Pais, John

Mathematics Department

Saint Louis College of Pharmacy

4588 Parkview Place

Saint Louis, MO 63110

paisj@medicine.wustl.edu



Elementary Functions via Dynamic Visualization



Arguably the most important single concept of twentieth century mathematics 

is the concept of function. Regarding the understanding of functions, an 

important insight is that any given function has several human-readable 

representations including a graphical representation, a computational 

representation, and a symbolic representation. However, since most 

functions ordinarily dealt with in mathematics are infinite (and hence 

abstract) objects, in general, no single human-readable representation nor 

combination of human-readable representations can provide a complete 

representation of such a function. By their very nature human-readable 

representations are finitary and as such can provide only a partial 

realization of any infinite abstract object. This means that in order to 

achieve mastery of the function concept it is essential that human learners 

acquire all available methods of representation together with the ability 

to fluidly move among them depending on the problem situation that presents 

itself. In fact, it is precisely this sort of mastery that constitutes the 

thrill of understanding a mathematical abstraction that apparently 

outstrips the finitary capabilities of the learner.



This multi-representation mastery is essential for understanding the 

abstract nature of the function concept and is simultaneously the source of 

the difficulty that makes this understanding a formidable task for the 

learner. The primary learning outcome for this interactive text is for the 

learner to obtain multi-representation mastery of the function concept in 

general, and of elementary functions (Courant and John) in particular. It 

is the intention of the author to focus narrowly on the concept of 

function, extracting various "function fibers" from topics in precalculus, 

calculus, and elementary physics. So, in as much as possible, the primary 

perspective is to explicitly view everything in sight as a function. The 

approach is to use Maple to create an interactive text emphasizing dynamic 

visualization (animation), guided discovery, and gradual abstraction. In 

addition, a guiding theme is to give conceptual precedence to visual 

representations, whenever possible.



 



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