tag:blogger.com,1999:blog-1075593398139537131.post1287745468730999836..comments2024-08-05T00:13:41.798-07:00Comments on Without Geometry, Life is Pointless: Proof Doesn't Begin with Geometry: Redefining ProofAvery Pickfordhttp://www.blogger.com/profile/10433339146333801163noreply@blogger.comBlogger8125tag:blogger.com,1999:blog-1075593398139537131.post-13647471677765654052015-06-02T14:10:10.567-07:002015-06-02T14:10:10.567-07:00(That blog link should be this which is to the spe...(That blog link should be <a href="http://y4ist.blogspot.fr/2015/05/same-difference.html" rel="nofollow">this</a> which is to the specific post.)Mr Gregghttps://www.blogger.com/profile/15114830266540886842noreply@blogger.comtag:blogger.com,1999:blog-1075593398139537131.post-34733329121017583882015-06-02T13:59:54.001-07:002015-06-02T13:59:54.001-07:00This is very interesting. I'm thinking lots ab...This is very interesting. I'm thinking lots about claims and justifying claims at the moment, and trying a few things out with my Year 4 class (Gr 3). <br /><br />I need to blog about this really, because there are lots of things to unpick. I agree with Joshua that understanding the "why" is the thing, having a sense of what is going on. But I also am beginning to think that the talk of moving statements from hunch to conjecture to proof (with all the other gradations in between) can work for young children; and also connects them with a whole history of mathematics.<br /><br />At the moment I'm more framing it as explanation, and looking for a kind of systematic thinking. A recent example is here <a href="http://y4ist.blogspot.fr/search/label/Maths" rel="nofollow">on our blog</a>; a lesson where the students tried to explain why it was true that if you add one to the minuend and subtrahend in a subtraction sum, you'll get the same difference. I was pretty pleased and convinced by the explanations that lots of the kids gave. I'm pondering how they could have gone further. (In this case even Euclid can only express it as an axiom: "If equals are added to equals the sums will be equal.")Mr Gregghttps://www.blogger.com/profile/15114830266540886842noreply@blogger.comtag:blogger.com,1999:blog-1075593398139537131.post-17902096556186799592013-11-16T18:54:56.120-08:002013-11-16T18:54:56.120-08:00Just saw your newest post, can't wait to find ...Just saw your newest post, can't wait to find a way to incorporate it into my classes. While researching Mastermind online, I found that it might have originated from a game called Bulls and Cows that uses the nine digits like you describe in your next post. I'm trying to come up with a catchier name.<br />Emilyhttps://www.blogger.com/profile/15708702453198417901noreply@blogger.comtag:blogger.com,1999:blog-1075593398139537131.post-8038137533125688902013-11-10T06:55:18.862-08:002013-11-10T06:55:18.862-08:00Do you use Mastermind to teach proof in your geome...Do you use Mastermind to teach proof in your geometry class? I would love to hear more about that.Emilyhttps://www.blogger.com/profile/15708702453198417901noreply@blogger.comtag:blogger.com,1999:blog-1075593398139537131.post-4391723763339128552013-11-06T18:47:12.455-08:002013-11-06T18:47:12.455-08:00Yeah, I agree that we're pretty much on the sa...Yeah, I agree that we're pretty much on the same page in terms of what to do, but maybe a little bit different in what parts of it we value most. I tend to use "convince a skeptical reader" more for the routine exercise-type work, as a guideline for how much justification they need to show and how thoroughly they need to explain their solution process. I use the "understand why" and "expose connections to other ideas" more when I want a proof-like answer to something that's usually a deeper problem.<br /><br />I think you're right about multiplication: if you're in 3rd or 4th grade, this kind of explanation would seem much more proof-like and deep and difficult. You're showing the connections between the distributive property, the place-value system, and the steps of the algorithm. Or something like that.Joshua Zuckerhttps://www.blogger.com/profile/04689961247338617418noreply@blogger.comtag:blogger.com,1999:blog-1075593398139537131.post-69160055990565111522013-11-05T19:05:54.074-08:002013-11-05T19:05:54.074-08:00@Josh: I agree with what you're saying about c...@Josh: I agree with what you're saying about continuously answering the why question, and I guess I see a pedagogical, historical, and philosophical motivation for the rest of the class playing this role. Pedagogically, I see how this format increases the level of discourse, both from the perspective of the student sharing his or her proof and the rest of the students. Historically, there are a number of instances of proofs living in a state of limbo between conjecture and theorem while peers worked through these ideas with a fine tooth comb. The proof of Fermat's Last Theorem is probably the most famous example. And philosophically, I want my students to understand that mathematics is a human construct. We get to decide (as a community) what is true and what is not true.<br /><br />That said, we're pretty much on the same page about bridging proof with explanations. It's still an open questions, though, as to where this ends. As I said in the post, my explanation for a multiplication problem feels different-not like a proof. But maybe that's only because, for me, these are things that I deeply understand and "proved" long ago.Avery Pickfordhttps://www.blogger.com/profile/10433339146333801163noreply@blogger.comtag:blogger.com,1999:blog-1075593398139537131.post-49032621077375858622013-11-05T13:52:06.417-08:002013-11-05T13:52:06.417-08:00I wish I had been able to attend your session. Th...I wish I had been able to attend your session. This is very interesting and I am looking forward to reading the rest of your tome.Robert Kaplinskyhttps://www.blogger.com/profile/12730219834465583755noreply@blogger.comtag:blogger.com,1999:blog-1075593398139537131.post-15489036716750340472013-11-05T09:50:41.071-08:002013-11-05T09:50:41.071-08:00I sometimes wonder about a good definition of proo...I sometimes wonder about a good definition of proof being "answering the 'why?' questions until the questions become silly". Maybe that has to do with having been the parent of kids who are very good at the 'why?' game, though, and trying to prove to them that it is, in fact, bedtime.<br /><br />I don't tend to be much in the "convincing a skeptical reader" group on proofs. Most of the time, most things that you're proving are things that you (and everyone) are already almost convinced by. So the main things I focus on in proof are those "why" questions -- why does this pattern hold? why is that angle always a right angle? -- and on the role of proof in showing connections between different ideas, such as that the parallel postulate is linked to the angle sum in a triangle. (There are certainly cases where the proof is of something we're less sure of the truth of, though, so in those cases the skeptical reader test is very important!)<br /><br />I think the "explain why" part of the role of proof is very well-served by the sort of less-formal modes of communication that you suggest here. The connections part sometimes requires more precise logic to make sure you're getting at the other ideas that are actually useful ingredients in understanding the idea in question.Joshua Zuckerhttps://www.blogger.com/profile/04689961247338617418noreply@blogger.com