Mathematics Content Courses Research Articles

Effects of an Additional Mathematics Content Course on Elementary Teachers' Mathematical Beliefs and Knowledge for Teaching

This longitudinal study examines the effects of changes in an elementary teacher preparation program on mathematics beliefs and content knowledge for teaching of two groups of prospective teachers (N = 276): (1) those who completed a program with three mathematics content courses and two mathematics methods courses and (2) those who completed a program with four mathematics content courses and a single mathematics methods course. The results reveal salient benefits of a second methods course that were not evident in the new program with only one methods course. Further, the addition of a fourth content course did not result in notable differences in mathematical knowledge for teaching. In addition, mathematical knowledge for teaching was positively linked to change in pedagogical beliefs about learners, further illuminating the interwoven nature of knowledge and beliefs.

An Examination of the Relationship Between Prospective Teachers' Dispositions and Achievement in a Mathematics Content Course for Elementary Education Majors

1. James E. R. Beyers[1][1] 1. 1The College of New Jersey, Ewing, USA 1. James E. R. Beyers, School of Education, The College of New Jersey, Ewing, NJ 08628, USA Email: beyers{at}tcnj.edu The purpose of this study was to examine the relationship between prospective teachers’ dispositions and their achievement in a mathematics content course for elementary education majors. Research suggests that components of student dispositions with respect to mathematics can mediate learning via how students may or may not take advantage of opportunities to learn. Consequently, one might expect that there would be measurable differences in the relationship between students’ dispositions and achievement in mathematics. A total of 107 prospective teachers who enrolled in a teacher preparation program at a midsized, mid-Atlantic university participated in this study by taking a researcher-designed survey, the Mathematics Dispositional Functions Inventory. Achievement was measured by the average of grades participants earned on two exams given in the mathematics course. Three achievement groups (high, moderate, and low) were determined based on the distribution of achievement scores. Results indicate that significant differences in prospective teachers’ dispositions with respect to mathematics exist according to achievement in mathematics in two of the three primary categories of dispositional functioning, affective and conative, but not in the third category, dispositional cognitive functioning. [1]: #aff-1

Mathematics Preservice Teachers Learning About English Language Learners Through Task-Based Interviews and Noticing

This article describes the 3rd cycle of an intervention in a mathematics content course that was designed to foster awareness among middle school mathematics preservice teachers (PSTs) of the challenges that English language learner (ELL) students face and the resources they draw on as they learn mathematics and communicate their thinking in English-only classrooms. Pairs of PSTs engaged 2 different ELL students in a videotaped task-based interview using 4 measurement tasks. Following each interview, the PSTs wrote a structured report guided by Mason's (2002) framework of noticing. The results of the intervention indicated that the PSTs went beyond awareness of ELLs' needs and challenges and also adopted strategies outlined in the literature that were aligned with best practices for teaching ELLs. The article also discusses the potential of the intervention and how it can be used by other mathematics educators.

Who teaches mathematics content courses for prospective elementary teachers in the United States? Results of a national survey

This article reports on a national survey of higher education institutions in the United States to answer the question, “Who teaches mathematics content courses for prospective elementary teachers, and what are these instructors’ academic and teaching backgrounds?” We surveyed 1,926 institutions, and a faculty member from each of 825 institutions (42.8 %) participated in the survey. The survey results demonstrate that the majority of institutions are not meeting the recommendations of the Conference Board of the Mathematical Sciences (The mathematical education of teachers. American Mathematical Society, Providence, RI, 2001), the National Council of Teachers of Mathematics (Professional standards for teaching mathematics. National Council of Teachers of Mathematics, Reston, VA, 2005), and the National Council on Teacher Quality (Greenberg and Walsh 2008) for prospective elementary teachers to take at least nine credits of mathematics content designed specifically for them. Additionally, most instructors for these courses do not have elementary teaching experience and have likely not had opportunities to think deeply about the important ideas in elementary mathematics, and most institutions do not provide training and/or support for these instructors.

The Potential of Prospective Teachers Experiencing Differentiated Instruction in a Mathematics Course

Differentiated instruction is a promising approach for supporting the diverse needs of students. To support prospective teachers with the challenge of implementing differentiated instruction in their future instruction and to meet their diverse needs as college students, we decided to incorporate differentiated instruction in our mathematics content course for them. The specific objective of this study was then to examine how the prospective elementary teachers experienced differentiated instruction in the mathematics content course, whether it met their varying instructional needs, and how it may impact their future mathematics teaching. Qualitative and quantitative analyses revealed that the prospective teachers found the differentiated instruction supportive of their diverse needs and that they plan to incorporate similar features in their future instruction. However, the prospective teachers' comments did reveal additional aspects about differentiated instruction to learn and strengthen. Suggestions for revising the mathematics content course and enhancing the differentiated instruction are provided along with directions for further research.

Fraction Operations: An Examination of Prospective Teachers’ Errors, Confidence, and Bias

Fractions are important in young students’ understanding of rational numbers and proportional reasoning. The teacher is fundamental in developing student understanding and competency in working with fractions. The present study spanned five years and investigated prospective teachers’ competency and confidence with fraction operations as they entered a college mathematics content course designed for teachers. Results indicate prospective teachers’ levels of competence vary by fraction operation. Many of the same error patterns exhibited in the 1980s and early 1990s are still exhibited today. Many prospective teachers inaccurately predict their performance on multiplying fractions with relatively prime denominators and dividing reciprocal fractions.

Motivation and Attitude of Preservice Elementary Teachers toward Mathematics

This study investigated preservice elementary teachers' achievement goal orientations for learning mathematics and the relationship of those goals to their attitudes toward mathematics. Self-report instruments were administered to assess three achievement goals—mastery, performance-approach, and performance-avoid, and three constructs of attitude—confidence in learning mathematics, usefulness of mathematics, and mathematics as a male domain. The preservice teachers were higher in mastery goals than in performance goals, and performance-avoid goals were higher than performance-approach goals. Mastery goals correlated positively to all three constructs of attitude. Since mathematics classes are traditionally performance-oriented, these results suggest a mismatch between personal and classroom goals that could result in negative attitudes and the adoption of maladaptive performance-avoid goals. These findings suggest that mathematics content courses for preservice elementary teachers should be taught in a classroom climate that supports and encourages mastery goals.

Extending Fibonacci numbers to negative subscripts through problem solving

This classroom note shows how Fibonacci numbers with negative subscripts emerge from a problem-solving context enhanced by the use of an electronic spreadsheet. It reflects the author's work with prospective K-12 teachers in a number of mathematics content courses.

A Local Instruction Theory for the Development of Number Sense

Gravemeijer's (1999, 2004) construct of a local instruction theory suggests a means of offering teachers a framework of reference for designing and engaging students in a set of sequenced, exemplary instructional activities that support students' mathematical development for a focused concept. In this paper we offer a local instruction theory to guide the design of a set of instructional activities in support of the development of number sense. We make explicit the goals, assumptions, underlying rationale, and related instructional activities and provide examples from a mathematics content course for preservice elementary teachers. In this way, we contribute to an elaboration of the construct of local instruction theory.

“What Was Really Accomplished Today?” Mathematics Content Specialists Observe a Class for Prospective K–8 Teachers

One of the important activities mathematics teacher educators engage in is the development of teachers at both the in-service and pre-service levels. Also of importance is the professional development of these professional developers. In the summer of 2004, a summer institute was held that allowed mathematics teacher educators watch the teaching of a mathematics content course for prospective K–8 teachers. This paper examines the manner in which a specific group of mathematics content specialists experienced this professional development.

A Learning Community for Pre-Service Elementary Teachers: A Collaboration Between Mathematics and Education

This article describes a collaboration between two faculty members, one from mathematics and the other from education, to strengthen the preparation of pre-service elementary school teachers for teaching mathematics. The model used is a learning community which combines a mathematics content course offered by the mathematics department, with a mathematics teaching methods course taught by the education department. This article details the design features of the learning community and its rationale and administrative implementation. Course materials and grading policy are discussed, as well as methods of assessment and interpretation of assessment. Of special note, this paper reports the benefits of the learning community as perceived by students and faculty.

The lived experiences of elementary prospective teachers in mathematics content coursework

Teaching mathematics to elementary children requires highly developed content knowledge, and questions of how teachers acquire this knowledge are of significant concern to teacher preparation programs. Even with research to the contrary, some policy makers in the USA propose additional advanced mathematics courses. One voice that is missing when considering this problem is the voice of the student (i.e. prospective teacher). To address this student‐vacant area of research and acknowledge the important role students should have in shaping research, we conducted a phenomenological inquiry into the experiences of elementary prospective teachers in mathematics content courses. The students described domains of mismatch between the courses and the professional program, affective reactions to the courses, and perceptions of effective classroom practices. These findings should inform decision making in the professional development of elementary teachers.

A Cross Discipline Study of Reformed Teaching by University Science and Mathematics Faculty

Researchers observed 28 university faculty in graduate and undergraduate science and mathematics courses in a three‐year, longitudinal study of teaching. Subjects were selected from five higher education institutions in the state where faculty had a two‐ to five‐year association with a reform‐based NSF professional development program. Observer field notes and a researcher‐designed observation instrument were used to describe and compare teaching practices. The participants in this study taught science and mathematics content courses for majors, non‐majors, as well as education majors. Our observation method created a descriptive profile that enabled comparisons across a variety of teaching contexts. The main results showed that the faculty lacked practices that supported development of divergent thinking, conceptual thinking, and metacognitive thinking skills. Observed differences in the profiles of instruction included greater emphasis on real‐world connections in science and greater use of cooperative problem solving in mathematics. A case study of a large lecture course describes reformed teaching practices in even the most challenging selling.

Prospective elementary teachers’ motivation to participate in whole-class discussions during mathematics content courses for teachers

Prospective elementary teachers’ (N = 148) motivation to participate in whole-class discussions during mathematics content courses for teachers, as expressed in their own words on an open-ended questionnaire, were studied. Results indicated that prospective teachers were motivated by positive utility values for participating (to achieve a short-term goal of learning mathematics or a long-term goal of becoming a teacher), to demonstrate competence (to achieve performance-approach goals), or to help others (to achieve social goals). Negative utility values for participating were expressed by those who preferred to learn through actively listening. Five motivational profiles, as composed of interactions among motivational values, beliefs, goals and self-reported participation practices, were prevalent in this sample. Self-reported variations among participants’ utility values and participation practices suggested that prospective teachers engaged differentially in opportunities to learn to communicate mathematically. Results provide pedagogical learner knowledge for mathematics teacher educators.

Infusing assessment into mathematics content courses for pre-service elementary school teachers

The present study aims to explore the use of assessment in mathematics content courses for future elementary school teachers. Analysis of self assessment data on mathematical understanding and peer assessment data on oral mathematical presentation showed that pre-service teachers had a balanced understanding of procedural knowledge and problem solving. Conceptual understanding was not in the structure of pre-service teachers’ mathematical knowledge. Understandings of conceptual knowledge, procedural knowledge, and problem solving had no meaningful effects on gains in mathematics performance. Aspects of oral mathematical presentation were associated with improved understanding of procedural knowledge and in particular conceptual knowledge. The result of the study calls for a conceptual approach to mathematical knowledge and sufficient mathematical problem solving in college-level mathematics content courses and in particular the infusion of assessment into college-level mathematics education for pre-service teachers.

Mathematics for Middle School Teachers: Choices, Successes, and Challenges

The opportunities for mathematics department faculty from institutions of higher education (IHEs) to work with middle school mathematics teachers are on the rise; the U.S. Department of Education has allocated almost $800 million for mathematics and science partnerships since 2004 that require collaboration between faculty from IHEs departments of arts and sciences and school districts. Changes in credentialing requirements due to the No Child Left Behind Act of 2001 mean that middle school mathematics teachers must take more mathematics content courses, yet current offerings for teachers in many mathematics departments typically focus on elementary or high school mathematics and often neglect the needs of middle school teachers. This article talks about the efforts of a group of mathematics faculty and school district personnel to develop mathematics courses that would help middle school teachers develop a Profound Understanding of Middle School Mathematics. An explanation of the curriculum structure – highlighted by the teachers’ responses – is given here.

Mathematical tasks designed to foster a reconceptualized view of early arithmetic

We discuss mathematical tasks used in a first mathematics content course for elementary teachers at our university to foster a deep conceptual understanding of early arithmetic, including basic concepts of number, number relationships and strategies, and coordinating units of different rank. Our approach is to immerse our students in a base 8 world for up to six weeks. A key aspect is that we develop base 8 vocabulary. We use base 8 analogs of instructional sequences developed in classroom teaching experiments in the elementary grades that have been proven successful to promote deep conceptual understandings of basic arithmetic and place-value numeration in young children. As a result, our students have unique opportunities to develop a reconceptualized view of early arithmetic and learn how it can be advanced.

Supporting Teacher Learning: Our Voyage with Knowing and Teaching Elementary Mathematics

When teaching mathematics content courses designed for preservice elementary teachers, we often meet resistance when we ask our students to learn mathematics conceptually; they consider us out of touch with what works in the classroom. Preservice elementary teachers question the need for conceptual understanding, feeling certain that elementary students cannot and will not understand mathematics in this way.

Supporting Teacher Learning: Using NSF–Funded Middle School Materials in a University Mathematics Content Course

As mathematics teacher educators, we grapple with determining the depth of mathematical knowledge required of prospective elementary school teachers in our mathematical content courses. We struggle with identifying the mathematics appropriate for these courses as well as making the mathematics relevant for these teachers. If these classes are housed in a department of mathematics, as ours are, this discussion must take into account the expectations of our departmental colleagues. We examined these issues in spring 2001, when we were asked to redesign a mathematics content course for preservice elementary school teachers. In this article, we describe some of our experiences, including our decision to use middle school mathematics curricular materials as the primary textbooks.

Technology Tips: Ferrari's Method and Technology

Welcome to another year of “Technology Tips.” I, Suzanne Harper, will be the primary editor for the 2005–2006 column issues, and it is my pleasure to introduce this year's new co-editor, Shannon Driskell. Shannon teaches undergraduate and graduate mathematics content courses at the University of Dayton in Ohio. Her main interests include the appropriate use of technology in K–12 mathematics, preservice and inservice teachers' pedagogical content knowledge, and the teaching and learning of geometry. We are always looking for new and interesting ways for teachers and students to use technology effectively. If you have a tip that can help other teachers learn how to use a technology application in the classroom, please send ideas to my contact address. I also would like to take the time to thank Hollylynne Stohl Lee for her amazing dedication and guidance as editor of the column last year.