Lily Droesch, Madison Ward, Matt Lane

   Mathematical Models

   Your project paper on the basic ideas of mathematical modeling is 
   reasonably good, but I think your oral presentation was stronger.  The
   major comments I have here are that it's not a good idea to try to
   include discussions of technical details from a source when they 
   go beyond what (it appears) you really understand (or what we have discussed
   in class).  See comment 5b below for more specifics here.  Also, the 
   paper does not have an effective overall conclusion.  It just sort of 
   "peters out" without tying together the points you have made.  I know it 
   can be hard to do that when you are "writing by committee." But the paper 
   as a whole could really have used a more focused summation at the end.  

   Your use of sources, citations, etc. are mostly correct, but see comment
   5a below

   Specific comments:

   1)  Page 1:  "The process of these models go in is a loop" -- that sentence
       does not really make sense (better without the "of" toward the beginning).
       But you also really need to explain what you mean here in more detail.  
       See suggestion on the hardcopy of the paper. 

   2)  Page 2:  I'm not really sure you understood the point of Wigner's
       "Unreasonable Effectiveness" paper.  He's saying that things developed
       within mathematics, without any reference to "real-world" problems
       often turn out to be useful later on even so.  In other words, mathematics
       is often useful even when it was only developed out of a sense of the 
       structure of the ideas involved.  The next underlined sentence is not
       clear -- don't know what you meant there.  Was something missing?

   3)  Page 3:  Exponential functions.  There are some typesetting issues
       in your formula f(x) = a . bx (the x needs to be shown as a superscript -- 
       above the main line of the formula).  Also, it's not the size of the 
       population that is proportional to the birth and death rates.  
       According to the most basic model, the rate of change of the population
       is proportional to the size of the population, and the constant of 
       proportionality is the difference between the the birth and death rates.

   4)  Bottom of Page 3:  Facts about the equations defining exponential 
       functions are standard general knowledge.  You don't need to attribute them
       to me, or to anyone else.  

   5)  Pages 4 and 5: The discussion of the Gao and Keinan study might have made 
       a nice addition to the topics we discussed in class.  However, there are
       two major problems:

       a)  I cannot identify which item in your list of references corresponds
           to this.  Is it listed?

       b)  More importantly, I think you tried to include too much technical
           detail from that source in what you wrote without really understanding
           what was going on.  Summarizing a technical discussion in a paper
           like this is hard because you need to convince the reader that you 
           know what's going on and that you have your own "take" on what is being
           discussed.  I believe Gao and Keinan's idea is that evidence from 
           human DNA ("genomic" evidence) shows that human populations have grown
           faster than exponentially.   In other words, they are saying that 
           a model that gives even faster growth is necessary to describe how
           human populations have changed over time.  The techniques they use involve some
           "heavy-duty" statistics, though.  The "p-values" that you are throwing
           in a couple of times would be a standard way to quantify how strong
           the evidence is for making a certain claim.  But we have not discussed
           that yet, and I'm not convinced from what you wrote that you really 
           understand that aspect of what they were saying, either.  So it would
           have been (much) better not to try to discuss what they said at this
           level of detail.  

  6)  Page 6:  Since C14 and N14 have the same atomic weight but N14 has a higher
      atomic number (number of protons), the process of radioactive decay of C14 
      is slightly unusual and it would have been good to say a bit more.  In effect, 
      one of the neutrons in a C14 nucleus is decaying to a proton by emitting 
      radiation.  (Most radioactive decay goes from a heavier "parent" nucleus to
      lighter "daughter" nuclei.) 

  7)  Some worked out examples of the use of radiocarbon or uranium/lead dating would
      have made a nice addition here.  You could illustrate all the features of 
      working with exponential models and solving equations using logarithms by
      doing that.  

  Grade:  90 (A-)