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\section{Mathematization Related Proofs Using the Pumping Lemma for Regular Languages} | |

We taught the introduction to the theory of computing course from Sipser's | |

third edition\cite{sipser2012introduction}, using chapters 1 through 5 and 7. The pumping lemmas were | |

given emphasis in class, help sessions and tutoring, in homework, exams, and | |

in review. We treated the pumping lemma in the context of logic, emphasizing | |

the inversion of quantifiers. We discovered that some students seem to | |

tire of attending to statements with more than one quantifier, consistent with | |

Devlin \cite{devlin2012mathematical}. We also treated the pumping lemma with diagrams of machines | |

from Sipser's book\cite{sipser2012introduction}. We encouraged student collaboration on all learning | |

activities, including homework. | |

For grading, we used only work (exams) known to be individual. | |

To encourage active learning \cite{prince2004does} beyond using the classroom response system, | |

we assigned participation in a discussion of a specified question, weekly. We | |

discovered that students preferred to have their contributions to these discussions | |

be anonymous to other students. | |

\section{Proofs by Induction} | |

We carried out a qualitative study, inspired by the ideas of Marton et al.\cite{marton1981phenomenography,svensson1997theoretical,marton1997learning,marton2005unit}] on phenomenography and variation theory. We are using qualitative | |

techniques because we seek to be able to describe the nature of the various | |

understandings achieved by the students, rather than the relative frequency | |

with which any particular understanding is obtained. Phenomenography and | |

its extension, variation theory, are applicable to this study, because the variety | |

of outcomes in student understanding can be used to guide future offerings of | |

the course. | |

Marton and other researchers, e.g., Bussey, using phenomenography and variation | |

theory \cite{bussey2013variation} direct attention to the information intended by the instructor | |

for delivery to the student and the information received by the student. These | |

are not necessarily the same: the student may take in material with a different | |

emphasis than intended by the lecturer, such as specifics of an example that are | |

unimportant for illustration of the intended point. Items that seem obvious to | |

the instructor might not be to the student. Bussey et al. observe that “variation | |

theory is a useful framework for guiding qualitative educational research studies | |

that attempt to identify gaps between teaching and learning.” \cite{bussey2013variation}We attempt | |

to address the third goal identified by Bussey et al., ``describe the variation in | |

student understanding of a given object of learning after the learning event | |

has taken place''. We want to identify what Suhonen et al. \cite{suhonen2007applications} call ``critical | |

aspects'', in the area of proof by mathematic induction, those that seem to be conceptually | |

difficult, or are seen not to have been grasped by the students. | |

We used semi-structured interviews with students to learn their conceptualizations | |

of proof. | |

\section{Conceptions of Domain, range, mapping, relation, function, | |

equivalence in Proofs} | |

\section{Conceptions of Definitions, Language, Reasoning in Proofs} | |

\section{Conceptions of Equivalence, Abstraction in Proofs} | |

\section{Instructive Problems} | |

``Instructive problem'':begin with a problem situation that embodies key aspects of the topic, and mathematical techniues are developed as reasonable responses'' This note was made the weekend I was reading research in collegiate mathematics education 3, and I left no reference. | |

\section{List of Questions for Initial (Before course) Assessment for Discrete Structures} | |

This list of questions is designed to elicit data about the mental representations students have prepared for encountering the teaching they are about to receive in the discrete structures course. | |

Discrete structures is the course in the curriculum that (re)acquaints students with proof, which will be used in other courses, including Algorithms and Introduction to the Theory of Computing, and possibly Data Structures, depending upon how it is taught, and possible Software Engineering, depending upon how it is taught. | |

The purpose of the instrument is to determine what level of skill with proof exists in the students, as they arrive. | |

The problems include problem situations posed in a variety of representations: words, and/or symbolically, and/or by figures and/or as pseudocode. | |

The students are asked to, in some cases, match the problems that are expressed in multiple ways, and in some cases provide the missing representation form. | |

This instrument was developed using inspiration from Gibson\cite{gibson1998students}, as well as Nelson\cite{nelson1993proofs}. | |

%these came out of chapter 4 | |

\section{Mathematical Definitions, Language, Reasoning} | |

\section{Equivalence Classes, Generic Particular, Abstraction in Proofs} |