This week’s article is a continuation on the fact that the NCTM is writing under the best possible conditions. There are always cases where students will not be on the same page. They will drown if you don’t give them something to keep them afloat. We hope that if we just throw them in, they will swim. However, in the case of mathematics and non-life-threatening situations, they will need to be thrown some kind of driftwood to keep their heads above the water.
If we throw our kids in, we can’t expect so much from them. Sure, you might have some kids who will grasp onto something, but you can’t expect perfection from young students. This is the age where they get to experiment and explore. They need to gain the skills and confidence demanded by proofs. They also need to be exposed to various proofs. Again, as teachers, we should pick and choose the best ways to present proofs to students. We don’t want to overwhelm students, but we also don’t want to coddle them. (Kids get enough of that now without teachers adding to it).
Generally, we tend to teach to one type of student, not keeping other learning styles in mind. In class, we always talk about the frog vs. bird approach where frogs hop around until they grasp at something and birds tend to look at the bigger picture from a distance. I would like to think of myself as a bit of both, but I know I’m more of a frog. If I can’t picture something, I’ll start making the picture and see where it’s going. If you can’t think of something right away, then DO something. You won’t go anywhere if you don’t start. Though, starting is the hardest, and you need some kind of entry point if you are to ever get moving on a problem. Kids need more exposure to math concepts, computation, and strategies if they are going to throw themselves at a problem.
As mathematicians, our methods of proving can be explained by truth tables in classical logic (which might be a little faulty…that may need some revision. I mean, just take a look at the conditional, or implies, operator…that brings along with it a whole mound of problems). Logic would be a great addition to introducing proofs. I wasn’t introduced to truth tables until my first year of college, and even then I didn’t make the connections between proofs and logic until I took a philosophy course in logic. Taking that course allowed me to make the connections required to understanding proof and why certain proof methods work. Logic allows you to better see the inner workings of things.
Overall, kids need more tools if they are going to have any hope of understanding proofs. The article we read doesn’t seem possible for every combination of classroom. It assumes prior knowledge that students may or may not have, and it’s a bit more sophisticated than an introductory proof. In addition, the proof was also very directed to a certain way of thinking and a certain answer. Like mentioned above, students do not think in the same way, so open-ended questions would allow for students to think in their own way. By closing off the questions, we are closing off students that think in other ways. We need proofs that will allow students to give more ideas and bounce their thoughts off of each other. Maybe, a good way to get kids thinking would be to have an open discussion where the kids offer their ideas and try to explain their ideas to their peers.
Proofs seem to rely on some “mastery” or prior knowledge. I believe that proofs should follow the skills that students have by high school and build on them. I also believe that as students continue to grow their skills, more proofs should be introduced to them. They need an idea as to what proofs look like before they can continue to write their own. It’s like writing a paper. You need vocabulary. You need to know sentence structure. You need to have your own style and voice. You need to know how to manipulate the words in order to sound cohesive. You need to make your points clear and focused and formulate good arguments based on evidence. (I would even go as far as comparing proofs to sonnets, having a particular structure, following certain rules and stringing together words strategically to make a point.)
We don’t want to give students the wrong idea about proof. We want to be clear about what proofs are. They are not just examples and solving problems. The best proofs offer a bit of frustration and a hint of adventure.
This week on tweets with Rebecca: I haven’t been great with Twitter. I’m going to stop fighting with my computer and use it more.
Anyway, I want to know: How would you present a proof to a high school student? How would you determine what would be the best proof for a given class?
Thanks for reading!