Reflections

Somehow summer classes have come and gone.  I know students take summer classes to learn information fast.  I hope that they really do learn.  I do know that I've learned so much more than I thought was possible, so I wanted to take some time to write what I thought went well this semester and what didn't go so well.  This way, if I get to teach calculus again, I can come back and remember what needs to be stressed more.

What I plan to keep:

• Bowman Dickson's method of introducing inflection points through the spread of an epidemic (inflection via infection)!  This worked especially well for my Business Calc students who are not quite as enthralled with pure mathematics as...some.
• Making students find the general antiderivatives for functions and then having them create a chart on the board.  Once a student found an antiderivative, s/he would pass the marker on to another student to find the next antiderivative on the chart.  This was inspired by Prof. Burdette from an article by Faculty Focus.
• Derivative cards.

What I need to change:

• MUCH better review of composite functions before we get too deep into calculus.  The Chain Rule and u-substitutions are just too hard to teach if we can't recognize composite functions as such.
• Critical points.  What are they and why are they important?  I didn't a good enough job emphasizing how important the locations are where the derivative is zero or undefined.  I assumed the geometric representation of critical values would be clear.  Never assume.
• More emphasis on the fact that integration is related to antidifferentiation.  Specifically, Leibniz's elongated s combined with a differential (dx, for example) is the COMMAND to antidifferentiate.  Once you do the command, the elongated s and dx disappear.  Notation, notation, notation.  It's important!  [What worked in private tutoring:  the integral sign is like the capital letter and the differential is like the period at the end of a sentence.  You need both; and combined they tell you to find the antiderivative.]

What I plan to focus on more with my algebra and pre-calc students to get them ready for calculus:

• Compositions.  I want to start early with this idea.  For example, what does √ mean?  Well, nothing, unless it has an argument.  Similarly, sin, ln, and ( )^2 are all meaningless without an input. 
• Exponents, baby.  My Calc I students seem to be fine on this, but my Business Calc students were way behind (in general).  I need students to know that  can be rewritten as x^(-1/4).  And I need them to be able to recognize this quickly.  Furthermore,  as crazy as this sounds, I need them to be able to add and subtract fractions so we can milk the Power Rule for all it's worth.  Please, for the love of all that is good and holy, don't pull out your calculator to figure out what (1/2)-1 is.  Please.
• I want to pound in the idea that the slope of a horizontal line is zero.  
• How do we find the change in a quantity? (Subtraction.)  Seems simple, but it's something I need to emphasize more.
• Difference quotients.  Less emphasis on computing a bunch of them, more emphasis on how it's just slope.  I don't want my students to memorize a formula, I want the formula to flow out of an understanding for how to mathematically write "How fast does y change with respect to x?"