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

After the pre-test, students in both control and test sections completed the same assignments on sequences and series in the online homework system. Students in the test section participated in additional in-class activities which emphasized comprehension of mathematical passages read by the students and completed several quizzes assessing reading comprehension of series convergence arguments. These passages were adapted from or excerpts from Stewart’s Calculus with Early Transcendentals (Stewart, 2008) and were able to be used as models for students’ own proofs. Assessments of students’ reading comprehension were designed using a model we adapted from Mejia-Ramos et al (2010).

The second in-class examination, covering convergence and divergence of series of constants, served as the post-test. The post-test required students to determine convergence of series and justify their arguments, but it did not directly test reading comprehension. All three 486 15TH Annual Conference on Research in Undergraduate Mathematics Education researchers will score and analyze error types for students in both sections.

After final grades were submitted, two students were interviewed from each section. The interview subjects were selected from the pool of volunteers as having roughly comparable scores on the pre-test. During the interview, subjects were asked to read an argument concerning the convergence of a series and were asked to explain the argument and to answer various questions about it.

Preliminary analysis Preliminary analysis based on the scoring of the pre-test and post-test by the class instructors shows no clear advantage on exam 2 to students who completed the reading comprehension activities and assessments. However, interview subjects who had completed the reading comprehension activities showed a greater degree of facility with the reading tasks requested during the interview than the subjects from the control section who had not completed any reading comprehension activities.

Further analysis of the pre-test and post-test will be conducted by the researchers. We will score the test papers from both sections with a common rubric and will compare scores with each other to look for agreement. We will then analyze the relative change from exam 1 to exam 2 for students from both sections, based on the uniform scoring of exams. Additionally we will code the types of errors seen, to look for any possible improvement in particular types of errors by students in the test section as compared to the control section.

Further analysis of the interviews will attempt to determine if the students from the test section read the mathematical argument differently from students in the control section, if they comprehended what they read differently, and if they can apply the general method in a new example (Mejia-Ramos et al, 2010). The researchers will look for instances that highlight how the student is reading the mathematics, such as evidence that they are able to re-state an argument in their own words or that they understand the big picture instead of just trying to read line-by-line, as noted in previous studies (Selden and Selden, 2003). The analysis will also look for evidence that students comprehend what they read. This evidence may come from students’ answers to the reading comprehension questions in the interview, or from whether or not they are able to use the argument as a resource.

Questions Can we improve the reading comprehension activities and quizzes to lead to better results?

Are there additional ways other than the assignments given to promote reading comprehension? Are there additional tasks or interview questions that can assess reading comprehension?

Are there other places in the calculus sequence where it would be valuable to promote better reading comprehension by our students?

References Alcock, L. & Simpson, A. (2004). Convergence of sequences and series: Interactions between visual reasoning and the learner's beliefs about their own role. Educational Studies in Mathematics, 57(1), 1-32.

15TH Annual Conference on Research in Undergraduate Mathematics Education 487 Alcock, L. & Simpson, A. (2005). Convergence of sequences and series 2: Interactions between nonvisual reasoning and the learner's beliefs about their own role. Educational Studies in Mathematics, 1, 77-100.

Brandau, L. (1990). Rewriting our stories of mathematics. In A. Sterrett (Ed.) Using Writing to Teach Mathematics (pp. 73-77). Washington, D.C.: Mathematical Association of America.

Burn, B. (2005). The vice: Some historically inspired and proof-generated steps to limits of sequences. Educational Studies in Mathematics, 60, 269-295.

Mejia-Ramos, J. Weber, K., Fuller, E., Samkoff, A., Search, R., Rhoads, K., (2010). Modeling the comprehension of proof in undergraduate mathematics. Proceedings of the 13th Annual Conference on Research in Undergraduate Mathematics Education.

Porter, M.K. (1996). The effects of writing to learn mathematics on conceptual understanding and procedural ability in introductory college calculus. Dissertation Abstracts International, 58, 2576A.

Porter, M. K. & Masingila, J. O. (2000). Examining the effects of writing on conceptual and procedural knowledge in calculus. Educational Studies in Mathematics, 42, 165-177.

Roh, K. H. (2008). Students’ images and their understanding of definitions of limit of a sequence. Educational Studies in Mathematics, 69, 217-233.

Roh, K. H. (2010). An empirical study of students’ understanding of a logical structure in the definition of limit via the ε-strip activity. Educational Studies in Mathematics, 73, 263Selden, A., & Selden, J. (2003). Validations of proofs written as texts: Can undergraduates tell whether an argument proves a theorem? Journal for Research in Mathematics Education, 34, 4-36.

Siegel,M., Borasi,R. & Fonzi, J. (1998). Supporting students' mathematical inquiries through reading. Journal for Research in Mathematics Education, 29(4), 378-413.

Stewart, J. (2008). Calculus: Early Transcendentals (6th ed.). Mason, OH: Cengage Learning.

Stickles, P. R & Stickles, J. A., Jr. (2008). Using reading assignments in teaching calculus.

Mathematics and Computer Education, 42(1), 6-10.

Watkins, A. E. (1979). The symbols and grammatical structures of mathematical english and the reading comprehension of college students. Journal for Research in Mathematics Education, 10 (3) 216-218.

488 15TH Annual Conference on Research in Undergraduate Mathematics Education Weinburg, A. & Wiesner, E. (2011). Understanding mathematics textbooks through readeroriented theory. Educational Studies in Mathematics, 76, 49–63.

15TH Annual Conference on Research in Undergraduate Mathematics Education 489 Improving Student Success in Developmental Algebra and Its Impact on Subsequent Mathematics Courses John C. Mayer and William O. Bond Mathematics Department, University of Alabama at Birmingham Preliminary Report1 Abstract. One direction taken by course reform over the past few years has been the use of computer-assisted instruction, often applied to large-enrollment service courses in mathematics, and justified in part by cost-effectiveness. Elementary algebra is typically taken by undergraduate students who do not place into a credit course. The goal of such a developmental algebra course has been to enhance students' “algebra skills,” for example, dealing procedurally with rational expressions. Higher-order thinking may be largely absent from such an approach.

Our motivating question is “What approach maximizes the student’s chance to succeed in subsequent courses?” In view of our theoretical perspective that an inquiry-based approach enhances learning, a subsidiary question is “Is it effective to blend a focus on skills development with a focus on problem-solving?” Results of the analysis, not yet complete, suggest that effectiveness is a matter of what student outcomes are valued, balanced against costeffectiveness.

Introduction. An elementary algebra course often is taken by undergraduate students who do not place into a credit-bearing course. Traditionally, the goal of such a developmental algebra course has been to enhance students' “algebra skills,” for example, dealing procedurally with rational numbers and expressions. While this is a form of active learning, higher-order thinking may be largely absent from such an approach. Our motivating question is “What pedagogical approach maximizes the student’s chance to succeed in subsequent courses?” In view of our theoretical perspective that an active learning approach enhances learning in STEM courses, a subsidiary question is “Is it possible to blend a focus on skills development (through computerassisted instruction) with a focus on problem-solving (through cooperative group learning)?” Research Question. Three studies (Mayer 2009, 2010, 2011) relevant to the current research compared treatments using quasi-experimental designs. The fundamental difference between the treatments in the two studies of a developmental algebra course (2010, 2011) was (1) incorporating one or more inquiry-based class meetings, or (2) incorporating lecture class meetings, both together with a common computer-assisted learning component. In the current research, which uses additional data gathered on the algebra student cohorts, we ask the question, “Does the treatment have a statistically significant effect on student success in the next mathematics course taken?” Theoretical Perspective. Our research is based on the premise that active learning (Prince

2004) promotes retention of knowledge, concept development, and problem-solving (Marrongelle and Rasmussen 2008). We take computer-assisted instruction, a form of active learning, as a ground – the figure is blending with another type of active learning: inquiry-based Keywords: inquiry-based learning, computer-assisted instruction, blended instruction developmental algebra, elementary algebra.

490 15TH Annual Conference on Research in Undergraduate Mathematics Education learning (IBL) in the form of collaborative small group work and whole-group sharing. We comment here only on the figure.

In their extensive report on the IBL Mathematics Project, Laursen (et al. 2011) identifies several features of IBL “typical of their project.” These features correlate well with the dimensions of the RTOP instrument for classroom observation (RTOP 2010, Sawada 2002).

Where Laursen identifies features of the course, we modify this and list features of the class

**meeting:**

1. The main work of the class meeting is problem-solving (e.g., Savin-Baden and Major 2004; Prince and Felder 2007).

2. Class goals emphasize development of skills such as problem-solving, communication, and mathematical habits of mind (e.g., Duch, et al. 2001; Perkins and Tishman 2001).

3. Most of the class time is spent on student-centered instructional activities, such as collaborative group work (e.g., Gillies 2007; Johnson, et al. 1998; Gautreau and Novemsky 1997; Cohen 1994).

4. The instructor’s main role is not lecturing, but guiding, asking questions, and giving feedback; student voices predominate in the classroom (Alrø and Skovsmose 2002).

5. Students and instructor share responsibility for learning, respectful listening, and constructive critique (e.g., Goodsell, et al. 1992; Lerman 2000; Prince 2004).

The inquiry-based treatments (identified as G, GG, or GL above) were designed to incorporate these features.

Prior Research and Relation to Literature. Three recent studies (Mayer et al. 2009, 2010, 2011), simultaneously compared different pedagogies over one semester. There are few such direct comparisons in the literature (examples: Doorn 2007, Gautreau 1997, Hoellwarth 2005;

literature review: Hough 2010a, 2010b). Nearly all previous studies have focused on courses at the calculus level and above (Hough 2011a). The results of the quasi-experimental studies of a finite mathematic course (2009), and of an elementary algebra course (2010, 2011) showed in all cases that students in the inquiry-based treatment(s) did significantly better (p0.05) comparing pre-test and post-test performance in the areas of problem identification, problemsolving, and explanation (see Figures 1 and 2). Moreover, students, regardless of treatment, performed statistically indistinguishably when compared on the basis of course test scores.

Outcomes of the first two studies by Mayer differed in gain in accuracy, pre-test to post-test: in the finite mathematics study, there was no significant difference between treatments, but in the first elementary algebra study there was a significant difference between treatments in favor of the inquiry-based treatment. In those studies, accuracy was assessed on a small set of openended problems. In the second elementary algebra study, the pre/post-test had both an openended and an objective portion. There was no significant difference among treatments in the second elementary algebra study with regard to the objective part of the pre/post-test. Mayer (2011) reported that students were distinctly more satisfied with a pedagogical approach that included at least some lecture meetings (see Figure 3).

Research Methodology. The methodology in (Mayer 2010, 2011) was quasi-experimental in that it sought to remove from consideration as many confounding factors as possible, to assign treatment on as random a basis as possible (constrained only by students being able to choose the time slot in which they take the course), and then to compare results for the same cohort of students. All students involved in the courses had identical computer-assisted instruction provided in a mathematics learning laboratory.

15TH Annual Conference on Research in Undergraduate Mathematics Education 491 This methodology was described completely in (Mayer 2010, 2011). For completeness herein, we briefly describe the experimental set-up. Students registered for one of three time periods in the Fall 2010 semester schedule for two 50-minute class meetings and one 50-minute required lab meeting. Students in each time slot were randomly assigned to one of the three

**treatments for the semester:**

(1) [GG] two sessions weekly of inquiry-based collaborative group work (random, weekly changing, groups of four) without prior instruction, on problems intended to motivate the topics to be covered in computer-assisted instruction;