In the last week I gave my students two different kinds of problems to solve. One involved a complicated picture where the students had to use various ways to find the measure of angles that allowed them to calculate other angles, and so on until they found the measure of about 10 different angles:
The students engaged immediately, and continued working until they found the answer. At first a number of them were confused about how to find the measure of the angles in the pentagon and hexagon, but they looked through their notes and found things to help them. Not all of them found the correct answers, but there was a lot of thinking going on.
Yesterday I gave the students a different kind of problem:
The students shut down. Why? I think the answer is feedback. When asked to prove something, the students can't really identify when they have the "right" answer. regarding a different problem, a student came up afterwards and said, "if it isn't true then the shape can't be a parallelogram, so it must be true." Of course, this is how we proceed to construct a proof by contradiction, so I told him his intuition was good. But he didn't quite understand that there was anything else to prove after making that statement. The conclusion was more obvious to him than the theorems I wanted him to use: alternate interior angles, side-angle-side, etc.
In the first problem, they have a situation where they don't know a piece of information and they try to find it. In the latter problem, they take a piece of information that seems obvious to them and are asked to "prove" it using things that are much less obvious.
One of the objectives of my proof website is to get students to develop a visual understanding of when a proof is complete:
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| CongruenceProof.com |
Still, I don't think this entirely solves the problem. Students don't like proofs because the don't "get" proofs. They don't see the point of an axiomatic approach to mathematics.
Proving things is the essence of mathematics. Prior to geometry students would never suspect this because they aren't really asked to articulate their reasoning. They simply learn algorithms. An algorithm is different than a proof because it always proceeds in the same way. A proof requires searching for connections.
Proofs are central to mathematics, but for most students a more important question is whether they are useful for anything else. The new common core suggests that they are not. Deductive proofs are mostly replaced by a focus on transformations. I happen to think that a focus on proofs develops very strong reasoning skills that transfer better to other applications than algorithmic skill.
I am currently undergoing a sort of writer's block. I don't know what to say because I don't even know what my opinion is. Should I give up on teachings proofs or should I continue to try to make them more accessible?
One idea is that I could focus on asking students to find fault with proofs instead of asking them to create their own. Maybe this kind of troubleshooting approach would develop the kind of reasoning skills that would transfer to more general applications and prepare them for higher level mathematics without requiring them to buy in as much to the importance of alternate interior angles and other fuzzy seeming theorems.
When I took geometry in High School I though proofs were incredibly stupid. I learned to do them, but didn't really understand what they were for. I didn't "get" proofs until I took abstract algebra in college. So my question would be, even if someone can effectively do proofs in high school geometry, is that an indication that they "get" them?
ReplyDeleteI'm old enough to remember doing delta-epsilon proofs in high school calculus. I even took introduction to analysis in college and still failed to truly understand proofs. It took finding some area of mathematics that totally grabbed my interest and gave me a desire to communicate why it worked. Then I understood why proofs were so important.
I started to get proofs in college course about basic set theory concepts. I didn't mind doing them in geometry, but I totally ignored the delta epsilon stuff in calculus. Maybe asking students to "get" proofs is too much at this stage. Perhaps just helping them to build a few logic skills is a better approach.
ReplyDeleteStill, once I started to understand proofs it had a major impact on my way of thinking. How much of this can I pass on to high school students?
I keep thinking that it would be nice to introduce proofs through some more tangible real life examples like I do with conditional statements. We talk about conditional statements with examples like: if I get doughnuts, then I'm happy. It's easy to tell that the converse isn't necessarily true, but the contrapositive is true. It seems like we should be able to introduce proving without pulling in the math. Maybe have the students detail their parents' postulates (house rules) and see what theorems we can prove from them and how to use syllogisms to prove them. I've never had time to develop this idea though and I also struggle with showing my students the importance of proofs.
ReplyDeleteMy students actually did pretty well with conditional statements. I had them make a lot of their own arguments using if-then, only, unless, etc. But I am starting to think that expecting them to "get" congruence proofs may be a bit too much at this stage. Perhaps it is enough to just help them develop enough logic skill so that when they come across proofs later on they won't be without any tools to deal with them.
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