Understanding verses Method

Hopefully I will find a better title for this thread soon

It is a discussion that has been going on in my head for 50 years and recently I think I made some progress in being able to articulate it. Lets see.

Probably in 11th grade math, I remember being introduced to the quadratic formula. And, as I always tell the story, the bright young math genious next to me says "don't memorize it. You can derive it any time you want". I have always used this story as an argument for private schools. I think in private schools you are more likely to be challenged to go beyond not only by the teachers but also by your fellow students. (Full disclosure: I went to public school for part of kindergarden, I taugh a 1/2 year in public school but otherwise all my education and 15 years of teaching have been in private schools)

Using the quadratic formula after deriving it using completing the square of a guadratic demostrates what I mean by understanding. I understand how 2 digit by 2 digit multiplication works even though I don't think about why I move the second line of the solution over one place. I understand how you get the answer to 3/4 Divided by 2/3 if you multiply 3/4 by 3/2. But wait, do I really? I can convert everything to 12's and see that the multiplication by reciprical gets the right answer. But does understanding just mean that once you have proved something then you "understand" it?

So often in teaching math I have felt defeated, given up trying to "explain" an idea and instead encouraged a student to just learn a solution algorithum or ruberic. Take the number over here and do this to it and then take that number over there etc. "Will that always work?" is the student's favorite question. I have to answer, no, but for the purposes of the test we are trying to help you pass, it will work on a vast majority of the problems that look like this. I was never as consistently been faced with this situation as I was in trying to help my own two daughters. Neither is a lover of math and again and again I had to give up trying to impart what I considered "understanding" and instead just try to drill into them a set of steps that solved the problems. Monkey see, monkey do.

What caused me to rethink this whole "understanding" issue was my thinking about how I was solving kenken puzzles. Normally, I can solve a puzzle buy just moving around and around. When I am able to fill in one cell, eventually that allows me to fill in another, etc. But, with the 8x8 harder ones that I am doing now, I really have to "make a guess". In practice this means that I know a cell could be a 1 or a 2. I choose to follow the 1 path first. I make sure that I have a "recovery" point so that I can retrun to the puzzle and have it look exactly as it did BEFORE I choose the 1 over the 2. Then I go on from there, hoping that I will either get to an inconsistency, which means I go back, undue everything I have done since I choose the 1 and now choose 2. Or, I continue on to the end, being able to solve the puzzle completely. With puzzle 71, I had to creat "recovery" points at least twice.

So, I started wondering if certain puzzles require you to guess, require you to choose one number and see if that one works. Maybe such puzzles have a fundamental difference from non-guess puzzles. I have decided that they do not. In fact, I am guessing all the time! It is just that I am able to keep the recovery point in my head! I'm able to "look ahead" as you would do in chess without having to write anything down. The only material difference with the situations in which I claim I am guess is that the sequence is too long for my small brain to hold. I need the tool using animal skills of pencil and paper to augent my memory. This leads me to say: "There is nothing fundamentally wrong with guessing". What I want the students not to do is write down a number UNLESS one of two things is true: either they have proved that it is the correct number for that cell OR they have a recovery disk and are sure they can get back to this point.

Side thought: In what endevors can you find the concept of system restore? Clealy it is a computer term. Most common with asking the opperating system to remember all changes from a certain point forward so they can be undone. It is certainly a database term. I think all programs employ it also! We make copies of our files, call them 1 2 3 etc and go back to the last one when the most recent one will not compile or work and we can't figure out why!!! Clearly restore is a puzzle and crossword term. Maybe in bridge? In go we call it a branch, an alternate move. Break Points in computer programming are related but different. CAD programs must have a restore feature. Word processors often have multiple levels of undo!

I'm afraid I have run out of time. I'll have to come back to this.