If ever I had an academic Achilles heel, mathematics would surely be it. Nothing makes my blood run cold like an indecipherable word problem, and the very term "calculus" is enough to give me nightmares. For others math is more difficult than terrifying, however, and word problems have traditionally been used to make math easier to understand. Word problems can often be easier to work through than jumbles of numbers and variables, but as an article in the latest edition of Science suggests, they may also prevent students from grasping the abstract concept behind the word problem.
Mathematics is a double-edged sword; without concrete examples math can be extremely difficult, but without an understanding of the abstract concept it is exceedingly arduous to apply mathematical concepts to new problems. Take, for example, a word problem that wants to ask what percentage of a road trip someone has completed. Making the question relevant might help the student figure out how to solve the problem, but if a word problem on the test is asking the same basic question with a different story (like what percent of a drink is alcohol) the new problem seems wholly unfamiliar. The context becomes more important than the abstract mathematical concept, and so the concept is very difficult to transfer to new situations. I know that I've had this problem myself; staring blankly at an exam, new word problems are frustratingly unfamiliar even if they are asking the same questions as those I was taught how to solve.
The authors of the paper suggest that teachers do what may seem to be counterintuitive to better ingrain mathematical concepts. Although many students (even myself) complain that it's difficult to see how mathematics is relevant, constantly presenting students with word problems appears to only show students how to solve that one species of problem. Teaching the more general concept with numbers and variables allows seems to allow students to achieve a better grasp of the mathematical concept. The problem with this, of course, is that the abstract concepts can be frustratingly difficult to learn and might require more time to acquire.
For someone who is extraordinarily bad at math like myself, this seems like a "Damned if you do, damned if you don't" situation. Word problems are useful if I want to know how to get a solution in a particular situation, but it's difficult to transfer the more general concept. On the other hand, abstract concepts can be frustratingly difficult to acquire without some sort of context to act as a roadmap. What's the answer? I don't know. A combined approach may or may not work, and I'm sure the effectiveness of any particular teacher also comes into play (and I've had some absolutely horrid math teachers over the years). Still, I doubt that there's a clear-cut best method that's exclusively "concrete" or "abstract," and finding the best way to connect the two will probably yield better results.
References;
Kaminski, J.A.; Sloutsky, V.M.; Heckler, A.F. (2008) "The Advantage of Abstract Examples in Learning Math." Science, Vol. 320 (5875), pp. 454-455
Education & Careers
Comments
There's been some discussion about this article over at God Plays Dice. I have serious doubts about whether their results can be generalized beyond the particular concepts they used in the experiment.
Posted by: Kurt | April 28, 2008 10:02 AM
What Kurt said. Familiarizing students with abstract concepts is one thing, but learning how to apply a concept to solve a real-world problem — finding the aspects of the problem which correspond to the features of the abstract mathematics which you've studied — is itself a skill which requires practice to master.
I can certainly sympathize. In fact, if I hadn't had avenues for exploring mathematics other than my schoolteachers — books, a few TV shows — my mathematical education would have been fatally useless. Somehow, I lucked into finding the essays of Isaac Asimov and the books of Keith Devlin, who fired my enthusiasm even if they didn't teach me things directly relevant to my schoolwork, and the video series like Project Mathematics and The Mechanical Universe which helped me figure out what this stuff I was supposedly learning in class was actually good for.
(It also helped a lot that Mom had gone all the way through business school, and Dad had made it as far as second-semester calculus before dropping out, so up until tenth grade or so I could count on homework help. Thanks, Mom and Dad! Naturally, once I got into multiple integrals and vectors and the other good stuff which constitutes the bread-and-butter of physics, the dynamic turned around — but the support was sure nice while it lasted.)
Posted by: Blake Stacey | April 28, 2008 12:08 PM
I third what Kurt said. As is pointed out in the discussion he links to, it's not even a very good concrete instantiation of the problem, so all the study shows is that the more abstract way used in the study sucks less than the more concrete way used in the study.
BTW, on a side track, while I know you've discussed having difficulties with mathematics, I'm not sure if you've discussed specifically what you've tried to do about it. Honestly, as a (primarily math) tutor, you seem from your blog like one of my favorite types of students to tutor, since you are clearly smart, and willing to put significant work in to understand things. Since Rutgers has a reasonably solid reputation for math, I would tend to assume that they'd have a free tutoring center on campus (like most schools I know of), and that the tutoring center would have reasonably good tutors in it. So, um, am I wrong about some part of that?
On another sidetrack, I'll be up in NYC for a geology internship this summer. Can I propose a meet up at the AMNH sometime for readers of this blog, or some such thing?
Anyway, I've been sufficiently offtopic now, so I guess I'll go back to lurking.
Posted by: Jay De Lanoy | April 28, 2008 12:50 PM
I have to write things down to solve problems. I like word problems, but I have to rewrite them to solve them.
The professor who taught me intoductory chemistry was outstanding. When we got to gas laws, we approached them from the point of view that we knew qualitatively what would happen, so we arranged the numbers such that the answer went in the proper direction. We know if we heat something it will get bigger, right? So we divide the high temperature by the low temperature which gives us a muliplier greater than one. If we cool something, it gets smaller, so we divide the little temperature by the big temperature which gives us a multiplier less than one. Basically move the numbers around to get the answer you want.
You are not good at math because you have never steeled your resolve and decided to put in the effort and get the help to allow you to get better. You may never be really good, but you can be a lot better than you are now, but only if you decide to be.
Posted by: Jim Thomerson | April 28, 2008 4:13 PM
Quoting Heinlein from memory, "Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable sub-human who has learned to wear shoes, bathe, and not make messes in the house." I am not quoting this to make you feel worse.
Seriously, mathematics without word problems is empty twiddling. The world is made out of word problems, and the act of abstracting the mathematics from them is the essence of science. Now, vertebrate zoologists famously can't count (a snake is a *tetrapod*?), but the only way one could doubt the relevance of mathematics to any branch of science is to deliberately avoid seeing the myriad places where it may apply (for fear, obviously, of being asked to apply it).
You should be glad you don't have the astronomers' problems. The mathematics that applies to almost everything an astronomer sees (and to absolutely everything in between) is that of plasma fluid dynamics. However, astronomers typically know literally less than nothing of plasma fluid dynamics; instead they learn a ludicrous parody of it called "MHD". Worse, the mathematics of real plasma fluid dynamics is almost completely intractable. If they did learn plasma fluid dynamics, they couldn't actually sit down and solve problems. Instead, they would have to run really enormous computer simulations or fiendishly difficult vacuum chamber experiments, which they would hate. So they don't. Instead, they pen press releases about unobservable Neutron Stars and Black Holes, and pray for insight into Dark Matter and Dark Energy (which are now said make up 98% of the universe, I kid you not!), and invent ammonia geysers. In the old days it was Angels, but only the words have changed.
Part of your problem is that mathematicians -- i.e., the people who teach it -- hate you and want you to fail; that maths could be useful for science galls them no end. The solution is ready to hand: any good Engineering college provides its own mathematics classes, geared toward doing useful things with it. Now, they might have their own reasons for wanting you to fail, but they don't do it by trying to make the maths look useless for real problems.n
Posted by: Nathan Myers | April 29, 2008 3:06 AM
Thanks for all the responses/advice, everyone. Only after I wrote this did I find out that the debate over concrete vs. abstract has been going on for a long time.
To answer some of the questions about my own problems, I wound out having to withdraw from precalculus last semester because the professor was horrible and I did not have the time to go to class and see a tutor for hours every week (I work in addition to my coursework). I thought things would be better this semester (I picked a different campus, different time, etc.) and wound up with the same guy. I'm taking the course by itself starting in about two weeks so I can concentrate on it.
Surprisingly, I have done well in math classes in the recent past. When I went to Union County College I did fairly well, and I at least felt like I understood what was going on. Maybe it was being in a smaller class (and not a huge lecture hall) or the professor, but with math-heavy courses I've found the smaller school to be far superior to Rutgers.
After precalc, statistics is the only other course I have to take that is purely math. I have no idea how I'll do, but I've got to do it.
Jay; As for a meetup this summer, you're more than welcome to try and organize one. I've made myself open to meeting people before but have generally not received much (if any) response, but I'd be more than happy to participate in a meet-up. Just let me know the details and I'll post something on here and see who's interested.
Posted by: Laelaps | April 29, 2008 8:54 AM
Good luck with the precalc and statistics. And, I have to say, I definitely think that community colleges are a great deal. Smaller classes, cheaper tuition, and at least the professors I had at mine were all excellent educators.
With regards to the meet-up, if I'm in charge, then I'll wait until I get the dates for a field trip for my internship nailed down. But my general idea is some Saturday in June (or maybe July), at AMNH, looking at cool paleo stuff. I should be able to figure out the details sometime after the end of finals and the start of the internship, so between mid-May and June, look for another sidetrack-y post on the topic.
Posted by: Jay De Lanoy | April 29, 2008 9:23 AM