Presently tertiary schools in German-speaking countries are experiencing a massive trend in mathematical bridging courses, as seen in the high attendance and wide variety of research presented at the two previous Kompetenzzentrum HochschulDidaktik Mathematik conferences in 2011 and 2013 on the transition to university level mathematics. Research on pre- and bridging courses covering “remedial” topics also indicated that students are still actively developing their conceptions of these topics during their university studies (Bausch, Biehler, Bruder, Fischer, Hochmuth, Koepf, Schreiber, & Wassong, 2014). However, the majority of these bridging courses follow format and content guidelines not so different from mathematical bridging courses as they existed thirty years ago in Germany (Kurz, 1985), suggesting there is still a great deal for us to know about which supports – in structure and content – are truly effective in bridging the transition to university mathematics. In order to understand this situation in greater depth, my research focuses on one specific remedial mathematics topic, exponential functions, critical to students at one type of tertiary school, the applied science university (Fachhochschule) focusing on business and finance.
Exponential functions are critical for the university study of financial mathematics, yet they persist as a challenging mathematical topic (Radley, 2004; Stango & Zinman, 2009). The challenge of designing supports for university financial mathematics students is exasperated by a scattered body of literature addressing student conceptions of and tasks developed to explore exponential functions (Confrey & Smith, 1995; Strom, 2008; Webb, van der Kooij, & Geist, 2011). That is, what we know about how students think about exponential functions and what tasks and pedagogical techniques foster such conceptions in limited, existing in relatively isolated pieces of research. What is known about the variety of conceptions students evoke about exponential functions is best summarized in a framework developed by Strom (2008). In my research I add to this existing inventory of conceptions regarding exponential function, with an emphasis on what university students’ notice about exponential functions as used in financial mathematics and how experiences in both a mathematical bridging course and the required financial mathematics course direct students’ noticing. I define noticing as Lobato, Hohensee, and Rhodehamel (2013) did in their Student Mathematical Noticing Framework: “selecting, interpreting, and working with particular mathematical features or regularities when multiple sources of information compete for one’s attention” (p. 809). Specifically, I ask the following research questions:
- In the context of financial mathematics, prior to and during an inquiry-oriented instructional sequence, what do students notice about exponential functions in a mathematical bridging course?
- In the context of financial mathematics, prior to and during an inquiry-oriented instructional sequence, what do students notice about exponential functions in their required financial mathematics course?
- How does student noticing regarding exponential functions during instructional sequence compare in the bridging course versus in the financial mathematics course?
As a first step to this process, I completed a thorough literature review of student conceptions of exponential functions and various tasks used to direct or expand student notions of exponential functions. These examples were used to develop a four-session instructional sequence exploring exponential functions in the context of financial mathematics. As I was the instructor of mathematical bridging course throughout the development of the tasks, I was able to test the tasks in the bridging course classroom on multiple iterations, in line with design-based research. Between iterations, the tasks were refined and the changes vetted by other mathematics education researchers. In particular, a full pilot round of data collection was completed in the summer semester 2013 testing several of the tasks in the sequence in a bridging course. An analysis of this pilot data was used in the make final refinements, resulting in the instructional sequence for exponential functions used for this research. The development of these tasks and their theoretical origins serves as a smaller but critical aspect of this larger research project.
Data for my research stems from two groups of first semester Fachhochschule students: one group pursuing a Bachelor’s degree in international financial management (IF: Internationales Finanzmanagement), the other pursuing a Bachelor’s degree in business law (WR: Wirtschaftsrecht). Despite the distinct fields of study, students from both groups must complete courses in financial mathematics in the first semester. These financial mathematics courses are nearly identical in content, and the expectations of the instructors who developed and instruct these courses on what their first semesters should know regarding exponential functions, both entering the Fachhochschule and as this understanding is applied to financial mathematics in their courses, are comparable. As such, these two groups were selected to do a comparison on how student noticing exponential functions during an instructional sequence taught during a mathematical bridging course versus during the required financial mathematics course. Specifically, the IF group was taught the instructional sequence in the mathematical bridging course, while the WR group was taught the instructional sequence as part of their required financial mathematics course. While participation was voluntary and students could opt out of the research course while still participating in the class during the instructional sequence, nearly all students consented to participate. Ultimately approximately 15 students participated regularly during the instructional sequence for the IF bridging course, while approximately 45 students participated regularly during the instructional sequence for the WR required financial mathematics course.
Data collection occurred with both groups in the first weeks of the winter semester 2014/2015. Prior to the beginning of the instructional sequence, students were administered a short-answer survey regarding their exponential functions. This survey was based on the literature, earlier iterations of teaching exponential functions to financial mathematics students in a bridging course, and a prior iteration of the survey with more open questions. The survey serves as a baseline for students’ individual notions of exponential functions. Then, in both the IF bridging course group and the WR financial mathematics course group, students were then asked to work in groups of 4-6 students (depending on the size of the class) for the four sessions of the instructional sequence. For data collection three video cameras were used in each session: one capturing whole-class discussions and two following small groups. As attendance of specific students fluctuated during the four sessions, it was not always possible to have small groups with the exact same students. However, as much consistency as possible was attempted in the small groups who were video recorded. Following each session, a debriefing was conducted between myself, the primary researcher and instructor for the exponential functions sequence in both groups, and another mathematics education researcher who also assisted in the data collection. All video recordings are then transcribed.
Transcripts then undergo an analysis using the Student Mathematical Noticing Framework developed by Lobato, Hohensee, and Rhodehamel (2013). This framework consists of four components: centers of focus (meaning aspects that students notice), focusing interactions (the discourse practices used to identify centers of focus), the mathematical tasks (which serve as the foundation for the centers of focus and focusing interactions), and the nature of the mathematical activity (particularly the norms of the classroom being analyzed). While the Student Mathematical Noticing Framework was primarily developed using a constructivist theory of learning, I personally align my beliefs with the emergent perspective (Cobb & Yackel, 1996). The emergent perspective acknowledges and aligns both psychological components of learning experienced by the individual and social components of learning experienced by the collective. As such, I have also left my analysis open to adapting and expanding the Student Mathematical Noticing Framework to encompass a more nuanced analysis of the social aspects of learning, such as adapting components of the Documenting Collective Activity Framework of Tabach et al. (2014). While the analysis of the pilot data did not consist of many instances of collective activity, this is at least partially due to the small number of participants in the pilot round, typically 2-6 on any given session.
While the analysis is still in progress, there are a few results that are already beginning to emerge. First, the initial notions of exponential functions documented by the surveys is similar for both students recommended to and participating in the mathematical bridging course and those attending only the required financial mathematics course. Also comparable were the various centers of focus that emerged during the instructional sequence in both the IF bridging course and the WR financial mathematics course. The IF bridging course did permit students significantly more opportunities to probe centers of focus brought up in both small-group and whole-class discussions than in the WR financial mathematics course, in particular in the collective development of normative ways of reasoning. In terms of the centers of focus that did arise, both groups focused on a notion of exponential function as yet not explored in existing literature, that being the distinction between exponential functions and power functions / polynomials. This center of focus arose in both groups and examples exist of why this distinction is particularly important in the context of financial mathematics.
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