NumberTheoryChapter1FollowUp

Proof by exhaustion (p.4) is a really nice strategy for students who are not ready to find and extend a pattern. Developmentally (and we're talking waaaay beyond Piaget's early developmental stages), many students are really not ready to *believe* the results of a postulational proof. When you are doing proofs at the middle/secondary levels, I highly recommend encouraging students to try out different types of proof and always promote "proof by exhaustion" as an acceptable and valid strategy for your students that may not be ready for the other approaches.

STORY #1: We read the "[|One Grain of Rice"] story by Demi in my class and then tried to determine (without a spreadsheet), what the total number of grains of rice would be that the raja would have to give to the young man in the story. [You can substitute the Persian story of the [|wheat and the chessboard problem] for the One Grain of Rice story...it's the same problem.] The story involves trying to imagine the quantity you would end up with if on one day you received one grain of rice, the second day you received double that (2 grains), the next day double that (4 grains), and that pattern continued for a set amount of time. We started the calculations as a class and figured out the number of grains per day for the first 7 days and then totalled the 7 days worth of grain. Then students were to figure out how much rice there would be if you continued at that rate for a total of 30 days. We started with what was essentially proof by exhaustion where we actually did all the math involved. Some students started to see a pattern and used postulational proof to determine the final number. Watching which students switched to the equation, and at what point they switched, as well as which ones continued with the laborious "exhaustion" process was quite revealing...I learned a lot about my students on that day.

One of my students who very quickly figured out an equation-type approach to the problem was very excited about his results and was definitely proud of his understanding. The next day, he told me that when he went home and told his parents about the story, the problem, and the answer, they didn't believe him. He said that his parents actually wrote the entire thing out, they did proof by exhaustion, to "check" his work! It just goes to show that proof by exhaustion is always a viable approach, no matter how old you are.

ANOTHER APPLICATION OF PROOFS TO MIDDLE/SECONDARY ED: Often times students will create their own "shortcuts" in math. The way that we traditionally write down and solve long division problems is all about shortcuts. Sometimes students will create their own shortcut. Instead of telling them it's wrong, we need to congratulate them on trying to really understand mathematics and for using "proof by induction" because that is typically where these shortcuts come from...they notice a pattern that is true for some numbers or some situations and they apply it to other similar situations. The problem is often that the "shortcut" they invented only works in certain situations and not for all whole numbers. Be gentle. Use proof by contradiction to show them when their shortcut doesn't work so that they don't have errors in their ways and help them understand when their shortcut does work. Anyone who still is inventing their own way to do math by the time they're in middle/secondary is someone who hasn't had their resiliency and/or creativity stifled yet. These are future mathematicians in the making. Encourage them and guide their way.