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Monday, October 15, 2012

Why Algebra

I've been taking notice more this year with all the Why Algebra back-and-forth of how indispensable it is for learning Geometry. I know I'm not going to win any converts by answering "why Algebra?" with "because Geometry" but those about to rage quit this post are probably not going to be persuaded by a blog post anyway.

I don't mean the tepid "Algebra!" problems with the snazzy xy logo the textbook offers. I assign some of these, don't get me wrong, but if this is the sum total of the algebra used in your geometry course, you're doing it wrong.



I mean in the process of exploring how measurements on a plane relate to each other, algebra is a weapon the kids should be deploying like on the daily, in the cycle/ladder of examples, conjecture, prove, extend. Kids' resistance to this leads me to believe they aren't often asked to do this.

Expressing your conjecture. Here are some examples.





The resistance at first will be formidable. They will truck along obligingly until the last part, which they will leave blank and wait for someone to tell them what to put. I have to restate just what I want them to do a bunch of different ways, and ignore that they are pleading for me to just do it for them with their pleady little faces, and have the patience to wait them out. "Okay you saw examples of what results when angle B is 36, 48, 55 degrees. What about any angle of x degrees?" "5 sides 3 triangles, 6 sides 4 triangles, 7 sides 5 triangles. n sides ??? Triangles?" "how did you turn a 25 into a 130, and a 41 into a 98?" They will look at you like you are a crazy person. Meet this with incredulity. "You did take and pass an algebra course last year. Yes?" They will look at you with a face all dark clouds that says, "you bitch." Meet this with unrestrained confidence. "You, you can do this. Take a breath and focus on it for a minute. I wouldn't ask you to do something I didn't know you can do." They will. After the first few it gets easier.

Using Algebra to Prove Things
It's a major way we can know something is always true. "Is that always true? How do you know?" are sentences I probably utter in my sleep by now.

Here are some examples





The lovely part about using Algebra to make sense of Geometry is it offers context to hang your algebra on. In that last example, Mat originally ended up with e = -360 because he forgot to distribute the negative. He knew something was wrong, so I asked him to write it on the board so we could help. One of his classmates spotted the error in short order. Sometimes the stuff from the previous course doesn't gel until you have to use it in the next course.

I guess my point is, maybe "Why Algebra?" is the wrong question. If we agree that Mathematics (actual Mathematics involving logical, abstract ways of thinking about how quantities and measurements fit together, generalization, observing and using patterns, making predictions) is a valuable thing for an educated person to learn about, the question seems kind of silly. Of course Algebra. But, maybe we don't all agree with that. Or maybe we should talk about whether the content of Algebra 2 and beyond is valuable for everyone to learn about. Which the CCSS seems to have decided "yes" without consulting anybody. I guess I don't think the question has been very well defined.

7 comments:

  1. Fascinating to come across your explanation of using a student-centered, meaning-making approach to math teaching right after seeing the reference to a new study on student-centered science teaching:

    http://phys.org/news/2012-10-advocates-student-centered-approach-science.html#jCp

    Both arguments bring to mind how important it is for students to experience and learn to deal with frustration and failure. The great flaw of the "empty vessel" formulation of the student learning process is that it is based on the illusions of right answers and singular methodologies.

    Students need to learn reasoning, not just facts, and then practice applying what they learn in a variety of contexts.

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  2. Patrick Callahan's Modest Proposal:

    http://www.youtube.com/watch?v=VlnBRkIOSrM

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  3. I love what you did with the Algebra in Geometry. I run into the same issues . . . becomes just simple solving equations with no true rigor or complex thinking. Algebraic proofs is an awesome direction to go . . . and it's gonna be part of my next unit! Thanks Kate!

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  4. Just found your blog today and I love where you are coming from. Looking forward to reading more.

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  5. Are you willing to share your exterior angle dynamic worksheet? I am using edmodo in my classes this year and I'm trying to find more effective online resources. Thanks!

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    Replies
    1. Sure. I can find it at school tomorrow. I didn't share these because they were mostly ripped off from key press. But I'll find you the link to the worksheet at least.

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  6. Sorry I forgot about this for a while. Here's the exterior angle:

    http://www.geogebratube.org/material/show/id/18464

    Here's the isosceles triangle:

    http://www.geogebratube.org/material/show/id/18474

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