tag:blogger.com,1999:blog-1075593398139537131.post5618021328522117731..comments2024-03-12T02:19:51.554-07:00Comments on Without Geometry, Life is Pointless: Minimally Defined Problems, Chapter 2Avery Pickfordhttp://www.blogger.com/profile/10433339146333801163noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-1075593398139537131.post-1678711629882951332022-07-03T01:55:20.847-07:002022-07-03T01:55:20.847-07:00Your Work On Blog Is Very Good Thank You And Keep ...Your Work On Blog Is Very Good Thank You And Keep it up, I Visited Daily On Your Website.<br /><a href="https://kepalabergetar24.com/tarik-aku-ke-syurga/" rel="nofollow">Tarik Aku Ke Syurga</a>Kepala bergetarhttps://www.blogger.com/profile/13234110921094506354noreply@blogger.comtag:blogger.com,1999:blog-1075593398139537131.post-67866290665002569022010-07-17T10:40:12.322-07:002010-07-17T10:40:12.322-07:00I am a senior, elementary education major at the U...I am a senior, elementary education major at the University of South Alabama. I have never been so excited to learn and teach math in my whole academic career until now. This new excitement culminated from taking Math for Elementary teachers 1 and 2. I got very lucky by enrolling with two teachers who don't care about the right answer, but instead focus on the means for discovering a way to find the right answer. They both encourage us to think about real applications of geometry and algebra. They help us see illustrations of what is actually happening-- and they do this by giving us chessboards, or the appropriate equivalent so that we can notice patterns and make predictions. I promise that my hatred of math would have been lifelong had I not experienced it as a problem solving experience versus a memorized way of following instructions to get the desired answer. I can't wait til I have more time to come back and explore this blog. What a killer resource!<br /><br />Thanks your thoughts and strategies.<br /><br />AnthonyAnthonyhttps://www.blogger.com/profile/08389518086169811140noreply@blogger.comtag:blogger.com,1999:blog-1075593398139537131.post-62033512445056465542010-07-16T13:54:12.908-07:002010-07-16T13:54:12.908-07:00I think my current comfort level with this ind of ...I think my current comfort level with this ind of thing is to begin with a situation that is pretty clearly crying out for a certain family of questions, so you're pretty sure to get someone asking the interesting and productive ones that they're ready to attack and learn something from. I also tend to use this sort of open-ended thing in a discussion, not on a paper the student would take home with them.<br /><br />For instance, in the chessboard example: "Here's a chessboard, what would you like to count?"<br /><br />As for your points above...<br />1. Motivation? In a class discussion, keep the questions attached to the kids' names. There's a lot of motivation there to ask a good question. I often end up with X's proof for Y's conjecture about Z's question, and those kids at least are staying motivated. <br /><br />Also I often try to toss out a few different puzzles that get at the same basic idea in different directions, and then let people pursue what interests them. Maybe the ultimate in this sort of thing is Harold Reiter's famous problem set with 10 problems whose answers are all 7 choose 4, even though at first they don't seem to have anything to do with each other. That makes for a great class discussion afterwards! I'm pretty sure you can find his set as a "Stretch" in a handbook from the early 2000s at the MATHCOUNTS web site, though my quick search fails to turn it up.<br /><br />2. Change the culture? Absolutely. You can start in small ways, by beginning with a question instead of an explanation. "What is the length of the third side of this triangle?" and then by the end of class you have picked up the law of sines or cosines or both, using standard tools like "make right triangles". Then instead of memorizing formulas, it becomes a problem-solving process. (There's still plenty of cultural obstacle to get kids to value the whole journey instead of just ignoring you until the last five minutes, writing down the summary, and memorizing it.)<br /><br />3. Sixth graders? Sure, just don't leave it TOO open-ended at first, and give them some guidance and structure. Show them how to take a too-easy question and extend it, and how to take a too-hard question and find a related simpler one. Those are probably even more valuable strategies than anything else you'd teach, anyway. For Perdita's example, "How many squares are in the next row?" is probably too easy, and so is "how many squares are in the nth row?" although there you'd have at least some interest in showing them how to express their description well in English and in math. So, the natural next question is "How many squares in total in the first n rows?" or "How many rectangles in the nth row?" and then ... wait a minute, what did we just discover, and why does that work!?!?<br /><br />4. Day-to-day? Point out that some days are for skills and some days are for problem-solving, and that both are valuable just like doing drills and scrimmages in soccer or basketball. Show them how skills help them solve problems and how problem-solving helps them develop skills without it seeming quite so boring (again, the athletics analogy probably works). Getting in the habit of introducing a new idea with a question, and drawing the answer out of the students as much as possible, might help too.<br /><br />5. I like solving hard problems, but surely the art of problem-posing is at least as important as the art of problem-solving, and our curriculum has been way too over-balanced away from problem creation. I mean, how many classrooms have you seen where the students have ANY time spent coming up with the questions?<br /><br /><br />Any other ideas?<br /><br />2.Joshua Zuckerhttps://www.blogger.com/profile/04689961247338617418noreply@blogger.comtag:blogger.com,1999:blog-1075593398139537131.post-76247569398339273442010-07-16T07:53:01.300-07:002010-07-16T07:53:01.300-07:00This is a very exciting conversation.
I've do...This is a very exciting conversation.<br /><br />I've done some experimenting with having students generate the problems. On the basis of this limited experience it seems to me that many of the concerns of folks you quoted above are real and substantial, but none of them are deal-breakers. Challenges to bear in mind when planning, in order to make the educational experience more powerful.<br /><br />I don't think everyone would necessarily come to the same conclusion, but my experiments have been most effective when I posed the first question, the one that gets everybody moving in the first place, but invited people to move on to their own questions if they got curious about something, and then listened for this and directed people to take on their own questions when they came up. The questions that people asked when they were already working were more focused, more mathematical, and people were more already-bought-into-them, than the questions people came up with when I just defined the objects of inquiry and then said, "what questions could you ask here?"Anonymousnoreply@blogger.com