a good day (off topic)

This morning both dogs got loose and went gallivanting uphill and down dale for several hours.

Finally, towards the end of the afternoon, they returned home safely without involving the local police, neighbors we have never met, or the volunteer dog-catcher person who lives a couple of streets over and  pursues her dog-catching vocation from her house, where she has erected a makeshift dog pound in the back yard.

rat psych - what to do about SAT math (part 2)

What makes SAT math “tricky” (part 1): here and here.

Picking up where I left off a little while ago, the answer to SAT math trickery, part of the answer, is extinction learning.

"Extinction learning" means learning that what you've previously learned no longer applies.

Say you're a rat in a cage and the sound of a buzzer means you're 2 seconds away from receiving an electric shock. You learn this lesson very, very well.

Then one day, things change.

Now, under the new regime, the sound of a buzzer means a piece of kibble, or perhaps a morsel of cheese.

You -- the rat whose life fortunes have taken such a dramatic turn for the better -- don't just go with the flow. You don't hear the buzzer and say to yourself: Buzzer means kibble---oh, boy!

No. You remember the dark days when buzzer meant shock, and it will take time and many repetitions to learn that buzzer means kibble now. You will learn that buzzer means kibble now, but you will never forget the old days; buzzer means shock will always be with you. There's a saying that once you've scared a person or an animal, you can't unscare him, and that's true for many things, including a. When you've learned something very, very well, you can't unlearn it; you can't not know. You just have to learn the new thing on top of the old.

So say you're a high school senior and you have spent the last 4 years of your life seeing the letter a only in the context of the quadratic equation in its standard form: ax² + bx + c = 0.

You are the rat, a is the buzzer, and buzzer means quadratic equation coefficient of x² in the standard form of the quadratic equation. By the time you've reached your 17th birthday, "a means quadratic equation coefficient of x² in the standard form of the quadratic equation" has been deeply imprinted into your brain; you have learned this so well you’ll remember it when you’re 80. There are probably people with dementia who still remember ax² + bx + c.

Now it's senior year and you're taking the SAT -- a math section -- and time is running out. Your eyes are bleeding from the protracted Ella Baker critical reading passage you’ve just hacked and bashed your way through, your future is being decided and your fate being sealed -- and all of a sudden, here you are, staring at the letter a inside a quadratic function.

And you blow it.

You don't see that this a isn't the a you know.

Which is exactly the effect the question has been written to produce.

Getting late – will finish this up tomorrow.

I'm a 10
rat psych: what to do about SAT math (part 1)
rat psych: what to do about SAT math (part 2)
rat psych: what to do about SAT math (part 3)
rat psych: careless reading errors on the SAT

rat psych - why SAT math is tricky, redux (part 1)

from perfectscoreproject:


SAT math questions use the phenomenon of associative interference against the test taker. That's what makes the questions tricky: each of the problems Debbie has posted on her site is designed to activate the wrong associations inside the student's mind. Why else choose the letter a in question #7? If your goal as a problem writer were to avoid associative interference, you would choose a different letter.

Put College Board math trickery together with the high-stakes, time-pressured, mentally grueling nature of the entire 4-hour ordeal, and you radically increase the odds that students will take the bait, especially students with high working memory. (pdf file)

As for people who breeze through the test racking up correct answers, I would be interested to see how they fare on find-the-missing-figure puzzles. I'm guessing many of them would do well. I have no idea whether aptitude for missing figure tests is associated with aptitude for math. I wouldn't be surprised to learn that it is, but I've never read anything about it one way or the other. The point is: doing SAT Math is about perception as much as anything else. SAT math is about finding the hidden right triangle in the not-drawn-to-scale figure that looks exactly like something else altogether because it is something else altogether, in real life. To do the problem you have to look at the figure, but when you look at the figure you have to not see the figure that's actually there on the page. You have to see the other, not-there figure. Finding a hidden figure that is not present on the page is a couple of quanta more challenging than finding a hidden figure that is present on the page, I think. ^*

Find the implied hidden figure!

Infer the implied hidden figure, and then solve a problem about it!

Apparently, people who breeze through the test racking up correct answers easily break free of the actual figure on the page (or so I gather).

For the rest of us, it is simply not possible to "break set" in the heat of the moment. It is not possible because breaking set in the heat of the moment is precisely what our brains are built not to do. Under pressure, normal human beings become less flexible, not more.

So, for the rest of us, the answer (one part of the answer) is "extinction learning," which is a critical component of SAT math test prep. All parents should know this.

Back in a bit.

I'm a 10
rat psych: what to do about SAT math (part 1)
rat psych: what to do about SAT math (part 2)
rat psych: what to do about SAT math (part 3)
rat psych: careless reading errors on the SAT

^*The hidden right triangle problems do not appear on 10 Real SATs. At least, I haven't come across one leafing through the book. For me, those are the hardest problems bar none, a reaction I've heard from numerous others.

writers should take the SAT

I was just talking to Debbie S, who reminded me that a passage from one of my books appeared on an SAT critical reading section. I think it was a section of Animals in Translation, but I don't recall at the moment and can't seem to scare up the email she sent me with the passage attached).

Meanwhile Debbie is no slouch in the professional writing department, either. Her book will be published by a major house, and her advance puts her in a small and select group.

We both have 10s.

I think other writers should take the SAT and see how they do. We can compile a database. I'm serious:  I'd love to see how 'real writers' do on the SAT essay. I'm guessing we'd see a lot of 10s.

Actually, I'd like to see professors take the SAT. I'd be willing to wager a small sum of money that college professors would consistently score lower than top-scoring high school students.

I'm not exactly sure why I think this, but I imagine it has to do with the K-12 grading I've been dealing with over the years.<sup>*

*</sup>grade deflation posts

awhile ago I was trying to figure out the difference between awhile and a while

Paul Brians explains:

When “awhile” is spelled as a single word, it is an adverb meaning “for a time” (“stay awhile”); but when “while” is the object of a prepositional phrase, like “Lend me your monkey wrench for a while” the “while” must be separated from the “a.” (But if the preposition “for” were lacking in this sentence, “awhile” could be used in this way: “Lend me your monkey wrench awhile.”)

High School Question

Now that the new Aspen X2 grading system is up and going and we're finally getting some grades, it appears that our son's English teacher (Am Lit) is one who likes to make a point by giving out lots of flunking grades. (What is it with English teachers? His freshman English teacher did this too.) You can find all sorts of horrible comments about him on RateMyTeachers. As far as I can tell, it's a game and that you have to pass the test. You have to make an appointment with the teacher and go in after school to show effort. I imagine that parents dare not go in to get answers. After starting the year with some of the highest grades, he has been hit with one flunking grade and one really bad grade. The teacher apparently doesn't hand back work with comments. He puts a code on the test so that if you go in to see him, he can use the code to refer to his notes. It has to be after school. I find that bizarre. Last year, when a paper interviewed the valedictorian, she specifically referred to this teacher and how she got flunking grades from him when she was a freshman. By the end of the year, she was proud that she managed to bring her grade up to an A. She thought that this taught her to work hard. This is a teacher who checks work before the due date and grades kids on whether they are procrastinating and waiting until the last minute.

So, my question is whether others have seen this sort of behavior and what solutions they came up with. My feeling is that our son has to play the game and jump through the hoops even though my first reaction is to go in and ask the teacher what the hell he is doing. Whatever it is, he has been doing it (and getting away with it) for many years.

meritocracy--

So the 12th Parliament of Singapore just opened, and Opposition MPs are talking about further improvements to the education system. (Finland was brought up as a model.)

In any case, I do have memories of Singaporean education that makes me exactly understand the issues presented in this cartoon.

Teach for America application deadlines are coming up. I am not sure if I have time to apply, or a chance even if I did apply. I grew up in a low-income family myself, and I have this fantasy of a system that would exploit both the strengths of Western and Asian education, while having none of their weaknesses.

I'm a 10

I probably used too many semicolons.

update: I left out the last 4 grammar questions. Probably failed to transfer my answers from the test booklet to the bubble sheet.

I'm a 10
rat psych: what to do about SAT math (part 1)
rat psych: what to do about SAT math (part 2)
rat psych: what to do about SAT math (part 3)
rat psych: careless reading errors on the SAT

The Conic Sections

Ah, the conic sections. Like many of the topics in the Pre-Calculus curriculum, this topic can take as much or as little time as you like. The basic ideas are a focus on the four conic sections Parabola, Circle, Ellipse, Hyperbola. There are many different ways to consider the shape and defining components of these mathematical objects.

Conics as a Slice of a Cone

The Greeks originally conceived of these as the shapes generated by slicing through a right circular cone. In A History of Greek Mathematics Vol. II (1921), Sir Thomas Heath says:

The question arises, how did Menaechmus come to think of obtaining curves by cutting a cone? On this we have no information whatever. (pg. 110)

Lost in the mists of time!



More after the jump...

help desk redux - precalculus or statistics?

I don't know if any of you are around during the day, but if you are I would be grateful for a quick read on this issue.

C. is set to move to a different math class.

He was planning to move to "Stat Honors," which unfortunately is the only class that fits his schedule. (Otherwise, we would have had him move to AP Stat.)

The calculus teacher, whose class he is leaving, thinks he should move to precalculus instead.

That makes sense except for the fact that he took precalculus last year, ending the year with a B average. The only precalculus course available is a lower level course than the one he's already taken.

That raises the college transcript issue: how do college admissions officers interpret a transcript showing two consecutive years of precalculus, with the second year being a lower level course than the first?

I have no idea.

There's also the question of the teacher. We don't know the teacher of the precalculus class; we do know the teacher of the stats class. C. took geometry with her, and my friend D's son, who is a math kid, is taking his 2nd course with her now. We are confident that C. will learn the material she's teaching.

We don't know the precalculus teacher.

One last thing: Ed is strongly opposed to hiring any more tutors. Precalculus tutors around these parts charge $150/hour. If you need a tutor to get your kid through 24 weeks of precalculus, you're talking about another $3600 on top of tuition.

I'm flummoxed, and I guess we need to make this decision today --- any thoughts?

Thank you!

Glenn Ellison on schools and the gender gap in math

I had read this research in Choke -- didn't know the author of the study was also the author of Hard Math for Middle School.

ABSTRACT
This paper uses a new data source, American Mathematics Competitions, to examine the gender gap among high school students at very high achievement levels. The data bring out several new facts. There is a large gender gap that widens dramatically at percentiles above those that can be examined using standard data sources. An analysis of unobserved heterogeneity indicates that there is only moderate variation in the gender gap across schools. The highest achieving girls in the U.S. are concentrated in a very small set of elite schools, suggesting that almost all girls with the ability to reach high math achievement levels are not doing so.

[snip]

[T]he highest-scoring boys and the highest-scoring girls appear to be drawn from very di fferent pools. Whereas the boys come from a variety of backgrounds, the top-scoring girls are almost exclusively drawn from a remarkably small set of super-elite schools: as many girls come from the top 20 AMC schools as from all other high schools in the U.S. combined. This suggests that almost all girls with extreme mathematical ability are not developing their talent to the degree necessary to do very well on the Olympiad contests.

[snip]

The nonrepresentativeness of the schools these girls come from is startling: the median CGMO [China Girls' Math Olympiad] team member comes from a school at the 99.3rd percentile among AMC participating schools, i.e. from one of the top 20 or so schools in the country. Only three come from schools that are not in the 99th percentile in most measures. And even those three are from schools that had at least one other student qualify for the 2008 USAMO and are at least in the 93rd percentile in terms of the number of high-scorers on the AMC 12. The male IMO team members, in contrast, come from a much broader set of schools. Some are from super-elite schools and most come from schools that do very well on the AMC 12, but the median student is just from a 93rd percentile school. The majority of the IMO team members had no schoolmates qualify to take the USAMO, whereas all CGMO team members had at least one schoolmate qualify and most had at least four.

[snip]

It may also be worth noting that almost all of the CGMO team members are Asian-American, which suggests that even within the super-elite schools the U.S. educational system may be missing the opportunity to bring many talented girls up to the highest level.
The Gender Gap in Secondary School Mathematics at High Achievement Levels: Evidence from the American Mathematics Competitions
Glenn Ellison MIT and NBER and Ashley Swanson MIT

Derek Owens - AP calculus

Crimson wife also points us to Derek Owens' course.

fractions in a 1940 arithmetic text

Terrific new article by Barry G: The Myth About Traditional Math Education.

The equal division of three cupcakes among four people, or the equal division of a 3 inch line into four parts, is an extension of the idea of division of a whole number by a lesser whole number, which students have already mastered. Students already know that if 12 cupcakes are equally divided among four people, then each person gets 12/4 cupcakes. This idea is extended by starting with the problem of 3 divided by 4, and expressing 3 as 12 fourths. The problem is now stated as 12 fourths divided among 3 people, so that each person receives 3 fourths. This idea is then applied to a line three inches long, so that fractions are ultimately related to a number line, and the final point made that fractions are a representation of division. This is a key concept and ultimately underscores an idea of representing a fractional part as a unit unto itself. That is, 3/4 of an inch can be thought of as a unit (i.e., there are four such units in a three inch line) which is a cornerstone idea when fractional division is studied later.

And here's Barry's chart showing the rise and fall in test scores:

best college email ever

I came downstairs this morning to find C. laughing over this email --

looks like University of Illinois!

Well, Joel. Your stats are very respectable. You've done some solid work, here. But it's not quite Ivy League, now, is it?

liveblogging Risky Business, part 2

Every parent should watch Risky Business on the eve of college apps. Especially if you saw it when it came out.

I have spent the last four years of my life busting my butt in this s***hole! I'm sorry. I don't think I can leave until I get just a little compassion from you.
Joel to the school nurse 

liveblogging Risky Business

omg

We're forcing C. to watch Risky Business with us tonight, and the movie opens with a dream about the SAT!

I had completely forgotten that.

Two scenes later, his mom asks him if he's gotten his SAT scores yet.

He has.

570 math, 560 verbal.

In the car on the way to the airport, his dad tells him he's set him up with an interview for Princeton.

not your father's SAT

I've been amazed by the difficulty of SAT math in its current incarnation; I remember SAT math as being pretty easy. Now it's hard.

I'd been wishing I could look at an old test -- and I seem to have misplaced the email Akil sent me that included an old test (must find email...) when I raised this issue before --

Anyway, long story short, I've carried on being mystified over the question of what has or has not happened to SAT math.

Suddenly, the other day, it hit me: 10 Real SATs! The book was published before CollegeBoard changed the test in 2005 (2006?) It has real SAT tests -- 10 of them! -- with real prior-to-2006 SAT math.

So I ordered it.

And ----- wow.

The math on the earlier tests is so much easier. Easier at the level of: I found myself doing the final, hardest problem in a section in my head, in bed, in a state of sleep deprivation, and after drinking a glass of wine.

I'll have to sit down and take a timed section and see what happens.

kp on the MathCounts course and SAT math

kp writes:

I took the AOPS Advanced MathCounts class this summer (I'm a coach and wanted to see what it would be like for my students and what new things I could learn from it.) I appreciated that the class taught the shortcuts but also focused on how the shortcuts worked and how you could adapt them when the problem was given a new twist. (For example, to find the number of factors a number has, first find its prime factorization, then add 1 to each of the exponents, then find the product of these numbers. They explained why this made sense, then assigned different variations of problems on this topic.)

I'm no expert on the SAT (I'm a middle school teacher), but there does seem to be a large overlap between hard middle school math and what is on the SAT. We sometimes use SAT practice problems in our MathCounts practices. Our district's merit scholars often participated in MathCounts in middle school. Perhaps that is because the type of kid who stays after school to do math is the type of kid who is also successful on the SAT, or perhaps it is because MathCounts helps to prepare them for the SAT.

The MathCounts course sounds like a blast.

Teaching the shortcuts is a great idea -- it's the shortcuts that help you see what's actually going on, I think.

I remember years ago reading an article -- it may have been a study -- about smart-works-hard type students versus the 'naturals.' The smart-works-hard types went on wild goose chases trying to solve problems, while the naturals produced short, elegant proofs and solutions. I laughed, reading that, having been on many a wild goose chase myself.

add this problem to the curriculum

re:

r² is a multiple of 24 and 10. What is the smallest value?

Here, from a few weeks ago, is Stanislaus Dehaene on multiplication:

[O]ur intuition of quantity is of very little use when trying to learn multiplication. Approximate addition can implemented by juxtaposition of magnitudes on the internal number line, but no such algorithm seems to be readily available for multiplication. The organization of our mental number line may therefore make it difficult, if not impossible, for us to acquire a systematic intuition of quantities that enter in a multiplicative relation. (This hypothesis is supported by the fact that patients can have severe deficits of multiplication while leaving number sense relatively intact; in particular, patient NAU (Dehaene & Cohen, 1991), who could still understand approximate quantities despite aphasia and acalculia, was totally unable to approximate multiplication problems).

This passage precisely captures my experience learning arithmetic.

Addition and subtraction make intuitive sense to me; multiplication and division do not. It's really that simple. For me, "math" - math as opposed to simple counting - begins with multiplication and division.

This observation brings me to a corollary: people always say kids fall off the math cliff when it's time to learn fractions, but I think the math cliff comes sooner. I think the math cliff is multiplication, only nobody knows it.

Nobody knows it because falling off a math cliff isn't like falling off a real cliff; with a math cliff, you can walk right over the edge and just hang there for awhile, suspended in mid air, like Wile E. Coyote.

The real drama comes after the fall, which is when kids finally get to fractions. Fractions aren't the cliff, and they aren't the fall. Fractions are the crash-landing at the bottom.

At least, that's my guess for the moment.

Setting metaphor aside, though, if it's true that we are not equipped with an intuitive understanding of multiplication and division, and I believe it is true, why don't more people know this?

Everyone knows fractions are hard; why doesn't everyone know multiplication and division are hard?

It's true most people perceive that certain aspects of multiplication and division are hard. Namely: memorizing the times tables is hard (for many children) and learning to do long division is hard (for many children). But I've never seen anyone take these facts to mean that there is something intrinsically challenging about multiplication and division in a way that is not the case with addition and subtraction.

Why?

I don't know, but I have some thoughts.

Which.... will have to wait. It's getting late, and I'm still trying to edit the body of this post into shape, so I'm going to set that aside and skip to the end, and just say that I think children should probably be taught to solve problems like the one above, which appeared on the October SAT. I'm pretty sure problems this can be used to find out whether students are suffering associative interference between addends and factors, which I bet an awful lot of students are no matter how quickly and accurately they can construct factor trees.

I'm also thinking more attention should be paid to teaching young children the terminology of arithmetic: addends, subtrahends, factors, and the like. I think -- I don't know -- that fluency with the terminology might help reduce associative interference. "All math looks alike": the 5 and the 2 in 5+2 look exactly like the 5 and the 2 in 5x2. But the words addend and factor have nothing in common whatsoever.

More later.