Mathematics Content Courses Research Articles

Expert mathematics teacher educators’ purposes and practices for providing prospective teachers with opportunities to develop pedagogical content knowledge in content courses

Enhancing teachers’ pedagogical content knowledge (PCK) is essential to improving the teaching and learning of mathematics. Mathematics teacher educators (MTEs) need to help prospective teachers enhance PCK (Marks in J Teach Educ 41(3):3–11, 1990. doi: 10.1177/002248719004100302 ; Mason 2008). However, we know very little about the practices of MTEs, especially in mathematics content courses, as these practices are not widely researched or disseminated (e.g., Bergsten and Grevholm, in: Jaworski, Wood (eds) The international handbook of mathematics teacher education, vol 4, Sense Publishers, Rotterdam, pp 223–246, 2008; Floden and Philipp, in: Lester, Ferrini-Mundy (eds) Proceedings of the NCTM research catalyst conference, National Council of Teachers of Mathematics, Reston, pp 171–176, 2003). This phenomenographical study offers empirical findings on commonly identified purposes across ten expert MTEs who provided K-8 prospective teachers with opportunities to develop PCK in their mathematics content courses. Furthermore, our emergent findings indicated that expert MTEs also provided opportunities for prospective teachers to develop orientations toward teaching the subject, which prompted framework adaptations and articulations of “orientations” as a construct discussed in the broader literature outside of mathematics teacher education research (Magnusson et al., in: Gess-Newsome, Lederman (eds) Examining pedagogical content knowledge, Kluwer, Dordrecht, pp 95–132, 1999). Research and practitioner implications from this study provide specific PCK-related learning opportunities of prospective teachers through the lenses of expert MTEs’ (personal and professional) purposes and reflections on teaching, as a foundation on which the field can continue building future research and MTEs can continue building their practice in mathematics content courses.

Mathematics teacher educators’ perspectives on their design of content courses for elementary preservice teachers

This descriptive case study examines six mathematics teacher educators’ (MTEs) perspectives on their design of content courses for elementary preservice teachers. By focusing on MTE’s design goals and considerations for their mathematics content courses, the means by which they achieved these course design goals, and the challenges they encountered as they carried out their design goals, this study adds to the image of mathematics content courses, and sheds light on the work of MTEs. Our findings indicate that MTEs design content courses that take a learner-centered approach to instruction and that in designing learner-centered content courses, MTEs face a variety of challenges. Implications for the professional development of MTEs are discussed.

Graduate Teaching Assistants' Enactment of Reasoning-and-Proving Tasks in a Content Course for Elementary Teachers

Graduate teaching assistants serve as instructors of record for numerous undergraduate courses every semester, including serving as teachers for mathematics content courses for elementary preservice teachers. In this study, we examine 6 teaching assistants' teaching practices in the context of a geometry content course for preservice teachers by focusing on their enactment of reasoning-and-proving tasks. Results indicate that teaching assistants engaged preservice teachers in a variety of reasoning-and-proving activities. For 42 of 82 tasks observed, preservice teachers' engagement in reasoningand-proving processes decreased relative to the potential for reasoning and proving in mathematical tasks. This investigation into teaching assistants' teaching practices identifies factors associated with their enactment of reasoning-and-proving tasks (e.g., generating student participation). This research has implications for professional development to support college mathematics instructors' teaching.

Prospective Elementary Teachers Making Sense of Multidigit Multiplication: Leveraging Resources

This study examines how collective activity related to multiplication evolved over several class sessions in an elementary mathematics content course that was designed to foster prospective elementary teachers' number-sense development. We document how the class drew on as-if-shared ideas to make sense of multidigit multiplication in terms of partial products and to reason flexibly about products. We document how the class overcame the challenge of accounting for partial products in multidigit multiplication, including particular activities and ways of reasoning that facilitated progress. The results provide insights into how prospective elementary teachers' understanding of multidigit multiplication can develop during a content course and how a sequence of instructional activities and practices can productively leverage the resources that they bring in support of that development.

Inviting Prospective Teachers to Share Rough Draft Mathematical Thinking

Engaging prospective elementary teachers (PTs) in participating productively by making their exploratory (rough draft) thinking public during class discussions remains a constant challenge for instructors of mathematics content courses for teachers, in part because of perspectives incoming PTs may hold about interacting in academic settings. In this article, we share the effects of an intervention designed to confront PTs' incoming perspectives. PTs were provided with opportunities to label the level of completeness and correctness of their thinking before they displayed and discussed their written work publicly during a mathematics content course for teachers. Results indicated that labeling their work increased PTs' level of comfort with sharing their thinking and awareness of the value of doing so. PTs also reported that the label served as a reflection tool. The label increased the PTs' productive disposition in terms of comfort level with taking intellectual risks when doing mathematics and reflecting on their work.

“Math Is All Around Us and … We Can Use It to Help Us”: Teacher Agency in Mathematics Education Through Critical Reflection

As social-justice mathematics teacher educators, we have always engaged and supported the prospective and practicing teachers in our classes in critical reflection about mathematics and the teaching and learning of mathematics (Felton, 2010, 2012; Felton & Koestler, 2012; Koestler, 2012). Although there are different notions, purposes, and traditions of reflection (Zeichner & Liston, 1996), critical reflection goes beyond teachers simply recounting their experiences and instead engages them in attempting to unpack and make sense of their experiences in terms of the social, political, and cultural contexts in which their work takes place. We believe this can support them in taking up more progressive pedagogies that can lead to more equitable and just mathematics education for all students.While we did not originally think about our work in terms of teacher agency, writing about this here and elsewhere (Felton & Koestler, 2012) has provided an opportunity to consider what role agency plays in our work, what agency in mathematics teacher education means to us, and how we may better support developing this form of agency in our future work as mathematics teacher educators. We have been collaborating as “critical friends” (Anderson, Herr, & Nihlen, 2007) for the past eight years by discussing the content and pedagogy of our courses, our theoretical frames, and our research methods. While, in this article we discuss data that were generated in one semester of a mathematics content course for prospective teachers taught by Felton, analyzing and theorizing about the data was a collaborative effort.

Implementation and outcomes of inquiry-based learning in mathematics content courses for pre-service teachers

This mixed-methods study describes classroom characteristics and student outcomes from university mathematics courses that are based in mathematics departments, targeted to future pre-tertiary teachers, and taught with inquiry-based learning (IBL) approaches. The study focused on three two-term sequences taught at two research universities, separately targeting elementary and secondary pre-service teachers. Classroom observation established that the courses were taught with student-centred methods that were comparable to those used in IBL courses for students in mathematics-intensive fields at the same institutions. To measure pre-service teachers' gains in mathematical knowledge for teaching, we administered the Learning Mathematics for Teaching (LMT) instrument developed by Hill, Ball and Schilling for in-service teacher professional development. Results from the LMT show that pre-service teachers made significant score gains from beginning to end of their course, while data from interviews and from surveys of learning gains show that pre-service teachers viewed their gains as relevant to their future teaching work. Measured changes on pre-/post-surveys of attitudes and beliefs were generally supportive of learning mathematics but modest in magnitude. The study is distinctive in applying the LMT to document pre-service teachers' growth in mathematical knowledge for teaching. The study also suggests IBL is an approach well suited to mathematics departments seeking to strengthen their pre-service teacher preparation offerings in ways consistent with research-based recommendations.

Examining and elaborating upon the nature of elementary prospective teachers’ conceptions of partitive division with fractions

The purpose of this study was to examine and elaborate upon elementary prospective teachers’ (PSTs) conceptions of partitive division with fractions. We examined the degree to which PSTs’ conceptions were connected (i.e., capable of translating between representations correctly; aware that partitive division generates a unit rate for its quotient) and flexible (i.e., capable of differentiating between opportunities to partition or iterate (or both) when solving a partitive division task; aware that partitioning or iterating (or both) could be associated with the operation of division, as appropriate). Seventeen PSTs participated in task-based interviews prior to instruction in a mathematics content course for teachers. These PSTs demonstrated disconnected conceptions of partitive division with fractions when they incorrectly translated between representations and either inconsistently or did not express awareness that the purpose of the task was to generate a unit rate. These PSTs demonstrated rigid conceptions of partitive division with fractions such that they did not express awareness that the process of iterating could be associated with the operation of division, even when they obtained a correct answer by iterating. Results extend prior research by looking beyond PSTs’ performance on tasks to elaborate upon PSTs’ conceptions of the operation of partitive division. This study contributes new insights into PSTs’ conceptions that can be used by mathematics teacher educators to inform the design of future instructional interventions.

Mathematics Perceptions of Pre-service Elementary School Teachers

The purpose of this study is to explore what mathematics perceptions are held by pre-service teachers. Sixty-eight pre-service elementary school teachers enrolled in a mathematics content course for elementary school teachers completed Mathematics Perceptions Survey (MPS) to measure their mathematics perceptions. All items were presented in the form of a 7-point rating scale. The questionnaire contained questions related to pre-service teachers’ perceptions of what mathematics is and how to do well in it, what mathematics solutions should be, how mathematics problems can be solved, how mathematics should be taught, and how mathematics is learned, mathematics genius to do mathematics, the nature of school mathematics, and student motivation. Findings indicated that the beliefs pre-service elementary school teachers reflected in the instrument items were in line with the current reform movements in mathematics education. The pre-service elementary school teachers generally indicated positive beliefs to the questionnaire items about mathematics. For instance, they gave importance to appreciate developing different ways of solutions to the same problem, why a solution to a mathematics problem works, think in mathematics can be creative and discover things by themselves, and apply mathematics knowledge and skills to real life. However, the results revealed that the pre-service elementary school teachers held several moderate and negative beliefs. The data indicate that pre-service elementary school teachers’ definition of mathematics reflects more of a subject that is all about symbols, numbers, equations, formulas, and procedures that are to be memorized. The results also indicated that pre-service elementary school teachers do not agree that the subject matter is interesting, and that learning the discipline will help them think clearly. Motivation is among the most powerful determinants of students’ success or failure in school. These results suggest that mathematics educators need to develop pre-service teachers’ motivation towards mathematics; help pre-service teachers understand and value the need for perseverance in solving mathematics problems; and help students realize the connections between the mathematics they learn in school and the mathematics-related fields that might interest them.

Using Mathematics Literature with Prospective Secondary Mathematics Teachers.

Literature in mathematics has been found to foster positive improvements in mathematics learning. This manuscript reports on a mathematics teacher educator’s use of literature via literature circles with 11 prospective secondary mathematics teachers in a mathematics content course. Using survey and reflection data, the author found that prospective teachers expressed either benefiting or not benefitting from the literature circle practice in this mathematics class. Finally, conclusions and recommendations for future work are shared.

Sustaining Social and Sociomathematical Norms with Prospective Elementary Teachers in a Mathematics Content Course

As students contribute to social and sociomathematical norms, they re-organize their own understandings. As such, the purpose of this article is to describe the ways in which prospective elementary teachers contributed to the norms that were established and re-established throughout an entire semester in an undergraduate mathematics content course. Findings illustrate that norms must be renegotiated between an instructor and prospective teachers when the mathematics content changes.

Journal Writing in a Mathematics Capstone Course for Prospective Secondary Teachers: Future Teachers Making Connections

In the context of a Capstone course for prospective secondary teachers (PSTs) of mathematics, this paper examines how a journal assignment was implemented and how it can help us understand PSTs’ views of mathematics and mathematics teaching. First, the course design and specifics of the assignment are described. Then, details about implementation of the assignment are included. Finally, affordances of incorporating this type of assignment are discussed with examples of PSTs’ responses. The identified benefits include PSTs: (a) connecting new mathematical ideas to their prior knowledge; (b) identifying what they do not know; and (c) articulating the importance of reasoning and proving in mathematics teaching and learning. These benefits support the notion that using a journal assignment in mathematics content courses designed for PSTs can help them develop a profound understanding of mathematics content and practices.

Differences in Beliefs Across a Series of Four Mathematics Content Courses

Differences in Beliefs Across a Series of Four Mathematics Content Courses

Toward Improving the Mathematics Preparation of Elementary Preservice Teachers

Research suggests the importance of mathematics knowledge for teaching (MKT) for enabling elementary school teachers to effectively teach mathematics. MKT involves both mathematical content knowledge (M‐CK) and mathematical pedagogical content knowledge (M‐PCK). However, there is no consensus on how best to prepare elementary preservice teachers (PSTs) to achieve M‐CK and M‐PCK. This study builds on research related to MKT by investigating influences of mathematics content courses designed specifically for elementary PSTs (IMPACT courses—Impact of Mathematics Pedagogy and Content on Teaching) on their attitudes (i.e., confidence and motivation) toward M‐CK and M‐PCK. Results suggest that the PSTs who participated in these IMPACT courses not only acquired high levels of confidence and motivation toward M‐CK, but also showed significant and greater gains in attitudes toward M‐PCK, after taking the required mathematics methods course, than their counterparts. Further, the findings suggest that these IMPACT courses provided a mathematical foundation that allowed the PSTs to engage in mathematics teaching methods better than those PSTs who did not have such a foundation. These results suggest potential course experiences that may enhance M‐CK and M‐PCK for elementary PSTs.

What Do Secondary Preservice Mathematics Teachers Need to Know? Content Courses Connecting Secondary and Tertiary Mathematics

T he recently released report The Mathematical Education of Teachers II (MET-II) [Conference Board of the Mathematical Sciences (CBMS), 2012] notes that, while most secondary mathematics teachers major in mathematics, many of the courses taken focus on mathematics needed for graduate study or careers in business. Prospective mathematics teachers could be better served by courses that “provide opportunities for future teachers to learn the mathematics they need to know to be wellprepared beginning teachers who will continue to learn new mathematical content and deepen their understanding of familiar topics” [CBMS, 2012, p. 5]. At present we appear to be in an exciting period of innovation and experimentation in the development of such courses, as mathematicians with interest in secondary education have created a wide array of new courses designed specifically for mathematics majors studying to become teachers. Teacher education programs include a variety of courses designed to improve teachers’ knowledge. These courses fall largely into two categories: mathematics courses offered by mathematics departments and teaching methods courses offered by education departments. Many teaching methods courses have begun to incorporate content knowledge, especially as it relates to the mathematics prospective teachers will teach and the specialized mathematical knowledge needed for the specific tasks of teaching [Ball, Thames, and Phelps, 2008]. Yet there does not appear to have been substantial change in the types of mathematics content courses that prospective teachers (as mathematics majors) take, with one exception. There are a growing number of courses offered by mathematics departments where prospective teachers learn new mathematics while simultaneously building connections between tertiary and secondary mathematics. These connections courses are presumed to help prospective secondary mathematics teachers (PSMT) situate and motivate the mathematics that they will teach. The impetus for the increasing number of connections courses has come about for at least two reasons. First, research over the past twenty-plus years has articulated the need for such knowledge, expanded our thinking about what mathematics teachers need to know, and pushed us to think about how this knowledge develops. Second, we see an increasing number of productive and sustained collaborations between mathematicians and mathematics educators, such that many in the Eileen Murray is a postdoctoral research fellow at Harvard University in the Graduate School of Education. Her email address is eileen_murray@harvard.edu.

Providing Space for Elementary Prospective Teachers' Viewpoints on Mathematics Content Courses: A Two-Dimensional Model of Learning

This multiple case study examined the experiences of two groups of elementary prospective teachers (N = 12) completing distinct mathematics content courses. The findings reveal perspectives on knowing, learning, and teaching mathematics as experienced in the context of these courses. One group characterized mathematics as a “record of knowledge,” difficult to understand and lacking in relevance; learning occurred through rote memorization and via external expertise, with teaching typified as explaining. The other group portrayed mathematics as process-focused, internally constructed, and relevant; learning took place through a focus on children's thinking, with teaching characterized as guiding and questioning. The prospective teachers' mathematical beliefs and affect also emerged as key findings, coupled with the quantitative data revealing differences in specialized content knowledge for teaching mathematics. Two salient dimensions emerged as promoting learning in the courses: caring classroom practices and curricular relevance. These factors are situated in a two-dimensional model offering considerations for practice.

Developing Preservice Teachers' Mathematical Knowledge for Teaching: Making Explicit Design Considerations for a Content Course

Research has highlighted the nature of the mathematical work in teachers' practice. However, preservice mathematics coursework often too narrowly focuses on the development of common content knowledge and not enough on the development of specialized content knowledge, a kind of mathematical knowledge that is specific to the work of teaching mathematics. We offer three design principles that have informed a mathematics content course for elementary preservice teachers, and we provide learning outcomes data that suggest the overall content course experience supports specialized content knowledge development. We provide relevant examples from our own work to illustrate how we have applied these design principles in our local context. Our aim is to begin a dialogue about principled design considerations for content courses for preservice teachers.

An exploratory study of pre-service middle school teachers’ knowledge of algebraic thinking

Using algebraic habits of mind as a framework, and focusing on thinking about functions and how they work, we examined the relationship between 18 pre-service middle school teachers’ ability to use the features of the algebraic thinking (AT) habit of mind “Building Rules to Represent Functions” and their ability to recognize and interpret the features of the same AT habit of mind in middle school students. We assessed the pre-service teachers’ own ability to use the AT habit of mind Building Rules to Represent Functions by examining their solutions to algebra-based tasks in a semester-long mathematics content course. We assessed the pre-service teachers’ ability to recognize and interpret students’ facility with the AT habit of mind Building Rules to Represent Functions by analyzing their interpretations of students’ written solutions to algebra-based tasks and their ability to plan and analyze AT interviews of middle school students during a concurrent field-based education course. The data revealed that the pre-service teachers had a limited ability to recognize the full richness of algebra-based tasks’ potential to elicit the features of Building Rules to Represent Functions in students. The pre-service teachers’ own overall AT ability to Build Rules to Represent Functions was related to their ability to recognize the overall ability of students to Build Rules to Represent Functions, as exhibited during one-on-one interviews, but not to their ability to recognize the overall ability to Build Rules to Represent Functions exhibited exclusively in students’ written work. Implications for mathematics teacher education are discussed.

The juxtaposition of instructor and student perspectives on mathematics courses for elementary teachers

This qualitative study examined perspectives of two key stakeholder groups, instructors and students, on mathematics content courses for prospective elementary teachers (Mathematics for Teachers [MFT] courses). A collective case study approach, which drew from the data of two cases in different but comparable settings, contributed to the robustness of the findings. Cross-case analysis of the interview data revealed several convergent themes: the role of affect in student learning, pedagogy and instructor disposition, connections to the elementary classroom, and mathematics content. The findings included both conflicting and complementary perspectives between the two key stakeholder groups. When juxtaposed, the multiple viewpoints offer insights into some of the central issues related to teaching and learning in MFT courses and suggest potential avenues for improving experiences in these courses.

Measuring mathematical knowledge for teaching: a longitudinal study using two measures

This longitudinal study examines growth in teacher knowledge as measured by two popular assessments—Learning Mathematics for Teaching (LMT) and Diagnostic Teacher Assessments in Mathematics and Science (DTAMS). Using data collected from 24 teachers, we compare the extent to which each assessment captured teacher learning during a K-8 mathematics content/pedagogy hybrid course and a general mathematics content course. Teachers made large gains on both measures, but the LMT better captured gains made during the hybrid course, whereas DTAMS better detected gains during the mathematics course. Patterns in teacher performance suggest substantive differences between specialized mathematical knowledge for teaching and everyday mathematics knowledge. The theoretical and practical implications of these findings are discussed.