Choice
Mathematics of Choice, that is. 
This is the text I use for my combinatorics elective. More or less. I wander far afield some days, and give them a “Games and Puzzles Day” each Friday. (It’s all math, and the kids know it, but they still think they are getting over by not doing ‘real work.’) We will get through a bit more than half the book in the term (1 term elective.)
So far we haven’t done very much. They struggled to articulate the difference between “do problems 20 through 30, how many problems?” and “read a chapter that begins on page 20, next chapter begins on page 30, how many pages?” But it only took a few minutes for everyone to catch on.
More below —>
I also let them read the introduction, and bang heads against some assorted combinatorial problems, and discuss (what I was looking for, they got. All the counting problems look like counting problems, but it is not easy to predict what is simple and what is not. Not yet.)
On our first puzzle day we played the adding up to 21 game. Add 1 or 2 to a running total, whoever says 21 wins. Then change it up, whoever says 21 loses. What were the winning strategies? Digression to teach them a little about congruence classes in modular arithmetic. And the game again, but add ½ or ¼, 6 loses.
We ask our students to take 4 years of math (NY State requires 3) For some seniors, continuing in precalculus or calculus at this point is too much. This is an easier way of earning their 7th math credit (1 term = 1 credit in NYC). For others, they love math and are taking this as an extra course. Two math-teamers are taking this to fill in their glaring gaps in combinatorics, which shows up on the national mathematics contests. And there are in-between kids who were just curious.
I’ll be blogging about this class and the problems and puzzles and games fairly regularly. It’s a load of fun.
Full title of the test: The Mathematics of Choice: How to Count Without Counting, Ivan Niven, Mathematical Association of America, Washington, 1965. It is part of the Anneli Lax New Mathematical Library.
JD2718 
The 21 game is similar to the one where you have ten sticks and can pickup (cross out) either 1 or 2 or 3 of them, and the goal is to pick up the last one. Every once in awhile I’ll put this one on the overhead and tell the kids how to play. They can choose to go first or let me. They usually choose me because they aren’t sure of the rules.
I have never lost on the first turn because I know the “secret”. So they I ask someone else to play, and they always say that they’ll go first because they think that will make them win. It never does.
I’ll play a third time, and win, and then stop and tell them that I am still king. LOTS of times kids come back the next day to tell me that they have it figured out, and ask me to play them on the overhead in front of the class. I win sometimes, even though they understand the rules because I (mistakenly!) put an extra “stick” on the overhead.
Good times!
Games, even non-counting games, are great for developing all sorts of interesting number sense. I know that counting around the Monopoly board (divided into 5’s and 10’s) made me a much better “adder with carries.” And then games like the sticks, counting to 21, etc do more, and explicitly, and the kids still voluntarily engage.
I also play krypto and set with them. Do you know these games?
Jonathan
IB a Math Teacher:
>> I win sometimes, even though they understand the rules because I >> (mistakenly!) put an extra “stick” on the overhead.
Oh man, that is cruel ;)