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From Mikusinski (and Adrian) With Love

You can have no end of fun poking little holes in my explanation of how multiplication is defined using induction.

I explained in a post before that, that I left out a number of steps, and I certainly rounded off some edges in the interest of trying to make it somewhat realistic. In general, though, I think the explanation was sound.

If you are interested, this is what I was working with, from Mikusinski (emphasis in original; also, I'm using the phrase "is an element of," rather than the symbol, because that symbol really messes with my layout):

The result of multiplication of two natural numbers x and y is called the product of x and y and is denoted by xy (or sometimes by x &bull y). The product of x and y is defined by induction as follows:

(&bull) x &bull 1 = x;
(&bull &bull) x(y + 1) = (xy) + x.

Let X be the set of all natural numbers y such that the product xy is uniquely defined for every natural number x. By (&bull) we see that 1 is an element of X. Consider some y such that y is an element of X. Then xy is uniquely defined and so is (xy) + x because of the uniqueness of addition. Thus, by (&bull &bull), the product x(y + 1) is uniquely defined, which proves that y + 1 is an element of X. By the Induction Principle the set X contains all natural numbers, so the product xy is uniquely defined for all pairs of natural numbers.

I do have at least one more post up my sleeve about all of this. And the reason I do is because I love talking and writing about this kind of stuff.

Hey, I even enjoy a good snark, even when I disagree with it (especially when it employs an analogy; I love analogies). This is from Adrian, defending the sacred unity of repeated addition and multiplication:

There is nothing more exactly identical in all of mathematics other than perhaps something like "0=0". Saying otherwise would be like saying that

"A triangle is not a three sided polygon -- a triangle is a triangle. That it has three sides quickly follows and 'it works', but it isn't part of the definition, and it is damaging and just false to say that 'a triangle has three sides'."

Preposterous.


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Transformations in the Coordinate Plane, I

I have mentioned here a couple of times how lower-level representations can be valuable both in and of themselves and as teaching aids.

For example, repeated addition (a lower-level representation of multiplication) can be used to find a product when a benchmark fact is known. And even something as simple as knowing how to count on your fingers (a lower-level representation of addition) is not scrubbed away suddenly when you learn how to add. Such knowledge can still have some value.

Intelligent adults still maintain the ability to count on their fingers—counting time intervals that cross over the hour (or the month of January), counting inclusive ranges (e.g., the range "8-12 years old" covers 5 ages, not 4), teaching their children addition and subtraction, etc.—when they lack a practiced algorithm to do otherwise; when they forget, under pressure, a method they have learned; or when they need to communicate using the less-efficient method.

One kind of lower-level representation that is pervasive in upper-elementary and middle-school teaching is the presentation of transformations in the coordinate plane as a kind of visuo-spatial training.

Take a look at the grid below. Students in sixth, seventh, and then eighth grade might be asked to name the coordinates of the point shown after it is rotated 90 degrees counterclockwise about the origin.

However, no "math" is used. Students are, as far as I've seen, expected to arrive at the correct answer by rotating the image mentally—or by drawing. I've yet to see a lesson in a purely middle-school text explaining how this mental rotation is to be done in any kind of systematic way.

grid1

But there is a "higher-level," mathematical way of figuring out these kinds of transformations that, because of education's obsessive fixation on—well, I'll leave that for my next post.

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Tim Russert

I love politics. And—I’ve alluded to this before--the reason I love politics is that politics is text.

One of my simple joys as a husband and a father of two sons and one daughter (oh, and a political junkie) was to come home from church every Sunday, curl up on the couch with a Dr. Pepper, and watch Tim Russert on the ol’ DVR.

I suppose I should say that I watched Meet the Press every Sunday, but that’s really not the truth. I watched Tim Russert.

I connected with his optimism, his stewardship, his work-ethic, his humor, his love of family, and especially his love of the work he did. Sometimes the connection was weak, sometimes strong, but I always thought of him as my connection to the political world—and in some respects to the real world--and I think many fathers, sons, wives, daughters, and soldiers thought the same, even if they couldn’t express it when he passed.

If it's Sunday, it's Meet the Press. But it won't ever be the same without Tim. I'm missing him terribly.

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Get-Off-My-Lawn Stupid

This is what an idiot sounds like. You just repeat the same thing over and over again (i.e., Saxon! Discovery! Singapore! TERC! Mastery! Appeasement!).


Until someone calls you on it. Then you look like what you are--an idiot. Take a look at the exchange starting at 2:14. Can we have more people speaking up please? These folks aren't really that scary, trust me, no matter how much they puff up their chests and talk about battles.

Lest I forget . . . If you can't or don't want to say anything, maybe you should just shutup. Hmm?


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Your Daily Dose of "Rationality"

Mike Goldenberg suspects that "anti-reformers" use the acronym TERC to evoke fears among parents that "Investigations" was created by Muslims.

Virtually any time INVESTIGATIONS is mentioned by a vehement anti-reformer, "TERC" is substituted for the actual name of the program, as if somehow the authors and publishers hadn't given it an actual name. Of course, there are at least two factors at work here, on my view. The more obvious one is sheer laziness. It's so much faster and easier to type "TERC," after all. And maybe some of these critics aren't able to spell "Investigations." But the more subtle effect, one that may be unconscious but which is consistent with people who call any reform math program, method, text, author, or advocate "fuzzy," and a host of similarly prejudicial epithets (and yes, I'm well aware that I return their fire in kind. However, I didn't start the mud-slinging, cheap name-calling, etc. The Mathematically Correct page that lists a host of such names was up before I'd ever heard of them. You can't make this stuff up), is that in the ear of the average parent, this math program sounds like "Turk Math." Not that any political conservative would want to trade on American fear and suspicion of Muslims, of course.

The rest of his article is a bunch of codswallop as well.

I will give him this, though. His writing does indeed balance out the get-off-my-lawn crazy from the fictitious "other side" of the debate, as represented by the cult crazy-heads at KTM and elsewhere.

All very entertaining at least. Not getting us anywhere, but entertaining.

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Scientific Illiteracy: A Core Family Value

A bill that backers touted as a way to spark freewheeling public school classroom discussions on evolution has lost much of its punch, the leader of the group that first promoted the measure said.

Gene Mills, executive director of the Louisiana Family Forum, said he has only lukewarm support for the legislation after it was changed and approved on Thursday by the Senate Education Committee.

State Sen. Ben Nevers, D-Bogalusa and sponsor of the plan, denied that his proposal suffered a setback when a last-minute committee amendment was added.

“It didn’t change the intent of the bill,” Nevers said.

The legislation is Senate Bill 561. It next faces action in the full Senate.

The bill, which was initially sought by Mills’ group, originally called for steps to widen classroom discussions on biological evolution, global warming, human cloning and other topics.

Backers said such a law is needed because teachers are afraid to stray from what textbooks say about evolution and other science topics.

Critics contend it would inject biblical talks into public schools.

Moments before the Senate committee approved the bill, the panel adopted an amendment suggested by opponents of the legislation.

It stripped references to evolution, global warming and other subjects from the proposal.

Instead, the proposed law calls for more general changes in science classes.

It now calls for state assistance so that students could pursue “critical thinking skills, logical analysis and open and objective discussions of scientific theories.”

(Here.)

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A History-Making Campaign

Excellent . . .


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Watch Out, MSM!

Bless his heart. Josh Marshall simply yearns to be a dimwit talking-head pundit on cable news so that he can stop actually working.

9:09 PM ... Obama's making a good point on the capital gains tax. But he's making it in a very bedraggled, painful, drawn out way. This is not good at all. All the right points are there but just not put well ... Charlie Gibson's 'history' of the capital gains tax? Please. There's a good answer to that. But he didn't seem to have it.

Oh, please, Josh, give us the answer . . . You must be hiding it from us. Here's an idea: Put it on TPMTv so we may all absorb its beauty both aurally and kinesthetically.

9:16 PM ... Did someone tell Charlie Gibson that he knows something about economics? There are a heck of a lot of people who make over $97,000 a year? Really? I think like 12% of the population makes more than $100,000 a year. And his capital gains point is a canard.

Did someone tell Josh Marshall that he is gifted at estimation? Twelve percent of 300 million is 36 million. Doesn't that sound like a "heck of a lot of people" to you?

9:24 PM ... I was disappointed that Charlie Gibson seems to spout off right-wing bromides as established facts. I was even more disappointed that Obama didn't seem able to knock them down.

LOL! Oh, boo hoo!

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Edu-News (04.14.2008)

Crestwood, Kentucky:

An Oldham County mother has filed a complaint with the state after learning that Crestwood Elementary officials put her 8-year-old autistic son in a small, empty room nearly 80 times last fall because of his behavior -- sometimes locking him in.

There's so much in the article to be mad about. By the time I finished reading it, I got the sense that the argument was centered around whether or not this "time-out room" had a lock on it and whether or not it was well-lit and carpeted. Maybe I'm crazy, but shouldn't the argument be about why this school has something called a "time-out room" in the first place?

Some video below.


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Edu-News (04.04.2008)

We've seen something similar to this before in Florida. Now we have two events to keep our eyes on.

Des Moines, Iowa:

Iowa has failed to provide public school students with an adequate education as required by the state's constitution and code, a lawsuit filed in Polk County District Court late Thursday alleges.

Three Iowa families filed the lawsuit against Gov. Chet Culver and Judy Jeffrey, Iowa Department of Education director.

An informative article.

And everyone's talking about the NAEP Writing results.

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Edu-News (03.28.2008)

There was a flurry of depressing education stories yesterday and today. So, by all means, check out this one first. Or wait, maybe it's actually more depressing than the others (emphasis is mine):

Salt Lake City, Utah:

As someone who has had sex with about 4,000 women in an acting career spanning more than 1,700 pornographic films, you could easily joke that Ron Jeremy likes to score . . . .

"Nobody can defend porn better than me, because I have a teaching license," said Jeremy, referring to his master's degree in special education from Queens College in New York. "Pornography is older than all of us, and it will still be around when we're all dead and buried. We're all voyeuristic by nature. If it goes out of style, fine. I'll take up tennis. But right now people enjoy it."

Tennis, yes. Please let it be tennis.

In other news, we have a sickout in New Orleans, a strike threat in Nashua, and possible position and program cuts in Mt. Olive, New Jersey.

And there's this craziness out of North Carolina (via The Chalkboard):

North Carolina teachers have used personal days to attend funerals and court hearings, assist friends during medical procedures and participate in training — a privilege that costs them $50 a day.

Teachers are now asking legislators to end the decades-old requirement that they pay a fee for taking those days off.


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Edu-News (03.25.2008)

Memphis, Tennessee:

The [Blue Ribbon] behavior plan has brought alternative schools for students who misbehave and more in-school suspensions and counseling. Most speakers have been critical of Blue Ribbon, saying it does not provide students with strong and rapid consequences for bad behavior.

Several have begged the board to consider bringing corporal punishment back into schools. The board abolished the practice starting with the 2005-06 school year . . . .

Board member Kenneth Whalum Jr., who lobbied the board for months before the town hall meetings were adopted, thanked the community Monday night for its support and said he would continue to fight to reinstate corporal punishment.

Emphasis is mine, but you can have some too. Full article here.

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Edu-News (03.20.2008)

Palm Beach County, Florida:

I would imagine that the majority of the people who will likely be encouraged by this news have seldom cheered for the group making it.

Calling Palm Beach County's high school graduation rate "shamefully low," the American Civil Liberties Union on Tuesday sued local educators to churn out more diplomas.

In a class-action lawsuit filed on behalf of parents and students, the ACLU accused the Palm Beach County School Board and Superintendent Art Johnson of failing to provide students with a high-quality education guaranteed under the state constitution.

The article also mentions graduation-rate disparities between different ethnic groups in Palm Beach County as a reason for the suit. Read the whole article here.

And be sure to take a look at Superintendent Johnson's response, in which he seems to sarcastically express support for the lawsuit. Ken's going to love this part:

The superintendent, who met with ACLU leaders a year ago about the issue, has suggested that the civil rights organization instead take aim at state legislators for insufficient classroom funding.

"I'm for the ACLU action if it focuses on bringing more resources to public education," Johnson said last year. "I wouldn't take it personally. I'd support any effort that would bring more resources to the district."


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Edu-News (03.19.2008)

Naperville, Illinois:

Just one item this afternoon, which bears striking similarities to yesterday's story out of New Jersey:

[Linda] Kane, who took over the Central Times in 1989, developed the monthly newspaper into one of the best in the U.S., earning nine National Scholastic Press Association Pacemakers. That award is given annually to the top 20 to 25 high school papers in the nation.

She also was known for being candid. But she became a little too outspoken for Naperville Unit School District 203 administrators early this month when she publicly criticized her principal after the newspaper published three controversial pieces Feb. 28. On Monday, after Kane declined administrators' request that she resign, they fired her as newspaper adviser.

The full story, along with links to the three "controversial" pieces can be found here.

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Taking Care of Business

First off, if you have not already heard of Erik Seligman's wonderful Math Mutations podcast, I strongly recommend you bookmark it.

Presenting interesting mathematics topics in audio only (though transcripts for each podcast are provided on the site) is certainly no small task, I would think. And Seligman does a wonderful job—each podcast is delightfully brief, intriguing, entertaining, and intelligent.

Secondly, I would like to extend a long overdue (nearly a year overdue) public thank-you to both Joanne Jacobs and the folks at NCLBlog for giving me some free advertising for a presentation I gave last year at NCTM's national conference in Atlanta. The topic was parent involvment, and I received a pretty good crowd (about 60 people) in spite of having to compete with more popular, more mathematical topics in the same time slot.

In the presentation, I attempted to combine a theoretical framework for understanding parent involvement with current research outside of that framework—all while not directly addressing either the theory or many of the nuances of the research.

It was, to say the least, a learning experience. I was disappointed, though not really surprised, to see that, despite my implicit goal of reframing the issue of parent involvement to focus on how schools could think about and reach out to parents, the teachers in the audience were much more interested in talking about ways to simply get parents to the school.

I think every edu-blogger should have the opportunity to have one or more humbling experiences similar to that.

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I'm Beginning to Think the Same Thing

From a brief but cutting piece by former Clinton appointee M.J. Rosenberg:

The race genie Ferraro and the other HRC [Hillary Rodham Clinton] deputies have let out of the bottle will not get Clinton nominated. That can't happen nor, obviously, can she be elected. But it sure can defeat Obama in November.

Is that intentional? Because I'm beginning to think that the one intolerable outcome for the HRC camp is the election of Obama. If that happens, Obama is the leader of our party and our country. If McCain wins, Bill Clinton remains the titular head of the party. Maybe this is all about not letting history turn the page.

There is a saying: don't let your old age disgrace your youth. This campaign is disgracing a fine Democratic administration. Sad.


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Watch and Listen Now!

I love politics. I really, truly do. And these clips are worth watching. The question for me is, Are we voting for a Senator-in-Chief or a President?

The first is that Republican ad put out by top notch Republican-imitator Hillary Clinton, trying to scare the living shit out of people. (I'm numb to that baloney, but to be fair, the ad seems to work well in Ohio, that one swing state that apparently can't make up its mind whether or not it's being screwed by Republicans. They should visit the people of Kansas, who, by and large, buy into that crap.)


Booogiee Wooogie Wooogie Wooooo! Next thing she's going to tell me is that if I question her, I'm going to see a mushroom cloud outside my bedroom window. Thanks so much; already had eight monotonous years of that tripe.

Here's Obama's response. A response to what he calls "a legitimate question":


And, of course, the four-years-younger and, apparently, wiser Bill Clinton weighs in:


And, by the by, how much time do you need to answer this question, ladies?

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My Guy

That's it, I've got it . . .


For months, there's been a tickle in my brain about the whole Hillary vs. Obama thing. Like there's some kind of cultural analogy that applies.

Tonight, I finally figured it out.

And, by the by, I'm still a reader and a "fan" of TPM, but it's off the blogroll. It was originally included because, from my read, it was relatively fair--what I thought was a good working example of someone dealing with the "text" of politics mostly fairly. I don't know what happened, but the fairness of TPM has been declining quite obviously for more than a year.

This was the final straw. I read it and knew it was sarcasm, but TPM has, last I checked, nearly a million readers per month who don't get it and will soak it in like it's important.

And, indeed, I was right. TPM got a lot of what I presume were negative responses to the post. But look at how TPM spins it. Not being helpful to the Democratic party's image as a collection of whiny, arrogant, pedants, Marshall responds like a whiny, arrogant, pedant:

A distressingly large number of readers weren't clear that this post was satire. But as long as we inhabit the same universe as Tim Russert, can we blame them?

"Distressingly," meaning, "I'm so distressed at how all my readers can be so stupid."

Sounds like a familiar strategy for addressing mistakes:

No sooner does Mrs. Clinton lose a state than her campaign belittles its voters as unrepresentative of the country.

And, of course, those same stupid voters--those whose intelligence doesn't match that of an upstart journalist--are, according to Marshall, all swayed by Tim Russert. They have no choice but to construe the world according to his "tough" questioning and are, faux-sadly, innocent victims.

Shut up.

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A Mathematical Magic Trick

I will soon get back into some serious writing (I think). In the meantime, learn this card trick and amaze your friends/students/children.

STEP 1: Shuffle a full deck of 52 cards (no Jokers). As you shuffle, ask your volunteer how many cards are in a full deck (52). Then ask, "What's half of 52?" (26)

STEP 2: Hold the deck in your hands with all cards face up. Deal 26 cards face up, one on top of the other. You can count while you do this, or you can have your volunteer count. The most important part of this step, however, is to remember the seventh card (face/number and suit). When you finish this step, you should have two sets of cards—the cards that are in your hand and the face-up pile that contains 26 cards, including that key card (the seventh card you dealt).

STEP 3: Without putting down the cards that are in your hand, take the face-up pile, turn it over so that all the cards are facedown, and set it aside. Turn the cards in your hand over so that they are all facedown in your hand.

You'll probably want to explain the next step to your volunteer before proceeding. However, such an explanation is not essential. What you will be doing is creating three "columns" of cards. Each column will have a face-up card at the top and anywhere from zero to nine facedown cards under it. Face cards are worth 10, Aces are worth 1, and the number cards are worth their face value (2s are each worth 2, 6s are each worth 6, etc.). For each column, you will be "making tens" using this equation:

Face-up Card Value + Number of Facedown Cards = 10

STEP 4: Deal the top card in your hand face up. The value of this card is the first addend in the equation above. This value will determine the number of facedown cards you deal beneath it. For example, if you deal an 8 face up, then you will deal 2 facedown cards beneath it (8 + 2 = 10). If you deal an Ace face up, then you will deal 9 facedown cards beneath it (1 + 9 = 10). And if you deal a 10 or a face card face up, then you will not deal any facedown cards beneath it (10 + 0 = 10). The image below shows an example of what your first column might look like. Note that it is not important to keep the facedown cards separate, but it is important to keep the face-up card visible.

8_first_column

STEP 5: Deal the next card in your hand face up to create a second column. Again, use the equation above to determine the number of facedown cards to deal beneath it. Repeat this process to create the third column. The image below shows what your three columns could look like after you've finished. To create the columns shown below, I dealt an 8 face up to start the first column, then 2 facedown cards beneath it, a King face up for the second column, an Ace face up to start the third column, then 9 facedown cards beneath it.

three_columns

STEP 6: Take the remaining cards in your hand and place them facedown on top of the pile that you set aside in Step 3. Pick up this pile and keep it facedown in your hand. Ask your volunteer to add up the face-up values in your three columns. In the example above, the sum would be 19 (8 + Ace + King = 8 + 1 + 10 = 19).

STEP 7: Deal the cards in your hand facedown, one on top of the other, and count the cards until you reach the sum found in Step 6. The last card you deal is that magic seventh card that you remembered way back in Step 2. In the example above, the 19th card would be the card that you remembered. Announce to your volunteer what this card is before turning it over.

I'll let you enjoy explaining the math behind this trick. And bonus points for finding the big flaw in this trick. And bonus bonus points if you can figure out a way to overcome said flaw.

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A Very Very Very Fine House

Our house is on the market. Some pics below. It's amazing how quickly you can start the process of selling a home.

front_medium

The front, obviously. We have bermuda grass, which goes dormant from about the middle of December to the middle of March in Central Texas. Luckily for us, the photographer convinced us to put in a different grass just for this picture. I believe the species is called photoshop green.


study_medium

Ah, yes. The comfy nook from which I do work, read my blogroll, and launch my immature vitriol into the vast empty-headedness of the so-called edusphere.


bfast2_medium

The dining, um, area. Baby seat removed for the picture. Designed that centerpiece myself, I did. Not a good shot of it.


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Living room. With the exception of the palm and the candles on top of the entertainment unit, everything in that room is older than the house itself.


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Master bedroom. The king sleigh bed is so sweet. The dresser with valets is an antique (I don't know what officially qualifies for "antique," but it's antique-ish) that I picked up from a colleague I worked with at Houghton in New Jersey. It belonged to his grandmother.


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This is the baby's room. The thing under the window is my eldest son's "train table." The red drawer contains all the track pieces, etc.


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This is the kids' loft area, right outside the master bedroom and the kids' room (which was not shown here). I think that little table with the white top used to be our kitchen table once upon a time. I sawed the legs off and put it upstairs for the kids. Easy clean up.

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