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Polynomial Interpolation 1 CREACOMP e−Schulung von Kreativität und Problemlösungskompetenz Prof. Dr. Bruno BUCHBERGER Prof. Dr. Erich Peter KLEMENT Prof. Dr. Günter PILZ und Mitarbeiter der Institute ALGEBRA, FLLL und RISC 0 CREACOMP Endbericht Inhalt CREACOMP ................................................................................................................1 Zusammenfassung ...........................................................................................1 Benutzung der CREACOMP Software .......................................................2 Kurzbeschreibung der Lerneinheiten ..........................................................3 Theorema ..................................................................................................3 Elementary Set Theory .............................................................................3 Relations ...................................................................................................4 Equivalence Relations ..............................................................................4 Factoring Integers .....................................................................................5 Polynomial Interpolation 1 .........................................................................5 Polynomial Interpolation 2 .........................................................................6 Real Sequences 1 .....................................................................................6 Real Sequences 2 .....................................................................................7 Continuous Functions ...............................................................................7 Markov Chains ..........................................................................................8 Gröbner Bases ..........................................................................................8 Cryptography ............................................................................................9 Fast Computations ....................................................................................10 Fuzzy Control ............................................................................................10 Clustering ..................................................................................................11 Imperative Program Verification ...............................................................12 Arbeitsweise ........................................................................................................12 Dissemination .....................................................................................................14 Finanzielle Abrechnung ...................................................................................15 Die CREACOMP CD ........................................................................................16 Anhang: Kommentierte Lerneinheiten ................................................................17 Equivalence Relations Polynomial Interpolation Public Key Crypotography Anhang: 2 Publikationen zu CREACOMP ...........................................................109 CREACOMP Endbericht 1 CREACOMP e−Schulung von Kreativität und Problemlösungskompetenz Endbericht über das Projekt 1 Zusammenfassung Das Projekt CREACOMP hatte das Ziel, Kreativität und úcomputational aspectsø in der Mathematik−Ausbildung, zusammen mit dem automatischen Beweisen der gewonnenen Sachverhalte zu entwickeln und zu schulen. Im ersten Teil sollen Studierende im wesentlichen mit dem an der JKU entwickelten Softwaresystem MeetMATH durch angeleitete Experimente mathematische Sätze und Lösungsverfahren (er)finden und erkennen und die Freude am eigenen Erkennen empfin− den. Im zweiten Teil sollen die gefundenen Sachverhalte mathematisch exakt formuliert und schließlich automatisch (!) bewiesen werden. Dieser Teil beruht auf dem ebenfalls an der JKU entwickelten System THEOREMA. Im Rahmen des Projektes CREACOMP wurde eine Integration der beiden Systeme MeetMATH und THEOREMA durchgeführt. Da sowohl MeetMATH als auch THEOREMA auf dem bekannten Mathematik−Softwaresystem Mathe− matica aufbauen, dient Mathematica auch als Plattform für CREACOMP. In 16 ausgewählten Lerneinheiten (jeweils für etwa eine Doppelstunde gedacht) wurde gezeigt, dass die Verschmelzung von MeetMATH und THEOREMA wirklich und sinnvoll möglich ist. Dies ermöglicht weltweit erstmalig eine vollständige Integration von Entdeckung, exakter Formulierung und Absicherung mathematischer Inhalte in Lerneinheiten des E−learn− ing. In der dem Endbericht beiliegenden CD sind alle diese Lerneinheiten (sowohl als Mathe− matica−Notebooks als auch in pdf−Form) enthalten. Weiters geben wir für jede dieser Units eine verbale Kurzfassung an. Sodann folgt eine Übersicht über die Arbeitsweise der CREA− COMP−Entwicklungsgruppe, Angaben über die Dissemination sowie die finanzielle Endab− rechnung. Schliesslich legen wir exemplarisch 3 kommentierte Units in Hardcopy bei (die natürlich nicht die wesentlichen Innovationen im e−Bereich zeigen können, dies versuchen wir in den Kommentaren zu kompensieren). Die Lerneinheiten werden z.T. bereits jetzt in Lehrveranstaltungen verwendet; diese Einsätze sollen noch wesentlich ausgeweitet werden. Als Zielgruppen standen uns vor Augen: Studierende, Lehrende und AbsolventInnen der JKU, sowie weiterbildungswillige Berufstätige. 2 CREACOMP Endbericht Das CREACOMP−Entwicklungsteam bietet auch eine Live−Präsentation an. Ebenso bieten wir wie im Projektantrag an, CREACOMP nach Ende des Projektes weiterzuentwickeln und aufgrund der Rückmeldungen zu verbessern. 2 Benutzung der CREACOMP Software CREACOMP besteht aus 16 großteils voneinander unabhängigen Lerneinheiten und ist als Standard|AddOn Paket für das bekannte Computeralgebra|System Mathematica konzipiert. Eine funktionsfähige Installation von Mathematica 5.2 (oder höher) ist als Systemvorausset− zung zu Grunde gelegt, es gibt keine weiteren Systemvoraussetzungen. Das CREACOMP| System kann entweder direkt von der CD gestartet werden oder der Inhalt der CD wird einmalig auf der Festplatte als Mathematica AddOn Paket installiert (Installationsanleitung liegt der CD bei!). Nach erfolgtem Setup kann CREACOMP wie jedes andere Mathematica| Paket durch den Standard|Befehl Needs in Mathematica geladen werden. Nach dem Laden von Needs@"CreaComp`"D startet CREACOMP mit der Standard CREACOMP|Navigation in einem eigenen Fenster. Die Navigationsstruktur und ein Autoren|Tool zum Erstellen der konkreten Navigation wurde aus MeetMATH übernommen. Die CREACOMP|Navigation besteht aus einer Java|Applika− tion, in der der Benutzer auf verschiedenen úLernpfadenø durch die Lerneinheiten wandern kann. Bild 1 zeigt die Standard CREACOMP|Navigation, in der die aus MeetMATH übernommenen Kategorien úRealWorldø und úMathWorldø als Menüs zur Verfügung stehen, in Untermenüs gruppiert finden sich die jeweiligen Lerneinheiten. Per Mausklick werden die Lerneinheiten als Mathematica|Dokumente (sogenannte úNotebooksø) geöffnet und können sodann vom Lernenden bearbeitet werden. Bild 1: Die Standard CREACOMP|Navigation CREACOMP Endbericht 3 Alternativ zur Standard|Navigation können die Lerneinheiten auch in einer Matrix|Struktur nach úRealWorldø und úMathWorldø gruppiert angezeigt werden. Auch aus der Matrix können die Einheiten per Mausklick geöffnet werden. Im derzeitigen Projektstadium von CREACOMP steht im Gegensatz zum Vorgänger− projekt MeetMATH die Verknüpfung von mathematischen Inhalten (úMathWorldø) mit Anwen− dungsbeispielen aus der Praxis (úRealWorldø) nicht im Vordergrund. Der Fokus liegt auf der experimentellen Aufbereitung von mathematischen Inhalten und deren Verknüpfung mit dem (automatisierten) Beweisen von mathematischen Theoremen. Demzufolge sind die derzeit verfügbaren Units ausschließlich in der Kategorie úMathWorldø zu finden, dazu passende úRealWorldø|Units können später hinzukommen. 3 Kurzbeschreibung der Lerneinheiten 3.0 Theorema Dies ist keine Lerneinheit im engeren Sinn, sondern hier sollen die Benutzer die Grundideen des Theorema Systems und dessen Benutzung kennenlernen. Die Einheit wurde von W.Windsteiger zusammengestellt. 3.1 Elementary Set Theory In dieser Lerneinheit werden die elementaren Sprachmittel der Mengenlehre vorgestellt. Die Einheit dient vor allem als Grundlage für andere Units, in denen die Sprache der Mengenle− hre zur Formulierung neuer Theorien verwendet wird. Es geht hier nicht um einen formal exakten Aufbau der Mengenlehre als die Basistheorie der Mathematik, unser Hauptaugen− merk liegt auf der Einführung und Erklärung der Sprchkonstrukte, die uns die Mengenlehre zur Verfügung stellt. Die meisten dieser Konstrukte sind dem Studierenden intuitiv vertraut, wir gehen aber auch auf formale Argumentation mit Mengen ein und verwenden dazu den Set Theory Beweiser, der in Theorema zur Verfügung steht. Es werden zuerst Darstellungsformen für Mengen diskutiert, die Problematik eines intuitiven Mengenbegriffs wird mit Hilfe des bekannten Russell’schen Paradoxon beleuchtet. Es wird an dieser Stelle aber kein axiomatischer Aufbau angestrebt, vielmehr werden die wichtigsten Mengenoperationen wie Vereinigung, Durchschnitt, Potenzmenge, etc. definiert, ohne dass auf deren Existenz im Detail eingegangen wird. Verschiedenste Eigenschaften, die das Zusammenspiel dieser Konstrukte beschreiben, werden mit Hilfe von Theorema exakt bewiesen, sodass dei Studierenden einen Eindruck des Beweisens in diesem wichti− gen mathematischen Teilgebiet vermittelt bekommen. Die Einheit wurde von W.Windsteiger zusammengestellt.
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