tag:blogger.com,1999:blog-7245208048685880741.post7918773945709262911..comments2018-01-19T19:23:14.570-05:00Comments on Rational Expressions: Proofs of the Pythagorean Theorem - What Am I Even Trying To Teach?Michael Pershannoreply@blogger.comBlogger6125tag:blogger.com,1999:blog-7245208048685880741.post-55716969350779947732015-03-06T21:07:42.823-05:002015-03-06T21:07:42.823-05:00Couple of interesting extensions to Pythagoras the...Couple of interesting extensions to Pythagoras theorem challenges<br />https://www.youtube.com/watch?v=YRdKI71tx-4<br />and<br />https://www.youtube.com/watch?v=li8g0FMD3wcCliff Packmanhttps://www.blogger.com/profile/13897336461097002538noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-1742561874943744152015-01-25T11:47:45.828-05:002015-01-25T11:47:45.828-05:00In case it is useful, this is the simplest example...In case it is useful, this is the simplest example I have seen in the proof-type family you are describing here:<br /><a href="http://3jlearneng.blogspot.com/2014/07/pattern-blocks-mini-follow-up.html" rel="nofollow">Same areas with pattern blocks.</a><br /><br />Since this version doesn't really involve calculating any areas, it was accessible to a 7 year old.<br />Joshua Greenehttps://www.blogger.com/profile/11702319994021721608noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-87335957930715719832015-01-21T20:53:46.188-05:002015-01-21T20:53:46.188-05:00Becoming more intellectually mature means they wou...Becoming more intellectually mature means they would be interested in Pythagorean theorem and/or feel that they're important? I am unsure about that...<br /><br />I see everything as an opportunity for the kids to learn. Generally in mathematics, my goal is typically getting them to learning reasoning in general - and not the specifics of how, for example, the Pythagorean theorem works. So, sure, they may not care about the proof of the Pythagorean theorem works, and why would they, but I would get them to think about this process of understanding proofs (or constructing their own) as an exercise in better understanding different ways of reasoning.Jim Paihttps://www.blogger.com/profile/10164577294310143232noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-38304053889825806172015-01-12T17:48:14.057-05:002015-01-12T17:48:14.057-05:00they aren't important to kids because kids are...<i>they aren't important to kids because kids aren't intellectually mature enough to grasp their interest or importance.</i><br /><br />No, I think they can be interested, and they can see the importance. I just don't think they can do it.<br /><br />You can't teach important and interesting philosophical issues to young kids. They don't yet have the mental equipment to balance conflicting concepts of "right", "just" etc. It's why we don't give 16 year olds the vote, for example.<br /><br />I maintain similarly that few students can properly hold proofs before about 14 or 15 years old. They can follow the steps but they don't get the underlying "why" which is at the heart of proof.<br /><br />Trying, for example, to get a 12 year old to form a proper proof on their own, even the simplest one, is an exercise in frustration. Yet the same student at 17 can be taught to crush them with ease, and actually enjoy doing so. Clearly at some time their mental equipment changes -- it most certainly is not that they have learned a few more things. <br /><br />Now we can disagree about when that changeover takes place, sure, but tying to teach things like proofs before that time is more or less useless. Markhttps://www.blogger.com/profile/14049701479076034749noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-89254818295347446482015-01-12T05:38:07.960-05:002015-01-12T05:38:07.960-05:00Thanks for the comment, Mark! I'm going to dis...Thanks for the comment, <b>Mark</b>! I'm going to disagree with you, but the points you raise are cogent and interesting.<br /><br />We both seem to agree that a mature math student should know proofs, and should also be able to make sense of proofs of the Pythagorean Theorem. That's good.<br /><br />You further argue that though proofs of the PT are important, they aren't important to kids because kids aren't intellectually mature enough to grasp their interest or importance.<br /><br />What it comes down to, probably, is that I think I can teach my kids to understand these proofs and you don't think that you can. Which is fine. Lots of math is too hard for my students, and maybe it's easier for yours for whatever reason.<br /><br />My issue is mostly with your argument, which is really an argument against learning anything at all. Students can't figure out the Pythagorean Theorem proofs? They also can't solve equations or divide. There's a million things that we don't understand? Students don't understand algebra or how area works. Math <i>itself</i> tends to be something that Maths teachers find important that students don't.<br /><br />That said: I agree that we need to decide what's worth our kids' time, and just because something is mathematically important doesn't mean that it's appropriate for our kids. Completely agreed.Michael Pershanhttps://www.blogger.com/profile/17046644130957574890noreply@blogger.comtag:blogger.com,1999:blog-7245208048685880741.post-91420683860549265162015-01-12T03:24:58.579-05:002015-01-12T03:24:58.579-05:00I used to show a proof of Pythagoras, but at the s...I used to show a proof of Pythagoras, but at the stage that students are shown it they actually can't grasp what a proof is. They can do all the steps, sure, but they don't have the ability to put it all together into what it means. So I stopped doing it, and gained back some time to teach them stuff they could grasp.<br /><br />There's a million things in my life I don't know how they work. Like the clutch on my car or how my wireless router works. Yet I can use them fine. Sometimes learning why things work is a waste of time, because we use them fine without understanding how they work, and we shouldn't fetish proofs for that reason.. <br /><br />My experience then is that proving Pythagoras strikes me as a thing Maths teachers think is important and students don't. <br /><br />I like proofs, and when students are intellectually mature enough to teach them I enjoy doing proofs with them. But your students may be struggling to make sense of your Pythagoras because they don't yet have the maturity to follow, rather than your inability to frame it correctly. Markhttps://www.blogger.com/profile/14049701479076034749noreply@blogger.com