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Learning to Elicit Student Thinking: The Role of Planning to Support Academically Rigorous Questioning Sequences during Instruction

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Title: Learning to Elicit Student Thinking: The Role of Planning to Support Academically Rigorous Questioning Sequences during Instruction
Language: English
Authors: Sheila Orr (ORCID 0000-0002-8818-2093), Kristen Bieda (ORCID 0000-0002-7407-5203)
Source: Journal of Mathematics Teacher Education. 2025 28(3):523-544.
Availability: Springer. Available from: Springer Nature. One New York Plaza, Suite 4600, New York, NY 10004. Tel: 800-777-4643; Tel: 212-460-1500; Fax: 212-460-1700; e-mail: customerservice@springernature.com; Web site: https://link.springer.com/
Peer Reviewed: Y
Page Count: 22
Publication Date: 2025
Sponsoring Agency: National Science Foundation (NSF), Division of Research on Learning in Formal and Informal Settings (DRL)
Contract Number: 1725910
1725920
Document Type: Journal Articles
Reports - Research
Tests/Questionnaires
Education Level: Higher Education
Postsecondary Education
Descriptors: Questioning Techniques, Teaching Methods, Thinking Skills, Preservice Teachers, Field Instruction, Field Experience Programs, Educational Planning, Lesson Plans
DOI: 10.1007/s10857-023-09603-5
ISSN: 1386-4416
1573-1820
Abstract: The use of questioning to elicit student thinking is one of the cornerstones of ambitious teaching; thus, supporting prospective teachers in this complex practice is a core objective of mathematics teacher education. In this paper, we examine the academic rigor of the sustained questioning sequences prospective teachers use during instruction to elicit student thinking and the possible factor lesson planning plays in enacted sequences involving academically rigorous questions. We analyzed the questions they asked during lessons in an early field experience and traced how prospective teachers prepared for those questions as represented in their lesson plans. From our analysis, we found prospective teachers used three types of sustained questioning sequences when eliciting student thinking in the enactment of the lesson. Further, we found evidence suggesting a connection between the questions asked during enactment of a lesson and the quality of questions prospective teachers incorporated into their lesson plan. These findings inform how mathematics teacher educators can support prospective teachers during lesson planning to prepare to elicit student thinking in productive ways.
Abstractor: As Provided
Entry Date: 2025
Accession Number: EJ1472072
Database: ERIC
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  Value: <anid>AN0185425510;oih01may.25;2025May28.06:55;v2.2.500</anid> <title id="AN0185425510-1">Learning to elicit student thinking: the role of planning to support academically rigorous questioning sequences during instruction </title> <p>The use of questioning to elicit student thinking is one of the cornerstones of ambitious teaching; thus, supporting prospective teachers in this complex practice is a core objective of mathematics teacher education. In this paper, we examine the academic rigor of the sustained questioning sequences prospective teachers use during instruction to elicit student thinking and the possible factor lesson planning plays in enacted sequences involving academically rigorous questions. We analyzed the questions they asked during lessons in an early field experience and traced how prospective teachers prepared for those questions as represented in their lesson plans. From our analysis, we found prospective teachers used three types of sustained questioning sequences when eliciting student thinking in the enactment of the lesson. Further, we found evidence suggesting a connection between the questions asked during enactment of a lesson and the quality of questions prospective teachers incorporated into their lesson plan. These findings inform how mathematics teacher educators can support prospective teachers during lesson planning to prepare to elicit student thinking in productive ways.</p> <p>Keywords: Secondary mathematics; Teacher preparation; Eliciting student thinking; High leverage Practices; Lesson planning; Teacher questions; Education Curriculum and Pedagogy Specialist Studies In Education</p> <p>Copyright comment Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</p> <hd id="AN0185425510-2">Introduction</hd> <p>Eliciting student thinking is an essential, necessary practice of teaching that promotes deep mathematics understanding for all students (AMTE, [<reflink idref="bib2" id="ref1">2</reflink>]; NCTM, [<reflink idref="bib32" id="ref2">32</reflink>]; TeachingWorks, [<reflink idref="bib47" id="ref3">47</reflink>]). The power of this practice is that it is elemental to student-centered instruction; eliciting student thinking is essential for knowing what a learner does or does not yet understand and enables effective decision making as to what instructional move to make next. As elaborated in the AMTE Standards for the Preparation of Teachers ([<reflink idref="bib2" id="ref4">2</reflink>]): "...well-prepared beginners endeavor to position students as authors of ideas—students who discuss, explain, and justify their reasoning using varied representations and tools" (p. 16). Yet, as further discussed in the AMTE Standards, the work of eliciting student thinking, especially in the context of leading whole-class discussions, is highly complex work that takes years to enact effectively and consistently.</p> <p>During whole-class discussions, one way teachers elicit student thinking is through the questions they pose. By "using questions to assess and advance students' reasoning and sense making about important mathematical ideas and relationships" (NCTM, [<reflink idref="bib32" id="ref5">32</reflink>], p. 10), teachers are engaged in an academically rigorous questioning practice (Boston, [<reflink idref="bib9" id="ref6">9</reflink>]). This practice is challenging because it requires teachers to not only know "what questions to ask the student in the moment" (Grossman et al., [<reflink idref="bib22" id="ref7">22</reflink>], p. 280), but also how to interpret and respond to students' answers (Shaughnessy & Boerst, [<reflink idref="bib40" id="ref8">40</reflink>]). To prepare prospective secondary teachers (PSTs) for the challenges of asking questions to elicit student thinking, including the potential of assessing and advancing students' thinking by following up with more questions, preparation programs should provide opportunities where PSTs not only have space to practice the kinds of questions they can ask to learn more about students' understanding (Freeburn & Arbaugh, [<reflink idref="bib17" id="ref9">17</reflink>]), but also grapple with the inevitable demands of deviating from the planned questions when students steer the conversation in a different mathematical direction (Peterson et al., [<reflink idref="bib36" id="ref10">36</reflink>]). In teacher preparation programs, the opportunities that PSTs have to manage this kind of work typically emerge in capstone field experiences, such as student teaching.</p> <p>This study had the opportunity to investigate how PSTs, who participated in a highly authentic, yet mediated, early field experience, faced the challenges of using questions to elicit student thinking. Guided by the curriculum enactment process (Remillard & Heck, [<reflink idref="bib38" id="ref11">38</reflink>]), our study explored the possible relationship between the ways PSTs elicited student thinking with questions during lesson enactment and the preparation as evidenced by anticipated teacher questions represented in the written lesson plans. The findings from this work can help us understand more about how PSTs' lesson planning supports them to manage using questions to elicit student thinking in actual classroom settings. As such, these findings have implications for how mathematics teacher educators (MTEs) can support PSTs preparation to engage in eliciting student thinking as "well-prepared beginners" (AMTE, [<reflink idref="bib2" id="ref12">2</reflink>]).</p> <hd id="AN0185425510-3">Literature review</hd> <p>To understand the ways that lesson planning functions as a site to support PSTs in eliciting student thinking with questions, we ground this study in the construct of teacher questions. In this section, we explore literature related to (<reflink idref="bib1" id="ref13">1</reflink>) the work of eliciting student thinking with questions, (<reflink idref="bib2" id="ref14">2</reflink>) supports to prepare PSTs to elicit student thinking with questions, (<reflink idref="bib3" id="ref15">3</reflink>) challenges faced by PSTs in eliciting student thinking with questions, and (<reflink idref="bib4" id="ref16">4</reflink>) ways lesson planning can support addressing challenges.</p> <hd id="AN0185425510-4">The work of eliciting student thinking with questions</hd> <p>The process of eliciting student thinking with questions require teachers to do the following: "(<reflink idref="bib1" id="ref17">1</reflink>) select a task that affords opportunities for eliciting, (<reflink idref="bib2" id="ref18">2</reflink>) anticipate student thinking, (<reflink idref="bib3" id="ref19">3</reflink>) know the learning goal and assess students' proximity to the goal, and (<reflink idref="bib4" id="ref20">4</reflink>) plan questions to deepen student understanding" (Orr & Bieda, [<reflink idref="bib34" id="ref21">34</reflink>], p. 1672). All of these actions are initiated during the planning process. During enactment, a teacher's ability to understand student thinking requires not only that they know how to phrase a question to get at students' ideas but also know which ideas are worth exploring (Grossman et al., [<reflink idref="bib22" id="ref22">22</reflink>]). A teacher's understanding of their lesson goal is necessary for making in-the-moment decisions as to which ideas should be pursued with further questioning and which ideas can be tabled. Once the teacher has taken action to elicit student thinking, they then have to respond, in-the-moment, to students' thinking. How they respond depends upon a number of factors, including their understanding that students do not always express their thinking clearly or completely (Shaughnessy & Boerst, [<reflink idref="bib40" id="ref23">40</reflink>]) and how the different solution methods connect to one another (McDonald et al., [<reflink idref="bib29" id="ref24">29</reflink>]). Sometimes responding to student thinking is asking follow-up questions, but it can also involve asking students to rephrase their thinking, opening the solution method up for class discussion, or allowing another student to suggest a different solution path (Grossman et al., [<reflink idref="bib22" id="ref25">22</reflink>]).</p> <p>Teachers can use a variety of different questions to elicit student thinking. Boaler and Brodie ([<reflink idref="bib7" id="ref26">7</reflink>]) discuss three types of questions teachers use that do promote rich discourse: exploring mathematical meanings or relationships, probing students to explain their thinking, and soliciting contributions from members of the class. However, teachers also often ask questions that require students to recall facts, procedures, or calculations (Bennett, [<reflink idref="bib4" id="ref27">4</reflink>]; Franke et al., [<reflink idref="bib16" id="ref28">16</reflink>]). These questions are often closed questions, which do not promote the rich classroom discourse. When teachers promote rich discussions through eliciting student thinking with questions, they (<reflink idref="bib1" id="ref29">1</reflink>) advance student understanding by asking questions that build on student thinking; (<reflink idref="bib2" id="ref30">2</reflink>) ask questions that require explanation and justification; (<reflink idref="bib3" id="ref31">3</reflink>) ask questions that surface mathematical ideas in an accessible way for students; and (<reflink idref="bib4" id="ref32">4</reflink>) allow for ample wait time (NCTM, [<reflink idref="bib32" id="ref33">32</reflink>]). These strategies are reflected in the criteria for high-level, academically rigorous teachers' questions on observation protocols of instructional quality such as the Instructional Quality Assessment (IQA), which states: "The teacher consistently asks academically relevant questions that provide opportunities for students to elaborate and explain their mathematical work and thinking...identify and describe the important mathematical ideas in the lesson..." (Boston & Candela, [<reflink idref="bib10" id="ref34">10</reflink>], p. 442).</p> <hd id="AN0185425510-5">Supports for PSTs learning to elicit student thinking with questions</hd> <p>In order to support PSTs in learning to elicit student thinking with questions, MTEs have developed and researched numerous ways to support PSTs in developing their practice. Some of these supports include experiences such as: (<reflink idref="bib1" id="ref35">1</reflink>) interviewing students, (<reflink idref="bib2" id="ref36">2</reflink>) simulations (both in-person and through virtual programs), (<reflink idref="bib3" id="ref37">3</reflink>) field placements, and (<reflink idref="bib4" id="ref38">4</reflink>) analyzing experiences after engaging as a student. In this section, we explore the different supports each of these activities offer to developing PSTs competencies with eliciting student thinking through asking questions.</p> <p>A widely used activity in supporting PSTs to elicit student thinking with questions is interviewing students. Typically, the PSTs conducted one-on-one interviews with students to learn more about the student's mathematical thinking (Kabar & Taşdan, [<reflink idref="bib25" id="ref39">25</reflink>]; Moyer & Milewicz, [<reflink idref="bib31" id="ref40">31</reflink>]; Schwartz, [<reflink idref="bib39" id="ref41">39</reflink>]; Sleep & Boerst, [<reflink idref="bib42" id="ref42">42</reflink>]; Weiland et al., [<reflink idref="bib49" id="ref43">49</reflink>]). Commonly, these interviews involve a protocol consisting of only a list of questions to ask (Sleep & Boerst, [<reflink idref="bib42" id="ref44">42</reflink>]). However, one study did include possible follow-up questions (Moyer & Milewicz, [<reflink idref="bib31" id="ref45">31</reflink>]). Often PSTs only conduct these interviews once, and typically ask surface-level questions (Moyer & Milewicz, [<reflink idref="bib31" id="ref46">31</reflink>]; Schwartz, [<reflink idref="bib39" id="ref47">39</reflink>]; Sleep & Boerst, [<reflink idref="bib42" id="ref48">42</reflink>]). However, in both Weiland et al. ([<reflink idref="bib49" id="ref49">49</reflink>]) and Kabar and Taşdan ([<reflink idref="bib25" id="ref50">25</reflink>]), the PSTs engaged in numerous interviews. Both of these studies found that the PSTs' questioning practices changed over time to include more follow-up, probing questions.</p> <p>Another activity, simulations, have been used to prepare PSTs to elicit student thinking with questions. Researchers have studied simulations as both in-person (Shaughnessy & Boerst, [<reflink idref="bib40" id="ref51">40</reflink>]; Shaughnessy et al., [<reflink idref="bib41" id="ref52">41</reflink>]) and virtual events (Conner et al., [<reflink idref="bib13" id="ref53">13</reflink>]; Walkoe & Levin, [<reflink idref="bib48" id="ref54">48</reflink>]). Conner et al. ([<reflink idref="bib13" id="ref55">13</reflink>]) found that PSTs who engaged in online simulations used questioning in three different ways: (<reflink idref="bib1" id="ref56">1</reflink>) questions which direct students to the correct answer, (<reflink idref="bib2" id="ref57">2</reflink>) questions built on or helped surface student thinking, and (<reflink idref="bib3" id="ref58">3</reflink>) a combination of the other two questioning strategies. Walkoe and Levin ([<reflink idref="bib48" id="ref59">48</reflink>]) found that when PSTs were prompted to analyze the questions they asked, they put more focus on the probing questions and questions that build on students' thinking compared to questions with only one correct answer. Similarly, research completed on PSTs eliciting student thinking with questions during in-person simulations shows that PSTs tend to start with asking a question that focuses on student thinking or the mathematical idea (Shaughnessy et al., [<reflink idref="bib41" id="ref60">41</reflink>]), and follow-up with more specific questions about the process the students used to solve the problem (Shaughnessy & Boerst, [<reflink idref="bib40" id="ref61">40</reflink>]).</p> <p>Field placements are another widely used activity of teacher preparation that can provide PSTs with opportunities to develop their practice of eliciting student thinking with questions (Brendefur & Frykholm, [<reflink idref="bib12" id="ref62">12</reflink>]; Hähkiöniemi, [<reflink idref="bib23" id="ref63">23</reflink>]; Lampert et al., [<reflink idref="bib28" id="ref64">28</reflink>]). Numerous researchers have found that field placements provide PSTs with opportunities to practice developing a lesson and asking questions of students. Yet, when PSTs develop their questioning skills through a field placement, the results may vary depending on the structure of the experience (Lampert & Ball, [<reflink idref="bib27" id="ref65">27</reflink>]). Brendefur and Frykholm ([<reflink idref="bib12" id="ref66">12</reflink>]) found that PSTs' beliefs around the structure and nature of questions can be highly impacted by the beliefs of their mentor teacher, illustrating how a PST "mirrored the practices of his cooperating teacher" and never attempted to implement the reform practices (p. 147).</p> <p>One innovation to help support PSTs to adopt ambitious teaching practices is the introduction of rehearsals into teacher preparation programs. Lampert et al. ([<reflink idref="bib28" id="ref67">28</reflink>]) discuss rehearsals as a space for the PSTs to practice a lesson and receive feedback from peers and MTEs. In almost all of the rehearsals conducted as part of the Lampert et al.'s study, there were interactions surrounding PSTs' use of eliciting student thinking with questions. These rehearsals provided PSTs with a chance to get support by the MTE providing a sample student response, suggesting a question to ask, or affirming the PSTs' questions (Lampert et al., [<reflink idref="bib28" id="ref68">28</reflink>], p. 235). This practice allowed for PSTs to try out eliciting student thinking with questions in a space supporting ambitious teaching practices.</p> <p>Finally, opportunities where MTEs engage PSTs in solving a mathematical task and then unpack how the discussion was facilitated through questioning is another activity that provides PSTs with opportunities to learn the practice of eliciting student thinking with questions (Blanton, [<reflink idref="bib6" id="ref69">6</reflink>]; Ghousseini & Herbst, [<reflink idref="bib19" id="ref70">19</reflink>]). This opportunity builds on the PSTs' experiences as students to develop their skills as teachers. Although the activity involves very little practice with asking questions, studies about this activity show that it can help PSTs develop their understanding of eliciting student thinking with questions. Blanton ([<reflink idref="bib6" id="ref71">6</reflink>]) showed that reflecting on a MTE's modeling of eliciting student thinking with questions as PSTs engage as students can shape their beliefs about collective knowledge, the importance of participating in discourse, and habits of discourse moves (Blanton, [<reflink idref="bib6" id="ref72">6</reflink>], p. 148). This shift in beliefs is important because it represents a change in how PSTs view mathematics classrooms and the practices involved in teaching mathematics. It may be that it is critical for PSTs to recognize the importance of discourse in a mathematics classroom to prioritize using discourse moves in their teaching.</p> <hd id="AN0185425510-6">Challenges of eliciting student thinking with questions</hd> <p>Despite these supports provided to PSTs in developing their practice to elicit student thinking with questions, this practice can still be very challenging for PSTs (Bennett, [<reflink idref="bib4" id="ref73">4</reflink>]; Inoue & Buczynski, [<reflink idref="bib24" id="ref74">24</reflink>]). Shaughnessy and Boerst ([<reflink idref="bib40" id="ref75">40</reflink>]) discuss one of the reasons eliciting student thinking in particular can be challenging for PSTs is due to the nature of the way people ask questions in everyday interactions. Unlike instructional situations, it is rare to ask questions where you know the answer in everyday interactions. This can result in the phenomena documented in literature of the teacher taking on too much of the mathematical thinking (Franke et al., [<reflink idref="bib16" id="ref76">16</reflink>]).</p> <p>A challenge PSTs face when eliciting student thinking with questions is how to connect the mathematical thinking of multiple students' responses. Franke et al. ([<reflink idref="bib15" id="ref77">15</reflink>]) argued that teachers find it difficult to follow-up on and pursue student thinking in a manner that supports students in explaining their own strategy or connecting it to another students' strategy. Research has shown PSTs also experience this challenge, particularly as they are learning how to effectively explain concepts (Inoue & Buczynski, [<reflink idref="bib24" id="ref78">24</reflink>]). Additionally, PSTs attempt to fill in student thinking or direct them to use a different method to approach the problem (Shaughnessy & Boerst, [<reflink idref="bib40" id="ref79">40</reflink>]). Sometimes PSTs are not even aware they are making these moves. Bennett ([<reflink idref="bib4" id="ref80">4</reflink>]) describes how two PSTs believed they were involving students and asking questions but were surprised to find out their observational data did not align with these perceptions. This lack of awareness is another challenge faced by PSTs in learning to elicit student thinking with questions. Sometimes this lack of awareness comes from PSTs having their perceptions obscured by the more immediate demands of the classroom (Bennett, [<reflink idref="bib4" id="ref81">4</reflink>]). Other times, this comes from a lack of awareness of the importance of different questioning patterns. Franke et al. ([<reflink idref="bib16" id="ref82">16</reflink>]) discuss how the use of single questions, no matter how specific, do not always uncover enough information and require teachers to make assumptions about the students' mathematical thinking. This can lead to teachers making "unhelpful subsequent instructional decisions such as deciding that posing further problems or questioning other students was unnecessary" (Franke et al., [<reflink idref="bib16" id="ref83">16</reflink>]).</p> <p>Inoue and Buczynski ([<reflink idref="bib24" id="ref84">24</reflink>]) discuss another of the challenges facing PSTs related to their ability to anticipate how students will respond to questions includes not only how to respond to a variety of responses but also how to respond when students do not respond to questions posed. In their study of elementary education PSTs at various points in a teacher preparation program, Inoue and Buczynski found when the student's response was outside an anticipated response, the PSTs are caught off guard and do not know how to respond. Highlighting how, PSTs often are unable to improvise and respond to the methods shared by students when the responses, or lack thereof, are not anticipated. Inoue and Buczynski discuss how one of the critical aspects of eliciting student thinking with questions is to have the confidence and flexibility to adjust to student thinking about work from there to achieve the instructional goal.</p> <p>The work of Gaspard and Gainsburg ([<reflink idref="bib18" id="ref85">18</reflink>]) lends support to the importance of PSTs' in-the-moment confidence to respond to unanticipated student thinking. They found, in a study of secondary education PSTs during their student teaching experience, that PSTs were avoiding questions which challenged students because the PSTs worried about students being confused or feeling discomfort at not being sure of an answer to the PSTs question. In their study, all the PSTs identified that students appeared frustrated with open-ended questions and needed more scaffolding to answer the questions. This resulted in the PSTs avoiding opening ended questions and "affording relief from the discomfort of what the prospective teachers perceived as occasions of strong student resistance and negativity." (Gaspard & Gainsburg, [<reflink idref="bib18" id="ref86">18</reflink>], p. 573).</p> <p>These previous analyses of PSTs' questioning practices tend to focus on individual questions (Moyer & Milewicz, [<reflink idref="bib31" id="ref87">31</reflink>]; Shaughnessy & Boerst, [<reflink idref="bib40" id="ref88">40</reflink>]) or the end goal of questioning (Nicol, [<reflink idref="bib33" id="ref89">33</reflink>]). For instance, Gaspard and Gainsburg ([<reflink idref="bib18" id="ref90">18</reflink>]) found the PSTs in their study changed the number of unpredictable questions asked over time. However, one unknown in their analysis is the types of questions that surrounded those with unpredictable answers. Additionally, questions with unpredictable answers open space for PSTs to face challenges when things do not go as planned, such as when none of the anticipated responses surface, and when students exhibit discomfort or choose not to respond to questions. Kennedy ([<reflink idref="bib26" id="ref91">26</reflink>]) refers to the kind of knowledge required to effectively respond in these moments as "craft knowledge" and found that teachers believe that craft knowledge emerges primarily from experience. If this is true, it presents a paradox for teacher education—if the challenges that PSTs' face largely involves responding to unexpected moments, does the improvement in PSTs' capacity to elicit student thinking with questions simply require more time facing unexpected moments again and again? In other words, is it futile for MTEs to attempt to support PSTs in developing more expertise in addressing uncertainties that emerge when eliciting student thinking with questions.</p> <hd id="AN0185425510-7">The role of lesson planning in eliciting student thinking with questions</hd> <p>The literature reviewed prior suggests that challenges arise when PSTs face unanticipated moments when eliciting student thinking with questions. One place of focus for supporting PSTs in these moments is lesson planning (Parrish et al., [<reflink idref="bib35" id="ref92">35</reflink>]; Purdum-Cassidy et al., [<reflink idref="bib37" id="ref93">37</reflink>]). Purdum-Cassidy et al. ([<reflink idref="bib37" id="ref94">37</reflink>]) explored how PSTs planned to elicit student thinking with questions in their field placements. They found a wide variety in the amount and types of questions planned. Overall, they found PSTs were likely to plan closed, procedural questions. However, when PSTs planned an open-ended question, all but one of the questions planned was conceptual. Similarly, through the use of curricular interpreting and responding (Dietiker et al., [<reflink idref="bib14" id="ref95">14</reflink>]), Parrish et al. ([<reflink idref="bib35" id="ref96">35</reflink>]) looked at the role of lesson planning in how PSTs planned to launch cognitively demanding tasks. This study found that PSTs planned to use questioning as a way to elicit students' understanding of contextual features, mathematical relationships, and/or common language. When examining the questions at a deeper level, Parrish et al. found that PSTs would plan both explicit questions to ask students as well as vague statements (e.g., "I will ask them questions to lead them to explain"). As neither study investigated PSTs' enactments of their lesson plans, questions remain about: (<reflink idref="bib1" id="ref97">1</reflink>) the relationship between the nature of the questions that were enacted during instruction and those that were planned and (<reflink idref="bib2" id="ref98">2</reflink>) how PSTs' preparation for eliciting student thinking using questions prepared them for when students responded in unanticipated ways.</p> <hd id="AN0185425510-8">Theoretical framework</hd> <p>As the existing studies on PSTs eliciting student thinking with questions highlight, this practice is extremely complex for novice teachers, whether being utilized through simulations or in face to face interactions with students (Shaughnessy et al., [<reflink idref="bib41" id="ref99">41</reflink>]). Despite the variety of different supports to help PSTs develop this practice, little work has examined whether there might be other activities or spaces, beyond just acquiring more experience or earlier experience in eliciting student thinking with questions, that can not only develop PSTs' understanding of the practice but also enhance PSTs' capacities to address the complexities and inherent uncertainties they encounter when implementing this practice in classrooms. Given that planning and enactment are both aspects of activities such as rehearsals and teaching in field placement experiences, not to mention their ubiquity as core activities of teaching that PSTs will do once they are practicing teachers, this study is informed by the <emph>curriculum enactment process</emph> (Remillard & Heck, [<reflink idref="bib38" id="ref100">38</reflink>]).</p> <hd id="AN0185425510-9">The curriculum enactment process</hd> <p>We draw upon the theory of the <emph>curriculum enactment process</emph> (Remillard & Heck, [<reflink idref="bib38" id="ref101">38</reflink>]) to inform the approach we took to investigate the role of PSTs' planning for questioning as an influence on the academic rigor of PSTs' enacted questioning. As shown in Fig. 1, the curriculum enactment process is a robust and complex process informed by inputs that are within, and outside of, a teacher's control. Within the <emph>operational curriculum</emph>, which Remillard and Heck ([<reflink idref="bib38" id="ref102">38</reflink>]) define as the teacher's transformation of elements of the official curriculum and guidelines from instructional materials into the curriculum that is experienced by students, lesson planning (the process resulting in the <emph>teacher-intended curriculum</emph>) plays a pivotal role in influencing, and being influenced by, the enacted curriculum and, ultimately, students' learning.</p> <p>Graph: Fig. 1 The curriculum enactment process. Note Reprinted from Remillard, J. T., & Heck, D. J. (2014). Conceptualizing the curriculum enactment process in mathematics education. ZDM, 46(<reflink idref="bib5" id="ref103">5</reflink>), p. 714</p> <p>In particular, the bidirectional arrow between student outcomes, which include students' abilities to contribute to classroom discourse (Remillard & Heck, [<reflink idref="bib38" id="ref104">38</reflink>], citing Herbel-Eisenmann & Otten, 2011) and the enacted curriculum intentionally reflects how students' engagement and interaction with tasks, engagement with other students, and engagement with the teacher during instruction contributes to the "ongoing construction of the enacted curriculum" (p. 713). Within the context of our study, we apply this conceptualization of the curriculum enactment process to posit that while the teacher-intended curriculum (e.g., lessons as planned by the PSTs) evolves during the curriculum enactment process, the various student outcomes that PSTs process while teaching inform the in-the-moment decisions they make that shape the enacted curriculum. Thus, understanding what influences the academic rigor of PSTs' enacted questions requires knowing not only about how the academic rigor of the questions PSTs planned to ask, but also the pedagogical choices they make as they attempt to elicit student thinking through questioning during a lesson.</p> <p>Remillard and Heck ([<reflink idref="bib38" id="ref105">38</reflink>]) remind the reader that the study of the teacher-intended curriculum is often difficult to access, as "it exists in its most detailed state in the teacher's mind" (p. 711). In this study, we rely on the physical artifacts of PSTs' lesson plans as representations with which to analyze the teacher-intended curriculum, particularly planned prompts to elicit student thinking with questions. Yet, we recognize that these written plans are still limited in providing us with information about PSTs' intentions for eliciting student thinking with questions even though PSTs' written lesson plans are typically far more extensive than those prepared by in-service teachers.</p> <hd id="AN0185425510-10">Academically rigorous eliciting student thinking with questions</hd> <p>The methods of our study is grounded in the conceptualization of academically rigorous questioning elaborated in the teacher questions rubric from the academic rigor portion of the IQA toolkit (Boston & Wolf, [<reflink idref="bib11" id="ref106">11</reflink>]). Based on the work of Stein et al. ([<reflink idref="bib45" id="ref107">45</reflink>]), the academic rigor portion of the IQA framework attends to how teachers maintain cognitive demand based on the mathematical task framework. The teacher question portion of the framework examines both the types of questions teachers use to elicit student thinking in both whole group and small group questions (Boston & Candela, [<reflink idref="bib10" id="ref108">10</reflink>]).</p> <p>Boston ([<reflink idref="bib9" id="ref109">9</reflink>], drawing on Boaler and Humphreys [<reflink idref="bib8" id="ref110">8</reflink>]), proposes six types of questions teachers engage with during whole-class discussion: (a) probing questions, (b) exploring mathematical meaning and relationships, (c) generating discussion, (d) procedural or factual, (e) other mathematical questions, and (f) non-mathematical questions. Probing questions are used to clarify students' mathematical thinking either for their benefit or the benefit of the class. Exploring mathematical meaning and relationships questions link ideas and point to the underlying mathematics. Generating discussion questions invite other members of the class to comment on the mathematics being discussed. Procedural or factual questions often are yes/no or require the recall of a memorized fact or procedure. Other mathematical questions are questions related to learning mathematics, but not the specific mathematical topic under discussion (e.g., which problem was the most challenging?). Finally, non-mathematical questions are all the other questions teachers may ask across a lesson that are not connected to mathematical content (e.g., is everyone in your group here today?). Although all questions have a purpose in the class, Boston describes high-quality eliciting student thinking with questions as the teacher asking "questions that provide opportunities for students to elaborate and explain their mathematical work and thinking (probing, generating discussion), identify and describe the important mathematical ideas in the lesson, or make connections between ideas, representations, or strategies (exploring mathematical meanings and relationships" (p. 11).</p> <hd id="AN0185425510-11">Summary</hd> <p>As described above, high-quality eliciting of student thinking with questions happens when teachers probe, generate discussion, and ask students to elaborate or explain mathematical ideas (Boston, [<reflink idref="bib9" id="ref111">9</reflink>]). We describe this practice as teachers engaging in <emph>academically rigorous (AR) questioning.</emph> We use the teacher question rubric from Boston and Candela ([<reflink idref="bib10" id="ref112">10</reflink>]) as a tool to examine teachers' high-quality eliciting student thinking with questions practice. Additionally, drawing on the curriculum enactment framework (Remillard & Heck, [<reflink idref="bib38" id="ref113">38</reflink>]), we posit that the choices teachers make as they engage in AR questioning connect to their understanding, formed during the planning process, of the importance of the following aspects of: (<reflink idref="bib1" id="ref114">1</reflink>) particular ideas and strategies related to the learning goal; (<reflink idref="bib2" id="ref115">2</reflink>) their awareness of whether particular ideas are commonly held or reflect naive conceptions; and (<reflink idref="bib3" id="ref116">3</reflink>) the breadth and robustness of their understanding of anticipated strategies students may use. Thus, our research is guided by the supposition that planning plays an important role in supporting PSTs in eliciting student thinking with questions. We seek to answer the following questions: (<reflink idref="bib1" id="ref117">1</reflink>) <emph>What approaches do PSTs use when attempting to enact academically rigorous moments of eliciting student thinking with questions in whole-class discussions? (<reflink idref="bib2" id="ref118">2</reflink>) How does their preparation as reflected in written lesson plans support moments of academically rigorous eliciting of student thinking with questions in classroom episodes?</emph></p> <hd id="AN0185425510-12">Methods</hd> <p>In this study, we used qualitative methods to gather and analyze data to understand the relationship between how PSTs elicited student thinking with questions during whole-class discussions and their preparation for those discussions. Since the difficulties PSTs have in developing a questioning practice has been well established in literature, we focused our analysis to closely examine episodes when PSTs attempted AR questions.</p> <hd id="AN0185425510-13">Research context and participants</hd> <p>The data were collected as part of a larger project studying the effects of the <emph>University Teaching Experience</emph> (UTE) model for secondary mathematics PSTs learning across three different teacher preparation programs. Grounded in Mezirow's ([<reflink idref="bib30" id="ref119">30</reflink>]) <emph>transformative learning theory</emph>, the UTE uses a practice-embedded approach (Gibbons et al., [<reflink idref="bib20" id="ref120">20</reflink>]) in which secondary mathematics PSTs engage in a mediated early field placement in a university undergraduate mathematics course (Bieda et al., [<reflink idref="bib5" id="ref121">5</reflink>]).</p> <p>This study comes from data collected during Fall 2018, involving seventeen PSTs at one site enrolled in an initial mathematics teaching methods course. All 17 PSTs were mathematics majors seeking a certification in teaching secondary mathematics. Concurrently with their methods course that semester, PSTs participated in the UTE which involved instructional responsibility for one section of a College Algebra course. With a curricular focus on linear, quadratic, logarithmic, logistic and polynomial functions, the content of the College Algebra section was similar to what PSTs will experience in a U.S. high school teaching setting. The mathematics teaching methods course is designed to complement the UTE to support PSTs in understanding the nature of teaching mathematics and enact teaching practices discussed in the methods course. Teaching practices discussed in the methods course include: developing lesson plans focused on learning goals and mathematical storylines, assessing the cognitive demand of tasks, understanding questioning practices (e.g., assessing and advancing question), and orchestrating whole-class discussions. Prior to this methods course and the UTE experience, PSTs had completed introductory education coursework on students as learners, multiple literacies, and the sociopolitical aspects of schooling.</p> <p>Over the course of the semester, the PSTs worked in small teams (7 pairs, and one group of three) to plan and teach lessons in the UTE. The PSTs taught two lessons; the first lesson was half a class period (~ 40 min) and the second was the entire class period (~ 80 min). The second lesson took place about a month after the first lesson. Prior to each lesson, the PSTs developed and received feedback by the MTE on their lesson plan. The lesson plans were based on a modified version of the Thinking Through a Lesson Plan Protocol (TTLP) (Smith et al., [<reflink idref="bib43" id="ref122">43</reflink>]). In our modified TTLP (included in Appendix A), the PSTs were provided the mathematical task and were asked to create two possible correct and two possible incorrect solution paths. Additionally, the PSTs generated learning goals and planned questions to elicit student thinking to: (<reflink idref="bib1" id="ref123">1</reflink>) assess student understanding, (<reflink idref="bib2" id="ref124">2</reflink>) extend students' understanding, and (<reflink idref="bib3" id="ref125">3</reflink>) support students in drawing connections across solutions. The feedback received on the lesson plan focused on both the mathematical and pedagogical aspects of the plan.</p> <p>In addition to support for planning, MTE and instructor of record for the College Algebra class, whose role was similar to a mentor teacher, provided pedagogical and technological support to the PST teaching pair. During lessons when they were not responsible for planning and enacting instruction, PSTs distributed themselves among tables to provide additional instructional support to College Algebra students.</p> <hd id="AN0185425510-14">Data sources</hd> <p>Data gathered included the transcripts of the enactment of the lessons taught by each PST teaching team (both the 40 min and 80 min lessons) as well as the final version of corresponding lesson plans. Each lesson enactment was video recorded using a Swivl robot (<ulink href="http://www.swivl.com">www.swivl.com</ulink>) with the audio later transcribed.</p> <hd id="AN0185425510-15">Data analysis</hd> <p>Since our goal was to understand the relationship between PSTs' enactment of academically rigorous questions in whole-class discussions and their preparation for questioning in those discussions, we conducted our data analysis in two parts. Given that there is a reciprocal relationship between the teacher-intended curriculum and enacted curriculum (Remillard & Heck, [<reflink idref="bib38" id="ref126">38</reflink>]), we chose to first examine the transcripts of the enacted lessons followed by tracing the enacted questions back to the lesson plan. This approach involved identifying episodes where students experienced high-quality questioning and then investigating the extent to which the teacher-intended curriculum (as represented in lesson plans) laid the foundation for the academically rigorous questions observed in the lesson episode. Our analysis could have been initiated by identifying questions in the planned lesson that reflected academically rigorous question types, followed by investigating whether those academically rigorous planned questions were enacted during the lesson. However, we opted not to take this approach because it assumes that PSTs (or any teacher) should follow their lesson plan as a script. In starting with the lesson enactment, we are centering how students experienced the lesson instead of how PSTs intended students to experience the lesson.</p> <hd id="AN0185425510-16">Enacted questions</hd> <p>We began our analysis by identifying episodes of eliciting student thinking using questions from phases of whole-class instruction. After transcribing these episodes, we coded them based on the IQA Academic Rigor for Teachers' Questions rubric (Boston, [<reflink idref="bib9" id="ref127">9</reflink>]). In addition to alignment with our theoretical framework, this tool was selected because it was designed to be used to analyze both classroom instruction and lesson plans, and reliability and validity of the rubric in these settings has already been established (Boston & Wolf, [<reflink idref="bib11" id="ref128">11</reflink>]). Further, the question types account for both initial eliciting of student thinking and follow-up questions. Enacted questions in the transcribed episodes were coded as one of six question types: probing, exploring mathematical meaning and relationships, generating discussion, procedural or factual, other mathematical, and non-mathematical (Boston, [<reflink idref="bib9" id="ref129">9</reflink>]). Examples of each type from our data are represented in Table 1.</p> <p>Table 1 Question analysis framework from the IQA academic rigor for teachers' questions rubric (Boston, [<reflink idref="bib9" id="ref130">9</reflink>])</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left"><p>Question type</p></th><th align="left"><p>Example</p></th></tr></thead><tbody><tr><td align="left"><p>Probing</p></td><td align="left"><p>Why does this work?</p></td></tr><tr><td align="left"><p>Exploring mathematical meaning and relationships</p></td><td align="left"><p>Why do you think that would work, to switch P and M, given what we've been doing with the other inverses?</p></td></tr><tr><td align="left"><p>Generating discussion</p></td><td align="left"><p>Looking at these examples here, take a second, look at them, and think, which of these relations or graphs are functions?</p></td></tr><tr><td align="left"><p>Procedural or factual</p></td><td align="left"><p>Is it invertible?</p></td></tr><tr><td align="left"><p>Other mathematical</p></td><td align="left"><p>Any questions on how we did those few steps?</p></td></tr><tr><td align="left"><p>Non-mathematical</p></td><td align="left"><p>Why would technology affect the prison rate?</p></td></tr></tbody></table> </ephtml> </p> <p>Since eliciting student thinking is more than a one-off question (Franke et al., [<reflink idref="bib16" id="ref131">16</reflink>]), we narrowed our data to examine only situations when the PSTs engaged in <emph>sustained questioning sequences</emph> during whole-class discussions. We define <emph>sustained questioning sequences</emph> as episodes where the PST initiated a question and then engaged in follow-up questioning exchanges with one or more students. A questioning sequence ends when the PST moves on to another topic or another instructor (MTE or College Algebra Instructor) intervenes to provide pedagogical support. Since we wanted to unpack PSTs practice during episodes of high-quality eliciting student thinking with questions, we further narrowed our data to sequences that included questions which are considered AR question types: probing, exploring mathematical meaning and relationships, and generating discussions (Boston, [<reflink idref="bib9" id="ref132">9</reflink>]). In order to determine how the different ways PSTs approached enacting questioning sequences that contained AR question types, we looked at the placement and number of AR questions in the sequence to find patterns in question enactment.</p> <hd id="AN0185425510-17">Planned questions</hd> <p>After narrowing the data to only look at patterns in questioning sequences containing AR question types, we traced the questions back into the lesson plan to understand what the preparation looked like for moments when high-quality eliciting was attempted. Analyzing lesson plans to characterize PSTs' preparation for eliciting student thinking with questions was a less straightforward task than analyzing the enacted questioning sequences. As mentioned earlier, we did not assume that there should be a one-to-one correspondence between the questions as planned and the questions that were enacted. This would be a form of scripted teaching practice, rather than a practice that is relational and responsive to students' needs. Instead, informed by the curriculum enactment process (Remillard & Heck, [<reflink idref="bib38" id="ref133">38</reflink>]), we aimed to characterize the types of teacher questions anticipated in written lesson plans and looked for patterns in types of anticipated questions and the quality of enacted questioning sequences.</p> <p>To accomplish this, we first determined the task being solved by the students from the episodes of eliciting student thinking using questions during whole-class instruction from the video-recorded lessons, and then identified the corresponding stage in the lesson (launch, explore, or discuss) where the task was planned to be taught. This allowed us to determine the portions of the lesson plan which corresponded to the moments questioning sequences which contained AR questions. Next, we further analyzed these portions by taking note of the planned questions that the PSTs actually enacted during the lesson, as well as any questions that the PSTs had an opportunity to enact given that the mathematical topic of the question emerged in the lesson. After identifying the planned-and-enacted questions and the potential questions in the lesson plans, we coded these questions using the same question analysis framework (Table 1) used for the transcripts of the lesson. This two-stage process, which includes analysis of potential questions not enacted in the lesson, allowed us to account for the potential benefit of thoughtful preparation in how PSTs responded during unanticipated or "on the fly" moments during instruction.</p> <p>Organizing our data in this way allowed us to look for how different questioning sequences evolved depending upon whether PSTs had asked questions that were planned or unplanned. Attending to this aspect of eliciting student thinking with questions, specifically PSTs' preparation to elicit student thinking with questions, can help establish whether PSTs intentions in their lesson plans are factors in the quality of their enacted questioning practice (Remillard & Heck, [<reflink idref="bib38" id="ref134">38</reflink>]). Additionally, the implications of these findings can inform MTEs' attention to anticipated teacher questions as to one area to focus on in their lesson planning feedback.</p> <hd id="AN0185425510-18">Findings</hd> <p>The primary goal of this study was to explore how the academically rigorous moments where PSTs elicited student thinking with questions during enacted lessons compared to the academic rigor of PSTs plans for eliciting student thinking as represented in written artifacts. We will begin by first characterizing the patterns which emerged around different types of sustained questioning sequences that PSTs enacted their UTE lessons (research question one), then we will share our analysis of the academic rigor of their planned questions during segments of the lesson plan that align with those enacted sustained questioning sequences (research question two).</p> <hd id="AN0185425510-19">Patterns in AR sustained questioning sequences</hd> <p>In our analysis, three distinct patterns of how PSTs enacted AR moments of eliciting student thinking with questions in whole-class discussions emerged. These three sustained questioning sequences PSTs enacted are: (<reflink idref="bib1" id="ref135">1</reflink>) AR maintained, (<reflink idref="bib2" id="ref136">2</reflink>) AR lowered, and (<reflink idref="bib3" id="ref137">3</reflink>) hybrid AR. In this section, we describe each of these types of sustained questioning sequences and provide examples of each type.</p> <hd id="AN0185425510-20">Academic rigor (AR) maintained</hd> <p>In sequences when the AR was maintained, PSTs began with an AR question, the students responded, and the PSTs continued to ask AR questions throughout the sequence. An example of this kind of sequence is highlighted in Table 2. As seen below, the PST starts the sequence by asking a generating discussion question to get the students thinking (Lines 1–2). After the students respond, the PST continues to ask a combination of probing (Line 4) and exploring mathematical meaning and relationship (Line 6–8) questions.</p> <p>Table 2 AR maintained example</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left"><p>Line</p></th><th align="left"><p>Speaker</p></th><th align="left"><p>Transcript</p></th></tr></thead><tbody><tr><td align="left"><p>1</p></td><td align="left"><p>PST:</p></td><td align="left"><p>Can somebody tell me what your line of thinking was, maybe, to get started</p></td></tr><tr><td align="left"><p>2</p></td><td align="left" /><td align="left"><p>on this? Yes</p></td></tr><tr><td align="left"><p>3</p></td><td align="left"><p>S:</p></td><td align="left"><p>You switch M and P and then solve for M</p></td></tr><tr><td align="left"><p>4</p></td><td align="left"><p>PST:</p></td><td align="left"><p>Okay. Why that decision or why that strategy?</p></td></tr><tr><td align="left"><p>5</p></td><td align="left"><p>S:</p></td><td align="left"><p>I don't know. I just know that's how you get the inverse—</p></td></tr><tr><td align="left"><p>6</p></td><td align="left"><p>PST:</p></td><td align="left"><p>You might have learned it before and remembered that. Why do you think</p></td></tr><tr><td align="left"><p>7</p></td><td align="left" /><td align="left"><p>that would work, to switch P and M, given what we've been doing with the</p></td></tr><tr><td align="left"><p>8</p></td><td align="left" /><td align="left"><p>other inverses? Yeah?</p></td></tr><tr><td align="left"><p>9</p></td><td align="left"><p>S:</p></td><td align="left"><p>Because then you would be getting the output as an input</p></td></tr><tr><td align="left"><p>10</p></td><td align="left"><p>PST:</p></td><td align="left"><p>Yeah, so, then you would get the output as the input</p></td></tr></tbody></table> </ephtml> </p> <hd id="AN0185425510-21">AR lowered</hd> <p>In sequences the AR was lowered, PSTs began with an AR question but the students did not respond to the question. In response, the PSTs lowered the AR of the questions for the remainder of the sequence (see Table 3). In this example, the PST starts with a generating discussion question (Line 1) but gets no responses from students (Lines 2 and 4). After the lack of student response, the PST switches to a procedural or factual question (Line 5). This lack of success in asking results in the PST continuing to ask questions of low AR (Lines 9 and 11–12).</p> <p>Table 3 AR lowered example</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left"><p>Line</p></th><th align="left"><p>Speaker</p></th><th align="left"><p>Transcript</p></th></tr></thead><tbody><tr><td align="left"><p>1</p></td><td align="left"><p>PST:</p></td><td align="left"><p>What do you guys already know about slope?</p></td></tr><tr><td align="left"><p>2</p></td><td align="left" /><td align="left"><p>[no response]</p></td></tr><tr><td align="left"><p>3</p></td><td align="left"><p>PST:</p></td><td align="left"><p>Does anyone have anything?</p></td></tr><tr><td align="left"><p>4</p></td><td align="left" /><td align="left"><p>[no response]</p></td></tr><tr><td align="left"><p>5</p></td><td align="left"><p>PST:</p></td><td align="left"><p>Does anyone know how we find the slope of a linear line?</p></td></tr><tr><td align="left"><p>6</p></td><td align="left"><p>S:</p></td><td align="left"><p>Rise over run</p></td></tr><tr><td align="left"><p>7</p></td><td align="left"><p>PST:</p></td><td align="left"><p>We think of this as the change in y over the change in x, also written as y2</p></td></tr><tr><td align="left"><p>8</p></td><td align="left" /><td align="left"><p>– y1 over × 2 − × 1. Another way is f of × 2 minus f of × 1 over × 2 minus × 1</p></td></tr><tr><td align="left"><p>9</p></td><td align="left" /><td align="left"><p>What graphs, other than linear graphs also have slopes?</p></td></tr><tr><td align="left"><p>10</p></td><td align="left" /><td align="left"><p>[not response]</p></td></tr><tr><td align="left"><p>11</p></td><td align="left"><p>PST:</p></td><td align="left"><p>We've seen graphs like absolute value graphs, or like this. Do these graphs</p></td></tr><tr><td align="left"><p>12</p></td><td align="left" /><td align="left"><p>have slopes?</p></td></tr><tr><td align="left"><p>13</p></td><td align="left" /><td align="left"><p>[Varying responses]</p></td></tr><tr><td align="left"><p>14</p></td><td align="left"><p>PST:</p></td><td align="left"><p>I heard some yeses, I heard some nos, but actually all graphs, all functions</p></td></tr><tr><td align="left"><p>15</p></td><td align="left" /><td align="left"><p>have slope, so these all have a slope, but it's important to remember the</p></td></tr><tr><td align="left"><p>16</p></td><td align="left" /><td align="left"><p>difference between a linear function, like this with a constant slope, where</p></td></tr><tr><td align="left"><p>17</p></td><td align="left" /><td align="left"><p>these over here are not constant</p></td></tr></tbody></table> </ephtml> </p> <hd id="AN0185425510-22">Hybrid AR</hd> <p>The final type, which we call the hybrid AR, is characterized by the PSTs initiating the sequence with a non-AR question. This lower rigor question elicits a response from students; once they have responded, the PST switches to using AR questions for the remainder of the sequence, thus increasing the rigor of the sequence. This sequence is represented in Table 4. Here we see the PST starting with a procedural or factual question (Lines 1 and 3). This "hooks" the students into participating in the discussion (Lines 2 and 4), as evident by multiple students responding at once. The PST follows up by switching to asking probing questions (Lines 5 and 7).</p> <p>Table 4 Hybrid AR example</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left"><p>Line</p></th><th align="left"><p>Speaker</p></th><th align="left"><p>Transcript</p></th></tr></thead><tbody><tr><td align="left"><p>1</p></td><td align="left"><p>PST:</p></td><td align="left"><p>Is that a one an exponential function or not?</p></td></tr><tr><td align="left"><p>2</p></td><td align="left"><p>Ss:</p></td><td align="left"><p>NO (multiple students respond in unison)</p></td></tr><tr><td align="left"><p>3</p></td><td align="left"><p>PST:</p></td><td align="left"><p>What is that type of function?</p></td></tr><tr><td align="left"><p>4</p></td><td align="left"><p>Ss:</p></td><td align="left"><p>Linear (multiple students in unison)</p></td></tr><tr><td align="left"><p>5</p></td><td align="left"><p>PST:</p></td><td align="left"><p>Linear, cool. How did you know that it was a linear function?</p></td></tr><tr><td align="left"><p>6</p></td><td align="left"><p>S:</p></td><td align="left"><p>There's an intercept</p></td></tr><tr><td align="left"><p>7</p></td><td align="left"><p>PST:</p></td><td align="left"><p>Gotcha, yeah. We can see the intercept. Yeah. Can you expand on that?</p></td></tr></tbody></table> </ephtml> </p> <p>Each of the three types of questioning sequences appeared in both rounds of teaching during the UTE. As shown in Fig. 2, we found that PSTs used sequences that involved AR questions (hybrid AR and AR maintained) in their lessons in both rounds of teaching in the UTE. We also found that PSTs engaged in AR-lowered sequences in each round, although the prevalence of this sequence type was more infrequent than the other two types.</p> <p>Graph: Fig. 2 Frequency of AR enacted questions in sequence by round</p> <p>Since the rounds are different lengths, we are unable to make claims about changes in questioning sequences between the rounds. Yet, we want to point your attention to the shift in the relative frequencies of the AR questioning strategies over time. We found that the shift in hybrid AR is larger than the proportion of increased time in the lesson. We suspect this shift may reflect PSTs adopting a new way to elicit student thinking with questions when students do not respond to the questions posed. However, our current data limit our ability to verify such claims.</p> <hd id="AN0185425510-23">Looking for clues in lesson plans to explore choices of question sequences enacted</hd> <p>Given our research focus, we expanded beyond analyzing only the enacted sustained questioning sequences to look at the ways the PSTs intended to elicit as reflected in their lesson plans. As lesson planning is one site where MTEs can offer feedback to support PSTs' practice, this analysis could offer insight to support, as Remillard and Heck ([<reflink idref="bib38" id="ref138">38</reflink>]) posit, whether the intention that PSTs set in their lesson plans has tangible effects on the quality of their enacted practice.</p> <p>To begin, we initially found that PSTs enacted far more questions than they planned (Fig. 3). This trend happened across all sustained questioning sequences. This finding is unsurprising, especially given that the UTE immerses PSTs in actual teaching situations where responding to students' learning needs is a goal. Unlike more scripted teaching interactions, and especially given their lack of teaching experience, we would expect that PSTs would need to ask more questions to elicit student thinking in ways that can guide their decision making.</p> <p>Graph: Fig. 3 Frequency of planned-and-enacted questions in sequences</p> <p>What can also be seen in Fig. 3 is that PSTs had the most planned questions for sequences in which the AR was maintained followed by the hybrid AR (see Fig. 3). To understand the extent to which the enacted questions were ones that were "enacted-as-planned" or ones that were developed in-the-moment, we categorized planned questions as whether they were enacted-as-planned, or they were planned-but-unenacted even though the opportunity was present. Figure 4 displays the results of this separation. This separation highlights that, although PSTs enacted considerably more questions than they planned, they did not always enact all the questions they planned, especially in sequences where AR was maintained. Further, our findings show that in sequences where AR was lowered, PSTs had significantly less questions planned.</p> <p>Graph: Fig. 4 Enacted-as-planned and planned-but-unenacted questions</p> <hd id="AN0185425510-24">Discussion and implications</hd> <p>Eliciting student thinking with questions is an important high-leverage practice of ambitious instruction (Ball & Forzani, [<reflink idref="bib3" id="ref139">3</reflink>]; Gotwals & Birmingham, [<reflink idref="bib21" id="ref140">21</reflink>]), yet, as illustrated in our data, learning to enact these practices in the context of authentic instructional situations goes beyond knowing <emph>what</emph> to ask or <emph>how to</emph> follow-up. By drawing on Remillard and Heck ([<reflink idref="bib38" id="ref141">38</reflink>]), this study adds to the field's understanding of how PSTs' engagement in academically rigorous questioning is associated with their anticipated questioning as represented in lesson plans. In particular, our study extends existing work on PSTs' eliciting student thinking with questions practice to the space of novice experience on the learning to teach continuum; the particular context of the UTE early field experience presented PSTs with authentic opportunities to engage in eliciting student thinking with questions.</p> <p>First, we want to highlight the contribution of our analytical construct of sustained question sequences. Previous analysis of prospective teachers' questioning has either focused on individual questions (Moyer & Milewicz, [<reflink idref="bib31" id="ref142">31</reflink>]; Shaughnessy & Boerst, [<reflink idref="bib40" id="ref143">40</reflink>]) or the end goal of questioning (Nicol, [<reflink idref="bib33" id="ref144">33</reflink>]). Rather than considering individual questions as part of an enacted eliciting practice, the construct of sustained question sequences expands our perspective on eliciting student thinking with questions to include related questions as a sequence. Taking sustained questioning sequences as a unit of analysis affords the opportunity to consider how questions connect together to see patterns that may be arising. For instance, Gaspard and Gainsburg ([<reflink idref="bib18" id="ref145">18</reflink>]) found the PSTs in their study changed the number of unpredictable questions asked over time. However, one unknown in their analysis is the types of questions that surrounded those with unpredictable answers. By utilizing a grain size of sustained questioning sequences for our unit of analysis, we were able to notice patterns, such as hybrid AR—where a lower AR question gets asked but then is followed up by a more rigorous question, that would not have been apparent with a unit of analysis at the level of a single question.</p> <p>In addition to being an analytical construct, sustained questioning sequences could also be used by MTEs when reviewing, interpreting and giving feedback on PSTs' planning for eliciting student thinking with questions. For instance, our study offers insight into PSTs preferences for utilizing hybrid AR questioning sequences during instruction. This practice echoes findings from Gaspard and Gainsburg ([<reflink idref="bib18" id="ref146">18</reflink>]), who learned, through interviews with PSTs, that PSTs asked less complicated questions believing that students' eagerness to respond to more simplistic questions reflected that the instruction was helping their students more. However, our construct of sustained questioning sequences allowed us to show that the PSTs in our study also asked questions that were lower rigor to encourage student responses. Once the students were "hooked," PSTs then raised the rigor of the questions.</p> <p>In considering this hybrid AR sequences, MTEs should recognize that while PSTs are (temporarily) reducing the rigor of questions to provide what they perceive is access for their students (Gaspard & Gainsburg, [<reflink idref="bib18" id="ref147">18</reflink>]), they also are reducing the risk to themselves as novices by not creating situations where students will not participate. More investigation is warranted as to whether explicitly encouraging PSTs to use a lower academic rigor question, one that might enhance the chances that students will participate, prior to asking a high AR question as a follow-up, could be a viable strategy to eliciting student thinking when students seem reluctant to share their thinking. Offering these strategies might help PSTs persist in developing this type of sustained questioning sequence as a possible avenue for future research not only into when PSTs utilize this method to hook students into a mathematical discussion but also why and how they use the method.</p> <p>Additionally, our analysis of the lesson plans in conjunction with the videos lends further support to existing work indicating the difficulties PSTs have when they find themselves in classroom situations they had not anticipated (see Gaspard & Gainsbourg, [<reflink idref="bib18" id="ref148">18</reflink>]; Inoue & Buczynski, [<reflink idref="bib24" id="ref149">24</reflink>]). In answering our second research question, we found that PSTs tended to lower the AR of sustained questioning sequences when they had significantly fewer AR questions planned around the concept. This is not surprising, as PSTs classroom inexperience often means that they face difficulties with anticipating students' responses (e.g., Arbaugh et al., [<reflink idref="bib1" id="ref150">1</reflink>]; Taylan, [<reflink idref="bib46" id="ref151">46</reflink>]). Improving the quality of PSTs' eliciting student thinking with questions likely means not only helping them to anticipate student responses, which typically improves with more classroom experience, but also helping them to draw on high AR questioning when the situation does not unfold as planned.</p> <p>Further, the findings of our study suggest the need for further research to examine whether MTEs feedback on lesson plans related to eliciting student thinking with questions could help PSTs overcome some of the challenges faced when eliciting student thinking with questions in the complex environment of classroom instruction. Considering the potential role MTEs' feedback has on PSTs' ability to anticipate, MTEs could provide feedback on lesson plans that not only focuses on anticipating different student solutions, but also attends to the questions PSTs plan to ask for each solution method (Smith & Stein, [<reflink idref="bib44" id="ref152">44</reflink>]). In providing this kind of feedback, MTEs would support the alignment between the planned questions and the intended learning goals for the lesson, enhancing the likelihood that PSTs maintain AR during (the inevitable) unplanned questioning sequences. If research shows this to be the case, improving PSTs' eliciting student thinking with questions practice may not need to solely involve more experience; targeted lesson plan feedback could accelerate the process of developing craft knowledge (Kennedy, [<reflink idref="bib26" id="ref153">26</reflink>]) that is used when effectively responding to the persistent challenges of eliciting student thinking in classrooms.</p> <hd id="AN0185425510-25">Conclusion</hd> <p>As a high-leverage teaching practice, eliciting student thinking with questions has received quite a bit of attention not only in documents outlining new teaching standards (NCTM, [<reflink idref="bib32" id="ref154">32</reflink>]) but also in research on ambitious teaching practice (Gotwals & Birmingham, [<reflink idref="bib21" id="ref155">21</reflink>]; Shaughnessy & Boerst, [<reflink idref="bib40" id="ref156">40</reflink>]; Shaughnessy et al., [<reflink idref="bib41" id="ref157">41</reflink>]). This study contributes to this growing body of the literature by presenting approaches that PSTs use when attempting academically rigorous questions in their initial efforts at this practice in classrooms with actual learners. Furthermore, our study invites reflection on what can be learned <emph>for</emph> teacher preparation when we focus research on moments where PSTs are engaging in ambitious pedagogy to explore how aspects of their preparation may have supported their practice in those moments. While this study focused on the connection between enacted practice that involved the use of academically rigorous questions and the quality of PST's lesson planning for those lesson episodes, lesson planning is but one of many typical activities of teacher preparation where MTEs can support PST learning of this ambitious practice. Further research could explore the relationship between activities such as rehearsals and post-lesson discussions and PSTs' enacted eliciting practice in the context of authentic learning settings, to provide insights regarding the specific kinds of feedback and in-the-moment mentoring that could support PSTs in becoming more masterful with this complex practice.</p> <hd id="AN0185425510-26">Acknowledgements</hd> <p>We would like to express our deepest thanks to Fran Arbaugh, Michelle Cirillo, and Kevin Voogt for their generosity with their time to review the manuscript of this work. This research was supported through funding from the National Science Foundation (DRLs 1725910, 1725920, 1726364; Bieda, Arbaugh, Cirillo PIs). Any opinions, conclusions, or recommendations contained herein are those of the presenters and do not necessarily reflect the views of NSF.</p> <hd id="AN0185425510-27">Appendix A</hd> <p></p> <hd id="AN0185425510-28">Lesson Preparation Template</hd> <p>Graph</p> <hd id="AN0185425510-29">Publisher's Note</hd> <p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p> <ref id="AN0185425510-30"> <title> References </title> <blist> <bibl id="bib1" idref="ref13" type="bt">1</bibl> <bibtext> Arbaugh F, Graysay D, Konuk N, Freeburn B. The three-minute-rehearsal cycle of enactment and investigation: Preservice secondary mathematics teachers learning to elicit and use evidence of student thinking. Mathematics Teacher Educator. 2019; 8; 1: 22-48. 10.5951/mathteaceduc.8.1.0022</bibtext> </blist> <blist> <bibl id="bib2" idref="ref1" type="bt">2</bibl> <bibtext> Association for Mathematics Teacher Educators. (2017). 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  Data: Learning to Elicit Student Thinking: The Role of Planning to Support Academically Rigorous Questioning Sequences during Instruction
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  Data: <searchLink fieldCode="AR" term="%22Sheila+Orr%22">Sheila Orr</searchLink> (ORCID <externalLink term="http://orcid.org/0000-0002-8818-2093">0000-0002-8818-2093</externalLink>)<br /><searchLink fieldCode="AR" term="%22Kristen+Bieda%22">Kristen Bieda</searchLink> (ORCID <externalLink term="http://orcid.org/0000-0002-7407-5203">0000-0002-7407-5203</externalLink>)
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  Data: <searchLink fieldCode="DE" term="%22Questioning+Techniques%22">Questioning Techniques</searchLink><br /><searchLink fieldCode="DE" term="%22Teaching+Methods%22">Teaching Methods</searchLink><br /><searchLink fieldCode="DE" term="%22Thinking+Skills%22">Thinking Skills</searchLink><br /><searchLink fieldCode="DE" term="%22Preservice+Teachers%22">Preservice Teachers</searchLink><br /><searchLink fieldCode="DE" term="%22Field+Instruction%22">Field Instruction</searchLink><br /><searchLink fieldCode="DE" term="%22Field+Experience+Programs%22">Field Experience Programs</searchLink><br /><searchLink fieldCode="DE" term="%22Educational+Planning%22">Educational Planning</searchLink><br /><searchLink fieldCode="DE" term="%22Lesson+Plans%22">Lesson Plans</searchLink>
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  Data: The use of questioning to elicit student thinking is one of the cornerstones of ambitious teaching; thus, supporting prospective teachers in this complex practice is a core objective of mathematics teacher education. In this paper, we examine the academic rigor of the sustained questioning sequences prospective teachers use during instruction to elicit student thinking and the possible factor lesson planning plays in enacted sequences involving academically rigorous questions. We analyzed the questions they asked during lessons in an early field experience and traced how prospective teachers prepared for those questions as represented in their lesson plans. From our analysis, we found prospective teachers used three types of sustained questioning sequences when eliciting student thinking in the enactment of the lesson. Further, we found evidence suggesting a connection between the questions asked during enactment of a lesson and the quality of questions prospective teachers incorporated into their lesson plan. These findings inform how mathematics teacher educators can support prospective teachers during lesson planning to prepare to elicit student thinking in productive ways.
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  Group: ID
  Data: EJ1472072
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1472072
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1007/s10857-023-09603-5
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 22
        StartPage: 523
    Subjects:
      – SubjectFull: Questioning Techniques
        Type: general
      – SubjectFull: Teaching Methods
        Type: general
      – SubjectFull: Thinking Skills
        Type: general
      – SubjectFull: Preservice Teachers
        Type: general
      – SubjectFull: Field Instruction
        Type: general
      – SubjectFull: Field Experience Programs
        Type: general
      – SubjectFull: Educational Planning
        Type: general
      – SubjectFull: Lesson Plans
        Type: general
    Titles:
      – TitleFull: Learning to Elicit Student Thinking: The Role of Planning to Support Academically Rigorous Questioning Sequences during Instruction
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Sheila Orr
      – PersonEntity:
          Name:
            NameFull: Kristen Bieda
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 01
              M: 05
              Type: published
              Y: 2025
          Identifiers:
            – Type: issn-print
              Value: 1386-4416
            – Type: issn-electronic
              Value: 1573-1820
          Numbering:
            – Type: volume
              Value: 28
            – Type: issue
              Value: 3
          Titles:
            – TitleFull: Journal of Mathematics Teacher Education
              Type: main
ResultId 1