An algebra problem
A story from the tutoring room, where I encountered a student taking first semester chemistry for science majors who was struggling with a simple density problem.
(Why they teach complicated algebra problems before they teach any chemistry is a mystery to me, but I no longer remember what pedagogy my chemistry classes followed.)
Given that our chemistry class has college algebra as a prerequisite, I think the problem itself is instructive.
After a significant amount of prompting on an easier problem, the student was able to get all of the given data for this problem into the appropriate chem-SI units of g and cm. This left something like the following problem, except written as a fraction: 2.705 = 276/(30.48 * 1219.2 * X)
Solve for X.
The student was,it appeared, utterly helpless when faced with all of those numbers.
My suggestion that he cross multiply the X and 2.705 met the sort of look I would expect if I had said it in Japanese. OK, clear fractions? Still no luck. I don't think he saw this as a fraction. Multiply both sides by X? Ah, progress. Now he could compute the right side and solve 2.705*X = Number. Just to check, I asked him about 5 = 7/(3*X)
No problem there, although slow as molasses doing it.
When I started telling this story to a chemistry colleague, she starting laughing so hard that she almost fell out of her chair before I was halfway through it. She regularly sees this at the start of every semester. Now I know why she says my students aren't like hers. Many (but not all) of the kids who make that kind of mistake are weeded out by pre-calc and trig before they get to me.
When I asked a couple of math colleagues about this problem, I learned that they include problems with "messy" numbers on pre-calc exams, but not in college algebra. They also said they suspect that not all pre-calc classes give messy application problems. So that is why I was not surprised that a student in Becky Hirta's Calculus Circus had trouble with a graph where the answer was not obviously going to come out in simple integer steps.
Footnote:
Two of my physics students messed up a problem essentially as follows:2.66 = 7.79+3.47*X
2.66/7.79 = 7.79/7.79 + 3.47*X
0.3415 = 3.47*X
To save you the effort, 2.66/7.79 is about 0.3415. Bet you didn't know that 7.79/7.79 was zero!
Yep, that is how at least two of them "canceled" that number that was added on the right. Again, my math colleagues tell me that this sort of error is not uncommon among students entering calculus 1, and that they are more likely to make it with numbers rather than symbols. I have to wonder if they would have subtracted 7.79 if it had been at the end2.66 = 3.47*X + 7.79
instead of at the beginning....
5 comments:
To me this whole discussion prompts an entirely different question - I had it drilled in to me in undergrad (and then attempted to encourage the same in students I TAed) that it's best to solve for X with everything in symbols and then plug the actual numbers in at the end, rather than carrying around a lot of digits as you do the algebra. But it seems that's not the canonical procedure for you? Are there differing schools on this? (Is it a chemistry vs. physics culture thing?)
This wasn't my student, and that was how he set it up: by plugging all of the known values into the equation.
I try to teach my physics students to do algebra first, but even students in calculus are uncomfortable with symbols and have trouble keeping track of which one is unknown when x is not the unknown.
Students coming out of college algebra are still incredibly weak at dealing with what the math folks call "literal equations" (ones without any numbers). They prefer to calculate or solve by graphing, with X as the only unknown. That student would have had his mind blown if I had suggested writing rho = m/(LWT) and then solving for T, because our chemistry classes just don't use equations that way.
After a 30 year career teaching hs mathematics I totally agree. Changing to a literal equation would blow their minds. Substituting in all known quantities is what most are taught. It's not so much the messy numbers here. You could change them to simple integers like 2, 3, 5, whatever. Many would still miss it. Graphing calculators hand messy but not always order of operations or understanding or dividing every term (every single one) by the same thing. Fun discussion for today! Thanks.
Wow - my 8th grade son makes those mistakes in Algebra. I hope he'll have it worked out by the time he gets to College.
When I taught calculus and precalc at a "small, PhD granting state institution", I often made deliberate errors in lectures so the students would have the pleasure of finding it out. After a gap of 25 years, I'm now teaching at an accredited, for profit, AA degree granting institution.
The first time I solved
2/7 + 3/5 = (2+3)/(7+5) I was briefly livid when no one in the class said anything. I didn't say anything, but I concluded that they weren't paying attention. Of course I was wrong; they were algebraically naive.
Since that time, it seems that the math abilities of the students have improved; I'll have to try it again and see if I get any notice.
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