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

However, in the process of raising this question to the class, Dr. Bond gained insight into the justification of the step in question by connecting the student’s proof to previously established 15TH Annual Conference on Research in Undergraduate Mathematics Education 469 result, if ab = ac then b = c. She then shared this realization with the class, stating, “my gut at the moment is that … what is, our cancelation property says that if ab equals ac then b equals c, right. And what was the contrapositive to this”? Having made this connection for herself, Dr.

Bond was then able to verify the steps of the student’s proof with the class.

Given Dr. Bond’s debriefing statement, it is clear that this result was not knowledge that she carried with her into class. Instead, Dr. Bond drew on her mathematical knowledge in order to carry out mathematical activity in the moment. As such, this example of proof analysis was categorized as an instance of mathematical activity for teaching. The instructional move implemented by Dr. Bond to connect her mathematical activity to the students’ was that of justification exhibition, and this instructional move resulted in a resolution of the proof.

Implications for Future Research I see this work as a first step in establishing the mathematical activity for teaching construct and the Mathematical Activity for Teaching framework. Both this construct and the framework can serve as analytic tools for better understanding the relationships between teacher activity and student activity in the classroom. Further, by investigating how mathematical knowledge for teaching supports mathematical activity for teaching, it may be possible to identify specific processes by which teacher knowledge can impact student learning.

Questions for the Audience

1. What other questions might this framework suggest?

2. One motivation for the mathematical activity for teaching construct was dissatisfaction with an acquisition interpretation of mathematical knowledge for teaching. Is this interpretation of mathematical knowledge for teaching consistent with its use in the literature?

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407.

Davis, B. (1997). Listening for differences: An evolving conception of mathematics teaching.

Journal for Research in Mathematics Education 28(3), 355-376.

Freudenthal, H. (1991). Revisiting Mathematics Education: China Lectures Dordrecht, Netherlands: Kluwer.

Johnson, E.M.S., & Larsen, S.P. (2011) Teacher listening: The role of knowledge of content and students. Journal of Mathematical Behavior, doi:10.1016/j.mathb.2011.07.003 Lakatos, I. (1976). Proofs and Refutations. Cambridge: Cambridge University Press.

Larsen, S., & Zandieh, M. (2007). Proofs and refutations in the undergraduate mathematics classroom. Educational Studies in Mathematics, 67(3), 205-216.

470 15TH Annual Conference on Research in Undergraduate Mathematics Education Rasmussen, C., & Marrongelle, K. (2006). Pedagogical content tools: Integrating student reasoning and mathematics in instruction. Journal for Research in Mathematics Education, 37(5), 388-420.

Speer, N. M., & Wagner, J. F. (2009). Knowledge needed by a teacher to provide analytic scaffolding during undergraduate mathematics classroom discussions. Journal for Research in Mathematics Education, 40(5), 530-562.

Yackel, E., Stephan, M., Rasmusen, C., & Underwood, D. (2003). Didactising: Continuing the work of Leen Streefland. Educational Studies in Mathematics 54, 101-126.

This report focuses on an ongoing project that is developing a calculus course required for all preservice elementary teachers at a large southeastern university. In the process of designing and implementing the new materials, several research-based tasks have been developed, tested and refined. We discuss the results of the implementation and the refined tasks. We specifically focus on the task developed for the introduction and development of students’ limit understanding. Preliminary results indicate that students in our classes have difficulty thinking about the big ideas of the calculus, including limit, and that participation in these tasks, although difficult, is providing a venue for preservice elementary teachers to think more like mathematicians and come to view mathematics as more than a set of procedures to be followed.

We hypothesize this experience will provide students with a stronger foundation as they begin their careers as elementary educators.

Keywords: Calculus, Limits, Preservice Teachers

## Introduction

We present preliminary results from an ongoing project developing a calculus course for preservice elementary teachers at a large southeastern university. In the process of designing and implementing the new materials, several research-based tasks have been developed, tested and refined. We discuss the refined tasks and the results of the implementation. We specifically focus on the task developed to introduce the limit concept. Preliminary results indicate that students in our classes have difficulty thinking about the big ideas of the calculus, including limit, but that participation in these tasks, although difficult, is providing a venue for preservice elementary teachers to reason more like mathematicians and view mathematics as more than a set of procedures to be followed. We hypothesize this experience will provide students with a stronger foundation as they begin their careers as elementary educators.Literature Review Mathematical knowledge for teaching. Studies abound that show prospective or practicing elementary teachers’ lack of: knowledge of mathematics (e.g., Ball, 1990; Fennema & Franke, 1992; Ma, 1999; Mewborn; 2001), productive beliefs about the discipline (Thompson, 1992; Phillip, 2007), and a sense of self-efficacy for teaching mathematics (Enochs, Smith, & Huinkee, 2000; Utley, Bryant, & Moseley, 2005; Utley & Moseley, 2006) and these studies have sparked great concern in education. More recently, the response to the question of teachers’ needed mathematical knowledge has moved toward the notion of teachers’ mathematical knowledge for teaching (Ball, Hill & Bass, 2005). As defined, this knowledge includes not only what is considered common content knowledge, but also specialized content knowledge, i.e., knowledge of mathematics that is specific to the needs of teachers (Ball, Thames & Phelps, 472 15TH Annual Conference on Research in Undergraduate Mathematics Education 2008). Additionally, and important to support our work, the MAA standards established in the Committee on the Undergraduate Program’s Curriculum Guide (2004) state we need to go further than just the basics in our education of elementary mathematics teachers.

Within these areas, recent research has begun to show that elementary teachers who demonstrate specialized content knowledge do positively impact student achievement (Hill, Rowan & Ball, 2005). In fact, the National Mathematics Advisory Panel (NMP) noted “teachers must know in detail and from a more advanced perspective the mathematical content they are responsible for teaching and the connections of that content to other important mathematics, both prior to and beyond the level they are assigned to teach” (National Mathematics Advisory Panel, 2008, p. xx). Our research addresses both the more advanced perspective and the connections to other important mathematics mentioned by the NMP.

Students as mathematicians. Some educators posit that mathematics students should approach school mathematics in a manner similar to how mathematicians do mathematics (e.g.

Papert, 1971; Seaman & Szydlik, 2007). In their study of the mathematical behavior of preservice elementary teachers, Seaman & Szydlik (2007) found, “teachers display a set of values and avenues for learning mathematics that is so different from that of the mathematical community and so impoverished, that their attempts to create fundamental mathematical understandings often meet with little success” (p. 179). However, as important rigorous mathematical practice is for students to participate in, there are necessary modifications. Wu (2006) calls mathematics education, “mathematical engineering, in the sense that it is the customization of basic mathematical principles to meet the needs of teachers and students” (p. 3) and stresses the importance of mathematicians partnering with educators in order to build appropriate mathematics for K-12 classrooms.

Student understanding of limit and designing a limit activity. Research on student understanding of limits has identified both common misconceptions students hold, as well as a number of features instructional activities for limit should include. For instance, students are likely to believe that a sequence cannot reach its limit and may confuse the limit with a bound (Davis & Vinner, 1983). Furthermore, students tend to hold intuitive, dynamic images of limit as evidenced by their language of a sequence “getting closer and closer” or “approaching” its limit (Mamona-Downs, 2002; Roh, 2008). Researchers have illustrated a number of components of limit activities in order to best avoid such misconceptions including beginning by helping students develop an intuitive sense of limit and structuring activities to coordinate with formal conceptions of limit (Mamona-Downs, 2002; Oehrtman, 2008; Roh, 2008).

Setting and Description of Research

**Setting. The project is a collaboration between individuals from three fields:**

mathematics, elementary education, and mathematics education. Each brings valuable background and perspective to the project. The setting is an elementary education preservice program that is “STEM-focused” and students are required to take 9 hours of undergraduate level mathematics and 3 hours of statistics in their course of study. Instructors in the pilot calculus class are emphasizing the big ideas of calculus as well as modeling the teaching strategies that they hope will be implemented by the future teachers. These strategies include: inquiry, collaboration, justification of ideas, and provision for diverse learners.

Research questions. We investigate the following two questions: What instructional sequence may provide preservice elementary teachers with an informal understanding as well as a basis for more formal understanding of limit? How do preservice elementary teachers understand limit of a sequence both informally and formally?

15TH Annual Conference on Research in Undergraduate Mathematics Education 473 Description of Research. Our work is primarily design-based research (Collins, Joseph, & Bielaczyc, 2004), in which we “carry out formative research to test and refine educational designs based on principles derived from prior research (p. 15). Specifically, we used research from mathematics education partnered with personal experience in calculus instruction to design the curriculum, sequence the instruction and design the specific tasks, teacher presentations, and assessments.

The task discussed introduces the concept of the limit of a sequence. We researched, developed the task, and tested it on several focus groups in spring 2011. We video-recorded these sessions, one of the researchers was the facilitator, and the other researchers took field notes. After each implementation, the research team met and revised the task. In fall 2011, we used the revised activity in the pilot class of 29 students. The class consists of 27 females and 2 male undergraduates, all freshmen or sophomores. The course is being team taught by two of the researchers (one from mathematics department, one from mathematics education). Another researcher attends, takes field notes, and video-records selected episodes. The implementation of the limit task was recorded during whole group instruction, and one small group discussion. Data are also presented from supplemental course material including field-notes and student work.

Preliminary Results Results lie in two areas: 1) new instructional sequences that are research-based, tested and refined, and 2) new evidence about student learning of advanced mathematics ideas. We have identified the primary notions we will emphasize in this course as function, limit, and derivative.

Thus, our research is focused on how students might learn these ideas.

Research-based instructional tasks and sequences. One of the tasks we have developed is called the “Sesame Street Activity.” The primary goal is to provide students with an experience where they are introduced to and begin thinking about limits informally. Space does not allow the inclusion of the full task, but the introduction and two of the questions students are

**asked to answer in groups follow:**

Big Bird and Count von Count are traveling back to Sesame Street when they come to a bridge. Just before the bridge there is a sign: Each step on my bridge must be special: Every step you take must be exactly half of the remaining distance you have left to cross.

5. Without computing, do you know if Big Bird will ever have a step size less than.000000001 meters? How about 10-100 meters? How could you find the number of steps?

6. Big Bird makes a shocking revelation: He claims that if you call out any number, as small as you like, if he follows Lord Zeno’s directions, after a certain step, the size of all his following steps will be smaller than your number. Test out Big Bird’s theory.

Task construction aligns with suggestions from research (Oehrtman, 2008; Roh, 2008), particularly structuring activities to support an informal sense of limit of a sequence that can be connected to a more formal definition of limit. While aspects of this task were successful, (e.g.

students gaining an informal sense of the limit of a sequence) other aspects proved problematic.

For instance, one question involved using logarithms to simplify an equation and students tended to focus heavily on procedural components of the question as opposed to the limiting idea.