tag:blogger.com,1999:blog-6651514617266100245.post537004588381823382..comments2024-01-03T04:58:04.221-05:00Comments on I Hope This Old Train Breaks Down...: On Open-Ended Non-QuestionsUnknownnoreply@blogger.comBlogger7125tag:blogger.com,1999:blog-6651514617266100245.post-55469620538363199192011-03-10T14:02:41.206-05:002011-03-10T14:02:41.206-05:00Sounds like a good start. The hurdle is shifting t...Sounds like a good start. The hurdle is shifting the kids' focus from finding "the answer" to describing the pattern/relationships/situation. Once they understand the framework and practice it a bit, they will surprise you by including things you haven't thought of.<br /><br />One thing I have seen a bit of in my kids, but want to focus on more explicitly, is trying to use models or strategies from earlier material in a different context. Can you still analyze this problem in terms of momentum even though it isn't a collision or an "explosion"? I have sometimes stopped the kids from going down a path like that too quickly because it isn't "a momentum problem." But when I catch myself and stay out of the way, then they really start to play and figure things out. A lot of the time it is perfectly fine to use momentum (etc), and I want to focus on that sort of thing explicitly so that they can get a feel for when it is possible or not, easy or not, etc.<br /><br />You can try doing some goal-less math problems on your own by just finding book problems that have enough information to play with if you get rid of the question. It's kind of fun to get into a "flow" type state of just seeing how much more you can show or find.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6651514617266100245.post-53934136172662483052011-03-08T21:56:06.318-05:002011-03-08T21:56:06.318-05:00@Jason Thanks for the link!
@kellyoshea Maybe gi...@Jason Thanks for the link! <br /><br />@kellyoshea Maybe give the kids a word description of a pattern (in context), and kids can:<br /><br />1. make a table (basic)<br />2. define all related variables and verbally state/summarize their dependencies (basic/intermediate)<br />3. graph relationships (intermediate? basic? maybe it depends on the topic)<br />4. write equations (intermediate / advanced, depending on the topic)<br />5. fully explain the meaning of the new numbers that emerged from Step #4 -- ie. the meanings of all calculated coefficients and constants. (intermediate / advanced)<br />6. state domain constraints of equations (intermediate / advanced)<br />7. further analyze inverse relationships, if one exists... or explain why the relationship has no inverse function. (advanced)<br />8. take two fully analyzed situations and draw comparison or combine them somehow to make further sense / draw further conclusion. Sort of like systems of equations type of stuff, where they can apply further math skills and make further interpretations. (advanced)<br /><br />Thoughts?untilnextstophttps://www.blogger.com/profile/15285583728476473117noreply@blogger.comtag:blogger.com,1999:blog-6651514617266100245.post-51124172219665953872011-03-08T06:39:18.582-05:002011-03-08T06:39:18.582-05:00I think in math, goal-less problems would be focus...I think in math, goal-less problems would be focused on defining relationships in a situation. It seems like if you gave a kid a situation (like the salt-shaker sliding along the table) and asked them how math related to it, the first impulse would be to just describe numbers or arithmetic. The first layer would be to describe relationships between concrete numbers, but the tricky part would be to push past that layer. So the next layer in the goal-less problem might be to define symbols for quantities about the situation (distance traveled, speed, change in time, friction force, angle of table, etc etc). And the next layer would be to come up with relationships between the symbols.<br /><br />Another key to goal-less problems is the use of multiple representations. So in math, you want to get the graphs, tables, equations, verbal descriptions, etc in there, too.<br /><br />I'd be interested to hear if you end up trying something like this! When we finish spring break, I'll run this by my colleague who teaches a section of physics with me but also teaches two math classes and see what he thinks about goal-less problems in algebra and calculus.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6651514617266100245.post-88036272731761330982011-03-06T17:28:15.803-05:002011-03-06T17:28:15.803-05:00@Mimi Avery over at Without Geometry Life is Point...@Mimi Avery over at Without Geometry Life is Pointless is trying "minimally defined problems." Here's an example. http://mathteacherorstudent.blogspot.com/2010/05/how-would-you-respond-to-minimally.htmlJason Buellhttps://www.blogger.com/profile/03029995715142652159noreply@blogger.comtag:blogger.com,1999:blog-6651514617266100245.post-69924298305757798492011-03-06T17:19:17.266-05:002011-03-06T17:19:17.266-05:00@betweenthenumbers A little more concretely, what ...@betweenthenumbers A little more concretely, what about something like "Sofia left her house at 3:30pm to walk towards Jason's house at a costant 1 m/s, and Jason left his house at 3:35pm to walk towards Sofia's. He initially walks at 1 m/s, but every 5 seconds he speeds up by 1 m/s, until he is sprinting at 7m/s. Their houses are located on opposite sides of the park as shown in the scale drawing below." This problem involves linear patterns and velocity vs. time, distance vs. time, and d=rt, and basic ratios (in reading a scale drawing). Common algebra concepts... But, this is not really that interesting of a problem. Sort of just sounds like a word problem that got chopped off. :(<br /><br />@ER Thanks! I've only done the "hat" thing a couple of times, but kids did love the idea of picking things out of a hat. :) But, the problem for me is that as soon as they sit down to work on the problem, they usually realize that it's just another way to trick them into doing a regular algebra problem, so the "wow" factor wears off real fast with my kids. :P --More power to you if you have better luck than me keeping them charmed!!untilnextstophttps://www.blogger.com/profile/15285583728476473117noreply@blogger.comtag:blogger.com,1999:blog-6651514617266100245.post-39084723523343906402011-03-06T14:41:03.579-05:002011-03-06T14:41:03.579-05:00Oooo, I LOVE your test problem. Put a spin on it, ...Oooo, I LOVE your test problem. Put a spin on it, have 3 or 4 different sets of 3 lines and have the kids pick out of a bag to see which one they get. Maybe not for a test problem, but for just a daily assessment.ERhttps://www.blogger.com/profile/03124846987033083264noreply@blogger.comtag:blogger.com,1999:blog-6651514617266100245.post-7055910587742146832011-03-06T12:17:01.418-05:002011-03-06T12:17:01.418-05:00initial thought, here. So, not sure how fully form...initial thought, here. So, not sure how fully formed it will be.<br /><br />I think that there is a way to do something like the goal-less problems in math, but it would look different than many of us are probably used to. I think the focus in a math class that uses goal-less problems would need to be on connecting ideas and using relationships rather than being extremely standards-driven. In other words, more focused on process standards than content standards. <br /><br />The idea/example that came to my head was with modeling. If that is the big idea for the unit/month/semester/course/unit-of-time then you could look at a lot of different types of situations through a modeling lens. And then when students looked at a new situation they could start out by modeling it (graphically, in a table) and build off of what they already know, with tools they already have to learn about this new situation.Anonymousnoreply@blogger.com