Instructor Guide

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Please consider including one or more “To the Instructor” sections. Perhaps many of your comments to your students pertaining to your own course set-up, grading structure, and other preferences could form a basis for giving instructors one possible way to structure the course. It would be good to also include other comments and alternate ways an instructor might use this text and structure their courses. Some advice for faculty who have not taught an IBL course before would be welcome.

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Comment from Reviewer 4: For example, if an instructor wanted to ensure that the students could complete Chapter 8 Cardinality, which earlier exercises or chapters could possibly be skipped?

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This will be addressed in Instructor Guide.

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Potentially add to Instructor Guide:

Evidence in favor of some form of active engagement of students is strong across STEM disciplines. \href{???}{Freeman et al. (2014)} conducted a meta-analysis of 225 studies of various forms of active learning, and found that students were 1.5 times more likely to fail in traditional courses as compared to active learning courses, and students in active learning courses outperformed students in traditional courses by 0.47 standard deviations on examinations and concept inventories. The following snippet from Freeman et al. (2014) captures the importance of utilizing active learning across STEM education: \begin{quote} \emph{``The results raise questions about the continued use of traditional lecturing as a control in research studies, and support active learning as the preferred, empirically validated teaching practice in regular classrooms."} \end{quote}

For IBL specifically, a research group from the University of Colorado Boulder led by Sandra Laursen conducted a comprehensive study of student outcomes in IBL undergraduate mathematics courses while linking these outcomes to students' and instructors' experiences of IBL (see Laursen et al. 2011; Laursen 2013; Kogan and Laursen 2014; Laursen et al. 2014). This quasi-experimental, longitudinal study examined over 100 courses at four different campuses over a period that spanned two years.

On average over 60\% of IBL class time was spent on student-centered activities including student-led presentations, discussion, and small-group work. In contrast, in non-IBL courses, 87\% of class time was devoted to students' listening to an instructor talk. In addition, the IBL sections were rated more highly for a supportive classroom environment and students conveyed that engaging in meaningful mathematical tasks while collaborating was essential to their learning. Below is a brief summary of some of the outcomes of Laursen et al.'s work. \begin{itemize} \item After an IBL or comparative course, IBL students reported higher learning gains than their non-IBL peers, across cognitive, affective, and collaborative domains of learning. \item In later courses, students who had taken an IBL course earned grades as good or better than those of students who took non-IBL sections, despite having ``covered" less material. \item Non-IBL courses show a marked gender gap: across the board, women reported lower learning gains and less supportive attitudes than did men (effect size 0.4--0.5). Women's confidence and sense of mastery of mathematics, and their interest in continued study of math were lower. This difference appears to be primarily affective, not due to real differences in women's mathematical preparation or achievement. \item This gender gap was erased in IBL classes: women's learning gains were equal to men's, and their confidence and intent to persist similar. IBL approaches leveled the playing field for women, fixing a course that is problematic for women yet with no harm to men. \end{itemize}

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Discuss how to interpret the occurrence of P(n) in the skeleton proof for induction. Don't have students write P(n)!

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Done or in progress.