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19 Aug 2005 - 15:43
Worse yet, the math is wrong. It's the usual mixup of percents with percentage points. Look at #5. Food is about 80%, Dandruff about 10%, but 80% is not 60% greater than 10%. 60% of 10% would be 6%. 80% is eight times bigger than 10% or 800%
Here's the original problem:
I'm sure it will come as a shock to no one that I was never taught how to compute a percent increase or decrease; nor was I taught, as far as I can remember, what the question 'How much greater is the percent of men who are willing to alert strangers to smudges on their faces than the percent of women who are willing to do so?' actually means. As a direct result, I managed to spend my entire adult life utterly confused about the Ultimate Meaning of news stories on Percentage Increase in Federal Spending On Education and the like. No more! Thanks to Algebra to Go & Russian Math, I now know what both questions mean, and how to answer them, at least in theory. By which I mean that percent increase/decrease and how-many-times-bigger still hold the status of New & Tenuous inside my head. Yes, I could demonstrate both on a Pop Quiz right this minute. But I'm not confident I'd be right. So when I read this question, my first thought was: 10 percent? Then I thought, Hunh. I only had to stare at the problem a couple seconds more to arrive at the conclusion that, OK, we're not talking percent increase here. Which was too bad, because of the two ideas, that's the concept I know best. The idea of how many 'times' bigger (or smaller) one number is than another is something I first learned literally one or two weeks ago. (I know; it's mortifying.) So 'how many times bigger?' is very new knowledge for me, new enough that I figured the folks at Everyday Math must KNOW.
This is a very interesting problem because it involves interpreting the mathematical meaning of words. The problems did not ask how many times greater or even percentage increase. The problem asked "how much greater". The author of the problem obviously intended that the student would interpret this as a simple difference problem by subtracting one percentage from another. The author intended that the student would treat each percentage number as a number with the same units that can be subtracted. The real difficulty comes when you write out the answer: 60% greater. The mathematical meaning of the answer is very clear: 60% greater means that you increase something by 60%. This is not what was done in this problem. The answer is incorrect because you cannot use the word "greater" as units to an answer the the problem.
sexism in Everyday Math
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I suppose in fuzzy math the difference between percent and percentage points is not important. Your site has links to very helful math resources. I wonder whether you are familiar with How to solve word problems in algebra by Mildred Johnson? Intimidating algebra word problems are made crystal clear. The reasoning is shown step by step. The book deal with 13 areas such as numbers, time, rate, and distance, mixtures, coins and so on. -- CharlesH - 18 Aug 2005
That book looks fabulous, and is in my Amazon cart right this minute, as a matter of fact. I think I saw it for the first time only recently, at a Barnes & Noble. Thanks! -- CatherineJohnson - 18 Aug 2005
This is a very interesting problem because it involves interpreting the mathematical meaning of words. The problems did not ask how many times greater or even percentage increase. The problem asked "how much greater". The author of the problem obviously intended that the student would interpret this as a simple difference problem by subtracting one percentage from another. The author intended that the student would treat each percentage number as a number with the same units that can be subtracted. The real difficulty comes when you write out the answer: 60% greater. The mathematical meaning of the answer is very clear: 60% greater means that you increase something by 60%. This is not what was done in this problem. The answer is incorrect because you cannot use the word "greater" as units to an answer the the problem. -- AnneDwyer - 19 Aug 2005
Oh Anne--thanks! Quoting Barbie, Math is hard! I'm no slouch when it comes to reading, but I was completely derailed by this. I could see, as (I'm grateful) you confirm, that the authors wanted kids to subtract. But after that the whole thing felt all wrong to me, and yet I couldn't explain why. I'm going to pull this up front. -- CatherineJohnson - 19 Aug 2005
There's something else even more fundamentally wrong about this question - it's not even really about math. To answer the question, it is basically asking the student to look at the given pie-charts, compare them against the 'scale' shown on the lower-right, then make an apprixmation of what the percentage breakdowns are based on how similar the pictures look. The fundamental concepts behind the question aren't actually being tested on the question. It doesn't require any knowledge of what a percentage is, that it represents a proportion of a larger sample, or how you go from the data to the chart. It just asks you to match the chart against the scale. A better way of testing these concepts would have been to start with the raw data, and then ask the students to calculate the percentage breakdowns. You could vary the total sample size to change the difficulty - a number like 20 breaks down easily into units of 5% points; a number like 37 is slightly more difficult because it has a remainder. (from here, it is easy segue into a discussion of prime and rational numbers for more advanced students). Once the percentage breakdowns are calculated, the next step is to see if they understand proportionality - now that they've translated 14/20 into 60%, can they understand what 60% of a whole is? To do this, you could either have a multiple-choice question showing five different pie-charts, and ask them to match their calculated percentages to the correct charts, or have them draw their own charts. There latter is a little more challenging, but it's also more ambiguous to have them draw the lines themselves (plus, not everyone is able to draw neatly). I'd leave this to the discretion of the teacher. The point is, even before we get to the question of 'percentage gains', the question is already skipping most of the mathematical concepts behind the charts. A good question both instructs and diagnoses - this one does neither. -- IndependentGeorge - 19 Aug 2005
Anne--if you don't mind, tell me if I'm doing this right. Say the percent of women willing to tell people they have dandruff is 10%. What would '60% greater' be? As I understand this, reverting to raw data, if 10 women were willing to tell strangers they had dandruff, then 16 women would be willing to tell people they have food in their teeth. That would be 60% more. I'm sure of that (let's hope).....but what is the correct language? The way I'm figuring it, '60% greater' equals '60 percent increase.' (Yes?) So was my original feeling, that 'percent increase' and 'percent greater' were two different things wrong? I think I'm getting back in the realm that confused me originally, trying to figure out a percent increase when comparing two percents....I wrote a blow-by-blow account in inflexible knowledge narrative -- CatherineJohnson - 19 Aug 2005