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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}
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\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{Math 4303\\Fundamental Mathematics from an Advanced Standpoint}\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.
\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}
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\begin{frame}
\begin{center}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}
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\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 (polynomial?) equations 
\item <3->To solve certain systems of linear differential equations
\end{itemize} 
\item<4-> Advanced standpoint:
\begin{itemize}
\item<5->$\cpl$ is also another field of numbers, albeit without a nice order like $\real$
\item<6->$\cpl$ is algebraically closed
\item<7->The complex plane
\end{itemize}
\end{itemize}
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\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.
\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}\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.
\item \textbf{Purposefulness:} Mathematics is goal-oriented, and every concept
or skill is there for a purpose. %Mathematics is not just fun and games.
\item \textbf{Coherence:} Mathematics is a tapestry in which all the concepts and
skills are interwoven. It is all of a piece.
\end{enumerate}
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\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
\[|f(w)-f(z)|=|w-z| \mbox{ for all } w,z\in\cpl.\]
\item<1>[Examples.]???  
\item<2->[Examples.]Translations, ???, Rotations, Reflections.
\end{description}
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\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 $f(z)=z_0+\overline{z} e^{i\theta}$.
\pause\item[Lemma.] Let $h:\cpl\to\cpl$ be an isometry that satisfies $h(0)=0$ and $h(1)=1$. Then $h(z)=z$ or $h(z)=\overline{z}$.
\end{description}
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\subsection{Definitions}
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\begin{center}
Wu's Principles and Student Beliefs:\\\textbf{2. Definitions}
\end{center}
\begin{center}
Example: Group Activity - From $\nat$ to $\zet$
\end{center}
\pause\begin{itemize}
\item Recall that a \emph{relation on a set} $X$ is a subset of the Cartesian product $X\times X$.\\[\baselineskip]

We define a relation $\sim$ on $\nat\times\nat$ as follows:
\[(p,q)\sim(p',q') \Leftrightarrow p+q'=p'+q.\]
Show that $\sim$ is an equivalence relation on $\nat\times\nat$.
\end{itemize}
\pause\emph{No student group notices that $X=\nat\times\nat$\ldots} 
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\subsection{Reasoning}
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\begin{center}
Wu's Principles and Student Beliefs:\\\textbf{3. Reasoning}
\end{center}
\begin{itemize}
\item Based on their own HS experience(?), students will not expect mathematical reasoning from their pupils.
\pause \item Students need to understand that reasoning with definitions will be the only way for them to ``check'' the Mathematics they will be teaching. 
\end{itemize}
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\begin{frame}
\begin{center}
Example: Discussion of a Test Problem\\(after finishing the Equations chapter)
\end{center}
\begin{itemize}
\item Solve $\log_2{x}-\log_2(1+\sqrt{x})=-1$.
\pause\item $\displaystyle{\log_2 \frac{x}{1+\sqrt{x}}=-1}$
\pause\item $\displaystyle{\frac{x}{1+\sqrt{x}}=\frac{1}{2}}$
\pause\item $ 2x-1=\sqrt{x}$
\pause\item $ 4x^2-4x+1=x$
\pause\item $ (4x-1)(x-1)=0$
\end{itemize}
\pause \emph{Why is $x=1$ a solution, while $x=1/4$ isn't?} 
\end{frame}

\begin{frame}
\begin{center}
Example: Discussion of a Test Problem\\(after finishing the Equations chapter)
\end{center}
\begin{itemize}
\item Solve $\log_2{x}-\log_2(1+\sqrt{x})=-1$.\\
\pause\quad\textcolor{red}{Domain of the equation: $\real^+$}
\pause\item $\Leftrightarrow \displaystyle{\log_2 \frac{x}{1+\sqrt{x}}=-1}$
\item $\Leftrightarrow\displaystyle{\frac{x}{1+\sqrt{x}}=\frac{1}{2}}$
\item $\Leftrightarrow 2x-1=\sqrt{x}$
\pause\item $\textcolor{red}{\Leftrightarrow}\ 4x^2-4x+1=x$ $\textcolor{red}{\ \ \wedge\ \ 2x-1\geq 0}$
\pause \item $\Leftrightarrow (4x-1)(x-1)=0$ $\textcolor{red}{\ \ \wedge\ \ 2x-1\geq 0}$
\item $\Leftrightarrow x=1$
\end{itemize}
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\subsection{Purposefulness}
\begin{frame}
\begin{center}
Wu's Principles and Student Beliefs:\\\textbf{4. Purposefulness}
\end{center}
\begin{itemize}
\item Intrinsic motivation seems to be a completely new concept to students.
\item Their HS and college experience for the most part seem to have been mathematically ``purposeless''.  
\end{itemize}
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\begin{frame}
\begin{center}
Example: Questions During the Trial Run
\end{center}
\begin{itemize}
\pause\item What is the purpose of the material you present?
\pause\item Does the material you present tell a story?
\pause\item Why do you present this example? What purpose does it serve?
\end{itemize}
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\subsection{Coherence}
\begin{frame}
\begin{center}
Wu's Principles and Student Beliefs:\\\textbf{5. Coherence}
\end{center}
\begin{itemize}
\item Students do not seem to have been exposed much to the idea that ``Mathematics is a tapestry in which all the concepts and skills are interwoven''. 
\item Mathematics seems to have been taught to them as a collection of more or less unrelated mathematical subjects.  
\end{itemize}
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\begin{center}
Topic: Algebra
\end{center}
\begin{itemize}
\item Being essentially an Algebra course, the course may contribute to the problem.
\item The connection between ``Modern Algebra'' and ``HS~Algebra'' is stressed.
\end{itemize}
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\begin{center}
A Link between Complex Numbers and Geometry:\\Classification of Isometries in the Complex Plane
\end{center}
\begin{itemize}
\item \textbf{Theorem.} \emph{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 $f(z)=z_0+\overline{z} e^{i\theta}$.}
\end{itemize}
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\section*{References}
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\begin{center}
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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\begin{center}
\textbf{Contact}
\end{center}
Helmut Knaust:\medskip\\{\tt hknaust@utep.edu}\\\href{http://helmut.knaust.info}{\tt http://helmut.knaust.info}\medskip\\
Presentation:\\\href{http://helmut.knaust.info/ppapers/GandC.pdf}{{\tt http://helmut.knaust.info/ppapers/GandC.pdf}}\medskip\\
Isometry Handout:\\\href{http://helmut.knaust.info/ppapers/iso.pdf}{{\tt http://helmut.knaust.info/ppapers/iso.pdf}}
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