the writing test and the math test

Chemprof (and others, I'm sure) pointed out in Comments that math/science professors value the SAT math test for the same reason I value the SAT writing test: both exams test standard mistakes that college students make.

Btw -- this is something I haven't gotten around to putting inside a post -- when I mention "the main errors student writers make," I'm referring to the Connor and Lunsford list of errors compiled in 1988, which is pretty close to the SAT list.

The Connor-Lunsford list is close to the SAT list except for the fact that Connor and Lunsford did not see ginormous numbers of parallel structure problems in the student papers they read, apparently. I find that hard to fathom. I personally do see ginormous numbers of non-parallel structures in the student writing that comes my way.

Faulty comparison, tested on the SAT, does not make the Connor-Lunsford list, either. (I'm not surprised by that.)

In any event, while musing about chemprof's observation (which I agree with, btw), an essential difference between the two tests, one that I hadn't focused on, suddenly leapt out at me: where SAT Math tests content and procedures students have been seeing in school for years,^* SAT Writing tests content students have never seen or even heard tell of unless their Spanish teacher happened to explain what a gerund is in Spanish class.

(I use that example because I asked C. this week whether he knew what a gerund was, and he said he did because he'd learned it in Spanish. I myself had no idea what a gerund was until this semester. Public schools don't teach formal grammar today and haven't taught formal grammar in decades.)

So....when you think about it....isn't the Writing Test a bit of an odd concept?

Students have never been taught grammar, and now they're being tested on grammar?

And why would I be in favor of testing students on content the schools don't teach?

Now I'm thinking: well, maybe I'm not!

Mulling over chemprof's comment, I realize that what I value about the writing test is almost exclusively the test prep kids do for the writing test. The fact of the writing test, the fact that that the writing test exists and students have to take it, gives parents an excuse to insist their kids learn some formal grammar before they graduate high school.

And that's pretty much it; that's what I value about the test.

So, since high scores on writing come entirely from test prep (at least in my experience), what does a high score on the writing test actually mean? Does a high score on the writing section tell us anything about the student's writing?

I don't know the answer to that, and I don't have a good guess.

Basically, I think it's a good thing for a student to recognize a comma splice in an SAT sentence regardless of whether he recognizes a comma splice in his own writing, and effective SAT prep can make that happen. This is a statement of value: I value knowledge of comma splices, and I want my kid to possess it.

^* Most of the content anyway. That's a subject for another post: these days the SAT now features counting problems, and students taking traditional algebra classes don't seem to have counting "units" in their courses (although chapters on counting  units are included in traditional texts). Ditto for the algebra 2 material on the SAT if a student has not taken algebra 2.

SAT math is tricky at all score levels

re: the a in f(x) = a - x² (question below), which appears in the College Board's Online SAT Test 5 and is designated "medium difficulty"

I should have made clear in the post I wrote about the post Debbie wrote^* that this question is not "tricky" for me. This question is easy for me, and the fact that this question is easy for me tells you nothing about whether I have mastery of quadratic equations and their graphical representations. At this point, I do not.

This question is easy for me because I have some basic understanding of shifts, because I have memorized the rules about shifts listed in all of the SAT test prep books (and most notably in Phillip Keller's book), and because I see the intended trick of the question the minute I look at it.

The fact that this question is easy for me probably does tell you I am pushing 700 on SAT math, which I am. I scored 680 on the real test, putting me at the 90th percentile for all test-takers; on sample sections my range is well into the 700s. I don't know the exact percentage of test-takers in the 90th percentile who get this question right and find it easy, but it's going to be very high. The top 10% of test takers is the group for whom this question is not tricky.

This question is tricky for test-takers scoring in the 500s, and the College Board knows it. They tell us they know it; they're not keeping it a secret. When the College Board assigns a "Medium" level of difficulty to an item, they are telling us that x number of kids scoring in the 500s will reliably get the question wrong, and x number of kids scoring in the 500s will reliably get the question right. That is the meaning of the words "medium level difficulty."

Kids scoring in the 500s are the ones you can depend upon to see a in f(x) = a - x² and think, in a certain percentage of cases, a-the-coefficient-of-x^2-in-ax²+bx+c=0. Those are the kids getting tripped up by this question.

Test-takers scoring in the mid-500s do not have mastery of quadratic equations and their graphical representations, and neither they nor anyone else is claiming they do. So when a designated percentage of kids scoring in the 500s get this question wrong, we learn nothing about them we did not already know. By the same token, when a designated percentage of kids scoring in the 500s get this question right, we also learn nothing about them. As a general rule, 500-scoring kids do not have mastery of quadratic equations and their graphical representations. I think that is a safe assumption to make.

Ditto for me. When I easily get this question right -- and, again, I stress the word easily -- no one knows anything about my level of mastery of quadratic equations and their graphical representations. I do not have mastery at present, yet for me this question is so easy it's a gimme. In Steve H's words,  I have (near)-mastery of the test, not the math.

So why is this question -- or, rather, questions just like it^**-- on the test?

Questions like this are on the test because they have appeared in the experimental sections of previously-administered SAT tests and have been found to reliably produce a certain level of error. The College Board needs test items that produce reliable levels of test-taker error in order to keep raw scores stable from test to test. They can't have everyone getting all the questions right; they can't have everyone getting all the questions wrong; and, in the big, mooshy middle, they need items that reliably produce a certain percentage of right and wrong answers.

Hence the a-question.

^* this is the mouse that lived in the house...
^** I don't know whether items in Online Course have been tested in Experimental sections. As I understand it, all items in the tests actually administered to real test-takers have indeed been tested in experimental sections taken by other test-takers.

the essay

C. was talking to his friend J. about their writing scores on the SAT.

After he hung up, he said, "I can't believe J. wrote the stupidest essay I've ever heard of and got a 9."

J.'s essay was about his family having to steal food in order to eat. Never happened.

"How come he wrote that?" I said.

"He spent too much time on the introduction and couldn't think of anything else to say."

Last year my neighbor's son got an 8 on an essay about why the essay question was stupid.

milestone

C. is applying to college today.

Assuming our electricity doesn't go out, that is. A small electrical fire from a fallen wire has been burning in the road next to our side yard since last night.

He wrote his what-I-did-with-my-summer-vacation answer on doing SAT math with his mom every day.

Susan S on her son's experience with competition math

re: the question of whether middle school competition math prepares students for ACT math:

I only have an anecdotal story concerning math contests and the ACT. When my son was taking the ACT in middle school at the beginning of the year, I hired a genius kid to tutor him on some of the things he had never experienced, like algebra 2 and some trig stuff. The tutor thought he would break 30, but he actually was in the middle 20s. Still not bad for a 13 yr. old.

As the year went on he made the Mathcounts team and had to practice the problems every week for a few months. He usually only finished half of them since we couldn't really help him. I signed him up for the actual Midwest Talent Search at the end of the year, but I didn't prep him this time or hire a tutor. His math score jumped to a 29 which placed him in the 99th percentile of the MTS, or Midwest Talent Search, kids and earned him a high scorer medal from Northwestern.

I have no idea if that means anything, but I thought it was interesting. He was only in accelerated algebra 1 at the time. Competition math just may help out in some way. It was the only thing I could think of at the time to explain the jump in scores.

If I had it to do over again, I would have had C. doing middle school competition math problems from the get-go -- and not just because SAT math is so closely related to middle school competition math.

C. and I both learned a tremendous amount prepping for SAT math. We learned a tremendous amount about algebra 1, geometry, and arithmetic, and we learned a tremendous amount about what we didn't know we didn't know.

puzzle math - from the comments

in an earlier Comments thread about SAT Math, gasstationwithoutpumps writes:

My son did no prep for the SAT when he took it in 6th grade and got a 720 on the math part. I'm quite sure that he'll do better on it when he takes it as high schooler next year without doing any SAT prep. The math questions just aren't all that hard.

to which SteveH responds:  

He had all of the material on the SAT by 6th grade? You are using this as normal?

gswp's reply:

No, he had not had all the SAT math. He'd had a very feeble algebra series in school (The Key to Algebra) and had self-taught geometry from a good book (he'd gotten through about half of Art of Problem Solving Geometry). The SAT supposedly tests much more than he had had at that point, but the questions are more like simple puzzles than testing content knowledge.

Gswp is exactly right! Setting aside the issue of the SAT using core features of cognitive function to produce error, which it does, SAT math is middle school puzzle math.

an SAT math tutor on seeing the 'trick'

A friend forwarded me an email from a parent whose high school age child rapidly sussed out SAT math and scored somewhere in the neighborhood of 800 with minimal test prep. The dad said his child has what he sees as a kind of "insight." After reading through some SAT math tests, the child could instantly see the 'trick' involved in the questions (trick used in the nonpejorative sense of the word). The dad said, too, that he believes this form of insight isn't necessarily present in students earning physics degrees from Columbia.

An SAT math tutor I know agreed and, when I asked why, wrote this response and gave me permission to post:

I've interviewed LOTS of math/physics/computer science majors from NYU and Columbia for potential teaching gigs who couldn't make sense of relatively simple SAT problems. Almost universally, the issue is that they WAY overcomplicate things.

Steve H mentioned Richard Feynman's hobby practicing "divergent thinking" problems. I think that's probably what this dad (and this tutor) are talking about.

Blue Book

October 2, 2011
the Blue Book

update update update

Quick note: C., who has left his calculus course for statistics (thank you all for the advice), just told us the calculus teacher is giving his students an essay test. Just to review: this is a calculus class populated by kids who learned next to no precalculus last year. They're in calculus, they don't know any precalculus, and...they're taking an essay test.

Ed said, "You should have stayed in the class. You could have raised your grade."

That was a joke. We both know it's not true. If C. had stayed in the class, he would have done worse on the essay test than he did on the math test, I'll wager.

Must get Allison's Comment about charismatic teachers pulled up front.

C. loves statistics. Loves! First math class he's ever felt that way about.

What if you're already a lean, mean educating machine?

I sit on a school board of a charter school that had its per pupil revenue cut again last year. We're under $6k per pupil, and we operate well on less money than the district schools have. Our school gives merit increases instead of the traditional step-salary schedule and this is now the second year that we have been unable to give any raises, excepting what is mandated by the state for the teacher retirement program.

Until the state raises PPR, however, the chances of teachers getting raises are minimal.  We ask a lot of our teachers and they deliver. How much longer will they be willing to do so? The surrounding district is still paying scheduled salary increases.

However...we do have a little bit of cash and so voted to allot some of it to provide a bonus for the teachers that have helped the school through the last few years when we were growing. Mind you, the amount of the bonus ranged between $700 -$1500. Hardly seems worth their hard work, yet as a Board, we want to acknowledge their dedication.

Last week, this arrived in my inbox:

Dear BOD Members,

I want to personally thank you for the bonus I will receive on October’s paycheck, as well as the hard work you put in to make XXX Schools what they are.  I recognize that money is hard to come by from the culmination of our economic situation and school growth.  Please know that I feel XXX is a fantastic place to be, and I look forward to helping us grow to be excellent in everything that we do.

Thanks again for thinking of me.

OK, so it's only one teacher, but I bet that's more personal thanks than the district board gets.

Net Price Calculator - get estimated college costs for YOUR child

After running the newly mandated Net Price Calculator for a dozen colleges using a few different scenarios, I've concluded that this can be a useful step in the initial phase of the college search process.

You can see the numbers I generated over at Cost of College.

Teachers' unions, explained

Making the rounds of some charter schools.
Enjoy.

New York SAT cheating scandal is expected to lead to more arrests

A former FBI chief is coming in to help clean up the SAT cheating mess.

Gaston Caperton, president of the College Board and a former governor of West Virginia, said that in addition to bringing in the former F.B.I. chief, Louis J. Freeh, as a consultant, the College Board was also considering additional safeguards over the next year, including bolstering identification requirements for students taking the SAT and taking digital photographs to ensure they are who they say they are.

Some educators think this action is long overdue and are calling for harsher penalties.

“The procedures E.T.S. uses to give the test are grossly inadequate in terms of security,” Bernard Kaplan, principal of Great Neck North, testified at the hearing. “Furthermore, E.T.S.’s response when the inevitable cheating occurs is grossly inadequate. Very simply, E.T.S. has made it very easy to cheat, very difficult to get caught.”

While the new security measures represent a change of tone for College Board and Educational Testing Service officials who previously insisted their system was adequate, some superintendents and principals said they did not go far enough. These officials have called for fingerprinting students, increasing stipends for proctors and imposing real consequences on those who cheat. Currently, if the testing service suspects cheating, the students’ scores are canceled and they are permitted to retake the test — with no notification to either their high school or colleges where they apply.

Educational Testing Service is already spending about 10% of its budget on security for College Board testing, but whatever they're doing may not be adequate.  Bernard Kaplan, principal of the high school where the cheating occurred, says this about the problem.

“It is ridiculously easy to take the test for someone else” 

Many young people have fake IDs, which are commonly  "being bought in bulk from vendors in China" and "nearly undetectable by bar employees".  I imagine SAT test proctors also find it hard to spot them.

Related:  Student cheating – the SAT, the Internet, and Ted Kennedy

(Cross-posted at Cost of College)

Is the Pope Jewish, II: classroom technology and the children of technocrats

In a post I wrote just two weeks ago, I discussed two New York Times articles about the failure of technology in the classroom to raise test scores in Arizona, and the failure of one of the most acclaimed educational technologies, Cognitive Tutor, to raise test scores in general. I concluded by asking:

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 the Pope is Jewish.

Sure enough, just one week later (last Wednesday), yet another NY Times article on online education appears, this one discussing how the Munster Indiana school district has jumped on the bandwagon:

Laura Norman used to ask her seventh-grade scientists to take out their textbooks and flip to Page Such-and-Such. Now, she tells them to take out their laptops.

The day all have seen coming — traditional textbooks being replaced by interactive computer programs — arrived this year in this traditional, well-regarded school district.

Munster's technological revolution was particularly sudden:

Unlike the tentative, incremental steps of digital initiatives at many schools nationwide, Munster made an all-in leap in a few frenetic months — removing all math and science textbooks for its 2,600 students in grades 5 to 12, and providing a window into the hurdles and hiccups of such an overhaul.

But Munster isn't the first to go digital:

Schools in Mooresville, N.C., for example, started moving away from printed textbooks four years ago, and now 90 percent of their curriculum is online.

and:

Munster’s is part of a new wave of digital overhauls in the two dozen states that have historically required schools to choose textbooks from government-approved lists. Florida, Louisiana, Utah and West Virginia approved multimedia textbooks for the first time for the 2011-12 school year, and Indiana went so far as to scrap its textbook-approval process altogether, partly because, officials said, the definition of a textbook will only continue to fracture.

The cost? Munster has paid $1.1 million for infrastructure while parents pay an annual $150 rental fee for laptops, Schools in general: are "spending an estimated $2.2 billion on educational software."

The benefits? No efficacy data is cited, of course. Students, to some extent, get to work at their own rates. And then there's this:

Angela Bartolomeo’s sixth graders spent a recent Wednesday rearranging terms of equations on an interactive Smart Board and dragging-and-dropping answers in ways that chalkboards never could. (In between, a cartoon character exclaimed that “Multiplying by 1 does not change the value of a number!” in his best superhero baritone.)

And this:

Ms. Norman, the seventh-grade science teacher, is using material from Discovery Education, which on that Wednesday included videos from Discovery’s “Mythbuster” series (commercial-free), an interactive glossary and other eye candy to help students investigate whether cellphones cause cancer. When Ms. Norman told the students to take out their ear buds to watch a video, two in the back yelped, “Cool!”

And this:

“With a textbook, you can only read what’s on the pages — here you can click on things and watch videos,” said Patrick Wu, a seventh grader. “It’s more fun to use a keyboard than a pencil. And my grades are better because I’m focusing more.”

Whether students are focusing on the right things is another matter. And wouldn't it be nice if there were explanations, rather than exclamations, regarding what happens when you multiply a number by 1? The basic problem with computerized instruction, as I noted earlier, is that it almost never provides perspicuous feedback. Answers are either right, or wrong, and that's it.

Perhaps no one knows the limitations of computer software better than computer software experts. Where are these people sending their kids to school?  An article in this weekend's New York Times provides a glimpse. Focusing on Silicon valley, it describes how the chief technology officer of eBay, along with "employees of Silicon Valley giants like Google, Apple, Yahoo and Hewlett-Packard," are sending their children to the area's Waldorf school:

The school’s chief teaching tools are anything but high-tech: pens and paper, knitting needles and, occasionally, mud. Not a computer to be found. No screens at all. They are not allowed in the classroom, and the school even frowns on their use at home.

Noting that "three-quarters of the students here have parents with a strong high-tech connection," the Times observes:

Schools nationwide have rushed to supply their classrooms with computers, and many policy makers say it is foolish to do otherwise. But the contrarian point of view can be found at the epicenter of the tech economy, where some parents and educators have a message: computers and schools don’t mix.

The article quotes Waldorf parent Alan Eagle, who "holds a computer science degree from Dartmouth and works in executive communications at Google, where he has written speeches for the chairman," and who "uses an iPad and a smartphone:"

“I fundamentally reject the notion you need technology aids in grammar school... The idea that an app on an iPad can better teach my kids to read or do arithmetic, that’s ridiculous.”

Of course, there is one particular way in which computers could be highly effective teaching tools: for instruction in computer programming. But has there has been a rise, or a decline, in computer programming instruction in the decades since schools began jumping on the edtech bandwagon? We might as well ask whether the pope is Jewish.

(Cross-posted at Out In Left Field)

Definitions, Precision, Coherence

What's missing from today's school math, and particularly, from middle school math?

Definitions, precision, and coherence.

Without a proper introduction to definitions, students don't get clarity in what a math statement is and what it is not. They don't get a sense of the abstractness of it. Without having actual definitions, they don't learn to work with definitions, so they can never learn how to use them to derive new things that are true of "all" of a given set. Without definitions, they cannot learn to REASON about mathematics because they have no basis for reasoning.

Without precision, students cannot make clear, unambiguous statements. They are not able to properly manipulate the symbols they are given, and they can't even say what their manipulations refer to and what they do not refer to. (As Wu says, In the same way that we do not ask “Is he six feet tall?” without saying who “he” is, we do not write down xyzrstuvw = a + 2b + 3cdefghijklmn
without first specifying what a, b, . . . , z stand for either (this is
an equality between what? Two random collections of symbols??
What does it mean??))

Without coherence, math is a series of unrelated facts designed merely to trick students. Without coherence, math is just a test of how big your working memory can be when you can never integrate any understanding together. There is no notion that the results you get follow from other results. Reasoning can't exist without coherence.

The reason SAT math feels so "tricky" to so many students is because they were never taught math in a coherent, reasoned way with definitions and precision. So the test seems to bejust a set of tricks designed to "catch" you, as opposed to a test of how what you know relates to other things you already know. That is why it seems to just test working memory. That is why it seems to have so much associative interference. The reason is because you were never taught that math was reasoned, with every piece of it following from the other pieces, a seamless whole that reinforced the same truths from a zillion different directions. And you were never taught what it meant.

summer boarding school and SAT prep

I talked to some friends I hadn't seen in awhile last night. They told me that over the summer they sent one of their kids to a summer boarding program where he prepared for the SAT.

His mom said he gained 60 points on reading, 90 points on writing, and 180 points on math.

I'm pretty sure this is the program: Wolfeboro The Summer Boarding School.

Steve H on SAT math and math prep

from the comments:

SAT math tries to trick students. You could say that the tricks relate directly to whether or not they really understand math. However, when you add in the time constraints, it really relates to preparation. Is preparation the same as mastery? Yes. Mastery of the test. Is this equivalent to mastery of math or whether you will do well in college math? Not necessarily. There are better ways of determining that than with the limited material included on the SAT. Why not just require students to take the Achievement Test? Look at the AP Calculus grade.

What is it about SAT-Math that is so important? They are trying to test something other than just math knowledge. They think that these tricky questions reflect on how well you think on your feet, but what it really does is test preparation and whether you have seen these questions before. The questions don't reflect on whether you have a wide body of knowledge and skills in math.

They create problems where you have to "see" the shortcut. You get problems with hidden 3-4-5 triangles. Add a time constraint and then what do you call those problems? It's not just about math knowledge and skills. The problem has to do with trying to determine the difference between aptitude and preparation. The tricks may have some basis in meaningful math, but that's not what they are trying to test.

It reminds me of questions companies like to ask at job interviews, like "Why is a manhole cover round", and "How many golf courses are there in the US.?" Preparation can make you look like you have a great aptitude. Preparation is directly related to math knowledge, and that is important, but identifying aptitude is an arms race for something like the SAT. That's causing the tricky problems, not any desire to test a breadth and depth of math knowledge.

In Dick Feynman's books, he talks about how he spent a lot of time in high school learning about all sorts of trick, lateral thinking problems. He would challenge people to ask him questions. There is nothing like preparation to make you look like a genius, although he really didn't need help with that. It really annoyed some of his colleagues.

My son will get to calculus in his junior year and he always gets A's. He still has to prepare for SAT-Math. He can't let others, with specific SAT-Math preparation, seem like they have a better aptitude than him.

and:

They try to trick students in most questions....What bothers me the most are the shortcut problems where using standard math techniques cause you to take too much time. This is supposed to identify aptitude, but it really tests preparation for the test.

There are also the problems where using a brute force or direct counting technique works better than any applied math technique. In some cases, there is no math to apply. One question on a sample PSAT test asked for the number of positive integers less than 1000 which don't have a '7' as one of the digits. (notice - "don't have" and positive integer) This simply checks how well you work under time pressure. Nobody expects you to apply any fancy math to this problem. One of the answers was the "have" solution. This tests preparation and practice, not aptitude or math ability. There may be a correlation between the test and aptitude or math ability, but not to the resolution colleges use it to select students. At the top levels, it correlates to preparation. That's not necessarily a bad thing, but there are better ways of figuring that out.

rat psych - "careless errors" in reading the SAT

During my year of living dangerously, doing SAT math prep off and on with C., I was chronically amazed stunned by the number and type of "careless errors" he and I both made taking timed sections of the test. In particular, I made repeated errors of "simple" reading, particularly when I was tired or the room was hot. I made so many reading errors that when I finally took the real test, I had no way to predict my math score at all: no way to estimate how many reading errors I had -- or had not -- made.

I eventually came up with a theory of careless errors, the details of which I've forgotten at the moment. I do recall that it had to do with working memory. Arguably the SAT tests working memory above all: all 10 sections put you into working memory blowout. I experienced working memory blowout so often that I began to notice a connection. As far as I can tell, you make more careless errors when your working memory is overtaxed (and you hit the limits of working memory much more quickly when you're sleep-deprived or overheated).

I've just come across a new study that I think confirms my subjective experience:

This study resolves two long-standing debates in the field. Does our working memory function like slots, and after our four slots [emphasis added] are filled with objects we cannot take in any more; or does it function like a pool that can accept more than four objects, but as the pool fills the information about each object gets thinner? And is the capacity limit a failure of perception, or of memory? [emphasis added]

“Our study shows that both the slot and pool models are true,” says Miller. “The two hemispheres of the visual brain work like slots, but within each slot, it’s a pool. We also found that the bottleneck is not in the remembering, it is in the perceiving.” [emphasis added] That is, when the capacity for each slot is exceeded, the information does not get encoded very well. The neural recordings showed information about the objects being lost even as the monkeys were viewing them, not later as they were remembering what they had seen.
Picower: 1 Skull + 2 Brains = 4 Objects in Mind

Failures of working memory are failures of perception!

Subjectively, that's what I experienced taking practice sections; that's what it felt like. Once I hit a certain level of tiredness, or heat, or working memory blow-out, I stopped being able to read.

The same thing happens on the reading and writing sections, too. The reading and writing sections are so taxing that you reach points where you simply cannot take in what the sentence or paragraph before you says. ^* I'm not talking about losing the ability to answer questions about the sentence or paragraph.

I'm talking about losing the ability just to read the words on the page.

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

^* I say "you" because I know I am not alone in this.

rat psych - what to do about SAT math (part 3)

Your typical high school student, I presume, has spent several years setting up equations and solving for x. At least, let's hope so. I certainly did.

The SAT uses this fact to elicit many wrong answers from test-takers who have worked a problem correctly. The student gets the solution right but the answer wrong because the answer isn't x. The answer is 3x, say, or xy. I seem to recall a problem or two where the answer was -x, for god's sake, but I might be making that up.

Other times the test will give you a value for x + y, say, and you're supposed to see that you should simply insert that value some place else in the problem, et voilà: the answer they're looking for pops up.

Here's a typical problem, medium difficulty (according to the College Board):

If 4(x + y)(x - y) = 40 and (x - y) = 20, what is the value of x + y?

A kid who's had no test prep at all will likely miss this question -- either miss it outright or take too much time spotting the solution, thus leaving him too little time to finish the test and increasing the likelihood he'll make "careless errors" on the questions he does get to because now he's working too fast trying to make up for the time he lost on the x + y problem.

For what it's worth, I think using x + y as the value, instead of x or y alone, is an interesting and instructive way to write a problem. (I'm curious what math people think). It seems to me that writing problems in which x + y is the salient unit may be a way of teaching what Ron Aharoni calls the fifth fundamental operation of arithmetic:

In addition to the four classical operations, there is a fifth one that is even more fundamental and important. That is, forming a unit, taking a part of the world and declaring it to be the “whole.” This operation is at the base of much of the mathematics of elementary school. First of all, in counting, when you have another such unit you say you have “two,” and so on. The operation of multiplication is based on taking a set, declaring that this is the unit, and repeating it. The concept of a fraction starts from having a whole, from which parts are taken. The decimal system is based on gathering tens of objects into one unit called a “10,” then recursively repeating it.

The forming of a unit, and the assigning of a name to it, is something that has to be learned and stressed explicitly. I met children who, in fifth grade, knew how to find a quarter of a class of 20, but had difficulty understanding how to find “three-quarters” of the class, having missed the stage of the corresponding process of repeating a unit in multiplication. What I Learned in Elementary School by Ron Aharoni

Maybe I'm wrong, but it seems to me that the x+y questions test math as opposed to obedience under pressure, which is what the Find xy questions test.

Still, there is no doubt in my mind that these questions elicit wrong answers from test takers who know the math involved, can do the math involved, and have a reasonable understanding of the math involved. Students who have spent years of their lives solving for x aren't going to break the Solve for x habit for the first time ever when they're working at breakneck speed and their eyes are bleeding from the Ella Baker passage.

Which brings me back to extinction learning. Test prep for SAT math involves spending a fair amount of time building new habits that conflict with ingrained old habits. You've been conditioned to solve for x; now you have to condition yourself not to solve for x. Also, you have to build as much speed as possible at not solving for x because you are never going to forget solve-for-x. The two impulses are inside your head, competing with each other, and the competition takes time (and probably eats up some precious working memory resources to boot).

Funny thing: during the time we spent doing SAT math prep around here, I overlearned don't solve for x to the degree that a couple of weeks before taking the real test I came across a practice problem that did ask the test-taker to solve for x. I was so surprised that I wasted several seconds reading and re-reading and re-reading again to make sure I hadn't misunderstood. You can't win.

For parents: your child needs to spend enough time not solving for x that he or she gets to be really, really fast at not solving for x.

Then he should be on the lookout for problems that say Solve for -x.


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

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.

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.

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. 

Exciting!

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

arrghh

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.

'10 Reasons to Skip the Expensive Colleges'

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.

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).

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).