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This article reports on a second national survey of higher education institutions in the USA to answer the question “Who teaches mathematics content courses for prospective elementary teachers, and what are these instructors’ academic and teaching backgrounds?” and addresses valuable information not collected with the first survey conducted in 2010. We surveyed 1740 institutions and a faculty member from each of 413 institutions (23.7%) participated in the survey. The survey results demonstrate that the majority of these institutions are not meeting the recommendations of the Conference Board of Mathematical Sciences (2012) and the Association of Mathematics Teacher Educators (2017) for prospective elementary teachers to take at least 12 semester-hour credits of mathematics content designed specifically for them. The data do indicate that there is movement toward more activity-based approaches for these courses as compared to the 2010 survey. Additionally, there is an increase in these courses having instructors with doctorates in mathematics education as well as an increase in instructors having grades 7–12 teaching experience. 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. While most institutions still do not provide training and/or support for these instructors, formalized support and training appear to be increasing since the 2010 survey.

Over the past two decades, the landscape of research on mathematics teacher educators (MTEs) has grown considerably. One particular area of interest has focused on MTE knowledge and the ways in which it is developed and used in teaching practice. However, studies have conceptualized MTE knowledge in different ways and have employed considerably different methodologies and approaches to its study. In an effort to understand this varied landscape, we conducted an extensive review of research on the nature and development of MTE knowledge. This review provides a broad descriptive analysis of the existing theoretical and empirical research on MTEs’ knowledge, explores the theoretical underpinnings of the existing frameworks for and studies on MTE knowledge, and considers implications for future research.

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.

This article discusses how teaching via problem solving helps enact the Mathematics Teaching Practices and supports students' learning and development of the Standards for Mathematical Practice. This approach involves selecting and implementing mathematical tasks that serve as vehicles for meeting the learning goals for the lesson. For the lesson discussed in this article, the learning goals are developing students' understanding of number theory concepts (factors; multiples; greatest common factor, or GCF; and least common multiple, or LCM). The authors seek to continue a yearlong goal of developing students' problem-solving skills.

In this study, we report on what types of mathematical knowledge for teaching teachers (MKTT) mathematics teacher educators (MTEs) use and develop when they work together and reflect on their teaching in a Community of Practice while helping prospective primary teachers (PTs) generate their own mathematical knowledge for teaching in learning mathematics via problem solving. Two novice MTEs worked with an experienced MTE and reflected on the process of learning to teach via problem solving and supporting PTs in developing deep understandings of foundational mathematical ideas. Taking a position of inquiry as stance, we examined our experiences teaching mathematics content courses for PTs via problem solving. We found that all of the MTEs used and developed some MKTT through (a) understanding and deciding on the mathematical goals of both the individual lessons and the two-course sequence as a whole, (b) choosing and facilitating tasks, and (c) using questions to scaffold PTs learning and engage them in mathematical processes such as making conjectures, justifying their reasoning, and proving or disproving conjectures.

Mathematical task design has been a central focus of the mathematics education research community over the last few years. In this study, six university teacher educators from six different US institutions formed a community of practice to explore key aspects of task design (planning, implementing, reflecting, and modifying) in the context of comparing fractions using reasoning and sense-making. By presenting results of their implementation of two tasks with 63 prospective elementary teachers across three institutions and their reflective analysis of the implementation, the authors highlight the importance of collecting and analyzing data and reflecting on this analysis to inform the redesign of tasks. The authors also found that considering different types of tasks (problem solving vs. problem posing) helps illuminate different aspects of prospective elementary teachers' understanding, which can inform task redesign. Finally the authors contribute to the knowledge base on reasoning strategies for comparing fractions and prospective elementary teachers’ knowledge of these strategies.

Prospective Teachers often enter their mathematics content courses believing that they know enough mathematics to teach elementary school. However, research has shown that much of PTs ' knowledge is procedurally based and lacks depth and conceptual understanding. One job of Mathematics Teacher Educators (MTEs) in mathematics content courses is to help PTs become more mathematically proficient, by relearning the mathematics that they believe they already know in deeper, more connected ways. We suggest that one way for MTEs to do this is to incorporate Transformative Learning Theory (TLT) into their mathematics content courses. TLT is an application of andragogy, which are the methods or techniques used to teach adults. Through TLT, learners participate in a process where they are presented with a disorienting dilemma that perturbs their prior understandings. Learners work through the dilemma by critically reflecting on their prior understandings and participating in rational discourse with others. Ultimately, learners are tasked with making connections between their prior understandings and their new knowledge. In this paper, we describe a cycle of transformative learning theory and give examples of incorporating TLT into mathematics content courses for PTs through lessons on proportional reasoning and whole number concepts. We conclude by discussing general considerations and resources for incorporating TLT into mathematics content courses and how this helps PTs develop the five strands of mathematical proficiency (National Research Council, 2001).

This article presents a research summary of prospective elementary teachers' (PTs') mathematical content knowledge in the area of fractions. The authors conducted an extensive review of the research literature and present the findings across three time frames: a historical look (pre-1998), a current perspective (1998-2011), and a look at the horizon (2011-2013). We discuss 43 articles written across these time frames that focus on PTs' fraction knowledge. Consistent across these papers is that PTs' fraction knowledge is relatively strong when it comes to performing procedures, but that they generally lack flexibility in moving away from procedures and using \"fraction number sense\" and have trouble understanding the meanings behind the procedures or why procedures work. Across the time frames, the trend in the research has moved from looking almost entirely at PTs' understanding of fraction operations, particularly multiplication and division, to a more balanced study of both their knowledge of operations and fraction concepts. What is lacking in the majority of these studies are ways to help improve upon PTs' fraction content knowledge. Findings from this summary suggest the need for a broader study of fractions in both content and methods courses for PTs, as well as research into how PTs' fraction content knowledge develops.

The growing number of studies on mathematics teacher educator knowledge have consistently argued that mathematics teacher educators require specialized knowledge in their work with prospective teachers (beyond the knowledge needed for teaching students), what researchers refer to as mathematical knowledge for teaching teachers. Drawing from existing research and aspects of our own work as mathematics teacher educators, we offer our own conceptualization of mathematical knowledge for teaching teachers and illustrate ways in which we as mathematics teacher educators use our own knowledge in teaching mathematics content to prospective teachers. We are particularly concerned with the knowledge mathematics teacher educators use to support prospective teachers ' relearning of mathematics, which involves prospective teachers ultimately reconstructing their previously developed knowledge of mathematics. We will illustrate ways in which we use various aspects of mathematical knowledge for teaching teachers to support prospective teachers' relearning of mathematics through the lens of three different tasks of teaching. We conclude with a discussion of the implications of our analysis for informing the growing knowledge base for mathematics teacher educators.

This study attempts to answer the question, What is the mathematical knowledge required by teachers of elementary mathematics content courses in the area of multiplication and division of fractions? Beginning in the mid-1980s, when Shulman (1986) introduced the idea of pedagogical content knowledge, researchers have been looking at the knowledge needed to teach in a variety of different content areas. One area that has garnered much of the research is that of mathematics. Researchers have developed frameworks for what they call mathematical knowledge for teaching, but there has been little work done looking at the knowledge requirements for teachers of teachers. This study attempts to fill this gap by determining some aspects of a framework for the mathematical knowledge required to teach prospective elementary teachers multiplication and division of fractions. In order to determine aspects of a framework for mathematics teacher educator knowledge in relation to multiplication and division of fractions, I interviewed, observed, and audiotaped three experienced teacher educators in different educational settings to determine the mathematical work of teaching prospective teachers fraction multiplication and division. My analysis focused on three of major tasks that came out of the work: introducing fraction multiplication, helping students make sense of fraction division, and assessing student understanding. Each of these tasks played a major role in the work of the teacher educators, and the knowledge required to perform these tasks was evident in varying degrees in each teacher educator. After analyzing the three mathematical tasks and the knowledge required by them, I was able to determine some components of a framework for the mathematical knowledge needed for teaching teachers multiplication and division of fractions. These aspects include: understanding multiple representations of fraction multiplication and division and how these representations relate to each other, to whole number ideas, and to the algorithms, deciding which aspects of the topics will help prospective teachers make the connections that they will need in order to teach these topics, especially since time often plays a factor in what gets taught in mathematics content classes for prospective teachers, setting specific goals of exactly what one wants one's students to know, rather than having a general goal of wanting prospective teachers to develop conceptual understanding of a topic, and being able to design and use assessments effectively to help decide if one is achieving one's goals. While each of the aspects described above are components of a framework for the mathematical knowledge needed by teacher educators, the three teacher educators in my study all lacked or were unable to demonstrate some of the knowledge components that would have helped them to meet their goals, despite having a wealth of experience teaching and designing mathematics content courses for prospective elementary teachers. One possible reason for this is that each of the teacher educators in my study were basically alone in their departments, without opportunities to collaborate or discuss these ideas with anyone else. These results suggest a need for better professional development for teacher educators in the field of mathematics education.