(Page 1 of 8) - Mathematics of College Algebra Students: The Interplay between Students’ Self-Efficacy and Formal Mathematical Beliefs authored by Cifarelli, Victor. and Goodson-Espy, Tracy.

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MATHEMATICS OF COLLEGE ALGEBRA STUDENTS: THE

INTERPLAY BETWEEN STUDENTS’ SELF-EFFICACY

AND FORMAL MATHEMATICAL BELIEFS

Victor V. Cifarelli

University of North Carolina at Charlotte

## email not listed ##

Tracy Goodson-Espy

Appalachian State University

## email not listed ##

This paper reports the results of a study of the beliefs of College Algebra students. Subjects
(N=195) came from College Algebra classes at two universities in the southern United States.
Data sources included a mathematical beliefs survey instrument and on-going individual
interviews conducted with 30 of the students. We address two types of mathematical beliefs,
formal mathematical and self-efficacy beliefs, and focus on the inter-connections of these as
students solve mathematics problems. Of particular interest is how the students' knowledge of
formal mathematical concepts co-exists with their efficacy beliefs in the course of on-going
mathematical activity. Drawing from episodes of interviews with one student, the analysis
explains the complexity of the students’ mathematical beliefs and how these impact the solvers’
on-going problem solving actions.

There is growing consensus that the traditional College Algebra (CA) course is not helping

students become quantitatively literate citizens (Hastings, 2006). Based on long-term
demographic studies, high D/F/W course rates and the failure of CA to provide students with
applicable skills, the Conference to Improve College Algebra has called for revamping the CA
course (Small, 2002). Other research conducted on CA students surveyed their mathematical
beliefs (Frank, 1986), documented their often fragmented conceptual understandings (Carlson,
1997), and examined the effectiveness of instructional strategies (Underwood-Gregg and
Yackel, 2000). However, few studies examined how mathematical beliefs influence the ways
students interpret and solve mathematics problems. More needs to be known about how beliefs
influence students’ initiative and efficacy in problem solving.

Purpose and Theory

Our goal is to improve our understanding of the interactions between a student’s

mathematical beliefs and his/her problem solving actions so that we can develop effective
intervention strategies. Drawing from the work of Cooney, Shealy, and Arvold (1998), we view
the learner’s mathematical beliefs as complex mental structures that aid his/her interpretations
in mathematical situations. Muis (2004) noted that previous research employed either a
qualitative approach, observing students’ problem solving, or a quantitative approach, using
students’ self-reported survey responses. Studies employing either methodology found
significant relationships between students’ mathematical beliefs, their engagement with
mathematical tasks, and achievement. Muis recommended that further studies employ both
qualitative and quantitative analyses in order to develop a more robust understanding of
interactions among beliefs, learning, and achievement. Our study included surveys and
repeated observations of students solving problems. This is compatible with Vergnaud’s (1984)
notion that solvers demonstrate their conceptions as “mathematical beliefs-in-action” as they
solve problems, and that beliefs serve as conceptual models where solution activity may
develop (Vergnaud, 1984, p.7). Often, CA students are viewed as lacking both formal
conceptual knowledge and self-efficacy in their actions. We believe this is not the case; the

Lamberg, T., & Wiest, L. R. (Eds.). (2007). Proceedings of the 29

annual meeting of the North

American Chapter of the International Group for the Psychology of Mathematics Education,
Stateline (Lake Tahoe), NV: University of Nevada, Reno.