On May 28, 2013, at 4:43 PM, "Louis Talman" <firstname.lastname@example.org> wrote:
> I don't know what is. And Robert acquiesces, suggesting that "a student of algebra" would never avoid algebra---leaving us to guess that he agrees that Wayne's strategy is Real Algebra.
When I use the phase "student of algebra" I mean a bonafide student of algebra. A student that is getting it. There has to be a way in these discussions to distinguish between students getting it and students not. How can you even discuss pedagogy if the student is just some random variable? A student of algebra is having the epiphanies we had, the least of which and the most fundamental is the personal recognition of algebraic reasoning and that these problems can be "solved". And I said that once you cross this point you no longer guess (I didn't say avoid). That blissful ignorance is gone forever. From this point forward, as far as you (the student) are concerned, problems are either solvable or not, and when they are not solvable, to you that means you just don't know "how". Only later, as an advanced student of algebra, will you realize that some of these problems are actually NOT solvable.
I actually didn't understand Wayne's response, at least in the context of "teaching algebra", that the students should plug in the values into the formulas and compare results. That isn't algebra. However, if I am given a problem with discrete values, that is generally what I do first. Run the numbers.
I ran numbers prior to knowing algebra and after knowing algebra. I can say without a doubt that running numbers never taught me algebra. I ran a lot of numbers prior to knowing algebra. I was/am a huge fan of "number" and when I first saw adding machines and then calculators I tested the machines, not the other way around. But it was all arithmetic. The closest it ever came to "algebra" was with situations involving simultaneous linear relationships, but the thinking was still not algebraic. Pre algebraic at best. And this goes for tables and graphs as well. Without algebra and the reasoned certainty it brings, you might as well be (and you will be) guessing.
Wayne does bring up an interesting point though. If students "know" the formula for the volume of a cylinder and are given two explicit examples of cylinders, then why wouldn't they immediately be able to answer which has the greatest volume? These students misunderstand more than just the formula for volume. I can't even say that their misunderstanding stops at math.
The only door to algebra is algebra. That is all I am saying. At least we all agree that what Richard continues to post here is not algebra. I have long moved past the question of "What the heck is it that Dan is teaching?" to "Why did Dan stop teaching algebra?" My conclusion, and this goes also to why Richard stopped teaching algebra, is that they are no longer teaching students of algebra. Without students of algebra how can you possibly be a teacher of algebra? Either you teach something that none of your students get OR as Dan and Richard have chosen to do, you teach something else. My only concerns are...
1. Don't call it algebra (that is a lie to both the students and their parents). 2. Are all of your students properly placed in this non-algebra class (could any of them have gotten algebra if given the chance)?