Learning Mathematics, John Mc-more-of-the-same McCain’s Free Pass

Learning of Mathematics Abstract Examples aid learning (psst: yes, math types already know this)

Abstract knowledge, such as mathematical knowledge, is often difficult to acquire and even more difficult to apply to novel situations (1-3). It is widely believed that a successful approach to this challenge is to present the learner with multiple concrete and highly familiar examples of the to-be-learned concept. For instance, a mathematics instructor teaching simple probability theory may present probabilities by randomly choosing a red marble from a bag containing red and blue marbles and by rolling a six-sided die. These concrete, familiar examples instantiate the concept of probability and may facilitate learning by connecting the learner’s existing knowledge with new, to-be-learned knowledge. Alternatively, the concept can be instantiated in a more abstract manner as the probability of choosing one of n things from a larger set of m things.

The belief in the effectiveness of multiple concrete instantiations is reasonable: A student who sees a variety of instantiations of a concept may be more likely to recognize a novel analogous situation and apply what was learned. Learning multiple instantiations of a concept may result in an abstract, schematic knowledge representation (1, 4), which, in turn, promotes knowledge transfer, or application of the learned concept to novel situations (1, 5). However, concrete information may compete for attention with deep to-be-learned structure (6-8). Specifically, transfer of conceptual knowledge is more likely to occur after learning a generic instantiation than after learning a concrete one (7).

Therefore, we ask: Is learning multiple concrete instantiations the most efficient route to promoting transfer of mathematical knowledge? Here, we tested a hypothesis that learning a single generic instantiation (that is, one that communicates minimal extraneous information) may result in better knowledge transfer than learning multiple concrete, contextualized instantiations.

In experiment 1, undergraduate college students learned one or more instantiations of a simple mathematical concept. They were then presented with a transfer task that was a novel instantiation of the learned concept. The to-be-learned concept was that of a commutative mathematical group of order three. This concept is a set of three elements, or equivalence classes, and an operation with the associative and commutative properties, an identity element, and inverses for each element. This concept was chosen because it involves the most basic properties of the real number system, yet it is simple, novel to the study participants, and can be easily instantiated in different ways.
[...]

These results indicate that learning one, two, or three concrete instantiations resulted in little or no transfer, whereas learning one generic instantiation resulted in significant transfer. If transfer from multiple instantiations depends on whether the learner abstracts and aligns the common structure from the learned instantiations (1, 4), then transfer failure suggests that participants may have been unable to recognize and align the underlying structure.
[...]
Because the difficulty of transferring knowledge acquired from concrete instantiations may stem from extraneous information diverting attention from the relevant mathematical structure, concrete instantiations are also likely to hinder transfer for young learners who are less able than adults to control their attentional focus. We have evidence that 11-year-olds transferred successfully from a generic instantiation, but not from a concrete one (12).

If a goal of teaching mathematics is to produce knowledge that students can apply to multiple situations, then presenting mathematical concepts through generic instantiations, such as traditional symbolic notation, may be more effective than a series of “good examples.” This is not to say that educational design should not incorporate contextualized examples. What we are suggesting is that grounding mathematics deeply in concrete contexts can potentially limit its applicability. Students might be better able to generalize mathematical concepts to various situations if the concepts have been introduced with the use of generic instantiations.

In other words, learning the abstract actually helps!

Politics
Oh, two of the three remaining presidential candidates are getting vetted, all right:

And now a new crop of candidates comes before the discerning eyes of the American people, (who are too busy to be discerning because they work more hours for less pay and so pass the job of discernment on to the four corporations. I mean free press). And once again the vetting has been thorough. Only this time there’s better graphics and Bill is craaazy. Whoo! I love me my vetting!!

After thousands of hours of news coverage we have learned that Hillary is a liar and Barack is a terrorist or something. I’m sure there will be stuff about their positions on the war, lobbying reform, the environment, the economy… I’m sure those details will receive exhaustive coverage. After all, that’s what the entire job they’re running for will be about. But vetting takes time… And Bill is craaazy!

John McCain on the other hand has locked up the Republican nomination and we know he was a war hero. But right now we’re too busy looking into Hillary’s lies and Barack’s weird name to get too deep into McCain. The only other thing about McCain is that he once called his wife the C word in public. Other than that he seems to have a pretty clean–[...]

Okay, I get it. We let all Republicans and most Democrats off the hook because they will roll over for the companies that own the media and in McCain’s case literally get blown by lobbyists, but come on! Are we not even remotely pretending anymore?

George W. Bush was a silver spoon dolt with no record to speak of other than bankruptcy and selling tropical plants and we let him sail into the White House but Barack talks about religious fundamentalism and guns being prevalent in poor areas and we roast him for weeks?

That isn’t just a bias, that’s deranged. Big money has seriously warped us. When there is a school shooting the news doesn’t talk about basic gun control, they talk about video games and their influence on kids. When we learn that the Vice President planted false news stories to take us to war we get outraged about Simon being too mean to contestants on fucking American Idol. And why? Because big money has log jammed our voice and we know it’s too hard to do anything about it. So the dollar drops, the insane war continues, polar bears drown, school shootings continue, the trade deficit soars, gas prices go through the roof and this country fades as quietly as someone falling asleep in front of the TV.

We’re not a stupid country; we’re comfortable and afraid. Afraid to hear the truth. Our leaders talk to us like a parent avoiding the subject of sex with an eight-year-old and we eat it up. And when someone like Obama talks to us like grown-ups we vet him. And vet him.

The same old game…the old Okie-Doke.

Not that this is anything new:

We vetted Reagan and found out he was an ex-cowboy who starred at Notre Dame and was able to communicate with primates ala Jane Goodall. And after Reagan was vetted he rewarded our free press (™) by taking away some regulations that were making it hard for them to vet and merge and take over effectively. And the fifteen or so corporations that owned the media were thankful.

Bush Sr. was put through the ringer as well. We learned that he was an all-glove, no-bat first baseman who had probably killed people in cool ways at the CIA. And best of all he had hung with Reagan. Another job well done by the eight or nine corporations that owned the media at that time.

Then reform-minded Clinton came along and the five or six corporations that owned the media got to work again. We learned that in 1964 he had ordered a ham sandwich and then we went through literally every second of his existence from that point on. We even hired a full-time prosecutor to investigate him 24 hours a day. Now that’s vetting. And while this was happening Clinton raised minimum wage and capital gains taxes and a slew of other things that made the four conglomerates that owned the media and most of Congress really mad. But finally the diligence of the American press and people paid off and we learned he had received a blow job. Once again America had vetted its leaders.

Next came W. Bush and the vetting kicked into high gear by the four companies that owned the press, W. Bush, Congress and mailboxes in the Bahamas. The American people did their part by listening to what that press said and believing them. Not always an easy task. We learned W. Bush was, like Reagan, a cowboy. And that he was a regular guy who, unlike Al Gore, loved Jesus. And oh yeah, the liberals whined about some other stuff like the fact that he deserted from the Air Force, bankrupted companies, had a huge gap in his resume during which he partied non-stop and had a bunch of DUIs…blah, blah. That aside he was the kind of guy you wanted to have a beer with which canceled out the swiftboating liberal noise.

So what is McMoreoftheSameMcCain up to? He says that Hamas wants Obama to win.

McCain spoke with bloggers this morning on a number of issues ranging from William Ayers to Rev. Wright to Tony Rezko. Jennifer Rubin noted that Hamas had endorsed Senator Obama and asked McCain whether Obama might have given “an unhelpful signal” to the terrorist group. McCain’s response:

All I can tell you Jennifer is that I think it’s very clear who Hamas wants to be the next president of the United States. So apparently has Danny Ortega and several others. I think that people should understand that I will be Hamas’s worst nightmare….If senator Obama is favored by Hamas I think people can make judgments accordingly.

Now, if McCain can just remember which of those groups are Shiite and which are Sunni; he seems to have trouble remembering such trivial details again and again.

But I suppose that to the Republicans, thinking that a President should know such things is being an elitist.

Humor
I miss watching Women’s Professional Beach Volleyball.

(larger)

Obama Supporters: Relax.

Workout notes 2650 yard swim (1.5 miles). Not that fast, but enough to get the circulation going. I did 500 free, 5 x 100 IM on the 2:30, 10 x 50 (25 drill, 25 swim), 5 x (100 free, 100 paddle) the 3 x 50 side to cool down.

Peoria Notes Spring is here, and there are bunnies all over the place (they are almost as numerous as the squirrels around here.

In the early morning hours, it is common to see 3-4 of them together. On my walks, I have seen groundhogs, rabbits, deer, wild turkeys, raccoons, and foxes.

Humor

Student stories. Yes, some of the little darlings lie. Here is my favorite “lying student story”: about 10 years ago, I had a bad student in my MTH 115 class (the calculus class that is stripped down of all theory, trig functions; in fact it is devoid of anything that might be challenging, and frankly, fun)

Well, the day before the Friday exam, I get a call from her. She was “stuck” in a town that was 2 hours away with “car trouble”. During the weekend I got reports on my answering machine about how the mechanic was doing this and that to the car.

Then on Monday, I get more calls on my answering machine from her saying that the car would be ready..so she was going to come in to take the exam!

So when she showed up, I asked for an explanation of why the car trouble kept her from being in class. “She smiled, and went on how she was stuck in that town because her car wouldn’t run”. So I said “so you were in XXX when you called?” She said “yes, that is why I couldn’t make it back.” I laughed and said “you know, of course, that the phone records were the calls came from, and they came from your dorm room.”

She said that “I wouldn’t understand the real reason she couldn’t take the exam”….she wasn’t even that embarrassed that she was caught lying.

Eventually, she dropped the course.

Election:
Many Obama supporters are getting anxious that the mainstream media (e. g., NY Times, The New Republic(an) Magazine) have turned their guns on Obama.

To that I say: “relax”. Remember that Obama is mostly running against the “status quo”, and those folks aren’t going to go away politely. It is up to us (his supporters) to lead the way; remember that we are the “change happens from the bottom up” campaign.

So let us duly make our calls and campaign! Even some of the mainstream media gets it that this race is over.

Other than ensuring the Greatest Show on Earth will continue, does it matter that Hillary Clinton defeated Barack Obama Tuesday in Pennsylvania by nine-plus points? Barack Obama is the nominee.

No matter how many kicks the rest of us find in such famously fun primary states as Indiana and South Dakota, it’s going to be McCain versus Obama in 2008.

I believe the cement set around the Clinton coffin last Friday. The Obama campaign announced it had received the support of former Sens. Sam Nunn of Georgia and David Boren of Oklahoma. [...]

And many of us get it that Obama is a better candidate against McCain than Clinton.

Ok, what about the Clinton is trying to damage Obama so she can run against McCain in 2012?

No way. Why exactly is Clinton going to be a stronger candidate in 2012 than she is now? Remember that she started this race at least tied with Obama in money raised, way ahead in super delegates, name recognition, and at least 20 point leads in the polls in states other than Illinois.

Now she is trailing in popular vote (no matter how badly the Clinton campaign lies; if you count people who caucused for Obama, Obama is ahead in that metric), pledged delegates, total delegates and in states won. In short, this “inexperienced” candidate has kicked her butt.

Her campaign is still deep in debt and she is still stiffing the folks that she owes money to, and she gets caught in a new lie almost daily.

More importantly, much of her strongest supporters (the current elderly, especially White and Hispanic women), well, sadly won’t be as numerous in 2012; that is a sad fact of life.

In short, barring BHO getting hit in the head by a falling meteor or ripping off his shirt in a rally and screaming “I love Osama Bin Laden”, it is over.

Bring on John “More of the Same” McCain.

The Conservative 527 Okie-Doke: Salvo of Blanks Fired at Obama

Well, it had to happen sooner of later: the Republican 527’s have started to fire at Barack Obama.

(For a discussion of 527’s and PACs, go here)

The latest 527 is the so called “National Committee”, which operates a site called “exposeobama.com”

The site is seeking funding to air this youtube video:

Oh no, there is violent crime in Chicago? That’s Obama’s fault, right? Oh wait, he is blamed for not voting to extend the death penalty to certain gang related crimes. Yes, that was in 2001.

There is one problem: Illinois suspended the death penalty in the January 2000 and it hasn’t been reinstated since!

Illinois Gov. George Ryan on Monday imposed a moratorium on the state’s death penalty. All lethal injections will be postponed indefinitely pending an investigation into why more executions have been overturned than carried out since 1977, when Illinois reinstated capital punishment.

“We have now freed more people than we have put to death under our system — 13 people have been exonerated and 12 have been put to death,” Ryan told CNN. “There is a flaw in the system, without question, and it needs to be studied.”

So Obama voted against a measure that couldn’t have done any good!

If this is the best that the Republicans can do, we are going to enjoy this November.

Update There are other articles about this:

Daily Kos: Jeffrey Feldman’s dairy (very good!)

Fact Check. They talk about the bill that Obama voted against and remind us that the creator of this (lame) attack ad was behind the Willie Horton ads (in the 1988 general election)

Conservative activist Floyd G. Brown, who had a hand in the 1988 “Willie Horton” attack ad, is seeking funds to show a new spot accusing Obama of being “weak” on Chicago gang killers in 2001 and suggesting he’d be weak on terrorism, too. Brown bases the claim on Obama’s vote against a bill to make gang killers automatically eligible for the death penalty.

We find that the ad misses the mark. The anti-gang activist who sponsored the death-penalty bill tells FactCheck.org that she doesn’t consider Obama weak on crime despite his opposition to her proposal. Chicago state Rep. Susana Mendoza said the ad makes her “sick to my stomach” and “completely mischaracterizes Senator Obama’s position against ruthless criminals.”

The record shows that Obama, while not a cheerleader for the death penalty, has supported it for a number of crimes – including terrorism. He voted for an Illinois law in 2003 that includes the death penalty for convicted terrorists. [....]

Mendoza’s bill, HB 1812, would have made anyone found guilty of a murder committed “in furtherance of the activities of an organized gang” eligible for the death penalty. It passed with large majorities in each house, despite Obama’s vote against it. Gov. George Ryan, a Republican, vetoed the bill Aug. 17, 2001. He said in his veto message that the bill was too broad, too vague and too likely to fall on minorities. He said that most gang killers were already eligible for the death penalty anyway:

Gov. Ryan, Aug. 17, 2001: Illinois has some of the toughest laws on the books to severely punish gang-related crimes. In fact, most gang-related murders would qualify for the imposition of the death penalty under the existing eligibility factors in our death penalty statue. Unfortunately, this still has not deterred gang members from killing.

That echoed Obama’s reasons for having voted against the bill. According to the transcript of the May 15 debate, Obama noted that a murderer already was eligible for death if the crime was found to be committed in “a cold, premeditated and calculated” manner and that “should be more than sufficient” to cover a gang killing. And he said he objected to the bill because it was likely to target minorities:

Obama, May 15, 2001: [What] I’m concerned about is that we use this term “gang activity” as a mechanism to target particular neighborhoods, particular individuals for, admittedly, heinous crimes that I think need to be punished to the fullest extent of the law irrespective of where they happen and irrespective of the particular criminal body that they’re working with.

The debate over the Mendoza bill arose after Gov. Ryan had suspended all executions in January 2000 because 13 death-row inmates were found to have been wrongly convicted in the previous 23 years. That total later rose to 17, giving Illinois the distinction of having the highest rate of overturned capital convictions of all 38 states with the death penalty, according to Ryan’s successor, Gov. Rod R. Blagojevich.
[...]

The ad suggests that Obama ignored an editorial cry for passage of the death-penalty bill, but that is not true. The ad’s graphics cite a Nov. 23, 2001, Sun-Times editorial, which indeed demanded action on gang violence, as the ad says. But the newspaper didn’t call for passage of the death-penalty bill. What the editorial actually urged was a local enforcement effort against gangs something like the federal government’s “unrelenting, multipronged attack” against terrorism following the events of September 11, 2001, just weeks earlier. The newspaper mentioned “the questioning of thousands of people and the thorough scouring of bank records” but did not call for any new legislation.

Furthermore, Obama’s vote took place on May 15, months before the Nov. 23 editorial appeared. In fact, the bill was dead by the time the editorial was published.

This is the Willie Horton ad:

I’ll let BHO have the last word: