kitchen table math, the sequel: 10/9/11 - 10/16/11

Saturday, October 15, 2011

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


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

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.


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.

what is SAT math?

I'll be interested to hear from all of you if I'm wrong about this, but as far as I can tell, SAT math is middle school competition math.

It's hard math for middle school.

If C. and I had worked our way through Art of Problem Solving's Competition Math for Middle School, neither of us would have gotten stuck on the factoring problem we both got stuck on:
r2 is a multiple of 24 and 10. What is the smallest value?
Competition Math for Middle School, a book for mathematically gifted middle schoolers, explicitly teaches the answer to this question:
441, 256, and 576 ... are all perfect squares. If a number is a perfect square, each of the prime factors will have an even exponent in its prime factorization.
J. Batterson, p. 139
Leafing through the two books, I am struck by the the amount of explicit, procedural teaching directed to mathematically gifted students. No one's asking them to figure these things out or to "problem solve." Instead, they're being directly told that all of the prime factors in a perfect square will have even exponents.

Ditto for the how many diagonals in an n-gon, a perennial favorite amongst SAT math writers these days, it seems.

correction: My wording above -- explicit, procedural teaching -- implies that these books teach procedures instead of concepts. That's not at all the case and isn't what I meant to convey.

What I meant to convey was the fact that gifted middle school students aren't being asked to figure out  concepts for themselves; they're being explicitly told how the problems work, and why.

Friday, October 14, 2011

Courtney James

re: what to do about C. and calculus, Crimson Wife writes:
I have no personal experience with him, but I have heard numerous raves about Courtney James' tutoring from fellow moms in our local homeschool support group. He tutors AP Calculus BC according to his website. 

help desk help desk help desk help desk help desk help help help


First of all: once again, I'm sorry to be MIA -- I'm looking forward to becoming a regular on my own blog some time again soon.

Second: help! help!

C's second calculus test came back today: a disappointment, and a Bad Sign. And I am fresh out of enthusiasm for dealing with another year of high school math. Meaning: I am fresh out of enthusiasm for dealing with high school math teaching and high school math grading and high school math hiring of high school math tutors to remediate high school math grading and high school non-learning of high school math and on and on and on and then further on and on some more.

I'm done.

Also: calculus tutors are few and far between. I found only one last year, and the lone session he had with C. didn't help. Other parents have told me they couldn't find calculus tutors, either, and for years I've been hearing things like: "My son did well in BC Calculus. We were lucky because his father can teach it." As I recall, the mom who told me her husband could re-teach BC Calculus at home also told me that her husband's first cousin was an economist who had won the Nobel Prize. I'm pretty sure I'm not making that up.

Nobody in this house can teach calculus. I haven't even finished taking algebra 2, and Ed has forgotten the calculus courses he took in high school and college and doesn't care to revisit them.

So here we are.

Also - and this is a repeat - last year's math class was a total, effing disaster. The teacher was and is seriously ill but is still teaching, and the kids were and are dropping like flies. A huge percentage of last year's AP Calculus AB students got 1s on the AP test. Ones. And when I say "huge percentage," I mean eighty-five or ninety percent: the guy broke the bell curve. This was the teacher C. had for pre-calc, so C. is bringing to this year's calculus course an epic level of non-preparedness.

The teacher C. has this year is supposed to be fantastic and in fact told us he was fantastic on Back to School Night (he actually said: "I am it," which was exhilarating at the time), but if the grades are not great then the learning is not great, either. Fantastic is as fantastic does.

Ed is thinking we should just transfer C. to AP Stat (or maybe it wouldn't even be AP Stat - maybe just "Stat Honors," a class in which, according to C., students recently took a test on bar charts after completing a bar chart project) and be done with it.

That's pretty much how I feel, too, but it leaves the problem of algebra 2 and calculus, neither of which C. will have learned in high school -- and neither of which I want him taking in college since he'll likely be attending a school with kids who are a lot better than he is in math. At this point it is crystal clear to me that there is no reason on Earth to take a math course in college, pay for it, get a bad grade in it, and not learn any math. And I'm thinking the same principle holds true for high school. Why spend a single second of your (child's) life in a non-required math course so he can not learn math?

Last data point: C. did fine on the first test in the class and thought he did well on this test, too. He did great on the diagnostic test going in, and is doing well in physics. His mistakes on this test appear to be mostly "careless errors" (I have a whole new take on the nature of careless error thanks to the SAT) and a failure to follow correct notation, etc. In short, he appears to understand the material, not that I would know.

Of course, if that's the case then he needs more practice - but how do we swing that? The teacher doesn't seem to use a textbook, and I'm not going to be able to scrounge useful practice sets this go-round as I did for the 3 years when C. was in middle school.

(Could I talk to the teacher? Why, yes indeed I could talk to the teacher. But seeing as how talking to a math teacher has yielded exactly zero results over lo these many years, talking to the teacher wasn't my first impulse. Writing a help desk post was my first impulse.)

So...any thoughts?

We had a great, great precalculus tutor last spring at the very end of the school year; if we'd hired him at the beginning of last year, C. might have learned pre-calculus. We could hire him again (I assume) and decree that this year C is actually going to learn pre-calculus, which means we could pay tuition for C. to study bar charts at his Jesuit high school and pay tutoring fees for him to study pre-calculus here at home --- who says Americans have to de-leverage?! If you've got kids in school and you want them to learn math, deleveraging is not for you!

Or....or what?

What else is out there?

He could enroll in a community college precalculus course -- when?

Now? While he's in high school? (While he's in high school commuting 15 miles there and back every day?)

Then take calculus over the summer?

Another question: are there solid precalculus and calculus courses online? I took all of the ALEKS geometry course and part of the ALEKS Algebra 1 course, and as much as I want to like ALEKS, as much as I do like the tone and feel of the site, I don't think ALEKS replaces a good teacher or a good textbook. But do others out there think that might be the way to go?

What about online schools and colleges? I know there are universities that offer online math courses that kids take when they've been expelled or can't physically attend their local schools for some reason. Is that a possibility? Any recommendations?

And while we're on the subject, can anyone out there explain to me how in this country does a smart student with no discernible learning, attentional, motivational, or emotional difficulties whose talents and interests lie in history/social science actually learn some damn math?

What does it take?

'10 Reasons to Skip the Expensive Colleges'

On the heels of CassyT's post on America's Ten Most Expensive Colleges, here are some ideas on why you might want to avoid them.

Thursday, October 13, 2011

America's Ten Most Expensive Colleges

America's Ten Most Expensive Colleges—And How Much Financial Aid They Provide
Via Good Education.

My husband is a 1985 graduate of #6 - Claremont McKenna,  where the average student currently pays $20,423 of the $55,865 sticker price.

Wednesday, October 12, 2011

Front page articles on the edtech bandwagon

Fast on the heels of a front page New York Times exposé on how education technology has failed to raise test scores comes a front page Education Week article on the virtues of replacing teacher-centered lessons at school with technology-centered lessons at home.  The technology in question is that of the Khan Academy, whose library of lectures and problem sets is impressive in its vastness but not in its instructional feedback. If you input a wrong answer to a math problem you are told that your answer is wrong, but not why, nor how to fix it. Despite this, the Khan Academy has empowered teachers like 10th grade biology teacher Susan Kramer to skip over direct, structured instruction, and instead to watch her students "weave through rows of desks, pretending to be proteins and picking up plastic-bead 'carbohydrates' and goofy 'phosphate' hats as they navigate their 'cell.'"

To be fair, the Khan Academy (1) hasn't been around that long and (2) is the creation of a former hedge fund manager with degrees in math, computer science, and engineering but not in, say, cognitive science and child development. As such, the Khan Academy hasn't profited from the "over 20 years of research into how students think and learn" that underpins more established educational software programs like Carnegie Mellon's Cognitive Tutor.

So it was a bit disconcerting to find, fast on the heels of the Edweek's Khan Academy article, a front page article in Sunday's New York Times on Cognitive Tutor, and how it, too, has turned out to have no statistically significant impact on test scores. While I'd never had a chance to try it out (unlike the Khan Academy, Cognitive Tutor gates access to demos and charges big bucks instead of nothing at all), I'd heard only good things about it, and J enjoyed soaring through its algebra lessons during middle school. But as soon as I read the Times' description of its pedagogy, its limitations became crystal clear:
When the screen says: “You are saving to buy a bicycle. You have $10, and each day you are able to save $2,” the student must convert the word problem into an algebraic expression. If he is stumped, he can click on the “Hint” button.

“Define a variable for the time from now,” the software advises. Still stumped? Click “Next Hint.”

“Use x to represent the time from now.” Aha. The student types “2x+10.”
A math buff would soar right through this; for anyone else, the hints seem way too much of a crutch. There's no mechanism here for ensuring that you're working things out to the best of your ability before resorting to "hint"---i.e., nothing to stop you from clicking "hint" the moment you're not sure what to do. And what if your answer is almost right: say you forgot to include the initial $10, or let x stand for hours rather than days? As far as I can tell (I've now tried it out a bit), you're either right or wrong, and that's it. The program simply isn't sophisticated enough to highlight exactly what needs adjustment. And there's a very simple reason for this. As I discovered in creating a software  program that highlights grammatical errors in English phrases and sentences, this kind of perspicuous feedback takes a huge amount of coding (of the sort that you don't find in any other language teaching software program, thank you very much). Programming in the analogous feedback for mathematical expressions and equations strikes me as even more prohibitive.

On closer inspection, therefore, Cognitive Tutor seems inevitably to foster--in all but the brightest, most motivated students (the ones most able to basically teach themselves)--far too passive of a learning environment for lasting learning. Indeed, the only truly active learning environment that I've ever seen in any software program for any academic subject is that which a computer programming language platform provides for--what else?--computer programming. Only here does the feedback--the error messages or the unexpected outputs--precisely reflect what you've done wrong.

Will these recent exposés about the limitations of educational technology for subjects other than computer science have any effect whatsoever on the edtech bandwagon?

We might as well ask whether recent cognitive science findings have had any effect on how schools teach "higher level thinking." Or whether mainstreaming kids on the autistic spectrum has had any effect on mandatory group work and personal reflections. Or whether parental concerns have had any effect on schools choosing Reform Math. Or whether, for that matter, the Pope is Jewish.

(Cross-posted at Out In Left Field).

Monday, October 10, 2011

Who is Christopher Columbus? (Ask an Aspie)

"I don't know."

The revelation in question happened in late August, but I share it today in honor of someone who appears to be fading from America's k12 classrooms.

I actually wasn't that surprised when, while reading with J about the Age of Exploration in The Story of the World, Volume 2, it emerged that he didn't know who Christopher Columbus was. After all, one of the main reasons I've been working my way through this four-volume series with him is that I know he's picked up very little world history in the course of his 15 1/2 years. But, while he's still mostly oblivious to the incidental factoids that float all around him, he's increasingly attending to school, and increasingly sitting in the same classes, doing the same assignments, and taking the same tests, as everyone else.

So while I'm guessing that most (all?) of his schoolmates not only have heard of Christopher Columbus, but also know something about what he's famous for, I'm also guesssing that none of them learned these things from a social studies class or reading assignment that made them their focus.

Indeed, in this age where it's anyone's guess which facts our schools are making it their responsibility to teach, it occurs to me that students like J--with their narrow interests and their tendency to tune out most of the ambient information that others soak up without deliberate instruction--are a valuable resource. Next time you wonder whether your school is actually teaching (rather than merely mentioning in passing) the Bill of Rights (say), or the Cold War, or the Silk Road, ask an Aspie. That is, look for a spaced-out, narowly focused child on the autistic spectrum who hasn't made the topic their personal specialty, and see what he or she can tell you about it.

(Cross-posted at Out in Left Field).