providing appropriate training and support (CPD) for teachers (whichis the responsibility of the DfES, the TTA and the Strategies).

(ii) We have not faced up to the conflicts between

(a) the official goal of improving the career structure for home-grown post-doctoral fellows (which was the apparent reason for increasing research funding as part of the Treasury’s response to the Roberts review);
(b) the effect of EU law (or its current interpretation) on the way the consequent substantial increase in EPSRC funding is being used;
(c) the local pressures on university departments which arise from this more generous EPSRC funding; and
(d) the effective pressures imposed by the HEFCE controlled research assessment exercise and EU employment law on university administrations and on academic appointment practices.

I take it back.

This is much dumber than the stuff we put out.

let me see if I've got this straight

Apparently, the problem with maths education in England is that there've been a number of government inquiries, followed by a number of government reforms, followed by no discernible improvement whatsoever.

How could that be?

These reports and the published government responses, have subsequently led to significant initiatives by government and its agencies. It would be comforting to conclude that “the nature of the problem has been understood and is being robustly tackled”.

And, apparently the reason nothing got better, was that the government inquiries didn't take the whole thing seriously enough:

....the rest of the introduction [of the DfES response to the Smith report] includes a succession of statements (such as that “achievement in mathematics at . . . KS3 is the highest it has ever been”), which indicate that the nature and seriousness of the problem have simply not been grasped (we give clear evidence of this relating to KS3 below). This negative impression is strengthened by such facts as that the flagship policy of establishing a “National Centre for Excellence in Mathematics Teaching” is being “implemented” with an emasculated budget.

OK, so here we have a close reading of the introduction to a response to a report. This thing is a report about the reports.

OK, why don't I just read ahead until I find some actual content.

I do like the scare quotes around the word 'implemented,' though.

this is interesting

....some of Smith’s recommendations (such as the need for a serious reduction in the proportion of mathematics time devoted to “data-handling”, and the urgent need to consider the introduction of “incentives” to increase numbers taking A level mathematics) have not been pursued in the way the mathematics and mathematics education communities had expected.

If I'm understandng this correctly, what we have here is an anti-Trailblazers moment.

Less data-handling.

The whole entire point of Trailblazers is all data-handling all the time; the curriculum was originally titled TIMS, for Teaching Integrated Mathematics and Science. It's just pure data, every step of the way.

Data and investigations.

this is good

The mathematical community constitutes an increasingly important “micro-culture” within modern society. Hence the different parts of this community need to be structured and sustained so that this micro-culture can “reproduce itself” in a routine and orderly way, passing on to the next generation that which is known to be of value, while at the same time facilitating the development and application of new methods and techniques to serve business, management and society in general. Instead the routine reproduction of mathematical culture in the UK has been allowed to decay.

[snip]

In the whole of the UK there were around 85 000 A level mathematics entries in 1989; 66 000 in 2001; and just 54 000 in 2002. This has led to a concomitant decline in the number of competent undergraduates and graduates in highly numerate disciplines, and hence to a shrinking of the basic “pool” from which competent workers in areas that increasingly require serious mathematical skills (including mathematics teachers) can subsequently be drawn.

This is so long I'm going to put the rest on a separate page.

More Maths In England Part 2




maths in England
maths in England, part 2
more maths in England, part 2
top students in England, US, & Singapore
why do kids like math?
another brilliant person who liked getting right answers (scroll down)
Catherine's cousin talks about Everyday Math

Call for national debate on maths teaching GUARDIAN
Where will the next generation of UK mathematicians come from? (GOVT REPORT: pdf file)





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EnglandVsAmericaVsSingapore 27 Jul 2005 - 14:08 CatherineJohnson

The British report, Where will the next generation of UK mathematicians come from?, (pdf file) includes this passage about the TIMSS study (Trends in International Mathematics and Science Study):

[a score of] 625 was fixed as the “Advanced benchmark”. In the “Comparison group” of countries [essentially, all countries with advanced economies], 13% of 14 year olds scored at this higher level – which might be taken as a rough indication of those who are well-positioned to subsequently study mathematics and other highly numerate subjects with some prospect of success post-16, or at university.

Naturally some countries in the “Comparison group” had a larger percentage performing at or above this level, while some fared worse. A mere 7% of the USA sample scored at or above this ”Advanced benchmark” level. And the International average was just 6%. But the results for England should have struck Ministers and officials as far more disturbing: the percentage of English 14 year olds scoring above the “Advanced benchmark” was just 5%!

I found this confusing, because the report tosses around a number of figures:

• 5 to 10% of any population are 'GATE' (gifted and talented) in maths

• the top 25% 'most able' of any population 'is likely to include most of those who have the potential ultimately to become competent workers in those areas that increasingly require serious mathematical skills – including mathematics teaching'

• and, finally, students who score 625 or above on TIMSS and are deemed to be 'well-positioned' to study math &/or 'other highly numerate subjects' (e.g., economics) in college

My question was: if the U.S. had 7% of its students in this above-625 level, and the developed world's average was 13%, how bad is our 7%?

And what was Singapore's number?

Well, I just found it, or something close to. It's high.

Forty-six percent of Singapore’s students were among the top 10 percent of all test takers, five times the 9 percent of U.S. students. Even a Singaporean student in the bottom quartile of Singaporean students outperformed more than two-thirds of U.S. students (Mullis, et al., 2000). In 2003, Singapore’s eighth-grade students retained the top average score among student from 46 countries (Mullis, et al., 2004).

I still don't know how a score above 625 relates to the various percentiles being bandied about. I'm assuming students in the top 10% on TIMSS received scores above 625, but I don't know. If that's true, it looks like almost half of Singapore's students could succeed in college-level mathematics or 'other highly numerate subjects.'

Only 7 to 9% of U.S. students are in a position to major in math, science, economics, or even the 'soft' sciences like experimental psychology & political science (which is pure math these days). I don't know anything about accounting, but these figures don't sound great for how many high school students are prepared to pursue accounting careers, either. And since calculus is still an entry requirement for business schools, we've got a pretty thin slice of the population on-track for B-school entry.

So I'm guessing we'll be seeing an upswing in applications to law school in 2011.

What the United States Can Learn From Singapore's World-Class Mathematics System (and what Singapore can learn from the United States): An Exploratory Study
(link takes you to recommended reading page, which includes a comment & an attached pdf file of the full report)

update

OK, I'm losing patience with online pdf files, so I'll post these links and go clean up my desk (and my floor).

The 'big' report on the 2003 TIMSS seems to be this one:
TIMSS 2003 Technical Report (pdf files for all chapters)
Martin, M.O., Mullis, I.V.S., & Chrostowski, S.J. (Eds.)(2004)

I haven't been able to track down the percentile that corresponds to a score of 625, although it strikes me that I may have 'sufficient information,' as the story problems put it, to figure it out myself. (If we know how many Singapore students scored in the top 10%, and we know the average score of Singapore students--does that do it? I don't know! I will have to investigate!)

I did find this table showing average scores for each country (England was ommitted for some unspecified reason, & I don't see France on here, either....we may be lousy at math in this country, but we also have a glaring deficiency in Information Architecture....).

Singapore's average score is 605; ours is 504.

update, update

OK, now I need some math help. (Apparently I am not in the 10% or 25% or heaven-only-knows-what percent who is in position to major in economics in college any time soon.)

If the average score of Singapore kids is 505, and 625 is a reasonable cut-off for students in position to major in 'highly numerate' subjects in college...that means that the 46% of Singapore kids who scored in the top 10% could not possibly all have scored above 625, right? (Unless the distribution were extremely odd, of course.)

Am I missing a step?

update 3

Good grief.

I used the wrong average for Singapore.

Their average is 605, not 505.

I realize an average is not a median, but setting that aside, and making the mean stand in for the median just this once.....they've got half of their kids scoring 605 or better, just 15 points shy of the 625 cut-off.

Incredible.

Here's Carolyn's comment:

I think you're confusing two different populations here -- one is the population of all kids who took the test, and the other is the population of kids in Singapore who took the test.

The information you're missing is the standard deviation for all these populations -- the 'spread' of the bell curve. You can't figure it out without that bit of info.

Assuming that the distribution for the Singaporeans is a bell curve centered at 605, with a spread-out standard deviation (i.e. a 'fat' rather than 'narrow' bell curve, it is possible that 46% of the Singapore population earned a score above 625.

And it's likely (even without knowing the standard deviations!) that 46% of the Singapore kids were in the top 10% of the population of all kids who took the test, simply because the Singapore average was so high.

a word problem only the top 10% of 9 year olds can do

maths in England
maths in England, part 2
more maths in England, part 2
top students in England, US, & Singapore
why do kids like math?
Catherine's cousin talks about Everyday Math

Call for national debate on maths teaching GUARDIAN
Where will the next generation of UK mathematicians come from? (GOVT REPORT: pdf file)

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HowAsiansAndWesternersThinkDifferently 27 Jul 2005 - 20:23 CatherineJohnson

I've mentioned Richard E. Nisbett's book The Geography of Thought a couple of times.

I can't possibly get into a whole long Thoughtfest about whether Asians actually do or do not think differently in some overarching way than Westerners....at least, not until I figure out reciprocals. (news flash: I've made progress on that front, thanks to Dan K!)

So here's what looks like a decent summary of the book (which I haven't read myself) in Education Review, and here's what looks like an interesting critique of the book at a blog I keep meaning to spend some time reading, Gene Expression.


warning: I've glanced at these 2 sites, & that's it. Both look interesting. End of message.


Nisbett is a psychologist who teaches at the University of Michigan. He's a serious guy, the recipient of a Guggenheim and a blurb from Howard Gardner, no less.

THE GEOGRAPHY OF THOUGHT is interesting to me, because of what Nisbett has to say about Asian superiority in math.

Most Americans (I'm willing to bet) think Asians are genetically superior in math. I've had 4th graders tell me Asians are genetically in math.

Nisbett says that not only are Asians not genetically superior at math, the only reason they're functionally superior at math is that, in essence, they're outworking us. Asian culture, in his view, does not particularly support mathematical thought, by which he means logical thought, or the logic of noncontradiction.

Most advances in mathematics were made by Westerners, few by Asians, and older generations in Asia in fact aren't particularly talented in math. (This is certainly something I heard from the Chinese mom I met at tennis lessons. Her husband, a Ph.D. mathematician, is to this day in awe of American mathematicians. I was shocked when I heard this, because I had the same Asian-math-awe everyone else does.)

excerpts from THE GEOGRAPHY OF THOUGHT:

The Greek faith in categories had scientific payoffs, immediately as well as later, for their intellectual heirs. Only the Greeks made classifications of the natural world sufficiently rigorous to permit a move from the sorts of folk-biological schemes that other peoples constructed to a single classification system that ultimately could result in theories with real explanatory power.

A group of mathematicians associated with Pythagoras is said to have thrown a man overboard because it was discovered that he had revealed the scandal of irrational numbers, such as the square root of 2, which just goes on and on without a predictable pattern: 1.4142135 ..... [yup, that bugs me, too] Whether this story is apocryphal or not, it is certainly the case that most Greek mathematicians did not regard irrational numbers as real numbers at all. The Greeks lived in a world of discrete particles and the continuous and unending nature of irrational numbers was so implausible that mathematicians could not take them seriously.

On the other hand, the Greeks were probably pleased by how it was they came to know that the square root of 2 is irrational, namely via a proof from contradiction....

The Greeks were focused on, you might even say obsessed by, the concept of contradiction. If one proposition was seen to be in a contradictory relation with another, then one of the propositions had to be rejected. The principle of noncontradiction lies at the base of propositional logic. ....The basic rules of logic, including syllogisms, were worked out by Aristotle. He is said to have invented logic because he was annoyed at hearing bad arguments in the political assembly and in the agora! Notice that logical analysis is a kind of continuation of the Greek tendency to decontextualize. Logic is applied by stripping away the meaning of statements and leaving only their formal structure intact. This makes it easier to see whether an argument is valid or not. Of course as modern East ASians are fond of pointing out, that sort of decontextualization is not without its dangers. Like the ancient Chinese, they strive to be reasonable, not rational.

Chinese philosopher Mo-tzu made serious strides in the direction of logical thought in the fifth century B.C., but he never formalized his system and logic died an early death in China. Except for that brief interlude, the Chinese lacked not only logic, but even a principle of contradiction. India did have a strong logical tradition, but the Chiense translations of Indian texts were full of errors and misunderstandings. Although the Chinese made substantial advances in algebra and arithmetic, they made little progress in geometry because proofs rely on formal logic, especially the notion of contradiction. (Algebra did not become deductive until Descartes. Our educational system retains the memory trace of their separation by teaching algebra and geometry as separate subjects.)

The Greeks were deeply concerned with foundational arguments in mathematics. Other peoples had recipes; only the Greeks had derivations. On the other hand, Greek logic and foundational concern may have presented as many obstacles as opportunities. The Greeks never developed the concept of zero, which is required both for algebra and for an Arabic-style place number system. Zero was considered by the Greeks, but rejected on the grounds that it represented a contradiction. Zero equals nonbeing and nonbeing cannot be! An understanding of zero, as well as of infinity and infinitesimals, ultimately had to be imported from the East.

pages 24-27

how Asians and Westerners think differently
describe this picture
how Asians and Westerners think differently, part 2
Harold Stevens, RIP
how Asians and Westerners think differently, part 3
creativity gap, part 2
don't know what we don't know

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DescribeThisPicture 27 Jul 2005 - 22:27 CatherineJohnson






Then go to the University of Michigan press release for The Geography of Thought to see what Nisbett found.


how Asians and Westerners think differently
how Asians and Westerners think differently, part 2
How Asians & westerners think differently, part 3
Harold Stevens, RIP
describe this picture
creativity gap, part 2





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OnlineTIMSSTest 27 Jul 2005 - 23:41 CatherineJohnson


This is a terrific resource. You can give your child 10, 15, or 20 questions from the 1995 & 1999 TIMSS tests. The web site scores them for you.




Explore Your Knowledge

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SampleEighthGradeTIMSSProblems 27 Jul 2005 - 23:59 CatherineJohnson



10 items

OK, I'm going to take this test.

I assume everyone can link to the same sample test, but I don't know for sure. The first question is about Penny & her bag of marbles.

oh, yay

I got all ten right, and my results around the world are just peachy. Penny and her marbles stumped 59% of U.S. students, 56% of international students (this is all intl students, I believe, including kids from very poor countries who've just started taking the TIMSS' test). Obviously, fractions are impossible. Although the Singapore Challenging Word Problems Grade 3 book made all the difference. That and Russian Math.





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CongressionalMathIncentives 28 Jul 2005 - 03:18 CarolynJohnston

OK, it's not my imagination, and it's not just happening in my little world either. We are starting to get officially panicky about the lack of kids going into math and science. Could it be Sputnik all over again? Let's hope.

This article discusses some bills coming up in the House of Representatives that would give incentives for students to take math and science majors in college.

The result is that recent graduates and current students could benefit from a series of bills that could become law including the Math and Science Incentive and the College Access and Opportunity acts. Wolf and U.S. Sen. John Warner (R-VA) as well as George Mason University President Alan Merten and University of Virginia School of Engineering and Applied Science Dean James Aylor are among those backing the new incentives.

Those students who majored in math, engineering and the physical sciences could have up to $5,000 in interest on their loans forgiven just as long as they work in their appointed fields for up to five years.

This opinion piece brings up the same issue that I was fussing about in this post: we need home-grown science and engineering talent to maintain our strength in technological innovation, particularly in the area of defense.

Two Metroplex companies -- Texas Instruments and Lockheed Martin -- can consume half the engineering and computer science students Texas produces.

[But] there's a separate issue for defense companies.

"The need to interest American students in engineering and science is even more imperative for Lockheed Martin because many of the technical positions on our defense contracts require a security clearance from the U.S. government," says Lockheed's Melissa Christensen. "Oftentimes, that clearance requires U.S. citizenship."

Everybody is concerned and paying attention to the problem, suddenly; Greenspan's testifying to the Senate about it, and Congress is preparing to throw money at it. Now all we have to do is the right thing by the kids we're educating, and I think the lack of American bodies will take care of itself. After all, during the internet boom of the 1990s, there was a huge demand for programmers, and people were there, both foreign and American, to take up the demand. If there's a demand for technical talent, motivation won't be a problem. Preparation will very likely be.


Whither American talent?
Paul Samuelson on the 'science gap'

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NewStudyOnManipulatives 28 Jul 2005 - 17:54 CatherineJohnson

I want to follow up on Carolyn's post on Congressional math incentives, but before I do that here's a reminder:

Anne Dwyer has posted new notes on her summer math class.

And...quickly checking her page just now, noticed this comment:

So, what have I learned so far?

• they like games where they compete with one another
• they prefer pencil and paper exercises
• they like to figure out puzzles

This is fascinating, because Kevin Killion, of Illinois Loop, just pointed me to a new article in Scientific American showing that manipulatives are less effective than pencil and paper with young children. I'll write a bit more on this later (bike ride time!), but here's the critical passage:

Teachers in preschool and elementary school classrooms around the world use "manipulatives"--blocks, rods and other objects designed to represent numerical quantity. The idea is that these concrete objects help children appreciate abstract mathematical principles. But if children do not understand the relation between the objects and what they represent, the use of manipulatives could be counterproductive. And some research does suggest that children often have problems understanding and using manipulatives.

Meredith Amaya of Northwestern University, Uttal and I are now testing the effect of experience with symbolic objects on young children's learning about letters and numbers. Using blocks designed to help teach math to young children, we taught six- and seven-year-olds to do subtraction problems that require borrowing (a form of problem that often gives young children difficulty). We taught a comparison group to do the same but using pencil and paper. Both groups learned to solve the problems equally well--but the group using the blocks took three times as long to do so. A girl who used the blocks offered us some advice after the study: "Have you ever thought of teaching kids to do these with paper and pencil? It's a lot easier."

Dual representation also comes into play in many books for young children. A very popular style of book contains a variety of manipulative features designed to encourage children to interact directly with the book itself--flaps that can be lifted to reveal pictures, levers that can be pulled to animate images, and so forth.

Graduate student Cynthia Chiong and I reasoned that these manipulative features might distract children from information presented in the book. Accordingly, we recently used different types of books to teach letters to 30-month-old children. One was a simple, old-fashioned alphabet book, with each letter clearly printed in simple black type accompanied by an appropriate picture--the traditional "A is for apple, B is for boy" type of book. Another book had a variety of manipulative features. The children who had been taught with the plain book subsequently recognized more letters than did those taught with the more complicated book. Presumably, the children could more readily focus their attention with the plain 2-D book, whereas with the other one their attention was drawn to the 3-D activities. Less may be more when it comes to educational books for young children.

This perfectly supports the study Carolyn mentioned way back when, showing that fraction manipulatives are good for middle schoolers.


CA state study on manipulatives
Fraction Manipulatives
Quick Thought about Fraction Manipulatives
Fraction Manipulatives Part 2
New Study on Manipulatives Part 2

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NewStudyOnManipulativesPart2 28 Jul 2005 - 20:18 CatherineJohnson

I'm reading the Scientific American article about manipulatives & symbolic representation now:

About 20 years ago I had one of those wonderful moments when research takes an unexpected but fruitful turn. I had been studying toddler memory and was beginning a new experiment with two-and-a-half- and three-year-olds. For the project, I had built a model of a room that was part of my lab. The real space was furnished like a standard living room, albeit a rather shabby one, with an upholstered couch, an armchair, a cabinet and so on. The miniature items were as similar as possible to their larger counterparts: they were the same shape and material, covered with the same fabric and arranged in the same positions. For the study, a child watched as we hid a miniature toy--a plastic dog we dubbed "Little Snoopy"--in the model, which we referred to as "Little Snoopy's room." We then encouraged the child to find "Big Snoopy," a large version of the toy "hiding in the same place in his big room." We wondered whether children could use their memory of the small room to figure out where to find the toy in the large one.

The three-year-olds were, as we had expected, very successful. After they observed the small toy being placed behind the miniature couch, they ran into the room and found the large toy behind the real couch. But the two-and-a-half-year-olds, much to my and their parents' surprise, failed abysmally. They cheerfully ran into the room to retrieve the large toy, but most of them had no idea where to look, even though they remembered where the tiny toy was hidden in the miniature room and could readily find it there.

Their failure to use what they knew about the model to draw an inference about the room indicated that they did not appreciate the relation between the model and room. I soon realized that my memory study was instead a study of symbolic understanding and that the younger children's failure might be telling us something interesting about how and when youngsters acquire the ability to understand that one object can stand for another.

here's the anti-constructivist moment:

[The] capacity [to] create and manipulate a wide variety of symbolic representations .... enables us to transmit information from one generation to another, making culture possible, and to learn vast amounts without having direct experience--we all know about dinosaurs despite never having met one. Because of the fundamental role of symbolization in almost everything we do, perhaps no aspect of human development is more important than becoming symbol-minded.

symbols aren't 'natural'

The first type of symbolic object infants and young children master is pictures. No symbols seem simpler to adults, but my colleagues and I have discovered that infants initially find pictures perplexing. The problem stems from the duality inherent in all symbolic objects: they are real in and of themselves and, at the same time, representations of something else. To understand them, the viewer must achieve dual representation: he or she must mentally represent the object as well as the relation between it and what it stands for.

A few years ago I became intrigued by anecdotes suggesting that infants do not appreciate the dual nature of pictures.

[snip]

.... the Beng babies, who had almost certainly never seen a picture before, manually explored the depicted objects just as the American babies had.

The confusion seems to be conceptual, not perceptual. Infants can perfectly well perceive the difference between objects and pictures. Given a choice between the two, infants choose the real thing. But they do not yet fully understand what pictures are and how they differ from the things depicted (the "referents") and so they explore: some actually lean over and put their lips on the nipple in a photograph of a bottle, for instance. They only do so, however, when the depicted object is highly similar to the object it represents, as in color photographs....

[snip]

it takes several years for the nature of pictures to be completely understood. John H. Flavell of Stanford University and his colleagues have found, for example, that until the age of four, many children think that turning a picture of a bowl of popcorn upside down will result in the depicted popcorn falling out of the bowl.

Andrew makes Barney fly

A couple of weeks ago Andrew (10, autistic, nonverbal) brought me Christopher's yellow plastic airplane, on top of which he'd mounted one of his Barney's, and handed the whole big package to me with an urgent look on his face. He was on a mission.

Martine came in and said, 'He wants you to make Barney fly.' She'd been sitting in the family room when Andrew had put his Barney on top of the plane, and then flung plane & Barney up into the air, apparently thinking Barney would fly around the room.

Andrew had been very unhappy with the outcome, and was now appealing to me. Clearly he believed that making Barney fly was one of those things, like operating the TIVO, only adults know how to do.

I was flattered, but also dumbfounded. What goes on inside this child's head? was my exact thought.

I was thinking....does he not understand gravity?

Does he not understand toys?

What's with this kid???!!

The Scientific American article makes me think that Andrew, although he can read, hasn't completely figured out the dual nature of symbolic representation.

He probably couldn't understand the plastic airplane as being TWO THINGS:

• an airplane
  
  AND
  
  
• a symbolic representation of an airplane

What I'd like to know is: what does he think about Barney?

is this a shoe?

Here's a little guy trying to put his foot inside a picture of a shoe.

lost in translation

I constantly have the experience of reading constructivist texts, noticing that the ideas they're advocating are good ones or at least not obviously bad ones.....and then, five seconds later, finding that they've taken a sound idea and just completely gummed it up in the application.

Assuming this work on manipulatives & symbolic representation is correct, the constructivist obsession with manipulatives looks to be another instance of a good idea lost in translation. Constructivism is majorly obsessed with manipulatives, that's for sure. I understand that the TERC curriculum is basically just a huge box of manipulatives, with no textbook or 'consumables'--workbooks--at all.

Following in Piaget's footsteps, constructivists believe children don't reach the stage of 'formal operations' until age 11; from 7 to 11 they're in the Period of Concrete Operations. (Often you'll see the word 'developmental' used to designate constructivist curricula. Apparently that's a reference to Piaget.)

Wayne Wickelgren says this is nonsense; children can handle abstract concepts long before age 11. But constructivists are the people time forgot, and they're still basing their pedagogy on work done in the 1950s.

That's bad enough in itself, seeing as how the field of cognitive science was just getting started around that time, and Piaget's work hasn't fared so well over the past 60 years.

But the more glaring misstep, it appears, is that they failed to grasp the nature of the concrete.

The reason constructivists think children should spend their grade school years working with manipulatives is that manipulatives are concrete. But they're not. Manipulatives are symbolic objects that require the child to have mastered the concept of dual representation.

Skinnies and bits are not concrete. They are symbolic representations of the Hindu-Arabic numeral system. Worse yet, they are more intellectually demanding, and hence more confusing, symbolic representations than pencil marks on paper.

They're harder to understand, not easier.

Lost in translation.

question

I hope I'll get a chance to talk to these researchers at some point.

My question is: why should pencil and paper be less challenging than manipulatives?

I can see why pencil and paper wouldn't be any more challenging than manipulatives, but why should pencil and paper be easier? Do pencil marks somehow not involve dual representation? That's what the authors seem to imply, but they don't say so directly.


CA state study on manipulatives
Fraction Manipulatives
Quick Thought about Fraction Manipulatives
Fraction Manipulatives Part 2
NewStudyOnManipulatives
New Study on Manipulatives Part 2





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AGoldenHello 28 Jul 2005 - 23:50 CatherineJohnson



If Parents Fret, Do SAT Tutors Cost $685/Hour? A) Yes (Update1)

Now that is a really golden hello.


via: Joannejacobs.com

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SummerDoldrums 29 Jul 2005 - 05:30 CarolynJohnston

I am inspired by Christopher's having wrestled with a two-part story problem and won. He's right to feel manly; doing two-step story problems is the modern equivalent of hunting mastodons.

But we are in the doldrums, at our house.

I had an email the other day from someone who asked for suggestions as to how she could make math more interesting for her son, who is going to be a freshman in high school. Anyone have any ideas for motivating a child in high school?

It's a tougher age than elementary and middle school. What you can do at that age depends heavily on whether they're already behind, and have to make up material they struggled with unsuccessfully in an earlier grade.

I haven't dwelt at all on trying to make math more interesting. I have tried instead to make my son's experiences with math successful, positive ones, trusting that it would then become something he felt good about doing. If he has to tackle something hard, I first try to build him up with easy problems (for a lot more on how to use that trick, which I think will work very well with a kid of any age, see this post). If he gets a 100 on his work for the night, it's a big deal; it gets stickers and goes up on the fridge. He always gets rewards of some sort for doing his math (and getting released from duty is a huge reward all by itself).

Ben and I are ploughing through Prentice-Hall Mathematics Course 1 this summer. My basic message to him, when he objects to doing any sort of homework (which he does frequently), has been: Too Bad, we all have to work a little bit; here's your carrot if you buckle down and get it done, and here's your stick if you don't. I figure I'm just being his frontal lobes until the day (if it ever comes) when he can use his own. Rational arguments about his future in the global technological marketplace don't seem to make much of a dent, at least not yet (I'm sure he'll thank me profusely when he's older, though).

This summer, having finally broken free of Everyday Math, we've made some real progress at a good pace. He's learned a lot of math, and been successful.

But he's developed an allergy, lately, to intellectual work. He fights me insistently when I try to get him to do his homework, even before he knows what he'll have to do (his reaction when he gets his assignment is usually: "That's it? That's all I have to do?" Sheesh).

(As an aside, I've found that with Ben it's essential to tell him, at the beginning of a lesson, exactly which questions he'll be expected to do that day. This means I can't leave the lesson open-ended, but it's okay. He copes a lot better if he knows the bite he's expected to chew off.)

I'm beginning to think we just need a vacation. We'll be taking one in a couple of weeks (I'll be taking my computer, but no real work), to Seattle.

I have a plan: I want to go on a Whale Watch tour.

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XtremeBehaviorismTeachingAndScripts 29 Jul 2005 - 17:01 CatherineJohnson

I just found a wonderful comment after the post on bullying:

smart constructivism

I haven't looked at the book, but I find the concept interesting. I believe that it takes a special skill to remember your own child accurately, through the lens of childhood, and if you can remember it, then you can teach children anything.

You can teach them math or history or art or how to be polite or how to handle a bully.

Teaching is a puzzle. It's a puzzle where you must navigate backwards in a maze. A child is at point K, but they are supposed to be at point Z. If you just show them again how to go from A to Z, you are missing the point of how they got to K.

And usually, kids made a rational mistake: they misunderstood something, or misheard something, and this thing is embedded in their minds. It leads them (Rationally) to this bad position K.

Teaching is about figuring out how someone got into that position, so you can FIX that misunderstanding. It's not enough to tell them that K is the wrong place; you have to help them never follow that wrong path in the first place.

The best way to help kids learn is to remember the typical misconceptions YOU had as a child, and ones similar to it, to try and understand why they would think what they think. Then, you can see how they are really very smart--just misguided.

a child must feel like himself

re: the aspergers/high functioning autism stuff: this kind of description is very similar to what behavioral psychologists teach to help children with anxiety and attachment disorders. I personally believe that there is a high correlation between attachment disorders and what's called asperger's, but I caution people to refrain from just teaching these techniques to children.

The problem with just teaching this techniques is that you need your children to feel like themselves. That may sound silly, but it isn't helpful to teach your child how to act. You may want them to learn how to behave, but they need an emotional makeup capable of backing up the behavior.

For a short term case like a bully, maybe it doesn't matter so much, but in terms of making friends, you need your child to have an emotional makeup that feels these behaviors are natural. If not, the other children will recognize that the behavior is still off, and worse, the child can often feel that they are not capable of making friends by being themselves but have to act like someone else. That's a painful experience for a child, and can do a lot of damage in the long run. Be careful at behavioral solutions that make a child feel that their personality isn't acceptable.

joannejacobs comment thread on bullying

Interesting comments on bullying at joannejacobs.com




how to stop a bully
Comments thread on bullying at joannejacobs.com

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SmartConstructivism 29 Jul 2005 - 20:36 CatherineJohnson

For awhile now I've been noticing that not infrequently I'll read a constructivist text and think: OK, that idea does not sound actually insane.

Then, with a sense of growing alarm, given the fact that I've just spent the last 3 months of my life banging on about constructivism here on the World Wide Web, I'll think: as a matter of fact, that idea sounds like an idea to which I myself subscribe.

Fortunately, the cognitive dissonance never lasts long, because the next paragraphs invariably put forth wing-y observations and grand, looping conclusions that do not follow logically from any known principle governing the natural world. Such as, to quote my number one most demented peer-reviewed constructivist nonsense on stilts prose passage of all time:


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


That's just nuts.

Still, I keep having these moments of recognition, stumbling over, however fleetingly, my own thinking & experience in constructivist texts. And from time to time this will have happened often enough that I'll have to stop and think: Wait a minute. Am I definitely against constructivism?

The answer is yes. As it turns out, there is an obvious and simple resolution to the problem of When Bad Constructivists Say Good Things, which is that there is smart constructivism, and stupid constructivism, the latter, I've just this moment discovered, being known as radical constructivism in the trade, which is what I intend to call it from now on.

Say it together, now:

radical constructivism!

There isn't a parent on the planet who's going to enjoy hearing that his school is implementing a radical constructivist curriculum, which is probably why the words radical constructivism are nowhere to be found on the web site of the NCTM.

So I'm going to be using radical constructivist from now on, and I'm going to say text instead of curriculum, for good measure. MATH TRAILBLAZERS: a radical constructivist text.

That's gonna make me popular.

What is smart constructivism, you ask?

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.**

According to smart constructivism, all knowledge is constructed, period.

There isn't active knowledge & passive knowledge, constructed knowledge and swallowed-whole knowledge, or any other kind of Correctly acquired knowledge versus Incorrectly acquired knowledge. Knowledge is knowledge; to get more of it, you have to build your new knowledge on the foundation of the old knowledge you already possess.

smart constructivism at Kitchen Table Math

Probably all of the commenters on ktm assume or understand this, and have thought about it. Here is Anne Dwyer's wonderful story about connecting a lesson with her daughter's pre-existing knowledge:

Erin is finally finishing up Primary Mathematics 1B. She is working on money. She has always had problems counting money. She can skip count by 5 and she can skip count by 10. But she couldn't skip count by 10 if she was on a number that ended in 5. I tried several methods to help her but nothing worked until we started working on the white board. I wrote out skip counting by 5 and underneath I wrote out skip counting by 10. And finally, in frustration, I suggested that when she had to increase the number by 10, she should skip count by 5 twice. Well, that just totally clicked with her. She got it and has never had another problem counting money. I never would have 'taught' it that way, but it totally worked for her.

And this comment from a ktm guest beautifully expresses the essence of smart constructivism:

Teaching is a puzzle. It's a puzzle where you must navigate backwards in a maze. A child is at point K, but they are supposed to be at point Z. If you just show them again how to go from A to Z, you are missing the point of how they got to K.

And usually, kids made a rational mistake: they misunderstood something, or misheard something, and this thing is embedded in their minds. It leads them (Rationally) to this bad position K.

Teaching is about figuring out how someone got into that position, so you can FIX that misunderstanding. It's not enough to tell them that K is the wrong place; you have to help them never follow that wrong path in the first place.

The best way to help kids learn is to remember the typical misconceptions YOU had as a child, and ones similar to it, to try and understand why they would think what they think. Then, you can see how they are really very smart--just misguided.

*Cobb, P., Yackel, E. & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23, 2-33.

** Bransford, John D., et al. (2000). How People Learn: Brain, Mind, Experience, and School Revised Edition. Washington, D.C.: National Academy Press. (pdf file)

keywords: radical constructivist radical constructivism lost in translation smart constructivism

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FlowChart 29 Jul 2005 - 21:51 CatherineJohnson


Yes, it's a Constructivist Flow Chart!







Actually, this web site is worth taking a look at for a quick overview of educational psychology. On the basis of very rapid skimming, I'd say that the author, Richard Hall, the associate Dean for Research, School of Management and Information Systems at the University of Missouri Rolla, seems to have some horse sense.

Topics covered:
active learning
assessment
behaviorist theory
constructivist theory
information processing theory
learning in groups
learning strategies
learning styles
metacognition
education, hypermedia, and the world wide web

update: definitely worth reading

I've just read the constructivist page closely, and this is quite a nice summary. When you put all 10 of these pages together, this is probably the most useful short, concise comparison-and-contrast discussion of contemporary ed psych topics I've come across.

I'm going to read all of them.

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MathTalkInTheCar 30 Jul 2005 - 06:24 CarolynJohnston

We took the kids to a bar tonight, as it happened. Colin (17) is into playing the bass these days; he has a band that he plays with during the school year. I have a friend at work who is a hot guitar player and who just joined a classic rock band, and he was playing his first gig tonight, and they were letting kids stay through the first set, so we went to see him. It was a long drive for us -- all the way out to Greeley. The place was an authentic roadhouse with motorcycles parked out front, and the food was good -- it was Cajun food, and very authentic given that we were not in Cajun country but in Greeley, Colorado, home of the Feedlot You Can Smell All The Way To Denver.

On the way home, Colin asked us about the difference between the median, the mean, and the mode of a data set, and what each of them is good for. This is, of course, the sort of thing we love to pontificate about. He then told us that he felt he had never really quite gotten the idea of a function, and asked us to explain it.

It's a smart kid who understands what he doesn't understand. Most adults can't do that very well.

Actually, most kids coming into calculus classes are confused by functions. A function is just a black box; you put in an input, and get out an output. What makes it a function is that, when you put in the same inputs, you always get the same outputs. You can't put the same number in the black box and get 2 one time, and 5 the next.

Most texts teach functions using formulas to define the functions; all the functions kids see look like f(x)=3x-5, or g(x)=x/6. But functions don't have to have formulas to go with them; they can defy description by a formula. The only rule is that if you put in the same input multiple times, you get the same output, every time.

The reason kids confuse formulas with functions is that it's hard to define functions that don't use formulas, even though in real life we encounter them all the time. When a function totally defies description with a formula, we often resort to trying to describe it with only a couple of numbers, such as the mean, median, and standard deviation (this is how the whole field of statistics arises).

We played a 'figure-out-the-function' game on the way home from Greeley. Bernie and I would think of a function, and Colin and Ben would give us numbers for inputs, and we would then tell them the output. They'd then try to guess the formula we were using to define the function.

They are both aces at extracting patterns. If anything, Ben would try to generalize from too little data; once he guessed, after one try, that the function was 'add 2'; he'd given me a 2, and I'd come back with 4 (the function I'd thought of was squaring; he got it on the next try). Bernie was giving Colin some functions that are so simple they trip up students with their obviousness, like the function that returns the same number you give it, and the one that returns '3' no matter what you give it. He gave Colin one function that was so bizarre you can't describe it with a pattern.

Ben knew more about functions than I thought, even piping up with "that's the constant function 3" at the appropriate moment. Did they do functions one day for 5 minutes in Everyday Math? Well, he was definitely on the ball that day.

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DanKOnMakingMathInteresting 30 Jul 2005 - 16:53 CatherineJohnson

Great comment from Dan K!

• Go bowling. Ignore the automated system, and keep score manually. Then, work through the calculation for some counter-factual cases (“What would my score have been if I hadn’t missed that @#$! spare in the fourth frame?”). Try to figure which one roll would have boosted your score the most if it would have knocked down all the pins.

• Check the standings. Develop the formula for computing “magic numbers” for clinching the division in baseball. Just please don’t tell me how small the Cardinals’ magic number is to eliminate the Cubs.

• Follow the market. Each person picks five stocks to watch. Invest your pretend portfolio in them. Track their performance throughout the month of August. Figure out how to plot their daily performance on a graph, comparing their performance to the Dow, the NASDAQ, and the S&P 500. Trade into other stocks along the way.

• Try Mathmania. Look for interesting problems in the Mathmania booklets put out by Highlights publishing. These periodicals are probably aimed at 4th or 5th graders, but you can upscale some of the problems by trying to describe them using algebra.

• Look at MATHCOUNTS. The MATHCOUNTS web site (www.mathcounts.org). They’ve got a “problem of the week” archive (with solutions!) that you can browse through. These problems are often topically related to current events. They’re designed to interest kids, so maybe some of them will succeed with your kid. MATHCOUNTS is for math-oriented middle schoolers, so it will challenge most high school students, too.

• Graph the logical flow. Develop a flow chart—or pseudo-code, if you’re already into programming—describing scoring in tennis. Nest a loop for point scoring within a loop for set scoring. Sometimes deuce is an infinite loop.

• Play Jeopardy. Write up your own problems and arrange them in categories. This could be a lot of work, depending on how hard you make the problems. Don’t be too strict about answering in the form of a question.

• WARNING: High risk of failure. Plan a math rally around the yard or neighborhood. Students must solve clues in the form of math problems to find out, say, which envelope to open to get the next clue. Then they must determine which direction to walk to find the next clue. If you open the wrong envelope (or box, or whatever), you lose points, but it then tells you what would have been correct, so you can get back on the right track. If this turns out to be fun, that’s great. If, however, the kid thinks it’s bogus, then you’ve invested a lot of time to end up looking pretty foolish.

I think I'm going to start a user page for this subject....I've forgotten how to set it up so anyone can edit it--Carolyn, do you want to do that?

I'm going to start it from the User Page Index.

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DontTeachInAMonotonePart1 30 Jul 2005 - 19:32 CarolynJohnston

This is the first in a two-part entry; the second part is here.

Bernie and I were talking yesterday about what makes math books really useful to us (we have to learn new things from math texts all the time in our line of work -- we like it).

Early texts in a math field, written when the topic is new and hot, generally have a lot of life; they start out talking about what motivates the creation of an idea -- they do examples -- they talk about what's known and what isn't. When the theory is better understood, the presentations get terser and cleaner, less emotional, and in a very real sense less useful. This is the higher life form to which mathematics texts seem to evolve, and I think it's a mistake.

It's been theorized that one of the purposes of our having emotions is that they mark more meaningful experiences in our memories,^* and this would imply that we can really get more out of those earlier, more lively textbooks. In those texts, the tone is more personal; the authors are generally telling you what's important, and why it's important, instead of presenting the whole field as impersonally and cleanly as possible. People just learn a lot better with a text like that.

The same problem occurs with K-12 texts: there are no markers to tell the reader what really matters for the children at a given level. A unit on naming polygons will follow a unit on subtraction with borrowing, with no indication that the unit on naming polygons is less important relative to the concept of subtraction with borrowing.

The material in beginning math texts is, of course, about as mature as it gets; as a culture, for example, we definitely understand how to factor quadratics. I think the constructivist movement recognizes that the 'dead tone' in a lot of math texts is a problem, but they've found the wrong remedy. Artificially forcing kids to derive from first principles concepts that have been understood for millennia is throwing out the baby with the bathwater.

If the topics in texts are presented in a monotone, then people approaching the material for the first time -- including the teachers who have to teach it -- have no way of knowing what's important and what isn't. For example, most people would probably agree that the ability to approach and solve story problems is more important than being able to calculate the median and mode of a set of data. If an adult can't do the latter, for example, then they can simply look it up; but if an adult can't figure out a simple story problem, then that's a fundamental intellectual incapacity.

But what is there in a typical middle school textbook to reflect the fact that story problem solving is a critical ability relative to calculating the mode and median? Topics are generally presented one unit after another. In one, you're solving mixture problems; in another, you're calculating modes. There are no marquee lights on the mixture problem unit saying, "THIS IS AN IMPORTANT THING TO KNOW HOW TO DO."

^*Descartes' Error: Emotion, Reason, and the Human Brain by Antonio Damasio


Don't teach in a monotone, part 2





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JDFisherOnTextbookFragmentation 31 Jul 2005 - 01:15 CatherineJohnson

J.D. Fisher of MathandText left a comment today that reminded me I'd wanted to point people to his post on textbook fragmentation, which is a HUGE, documented factor in bad math ed here in the U.S.

One reason publishers maintain a great deal of fragmentation in elementary basal mathematics texts, for instance, is that such a structure allows adoption committees and other, similar decision-making bodies, to quickly judge, with great confidence, that a text has indeed covered all of the requisite state standards.

But this structure also has the effect of 'un-prioritizing' content. Simple ideas and less relevant topics are given the same priority and the same space as more robust, more relevant topics.

And check out his excerpt of a 2005 math textbook's TOC:

1 Place Value Through Hundred Thousands
2 Place Value and Exponents
3 Place Value Through Hundred Billions
4 Compare, Order, and Round Whole Numbers
5 Place Value Through Thousandths
6 Problem-Solving Strategy: Find a Pattern
7 Compare, Order, and Round Decimals

The blue lessons (with the possible exception of Lesson 2) represent the exact same concept applied to larger and larger--and then much smaller--numbers. The red lessons are also closely related, but are separated by two somewhat unrelated lessons.

writing is organizing

People tell you writing is rewriting, which is true, but the main reason for all the rewriting is that what writing really is, is organizing. Ed had this insight today when I read him a line from a terrific critique of constructivism by two cognitive scientists, and it was a Brand New Thought for both of us. More on this later.

In the meantime, I can tell you that I've had a visceral understanding of just how dangerous unprioritized content is ever since I listened to Temple's stories about what happens to animals in a meatpacking plant once the employees have lost sight of the difference between the big stuff and the small stuff.

More on that later, too.

PowerPoint makes you dumb

(although, in the case of dimensional analysis, I am going to be relying on PowerPoint to make me smart)

I have zero time at this moment (or possibly ever) to read Edward Tufte's discussion of Boeing's PowerPoint presentation on the space shuttle Columbia, but I'm hoping maybe J.D. will take a look and fill us in. As I understand it, Tufte argues that PowerPoint's built-in bulleting structure equalized or even 'unprioritized' the 'possible tile damage.' That's my impression.

Whether or not I've got the jist, I can easily imagine a poorly structured, unprioritizing report resulting in catastophic failure. Easily.

Getting back to children and math, a severely fragmented textbook is going to be at a bare minimum a catastrophic obstacle to learning.

Of that, I'm sure.

update

I've just tracked down Edwart Tufte's long essay, The Cognitive Style of PowerPoint.

And a blog called The Talent Show has a lengthy excerpt from the TIMES article on PowerPoint's role in the Columbia disaster that's worth quoting in full here, too:

In August, the Columbia Accident Investigation Board at NASA released Volume 1 of its report on why the space shuttle crashed. As expected, the ship's foam insulation was the main cause of the disaster. But the board also fingered another unusual culprit: PowerPoint, Microsoft's well-known ''slideware'' program.

NASA, the board argued, had become too reliant on presenting complex information via PowerPoint, instead of by means of traditional ink-and-paper technical reports. When NASA engineers assessed possible wing damage during the mission, they presented the findings in a confusing PowerPoint slide -- so crammed with nested bullet points and irregular short forms that it was nearly impossible to untangle. ''It is easy to understand how a senior manager might read this PowerPoint slide and not realize that it addresses a life-threatening situation,'' the board sternly noted.

PowerPoint is the world's most popular tool for presenting information. There are 400 million copies in circulation, and almost no corporate decision takes place without it. But what if PowerPoint is actually making us stupider?

This year, Edward Tufte -- the famous theorist of information presentation -- made precisely that argument in a blistering screed called The Cognitive Style of PowerPoint. In his slim 28-page pamphlet, Tufte claimed that Microsoft's ubiquitous software forces people to mutilate data beyond comprehension. For example, the low resolution of a PowerPoint slide means that it usually contains only about 40 words, or barely eight seconds of reading. PowerPoint also encourages users to rely on bulleted lists, a ''faux analytical'' technique, Tufte wrote, that dodges the speaker's responsibility to tie his information together. And perhaps worst of all is how PowerPoint renders charts. Charts in newspapers like The Wall Street Journal contain up to 120 elements on average, allowing readers to compare large groupings of data. But, as Tufte found, PowerPoint users typically produce charts with only 12 elements. Ultimately, Tufte concluded, PowerPoint is infused with ''an attitude of commercialism that turns everything into a sales pitch.''

(btw, these are the same problems we face writing for the web....

update 2

I'm pulling J.D.'s comment up front:

Mr. Tufte butters his bread by analyzing, among other things, the contexts under which information is presented. He is likely correct in his critique of Powerpoint as a tool for information sharing.

For anyone, even the TIMES, to suggest or insinuate that this technology was responsible for seven deaths is, I think, irresponsible.

I agree, and I certainly don't want to be seen to be blaming PowerPoint for 7 deaths.

I'll also add that the 'PowerPoint makes you stupid' heading is a joke! (That's the heading used by the blog I mentioned.) I don't remotely feel that bulleted points make people stupid, and as a matter of fact I do feel that bulleted points frequently make people much more clear.

It would be extremely difficult to make sense on the web without them.

screenplays are structure, fyi

Back when I first started out, I thought writing was good sentences if you were writing nonfiction, and good dialogue if you were writing fiction.

Wrong.

SCREENPLAYS ARE STRUCTURE," shouts William Goldman in Adventures in the Screen Trade. "The essential opening labor a screenwriter must execute is, of course, deciding what the proper structure should be for the particular screenplay you are writing."

This, he believes, is "the single most important lesson to be learned about writing for films... Yes, nifty dialog helps one hell of a lot; sure, it's nice if you can bring your characters to life. But you can have terrific characters spouting just swell talk to each other, and if the structure is unsound, forget it."

Real Craft

He's right.

And, on the same page, here's Syd Field:

In The Screenwriter's Workbook, Syd Field seconds Goldman. "Structure is the most important element in the screenplay. It is the force that holds everything together; it is the skeleton, the spine, the foundation."

People look at Saxon Math and think it's prosaic, obvious, behaviorist.

But what's brilliant about Saxon is mostly invisible.

It's the structure.

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RoadsideFragmentation 31 Jul 2005 - 02:01 CatherineJohnson

Check out the cool image J.D. found to illustration fragmentation:





I feel a traffic mishap coming on just looking at this.

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DontTeachInAMonotonePart2 31 Jul 2005 - 02:29 CarolynJohnston

This is the conclusion of a two-part post. The first part is here.

The equivalent of marquee lights in textbooks would actually be, from the children's perspective, time spent on a topic. We need more time and lots of practice when learning the mixture problems relative to learning the statistical markers. The teacher has to convey the relative importance of topics to the children on a daily basis; kids can't learn at uniform power all the time any more than adults can.

For the teachers, who generally do not see elementary math as a seamless whole and need real support from their texts to start with, the equivalent of marquee lights would be clear instructions in the teacher's material as to what is critical and what isn't.

If the kids aren't getting story problems, the teachers should be told: stick with them, spend more time on them, sacrifice the mode and median stuff. They need to be told it's okay to sacrifice the less important units if the more important ones need more time, that they don't have to rush to stuff every minor and major bit of material in the book into the children in order to succeed in their quest.

Another problem; US curricula tend to impose demands for mastery uniformly. In mastery teaching classrooms, every topic is drilled to mastery before moving on; in classrooms that adopt the 'spiraling approach', mastery is never required in any unit (if a teacher tells you, after Junior fails Unit 6, "don't worry, it'll be covered again later", then watch out -- you've got one of these spiraling curricula).

The truth is that some topics need mastery on the spot, before moving on; some topics don't really require mastery at all. A teacher will need to be told which are which, and to impose appropriate requirements.

One way to achieve this is to leave the less critical material out of textbooks entirely. Textbooks in other countries do this pretty frequently; Singapore texts, for example, focus very heavily on computation, story problems, and the techniques needed to solve them at an incredibly early age. This is probably why they get such awesome performance, and why we are amazed at the sophistication of those beginner-level problems.

Here are charts from one of Catherine's Pantheon articles (pdf file) that tell the whole story very clearly. The charts show topics typically covered, versus the grades in which they are covered. Look at this top one, which describes topics covered in the US: 14 topics are covered in first grade, 15 in second, and 18 in third.





Now look at the same grades from countries that scored high on the TIMSS: 3 topics are covered in depth in first and second grades, and 7 in the third. That's an incredible difference. The trend toward focusing on topics continues into the later grades, but the difference is less startling.





What exactly are the extra topics we cover in the US in early grades? The chart says that we do lots of estimating, and geometry and geometric transformations, and (here's what gets me) we cover the heck out of pattern detection and functions . No wonder Ben and Colin are aces at pattern-detection; they've been covering it every year since first grade! But it's ironic that, with all the coverage of functions, kids still come out of K-12 with no real feeling for what functions are.

To finish, I offer two thoughts for curriculum-builders in the early grades: we could either leave out those extra topics completely, or write lively and interesting texts and teacher's guides that really tell readers (parents, children, and teachers) very clearly what's critical and what isn't.

Regarding leaving out material (which would be the optimal solution in my opinion), JdFisher reminds me that there's a very good reason why all those topics are in there: to meet various states' demands for coverage of material.

This is probably, in fact, why the U.S. chart looks the way it does, with everything being covered everywhere: to sell a lot of books in all the states, a textbook writer has to include state-mandated material from all the states. This is actually a good reason, I think, to consider adopting a federal-level curriculum.

But if the teachers don't know what can reasonably be left out, they'll go through slavishly every year and cover everything in the book in a monotone.

Finally, I think there is a gold mine of information in those two charts about what American curriculum builders could be doing better. Look for me to be doing more digging.


Don't teach in a monotone, part 1

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CognitiveScienceTreasureTrove 31 Jul 2005 - 20:11 CatherineJohnson



What a find!

10 classic articles from the journal Cognitive Science.

I'm starting with number 1:

1. Johnson-Laird, P. N. (1980). Mental models in cognitive science. Cognitive Science, 4, 71-115.

This article postulates that mental models differ from visual images and from propositional representations, and it presents evidence that corroborates the differences. It argues that reasoners use propositional representations of, say, spatial descriptions to construct mental models. It also argues that mental models rather than formal logic underlie syllogistic inference, e.g., some of the parents are drivers, all of the drivers are scientists, therefore, some of the parents are scientists. The article was the first in a journal to present a case for mental models as the end result of comprehension and as the starting point of deductive reasoning. This idea led to many subsequent investigations (see the mental models Website).

I've been struggling with the question of:

What is conceptual understanding of mathematics, anyway?

For some reason, I've gotten the sense that conceptual understanding = visual representation. This notion has tripped me up, because I'm pretty much maxed out on visual models of mathematical knowledge at this point. It's easy to understand addition and subtraction by looking at a visual model, but I don't readily grasp the one visual model I've seen of the multiplication of fractions, and I can't even imagine a visual model for multiplying-by-the-reciprocal (although Dan K has left a possibility in a Comments thread).

fyi, I was just talking to Carolyn about this, and it turns out she doesn't have a visual representation inside her head of dividing a fraction by multiplying by the reciprocal.

I can't tell you how liberating that is!

If Carolyn isn't walking around with a multiply-by-the-reciprocal picture inside her head, I certainly am not going to devote one second more trying to come up with one myself. Forget it!

Anyway, I've been feeling that it's time for me to 'move on,' and that in fact I am moving on. But I haven't trusted the feeling, because I'm stuck on, & stumped by, visuals.

Reading this abstract, I had a moment of recognition. Mental representations sound like what I've been reaching for, and like what I'm starting to develop. (And the new 'knowledge' I'm developing is not at all like a 'proposition' from which I can 'deduce' follow-up truths & principles. It's much more.....'holistic' than that?? Is that the word? I don't know, but early on Carolyn told me that her knowledge of math is a 'seamless whole,' and I'm starting to have a bit of that feeling. Some kind of 'knowing' seems to be taking form inside my mind that isn't a picture and isn't a logical proposition, either.)

So I'm looking forward to reading this article. I suspect it's going to give me confidence that whatever it is that's taking shape inside my head is real; that I'm making progress, not just stalling out in 5th grade math.

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ChannellingJohnDewey 31 Jul 2005 - 21:43 CatherineJohnson

Why is constructivism here?

Good question. I know even less about history than I do about math, but fortunately I'm married to a historian, which comes in handy.

Constructivism is here, I gather, thanks to Jean Jacques Rousseau. One of these days I'll make Ed sit down and write a short explanation of just how & why, exactly, Rousseau managed to come up with an idea that is running math education in the New World today, 150 years later.





More recently, in this country, the Prime Mover was John Dewey:

Only by wrestling with the conditions of the problem at hand, seeking and finding his own solution (not in isolation but in correspondence with the teacher and other pupils) does one learn."
John Dewey, How We Think, 1910

Sound familiar?

UPDATE 11-20-2006: No! The prime mover was not John Dewey!

John Dewey was the loser, not the winner!

Ellen Lagemann explains ... with admirable precision: ‘I have often argued to students, only in part to be perverse, that one cannot understand the history of education in the United States during the twentieth century unless one realizes that Edward L. Thorndike won and John Dewey lost.’

source:
Progressivism, Schools and Schools of Education: An American Romance (pdf file)
David F. Labaree
Paedagogica Historica,
Vol. 41, Nos. 1&2, February 2005, pp. 275–288

It's always worse than you think.

And here is an Inquiry Circle, which I found at the Inquiry Page!





Wait.

Isn't it supposed to be a spiral?

update

I may have located our Opposite Number.

from The Inquiry Page:

The Inquiry Page is more than a website. It's a dynamic virtual community where inquiry-based education can be discussed, resources and experiences shared, and innovative approaches explored in a collaborative environment.

Here you can search a growing database of inquiry units, and you can also build your own inquiry units. You can see pictures of inquiry-based activities and learn more about some of our partners who use inquiry methods. Learn how to assess and evaluate inquiry-based education or look for more inquiry resources to support what you're doing. Or you can simply find out more about what inquiry and The Inquiry Page are all about.

That's a whole lot of Inquiry. So right off the bat, they've got us beat, because there's nobody here at KTM with the chops to squeeze 8 'direct instructions' into 4 sentences.

Based on John Dewey's philosophy that education begins with the curiosity of the learner, we use a spiral path of inquiry: asking questions, investigating solutions, creating new knowledge as we gather information, discussing our discoveries and experiences, and reflecting on our new-found knowledge.

Problem identified.

The reason I have screaming, yelling, crying, and playing out the clock here at home is that I've been using direct instruction. If I'd been using a spiral path of inquiry we'd be having a gas.

Each step in this process naturally leads to the next: inspiring new questions, investigations, and opportunities for authentic "teachable moments."

I'll just bet.


johndewey

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WichitaBoyOnMath 31 Jul 2005 - 22:20 CatherineJohnson

We have an embarrassment of riches! At least 2 great comments from WichitaBoy, and Ed sat down and wrote out his constructivism-as-psychoanalysis thoughts, too.

Here's one of WichitaBoy's observations:

"Writing is organizing." Now there's a great thought I can take to the bank.

Here's one back for you: professional mathematics is organizing. You have vague thoughts, you notice a vague pattern, and you try to organize your thoughts, to nail down the pattern, to really clarify what's going on beneath the hood. When you've nailed it completely, when you understand with perfect perspicacity the essence of the pattern, then you've got a proof of a new theorem. If you've really organized it, you've got a theorem that goes in the "Book of God".

There's this, too, in response to my saying that what is brilliant about Saxon Math--the structure--is largely invisible:

Read Confucius or Socrates. The ideal teacher should be able to fade into the background like the Cheshire cat. And so with the ideal textbook.

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ConstructivismAsPsychoanalysis 31 Jul 2005 - 22:26 CatherineJohnson

from Ed Berenson:

The radical constructivist notion that students have to develop their own knowledge seems analogous to the psychoanalysts’ conceit that patients have to arrive at insights about themselves, guided only lightly by the doctor. Analysts and therapists routinely say that they can’t just tell you what’s wrong, because what’s told won’t sink in. You need to think and feel your own way into the heart of your neuroses. 

There is, of course, a measure of truth in this idea: if your problem is denial, and the analyst tells you as much up front, you will, well, deny it.  But beyond obvious things like this, you don’t have to be a cynic to see how self-serving such an idea is. It takes longer to come to insights on your own, and that suits the doctor, paid by the hour, just fine. 

Educationists have brought this therapeutic model into the classroom. Teachers can’t just give students information, so the constructivists say; kids have to develop insights on their own. Only then will they genuinely know that material at hand. But arriving at insights takes a long time, and often those insights won’t surface by themselves. To wait for them is to draw out the process, preventing kids from moving on. 

It would be sensible for psychoanalysts to tell the patients certain things when they’re ready to hear them, without having to wait for patients to figure them out. The same holds for teachers and students. Why not present material in a way that builds on prior knowledge so that students will learn new ideas and information when they’re ready? 

One quality of a good teacher is to know when kids are poised to progress. Such readiness happens a lot sooner than the self-generated knowledge that constructivists falsely believe to be the only genuine kind.  

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ConstructivistChecklist 31 Jul 2005 - 23:03 CatherineJohnson





created by Elizabeth Murphy, professor, writer, researcher, and inquirer.
Constructivism: From Philosophy to Practice

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