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\title{Constructing the Real Numbers IBL-Style}
\institute{{\large Helmut Knaust}\medskip\\Department of Mathematical Sciences\\The University of Texas at El Paso\medskip\\\tt{hknaust@utep.edu}}
\date{{\scriptsize April 8, 2022}}
\newcommand{\field}[1]{\mathbb{#1}}
\newcommand{\nat}{\field{N}}
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\begin{document}
\begin{frame}
\titlepage
\end{frame}
\section{Introduction}
\subsection{UTEP}
\begin{frame}
\begin{columns}
\column{5cm} 
UTEP and Its Students\bigskip
\begin{itemize}
\item $\sim$25,000 students
total
\item Close to 3,000 graduate students
\item 83\% Hispanic
\item UTEP has Carnegie R1 status
\end{itemize}
\column{4cm}
\begin{figure}
\includegraphics[height=7cm]{UTEP.png}
\end{figure}
\end{columns}
\end{frame}


\subsection{Closed Loop}
\begin{frame}
The Closed Loop
\begin{columns}
\column{4cm} 
\begin{itemize}
\item The overwhelming majority of UTEP's students graduate from local public schools. 
\item A large majority of local teachers are UTEP alumni.
\end{itemize}
\column{4cm}
\begin{figure}
\includegraphics[height=4cm]{UTEPmap.png}
\end{figure}
\end{columns}
\end{frame}

\subsection{MATM}

\begin{frame}
\textbf{Master of Arts in Teaching Mathematics}
\begin{itemize}
\item<1-> For high school teachers with some teaching experience (three years or more preferred).
\item<2-> Teachers usually take two courses per semester --- taught during two evenings from 5--8 p.m. 
\item<3-> Participants take 18 graduate hours of Mathematics Education courses --- taught in our department. 
\item<4-> Participants take 18 graduate hours of Mathematics courses --- designed for MATM students.
\end{itemize}
\end{frame}

\begin{frame}
\textbf{My Contributions to the MATM Program}
\begin{itemize}
\item<1-> ``Technology in the Mathematics Classroom,'' a math education course, alternating with two other faculty members
\item<2-> An ``exemplary'' applied mathematics course on Discrete Wavelet Transforms
\item<3-> A ``second'' Analysis course using Inquiry-based Learning
\end{itemize}
\end{frame}

\section{Topics in Advanced Calculus}
\subsection{The course}
\begin{frame}
\textbf{Topics in Advanced Calculus}
\begin{itemize}
\item<1-> Prerequisite: A first undergraduate Analysis course --- Analysis on the real line (using Gaughan or Abbott) 
\item<2-> Topic: The axiomatic development of the real number system
\item<3-> Method: A Moore-style theorem sequence
\end{itemize}
\end{frame}


\begin{frame}
\textbf{Why numbers?}
\begin{itemize}
\item<1-> An accessible topic 
\item<2-> Numbers feature prominently in the school curriculum
\item<3-> An example of working with an axiomatic system
\end{itemize}
\end{frame}


\begin{frame}
\textbf{Why IBL?}
\begin{itemize}
\item<1-> The only course in the graduate program to use IBL formally
\item<2-> As a model: Give teachers a chance to explore. If they like it, they will let their students explore... 
\end{itemize}
\end{frame}

\subsection{Natural Numbers}
\begin{frame}
\textbf{The axiomatic system}

Richard Dedekind
started by giving the following definition of the set of
\textbf{Natural Numbers}.

\begin{block}{Axioms for $\nat$}
The natural numbers are a set $\nat$ containing a special
element called $0$, and a function $S:\nat\to\nat$ satisfying the
following axioms:
\begin{description}
\item[D1]<2-> $S$ is injective.
\item[D2]<3-> $S(\nat)=\nat\setminus \{0\}$.
\item[D3]<4-> If a subset $M$ of\/ $\nat$ contains 0 and satisfies
$S(M)\subseteq M$, then $M=\nat$.
\end{description}
The function $S$ is called the successor function.
\end{block}
\end{frame}

\begin{frame}
\textbf{First steps}

After letting students prove the \textbf{Principle of Induction}, \textbf{addition} is defined recursively:\medskip

\begin{block}{Addition on $\nat$}
For a fixed but arbitrary $m\in\nat$ we define
\begin{eqnarray*}
m+0&:=&m\\
m+S(n)&:=&S(m+n) \mbox{\qquad for all  }n\in\nat
\end{eqnarray*}
\end{block}
\end{frame}

\begin{frame}
The standard \textbf{arithmetic properties} of the natural numbers are established. \medskip

One target theorem  or this section is the \textbf{totality }of the order: 
\begin{block}{The Order on $\nat$ is total}

\[\mbox{For all }m,n\in\nat,\  m\leq n \mbox{ or } n\leq m.\]
\end{block}
\end{frame}


\begin{frame}
The \textbf{general recursion principle} is placed at the end of the section on natural numbers:

\begin{block}{Recursion Principle}
Let $A$ be an arbitrary set, and let $a\in A$ and a function
$f:A\to A$ be given. Then there exists a unique map
$\varphi:\nat\to A$ satisfying \begin{enumerate}
\item $\varphi(0)=a$, and \item $\varphi\circ S=f\circ\varphi$.
\end{enumerate}
\end{block}
\end{frame}

\begin{frame}
The students then use the recursion principle to prove, among other results,  the essential \textbf{uniqueness of the natural numbers}:

\begin{block}{Uniqueness of $\nat$}
Suppose that $\nat$, $S:\nat\to\nat$ and $0$ satisfy
Axioms (D1)--(D3), and that $\nat'$, $S':\nat'\to\nat'$ and $0'$
satisfy Axioms (D1)--(D3) as well.

Then there is a bijection $\varphi:\nat\to\nat'$ such that
\begin{enumerate}\item $\varphi(0)=0'$, and
\item $\varphi\circ S=S'\circ\varphi$.
\end{enumerate}
\end{block}
\end{frame}

\subsection{From the Natural Numbers to the Reals}
\begin{frame}
The course spends quite some time on the development of the \textbf{integers} $\zet$ as well, but does not spend much time on the set of \textbf{rational numbers} $\rat$.\medskip

The crucial definition for the construction of the real numbers  is \textbf{Dedekind cut}:\medskip

\begin{block}{Dedekind Cut}
A partition $(L,U)$ of $\rat$ is called a \textit{Dedekind cut},
if the following properties hold:
\begin{enumerate}
    \item If $a\in L$ and $b\in U$, then $a<b$.
    \item $U$ has no minimal element.
\end{enumerate}
\end{block}
The real numbers $\real$ is then defined as the set of all Dedekind cuts.
\end{frame}

\subsection{The Completeness of the Set of Real Numbers}
\begin{frame}
While establishing the arithmetic properties of Dedekind cuts is unpleasant (and only partially established by the students), defining the order on Dedekind cuts is \textbf{simple},

\begin{block}{Order on Dedekind cuts}
\[(L_1,U_1)\leq (L_2,U_2) \quad\mbox{ if }\quad L_1\subseteq L_2.\]
\end{block}
and the target theorem of this section has a \textbf{``beautiful''} proof:
\begin{block}{Completeness of $\real$}

Show that $\real$, the set of all Dedekind cuts, satisfies the following property:
All subsets of $\real$ that are bounded from above have a least upper bound.
\end{block}
\end{frame}

\begin{frame}
The Heroes
\begin{columns}
\column{4cm} 
\begin{center}
\begin{figure}
\includegraphics[width=4cm]{ELandau.png}
Edmund Landau (1877-1938)
\end{figure}
\end{center}
\column{4cm}
\begin{center}
\begin{figure}
\includegraphics[width=4cm]{Dedekind.jpg}
\end{figure}
Richard Dedekind (1831-1916)
\end{center}\end{columns}

\end{frame}

\section{Observations}
\begin{frame}
\begin{itemize}
\item<1-> The teachers are struggling with the material; often they need more than one attempt at the blackboard. 
\item<2-> If there are no more solutions to present during a class period, students work in pairs on assigned problems.
\item<3-> All teachers are improving substantially throughout the semester.
\item<4-> Each teacher proves at least one not-so-easy problem.\bigskip
\item<5-> The material takes up 2/3 to 3/4 of a standard long semester. 
\end{itemize}
\end{frame}

\subsection{Remote Learning}
\begin{frame}
In Spring 2021 the course moved online. 
\begin{itemize}
\item<1-> The students presented their solutions on an asynchronous bulletin board with basic \LaTeX\ support.
\item<2-> The normal class time was used for online office hours.
\item<3-> Every student could only start two problems per week.
\item<4-> This is a viable alternative to teaching this course in person. 
\end{itemize}


\end{frame}


\section{References}
\begin{frame}
References:
\begin{itemize}
\item Stephen Abbott: Understanding Analysis. Springer, 2nd edition (2015) 
\item Heinz-Dieter Ebbinghaus, et al.: Zahlen, Springer (1983)
\item  Edward D. Gaughan: Introduction to Analysis, AMS, 5th edition (2009)
\item Edmund Landau: Grundlagen der Analysis. Chelsea, 4th edition (1965)
\end{itemize}\medskip
The complete \emph{Theorem sequence}\footnote{\includegraphics[width=1cm]{cc.png} 
This work is licensed under a {\href{http://creativecommons.org/licenses/by/4.0/}{Creative Commons Attribution 4.0 International License}.}} is available  from {\href{mailto:hknaust@utep.edu}{\texttt{hknaust@utep.edu}}, or directly at  {\href{http://helmut.knaust.info//presentations/AnalysisII.pdf}
{\texttt{helmut.knaust.info/presentations/AnalysisII.pdf}}}}.\bigskip
\end{frame}
\end{document}
