As of late, she has noticed that her students, who usually did quite well in math classes in high school and even college seem to still be innumerate when they reach her. I ask the percentage change between 8 million and 10 million and they give an answer of 200% and don't bat an eye. "Don't you see that that makes no sense?" she asks them. "Math is hard," they answer. If they have a formula, they can plug and chug an answer, but seem incapable of thinking about what it means.
Q: How is it Johnny can DO math (sometimes), but can't understand it?
On one level, I think you've answered your own question. They can plug and chug an answer. This is probably all they've ever needed to do to get good math grades. I suspect it's probably easier for teachers to give clear-cut, apply-this-formula questions, than questions which test for a deeper understanding. Plug and chug questions are easier to grade, and easier for the students to 'know what you want us to do'. Trying to test whether they understand is a lot more difficult and requires judgement. It also runs afoul of (administrators who have a boner for) Bloom's taxonomy of verbs, who think that if you can't measure understanding, then it has no educational value.
ReplyDeleteI'm about to start a relatively mathematical section of my Hamster Husbandry for Criminologists course. I had taught this for several years before I realized the stumbling block was that they couldn't see an equation as anything but a template into which one plugs as a prelude to chugging. They couldn't see equations as descriptions of how variables (ie real world stuff) are related to each other until I spelled this out for them. In fairness though, abstract reasoning is a skill that takes a lot of practice.
Also in fairness, percentages are pretty treacherous, because the decimal point it two places over from where it should be. The percentage change you refer to depends on whether you are increasing (25%) or decreasing (20%), though I admit, 200% is way off.
But I'm rambling. I need more coffee.
That sounds familiar. I often noticed that while I was teaching.
DeleteIt isn't just that they see an equation as a template. They often don't understand the form.
For example, the expression for a straight line (i. e., y = mx + b) could become completely incomprehensible to them if one changed the variables used. So, if I used, say, y = (alpha * x) + beta instead and explained the expressions were identical, they'd become confused. The fact that m and alpha represent the same quantity, and have the same function, went completely over their heads.
Writing out one's calculations to show how they obtained their answers was considered an archaic activity. They simply needed to know what the equation was and then keyed in the numbers into their calculators. If the batteries in those little boxes ever died, they wouldn't have had any idea how to get the final result. (Don't even think about suggesting using a slide rule or log tables.)
I feel your pain. I am teaching a "graduate" level numerical analysis class. One problem was to find the "best fit line" to some data (linear regression); it turns out that the data was world population from 2000 to 2010.
DeleteHe came up with a line of negative slope, and duly plotted the negative slope line along with the data, which, of course, trended upward every year. This was on a TAKE HOME EXAM.
Part of it is, as Rosencrantz Andor Guildenstern pointed out, due to the fact that most people never have to develop an equation.
ReplyDeleteCertainly, when I was learning math about 10 years before your students, I was discouraged from asking questions on WHY an equation was formatted the way it was, or WHY we needed to do a problem in this particular way.
When I accidentally taught myself how to calculate percentages accurately in my head, I was marked wrong on math tests, because the way I calculated them (yes, I showed my work) was not how we were shown to do it in the book. My answer was correct, but I had not followed the correct equation.
Add to it the joys of teaching to the standardized tests of No Child Left Behind, and there is no reason to teach higher level analyzing. Can they do basics with a calculator? Then good enough.
You know, of course, this is only going to get worse. Pretty soon, the students won't have to know how to push a button:
ReplyDeletehttp://twistedsifter.com/videos/camera-calculator-app-instantly-solves-math-problems/
I am guilty of this. Or maybe I am guilty of this when it comes to things like calculus and physics. My better half and I were dating in college. We were both in physics. I was in what I now call "physics for dummies," while they were in calc. based physics. They could not help me on my homework, because I had equations and I had NO idea where the equations came from, while they derived their equations and they looked nothing like mine. I struggled with calculus, I just didn't get it. But throw me in a stats class, or chemistry and I can happily work with the math because I understand the relationships and the point of what I am doing. I still don't get calc and I am okay with that.
ReplyDeleteI hate the way math is taught now and I am scared at the reliance on calculators.
I recall being forced to memorize the quadratic formula and always wanting to know how and why it works to solve quadratic equations, but none of my math teachers would ever tell me. It was just something to memorize, but it made it impossible for me to understand or even check my work.
ReplyDeleteWhen I got to college and was taught to solve quadratic equations by factoring, it was the easiest thing in the world. In fact I enjoyed it and even excelled at it, and actually found the abstract algebra and finite fields and modular arithmetic to be recreational. Quite a difference for someone who failed the same high school math class three years in a row.
Everyone uses base 10 blocks with currency - making change, determining how many 10 cent apples your $2 can buy - but when Common Core math tried to impart the same to grade school students, parents kicked and screamed. Memorization and long vertical arithmetic are the only acceptable way; anything that teaches kids to understand arithmetic rather than just memorize rules is completely unacceptable to parents. Heaven forbid their kids learn something that they did not.
I think I need someone to show me how to do Common Core math, before the offspring starts learning it. I have had students try to teach me (they were ed majors) but I just don't get it. I am very visual when it comes to learning math and I need lots of practice.
DeleteI'm kind of hoping that the Common Core will help with this kind of problem, not only in math, but also in English/writing (where the same rote-answer vs. critically-thought-out-solution-to-a-problem issue applies, though in slightly different ways -- e.g. memorizing MLA or APA style vs. understanding what a citation needs to accomplish and arranging the necessary information in the way that best accomplishes that goal in the context of the particular sentence/paragraph). I understand why some teachers are unhappy about the common core: apparently it was designed in large part by people who were not current classroom teachers, with very little input from current classroom teachers. Also, there's a very real danger that, although the overall design seems to allow for teachers to come up with their own solutions to reaching learning goals, the whole thing will, instead, become another huge profit engine for Pearson et al., who will produce both the tests and prepackaged curricula geared to those tests (and, in a high-stakes assessment environment for both students and teachers, there will be tremendous disincentives against individual/local experimentation/innovation). But, from what I've seen, it does seem designed to help students think about what they're doing, and why, rather than just follow rote formulae. That would be a good thing (and I think it would carry over from math to English and vice versa; even if they made no changes in the English curriculum, I'd still rather teach students who'd had to derive their own formulae in math. The habits of mind involved would make things easier in my classroom as well.)
DeleteNot to say that the Common Core math questions are well-phrased, however, pretty much anyone can understand that adding
Delete12
8
15
6
+ 4
____________
can also be done by factoring "bundles" of base 10, exactly the way we think of currency:
12 + 8 + 15 + 6 + 4
becomes
20 + 15 + 6 + 4
and then
35 + 6 + 4
and then
35 + 10
and finally
45
Many people can even do this sort of arithmetic in their heads because we are so familiar with base 10. But for whatever reason, parents seem to prefer the long vertical way with lots of carrying and so forth. Not sure why. Common Core has a lot of potential to make students comfortable with arithmetic - I wish I had learned that way.
Patty:
DeleteThere's a saying in industry: if it works, don't fix it.
There's another saying: do what you've always done, and you'll have what you've always had, which in our case, is students who are pathetically unskilled in mathematics, lagging far behind the other developed countries. So, if you think math education is working very well and the statistics are just a huge lie, then of course there is no reason to change math education. Statistics do not support this: "Among the 34 OECD countries, the United States performed below average in mathematics in
Delete2012 and is ranked 27th" http://www.oecd.org/pisa/keyfindings/PISA-2012-results-US.pdf
Units were another thing that tended to stump my students.
ReplyDeleteI learned early on in freshman year statics that units in calculations have to be consistent. I got into the habit of writing them down along with the numerical values and then checking that they made sense at the end.
Move ahead more than 15 years to when I started teaching. Did I ever get an earful from my students when I insisted on it and, worse, *penalized* them for not checking their units. "OK," I figured and thought about the old Fram oil filter ad slogan: "You can pay me now or you can pay me later." And pay they did when I took off marks for the *incorrect* results, results which might have been right if they bothered to check if their units were consistent.
Then again, when I started as an undergrad, calculators of any kind were rare and I used a slide rule and log tables. One had to think one's way through a calculation and writing down details and interim results was more the rule than the exception.
By the way, reminding students that not watching their units could have serious consequences doesn't help, even if it was in the news. (The loss of the Mars Climate Orbite was believed to have been caused by an error in converting between Imperial and SI units. A similar such error was thought to have contributed to the Air Canada "Gimli Glider" incident.) Did telling them about it help? Of course not. They would *never* make such mistakes, would they? (Yeah, right.)
Could not understand math until I took a logic course my senior year in college...
ReplyDeleteJohnny isn't doing math. He's cheating.
ReplyDelete