My Philosophy of Teaching Calculus to Beginners
I believe that the classical (and correct) sequence of ideas that seems to be used universally in introducing calculus is not necessarily the best way for everyone to learn it. It is well accepted that Newton and Leibnez did not think about many of the details of calculus in the way that we think about them today. They may truly have understood continuity and limits, but if they did, they did not communicate their understanding very clearly. The ideas of continuity and limits were not formalized until later mathematicians – notably Augustin-Luis Cauchy, 1789-1857 (The History of the Calculus and Its Conceptual Development by Carl Boyer, Dover Publications, 1949, pgs. 274-275).
By today’s standards both Newton and Leibniz made conceptual jumps and assumptions that, while they turned out to be correct, could not be rigorously proved while they were living. The point here is that calculus did not historically evolve in the same highly polished and proper (rigorous) manner in which it is now taught. (Fluxions are so 1600!) Calculus did not “jump out of the heads of Newton and Leibniz” in the form that it is known today.
It is possible that Mr. Newton, the mathematician, made his initial discoveries when seeking the slope of a tangent to a curve at a specified point, but there is little utility for such knowledge.
It is more probable that Mr. Newton, the scientist, was struggling with the concept of instantaneous speed
or possibly max and min problems when he started thinking about the ideas that eventually led to what we refer to as calculus.
As the title indicates this book is not intended to be a comprehensive a calculus book. It is a sequencing and presentation of ideas whose purpose is to help needy students transition into calculus by helping to create “student readiness” for those students who could benefit from studying such materials. “Student readiness” is a topic that is not properly appreciated by many textbook authors and even less by publishers. I am suggesting that these materials could be “required or recommended reading” for those students planning on taking “real calculus” the following term or as correlated reading while currently taking a formal calculus class.
Some 25+ years ago I read an interesting book titled They’re Not Dumb They Are Different by Sheila Tobias, Research Corporation,1990. It was about a very tiny study of bright university post graduates who, as freshmen, had either avoided science classes or who had transferred out of them. As post-graduates they were monetarily bribed to go back and take an introductory science class in either physics or chemistry (their choice). They were asked to keep notes on their reactions to the curriculum as they went through the class. Most of that book is not relevant to this book but following are some quotes from or about the study participants which I believe are helpful to explaining the philosophy of this book.
Eric: The greatest stumbling block to understanding” was the lack of identifiable goals and the absence of linkage between concepts. (pg. 29)
Jacki: If he (the teacher) could tell us what’s coming next, why we moved from projectile to circular motion… I would find it easier to concentrate… I always wanted to know how to connect the small parts of a large subject. In humanities classes, I searched for themes in novel, connections in history, and organizing principals in poetry (pg. 34)…. I never really knew where we were heading… He (the teacher) knows the whole picture but we don’t. pg. 38
Michelle: My curiosity simply did not extend to the quantitative solution… I was more interested in the “why” (pg. 40)
Tom: more attention was given to Avagadro’s number than to Avogadro’s insight. …how and why he came up with that number. (pg. 43)
Stephanie: In the humanities and social sciences we are taught to ask “why” questions. In chemistry I felt we were only being taught to ask “how.” (pg. 58)
Although the statements above refer to introductory chemistry and physics curriculum I believe that there are lessons to be learned there for math teachers as well. Most of the sequencing decisions in this book have been made with the quotes above in mind. The traditional calculus curriculum is undeniably the best in terms of logic and efficiency but that same logic and efficiency is wasted if a student is not “ready” to receive that instruction. Also having a 1978 vintage math degree I was astounded to find out that my son’s calculator had a “derivative button” and an “ integral button” giving exact answers and not decimal approximations. The implication for a teacher seems to be that understanding of concepts is as much or more important than symbol manipulation. As a result I have spent more page space on concepts, linkage, analysis, and problem analysis than on “symbol manipulation.”
The sequencing and presentation of concepts in this book has been designed with the following beliefs in mind:
1.) “Tell the students what you are going to teach them (anticipatory set) , teach them (communicate course content) , tell them what you just taught them (review).” 2.) Explain or demonstrate why a skill is important before teaching the skill. Why is finding the slope of a tangent to a curve at a given point important? Why is finding area under a curve important? What is the utility of those skills? 3.) Try to show how and why a skill or idea evolves from or connects to previous math instruction or history. If you wish to find how much work is being done, why do you instead find the area under a curve.
To the typical student, finding the area under a curve when you really wish to determine the amount of work being done is like Athena jumping out of the head of Zeus. Deus ex machina!! What’s the connection? It may be intuitive to a teacher, but it certainly is not intuitive or obvious to most students. 4.) Teach the big picture and big ideas first, deferring specifics for later. This is a considerable paradigm shift. 5.) Repeat, rephrase, reteach ideas and concepts whenever relevant and helpful. Any teacher who feels that review and reinforcement is a waste of time probably needs to take a graduate class in topology to remember how frustrating and demoralizing learning math can be. 6.) Whenever possible, keep all ideas leading to a new concept on one page or a two-page spread so that visual learners can synthesize all those ideas without having to flip pages. 7.) Compare and categorize similar ideas and contrast dissimilar ideas whenever possible.
8.) New ideas and concepts are almost always accompanied with new vocabulary. Always introduce and teach new vocabulary very carefully. Teach and reteach technical terms several times before using them casually. 9.) Finally, never tell when you can show. Even though it was crucial to the development of calculus to discount the senses in favor of definitions and logic, I believe that in education the more visual the instruction the better. I use many, many tables to show patterns, as many relevant images as I could find, illustrative cartoons as well as traditional graphs.
These rules give up the efficiency and benefits of the traditional,logical and impeccably sequenced curriculum in order to receive the benefits resulting from giving the students previews and “big picture overviews” of the material. These rules also sacrifice (temporarily) rigor for curricular flow. Not everybody needs such materials. Many students, however, could greatly benefit from the unorthodox progression of the following materials, which delay for three chapters the specifics of calculus all the while teaching and re-teaching in generality the ideas and vocabulary which are at the heart of the calculus curriculum: limits, derivatives, instantaneous rates, etc.