Gifted Students Left Behind Too?

My son has a presentation tonight—a project for his QUEST class—so this article seemed pertinent.

Nationally, about 3 million kindergarten through 12th-grade students are identified as gifted, but 80% of them do not receive specialized instruction, experts say. Studies have found that 5% to 20% of students who drop out are gifted.

There is no federal law mandating special programs for gifted children, though many educators argue that these students -- whose curiosity and creativity often coexist with emotional and social problems -- deserve the same status as those with special needs.

I was in a gifted program throughout elementary school, and I hated it. The only thing I wanted to do in, say, fourth grade was learn about fifth- and sixth- and seventh-grade stuff. But the only thing I ever did was "explore" the fourth-grade stuff by being forced to spend my time doing cut-and-paste "projects." I was already bored with the material. How was making a diorama about it going to change that?

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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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Experimental Probability, Part I

Experimental probability is usually introduced to students somewhere between the fifth and seventh grades and could be defined as follows:

In the context of a probability experiment, the experimental probability of an event is the ratio of the observed outcomes in the event to the total number of trials in the experiment.

Suppose a person rolls a number cube (a die) 50 times. In that case, rolling the die is the probability experiment, and the total number of trials in the experiment is 50. The table below shows possible results of that probability experiment:


So, consider the event "rolling an even number." According to the probability experiment results shown above, there were 24 outcomes observed in this event—the person rolled a two 7 times, a four 8 times, and a six 9 times: 7 + 8 + 9 = 24. Thus, one could say that the experimental probability of the event "rolling an even number" is 24/50, or 12/25, or 0.48, or 48%.

Now consider the event "rolling a 2." According to the probability experiment results shown above, there were 7 outcomes observed in this event—the person rolled a two 7 times. Thus, one could say that the experimental probability of the event "rolling a 2" is 7/50, or 0.14, or 14%.

When it is introduced, experimental probability is usually distinguished from theoretical probability, which could be defined as follows:

In the context of a probability experiment, the theoretical probability of an event is the ratio of the number of possible outcomes in the event to the total number of possible outcomes in the experiment.

Consider again the probability experiment described above—rolling a number cube, or die—and consider the first event we discussed above, "rolling an even number." The number of possible outcomes in that event is 3—there are 3 ways to roll an even number: by rolling a 2, 4, or 6. And the total number of possible outcomes in the experiment is 6—the person could roll a 1, 2, 3, 4, 5, or 6. Thus, one could say that the theoretical probability of the event "rolling an even number" is 3/6, or ½, or 0.50, or 50%.

Now consider again the event "rolling a 2." The number of possible outcomes in that event is 1—there is 1 way to roll a 2—and the total number of possible outcomes in the experiment is 6. Thus, one could say that the theoretical probability of the event "rolling a 2" is 1/6, or about 16.7%.

The key differences to note, at least for this post, between the concepts of experimental and theoretical probability as they are introduced to students are that where experimental probability employs "number of observed outcomes in an event," theoretical probability employs "number of possible outcomes in an event," and where experimental probability uses "total number of trials in an experiment," theoretical probability uses "total number of possible outcomes in an experiment." Obviously, these different definitions give us different probabilities for the same events. In the context of the probability experiment described above, the experimental probability of rolling an even number is 48%, whereas the theoretical probability of the same event is 50%. The experimental probability of rolling a 2 is 14%, whereas the theoretical probability of the same event is about 16.7%.

The Argument

Given that background information, consider the following argument put to me a few months ago, in reference to a word problem like the following:

A bag contains ten tiles numbered 1–10. Megan chooses a tile from the bag, records the number on it, and then replaces it seven times. She chooses the number 4 twice, the number 6 once, the number 2 three times, and the number 5 once. Based on these results, what is the experimental probability that the next number Megan chooses will be a 2?

Quite simply, the argument was this: Experimental probability can not be used to discuss the probability of a future event such as "the next number Megan chooses" because experimental probability uses "total number of trials in an experiment," and, for future events, there are no trials to consider.

Well, such an idea contradicts at least one published source:

Megan plays on the high school's varsity softball team. She has been at bat 35 times this season. She gets a hit 9 times. What is the experimental probability that she gets a hit her next time at bat?

So, what do you think?

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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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Dichotomies Are the Reason Daddy Drinks

This happens to me way too often. I stumble across some education "research," read the abstract, print it, and quickly become bored with it.

Then I feel guilty for wasting paper, struggle through the entire piece (PDF), and, usually, find some ideas that seem to be worth further exploration:

There appears to be an inclination within the education community to dichotomise and an associated tendency to (i) ignore the connectedness of the dichotomous categories, and (ii) on occasion, to privilege one category while denigrating the other . . . .

This paper addresses five of these dichotomies: Teaching and Learning; Abstract and Contextualized mathematical activity; Teacher-Centred and Student-Centred classrooms; the teacher's contemporary dilemma: to Tell or Not to Tell; and the related issue for students: to Listen or to Speak. (376)

[The Teaching and Learning dichotomy] is a particularly insidious consequence of the constraints that language (and the English language, in particular) imposes on our theorizing . . . . this is particularly evident in the various translations of Vygotsky, in which the Russian word 'obuchenie' has been represented as either teaching or learning in different translations (Clarke, 2001). The integration of teaching and learning as components of a jointly enacted single activity also occurs in several other languages, including Chinese, Japanese and Dutch. (378)

Differences in the use of abstract and contextualised tasks seem strongly connected to a perceived need in Western classrooms to present mathematics as relevant to students. (379)

In the Swedish classroom, the students demanded that the teacher justify the relevance of what was being taught . . . . Despite the teacher's efforts, students were outspoken in their lack of belief in the relevance of the mathematics they were studying. . . .

By contrast, in the classroom in Shanghai, mathematics tasks tended to be very abstract in character and the teacher made no effort to demonstrate or argue for the real world applicability of the mathematics being studied. . . . However, in the post-lesson interviews, the Chinese students consistently expressed strong beliefs in the utility of mathematics in general and in relation to the specific mathematics they were studying [ed: There is a bit of apples-and-oranges at work here, so be warned. "Demanding" relevance of what you're studying and "believing" that what you're studying has relevance are two completely different things.]. . . . . Svan has christened this the "Expanded Relevance Paradox" (Svan & Clarke, in preparation) and means, by this term, to refer to the paradoxical character of application-oriented mathematics teaching associated with subjective irrelevance and pure mathematics-oriented mathematics teaching associated with subjective relevance . . . . Mathematical tasks are a constituent element of the social activity in which students engage. Attempts to increase the 'relevance' of these tasks through a figurative contextualisation may be counter-productive if these efforts are perceived by students to be artificial and are interpreted as reifying the very distinctions they seek to dissolve. (380)

This one was really good. Hell, it's been "good" for a long time. (emphasis mine):

One common interpretation of the constructivist manifesto (i.e., that "knowledge is the result of a learner's activity rather than of the passive reception of information or instruction," von Glasersfeld, 1991, p. xiv) has been that it became no longer legitimate for teachers to "tell" students anything. This position is not a logical consequence of adherence to constructivist learning theory, which suggests that students inevitably construct their own mathematics, whatever the classroom situation (Cobb, 1995). However, Telling or Not-Telling have been constructed oppositionally with such success that publications on contemporary pedagogy, . . . while usefully discussing many pedagogical strategies, see no need to address any strategies that might be construed as analogous to "telling" and even articles that purport to address the issue (such as Chazan and Ball, 1999) offer teachers little insight into how (and, as importantly, when) their mathematical knowledge might be articulated explicitly to the benefit of their students.

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Wu and I

Allison pointed me to some notes from a presentation by Hung-Hsi Wu at NCTM's 2007 meeting, and I've been meaning to write about them.

I gave a presentation at that conference, so during Wu's talk I was, if I remember correctly, holed up in my hotel room frantically reviewing my notes—a shame, considering that what appears to be the central point of Wu's presentation was, with fewer differences than similarities, the idea that I tried to flesh out for textbooks when I first started writing this blog three years ago. (A further shame is that many of my posts which directly addressed that idea are gone forever.)

Wu's idea below, minus the brilliant parallel to engineering, is essentially what I have referred to as the Boundary Principle (emphasis in original):

Engineering cannot cater to any human need no mater [sic] how scientifically absurd, any more than it should produce any product that is useless though scientifically correct.

Engineering must mediate between two extremes:

(1) inviolable scientific principles.
(2) user-friendliness of the final product.

Mathematics education, as Wu explains, is constrained by the same impossibility on the one hand and impracticality on the other. That is, it is not possible for mathematics education to serve every student desire, nor practical for it to attempt to do so. Rather, mathematics education (what Wu refers to as "mathematical engineering") must strike a principled compromise between its often conflicting fealties to both mathematics students and to mathematics itself.

Wu goes on to list five "inviolable scientific principles" for mathematical engineering, four of which fit reasonably well under my three for textbooks—accuracy, coherence, and language:

Precision: Mathematical statements are clear and unambiguous. At any moment, it is clear what is known and what is not known.

Definitions: Bedrock of mathematical structure (no definitions, no mathematics).

Reasoning: Lifeblood of mathematics; core of problem solving.

Coherence: Every concept and skill builds on previous knowledge and is part of an unfolding story.

Purposefulness: Mathematics is goal-oriented. It solves specific problems.

I would disagree that the last of these rises to the importance of being a principle, and Wu's contention that the damaging split (visit the links in Allison's post [linked above] for more details) is between mathematicians on the one hand and mathematics educators on the other is, I think, a bit simplistic. Nevertheless, I was heartened to see so many similarities between Wu's ideas and my own.

And here's just a fantastic section from Wu's article "How mathematicians can contribute to K-12 mathematics education" (again, see Allison), which, to my mind, connects with some ideas here, here, here, definitely here, and certainly here and here:

At present, what most children get from their classroom instruction on fractions is a fragmented picture of a fraction with all these different "personalities" lurking around and coming forward seemingly randomly. What a large part of this research does is to address this fragmentation by emphasizing the cognitive connections between these "personalities". It does so by helping children construct their intuitive knowledge of the different "personalities" of a fraction through the use of problems, hands-on activities, and contextual presentations.

This is a good first step, and yet, if we think through students' mathematical needs beyond grade 7, then we may come to the conclusion that establishing cognitive connections does not go far enough. What students need is an unambiguous definition of a fraction which tells them what a fraction really is. They also need to be exposed to direct, mathematical, connections between this definition and the other "personalities" of a fraction. They have to learn that mathematics is simple and understandable, in the sense that if they can hold onto one clear meaning of a fraction and can reason for themselves, then they can learn all about fractions without ever being surprised by any of these other "personalities".

Update: I promised to post this great quote from Allison on the Wu presentation/paper. It's a fantastic quote, and it's like a bottom line for people who read this post (or read this blog) and say, "Harumphh, he's so abstract":

The point is to simplify without lying, misleading, or otherwise undermining future growth, so that the presentation true for the 4th grader is still true for the 12th, but the 12th can handle more, and can see the connections to the stuff already in his mind. Mastery is then possible.

Update II: One reason I like Allison's statement so much--aside from the fact that it's pretty much spot on in interpretation--is that she uses the word lying. Exactly. I can go into detail later, but that's exactly what a lot of mathematics education is today--straight-up lying to kids' faces.

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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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Ordinals and Denominators

At the moment, what is fascinating about the book are the little interesting nuggets.

And, at the risk of being exposed to a collective raising of readers' eyebrows at its obviousness, I present below one of those nuggets—about fractions—that I came across tonight.

Every third lot wins, every fifth man is Chinese. It seems that this is the origin of the ordinal numbers as a means to indicate denominators of fractions: Counting 1, 2, 3, . . . , 10 to count out the tenth; all these "tenth" people or objects together form a (one, the) tenth of the whole. Thus the tenth part is in fact the last of all of them. In an obsolete terminology nine parts means 9/10, the remainder that is left if the tenth is counted out. "Decimate" originally meant counting out the tenth (to be shot).

At this point in his writing about fractions, Freudenthal is not making teaching suggestions (thankfully, given that last sentence), but simply conducting a survey—a thorough survey, it seems—of the different ways fractions (have) come up in informal contexts. Still, the paragraph above brings up an interesting (to me) perspective and mathematical connection that may be didactically useful.

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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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Morning Edu-News (03.18.2008)

Sparta, New Jersey:

This one really hits home for me:

Anthony Maitilasso, a Sparta High School junior, spoke up at a school board meeting and was called to the principal's office a few days later to talk about it . . . Maitilasso, 16, said school officials wanted him to retract statements he made at the board meeting, saying he heard that a teacher was seen using drugs at the school more than a year ago.

Acting Principal James Bevere said Maitilasso was brought to the office simply to refute his assertion during the meeting that school officials blocked a story from running in the school paper about the possibility of having random drug testing for teachers. . . .

While Maitilasso wasn't disciplined, he characterized the tone of school officials as "intimidating." Maitilasso said he also received a call from Sparta Police, who wanted to talk to him about his statements to the board.

When I was a junior in high school, I was the chief editor of my high school's newspaper. I wrote an editorial titled "Nothing More Like a Fool"—which is the first part of a famous quote about drunkenness from an even more famous drunk, Benjamin Franklin.

The point of the editorial was to call attention to what was (and probably still is) a typical setup for small Midwestern towns: no grocery store, no library, five startup-and-shutdown restaurants that for some reason can't afford the rent on Main Street to stay in business, but at least two bars and at least one liquor store that have been in business for decades.

The reaction to that editorial—as it was reported to me by some teachers in the school—was ferocious. The owners of each establishment (Dean Martin was the name of one of them. How hilarious is that?), as a group, stormed into the principal's office one day, demanding that I be forced to retract the editorial and apologize.

The principal at the time, Floyd Harris, did ask me to retract and apologize. I refused to do so. The school paper was cancelled. (Side note: I helped to deliver newspapers in the town at the time [yes, I was a paperboy], and Dean Martin, with his wife Kathy, school newspapers in hand, actually approached me—a 16-year-old boy—the very day that the editorial was published, on Main Street, and quite angrily told me that they were going to cancel their subscription. After that incident, I thought to myself, "Wow, did I hurt their business in some way by suggesting in a school newspaper, whose audience is composed mainly of young women and men not yet of legal drinking age, that people ought to step back and think about what it means to live in a community seemingly centered around alcohol?" I'm guessing that maybe I did.)

You can find some clarifications (and more muddying of the waters) here. I'm going to be keeping up with this one.

Oshkosh, Wisconsin:

Innovative Curriculum Substitutes the Word 'Brain' for 'Eyes,' Receives Funding:

I know if I give the students something specific to do – 'touch this,' 'say that' – it puts all of their brains on the same page," she said. "I make choices about the movements I will have them do and I know the reason I'm doing the things I'm doing with them."

I have to hope that it's just the reporter's word limit (or angle or topic) that makes the whole thing sound ridiculous.

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Not Choosing Sides

If I have not already forewarned readers and visitors that in the coming weeks I may annoy them with all things Freudenthal, well, let me do so now.

From Chapter 2, this paragraph gets at what is a core tension in debates about mathematics education:

In order to have some X conceived, one teaches, or tries to teach, the concept of X. In order to have numbers, groups, linear spaces, relations conceived, one instills the concepts of number, group, linear space, relation, or rather one tries to. It is quite obvious, indeed, that at the target ages where this is tried, it is not feasible. For this reason, then, one tries to materialise the bare concepts (in an "embodiment"). These concretisations, however, are usually false; they are much too rough to reflect the essentials of the concepts that are to be embodied, even if by a variety of embodiments one wishes to account for more than one facet. Their level is too low, far below that of the target concept. Didactically, it means the cart before the horse: teaching abstractions by concretising them.

Though I am extremely skeptical about Freudenthal's remedy for this tension, didactical phenomenology (more on that later), it is encouraging to see that he does not simply give up and choose a side.

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Morning Edu-News (3.16.08)

Jefferson Parish, Louisiana:

Louisiana students who attend public schools in Jefferson Parish must work harder than almost all their peers in other parts of the state to earn the same grades, according to an article in Sunday's Times-Picayune.

"We looked at more than 25 schools and school systems as well as school systems in other states, and only Lafourche Parish has the same grading scale as Jefferson," said Richard Carpenter, deputy superintendent for Jefferson Parish public schools.

In Jefferson Parish, an A is 95–100, a B is 88–94, a C is 78–87, a D is 70–77 and an F is 69 and below. Grading scales in other school districts in the New Orleans area, as well as several parochial schools, are one to five points lower. Independent private schools were not surveyed.

A 21-member committee assigned to look into the issue has warned that the disparity makes it more difficult for Jefferson Parish students to compete for scholarships. A recommendation for adjusting the grading scale may come in April.

Baltimore, Maryland:

"If you're driving and you feel you're in control, you're not going fast enough," says Andres Alonso, the CEO of Baltimore's public school system since July of 2007. Alonso was quoting Mario Andretti in describing his approach to turning around the city's schools.

Here again we see the concept of decentralization that we took note of here.

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This Just In: School Reform Is Complex

The online version of the Denver Post is really standing out among other papers lately as a provider of interesting education articles.

The latest, a brief interview with former Denver superintendent Jerry Wartgow about his new book, Why School Reform Is Failing and What We Need to Do About It: 10 Lessons from the Trenches, was enough to make me think about giving Amazon even more of my monthly salary.

Some choice quotes from the interview:

Q: But people say there should be a sense of urgency about the necessity of school reforms.

A: There is a sense of urgency. There was urgency when I was there, there was a sense of urgency in 1990 and 1980. My point is, yes, there is a sense of urgency. With all of the studies and reforms and legislation and regulations and billions of dollars, we have made no progress. We ought to stop and reflect upon what the real issues are before we just continue.

Q: What reforms do you think work?

A: I am convinced that it almost doesn't make a difference about which one you select as long as you select one and stay with it.

Q: In Denver's case, you are saying people should get behind the reforms and let them stay in place for a while?

A: Yes. My other big theme is the three Ls are strangling the three Rs. The three Ls are legislation, litigation and labor agreements. Those three combined have created such a bureaucratic mess that it is literally impossible for most schools to do a good job and carry out the responsibilities.

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Meet Hans Freudenthal

I finally did get my book, though I must confess I haven't made it very far yet. Great first sentence, though: "Men die, systems last."

Okay, I have made it further than that. Anyway . . .

Freudenthal's thinking about mathematics education is interesting to me for two reasons: (1) it seems to deliberately avoid (or perhaps be unconscious of) what I have been thinking of lately as "didactic maternalism" and "didactic paternalism."

With apologies to women and men everywhere for such a gross generalization, I would link the former with the sanctification of error, an obsession with "behavior," and an unhealthy animosity to uncomfortable change; the latter (which will be much more familiar with "edusphere" readers) with stoic arrogance, intimidation, and an unhealthy obsession with individuality at the expense of everyone else (i.e., parochialism); (2) it seems to recognize the complexity of the subject of mathematics education, which is pretty rare on both of the made-up sides of our education debate in the U.S.

Here's a nice quote I wanted to share:

In order to write a phenomenology of mathematical structures, a knowledge of mathematics and its applications suffices; a didactical phenomenology asks in addition for a knowledge of instruction; a genetic phenomenology is a piece of psychology.

All the psychological investigations of this kind which I know about suffer from one fundamental deficiency: investigations on mathematical acquisitions (at certain ages) have involved the related mathematical structures in a naive way—that is, they lack any preceding phenomenological analysis—and as a consequence, are full of superficial and even wrong interpretations. The lack of a preceding didactical phenomenology, on the other hand, is the reason why such investigations are designed in almost all cases as isolated snapshots rather than as stages in a developmental process.

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Innovation Zones

Thursday, March 13, 2008 -- Denver, Colorado

I won't pretend to know much about the technicalities or legalities of district/school relationships nor anything about the idiosyncracies of Colorado law, but the news out of Denver today certainly grabbed my attention:

A panel of Colorado lawmakers Thursday unanimously backed what is perhaps the most-watched education legislation of the year, a bill that would let clusters of schools break from district rules and state law to form "innovation zones."

Schools could make their own decisions on spending, the length of the school day and year, course content, hiring and teacher pay if the measure becomes law.

Of course, the actual wording of the legislation (Senate Bill 08-130 [PDF]) is less dramatic and likely a smidge more accurate:

[The bill] allows a public school or group of public schools to submit to its school district board of education ("local board") an innovation plan ("plan") to allow the school or group of schools to implement innovations within the school or group of schools, including but not limited to innovations in delivery of educational services, personnel administration and decision-making, and budgeting. Requires the local board to review each submitted plan and approve the school as an innovation school or the group of schools as an innovation school zone or reject the plan.

Language in the bill that allowed schools identified as innovation schools to waive union bargaining agreement provisions seems to have been watered down to garner union support for the bill. Two of these amendments are shown below. (Emphasis is mine.)

Original Bill

Amended Bill

On and after the date on which the state board designates a school district as a district of innovation, any collective bargaining agreement initially entered into or renewed by the local school board of the district of innovation shall include a term that allows each innovation school and each innovation school zone in the school district to be removed from the collective bargaining agreement

On and after the date on which the state board designates a school district as a district of innovation, any collective bargaining agreement initially entered into or renewed by the local school board of the district of innovation shall include a term that allows each innovation school and each innovation school zone in the school district to waive any provisions of the collective bargaining agreement identified in the innovation plan as needing to be waived for the innovation school or the innovation school zone to implement its identified innovations.

This new language makes the wholesale disposal of any collective bargaining agreement nearly impossible and provides a way to challenge a school's decision to waive any provision of such an agreement.

Original Bill	Amended Bill
For an innovation school, removal from the collective bargaining agreement shall be based on obtaining the approval by means of a secret ballot vote of a majority of the personnel who are employed at the innovation school.	For an innovation school, waiver of one or more of the provisions of the collective bargaining agreement shall be based on obtaining the approval, by means of a secret ballot vote, of at least sixty percent of the members of the collective bargaining unit who are employed at the innovation school.

And, of course, this changes 51% to 60%.

The bill clears the way for what might become either a successful or unsuccessful large-scale school-autonomy experiment in Colorado. Education groups across the nation will be following the results closely:

"I was texting the whole time," said Alex Medler, vice president of research and analysis for the Colorado Children's Campaign.

Medler said he expects school boards would watch autonomous schools closely to see how creative teaching plans affect the dropout rate and achievement gap between white students and minorities.

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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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Historical Phenomenology of Mathematics

One of these days, I'll have to pick up the book. For now, I'll write about the ideas, using incomplete information.

First, a definition from the man himself, the late Hans Freudenthal. This is his description of historical phenomenology as it concerns mathematics:

Our mathematical concepts, structures, ideas have been invented as tools to organise the phenomena of the physical, social and mental world. Phenomenology of a mathematical concept, structure, or idea means describing it in its relation to the phenomena for which it was created.

In a previous post, I shared an example of what seems to be an early conception of the median, quoted in a paper by Bakker and Gravemeijer titled An Historical Phenomenology of Mean and Median. The example comes from a book by Edward Wright, cartographer and mathematician, circa 1599:

Neither if there be disagreement betwixt observations, are they all by and by to be rejected; but as when many arrows are shot at a mark, and the mark afterwards away, he may be thought to work according to reason, who to find out the place where the mark stood, shall seek out the middle place amongst all the arrows: so amongst many different observations, the middlemost is likest to come nearest the truth.

For a more accessible example of "Wright's median," one can think of a guess-the-number contest at a fair in which participants are invited to publicly guess (write down) the number of, say, jellybeans in a jar to win a prize. Here, each guess is like an arrow, and the exact number of jellybeans like the missing bullseye. Assuming that every participant had access to the same information (they all saw the same jar of jellybeans) and that the distribution of their "errors" was relatively normal, the median (or mean) of all the guesses would likely fall close to the exact number of jellybeans in the jar:

Treynor, former editor of Financial Analysts Journal, told us that when he taught finance, he would pass a jar of beans among his students and have them guess the number. As I wrote: "The guesses would vary wildly, but always, when the number guessed in total was divided by the number of students guessing, the result was within 3% of the correct number, he said. As there were 52 of us assembled, and a bowl of peppermint candies on the table, we tried the experiment. A low guess of 32 was recorded, a high of 71. The median guess was 46, the mean was 45. The correct total was 46, a number only one of the 52 had guessed."

More later. Bye!

Reference: Bakker & Gravemeijer. 2006. An Historical Phenomenology of Mean and Median Educational Studies in Mathematics. 62: 149-168.

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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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The Wright Median

Historical and didactical phenomenology are both mouthfuls. Interesting, though. More on these later.

The paragraph below seems to refer to the concept of the median, and was written by Edward Wright, cartographer and mathematician, in about 1599:

Neither if there be disagreement betwixt observations, are they all by and by to be rejected; but as when many arrows are shot at a mark, and the mark afterwards away, he may be thought to work according to reason, who to find out the place where the mark stood, shall seek out the middle place amongst all the arrows: so amongst many different observations, the middlemost is likest to come nearest the truth. (Eisenhart, 1974, p. 52, spelling modernized)

Reference: Eisenhart, C.: 1974, 'The development of the concept of the best mean of a set of measurements from antiquity to the present day', 1971 ASA Presidential Address. Unpublished manuscript.

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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.


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.


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.


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.



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.



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



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.