Saturday, December 31, 2005
Ack – cont.
Another lesson on the to-do list is prepping a math chapter for the 6th graders. This is from the infamous “Continued Math” series. I have serious problems with this book, including:
- It is very wordy. I think I understand the rational, might as well polish up on those reading skills. Problem is many (most?) of the students at my school speak English as a second language. It seems unfair that in a subject that has a universal language, one that has a concise set of symbology, these kids are getting hit over the head with their lack of language skill, again.
- Because it is wordy the book is huge, as are all modern text books. (It is almost as if the text book publishers get paid by the page. Oh wait, they do.) This one is broken up into little booklets, which encourages an attitude of “finished that chapter, never need to remember that again!”.
- Because it is wordy, huge and broken up into many separate booklets it is pretty much impossible to find a specific item unless you have been through the whole book, and have an excellent memory, which is sadly no longer true for me.
- But my REAL gripe is the total reliance on the “constructivist” school of education where students learn by exploration and discovery. My favorite example is an activity where students “discover” that the sum of the interior angles of a polygon is equal to (n-2)*180 degrees, where n is the number of sides of the polygon. How do they discover this? By drawing polygons and measuring the angles with a protractor. So this is good exercise with the protractor but the mathematical proof of the “discovery” is never mentioned. Argh! One of the most elegant and simple constructions in all of geometry, that the sum of the interior angles of a triangle is 180 degrees and that all polygons can be broken up into triangles. No logic, no deduction, no chains of reasoning.
I get quite a bit of leeway in how I present material, the kids I work with are either a slam dunks to pass any and all standardized tests; or don’t have a chance at all. Plus I am a volunteer, so there is no financial incentive to do things the “right” way. I see no evidence that a constructive approach works better than a traditional approach. I am giving it a try; we will scale drawings with rubber band pantographs and discover scaling laws. But don’t be surprised if I happen to blurt out a few truths along the way that someone else discovered, say a few millennia ago.
- It is very wordy. I think I understand the rational, might as well polish up on those reading skills. Problem is many (most?) of the students at my school speak English as a second language. It seems unfair that in a subject that has a universal language, one that has a concise set of symbology, these kids are getting hit over the head with their lack of language skill, again.
- Because it is wordy the book is huge, as are all modern text books. (It is almost as if the text book publishers get paid by the page. Oh wait, they do.) This one is broken up into little booklets, which encourages an attitude of “finished that chapter, never need to remember that again!”.
- Because it is wordy, huge and broken up into many separate booklets it is pretty much impossible to find a specific item unless you have been through the whole book, and have an excellent memory, which is sadly no longer true for me.
- But my REAL gripe is the total reliance on the “constructivist” school of education where students learn by exploration and discovery. My favorite example is an activity where students “discover” that the sum of the interior angles of a polygon is equal to (n-2)*180 degrees, where n is the number of sides of the polygon. How do they discover this? By drawing polygons and measuring the angles with a protractor. So this is good exercise with the protractor but the mathematical proof of the “discovery” is never mentioned. Argh! One of the most elegant and simple constructions in all of geometry, that the sum of the interior angles of a triangle is 180 degrees and that all polygons can be broken up into triangles. No logic, no deduction, no chains of reasoning.
I get quite a bit of leeway in how I present material, the kids I work with are either a slam dunks to pass any and all standardized tests; or don’t have a chance at all. Plus I am a volunteer, so there is no financial incentive to do things the “right” way. I see no evidence that a constructive approach works better than a traditional approach. I am giving it a try; we will scale drawings with rubber band pantographs and discover scaling laws. But don’t be surprised if I happen to blurt out a few truths along the way that someone else discovered, say a few millennia ago.
// posted by Old Math @ 11:01 AM 
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Regarding your thoughts about contructivist approaches, whether it "works" better than traditional approaches depends heavily on what you mean by "it works", which depends on what you want your kids to be able to do once the course is over. If you want them to have experience with experimentation and conjecture, then constructivism "works" pretty well. If it's only facts you want (not saying this is a bad thing) then traditional approaches might work better. Or they might not, depending on the kind of students you've got. (Kinesthetic learners in my experience do better with discovery learning.)
But you're right that it's a pain when a textbook uses a one-size-fits-all approach. The #1 thing teachers need in the classroom is the ability and freedom to be flexible and use their instincts and skills when a particular approach isnt' working. That goes as much for constructivist textbooks as it does for traditional ones.
Also, I agree with you that a discovery learning example that doesn't follow through with a proof of some sort is silly, and frustrating for studets who end up learning just the facts and not the "why" behind them (which incidentally is the very thing constructivists hate about traditional pedagogies). It's easy enough to notice the pattern that the angle sum is (n-2)*180, so why not finish the deal and have students "discover" a proof? That's pretty easy too, right?
But you're right that it's a pain when a textbook uses a one-size-fits-all approach. The #1 thing teachers need in the classroom is the ability and freedom to be flexible and use their instincts and skills when a particular approach isnt' working. That goes as much for constructivist textbooks as it does for traditional ones.
Also, I agree with you that a discovery learning example that doesn't follow through with a proof of some sort is silly, and frustrating for studets who end up learning just the facts and not the "why" behind them (which incidentally is the very thing constructivists hate about traditional pedagogies). It's easy enough to notice the pattern that the angle sum is (n-2)*180, so why not finish the deal and have students "discover" a proof? That's pretty easy too, right?
# posted by 
: 1:30 PM
: 1:30 PM
good points.
By "it works" I mean giving students the ability to do mathematics. Doing mathematics means (among other things) using logic, making generalizations and seeing patterns. The number of facts in math at these grade levels is very small, but the facts are very important. Constructive methods do not seem to do very well at those things, there is no emphasis on logical reasoning at all.
The constructivist method for teaching multiplication does not seem "to work" in the sense that many students have trouble multiplying. They understand the idea of multiplication but they have not learned the basic facts.
I would be interested to see how a constructivist Calculus I class would go.
By "it works" I mean giving students the ability to do mathematics. Doing mathematics means (among other things) using logic, making generalizations and seeing patterns. The number of facts in math at these grade levels is very small, but the facts are very important. Constructive methods do not seem to do very well at those things, there is no emphasis on logical reasoning at all.
The constructivist method for teaching multiplication does not seem "to work" in the sense that many students have trouble multiplying. They understand the idea of multiplication but they have not learned the basic facts.
I would be interested to see how a constructivist Calculus I class would go.
I've thought a lot about that, since I teach a lot of calculus and I like and use certain aspects of constructivism in my calculus classes already. I think a successful constructivist approach would produce students who are very skilled at the concepts of calculus -- a derivative is a rate of change, we calculate a derivative by a limit of slopes of secant lines, etc. -- but would not have amazing technical skill at computing derivatives and integrals algebraically. I personally don't think that's necessarily a bad thing, depending on your audience. My audience is mainly business and finance majors, and those folks need conceptual knowledge more than algebraic skill.
But I will say that in order to have a successful constructivist calculus class, your students HAVE to have mastered the basic mechanical skills of algebra and geometry, and I think traditional approach works a lot better for that kind of thing. I have students now who cannot find the slope of a line if I give them two points that are on it, or they think it's run divided by rise, or something -- and don't even get me started on their lack of ability with factoring and solving algebra equations. My students' overall lack of mechanical skill renders the idea of a constructivist approach to calculus moot.
But it also makes me wonder just how well the traditional approach "works" after all, because that is definitely the approach they are getting in high school, and it sure doesn't look like it's working to me.
Personally: (1) I wish we'd stop thinking of constructivist and traditional approaches as two completely separate things, and start thinking of them as two types of pedagogical tools that are complementary; and (2) I'd like to see an all-traditional approach to math in K-12 and then an all-constructivist approach in college. :)
But I will say that in order to have a successful constructivist calculus class, your students HAVE to have mastered the basic mechanical skills of algebra and geometry, and I think traditional approach works a lot better for that kind of thing. I have students now who cannot find the slope of a line if I give them two points that are on it, or they think it's run divided by rise, or something -- and don't even get me started on their lack of ability with factoring and solving algebra equations. My students' overall lack of mechanical skill renders the idea of a constructivist approach to calculus moot.
But it also makes me wonder just how well the traditional approach "works" after all, because that is definitely the approach they are getting in high school, and it sure doesn't look like it's working to me.
Personally: (1) I wish we'd stop thinking of constructivist and traditional approaches as two completely separate things, and start thinking of them as two types of pedagogical tools that are complementary; and (2) I'd like to see an all-traditional approach to math in K-12 and then an all-constructivist approach in college. :)
# posted by : 2:10 PM
: 2:10 PM
I just try to make maths "fun" by throwing snippets at them for them to think about, see
http://www.savory.de/maths0.htm
Even if they only learn the snippets and not the reasoning behind, at least its a step :-)
Stu
http://www.savory.de/maths0.htm
Even if they only learn the snippets and not the reasoning behind, at least its a step :-)
Stu
# posted by : 1:50 AM
: 1:50 AM
About Polygons: isn't it easier to say the outside angle is just 360/N and subtract that from 180 of you want the inside angle?
# posted by : 12:27 AM
: 12:27 AM
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