Example I
40% of the the Freshman Calculus class at Turing University withdrew. If 240 students left, how many were in the class to start?
Solution without explanation or discussion:
0.4x = 240 ⇒ x = 600
Example II
40% of the the Freshman Calculus class at Turing University withdrew. If 240 students were left, how many were in the class to start?
Solution without explanation or discussion:
0.6x = 240 ⇒ x = 400
Thinking that the issues in the problems above are more language-dependent than based on learning key mathematics principles or effective methods? I would expect that many would say that using the word "left" in both problems was unnecessarily devious and that clearer language should be used to demonstrate the mathematics here. Perhaps, but when I taught these types of problems I would frequently juxtapose these types of questions and intentionally use such ambiguous language to generate discussion - creating disequilibrium so to speak. If nothing else, the students may become more critical readers! Further, the idea of using similar but contrasting questions is an important heuristic IMO.
Even though I've been a strong advocate for a standardized math curriculum across the grades, I fully understand that the methods used to present this curriculum are even more crucial. Instructional methods and strategies are often unpopular topics because they seem to infringe on individual teacher's style and creativity. BUT we also know that some methods are simply more effective than others in reaching the maximum number of students (who are actually listening and participating!). I firmly believe there are some basic pedagogical principles of teaching math, most of which are already known to and being used by experienced teachers.
Percent word problems are easy for a few and confusing to many because of the wide variety of different types.
Here are brief descriptions of some methods I've developed and used in nearly four decades in the classroom.
I. (See diagram at top of page)
The Pie Chart builds a strong visual model to represent the relationships between the parts and the whole and the "whole equals 100%" concept. How many of you use this or a similar model ? Please share! There's more to teaching this than drawing a picture but some students have told me that the image stays longer in their brain. I learn differently myself but I came to learn the importance of Multiple Representations to reach the maximum number of students.
II. "IS OVER OF" vs. "OF MEANS TIMES"
The latter is generally more powerful once the student is in Prealgebra but, of course, the word "OF" does not appear in every percent so many different variations must be given to students and practiced practiced practiced practiced over time. The first method can be modified as a shortcut in my opinion to find a missing percent and that may be its greatest value. However many middle schoolers use proportions for solving ALL percent problems. I personally do NOT recommend this!
Well, I could expound on each of these methods ad nauseam and bore most of you, but I think I will stop here and open the dialg for anyone who has strong emotions about teaching/learning per cents...
Sunday, September 13, 2009
Demystifying Harder Per Cent Word Problems for Middle Schoolers and SATs - Part I
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9 comments:
I think your second diagram is wrong. It's still 40% of the students who leave and 60% who remain, it's just which of those is the 240 number that differs.
Thanks, mathmom, for catching my careless error! I've corrected it...
Can you comment briefly on the use of "is over of" and "of means times"? I was taught to do percent problems with the "of means times" method, but Saxon teaches the "is over of" method. Some of my children are confused by the Saxon way, and understand quickly when I explain it the way I was taught. Others of my children prefer Saxon's method, telling me it makes more sense to them. Are the two methods equally valid?
S:
20 IS what percent OF 5?
Whatever number or variable comes AFTER "OF" goes in the denominator. The other quantity (close to the word "IS") goes in the numerator and a proportion is set up:
"20 over 5" EQUALS "N over 100."
This "IS OVER OF" method has been taught for years but, IMO, students should be seeing "OF MEANS MULTIPLY" early on, since, as you know, it's far more effective for a variety of problems. I'll probably say more about this...
My own kids get "OF means multiply" from 1st grade on. Miquon Math uses that for fractions: 1/4 of 8, which is transformed into 1/4 x 8 within a few workbook pages, but we still read the times symbol as "of" all the way through the book.
In my classes, I find that a few of the other students have been taught this, but many have not, so I stress it as often as it comes up in fraction, decimal, and percent problems.
Dave, I would prefer to see a merging of your methods I and II, which is another way of saying what you’ve said many times—multiple representations, multiple explanations.
I am not fond of rules based on keywords, such as “ALTOGETHER means add” or “OF means multiply” although I have used the latter.
The object of the preposition “of” is the quantity that serves as the basic unit, what is the whole or 100%. This is true for “is over of” as well as “of means multiply,” as in “half of 10” or “250% of 20” or “8 of 10” (as in 8 of 10 teachers prefer…”) or “20 is what percent of 5.” In the first two examples we multiply, in the second two we divide, but the number following the word “of” is always the reference quantity, the whole or 100%.
We multiply when we take “portions of” the whole, as in one half or 200%. The portion is some kind of ratio—fraction, decimal, or percent, that defines a relationship between two quantities. We divide when we compare one quantity to the reference quantity. Even if the quantity to be compared is not a whole number, we treat it as a fixed value not as a portion. To easily handle “portions of” a quantity and “comparison of” two quantities, students need to understand what you say the Pie Chart shows: “the relationships between the parts and the whole and the ‘whole equals 100%’ concept.” Even as rules are taught, I think the emphasis should be on the concepts along the lines of 1) taking portions of a quantity, and 2) comparing two fixed quantities.
Profound and important distinctions between the two methods, Burt! Thank you. The part-whole discussion could lead to a series of posts and discussions. I was only scratching the surface.
Students need to come to the realization that percent problems are nothing more than fraction/ratio problems. The underlying difficulties always come back to fractional parts and ratios.
I also do not like using keywords as a substitute for understanding, however, for the less intuitive student, these can be survival techniques in the beginning. If the fraction or percent (same thing!) is given, then N% of something can always be set up as a multiplicative sentence. If the percent or fraction is not given, one can still use the multiplicative model and solve an equation, but the student also has the option of setting up a ratio, a quicker method. Students who struggle with math have more difficulty in being given choices such as this because they are learning mechanically at first with minimal conceptualization. Two methods can seem overwhelming to them but, over time, they can adapt. The visual representation is still the best, IMO, for helping these students make sense of it all. In fact, they should be required to SHOW the part_whole relationships!
Teaching percent to struggling learners is a challenge because the concept of what a percentage represents must be taught first. I find that teaching percents, fractions, and decimals at the same time helps students understand that all can represent a part of the whole in a different way. In a recent lesson, I asked my high school algebra students what 25% of 4 is. I then asked what 1/4 of 4 is. None of the students could answer either question. I realized that none of them understood what a percent or fraction represents. Tough lesson.
Jmoser--
I admire your attempt to show them that all three forms are equivalent. I've done the same with some success but we always returned to the pie chart or some other visual model.
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