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\title{Transformation-based Geometry\\Through Complex Numbers: \\Teaching a Capstone Course\\for Future High School Teachers}
\author{Helmut Knaust\\Department of Mathematical Sciences\\The University of Texas at El Paso}
%{\tt{hknaust@utep.edu}}}
\date{January 11, 2025}
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\begin{document}
\begin{frame}
\titlepage
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\begin{frame}
\begin{center}Overview\end{center}
\tableofcontents[hideallsubsections]
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\section{UTEP}
\begin{frame}
\begin{columns}
\column{6cm} 
UTEP and Its Students\bigskip
\begin{itemize}
\item 25,000+ students
total
\item $\sim$ 3,700 graduate students
\pause\item 84\% Hispanic
\pause\item UTEP has Carnegie-R1 status
\end{itemize}
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\begin{figure}
\includegraphics[height=5cm]{files/UTEP.png}
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\section{Course Design and Contents}
\begin{frame}
\begin{center}\textbf{Fundamental Mathematics from an Advanced Standpoint}\vspace{-1em}\end{center}
\begin{itemize}
\item<1->  The only course in our department taken by future secondary Mathematics teachers that specifically addresses High School Mathematics.
\item <2-> The prerequisite is an \textit{Introduction to Proof} class. Some students may have taken a number theory and/or geometry class.
\item <3-> The course is almost exclusively taken by juniors and seniors.
\item <4-> The vast majority of students will be student-teaching within a year.
\end{itemize}
\end{frame}

\begin{frame}
\begin{center}\textbf{Course Contents}\end{center}
\begin{itemize}
\item From the counting numbers to the real numbers
\item \textbf{Complex numbers}
\item Functions
\item Algebra (= Solving Equations)
\end{itemize}
\end{frame}

\subsection{Advanced Standpoint}
\begin{frame}
\begin{center}\textbf{Fundamental vs. Advanced Standpoint\\Example: Complex Numbers}\end{center}
\begin{itemize}
\item <1->In what context have students seen complex numbers?
\begin{itemize}
\item <2->To solve quadratic equations 
\item <3->To solve certain systems of linear differential equations
\end{itemize} 
\item<4-> Advanced standpoint:
\begin{itemize}
\item<5->$\cpl$ is a field of numbers just like $\real$, albeit without a nice order
\item<6->$\cpl$ is algebraically closed: The Fundamental Theorem of Algebra
\item<7->The complex plane
\end{itemize}
\end{itemize}
\end{frame}

\section{Conceptual Framework}
\subsection{Wu's Principles}
\begin{frame}
\begin{center}\textbf{Basic Characteristics\\of a ``Working'' Mathematics Classroom\\by H. Wu}\end{center}
\begin{enumerate}
\pause\item \textbf{Precision:} Mathematical statements are clear and unambiguous. At 
 any moment, it is clear what is known and what is not known.
\pause\item \textbf{Definitions:} They are the bedrock of the mathematical structure.
They are the platform that supports reasoning. No definitions, no
mathematics.
\end{enumerate}
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\begin{frame}
\begin{center}\textbf{Basic Characteristics of a Mathematics Classroom\\by H. Wu} II\end{center}
\begin{enumerate}\addtocounter{enumi}{3}
\item \textbf{Reasoning:} The lifeblood of mathematics. The engine that drives
problem solving. Its absence is the root cause of teaching---and learning---by rote.
\pause\item \textbf{Purposefulness:} Mathematics is goal-oriented, and every concept
or skill is there for a purpose. %Mathematics is not just fun and games.
\pause\item \textbf{Coherence:} Mathematics is a tapestry in which all the concepts and
skills are interwoven. It is all of a piece.
\end{enumerate}
\end{frame}

\section{Complex Numbers and Geometry}
\begin{frame}
\begin{center}\textbf{An Example of ``Coherence'' --\\The First Baby Step of the Erlangen Program}\end{center}
\begin{description}
\item[Definition.] A function $f:\cpl\to\cpl$ is called an \textit{isometry} if it satisfies
\[|f(w)-f(z)|=|w-z| \mbox{ for all } w,z\in\cpl.\]
\pause
\item<2>[Examples.]???  
\item<3>[Examples.]Translations, ???, Rotations, Reflections.
\end{description}
\end{frame}

\begin{frame}
\begin{center}
\textbf{Erlangen Program II}
\end{center}
\begin{description}
\item[Theorem.] Let $f:\cpl\to\cpl$ be an arbitrary isometry. Then there exist $z_0\in\cpl$ and $\theta\in\real$ such that $f(z)=z_0+z e^{i\theta}$ or $f(z)=z_0+\overline{z} e^{i\theta}$. \medskip
\end{description}
\pause\emph{Starting  with an arbitrary isometry $f$, the isometry $g(z)=f(z)-f(0)$ has 0 as a fixed point. \pause It follows from the isometry condition that then $|g(1)|=1$, and so for a suitable rotation around the origin, the isometry $h(z)=g(z)e^{-i\theta}$ satisfies $h(0)=0$ and $h(1)=1$.} 
\end{frame}

\begin{frame}
\begin{center}
\textbf{Erlangen Program III}
\end{center}
\begin{description}
\item[Lemma.] Let $h:\cpl\to\cpl$ be an isometry that satisfies $h(0)=0$ and $h(1)=1$. Then $h(z)=z$ or $h(z)=\overline{z}$.\medskip
\end{description}
\pause\emph{
Reversing the steps on the previous slide we obtain the desired result.}
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\section{$\cpl$ as a Set of Matrices}
\begin{frame}
\begin{center} 
\textbf{Multiple Representations of $\cpl$}
\end{center}
\begin{itemize}
\item Algebraic representation as numbers of the form $a+b i$
\item Complex numbers in polar form: $z=r e^{i\theta}$
\item The complex plane 
\item The Riemann sphere via stereographic projection
\item The complex numbers as certain real $2\times 2$ matrices
\end{itemize}
\end{frame}

\begin{frame}
\begin{center}
\textbf{An Example of ``Purposefulness'' --\\$\cpl$ as real $2\times 2$ Matrices}
\end{center}
\begin{description}
\item[Theorem.] The matrices of the form 
\[
\left(\begin{array}{rr} a&b\\-b&a\end{array}\right) \] (with $a,b\in \real$)
form a field isomorphic to $\cpl$.\medskip
\end{description}
\pause \emph{Here $1$ corresponds to the identity matrix and $\pm i$ corresponds to \ilf{\left(\begin{array}{rr} 0&\pm 1\\\mp 1&0\end{array}\right) }.}
\end{frame}

\begin{frame}
\begin{center}
\textbf{$\quat$ as Complex $2\times 2$ Matrices}
\end{center}
\begin{description}
\item[Theorem.] The matrices of the form 
\[
\left(\begin{array}{rr} z&-\overline{w}\\w&\overline{z}\end{array}\right) \] (with $z,w\in \cpl$)
form a 4-dimensional (over $\real$) division algebra, known as $\quat$, the set of \textbf{quaternions}.\medskip
\end{description}
\pause \emph{Here the basis elements are $\vv{1}$, \ilf{\vv{i}=\left(\begin{array}{rr} 0&-1\\1&0\end{array}\right) } ,  \ilf{\vv{j}=\left(\begin{array}{rr} i&0\\0&-i\end{array}\right) }, and  \ilf{\vv{k}=\left(\begin{array}{rr} 0&i\\i&0\end{array}\right) }.}
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\section*{References}
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\begin{center}
\textbf{References}
\end{center}
\begin{enumerate}
\item Hung-Hsi Wu, \emph{The Mis-Education of Mathematics Teachers}. Notices of the AMS 58 (2011). 
\item Matthew Winsor, \emph{One Model of a Capstone Course for Preservice High School Mathematics Teachers}. Primus 19 (2009), pp.~510--518.
\item Tristan Needham, \emph{Visual Complex Analysis}, Clarendon Press, Oxford (1997).
\item Heinz-Dieter Ebbinghaus, et al. \emph{Zahlen}, Springer-Verlag, Berlin (1983).
\end{enumerate}
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\section*{Contact}
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\textbf{Contact}
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Helmut Knaust:\medskip\\{\tt hknaust@utep.edu}\\\href{http://helmut.knaust.info}{\tt http://helmut.knaust.info}\medskip\\
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