1. "Infinitesimal" change ratio 
                     2. "Symbolic" derivation 
                     3. "Logical" epsilon-delta argument 
                     4. "Geometric": slope of attached tangent line 
                     5. "Rate": instantaneous speed 
                     6. "Approximation": best linear approximation 
                     7. "Microscopic" view of higher and higher power 

 There are rich pedagogical questions to deal with regarding this list alone for the context of a specific course, in a specific place, at a specific time, for a specific population of students: 

                     Mathematically, which one first? 
                     In terms of ease of understanding, which one first? 
                     In terms of ultimate applications, which must be addressed? 
                     How many the first time through? 
                     .....? 
                     .....? 

 In the learning context that I deal with, a key observation that leads me to choose 4. as the primary focus is that human reasoners (my students) are accustomed to using all their faculties to understand the world. In particular, they often effortlessly absorb sophisticated visual information from a diagram and automatically generate a linguistic representation of the information. This is nicely described by Barwise and Etchemendy in [1]. 

                     "...even a relatively simple picture or diagram can 
                      support countless facts, facts that can be read off 
                      the diagram. Thus a diagram can represent in compact 
                      form what would take countless sentences to express." 

This is a very powerful human mode of representing, processing, and understanding information, which has yet to be fully exploited in helping human learners acquire knowledge. 

But what to do to exploit this insight, educationally? Barwise and Etchemendy have developed an approach to teaching (and doing) logic that uses both linguistic and visual representations, which has  resulted in their (excellent) courseware: "Tarski's World" and "Hyperproof". 

In response to my learning context, I am developing an interactive Maple-based text stressing student-active learning, visualization, writing to learn, guided discovery, gradual abstraction, and mathematical understanding. 

In [3] William Thurston challenges all mathematicians to recognize and focus on this human perspective. 

                     "How do mathematicians advance human understanding of 
                      mathematics? We are not trying to meet some abstract 
                      production quota of definitions, theorems, and proofs. 
                      The measure of our success is whether what we do enables 
                      people to understand and think more clearly and effectively 
                      about mathematics." 

 In the learning context I am trying to create I also find George Polya's [2] perspective extremely helpful. 

                      "...learning begins with action and perception, 
                       proceeds from thence to words and concepts, 
                       and should end in desirable mental habits... 
                       For efficient learning an exploratory phase 
                       should .. [be the first phase in which learners] 
                       discover by themselves as much as is feasible 
                       under the given circumstances." 

In addition, I find George Polya's [2] Ten Commandments for Teachers to be a very useful set of guiding principles that help me keep pedagogically focused (can you guess the 11th?): 

                      1. Be interested in your subject. 
                      2. Know your subject. 
                      3. Know about the ways of learning: the best way 
                          to learn anything is to discover it for yourself. 
                      4. Try to read the faces of your students, try to 
                          see their expectations and difficulties, put 
                          yourself in their place. 
                      5. Give them not only information, but "know-how", 
                          attitudes of mind, the habit of methodical work. 
                      6. Let them learn guessing. 
                      7. Let them learn proving. 
                      8. Look out for such features of the problem at hand 
                          as may be useful in solving the problems to come, 
                          try to reveal the general pattern. 
                      9. Do not give away your whole secret at once. Let 
                          the students guess before you tell it. Let them 
                          find out by themselves as much as is feasible. 
                    10. Suggest it, do not force it down their throats. 

 Alright then, what is my definition of the activity I am engaged in that I refer to as "calculus reform"? I guess I will plant my feet and say that I am trying to create a Polya-Thurston Learning Context. But more precisely what is this and how do I do it? I don't know. Its complicated. I am still trying. How about technology? It is a tool and not a magic bullet. It is very powerful, but using it effectively and appropriately requires much more effort and creative insight on my part. It requires me to reform the way I think about how I can help human learners acquire mathematical knowledge. When I focus on this question and try to get it right, technology issues seem to take care of themselves. 

 References

[1]  Jon Barwise and John Etchemendy. Visual Information and Valid Reasoning. In Walter Zimmermann and Steve Cunningham (eds.), Visualization in Teaching and Learning Mathematics, 9-24, The Mathematical Association of America, Washington, DC, 1991. 

[2]  H. Taylor and L. Taylor. George Polya: Master of Discovery. Palo Alto: Dale Seymour Publications, 1993. 

[3]  William P. Thurston. On Proof and Progress in Mathematics. Bulletin of the American Mathematical Society 30 (2), 161-177, 1994.