Moments ago, @dandersod tweeted about an interesting, but non-intuitive probability problem. There are plenty in this genre such as the Monty Hall Problem and many in other areas like the uncountability of irrationals, area/perimeter relationships (@CmonMattTHINK), etc.
I think these problems are like lifting the wolverine's lip to show the teeth. Some kids will be like, "Wow! Those are sharp! So cool! I wonder what that wolverine would do to a rabbit..." Others will just be, "AHHH! Run away! Don't ever make me go near that thing ever again!"
These kinds of problems really separate those who end up loving math from those who do not. I present one or a dozen of these throughout the school year to my various math classes. Some of the kids LOVE it and try to figure out why it is and will read up on it when they go home and have all sorts of questions and really work to pay attention to the "proof." Others will just throw their hands up and say, "See! This is why I hate math!"
Basically, it polarizes the classroom. Those who enjoy thinking this way will get excited and encouraged to find out more. Those who already hate it will have fodder for their fears.
So, I guess what I'm asking here is: Is it a good idea to put forth these "paradoxical" problems in class?
I vote for "yes". I'm in agreement that it will cause some initial discomfort for kids who thought they had it all figured out. But I believe strongly that school should be a place where you feel some mental discomfort from time to time. It's a little like growing pains, some people get them from their bones growing faster than their muscles and ligaments expected, but in the end they are taller/stronger/faster. I understand your hesitance with turning kids off, but maybe a short term pain can turn into a long term benefit.
ReplyDeletebtw, I like "fake proofs" too.
http://www.sandeepkejriwal.com/images/triangles.gif
and
http://www.mathmojo.com/interestinglessons/1equals2/1equals2.html
Mentally knock your students for a loop from time to time, it should keep them intellectually keen.
I think as long as the problems are at the proper level--that is, so that the kids who would normally be scared have at least an "entry point" to play with the ideas, and a chance to see that the wolverine is actually rather cuddly--they are a good idea. In small doses.
ReplyDelete- Matt E
Based on my experience, the major challenge in science education is getting students to *observe* and to *reason* based on what is observed. In mathematics in particular (but also in politics and any other science), it's just as important to determine what one *cannot* conclude from a set of observations. It's easy to make quick generalizations and assumptions based on a few observations (say, from how to add integers to how to add fractions). Instead of sending the message to students that "Everything you think you know is wrong!", we should be encouraging them to ask the questions: "In what situations does what you know really apply? What happens in other situations?"
ReplyDeleteI think a fear of math comes from a pair of assumptions: (1) there is only one right answer to any question, and (2) I will never figure out what it is. While the first assumption is essentially correct, I think the purpose of math classes is to provide tools and skills to demolish the second assumption. With that in mind, "counter-intuitive" examples should be brought up as additional data to observe, not strictly as a jolt to the senses. (Some students will appreciate the jolt, however; many mathematicians do. That's an issue of how we deal with individual students, I think, not entire classrooms.)
@Torquedu raised a point (in the tweet linked at his name) about grading. I do not usually assess these deviations from the curriculum/standards in the hopes that the intrinsic curiosity is enough to have them learn it. Of course, once I say, "We're not testing on this" half the class shuts off and quits paying attention anyways. Since it's not curriculum related, is it "fair" to test over tangential discussions like these?
ReplyDeleteSo, back to @torquedu's comment: I agree, if they're not being assessed on the material, the wolverine can't directly bite the student, so if they're not interested and I'm not testing, it's not a direct impact. On the other hand, I think kids have this pedestal view of Math and anything they connect with it in their head can scare them away.
While I love enriching the understanding of those interested in math and math-related subjects, I want to bring in those who shy away from it in general, too. I think if I took things to their logical extremes, I'd rather not scare anyone away than intrigue the ones who are already interested.
I posted on twitter that I thought it was our grading schemes that made the teeth of the wolverine so sharp (i.e., dangerous). What I had in mind was that when good grades are only attached to 'right' answers, then it's really no wonder that our kids with math self-esteem issues would run like hell - there's too much to lose. However, if the grades are attached to taking multiple approaches to a problem, following an analyze-the-problem, create-a-plan, carry-out-the-plan method of problem solving, drawing connections to similar problems and background knowledge, and all of the other things that we say we want our kids to do, then there's really nothing to be lost by playing with a difficult problem. I think this is a great illustration of how our assessments and our values are often mis-matched.
ReplyDeleteIn direct response to the question: if you're willing to give them the time and support to muddle through, then exposure to non-intuitive problems is essential. One of the primary strengths of mathematics is its ability to expose the shortcomings of common-sense.
Aha. Then I think we are agreed, Theron.
ReplyDelete