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.

proof pic1

 

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?

mistake2   mistake3

 

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?

IMG_3019 IMG_3020 IMG_3021 IMG_3022 IMG_3023 IMG_3024

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.

 

 

 

 

 

 

Your thoughts?

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?