Intuiting Mathematical Objects using Kinetigrams - Kinetigrams: Absorption Models I

John Pais

Next, we have a kinetigram definition for a situation in which both absorption and excretion processes are modeled. In contrast to our definitions above, the dissolution process is not directly modeled, however it is accounted for in terms of a simplifying assumption. (Please be patient while the 218 KB animated gif in Panel 1 loads.)

Kinetigram Definition 2:  Panels 1 & 2
     Panel 1
          Panel 2

This kinetigram analytically specifies a first-order absorption model as a linear dynamical system by means of the rate equations, while only implicitly specifying the solution functions for a particular model with given rate constants k1 and k2. Instead of first launching into an explicit derivation of these solution functions, the learner can immediately see several helpful instances of variation. This order of presentation permits one to begin constructing an intuitive mental picture of how a set of three solution functions (the mathematical objects under development) might look. Based upon changes in the analytic specification of the rate equations, the corresponding variation from solution set to solution set can be seen. With this initial understanding in mind, it seems that now the learner is in a position to be very curious about an explicit analytic description of the solution functions, and in particular, how the graphs of these solution functions look and relate to the information already gleaned from the kinetigram.

Once formulas for the solution functions are in hand (see below), one can illustrate the interplay between the physical situation and the mathematical model by asking students, "What happens if the rate constants are the same for both the first and second reactions?" Evidently, if k1 = k2 then we can't use the solution functions described by these formulas. What does this mean physically? Is the mathematics telling us something interesting about the science or vice versa?

Apparently, the experienced mathematician looks at the rate equations and/or the solution functions and automatically intuits (sees) instances of variation like those in Kinetigram Definition 2, Panel 1. But how does the student inexperienced in mathematical visualization develop such a personal intuitive skill? Using currently available technology, we can create intuitively oriented representations, such as kinetigram definitions, that can rapidly stimulate inexperienced learners to begin generating their own mathematical visualizations to aid in the immediate intuitive unfolding of the mathematical objects under consideration.

Here is a Java version of Kinetigram Definition 2,
Panels 1 and 2:
Absorption Models I