Re: [Edu-sig] a non-rhetorical question

On Sun, 2007-07-08 at 21:46 +0200, Laura Creighton wrote:
Do you have many students who are good at geometry and still rotten at algebra? Also what do they say when you ask them 'what don't you understand here?'
I wouldn't say rotten, but it's not unusual to have students who do significantly better in geometry than algebra. Again, it could just be a year of brain maturity that makes that so. "What don't you understand?" Usually if a kid can answer that they don't have a problem: Ex: 2x + 3y + 5x = 7x + 3y Kid 1: "Why don't you understand?" "Where did the 7x come from?" "From combining like terms. I added 2x and 5x." "What are like terms?" "Terms with the same variables to the same powers. 2x and 5x both have x to the first power and no other variables." versus Kid 2: "Why don't you understand?" "I just don't get it." Not a great example, because most students can do a little better with like terms than Kid 2. But the point is that the "don't get it" kids I'm talking about can't really tell you what they don't get. They see a string of symbols on line one and another on line two and claim to see no connection between them. And though I can often ask a series of questions to determine what they don't get and to explain it - "Do you see where the 3y comes from? Good. How about the 7x? Okay, do you see the 2x and 5x? etc." - they'll still say they don't get it. That was probably a longer yet less complete answer than you were looking for. Nonetheless, I hope it helps. Jay

On the algebra/geometry questions: As a longtime math professor in a past life, it is my observation of students going forward from high school, and of students long ago when I was in high school myself, and tutored a lot of peers, that there are several effects in the algebra/geometry discussion. It seems to me that there are many students who process visual and spatial information very well, who just 'get' geometry, and may have a very hard time with the pure symbol manipulation in algebra. Of course constructing a geometric proof does involve getting into symbolism, and some who learn to get motivation for that from the spatial images in geometry, then have an easier time with algebra. I have no hard data, but only a great deal of examples. I also hypothesize that some students who are capable of learning symbol manipulation, but need practice, do well getting that experience in algebra, and then are less overwhelmed taking geometry later by the added requirement to have spatial intuition. As to the natural order of brain maturation for processing symbols vs spatial relationships, I leave that to others. Andy Harrington Jay Bloodworth wrote:
On Sun, 2007-07-08 at 21:46 +0200, Laura Creighton wrote:
Do you have many students who are good at geometry and still rotten at algebra? Also what do they say when you ask them 'what don't you understand here?'
I wouldn't say rotten, but it's not unusual to have students who do significantly better in geometry than algebra. Again, it could just be a year of brain maturity that makes that so.
"What don't you understand?" Usually if a kid can answer that they don't have a problem:
Ex: 2x + 3y + 5x = 7x + 3y
Kid 1: "Why don't you understand?" "Where did the 7x come from?" "From combining like terms. I added 2x and 5x." "What are like terms?" "Terms with the same variables to the same powers. 2x and 5x both have x to the first power and no other variables."
versus
Kid 2: "Why don't you understand?" "I just don't get it."
Not a great example, because most students can do a little better with like terms than Kid 2. But the point is that the "don't get it" kids I'm talking about can't really tell you what they don't get. They see a string of symbols on line one and another on line two and claim to see no connection between them. And though I can often ask a series of questions to determine what they don't get and to explain it - "Do you see where the 3y comes from? Good. How about the 7x? Okay, do you see the 2x and 5x? etc." - they'll still say they don't get it.
That was probably a longer yet less complete answer than you were looking for. Nonetheless, I hope it helps.
Jay
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On 7/9/07, Andrew Harrington <aharrin@luc.edu> wrote: <<SNIP>>
I also hypothesize that some students who are capable of learning symbol manipulation, but need practice, do well getting that experience in algebra, and then are less overwhelmed taking geometry later by the added requirement to have spatial intuition.
As to the natural order of brain maturation for processing symbols vs spatial relationships, I leave that to others.
Andy Harrington
Yes indeed. And I think many of us are making the point that students develop differently, such that they might use their strengths to address their weaknesses (with guidance from a teacher/mentor should they be lucky enough to have one). Lexical: algebra, computer programming, models, controllers Graphical: geometry, (interactive) views of models Going back and forth between the two, seeing how intensively lexical expressions, of polyhedra in terms of edge-connected vectors say, relate to purely geometric views of same (in a ray tracer, game engine or VRML viewer), is a way to help the brain mature I'd say. We're connecting the dots across the lexical and the graphical, building a bridge strongly anchored on both sides. This was a major theme in my presentation at Europython in Lithuania yesterday: http://controlroom.blogspot.com/2007/07/connecting-dots.html Kirby

On Tue, 2007-07-10 at 01:22 +0300, kirby urner wrote:
Yes indeed. And I think many of us are making the point that students develop differently, such that they might use their strengths to address their weaknesses (with guidance from a teacher/mentor should they be lucky enough to have one).
I agree with what you're saying in principle. I'm just not sure that talents in one area are always infinitely marshalable to the service of other tasks. For example, I'm good at math but suck at basketball. But I doubt that even the genetically engineered lovechild of Jaime Escalante and Phil Jackson could turn my mathematics talent into success on the basketball court to any significant extent. So maybe basketball and math is too big a gap to bridge. But surely algebra and geometry is doable, right? Certainly, there are numberless fascinating connections that might be productively studied in either course. I'm just not sure that geometric models are likely to help with the difficulties struggling students often have. Two examples: * It's like pulling teeth to get students to respect order of operations and remember consistently that -3^2 = -9. Where's the geometric model for that? * (a + b)^2 = a^2 + 2ab + b^2, not a^2 + b^2. Here there is the standard geometric area model for multiplication. Sometimes I present multiplication with the model, sometimes not. It doesn't seem to change the error rate. There are certainly many beautiful connections among the branches of math and computer science, and I agree that mathematics curricula should include some of them. I'm just skeptical that these connections are the golden door to learning formal math. Information and computational theory teaches us that data and algorithms have a certain irreducible complexity. If your Turing machine doesn't have enough states or enough tape, there are things it just can't do. I don't think in undamaged human brains difficulties in learning algebra can be traced to such gross level deficiencies in computational power, but I do think computer science may have something to say here. Not every Turing machine is a universal Turing machine. If your algebra machine is working, I believe it's doubtful your geometry machine will work in the pinch. Kirby, I have skimmingly followed your work the past couple of years, and I don't mean for any of this to be a criticism of what you're doing in particular, just the general notion that models and applications are the answer to everything in education. Jay
participants (3)
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Andrew Harrington
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Jay Bloodworth
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kirby urner