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.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.
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
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
I agree that "the x+y questions test math." I don't think that it takes a lot of test prep to learn this, if the math was learned in the first place.
ReplyDeleteIf only procedural algorithms were learned by rote, without understanding, then it might take a very long time to learn more general procedures.
I think that questions like this test two things: ability to do math and ability to read questions carefully. Both are useful skills for a college admissions test to be trying to measure.
I guess I see your concerns about SAT math as more indicative of the poor way that math is taught, with much too much emphasis on solving for x. One of the worst cases I remember was a student who could only solve for x if it were on the left hand side. If she got something like 20 = 4x she didn't know what to do. After 15 minutes of trying to convince her that she could just divide by 4, I finally realized it would be easier to teach her that she could switch the two sides of the equation.
ReplyDeleteHaving said that, while she could solve for x, she didn't know how to do math in general, and that is part of what the SAT is testing for. They are TRYING to trip up students who are too procedural, because those students are not going to be able to handle the math they need to do for STEM majors at institutions where everyone has math SATs of 750 or more. This can become a problem for students who are not mathy kids but who need to do as well as they can for college, where they will proceed to do as little math as possible.
As for the bad drawings, there may be more than there were ten years ago, but I do remember these diagrams from the SAT back in the 80s. I think the larger number of them may just be due to more emphasis on geometry in general.
"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."
ReplyDeleteI laughed out loud when I read that...
No, a kid who understands symbols, functions, and the law of associativity, commutativity, and distribution will confidently understand "what is x+y" with no test prep.
ReplyDeleteYes, the x+y question is a good one--and iirc, you missed it when it was given in the form of r+s, asking to find a specified length in a picture showing two semicircles.
But I disagree here:
"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."
I don't know what that "reasonable" means, so maybe your wiggle room here is the size of Kansas. But I'd say someone who got this wrong didn't have a "reasonable" understanding. They are a superficial understanding that showed under pressure to be minimal.
That test takers get this wrong because they were mistaught is true. The "solve for x" idiocy is a disaster that prevents students from understanding expressions, equality, and the basic laws of mathematical manipulation.
But the SAT isn't trying to test "did you learn what your school taught you", are they?
Worse, I think the test prep here is red herring.I suggest that *test prep* --in the sense of a formal course not in mathematics but a longish term of studying to game the test-- probably undermines this understanding.
I am beginning to think that test prep as I defined it above is like a set of training wheels on a bike: they can get you up and moving, but they hinder actual bike riding. The training wheels are a crutch at odds with the actual goal of mastery of a bike because they don't teach you the necessary skills to reach that mastery: namely, learning to balance.
Instead they are devices that *limit your risk*, but that is NOT the same goal as learning to ride. You can't win a bike race with training wheels on. Test prep does the same thing. You won't fall down and bomb the thing, but you can't win the race with their strategies, either.
re: "solve for x":
ReplyDeleteWu gave a talk here
http://math.berkeley.edu/~wu/Bethel-5.pdf
excerpt:
"Textbooks tell you how to “solve” x^2 - x - 1 = 0
by completing the square.
...(see slide show for the equations)
The whole process is highly unsavory: It is a case of wanton manipulations of symbols with no reasoning whatsoever.
First, x is just a symbol, but the equality x^2 - x = 1 means the two symbols x^2 and -x combined is equal to the number 1.
How can a number be equal to a bunch of symbols?
The passage from x^2 -x = 1 to (x^2 -x + 1/4) = 1 + 1/4
is usually justified by “equals added to equals are equal”, which is in turn justified by some metaphors such as adding 1/4 to two sides of a balance, with x^2 -x on one side and 1 on the other.
Finally, even ignoring all the questionable steps, how do we know
1/2* (1 +- sqrt(5)) are solutions of x^2 -x -1?
Allison:
ReplyDeleteMost verbal SAT prep works exactly the same way. It's a bunch of tricks and strategies that are designed to help kids eliminate wrong answers but that teach them absolutely nothing about how to actually derive the correct answer from the text itself or how to reason one's way from point A to point B. A kid who's developed those kinds of reasoning and comprehension abilities through years of reading challenging, adult-level material likely has most of those skills whether or not they've ever had any test prep.
People who complain that the SAT phrases questions in unfamiliar ways are completely missing the point: the whole GOAL of the test is to see whether someone's understanding of fundamental mathematical and reading concepts is such that they can apply their knowledge to a totally novel situation. If a kid can only solve for x or understand a problem if it's written in a specific way, the underlying knowledge just isn't totally there. It's the same thing that kids go through trying to memorize 5,000 words, then getting tripped up on a question that requires them to use a common second meaning of a very common word (e.g. "conviction") in a way that doesn't immediately jump to mind for someone who doesn't have a lot of exposure to writing in general.
"..use a common second meaning of a very common word (e.g. "conviction") in a way that doesn't immediately jump to mind for someone who doesn't have a lot of exposure to writing in general."
ReplyDeleteLearning definitions directly should cover simple variations in meaning. With learning definitions by reading, I found that I learned the wrong spin to lots of words, and that gave me problems on standardized tests. I find that if I know the official, general definitions of a word, I'm more able to extrapolate to other usages. My son reads a lot, but I am often surprised by what words he doesn't know, or how he misunderstands words just like I did. Our solution is not to get him to read more. It's to have him study word lists and root definitions. This is a sensitive issue for me because our K-8 schools make a BIG DEAL about reading a lot - as if that absolves them of any need for directly tackling things like spelling, grammar, and definitions. Besides, they were never very good about selecting good versus bad reading material.
"If 4(x + y)(x - y) = 40 and (x - y) = 20, what is the value of x + y?"
ReplyDeleteThere is also wiggle room in declaring that this is just testing a true understanding of math. But what is the roll of time in this case? The SAT goes out of its way to find problems that people haven't seen before. With the time constraint, it's not just a matter of whether you know math or not. It's whether you have seen it before so that you can finish the problem in time. The issue is whether they are testing aptitude or preparation.
However, this is a very common problem for SAT and anyone who does some SAT prep will be ready. Does this problem reflect a proper understanding of math if you have 1.25 minutes to finish? That's not what they are trying to test.
I came across another problem of this ilk at PWN the SAT. It was:
X^2 + Y^2 = 14; XY = 3; What is (X+Y)^2?
To solve this in the allotted time, you need more than an understanding of math. You need to understand the SAT.
First, this is not testing a general math approach to an unknown problem. It's testing your ability to solve a SAT math problem. You know, a priori, that there must be a quick solution. That's not math. You also know from doing other SAT tests that you must be able to combine things together to find the answer. Don't solve for X or Y. That's the only way to do things fast. You can't factor the first equation, and you can't do anything to the second, but you can expand (X+Y)^2. You should know that the expansion will give you an XY term. Substitution! In the real math world, it's never that simple, and in the real world, being methodical is much more important than speed.
Does this relate to a proper understanding of math? Of course it does. With the time limit, is there something else going on here? Of course there is.
At a SAT score of (my guess) 600 or less, you can make a big argument for a direct correlation between knowing math and your SAT score. Above that point, it's more about specific preparation for SAT-Math. There is quibbling room, but that gets squeezed down to nothing when you add in the time factor. As students prepare more, ETS goes out of its way to find some unique new special case or variation. Does it reflect a real understanding of math? That's not what's going on here.
I think the SAT is trying to test what your school taught you -- if you were in an honors class rather than gen ed Algebra for All.
ReplyDeleteThe problem Steve posted is instantly recognizable to someone that is taught on today's honors level..x^2+y^2=14,xy=3; what is (x+y)^2..easy to say oh, I recognize that: (x+y)^2 is always x^2+2xy+b^2 and I just mentally substitute and solve. It's Dolciani Alg I, Ch 5 in my edition and there are two more factoring patterns that should be committed to memory there -- and everyone did that back in my day. Without taking the course to that level, a student may just look at the problem posed and say..ok I have the equations of a circle and a line. Now what?
Algebra for All avoids symbolism. Even for reciprocals, you'll never see examples generalized to a*1/a=1 or 1/a*a=1.
Catherine, you mentioned you didn't take Alg II. If you had, the medium problem would be easier as you would have learned the Substition Axiom of Equality as your teacher showed you proofs and you worked exercises and problems. (Substition Axiom of Equality: For all real numbers a,b,c, and d: (1) if a=b and a+c=d, then b+c=d; (2) if a=b and ac=d, then bc=d. A NY student would have learned this in Integrated Alg I in the binary operations unit, if it was taught as math rather than a procedure (our exp was gen ed taught it as a procedure, accel taught it as math however ymmv).
The conclusion I draw is again, Algebra for All is just as bad as Fully Included Elementary School math classes. Very little is done. Parents have a tiny lifeline when the counselor sends home a JHU-CTY flyer if they aren't in the know.
"The problem Steve posted is instantly recognizable to someone that is taught on today's honors level..x^2+y^2=14,xy=3; what is (x+y)^2.."
ReplyDeleteIt takes more than good math classes to get a good SAT score. This problem is easy if you know it's a SAT problem and are prepared. It's easy to point to any one SAT problem and find real math, but the time factor is huge. I was always annoyed to find test problems like this that were special cases. Who would ever think that there would be a simple substitution? ... those taking a SAT test. This is not math without a context. The context is that you have 1.25 minutes and you know that there has to be an easy subsititution. Why would a general approach to a math problem include a search for 3-4-5 triangles? The only reason you would look for that is because it's the SAT test. They can't test to see if you know a general solution approach to a problem that has no trick and that uses not-nice numbers.
My son figured out this problem in 15 seconds, but he knew the context. I then gave him another problem that didn't substitute easily. How long do you spend on finding a simple substitution in the real world? What does it mean if you can quickly find the substitution on the test? Does it mean that you have aptitude, test preparation, or that you really know math? A proper understanding of math requires the demonstration of problem solving when you don't have special cases. Can students find a solution if they have x-y=3 rather than xy=3? Do they "see" that they now have a circle and a line and that the solutions are the intersections? What do the different solutions mean? What if they had x-y=10?
I was reading a math book yesterday in the geometry section and realized that there are so many different variations and ways to come up with tricky problems. You can look at any one of these problems and point back to a specific book, chapter, and page where it may be mentioned, but it's quite another thing to claim that students are then expected to solve special cases in a very short amount of time. They are testing some vague idea of aptitude, not real math ability. However, they are in an arms race with those who prepare well.
The context of SAT math is to spot the special cases (tricks). This is the only way ETS can keep the bell curve from being truncated at the top. Their goal is to separate students in the context of a limited domain of material and a small amount of time for each question.
Below 600, I expect a linear correlation of SAT score and math ability, but above 600, there is probably a linear correlation with specific SAT math preparation.
I disagree with SteveH about SAT prep being most useful above 600. I think that it is most useful for getting students from 450 to 550, not for moving students nearer the extremes.
ReplyDeleteThe math question "x^2+y^2=14,xy=3; what is (x+y)^2?" does not require any SAT prep for a top math student—the method for solving it is obvious. A struggling student may benefit from specific training in figuring how to do it, but that is not going to make much difference for the top students.
Certainly the SAT could ask harder questions (like he suggests, using x-y=3 rather than xy=3), but then they would either have to ask fewer questions (increasing the random noise in the measurement) or make the test longer (and we've already heard complaints here about fatigue during the test).
Are you saying that a "top math student" need not prepare for SAT math?
ReplyDeleteI disagree. A top score depends on speed and speed depends on preparing for all of the special case (not general case) problems. My son (a top math student) can look at any problem and know that he can solve it ... but not in the time allotted. This is not just about getting faster with a general approach to solving math problems. It's about getting faster with all of the special case problems of a SAT math test.
Specific SAT math prep might help a 450 student, but the help will be very limited. Improved test scores will depend more on learning more math. The 450 student won't "see" the xy term in (x+y)^2.
SteveH asked 'Are you saying that a "top math student" need not prepare for SAT math?'
ReplyDeleteYes, that is what I was saying.
A student who struggles a bit with the math may get some advantage from doing prep and learning special cases, but the top students don't need the extra study.
Perhaps we are seeing different students with different skill sets.
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.
The kids who are making 780-800 don't need the extra prep, because the things they do for fun -- recreational math puzzles, math contests, computer programming, etc. -- already prepare them far beyond anything the SAT will test them on.
ReplyDelete"My son did no prep for the SAT when he took it in 6th grade and got a 720 on the math part."
ReplyDelete6th grade? He had all of the material on the SAT by 6th grade? You are using this as normal?
"The kids who are making 780-800 don't need the extra prep, because the things they do for fun --"
So it does take more than just the regular classes to do well. "Math contests" are not test prep? Duh! case closed.
Math contests are not test prep for the purpose of prepping for the SAT, and if you don't see that, I don't know how to explain it to you, because we are speaking in different languages.
ReplyDeleteI have on my bookshelves probably 50-100 recreational math books that I've been collecting since I was about 10. Did they prep me for the SAT? Well, maybe, but that's not why I was reading them. I read (past and present tense) them because I enjoy them.
I also did my share of math contests when I was in junior high and high school, but these were put on by the local U, not international contests, so all we did was show up and do the contest; we didn't prep for anything.
--So it does take more than just the regular classes to do well.
ReplyDeleteThe purpose of the SAT is to distinguish who has mastery and who doesn't, on a scale that makes more than a binary subdivision into these groups.
The regular k-12 curriculum doesn't teach to mastery, so by defn, it takes "more" to do well on the SAT.
Whether that "more" was an internally driven desire to learn more math, a natural inclination to mathematics that meant they "got" it when they saw it, or someone pushing a student to mastery by afterschooling or math competition competing isn't something the SAT disambiguates.
Real mastery requires speed and accuracy. It requires reading what's in front of you and making no assumptions. It requires "seeing" what's relevant. It's a level of maturity in problem solving. Someone recently had a GREAT definition of it, but I've forgotten their wording, unfortunately.
"knowing it's an SAT math question" changed absolutely nothing about how I approached a problem, and didn't have to.
"Real mastery requires speed and accuracy...Someone recently had a GREAT definition of it, but I've forgotten their wording, unfortunately."
ReplyDeleteDefinition of fluency (specifically in regards to mastering math facts) is:
1. Efficiency-speed (4 seconds)
2. Accuracy
3. Flexibility
Does that help?
The purpose of the SAT is to distinguish who has mastery and who doesn't, on a scale that makes more than a binary subdivision into these groups.
ReplyDeleteHaving taken every math section available, and having spent a fair amount of time talking to SAT math tutors, I would say that the SAT has a couple of different purposes.
One purpose is to produce as many errors as possible regardless of a test-taker's level of mastery.
You really need to look at this test through the lens of learning, memory, and cognitive science.
One aspect of the test I haven't managed to post much about is the working-memory-blowout aspect: some of the 'hard' problems are hard only because they have more components to keep track of, which means more components than working memory can hold at one time.
The problems are printed in tiny font, and you aren't allowed scratch paper, so you have pretty close to no ability to use paper and pencil as an aid to working memory.
This means you spend more time swapping problem elements in and out of WM than you would with decent font size and a full piece of paper to work with -- increasing the odds you'll lose track of an element and make a "careless error."
Working memory is extremely limited in all people; a math test that makes problems hard by making them too long to remember isn't a math test. It's a neuropsych test, and if you're going to give a neuropsych test, just give the n-back to everyone and be done with it.
I disagree with SteveH about SAT prep being most useful above 600. I think that it is most useful for getting students from 450 to 550, not for moving students nearer the extremes.
ReplyDeleteI prepped myself. Lowest starting score was 570; I tested at 680 when I took the test (and that was with a calculator mishap).
Test prep is definitely useful above 600. The high-end tutors here produce gains in precisely that range.
Math contests are not test prep for the purpose of prepping for the SAT, and if you don't see that, I don't know how to explain it to you, because we are speaking in different languages.
ReplyDeleteDidn't Glenn Ellison confirm that SAT math is quite close to middle school competition math?
I'm pretty sure he did.
If I were a parent of a middle school child, I would definitely be having him or her work with Glenn Ellison's book and/or the Art of Problem Solving book.
Below 600, I expect a linear correlation of SAT score and math ability, but above 600, there is probably a linear correlation with specific SAT math preparation.
ReplyDeleteI'd love to know if that's true.
I need to look more closely at the 'easy' and 'medium' questions. Some of them have way too many variables to deal with for the tiny little space you have to write in ----
Catherine wrote
ReplyDelete"The problems are printed in tiny font, and you aren't allowed scratch paper, so you have pretty close to no ability to use paper and pencil as an aid to working memory."
My understanding was that you can write on the test book and that there is plenty of white space there for doing scratch work. I've not taken an SAT lately, so I can't verify this from personal experience.
If you have bad eyesight or difficulty writing small enough, then perhaps you could request accommodations that have the test in a bigger font with more paper. I believe that such accommodations are given to those with poor vision, but documenting need for accommodations is supposedly a bit of hassle and may take months to be approved.
SteveH asked "He had all of the material on the SAT by 6th grade? You are using this as normal?"
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.
No, I don't regard his scores as normal, nor was I claiming that. I was just providing a counterexample to the insistence that test prep is essential for getting a good math score.
Since then he has had geometry, Algebra 2, and Precalculus. He'll have had single-variable calculus before he takes the SAT for real. I suspect that the extra content of the math courses won't help him at all on the SAT, but he may have gotten a bit better at problem solving, since the AoPS precalculus and calculus classes have a lot of problem-solving practice (though generally of the think-for-a-while type, not speed problems).
--One purpose is to produce as many errors as possible regardless of a test-taker's level of mastery.
ReplyDeleteNo, it's to produce errors that FORCE the distinction between those who have mastery and those who don't. The forced errors exploit the lack of flexibility and speed, and lack of deep structure understanding.
--Working memory is extremely limited in all people; a math test that makes problems hard by making them too long to remember isn't a math test
The problems aren't hard if you immediately see them. If you've got the deep structure understanding chunked, you're not taking working memory. If you need to do work to expand (x+y)^2 and substitute in values, yes, you're taxing memory. But the right way to do that problem was not to need to do work to do the expansion in the first place, but to just know it.
Hey Cassy - I've never seen 'flexibility' listed by the precision teaching people.
ReplyDeleteThe list for fluency that I've always used is:
1. speed
2. accuracy
3. automaticity
Automaticity is a huge part of fluency.
Developing automaticity for SAT math is a big part of test prep.
I developed so much fluency in NOT solving for X (instead solving for xy or x+y or whatever) that when I hit a problem where I was supposed to solve for x (literally x - that was the variable used), I spent several seconds checking and re-checking to make sure I was really, truly supposed to solve for x.
SAT math is a genre; it's its own thing. When you've done as much SAT math as I have, you recognize it: you know when you're seeing an SAT math problem and when you're not.
Test prep means becoming as fluent as you possibly can in SAT math per se.
fyi: In high school I took no more math than what would today be covered in algebra 1, if that, and I took a rural IL geometry course. I've worked my way through all of Saxon Math books 1 & 2.
My SAT score--680--puts me in the 90th percentile of the country.
My understanding was that you can write on the test book and that there is plenty of white space there for doing scratch work.
ReplyDeleteThat's not the case. You have a very, very small piece of real estate in which to work. If you make a mistake, you have to erase all of your work; you can't move on to a clean space; and you have to do this under time pressure. Every second counts, literally.
btw, when RH looked at C's calculus (precalculus, actually) test, one of the major points she made had to do with use of white space and with working memory. I wonder if she'd let me post her comments....The SAT makes it impossible to follow RH's advice.
Tiny font size is a problem for everyone, not just for middle-aged people whose eyesight has changed. I'll try to dig up the research on this.
What happens with tiny font size is that, because you have no white space, you are severely limited in your ability to pull out the critical elements from the problem and line them up in your "problem space" (if that's the term). I think test-takers should do this anyway, and just take the risk they'll run out of space, but it's a gamble.
If you don't pull out the problem elements, you have to constantly re-read the tiny print in order to re-find the elements, which is highly costly in terms of time.
Tutors teach you to underline the critical elements of problems, which is essential, but you're still dealing with such tiny print that underlines don't ease the WM burden.
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.
ReplyDeleteAs 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 MAPS 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.
Susan
Oops, that acronym that I used (MAPS) is supposed to be MTS, or Midwest Talent Search.
ReplyDeleteNot enough coffee...
Susan
No, it's to produce errors that FORCE the distinction between those who have mastery and those who don't.
ReplyDeleteThis account doesn't jibe with the way in which questions are chosen.
Essentially, questions are chosen by 'popular vote.'
Each SAT test includes an experimental section; the test taker does 10 test sections, but only 9 count. The College Board selects questions for future tests from the experimental section based on how many kids got which questions right. A "hard" question is a question X percentage of students got right; a "medium" question is a question more-than-X students got right; an "easy" question is a question many-more-than-X students got right.
That's all there is to it in terms of the final selection process; there is no independent criteria representing mastery vs nonmastery.
This method of selecting problems means that the parabola questions can be easier than the algebra 1 questions. The reason the parabola questions are easier is that many test takers have not taken algebra 2 and cannot answer a parabola question. Thus the high fail-rate on parabola questions puts parabola questions in the "hard" category.
fyi: I am far from mastery on parabolas and quadratic equations. (I think I'm probably reasonably close to mastery on linear equations & their graphs...)
I learned just about everything I needed to know to get quadratic equation-parabola problems right from Phillip Keller's very brief explanation in his book of how to get SAT parabola questions right.
That's not mastery.
That's learning how the SAT asks the "hard" questions and practicing doing those questions until you're really fast at doing them.
For the record: SAT quadratic equation/parabola questions always ask you how the graph will shift when you change the equation. Up or down; left or right. You can learn how the graphs shift as a matter of procedure with no conceptual understanding at all.
The SAT test produces errors at all levels, not just the "hard" level.
ReplyDeleteI think (would have to check) it's the "easy" questions that ask you to find xy or x+y instead of x.
The test isn't designed to produce errors just at the top of the distribution; it's designed to produce errors at all levels.
gasstation wrote: 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.
ReplyDeleteRIGHT!!!
gasstation is right.
SAT math is puzzle math; that's why it's fun (and it is fun - I had a blast prepping for the test).
Is puzzle math the same thing as "math math"?
Are mathematicians and physicists good at puzzle math by definition?
(I don't know the answer to that question -- but I do know that there are certainly levels of mastery, or near mastery, that do not make you a whiz at puzzle math.)
They are TRYING to trip up students who are too procedural
ReplyDeleteI just don't think the evidence of the actual test supports this.
You can solve the quadratic equation/parabola questions using only procedural knowledge. I know because I've done it and so has C.
I think it's right to say that you can solve the counting problems procedurally, too. Once you learn the logic of listing the alternatives, you get everything right.
C. has extremely limited understanding of counting, and once an SAT math tutor taught him the standard SAT math tutor way of solving SAT counting problems, he got all of them right.
SAT problems are always chosen so as to produce the same curve, and the selections are based on how many students got each question right or wrong when the questions appeared on experimental sections.
Catherine,
ReplyDeleteA student *can* solve all of the counting problems procedurally, but chances are, they will run out of time or make an error--those errors you might call "careless" will be more and more likely the more times you proceed procedurally. Kids not resorting to the procedural method won't make those errors and won't need as much time to check their work.
You don't need to work especially hard to select problems to produce the same curve--that's how big distributions work. it's the same reason a straight-scaled class in college produces a bell curve in grades anyway.
and re: how procedural knowledge was enough for the parabola problem,
you got flummoxed by the "a", right? that was a parabola problem. you might have gotten it right in the end, but you were twisted up and wasted time on it at the very least.
chances are, they will run out of time or make an error
ReplyDeleteThat's the way it seems, logically.
But if you talk to SAT math tutors who are working with kids scoring 650 and above, it's not the way it works in reality.
They more often get the answer right via listing.
LOTS of what seems true of the SAT turns out not to be true when you actually deal with the test. It's counterintuitive.
I thought what you thought when I started.
you got flummoxed by the "a", right?
ReplyDeleteNo.
That was DS's example, not mine.
I'm talking about the shift problems. I do them purely procedurally, and I always get them right, and I get them right in about 2 seconds flat.
My guess is that at some point enough ordinary, middle-class students who can't afford $150/hour SAT tutors ($150 is the starter price, btw) & don't know about Phillip Keller's book will have figured out the shift procedure that those problems will disappear from the test.
ReplyDelete