Critical thinking with my students

I came across this video ( http://www.youtube.com/watch?v=vKA4w2O61Xo#t=237 ). After viewing this, I made the connections between what I saw and what had been discussed in class. So I decide to try this with my classes that I teach. I gave the students the same introduction as they did in the video using the same rule he gave to the people. So I told I have a rule and the numbers 2,4,8 followed that rule.

I also tried this with my sister who is also an English teacher to get an idea how this would work with my students. My sister was able to do figure it out in three tries. She started tried 3,6,12 and then 1,2,3. After she found that those worked, she tried 3,2,1 and she concluded the answer from there.

Then I told students to give me three numbers and I would tell them if they followed the rule or not. To help students to see this, I also wrote the numbers on the board so they could refer back to them. When working with my students, there were several things that I noticed with the students.

The first thing I noticed is they gave more right answers than wrong. When we discussed this after the activity, they wanted their sequence to be correct and not wrong. Students want the right answers, but as mathematics, the null hypothesis is generally more important to us than the hypothesis. One example we worry about is dividing by zero. We stress to students that we can not divide by zero so we look for instances where we will end up dividing by zero.

The second thing I noticed was students mimicked what answers the other students gave as answers. Once they saw that 11,12,13 worked, they would use three consecutive numbers. Now, they were not sure why they worked, but they didn't want to get the answer wrong. They also could not get the idea out of their minds that of the order. Most of the answers at the beginning were just ascending order. The students could not see that they could change the order of the numbers.

The biggest discussion came with one of the classes. Earlier in the discussion, 3,6,9 came up a correct solution. Now, as the activity, went on a couple of students had figured out the solution, but I told them not say anything to the class. They gave 9,6,3 as a solution and this caused a big stir in the class. Many students wanted to say that these were the same sequence. They focused on the numbers and not the order. As the students started talking back and forth, they realized what the rule was.

Based the results of this activity and the lack of reasoning in these sequences, I decided that I would do this for the rest of the quarter. I would spend time with every class doing an activity like this at the beginning of class. So the next class, I used the rule of even odd even and gave the students the sequence of 4,5,6. This time around the students do much better thinking abut the types of sequences they used. In fact, one of the students started with 6,5,4 just to see if ascending had anything to do with it. I'm looking forward to doing this with my students for the rest of the quarter.

UPDATE:
So I was able to continue doing this with my class three more times (only 3 weeks left in the quarter).  I tried a few more rules: even, odd, even; one number is prime; the sum of the numbers is even.  With each class, students were able to come up with the rules faster and faster.  When a student mentioned a rule they thought worked, I would ask that student to share with the class their thought process and what made them believe that this was their rule.  Some students still did not understand why they had to do this.  They just wanted to give me a sequence that worked.  Overall, I could see much that students were thinking through the process with their sequences they proposed.