seeing the forest for the trees

Today in calculus we were discussing differential equations, an important topic in the AP curriculum.  I have posted a sample of a question here.  There is a similar question on the test almost every year, and you can see all kinds of questions here.

What makes this question so important is that students are asked to apply the Fundamental Theorem of Calculus.  As it's name implies, it is the fundamental centerpiece of my AB Calculus curriculum and describes the relationship between two seemingly disparate processes, differentiation and integration. 

In the last minute of class, I asked my students why we were solving problems of this type.  I am sad to report that most of them did not know.  They had not yet made the connection between today's work and a theorem we learned last semester.  In the context of education, I think helping students make such connections is one of the most important things teachers do.  In every curriculum, there are big, hairy ideas and tiny details.  Differentiating between them is vital to a comprehensive understanding of mathematics at every level. 

When kids are learning, every single day is a tree.  As a result there are seemingly 100+ identical trees.  But that's not really true.  There are mighty oaks, sweeping willows, and scrawny weeds.  The best students are terrific gardeners.  I'm suggesting you ask your students why what you are studying this week is important and report back on the results.  I'm betting many of them don't really know.  If they don't, maybe you can take them up in the helicopter so they can see the entire forest once in a while. 

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