This week, the Calculus students were continuing their work on optimization, now looking at the many different scenarios in which it is applicable. Basically, the students tend to struggle with turning the word problems in to the two equations they need to start the whole process. I spent a lot of time drawing pictures and walking through the basic logic of the situations; stuff that is more geometry and arithmetic than Calculus. I enjoyed going over the concepts with students, even had a lengthy discussion with a student about which method to use when deriving fractions and when. It's always interesting to get a closer look at how students think... Overall, a good week of observations.
Optimization was the topic when I observed this week, which is basically about finding the x value that makes the maximum or minimum y value given two equations with one other variable. This makes for 3 total variables with two equations, and since the students know they need 3 equations to do substitution for 3 variables, they're all lost when they start. The key is explaining to them that they can use calculus to eliminate the variable being optimized, and then they can treat most of the work like basic algebraic substitution.
It's interesting to see how the students will create a mental block when something looks extremely complex, and start asking questions about things they learned a month ago. They tend to let a lot of variables overwhelm them, so I have to be patient and make sure to re-explain older concepts when they get confused. However, it's actually pretty entertaining sometimes when I ask a student "...and what does y equal?" and they can't answer me, even though in the typed question itself it is stated that "y=3". Once again, I'm just seeing some of the struggles that math teachers frequently experience. It's good practice, and I'm still enjoying it each week.
The students in Calculus were learning how to relate values of functions to their derivatives, and the derivativesof their derivatives (acceleration), so it was a really rough week for some of them. These concepts are mostly based in one's ability to visualize graphs and interpret data in their head, and calls for complete mastery of previous derivative information. The students can get easily confused by the difference between a decreasing slope and anegative slope. As it's also easy to mix things up as the person explaining the material, I have to be very careful with my wording, and do my best to help paint the picture the students need. It's almost impossible to explain complex calculus without drawing the graphs themselves (though technically this is simple in comparison to what they will do later), so I've been doing a lot of board work and arm waving. Though it's difficult sometimes to really get the right point across, it's extremely satisfying when the students truly understand because it's more obvious in these cases that the visuals were the key. This was yet another good week for practicing explaining complex concepts to students in unambiguous ways, and it's helping me expand my repertoire of strategies for getting the student to finish my thought for me. That's often the best way to know they got it.