updating for table of contents improvement

ch3.tex

@@ -180,7 +180,7 @@ In this way, the meaning is socially constructed.
 
 
 
-\section{Perspective and Epistemology from Mathematics Education}
+\section{Heritage from Mathematics Education}
 Part of the research perspective is formed by the goals for what students learning proof should know: according to Ball et al.\cite[p, 32 -- 34]{loewenberg2003mathematical} "These activities -- mathematical representation, attentive use of mathematical language and definitions, articulated and reasoned claims, rationally negotiated disagreement, generalizing ideas, and recognizing patterns -- are examples of what we mean by \textit{mathematical practices}. \dots These practices and others are essential for anyone learning and doing mathematics proficiently. \ldots investing in understanding these 'process' dimensions of mathematics could have a high payoff for improving the ability of the nations' schools to help all students develop mathematical proficiency".
 Ball goes on to say\cite[p. 37]{loewenberg2003mathematical} "Another critical practice -- the fluent use of symbolic notations -- is included in the domain of representational practice. Mathematics employs a unique and highly developer symbolic language upon which many forms of mathematical work and thinking depend. Symbolic notation allows for precision in expression. It is also efficient -- it compresses complex ideas into a form that makes them easier to comprehend and manipulate. Mathematics learning and use is critically dependent upon one's being able to fluently and flexibly encode ideas and relationships. Equally important is the ability to accurately decode what others have written."
 

ch4.tex

@@ -35,8 +35,10 @@ though some graduate students also volunteered. Some students were
 domestic, and some international. Some students were African-American,
 some Asian, some Caucasian, some Latino/a, some with learning disabilities
 such as being diagnosed as on the autistic spectrum.
-\subsection{Proofs Using the Pumping Lemma for Regular Languages}
-In a recent course offering to forty-two students, of whom thirty-four were men and eight women,
+%\subsection{Proofs Using the Pumping Lemma for Regular Languages}
+The participants for the study of proofs using the pumping lemma for regular languages were
+%In a recent course offering to 
+forty-two students, of whom thirty-four were men and eight women,
 forty-one traditional aged, 
 %one former Marine somewhat older, one collegiate athlete (a
 %woman),
@@ -48,8 +50,12 @@ lemma. Half the students (21/42) selected this problem. These were 17 men
 and 4 women. Three quarters of those (15/42) selecting the pumping lemma
 got it wrong. These students, who chose the pumping lemma problem and
 subsequently erred on it, form the population of our study.
-\subsection{Proofs by Mathematic Induction}
-We studied students who were taking, or who had recently taken, a course
+%\subsection{Proofs by Mathematic Induction}
+The participants for the study of proof by mathematic induction 
+%We studied students who 
+were taking, or
+% who
+ had recently taken, a course
 on Discrete Systems required of all computer science, and computer science and
 engineering students.
 Volunteers were solicited from all students attending the Discrete Systems
@@ -58,8 +64,10 @@ Interviews of eleven students were transcribed for this study. Participants
 included 2 women and 9 men. Two were international students, a third was a
 recent immigrant.
 
-\subsection{Domain, Range, Mapping, Relation, Function, Equivalence in Proofs}
-Students taking, or having taken, discrete systems, especially students who
+%\subsection{Domain, Range, Mapping, Relation, Function, Equivalence in Proofs}
+For the study about domain, range, mapping, relation, function and equivalence in proofs, students
+%Students 
+taking, or having taken, discrete systems, especially students who
 had sought help while taking introductory object oriented programming volunteered
 to be interviewed.