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

15TH Annual Conference on Research in Undergraduate Mathematics Education 419 Preliminary Report 8 Sierpinska, A., Trgalova, J., Hillel, J., & Dreyfus, T., (1999). Teaching and Learning Linear Algebra with Cabri. Research Forum paper, in The Proceedings of PME 23, Haifa University, Israel, Volume 1, 119–134.

Stewart, S., & Thomas, M.O.J. (2009). A framework for mathematical thinking: the case of linear algebra. International Journal of Mathematical Education in Science and Technology,40(7), 951-961.

Wilensky, U. (1991). Abstract mediations on the concrete and concrete implications for mathematics education. In I. Harel, & S. Papert (Eds.), Constructivism (pp. 193-204).

Norwood, NJ: Ablex Publishing Corporation.

420 15TH Annual Conference on Research in Undergraduate Mathematics Education Title: The Role of Technology in Constructing Collaborative Learning Spaces Preliminary Research Report Authors: Brian Fisher and Timothy Lucas, Pepperdine University Abstract: Traditionally, research on technology in mathematics education focuses on interactions between the user and the technology, but little is known is about how technology can facilitate interaction among students. In this preliminary report we will explore the role that iPads versus traditional laptops play in shaping the learning spaces in which students explore concepts in business calculus. We will report on classroom observations and a series of small-group interviews in which students explore the concepts of local and global extrema. Our preliminary results are that the introducing the iPad, a portable device with intuitive applications, enhances collaboration by allowing students to transition back and forth from private to public learning spaces.

Keywords: learning spaces, classroom technology, iPad, social constructivism, business calculus Proposal: For the past half-century mathematics educators have been contemplating the role of technology in mathematics education. Recent decades have seen signiﬁcant growth in student

**access to technology in the classroom. Among the key strands of research are:**

• Handheld devices and calculators, e.g. (Burrill et al., 2002).

• Technology designed to accumulate real data for student exploration, e.g. (Konold & Pollatsek, 2002).

• Dynamic geometry software and other microworlds, e.g. (Jones, 2000).

Like the strands mentioned above, the bulk of research on technology in mathematics education focuses on interactions between the user and the technology. Little is known about how individuals use technology to interact with one another. However, the current generation of undergraduates is likely to incorporate technology throughout their social interactions with each other. In this preliminary report we will explore how students use iPads while negotiating mathematical meaning in a community of learners.

There are many ways that technology can facilitate learning, but our goal is to understand the role of technology in facilitating joint explorations of mathematical concepts. We view a students understanding of mathematics to be directly impacted by both the medium in which the student encounters the concept and the interactions of the student with others in his/her learning community.

Our study of interaction leads us to draw, primarily, from the perspective of social constructivism, which views learning as an inherently social process, e.g.(Vygotsky, 1978; Cobb & Yackel, 1996;

Stephan & Rasmussen, 2002). However, we view technology as one of many ways in which a student may physically interact with a mathematical concept, and we view these interactions via technology as a signiﬁcant element of our students understanding of mathematics. This viewpoint u˜ leads us to take the perspective of embodied cognition (Lakoff & N´ nez, 2000) in the sense that we cannot divorce the ways a student may physically interact with concept from their perception of the concept. By taking this perspective we are emphasizing the physical role of technology within 15TH Annual Conference on Research in Undergraduate Mathematics Education 421 student interactions and, in particular, students abilities to convey to their peers their embodied understanding of a concept developed using technology.

The motivation for this study originated with a university wide study of the effectiveness of the iPad as a classroom tool. In the fall of 2010, Pepperdine University distributed iPads to one section of Business Calculus along with two applications, Numbers (spreadsheet) and Graphing Calculator HD. Students used the iPads both inside and outside the classroom for the entire semester. In contrast, a second section of the course used laptops throughout the course with Excel and a java graphing applet. Much of the course is designed around activities that allow students to reconstruct mathematical principles within a small group setting. The university study focused on the effect the iPad had on student performance on speciﬁc learning outcomes, but during that fall study we became aware of how the iPads were changing the social dynamic in the classroom. This prompted a revised study that focused on recording student interaction in two sections of Business Calculus in the fall of 2011.

In order to analyze the role technology plays in collaborations we adapted Granott’s framework for student interaction (Granott, 1993). Granott’s two dimensional model is contructed from the relative expertise of the students in a group and the degree of interactions among the group. Our framework incorporates a third dimension which measures the depth of conversation amongst the students. We also chose to borrow the notion of public and private spaces from a study that contrasts a class that uses private handheld devices with one that incorporates public handheld devices that connect to shared LCD displays (Liu et al., 2009). This language of private versus public spaces allows us to describe the role that iPads and laptops play in constructing learning spaces.

In Figure 1 we present some diagrams of student behavior that depict the three dimensions of student interaction. The ﬁrst group of diagrams depicts students working in parallel, either in isolation from one another or with some discussion that is limited to simply verifying answers.

Here the students use technology entirely as a private space to interact with the mathematics. The second group of diagrams demonstrates how students may choose to use the technology as a public learning space. Within this public space, a strong student may use the technology as a teaching tool or two or more students at similar levels may use the technology to collaborate. In those cases the conversations about the mathematics may be richer and more meaningful.

**We are currently using the following qualitative methods to conduct this study:**

1. Classroom Observations: We will record student behavior during in-class activities using the three-dimensional framework outlined above.

2. Group interviews: We will conduct a series of small-group interviews focusing on the concepts of local and global extrema. Students often approach these concepts from a purely computational perspective, but would beneﬁt from the use of technology to visualize the problem. We will observe how students incorporate technology while negotiating the problem with their classmates.

From our study in the fall of 2010 we have already seen evidence of how students can transform the private space on their iPad into a public space. For example, we observed a lesson on limits that requires the use of spreadsheet and graphing calculator. During that lesson we witnessed that the size and portability of the iPad allowed students to share their screens as part of their dialogue.

The fact that the class is using a uniform device also facilitated students assisting each other in the learning process. Throughout the class activities the students were fully engaged and did not stray 422 15TH Annual Conference on Research in Undergraduate Mathematics Education to online distractions. In contrast, students with personal laptops had trouble working as a team due to the physical barriers that their screens presented. Students using laptops often chose not to share their screens with others unless there was a speciﬁc request from another group member.

The private spaces created by laptops also tempted several students strayed to Facebook. Our taskbased interviews revealed that students working with the iPad immediately incorporated graphs into their calculations of maximums and minimums. The students with laptops were reluctant to turn them on and only did so when the problems became too complex to solve by hand.

Based on our experiences this semester, we would like to ask for feedback on future analysis

**of our data. We ask the audience to consider the following questions:**

• Is there relevant literature that we have not considered?

• Are there other means of interpreting the data that we have not considered?

• As we re-examine the videos, are there other types of interactions that we might observe?

• The university conducted a survey of general technology use for the students involved in the study. Should we use these surveys to classify students by technological comfort and track how that inﬂuences student interaction with the technology and each other?

• The criteria for the university-wide study included having one section taught with iPads and one section taught without. Is the comparison between the iPad section and the section where students use personal laptops of interest to the mathematical education community?

15TH Annual Conference on Research in Undergraduate Mathematics Education 423 References Burrill, G., Allison, J., Breaux, G., Kastberg, S., Leatham, K., & Sanchez, W. (2002). Handheld graphing technology at the secondary level: Research ﬁndings and implications for classroom practice. Texas Instruments.

Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational Psychologist, 31(3), 175–190.

Granott, N. (1993). Patterns of interaction in the co-construction of knowledge: Separate minds, joint effort, and weird creatures. In R. H. Wozniak, & K. W. Fisher (Eds.) Development in context: Acting and thinking in speciﬁc environments, (pp. 183 – 207). Lawrence Erlbaum Associates.

Jones, K. (2000). Providing a foundation for deductive reasoning: Students’ interpretations when using dynamic geometry software and their evolving mathematical explanations. Educational Studies in Mathematics, 44, 55–85.

Konold, C., & Pollatsek, A. (2002). Data analysis as the search for signals in noisy processes.

Journal for Research in Mathematics Education, 33(4), 259–289.

u˜ Lakoff, G., & N´ nez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books.

Liu, C.-C., Chung, C.-W., Chen, N.-S., & Liu, B.-J. (2009). Analysis of peer interaction in learning activities with personal handhelds and shared displays. Educational Technology & Society, 12(3), 127 – 142.

Stephan, M., & Rasmussen, C. (2002). Classroom mathematical practices in differential equations.

The Journal of Mathematical Behavior, 21(4), 459 – 490.

Vygotsky, L. S. (1978). Mind in society: Development of higher psychological processes. Cambridge, MA: Harvard University Press.

424 15TH Annual Conference on Research in Undergraduate Mathematics Education Student Note Taking Behavior in Proof-based Mathematics Classes Tim Fukawa-Connelly1, Aaron Weinberg2, Emilie Wieser2, Sarah Berube1 and Kyle Gray1

1. Department of Mathematics and Statistics, The University of New Hampshire

2. Department of Mathematics, Ithaca College

**Abstract:**

There is a need to explain the relationship between teaching (classroom activities) and learning. This study is one attempt to explore student note-taking as a form of mediation between teaching and learning outcomes. We will adapt the theoretical framework described by Weinberg and Wiesner (2010), who applied ideas of literary criticism to describe factors that impact the ways students read and understand mathematics textbooks. The two concepts we will use are those of the implied reader and reading models.

We are investigating student note-taking in the context of a proof-based abstract algebra class that is taught, primarily, via a lecture. We are recording the lectures and creating a set of expert-notes that are then compared with the student notes. We then interview the students to better understand the decisions that they make vis-à-vis note taking and how they “read” the text of a lecture.

Keywords: codes, behaviors, competencies, students reading of lecture, proof-based mathematics

0. Introduction and Background The focus of most upper-level mathematics courses is on presentations of definitions, theorems and proofs of key results. Although some “inquiry-based” curricula have been designed as alternatives to the standard curriculum (e.g. Davison & Gulick, 1976; Dubinsky & Leron, 1994; Larsen, 2004), most of these courses are still lecture-focused. In order to understand how students learn the material from these lectures, it is important to understand how various aspects of the lecture relate to what the students “take away.” While there are studies relating the taking of notes with later scores (Bligh, 2000;

Johnstone & Su, 1994; Kiewra et al., 1991), these studies are focused on recall and subsequent exam performance, primarily in lower-level undergraduate courses. There are studies of how students make sense of presented proofs (Mejia-Ramos, et al., 2010) and studies of how students read textbooks (e.g. Weinberg et al., 2011), but there is no corresponding study of how students “read” the text of a lecture. In this vein, research on student learning, even of topics directly related to undergraduate coursework, is often done without reference to the teaching that the students experienced and how the students made sense of their classroom experience (e.g., Mejia-Ramos, et al., 2010. As a result, there is a need to explain the relationship between teaching (classroom activities) and learning. This study is one attempt to explore student notetaking as a form of mediation between teaching and learning outcomes.