This year, I began tutoring Differential and Integral Calculus. Let's be honest: for newcomers, it is often a challenge to wrap one's head around the concepts presented. Approaching? Limits? Derivatives? Integrals? What are these things and what do the represent?
My first foray into the topic was not a stellar one. I flunked Cal I the first time. Managed a C on round two and barely escaped Cal II. My first Vector Calc class was a little better, getting a B. But then it was ten years between that and the second instalment of Vector Calculus. At each turn, there was so much that I just couldn't latch onto, until recently, when it all kind of clicked. It's weird. But that's a ten year span. Mind you I was doing something entirely unrelated to Math in that time, rarely, if ever lifting a pen to differentiate or integrate anything for any reason.
For my students, I really want to avoid that experience: of not really understanding, of feeling completely lost or confused. To that end, I never really clicked with the introductions to integral calculus and I'd like to have a way of introducing the concept in a way that gives a more intuitive feel for what is going on.
Does anyone have any hot tips, clues, alternatives to the usual route that helped them, that made the ideas more concrete? Resources to share? NB: I'm looking for unusual or non-traditional ways to introduce the topic.