# «PROCEEDINGS OF THE 15TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION EDITORS STACY BROWN SEAN LARSEN KAREN MARRONGELLE ...»

Student Thinking About Limit Research indicates that calculus students have difficulties with the concept of limit (Oehrtman, 2002; Oehrtman, 2008; Bezuidenhout, 2001; Williams, 1991). Researchers have found that firstyear university students’ knowledge and understanding are based on isolated facts and procedures (Bezuidenhout, 2001). Research has shown that students see limits as a boundary that cannot be passed (Williams, 1991; Davis & Vinner, 1986). Limit is also seen as an approximate value obtained through an evaluative process or by imagining points on a graph getting closer to the limit (Williams, 1991). Some students believe the limit is as an infinite process (Williams, 1991; Orton, 1983). Others see limit as a value reached at the end of a process (Orton, 1983; Davis & Vinner, 1986). Some of these conceptions are combined in a dynamic viewpoint where the limit can be deduced by finding function values closer and closer to a given point (Williams, 1991). Students also tend to show conflicting conceptions of limit, continuity, and differentiability (Benzuidenhout, 2001). These conflicting conceptions may be reflective of the informal mental models students have formed from prior experiences, including nonmathematical intuitions of limit (Williams, 1991; Oehrtman, 2002). Prior experience appears to play a role in the choice of finding a limit as well.

A fair amount of what we know about student thinking about limits was generated with data from written surveys. This study focuses on the representation of the questions asked to illicit student conceptions. This study focused on student understanding of limit using data generated from tasks from multiple sources (see below for details about the research design) and using multiple question formats. The goals were to examine student responses to differently formatted questions and investigate interactions between question format and the knowledge of limit students displayed in those question formats.

Research Design Survey data was collected mid-semester from 111 students in a first semester calculus course at a public university in the northeastern United States. Some of the questions were adopted or adapted from other researchers’ studies on student thinking of limit (Benzuidenhout, 2001;

Oehrtman, 2002; Williams, 1991) and some were created. Students were asked to explain their definition and meaning of limit in various context and representations. Students were asked to describe what limit means at the beginning of the questionnaire and also at the end as well. Two similar multiple-choice/multiple-answer mathematical notation questions were given using different limits to see if students would give consistent answers. Two graphical representations of limit that addressed the same concepts as the multiple choice questions were given in order to see if students could answer consistently across various representations. Lastly, a true/false multiple-answer question was given to see what definition of limit students’ hold and with a final question asking which definition would best describe their definition of limit. Responses were coded using categories from other researchers’ studies where possible (Benzuidenhout, 2001) 15TH Annual Conference on Research in Undergraduate Mathematics Education 349 and using a Grounded Theory (Strauss & Corbin, 1990) approach in other cases. Responses were examined for correctness, inconsistencies between answers and between questions asked with different representations. Definition questions were checked for consistency throughout the questionnaire.

Data Students showed a much higher correct response rate for graphical tasks than mathematical notation or definition tasks. The questions are appended to this report.

Mathematical Notation Tasks Graphical Tasks Definition Tasks 19.8% - correct (Q3) 79.3% - correct (Q5) 21.6% - correct (Q8) 25.2% - correct (Q4) 85.6% - correct (Q6) Students answering both mathematical notation tasks (Q3 & Q4) correctly were coded for their response to the graphical tasks (Q5 & Q6). Similarly the students answering the graphical tasks correctly were coded for their response to the mathematical notation tasks. The results are given

**below:**

Both notation tasks correct = 14 Both graphical tasks correct = 66 12 students correctly answered 12 students correctly answered graphical tasks mathematical notation tasks 2 students did not correctly 54 students did not correctly answer graphical tasks answer mathematical notation tasks Responses to the mathematic notation tasks were coded for contradictory responses. Of the 111 students, 20 (18%) students had contradictory responses between questions 3 and 4. The response to question 4 was examined for mutual contradiction similar to the Benzuidenhout (2001) study where the researcher claimed that these contradictory responses indicated that students have an underdeveloped concept of limit. The contradictions from the current study are

**listed below:**

• 19 students selected A but not B;

• 19 students selected A but not C;

• 29 students selected C but not B;

• 24 students selected E but not A;

• 20 students selected E but not B;

Conclusions & Implications There was a significant difference in the number of correct responses to limit questions based on the representation of the question. This raises the question of which question type provides the more accurate information about student thinking. Additional studies are needed to further investigate these patterns and links between student thinking and question format. There were also some interesting patterns apparent in the more detailed analysis of student responses. In particular, the student response to mathematical notation and definition tasks were low even though those students had correctly responded to graphical tasks about limit. This difference could be due to student prior experience with graphical questions. Further research is needed to determine what it is that students who answer the graphical questions correctly understand about limit and why that understanding is not being demonstrated on the notation-type tasks. It could be that students may know how to respond to graphical tasks without having a solid conceptual foundation about limit or it could be that students are able to demonstrate their solid understanding of the ideas when interpreting a graph but are, for some reason, unable to do so 350 15TH Annual Conference on Research in Undergraduate Mathematics Education when reading the notation-type questions. There were inconsistencies among student answers in the mathematical notation representation questions. The contradictions between questions 3 and 4 were significant. There were also many mutually contradictory answers on question 4. Many researchers have suggested that students have multiple models of limit. The responses to question 7 seem to indicate that they do have a wide range of ideas about limit and have not learned a more formal definition of limit. Many students did select the formal model as a true definition for limit, but they did not exclusively select the formal definition. The choice of types of questions and representations used by researchers and instructors may have a significant impact on what knowledge of limits we ascribe to students.

Questions for discussion during preliminary report

1. If interviews were to be conducted with students who took the survey, what questions might help uncover the sources of the discrepancies of how they respond to questions?

2. Are there additional questions or question formats that should be included in future surveys if the goal is to further examine these patterns in student responses?

3. The next phase of this project is to examine college mathematics instructors' knowledge of student thinking about limit. What questions might be asked of these instructors to tap into their knowledge of the student thinking, including their knowledge of the impact of these format differences on students' performance on tasks?

References

• Bezuidenhout, J. (2001). Limits and continuity: Some conceptions of first-year students.

International Journal of Mathematical Education in Science and Technology, 32(4), 487-500.

• Davis, R., & Vinner, S. (1986). The notion of limit: Some seemingly unavoidable misconception stages. Journal of Mathematical Behavior, 5, 281-303

• Garthwaite, P. H., Kadane, J. B., O'Hagan, A. (2005) Statistical methods for eliciting probability distributions. Journal of the American Statistical Association, 100(470), 680-701.

• Koriat, A., Lichtenstein, S., Fischhoff, B. (1980). Reasons for confidence. Journal of Experimental Psychology: Human Learning and Memory, 6(2), 107-118.

• Nickerson, Raymond S. (1998). Confirmation bias: A ubiquitous phenomenon in many guises, Review of General Psychology (Educational Publishing Foundation) 2(2), 175–220.

• Oehrtman, M. (2002). Collapsing dimensions, physical limitations, and other student metaphors for limit concepts: An instrumentalist investigation into calculus students’ spontaneous reasoning. PhD Thesis, The University of Texas at Austin.

• Oehrtman, M. (2008). Layers of abstraction: Theory and design for the instruction of limit concepts. In M. Carlson & C. Rasmussen (Eds.), Making the connection: Research and

**practice in undergraduate mathematics, MAA Notes Volume, 73, 65-80. Washington, DC:**

Mathematical Association of America.

• Orton, A. (1983). Students’ understanding of differentiation. Educational Studies in Mathematics, 14, 235-250.

• Schuman, H., Presser, S. (1981). Questions and answers in attitude surveys. New York:

Academic Press Inc.

• Strauss, A., & Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures and techniques. Newbury Park: Sage.

• Tanur, J. M. (1992) Questions about questions. New York: Russell Sage Foundation.

• 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 353 Doing Mathematics: Perspectives from Mathematicians and Mathematics Educators

Preliminary Research Report Abstract: Learner-centered teaching strategies such as inquiry-based learning ask students to actively engage in the material they are learning, to do mathematics in order to learn mathematics. A teacher’s interpretation of the meaning of “doing mathematics” is related to his or her beliefs about mathematics and about mathematics teaching. In this exploratory study, we report the results of interviews with sixteen university level mathematics and mathematics education faculty regarding their perspectives on the meaning of doing mathematics within the context of a calculus course, a proof-oriented course, and their own mathematical experiences.

Key words: teacher beliefs, mathematical tasks, communication 1 Introduction One of the foci of the recent mathematics education reform effort has been to shift students’ classroom experience to a more learner-centered model. In terms of undergraduate mathematics education, recent research has focused on the impact of inquiry-based learning. Inquiry-based learning refers to “teaching and learning approaches that engage undergraduates in learning new mathematics by exploring mathematical problems, proposing and testing conjectures, developing proofs or solutions, and explaining their ideas” (Hassi et al, 2011, p. 73). Proponents often contrast this approach with lecture, pointing out that “sitting still, listening to someone talk, and attempting to transcribe what they have said into a notebook is a very poor substitute for actively engaging with the material and hand, for doing mathematics” (Bressoud, 2011). Notice the phrase “doing mathematics.” In inquiry-based learning, students are active participants that do mathematics in order to learn mathematics. In this study, we explore different faculty perspectives on “doing mathematics.” In particular, is there a consensus among university level mathematicians and mathematics educators regarding the meaning of doing mathematics? Further, does the notion of doing mathematics depend on the course, or is it independent of the mathematical content?

2 Previous related research Faculty perspectives on the notion of doing mathematics are connected to teacher beliefs regarding mathematics and mathematics teaching. Philipp (2007) provides an overview of research involving mathematics teachers’ beliefs; we will highlight a few key points from that chapter. First, beliefs are fairly stable and resistant to change. Beliefs act as a ﬁlter for what we see, making change difﬁcult without observation and reﬂection on practice. Second, teacher beliefs regarding mathematics and mathematics teaching correlate with instructional practice. For example, if a teachers’ beliefs about mathematics have a calculational orientation, their classroom practice will tend to focus on developing procedural skills. On the other hand, researchers have also observed apparent inconsistencies between a teacher’s stated beliefs and their actual classroom practice. In some cases, these inconsistencies can be explained by closer examination of the context. Finally, we should point out that the research regarding teacher beliefs summarized in Philipp (2007) involves preservice and 354 15TH Annual Conference on Research in Undergraduate Mathematics Education inservice K-12 teachers. We are unaware of similar research regarding the beliefs of college level mathematics instructors.

The phrase “doing mathematics” implies some type of activity. From this perspective, a variety of theoretical research articles attempt to categorize and describe mathematical tasks. For instance, Stein, Smith, Henningsen, and Silver (2000) group mathematical tasks into four categories. Tasks with lower level cognitive demand include memorization tasks and procedural tasks without connections. Tasks with higher level cognitive demand include procedures with connections as well

**as “doing mathematics” tasks. More speciﬁcally:**