From e09cd770a38f77839a34eefc5a47a4b8eefa076b Mon Sep 17 00:00:00 2001 From: Elias Fierke Date: Tue, 14 Jul 2026 13:40:56 +0200 Subject: [PATCH] Einleitung verfasst --- content/Einleitung.tex | 35 ++++++++++++++--------------------- galton.pdf | Bin 499829 -> 499010 bytes 2 files changed, 14 insertions(+), 21 deletions(-) diff --git a/content/Einleitung.tex b/content/Einleitung.tex index aa755a2..3b13206 100644 --- a/content/Einleitung.tex +++ b/content/Einleitung.tex @@ -1,23 +1,16 @@ -\textbf{Wichtig:} Dieses Dokument hier soll nicht mehr als 30 Seiten umfassen (mehr bitte nur nach Absprache). -\\[1em] -% großer Abstand -Ziel des Seminars ist die Ausarbeitung eines \glqq rundum-sorglos-Pakets\grqq \ für eine Einheit in einem Matheclub, einer Mathe-AG oder einer ähnlichen Veranstaltung an einem Gymnasium. Unter einer Einheit ist nicht zwingend eine Doppelstunde zu verstehen, aber eine Doppelstunde soll mindestens mit Inhalt gefüllt werden. +Das Galtonbrett zählt zu den bekanntesten Demonstrationsmodellen der Wahrscheinlichkeitsrechnung. Eine Vielzahl kleiner Kugeln fällt durch ein regelmäßiges Gitter von Stiften, wobei an jedem Hindernis zufällig entschieden wird, ob die Kugel nach rechts oder nach links fällt. Das Galtonbrett verbindet aufgrund des entstehenden charakteristischen Musters trotz der Zufälligkeit jeder einzelnen Entscheidung auf anschauliche Weise Zufall und Gesetzmäßigkeit. + +Aus mathematischer Sicht bildet das Galtonbrett die Binomialverteilung ab, wobei sich die Verteilung mit zunehmender Anzahl an Stiftreihen einer Normalverteilung annähert. Das Verhalten steht in engem Zusammenhang mit dem zentralen Grenzwertsatz. Zur Beschreibung dieser Annäherung spielen außerdem die Fakultätsfunktion sowie die zugehörige Approximation durch die Stirling-Formel eine wichtige Rolle. + +Das Galtonbrett stellt einen hohen didaktischen Wert dar, indem es ein anschauliches Experiment zu Grundkonzepten der Stochastik und Analysis und deren Verknüpfung liefert. Das Modell eignet sich besonders für den Einsatz in der Sekundarstufe II in Mathe-AGs und -clubs, in denen über den Unterricht hinausgehende Inhalte behandelt werden können. Insbesondere die Verbindung von praktischem Experiment, Simulation und theoretischer Herleitung ermöglicht einen vertiefern Zugang zur Wahrscheinlichkeitstheorie. + +Ziel der Ausarbeitung ist es, die mathematischen Grundlagen des Galtonbretts auf Hochschulniveau darzustellen und deren Zusammenhänge verständlich herzuleiten. Anschließend werden diese Inhalte auf das mathematische Vorwissen der Klassenstufe 10 bis 12 übertragen und didaktisch aufbereitet. Die entstehenden Arbeitsmaterialien ermöglichen den Schüler*innen einen vertieften Einblick in die mathematischen Hintergründe des Galtonbretts. + +%\begin{enumerate} +% \item Es handelt sich um ein Zusatzangebot (Freizeit!) für die Schülerinnen und Schüler. Der Spaß an der Mathematik darf also auf keinen Fall zu kurz kommen. +% \item Mathclubs sind in der Regel nach Jahrgängen aufgeteilt. Bei uns sollen als Zielgruppe die Klassenstufen 9-12 angenommen werden. Wählen Sie sinnvoll eine Klassenstufe aus, für die sich ihr Thema eignet. Welche Voraussetzungen bringen die Schülerinnen und Schüler mit? +% \item Gehen Sie davon aus, dass Sie es mit motivierten Schülerinnen und Schülern zu tun haben. Die Kinder erwarten klare Argumente, Beweise, etc., die auch über den Lehrplan (deutlich) hinaus gehen. +% \item Überlegen Sie, woran Sie Spaß hätten. Seien Sie kreativ! +%\end{enumerate} -Besonders wichtig ist es die folgenden Punkte zu beachten: - -\begin{enumerate} - \item Es handelt sich um ein Zusatzangebot (Freizeit!) für die Schülerinnen und Schüler. Der Spaß an der Mathematik darf also auf keinen Fall zu kurz kommen. - \item Mathclubs sind in der Regel nach Jahrgängen aufgeteilt. Bei uns sollen als Zielgruppe die Klassenstufen 9-12 angenommen werden. Wählen Sie sinnvoll eine Klassenstufe aus, für die sich ihr Thema eignet. Welche Voraussetzungen bringen die Schülerinnen und Schüler mit? - \item Gehen Sie davon aus, dass Sie es mit motivierten Schülerinnen und Schülern zu tun haben. Die Kinder erwarten klare Argumente, Beweise, etc., die auch über den Lehrplan (deutlich) hinaus gehen. - \item Überlegen Sie, woran Sie Spaß hätten. Seien Sie kreativ! -\end{enumerate} - - -Wir wollen eine Datenbank für Mathe-Clubs aufbauen. Daher ist eine einheitliche Struktur aller Datensätze zwingend erforderlich. Es sind somit einige technische Punkte zu beachten: - -\begin{itemize} - \item Halten Sie sich bitte streng an die vorgegebene Latex-Vorlage (inklusive Ordner- und Dateistruktur). -\end{itemize} - -Die Einleitung soll einen kurzen Überblick über den Inhalt des Dokuments geben. Sie soll den Leser neugierig machen und ihn dazu bringen, weiterzulesen. 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