tag:blogger.com,1999:blog-3784276984960421233.comments2017-08-21T16:16:37.838-07:00My Math Education BlogHenri Picciottohttps://plus.google.com/107858350012538689018noreply@blogger.comBlogger84125tag:blogger.com,1999:blog-3784276984960421233.post-46700716877485356662017-08-21T16:16:37.838-07:002017-08-21T16:16:37.838-07:00Alas, the decline in the amount of geometry in the...Alas, the decline in the amount of geometry in the US has been going on a long time... I wrote about this in my mega-article on Common Core. (http://www.mathedpage.org/teaching/common-core/) US pragmatism and anti-intellectualism may lead us to continue in this downward slide... Note that the wonderful geometric construction app Euclidea seems to originate in Russia.<br /><br />Still, a unit on construction does help teach some of the surviving topics, such as congruent triangles, and as you can see in my Transformational Geometry page (http://www.mathedpage.org/transformations/), it is one key to a strong transformational approach to proof. So I say: make room for it!<br /><br />As for Geometry 2, I'm all for it. See my Space course (http://www.mathedpage.org/space)Henri Picciottohttps://www.blogger.com/profile/06875198126877279937noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-1665167368313849342017-08-21T11:02:36.645-07:002017-08-21T11:02:36.645-07:00I think this is great but I can't imagine most...I think this is great but I can't imagine most school's fitting constructions back into the curriculum given the current time constraints. When you look through the modern geometry course, one sees a bit of trigonometry, 3D volumes, analytic geometry and sometimes an unrelated statistics unit tacked on as well.<br /><br />This has come at the cost of de-emphasizing Euclidean synthetic techniques including constructions and proofs. I've gone through some thought experiments and I think you could just as easily do a Geometry II course with all the material that can't be covered and in many ways its a more natural narrative arc than the topics in Algebra II.<br /><br />Another fun idea, what if we delayed computational math until late elementary school and focused on Geometry for the beginning years?<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />Benjamin Leishttps://www.blogger.com/profile/10974191081762367425noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-8639288625990695932017-05-11T07:05:01.372-07:002017-05-11T07:05:01.372-07:00I know! Pattern blocks have been imitated, but nev...I know! Pattern blocks have been imitated, but never equalled.Henri Picciottohttps://www.blogger.com/profile/06875198126877279937noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-88199136715297850642017-05-10T14:12:17.921-07:002017-05-10T14:12:17.921-07:00This is all fascinating. I'd like to know how ...This is all fascinating. I'd like to know how EDC came up with the 6 pattern blocks too. (They were a good choice.)Simon Gregghttps://www.blogger.com/profile/07751362728185120933noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-44052649821740684002017-05-07T21:22:57.336-07:002017-05-07T21:22:57.336-07:00I've written a little bit on the history of al...I've written a little bit on the history of algebra manipulatives:<br />http://www.mathedpage.org/manipulatives/alg-manip.html<br />(Scroll to the end.)<br /><br />Other than the Lab Gear, I've been mostly a user, not an inventor, though as you know, I did come up with many clever uses for the geoboard, the circle geoboard, and pattern blocks, in addition to the puzzles mentioned in this post. Some of my ideas were ground-breaking, e.g. geoboard squares leading to the Pythagorean theorem. Also: I helped create great activities for Zome, under George Hart's leadership. (Zome history can be looked up online.)<br /><br />EDC invented pattern blocks, and created tremendous sets of mirror puzzles, and tangram puzzles, all of which really shaped my curricular aesthetic back in the 1970's. And yes, Marilyn knows a lot of the early history! I was already a fan of hers in 1975!Henri Picciottohttps://www.blogger.com/profile/06875198126877279937noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-29289070124196507932017-05-07T18:56:46.020-07:002017-05-07T18:56:46.020-07:00This is fascinating. I knew about Logo, but I didn...This is fascinating. I knew about Logo, but I didn't know that 'low threshold/high ceiling' originated there.<br /><br />It doesn't need to be you, but someone should write a history of math manipulatives. I'll read anything about how you invented all the tools you invented, their relation to what came before or after, etc. <br /><br />Seriously: anything. It's fascinating stuff. Someone should just get you and Marilyn Burns in a room with a voice recorder, some coffee, and lock the door until all this history has come out.Michael Pershanhttps://www.blogger.com/profile/17046644130957574890noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-22671576335823836862017-05-07T17:35:58.425-07:002017-05-07T17:35:58.425-07:00Here is a short version:
Logo was a computer lang...Here is a short version:<br /><br />Logo was a computer language that swept through elementary schools in the early days of educational computing. Seymour Papert, the MIT professor who was the founder and leader of the Logo movement, was a prophet of educational transformation through technological change. His utopian vision has failed to materialize, to say the least, but for some of us it triggered a fundamental shift in perspective. The concept of "objects to think with" (in his book Mindstorms) readily expands to "objects to talk and write about", and is in a lot of ways foundational to my tool-rich educational vision.<br /><br />Logo was an environment that empowered students in multiple ways: it offered students opportunities to experiment, to explore and solve problems their own way, to set their own goals and ask their own questions. Starting on day one, all students were able to achieve interesting results. That was dubbed "low threshold". But Logo was a full programming language, meaning that just about any project a student could conceive, they could pursue. "High ceiling"! (Or even "no ceiling"!)<br /><br />The fact that the environment was highly visual and geometric was a plus, and it opened up opportunities to learn a lot of geometry, from very basic ideas about angles, to college-level math. (See the amazing book _Turtle Geometry_, from MIT Press, by Abelson and DiSessa.)<br /><br />Logo has many descendant languages. Scratch is probably the best known. It added tremendous possibilities for animation and more, but unlike Logo, it is not (nor is it intended to be) a full computer language. Snap (from UC Berkeley) is directly inspired by Scratch. I can do everything Scratch can do, but it makes it possible to program anything -- no ceiling!<br /><br />Anyway, when the "low floor, high ceiling" language took off with Jo Boaler's help, the elders among us know where the phrase came from!Henri Picciottohttps://www.blogger.com/profile/06875198126877279937noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-84622076423656854122017-05-07T10:53:22.835-07:002017-05-07T10:53:22.835-07:00That way of describing such activities originated ...<i>That way of describing such activities originated in the 1980's Logo movement, but that is another story.</i><br /><br />I would love to hear that story, some time.Michael Pershanhttps://www.blogger.com/profile/17046644130957574890noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-76563508406331879012017-04-11T21:31:25.942-07:002017-04-11T21:31:25.942-07:00Happy it's been helpful! Yes, of course what I...Happy it's been helpful! Yes, of course what I learned was not about math, at least not directly.Henri Picciottohttps://www.blogger.com/profile/06875198126877279937noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-9854944037698134182017-04-11T18:41:00.636-07:002017-04-11T18:41:00.636-07:00Thank you, Henri, for this post. I don't even ...Thank you, Henri, for this post. I don't even have a math classroom right now, but you have me excited and thinking how this applies to raising my kids, tutoring students, and teaching yoga. Pam Armstronghttp://www.mathymoments.comnoreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-5345536834764345442017-03-08T11:04:26.798-08:002017-03-08T11:04:26.798-08:00Hi,
I really enjoyed thinking about some of the su...Hi,<br />I really enjoyed thinking about some of the subproblems here. You'll have to imagine my yellow-pad full of tangents.<br /><br />Among other ones that might be of interest: Why is there a triangle of area 15 at all as opposed to some irrational value, do all the possible triangles on the board have interesting areas? Yes: they're either integers or multiples of 1/2. This flows out of boxing the triangles in and calculating the area via subtraction of the right triangles on the edges (which all are also integral or multiples of 1/2).<br /><br />Find the areas via integer factorizations:<br /><br />For instance on an 7 x 6 box that encloses the triangle in your picture you get the equation for the area of any triangle of this basic form as 7 x 6 - 1/2xy - 1/2(7-x)6 - 1/2(6-y)7 = 21 - 1/2(7 - x)(6 - y)<br /><br />If you set it it any particular desired value like 15 then you just have to check the factorizations:<br />So for 15 you get 12 = (7 - x)(6 - y) and you need to check (1,12)(2,6),(3,4),(4,3),(6,2) and (12,1)<br />(3,4) corresponds to your picture. All the other possible ones flow out in the same way and then you obviously rotate or reflect them.<br /><br />It was then fun to think about the super-obtuse triangles that don't fit this model i.e. two vertices must be at opposite corners with the 3rd on the inside of the box.<br /><br />Finally, I thought checking all the possible areas for a given box was fun too. It relates mostly to the number of factorizations for each size biased by cutoffs where the factorizations are not possible. <br /><br />ThanksBenjamin Leishttps://www.blogger.com/profile/10974191081762367425noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-429383246598781712017-03-01T16:09:47.862-08:002017-03-01T16:09:47.862-08:00By all means! You can e-mail me by clicking on the...By all means! You can e-mail me by clicking on the link at the top of most of the pages on my site.Henri Picciottohttps://www.blogger.com/profile/06875198126877279937noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-50698703623060925172017-03-01T16:05:17.096-08:002017-03-01T16:05:17.096-08:00This is a great response! Thanks, Henri! That comp...This is a great response! Thanks, Henri! That completely answered my question! I might be in touch later to ask you more questions if that is okay. Jonathan Schoolcrafthttps://www.blogger.com/profile/07465744000185670168noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-12615986714323021042017-02-28T16:55:16.529-08:002017-02-28T16:55:16.529-08:00This is a tough question, and like everything invo...This is a tough question, and like everything involving school and department culture, there's no simple answer. <br /><br />First of all, students get credit for the homework just for doing it, even if all their answers are wrong. All I ask is evidence of having tried. So there is less incentive to copy. Second, I do walk around while students are going over the homework, so if someone is just copying it then, I would almost certainly catch them in the act. Third, my homework assignments are short, accessible, and lagged, so that too reduces the incentive to copy. Fourth, the students know that what I say in the post is true (homework is where you see if you can do it on your own) and they know that this is an important part of the learning process. Going over the homework is hugely valued time among my students for this very reason.<br /><br />That said, if you're comparing with not assigning homework at all, there's nothing to lose! Those students who copy will not gain anything, but those who do the work will get the benefits.Henri Picciottohttps://www.blogger.com/profile/06875198126877279937noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-55988928514071253182017-02-28T16:12:07.647-08:002017-02-28T16:12:07.647-08:00Hi Henri,
I really enjoyed your post. I have had...Hi Henri, <br /><br />I really enjoyed your post. I have had a lot of internal conflicts with this very issue (as I'm sure many other teachers have). I think that I am going to start assigning homework next school year and using the lagging approach that you (and others) in the MTBoS have discussed. I have a question, though. When students are working in groups, how do you keep them from just copying the homework? I really like your idea; however, I can see students just copying. Thanks! Jonathan Schoolcrafthttps://www.blogger.com/profile/07465744000185670168noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-57303577280370368502017-02-02T21:00:18.456-08:002017-02-02T21:00:18.456-08:00Yeah, when we do all the talking, we can fall into...Yeah, when we do all the talking, we can fall into the delusion that the students are all listening, and that they all understand what we're saying. Seeing them work certainly disproves that! And it's so much more useful to know what they don't know early on, rather than wait for a test.Henri Picciottohttps://www.blogger.com/profile/06875198126877279937noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-19526621603013332882017-02-02T19:07:53.148-08:002017-02-02T19:07:53.148-08:00Approaches 2 and 3 also open up opportunities for ...Approaches 2 and 3 also open up opportunities for kid watching. During a similar lesson last year I found two students who could not rotate or flip to put the original shape back together. I also had more than one student who looked at their rearranged shapes and sai they were bigger now. They thought that the actual area had changed. What are students missing through lack of hands on and talk? A lot more than they ever can from a formulaic teacher driven lesson I think.Ann Bakerhttps://www.blogger.com/profile/10641665468658263016noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-20664697754776346812017-02-02T07:37:19.935-08:002017-02-02T07:37:19.935-08:00For sure!For sure!Michael Pershanhttps://www.blogger.com/profile/17046644130957574890noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-6830835607173974152017-02-02T06:07:40.743-08:002017-02-02T06:07:40.743-08:00I have no problem at all with Approach 3. It is no...I have no problem at all with Approach 3. It is not even ruled out by Approach 2. If you think dividing along the midline is important, you can always suggest that later. But really, this is a quibble. Your approach is based on your experience with students at your school. You trust the students to engage with the problem in their own way. You provide some leadership as the teacher, which fulfills your responsibility without smothering them. But in case it wasn't clear, I was not suggesting that it's right to stop at "whatever works". The advantage of students using familiar shapes is that, well, they're familiar. The whole point I was trying to make was that giving students a chance to think about the problem does not prevent the teacher from offering their own strategy. In fact it enhances that.Henri Picciottohttps://www.blogger.com/profile/06875198126877279937noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-57771882832598490722017-02-02T03:27:28.316-08:002017-02-02T03:27:28.316-08:00This is one of those eerie moments when someone...This is one of those eerie moments when someone's blogging about the lesson I taught just hours ago.<br /><br />Approach 3:<br /><br />Provide students an example and explanation of a midline cut, and give them time to practice making those midline cuts on various shapes. ("Draw a bunch of shapes with H Picciotto's Shape Tracer tool, and then draw the midline cuts.") <br /><br />Then, I gave an assortment of triangles, parallelograms and trapezoids and asked students to find their area. I said that they should solve them however they'd like, but if they were stuck my advice today would be to draw a midline.<br /><br />This is like Approach 2, but I think with more support. I'm not asking students to discover the midline cut. (Experience teaching this course shows me that kids often think to chop off shapes they recognize like triangles and rectangles, but the midline cut is harder to see.)<br /><br />There's room between "tell kids a formula" and "let kids figure out whatever works." We can tell kids strategies that, while still being specific, are more generally useful (and simpler to remember) than an area formula.Michael Pershanhttps://www.blogger.com/profile/17046644130957574890noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-8008543004224061742017-01-29T10:23:58.837-08:002017-01-29T10:23:58.837-08:00I don't think I realized how helpful reading A...I don't think I realized how helpful reading Algebra could be until I started teaching newcomer students with interrupted education (SIFE). Now I think I'll start teaching it to all students! It also helps develop/discover the concept of GEMA as well. Hwanghttps://www.blogger.com/profile/03731857134746207001noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-86329502542063287642016-12-06T08:41:48.427-08:002016-12-06T08:41:48.427-08:00Thank you, Henri! I especially appreciate the remi...Thank you, Henri! I especially appreciate the reminders about ways to slow down classroom discourse and increase engagement for students who are shy or have slower processing.Unknownhttps://www.blogger.com/profile/17679777447094859338noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-67360680129000731972016-10-02T00:03:01.626-07:002016-10-02T00:03:01.626-07:00This is an important topic. Thanks for sharing th...This is an important topic. Thanks for sharing these ideas. Students really struggle when they are asked to express their mathematical thinking in writing, and are often at a loss. They default to writing about a mathematician or about a historical topic instead, as this is more comfortable and familiar process for them. I have found that they need a ton of scaffolding and support at first. This includes fill-in-the-blank sentences, word banks, leading questions, super-short assignments to begin with, etc. But they do get better at this as they become more comfortable with the process, and many can give up these crutches one by one. <br /><br />The idea for analyzing technical writing is excellent. I will definitely try this with my students this year. <br /><br />Thanks as always!Nathttps://www.blogger.com/profile/10678868201951749642noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-15527577297944512542016-09-29T07:00:04.678-07:002016-09-29T07:00:04.678-07:00Great summary! I've also found analyzing sampl...Great summary! I've also found analyzing samples of student work to be really helpful when students are just starting to write in math. Peer editing is another way to help students get specific feedback, see examples of how others write, and have an authentic audience. Anna Blinsteinhttps://www.blogger.com/profile/13960574914938362477noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-34568591151579872052016-09-23T07:42:52.052-07:002016-09-23T07:42:52.052-07:00!!!!!!WilliamKinghttps://www.blogger.com/profile/11551278829221366384noreply@blogger.com