Developing Student Teachers' Skills at Eliciting Students' Mathematical Thinking Using the Coaching Cycle
Saved in:
| Title: | Developing Student Teachers' Skills at Eliciting Students' Mathematical Thinking Using the Coaching Cycle |
|---|---|
| Language: | English |
| Authors: | Reinke, Luke T. (ORCID |
| Source: | Teacher Educator. 2022 57(2):215-237. |
| Availability: | Routledge. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals |
| Peer Reviewed: | Y |
| Page Count: | 23 |
| Publication Date: | 2022 |
| Document Type: | Journal Articles Reports - Research |
| Education Level: | Higher Education Postsecondary Education |
| Descriptors: | Student Teachers, Teaching Skills, Skill Development, Thinking Skills, Coaching (Performance), Prompting, Mathematics Education |
| DOI: | 10.1080/08878730.2021.1990454 |
| ISSN: | 0887-8730 |
| Abstract: | Eliciting students' thinking is a key component of effective student-centered mathematics instruction, and research demonstrates promising approaches to developing this skill in pre-service teachers [PSTs] during methods courses. However, few studies have provided insight into how PSTs can be supported in continuing to develop the skill of eliciting thinking during student teaching internships. This case study describes the development of two PSTs as they worked to fully elicit their students' thinking during their internship, with the support of a coach. The PSTs' development generally aligned with an emergent trajectory synthesized from the literature; however, additional specificity was needed to fully interpret the data. Both PSTs learned to effectively probe the thinking of students who provided correct answers to initial prompts but made less progress in probing students' incorrect answers. The article concludes with an analysis of the learning opportunities presented by the coaching moves, as well as missed opportunities. |
| Abstractor: | As Provided |
| Entry Date: | 2022 |
| Accession Number: | EJ1345214 |
| Database: | ERIC |
|
Full text is not displayed to guests.
Login for full access.
|
|
| FullText | Links: – Type: pdflink Url: https://content.ebscohost.com/cds/retrieve?content=AQICAHj0k_4E0hTGH8RJwT4gCJyBsGNe_WN95AvKlDbXJGqwxwFYsN2X8xNiXVLv6q9ospvCAAAA4zCB4AYJKoZIhvcNAQcGoIHSMIHPAgEAMIHJBgkqhkiG9w0BBwEwHgYJYIZIAWUDBAEuMBEEDBkKgnRkPmMI1WzIuwIBEICBm-bpp6ocTNPwMLZEONQJUl9fPbzGLcBUVgKhOxCfKMLZ-Frs0lxLBPtn9xMzoknnf5SF812vzxcN1ZGAv8fUGayErMkiSEzHj_jDxT_iMfncmdnYDk3EFZEJ9WcLFL32lECZ3B6AKPeWc-rMatDfb8El_SEqnjhtGVXFrkMJcLrQ9kjXFpZZr3Ei7XFCp1WNYnd9FG4IM6C-PihY Text: Availability: 1 Value: <anid>AN0156414408;gxz01apr.22;2022Apr22.02:55;v2.2.500</anid> <title id="AN0156414408-1">Developing Student Teachers' Skills at Eliciting Students' Mathematical Thinking Using the Coaching Cycle </title> <p>Eliciting students' thinking is a key component of effective student-centered mathematics instruction, and research demonstrates promising approaches to developing this skill in pre-service teachers [PSTs] during methods courses. However, few studies have provided insight into how PSTs can be supported in continuing to develop the skill of eliciting thinking during student teaching internships. This case study describes the development of two PSTs as they worked to fully elicit their students' thinking during their internship, with the support of a coach. The PSTs' development generally aligned with an emergent trajectory synthesized from the literature; however, additional specificity was needed to fully interpret the data. Both PSTs learned to effectively probe the thinking of students who provided correct answers to initial prompts but made less progress in probing students' incorrect answers. The article concludes with an analysis of the learning opportunities presented by the coaching moves, as well as missed opportunities.</p> <p>Mathematics educators have defined mathematically proficient students as those who demonstrate conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive dispositions (see National Research Council, [<reflink idref="bib32" id="ref1">32</reflink>]). Given evidence that traditional models of mathematics teaching, which emphasize the memorization of procedures, do not lead to mathematical proficiency for most students (Tarr et al., [<reflink idref="bib44" id="ref2">44</reflink>]; U.S. Department of Education, [<reflink idref="bib46" id="ref3">46</reflink>]), mathematics educators have developed a new vision for teaching and learning, frequently referred to as ambitious instruction (Lampert et al., [<reflink idref="bib25" id="ref4">25</reflink>]). The goals of this form of instruction are indeed ambitious: that a collection of diverse learners simultaneously engage in rigorous mathematical work, reason about what strategies or processes make sense to solve problems, and demonstrate conceptual understanding as they articulate why these strategies make mathematical sense.</p> <p>The term ambitious instruction also highlights the increased demand on students and teachers. Students are asked to take responsibility for grappling with cognitively demanding tasks and explaining their thinking so others can understand. Teachers must enact what have been called "high-leverage practices" (TeachingWorks, [<reflink idref="bib45" id="ref5">45</reflink>]), including observing and eliciting students' thinking, interpreting this thinking in relation to their own mathematical understanding, and responding in ways that advance student learning without taking over the mathematical work. To support pre-service teachers [PSTs] in developing the capacity to deliver ambitious instruction, many teacher education programs have adopted a practice-based approach (Ball et al., [<reflink idref="bib2" id="ref6">2</reflink>]; McDonald et al., [<reflink idref="bib29" id="ref7">29</reflink>]). Not limited to programs that aim to support PSTs in developing ambitious mathematics instruction, practice-based approaches involve the identification of desired teacher practices and opportunities for PSTs to develop these practices during their teacher preparation programs.</p> <p>In this study, we focus on the high-leverage practice of eliciting student thinking. Eliciting, or uncovering, students' mathematical thinking through initial mathematical tasks and follow-up questioning (also known as probing) is a prerequisite to interpreting students' thinking and responding to advance learning (Jacobs et al., [<reflink idref="bib20" id="ref8">20</reflink>]). Researchers have found that PSTs have difficulties interpreting and eliciting their students' thinking (Moyer &amp; Milewicz, [<reflink idref="bib31" id="ref9">31</reflink>]; Shaughnessy &amp; Boerst, [<reflink idref="bib37" id="ref10">37</reflink>]; Sleep &amp; Boerst, [<reflink idref="bib39" id="ref11">39</reflink>]) yet they can progress toward effective eliciting through reflecting on one-on-one interviews and expert feedback (Weiland et al., [<reflink idref="bib47" id="ref12">47</reflink>]) as well as cycles of planning, rehearsal, enactment and feedback during methods courses (Hallman-Thrasher, [<reflink idref="bib16" id="ref13">16</reflink>]; Kazemi &amp; Waege, [<reflink idref="bib23" id="ref14">23</reflink>]). Even after these activities, however, PSTs demonstrate areas for continued growth (Hallman-Thrasher, [<reflink idref="bib16" id="ref15">16</reflink>]; Kazemi &amp; Waege, [<reflink idref="bib23" id="ref16">23</reflink>]; Weiland et al., [<reflink idref="bib47" id="ref17">47</reflink>]).</p> <p>For PSTs no longer engaged in coursework, student teaching internships present essential opportunities for continuing to develop skill with ambitious practices in an authentic setting. However, our review of the literature identified no studies documenting efforts to develop the practice of eliciting students' thinking during later stages of internships, when PSTs are no longer enrolled in methods courses. The absence of this research presents several broad questions. How can PSTs be supported in continuing to develop ambitious practices even while they are engaged in student-teaching internships within complex, authentic classroom settings? What are the potentials and limitations of traditional structures of feedback for continuing to develop PSTs' skills?</p> <p>This practitioner research case study begins to answer these questions by focusing on one practice, eliciting and interpreting students' thinking, and one feedback structure, formal coaching cycles composed of pre-observation meetings, observations, and post-observation feedback. We formed a team to support a mathematics teacher educator (first author) in studying multiple coaching cycles with two elementary PSTs over the course of an extensive internship. More specifically, the study examined two research questions:</p> <p></p> <ulist> <item> What trajectory or trajectories did elementary PSTs progress along as they learned to elicit and interpret student thinking during student teaching?</item> <p></p> <item> How did the coach support (or fail to support) PSTs in learning to elicit and interpret student thinking?</item> </ulist> <p>The answers to these questions have the potential to inform practitioners who design and enact supports aimed at developing PSTs' and teachers' capacity for enacting ambitious instructional practices as well as researchers who study these efforts.</p> <hd id="AN0156414408-2">Synthesis of the literature</hd> <p></p> <hd id="AN0156414408-3">Eliciting and interpreting student thinking</hd> <p>Eliciting student thinking is one of a set of instructional practices that have been empirically linked to student achievement (Hattie, [<reflink idref="bib17" id="ref18">17</reflink>]; Lampert et al., [<reflink idref="bib25" id="ref19">25</reflink>]; National Council of Teachers of Mathematics [NCTM], [<reflink idref="bib33" id="ref20">33</reflink>]).It is an essential practice for teachers to develop because students come to learning experiences with preconceptions, and those preconceptions can either act as foundations or barriers for understanding target concepts (Fuson et al., [<reflink idref="bib12" id="ref21">12</reflink>]). Teachers who elicit students' thinking can use those preconceptions as a basis for learning and address counterproductive preconceptions.</p> <p>Furthermore, eliciting and interpreting thinking are essential components of formative assessment, or the use of evidence to adapt instruction according to students' needs (Wiliam, [<reflink idref="bib49" id="ref22">49</reflink>]). Professional development aimed at helping teachers enact instruction responsive to students' thinking has been linked to gains in student outcomes (Bobis et al., [<reflink idref="bib5" id="ref23">5</reflink>]; Carpenter et al., [<reflink idref="bib7" id="ref24">7</reflink>]; Higgins &amp; Parsons, [<reflink idref="bib18" id="ref25">18</reflink>]). Effective teachers attend to students' ideas, interpret those ideas, and respond in ways that advance learning (Jacobs et al., [<reflink idref="bib20" id="ref26">20</reflink>]; NCTM, [<reflink idref="bib33" id="ref27">33</reflink>]).</p> <p>The practice of eliciting and interpreting thinking represents a subcategory of the broader practice of teacher questioning, an essential component of instruction that aims to advance students' understanding through classroom dialogue. Researchers who study teacher questioning tend to characterize patterns in discourse by categorizing the types of questions that teachers ask (Boaler &amp; Brodie, [<reflink idref="bib4" id="ref28">4</reflink>]; Chapin &amp; O'Connor, [<reflink idref="bib8" id="ref29">8</reflink>]). Questions aimed at eliciting student thinking represent one of these categories of questions.</p> <p>Scholars have decomposed the practice of eliciting and interpreting student thinking into component elements as shown in Table 1 (Shaughnessy &amp; Boerst, [<reflink idref="bib37" id="ref30">37</reflink>]; TeachingWorks, [<reflink idref="bib45" id="ref31">45</reflink>]). These decompositions can be synthesized as shown in Figure 1. First, teachers <emph>pose initial tasks or problems</emph> aimed at eliciting students' thinking (TW1). After students respond to the tasks, teachers must elicit students' processes (which we also refer to as strategies) for solving the task (S&amp;B1), and attempt to interpret the students' thinking (TW2). Often, the teacher must develop and ask follow-up questions about the students' process (S&amp;B3 and TW3). These follow-up questions can be aimed at gathering clarification on the process (S&amp;B1) or can probe for understanding of the mathematics (S&amp;B2). If follow-up questions are asked effectively, the teacher gathers enough information to be able to interpret students' thinking (TW2). As shown in Figure 1, the practices of eliciting and interpreting thinking are subcomponents of professional noticing (Jacobs et al., [<reflink idref="bib20" id="ref32">20</reflink>]), defined as attending to students' ideas, interpreting those ideas, and finally making a decision about how to respond to those ideas in the moment. Teachers elicit students' thinking as a prerequisite to attending to that thinking; they continue to ask probing questions in order to be able to interpret those ideas.</p> <p>Graph: Figure 1. A synthesis of decompositions of eliciting and interpreting student thinking.</p> <p>Table 1. Components of eliciting and interpreting students' thinking.</p> <p> <ephtml> &lt;table&gt;&lt;thead&gt;&lt;tr&gt;&lt;td&gt;TeachingWorks (2019)&lt;/td&gt;&lt;td&gt;Shaughnessy and Boerst (&lt;xref ref-type="bibr" rid="bibr37"&gt;2018&lt;/xref&gt;&lt;bold&gt;)&lt;/bold&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/thead&gt;&lt;tbody valign="top"&gt;&lt;tr&gt;&lt;td&gt;Formulating and posing initial questions (TW1)&lt;/td&gt;&lt;td&gt;Eliciting students' process (S&amp;B1)&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Listening to and interpreting students' thinking (TW2)&lt;/td&gt;&lt;td&gt;Probing students' understanding of key mathematical ideas (S&amp;B2)&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Developing additional questions/tasks (TW3)&lt;/td&gt;&lt;td&gt;Attending to students' ideas (S&amp;B3)&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td /&gt;&lt;td&gt;Deploying other moves that support learning (S&amp;B4)&lt;/td&gt;&lt;/tr&gt;&lt;/tbody&gt;&lt;/table&gt; </ephtml> </p> <p>Research on professional noticing has demonstrated the difficulties that teachers, and PSTs in particular, have demonstrated in regard to attending to, interpreting and responding to student thinking (Amador et al., [<reflink idref="bib1" id="ref33">1</reflink>]; Jacobs et al., [<reflink idref="bib20" id="ref34">20</reflink>]; Kilic, [<reflink idref="bib24" id="ref35">24</reflink>]; Simpson &amp; Haltiwanger, [<reflink idref="bib38" id="ref36">38</reflink>]; Son, [<reflink idref="bib40" id="ref37">40</reflink>]). These difficulties can be seen even at the first stages of the process as they elicit students' thinking. Shaughnessy and Boerst ([<reflink idref="bib37" id="ref38">37</reflink>]) found that beginning PSTs were capable of eliciting students' processes for solving tasks and attending to students' ideas; but, instead of probing for understanding of mathematical ideas with follow-up questions, PSTs tended to fill in student thinking based on their own assumptions. PSTs also have difficulty formulating eliciting questions on their own (Sleep &amp; Boerst, [<reflink idref="bib39" id="ref39">39</reflink>]). When students provide initially incorrect answers, some PSTs ask a series of leading questions that center the PSTs' desired strategy or way of thinking rather than students' thinking, a practice sometimes referred to as funneling (Wood, [<reflink idref="bib50" id="ref40">50</reflink>]); or, they abandon questioning altogether (Moyer &amp; Milewicz, [<reflink idref="bib31" id="ref41">31</reflink>]; Sleep &amp; Boerst, [<reflink idref="bib39" id="ref42">39</reflink>]; Weiland et al., [<reflink idref="bib47" id="ref43">47</reflink>]) and resort to teacher-centered responses like telling and showing (Son, [<reflink idref="bib41" id="ref44">41</reflink>]; Son &amp; Sinclair, [<reflink idref="bib42" id="ref45">42</reflink>]). Alternatively, when students provide correct answers, some PSTs simply decide not (or forget) to elicit the students' process and move to the next problem or task (Moyer &amp; Milewicz, [<reflink idref="bib31" id="ref46">31</reflink>]; Weiland et al., [<reflink idref="bib47" id="ref47">47</reflink>]), which is referred to as a checklist approach.</p> <p>Researchers have also documented the ways in which PSTs professional noticing, and skill at eliciting thinking specifically, improved through activities in methods classes. Schack et al. ([<reflink idref="bib36" id="ref48">36</reflink>]), for instance, demonstrated that PSTs' professional noticing abilities were significantly developed through a 5-session video-based module. Specific to eliciting thinking, Weiland et al. ([<reflink idref="bib47" id="ref49">47</reflink>]) showed, through repeated clinical interviews and reflections, that PSTs increased their tendency to ask competent follow up questions that encouraged students to elaborate on or extend their own thinking; however, PSTs still missed opportunities to elicit thinking and asked leading questions even toward the end of the course. Kazemi and Waege ([<reflink idref="bib23" id="ref50">23</reflink>]) documented PST growth over a semester of repeated cycles of planning, rehearsal with other PSTs, controlled enactments (co-teaching in small groups) and feedback and found that three PSTs who tended not to follow-up on student responses during early enactments developed their capacity to elicit students' elaboration on those ideas. However, only one PST progressed to the point where she asked questions aimed at getting students to develop mathematical justifications. Hallman-Thrasher ([<reflink idref="bib16" id="ref51">16</reflink>]) found that, through cycles of controlled enactment (co-teaching pairs of students) and feedback, PSTs improved at responding to unanticipated solutions by following up to elicit elaboration. Surprisingly, some PSTs also developed the undesired tendency to adopt a checklist approach in response to student strategies that they had anticipated. Clearly, developing the capacity for effective eliciting student thinking is complex work, and the field would benefit from a study of the possibilities and challenges of supporting PSTs in this endeavor once they progress to full-time student teaching internships or initial teaching.</p> <hd id="AN0156414408-4">Field-based coaching to develop ambitious practices</hd> <p>From the situated learning perspective (Lave &amp; Wenger, [<reflink idref="bib27" id="ref52">27</reflink>]) that grounds our study, student teaching represents an important stage in the development toward full participation in the teaching community. During these internships, PSTs take over teaching responsibility in an actual classroom, under the apprenticeship of a mentor teacher and one or more mentors from the teacher preparation program. The PST learns to be a teacher through their participation in the school community and their interactions with these mentors. One model of support commonly used during this apprenticeship stage is the formal coaching cycle, through which a mentor (or coach) meets with the PST to discuss a lesson plan, observes the lesson, then provides feedback (Stahl et al., [<reflink idref="bib43" id="ref53">43</reflink>]). During meetings with teachers, coaches engage in a variety of coaching moves, including reflective questioning, offering suggestions, listening and revoicing, and analyzing student data (Jakopovic, [<reflink idref="bib21" id="ref54">21</reflink>]). In our review of the literature, we found no studies documenting the use of formal coaching cycles to develop PSTs' (or teachers') skill at eliciting thinking specifically.</p> <p>However, we found two studies that suggest that coaching can be an effective support for in-service teachers' development around questioning more broadly, even as they deal with the complexities of an authentic teaching environment. Polly ([<reflink idref="bib34" id="ref55">34</reflink>]) documented that coaching through a variety of supports (e.g. planning and task selection, co-teaching) helped four in-service teachers shift their questioning patterns to ask more probing and open questions; post-lesson feedback was not a primary support in this study. In contrast, Bennett ([<reflink idref="bib3" id="ref56">3</reflink>]) documented two teachers' growth as he supported them by sharing post-lesson feedback: specifically, the frequencies of different types of questions the teachers asked, including follow-up questions and questions probing for understanding. Through this support, the frequency with which teachers asked questions in each of these categories increased. More research is needed to document how coaches' efforts with individual teachers and PSTs support the development of ambitious practices (Gibbons &amp; Cobb, [<reflink idref="bib14" id="ref57">14</reflink>]),particularly eliciting and interpreting students' thinking.</p> <p>As Gibbons and Cobb ([<reflink idref="bib13" id="ref58">13</reflink>]) point out, effective coaches first characterize a path or trajectory through which teachers typically pass, then they use that trajectory to assess teachers' performance and respond in ways to support their development. Accordingly, we reviewed the literature to develop an emergent trajectory for learning to elicit students' thinking.</p> <hd id="AN0156414408-5">Trajectories for developing high-leverage practices</hd> <p>Finding no existing trajectory specifically focused on PSTs' or teachers' development of the skill of eliciting and interpreting thinking across time, we consulted research on the broader bodies of literature related to teacher questioning (Hufferd-Ackles et al., [<reflink idref="bib19" id="ref59">19</reflink>]) and professional noticing (Jacobs et al., [<reflink idref="bib20" id="ref60">20</reflink>]) to hypothesize a developmental trajectory.</p> <p>Hufferd-Ackles et al. ([<reflink idref="bib19" id="ref61">19</reflink>]) describe a set of developmental trajectories that describe teachers' development of ambitious instructional practices, which they refer to as the creation of a math-talk learning community. One of these trajectories addresses teacher questioning and the eliciting of student ideas. Polly ([<reflink idref="bib34" id="ref62">34</reflink>]) adapted this framework to develop a heuristic for describing the depth to which teachers elicited student thinking. Teachers at Levels 0 and 1 fail to elicit students' processes. At Level 2, teachers elicit and ask follow up questions about processes, but do not ask deeper questions about students' mathematical thinking. Teachers functioning at Level 3 ask questions about students' mathematical thinking and rationale.</p> <p>Jacobs et al. ([<reflink idref="bib20" id="ref63">20</reflink>]), on the other hand, propose a trajectory for developing professional noticing. They argue that novices first develop the capacity to attend to students' mathematical thinking (as identified in Figure 1), which is necessary for interpreting student thinking. Eventually, teachers can learn to respond to students' mathematical thinking in productive ways that lead to gains in student understanding. Comparing these two trajectories, it appeared that the decomposition of eliciting thinking and professional noticing, as shown in Figure 1, was fitting to use as an emergent developmental trajectory for PSTs' learning to elicit and interpret students' thinking. First, they learn to elicit the students' process, then to ask effective follow-up questions, and these skills would allow them to interpret students' thinking correctly, which is necessary for responding to that thinking in a way that forwards the students' learning.</p> <p>In summary, attention to student thinking is essential, and in order to attend to students' thinking, teachers must elicit student thinking effectively. But PSTs demonstrate difficulty eliciting thinking, even after repeated one-on-one interviews and cycles of rehearsal and enactment in controlled settings during methods courses. More research is needed on how to support novice teachers as they interact with students in diverse, challenging settings like those they experience at later points in field-based internships (Kazemi et al., [<reflink idref="bib22" id="ref64">22</reflink>]). In this study, the unique situation of a mathematics teacher educator assigned to coach student teachers during a six-month internship allowed for a study of how mentors can work with student teachers to develop their capacity for eliciting students' mathematical thinking.</p> <hd id="AN0156414408-6">Methodology</hd> <p>Consistent with other studies aiming to gain in-depth understanding of the complex phenomenon of learning to teach, we employed a case study design (Yin, [<reflink idref="bib51" id="ref65">51</reflink>]). The study also fulfills Cochran-Smith and Donnell's ([<reflink idref="bib10" id="ref66">10</reflink>]) description of two critical features of practitioner research. The study was intentional in that it was planned and designed prior to the coaching; it was also systematic, as multiple data points were collected and analyzed in a variety of ways.</p> <hd id="AN0156414408-7">Study context</hd> <p>Two undergraduate PSTs, whom we refer to as Grace and Jennifer, were coached by the first author during an extended field-based internship at a university in the southeast US.[<reflink idref="bib1" id="ref67">1</reflink>] Both PSTs were White females between 20-25 years of age. They interned in first grade classrooms at a public elementary school in a local urban school district; the students at the school were mostly Black (59%) or Hispanic (28%) from families categorized by the district as medium (68%) or low (24%) socioeconomic status. The coaching took place during the beginning phase of a college-wide initiative to increase alignment across coursework and field experiences. Selected faculty members were assigned to coach PSTs around three high-leverage practices: eliciting and interpreting student thinking, setting up and managing small group work, and facilitating whole class discussions. The coach had not taught either of the PSTs in their previous methods courses. They were assigned by the office that coordinated student teacher supervision and prior instructor/student contact was not considered. Although he coached four students during the academic year, only the two PSTs assigned to the school district from which we had permission to do research were asked to participate in the study and both agreed.</p> <p>The three high-leverage practices were the focus of a two-day professional development where university faculty (including the coach) and cooperating teachers jointly decomposed the three practices and developed observation "look fors" that served as indicators of productive implementation of the practices. Many of the look fors correspond with aspects of the research-based decomposition published by TeachingWorks ([<reflink idref="bib45" id="ref68">45</reflink>]). For example, the look for "Candidate asks questions that require students to share or justify their thinking" corresponds to the TeachingWorks practice of "Developing general, open-ended questions," and the look for "Candidate gives students time to think and respond using content-specific language" corresponds with "Giving the student plenty of time to think." During the subsequent academic year, cooperating teachers, university supervisors, and faculty coaches observed the PSTs and provided feedback focused on these practices. Pre-observation and post-observations meeting protocols were provided, and coaches were expected to use the observation look-fors as tools for assessing the PSTs. Cooperating teachers were encouragedto model the practices during the beginning phases of the internship. Faculty coaches functioned purely as coaches, and evaluations were provided by the university supervisors.</p> <p>PSTs did not receive instruction on the high-leverage practices during student teaching, outside of the coaching from cooperating teachers, university supervisors and faculty coaches. Prior to student teaching, however, the PSTs took two elementary mathematics methods courses, one focused on teaching students grades K-2, the other focused on upper elementary mathematics. In the K-2 course, a number of assignments aimed at developing PSTs' capacity for eliciting student thinking, including a clinical interview to elicit a student's understanding of early number concepts and a small group activity where PSTs elicited student thinking to assess proficiency with addition and subtraction. Field experiences in the second course were less focused on eliciting the thinking of individual students and more on eliciting thinking during whole class discussions. During their internship year, PSTs spent one day per week in the first-grade classrooms in the fall, then completed 16-weeks of full-time student teaching in the same classroom during the spring.</p> <hd id="AN0156414408-8">Positionality</hd> <p>The coach was an early-career assistant professor of elementary mathematics education with no prior field-based coaching experience. He engaged in practitioner research as a way of systematically reflecting on and improving his coaching. Being familiar with the research into ambitious instruction and formative assessment, he sees eliciting student thinking to be a foundational teaching practice, necessary to support students in developing mathematical understanding. Because the primary goal was to learn from the experience, rather than prove the effectiveness of the intervention, he was committed to collecting and analyzing data to determine how the PSTs developed through the coaching; mistakes and missed coaching opportunities were welcomed and seen as valuable opportunities for growth. He recruited a team to support the research with the aim of enhancing the validity, which, in practitioner research can be described as trustworthiness (Cochran-Smith &amp; Donnell, [<reflink idref="bib10" id="ref69">10</reflink>]; Lincoln &amp; Guba, [<reflink idref="bib28" id="ref70">28</reflink>]).</p> <hd id="AN0156414408-9">Data and analytical methods</hd> <p>The coach observed each PST three times, once during the fall semester and twice in the spring; each time the observation was preceded by a pre-observation meeting one or two days before the observation and a post-observation meeting either on the same day or the day after. Before each observation, the coach and the PST met to discuss the plans and goals for the lesson. The coach also elicited PSTs' understandings of the relevant learning trajectories. For instance, prior to the lesson on addition, the coach activated the PSTs' knowledge of the trajectory of addition strategies, including counting all, counting on, using derived facts, and recall (Carpenter et al., [<reflink idref="bib6" id="ref71">6</reflink>]). During the observation, the coach took detailed notes, often writing verbatim what the PSTs and students said as closely as possible so that this data could be shared with the PSTs during post-observation meetings. The coach also documented non-verbal actions and notes of his interpretations, in brackets, as in the following note describing a card game: "[Another student comes in and has 8,7,4,5] T: we're going to start with Leila.... [Switches out a 5 with a 5.] [Didn't ask why]." Here, the note "[Didn't ask why]" indicates that the PST did not ask the student why she switched out a 5 with a 5, a move that provided no benefit toward the goal of the game. A few lines later, the notes say "[I would ask: why did you switch that one?]" indicating the coach's own interpretation. Due to the difficulty of securing consent for students to be recorded, we were not able to video or audio record observations. At times, the coach made recommendations and modeled instruction during the lesson; these instances were also noted in brackets. As such, the coaching represented a blend between traditional cycles of pre- and post- observation meetings as well as some in-the-moment coaching.</p> <p>Post-observation meetings followed a semi-structured protocol and were focused on the three focus practices: eliciting students' thinking, setting up and managing small group work, and facilitating whole class discussions. Because both PSTs taught in first-grade classrooms in the same school and the coach visited the PSTs on the same day, the PSTs taught lessons on the same topic each visit, sometimes using the same instructional activity.</p> <p>Both pre- and post-observation meetings were audio recorded and transcribed (with the exception of the first pre-observation meeting for one of the PSTs, which was not recorded due to technical difficulty.) The transcripts were analyzed by the coach, another full-time faculty member, and one graduate assistant. During the first stage of analysis, each researcher used descriptive open coding (Miles et al., [<reflink idref="bib30" id="ref72">30</reflink>]) of the first post-observation meeting transcript to label milestones of learning how to elicit student thinking. Some of these codes, such as asking follow-up questions, corresponded with the constructs identified in the decompositions of eliciting and interpreting student thinking (Shaughnessy &amp; Boerst, [<reflink idref="bib37" id="ref73">37</reflink>]; TeachingWorks, [<reflink idref="bib45" id="ref74">45</reflink>]).Others corresponded with Hufferd-Ackles et al.'s ([<reflink idref="bib19" id="ref75">19</reflink>]) trajectory of developing questioning techniques (e.g. follow-up questions represent Level 2). Although we had this literature in mind, we engaged in an open coding process to allow for other codes to emerge. We also labeled the various types of coaching moves to support the development of the PSTs' eliciting skills. By discussing these emergent codes we began the process of creating a list of generalizable codes that could be applied across transcripts. The list of codes was divided into two groups: elements of the PSTs' developmental trajectory and coach moves. <emph>Funneling</emph> (meaning asking a series of leading questions not focused on student thinking [Wood, [<reflink idref="bib50" id="ref76">50</reflink>]]), and <emph>follow-up questions</emph> to elicit students' thinking were examples of the codes created under the PST developmental trajectory category. <emph>Prompting reflection on data</emph> and <emph>asking what to do differently</emph> next time were examples of coach moves.</p> <p>Separately, the researchers continued to read each transcript turn-by-turn, assigning one or more codes and creating new codes when the existing codes were insufficient for describing the observation. The team resolved disagreements through discussion as a reliability check (Miles et al., [<reflink idref="bib30" id="ref77">30</reflink>]). The master code list was edited as codes were removed, added, and combined. This process was repeated after each transcript was analyzed, and the relevant portions of previously coded transcript were re-coded when codes changed. A codebook was created to articulate the boundaries for each code (Creswell &amp; Poth, [<reflink idref="bib11" id="ref78">11</reflink>]). Thirty-five coaching moves, or practices, were identified, such as <emph>prompting reflection on whether or how a goal was achieved</emph>, <emph>probing for specificity around a claim</emph>, and <emph>suggesting what to do next time</emph>.</p> <p>Once all the transcripts were coded, the research team read within and across codes and performed analytical memoing (Miles et al., [<reflink idref="bib30" id="ref79">30</reflink>]) to identify patterns. For example, we noted that as PSTs funneled more frequently, they probed and elicited thinking less, and vice versa.</p> <p>After the memos were discussed, the lead author performed a separate analysis through which he coded each of the detailed observation notes with the elements of eliciting student thinking from Figure 1, which aligned well with the observations as indicated by the analytical memoing. He also noted whether these moves were made in response to students' initially correct or incorrect responses, an important distinction that arose from the analytical memos. The other faculty member reviewed those codes, and disagreements were discussed and resolved. This analysis of the notes provided a reliability check on whether the patterns noticed by the coach and discussed during the post-observation meetings were confirmed in the observation data.</p> <p>From the memos, we created a visual to describe the emergent PST developmental trajectory and the related coaching moves (similar to Figure 2, which is described in the next section). Transcripts and observation notes were reviewed to validate the trajectory; researchers looked for evidence to both support and refute the progression. In the next section, we present the findings summarized by the visual trajectory and provide illustrative examples.</p> <p>Graph: Figure 2. The emergent trajectory of learning to elicit and interpret student thinking.</p> <hd id="AN0156414408-10">Findings</hd> <p>Supported by the coach, both PSTs followed a similar trajectory in developing their practice of eliciting student thinking. Although the observed trajectory aligned to the trajectory synthesized from the literature, we represent the observed trajectory with two branches as shown in Figure 2, because the PSTs followed different trajectories in their development for responding to correct or incorrect answers. The boxes represent the specific moves the PSTs made as they attempted to elicit thinking. The arrows labeled Grace and Jennifer represent the PSTs' trajectories, with solid lines representing demonstrated effective practice and the dotted lines representing areas identified as needing continued growth. We first present the findings related to the top branch, which represents the PSTs' moves to elicit the thinking of students who provide a mathematically correct answer to the initial task or question.</p> <hd id="AN0156414408-11">PST responses to correct answers</hd> <p>As shown in Figure 2, Grace first demonstrated a checklist approach in responding to students' correct answers to initial tasks, affirming student responses then moving to another student. However, after a coach intervention during observation 1 she quickly moved to asking initial eliciting questions, which is where Jennifer began. By the second observation, both PSTs frequently asked follow-up questions with the goal of interpreting the thinking of students who answered initial prompts correctly. These patterns are also evident in Table 2, where we present the frequency with which the PSTs asked follow-up questions to elicit students' explanations after correct and incorrect responses. The table provides the frequencies identified from the observation notes; the intent is not to represent exact frequencies that occurred in the lesson as we were not able to record audio or video. Below, we present illustrative examples that show what this questioning looked like across varying mathematical content, and we describe the ways the coach supported the PSTs in their development. Then, we transition to the trajectory along the lower branch, which describes PST responses when students answered the initial task or question incorrectly.</p> <p>Table 2. Frequency of PST eliciting students' explanations after initial responses as captured in the observation notes.</p> <p> <ephtml> &lt;table&gt;&lt;thead&gt;&lt;tr&gt;&lt;td /&gt;&lt;td&gt;Number of instances in which PST elicited explanation after correct response&lt;/td&gt;&lt;td&gt;Number of instances in which PST did not elicit explanation after correct response&lt;/td&gt;&lt;td&gt;Number of instances in which PST elicited explanation after incorrect response&lt;/td&gt;&lt;td&gt;Number of instances in which PST did not elicit explanation after incorrect response&lt;/td&gt;&lt;/tr&gt;&lt;/thead&gt;&lt;tbody valign="top"&gt;&lt;tr&gt;&lt;td&gt;Grace&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Observation 1 (before coach intervention)&lt;/td&gt;&lt;td&gt;0&lt;/td&gt;&lt;td&gt;16&lt;/td&gt;&lt;td&gt;1&lt;/td&gt;&lt;td&gt;4&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Observation 1 (after coach intervention)&lt;/td&gt;&lt;td&gt;8&lt;/td&gt;&lt;td&gt;0&lt;/td&gt;&lt;td&gt;1&lt;/td&gt;&lt;td&gt;2&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Observation 2&lt;/td&gt;&lt;td&gt;26&lt;/td&gt;&lt;td&gt;19&lt;/td&gt;&lt;td&gt;6&lt;/td&gt;&lt;td&gt;8&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Observation 3&lt;/td&gt;&lt;td&gt;Observation notes did not capture this type of data during this observation&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Jennifer&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Observation 1&lt;/td&gt;&lt;td&gt;9&lt;/td&gt;&lt;td&gt;5&lt;/td&gt;&lt;td&gt;0&lt;/td&gt;&lt;td&gt;1&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Observation 2&lt;/td&gt;&lt;td&gt;12&lt;/td&gt;&lt;td&gt;11&lt;/td&gt;&lt;td&gt;1&lt;/td&gt;&lt;td&gt;6&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Observation 3&lt;/td&gt;&lt;td&gt;Observation notes did not capture this type of data during this observation&lt;/td&gt;&lt;/tr&gt;&lt;/tbody&gt;&lt;/table&gt; </ephtml> </p> <hd id="AN0156414408-12">Grace's responses to correct answers (observation 1)</hd> <p>During the first observation, which occurred in November, before the full-time student internship, Grace planned to work on making tens with her students. During the pre-conference, the coach and Grace set a goal related to eliciting thinking, to "see what strategies students are using," and discussed a trajectory of addition strategies. During the lesson, she led small groups in playing a card game. Students attempted to make tens by swapping cards from their hand with cards laid out by the PSTs. As shown in the first column of Table 2, Grace initially failed to elicit the students' processes after correct answers, In rounds 1 and 2, Grace did not ask questions to elicit student thinking if students responded correctly to her initial question, which was usually "do you have a match?" Instead, the notes indicate that on at least 16 occasions, she either affirmed the correct responses and moved to the next student's turn, or followed up by asking students to formalize the relationship as a verbal equation. Although it is possible that Grace was able to notice students' processes without explicitly asking each time, on six of the occasions, the coach marked in the notes that he would have asked follow-up questions to determine the students' process and only once identified that he could determine the strategy without additional information. Furthermore, these instances present missed opportunities for students to practice explaining their thinking, which is an important skill in and of itself.</p> <p>During a transition while groups rotated, the coach made an in-the-moment suggestion, reminding Grace of the goal of eliciting thinking and recommending two eliciting questions: "how do you know that" and "why did you make the switch?" In response, on eight occasions where students answered correctly, Grace asked a question aimed at eliciting a process. However, she was only definitively successful in eliciting the students' description of their process twice. On two occasions, the notes indicate that she failed to ask adequate follow-up questions. In the remaining five instances it was unclear from the notes whether she elicited the student's strategy. An illustrative example of an initial eliciting question followed by inadequate follow-up comes from the field notes:</p> <p>Grace: Tell us how you knew you had a match.</p> <p>Student: Five and five equals ten and nine plus one equals 10.</p> <p>Grace: Tell us how you know nine plus one equals 10.</p> <p>Student: If I have five and then I can do five more and get 10.</p> <p>Here, because Grace did not probe further, she was left without evidence of what type of strategy the student used to determine the sum of five and five. To determine whether the student was demonstrating automaticity, she could have asked, "did you count or did you just know that 5 plus 5 equals ten?" This was illustrative of Grace's practice at the end of the first observation: she asked initial eliciting questions but was inconsistent in asking follow-up questions necessary to determine the students' strategy.</p> <p>Despite her tendency not to ask follow-up questions, she did demonstrate the ability to interpret where some of the students were on the trajectory of addition strategies, as evidenced in the post-interview, when she identified several students who were counting all and named two students who demonstrated the ability to recall addition facts. She even described how she modeled how to count-on for a student who was successfully counting all.</p> <hd id="AN0156414408-13">Coaching to reinforce probing</hd> <p>To address the second research question, we examined the labels we attached to the moves the coach made most frequently to support the PSTs' emerging practice related to eliciting thinking. The coach <emph>pointed out progress</emph> four times during the meeting, telling Grace for example, that "in rounds one and two [of the game] I didn't see it, but then in round three things changed quite a bit." He <emph>presented data</emph>, naming or highlighting instances where she provided wait time and asked questions aimed at eliciting students' processes. He also read back some of the initial exchanges where she did not elicit thinking, <emph>asked her what she would do next time</emph>, and asked what the intent behind her change in practice was. Her response demonstrated her understanding of the value of eliciting questions to determine placement on the trajectory of addition strategies:</p> <p>I felt like the first set, I was just giving it to them, they weren't doing much thinking.... And then the second or third time around they had to think about it for themselves and they actually had to count. Plus, I also can see if they had counting on or counting all. Some had counting all.</p> <p>Later in the coaching session, the coach implicitly linked the idea of probing student thinking to the act of responding to advance that thinking, by asking Grace what she could do to help students get from "counting all" to "recall." In response, Grace recalled the case where she had introduced "counting on" to a student, which was the one instance in the notes where she responded in a way that aimed to help the student advance on the trajectory.</p> <hd id="AN0156414408-14">Grace's progress related to follow-up questions (observation 2)</hd> <p>During the second lesson involving small group work focused on the classification of two- and three-dimensional shapes, which occurred three months after the first observation once the PSTs had started taking over whole-class instruction, Grace continued to ask eliciting questions after initially correct answers, doing so at least 26 times (see Table 2). On most occasions, the students' responses to a single eliciting question about how they knew a shape was two- or three-dimensional was sufficient to understand their thinking. On six occasions, Grace demonstrated progress in asking follow-up questions to clarify students' reasoning. For instance, when a student explained that she knew a pyramid was three-dimensional because "it's not flat," the PST asked, "What do you mean by flat?" Similarly, when she asked about the difference between a sphere and a cylinder, and a student responded that "a cylinder has two sides," she followed up saying "Two sides? Point at the sides." On 19 occasions in the notes, Grace did not elicit further explanation after students provided correct responses, which is reasonable, as some questions were recall questions aimed at reviewing the names of shapes, for instance. She did however, show an inclination to probe to provide students the opportunity to explain their thinking.</p> <hd id="AN0156414408-15">Jennifer's responses to correct answers and subsequent coaching</hd> <p>Jennifer, on the other hand, asked initial eliciting questions like "how do you know?" and follow-up probes in response to students' correct answers from the beginning of the first observation, doing so nine times. In that lesson, which involved the same making-tens card game that Grace used, Jennifer also demonstrated the ability to ask follow-up questions, although one question was sometimes sufficient for determining the students' strategies. For instance, when a student responded that he counted on his fingers, she asked "did you start with 7 or did you start with 3?" The answer to this question provided information needed to determine whether the student was counting on from the first number or counting on from the larger number.</p> <p>Jennifer also demonstrated an emergent capacity to respond to advance student thinking. On three occasions, when students answered correctly, she asked "how many would you need to get ten," which prompted students to answer a more complex problem. There was, however, no evidence of Jennifer responding to students in ways that supported them in moving from counting all or counting on from the first number to more sophisticated strategies.</p> <p>Like with Grace, the coach read back data and named the follow-up questions Jennifer asked as probes. He also asked Jennifer to identify students' placement on a trajectory of strategies and described moves she could make to nudge students forward. By referencing the trajectory of addition strategies in this way, he implicitly provided a rationale for why teachers need to elicit student thinking: to make a decision about how to support them in advancing their thinking, a point taken up in the discussion. The pair also discussed how she need not elicit students' processes each time, once their preferred strategy had been determined.</p> <hd id="AN0156414408-16">PST responses to incorrect answers</hd> <p>Although both PSTs developed the practice of probing on correct responses to be able to interpret student thinking, the same was not true when students offered mathematically incorrect responses to the PSTs' initial questions.</p> <hd id="AN0156414408-17">Initial patterns in responding to incorrect answers (observation 1)</hd> <p>During the first observation of the game involving making tens, neither PST showed a tendency to probe incorrect responses. Grace failed to elicit students' strategies on incorrect answers six times (four times prior to the coach intervention and twice after), although she did probe once to determine that a student did not understand the rules. Instead of eliciting, Grace demonstrated a tendency to respond to incorrect responses with a series of leading questions aimed at getting the student to say the correct answer, a practice referred to as funneling (Wood, [<reflink idref="bib50" id="ref80">50</reflink>]). An example comes from a case where the student had two fives, which represented a ten-pair:</p> <p>Grace: Do you have a match?</p> <p>Student: [shakes his head.]</p> <p>Grace: No? you sure?</p> <p>Another student: [gasp] He does!</p> <p>Grace: Do you want a hint? [puts the 5 next to the other 5]. What's 5 + 5?</p> <p>Student: Ten</p> <p>Grace: So which do you want to switch out?</p> <p>In this instance, the PST asked a follow-up question ("are you sure?") that had the potential to signal to the student that his initial response was incorrect and did not elicit the student's thinking about why he did not have a match.</p> <p>Jennifer also failed to elicit a student's process in the only example of an incorrect answer in the notes. In this case, a student said that the sum of 9 and 5 was 15, and Jennifer replied, "Is it 15, because 10 and 5 make 15." The student replied that the answer was 14, and the coach was not able to discern why the student initially said 15.</p> <hd id="AN0156414408-18">Initial coaching</hd> <p>Noting the PSTs' tendency to respond to incorrect answers in ways that did not elicit the reasoning behind these answers, the coach engaged in several moves to address this issue. As demonstrated in the following excerpt, he read back the PSTs' questions and student responses, naming and defining funneling during the first post-observation meeting with both PSTs.</p> <p>Coach: When he says 15 from [adding] nine and five and you ask "is it 15? because 10 and five make 15," you're sort of... So what, you just made a face. What are you, what are you making that face?</p> <p>Jennifer: I felt bad that I said that I was trying to make him think of one less than 15. It sounds bad now that I hear it.</p> <p>Coach: I mean, all this is the learning process and, for where you're at, you are asking really nice questions. So, you never have to feel bad. It's always like, okay, I can learn from that. This is a really cool opportunity I can learn from. There's a distinction between questions that you ask to find out more about how they're thinking and questions that we call funneling questions which are like, you're trying to get them to give you the right answer and you're forcing them down a particular path.</p> <p>After naming and defining funneling, he states the rationale for avoiding this pattern of questioning more explicitly to Grace, saying, "if you're funneling you can't really figure out how [the students] are thinking about it."</p> <hd id="AN0156414408-19">Beginning to probe incorrect answers (observation 2)</hd> <p>In the notes from the second observation involving shapes, Grace asked probing questions in response to an incorrect answer six times, demonstrating development of an inclination to elicit students' partial understandings. For example, when asking a student how many faces a rectangular prism has, the student responded incorrectly, saying "four." Without saying whether the response was correct or incorrect, the PST responded by handing the student the rectangular prism and asked the student to count the faces. This move allowed the PST to see what the student was counting as faces and provided a scaffold if the student needed support in counting them. On eight occasions, however, Grace engaged in a funneling pattern or directly instructed students after they answered incorrectly, without eliciting their thinking.</p> <p>Jennifer showed less growth in responding to incorrect answers during the second observation. She probed an incorrect response one time; however, on six occasions, she did not ask questions to determine the reason for the error. On three occasions, her response to an incorrect answer was to repeat the answer back as a question. For instance, when a student said that a cube had four sides, Jennifer replied "Does it have four sides? Think about it again." This questioning pattern avoided funneling but did not help the PST determine what the student considered a side and why he said there are four. On another three occasions, she provided instruction, without eliciting information about why the student provided the incorrect answer.</p> <hd id="AN0156414408-20">Coaching to reinforce probing on incorrect answers</hd> <p>To reinforce Grace's emerging practice of probing on incorrect answers, the coach read back excerpts, asking the PST about her intent, then contrasted her "diagnosing"[<reflink idref="bib2" id="ref81">2</reflink>] moves with "funneling" and noted that probing incorrect answers was a goal set after the last observation.</p> <p>In contrast, to support Jennifer in her development, during the post-observation meeting for the second observation, the coach read back a classroom interaction, and asked the PST what she thought about her questioning. When she explained that she could have probed more on one student (Jillian) but did not because she wanted to "keep it moving", the coach praised her questioning generally, then focused her attention on a missed opportunity to probe the thinking of another student, Sean.</p> <p>[Sean] says "because it has two vertices" and you said, "wait, does it have vertices?" That is signaling to him that he's wrong. And it would be interesting to ask him, "what do you mean vertices?" Or "where do you mean vertices?" So, it's like it's an opportunity to probe, to figure out what he's not getting.</p> <p>The coach then identified probing on incorrect answers as an appropriate "next step" to work on. For both PSTs, the coach highlighted moves like asking eliciting questions and funneling during the second post-observation interview, but not as frequently as he did during the first.</p> <hd id="AN0156414408-21">Continued difficulty probing incorrect answers (observation 3)</hd> <p>During the third observation of Grace, the coach remarked that she was "consistently asking the 'how did you get that'-like probing questions," which suggests that she was asking questions to elicit students' thinking. The notes from the session do not allow for quantitative measures, because the coach was forced to take notes by hand due to a technology failure. He also took a more proactive role in the instruction, so the written notes were not as detailed. He took a more proactive role because Grace was not able to interpret the thinking of a student who struggled to add numbers greater than 10 using a hundreds chart. The student was attempting to count on but stopped counting at an incorrect number, and Grace was not able to fully elicit the student's process to understand why. To model the act of probing, the coach stepped in and posed a few tasks to the student. (Up to that point, the coach had only observed and had not interrupted the PSTs instruction.) As the PST worked with the other students in the group, the coach worked with the student, asking her to count out loud. The coach recounted the interaction with the PST in the post-observation meeting, saying: "I said, [...] 'are you saying something to yourself as you're counting?' And she said yes. And I said, 'can you tell me that?' And so, when she was doing three plus four, she would start at three and she would go 'four, five, six, seven.'" By having a student count out loud, the coach was able to probe deeper to determine why her strategy failed when she tried to add larger numbers: she was unable to keep track of the second addend. The coach then pushed Grace to interpret the student's thinking and identify a developmentally appropriate strategy to lead the student toward. He also described the need for the student to work with concrete manipulatives to begin unitizing tens.</p> <p>Recounting the interaction with the student, and especially the additional probing that the coach engaged in, made an impression on the PST. This was evident at the end of the interview, when the coach asked what her big takeaways were from the coaching session. She responded with "I learned a lot more how to ask them to [...] show me how you're counting in your head. [...] I really am going to do that like Monday, probably today, honestly, if I can."</p> <p>In terms of the developmental trajectory in Figure 2, Grace asked eliciting questions in response to incorrect answers (as evidenced by the notes from observation 2 and by coach report in observation 3) and was developing her capacity to probe to the point of interpreting student thinking. As demonstrated in this interaction, in cases where she was unable to interpret students' ideas, she was also unable to respond in ways that productively advanced student thinking. Much of the post-conference discussion focused on the student described above, and the coach presented less data and highlighted moves that Grace made much less frequently. Perhaps because this was their last meeting, he was also much more direct, making more suggestions about what she should do in the future.</p> <p>During the third observation of Jennifer, the pair shifted their focus to the practice of orienting students to each other's thinking. The coach did more modeling of talk moves to orient students to each other's thinking, and because his notes were focused on these moments as well, there were fewer records of the verbatim dialogue. Consequently, there is not evidence as to whether she did or did not probe students' initial incorrect responses in the form of recorded discourse. He noted during the post-conference that she was still funneling at times when students demonstrate alternate or partial understandings. Like Grace, Jennifer noted this as a major takeaway. From these data, we propose that Jennifer was still in the initial phases of developing her capacity to elicit the thinking of students who initially provided incorrect answers during the final observation.</p> <hd id="AN0156414408-22">Summary of the findings</hd> <p>Our first research question focused on the trajectories PSTs progressed along as they learned to elicit and interpret student thinking during student teaching. As shown in Figure 2 and Table 2, when students provided correct answers to initial prompts, both PSTs demonstrated the capacity to probe deeper to interpret students' thinking by the third observation. However, when students provided incorrect answers to initial tasks or questions, neither PST consistently probed to the extent necessary to interpret students' thinking in relation to established learning trajectories.</p> <p>The second research question examined how the coach supported student teachers in learning to elicit and interpret student thinking. The ten most frequently used coaching moves aimed at supporting PSTs' development in eliciting student thinking are shown in Table 3. The coach frequently made direct suggestions of what to do next time the students were in a similar situation; most of these instances occurred in the meeting after the third observation (8 instances with Grace and 17 with Jennifer) when he seemed to be giving advice knowing he would not observe them again. He also consistently named and/or highlighted whether the PST elicited and probed or conversely, when they funneled students to the correct answer; this was most frequent in the meeting following the first observation, then was less prevalent after. Often, he named these teaching moves during the many instances when he presented data to the PSTs by reading exchanges between the PST and the students. He also provided meta-commentary on the PSTs progress, provided rationale for why particular teaching moves are important, and prompted PSTs to interpret student thinking, either after presenting data or simply by asking the PSTs what they learned about students' thinking over the course of the lesson. Often, this involved asking the PSTs where particular students were operating in terms of the learning progressions. Although we are careful not to make causal claims between the types of coaching moves used most frequently and PSTs progress along the trajectory of learning to elicit students' thinking, these different coaching moves have the potential to support PSTs in the work of developing their capacity for eliciting and interpreting student thinking in a variety of ways, a topic we take up in the following discussion.</p> <p>Table 3. Coaching moves made to support PSTs in eliciting student thinking.</p> <p> <ephtml> &lt;table&gt;&lt;thead&gt;&lt;tr&gt;&lt;td&gt;Coaching move&lt;/td&gt;&lt;td&gt;Total number of occurrences&lt;/td&gt;&lt;td&gt;Number of post-observation meetings with at least one occurrence&lt;/td&gt;&lt;/tr&gt;&lt;/thead&gt;&lt;tbody valign="top"&gt;&lt;tr&gt;&lt;td&gt;Suggesting what to do next time&lt;/td&gt;&lt;td char="."&gt;46&lt;/td&gt;&lt;td char="."&gt;6&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Naming/highlighting teaching moves (e.g. eliciting and funneling)&lt;/td&gt;&lt;td char="."&gt;33&lt;/td&gt;&lt;td char="."&gt;6&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Presenting data&lt;/td&gt;&lt;td char="."&gt;25&lt;/td&gt;&lt;td char="."&gt;6&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Pointing out progress in learning to elicit thinking&lt;/td&gt;&lt;td char="."&gt;15&lt;/td&gt;&lt;td char="."&gt;4&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Providing rationale for teaching move&lt;/td&gt;&lt;td char="."&gt;14&lt;/td&gt;&lt;td char="."&gt;6&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Prompting for interpreting student thinking&lt;/td&gt;&lt;td char="."&gt;14&lt;/td&gt;&lt;td char="."&gt;5&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Prompting reflection on data&lt;/td&gt;&lt;td char="."&gt;13&lt;/td&gt;&lt;td char="."&gt;5&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Asking what to do differently next time&lt;/td&gt;&lt;td char="."&gt;12&lt;/td&gt;&lt;td char="."&gt;3&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Probing for specificity around a claim about student thinking&lt;/td&gt;&lt;td char="."&gt;12&lt;/td&gt;&lt;td char="."&gt;4&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Pointing out missed opportunities&lt;/td&gt;&lt;td char="."&gt;9&lt;/td&gt;&lt;td char="."&gt;4&lt;/td&gt;&lt;/tr&gt;&lt;/tbody&gt;&lt;/table&gt; </ephtml> </p> <hd id="AN0156414408-23">Discussion</hd> <p>Findings from this study contribute to the existing literature about supporting the development of PSTs' capacity to elicit and interpret students' thinking in several ways. First, the emergent developmental trajectory synthesized from the literature (Figure 1) proved to be a useful developmental trajectory for identifying long-term goals for the PSTs' development, assessing their current practices, and identifying next steps (Gibbons &amp; Cobb, [<reflink idref="bib13" id="ref82">13</reflink>]). However, we found that because PSTs' responses to correct and incorrect answers differed, the observed trajectory in Figure 2 provided helpful specificity for our analysis and for future coaching.</p> <p>Like other researchers (e.g., Moyer &amp; Milewicz, [<reflink idref="bib31" id="ref83">31</reflink>]; Sleep &amp; Boerst, [<reflink idref="bib39" id="ref84">39</reflink>]; Weiland et al., [<reflink idref="bib47" id="ref85">47</reflink>]), we found that when students provided incorrect answers, the PSTs frequently responded by asking leading questions, rather than eliciting students' thinking. This study extends those findings by documenting that, even with targeted coaching, this pattern was more difficult to interrupt than failing to probe correct answers. We examined the meeting transcripts in search of data explaining why these PSTs had more difficulty developing the practice of eliciting the thinking of students who initially answered incorrectly. Some insight came from Jennifer's comment: "maybe I'm trying too hard to push them into what I want them to think." Funneling incorrect answers is natural for novice teachers who were likely taught in classrooms where teachers followed the normative practice in U.S. classrooms involving "a series of low level questions in rapid-fire sequence, with the focus primarily on the correct answer, rather than on understanding" (Weiss et al., [<reflink idref="bib48" id="ref86">48</reflink>], p. 65).</p> <p>Given that Grace struggled with probing adequately on incorrect responses and that Jennifer was at the beginning stages of developing this practice during the final observation, we can conclude that the coaching moves made most consistently, reading back data, naming funneling, and suggesting what to do next time, were not adequate to support PSTs in developing the capacity to probe to the point of interpreting students' strengths and areas for growth. In response to these findings, we looked to the data and literature to develop hypotheses about what sorts of coaching support could have provided the needed support. During the last observation of Grace, the coach stepped in and modeled how to probe deeper with a follow-up question. This form of modeling is consistent with the recommendations for practice-based teacher preparation (McDonald et al., [<reflink idref="bib29" id="ref87">29</reflink>]), and the absence of modeling early in the coaching trajectory may explain why students kept asking leading questions.</p> <p>In terms of the developmental trajectory offered by Hufferd-Ackles et al. ([<reflink idref="bib19" id="ref88">19</reflink>]), the candidates progressed to exhibit Level 2 questioning, particularly for correct answers, but codes related to Level 3, asking for students' rationale or deeper mathematical thinking, did not emerge. The absence of coaching interactions around Level 3 questioning is significant and shows room for coaching in this area. Because the candidates were still developing their capacity for Level 2 questioning, it is reasonable that coaching for even deeper eliciting would come later and may require more than three observations. It is also important to consider the mathematical content and the tasks presented to students as factors contributing to PSTs' ability to elicit student thinking. In the first lesson, for instance, the opportunity for PSTs to ask Level 3 questions was less prevalent because they were working on fluency. Further study is necessary to investigate the moves that might support candidates in this development and the ways that the mathematical content and tasks contribute to PSTs' ability to elicit student thinking.</p> <p>In looking at the frequency of the coaching moves, we found it striking that the coach directly suggested what to do next time a total of 46 times, and asked PSTs to explicitly reflect on what they would do differently after sharing data 12 times. These practices are similar, in that they consider alternative actions, but they are very different in terms of the potential for PST development. Asking PSTs what they might do differently is recommended in practice-based teacher education literature, because it provides the opportunity for the PST to rehearse a specific response to a student in a controlled setting without the pressures of time and managing multiple students (Grossman &amp; McDonald, [<reflink idref="bib15" id="ref89">15</reflink>]), and it also allows the PST to receive expert feedback related to that response. Future research is needed to determine whether earlier modeling and more frequently asking students what they would do differently would have supported the PSTs in more effectively probing students' initial incorrect answers.</p> <p>Finally, we looked to the data to determine the ways and extent to which the PSTs and coach expressed the rationale for probing on incorrect answers specifically. In the earlier observations, either the coach or PST provided three rationales: so the teacher can uncover the students' thinking and know where they were on the trajectory, so other students would have access to their thinking, and so that the student might correct themselves. However, these rationales omit the critical reason for uncovering the students' thinking when they respond incorrectly: to respond in a principled way to advance students' understanding (Jacobs et al., [<reflink idref="bib20" id="ref90">20</reflink>]; Rowland, 2013). Although the coach asked the PSTs what they could do to advance student thinking on several occasions, the absence of explicit discussion about this fundamental rationale is another potential reason why the PSTs had difficulty refraining from asking leading questions.</p> <p>This study also adds to the literature by documenting the ways that the coach most often addressed the PSTs' tendency to funnel and the PSTs'responses. Like Bennett ([<reflink idref="bib3" id="ref91">3</reflink>]), the coach met with PSTs and presented data about their teaching back to them, in this case by presenting data and explicitly naming funneling and identifying when it occurred. Each of these moves seems to provide an important role in supporting the PST to learn to elicit student thinking. Reading back the PST/student exchanges presents PSTs the opportunity to reflect on their own teaching and helps the coach understand what the PST initially notices related to their questions. Next, by naming funneling, the coach provided labels that the PST could use to cue themselves in the moment and a shared language for the PST and coach to use during further coaching sessions. By providing (and labelling) a more productive alternative (probing/diagnosing), the coach avoided common pitfalls of educational reform by suggesting only "what not to do" and contributing "nothing toward examining what teachers should or could do" (Chazan &amp; Ball, [<reflink idref="bib9" id="ref92">9</reflink>], p. 2).</p> <hd id="AN0156414408-24">Implications</hd> <p>This study used observation and interview data at three different points of time to document the development of PSTs and identify the coaching moves that were used to support their development. An emergent developmental trajectory, representing an elaboration on the trajectory synthesized from the literature, provided a useful analytical lens. Future research is necessary to determine the extent to which this trajectory is generalizable to other PSTs, novice teachers, or more experienced teachers who are interested in transitioning to more ambitious instruction. Future studies should include a more diverse population of student teachers and multiple coach/PST pairs.</p> <p>The PSTs' starting points and trajectories in this study were influenced by their methods classes and the coach's support. Had they had more opportunities to rehearse and practice eliciting thinking in their methods classes and initial field experiences, it is likely that the PSTs would have entered their internships with a greater capacity to elicit students' thinking. Like others (e.g., Lampert et al., [<reflink idref="bib26" id="ref93">26</reflink>]), we recommend that these opportunities are presented in methods classes so that PSTs are more prepared for student teaching and can work on developing other practices during internships, such as orienting students' to each other's thinking. Rehearsals during student teaching seminars would also provide a fruitful way to review and continue to develop high-leverage teaching practices during that phase.</p> <p>In this study, our findings were limited by the type of data collected. Any evidence of PSTs' development was collected through the coach's observation notes and interactions presented to PSTs in the post-conference interviews; future studies should utilize video data to allow the researchers to objectively characterize PST progress without relying solely on the coach's note-taking. Future studies should also examine if PSTs actions differ based on the content being taught. In this present study, the PSTs and the coach scheduled coaching sessions based on specific times and dates rather than the mathematics concepts or tasks being taught.</p> <p>Finally, the finding that the PSTs demonstrated different trajectories in responding to students who initially provided correct or incorrect answers has implications for coaching. Although the coach noticed this phenomenon during the coaching, he did not adequately support the PSTs in successfully probing on incorrect answers. Modeling and explicitly discussing the rationale for probing on incorrect answers are recommended practices for coaching that require further study. It is crucially important that we support teachers and PSTs in probing student responses that do not initially make sense, so that the understandings that students possess can be identified and leveraged for the development of further knowledge.</p> <hd id="AN0156414408-25">Disclosure statement</hd> <p>No potential conflict of interest was reported by the authors.</p> <ref id="AN0156414408-26"> <title> References </title> <blist> <bibl id="bib1" idref="ref33" type="bt">1</bibl> <bibtext> Amador, J. M., Carter, I., &amp; Hudson, R. A. (2016). Analyzing preservice mathematics teachers' professional noticing. Action in Teacher Education, 38 (4), 371 – 383. https://doi.org/10.1080/01626620.2015.1119764</bibtext> </blist> <blist> <bibl id="bib2" idref="ref6" type="bt">2</bibl> <bibtext> Ball, D. L., Sleep, L., Boerst, T., &amp; Bass, H. (2009). Combining the development of practice and the practice of development in teacher education. The Elementary School Journal, 109 (5), 458 – 474. https://doi.org/10.1086/596996</bibtext> </blist> <blist> <bibl id="bib3" idref="ref56" type="bt">3</bibl> <bibtext> Bennett, C. A. (2010). It's hard getting kids to talk about math": Helping new teachers improve mathematical discourse. Action in Teacher Education, 32 (3), 79 – 89. https://doi.org/10.1080/01626620.2010.10463561</bibtext> </blist> <blist> <bibl id="bib4" idref="ref28" type="bt">4</bibl> <bibtext> Boaler, J., &amp; Brodie, K. (2004). The importance, nature and impact of teacher questions. In Proceedings of the 26th Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 773 – 781).</bibtext> </blist> <blist> <bibl id="bib5" idref="ref23" type="bt">5</bibl> <bibtext> Bobis, J., Clarke, B., Clarke, D., Thomas, G., Wright, B., Young-Loveridge, J., &amp; Gould, P. (2005). Supporting teachers in the development of young children's mathematical thinking: Three large scale cases. Mathematics Education Research Journal, 16 (3), 27 – 57. https://doi.org/10.1007/BF03217400</bibtext> </blist> <blist> <bibl id="bib6" idref="ref71" type="bt">6</bibl> <bibtext> Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., &amp; Empson, S. B. (2014). Children's mathematics: Cognitively guided instruction (2nd ed.). Heinemann.</bibtext> </blist> <blist> <bibl id="bib7" idref="ref24" type="bt">7</bibl> <bibtext> Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C. P., &amp; Loef, M. (1989). Using knowledge of children's mathematics thinking in classroom teaching: An experimental study. American Educational Research Journal, 26 (4), 499 – 531. https://doi.org/10.3102/00028312026004499</bibtext> </blist> <blist> <bibl id="bib8" idref="ref29" type="bt">8</bibl> <bibtext> Chapin, S., &amp; O'Connor, C. (2007). Academically productive talk: Supporting students' learning in mathematics. In W. G. Martin, M. E. Strutchens, &amp; P. Elliott (Eds.), The learning of mathematics: Sixty-ninth yearbook (pp. 113 – 128). National Council of Teachers of Mathematics.</bibtext> </blist> <blist> <bibl id="bib9" idref="ref92" type="bt">9</bibl> <bibtext> Chazan, D., &amp; Ball, D. L. (1999). Beyond being told not to tell. For the Learning of Mathematics, 19 (2), 2 – 10.</bibtext> </blist> <blist> <bibtext> Cochran-Smith, M., &amp; Donnell, K. (2006). Practitioner inquiry: Blurring the boundaries of research and practice. In J. Green, G. Camilli, &amp; P. B. Elmore (Eds.), Handbook of complementary methods in education research (pp. 503 – 518). Lawrence Erlbaum.</bibtext> </blist> <blist> <bibtext> Creswell, J., &amp; Poth, C. (2018). Qualitative inquiry &amp; research design: Choosing among five approaches. SAGE.</bibtext> </blist> <blist> <bibtext> Fuson, K. C., Kalchman, M., &amp; Bransford, J. D. (2005). Mathematical understanding: An introduction. In M. Donovan &amp; J. Bransford (Eds.), How students learn: Mathematics in the classroom (pp. 217 – 256). National Academies Press.</bibtext> </blist> <blist> <bibtext> Gibbons, L. K., &amp; Cobb, P. (2016). Content-focused coaching: Five key practices. The Elementary School Journal, 117 (2), 237 – 260. https://doi.org/10.1086/688906</bibtext> </blist> <blist> <bibtext> Gibbons, L. K., &amp; Cobb, P. (2017). Focusing on teacher learning opportunities to identify potentially productive coaching activities. Journal of Teacher Education, 68 (4), 411 – 425. https://doi.org/10.1177/0022487117702579</bibtext> </blist> <blist> <bibtext> Grossman, P., &amp; McDonald, M. (2008). Back to the future: Directions for research in teaching and teacher education. American Educational Research Journal, 45 (1), 184 – 205. https://doi.org/10.3102/0002831207312906</bibtext> </blist> <blist> <bibtext> Hallman-Thrasher, A. (2017). Prospective elementary teachers' responses to unanticipated incorrect solutions to problem-solving tasks. Journal of Mathematics Teacher Education, 20 (6), 519 – 555. https://doi.org/10.1007/s10857-015-9330-y</bibtext> </blist> <blist> <bibtext> Hattie, J. (2008). Visible learning. A synthesis of 800 meta analyses related to student achievement. Routledge.</bibtext> </blist> <blist> <bibtext> Higgins, J., &amp; Parsons, R. (2009). A successful professional development model in mathematics: A system-wide New Zealand case. Journal of Teacher Education, 60 (3), 231 – 242. https://doi.org/10.1177/0022487109336894</bibtext> </blist> <blist> <bibtext> Hufferd-Ackles, K., Fuson, K. C., &amp; Sherin, M. G. (2004). Describing levels and components of a math-talk learning community. Journal for Research in Mathematics Education, 35 (2), 81 – 116. https://doi.org/10.2307/30034933</bibtext> </blist> <blist> <bibtext> Jacobs, V. R., Lamb, L. L., &amp; Philipp, R. A. (2010). Professional noticing of children's mathematical thinking. Journal for Research in Mathematics Education, 41 (2), 169 – 202. https://doi.org/10.5951/jresematheduc.41.2.0169</bibtext> </blist> <blist> <bibtext> Jakopovic, P. M. (2017). Understanding what makes for productive coaching moves to help teachers attend to mathematical tasks of teaching [Ph.D thesis]. The University of Nebraska - Lincoln.</bibtext> </blist> <blist> <bibtext> Kazemi, E., Ghousseini, H., Cunard, A., &amp; Turrou, A. C. (2016). Getting inside rehearsals: Insights from teacher educators to support work on complex practice. Journal of Teacher Education, 67 (1), 18 – 31. https://doi.org/10.1177/0022487115615191</bibtext> </blist> <blist> <bibtext> Kazemi, E., &amp; Waege, K. (2015). Learning to teach within practice-based methods courses. Mathematics Teacher Education and Development, 17 (2), 125 – 145.</bibtext> </blist> <blist> <bibtext> Kilic, H. (2018). Pre-service mathematics teachers' noticing skills and scaffolding practices. International Journal of Science and Mathematics Education, 16 (2), 377 – 400. https://doi.org/10.1007/s10763-016-9784-0</bibtext> </blist> <blist> <bibtext> Lampert, M., Beasley, H., Ghousseini, H., Kazemi, E., &amp; Franke, M. (2010). Using designed instructional activities to enable novices to manage ambitious mathematics teaching. In M. K. Stein, &amp; L. Kucan (Eds.), Instructional explanations in the disciplines (pp. 129 – 141). Springer.</bibtext> </blist> <blist> <bibtext> Lampert, M., Franke, M. L., Kazemi, E., Ghousseini, H., Turrou, A. C., Beasley, H., Cunard, A., &amp; Crowe, K. (2013). Keeping it complex: Using rehearsals to support novice teacher learning of ambitious teaching. Journal of Teacher Education, 64 (3), 226 – 243. https://doi.org/10.1177/0022487112473837</bibtext> </blist> <blist> <bibtext> Lave, J., &amp; Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge University Press.</bibtext> </blist> <blist> <bibtext> Lincoln, Y., &amp; Guba, E. (1990). Naturalistic inquiry. Sage.</bibtext> </blist> <blist> <bibtext> McDonald, M., Kazemi, E., &amp; Kavanagh, S. S. (2013). Core practices and pedagogies of teacher education: A call for a common language and collective activity. Journal of Teacher Education, 64 (5), 378 – 386. https://doi.org/10.1177/0022487113493807</bibtext> </blist> <blist> <bibtext> Miles, M. B., Huberman, A. M., &amp; Saldana, J. (2014). Qualitative data analysis: A methods sourcebook (3rd ed.). SAGE Publications Ltd.</bibtext> </blist> <blist> <bibtext> Moyer, P. S., &amp; Milewicz, E. (2002). Learning to question: Categories of questioning used by preservice teachers during diagnostic mathematics interviews. Journal of Mathematics Teacher Education, 5 (4), 293 – 315. https://doi.org/10.1023/A:1021251912775</bibtext> </blist> <blist> <bibtext> National Research Council. (2001). Adding it up: Helping children learn mathematics. In J. Kilpatrick, J. Swafford, &amp; B. Findell (Eds.), Mathematics learning study committee, center for education, division of behavioral and social sciences and education. National Academies Press.</bibtext> </blist> <blist> <bibtext> NCTM. (2014). Principles to actions: Ensuring mathematical success for all students. NCTM.</bibtext> </blist> <blist> <bibtext> Polly, D. (2012). Supporting mathematics instruction with an expert coaching model. Mathematics Teacher Education and Development, 14 (1), 78 – 93.</bibtext> </blist> <blist> <bibtext> Rowland, T. (2013). The knowledge quartet: The genesis and application of a framework for analysing mathematics teaching and deepening teachers' mathematics knowledge. Sisyphus: Journal of Education, 1 (3), 15 – 43.</bibtext> </blist> <blist> <bibtext> Schack, E. O., Fisher, M. H., Thomas, J. N., Eisenhardt, S., Tassell, J., &amp; Yoder, M. (2013). Prospective elementary school teachers' professional noticing of children's early numeracy. Journal of Mathematics Teacher Education, 16 (5), 379 – 397. https://doi.org/10.1007/s10857-013-9240-9</bibtext> </blist> <blist> <bibtext> Shaughnessy, M., &amp; Boerst, T. A. (2018). Uncovering the skills that preservice teachers bring to teacher education: The practice of eliciting a student's thinking. Journal of Teacher Education, 69 (1), 40 – 55. https://doi.org/10.1177/0022487117702574</bibtext> </blist> <blist> <bibtext> Simpson, A., &amp; Haltiwanger, L. (2017). This is the first time I've done this": Exploring secondary prospective mathematics teachers' noticing of students' mathematical thinking. Journal of Mathematics Teacher Education, 20 (4), 335 – 355. https://doi.org/10.1007/s10857-016-9352-0</bibtext> </blist> <blist> <bibtext> Sleep, L., &amp; Boerst, T. A. (2012). Preparing beginning teachers to elicit and interpret students' mathematical thinking. Teaching and Teacher Education, 28 (7), 1038 – 1048. https://doi.org/10.1016/j.tate.2012.04.005</bibtext> </blist> <blist> <bibtext> Son, J.-W. (2013). How preservice teachers interpret and respond to student errors: Ratio and proportion in similar rectangles. Educational Studies in Mathematics, 84 (1), 49 – 70. https://doi.org/10.1007/s10649-013-9475-5</bibtext> </blist> <blist> <bibtext> Son, J.-W. (2016). Preservice teachers' response and feedback type to correct and incorrect student-invented strategies for subtracting whole numbers. The Journal of Mathematical Behavior, 42, 49 – 68. https://doi.org/10.1016/j.jmathb.2016.02.003</bibtext> </blist> <blist> <bibtext> Son, J.-W., &amp; Sinclair, N. (2010). How preservice teachers interpret and respond to student geometric errors. School Science and Mathematics, 110 (1), 31 – 46. https://doi.org/10.1111/j.1949-8594.2009.00005.x</bibtext> </blist> <blist> <bibtext> Stahl, G., Sharplin, E., &amp; Kehrwald, B. (2018). A review of the literature on coaching in pre-service teacher education. In Real-time coaching and pre-service teacher education (pp. 13 – 30). Springer Nature.</bibtext> </blist> <blist> <bibtext> Tarr, J. E., Reys, R., Reys, B., Chávez, Ó., Shih, J., &amp; Osterlind, S. (2008). The impact of middle school mathematics curricula and the classroom learning environment on student achievement. Journal for Research in Mathematics Education, 39 (3), 247 – 280.</bibtext> </blist> <blist> <bibtext> TeachingWorks. (2019). Eliciting and interpreting student thinking decomposition. https://library.teachingworks.org/wp-content/uploads/Eliciting_Decomposition.pdf</bibtext> </blist> <blist> <bibtext> U.S. Department of Education. (2008). Foundations for success: The final report of the national mathematics advisory panel. <ulink href="http://www.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf">www.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf</ulink></bibtext> </blist> <blist> <bibtext> Weiland, I., Hudson, R., &amp; Amador, J. (2014). Preservice formative assessment interviews: The development of competent questioning. International Journal of Science and Mathematics Education, 12 (2), 329 – 352. https://doi.org/10.1007/s10763-013-9402-3</bibtext> </blist> <blist> <bibtext> Weiss, I. R., Pasley, J. D., Smith, P. S., Banilower, E. R., &amp; Heck, D. J. (2003). Looking inside the classroom. Horizon Research Inc. <ulink href="http://secure.horizon-research.com/insidetheclassroom/reports/looking/complete.pdf">http://secure.horizon-research.com/insidetheclassroom/reports/looking/complete.pdf</ulink></bibtext> </blist> <blist> <bibtext> Wiliam, D. (2010). An integrative summary of the research literature and implications for a new theory of formative assessment. In H. Andrade &amp; G. Cizek (Eds.), Handbook of formative assessment (pp. 18 – 40). Routledge.</bibtext> </blist> <blist> <bibtext> Wood, T. (1998). Alternative patterns of communication in mathematics class: Funneling or focusing?. In H. Steinbring, M. G. B. Bussi, &amp; A. Sierpinska (Eds.), Language and communication in the mathematics classroom (pp. 167 – 178). National Council of Teachers of Mathematics.</bibtext> </blist> <blist> <bibtext> Yin, R. K. (2014). Case study research: Design and methods (5th ed.). SAGE.</bibtext> </blist> </ref> <ref id="AN0156414408-27"> <title> Footnotes </title> <blist> <bibtext> During recruitment by one of the authors, it was made clear that their participation in the study was voluntary. The coaching was non-evaluative and no grades were assigned, minimizing the chance that they felt coerced to participate.</bibtext> </blist> <blist> <bibtext> At a colleague's suggestion, the coach has begun to use the term "uncover" rather than "diagnose," to help PSTs see that these eliciting questions are meant to uncover partial understandings upon which a deeper understanding can be built, as opposed to diagnosing a deficiency</bibtext> </blist> </ref> <aug> <p>By Luke T. Reinke; Leslie W. Schmidt; Adam Myers and Drew Polly</p> <p>Reported by Author; Author; Author; Author</p> </aug> <nolink nlid="nl1" bibid="bib32" firstref="ref1"></nolink> <nolink nlid="nl2" bibid="bib44" firstref="ref2"></nolink> <nolink nlid="nl3" bibid="bib46" firstref="ref3"></nolink> <nolink nlid="nl4" bibid="bib25" firstref="ref4"></nolink> <nolink nlid="nl5" bibid="bib45" firstref="ref5"></nolink> <nolink nlid="nl6" bibid="bib29" firstref="ref7"></nolink> <nolink nlid="nl7" bibid="bib20" firstref="ref8"></nolink> <nolink nlid="nl8" bibid="bib31" firstref="ref9"></nolink> <nolink nlid="nl9" bibid="bib37" firstref="ref10"></nolink> <nolink nlid="nl10" bibid="bib39" firstref="ref11"></nolink> <nolink nlid="nl11" bibid="bib47" firstref="ref12"></nolink> <nolink nlid="nl12" bibid="bib16" firstref="ref13"></nolink> <nolink nlid="nl13" bibid="bib23" firstref="ref14"></nolink> <nolink nlid="nl14" bibid="bib17" firstref="ref18"></nolink> <nolink nlid="nl15" bibid="bib33" firstref="ref20"></nolink> <nolink nlid="nl16" bibid="bib12" firstref="ref21"></nolink> <nolink nlid="nl17" bibid="bib49" firstref="ref22"></nolink> <nolink nlid="nl18" bibid="bib18" firstref="ref25"></nolink> <nolink nlid="nl19" bibid="bib24" firstref="ref35"></nolink> <nolink nlid="nl20" bibid="bib38" firstref="ref36"></nolink> <nolink nlid="nl21" bibid="bib40" firstref="ref37"></nolink> <nolink nlid="nl22" bibid="bib50" firstref="ref40"></nolink> <nolink nlid="nl23" bibid="bib41" firstref="ref44"></nolink> <nolink nlid="nl24" bibid="bib42" firstref="ref45"></nolink> <nolink nlid="nl25" bibid="bib36" firstref="ref48"></nolink> <nolink nlid="nl26" bibid="bib27" firstref="ref52"></nolink> <nolink nlid="nl27" bibid="bib43" firstref="ref53"></nolink> <nolink nlid="nl28" bibid="bib21" firstref="ref54"></nolink> <nolink nlid="nl29" bibid="bib34" firstref="ref55"></nolink> <nolink nlid="nl30" bibid="bib14" firstref="ref57"></nolink> <nolink nlid="nl31" bibid="bib13" firstref="ref58"></nolink> <nolink nlid="nl32" bibid="bib19" firstref="ref59"></nolink> <nolink nlid="nl33" bibid="bib22" firstref="ref64"></nolink> <nolink nlid="nl34" bibid="bib51" firstref="ref65"></nolink> <nolink nlid="nl35" bibid="bib10" firstref="ref66"></nolink> <nolink nlid="nl36" bibid="bib28" firstref="ref70"></nolink> <nolink nlid="nl37" bibid="bib30" firstref="ref72"></nolink> <nolink nlid="nl38" bibid="bib11" firstref="ref78"></nolink> <nolink nlid="nl39" bibid="bib48" firstref="ref86"></nolink> <nolink nlid="nl40" bibid="bib15" firstref="ref89"></nolink> <nolink nlid="nl41" bibid="bib26" firstref="ref93"></nolink> |
|---|---|
| Header | DbId: eric DbLabel: ERIC An: EJ1345214 AccessLevel: 3 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
| IllustrationInfo | |
| Items | – Name: Title Label: Title Group: Ti Data: Developing Student Teachers' Skills at Eliciting Students' Mathematical Thinking Using the Coaching Cycle – Name: Language Label: Language Group: Lang Data: English – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Reinke%2C+Luke+T%2E%22">Reinke, Luke T.</searchLink> (ORCID <externalLink term="http://orcid.org/0000-0003-0747-0040">0000-0003-0747-0040</externalLink>)<br /><searchLink fieldCode="AR" term="%22Schmidt%2C+Leslie+W%2E%22">Schmidt, Leslie W.</searchLink> (ORCID <externalLink term="http://orcid.org/0000-0002-8040-9185">0000-0002-8040-9185</externalLink>)<br /><searchLink fieldCode="AR" term="%22Myers%2C+Adam%22">Myers, Adam</searchLink> (ORCID <externalLink term="http://orcid.org/0000-0002-3132-8506">0000-0002-3132-8506</externalLink>)<br /><searchLink fieldCode="AR" term="%22Polly%2C+Drew%22">Polly, Drew</searchLink> (ORCID <externalLink term="http://orcid.org/0000-0003-2370-4409">0000-0003-2370-4409</externalLink>) – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="SO" term="%22Teacher+Educator%22"><i>Teacher Educator</i></searchLink>. 2022 57(2):215-237. – Name: Avail Label: Availability Group: Avail Data: Routledge. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals – Name: PeerReviewed Label: Peer Reviewed Group: SrcInfo Data: Y – Name: Pages Label: Page Count Group: Src Data: 23 – Name: DatePubCY Label: Publication Date Group: Date Data: 2022 – Name: TypeDocument Label: Document Type Group: TypDoc Data: Journal Articles<br />Reports - Research – Name: Audience Label: Education Level Group: Audnce Data: <searchLink fieldCode="EL" term="%22Higher+Education%22">Higher Education</searchLink><br /><searchLink fieldCode="EL" term="%22Postsecondary+Education%22">Postsecondary Education</searchLink> – Name: Subject Label: Descriptors Group: Su Data: <searchLink fieldCode="DE" term="%22Student+Teachers%22">Student Teachers</searchLink><br /><searchLink fieldCode="DE" term="%22Teaching+Skills%22">Teaching Skills</searchLink><br /><searchLink fieldCode="DE" term="%22Skill+Development%22">Skill Development</searchLink><br /><searchLink fieldCode="DE" term="%22Thinking+Skills%22">Thinking Skills</searchLink><br /><searchLink fieldCode="DE" term="%22Coaching+%28Performance%29%22">Coaching (Performance)</searchLink><br /><searchLink fieldCode="DE" term="%22Prompting%22">Prompting</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Education%22">Mathematics Education</searchLink> – Name: DOI Label: DOI Group: ID Data: 10.1080/08878730.2021.1990454 – Name: ISSN Label: ISSN Group: ISSN Data: 0887-8730 – Name: Abstract Label: Abstract Group: Ab Data: Eliciting students' thinking is a key component of effective student-centered mathematics instruction, and research demonstrates promising approaches to developing this skill in pre-service teachers [PSTs] during methods courses. However, few studies have provided insight into how PSTs can be supported in continuing to develop the skill of eliciting thinking during student teaching internships. This case study describes the development of two PSTs as they worked to fully elicit their students' thinking during their internship, with the support of a coach. The PSTs' development generally aligned with an emergent trajectory synthesized from the literature; however, additional specificity was needed to fully interpret the data. Both PSTs learned to effectively probe the thinking of students who provided correct answers to initial prompts but made less progress in probing students' incorrect answers. The article concludes with an analysis of the learning opportunities presented by the coaching moves, as well as missed opportunities. – Name: AbstractInfo Label: Abstractor Group: Ab Data: As Provided – Name: DateEntry Label: Entry Date Group: Date Data: 2022 – Name: AN Label: Accession Number Group: ID Data: EJ1345214 |
| PLink | https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1345214 |
| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1080/08878730.2021.1990454 Languages: – Text: English PhysicalDescription: Pagination: PageCount: 23 StartPage: 215 Subjects: – SubjectFull: Student Teachers Type: general – SubjectFull: Teaching Skills Type: general – SubjectFull: Skill Development Type: general – SubjectFull: Thinking Skills Type: general – SubjectFull: Coaching (Performance) Type: general – SubjectFull: Prompting Type: general – SubjectFull: Mathematics Education Type: general Titles: – TitleFull: Developing Student Teachers' Skills at Eliciting Students' Mathematical Thinking Using the Coaching Cycle Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Reinke, Luke T. – PersonEntity: Name: NameFull: Schmidt, Leslie W. – PersonEntity: Name: NameFull: Myers, Adam – PersonEntity: Name: NameFull: Polly, Drew IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 01 Type: published Y: 2022 Identifiers: – Type: issn-print Value: 0887-8730 Numbering: – Type: volume Value: 57 – Type: issue Value: 2 Titles: – TitleFull: Teacher Educator Type: main |
| ResultId | 1 |
