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\title{A Capstone Course \\for \\Future High School Mathematics Teachers}
\institute{{\large Helmut Knaust}\medskip\\Department of Mathematical Sciences\\The University of Texas at El Paso\\El Paso TX 79968-0514\medskip\\\tt{hknaust@utep.edu}}
\date{{\small Roswell NM\\May 20, 2011}}
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\begin{document}
\begin{frame}
\titlepage
\end{frame}

\begin{frame}
\begin{center}Overview\end{center}
\tableofcontents[hideallsubsections]
\end{frame}

\section{Conceptual Framework}
\subsection{Wu's Principles}
\begin{frame}
\begin{center}Basic Characteristics Important for K--12 Math Teaching\\by H. Wu\end{center}
\begin{itemize}
\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.
\pause\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.
\end{itemize}
\end{frame}

\begin{frame}
\begin{center}Basic Characteristics Important for K--12 Math Teaching\\by H. Wu\end{center}
\begin{itemize}
\pause\item \textbf{Coherence:} Mathematics is a tapestry in which all the concepts and
skills are interwoven. It is all of a piece.
\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.
\end{itemize}
\end{frame}

\subsection{Common Core State Standards}
\begin{frame}
\begin{center}Common Core State Standards:\\
Mathematical Practices (for Pupils)\end{center}
\begin{enumerate}
\item Make sense of problems and persevere in solving them.
\item Reason abstractly and quantitatively.
\item Construct viable arguments and critique the reasoning of others.
\item Model with mathematics.
\item Use appropriate tools strategically.
\item Attend to precision.
\item Look for and make use of structure.
\item Look for and express regularity in repeated reasoning
\end{enumerate}
\end{frame}

\section{Course Structure}
\subsection{Textbooks}
\begin{frame}
\begin{figure}
\includegraphics<1>[width=4cm]{files/MfHST.jpg}
\end{figure}
\begin{center}
Zalman Usiskin, Anthony L. Peressini,\\Elena Marchisotto, and Dick Stanley:\smallskip\\Mathematics for High School Teachers.
\end{center}
\end{frame}

\begin{frame}
\begin{figure}
\includegraphics<1>[width=4cm]{files/MSST.jpg}
\end{figure}
\begin{center}
Elizabeth G. Bremigan, Ralph J. Bremigan, and John D. Lorch:\smallskip\\
Mathematics for Secondary School Teachers
\end{center}
\end{frame}

\begin{frame}
\begin{figure}
\includegraphics<1>[width=3cm]{files/FK.jpg}
\end{figure}
\begin{center}
Felix Klein:\smallskip\\Elementary Mathematics from an Advanced Standpoint\\
Arithmetic, Algebra, Analysis\smallskip\\\emph{First published in 1908}
\end{center}
\end{frame}

\subsection{Course Content}
\begin{frame}
\begin{columns}[c]
\column{4cm}
\includegraphics[width=4cm]{files/Alice3.jpg}
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\begin{center}
Course Content:
\end{center}
\begin{itemize}
\pause\item From the Natural Numbers to the Real Numbers
\pause\item Complex Numbers
\pause\item Functions
\pause\item Equations and Inequalities
\end{itemize}
\pause
\begin{center}
Prerequisite:
\begin{itemize}
\item Introduction to Logic and Proof
\end{itemize}
\end{center}
\end{columns}
\end{frame}

\section{Wu's Principles}
\subsection{Precision}
\begin{frame}
\begin{center}
Wu's Principles and Student Beliefs:\\1. Precision 
\end{center}
\begin{itemize}
\item Students make progress during the course in understanding that they need to transition from being ``Math learners'' to ``Math talkers''.
\end{itemize}
\end{frame}


\begin{frame}
\begin{center}
Student Group Activity I
\end{center}
\begin{itemize}
\item Teach a Lesson
\pause\begin{itemize}
\item Each student group teaches 1--2 sections in the Equations chapter
\item Fellow classmates present homework solutions - lesson presenters critique the solutions. 
\item Duration: 50--80 minutes\smallskip
\pause\item Each group needs to complete a trial run with the instructor before teaching the lesson 
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}
\begin{center}
Student Group Activity II
\end{center}
\begin{itemize}
\item Final Paper \& Presentation
\pause\begin{itemize}
\item Countability of algebraic numbers 
\item Algebraic numbers form a field
\item	Stereographic projection 
\item Transcendence of the number $e$ (\`a la Felix Klein)
\item	Cardano-Tartaglia  method for solving cubic equations (incl.~\emph{Casus Irreducibilis}) 
\item Newton's Method
\end{itemize}
\end{itemize}
\end{frame}

\subsection{Definitions}
\begin{frame}
\begin{center}
Wu's Principles and Student Beliefs:\\2. Definitions
\end{center}
\begin{itemize}
\item Students do not understand the role of definitions in Mathematics and/or believe that definitions are not important for HS Mathematics. 
\item Students are not good as ``digesting'' definitions.
\end{itemize}
\end{frame}

\begin{frame}
\begin{center}
Example: Class Activity
\end{center}
\begin{itemize}
\item Define ``chair''.
\item Define ``prime number''.
\end{itemize}
\pause\emph{Subsequent discussion of the differences between the two definitions\ldots\\
Students say that mathematical definitions are more precise than other definitions; the words ``descriptive vs. normative'' do not surface.}
\end{frame}

\begin{frame}
\begin{center}
Example: Group Activity
\end{center}
\begin{itemize}
\item Recall that a \emph{relation on a set} $X$ is a subset of the Cartesian product $X\times X$.

We define a relation $\sim$ on $\nat\times\nat$ as follows:
\[(p,q)\sim(p',q') \Leftrightarrow p+q'=p'+q.\]
Show that $\sim$ defines an equivalence relation on $\nat\times\nat$.
\end{itemize}
\pause\emph{No student group notices that $X=\nat\times\nat$\ldots} 
\end{frame}

\subsection{Reasoning}
\begin{frame}
\begin{center}
Wu's Principles and Student Beliefs:\\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 do not understand that reasoning with definitions will be the only way for them to ``check'' the Mathematics they will be teaching. 
\end{itemize}
\end{frame}

\begin{frame}
\begin{center}
Example: Class Discussion\\(after discussing and practicing the Division Algorithm) 
\end{center}
\begin{itemize}
\item Instructor: Why do rational numbers have terminating or periodic decimal expansions?
\pause\item Students: Silence\ldots
\pause\item Students: Silence\ldots
\pause\item Student: Our teacher told us.
\end{itemize}
\pause \emph{Last semester was the first time that the instructor prompt ``How do you compute the decimal expansion of a rational number?'' enabled a couple of students to come up with a line of reasoning\ldots} 
\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\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 $\textcolor{red}{\Rightarrow}\ 4x^2-4x+1=x$
\item $\Leftrightarrow (4x-1)(x-1)=0$
\end{itemize}
\emph{Why is $x=1$ a solution, while $x=1/4$ isn't?} 
\end{frame}

\subsection{Coherence}
\begin{frame}
\begin{center}
Wu's Principles and Student Beliefs:\\4. 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}
\end{frame}

\begin{frame}
\begin{center}
Example: Algebra
\end{center}
\begin{itemize}
\item Being essentially an Algebra course, the course probably contributes to the problem.
\item The connection between ``Modern Algebra'' and ``HS~Algebra'' is stressed.
\end{itemize}
\end{frame}

\subsection{Purposefulness}
\begin{frame}
\begin{center}
Wu's Principles and Student Beliefs:\\5. 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}
\end{frame}

\begin{frame}
\begin{center}
Example: Questions for Discussion
\end{center}
\begin{itemize}
\pause\item What are real numbers good for? Where/when in the HS curriculum do we need real numbers?
\pause\item What were complex numbers invented for?
\pause\item Why are surjectivity and injectivity important properties of functions?
\end{itemize}
\end{frame}

\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}
\end{frame}

\section{References}
\begin{frame}
\begin{center}
References
\end{center}
\begin{enumerate}
\item Matthew Winsor, \emph{One Model of a Capstone Course for Preservice High School Mathematics Teachers}. Primus 19 (2009), pp.~510--518.
\item Hung-Hsi Wu, \emph{The Mathematics K--12 Teachers Need to Know}. \\{\small{\texttt{http://math.berkeley.edu/\~{}wu/Schoolmathematics1.pdf}}}
\item Hung-Hsi Wu, \emph{The Mis-Education of Mathematics Teachers}. Notices of the AMS 58 (2011). 
\item Jo Boaler, \emph{What's Math Got to Do with It? How Parents and Teachers Can Help Children Learn Their Least Favorite Subject}. Penguin 2008.
\end{enumerate}
\end{frame}
\end{document}0