Learning Calclulus The Way It Is Taught Now Is Unsettling

[Drafter]

In my endeavor of self learning calculus, I realized that there was something a little off about the subject. I would watch Khan Academy, MIT Opencourseware, and even cracked open my college textbook of calculus before I could even enroll in my Calc 1 class.I noticed that all of them introduced Calculus without mechanics. In my opinion, teaching calculus without an intuitive application approach was just unsettling to me. Everyone has that nagging question as to WHY WHY WHY am I learning limits, and derivatives etc. Teaching the calculus with mechanics consolidates the "why" a lot. Sure, one could say that a general physics course covers that. But even if one just starts learning the calculus in a math course, the intuitive physical approach makes things come together a lot easier. Then, a follow up of concise rigorous definitions should come through to make things rock solid and clear up any sort of "what ifs" after the "whys" are answered.I don't see how so many books and professors focus on formalism and rigor while completely ignoring the intuitive "physical" approach in the first place??I picked up a couple of old calculus books from the library, and compared them concept by concept, with books published now, and it really makes me wonder what in the world goes through people's heads to write such books and somehow get it published to students!? The text is always dry, mundane, and impersonal. For an introduction to the end of a unit, one is spatted out formulas with concise formal definitions of the concept, and they seem to be written in a borderline non-human tone. It almost makes me wonder if the people who write these books are machines?

Well, that's just my quick rant on books or colleges having higher math completely ignore the application viewpoint of this subject, particularly when a concept is being introduced. While putting rigor and formalism on the pedestal in education.

Do you guys agree or disagree with my arguement? Let me know!

 

[wabbit]

There's nothing wrong with mechanics but what's so special about it wrt calculus besides history ? Calculus is far more general than mechanics, it applies as well to economics, biology, chemistry, and many other things - including of course, well, mathematics.
I can understand however your complaint about dry textbooks not providing intuition or applications, I also think this isn't a good way to introduce things - which texts in particular do you have in mind ? I didn't realize there was a move in recent textbooks towards abstraction and formalism, is this general or specific to a certain country ?

 

[disregardthat]

What's so bad about taking a mechanics course on the side, thereby learning some of the applications you have in mind. As wabbit points out, mechanics is not unique as an application of calculus, so it wouldn't make much sense to force everyone to learn that, when they perhaps are focusing on a different application in their studies. In any case, in my opinion it is best to leave calculus as "dry and mathematical" as possible, and leave the applications to other courses.

With your second point I can sympathize more (although I would personally have nothing against a more formally oriented course). This is partly speculation, but I believe that higher level courses are more commonly put as obligatory in various lines of studies. For example, here an analysis course was recently made obligatory in (math oriented) pedagogical courses. This all requires a high level of formal understanding of calculus, which may be the reason for a change in trends.

 

[Drafter]

There's nothing wrong with mechanics but what's so special about it wrt calculus besides history ? Calculus is far more general than mechanics, it applies as well to economics, biology, chemistry, and many other things - including of course, well, mathematics.
I can understand however your complaint about dry textbooks not providing intuition or applications, I also think this isn't a good way to introduce things - which texts in particular do you have in mind ? I didn't realize there was a move in recent textbooks towards abstraction and formalism, is this general or specific to a certain country ?

Calculus: An Intuitive and Physical Approach by Morris Kline does an excellent job at introducing calculus with applications using simple mechanics. Then he formalizes things afterwards.

He also is so staunchly opposed to the dry mathy world that education has become that he dedicated an entire book on the subject. A devestating critique to all educational institutions everywhere. The book is called, Why The Professor Can't Teach: Mathematics and The Dilemma of University Education. Great read.

 

[wabbit]

Sounds like a good read, thanks. I disagree with the identification of "dry" with "mathy" but not with the general sentiment. To qualify what I said, I am not against using mechanics as one pedagogical anchor of intuition, I just think it's a limiting perspective, and not the only way to gain intuition.

 

[DrewD]

Calculus: An Intuitive and Physical Approach by Morris Kline does an excellent job at introducing calculus with applications using simple mechanics. Then he formalizes things afterwards.

I personally like this book and if you like it too go ahead and read it. You're also in luck that it is so cheap!

I teach calculus (granted for only a few years) and have not yet had a student that found that text more intuitive than any other option I presented. Again, I am not complaining about Kline's book, but I find it much more dry than most college calculus texts published in the last decade. My students complain that there are so few diagrams and not many worked examples. Kline also assumes a level of geometry and algebra that most high school students don't have now.

 

[NascentOxygen]

Perhaps a focus on mechanics may not help---and might be a turn-off to---many female students. They typically lack the experience and familarity with mechanics that males accept as second nature.

 

[mathwonk]

Different people appreciate different applications, as mentioned here, so focusing on mechanics would only help a subset. There are many books, taking different tacks. The most common one is to use geometry to motivate calculus since most people have studied geometry. Indeed the original definition of a tangent line using lmits can be extracted from Euclid's Prop 16, Book III, a source possibly used by Newton for his inspiration.

However there is a book that explains the ideas of calculus, using many different applications and examples, so that one does not get the idea that calculus applies only to one application. I recommend it:

https://www.amazon.com/dp/9810249047/?tag=pfamazon01-20The standard calculus book for a long time was the one by George B Thomas, before his modern co authors began to change it. Not as sophisticated as some today, and more engineering oriented, but very solid. It has plenty of physical applications. Here are some early editions:

http://www.abebooks.com/servlet/SearchResults?bi=0&bx=off&ds=30&kn=george+thomas,+calculus&recentlyadded=all&sortby=17&sts=t&yrh=1965&yrl=1940

 

[WannabeNewton]

Perhaps a focus on mechanics may not help---and might be a turn-off to---many female students. They typically lack the experience and familarity with mechanics that males accept as second nature.

Is this meant to be a sarcastic remark or...? Because if not I am at a loss for words. This sounds more like the title of an Onion article.

 

[axmls]

Spivak's Calculus is the least dry textbook I know in my opinion, and it has exactly zero real-world applications that I know of mentioned in the book. Arguably, it's more of an analysis book than an intro calculus book, but different people learn differently. Mechanics might be less intuitive to some people than calculus.

 

[Loststudent22]

My professor just mentioned they don't have nearly enough time to show you what its all good for but only to teach the mechanics of it for itself. If you want to know the applications for it that information is easily available.

 

[symbolipoint]

Perhaps a focus on mechanics may not help---and might be a turn-off to---many female students. They typically lack the experience and familarity with mechanics that males accept as second nature.

Maybe this depends on what type or level of "mechanics" and on the nature of the particular female. Not too necessary to discount the understanding that any female has for mechanical things.

 

[russ_watters]

There's nothing wrong with mechanics but what's so special about it wrt calculus besides history ?

I found calculus to be the easiest of my high school math classes and I credit the fact that I took physics at the same time - and the two became linked - for a lot of that. The practical application helped me understand the relationship between the operations and added back-check capabilities for the problems. Whether it is essential for it to be Newtonian mechanics or not, it worked-out that way because of my school's course chronology, but I think the fact that their histories were tied together probably helped as well.