A friend of mine teaches chemistry at a community college in South Carolina and is pulling out his remaining hair. It appears that in his introductory chemistry and remedial classes the vast majority of students (recent high school graduates) is unable to do simple calculations involving percentages. Forget about dimensional analysis.
Something must have gone awfully wrong with these students' math instruction in high school and probably even earlier in elementary school. I am on the case investigating.
I teach math to eighth graders and know that teaching math successfully need not be rocket science. Most of my students can now convert fractions to decimals and percents (and vice-versa) in their sleep. They can also solve the three different types of percent word problems (unkown rate, whole and part) in their sleep.
The following is an aspect of how I teach percentages after teaching the concept that a percentage is an equivalent fraction with a denominator of one hundred.
I have my students create a table with three columns. Each column is labeled fraction, decimal, percent. This visualizes the operations and helps reduce confusion. They can go from left to right to start with fractions and end up with percents, or from right to left to start with percents and end up with fractions.
Converting percents to fractions generally poses no problems when the percent is a whole number, e.g. 47% --> .47 --> 47/100. A special problem arises when the percent is a fraction like 5-1/4 %. This is where the students need to realize that an additional step is required. Many students want to enter 5.25 in the decimal column. The task of the teacher is to focus on this problem and to show that the students must first convert to 5.25%, then to the decimal .0525 and on to the fraction.
A good way to drive home this point is to contrast the conversion of 5-1/4 (without %) and 5-1/4% to decimals and writing 5-1/4% and 5.25% in the percent column. Only now can 5.25% leave the percent column and move to the decimal column to become .0525. Bingo!
I also have my students solve percent problems as proportions and equations. I drill them in identifying the rate (percent), part and whole in any percent problem. For equations I have them use the rb=a formula. r=rate (percent), b=base (whole) and a=amount (part). This is also a nice algebra exercise since they need to solve for a, b or r as the case may be.
I once again have them make a table with three columns and several rows. The columns are labeled r, b and a.
Then I have them look at various percent problems and ask them to identify what's known and unknown. If the percent is unknown, a question mark goes into the box of the r-column and what's known (whole and part) goes into the respective boxes of the b-column and a-column.
If a (the part) or b (the whole) is unknown, the procedure is repeated accordingly. This works amazingly well and the visualization once again helps to minimize confusion.
An added advantage is that the students learn how to solve for the various variables and how to manipulate the rb=a formula, e.g. r=a/b, b=a/r.
It is amazing, though, how many of these 8th graders find this manipulation of variables difficult. Adults might consider this manipulation child's play. But it really represents a huge jump to this age group to go from the concreteness of numbers to the abstractness of variables (letters).
This concludes the lesson on how to teach percents for understanding. Instructivist will give other lessons on how to teach math successfully to middle graders as he sees fit.
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8 comments:
Teachers who blog http://teacherswhoblog.blogspot.com/2005/03/question-4.html is surveying why teachers blog.
I posted this:
6/14/2005 1:55 PM
Instructivist said...
I blog to make my minute contribution to combating the regnant progressive/constructivist ed credo. This credo dumbs down education and is especially harmful to the disadvantaged.
One Bill D. gave his reasons for blogging:
6/15/2005 9:30 PM
Bill D. said...
I use my blog to combat ignoramuses like "instructivist" who are destroying education. Dumbing down education...please. Its the test nazis and the traditionalis who refuse to allow education achieve higher potential "Destructivist' is more like it. They destroy student's souls. As yo can see., I often use the blog to argue I also see great potetial for student/teacher interaction through blogging.
Looks like this creative worker needs some spelling help from an instructivist.
ok
It's necessary, I think, to make it instinctive in students to understand percent as a "number over 100."
From there, I've always liked the idea of moving the decimal point backwards.
5 1/4 = 5.25
5 1/4% = 5.25%
5.25% = 5.25/100
Move the decimal point to the left twice (the power of ten you are dividing by), write in the necessary zeros, and remove the percent symbol:
5.25% = 0.0525
Instructivist -
Enjoy the hate mail. The level of thinking demonstrated by Bill D's comment shows clearly that he has no intellectual defense against your ideas.
~Quincy~
I am a high school chemistry teacher at Bowie High School in Bowie, MD. I have students in their junior and senior year who have received straight As in math for their entire middle and high school careers. They can't do simple arithmetic. They don't know the 'why' of a mathematical operation and so do not remember the 'how'. In fact, they have never been taught 'why' they had to learn anything in math other than counting for the use of money. I kid you not.
I have to teach math for 4 to 6 weeks every year. The students have been taught the wrong rounding rules by their math teachers. They also can't find a percentage or know what it means, understand significant figures, do scientific notation, know anything about the laws of exponents, find the equation of a line or graph an equation without their $90 graphic calculators (which they don't understand how to use), solve for a variable in an equation (density = mass / volume is an example), convert inside the metric system, covert in the English system, understand the concept of conversion factors, or do anything without extensive direct instruction with days of practice and activities in the mathematical concept.
Here is an example lab that was so so sad. Many of you may remember the Measure the Thickness of Al Foil activity where you combine the equations for density with the equation for the volume of a box. The intent of the activity is to show the new chemistry student that math can be used to indirectly measure something that cannot be measured with the available tools. I can't just jump in and do the activity with the students. The issue becomes not the intent of the lab but how to do ANY of the math. I have to spend at least two weeks of work just to get to the lab so the point of the lab is not lost.
What in the ever loving heck is going on in the math classes in my country? This is insane. You ought to see when I have a parent conference of an honor student and point out that the kid they say is not good at math (a backdoor plea to lower the standards) had received As and Bs in all their math classes and I pretend to act incredulous that they would offer such an excuse for their obviously highly mathematically competent child. The parent and admin damn near chew a hole in their cheek. I just sit there and look all innocent. Admin is way on to me...but they never say anything. They do glare and act cold.
Want to here more math silliness with honor students? Have tons of stories.
This is quite similar to the way Saxon Math does it, and it seems to work pretty well.
Ragnarok
It goes all the way up the line. When I was a college professor (at an expensive, selective school) teaching an upper-level molecular genetics course, I observed juniors and seniors, who had somehow gotten As and Bs in freshman calculus, who were mystified by the simple Algebra I equation-solving involved in teaching DNA renaturation kinetics. It's a wonder we manage to produce any scientists and engineers at all in this country.
It's necessary, I think, to make it instinctive in students to understand percent as a "number over 100."
From there, I've always liked the idea of moving the decimal point backwards.
5 1/4 = 5.25
5 1/4% = 5.25%
5.25% = 5.25/100
Move the decimal point to the left twice (the power of ten you are dividing by), write in the necessary zeros, and remove the percent symbol:
5.25% = 0.0525
Thank you for this very helpful suggestion.
This illustrates the solution very nicely.
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