tag:blogger.com,1999:blog-1075593398139537131.post8969901986479652057..comments2020-09-29T04:38:19.244-07:00Comments on Without Geometry, Life is Pointless: What Makes a Problem GreatAvery Pickfordhttp://www.blogger.com/profile/10433339146333801163noreply@blogger.comBlogger6125tag:blogger.com,1999:blog-1075593398139537131.post-17422271225407316392010-09-27T13:34:32.865-07:002010-09-27T13:34:32.865-07:00@CA NYC: Welcome! Based on your name, I guess I&#...@CA NYC: Welcome! Based on your name, I guess I'm NYC CA. Both the NCTM and Common Core Standards talk about process and habits and I am sure that their list and my list both come from some of the same sources. My issues with the NCTM and common core habits have always been that 1, they don't talk about ways to teach these things (and unlike fractions, teachers have little experience learning these things explicitly and there are very few existing materials/lessons for teaching habits, nevermind the fact that I've yet to see a district pacing guide that builds this time in) and 2, every standardized test I have ever seen has devalued these processes by saying that they are "inherently assessed in the content questions." I find this a total cop out.<br /><br />I'm trying to give a more detailed description of what it means to, for example, "Make sense of a problem". Hence the long and detailed list.<br />Then I'm building in class time to explicitly teach these processes/habits (which I fortunately have the luxury to do).<br />Then I'm assessing kids specifically on these processes/habits (first quiz will be tomorrow...I'm interested to see how things go).<br />Then I will attempt to get kids to use these habits effectively on problems where I haven't told them to use a specific habit of mind.Avery Pickfordhttps://www.blogger.com/profile/10433339146333801163noreply@blogger.comtag:blogger.com,1999:blog-1075593398139537131.post-41177107375279298762010-09-27T07:54:32.892-07:002010-09-27T07:54:32.892-07:00I'm wondering here about California's adop...I'm wondering here about California's adoption of the national Common Core State Standards. The eight Standards of Mathematical Practice get to a lot of the habits of mind NCTM created and you talk about. <br /><br />1. Make sense of problems and persevere in solving them.<br />2. Reason abstractly and quantitatively.<br />3. Construct viable arguments and critique the reasoning of others.<br />4. Model with mathematics.<br />5. Use appropriate tools strategically.<br />6. Attend to precision.<br />7. Look for and make use of structure.<br />8. Look for and express regularity in repeated reasoning.<br /><br />The language isn't as classroom friendly as the lists above but the concepts are similar.<br /><br />Are these national standards playing into your work/planning/ thinking at all?CA NYChttps://www.blogger.com/profile/17579972135813888265noreply@blogger.comtag:blogger.com,1999:blog-1075593398139537131.post-20108520314648426002010-09-21T06:49:05.629-07:002010-09-21T06:49:05.629-07:00I have been talking with teachers about this very ...I have been talking with teachers about this very topic. The idea of learning through problem solving, rather than learning for problem solving is one of my main themes. Here is what I suggest makes a good problem. It's quite similar to yours, and is based on the work of Marilyn Burns, Marian Small, John Van de Walle, Dan Meyer, and me.<br /><br />A good problem is:<br /><br />Given at the beginning of the learning, not at the end.<br /><br />Non-routine. Students can not answer it immediately. <br /><br />Allows every student an entry into it. <br /><br />Is compelling enough that students are engaged in the problem.<br /><br />Is interesting enough that students are compelled to persevere.<br /><br />Fosters discussion and debate.<br /><br />Invites multiple methods of solution.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-1075593398139537131.post-76984516200345135402010-06-11T13:34:26.705-07:002010-06-11T13:34:26.705-07:00This is one of the best definitions of what a prob...This is one of the best definitions of what a problem-solving approach to mathematics should be about. It fits right in with a lot of things going on with the math circles movement here in the Bay Area, such as http://mathteacherscircle.org . We're definitely going to link to this post from our discussion of what a math teachers' circle is!<br /><br />I wonder if you'd like to join us for one of our workshops this summer? I think you'd get some good problems from us and we'd get some great ideas from you on how to improve our pedagogy.<br /><br />We'd be happy to have you as a visitor in the mornings of the week of 6/28 when we host teams from around the country that are interested in the math circles approach to professional development, or, even better, any time during the week of 7/6 at our workshop for local middle school teachers.<br /><br />You can reach me at Joshua dot Zucker at gmail.Joshua Zuckerhttps://www.blogger.com/profile/04689961247338617418noreply@blogger.comtag:blogger.com,1999:blog-1075593398139537131.post-28657731504948246582010-05-30T12:01:33.317-07:002010-05-30T12:01:33.317-07:00@Benedict: Thanks for the contributions! Could you...@Benedict: Thanks for the contributions! Could you talk more about the characteristics that make these problems good/interesting to you? Imagine I were going to the great repository of math problems to find you a new problem to work on, but I wanted to make sure you'd be interested.Avery Pickfordhttps://www.blogger.com/profile/10433339146333801163noreply@blogger.comtag:blogger.com,1999:blog-1075593398139537131.post-47728267182307058012010-05-29T11:37:44.689-07:002010-05-29T11:37:44.689-07:00Two problems:
1. How many coplanar points does i...Two problems:<br /><br />1. How many coplanar points does it take so that there is precisely one unique conic section that contains all those points? (I've found two elegant ways of finding the answer for a circle; I suspect it depends upon the placement of the points for others.)<br /><br />2. What is the radius of the biggest circle one can "drop" into a concave-up parabola (with directrix-focus distance of 2, let's say) such that the circle will touch the bottom of the parabola? (This is not a great problem by any means, but fun and pleasing.)Benedicthttps://www.blogger.com/profile/07517537478049704680noreply@blogger.com