I recall Al Roaker of the Today Show recently making some kind of public service statement about how people should consider a career in teaching. Perhaps President Obama has also made such a comment about how we need teachers. I did 15 substitute teaching assignments in the Winter and Spring of 1986. I had been out of the active duty Naval Reserves since June of 1976. I had used the GI Bill to go back to school to study Accounting as well as some computer programming classes and still hadn't settled on a long term career. I had renewed my teaching certificate and had even been a full time junior high math teacher for 12 weeks in the Fall of 1972, burned out and then just quit that job. So I thought I'd just substitute for Math classes only to get a little money. I registered with three different districts. Eventually I let all three districts know that I was no longer available to do any substitute teaching. Teaching requires a certain tolerance for failure and I am just not made that way. Anyway my last assignment was with the Ann Arbor Public Schools at Tappan Junior High. It was an older brick building on East Stadium and may not even exist anymore. I had just suffered a not so successful class the prior period. It was a huge room with ascending seating like in a theater so it wasn't just 35 or 40 students. There were easily twice that number. And as the class progressed the students started talking among themselves and it got louder and louder until finally near the end of the period the teacher next door entered through a door that was located right at the left front of the room and I presume it connected to his classroom. Anyway the kids didn't notice him at first. He just stood there and gazed around the room noting who was talking and who was paying attention. He didn't look at me and I took that as he wasn't blaming me. He just kept gazing and little by little the kids took note of him and eventually quieted down. It was like Moses parting the Red Sea. I did not have that kind of power and I never mastered the stink eye dirty look stare. It just wasn't me.
In my bones I knew I wasn't going to take any more substitute teaching assignments after that. So when I got to this next class I felt free to just depart from whatever the lesson plan was and tell the kids about the 12 marble problem and solution. I had the best experience showing this class of 7th graders at Tappan Junior High the solution to the 12 marble problem. I started off by telling them that when I was in the 7th grade my math teacher, Mr. Morgan, gave us this 12 marble problem and nobody could come up with a solution. And in hindsight even Mr. Morgan didn't have the correct solution. That really got their attention. Even the Math teacher didn't get it. There are 12 marbles and one of the marbles is either heavier or lighter than the other eleven. You can only use an apothecary scale three times to identify the odd marble and tell whether it's heavier or lighter than the others. An apothecary scale is like a teeter totter. One side goes up and the other side goes down or they balance if equal weights are placed on both sides of the scale. Like the scale that the statue of Lady Justice holds in front of the courthouse. She's blindfolded because justice is supposed to be blind. Mr. Morgan started out placing an UP arrow over the even numbers and a DOWN arrow over the odd numbers and saying each marble had an equal chance of being either heavier or lighter than the others. He was giving himself one free weighing by doing that. The only way he could do that is if all six odd numbered marbles were put on one side and all six even numbered marbles were put on the other side and then the even numbered side went up and the side with all the odd numbered marbles went down. I did not explain what Mr. Morgan did. All I said is that he didn't solve it. Everyone was paying attention. There was no talking. And about 30 seconds after I finished the solution the bell for the next period rang. This is the way all classes should be. I was a teaching fellow in the Math Department at Eastern Michigan University for two semesters right after I graduated in August of 1971 and every class was good. I taught two sections of Algebra each semester. My older stepbrother, John, taught junior high math for two years before he got out of teaching. So what needs to be answered is why don't people stay in teaching? Don't just say we need more teachers. We need to retain teachers. They are a scarce resource as are nurses and all college graduates. Punishing them with crushing debt makes no sense. Nor does exclusionary zoning make any sense. We need to support and include each other not exclude each other. President Obama eloquently stated we should turn to one another and not on one another. Then not a peep out of the guy about ending exclusionary zoning. Also the key to retaining teachers is retaining students. When students are miserable then teachers will be miserable. So the key is making students happy. Problem solved. Otherwise it's incarceration with release upon graduation. Same for the unemployed. Either make employment appealing or it's incarceration with release upon retirement.
Here's the 12 marble solution. Let's assume you have some way of keeping track of the 12 marbles. Number them with a Sharpie, put them in an egg carton or whatever. Also, without loss of generality, when I say the left side goes down well it could just as easily have gone up. The logic is the same. When you do a weighing either one side goes up, the other side goes down or the two sides balance.
FIRST WEIGHING: Put 1,2,3,4 on the left side and 5,6,7,8 on the right side. Suppose the left side goes down and the right side goes up. We know that 9,10,11 and 12 or OK. Either 1,2,3, or 4 is heavy or 5,6,7 or 8 is light.
SECOND WEIGHING: Put 1,2 and 5 on the left side. Put 3,4 and 6 on the right side. 7 and 8 are just off to the side.
THIRD WEIGHING: If the left side goes down on the second weighing then either 1 or 2 is heavy or 6 is light. In that case the third weighing would put 1 on the left side and 2 on the right side. Whichever side goes down is the heavy marble. If they balance then 6 is the light marble.
If the second weighing balanced then either 7 or 8 is light. So put 7 on the left side and 8 on the right side. Whichever side goes up contains the light marble.
ALTERNATE SENARIO ON THE FIRST WEIGHING: Suppose the first weighing balances. Then we know the odd marble is not 1 through 8. The odd marble must be 9,10,11 or 12.
ATERNATE SECOND WEIGHING: So put 9 and 10 on one side. Do not weigh 12. Put 11 and any other of 1-8 on the right side.
A)The weighing does not balance. If the left side goes down and the right side goes up then we know either 9 or 10 is heavy or 11 is light. So for the alternate third weighing we put 9 on the left side and 10 on the right side. Whichever side goes down is the heavy marble. If they balance then 11 is the light marble.
B) Suppose the alternate second weighing balances. Then 12 must be the odd marble. So the third weighing would be 12 against any other marble to find out if 12 is heavier or lighter than the other marble.