Tuesday, February 23, 2010

Apropos of nothing...
There were two candidates in an election. Total of 2.8 million votes were cast. Candidate I earned 28,000 more votes than Candidate II. All votes were either for Candidate I or Candidate II. What percentage of the total votes did Candidate I earn?
The Korean volunteers as an SAT tutor for a 10th grader who gets As and Bs. (The Korean reviews his students' grade sheets himself.) In the 30 minutes that the Korean and the student spent on the question above, the student --

- could not understand that Candidate I earned 14,000 more votes than 50 percent.
- did not know how many zeroes there were in 2.8 million.
- could not subtract 28,000 from 2.8 million without a calculator.
- could not convert "Candidate I earned 28,000 more votes than Candidate II" to "x = y + 28,000".
- could not solve for x and y with the two equations "x + y = 2,800,000" and "x = y + 28,000".
- could not divide 2,772,000 by two without a calculator.

Again, the Korean's 10th grader student gets As and Bs. An average 5th grader in Korea could solve this problem. Math education in New York City public school system is criminal.

1. I would've able to solve that problem in 6th grade in public school in Oregon. Didn't even need to understand algebra--would've been able to handle everything but percentage in fifth grade, sixth grade would've covered percentages. Granted, I would've preferred to use a calculator to get the percentage (long division=no fun), but I would've been able to do it by sixth grade, once I learned about percentages/ratios.

Is your student also getting B's in math? And is that student just taking basic algebra now?

Granted, I had a lot of personal initiative as a student, but I was never tracked into upper tier mathematics programs until 7th grade and so I was just learning what everyone else was learning until sixth grade.

2. I hope you at least taught the student to eliminate obviously wrong answer choices.

That might be the only way to get by. And once he/she does, I hope he/she is not my doctor, airline pilot, or dentist.

3. Those numbers are frankly too big and time consuming to be dividing by hand. Tests are all about speed and accuracy, something that calculators provide. However the fact that the kid didnt know how many zeroes in 2.8million is quite er... sad to say the least.

4. Solving for x initially rather than y would have eliminated the need to divide 2 into all those sevens in 2,772,000.

x + y = 2,800,000
y = x - 28,000
x + x - 28,000 = 2,800,000
2x = 2,828,000
x = 1,414,000

1,414,000 = per.(2,800,000)
1,414,000/2,800,000 = per.
per. = 0.505 = 50.5%

5. Whoa. I went to a high school fifteen years ago that was ranked just worse than 100 in the state of Illinois, and I think most the students that got As and Bs in that school could most certainly have solved that in 6th or 7th grade. Either things have gotten much worse, or NYC is especially bad.

6. Actually, now that I think about it, in most inner city schools, most of the teachers' focus is on maintaining order in the classroom. If this student goes to a school like that, then this isn't a surprise, nor is it anything new.

7. As a recovering GRE math instructor myself, I'd say you guys are going about it the long way. SAT and GRE math questions are designed to have a shortcut. That 2.8 and 28K have an inherent relationship to each other and that the question is calling for a percent, not a number of votes, are two clues.

In this case, it's that 28,000 (28 followed by three zeroes) is 1/100 or 1% of 2.8 million (28 followed by five zeroes).

So the difference between them is 1%, half of that being 0.5%. 50% + 0.5% is 50.5%.

8. @kushibo: The kid needs to be able to do it the long way before understanding why your shortcuts work. In any case, I did just fine on the SAT- a 1420 in 7th grade and then a 1590 in high school- by doing things "the long way" at a decent pace.

9. I think the easiest way to solve the problem is to convert to percents right off the bat, which eliminates the problem of having to divide numbers like 2,772,000 by 2. While there's no excuse for a high schooler not to be able to do that without a calculator, SAT problems are generally written in ways that they can be simplified beyond just brute force, so on the SAT, students have to be able to be able to quickly spot patterns and gimmicks in the questions to minimize work. (The SAT would never ask a question where the candidates won 2.8 million votes and candidate 1 won a random number like 47,500 votes more than candidate 2, for example.)

Total votes candidate 1 won more than candidate 2: 28,000

Percent more of the vote candidate 1 won than candidate 2: 28,000 / 2.8 million = 1%

Splitting that 1% evenly from 50/50 --> 50.5% for candidate 1, 49.5% for candidate 2.

In general, the state of the educational system really is sad. Have you seen the ongoing blog running in the Times about teaching kids math? (http://opinionator.blogs.nytimes.com/2010/02/21/division-and-its-discontents/) Read the story about the Verizon customer service representatives and the full transcript (http://verizonmath.blogspot.com/2006/12/transcription-jt.html) and weep for the sorry state of simple math understanding...

10. Wanda, that would depend on how much remedial stuff he/she can master. If not much, then it's time to go to the answer elimination method, narrowing the odds.

Making sure the kid knows how to write out numbers and how many powers of 10 there are in 10, 100, 1000, ... , 1,000,000 is some basic remedial stuff that is especially useful on the SAT/GRE.

The thing is that in the vast constellation of ETS math, the SAT and GRE tend to focus on a few stars. Over and over again.

11. *sees math problem; brain automatically seizes up like an overheated engine*

Once again, it's proven how much I suck at math. *sigh*

Don't worry, people! You won't have to encounter the likes of me at your health care centers!

12. Peta-Ann Smith I would rather see your sense of humor at my health care center rather than any "mad math skills!" :-p

13. @Juan: Basic math skills (we're talking 5th, 6th grade level stuff here) and a sense of humor are not exclusive. Math is not scary. (At least not until you get to linear algebra and complex analysis.)

@kushibo: I see your point about the student who needs to improve his scores in a rush. Seeing someone have to resort to those kinds of tricks (ruling out obviously wrong answers) because he can't do simple algebra is incredibly frustrating. I would also want to teach the kid some real math.

14. Wanda wrote:
@kushibo: I see your point about the student who needs to improve his scores in a rush. Seeing someone have to resort to those kinds of tricks (ruling out obviously wrong answers) because he can't do simple algebra is incredibly frustrating. I would also want to teach the kid some real math.

나도, hence the recovering GRE math instructor label.

Teaching GRE in California was frustrating at time, dealing with kids who skated through school and expected their parents' money to give them a boost. At least in Korea you had people who were working hard and were just being obstructed by an unnecessary and unfair test.

Ah, don't get me started.

WORD VERIFICATIOIN: aushoot, sort of "Ah, shoot!"

15. @Wanda, never said they were exclusive :)
I was just pointing out in real life things like humor (which peta-ann seems to have in abundance) may be more relevant.

Now you've got me started :(

As someone who has to deal with numbers and quantitative methods everyday, I sometimes find myself wondering if I am looking at the real picture, or just using tools of the trade to procreate more bullshit.
I may look like I am performing heavy duty mental lifting because I am crunching numbers through uber complicated models, but in reality there is little critical thinking involved. What I see often around me (including me) are people who know the tools but are too lazy to truly think and become relevant to the real world.

Call it heuristics, call it surviving by being mentally efficient, call it what you like, but the fact remains we tend to value tools and people who learn them well a lot more than the ability to think about an issue critically and from different perspectives.
Why? It takes too much effort, so we teach short cuts, pre-packaged tools, mental crutches to speed up the process and lessen the pain. While these tools, short cuts, and mental gimmiks are important and are great in lessening the burden of our everyday and professional lives, too much of a reliance on them, especially at a young age, stifles the spark and will to truly think and look at the world with a healthy dose of skepticism.

tK, as can be seen from his previous posts does not really care to differentiate critical thinking and dismisses it. I would agree to a certain degree as much of the supposed "critical thinking" spouted by "experts" are nothing more than more packaged tools. tK being born with his critical reasoning spark, thinks it is something natural, that all people are curious, skeptical, objective and have the energy and will power to wrap their mind around an issue from multiple perspectives, to think out of the box.

While Kushibo's solution method is awesome and I would definitely use his form of "short cut" when in a test situation, I for one would nudge my student to first think and then learn the "tools." We so often just teach the "tools" and forget to stimulate the thinking aspect.

We tell our young ones that they need to study really hard and learn (which is of course important!) but we also imply that it is okay to be lazy when it comes to thinking. And believe me, it is perfectly possible to learn math with minimal thinking involved (I've done it) if what we are trying to teach and learn are nothing more than mere tools, algorigthms. Higher thinking? What a joke.

To come back to a full circle from my 삼천포 like tirade, yes I would definitely like my health care center person to have a sense of humor like Peta-Ann, and hopefully a decent enough intellect and expertise to look after my health problems. However, that person does not need to know the finer points of math test time shaving techniques ;)

16. I was scared for a second: I was getting a very different answer than everyone else's, and thought my math skills had atrophied dramatically (which is particularly disturbing since I'm in engineering). Then I realized I'd misread the problem: I thought it was asking for the percentage difference between the votes of Candidate I and Candidate II (which is about 2%), not the total percentage of Candidate I's votes. So apparently it's only my critical reading skills that have atrophied. :)

To make a more serious point, I was able to recognize right away that 28,000 and 2.8 million only differ by orders of magnitude, and divide by 10^3 (1000 for you non-math folks) to simplify the math somewhat. Kids should be taught to see the relationships between numbers so they can recognize these kinds of shortcuts when they appear. I've always had somewhat of an innate ability to see these relationships, but I think I'm in the minority.

17. The Korean initially tried the shortcut, but again, even after 10 minutes the student could not get the idea that Candidate I earned 14,000 more votes than 50 percent.

So the Korean told him to do it his way. He managed to set up "x + y = 2.8 million", but could not figure out how many zeros there were in 2.8 million. The Korean thought it was worthwhile for him to do it his way, except he could not set up the second equation. He got the idea after several minutes of banging his head against the wall. Then he could not solve for the variables. And he could not do basic arithmetic.

Overall, the Korean is a much better English tutor than a math tutor. He simply does not know how to teach anyone math other than beating the shit out of him/her, which is (regrettably) not allowed in America.

18. Hmmm... maybe he doesn't do well with abstract math symbols?

There are two monkeys in an all you can eat banana buffet.

There are a total of 2,800 banans in total. (Explain that for the purpose of this problem you can get rid of the three zeros for calcualation convenience sakes)

Monkey 1 and monkey 2 together eats all 2,800 bananas.

We don't know how many bananas each monkey ate yet so we will call the unknown number that monkey 1 ate X and the unknown number that monkey 2 ate Y. If the two monkeys ate the same number of bananas it would be the two same unknown number, so two Xs. Algerbraically 2X = 2,800. To solve for the unknown number just divide 2,800 by 2.

Here the monkeys obviously did not eat the same number of bananas so it's X + Y = 2,800.

Darn~! But you do know that greedy monkey 1 ate 28 more bananas than mokey 2. Okay now, let's think here for a moment. Monkey 1 ate 28 more bananas. Hmmm.. So if monkey 1 wasn't greedy and ate 28 more bananas, the two monkeys would have eaten the same number of bananas!
Algebraically, monkey 1 ate 28 more bananas than monkey 2. Remember monkey 2's unknown banana consumption was Y. So the number monkey 1 ate is X equal to = Y + 28. Let's update our list so we have everything straight.

X = consumption by monkey 1
Y = consumption by monkey 2
X + Y = 2,800; total consumed
X = Y + 28; number consumbed more by monkey 1

Since we know what X is now (to a certain degree) lets plug in X.

(Y + 28) + Y = 2,800

Hey we have two Ys here! So,

2Y + 28 = 2,800

We do a little algebra magic, really simple stuff.

2Y = 2,800-28 = 2772
Y = 2772/2 = 1,386
so Y the number that monkey 2 ate is 1,386.
and X the number that monkey 1 ate is (Y + 28) = 1,386 + 28 = 1,414.

Since the question wants to know what percent of the total bananas monkey 1 ate.
Total # of bananas = 2,800
Total # of bananas digested by monkey 1, X = 1,414

We divide the number consumbed by monkey 1 by the total number and voila 1,414/2,800 = 0.505!
To show that in percentages, move the little dot two places and you get 50.5%~~~

Really simple stuff... explained so that a fifth grader (with very basic algebra knowledge) would understand.
Oh wait... tK mentioned a 10th grader!?

19. I could of done that problem easily in 5th grade, would of abstracted on it some in 7th grade (there wasn't a 6th grade).

Recently had an interesting conversation with a Korean friend of mine. He didn't understand the concept of nuclear fission and neutron decay, he had just graduated college as an architectural engineer. That is something I learned in junior high. Science education in Korean school system is criminal...

You can make examples in both directions all day long, neither of which would actually mean anything.

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