tag:blogger.com,1999:blog-8537494321067959493.post3271918110418339001..comments2020-03-29T14:49:53.024-07:00Comments on BorschtWithAnna: My issue with hintsAnna Blinsteinhttp://www.blogger.com/profile/13960574914938362477noreply@blogger.comBlogger13125tag:blogger.com,1999:blog-8537494321067959493.post-30712540595290635772015-05-03T10:23:09.803-07:002015-05-03T10:23:09.803-07:00A good hint should make the problem more interesti...A good hint should make the problem more interesting to the student. The hint is dependent on the problem and the student. This problem, I think, is likely more interesting if we avoid algebra & systems of equations. Perhaps give a hint that leads towards a less formal approach.<br /><br />Would it be reasonable to ask something like "can you tell which of the bales is the heaviest"? This gets the thinking back to bales of hay and encourages reasoning and analysis rather than premature symbol manipulation. The student will immediately be able to start on this new question. Quickly narrow it down to one of two bales. And likely be motivated to figure out which of these two is the heaviest of the five. This of, course requires looking at a couple of the other pairs (you now have five triangles version of the problem without equations or symbols). Hopefully the student begins to think about "how much heavier", "what is the next heaviest", etc. <br /><br />I guess I feel like the key to having good hints is having a good problem or collection or problem(s). If it is a good problem, you need to think about hints before you even ask the question. Should I include a hint with: part a) which is the heaviest bale. Or should I remove a hint by just stating that the weight of each pair is know, without providing those weights. Etc, etc. Anonymoushttps://www.blogger.com/profile/16951223418671175516noreply@blogger.comtag:blogger.com,1999:blog-8537494321067959493.post-61537389889034147052015-05-03T05:57:44.594-07:002015-05-03T05:57:44.594-07:00I think that I would still like those specific str...<i>I think that I would still like those specific strategies to be developed by students as much as possible</i><br /><br />This might be the root of our disagreement! I don't have much of a preference for students to invent the content or strategies of a math class. Michael Pershanhttps://www.blogger.com/profile/17046644130957574890noreply@blogger.comtag:blogger.com,1999:blog-8537494321067959493.post-20388601476613292522015-05-01T17:28:46.676-07:002015-05-01T17:28:46.676-07:00I'm still mulling this over and may have more ...I'm still mulling this over and may have more to say :) it's a tough topicAnna Blinsteinhttps://www.blogger.com/profile/13960574914938362477noreply@blogger.comtag:blogger.com,1999:blog-8537494321067959493.post-74540074769109888572015-05-01T17:27:48.946-07:002015-05-01T17:27:48.946-07:00Thanks, Jen!Thanks, Jen!Anna Blinsteinhttps://www.blogger.com/profile/13960574914938362477noreply@blogger.comtag:blogger.com,1999:blog-8537494321067959493.post-2551017102950668902015-04-30T17:06:48.141-07:002015-04-30T17:06:48.141-07:00Oh, and I think that I will offer the ambiguous ve...Oh, and I think that I will offer the ambiguous versions as optional extensions for my students - fun!Anna Blinsteinhttps://www.blogger.com/profile/13960574914938362477noreply@blogger.comtag:blogger.com,1999:blog-8537494321067959493.post-67039397804309743792015-04-30T17:03:28.277-07:002015-04-30T17:03:28.277-07:00For older students (this was a task I gave to 10th...For older students (this was a task I gave to 10th graders), it may be reasonable to expect that some would create an easier example to play with on their own and then investigate its patterns and transition to harder versions rather than having those scaffolds provided by the teacher. For younger students, I think that this is a really helpful way to frame the problem, and thank you for sharing those versions!Anna Blinsteinhttps://www.blogger.com/profile/13960574914938362477noreply@blogger.comtag:blogger.com,1999:blog-8537494321067959493.post-37936040416907403802015-04-30T17:01:49.215-07:002015-04-30T17:01:49.215-07:00Hey Michael - that's a great point. I think th...Hey Michael - that's a great point. I think that I would still like those specific strategies to be developed by students as much as possible, just like I would like content knowledge to be developed by them. So instead of telling them to play with the diagram, I would give problems where that would naturally fall out as a strategy for some students and have those students share their approach with the class so that over time, the class is developing a set of strategies together that they own. Now, if no one comes up with a strategy that I think is useful, I don't have an issue with providing it, but I would like for most of the strategies to be student-produced. Do you think that this is the root of our disagreement?Anna Blinsteinhttps://www.blogger.com/profile/13960574914938362477noreply@blogger.comtag:blogger.com,1999:blog-8537494321067959493.post-80936583273490890862015-04-30T12:18:47.826-07:002015-04-30T12:18:47.826-07:00We might begin with an easier example, say with 3 ...We might begin with an easier example, say with 3 bales, A, B and C, A+B=5, B+C=9 and A+C=8. That's already challenging to someone who's never solved problems of this nature. It's possible for students to solve it without applying algebraic skills, then to develop a sense of the patterns involved without being given any hints, say by comparing B+C=9 and A+C=8 and realising that B>A.<br /><br />Then transitioning to harder versions is only a small step up:<br />http://fivetriangles.blogspot.com/2013/03/50-mail-bags.html<br /><br />Same concept, but pretty difficult, because they involve an ambiguity:<br />http://fivetriangles.blogspot.com/2015/03/225-integer-pairs.html<br />http://fivetriangles.blogspot.com/2013/05/69-weights_8.htmlFive Triangleshttps://www.blogger.com/profile/12846752710456413605noreply@blogger.comtag:blogger.com,1999:blog-8537494321067959493.post-35022102730566386042015-04-30T07:56:05.318-07:002015-04-30T07:56:05.318-07:00"I make a distinction between teaching a spec...<i>"I make a distinction between teaching a specific procedure or specific content...versus when you are asking students to work on a more open problem in which they are meant to develop problem-solving and sense-making."</i><br /><br />This is a fine distinction, but 95% of my teaching takes place in between these two poles.<br /><br />This year I'm teaching 9th grade geometry, and I often want them to solve geometry problems that they've never seen before. I want them to learn not just their general problem-solving tools, but I also want them to use their specific geometry problem-solving strategies. Something like, "If you're just starting off on a polygon problem, checking out the angles can help you get oriented." Or, even more broadly, "Have you tried playing with the diagram?"<br /><br />This is not a specific procedure. There is a lot of student thinking and sense-making for students to do after this. And depending on the particulars of the student, the problem, the situation I might choose to not share this. But I think this lies in between the two poles you outlined and it's that vast middle ground that I'm interested in.<br /> Michael Pershanhttps://www.blogger.com/profile/17046644130957574890noreply@blogger.comtag:blogger.com,1999:blog-8537494321067959493.post-15189100960882459882015-04-30T03:48:25.051-07:002015-04-30T03:48:25.051-07:00Great distinction between the two types of hints -...Great distinction between the two types of hints - thanks for sharing!<br />Ruthhttps://www.blogger.com/profile/13456334019433707669noreply@blogger.comtag:blogger.com,1999:blog-8537494321067959493.post-65344542090911049562015-04-30T03:25:15.403-07:002015-04-30T03:25:15.403-07:00I really appreciate that way of thinking about it....I really appreciate that way of thinking about it. Thank you for sharing specifics! Jasminehttps://www.blogger.com/profile/14163491309269691356noreply@blogger.comtag:blogger.com,1999:blog-8537494321067959493.post-59161450976545239432015-04-30T03:24:39.757-07:002015-04-30T03:24:39.757-07:00I really appreciate that way of thinking about it....I really appreciate that way of thinking about it. Thank you for sharing specifics! Jasminehttps://www.blogger.com/profile/14163491309269691356noreply@blogger.comtag:blogger.com,1999:blog-8537494321067959493.post-63298333080890133182015-04-30T03:13:56.484-07:002015-04-30T03:13:56.484-07:00That was great, thanks!That was great, thanks!JFairbankshttps://www.blogger.com/profile/09111997601925361204noreply@blogger.com