tag:blogger.com,1999:blog-3784276984960421233.comments2017-11-20T19:12:44.334-08:00My Math Education BlogHenri Picciottohttps://plus.google.com/107858350012538689018noreply@blogger.comBlogger94125tag:blogger.com,1999:blog-3784276984960421233.post-81332149405728863162017-11-20T17:51:55.891-08:002017-11-20T17:51:55.891-08:00What an excellent assessment of the situation! Tha...What an excellent assessment of the situation! Thanks Henri for your thoughtful comments. Math Teacherhttps://www.blogger.com/profile/04420227135918089469noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-61626960780008965712017-11-14T14:55:35.504-08:002017-11-14T14:55:35.504-08:00My students got the puzzles relatively quickly, bu...My students got the puzzles relatively quickly, but were slower to the underlying idea you saw. I made the pieces close to square to make the puzzles less obvious, and the students who made the 9 with 1, 3, 5 got it faster than the people who just said they have the 9 already. Interesting. Want to try with elementary, but haven't had the chance yet.John Goldenhttps://www.blogger.com/profile/18212162438307044259noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-52176071024857537522017-11-13T16:57:02.998-08:002017-11-13T16:57:02.998-08:00I think you're quite right: beginners need to ...I think you're quite right: beginners need to know there is a solution. It takes much greater mathematical maturity to jump in without knowing that. For one thing, with more maturity comes the understanding that one can change the question in order to get an interesting answer. <br /><br />I enjoyed your "make all the squares by combining these odd-area polyominoes". It was easy for me, as I knew enough to quickly choose my pieces, but it was probably well calibrated for your students. How did they do?Henri Picciottohttps://www.blogger.com/profile/06875198126877279937noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-5406708140394653282017-11-12T08:07:35.449-08:002017-11-12T08:07:35.449-08:00Thanks so much for these two posts. Interesting in...Thanks so much for these two posts. Interesting in the details and the bigger implications for curriculum writing. What strikes me is that these puzzles do such a good job of emulating what mathematicians do. There's a time for open play, but so often we're chasing a conjecture, following the rules of the particular situation. We don't know how to get there, but have faith that there is a path. My students were working on halving puzzles this week (dividing a shape into two congruent halves) and some needed to know that there was a solution. They weren't comfortable really trying until they knew there was an answer. Which we don't have while we're being mathematicians, but maybe necessary as a support for learners? <br /><br />In short, thanks for all the amazing work!John Goldenhttps://www.blogger.com/profile/18212162438307044259noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-57291708659840532022017-11-09T13:47:29.118-08:002017-11-09T13:47:29.118-08:00Yes, there were many great small publishers back i...Yes, there were many great small publishers back in the day. They've been swallowed up by giant corporations with little interest in anything besides huge textbook sales.Henri Picciottohttps://www.blogger.com/profile/06875198126877279937noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-23824024503674426512017-11-09T10:55:51.202-08:002017-11-09T10:55:51.202-08:00Henri,
Funny, I taught 42 years before retiring! ...Henri,<br />Funny, I taught 42 years before retiring! I think math could use another Dale Seymour Publication type blitz where lots of fun problem solving is sent out to math educators. Keep it Up<br />Richard "Dick" Seitz401 Montana Seitz'shttps://www.blogger.com/profile/05724976490144258503noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-54540017681780810582017-11-05T14:27:20.816-08:002017-11-05T14:27:20.816-08:00Thanks for writing this. You're not uniquely p...Thanks for writing this. You're not uniquely positioned to see connections, but rare at least!John Goldenhttps://www.blogger.com/profile/18212162438307044259noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-61526711487180509042017-10-08T14:14:23.506-07:002017-10-08T14:14:23.506-07:00Perhaps this is related to trying to find a busine...Perhaps this is related to trying to find a business model that works financially? But yes, I agree with you.Henri Picciottohttps://www.blogger.com/profile/06875198126877279937noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-64616906280036567402017-10-08T13:18:32.054-07:002017-10-08T13:18:32.054-07:00The paywall bothers me the most. Not only will the...The paywall bothers me the most. Not only will they not publish your article if it was blogged, but then they lock it up, away from most teachers. The start of my disillusionment was locking up the standards, which they wanted everyone to see. I think teachers & researchers would support them in greater numbers if they could access and use the amazing resources NCTM has piled up.John Goldenhttps://www.blogger.com/profile/18212162438307044259noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-19533022757395369132017-09-06T21:06:54.856-07:002017-09-06T21:06:54.856-07:00Thanks for posting this! My first year, I started ...Thanks for posting this! My first year, I started a few days after school started so I never had a real first day. This year I was unsure of how to begin. This blog was helpful. I love the activities! Ashleyhttps://www.blogger.com/profile/08911422717153413509noreply@blogger.comtag: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.com