@article{johnson_andrews-larson_keene_melhuish_keller_fortune_2020, title={Inquiry and Gender Inequity in the Undergraduate Mathematics Classroom}, volume={51}, ISSN={["1945-2306"]}, DOI={10.5951/jresematheduc-2020-0043}, abstractNote={Our field has generally reached a consensus that active-learning approaches improve student success; however, there is a need to explore the ways that particular instructional approaches affect various student groups. We examined the relationship between gender and student learning outcomes in one context: inquiry-oriented abstract algebra. Using hierarchical linear modeling, we analyzed content assessment data from 522 students. We detected a gender performance difference (with men outperforming women) in the inquiry-oriented classes that was not present in other classes. We take the differential result between men and women to be evidence of gender inequity in our context. In response to these findings, we present avenues for future research on the gendered experiences of students in such classes.}, number={4}, journal={JOURNAL FOR RESEARCH IN MATHEMATICS EDUCATION}, author={Johnson, Estrella and Andrews-Larson, Christine and Keene, Karen and Melhuish, Kathleen and Keller, Rachel and Fortune, Nicholas}, year={2020}, month={Jul}, pages={504–516} }
 @article{rasmussen_keene_2019, title={Knowing solutions to differential equations with rate of change as a function: Waypoints in the journey}, volume={56}, ISSN={["1873-8028"]}, DOI={10.1016/j.jmathb.2019.03.002}, abstractNote={In this paper we illustrate five qualitatively different ways in which students might reason with rate of change as a conceptual tool in order to graphically determine solutions to first order autonomous differential equations. These five different ways of reasoning about rate of change offer an empirically-grounded theoretical account of the waypoints through which student reasoning may progress in increasingly sophisticated ways. The term waypoint comes from the literature on learning progressions, which seeks to identify conceptual landmarks that differentiate ways of reasoning that students are likely to use as they engage in mathematical tasks and solve problems. For each waypoint we give an illustrative example, the associated task and goal, typical inscriptions that students use as a source for reasoning about rate of change, and typical target inscriptions that students use to depict solutions. The paper concludes with implications for research and curriculum development.}, journal={JOURNAL OF MATHEMATICAL BEHAVIOR}, author={Rasmussen, Chris and Keene, Karen}, year={2019}, month={Dec} }
 @article{rasmussen_dunmyre_fortune_keene_2019, title={Modeling as a Means to Develop New Ideas: The Case of Reinventing a Bifurcation Diagram}, volume={29}, ISSN={1051-1970 1935-4053}, url={http://dx.doi.org/10.1080/10511970.2018.1472160}, DOI={10.1080/10511970.2018.1472160}, abstractNote={Abstract This article provides an overview of a modeling sequence that culminates in student reinvention of a bifurcation diagram. The sequence is the result of years of classroom-based research and curriculum development grounded in the instructional design theory of Realistic Mathematics Education. The sequence of modeling tasks and examples of student work tendered offers a fresh perspective on modeling and how an inquiry-oriented sequence of tasks can lead to the reinvention of significant mathematics.}, number={6}, journal={PRIMUS}, publisher={Informa UK Limited}, author={Rasmussen, Chris and Dunmyre, Justin and Fortune, Nicholas and Keene, Karen}, year={2019}, month={Jan}, pages={509–526} }
 @article{kuster_johnson_keene_andrews-larson_2017, title={Inquiry-Oriented Instruction: A Conceptualization of the Instructional Principles}, volume={28}, ISSN={1051-1970 1935-4053}, url={http://dx.doi.org/10.1080/10511970.2017.1338807}, DOI={10.1080/10511970.2017.1338807}, abstractNote={Abstract Research has highlighted that inquiry-based learning (IBL) instruction leads to many positive student outcomes in undergraduate mathematics. Although this research points to the value of IBL instruction, the practices of IBL instructors are not well-understood. Here, we offer a characterization of a particular form of IBL instruction: inquiry-oriented instruction. This characterization draws on K-16 research literature in order to explicate the instructional principles central to inquiry-oriented instruction. As a result, this conceptualization of inquiry-oriented instruction makes connections across research communities and provides a characterization that is not limited to undergraduate, secondary, or elementary mathematics education.}, number={1}, journal={PRIMUS}, publisher={Informa UK Limited}, author={Kuster, George and Johnson, Estrella and Keene, Karen and Andrews-Larson, Christine}, year={2017}, month={Aug}, pages={13–30} }
 @article{keene_hall_duca_2014, title={Sequence limits in calculus: using design research and building on intuition to support instruction}, volume={46}, ISSN={1863-9690 1863-9704}, url={http://dx.doi.org/10.1007/S11858-014-0597-8}, DOI={10.1007/S11858-014-0597-8}, number={4}, journal={ZDM}, publisher={Springer Science and Business Media LLC}, author={Keene, Karen Allen and Hall, William and Duca, Alina}, year={2014}, month={Jun}, pages={561–574} }
 @article{keene_rasmussen_stephan_2012, title={Gestures and a chain of signification: the case of equilibrium solutions}, volume={24}, ISSN={1033-2170 2211-050X}, url={http://dx.doi.org/10.1007/S13394-012-0054-3}, DOI={10.1007/S13394-012-0054-3}, number={3}, journal={Mathematics Education Research Journal}, publisher={Springer Science and Business Media LLC}, author={Keene, Karen Allen and Rasmussen, Chris and Stephan, Michelle}, year={2012}, month={Aug}, pages={347–369} }
 @inproceedings{keene_glass_kim_2011, title={Identifying and assessing relational understanding in ordinary differential equations}, DOI={10.1109/fie.2011.6143074}, abstractNote={Is it possible to assess conceptual understanding of ordinary differential equations and their solutions? There is significant tension between students learning mathematics for understanding and students learning to drill a set of algorithms to solve standard ODE exercises. This paper presents the Framework for Relational Understanding of Procedures, a categorization of assessable conceptual knowledge. Example conceptual knowledge we examine in this project include relating a graphical representation of an ODE to a symbolic one, checking a result, and knowing why a particular solution method might be applicable. We applied this framework to three techniques taught in Ordinary Differential Equations: separation of variables, solving a first order linear ordinary differential equation, and Euler's method and developed a set of assessment items. These assessment questions were then transformed into knowledge pieces and moved into an online platform and elaborated. Thus, the assessment can be administered either in traditional pencil-and-paper form or through a learn-as-you-assess online web site.}, booktitle={2011 frontiers in education conference (fie)}, author={Keene, K. A. and Glass, M. and Kim, J. H.}, year={2011} }
 @inproceedings{keene_dietz_holstein_craig_2011, title={Mathematics instruction using decision science and engineering tools}, DOI={10.1109/fie.2011.6143009}, abstractNote={MINDSET (Mathematics INstruction using Decision Science and Engineering Tools) is a multiyear project involving collaboration between engineers, educators, and mathematicians at three universities in two states. Through a partnership with secondary teachers, the MINDSET project staff has created a ground-breaking new mathematics curriculum for fourth-year high school students using mathematical-based decision-making tools from the fields of operations research and industrial engineering. High school students are presented a series of real-world problem contexts with the purpose of making the underlying mathematics more relevant to them and allowing them to model the situations and actively do mathematics. This innovative new integration of mathematics and engineering at the high school level provides new opportunities for students, who have not been successful in the abstract world of mathematics, to learn to enjoy and to want to pursue a more applied form of mathematics. This paper reports the results of the nationwide implementation of this engineering-based mathematics course.}, booktitle={2011 frontiers in education conference (fie)}, author={Keene, K. and Dietz, R. and Holstein, K. and Craig, A.}, year={2011} }
 @article{keene_2007, title={A characterization of dynamic reasoning: Reasoning with time as parameter}, volume={26}, ISSN={0732-3123}, url={http://dx.doi.org/10.1016/j.jmathb.2007.09.003}, DOI={10.1016/j.jmathb.2007.09.003}, abstractNote={Students incorporate and use the implicit and explicit parameter time to support their mathematical reasoning and deepen their understandings as they participate in a differential equations class during instruction on solutions to systems of differential equations. Therefore, dynamic reasoning is defined as developing and using conceptualizations about time as a parameter that implicitly or explicitly coordinates with other quantities to understand and solve problems. Students participate in the following types of mathematical activity related to dynamic reasoning: making time an explicit quantity, using the metaphor of time as “unidimensional space”, using time to reason both quantitatively and qualitatively, using three-dimensional visualization of time related functions, fusing context and representation of time related functions, and using the fictive motion metaphor for function. The purpose of this article is to present a characterization of dynamic reasoning and promote more explicit attention to this type of reasoning by teachers in K-16 mathematics in order to improve student understanding in time related areas of mathematics.}, number={3}, journal={The Journal of Mathematical Behavior}, publisher={Elsevier BV}, author={Keene, Karen Allen}, year={2007}, month={Jan}, pages={230–246} }