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What we find most interesting in our data is the difficulty students have with the IVT, even though they seem to understand the concepts behind it. Much of the research in our RUME community supports that if the conceptual understanding is present, then the rest of the work (application problems, etc.) should follow without too much difficulty. We realize we are overTH Annual Conference on Research in Undergraduate Mathematics Education 539 simplifying the research, but we want to stress that, in this case, a conceptual understanding of the theorem was not enough to allow our students to move forward. We still have more work to do to determine what exactly the deficiency is and what solutions to these problems are.

Implications for Teaching and Future Research:

As teachers, identifying our students’ misconceptions in understanding fundamental mathematical theorems and concepts will help us to better teach these concepts in ways that address the common misconceptions, thereby improving student understanding. Not only will this help us to reach future students when presenting the IVT, but it will also help us to reach our current students in other topics in the class, such as the Mean Value Theorem. Later this semester, we plan to collect similar data regarding student understanding of the Mean Value Theorem. Based on results from our initial study of the IVT, we already know some of the common underlying issues students will have.

Questions for the Audience

1. We believe that it is important for our students to be able to express their ideas using correct mathematical language and notation, but our students do not always see the need.

How do we convince our students to buy into this idea and understand the importance of using mathematical language correctly?

2. One weakness that we see in our results is convincing the reader that students do understand the concept of the IVT, even though they cannot express it well using the appropriate mathematics. What data could we collect to convince readers of this?

3. What changes or additions should we make in the next round of data collection?

540 15TH Annual Conference on Research in Undergraduate Mathematics Education References Abramovitz, B, Berezina, M, Berman, A and Shvartsman, L. (2007). Lagrange's Theorem: What Does the Theorem Mean?, in Proceedings of CERME 5, 2231–2240. Cyprus: University of Cyprus, Nicosia.

Abramovitz, B., Berezina, M., Berman, A., & Shvartsman, L., (2009). How to Understand a Theorem, International Journal of Mathematical Education in Science and Technology, 40, 5, pp. 577-586 Carlson, M. (1998). A Cross-Sectional Investigation of the Development of the Function Concept, Research in Collegiate Mathematics Education III, Conference Board of the Mathematical Sciences, Issues in Mathematics Education Volume 7; American Mathematical Society, 114-163 Corbin, J., & Strauss, A., (2008). Basics of Qualitative Research: Techniques and Procedures for Developing Grounded Theory, 3rd Edition, Sage Publications Cottrill, J., Dubinsky, E., Nichols, D., Schwingendorf, K., Thomas, K., & Vidakovic, D. (1996).

Understanding the Limit Concept: Beginning with a Coordinated Process Schema, Journal of Mathematical Behavior, 15, 167-192.

Monk, S. (1992). Students’ Understanding of a Function Given by a Physical Model: The concept of function, aspects of epistemology and pedagogy, The Concept of Function: Aspects of Epistemology and Pedagogy, MAA Notes, 25, 175-194.

Oehrtman, M. (2009). Collapsing dimensions, physical limitation, and other student metaphors for limit concepts. Journal for Research in Mathematics Education, 40, 396–426.

Piaget, J. (1970). Structuralism. New York: Basic Books, Inc.

Piaget, J. (1975). The equilibration of cognitive structures (T. Brown & K. J. Thampy, Trans.).

Chicago: The University of Chicago Press.

Stewart, J. (2007) Essential Calculus: Early Transcendentals, 6th Edition, Brooks Cole, Belmont, CA Williams, S. (1991). Models of Limit Held by College Calculus Students, Journal for Research in Mathematics Education, 22, 219-236.

15TH Annual Conference on Research in Undergraduate Mathematics Education 541 Examining  Students’  Mathematical  Transition  Between  Secondary  School  and   University  –  The  Case  of  Linear  Independence  and  Dependence     Natalie  E.  Selinski   University  of  Kassel  –  Germany   NSelinski@hotmail.com     To  understand  the  mathematical  transition  students  make  between  secondary  school  and   the  university  requires  an  in ­depth  look  at  the  mathematical  topics  students  learn  at  the  time   of  this  transition  and  the  contextual,  institutional  changes  that  simultaneously  occur.  This   preliminary  presentation  explores  how  linear  algebra  students  at  both  the  secondary  school   and  university  in  Germany  understand  vectors  and  linear  independence  and  dependence  in   the  course  of  video ­recorded,  think ­aloud  problem ­solving  interviews.  Analysis  of  these   interviews  indicate  not  only  differences  in  mathematical  content  and  sophistication  between   secondary  school  and  university  students,  but  also  in  students’  disposition,  particularly   towards  new  mathematical  experiences.  A  look  at  more  informal  data  about  the  various   institutional  environments,  secondary  school  and  university,  provides  a  potential  reason  for   these  differences.  This  report  concludes  with  a  discussion  on  how  to  create  a  blended   analysis  of  these  individual  understandings  and  dispositions  and  their  relationship  with  the   institutional  context  as  a  better  means  of  understanding  the  transition  to  university ­level   mathematics.     Keywords:  transition  to  university  mathematics,  linear  algebra,  conceptual  understanding,   institutional  environments   542 15TH Annual Conference on Research in Undergraduate Mathematics Education Examining Students’ Mathematical Transition Between Secondary School and University – The Case of Linear Independence and Dependence The gap between secondary school mathematics and university mathematics has proved to be a particularly difficult challenge for students (cf. De Guzmann, Hodgson, Robert, & Villani, 1998; Tall, 1991). To understand the mathematical transition students make between secondary school and the university requires an in-depth look at the mathematical topics students learn at the time of this transition and the contextual, institutional changes that simultaneously occur. In particular, there are certain courses that fall exactly during this transition. In Germany, linear algebra is one such course, with foundational linear algebra topics like vectors and linear independence being introduced in the last years of secondary school then revisited and built upon in the first year at the university.

This study begins by asking how do students think about and work with the ideas of vectors, linear independence, and linear dependence at the secondary school and university levels and what differences these two distinct groups of students have in viewing and working with these concepts.

The initial results of this analysis suggest differences not only in how these distinct groups view these concepts, but also in how students approach tasks that require the students to work with these concepts in novel or more unfamiliar settings and their disposition towards these new mathematical experiences. This begs the question: how do we account for the differences between the secondary school students and university students? The study conjectures that these differences come from not only the level and sophistication of the mathematical content of their courses, but also from the differences in the institutional settings.

Literature There is a growing body of work regarding student reasoning in the context of linear algebra, the most comprehensive of which is an edited volume by Dorier (2000). Within this volume, Hillel (2000) observes that at the US university linear algebra is often the first mathematics course that students encounter as a mathematical theory, with formal definitions and proofs and built up systematically. Furthermore, Hillel details some of the difficulties students have in terms of understanding different but related modes of description of mathematical objects in linear algebra such as abstract, algebraic and geometric notions of vectors. More recent work regarding student difficulties in learning linear algebra include students’ conception of the equal sign in matrix equations and early notions of eigenvalues (Larson, Zandieh, Rasmussen, & Henderson, 2009) and connections students make between fundamental concepts in linear algebra (Selinski, 2010).

However, in each of these contexts, because students often have encountered linear algebra for the first time at the university and only university students are addressed, these reports do not touch directly on the transition from secondary school to university mathematics nor do they examine students understanding of vectors and linear independence in depth.

This study aims to build from these works by examining how students think about vectors, linear independence and linear dependence. In terms of these key linear algebra concepts, we focus on the importance of a flexible understanding of mathematical concepts as detailed by Tall and Vinner (1981). Furthermore, as with Dahlberg and Housman (1997), we explore the significance of example generation for student reasoning and concept understanding. This report uses this strong foundation on students understanding of concepts as a means for exploring the difficulties in transitioning from notions of mathematical concepts from the secondary school to the university level.

15TH Annual Conference on Research in Undergraduate Mathematics Education 543 One piece of literature that may help us understand how these more individualistic understandings relate to the institutions and learning environments, and thus to this transition, would be Cobb and Yackel’s (1996) elaborated interpretive framework for the emergent perspective. This framework sees the “individual students’ activities… located in the broader institutional setting” (p. 181) and create a framework for understanding the reflexive relationship between individual psychological and more sociocultural perspectives.

Methods Data for this report comes from a year-long project examining how students learn linear algebra at the end of their secondary schooling at the German Gymnasium (upper-level high school) and in their first year at the German university. As a part of this project, six secondary school students and five university students participated in individual, semi-structured, think-aloud problem-solving interviews (Bernard, 1988) that were approximately 60 to 90 minutes long. The interviews were video-recorded, and the analysis of the data involves repeatedly reviewing these videos, selective transcriptions of the videos, and copies of students written work created during the course of the interviews. This report will focus on the two questions posed in these interviews,

which asked:

–  –  –

Additional follow-up questions were also asked to clarify how students thought of vectors, linear independence and linear dependence, and how these understandings were reflected in their creation of examples.

These interviews were then reviewed and selectively transcribed. The initial analysis paid extra attention to the differences between these distinct groups of students.

Furthermore, in order to account for the different environments in which the students learned linear algebra, more informal data was collected about the Gymnasium and German university. Data about the institutional environments comes from notes completed while observing classes at the secondary school and lectures, homework sessions, and informal study groups of students at the university. Further data comes from notes while discussing the expectations of learning in these environments with instructors from both institutions.

Preliminary Results:

Preliminary results suggest that most students at the Gymnasium had well-established geometric and algebraic notions of vectors and linear independence and dependence. Similar understandings were given by all university students, which the students cited originated or built from their studies of linear algebra at the Gymnasium, before university. Surprisingly, despite the strongly formal approach to instruction of linear algebra at the university, few university students were able to cite or work with abstract notions of vectors or linear independence, rather opting for algebraic and geometric descriptions first seen at the Gymnasium.

544 15TH Annual Conference on Research in Undergraduate Mathematics Education A more surprising result was not initially seen until the students were pushed into more unfamiliar mathematical situations. For example, when Gymnasium students were asked to generate an example of linearly independent vectors in R4 or the university students were asked to create a similar example but not in Rn. About half of the Gymnasium students struggled with sets of vectors in R4, citing that they could not see R4, so a set of such vectors does not exist. This reasoning came quickly and without question in the course of the interview. Compare this to the three university students who could not produce an example not in Rn. Each of these students paused and struggled with the problem, and when they could not produce an example, the students reasoned that did not mean such a thing did not exist. Rather, it meant that they had not previously seen such an example before or could not understand how to create an example with their personal understanding of vectors, vector spaces, and linear independence.

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