Let me start with the Mini-Conference session on a Transformational Approach to Geometry by Lew Douglas and Henri Picciotto.
One of my colleagues asked our Principal if he could go and he thought it would be a good idea if one of our Geometry team members went with him - so I ended up being able to go "early" to this wonderful session.
At the beginning Henri and Lew presented the idea that we'd have to (at least partially) re-define the way that we define specific Geometric terms and we would have to become more comfortable with students giving informal arguments because we should reserve formal proofs for the really important things (yay!). I whole-heartedly agree.
When I took Geometry we spent almost all of our time proving. While it was fun for me (I guess it shouldn't have been a really big surprise to me that I became a math teacher!) I now wonder about my classmates. How many of them were turned off by the 180 degree turn into the land of proofs?
Students should always have to justify their reasoning (SMP3), construct viable arguments and critique the reasoning of others in all of their classes. Math makes people reasonable, right?
But does it have to be a 2-column proof every time? Can't a student explain something to me and that's enough? If students are using language to convey meaning in a meaningful way then I feel plenty satisfied with that.
I'm a big fan of flow proofs so oftentimes when I begin to venture into the land of formalization I use flow proofs to soften the blow. A tiny bit visual and a tiny bit less formal than the 2-column cousin. I'm not saying we shouldn't ask students to write about mathematics and I'm not asking or advocating that 2-column proofs be stricken from classrooms but it was at least a little bit vindicating to hear others say that we should reserve the formalization for when we really need it.
Was that their main point? No. Transformational Geometry takes a different way of thinking about justifications for proof and it also takes a more relational-thinking approach as opposed to always thinking about properties and definitions. Again, not that you don't use properties and definitions but it just takes a back seat to relations of objects.
What would happen if this was an angle bisector? What does this mean for reflections? What do I know about points that are reflections? What properties are true for reflected images? I really think this is a great way to get students to think about the relationships that different Geometric objects have with each other and to get students to create their own line of questioning in order to get themselves to the right place. I can't verify that this will but I'd like to think that moving towards using Transformations as a basis for getting students to justify their reasoning by using their language and understanding as opposed to forcing them to use words that aren't naturally their own.
Just an opinion. Agree or disagree - it's up to you. Also, it would probably be rare (possible, not probable) that you sat in the same session that I did and it's my right to disagree with myself tomorrow even! But these are the ravings of a almost-on-the-final-stretch-can't-wait-for-all-of-my-finals-to-be-over-teacher and it's getting late. I apologize in advanced for any inadvertent craziness I may have typed.
One of my colleagues asked our Principal if he could go and he thought it would be a good idea if one of our Geometry team members went with him - so I ended up being able to go "early" to this wonderful session.
At the beginning Henri and Lew presented the idea that we'd have to (at least partially) re-define the way that we define specific Geometric terms and we would have to become more comfortable with students giving informal arguments because we should reserve formal proofs for the really important things (yay!). I whole-heartedly agree.
When I took Geometry we spent almost all of our time proving. While it was fun for me (I guess it shouldn't have been a really big surprise to me that I became a math teacher!) I now wonder about my classmates. How many of them were turned off by the 180 degree turn into the land of proofs?
Students should always have to justify their reasoning (SMP3), construct viable arguments and critique the reasoning of others in all of their classes. Math makes people reasonable, right?
But does it have to be a 2-column proof every time? Can't a student explain something to me and that's enough? If students are using language to convey meaning in a meaningful way then I feel plenty satisfied with that.
I'm a big fan of flow proofs so oftentimes when I begin to venture into the land of formalization I use flow proofs to soften the blow. A tiny bit visual and a tiny bit less formal than the 2-column cousin. I'm not saying we shouldn't ask students to write about mathematics and I'm not asking or advocating that 2-column proofs be stricken from classrooms but it was at least a little bit vindicating to hear others say that we should reserve the formalization for when we really need it.
Was that their main point? No. Transformational Geometry takes a different way of thinking about justifications for proof and it also takes a more relational-thinking approach as opposed to always thinking about properties and definitions. Again, not that you don't use properties and definitions but it just takes a back seat to relations of objects.
What would happen if this was an angle bisector? What does this mean for reflections? What do I know about points that are reflections? What properties are true for reflected images? I really think this is a great way to get students to think about the relationships that different Geometric objects have with each other and to get students to create their own line of questioning in order to get themselves to the right place. I can't verify that this will but I'd like to think that moving towards using Transformations as a basis for getting students to justify their reasoning by using their language and understanding as opposed to forcing them to use words that aren't naturally their own.
Just an opinion. Agree or disagree - it's up to you. Also, it would probably be rare (possible, not probable) that you sat in the same session that I did and it's my right to disagree with myself tomorrow even! But these are the ravings of a almost-on-the-final-stretch-can't-wait-for-all-of-my-finals-to-be-over-teacher and it's getting late. I apologize in advanced for any inadvertent craziness I may have typed.
