Towards the end of the summer I was brainstorming this idea of teaching Calculus backwards, starting with applications and graphing calculators, then manual Calculus skills, then finally tying those manual Calculus skills to various limits. It is now a little more than a month in, and I have to say that although I cannot compare this approach to a traditional curriculum because I've never taught Calculus the traditional way, I love the way that I am doing it!!!
After a unit on analyzing numerical and graphical derivatives, my kids have a very strong foundation of how to read/interpret graphs visually and how to sketch related derivative graphs carefully. Some of them even have a surprising ability to draw original functions given the shapes of the derivative graphs, even though we really didn't practice that as a class (not yet... I am saving it as part of our intuitive intro to integral Calculus).
And the best part of spending the first part of the year on application is that kids are now itching for algebra, because all of the visual introduction has really primed them for more specificity. They were so, SO excited today when I set them loose on an exploration regarding the power rule of differentiation. After just observing three examples that they generated via explorations on their calculator, they were fully able to generalize the pattern in partner pairs, and by the time they got to the back side of the worksheet, they thought it was just SO COOL that they could now differentiate by hand, without a calculator. It was so AWESOME to see. I've never seen kids so excited to differentiate by hand before! And, when they weren't sure whether they were doing something by hand correctly (ie. when they needed to differentiate a term already with a negative exponent on x), I would encourage them to put in an x value into their resulting f' formula and to check it against the dy/dx value that the calculator returns. Their tech skills from Unit 1 are now becoming a powerful tool for self-monitoring their process.
And, because they've already done some mixed applications with f(x) and f'(x) analyses, I can immediately set them up tomorrow to answering a question such as, "When will the instantaneous rate reach 2 along this graph?" and the skills that we have already worked on for choosing the correct tool between f and f' will come in handy then, to smoothly transition them into the new sets of manual skills (I hope!).
I am so excited! Also, in context, we are reviewing various algebra skills incrementally. On their next quiz/practice quiz, they'll be expected to show work to find standard-form equations of polynomials by hand, and then to differentiate them in order to answer follow-up questions. My plan is to loop in all of the manual algebra modeling skills alongside the new derivative and integral skills, so that the kids would always know that they need to keep up with all of the transformational concepts from Algebra 2 and Precalc, as part of the norm for our class.
So, Calculus is pretty much my favorite class to teach right now. So incredibly fun!!!! I cannot wait until we dig into more of these derivative skills, in order for us to talk about some juicy applications that require both algebraic and geometric analysis. YES!!!!!