Not. So. Hot.
Then, as I helped students in our Math Lab, I realized something--recursive definitions are not that obvious to students.
I think what happened here was a classic case of it's-so-obvious-to-the-teacher-she-automatically-thinks-it's-obvious-to-everyone-else. We've all had teachers like this. My absolute favorite prof from grad school loved the phrase, "Oh, this is kindergarten stuff!" Which usually had one of two effects on me: (1) Ahhhh!! This is NOT kindergarten stuff! I just spent the majority of my weekend trying to figure this out! (2) Where in the world did you go to kindergarten? Remind me to send my kids there.
But I digress.
What hit me was that when I see something like:
I automatically think, "If I want to find a certain term, I need to sum up the two previous terms." Furthermore, I know that
means the same thing as the previous equation.
On the other hand, when my students saw a recursive definition, I'm pretty sure they thought, "WTF. Skip it."
So, this semester I paid much more attention to these types of sequences. The very first thing I did regarding recursive definitions was show a slide with this at the top:
I asked students to fill in the blanks and then asked them three questions:
- What do the dot, dot, dots mean?
- What do we call the term before a_n?
- What about the term after a_n?
We then did some examples with
an=an−1+an−2
I made them write "The fourth term is equal to the third term plus the second term." And so on.
Then came
which everyone was convinced was a totally new problem (darn you, indices!). But, once we did the same examples (finding a_4, etc.), I think/hope all minds were changed.
After working some specific examples, where initial values were given, I gave them an exit ticket of something like:
List the first five terms of the sequence defined by:
a_1 is the number of boys in the room; a_2 is the number of girls;
a_1 is the number of boys in the room; a_2 is the number of girls;
I think about 80% of students got it with zero help from me. Not perfect, but I'll take it this time around!
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