I've been reading The Design of Everyday Things, which I recommend as a useful and interesting way of thinking about all sorts of minor frustrations in daily life. It's also applicable to teaching - I've definitely noticed many student problems that have more to do with misunderstanding the nature and purpose of the assignment, than with misunderstanding the concepts involved. I was blown away when I realized that not everyone automatically interprets an expression like "Nc(M)" to mean a quantity Nc that is a function of a variable M, but apparently a dedicated mathphobe in the California public school system can easily escape such basic mathematical acculturation. If we don't provide some other cue to them that we expect to see several different values of Nc, they will get confused and make mistakes.
Anyway, here's what Donald Norman has to say about math education:
With badly designed objects - constructed so as to lead to misunderstanding - faulty mental models, and poor feedback, no wonder people feel guilty when they have trouble using objects, especially when the perceive (even if incorrectly) that nobody else is having the same problems. Or consider the normal mathematics curriculum, which continues relentlessly on its way, each new lesson assuming full knowledge and understanding of all that has passed before. Even though each point may be simple, once you fall behind it is hard to catch up. The result: mathematics phobia. Not because the material is difficult, but because it is taught so that difficulty in one stage hinders further progress. The problem is that once failure starts, it soon generalizes by self-blame to all of mathematics. Similar processes are at work with technology. The vicious cycle starts: if you fail at something, you think it is your fault. Therefore you think you can't do that task. As a result, next time you have to do the task, you believe you can't so you don't even try. The result is that you can't, just as you thought. You're trapped in a self-fulfilling prophecy.
Mathphobes and math teachers in the audience, does this describe your experience? I find it hard to imagine a math curriculum that doesn't build on previous lessons, but I can certainly imagine a setup that allows more time and flexibility for teachers to address sticking points before they turn into nightmares.
A couple of old-ish (in blog time, anyway), related discussions from the geoblogosphere, one at Sismordia:
A similar debate is happening at the moment in the Geophysics circles at our lab. We are being asked to open up geophysics teaching to students from a variety of backgrounds, including a number who have little or no mathematical training. The question is how to follow these courses with a master's or PhD program that necessarily involves a great deal of mathematical analysis?
And the other at All of My Faults are Stress-Related with some useful links.
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Maria, I'm not sure that last link (to Kim's site, I think) is working correctly.
Keeping in mind my math teaching experience consists of teaching junior high math on a limited basis and homeschooling my older son in elementary, I'd say it is possible to teach math with a limited dependence on previous knowledge. However, it relies on teaching everything on a very conceptual level (i.e. that math is about relationships and how operations represent changes in those relationships...how I taught at home) rather than teaching math as a process to solve for the "right answer"...which is most of what I saw in the classroom setting.
But that is just my opinion. :-)
Problem solving can be great for teaching some kinds of math and to certain people. I did fine until the last part of 2nd year calculus, when working problems without knowing what it really meant didn't work for me. It finally made sense some when taking Ground Water Hydrology as an undergra, and then really made sense when taking Thermodynamics in grad school. All previous calculus math classes had really not shown what integration was really all about - just do this, do that, and you'll get the right answer. Sure...
What Cherish said. :) I was a lifelong mathphobe and recently have been giving a lot of thought to where that ingrained fear (and, let's face it, revulsion) came from -- because it's a very common problem, and one that should really no longer be blithely accepted in US society. It's all about the connections. Subjects in school, particularly in later levels, are taught completely separately, and it's not until later that one begins to see how they all connect with each other.
Conceptual approaches to teaching math get slammed from time to time because of course, math's real power lies in the abstraction. But you've got to understand how the two are connected, not just understand each separately.
FInally a word about failure: part of the problem is that our culture fears failure (or just plain looking "stupid") so much that it becomes a detriment to true learning. Classes that emphasize solving for the "right" answer, as opposed to the process of discovering the right answer, just make this attitude worse. Failure is how we learn, after all. Perhaps we need to change our ideas about failing first....
Part of the problem seems to be a "one size fits all" approach to math education, where students who don't master the concepts aren't give the extra attention they need. A couple of years ago there was an article in the LA Times about students who were having a difficult time passing the exam for California's new algebra requirement. Part of the problem seemed to be students who were left behind in math in elementary school. That meant that there were some students who hadn't mastered fractions taking the same algebra class over and over again - and eventually dropping out.
What seems like part of the problem is that many (most?) people buy into the idea that math is "hard" and that not everyone has the ability to do it. So instead of blaming inadequate teaching, the blame falls on the student who "just can't do math."
Given that my maths education was 30 years ago on a different continent I would have expected it to have been written Nc = f(M) and I would have expected some comment about the nature of 'c' (integer, countinuous, etc.). But I'm stretching the brain cells a bit here.
One of the biggest problems I've always had with math classes, especially in college, is that I didn't know why I was supposed to care about the math. I took the courses through the math department. They seem to assume that all the students are interested for the mathematical value of the material. I only wanted to learn it if I could apply it to science.
I also had a hard time understanding concepts if I didn't have a mental image to go along with them. For example, I didn't know what a gradient was until the very end of my vector calculus class. I could find the gradient for you, but it meant nothing. Then someone showed me a topographical map and I immediately understood what this gradient thing they kept talking about was and why it was important.
I don't know how universal these ideas are, but many students may understand concepts more quickly and completely if the concepts are related to their interest and previous knowledge.
Nicole, I had the same problem as you when I began calc. However, I lucked out and had a topologist as my vector calc prof. What a difference! Being a topologist (from Boston), he was always "drawring pictures". Everything he did had pictures to go with it, and it made so much more sense than a lot of my previous math. When I started taking more higher level math and physics and didn't understand what was going on, I began asking the question, "What would this look like?" or asking for a more conceptual explanation. Once I had that, it was a lot easier to grasp how the details fit into the big picture, but there are a lot of teachers who get very caught up in the details and fail to explain why what you're doing is necessary or how it fits into the scheme of things. I don't know how many classes I had that nothing seemed to make sense while I was taking the class, but later, whenl I was taking another class and had to apply what I had learned, it fell into place.
Hmmm...either I wasn't paying attention when I typed my name, or the internet doesn't like me today.
Math is totally regular, right?
On math professor shows another this statement:
y = sin(x)
The other says, I see the function n(x), but what are s and i?
Yeah, recovering mathamaphobe here.
I had it bad too, wouldn't even count my change after making a purchase. I eventually embraced the subject when I became interested in geology and found that I could do it if I put my mind to it.
Some things to get off my chest about teaching before my rant begins in earnest:
-- If you want to do or just understand some science and be a responsible citizen, you have to have math. Period.
-- In my day at least, not enough attention was paid to how to do proofs (lots of plug-n-chug though).
-- This is a big pet peeve: writers who assume the math (or whatever) is as fresh in your mind as it is theirs. Would it really kill a body to go ahead and fill in the gaps fercristsake? Personally, when I'm coming to a subject cold for whatever reason (for example, it could be a bad breakfast, or time away from the subject) I find it an aid to learning when the material is presented more gradationally and less in big quantum jumps. At least it's easier to pick up a lost thread that way. Cohesion is important.
-- I'm inclined to agree with Steve Gunnell about the function notation, BTW, but it's been a while for me too.
-- Also agree about the visual aids.
I think there's a cultural issue about math and science that tends to be under-appreciated in ivory towers. This is not a post modernist observation. The most obvious example is that other peoples' love and enthusiasm for a subject can be a powerful motivator. Math and science are and rightly should be highly competitive, but you're only hurting your subject if you let a sense of community among novices or amateurs be destroyed in the process. I've seen 98 pound weaklings get better treatment by jocks in health organizations.
There are more things to making learning and participating satisfying than just the satisfaction of solving a puzzle. I say that even though I think problem solving needs to be more effectively celebrated. Your mental environment for learning is influenced by the complex social ambience of the subject at hand, things as obvious as the lighting in your classroom and as subtle as a turn of phrase. And there's satisfaction in being included in the company of problem solvers. Think of different institutions, clubs, churches, rock bands, whatever. They all call to mind their own styles and �elan. What does math call to mind? A exercise in unpleasantness? A migraine like a railroad spike shot between the eyes? Or maybe just as bad, some weird clown dressed in a lab coat on TV going on and on about boogers? Yeah there's some edumakation for ya...
A while back GG did a piece on the vocabulary and language of geology. I found that inspirational.
Radge Havers says, "I'm inclined to agree with Steve Gunnell about the function notation, BTW, but it's been a while for me too."
I agree, too: the function notation in the main post meant nothing to me; I had a lot of math in the 60's and 70's - so I guess the notation has changed over time, and it could be different on different continents, maybe.