Exploring the Mathematical Knowledge Needed for Teaching Teachers

Conceptualizing Knowledge

To explore the forms of knowledge that MTEs draw on in their work with preservice teachers, it is necessary to consider what constitutes knowledge. Acknowledging that there are various perspectives, the authors follow Hiebert and Morris (2009) in positing that knowledge includes “know-how for teaching based on past experiences, empirical data, and well-reasoned arguments and predictions” (p. 476). Moreover, some researchers suggest that knowledge supporting the practice of teaching is highly situated and intimately related to individual practice (Cochran-Smith & Lytle, 1999). In their review of existing research about knowledge and teacher learning, Cochran-Smith and Lytle (1999) describe three prominent perspectives, including knowledge for practice, knowledge of practice, and knowledge in practice. Given this study’s focus on using particular MTEs’ practice to understand the knowledge demands of their work, the perspective taken in this study is most closely aligned with knowledge in practice, which promotes the idea that practical knowledge is developed through reflection on personal teaching experiences.

Research focused on describing MTE knowledge in practice offers some insights into the forms of knowledge drawn on by MTEs. By analyzing aspects of his own developmental trajectory, Tzur (2001), for example, proposed a series of four levels of foci that teacher educators progress through on their way to become mentors. He suggested that teacher educators begin by learning math (Focus 1), learning to teach math (Focus 2), learning to teach teachers (Focus 3), and finally learning to teach teacher educators (Focus 4). Although there may be some overlap in the foci, the questions that the MTE answers evolve as they progress through the levels. Whereas initially an MTE may attend to questions about the mathematics, as they move through levels, the questions come to emphasize the meaning of learning mathematics and then the meaning of mathematics teaching. Drawing from Tzur’s exploration of his own craft knowledge, he proposed a general hypothetical learning trajectory for teacher educators.

Consistent with other researchers who emphasize the importance of craft knowledge or knowledge in practice (cf. Hiebert, Morris, Berk, & Jansen, 2007; Schön, 1987), Bergsten and Grevholm’s (2008) research suggests knowledge MTEs draw on to prepare teachers to continue to learn from their own practice. With no clear synthesis of what teachers must know and be able to do, Bergsten and Grevholm argued that one potentially productive avenue for preparing teachers is to focus on helping them become reflective practitioners (Dewey, 1933/1960) who not only reflect on their own practice in generative ways but also purposefully investigate their practice. Bergsten and Grevholm described how MTEs can leverage preservice teachers’ learning to forge links between theoretical tools and the activities MTEs use in their teacher education work. Their research implies MTEs draw on knowledge about the content of teachers’ learning and teaching practice in ways that support teachers in making connections between teachers’ own learning and teaching experiences. Mason (1998, 2010) enhances this perspective by describing how MTEs might guide and help develop what preservice teachers attend to when they analyze their own practice, thus implying the use of certain forms of knowledge to carry out this work.

Building on the aforementioned research, this study uses particular MTEs’ knowledge in practice as a tool for understanding the knowledge demands of work with preservice teachers and how this knowledge is different from that used by K-12 teachers. Whereas the aforementioned research on MTEs from a knowledge in practice perspective begins to illuminate the forms of knowledge MTEs draw on in their work, such research is limited in that it does not reveal how MTE knowledge is used in actual practice.

Knowledge Needed for Teaching Mathematics to Teachers

Given the knowledge used by MTEs in their work with teachers as suggested in the aforementioned research, the knowledge used by MTEs is different from that of K-12 teachers. Researchers generally agree that the work of MTEs involves working with practicing teachers and/or preservice teachers to develop and improve the teaching of mathematics (Jaworski, 2008). As there is considerable diversity in the nature of MTEs’ work, the range of expertise shared by MTEs is similarly diverse. MTEs can be considered as individuals with one or more of the following types of expertise: those with mathematics expertise, those with pedagogical expertise, and those with expertise derived from their experiences as schoolteachers (Bergsten & Grevholm, 2008). The MTEs included in this analysis have at least two kinds of the expertise described by Bergsten and Grevholm (2008). Indeed, as Thames (2008) suggested, such diversity in MTE expertise might be used to understand the knowledge demands in the work of teaching teachers.

The challenges MTEs face in supporting preservice teachers’ development of mathematical knowledge for teaching parallel the challenges K-12 teachers face in teaching students, implying MTEs and K-12 teachers draw on similar forms of knowledge in their work. For example, many preservice teachers may not consider analyzing student solutions or revising mathematical definitions, for example, as part of the domain of mathematical knowledge they need to learn as a teacher. This, in turn, implies a need for MTEs to be knowledgeable of ways to foster such changes in the ways in which preservice teachers learn and understand the mathematics needed for teaching. If preservice teachers need to develop both common content knowledge (e.g., using percentages to compute amounts of discounts) and specialized content knowledge (e.g., evaluating the validity of the mathematics in solution methods; Ball et al., 2008), so too do MTEs need to possess such mathematical knowledge. To support preservice teachers’ thinking at a high level, MTEs need to know not only the content that preservice teachers need to know but also the ways in which preservice teachers engage with such content to anticipate the questions, misconceptions, and challenges preservice teachers may have with learning this content. Thus the knowledge required to manage such instructional challenges is similar for both MTEs and K-12 teachers.

However, there are important differences between the knowledge needed for teaching K-12 students and that needed for teaching teachers. In his work focused on teacher educators, Mason (1998, 2010) suggested that the work of MTEs is similar to that of teachers and that, in addition, the work of MTEs involves helping preservice teachers recognize how to relate what they are learning to teaching. Although not an explicit focus on knowledge, Mason (1998) suggested that the work of MTEs involves developing and enhancing different levels of awareness in preservice teachers as opposed to simply helping them learn the content that needs to be learned: “Teaching is fundamentally about attention, producing shifts in the locus, focus, and structure of attention” (p. 244). Mason (1998) argued that preservice teachers need to be able to engineer instructional situations in which K-12 students experience a shift in their attention where they (i.e., K-12 students) become aware of ideas and concepts of which they were previously unaware. Consequently, the work for MTEs is to develop preservice teachers’ understanding of certain ideas and concepts, and develop preservice teachers’ awareness of how to connect what they are learning to teaching—work that is different from what teachers do with K-12 students. For example, MTEs must engage preservice teachers in mathematical explorations in ways that emphasize how their learning of mathematics might influence their future work of teaching students. MTEs also must focus on enhancing preservice teachers’ awareness of the underlying structure of mathematical concepts so that preservice teachers may, in turn, support students’ understandings of mathematical concepts. In addition, MTEs must not only develop preservice teachers’ ability to evaluate the transparency of mathematical ideas in mathematical representations for themselves as learners but also support preservice teachers in recognizing why evaluating the transparency of mathematical representations is important for planning lessons and selecting representations that will support the development of students’ understandings. Such practices of MTEs have the potential to enhance preservice teachers’ awareness, and connect their learning to teaching practice, work that is different from that of K-12 teachers.

Thus, in addition to understanding the mathematics content and supporting preservice teachers as they engage with the content, MTEs have to have knowledge of what is involved in teaching mathematics to students to develop preservice teachers’ understanding of what will be required of themselves as they teach mathematics to students. In other words, MTEs need mathematical knowledge for teaching teachers. According to Zopf (2010), mathematical knowledge needed for teaching teachers is different from teaching mathematics to students in the following ways. First, students have informal understandings of mathematics whereas preservice teachers have more formal, albeit often limited and procedural, understandings of mathematics. Second, the mathematics content that is being taught is different: teachers teach mathematics whereas MTEs teach mathematical knowledge for teaching. Finally, the purposes of teaching students and purposes of teaching preservice teachers are different. Whereas students often learn mathematics to participate in school and society writ large, preservice teachers learn mathematical knowledge for teaching to teach students. Thus, MTEs need to have knowledge of not only mathematics content but also about how preservice teachers’ content learning connects to teaching practice.

Despite the extant research on this phenomenon, the knowledge demands of MTEs’ work with preservice teachers have yet to be adequately explored. Indeed, the field of teacher education lacks an evidence-based understanding of the knowledge MTEs need to carry out their work. Thus, this study focuses on how particular MTEs use knowledge in their practice and uses this analysis as a tool for understanding the knowledge demands of work with preservice elementary teachers and how this knowledge is different from that required to teach K-12 students.

Overview of Data Corpus

Data for this study come from a multimedia database of teaching and learning artifacts from five iterations of a university-based mathematics content course for preservice elementary teachers, which is the first of two required content courses preservice teachers take during their first or second years at the university. There are 29 class periods in each semester and each class period is 120 min long. In any given semester, 15 to 30 preservice teachers enroll in each of the two sections of the course. The content course is designed around developing mathematical knowledge needed for teaching (Ball et al., 2008) in the area of place value and number operations, rational numbers, and number theory. The course also engages preservice teachers in explaining, representing, and understanding and reacting to mathematical thinking that is different from their own. The nature of the learning environment in the content course is collaborative and interactive. In a typical class, preservice teachers first solve tasks individually by formulating questions and choosing an appropriate strategy, then work collaboratively on the task with their peers, and finally formulate a solution and, when appropriate, different strategies to solve the problem. This process is typically followed by a whole-class discussion on the negotiation of what counts as an acceptable solution. Four MTEs with varying degrees of mathematical and pedagogical expertise taught at least one of the course iterations, and these MTEs have expertise that falls into one or more of the three categories of MTE expertise (i.e., mathematical, pedagogical, school teaching) (Bergsten & Grevholm, 2008). A list of the five course iterations with instructor information can be found in Table 1. All types of data mentioned above were collected in all iterations of the content course with the exception of the last two iterations of the course that were not videotaped.

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Table 1.

Sequencing of the Five Course Iterations Including Course Instructors.

Course Iteration	Section 1	Section 2
Spring 2007	MTE and graduate student	N/A
Fall 2008	MTE and graduate student	MTE and graduate student
Spring 2009	MTE and graduate student	N/A
Fall 2009	MTE and graduate student	N/A
Spring 2010	MTE and graduate student	N/A

Note. MTE = mathematics teacher educator; N/A = not available.

MTEs and graduate students attend a planning group on a weekly basis for about 3 hr, during which time the group debriefs the previous week’s class session, comparing the initial lesson plan with the enacted lesson, and creates or modifies a lesson plan for the next class session. MTEs discuss and finalize the goals of the lesson, mathematics problems, and the sequence of the problems. Given variations in preservice teachers’ mathematical knowledge and engagement and MTEs’ expertise, the enacted lesson could be different across MTEs’ course sections by virtue of how MTEs interact with preservice teachers. In each class, MTEs work collaboratively to implement the lessons. For example, although MTEs might take turns to lead whole-class discussions, they both interact with preservice teachers during small group activities.

MTEs’ instructional interactions and preservice teachers’ learning were extensively documented in a multimedia database. The database includes detailed lesson plans, which provide both a description of the evolving course content as well as rationales for design and instructional decisions; PowerPoint slides used during each lesson; over 200 hr of videotaped class sessions; over 100 hr of audiotaped small-group discussions; transcripts of all video and audio data; photographs of preservice teachers’ board work generated during class; digitized copies of preservice teachers’ classwork, exams, and course notebooks; and audio recordings of planning meetings.

This database is well suited for this study for the following reasons. The extensive records of practice and various artifacts allow analysis of the range of MTEs’ work, the variation in MTE experience makes different knowledge forms more visible, and finally, the records of the collective work of the interdisciplinary planning group make explicit certain forms of knowledge that might otherwise remain implicit. Indeed, as Thames (2008) suggested, having access to a more varied range of teaching practice might be used to analyze the work entailed in teaching. For example, although MTEs with more pedagogical and school teaching expertise may be able to anticipate and manage multiple and simultaneous instructional interactions in teaching, drawing on their extensive and varied repertoire of strategies and experiences, MTEs with different expertise lack these same resources (Wenger, 1998). Thus, MTEs with different expertise may engage in certain instructional interactions that elucidate knowledge that is different from the knowledge MTEs with different expertise may use in their work with preservice teachers. The database includes examples of instructional interactions of MTEs with a range of expertise.

Data Analysis

The authors investigated the instructional interactions of different MTEs using artifacts from the aforementioned database and focused on what actual instruction suggested in terms of the knowledge MTEs draw on as they teach mathematics to preservice teachers. Similar to other analyses of mathematics teaching (Ball & Bass, 2003; Ball et al., 2008) and knowledge use in other professions (e.g., Hoyles, Noss, & Pozzi, 2001), the authors use a practice-based theory of mathematical knowledge for teaching. Ball and Bass (2003) developed a practice-based approach to examine the mathematical demands of the work of teaching, an approach that is grounded in the discipline of mathematics. Thames (2008) provided further specification of the methods and design of such an analysis. Specifically, the type of analysis used in the current study is “top-down” in that it operates with a set of hypotheses about the particular practices of teaching mathematics to preservice teachers and also “bottom-up” in that it closely examines what is happening in the university classroom. As such, this analysis is empirically grounded, constantly basing the hypotheses and claims on evidence from the data. The authors specifically focus on MTEs’ work around developing specialized content knowledge as one hypothesis is that K-12 students do not have to develop specialized content knowledge as preservice teachers do (Ball et al., 2008). Thus, identifying knowledge forms that are specific to the work of teaching mathematics to preservice teachers may be possible by analyzing the enactment of specialized content knowledge tasks across different iterations of the mathematics content course.

As part of the analysis, the authors first developed two maps from the database artifacts. One is the “task map” created from the enacted lesson plans and PowerPoint slides. A “task map” (see Table 2) includes the name of the mathematical tasks and content strand addressed in the task (e.g., whole numbers and operations, rational numbers), and in some instances the intended learning goals. The other is the “classroom event map” produced from classroom video records, which documents instructional activity or tasks implemented, as well as time spent in small- and whole-group discussions for each task. These two maps allowed the authors to effectively locate MTEs’ instructional interactions around certain types of instructional activities in the video artifacts.

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Table 2.

An Example of a Task Map.

	13 September	18 September	20 September	25 September	27 September
Date	Class 6	Class 7	Class 8	Class 9	Class 10
9:00	Evaluating explanations	Definition of factors and multiples: small-group discussion	Homework review	GCF and LCM	Place value: introduction
9:05
9:10
9:15					Place value: representing place value
9:20			Factorizations: all the factorizations of 60
9:25	The locker problem: set up				Place value: analysis of students’ error
9:30	The locker problem: small-group discussion	Definition of factors and multiples: whole-group discussion
9:35
9:40
9:45
9:50		The locker problem: more questions
9:55			Prime factorizations		Number system
10:00
10:05
10:10	The locker problem: definition of factor, multiple		Prime: The sieve of Eratotstenes
10:15
10:20					Represent rational numbers in different ways
10:25			GCF
10:30
10:35
10:40			Homework review: last digit check for even numbers
10:45

Note. GCF = greatest common factor; LCM = least common multiple.

Drawing from Thames (2008), the following considerations were made for parsing the data and in selecting instructional interactions to analyze:

Two coders (i.e., authors) identified the occurrences of instruction on tasks of teaching that involved the development of specialized content knowledge, which includes tasks focused on analyzing student errors, mapping between representations and the underlying mathematical ideas, and making mathematical arguments. Bounding the content to be analyzed allowed the coders to rule out more than 50% of the video artifacts.

Next, the coders independently identified MTE instructional moves around the identified specialized content knowledge tasks. The goal of this analysis was to select rich episodes less than 15 min in duration from several hour-long video artifacts related to each identified task. For example, videos involving the task of mapping between representations are about 9 hr in total duration, and from those data, the coders identified seven rich episodes ranging from 3 min to 13 min in duration. In doing so, the analysis focused on different grain sizes of instructional practice, zooming in on interactions with preservice teachers and zooming out to focus on a lesson or a sequence of tasks and activities across lessons. For example, the coders zoomed in on particular instructional interactions such as those involving an MTE and a preservice teacher or a small group of preservice teachers. The coders then identified several instances of such interactions related to a given task or activity, and then zoomed out to the level of class session. These analyses were conducted in light of the intended goals of the tasks and activities related to these different interactions and class sessions. The purpose of this zooming in and out on instructional practice was to understand how different MTEs drew on knowledge in their instructional interactions with different groups of preservice teachers. In addition, if possible, occurrences of different MTEs enacting the same task were compared. For example, small-group discussions on a task about analyzing student errors were selected from two MTEs’ with different areas of expertise: a mathematician (mathematical and pedagogical expertise) and a mathematics educator (mathematical and school teaching expertise). The comparison focused on what was made explicit in both MTEs’ instructional interactions with preservice teachers, and what was made explicit in one MTE’s instructional interactions that was not made explicit in the other MTE’s instructional interactions. The mathematician who has considerable pedagogical expertise frequently connected to elementary mathematics curricula, for example, whereas the mathematics educator did not make such connections. Comparing different MTE’s instructional interactions around similar mathematics tasks prompted questions, such as, why can the MTE make explicit a certain instructional interaction? What knowledge is exhibited in this interaction? What knowledge is not exhibited when the same instructional interaction was not made explicit by the MTE? Such comparisons can illuminate the potential knowledge unique to MTEs.

Evaluation involving agreement about the existence of an episode that suggested interactions specific to MTEs’ work was conducted. Specifically, the extent to which the coders agreed on instances of an instructional interaction that was specific to teaching teachers was analyzed. A high inter-rater reliability (K = .91) was obtained, which indicates that using lesson artifacts (i.e., enacted lesson plans, learning goals) and videotaped class sessions was a reliable means of identifying MTE instructional interactions involving knowledge that was specific to teaching teachers. Table 3 shows the number of episodes identified for each type of task.

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Table 3.

Frequency of Episodes Identified for Different SCK Tasks.

SCK tasks	No. of episodes identified
Analyzing student errors	7
Representing place value	2
Mapping between representations	7
Making mathematical arguments	9

Note. SCK = specialized content knowledge.

A second evaluation was conducted and concerned the description and potential purposes of the instructional interactions in the identified episodes. Discussions of the potential pedagogical and mathematical purposes of observed MTE instructional moves can lead to identifying potential teaching practices (Thames, 2008). Specifically, the two coders examined the work of the MTEs in those instances, answering the following questions: What are they doing? What may be potential pedagogical purposes of such instructional moves and decisions in those instances? What may be potential mathematical purposes of such instructional moves and decisions in those instances? The initial inter-rater reliability score (K) was .85. All discrepancies were subsequently resolved through discussions between the two coders.

The last step of the analysis was to select and scrutinize episodes that suggested instructional interactions unique to MTEs’ work with preservice teachers; the selected interactions would not be likely to happen in K-12 classrooms. Specifically, the coders answered the following questions: What MTE instructional interactions with preservice teachers are not likely to occur in the K-12 classroom? What did MTEs say and do when interacting with preservice teachers in these instances? What might this indicate about the knowledge demands of MTEs’ work with preservice teachers? Three episodes highlighting MTE knowledge specific to teaching teachers were identified. To be sure, the purpose of this analysis was not to illustrate how prevalent different types of instructional interactions or knowledge form use were across the four MTEs in this study. Rather, the authors aimed to conduct a thorough analysis of the three episodes that emerged from the data to gain further insights into the knowledge demands of work with preservice teachers and how this knowledge is different from that required to teach K-12 students.

Connecting Student Errors to Instructional Moves

In the content course, one task focused on developing specialized content knowledge involves analyzing student errors with multiplication. The goals for preservice teachers associated with the task of analyzing student errors were to identify and provide mathematical explanations of the error and articulate potential reasons for the error. An implicit goal associated with the task of analyzing student errors that was articulated in planning meetings was to connect what preservice teachers were learning in the content course to the work that teachers do in the classroom. The following transcript illustrates what preservice teachers and an MTE are doing and saying around this task in one class session:

In this example, the MTE introduces the potential for uncertainty in teaching as to whether the student errors in this task represent random errors from students or whether the errors are indicative of a lack of more conceptual misunderstanding on the part of the student. Specifically, the MTE suggests, in the classroom, to use multiple problems with students to determine whether a student is consistently making the same mistake. In doing so, as a teacher, “You might be more sure your answer is correct.” In this way, the MTE is suggesting to preservice teachers an instructional means by which they can determine whether this particular student error is random or more systematic. In doing so, the MTE is providing preservice teachers with an instructional move for uncovering student thinking, in this case determining whether student errors are due to random error or are indicative of an underlying lack of understanding on the part of the student.

Taken from a class session similarly focused on analyzing student errors, the following transcript illustrates what preservice teachers and a different MTE are doing and saying around the same task of analyzing student errors:

In this example, the MTE solicited two different explanations of the student error, both of which resulted in the same answer. He then posed an instructional scenario, suggesting that one way to determine which explanation correctly describes the student error is to generate an additional problem for the student to solve to see the way in which the student would solve the new problem. As with the previous example, the MTE is providing preservice teachers with an instructional move for determining whether student errors are due to random error or are indicative of an underlying lack of understanding on the part of the student.

These two episodes involving student errors illustrate particular knowledge forms that are drawn on by these MTEs to support preservice teachers’ exploration and understanding of student errors. The MTEs exhibit knowledge of the student errors and knowledge of possible instructional means by which to determine potential reasons for the errors. They use these knowledge forms to elicit the exploration of the nature of student errors, and connect preservice teachers’ learning to aspects of teaching students. Indeed, although K-12 teachers use knowledge of student errors to understand students’ thinking in their teaching practice, they arguably do not need knowledge of ways to facilitate the exploration of the nature of those errors.

Connecting Algorithms to the K-6 Curriculum

In the content course, preservice teachers studied multiple algorithms for the four basic operations. One of the goals for preservice teachers involving the use of multiple algorithms for multiplication was to promote conceptual understanding of the conventional multiplication algorithm (e.g., why it works, what it means to “carry” a number, why do you leave the ones place empty, and so on) so that they can make sense of the various algorithms that students might use in the classroom. The following transcript presents a whole-class discussion of two multiplication algorithms (i.e., partial products algorithm and conventional algorithm):

In this example, the MTE articulated the roles of these two algorithms in the current K-6 curriculum. Specifically, the MTE indicated the learning sequence related to these two algorithms, which connected preservice teachers’ content learning to teaching practice. She then pointed out the advantage of the partial products algorithm by highlighting the transparency of the algorithm. For example, she indicated that 5 times 5 was 25, and the 2 that is carried is 20. In the partial products algorithm, 25 was explicitly listed as one of the products. Moreover, the MTE exhibited knowledge of different variations of the conventional algorithm in the curriculum (e.g., slashing out the last zero in the second product, or moving the second product one place over). This information from different curricula seemed to evoke preservice teachers’ K-12 learning experiences, which allowed the MTE to effectively expose preservice teachers’ lack of understanding of the role of place value in the conventional algorithm.

This episode illustrates how an MTE’s knowledge about the K-6 curriculum is used differently from that of K-12 teachers. Whereas K-12 teachers use their knowledge of the curriculum to inform the basis for the tasks and activities for students, this MTE used knowledge of the curriculum to connect preservice teachers’ mathematics content learning to the K-6 curriculum.

Connecting Research to Mathematics Content Learning

In the content course, preservice teachers worked on constructing numbers using various manipulatives, such as base-10 blocks, bundling sticks, and snap cubes. The goal for preservice teachers using base-10 blocks was to model numbers to make visible the role of place value in different multi-digit numbers. The following transcript presents an introduction to a class about place value and representing numbers using base-10 blocks:

In this example, albeit briefly, the MTE cited research about learning numbers with various representations to explain the importance of learning about place value to preservice teachers. In addition, the MTE also explained that various representations could make explicit the concept of place value in the base-10 number system. K-12 teachers may use research about K-12 students’ learning to inform their teaching practice. In contrast, MTEs, as this episode suggests, do not directly use research about K-12 students’ learning to guide their own teaching of preservice teachers. However, MTEs need to be able to make explicit the implications of research about K-12 students’ learning for their preservice teachers’ content learning.

Unpacking the Knowledge Used by MTEs

In the first example, the MTEs exhibited knowledge of the nature of student errors and how that knowledge is used in practice with preservice teachers in ways that are different from how such knowledge is used by K-12 teachers. Whereas K-12 teachers may have knowledge of student errors and potential reasons for the errors, they do not engage their students in explorations of student errors. Instead, K-12 teachers use this knowledge to directly inform their instructional interactions with students. MTEs in this analysis, however, used their knowledge of the nature of student errors to connect preservice teachers’ exploration of the errors to teaching practice. Specifically, the MTEs suggested an instructional move that preservice teachers could use in their future teaching to determine whether a student error is random or evidence of more systemic errors in students’ thinking. By suggesting an instructional move that could be used in teaching, the MTE is connecting the mathematical work of analyzing student errors to teaching practice. In this example, an MTE would need to not only be able to recognize and understand the nature of the student error but also be knowledgeable of students’ thinking, more specifically, the factors that affect students’ mathematical thinking. The MTE would have to know potential factors that may contribute to the development of the error, such as a lack of understanding of the role of place value in a multiplication algorithm, or due to a random misstep or number misplacement on the part of the student. Thus, the MTEs are exhibiting knowledge about ways of recognizing and making explicit how analyzing student errors relates to teaching mathematics to students.

In the second example, the MTE exhibited knowledge of the K-6 curriculum and knowledge of certain multiplication algorithms, and used that knowledge in ways different from how K-12 teachers may use such knowledge. K-12 teachers need to have knowledge of certain multiplication algorithms and how these algorithms fit within the K-6 curriculum, which they then may use to sequence students’ learning. In contrast, the MTE used their knowledge of two multiplication algorithms as part of the K-6 curriculum to not only connect preservice teachers’ content learning to teaching practice (e.g., pointed out the advantage of the partial products algorithm by highlighting the transparency of the algorithm) but to also expose preservice teachers’ own misunderstandings of the role of place value in the multiplication algorithm. Specifically, the MTE was knowledgeable of different variations in how the steps of the conventional algorithm may have been taught to preservice teachers. Indeed, Mason (1998) suggested that the work of MTEs is often challenging, because preservice teachers often enter their coursework generally unaware as their awareness has been clouded by their past procedurally focused experiences in learning mathematics. Thus, the MTE is exhibiting knowledge of how students’ learning of multiplication can be supported by a particular algorithm (i.e., partial products algorithm).

Finally, in the third example, the MTE exhibited knowledge of research about student learning, and used that knowledge to connect research to preservice teachers’ content learning. Teachers may also be knowledgeable of research about student learning. However, the key difference between MTEs and K-12 teachers in terms of such knowledge is how the knowledge is used in practice. Teachers may use knowledge about research-based ways of supporting students’ understanding of the base-10 number system, for example, to directly inform their instructional interactions, either in their sequencing of instruction or guide their instruction in other ways. The MTE in the third example, however, did not directly use research about K-12 students’ learning to guide their own teaching of preservice teachers, but rather to make explicit the implications of research for preservice teachers’ current learning. Thus, the MTE, as the example suggests, is exhibiting knowledge of mathematics education research in ways that connect to preservice teachers’ learning of content.

Taken together, these examples suggest that the mathematical knowledge needed for teaching teachers is different from that which teachers have to know to teach students. MTEs need to know not only the content preservice teachers have to know (e.g., specialized content knowledge) but also ways in which preservice teachers’ learning relates to teaching mathematics and how to facilitate preservice teachers’ recognition of such connections. In short, the forms of knowledge used by MTEs in this analysis included knowledge of certain concepts related to preservice teachers’ mathematics learning and how those concepts connect to teaching practice in K-12 classrooms.

A Process for Identifying Components of a Knowledge Base for Teacher Education

Applying the knowledge in practice perspective to an analysis of teacher educators’ work can highlight the forms of knowledge teacher educators draw on in their work with teachers. Although this study focuses on teacher educators in the domain of mathematics, the approach used in this study can be applied to other domains to understand the knowledge demands of teacher educators’ work. Furthermore, by making visible the process used in this study, other researchers can use similar processes to understand the knowledge teacher educators use in their interactions with teachers. In doing so, the field of teacher education can begin to build an evidentiary base for understanding the knowledge demands of teacher educators’ work.

As Morris and Hiebert (2009) argued, the domain of mathematics education currently lacks a knowledge base for teacher education. Building such a knowledge base requires careful attention to the ways in which knowledge is generated, recorded, and vetted, which can take the form of shared learning goals, tangible products, the iterative testing of small changes to the curriculum, and multiple sources of innovation that involves sharing information across local systems about what is effective (e.g., planning groups that engage in collective work around courses for preservice teachers; Morris & Hiebert, 2009).

The current analysis involved a process for identifying knowledge forms used by MTEs and thus fits well with one of the four features of the knowledge building process, namely, having multiple sources of innovation. Morris and Hiebert (2009) offered examples of the multiple sources innovation from which they drew, including teacher educator reflections of lesson plans they use in their teaching of preservice teachers and analyses of preservice teachers’ questions and written work. In the current analysis, having observations of different MTEs with different categories of expertise (i.e., those with mathematical, pedagogical or school teaching experience) implementing the same tasks across different course iterations constitutes a source of innovation. Indeed, MTEs bring a host of resources (e.g., experience, conceptions, knowledge) to bear on their instruction and draw on different forms of knowledge in light of their expertise(s). As Morris and Hiebert argued, a knowledge base for mathematics teacher education includes not only a focus on developing a coherent curriculum for preparing teachers but also attention to understanding the knowledge demands of MTEs’ work to determine ways of supporting them in their interactions with preservice teachers.

Although the process explored in this study is one potential approach for understanding the work of MTEs, there are ways to build from this process. For example, including reflections from MTEs, before and after their teaching, on the identified instructional interactions could provide insights into the potential mathematical and pedagogical purposes of those interactions. Including such reflections could also identify other forms of knowledge in use that may not be as explicit in observations of their teaching. Still, building from this process, one could examine the relationship between the lesson plans used by MTEs in relation to their lesson implementation of the lesson as a way to identify potential components of a knowledge base for MTEs that may be missing. If, for example, the lesson plans are not supportive of MTEs’ instruction in certain ways, it could point to knowledge components that MTEs require, but that are not available in the information in the lesson plans. Indeed, lesson plans serve not only as plans for instruction but also as reifications of the knowledge of a group or community of MTEs (Hiebert & Morris, 2012). Doing so can build a more robust knowledge base for mathematics teacher education, a task to which researchers have increasingly been drawing attention (Morris & Hiebert, 2009)