For example, teachers traditionally introduced the Pythagorean theorem by drawing a right triangle on the blackboard, adding squares on its sides, and then explaining, perhaps even proving, that the area of the largest square exactly equals the combined
areas of the two smaller squares.
According to fuzzy math, this is a terrible way to teach the theorem. Students must be allowed to discover it for themselves. As Cheney describes it, they cut from graph paper squares with sides ranging from two to fifteen units. (Such pieces are known as "manipulatives.") Then they play the following "game." Using the edges of the squares, they form triangles of various shapes. The "winner" is the first to discover that if the area of one square exactly equals the combined areas of the other two squares, the triangle must have a right angle with the largest square on its hypotenuse. For example, a triangle of sides 3,4,5. Students who never discover the theorem are said to have "lost" the game. In this manner, with no help from teacher, the children are supposed to discover that with right triangles a^2+b^2=c^2.
"Constructivism" is the term for this kind of learning. It may take a group several days to "construct" the Pythagorean theorem. Even worse, the paper game may bore a group of students more than hearing a good teacher explain the theorem on the blackboard
This is topical since I'm thinking about doing a Pythagorean Theorem day again this year. My attempt last year: http://mymathclub.blogspot.com/2015/01/16-winter-session-starts-pythagorean.html somewhat followed the design Gardner derides. Among other ways, I had the kids cut out triangles and try to find two squares with equivalent area that would then reduce into one of the easier proofs.
Generally, I don't think it worked as well I wanted it to either. Only a few kids found the necessary equivalent squares and even then it took some guiding to reach any insights.
So what I could do better this time around bearing in mind my primary goal is to engage with why the Pythagorean Theorem works?
After doing a cursory look on the web I've thought of the following ideas so far:
- Remember to emphasize the geometric nature of square numbers i.e. that they are the area of squares before starting.
- Start with a measuring exercise and observe patterns? I could preprint various pythagorean triple triangles to make things easier to spot.
- Do a short brainstorm session and the introduce a few proofs directly.
- Concentrate on extension problems rather than the proofs.
- Focus on the Pythagorean triples and finding them and looking for patterns?
This feels like I haven't quite found the right answer yet....