I am taking “remedial” Calculus II alongside Numerical Computing this semester. My Calc course is “remedial” in that I haven’t seen any Math over the reals for about 4 years (took Discrete Math and Linear Algebra, which both focus on integers) and this semester I am overloading on real numbers (and even complex numbers) just when I had forgotten they even existed 🙂
That said, after spending some time in the humanities (where writing quality is high) and much time in Computer Science (where literacy is defined as being able to read code), coming back to traditional math textbooks has been quite a culture shock. They are so horribly written, it really blows my mind.
So, in response to my horrible Calculus II textbook (published at NYU only for NYU classes, this book features minimal explanation and the maximum amount of notation), I have been using it only for the homework problems and using instead James Stewart’s excellent book, Calculus: Early Transcendentals for rigorous proofs of concepts (because Stewart really does present them nicely), and the lighter but infinitely more illuminating Calculus Made Easy, by Silvanus Thompson.
A somewhat controversial book, Calculus Made Easy chooses to skip the notation-laden explanations of Calculus concepts provided by typical textbooks, and opts instead of a clear, textual elucidation of core concepts in the context of their applications. The philosophy of the book is well-described by this excerpt from the Epilogue.
I think this is wonderful writing, however damning it may be:
It may be confidently assumed that when this tractate Calculus Made Easy falls into the hands of the professional mathematicians, they will (if not too lazy) rise up as one man, and damn it as being a thoroughly bad book. Of that there can be, from their point of view, no possible manner of doubt whatever. It commits several most grievous and deplorable errors.
First, it shows how ridiculously easy most of the operations of the calculus really are.
Secondly, it gives away so many trade secrets. By showing you that what one fool can do, other fools can do also, it lets you see that these mathematical swells, who pride themselves on having mastered such an awfully difficult subject as the calculus, have no such great reason to be puffed up. They like you to think how terribly difficult it is, and don’t want that superstition to be rudely dissipated.
Thirdly, among the dreadful things they will say about “So Easy” is this: that there is an utter failure on the part of the author to demonstrate with rigid and satisfactory completeness the validity of sundry methods which he has presented in simple fashion, and has even dared to use in solving problems! But why should he not? You don’t forbid the use of a watch to every person who does not know how to make one? You don’t object to the musician playing on a violin that he has not himself constructed. You don’t teach the rules of syntax to children until they have already become fluent in the use of speech. It would be equally absurd to require general rigid demonstrations to be expounded to beginners of the calculus.
One thing will the professed mathematicians say about this thoroughly bad and vicious book: that the reason why it is so easy is because the author has left out all the things that are really difficult. And the ghastly fact about this accusation is that — it is true! That is, indeed, why the book has been written — written for the legion of innocents who have hitherto been deterred from acquiring elements of the calculus by the stupid way in which its teaching is almost always presented.
I should note that my Calculus professor is actually quite good, and provides very nice explanations of complex topics, usually beginning with an elucidation of the general idea, and then going on to the formalities. But our assigned textbook is not nearly as clear, and many professors I’ve had in the past have lived entirely inside their constructed notational apparatus.
This reminds me of an old joke I heard awhile back:
A math professor begins his lecture by writing on the blackboard. He only pauses for brief moments of notational explanation, but continues writing and writing, one symbol after the other, for thirty minutes on end. He fills up six blackboards full of derivation, algebraic manipulation, and what have you. At the end, he smiles and draws the open box, indicating the completion of the proof. “Is that clear?” the professor asks. Blank stares all around.
At that point, the professor stops himself. “Oh, no, I believe I’ve made a mistake.” He then looks at the six boards of writing, and begins pointing at certain sections while nodding his head, clearly doing calculations internally. He then paces back and forth across the front of the classroom, with his head bent down and his fist to his chin. For five full minutes, he paces and nods, thinking about the proof just presented.
Then he stops pacing, looks at the students, and says, “Ah, yes, yes. It’s clear.”
Here’s an interesting read, by the way. Came as especially relevant to me, as I “rediscover” math for math’s sake.