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18 Aug 2005 - 20:25
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 approximation 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 70%, can they understand what 70% 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. [Edited by author because 14/20 = 70%, not 60%. Oy.] -- 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
Oh wow--just saw Independent George's response--back shortly! -- CatherineJohnson - 19 Aug 2005
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. Well you've hit on one of my constant complaints about fuzzy math, which is the huge amount of unrecognized & unacknowledged "rote" work. It's the same thing with all the calculator use. How is a calculator not 'rote'? For many calculator-raised kids, numbers are just sequences of buttons. When a college kid uses his calculator to work 'What is 67 10' you're not talking conceptual knowledge. -- CatherineJohnson - 19 Aug 2005
Catherine, About caluculating percentage increase: you successfully calculted the percentage increase of women who would be willing to tell someone they had dandruff. About whether percentage increase = percentage greater: This requires a longer answer. When a number is expressed as a percentage or a ratio, it has no units. In order to be understood, it must be paired with the original units. So each of the percentages in this problem must be expressed as a percentage of what: percentage of women who are willing to tell someone they have food in their teeth. Now, when you manipulate percentages as in calculating an increase or decrease, you have to go back to the original numbers to make sure that you are using the same units. That is why you strictly dealt with women and dandruff. When you a talking about a phrase like "percentage greater", you have two problems. The first problem is that greater is usually not used alone in a math problem of this type. Something is usually greater than or a certain number of times greater. So I would not equate percentage increase with percentage greater and I would actually not use percentage greater. BUT the real problem with this problem is that this equation is supposedly corrrect: 80% of woman would tell someone they had food in their teeth - 20% of women would tell someone they had dandruff = 60% greater This equality is not only incorrect, it is totally illogical. As is the whole problem. -- AnneDwyer - 19 Aug 2005
(Hi, I'm back, momentarily. I've already been at work for 3 weeks. Schools start early here NW of Houston and I've hardly caught my breath!) Anne, I love it ("Something is usually greater than or a certain number of times greater. (Bold is mine.) This sentence just nails it! OK, you've got to remember that I teach 5th grade, so I'm going to put my 5th grade teacher's two-cent explanation in this. It will not be short, but will hopefully be simple. The "language" is critical. This is what we have to watch: What words are used?,, and with what other words are they used? We get used to interchanging words when we are using real, exact numbers to represent items (people, things, units). We interchange "more" for "greater". Try this little problem: "8 is 6 more than 2. How much greater? 6 greater, so 8 is 6 greater than 2" or "8 is 6 more than 2". This is a simple "difference" problem. From the time children are young we use these two words "more" and "greater" interchangeably. You probably grew up hearing those two words used interchangeably. In fact, we're so used to hearing them interchanged that we hear one and think the other. We know we are solving a difference problem, which involves subtraction. (The same thing would be true for the words "less than" and "fewer than".) We can also use the word "increase" or "decrease" in this same little problem. We could say we have an increase of 6 (if starting at 2) or a decrease of 6 (if starting at 8). This simple little problem I've suggested calls on us to do a simple comparison. We can compare numbers by finding differences (using subtraction). Our children compare numbers using signs for "greater than" and "less than" (<, >) all the time. (80% > 20%) What's the difference? (60%) NOW THE TRICKY PART COMES WHEN WE ARE ASKED TO COMPARE NUMBERS IN A DIFFERENT WAY. We find a problem that asks "How many TIMES greater?" and we see the word "greater" and we think "more". But we might fail to notice the use of the word "times". Our brains just run to interchange the words "more" and"greater". When we are asked "how many times more" or "how many times greater", we are also being asked to compare. BUT we are NOT being asked to find a simple difference ( using subtraction). Also, added into this mix, when you begin to talk percent, you must be very careful about using "percent" and "times" and "increase" in the same sentence. In the original problem that began this whole thread, I think 80% can be 60% more (greater) than 20% when used as a simple difference problem. And I think it was a simple HOW MUCH GREATER (MORE) IS _ THAN _? kind of a question. The word "times" does not appear anywhere in this statement and so I think that 60% is the difference. There is a difference of 60%; one is 60% more than the other, or one is 60%less. IF we have this question: "How many times more . . .?" or "Eighty percent is how many times more than 20%?" our reasoning must change, for we are making another kind of comparison We must multiple to get the proper answer. So 80% is 4 times 20%. It is 4 times greater than 20%. And this question is not asking for percent greater; it is asking for how many times greater. However, there is one more question that I think some of us may be thinking about and we're trying to force it upon this problem. It addresses the increase (or it could be the decrease) and asks us to use a percent to relate a second number(or percent)to an original or previous number (or percent). (I'm going to use a different example first and then come back to our 80% and 20% later. Bear with me.) Let's say this is our problem: In 1999, 20% of the students made 100 on a certain test, and in 2000, 40% of the students made a 100 on the test. The increase was what percent of the original 20%? (That's the question we are looking for.) Doing the math we see that from 20% to 40% is adding another 20%, which is a 100% increase. (20% is 100% of 20%) Now, if 60% of the students made 100 in 2001, the 60% would be an increase of 200% over the original 20%. And if 80% of the students made 100 in 2002, that would be an increase of 300% of the original 20%. So in this example, 80% is an increase of 300% of the original 20%. Now let's go back and see if we can use this in our starting problem. It will be a strain to do so because these two percents (80% and 20%) are answers to two different questions, but they are both representing women's answers, so we will try. Our question will be worded something like this: "The 80% who answered 'yes' to question 2 is an increase of what percent of the original 20% who answered 'yes' to question 3." This really is indeed a stretch and would make more sense to us if Question 3 had been asked before Question 2. Note the precise use the words "increase" and "percent" in the same sentence, which causes us to recognize what reasoning we must do. This question is very different from a question which simply asks us to compare a number (or percent) which is GREATER THAN another number (or percent). And it is different from another question which might ask us to determine HOW MANY TIMES GREATER one number (percent) is than another. This final question asks: "The increase is what percent of the original number (or percent)." I have another hypothetical example, but I feel like I'm about to beat a good horse to death. I know that I often tend to make my comments terribly long so I'm going to stop and grade 5th grade math and science papers. I'm hoping I'll have time to drop in once and a while. I'd love to try to help students understand Math or help parents as they try to help students understand. That's my real desire. I check your website often but haven't had time to get seriously into any of your threads these last few weeks. Maybe that will change. Think of me while you continue your last relaxing days of summer. God bless you all for what you are doing. -- CarolynMorgan - 20 Aug 2005
Hi you guys--thank you! I haven't taken the time yet to read this closely, but I will! Meanwhile, I have another question from Russian Math that is exactly in this neighborhood--I'll get it posted shortly! -- CatherineJohnson - 20 Aug 2005
Carolyn--welcome back! Are you completely overrun with work & teaching?? We miss you! -- CatherineJohnson - 20 Aug 2005
OH, you're back in school! Also, DON'T WORRY ABOUT LONG COMMENTS!!!! THAT'S WHAT WE NEED! (This web site is completely different from a 'regular' blog--we're trying to learn some math around here! I'm trying to learn math AND math ed; Carolyn's trying to learn math ed. So fire away!) -- CatherineJohnson - 20 Aug 2005
Yes, my days are very long. I'm up every morning at 4:40, leave for school at 5:50, get to work at 6:30 and don't leave school until 6:30 or later in the evening. Even then, I'm often not really finished. I take work home, which may or may not get done. It usually stacks up by the weekend, which means more work over the weekend. That's why you may or may not hear as much from me. -- CarolynMorgan - 21 Aug 2005
Carolyn, I was very interested to read your explanation of the answer to the problem. However, I feel that I have to comment on two aspects of your explanation. You said that In 1999, 20% of the students made 100 on a certain test, and in 2000, 40% of the students made a 100 on the test. The increase was what percent of the original 20%? (That's the question we are looking for.) Doing the math we see that from 20% to 40% is adding another 20%, which is a 100% increase. (20% is 100% of 20%) While it is true that the percentage has increased by 100%, it is not necessarily true that the number of students who made 100 on the test has increased by 100%. This is only true if the number of test takers is the same. For example, if 20% out of 300 made 100 in one year and 40% out of 600 made 100 in the next year, this would be an increase of 400% in the number of student who made 100, even though the actually percentage increased by 100%. Likewise, if 20% out of 300 made 100 in one year and 40% out of 150 made 100 in the next year, there is no increase in the number of children who made 100 in the two different years. That is why I believe it is so important to go back to the orginal data when manipulating percentages. Additionally, you said, Now let's go back and see if we can use this in our starting problem. It will be a strain to do so because these two percents (80% and 20%) are answers to two different questions, but they are both representing women's answers, so we will try. Again, you can only do this by assuming that the same women who said yes to dandruff are the same women who said yes to teeth. Let's look at a hypothetical example: Say you have a sample of 10 women. 8 out of the ten say they would tell someone they had food between their teeth but would never tell them they had dandruff. 2 out of ten say they would tell someone they had dandruff but would never tell them they had food between their teeth. You will still have the 80% and 20% but you now have a mutually exclusive data set: the two do not overlap and you can't subtract them. Now I am beating a dead horse but my real purpose is to show that an ambigous, pc problem like this wastes our children's time. While the children in Asian countries are getting a good foundation in elementary school math with problems that can be justified by mathematical proofs, we are mixing social studies into our teaching of ratios and percentages. -- AnneDwyer - 21 Aug 2005
Anne, Yes, you are correct. There are always two things to look at: changes in percentages and the total real numbers. I was addressing the changes in percents. Thanks for making it clear that a change in the number of students skews the actually numbers. We had a case just like that here in Texas only a few years ago. Our test scores appeared to rise and schools all over the Houston area were being praised for their great gains! Until someone pointed out that schools/districts were no longer required to include in the test results those scores of certain groups of children (the ethnic groups who had been pulling down the scores). (I want to clear this up. We have migrant workers who move around and we have illegals who move through, attending schools temporarily; both of these groups speak only a little English. These are the ethnic groups I am speaking of.) As for the women in our graphs, I suppose I was assuming that they asked the same women the same 3 questions at the same time. And yes, I was troubled by comparing and subtracting the two different answers of two different questions to find a simple difference. I hoped I had made that clear, but probably didn't. I was really only stating that in a simple comparison we can subtract (even percentages) to find a difference and in reading the question I think that is all they were asking students to do. But comparing percents is trickier than using whole numbers, I agree, for the reasons you have explained. Yes, indeed, what was probably an attempt to teach students how to read pie graphs and find percents out of a hundred, was turned into a ridiculous PC problem. Who cares what women or men are likely to say or when they are most likely to say it? Someone was paid to think up and design this problem! Thanks for you help. -- CarolynMorgan - 21 Aug 2005