filling in the gaps

One of the biggest disadvantages of having taken off a number of years between my Masters and PhD projects is that I had time to forget much of what I had learned in the way of mathematics and its application to the other sciences (use it or lose it!). As a result I have found myself skipping over the formulas and calculations presented in many of the papers I’ve read. This is, of course, a very bad habit, because if one doesn’t think about what those various symbols mean one doesn’t understand what the author of the paper was attempting to communicate.

Sometimes the text contains enough detail that one can manage without that bit of information; other times they let the equation stand alone as the most precise way to express the relationships in question. One solution I’ve used in the past to help understand equations given in a paper is to create an Excel spreadsheet in which I can set up formulas which point to other cells into which I can enter their numeric data to see if I can get the formulas to yield the same results that the authors reported.

However, this technique doesn’t lend itself very well to understanding general equations for which we have no actual numbers to substitute into the equations to solve them. Today I decided to set up a Word document to help me dissect and understand the formulas. The format I’ve decided upon is to use the Outline view mode with the first rank of the outline giving the name/number of the equation and the source paper in which I found it (e.g. Equation (2) from (Tirone and Ganguly, 2010*). Then, below that heading I add a paragraph which gives the equation (e.g. r_i = k₁√D(t-t_n(i))). Below that I add a series of level-two headers for (1) what the equation means in plain English (e.g. “The crystal radius is equal to the quantity of a dimensionless constant times the square root of the coefficient of intergranular diffusion through the matrix of the growing crystal times the amount of time which has elapsed since the crystal started growing.”) followed by (2) a “because” header which lists the definition of each of the variables

In this example:

r = crystal radius

_i = specific (growing) crystal in question

k₁ = a dimensionless constant

D = coefficient of intergranular diffusion through the matrix of the growing crystal (_i)

t = time

t_n(i) = the nucleation time of the crystal

Followed by (3) a summary of what one would use the equation to do (e.g. Calculate the relationship between the size of the crystals and how long it took them to grow.)

Setting this up and filling it in for the equations I’ve encountered in today’s 1000 words of geologic literature has really underscored my need to refresh my memory about basic chemistry. The equation (3) from the same paper as the above example was no where near as easy to fill in, because the authors first stated “The diffusion coefficient is expressed as a function of temperature according to the Arrhenius relation D=D_oexp(−E/RT)” after which they gave a new equation created by substituting the Arrhenius equation into their equation (2). In the text which followed they did not define any of the new terms which come from the Arrhenius equation. Presumably because the reader is assumed to already be familiar with said equation.

Therefore I took down my undergraduate Chemistry text book and looked up the Arrhenius equation. I found it necessary to read a couple of sections leading up to that equation as well as the section in which it is introduced in order to feel like I understood what the textbook was describing. Next I will need to figure out how the version of the equation the authors of the paper presented relates to the version in my textbook.

Is any of this really necessary? Could I just go through life not really understanding the equations I see and just jump to the part of the article where the authors describe what the results of their calculations mean? Perhaps I could. However, I think I will be a better scientist as a result of my going back and filling in these gaps in what I retained from my undergraduate education as I find them.

*Tirone, M. and Ganguly, J. (in press (downloaded Feb 2010)). "Garnet compositions as recorders of P-T-t history of metamorphic rocks." Gondwana Research.