PLS READ THE COMMENTS. VERY ASTUTE OBSERVATIONS FROM OUR REGULAR CONTRIBUTORS TO WHICH I RESPONDED IN GREAT DETAIL. THIS MAY CONTINUE...
For now, I'll just leave the title as the problem to be discussed. Please consider how middle and high school students would approach this. How many might incorrectly guess that 60% of the seniors are girls and 40% are boys. If the problem were rephrased as a multiple-choice question, this type of error would occur frequently from my experience. What causes the confusion? Is it just the wording of the question or is there also an underlying issue regarding conceptual understanding of percents? Could it be that the question is asking for a percent and hasn't provided any actual numbers of students? What are the most effective instructional methods and strategies to help students overcome these issues? Certainly, algebraic methods would be a direct approach, but what foundation skills and concepts should middle schoolers develop even before setting up algebraic expressions?
Sunday, July 29, 2007
There are 20% more girls than boys in the senior class. What % of the seniors are girls? The confounding semantics of percents...
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14 comments:
I think virtually no middle schoolers would answer this correctly. I think the thing that would confound them most would be "20% more girls than boys". The semantics of percent more or less are confusing. The fact that any real number of students is not given is also a contributing factor in the difficulty of this problem. Finally, the fact that the answer does not work out to be a whole number makes it difficult at this level, where they might be trying a guess-and-check strategy instead of setting it up algebraically.
As to what to teach to give them foundations:
1) teach and re-teach what 20% more or less means
2) teach how to deal with a percent problem with no total number given by using convenient numbers.
In my past experience, these are the types of responses I typically see:
--Girls 70%, Boys 50%.
--there are 70 girls and 50 boys.
-- I can't do the problem, no numbers were given.
I think students get caught up in the wording (does this mean that # girls is 20% more than # of boys or does this mean that the % of girls is 20% higher than % of boys) due to their lack of understanding of percents.
Before moving to an abstract approach, I ask my students to pick some numbers. We look for a pattern - only then do we move to a generalized algebraic approach. I would love to hear as to how others would help students understand this type of problem.
mathmom, jackie--
I think we agree that middle school students would have a hard time with this for all of the reasons mentioned. The irony is that high school students also struggle with these kinds of problems! Those who can set it up algebraically and solve are generally successful, but only a few can handle these conceptually.
Choosing convenient numbers is a powerful teaching strategy for all. Once the student recognizes that that we can choose 100 boys, then there are 120 girls for a total of 220 seniors. The girls represent 120/220 of the class or 6/11 or
54 6/11%. The boys represent 100/220 or 5/11 or
45 5/11% of the class. Students should verify that these percents total 100%. It will still bother them that the difference of these percents is only 9 1/11% and not 20%.
A conceptual approach usually involves fractional thinking which is challenging for this generation. Note that 20% or 1/5 of 5/11 equals 1/11. Thus 6/11 really is 20% more than 5/11!
The key to all of this is to help students focus on the BASE of each percentage. Some students argue that the answers should be 60% (girls) and 40% (boys), since they differ by 20% and add up to 100%. But 20% more than 40 is 48, not 60. As educators we need to help students recognize that in the phrase "the number of girls is 20% more than the boys", the 20% is BASED ON THE BOYS not THE CLASS! These ideas are subtle and they are really about part/whole relationships which is the overarching idea here. We think it's the percents that cause the problem but if the question had been phrased in terms of fractional ratios, would it have been easier? These important part:whole concepts must be developed in middle school alongside the mechanical skills with fractions, ratios and percents. Just my opinion of course but I have a lot of personal evidence to back it up!
1/5 of 5/11 is 1/11. I could spend an entire class period on that sentence alone!
Sadly, many students I see arrive in high school with little understanding of fractions, decimals and/or percents - let alone that there is a relationship there.
As an example: in a discussion with my summer school class last week, we were discussing the meaning of "bisect". A student was arguing that a chord was a diameter because it was perpendicular to another chord and split this second chord into two "almost" even pieces (17 inches and 18 inches). That these two segments were close in length was "good enough" to say it was bisected. So our circle work was put on hold to discuss the meaning of 1/2. Ten minutes of students discussing and trying to convince this student that 17 inches and 18 inches were not equal parts. This student will be a senior in high school in three weeks.
Sadly, it wasn't his understanding of the geometry concepts that was a barrier, it was his number sense.
Dave - as for your comment, I think many students will get hung up by the percent being a fraction. The will know that the number of girls/boys must be a positive integer and then they project this restriction onto the percent.
jackie--
Yes, the fractional percent would definitely be a block for many. In 8th grade I had to memorize the percent equivalents of 1/3/, 1/6, 5/6, 1/8, 3/8, and a few others. Those days are long gone. Of course it didn't mean that I thoroughly grasped that 1/2% means one-half of one percent or 1/200.
All hs math teachers I have spoken to decry both the lack of student facility with fractions and lack of 'fraction sense'. I don't think we will solve this problem, tonight!
Dave
As Shakespeare wrote, "The play is the thing." And by "the play," of course, he meant "the perspective one adopts in solving math word problems."
One of the fundamental conceptual confusions at the heart of this problem is the result of the way many textbooks (and teachers) inflexibly present percent increase and decrease as (a) involving a single amount and (b) going up or going down.
Because the problem involves two different amounts (girls and boys instead of girls and girls) and that "action" wording is not present in the problem, students fail to access the simple and correct percent increase set up:
increase / original amount = 20/100
OR
girls - boys / boys = 20/100
y - x / x = 20/100
It would help, I think, if textbooks spent some time showing that not only can percent increase or decrease be set up like this, but "percent more" and "percent less" can as well.
I sent this problem to my kids at camp. My 13yo who just completed Algebra I got the correct answer on his first try. He didn't say how he did it, but it was a letter home from camp, so I didn't expect much. :) He's very mathematically inclined, into contest math, and probably not typical of other students completing Algebra I, but there's a single data point for you anyhow. My 11yo hasn't sent back an attempt at it yet; don't know if he will.
Mathmom,
You just made my day - you sent your son a math problem at summer camp, he tried it, and he wrote back with his response. The fact that he arrived at the correct answer is just a bonus in my opinion!
Jackie, he's a math geek like his mom, and I bribed him to do it (said I'd put something extra in his next package if he got it right -- now I have to find something cool to put in.) :)
This problem is harder than ones where a number is given because it requires reading. Carefully, with attention to detail. Test-taking strategies for your typical "high stakes" test will lead to simply guess, and guess wrong, what is being asked.
I think the in your own words approach, which I borrowed from a college lit teacher to use in my college physics class as an experiment, is what you need to think about here.
That is bascially what Mr. Person did, although the step of saying "number of girls minus number of boys is (equals) 20% of (0.20 times) the number of boys, where I will let x = number of boys because that looks easiest" was skipped. That would be an example of putting "more girls than boys" into my own words.
Dave Marain made it concrete, thereby avoiding the use of algebra. Rather than digress into fractions (which some K-5 curricula mangle horribly), I would have gone from "assume 100 boys" to "assume B boys". Then the exact same steps would lead to G =1.2*B, total = B+G = 2.2*B, G/total = 1.2/2.2. QED. Since the class ought to be able to solve the problem where the actual number of boys is given, they might then start to see how to do it with algebra.
A key thing common to both approaches was using an unknown variable (x, B, whatever) to replace a word rather than try to figure out (or guess) an equation that has x as the thing you want for the final answer. Students tend to think, incorreclty, that the latter is what we do.
Beautiful analysis, Dr. Plon...
In the end, an algebraic approach does make the procedure simpler, and, thankfully, that was utilized by several students. Middle school students need a far more careful development of incrementally developed questions until they're ready for this one (and some may never be given some of the reading comprehension issues). Thanks for the insightful comment.
Dave Marain
Very intriguing discussion. I don't think the semantics of percents can be emphasized enough. Like the semantics of fractions, there is so much going on in general and in this problem in particular that it seems like a tragic error to put the onus on the students. There's lots of blame to go around: we can start with the weak foundation many elementary school teachers have in rational numbers and their various forms. I've observed many classrooms where fractions, decimals, and percents were taught as if they were three utterly different things, unrelated to one another except in the most peripheral ways. It's unsurprising that kids rarely see them as three ways to represent the same thing, each with advantages and challenges, each particularly helpful depending on the problem and the tools one has available to work with.
We can also blame textbooks that shy away from "awkward" numbers like percents that entail mixed numbers or which present them without real-world contexts to show that such numbers arise outside of "calculation only" problems.
And this brings us to the case in point. While I knew that the right answer wasn't 40% boys and 60% girls, it would be fun to think about whether you can rewrite the problem so that WOULD be the correct response. Contrasting that with what's actually asked might be useful to students: they really need semantic skills that aren't taught frequently enough in either math or English classes, from what I've seen. These sort of subtleties inform a lot of the questions in the "English" (not the reading comprehension) section of the ACT exam, and many students lose points because they don't see real differences among alternatives.
Finally, since I thought about this as a problem with x boys and 6x/5 girls, I saw elevenths coming into play. That led me to choose a class total of 99 students, rather than 100 (which clearly wouldn't work here) with 9 students as 1/11th of the class. That quickly gave me 45 boys and 54 girls, with 9 being 1/5th or 20% of 45. It is still a pain to divide 99 into 5400 to get the percentage, and of course no way to avoid that mixed number in the answer.
I very much enjoyed the other suggestions here, like starting with 100 boys, or a more purely algebraic approach. I hope that no one sees this problem as "obvious," however, and that we don't judge students too harshly based on their likely difficulties with it. If they can do similar problems with less awkward numbers, they probably can learn to do this one. But if they're dropped into this one first, they're going to be made to feel rather foolish, and I think that's not a great idea.
mike--
Wonderful comment! Thanks for contributing. I haven't communicated with you in a long time.
Now you know why I started this blog rather than continuing to contribute to other forums. An extensive dialogue about a math problem without politics!
A few points:
(1) I know you realized that this question was never intended to be an entry level problem for middle schoolers! It was a high-end problem that required extensive experience with a variety of percent questions of increasing difficulty. However, the central idea of 'PERCENT MORE' is so crucial for students to grasp, it must be learned before 9th grade.
(2) The issue of wording is also central to this discussion. Would students have had more success if the question was worded:
"The number of girls is 20% more than the number of boys." I think so but I'd have to test it. The way I worded the original question really sets the student up to think 60% and 40% if one is not experienced.
(3) Talk about choosing simple numbers - how about starting with 5 boys! If students felt comfortable with 20% as 1/5, it makes sense:
Boys = 5; Girls = 6; Seniors = 11
Voila! Not one student thought of that!
(4) Many comments and discussions about this on other blogs referred to the issue of the whole being more than 100%, which is unusual for students. However, in every '% increase' or '% more' problem, that's exactly what happens, since the original amount is the BASE or 100%. This concept needs to be thoroughly developed. I wonder how Singapore Math does it! Considering how sophisticated their ratio problems are in 6th grade, I suspect they do a decent job...
Mike, visit us again!
Thanks for the kind words and warm welcome, Dave. Feel free to visit me at rationalmathed.blogspot.com. Not much of a conversation, but I get to spout off on this and that, more pedagogical issues than anything else of interest (though I'll plead guilty in advance to waxing political). I've been lazy lately over there. I did link to you, though, and will likely put this problem and discussion over there, because as you rightly say, it's a richer conversation without all the silly bickering.
I can never weigh in with quite the same degree of confidence as others seem to have about exactly when something "must" be mastered. I keep thinking that it depends on the flexibility of the curriculum, the school, and the teachers (not to mention the students): some kids may be ready for things later than we'd like, but once they're ready, they can run with it if we don't bar the way or undermine them while they're gaining mathematical and personal maturity.
I do think the wording you suggest in your reply to me would be less difficult. Not a slam dunk, but more user-friendly. And I favor user-friendly wording over "clever" wording that we can see leads students unnecessarily into error. My argument against the latter is simple: in the real world, if you wrote instructions to someone on the job that you knew had a propensity for misinterpretation, whose butt should get fired? I don't think it's the worker who falls into the potential trap, but rather the person who should have known better and written as unambiguously as possible.
Have folks seen Edward Zaccaro's THE 10 THINGS ALL FUTURE MATHEMATICIANS AND SCIENTISTS MUST KNOW (But are Rarely Taught)? Interesting book. Lots of nice little problems. A bit of hyperbole in that title, of course, but what the heck? It's worth tracking down (I got a copy through interlibrary loan). In any event, he writes at length about a number of tragedies and disasters due to various sorts of design errors, miscalculations, etc., that are very thought- provoking. Definitely worth considering in relation to the question of where the responsibility lies for writing problems that aren't subject to REASONABLE and PREDICTABLE misinterpretation.
Mind you, I'm not letting kids off the hook for simply not thinking, for telling us that if you go from A to B at 40 mph and return from B to A at 30 mph that your average speed is 35 mph. That's not ambiguous: it's just lack of thought. But I suspect it's possible to set up a series of problems that would reduce the likelihood of students drawing such silly conclusions, and it's worth considering whether we get more kids on board by first tricking them and then guiding them out of error, or by incrementally leading them through a topic so as to reduce the likelihood of falling into such traps. I suspect a steady diet of either would be a mistake, but that surely a steady diet of "Gotcha's" would be a turn-off for many students. Teachers of a certain mind-set seem to think there's nothing more fun than making kids look dumb: perhaps there's a rationalization that doing this makes the student appreciate how much s/he needs the teacher's invaluable expertise, but personally the teachers I had who worked that way were not amongst my favorites.
The best work I've seen on the semantics of fractions is by Stellan Ohlsson. He published a number of things in the late '80s and early '90s (he's moved on to other issues) worth tracking down. Some are on-line. Reading his work would be rough sledding for some teachers, but it would be worth having someone boil his work down to more accessible language in order to show K-12 teachers how complex fractions are and why there is much more to them than meets the eye. They may know this, but Ohlsson really gets into levels that I suspect few of them have considered. I know I hadn't when I read him in 1992.
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