How easy should examples be?

Really quite a few of my spring semester students told me my examples were too easy. I am still baffled by this, but I work fairly difficult examples in class now, and very rarely ask simple ones.

One reason to avoid simple examples: it can be awkward if the person cannot actually solve it. During class this is no problem for me, I am very patient and gentle, and don’t mind catching the person up. They are on the spot and generally try to catch up quickly. However, outside of class it has become clear to me that some people cannot answer “how many fingers am I holding up” even when given ten tries. I assume they can count, but for whatever reason cannot answer questions.

However, almost none of my students have much practice at the “if there is a hard problem you cannot yet solve, there is a simpler problem you cannot yet solve” with I guess the implicit assumption that you’ll make progress faster on the easier one, and it will transfer to the harder one. I try to give the small problems, but people are usually bored and the class divides very clearly into those that can’t solve it instantly and those that can, making those that can’t feel stupid and disconnected, and those that can overconfident and bored. Crazy. Me, it just makes me think, oh I get that one, what is next?

However, this semester I did manage to ask too hard of questions a few times. How many ways are there to get a pair in a three card hand from a standard 52 card deck? Apparently way too hard, but as far as I can tell, on the homework and on the exam. So, had I asked the easy problem, probably most would have gotten it, and I would have assumed this part was easy. Would the full version have been easy after the easy version? Maybe, maybe not.

At any rate, I’m still working on this proximal zone of learning thing. Pretty tricky stuff. It is particularly irritating that Zem seems to have mastered it in her talks: she gives very easy examples that connect previous knowledge to current knowledge.

I try that in class, referring to other UK course, referring to other major aspects of learning or life, and most tune me out. Maybe incorporating new knowledge into their “real” data banks or even into their other courses’ data banks runs too high a risk of ASK and anxiety. If I teach them something new about life, then suddenly they don’t know everything about life, and that would be pretty scary. Maybe something they learned earlier was wrong, and that makes them angry and defensive. Safer to let sleeping knowledge lie.

I dunno. I think I do want to make my students better people, thinking people, caring people who understand themselves, others, and heaven forbid, mathematics. At any rate in fall, I’ll just settle for not being an awful teacher for College Algebra.