Doctor of Philosophy / Department of Curriculum and Instruction / Jennifer M. Bay-Williams / The purpose of this study was to assess the impact of student attitudes in a college algebra mathematics classroom when lessons are primarily composed of standards-based pedagogy. National reports advocate for a change in teaching K-12. Nowhere is this more needed than in community colleges where students are taught in traditional formats and rarely challenged to make connections between mathematics and their personal experiences. A thorough review of the literature shows the need for mathematics reform at every level, including the college mathematics classroom. There are several national reports, Principles and Standards for School Mathematics, Adding it Up, How People Learn, and Undergraduate Programs and Courses in the Mathematical Sciences, that have been published to address the need to change mathematics teaching and learning. They are advocates for the implementation of standards-based instruction into the mathematics classroom. This study focused on students’ perceptions about the nature of mathematics and learning mathematics, specifically, does such a learning environment impact students’ perceptions of being a student of mathematics in the areas of confidence, anxiety, enjoyment, and motivation, and relevance of mathematics in personal and professional experiences. Over the course of one semester, two sections of college algebra students participated in the study. By using both qualitative and quantitative data collection methods, the study was able to see if there was an impact in student attitudes toward mathematics. The standards-based pedagogy used in this study was cooperative learning, problem solving, discourse, and the graphing calculator. Changes in attitude were determined by attitudinal surveys, student questionnaires, observations, and focus groups. College algebra students had a statistically significant change in their enjoyment of mathematics. Although the other attitudes, confidence, motivation, and value did not have a statistically significant change, the qualitative data indicates a change in these attitudes did occur. This study identified that cooperative learning, problem-solving, discourse, and graphing calculators increased student confidence in doing mathematics because they felt more competent in working problems on exams. Students also found the class enjoyable, anxiety was reduced as students became more familiar with the instructional strategies, and students recognized the value of mathematics for job skills and personal business.

Doctor of Philosophy / Department of Educational Leadership / Sarah Jane Fishback / This study explored the effects of personal epistemological beliefs about mathematics and beliefs about the ability to do well in mathematics on achievement in a college-level, developmental mathematics class. The influences of gender, age, and ethnicity on these beliefs as they relate to mathematics achievement were also explored. The Mathematics Belief Scales (MBS) was adapted from the Indiana Mathematics Belief Scales and Self-Description Questionnaire III to measure beliefs about the time it takes to solve mathematics problems, the importance of conceptual understanding in mathematics, the procedural emphasis in mathematics, the usefulness of mathematics, and self-concept about mathematics. MBS was administered to 159 participants enrolled in Intermediate Algebra over two semesters at an urban, state-supported mid-western university and two small private mid-western universities. Responses to the surveys and scores on the final exams for the Intermediate Algebra courses were analyzed using descriptive statistics, the Pearson product-moment correlations, analysis of variance techniques, and hierarchical regression analysis. Results indicated that students generally held nonavailing beliefs about mathematics and mathematics self-concept. Students typically believed that mathematical problems should be solved within ten minutes. Students generally did not believe that math problems can be solved with logic and reason instead of learned math rules. Over 40% of the students did not believe that mathematics beyond basic mathematics was useful to everyday life. Students were also generally not confident in their ability to solve mathematics problems. Additionally, men’s self-concept was significantly higher than women’s self-concept. Adult learners’ self-concept was also significantly higher than traditional age students’ self-concept. Hierarchical regression analyses revealed that the importance of understanding mathematical concepts positively influenced final exam scores for men more so than women and self-concept positively influenced final exam scores for women more so than men. These results indicate a need for academic experiences at the college-level that will challenge students’ current belief system and provide an environment that is supportive and conducive to building individual self-confidence.

Doctor of Philosophy / Curriculum and Instruction Programs / Margaret G. Shroyer / This study was designed to investigate teaching practices of mathematics teachers in rural high schools in Kansas in the context of the NCTM Principles and Standards. National reports advocate for change in the mathematics classroom while state assessments force teachers to focus on test scores. This study investigated the extent to which teachers whose students experienced repeated success on state assessments integrated the NCTM Process and Content Standards into the mathematics classroom. Those data were then compared with the teaching practices in schools whose students repeatedly did poorly on state assessments. This two-phase study used both quantitative and qualitative data from four main sources: survey, interview, observation, and collection of artifacts. Phase I surveyed all mathematics teachers in high performing and low performing rural high schools throughout the state of Kansas. Data collected in Phase I were used to examine differences and similarities in teaching practices of teachers from high and low performing schools. During Phase II qualitative data were collected and analyzed to further explore any existing patterns among high performing and low performing schools. Results from teachers in high and low performing schools were compared and contrasted to determine if there were differences between the teaching practices that were demonstrated by each group of teachers. Results of surveys, interviews, observations, and artifacts revealed teachers in high performing schools used a variety of different representations to teach and assess a topic while those teachers from low performing schools used one or two representations. Students from high performing schools had more frequent opportunities to communicate with the teacher to gain additional assistance in learning the mathematics content. Teachers in high performing schools also used formal assessment strategies as part of the learning process more consistently than their counterparts from low performing schools. Results from interviews, observations, and artifacts reveal that teachers in high and low performing schools implement teaching practices aligned with the algebra content standards in a very similar manner.

Doctor of Philosophy / Curriculum and Instruction Programs / Diane McGrath / The goal of this study was to explore the use of interdisciplinary PBL projects for teaching mathematical concepts according to NCTM (2000) goals for mathematics instruction. This study sought to answer the question: what are the teaching issues and evidence of student learning of mathematical concepts over a series of three interdisciplinary PBL projects involving STELLA™ modeling software which are designed to engage students, integrate technology, and provide a context for learning mathematics based on the 5 NCTM (2000) goals? HyperStudio™ was used as a communication tool with which students built artifacts of understanding. This study was a naturalistic case study employing videotaped observations, interviews, student-peer reviews and student generated artifacts of learning as data sources. Data were categorized into two variable clusters: Teaching and Learning. Implementation issues for three computer-based PBL simulations are discussed. Themes that emerged from analysis of the data are grouped into teaching themes and learning themes. Themes relating to teaching include the struggle to form a community of learners, relevancy of the simulations to middle school students, need for group-worthy projects, helping students balance creativity and content, lesson adaptation, and critical review and student reflection on constructive feedback. Findings of the study suggest the students were able to meet a majority of the expected content goals. Themes relating to learning include the struggle to find a balance between creativity and content, ownership and control, engagement with the simulations, and students’ ability to think and express themselves mathematically. Recommendations are made for teachers who wish to implement PBL, simulations, and similar teaching strategies and for researchers who are studying similar learning environments.

Doctor of Philosophy / Department of Secondary Education / Jacqueline D. Spears / Moses & Cobb (2001) argue that algebra is a “civil right” and assert that limited algebraic understanding has an unfavorable impact on African American students’ entry into post-secondary education. Gay (2000) outlines six pedagogical methods, known as culturally responsive teaching (CRT), which emphasize the importance of teachers creating learning environments that relate to the personal experiences and cultural perspectives of minority students. The National Council of Teachers of Mathematics (NCTM) prescribes five process standards (communication, problem solving, connections, representation, and reasoning and proof) and the Equity Principle (includes setting high expectations, responding to the needs of culturally and linguistically diverse students, and providing support) for effective mathematics instruction. CRT, the NCTM Process Standards, and the NCTM Equity Principle served as the conceptual framework for this mixed-method study. Thirty-four teachers from two elementary and two middle schools in one school district in the Midwest responded to The Powell Teaching Mathematics Index (PTMI), a five-option Likert survey that explored teachers’ current “use” and “desire” to use CRT methods, NCTM process standards, NCTM Equity Principle, and teachers’ personal efficacy in learning and teaching mathematics in general and in algebra. Results from the PTMI revealed that teachers had a “desire” to use CRT in mathematics with AA students (M=4.41, SD=0.70); and although there was more variance among respondents, teachers also reported a “desire” to use process standards in algebra with AA students (M=3.94, SD=1.03). One bivariate correlation revealed a relationship between “use” of process standards in general and “efficacy” (r =0.681, p[less than or equal to]0.01). Eight volunteer teachers participated in a professional development workshop on CRT and integrated one of the six pedagogical methods into their classrooms for one month. Teachers reported “strengths” from the implementation phase as: increased student engagement, transition from teacher-directed to student-directed learning and an increase in student confidence in mathematics. Implementation “strains” were reported as: a time consuming process, difficulty in providing individual attention and an increase in classroom noise level. Findings have implications for teacher education programs, local school district and teacher networks.

Doctor of Philosophy / Curriculum and Instruction Programs / Jennifer M. Bay-Williams / This study examined end-of-the-year seventh grade students’ rational number knowledge using comparison tasks and rational number subconstruct tasks. Comparison tasks included: comparing two decimals, comparing two fractions and comparing a fraction and a decimal. The subconstructs of rational number addressed in this research include: part-whole, measure, quotient, operator, and ratio. Between eighty-six and one-hundred-one students were assessed using a written instrument divided into three sections. Nine students were interviewed following the written instrument to probe for further understanding. Students were classified by error patterns using decimal comparison tasks. Students were initially to be classified into four groups according to the error pattern: whole number rule (WNR), zero rule (ZR), fraction rule (FR) or apparent expert (AE). However, two new patterns emerged: ignore zero rule (IZR) and money rule (MR). Students’ knowledge of the subconstructs of rational numbers was analyzed for the students as a whole, but also analyzed by classification to look for patterns within small groups of students and by individual students to create a thick, rich description of what students know about rational numbers. Students classified as WNR struggled across almost all of the tasks. ZR students performed in many ways similar to WNR but in other ways performed better. FR and MR students had more success across all tasks compared to WNR and ZR. On average apparent experts performed significantly better than those students classified by errors. However, further analysis revealed hidden misconceptions and deficiencies for a number of apparent experts. Results point to the need to make teachers more aware of the misconceptions and deficiencies because in many ways errors reflect the school experiences of students.

Doctor of Philosophy / Curriculum and Instruction Programs / Margaret G. Shroyer / While there has been much quantitative research done in the area of attitudes and self-efficacy beliefs, this study sought hear the voices of the middle school child. Therefore, this qualitative study investigated the attitudes toward mathematics and mathematics self-efficacy beliefs of middle school students in one middle school in western Kansas. The conceptual framework for this study is supported by the research of Albert Bandura on Social Cognitive Theory. This study used a naturalistic inquiry approach and data were collected from multiple sources, including short-answer questionnaires, classroom observations, and one-on-one interviews. Coded data were examined for patterns, themes, and relationships. Middle school students in this study exhibited positive, negative, and variable attitudes toward mathematics, and both positive and negative mathematics self-efficacy beliefs. Students attribute their high mathematics self-efficacy beliefs to the teacher or the high grades they receive on daily assignments, as well as the scores they receive on state and local assessments. Conversely, middle school students have low mathematics self-efficacy beliefs when they feel unsuccessful or distressed, and they attribute those beliefs to the low grades they receive on daily assignments and assessments, as well as the distress of not understanding the mathematics. Middle school students told their mathematical stories of the change in attitudes toward mathematics and mathematics self-efficacy beliefs, and attributed positive changes to the mathematics teacher. Negative changes in attitudes toward mathematics and mathematics self-efficacy beliefs were attributed to the amount of homework expected at the middle school level, as well as the lack of hands-on activities. The influence of the teacher, grades, and hands-on activities impact middle school students’ attitudes toward mathematics and mathematics self-efficacy beliefs. There is a relationship between attitudes toward mathematics and mathematics self-efficacy beliefs. Low mathematics self-efficacy beliefs and poor attitudes toward mathematics are related since low mathematics self-efficacy beliefs and poor attitudes toward mathematics are highly connected. Conversely, high mathematics self-efficacy beliefs and good attitudes toward mathematics are highly related. Middle school students’ experiences impact both mathematics self-efficacy beliefs and attitudes toward mathematics. Students’ mathematics self-efficacy beliefs impact their attitudes toward mathematics.

Doctor of Philosophy / Curriculum and Instruction Programs / Gail Shroyer / David Allen / This study was designed to gather input from early career elementary teachers with the goal of finding ways to improve elementary mathematics methods courses. Multiple areas were explored including the degree to which respondents’ elementary mathematics methods course focused on the NCTM Process Standards, the teachers’ current standards-based teaching practices, the degree to which various pedagogical strategies from mathematics methods courses prepared preservice teachers for the classroom, and early career teachers’ suggestions for improving methods courses. Both qualitative and quantitative methodologies were used in this survey study as questions were of both closed and open format. Data from closed-response questions were used to determine the frequency, central tendencies and variability in standards-based preparation and teaching practices of the early career teachers. Open-ended responses were analyzed to determine patterns and categories relating to the support of, or suggestions for improving, elementary mathematics methods courses. Though teachers did not report a wide variation in the incorporation of the NCTM Process Standards in their teaching practices, some differences were worth noting. Problem Solving appeared to be the most used with the least variability in its frequency of use. Reasoning, in general, appeared to be used the least frequently and with the most variability. Some aspects of Communication, Connections and Representation were widely used and some were used less frequently. From a choice of eight methods teaching practices, ‘Observing in actual classrooms or working with individual students’ and ‘Planning and teaching in actual classrooms’ were considered by early career teachers to be the most beneficial aspects of methods courses.