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

(2) [LL] two sessions weekly of traditional summary lecture with teacher-presented examples on the topics to be covered in computer-assisted instruction, and (3) [GL] a blend of treatments (1) and (2), with one weekly meeting traditional lecture, and one weekly meeting inquiry-based group work.

Students registered for one of four time periods in the Fall 2009 semester schedule for one 50minute class meeting and one 50-minute required lab meeting. Students in each time slot were randomly assigned to one of the two treatments for the semester, similar to (1) designated [G] and (2) designated [L], above, with just one class meeting per week. Each instructor involved taught all treatments, and all instructors had previous experience in both didactic and inquirybased teaching. Each instructor also met with his/her class in the mathematics computer lab.

Data gathered during the experiments in Fall 2009 and Fall 2010, and reported by Mayer (2010, 2011) on the two cohorts of elementary algebra students, included (1) course grades and test scores, (2) pre-test and post-test of content knowledge based upon a test which incorporated three open-ended problems, (3) for the 2010 cohort only, pre-test and post-test of content knowledge based upon a test consisting of 25 objective questions, (4) student course evaluations using the online IDEA system (IDEA 2010), and (5) RTOP observations of the instructors (RTOP 2010, Sawada 2002).

For this study, in Summer 2011, (6) data on performance of students in the next mathematics course taken after the elementary algebra course was collected from the university data base. At the time of submission of this paper, student performance in subsequent courses was available for Spring 2010, Summer 2010, Fall 2010, and Spring 2011. Thus, we have more data on performance in subsequent courses for the Fall 2009 experimental cohort than for the Fall 2010 cohort. By the time of the RUME 2012 meeting, we expect to have data for Summer 2011 analyzed, and possibly also for Fall 2011.

Results of the Research. Analysis of student success in subsequent courses, as measured by students’ final grade in the next course, was analyzed by using the comparisons of means independent t-test with an alpha of 0.05. Students’ grades in subsequent courses were coded as follows: A-5, B-4, C-3, D-2, and F-1. Figure 4 depicts statistics on students’ grades for the Fall 2009 cohort in their subsequent math course making no distinction between subsequent courses.

There was no significant difference between student grades in the next course based on the MA098 treatment (G or L) they received. Figure 5 breaks down the Fall 2009 cohort based on the specific subsequent course taken: MA110 is finite mathematics and is taught only in an inquiry-based/computer-assisted format and MA102 is Intermediate Algebra, taught only in a lecture/computer-assisted format. There was no significant difference between MA098 treatment groups for either MA110 or MA102 as the next course, though the MA098(L)MA102 trajectory narrowly missed significance. There were three treatments in the Fall 2010 cohort: GG, LL, and GL. Figure 6 shows data on how these treatment groups compared pair-wise based on student success in subsequent courses, making no distinction 492 15TH Annual Conference on Research in Undergraduate Mathematics Education between the next two possible courses. There were no significant differences between any of the three MA098 treatments as measured by final grades in subsequent courses. In summary, we found no differences in success in subsequent courses ascribable to treatment in MA098.

Questions for Further Research/Analysis. We will be analyzing data about subsequent courses for the 2010 cohort to include in our final report. We would like the audience’s input on

**the following:**

1. What additional data on students would be useful if we want to try to understand the differences between students going on to MA110 (a terminal mathematics course) versus students going on to MA102 (a pre-requisite for pre-calculus algebra)?

2. What would be a reliable and rigorous way to determine what impact the treatment has on a student’s course trajectory?

Implications for Practice. We now teach all regular sections of elementary algebra following the blended treatment of the Fall 2010 experimental cohort: three class meetings weekly, one inquiry-based, one lecture, and one in the lab. We made our decision to change MA098 instruction prior to analyzing student success in subsequence courses based upon gains on openended problems and student satisfaction. In view of the inherent coherence of algebra-related topics cutting across courses (Oehtrman, 2008), we expect to extend this study in subsequent years to credit courses such as intermediate algebra, pre-calculus algebra, and pre-calculus trigonometry, all of which presently incorporate computer-assisted instruction together with one weekly lecture meeting, and all in the course trajectory leading to calculus.

494 15TH Annual Conference on Research in Undergraduate Mathematics Education

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496 15TH Annual Conference on Research in Undergraduate Mathematics Education Factors influencing students’ propensity for semantic and syntactic reasoning in proof writing: A case study Juan Pablo Mejia-Ramos1, Keith Weber1, Evan Fuller2, Aron Samkoff1, Kevin Calkins1 1- Rutgers University 2- Montclair State University Abstract. We present a case study of an individual student who consistently uses semantic reasoning to write proofs in calculus but infrequently uses semantic reasoning to write proofs in linear algebra. The differences in these reasoning styles can be partially attributed to his familiarity with the content, the teaching styles of the professors who taught him, and the time he was given to complete the tasks. These results suggest that there are factors, including domain, instruction, and methodological constraints, that researchers should consider when ascribing to students a proving style that have been ignored in previous research.

Keywords: Proof; Proving styles; Semantic proof productions; Syntactic proof productions Introduction and research questions In recent years, mathematics educators have noted that there are two qualitatively distinct ways to produce formal mathematical proofs (e.g., Raman, 2003; Vinner, 1991; Weber & Alcock, 2004). A prover can concentrate on the formal and logical aspect of proving, starting with appropriate definitions and hypotheses, carefully formulating what needs to be proven, and applying theorems and other valid rules of inference to these starting points until the desired conclusion is reached. This is sometimes referred to as a syntactic proof production and reasoning in this way is referred to as syntactic reasoning.

Alternatively, a prover can try to represent relevant mathematical objects, explore their properties, and see why the theorem is true using informal representations of mathematical concepts, such as exploring prototypical examples, diagrams, or graphs, and using this insight as the basis for constructing a formal proof. This is referred to as a semantic proof production and reasoning in this way is referred to as semantic reasoning.

Researchers have recently advanced a number of intriguing hypotheses about these constructs (and related constructs). First, based primarily on case studies, some researchers have hypothesized that some students rely predominantly on one form of reasoning in most of their proof production tasks—that is, we can reasonably refer to some students as syntactic provers or semantic provers (e.g., Alcock & Inglis, 2008;

Alcock & Simpson, 2004, 2005; Alcock & Weber, 2010a, 2010b; Burton, 2004;

Moutsios-Rentzos, 2009; Pinto & Tall, 1999, 2002; Weber, 2009). From hereon, we refer to students’ propensity to use semantic or syntactic reasoning as their proving style.

Second, again based on case studies, some researchers have speculated that there is not a strong correlation between proving styles and success in proof writing in advanced mathematics (e.g., Alcock & Simpson, 2004, 2005; Pinto & Tall, 1999; Weber, Alcock, & Radu, 2005). We have recently begun a NSF-funded large-scale study on examining 15TH Annual Conference on Research in Undergraduate Mathematics Education 497 the proving processes of 100 mathematics majors to assess the viability of these hypotheses.

A primary purpose of this preliminary report is to gain feedback on research goals and methodologies of our project from the undergraduate mathematics education research community. However, we also want to present a research finding. While numerous case studies have illuminated a consistency in individual students’ propensity to use syntactic or semantic reasoning in proof writing (e.g., Alcock & Inglis, 2008; Alcock & Simpson, 2004, 2005; Alcock & Weber, 2010a, 2010b; Burton, 2004; Pinto & Tall, 1999, 2002;