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Wednesday, November 14, 2007

Re: “You’ve Got A Problem on Your Hands.”

posted by on November 14 at 15:09 PM

We’ve already established that Seattle Weekly’s Laura Onstot is not good at math. (She reversed the numbers at the secretary of state’s election site and concluded, contrary to all other media reports, that the simple majority for schools measure was losing when it was actually winning.) However, she apparently wants to make that really, really clear. In a post about how tough the WASL is (Annie, wanna weigh in here?) she posed this question as an example of a really hard math question:

test.jpg

Kent is using the scale to compare the weight of various solids.

How many spheres will balance one cube?

A. 2
B. 3
C. 4
D. 5

“Remember,” she notes, “you’re being timed.”

Now, I’m certainly not a fan of high-stakes standardized tests like the WASL, and I do think there should be different standards for ESL students and those with learning disabilities. But I’m a little shocked that 42 percent of high school kids didn’t answer that question correctly. I haven’t had any kind of math at all in more than a dozen years, but I figured it out pretty fast. It’s B, 3. I’ve buried the math below the jump.

Here's how I did it (I'm sure there's a much simpler way, but like I said, no math for a while). First, substitute letters for shapes. Cubes are x. triangles are y, and spheres are z. So the top scale can be represented like this:

4x+2y+3z=3x+4y+5z.

Subtracting from either side (and eliminating some steps), you get:

x=2y+2z
2y=x-2z
2z=x-2y

Now go to the second scale. Take out one sphere on both sides (z) because they cancel out. So you have: x=4y+2z. Put in terms of just x and z (the cube and the spheres) it's like this:

x=2(2-2z)+z

or:
2x=x+3z

therefore:

x=3z.

So the correct answer is, three spheres (z) balance one cube (x).

Unless, of course, my math is totally off.

RSS icon Comments

1

It is simple and I could do it in 6th grade. It's just solving 2 equations simultaneously. The only hard part is substituting variables for the pictures.

Posted by Andrew | November 14, 2007 3:22 PM
2

Yup, looks right to me. Cute little problem.

Posted by gfish | November 14, 2007 3:22 PM
3

As a former SAT teacher, I can assure you that this problem would be impossible for most of my students. It's not that they're dumb. The problem is, these problems are very different from the ones they get in actual math class. High school students do math in a pretty rote form. When you throw them a curve ball, even a curve ball that by all accounts they should understand, they fall apart.

It doesn't help that that picture is as blurry as all get out.

Posted by arduous | November 14, 2007 3:27 PM
4

be thankful that you had good, solid middle class educations. the kind of educations that made it possible to do such math in the sixth grade. not all kids, in our economically diverse school districts, were or are as lucky.

if the 42% number shocks you, than it is time to read Kozol's Savage inequalites or pedagogy of the opressed one more time.

Posted by SeMe | November 14, 2007 3:29 PM
5

laura is very nice, and rather intelligent as well. you should be nicer to her. i'm sure she would rather work at the stranger anyway....

Posted by infrequent | November 14, 2007 3:34 PM
6

There's more than that, @4. I think that a significant number of people seem to decide at some point that they are not Math People, and, having decided that, just instinctively see a problem like this and just say "Nope! Math! Can't do it!" Even many reasonably intelligent, well-educated I know are like this, particularly people who majored in non-technical fields.

Posted by tsm | November 14, 2007 3:36 PM
7

The only error I see is in this line:

x=2(2-2z)+z ... should be x=2(x-2z)+z

And that's merely a typo.

Posted by T | November 14, 2007 3:36 PM
8

"reasonably intelligent, well-educated people I know"

I guess we people who majored in technical fields can't proofread.

Posted by tsm | November 14, 2007 3:37 PM
9

No, you learn proofreading when you work on the student newspaper. Most techs don't do that.

Posted by Will in Seattle | November 14, 2007 3:39 PM
10

25% of people believe 9/11 was part of biblically-prophecized end times. So the idea that 42% of students couldn't solve this problem doesn't seem that hard to believe.

Posted by Big Sven | November 14, 2007 3:41 PM
11

I'm a former SAT teacher as well, and I think this question is quite elegant. It tests two things well: 1) Do you really understand the definition of a variable? and 2) Can you solve simultaneous equations? The actual manipulation of numbers is minimal, which lessens the chance that kids will get it wrong because of nervous errors in addition or subtraction.

Posted by annie | November 14, 2007 3:45 PM
12

ECB, are you sure you didn't receive some help with this?

Before you get all uppity with Laura Onstot, recall that you've struggled in the past with the elementary arithmetic concepts of set, subset, intersection, and union.

Posted by Luigi Giovanni | November 14, 2007 4:04 PM
13

i'd like to be the first to say that i couldn't have even begin to solve that problem on my own, and then i got lost during the explanation.

i'm fucked.

Posted by greg | November 14, 2007 4:09 PM
14

Count me among the 42% who would have missed this.

My inability to even realize that I should change the pictures to X, Y and Z certainly doesn't mean that I fall apart when a "curve ball" gets thrown at me. It means that my brain is wired to freak out when it sees math problems like this and loses all interest and the ability to go any further. I looked at that problem and saw a bunch of shapes that, if I really cared, I *might* have started counting and hoped I could have come up with a pretty educated guess.

It's not because I'm stupid. It's because my brain works differently from the "average" person's. (I can handle "curve balls" and not "fall apart" if they don't involve math.)So my question is: What does all this matter? Does it prove the girl at the weekly is stupid? no. Does the 42% number ECB threw out there from where ever she got it mean anything? No.

It's just a question that I am sure instills fear in kids today that if they miss it, they're doomed to a life of no good.

Speaking as someone who grew up in NY -- a state that has forced students to pass standardized tests each and every year in a variety of subjects since my Grandmother's day -- I know that's not true (I bombed my Trig Regents and I'm a successful writer). This kind of testing/question shows nothing and does no good.

Posted by Charlie | November 14, 2007 4:10 PM
15
I havenít had any kind of math at all in more than a dozen years, but I figured it out pretty fast. Itís B, 3.

Well, look at that -- girls can do math after all.

Posted by Judah | November 14, 2007 4:10 PM
16

Using C for cube, T for triangle, and S for spheres:

Top shows 1C = 2S + 2T
Bottom shows 1C = 1S +4T

Which means 2S + 2T = 1S + 4T
and 1S = 2T

Substitute 1S for 2T in any of the originals, and you've got 1 cube = 3 spheres.

Posted by JenK | November 14, 2007 4:10 PM
17

For an example of comment #6, see comment #14.

I didn't even get variables explicitly involved. I just looked at the second picture and said "Okay, one cube equals four pyramids and a sphere..." and went from there.

Posted by Ben | November 14, 2007 4:27 PM
18

It takes a bit of work, but it's do-able. Even in Vegas, half the kids failed the standardized tests for math and writing, and the math test was basic crap like 23x32=???, while the writing test was basic shit, like 'Why are apples good for you? Write a basic five paragraph essay, max one page.' How do you fuck that up?

Posted by Gomez | November 14, 2007 4:31 PM
19

Er, assuming of course that the apples questionwas preceded by a 2 paragraph passage explaining why apples good for you.

Posted by Gomez | November 14, 2007 4:33 PM
20

I failed this miserably, even after passing the LSAT. Easy to see you need to reduce it to an algebraic equation. I couldn't, for the life of me, resolve it though.

I submit that this is difficult math and further submit that in my entire adult life, I've never had solve a problem like this in real life. I would, of course, use an electronic scale. Next they'll ask me to do my taxes with an abacus.

Posted by Medina | November 14, 2007 4:36 PM
21

The whole "lets turn every shape into a letter" thing shuts down a lot of people's brains. Nor is it really necessary for the math, it just makes writing it shorter. Here's an equivalent non-algebraic explanation.

Between the two pictures, from the top to the bottom one, we see that three cubes, two triangles, and two balls were removed from the left side, and three cubes and three balls were removed from the right side. Since the scales were still balanced, that means the stuff that was taken from either side weighed the same.

3 cubes         3 cubes
2 triangles  =  0 triangles
2 balls         3 balls

So, 1 ball weighs the same as two triangles. So to see how many balls balance one cube, just use the bottom picture and take one ball off either side. We're left with

1 cube  =  4 triangles
           1 ball

Turn each pair of triangles into balls, so 2 pairs of triangles becomes 2 balls. So 1 cube equals 3 balls.

Algebra just makes it easier to write down (and remember) formulas. It doesn't really make math itself easier.

Posted by K | November 14, 2007 4:36 PM
22

I want to point out that this is a special case: you can't usually solve three-variable problems with only two equations.

Posted by Big Sven | November 14, 2007 4:39 PM
23

Or ... we should realize that teaching "how to solve problems" is a more recent scholastic innovation, and the distance one has from school has a correlation with one's ability to solve such problems quickly.

When you're used to taking tests like this, everything looks like a number 2 pencil.

Posted by Will in Seattle | November 14, 2007 4:40 PM
24

And the republicians think that 1.5 Trillion = 50 billion. No wonder we are fucked.

Posted by xx | November 14, 2007 4:44 PM
25

Big Sven,

You can solve for the ratio of two variables with three equations, which is what you're doing here.

Anyway, this problem is a slight step up from very basic algebra. It's a valid question if you think any high school graduate should be able to use algebra in real life when required to.

Posted by MHD | November 14, 2007 4:44 PM
26

I always find it interesting how people solve math problems in different ways.

While I tend to be very math oriented, I solved this problem without any substitution of X, Y, and Z or other figures. I just subtracted blocks, triangles, and circles in the upper picture so that the left side had 1 block and 1 circle. Then set the resulting right side on top equal to right side on the bottom and subtracted out duplications on both sides, bringing 1 circle is equal to two triangles, then substitute the circles for the triangles on the bottom and subtract a circle from both sides and you're there.

Follow-up question. What's 12 x 14? How did you calculate it? Multiply 12 by 10 first, then by 4 and add? Multiply 14 by 10 first, then by 2 and add? Approximate by squaring 13, then pinpointing the answer by looking at 2 x 4? Reaching for the calculator . . .

Posted by NealH | November 14, 2007 4:45 PM
27

@22: That's because you're not solving it here, you're just reducing it. You can't actually *solve* the equations, until someone gives you the real weight of one of the objects.

Posted by K | November 14, 2007 4:47 PM
28

Perhaps I should have closed mrskin.com while attempting to solve this problem. Multi-tasking decreases algebraic skills.

Posted by Medina | November 14, 2007 4:50 PM
29

I couldnt make out what the shapes were because of the poor quality of the scan.

Posted by Bellevue Ave | November 14, 2007 5:05 PM
30

well hey, nice post if you feel the need to publicly validate that you're smarter than someone else. gold star!

Posted by smarty pants | November 14, 2007 5:16 PM
31

I hate being wrong. First the outcome of "Prop 1", now this. Twice in one year. 2007 is turning out to be a real cock-up.

Posted by Big Sven | November 14, 2007 5:19 PM
32

I'm not sure if I'm part of the 42% or not. I was taking 'honors' math courses in high school, but it has been a few years since I graduated. I got totally befuddled trying to solve it myself, but I could follow every one of the solutions step by step, and could fill in the intermediat stuff once someone pointed me in the right direction.

Posted by Jason | November 14, 2007 5:21 PM
33

No friggin' clue. My brain shuts down when I see stuff like this. I feel anxious just looking at it.

But hey, I did Honours English and graduated with distinction. Give me a poem or a painting any day!

Posted by Irena | November 14, 2007 6:10 PM
34

Someone explain to me why Washington high school students have to learn algebra in order to graduate.

Let's just keep raising the standards higher and higher so more and more kids drop out without learning anything that will actually enable them to get jobs.

Posted by keshmeshi | November 14, 2007 6:10 PM
35

@34 - math is one of those things that will actually enable them to get jobs. And it will be increasingly necessary. There's no particular reason that Washington high school students can't learn math; kids in public schools in almost every other developed nation on earth somehow manage to do so at a higher rate than those in the US.

Posted by tsm | November 14, 2007 6:16 PM
36

An interesting fact not related to the actual math or relevant at all: Everyone on here identified the cube and the sphere by their three dimensional name, all but one called the Pyramids triangles. I did this myself at first. I wonder why that is.

I don't think it's the quality of the image, since I knew that they were 3-D. Erica calling them triangles first may have affected some people, but I didn't read her explanation first.

Posted by Chris in Tampa | November 14, 2007 6:21 PM
37

Welcome to the real world, ECB. Perhaps now you can understand why some of us are distrurbed that we are handing out high school diplomas to kids who can't do sixth grade math. Perhaps now you understand that people who say "I'm opposed to high-stakes testing" really mean "Not only would my kid not get a high school diploma if he had to master sixth grade math, but I wouldn't get one either."

The WASL is not timed. You can take re-take it if you need to. It has many questions which, like this one, test the ability to apply knowledge rather than just give rote answers. In short, it's a very well-designed test.

Posted by David Wright | November 14, 2007 6:24 PM
38

BTW, if I sound defensive, I am! Still traumatized by high school, I guess.

Posted by Irena | November 14, 2007 6:27 PM
39
I submit that this is difficult math and further submit that in my entire adult life, I've never had solve a problem like this in real life.

This all-too-prevalent attitude of "When am I gonna use this stuff anyway?" we have towards education is deeply depressing. In my entire career, the early history of the United States has never come up in my work; this does not mean that it is useless to me to know it. For that matter, more involved math can indeed show up in your everyday life; it's just that if you don't know it, you never see how it is applicable to you.

And even if you don't use math in your career, you'd better hope that the engineers, scientists, and software developers that create and refine all the technology around you didn't sleep through this in school.

Posted by tsm | November 14, 2007 6:29 PM
40

Charlie, Medina, Irena: I am very glad that all of you have managed to become valuable, productive, members of society without having mastered sixth grade math. Really, I am. But I don't think any of you should have high school diplomas.

A high school diploma isn't supposed to mean that you are a valuable, productive member of society: if it were, we should hand them out on the streets for doing good deeds. It isn't supposed to mean that you sat on your butt in a classrom for four years: why would we want to give you an award for that? It's supposed to mean that you have mastered the core high school curriculum.

Posted by David Wright | November 14, 2007 6:39 PM
41

Maybe my fancy southern education is spoiling me here, but I think it's pretty silly to believe that high schools exist for any reason other than to keep most teenagers out of the way as much as possible for as long as possible. I mean, ideally, sure, everyone can apply algebra and knows what the difference between a conspiracy theory and a scientific theory is. But in reality, that's what college is for.

Posted by Chris in Tampa | November 14, 2007 7:15 PM
42

David @40 -- So what would happen to us, and to all the kids who are like us, if you were king of the world?

Posted by Irena | November 14, 2007 7:51 PM
43

@17 -- I did the same thing and still got it wrong. Came up with 5. Argh.

Posted by idaho | November 14, 2007 8:09 PM
44

Okay, what am I doing wrong? Using the bottom scale, taking away the sphere on the left means that one cube equals 4 triangles (aka pyramids) and 1 sphere. Getting rid of all the extras on the top scale means 1 cube also equals 4 triangles and 1 sphere. So combining those two, we find that 1 sphere equals 2 triangles. Therefore, ohhhhhhhhh, found my mistake. I was turning that into 1 triangle into equaling 2 speres. See, even if we comprehend how to do the problem doesn't mean one's brain does all the addition and subtraction *right*. So are we testing comprehension here or execution? That's why I always liked getting partial credit on math tests.

Posted by idaho | November 14, 2007 8:16 PM
45

I love a few things about this problem;

1. it's algebra with shapes
2. you can apply this problem to real life
3. math isnt hard until you get to a point somewhere beyond calculus like topographical geometry.

Posted by Bellevue Ave | November 14, 2007 9:37 PM
46

Irena: Well, if you wanted very much to do something that required mastering the core high school curriculum, you would presumably work very hard to do so. And if what you wanted very much to do didn't require mastering the core high school curriculum, you presumably wouldn't bother. And any college or employer who wanted to know whether an applicant had mastered the core high school curriculum could, you know, look to see whether that person had a high school diploma and know that's what it meant.

You seem to imply that we get some societal benefit from handing out certifications which we all know doen't reliably certify what they claim to. Just what benefit would that be?

Posted by David Wright | November 14, 2007 10:09 PM
47

I also like how a fellow slogger helped me with my first quarter back into real algebra

Posted by Bellevue Ave | November 14, 2007 10:11 PM
48

We're all hosed anyway. Here's a sample Chinese university maths entrance test. I thought I was good at maths until I tried this. Solved, yes. Quickly, no. http://news.bbc.co.uk/2/hi/uk_news/education/6589301.stm

Posted by Paul Barclay | November 14, 2007 10:58 PM
49

David, I accept your frustration with an education system that graduates students who can't do algebra (or interpret literary symbolism, for that matter). And it would be nice to live in a world where all the pieces fit snugly together, and where the paper certificates we attach to people were a better guarantee. It would be especially nice if everyone could just conform to the standards laid out by the designers of our institutions, because those designs are really very elegant. But here are all these messy people who just don't fit in.

Well, oddly enough, the professionals I came into contact with as a university student seemed to understand that, and never bothered to check my algebra skills before I signed up to study languages and literature. And, funny, I don't recall ever having to parse a sonnet to get a job as an administrative assistant in a high-tech firm. And yet without that high school diploma, both of those doors would have been closed to me.

Your view of how our society should operate sounds really good on paper, but I'm afraid it might not deliver in its application. What do you intend to do with all those students who don't fit your standards for graduation? Let them scramble for menial jobs because they had bad math teachers?

Seems like a terrible waste to me.

Posted by Irena | November 15, 2007 12:09 AM
50

Hmmm..i wouldn't have gotten it at all. And I got A's in all my Algebra classes in high school. I wouldn't have known to sub an equation for the shapes unless the problem gave me some sort of hint about it. Guess I can't do math, oh wait, I did take 3 quarters of Calculus in college as well. Hmmm..lol.

Posted by Brian in Seattle | November 15, 2007 12:35 AM
51

Irena: You seem to assume that, even if a high-school diploma meant something different than it does today, it would still be used as a prerequisite for the same things. That seems rather unlikely.

Right now, a high school diploma doesn't mean much more than "can sit through 6 hours of instruction a day without being too disruptive". So applicants who need to show more than that must take additional tests: colleges require SAT score, businesses administer literacy tests, etc. Those private certifications do their jobs fairly well.

It's okay with me if we simply eliminate high school diplomas and let the private certifications take over. Alternatively, we could use the cultural significance of "graduating from high school" to push students to master skills they otherwise wouldn't, by using standardized tests to change a high school diploma from an "I can sit still" certificate to an "I have mastered a core curriculum" certificate; this is the plan our society is currently trying. Yet another possibility is for us to follow the British school system and offer students an array of tests which will certify them to varying degrees in specific subjects, instead of an all-round certification. ("I got my A-levels in English and B-levels in maths. Does that qualify me for this job?")

But it is really useless for us to continue to issue "I can sit still" certificates in the perverse hope that some misguided employer will mistake one for a real qualification and hire an unqualified applicant into a non-menial job, in a bizarre and doomed attempt at some kind of pseudo-"affirmative action". People are going to find the societal roles for which they are qualified. Our educational certification system can either play a useful role in that sorting or it can be a useless side-show. But what it can't do is magically turn a useless certificate into a useful qualification.

Posted by David Wright | November 15, 2007 1:02 AM
52

@16 - jenK for the win, with style and ease.

@26 - I knew 12x12=144 and added 24.

@36 - Me too. Very curious. Maybe because Erica stated "spheres and cubes". Had she not defined those terms we might be discussing this problem as circles and squares.

Posted by Paulus | November 15, 2007 2:28 AM
53

That looks a lot more like a lever than a scale and the distance from the fulcrum would matter. You would need A) a ruler, and B) AP level Physics, to solve this problem. Maybe Golob could weigh in?

Posted by fubar | November 15, 2007 4:53 AM
54

@53: levers are easy, just a cross product of the leverarm (length to fulcrum) by the force (weight) of the blocks. Since the applied force is perpendicular to the lever arm, the sin is 1 and you just multiply weight by the leverarm to determine force. Find where they equal and !presto!, there's your balance point.

Posted by LT L | November 15, 2007 5:07 AM
55

paul, that problem isnt hard at all. it is actually easy because of the values they gave you and all you need to know is your trig functions.

Posted by Bellevue Ave | November 15, 2007 8:08 AM
56

Paul. that problem is easy because of the values given and using your basic trig functions and identities.

Posted by Bellevue Ave | November 15, 2007 8:12 AM
57

@37 said: Perhaps now you understand that people who say "I'm opposed to high-stakes testing" really mean . . . [etc.]

Um, actually, no. The current mania for testing has been a disaster for our educational system. It's like weighing the pig twice a day but never feeding it.

Posted by Margaret L. | November 15, 2007 9:43 AM
58

@48 I could solve those questions without having to find a paper to write on. They do not deserve to be questions that judge entry into a university.
And no I'm not insanely awesome at math, I'm just slightly above average.
It looked like complicated bullshit to me until I actually I read what it was saying.

Full disclosure: I'm Indian :)

Posted by arandomdude | November 15, 2007 10:12 AM
59

You want to see examples of difficult questions go look up pre 1970's state level exams. Those were the ones that people who got 70-80% on were damn well geniuses.

Posted by arandomdude | November 15, 2007 10:15 AM
60

the tests are a problems for many reasons, but first and foremost they reveal that the students are not learning.

Posted by infrequent | November 15, 2007 10:23 AM
61

"ECB" loves to jiggle her junk and do the "I'm bigger" thing with her fellow female journalists.

Posted by Stephanie | November 15, 2007 12:08 PM
62

Maybe I'm just very young, but speaking as someone who has taken (and passed!) the WASL, and is an alum of the Seattle Public Schools, I could give a few reasons why students would get this question wrong, based on my experiences in local public schools. Maybe it's because, in the span of one year, my class had four different teachers and two different assistants for the "Integrated 1" level of math...the level where you learn basic algebra. Maybe it's because my 6th grade math teacher was fired for having porn (which my classmates actually discovered) on class computers, or maybe it's because I have never been in a math class with fewer than 30 students. So when I look at this problem and go "Hmm, that looks complicated, I'll just guess," perhaps it's just a reflection of the apathy in the whole system.

Posted by amp | November 15, 2007 1:46 PM
63

I wish more liberal arts types would have as much regard for science and mathematics as I (an engineer and science-y type) have for art and music and literature.

But not romantic poetry. BO-RING!

(only half joking on that last part.)

More seriously, the most interesting people I talk to are people who are literate on a wide variety of subjects, rather than stovepiping their interests based on what they learned in high school.

Posted by Lee Gibson | November 21, 2007 11:08 PM

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