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## Do These Properties Guarantee Congruence?

The activity is from a Shell Center task, and the student work is from my own class. We’re missing a few kids, but this is representative of the whole group’s work.

Questions:

1. What do you notice? Anything interesting?
2. What categories of student responses do you see?
3. What sort of feedback would you give to push their mathematical thinking further?

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## “What feedback would you give?”, continued

In a previous post, lots of commenters said that they didn’t feel that you could give helpful, written feedback because there wasn’t enough evidence of student thinking on the quiz. Given that complaint, it might be interested to see how those same teachers would give written feedback on a quiz that gives significantly more evidence of how a student is thinking.

Here’s another quiz: what sort of written feedback would you give? (The checkmarks are from the student, who was provided with an answer key and checked her own work, ala this.)

As before, imagine that you don’t have to write a grade on this paper. Some things I’m wondering about:

• Would you give comments on every solution, or only some of them?
• Will you ask kids to “explain why you said _______”? When is it helpful to ask for an explanation? When isn’t it?
• Will you give your kids specific next steps, or will you mostly point out the good and the bad of their work?
• Will you throw up your hands and say “You really need to have a conversation with the kid!” for this sort of quiz also?

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## Troubles With Proofs

What mistakes would you expect to see in the proofs of the problem below? Take a moment and make some predictions. You might find it helpful to know that this was part of an end-of-the-year exam, and that kids were able to use whatever proof representation they wanted. In other words, they could write two-column proofs, paragraph proofs or flowchart.

Here are the mistakes, pulled from the class set. Or maybe you feel uncomfy describing these proofs as “mistakes”? Maybe it’s better to say that they contain mistakes? Or that they are proofs that aren’t where we probably want them to be?

Do you feel comfy calling these “mistakes”? What would you call them?

What are the next steps for these kids? What would you recommend that their teacher do?

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## If ABCD is a parallelogram…Prove that angle A is congruent to angle E

These are from my classroom, and a little bit of context might be helpful:

• This is from a very strong Honors class.
• Proof, right now, means something less formal in our Geometry class than it might mean in yours’. Proof doesn’t mean “Statements/Reasons.” Proof means offering an explanation for why something is true.
• We do this because it’s just as rigorous without crushing the souls of anyone in the classroom. Look at the array and variety of reasoning going on in these proofs. By keeping things less formal, we’ve got enough breathing room to actually do some Geometric thinking.
• As the year has been going on, though, we’re getting more and more rigorous. These exercises help reveal some sloppiness in the kids reasoning. These proofs fail by their own standards of explanation. I’m thinking that I’ll be printing these out and handing them back for discussion. This is exactly what my English teachers friend does with essays.

Feel free to comment wildly here, either on my standards, some of my bulletted statements, or about any of the student work.

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## Problems with Proofs

The proof above isn’t great. In the comments, take on any of the following questions (or any others):

1. Sometimes kids slap stuff together when they’re confused, and other times they’re substantively mistaken. Which is this, and what evidence supports your position?
2. How would you help this student recognize that the logic in his proof doesn’t flow?
3. What would your next steps be in working with this student? What sorts of problems would you ask him/her to solve?
4. From the picture above, what evidence of knowledge do we have?