tag:blogger.com,1999:blog-3784276984960421233.comments2017-03-08T11:04:26.798-08:00My Math Education BlogHenri Picciottohttps://plus.google.com/107858350012538689018noreply@blogger.comBlogger74125tag: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 Leishttp://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 Picciottohttp://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 Schoolcrafthttp://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 Picciottohttp://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 Schoolcrafthttp://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 Picciottohttp://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 Bakerhttp://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 Pershanhttp://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 Picciottohttp://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 Pershanhttp://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. Hwanghttp://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.Unknownhttp://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!Nathttp://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 Blinsteinhttp://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!!!!!!WilliamKinghttp://www.blogger.com/profile/11551278829221366384noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-77560627893553166802016-09-19T21:41:28.222-07:002016-09-19T21:41:28.222-07:00Yes. Click on "More on Extending Exposure&quo...Yes. Click on "More on Extending Exposure" and "Implementation Advice" to see how I did it.Henri Picciottohttp://www.blogger.com/profile/06875198126877279937noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-19219894626846195272016-09-19T21:07:14.260-07:002016-09-19T21:07:14.260-07:00Hello..This is the first I have heard of lagging h...Hello..This is the first I have heard of lagging homework, and this year I am starting something similar to the other reply in terms of 1 week and 1 month reviews. I just read your lagging homework and philosophy about quizzes. As far as quizzes based on what I read, were the quizzes were "lagged" also? Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-88078070746102514152016-07-12T20:55:54.747-07:002016-07-12T20:55:54.747-07:00Hi.
If your students all always get A's in a ...Hi.<br /><br />If your students all always get A's in a certain course, you are probably not challenging them enough. It is almost certain that if you covered more material, or gave harder problems, some of your students would do a lot better than others, and it would be impossible to justify giving the same grades to all of them. If on the other hand you didn't cover more material or give harder problems, you're robbing your stronger students, and in fact all your students because you're not stretching them or giving them something to aspire too. <br /><br />Your belief that there is an objective 95% on a logarithm test makes no sense to me. The same test could have questions weighted differently, or have partial credit assigned differently, or have more of this type of question and less of that kind, and so on. Even if that were not the case, one could argue that your test is too easy or too hard -- that is strictly a matter of context and selection of goals. One teacher may think that students correctly switching to another base via a memorized formula is important in Algebra 2, another teacher may think it's fine to use any base as long as it makes sense to the student and they can apply it to real world problems, a third teacher may not have an opinion this, and just want their students to be able to answer SAT questions about logs. The same test would yield different results in each of these teachers' classes. Just because you give a percent score doesn't make it scientific. Grading is extremely subjective, though it almost certainly reflects how you see your students.<br /><br />Finally, if _all_ Mr. X's students were _consistently_ getting C's, or A's, most administrators would think something is wrong. It can happen on one test, but not throughout a course. Mr. X would be under pressure to change their teaching, or their assessments, or the way they arrive at the grade, so that the grades can be used to compare the students. <br /><br />Still, you're right, grades can indeed help see a student's progress. But I don't think that can be separated from the reality that they are strictly a relative measure.Henri Picciottohttp://www.blogger.com/profile/06875198126877279937noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-51454838447233422852016-07-12T13:32:32.241-07:002016-07-12T13:32:32.241-07:00Henri, thank you again for coming out to do your s...Henri, thank you again for coming out to do your summer workshop on our Head-Royce campus. May the learning never stop! <br />I find myself agreeing with the many points you make about assessments and grades in your blog series. Regarding this post in particular, I am struggling to agree that grades are ONLY about comparing students to each other. To some extent, grades can help students compare their current selves to their previous selves. A student who receives Cs in 9th grade math, Bs in 10th grade math, and As in 11th grade math has likely made significant progress as a math student (assuming there was some degree of consistency among members of the math department.) More importantly, I like to think that grades can summarize how my students performed against absolute standards of scholarship. If I select a finite set of conceptual understandings, math skills, and problem-solving dispositions, then I can strive to create assessments that measure student progress in these areas. For example, a student can score 95/100 points on my Logarithms test regardless of what any other student scores. In fact, my student can score 95/100 even if no other student takes the test. Furthermore, ALL of my students could conceivably score 95/100 pts. When I publicly post to parents and colleges that all of my students received an "A", then I am not using grades to rank students, at least not in a way that helps the sorting process. Rather, I am comparing each of them to my absolute standard of mastery in the area of logarithms. Sure, the colleges might later compare my "A" student at Head-Royce to an "A-" students at Urban School, but I the teacher am not using grades to compare or rank students.<br />At least, this is what I think I am doing. Am I mistaken or naive in my use of grades?Unknownhttp://www.blogger.com/profile/00928840322893608311noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-87920979505944568612016-06-18T02:15:08.851-07:002016-06-18T02:15:08.851-07:00This comment has been removed by a blog administrator.Aurang Zaibhttp://www.molaritycalculator.comnoreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-27392036758330719932013-08-06T20:15:27.738-07:002013-08-06T20:15:27.738-07:00A correspondent writes: "Some years ago Maril...A correspondent writes: "Some years ago Marilyn Burns wrote that homework could benefit from being something unique to the home. Volume is a good exampleâ€”developing understanding, comparisons, etc. can be done easily at the sink with water or in the garden with sand or dirt. Great for home but harder in the classroom." Good point! But perhaps more difficult to implement in high school?Henri Picciottohttp://www.blogger.com/profile/06875198126877279937noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-59917891964183968432013-07-29T22:12:01.825-07:002013-07-29T22:12:01.825-07:00My pleasure! It's always nice to hear that som...My pleasure! It's always nice to hear that someone out there finds those useful!Henri Picciottohttp://www.blogger.com/profile/06875198126877279937noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-38225968609818976492013-07-29T04:42:07.060-07:002013-07-29T04:42:07.060-07:00Henri,
Thank you for sharing your thoughts and res...Henri,<br />Thank you for sharing your thoughts and resources.<br />JustineAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-19263122995035303982013-07-01T06:55:52.648-07:002013-07-01T06:55:52.648-07:00You can check out the materials I've created a...You can check out the materials I've created at my site: https://sites.google.com/site/beautyrigorsurprise/home/courses/elementary-euclidean-geometry. (Google Drive is a nice way to distribute notes and assignments.) I'd love to hear any suggestions you might have.<br /><br />I plan to ditch the textbook. Hopefully I can get the other teachers in my corporation on board. The wheels are in motion . . .F. C. Mason Jr.http://www.blogger.com/profile/00209597695197799059noreply@blogger.comtag:blogger.com,1999:blog-3784276984960421233.post-85871795140767083022013-07-01T06:46:36.585-07:002013-07-01T06:46:36.585-07:00I'd love to hear how this plays out. I have no...I'd love to hear how this plays out. I have not thought this through at this level of detail, so for me, your steps 3 and 4 seem to require the most working out. In particular, I wonder if the postulate you state in Step 2 is sufficient. Please keep me posted as you implement your plan!Henri Picciottohttp://www.blogger.com/profile/06875198126877279937noreply@blogger.com