End of induction..."it's cold or hot, not lukewarm", sets and "The Fly in the Ceiling"

 Another Monday. Many of us were on campus before 7:30 am. The subway was not bad, I enjoyed my reading. 

Some students arrived with a cup of coffee in hand. 

We continued discussing mathematical induction. I asked them what they remember about proving by induction. Phrases such as basis step and inductive hypothesis were heard. 

We picked up on the example we left of proving the formula for the geometric series. Everyone got that P(0) was true. 

The step of using the inductive hypothesis P(k) to show P(k+1) was explained several times. We wrote the inductive step on a green box and what we needed to prove P(k+1) on a pick box (yes, with my colorful Japanese chacks.) I then stressed that if they really understand that they need to use the inductive hypothesis to show P(k+1), they pretty much got the concept.  I encouraged them vehemently to ask questions now, in the present. One student said that he was trying to understand how I used the formula of the inductive hypothesis on the sum involved in P(k+1).  I explained it, and he said "ok, so so".  I replied that "this step has to be hot or cold in terms of understanding, not lukewarm" This is something that you either understand or you don't. I believe this shocked them somehow, but in a good way. I also asked: "would be able to reproduce this proof in the future without memorizing it?". Once I got an affirmative answer from the class, I did not move on. 

I started introducing the basic definition of sets and then:

• empty set
• power set
• size of a set or cardinality
• finite and infinite sets
• Venn diagrams
• Cartesian products. Mentioned Rene Descartes (sorry for missing the accents), yes! Remember this book for children's book "The Fly on Ceiling"...it would be nice if I can get some copies for them!
• Operation of sets (unions and intersections)

The class was 100% chalk board...no a single slide projected, such a pleasure!

Spring break is approaching (they have an exam right before the break) and I told them that I felt so sad they would be so many days without math. Should I send homework for Spring break? 

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