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Teacher Questioning Practices over a Sequence of Consecutive Lessons: A Case Study of Two Mathematics Teachers Lianchun Dong 1 , David Clarke 2 , Yiming Cao 3,4, *, Lidong Wang 5, * and Wee Tiong Seah 2 1 2 3 4 5

*

College of Science, Minzu University of China, No. 27 Zhongguancun South Avenue, Haidian District, Beijing 100081, China; [email protected] Melbourne Graduate School of Education, University of Melbourne, 234 Queensberry Street, Carlton 3053, Australia; [email protected] (D.C.); [email protected] (W.T.S.) School of Mathematical Sciences, Beijing Normal University, No. 19 XinJieKouWai Street, HaiDian District, Beijing 100875, China International Center for Research in Mathematics Education, Beijing Normal University, No. 19 XinJieKouWai Street, HaiDian District, Beijing 100875, China Collaborative Innovation Center of Assessment for Basic Education Quality, Beijing Normal University, No. 19 XinJieKouWai Street, HaiDian District, Beijing 100875, China Correspondence: [email protected] (Y.C.); [email protected] (L.W.); Tel.: +86-10-5880-1986

Received: 15 October 2018; Accepted: 20 December 2018; Published: 28 December 2018

Abstract: This study examined teacher questioning practices over a sequence of consecutive lessons in China. Based on the IRF (initiation–response–follow-up) framework, a comprehensive coding system was developed to analyze what kinds of verbal questions were initiated by the teachers to elicit mathematical information and in what ways the teachers made use of students’ verbal contributions. This study finds that all participating teachers’ questioning practices showed both variations and consistencies over the lesson sequence. It is argued that the act of asking questions in classroom interaction not only includes the teachers’ conscious planning so as to accomplish pedagogical goals, but also involves the teachers’ unconscious routine in how to build up on students’ thinking. Keywords: teacher questioning; consecutive lessons; mathematics; China

1. Introduction The ultimate goal of school education is to support and develop the sustainability of students’ thinking and understanding. In the classroom, teacher questioning practices help to build an environment for students’ sustainable learning and development. However, teachers’ effective and efficient employment of questioning strategies in classrooms has been one of the most challenging techniques. Some researchers (e.g., Reference [1]) argued that expert teachers’ questioning practices vary a lot to suit particular contexts and teaching purposes. In other words, effective questioning strategies do not represent a fixed set of techniques, but flexible adaptation, adjustment, and alteration of various questioning strategies according to actual teaching situations. However, very few empirical studies examined the ways in which teachers might vary their practices of asking questions according to different situations (e.g., in different lessons with different learning contents), which hindered our understanding of the instructional flexibility in classroom questioning. To find out how teachers might change their questioning practices in different lessons, this study conducted a detailed analysis of two mathematics teachers’ employment of questioning strategies over a unit of consecutive lessons.

Sustainability 2019, 11, 139; doi:10.3390/su11010139

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Sustainability 2019, 11, 139

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2. Literature Review 2.1. The Early Debate on Effective Question Types Many attempts have been made to identify the effective question types that could improve student achievement. These attempts lead to the development of classification systems of teacher questions. One of the most influential criteria used to classify teacher questions was the adaptation of Bloom’s taxonomy for question types: classifying teacher questions according to the level of cognitive process needed to achieve the answers [2]. In addition, the premise that certain types of question work better than others in classrooms sparks the debate regarding the superiority of higher-cognitive questions over lower-cognitive questions to influence student achievement (e.g., References [3,4]). Many researchers argued higher-cognitive questions contributed to students’ high-order thinking and deep understanding. However, a great many studies reported that teachers asked predominantly lower-cognitive questions rather than higher-cognitive questions in classroom interaction [5]. It seems to be puzzling that many teachers do not prefer to ask questions that could promote students’ better understanding. One of the possible explanations might be that teachers’ selection of question types is influenced by many other factors. For example, Gall pointed out that the effectiveness of teacher questions depends significantly on the student capabilities: lower-cognitive questions were more effective for students with low-level capabilities, whereas the students with average- or high-level capabilities could learn better from higher-cognitive questions [5]. In addition, the conceptions of questioning practices’ effectiveness also go beyond preparing a list of effective question types. Wilen and Clegg concluded that effective questioning practices do not mean simply selecting certain types of questions, but rather involve the ways of posing questions and responding to student answers [6]. In other words, effective questioning is a dynamic process involving flexible strategies of asking questions and dealing with students’ responses. 2.2. Sequences of Questions Recent studies argued that no teacher question types should be glorified in classroom teaching because any question had a place in enhancing students’ learning by interaction [7]. Some researchers claimed that more attention ought to be given to the sequence of discourse in which the teachers pose questions in classroom instruction [8,9]. Fine-grained analysis of the guiding process resulted in the identification of two different patterns of questioning: funneling and focusing [10–12]. In the funneling pattern, the teacher typically raises a sequence of questions as guidance towards the correct answers and students merely need to give the answers to each question without necessarily understanding the connections among the questions [10,12]. In contrast, the focusing pattern allows students to articulate their own mathematical thinking and to reflect on the meaning of teacher–student dialogues [11]. A prevalent structure in classroom discourse is referred to as the IRF sequence (teacher initiation—student response—teacher feedback; Sinclair and Coulthard [13]) or IRE, where E stands for evaluation [14], or “triadic dialogue” [15]. This triadic structure has been well-documented in the previous studies [16,17]. The initiation is typically enacted by the teacher questioning, followed by students’ reply and subsequently the teacher’s feedback or follow-up [18]. In recent years, the third move in IRF has attracted increasing attention from educational researchers in their attempts to investigate the value of IRF for teaching and learning. Some researchers [19,20] also pointed out that IRF proved to be productive in fostering collaborative work, during which the follow-up move plays a significant role. 2.3. Factors Influencing Teacher Questioning Practices Recent research added evidence regarding the sophisticated nature of effective questioning by highlighting the changes of teacher questions in different contexts. Some researchers argued that teacher questioning practice is directly reflective of instructional goals. Hiebert and Wearne pointed


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out the connections between instructional goals and teacher questions in mathematics classrooms, claiming that teachers raised questions with the intention to direct students’ attention to interpret and solve mathematics problems in the expected ways according to the instructional goals [21]. Meanwhile, Nathan and Kim also highlighted the effects of teachers’ curricular goals on classroom interactions [22]. They examined information flow and scaffolding in mathematics classrooms and demonstrated that classroom questioning practices could be shaped by teachers’ interpretation of curricular goals. Shi argued that in mainland China the analysis of teacher questioning practices could be used as a framework to interpret the teachers’ pedagogical goals in classroom instruction [23]. By examining five aspects of classrooms, namely, teachers’ questions, student responses, teachers’ responses to student answers, student activities, and teachers’ balancing on student answering and student discussion, Shi found that teachers adjusted the cognitive requirements of classroom questions so as to ensure the completion of lesson goals [23]. In addition, Nathan and Kim reported that teacher questions in mathematics classrooms are highly responsive to the perceived state and current needs of students [21]. In the case of inaccurate or incomplete student responses, the teacher tended to reduce the level of cognitive complexity of a subsequent question, whereas teacher subsequent questions became more complex when students’ responses were mathematically accurate. 2.4. Coding Teacher Questioning Practices Some coding systems were developed in order to describe the complexity of teacher questioning in mathematics classrooms (e.g., References [21,23,24]). Instead of looking at the cognitive level of questions as is used by the earlier researchers, these coding systems focused on the teacher’s intentions when asking questions. For example, Hiebert and Wearne grouped teacher questions into four broad categories, namely requesting recitation of previously taught facts or procedures, asking students to describe invented solution strategies, requesting students to generate problems, and asking students to explain why things work the way they do [21]. Similarly, in Boaler and Brodie’s classification system [24], all the categories were named with verbs to describe teacher intentions when asking questions: (1) gathering information, leading students through a method; (2) inserting terminology; (3) exploring mathematical meanings and/or relationships; (4) probing, getting students to explain their thinking; (5) generating discussion; (6) linking and applying; (7) extending thinking; (8) orienting and focusing; and (9) establishing context. When coding teacher questions, a distinction between initiation questions and follow-up questions is highlighted. As opposed to initiation questions which signal the opening of the teacher-student conversation, follow-up questions refer to those teacher questions asked in response to a student utterance, such as a student’s answer or response to teacher questions posed earlier [25]. The investigation of follow-up questions is significant because teachers were reported to experience more challenges when properly following up students’ responses than posing proper initiation questions [8]. 2.5. The New Lens: A Unit of Consecutive Lessons In recent years, researchers highlighted the necessity of examining mathematics teachers’ practices over a unit of consecutive lessons rather than a single lesson (see, for example, References [24,26,27]). In particular, Koizumi pointed out teachers might prepare each lesson as one part of a sequence of lessons which could be regarded as a unit and thus different lesson within one unit usually had different instructional goals and structures which would influence classroom questioning practices [28]. However, much previous research has focused on single lessons, leading to an inability to comprehensively consider the lesson context as a factor in understanding teacher questioning. In one of the few studies investigating teacher questioning in consecutive lessons, Koizumi focused on teacher questioning strategies in introductory stages in mathematics lesson sequences in Japan and


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Germany, concluding that the context of consecutive lessons enables the better observation of cultural specificity in teacher questioning practices [28]. 2.6. Purpose of this Study Despite the fruitful findings mentioned above, there is still a lack of comprehensive understanding of teacher questioning in mathematics classrooms. Especially, the instructional flexibility in teacher questioning practices is rarely investigated. Very little is known about the ways in which teachers flexibly vary their questioning practices in response to instructional contexts. Very few studies have been conducted to investigate how teachers change their ways of asking follow-up questions to probe or facilitate students’ learning. However, the previous studies demonstrated that a sequence of consecutive lessons could show more variation in teachers’ teaching practices and also offer greater insight into the teachers’ purposeful deployment of particular question types, reflective of their changing instructional purposes over the course of a lesson sequence. In other words, only in the sequence of consecutive lessons can we do justice to the situated nature of teacher questioning and give appropriate recognition to the strategic adjustment that teachers make in their questioning practices in response to changing classroom conditions. This current study has been designed to investigate the flexibility in teacher questioning practices through the new lens: a unit of consecutive lessons. It aims to address the following research questions: (1) In each participating teacher’s classroom, what variations in teacher questioning practices were evident across a sequence of consecutive lessons? (2) In each participating teacher’s classroom, what consistencies in teacher questioning practices were evident across a sequence of consecutive lessons? 3. Methodology This study aims to reveal detailed and in-depth features of teacher questioning practices in mathematics classrooms. Such a detailed investigation required the employment of a case study approach [29]. Each case concerns a single mathematics teacher and their class. 3.1. Participants Two mathematics teachers from China participated in this study. One of them is from Beijing in the northern region of China, while the other one is from the Jiangsu Province, the southern part of China. Both participating teachers were considered competent by local standards, and were both experienced in teaching the respective grade levels being studied. The information about the participants is shown in the Table 1. Table 1. Details of the participating teachers and the classes. Teacher

Gender

Year of Experience

Site

School/Class

CHN1 CHN2

Female Male

More than 5 years More than 10 years

Beijing Haimen, Jiangsu Province

Urban public school, 36 students Urban public school, 46 students

3.2. Data Collection For each teacher, one unit of consecutive lessons was videotaped through three cameras, separately focusing on the teacher, the whole class, and a group of focus students. The three-camera video recordings increased the possibility of continuously documenting all teacher–student interaction in classrooms, contributing to a better observation of teacher questioning practices in the classrooms. The details of the video-recorded lessons for each participating teacher are listed in the Table 2.


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Sustainability 2018, 10, x FOR PEER REVIEW

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Table 2. Lesson topics delivered by the participating teachers. Table 2. Lesson topics delivered by the participating teachers. Lesson Topics

Lesson Topics

L1: Solving right triangles; L2: Solving non-right triangles; L3: Applications: CHN1:CHN1: L1: Solving triangles; L2: Solving non-right triangles; L3: Applications: measurements Solving rightright triangles measurements of height and distance; L4: Applications: slope angles and Solving right of height and distance; L4: Applications: slopeL5: angles marine problems; navigation; Mixed marine navigation; Mixedand application L6:L5: Chapter review

triangles

application problems; L6: Chapter review L1: Introduction to quadratic functions; L2: The graph and properties of y = ax2 ; 2 + k; L4: The graph2and properties of L3: The graph and of and y = ax L1: Introduction to quadratic functions; L2:properties The graph properties of y = ax ; L3: The 2 ; L5: The graph and properties of y = a(x − h)2 2+ k; L6: The graph a(x −k;h)L4: 2+ CHN2: CHN2: Quadratic graph andfunctions properties of yy==ax The graph2and properties of y = a(x − h) ; L5: The and properties of y = ax + bx + c; L7: Finding the equation of quadratic Quadratic graph and properties of y = a(x − h)2 + k; L6: The graph and properties of y = ax2 + bx + c; L7: functions; L8: Quadratic functions and quadratic equations; L9: Quadratic functions Finding the equation of quadratic functions; Quadratic functions quadratic functions andL8: real-world problems; L10: and Chapter review equations; L9: Quadratic functions and real-world problems; L10: Chapter review

3.3. Data Analysis 3.3. Data Analysis 3.3.1. The Definition of the Teacher Questions 3.3.1. The Definition of the Teacher Questions Three types of occasions when the teacher interacted with students by using questions were Three types of occasions the teacher interacted with students using did questions were these identified first. Where the when student/s replied to teacher questions and thebyteacher not respond, identified first. Where student/sasreplied to teacher questions and There the teacher did not of respond, interactions were the categorized question–answer (Q&A) pairs. are two types IRF (teacher these initiation—student interactions were categorized as question–answer (Q&A) pairs. There are two types of IRF asks response—teacher follow up) sequences: (1) IRF (single) in which the teacher (teacher initiation—student response—teacher follow up) sequences: (1) IRF (single) in which the a question and then gives closed follow-up moves (such as evaluation) to students so as to accomplish teacher a question and then closed follow-up movesthe (such as evaluation) to students so as gives theasks current discussion, andgives (2) IRF (multiple) in which teacher asks a question and then to accomplish the current discussion, and (2) IRF (multiple) in which the teacher asks a question open follow-up moves such as clarification or elaborations that require a further student and response. then gives open follow-up moves such as clarification or elaborations that require a further student The second type could potentially extend the discussion and the associated IRF sequence. response. The second type could potentially extend the discussion and the associated IRF sequence. When analyzing teacher questions, a distinction was made between initiation questions and When analyzing teacher questions, a distinction was made between initiation and follow-up questions. Initiation questions are those questions asked by teachersquestions for the purpose of follow-up questions. Initiation questions are those questions asked by teachers for the purpose of starting a conversation or discussion. In contrast, follow-up questions are those questions asked for starting conversation or discussion. contrast, follow-up questions are thoseanswer questions asked forto the theapurposes of responding to aInstudent utterance, such as a student’s or response the purposes of responding to a student utterance, such as a student’s answer or response to the teacher’s previous question. In some cases, students stayed silent for 2 seconds or longer after the teacher’s previous question. In some cases, students stayed silent for 2 seconds or longer after the teacher posed a question and afterwards the teacher gave some clues by asking further questions. teacher posed questionare andalso afterwards theasteacher gavequestions, some clues by asking further The latteraquestions categorized follow-up because a silent periodquestions. of 2 seconds or The latter questions are also categorized as follow-up questions, because a silent period of 2 seconds longer could be recognized as a special type of student responses. In such cases, student responses or longer could be recognized as a special type of student responses. In such cases, student responses will be shown as “[silent]” in the transcriptions. will be shown as “[silent]” the transcriptions. In this study, thein Q&A pair and IRF (single) sequence contain teacher initiation questions only In this study, the Q&A pair and IRF (single) sequence containstudent teacherresponse, initiationand questions and the IRF sequence includes the teacher initiation question, teacher only follow-up and the IRF sequence includes the teacher initiation question, student response, and teacher followquestion, as is shown in the Figure 1. up question, as is shown in the Figure 1.

Figure 1. The framework to categorize teacher questions. Note: FQ-IRF-m = follow-up questions in Figure 1. The framework to categorize teacher questions. Note: FQ-IRF-m = follow-up questions in IRF (multiple) sequences; FQ = follow-up questions; IQ-IRF-m = initiation questions in IRF (multiple) IRF (multiple) sequences; FQ = follow-up questions; IQ-IRF-m = initiation questions in IRF (multiple) sequences; IQ-QA&IRF-s = initiation questions in Q&A question pairs and IRF (single) sequences; IQ = sequences; IQ-QA&IRF-s = initiation questions in Q&A question pairs and IRF (single) sequences; IQ initiation questions. = initiation questions.


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3.3.2. Coding Systems A coding system was developed to categorize the initiation questions and follow-up questions. Instead of inventing the name of each possible category in advance, those questions documented in our data were analyzed first and then attempts were made to provide names to describe these different kinds of questions. The naming of some question categories in this study was also informed by the research of Boaler and Brodie [24] and Hiebert and Wearne [21], whose classification schemes included question categories closely aligned with some of the question types identified in this study. The categories developed in the above to studies were introduced in the section of literature review. However, because the coding systems developed by these researchers did not explicitly distinguish the initiation questions from the follow-up questions, new coding systems were developed in this study. For example, the category of Information extraction (see Table 3) was mentioned by Boaler and Brodie [24] who used the category of Gathering information, though in this study Information extraction questions only cover questions asked in the initiation stage of teacher–student interaction. In other words, the Gathering information questions by the definition of Boaler and Brodie [24] might be asked in both the initiation and follow-up stages of teacher–student interaction, whereas in the current study, the category Information extraction question only referred to the questions asked in the initiation stage. Those Gathering information questions asked in the follow-up stage would be coded with follow-up question categories depending on the actual purpose of the follow-up questions. The coding systems are presented in Table 3, in which each question category is associated with a three-letter code, accompanied by a description. Each three-letter code is used in the following sections.


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Table 3. Sub-categories for initiation questions and follow-up questions. Regrouped Codes

Initial Codes and Descriptions Initiation Questions

Reflection (Ref.)

Initiation questions requiring students’ reflection after mathematical activities.

Build-up (Bui.)

Conjecture: Initiation questions requiring students to come up with suppositions or presumptions about patterns. Variations: Initiation questions requiring students to consider a problem for which certain aspects vary while the other aspects are kept the same as a previous problem. Link/application: Initiation questions requiring students to provide examples of mathematics knowledge, or to apply mathematical knowledge. Comparison: Initiation questions requiring students to make comparisons between descriptions, graphs, etc. Evaluation: Initiation questions used to elicit students’ opinions.

Open-up (Ope.)

Explanation: Initiation questions used to elicit students’ thinking or interpretation. Strategy/procedure: Initiation questions requiring students to describe their strategies, procedures or the process of solving problems.

Result (Res.)

Initiation questions requiring the results of mathematical operation.

Tracking (Tra.)

Understanding check: Initiation questions used to check whether students understand the teacher’s or students’ statements. Progress monitoring: Initiation questions requiring students to monitor and regulate the process of reasoning and problem solving.

Information processing (Inf.)

Information extraction: Initiation questions requiring students to identify and select information from text descriptions, graphs, etc. Generation: Initiation questions asking students to generate a problem/scenario to satisfy given requirements.

Review (Rev.)

Initiation questions used to elicit the previously learnt knowledge. Follow-up Questions

Inviting comments (Inv.)

Agreement request: Follow-up questions that are used to elicit students’ agreement on comments given by some students. Supplement: Follow-up questions used to request for a larger variety of opinions or comments.

Probing (Pro.)

Clarification: Follow-up questions requiring students to show more details about their answers, solutions, or comments. Justification: Follow-up questions requiring students to justify their answers, where the answers are responses to the teacher’s initiation question or follow-up question in the last turn. Elaboration: Follow-up questions used to guide students towards a more comprehensive response by building on students’ existing responses.

Redirecting (Dir.)

Cueing: Follow-up questions used to redirect students to focus on key elements or aspects of the problem situation so as to enable students’ problem-solving. Refocusing: Follow-up questions used to redirect students to refocus on the essential points of the problem under discussion, especially when students’ thinking is off the right track.

Extension (Ext.)

Follow-up questions used to extend the topics under discussion to other topics.

Reformulation request (Rer.)

Follow-up questions requiring one student to reformulate his or her answer.

Repeat (Rep.)

Follow-up questions where the teacher repeats or rephrases the question asked in the last turn.


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3.3.3. Interrater Reliability One lesson was selected separately from teacher CHN1’s class and CHN2’s class for a reliability check, in which three coders (one of whom was the first author) in the field of education research independently coded the questions in the transcripts of the selected lessons. Due to the ethical requirement restricting video access to the researcher, the classroom videos could not be watched by any coders other than the first author. However, coding of teacher questioning could be conducted using the lesson transcripts and also all teaching and learning materials including lesson plans, PowerPoint slides, and copies of student worksheets were available to both coders. Also, teacher questionnaires and classroom observation tables were also given to both coders so as to allow each coder the best possible basis on which to make sense of the instructional settings. An agreement of 80% was achieved between the three coders’ coding results. Any inconsistent coding results were discussed, and a consensus coding result was determined by all the coders and the authors of this paper. Where coding differences arose from differences in the interpretations of the coding scheme, rather than differences in the interpretation of the data, these differences were resolved by refining the categories’ descriptions to achieve consensus as to the meaning of the code/category. Afterwards, the first author coded all the remaining lessons with the revised coding systems. The questions identified within the lesson transcripts were not coded in isolation, disconnected from the interactions of which they were a part. They were instead all coded within the classroom transcripts, where the teacher and students engaged in questioning and answering practices, such that the social context of each exchange could be taken into account, together with actions that immediately preceded or followed the coded exchange. The classroom videos were revisited and checked whenever there was ambiguity. 4. Findings 4.1. The Variations in the Number of Teacher Questions The number of questions asked by both teachers varied substantially from one lesson to another across the lesson sequence. Figure 2a,b show the total number of teacher questions in one lesson and the average number of teacher questions in each minute in the classes of the teacher CHN1 and CHN2. Both numbers experienced an overall falling trend across the lesson sequence taught by the teacher CHN1. Sustainability 2018,sequence, 10, x FOR PEER 8 ofones. 18 Over the lesson she REVIEW tended to ask fewer questions in the later lessons than in the earlier

(a) Teacher CHN1

(b) Teacher CHN2

Figure2.2.The Thetotal totalnumber numberof of questions questions asked lesson. Note: Figure asked by by the the teacher teacherCHN1 CHN1and andCHN2 CHN2inineach each lesson. Note: thetotal total number number of questions; = the average number of questions askedasked in QsQs= =the questions;Qs Qsininevery everyminute minute = the average number of questions = = thethe total number of of Q&A pairs and thethe IRF (single) ineach eachIRF IRF(multiple) (multiple)sequence; sequence;QA&IRF-s QA&IRF-s total number Q&A pairs and IRF (single) sequences;IRF-m IRF-m==the thenumber number of of the the IRF IRF (multiple) sequences; (multiple) sequences. sequences.

4.2. The Consistencies in the Distribution of Questions For each participating teacher, the questions asked to initiate the IRF (multiple) sequences consistently took up a regular proportion of the total number of questions in a lesson. This means both participating teachers used questions to start the IRF (multiple) sequences in a consistent way


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For the teacher CHN2, the total number of teacher questions in one lesson and the average number of teacher questions in each minute both remained stable in the first six lessons, except lesson 3 where there was an obvious drop. The last four lessons saw a significant increase in the number of teacher questions. Compared with the teacher CHN1, teacher CHN2 preferred to ask more questions in the second half of the teaching unit than the first half. 4.2. The Consistencies in the Distribution of Questions For each participating teacher, the questions asked to initiate the IRF (multiple) sequences consistently took up a regular proportion of the total number of questions in a lesson. This means both participating teachers used questions to start the IRF (multiple) sequences in a consistent way across the unit of consecutive lessons. Besides, each IRF (multiple) sequence on average includes one initiation question and two follow-up questions on average for each participating teacher. This means both participating teachers tended to maintain the length of IRF (multiple) sequences in different lessons. 4.2.1. The Case of Teacher CHN1 Figure 3a shows the distribution of questions in three occasions, namely, (1) Q&A pairs and IRF (single) sequences, (2) an initiation phase of the IRF (multiple) sequences, and (3) a follow-up phase of Sustainability 2018, 10, sequences. x FOR PEER REVIEW 9 of 18 the IRF (multiple)

(a) Three types of questions

(b) The average number of follow-up questions in each IRF (multiple) sequence

Figure of questions in theinclass teacher CHN1. Note: Qs-QA&IRF-s = all questions Figure3.3.The Thedistribution distribution of questions the of class of teacher CHN1. Note: Qs-QA&IRF-s = all in Q&A pairs and IRF (single) sequences; Qs-IRF-m =Qs-IRF-m all questions IRF (multiple) questions in Q&A pairs and IRF (single) sequences; = allinquestions in IRFsequences; (multiple) FQ = follow-up sequences; FQ =questions. follow-up questions;

This This illustrated illustrated that that the the proportion proportion occupied occupied by by the the initiation initiation questions questions in in the the IRF IRF (multiple) (multiple) sequences, in relation to the total number of the questions in one lesson, stayed stable 20% sequences, in relation to the total number of the questions in one lesson, stayed stable at 20%atacross across all the lessons, regardless of how many questions were recorded in the lesson. A typical all the lessons, regardless of how many questions were recorded in the lesson. A typical IRF IRF (multiple) sequence recorded this study includedananinitiation initiationquestion questionfollowed followed by by aa couple (multiple) sequence recorded in in this study included couple of of follow-up questions. follow-up questions. Figure Figure 3b 3b presents presents the the average average number number of of follow-up follow-up questions questions asked asked in in each each IRF IRF (multiple) (multiple) sequence. average number of follow-up questions recorded in each IRF sequence. Across Acrossthe thelesson lessonsequence, sequence,the the average number of follow-up questions recorded in each IRF (multiple) sequence stayed relatively stable at around 2. In other words, in the IRF (multiple) sequence, teacher CHN1 tended to ask two follow-up questions after every initiation question. An episode was selected to show how the teacher asked two follow-up questions after an initiation question. Episode 1: CHN1-Lesson 6, 0:02:49-0:04:04

questionsFQin= Q&A pairsquestions. and IRF (single) sequences; Qs-IRF-m = all questions in IRF (multiple) sequences; follow-up sequences; FQ = follow-up This illustrated that questions. the proportion occupied by the initiation questions in the IRF (multiple)

This illustrated that thetoproportion occupied by questions the initiation questions in the IRF (multiple) sequences, in relation the total number of the in one lesson, stayed stable at 20% across This illustrated that the proportion occupied by the initiation questions in the IRF sequences, in relation to the total number of the questions in one lesson, stayed stable at 20% across all the lessons, regardless of how many questions were recorded in the lesson. A(multiple) typical IRF in relation to the total number of the questions in one lesson, stayed stable at 20% allsequences, the(multiple) lessons, regardless of how many questions were recorded in the lesson. A typical IRF sequence recorded in this study included an initiation question followed by a across couple of all the lessons, regardless of how many questions were recorded in the lesson. A typical IRF 10 of 18 (multiple) sequence recorded in this study included an initiation question followed by a couple of follow-up questions. Sustainability 2019, 11, 139 (multiple) sequence recorded this study included an initiation question followed by aIRF couple of follow-up questions. Figure 3b presents theinaverage number of follow-up questions asked in each (multiple) follow-up questions. Figure 3b presents thelesson average numberthe of average follow-up questions asked inquestions each IRFrecorded (multiple) sequence. Across the sequence, number of follow-up in each Figure 3b presents averagerelatively number of follow-up asked IRF (multiple) sequence. Across thesequence lessonthe sequence, the average number of follow-up questions recorded in each (multiple) sequence stayed relatively stable atstable around Inquestions other in in theeach IRF (multiple) sequence, IRF (multiple) stayed at 2. around 2. Inwords, other words, in the IRF (multiple) sequence. Across the to lesson average number of follow-up recorded inepisode each Anwas IRF (multiple) sequence stayed relatively stable atfollow-up around 2.questions Inevery other initiation words, in question. the IRF (multiple) sequence, teacher CHN1 tended to the ask two afterquestions every initiation question. teacher CHN1 tended ask sequence, two follow-up questions after An IRFepisode (multiple) stayed stable at around 2. In other words, in thequestion. IRF (multiple) sequence, CHN1 tended torelatively ask twothe follow-up questions after every initiation An wassequence selected to show how teacher asked two follow-up questions after an initiation selected toteacher show how the teacher asked two follow-up questions after an initiation question. sequence, teacher CHN1 tended to ask two follow-up questions after every initiation episode was selected to show how the teacher asked two follow-up questions after anquestion. initiationAn question. episode selected to show how the teacher asked two follow-up questions after an initiation Episode 1: was CHN1-Lesson 6, 0:02:49-0:04:04 question. Episode 1: CHN1-Lesson 6, 0:02:49-0:04:04 question. Episode 1: CHN1-Lesson 6, 0:02:49-0:04:04 T: Episode For T:solving rightright there two depending cases depending on what For solving triangles, there are are two cases on what are given. What are given. 1: CHN1-Lesson 6,triangles, 0:02:49-0:04:04 T: For solving right triangles, there are two cases depending on what are given. What are thetwo twocases? cases? Initiation question What are the T: are For solving rightInitiation triangles, question there are two cases depending on what are given. What the[silent] two cases? Ss: Ss: Ss:[silent] are the two cases? Initiation question [silent] T: Given…? Follow-up question Ss: [silent] Given…? question Ss: . .Two side lengths. T: T:Given . ? Follow-up Given…? Follow-up question Ss:T: Two side lengths. T: Given two side lengths and…? Follow-up question Ss: Two side lengths. TwoGiven sideside lengths. T: Ss:Given two and…? Ss: onelengths side length andFollow-up one angle question size. T: Given Given two side lengths and…? Follow-up question Ss: one side length and one angle size. T: two Given side length T: Given sideone lengths and .and . . ?one angle size. Given one side length and one angle size. T: Ss:Given one side length and one angle size. Ss: GivenInone length one led angle thisside episode, theand teacher the size. whole T: Given one side length and one angle size. class to recall the cases of problems in solving right In this episode, the teacher led the whole class recall the of class problems in to solving right triangles. teacher had initiated firsttoquestion, thecases whole seemed be a bit reluctant T: Given oneAfter sidethe length and one angle the size. In this episode, the teacher led the whole class to recall the cases of problems in solving Sustainability 2018, 10, x FOR PEER REVIEW 10 of 18 triangles. After the teacher had initiated the first question, the whole class seemed to be a bit reluctant to respond. Therefore, the teacher repeated the question and the choral response was elicitedright but the triangles. After the teacher had initiated the first question, the whole class seemed to be a bit reluctant to respond. the question and choral elicited but the classright Inresponses this Therefore, episode, the teacher teacherrepeated led thethe whole class to the recall cases ofwas problems solving were not complete. Then teacher requested forthe a response supplement from thein whole to respond. Therefore, the CHN2: teacher repeated question and the choral was but 4.2.2. The Case of responses were notTeacher complete. Then the teacher requested for the a supplement the elicited whole whose responses achieved the teacher’s eventual expectations. triangles. After the teacher had initiated thethe first question, wholeresponse classfrom seemed to be aclass bitthe reluctant responses were not complete. Then the teacher requested for a supplement from the whole class whose responses achieved the teacher’s eventual expectations. to respond. Therefore, thebeteacher repeated theproportion question and the choral response wasquestions elicited but In Figure 4a, it can seen that that the occupied by the initiation in the the whose responses achieved the teacher’s eventual expectations.

responses weresequences, not complete. Then thetoteacher requested a supplement from wholestayed class whose IRF (multiple) in relation the total numberfor of the questions in onethe lesson, stable responses achieved the teacher’s eventual expectations. at around 20% across all the lessons.

In addition, for all the lessons except lesson 3, teacher CHN2 asked about two follow-up 4.2.2. The Case of Teacher CHN2 questions on average in each IRF (multiple) sequence (as is shown in Figure 4b). In lesson 3, an In Figure it can be seen that the the initiation questions in the IRF average of 3.44a,questions was that recorded inproportion each IRF occupied (multiple)bysequence. Given that the fewest (multiple) sequences, in relation to the total number of the questions in one lesson, stayed stable at questions were observed in lesson 3 (as is shown in Figure 2), it can be inferred that the teacher CHN2 around 20% across all the lessons. might ask fewer questions in lesson 3 but engaged students in longer IRF (multiple) sequences.

(a) Three types of questions

(b) The average number of follow-up questions in each IRF (multiple) sequence

Figure 4.4.The Thedistribution distribution of questions inclass the of class of teacher CHN2. Note: Qs-QA&IRF-s = all Figure of questions in the teacher CHN2. Note: Qs-QA&IRF-s = all questions questions in Q&A pairs and sequences; IRF (single) sequences; IRF (multiple) in Q&A pairs and IRF (single) Qs-IRF-m = allQs-IRF-m questions =inall IRFquestions (multiple)insequences; FQ = sequences;questions; FQ = follow-up FQs in each IRFaverage (multiple) = theof average number of follow-up follow-up FQs inquestions; each IRF (multiple) = the number follow-up questions in each questions in each IRF (multiple) sequence. IRF (multiple) sequence.

addition, all the lessons except lesson 3, teacher CHN2 askedQuestions about two follow-up questions 4.3. AInMixture of for Variations and Consistencies in the Functions of Teacher on average in each IRF (multiple) sequence (as is shown in Figure 4b). In lesson 3, an average of In each participating teacher’s class, the question types recorded in the Q&A question pairs and 3.4 questions was recorded in each IRF (multiple) sequence. Given that the fewest questions were the IRF (single) sequences changed from one lesson to another and a similar change was identified regarding the questions types in the initiation questions of the IRF (multiple) sequences. In contrast, the question types documented in the follow-up questions of the IRF (multiple) sequences were relatively consistent across the lesson sequence.


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observed in lesson 3 (as is shown in Figure 2), it can be inferred that the teacher CHN2 might ask fewer questions in lesson 3 but engaged students in longer IRF (multiple) sequences. 4.3. A Mixture of Variations and Consistencies in the Functions of Teacher Questions In each participating teacher’s class, the question types recorded in the Q&A question pairs and the IRF (single) sequences changed from one lesson to another and a similar change was identified regarding the questions types in the initiation questions of the IRF (multiple) sequences. In contrast, the question types documented in the follow-up questions of the IRF (multiple) sequences were relatively consistent across the lesson sequence. 4.3.1. The Case of Teacher CHN1 In Figure 5, the left graph shows the coding of questions asked in the Q&A question pairs and the IRF (single) sequences, with the middle and the right graphs separately for the coding of initiation questions in the IRF (multiple) sequences, and the coding of follow-up questions in the IRF (multiple) Sustainability 10, xrefer FOR PEER REVIEW 11 of 18 sequences.2018, Please to Table 3 for the explanation of three-letter codes in the graphs.

Figure 5. The categorization of questions asked by the teacher CHN1 over the lesson sequence. Figure 5. The categorization of questions of asked by theasked teacher over and the lesson sequence. Note: Note: Qs-QA&IRF-s = the categorization questions inCHN1 Q&A pairs IRF (single) sequences; Qs-QA&IRF-s = the categorization of questions asked in Q&A pairs and IRF (single) sequences; IQ-IRF-m = the categorization of initiation questions asked in IRF (multiple) sequences; FQ = IQthe IRF-m = the categorization of initiation questions asked insequences. IRF (multiple) sequences; FQ = the categorization of follow-up questions asked in IRF (multiple) categorization of follow-up questions asked in IRF (multiple) sequences.

As is shown in Figure 5, the employment of initiation questions varied greatly in the constitution of subcategories and these5,categories’ proportions acrossquestions the consecutive lessons. example, in As is shown in Figure the employment of initiation varied greatly in For the constitution lessons 1, 2, and 6, thethese Review questionsproportions were observed to take significantlessons. proportions (about 25%) of subcategories and categories’ across the up consecutive For example, in of all the in questions Q&A question and IRF (single) sequencesproportions (shown in the left graph lessons 1,teacher 2, and 6,questions the Review were pairs observed to take up significant (about 25%) of all Figure 8), butquestions in the lessons 4 and 5 this pairs question was notsequences observed(shown at all ininQ&A question of the teacher in Q&A question and type IRF (single) the left graph pairs and 8), IRFbut (single) Similarly, is shown in was the middle graph of 5, thequestion Build-up of Figure in thesequences. lessons 4 and 5 this as question type not observed at Figure all in Q&A questions occupied more than 30% of all theas initiation questions in IRFgraph (multiple) sequences lesson pairs and IRF (single) sequences. Similarly, is shown in the middle of Figure 5, the in Build-up 4, while this question type was not observed in IRF (multiple) sequences in lesson 1. questions occupied more than 30% of all the initiation questions in IRF (multiple) sequences in lesson By contrast, the employment the follow-up types was relatively consistent across the 4, while this question type was notofobserved in IRFquestion (multiple) sequences in lesson 1. consecutive lessons, regardless of of where the lessonquestion is located in the teaching sequence. In all lessons By contrast, the employment the follow-up types was relatively consistent across the taught by thelessons, teacherregardless CHN1, theof follow-up questions of mainly Probing questionsInand Inviting consecutive where the lesson isconsisted located in the teaching sequence. all lessons comments other words, during extended in the questions IRF (multiple) taught by questions. the teacherInCHN1, the follow-up questions questioning consisted ofexchanges mainly Probing and sequences, the participating ask a particular of follow-up question types to Inviting comments questions.teachers In othertended words,to during extendedgroup questioning exchanges in the IRF (multiple) sequences, the participating teachers tended to ask a particular group of frequently follow-up build on students’ responses. Although each teacher employed some question types more question types to whole build on students’ofresponses. Although teacher some question types than others, the sequence consecutive lessonseach shared theseemployed frequently used questions in more frequently than others, the whole of consecutive lessons shared these frequently common. This might suggest that eachsequence teacher might employ follow-up questions habitually used with questions in common. This might suggest each employ sequence. follow-upAn questions certain strategies, less dependent on where thethat lessons areteacher locatedmight in the teaching episode habitually with certain strategies, less dependent on where the lessons are located in the teaching sequence. An episode is provided in the following paragraphs to demonstrate the regular use of some particular follow-up questions types. In Episode 2, the teacher was leading the whole class to analyze how to solve the second part of the task (as is shown in Figure 6), namely to find the tangent of angle EDC, which was not in any


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is provided in the following paragraphs to demonstrate the regular use of some particular follow-up questions types. In Episode 2, the teacher was leading the whole class to analyze how to solve the second part of the task (as is shown in Figure 6), namely to find the tangent of angle EDC, which was not in any right triangle in the given diagram. Before this example, the ideas of constructing and converting had been introduced as two main directions for solving similar questions involving non-right (or oblique) triangles. Based on these two ideas, here the teacher asked the students to think about the possible Sustainability 2018, 10,did x FOR REVIEW 12 of 18 strategies. She notPEER select any individual student or ask volunteers to answer the questions, but posed the questions to the whole class and elicited the choral responses. In the follow-up phase of the conversation, the teacher consistently employed two types of questions, namely Probing and Inviting Episode 2: CHN1-Lesson 2, 0:13:19-0:14:35 comments questions. Sustainability 2018, 10, x FOR PEER REVIEW Sustainability 2018, 10, x FOR PEER REVIEW Sustainability 2018, 10, x FOR PEER REVIEW Episode 2:10, CHN1-Lesson 2, 0:13:19-0:14:35 Sustainability 2018, x FOR PEER REVIEW

Episode 2: CHN1-Lesson 2, 0:13:19-0:14:35 Episode 2: CHN1-Lesson 2, 0:13:19-0:14:35 Sustainability 2018, 10, x FOR PEER REVIEW Episode 2: CHN1-Lesson 2, 0:13:19-0:14:35

12 of 18 12 of 18 12 of 18 12 of 18 12 of 18

Episode 2: CHN1-Lesson 2, 0:13:19-0:14:35

Figure 6. 6. Teacher CHN1’s PowerPoint slide used forfor example 22018, in lesson 2. PEER REVIEW Figure Teacher CHN1’s PowerPoint slide used example 2 in lesson Sustainability 10, x FOR 2.

T: So 2: what should you do? There are two options. One is what? Episode CHN1-Lesson 2, 0:13:19-0:14:35 Episode 2: lesson CHN1-Lesson 2, 0:13:19-0:14:35 2 in Figure 6. Teacher CHN1’s PowerPoint slide used for example IQ: Open-up (Strategy/procedure) Figure 6. Teacher CHN1’s PowerPoint slide used for example 2 in lesson 2. Constructing. T:Ss: So whatSoshould you do? There are twotwo options. One what should ou do? There are Oneisis what? what? 2. T: Figure 6. options. Teacher CHN1’s PowerPoint slide used for example 2 in lesson 2. T: Constructing. If you like the constructing way, willforyou have 2toinconstruct Figure 6. Teacher CHN1’s PowerPoint slide used example lesson 2. a perpendicular T: So what you (Strategy/procedure) do? There are two options. One is what? IQ:should Open-up line? FQ: ProbingT:(Clarification) So what should you do? There are two options. One is what? IQ: Open-up Ss: (Strategy/procedure) Constructing. T: So what should you6.do? ThereCHN1’s are two options. Oneslide is what? Ss: Constructing. Figure Teacher PowerPoint used for example 2 in lesson 2. Ss: Yes. (Strategy/procedure) Ss: Constructing. T: Constructing. IfIQ: youOpen-up like the constructing way, will you have to construct a perpendicular IQ: Open-up (Strategy/procedure) T: Constructing. If you like the constructing will ayou have toif construct perpendicular line? T： Then we can put it [the angle EDC] into a right triangle. What you don’tawant to Constructing. T: Constructino. the Ss: constructing way, will you have to way, construct perpendicular line?Iou FQ:like Probing (Clarification) So what should you do? There are two options. One is what? Ss: T:Constructing. construct perpendicular line? Iou What could you do? Inviting comments T: Constructing. like the constructing way,FQ: will you have to construct a perpendicular line? Ss: FQ: Probing (Clarification) Yes. the Open-up (Strategy/procedure) T: IQ: Constructing. If you like the constructing way, will you have to construct a perpendicular (Clarification) Probing Ss: Yes. T： Then we can put it line? [the FQ: angle EDC] into a right triangle. What if you don’t want to (Supplement) Ss: Ss:Yes. Constructing. line? FQ: Probing (Clarification) Ss: we can put it the [the perpendicular angleYes. EDC] into a right Whatdo? if you want tocomments T： Then construct line? Whattriangle. could you FQ:don’t Inviting Ss: Yes. Converting. Constructing. you like put the constructing way, willangle you have to construct into a perpendicular Ss: T:Then T:construct we Ifcan itcould [the EDC] a What right triangle. angle EDC] into a right triangle. io oou don’t want T： Then we can put it [the the perpendicular line? What you do? FQ: Inviting comments (Supplement) T: Then Converting. Weitcan also convert it toaan equivalent, right? Do we have thetoequivalent line? (Clarification) T： weFQ: canProbing put [the angle EDC] into right triangle. What if you don’t want Figure 6. Teacher CHN1’s PowerPoint slide used fo to construct the perpendicular line? What could you do? FQ: Inviting comments (Supplement) Ss: Converting. if angle you want to construct thediagram? perpendicular line? could comments you do? of Yes. that [angle EDC] here the FQ:do? Probing (Clarification) Ss:What construct thedon’t perpendicular line?in What could you FQ:What Inviting Ss: Converting. T: Converting. We can(Supplement) also convert it to an equivalent, right? Do we have the equivalent T:if you So what you do? There are two options. One is what? Then C. we can put it [the angle EDC] into a right triangle. What don’tshould want to Ss:T： Angle (Supplement) Ss: Converting. T: Converting. We angle can also convert ithere to an Do we have the equivalent of that [angle EDC] in equivalent, the diagram?right? FQ: Probing (Clarification) Open-up (Strategy/procedure) construct the perpendicular line? Why? What could you do? FQ:ofIQ: Inviting comments Ss: T: Converting. Good. Is angle C T: the equivalent? Is the side length DE equal to that of EC? can also convert it to an equivalent, right? Do we have the equivalent of thatConverting. angle [angleC.EDC] here inConverting. the diagram?We FQ: Probing (Clarification) Ss: Ss: Angle Ss: Constructing. (Supplement) T: Converting. We is can also convert it to an equivalent, right? Do we have the equivalent the reason? FQ: Probing ofWe that angle [angle EDC] here in length the diagram? FQ: Probing Ss: T:Angle C. Good.What T:Reason? Is angle C the equivalent? Why? Is(Justification) the side of equal to that(Clarification) of EC? also convert it DE an equivalent, right? way, will you have T: toConstructino. Iou like the constructing Converting. Ss:Converting. ofThe that angle [angle EDC] here in thecan diagram? FQ:triangle Probing (Clarification) Ss: median on the hypotenuse of a right equals Angle C. T: Good. Is angle C the equivalent? Why? the side (Justification) length of DE equalone-half to that of the EC?hypotenuse. Reason? What isSs: the reason? FQ:IsProbing line? FQ: Probing (Clarification) Converting. We can alsoofconvert it to[angle an equivalent, right? Dodiagram? we have the equivalent Ss: Angle C. we have thereason? equivalent that angle EDC] intriangle the T: T:Do Good. The median onProbing the hypotenuse of a here right equals one-half the to that of EC? Good. Is angle the equivalent? Why? Is the side length of DE equal Reason? What is the FQ: (Justification) Ss: The median onT:the hypotenuse of aCright triangle equals one-half the hypotenuse. Ss: to Yes. of that angle [angle EDC] here inWhy? the diagram? FQ:length Probing T: Good. Is angle C the equivalent? Is the side of (Clarification) DE equal that of EC? Reason? What is the reason? FQ: Probing (Justification) hypotenuse. In this way, we can find it [tan∠DEC] by converting it to tanC. Ss: Ss: The median on the hypotenuse of a right triangle equals one-half the hypotenuse. T: Good. The median on the hypotenuse of a right triangle equals one-half the Angle C.

T： Then we can put it [the angle EDC] into a right triang Ss:Reason? AngleWhat C. is the reason? FQ: Probing (Justification) The median the hypotenuse ofequals aconverting right one-half triangle equals one-half hypotenuse. T: T:Good. The median onInSs: the of a itWhy? right triangle the hypotenuse. this hypotenuse way, we canon find [tan∠DEC] by it toof tanC. Is angle C equivalent? Islenoth the side DEofthe equal tothethat ofline? EC?What could you do construct perpendicular Is angle thethe equivalent? Who?triangle Is the side of DElength equal to that EC? Ss: T:Good. TheGood. median on theChypotenuse of a right equals one-half the hypotenuse. T: Good. The median on the hypotenuse of a right triangle equals one-half the hypotenuse. In this way, we can find it [tan∠DEC] by converting it to tanC. 4.3.2. The Case of Teacher CHN2 Reason? What theon reason? FQ: Probingof (Justification) T:4.3.2. Good. The median the hypotenuse a right triangle equals (Supplement) one-half the The Case of Teacher CHN2 Reason? What isisthe reason? hypotenuse. In this way, we can find it [tan∠DEC] by converting it to tanC. hypotenuse. In this way, we can find it [tan∠DEC] by converting it to tanC. The median on the hypotenuse of a right triangle equals one-half the hypotenuse. Ss: Ss: Converting. As is shown in Figure 7, the employment of initiation questions varied greatly in the constitution 4.3.2. The of median Teacher CHN2 Ss:Case The on the hypotenuse of a right triangle questions equals one-half the hypotenuse. As is shown inmedian Figure 7,onthethe employment of initiation varied greatly in the T: Good. The hypotenuse of a right triangle equals one-half theconstitution T: Converting. We can example, also convert of subcategories and 4.3.2. theseThe categories’ proportions For init to an equivalent, rig Case of Teacher CHN2 across the consecutive lessons. T:is of Good. The median on thewe hypotenuse ofquestions a right triangle equalsitinone-half the hypotenuse. thisdiagram? FQ: Probin As shown in Figure 7, the employment of initiation varied greatly the constitution subcategories and these categories’ proportions across the consecutive lessons. For example, in In 4.3.2. The Case of Teacher CHN2 hypotenuse. In this way, can find it [tan∠DEC] by converting to tanC. of that angle [angle EDC] here in the lesson 5, the Result questions were observed to7,take up significant proportions (about 30%) of allinthe As is shown in Figure the employment of initiation questions varied greatly the constitution of subcategories categories’ proportions across theup consecutive For example, in of all the lesson thethese Result questions observed to take proportions (about 30%) way,5,and we can find it [tan∠were DEC] by converting it significant to tanC. lessons. Angle C.the As isquestions shown in in Figure 7, question the employment of initiation questions variedSs: greatly ininthe constitution teacher Q&A pairs and IRF (single) sequences (shown left graph of Figure of subcategories and these categories’ proportions across the consecutive lessons. For example, in lesson 5,4.3.2. the Result questions observed take upIRF significant (about of all the teacher inwere Q&ACHN2 question to pairs and (single) proportions sequences (shown in30%) theIsleft graph of Figure Thequestions Case of Teacher T: Good. angle C the Why? Is the side leng of subcategories and these6 categories’ proportions across the consecutive lessons. For example, inequivalent? 7), but in lessons 4 and this question type was not observed at all in Q&A question pairs and IRF lesson 5, the Result were observed up significant proportions (about 30%) of all the Thein Case ofquestion Teacher CHN2 teacher4.3.2. questions in Q&A and IRF questions (single) sequences (shown the left graph of Figure 7), but lessons 4 and 6pairs this question type was not observed attoin alltake in Q&A question pairs IRF Reason? What is all the and reason? FQ: Probing (Justification) lesson 5,As the were observed to take up middle significant proportions (about 30%) of the is Result shownquestions inSimilarly, Figure 7, the employment ofquestion initiation questions varied greatly the constitution (single) sequences. as shown in the graph Figure 7, in the teacher questions innot Q&A and IRF (single) (shown inquestions the left graph of Figure 7), but in lessons 4sequences. and 6 this question type observed atpairs allgraph in Q&A question pairs andReflection IRF questions (single) Similarly, asisiswas shown in the middle of of Figure 7,sequences the Reflection Ss: The median on the hypotenuse of a right triangle equa teacher questions in Q&A question pairs and IRF (single) sequences (shown in the left graph of Figure As is shown in Figure 7, the employment of initiation questions varied greatly in the constitution of subcategories and 7), these categories’ across the consecutive lessons. For example, in but inalllessons 4proportions and 6questions this question type was notReflection observed at lessons allin inlessons Q&A question occupied more than 20% ofof all the initiation questions IRF (multiple) sequences 4the (single) sequences. Similarly, as is shown the middle graph ofinIRF Figure 7, the questions occupied more than 20% thein initiation in (multiple) sequences in 4 and 5,and 5,pairs and IRF T: Good. The median on hypotenuse of a right t 7),of but in lessons 4 and 6 this question type was not observed at all in Q&A question pairs and IRF lesson 5, the Resultand questions were observed to takeas upissignificant proportions (about 30%) of all the subcategories these categories’ proportions across the consecutive in questions sequences. Similarly, shown in the middle graph Figure the Reflection occupied more 20% oftype all(single) the initiation questions IRF(multiple) (multiple) sequences 4 of and 5, For7,example, while this question type wasnot notobserved observed sequences ininlesson 1.lessons. while thisthan question was ininIRF IRF (multiple) sequences inlessons lesson 1. hypotenuse. In this way, we can find it [tan∠DEC] by c (single) sequences. Similarly, as is shown in the middle graph of Figure 7, the Reflection questions teacher questions in Q&A question pairs and IRF sequences (shown in the left graph of Figure occupied more 20% of (single) all sequences the initiation questions while this question type was not observed inthan IRF (multiple) in lesson 1. in IRF (multiple) sequences in lessons 4 and 5, occupied than 20% of6all the initiation questions in IRF (multiple) sequences in lessons 4 and 5,IRF 7), but more in lessons 4 and this in (multiple) Q&A question pairs and while thisquestion questiontype typewas wasnot notobserved observedatinall IRF sequences in lesson 1. 4.3.2. The Case of Teacher CHN2 while this question type was not observed in IRF (multiple) in lesson 1. (single) sequences. Similarly, as is shown in the middle sequences graph of Figure 7, the Reflection questions occupied more than 20% of all the initiation questions in IRF (multiple) sequences in in lessons and As is shown Figure4 7, the5,employment of initiation quest while this question type was not observed in IRF (multiple) sequences in lesson 1. and these categories’ proportions across the of subcategories


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lesson 5, the Result questions were observed to take up significant proportions (about 30%) of all the teacher questions in Q&A question pairs and IRF (single) sequences (shown in the left graph of Figure 7), but in lessons 4 and 6 this question type was not observed at all in Q&A question pairs and IRF (single) sequences. Similarly, as is shown in the middle graph of Figure 7, the Reflection questions Sustainabilitymore 2018, 10, x FOR REVIEW of 18 occupied than 20%PEER of all the initiation questions in IRF (multiple) sequences in lessons 413and 5, while this question type was not observed in IRF (multiple) sequences in lesson 1.

Figure 7. The categorization of questions asked by the teacher CHN2 over the lesson sequence. FigureQs-QA&IRF-s 7. The categorization of questions of asked by theasked teacher overand the lesson sequence. Note: Note: = the categorization questions in CHN2 Q&A pairs IRF (single) sequences; Qs-QA&IRF-s the categorization of questions asked in Q&Ainpairs and IRF (single) sequences; IQIQ-IRF-m = the= categorization of initiation questions asked IRF (multiple) sequences; FQ = the IRF-m = the categorization of initiation questions asked in IRF (multiple) sequences; FQ = the categorization of follow-up questions asked in IRF (multiple) sequences. categorization of follow-up questions asked in IRF (multiple) sequences.

By contrast, the employment of the follow-up question types was relatively consistent across the By contrast, the regardless employment the follow-up types was relatively consistent across the consecutive lessons, of of where the lessonquestion is located in the teaching sequence. In all lessons consecutive where the lesson isconsisted located in teaching sequence. In all lessons taught by thelessons, teacher regardless CHN2, theof follow-up questions of the mainly Redirecting questions, probing taught by the teacher CHN2, the follow-up questions consisted of mainly Redirecting questions, questions, and Inviting comments questions. An episode is provided in the following paragraphs to probing questions, and use Inviting comments questions. An episodetypes. is provided in the following demonstrate the regular of some particular follow-up questions paragraphs to demonstrate the regular usestudent of sometoparticular questions In this episode, the teacher asked the recall the follow-up conclusions that hadtypes. been achieved in In this episode, the teacher asked the student to recall the conclusions that had been in achieved in the last few lessons and to put these conclusions together in the task sheet (as is shown Figure 8). the last few lessons and to put conclusions together in the task sheet translation. (as is shown The in Figure 8). The student (Li) answered thethese question from the perspective of graph teacher The student (Li) answered the question from the perspective of graph translation. The teacher interrupted this student’s talk and redirected the student to the right track. On the right track, interrupted this student’s talk and of redirected student to the On the right the the student described the direction opening the independently andright thentrack. the vertex with the track, teacher’s student described the direction of aopening independently and for then vertex with theparabolas’ teacher’s assistance. Then the teacher asked further question to request thethe description of the assistance. Then the teacher asked a further question to request for the description of the parabolas’ vertices by looking at the coordinates. Although the student failed to give answers to the teacher’s last vertices by themade coordinates. Although the student to give answers of tomathematical the teacher’s question, thelooking teacherathad a lot of efforts in facilitating thisfailed student’s articulation last question, made a lot of efforts in this student’s articulation of knowledge. In the this teacher episode,had all the follow-up questions fellfacilitating into three categories, namely Redirecting mathematical knowledge. In this episode, all the follow-up questions fell into three categories, questions, Probing questions, and Inviting comments questions. namely Redirecting questions, Probing questions, and Inviting comments questions. Episode 3: CHN2-Lesson 5, 0:05:06-0:06:31

Figure 8. The mind map used in teacher CHN2’s lesson 5.

vertices by looking at the coordinates. Although the student failed to give answers to the teacher’s last question, the teacher had made a lot of efforts in facilitating this student’s articulation of mathematical knowledge. In this episode, all the follow-up questions fell into three categories, namely Redirecting questions, Probing questions, and Inviting comments questions. Sustainability 11, 139 Episode 3: 2019, CHN2-Lesson 5, 0:05:06-0:06:31

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Figure 8. The mind map used in teacher CHN2’s lesson 5.

T: The question is, for5,these three special types of parabolas, if we look at the direction Episode 3: CHN2-Lesson 0:05:06-0:06:31 of opening, the vertex, and the axis of symmetry, and then think about the effects Sustainability 2018, 10, x FOR PEER REVIEW

T:

Sustainability 2018, 10, x FOR PEER REVIEW14 of 18

The question is, for these three special types of parabolas, if we look at the direction of opening, Sustainability 2018, 10, x FOR PEER REVIEW 14 of 18 thethat vertex, and theofaxis symmetry, think What about the effects that values of a,on h, the andparabolas. What are thatare thethe values of a,the h, and k have the values a, h,ofand k have onand thethen parabolas. corresponding

Sustainability 2018, 10, x FOR PEER REVIEW 14 of 18 relationships? Li? IQ: Review relationships? Li?k IQ: Review that the kvalues of h, parabolas. and have on the are parabolas. What are therelationships? corresponding have ona, the What the corresponding Li? Sustainability 2018, 10, x FOR PEER REVIEW 14 of 18 Sustainability 2018, 10, x FOR PEER REVIEW S: have When is greatery than to the left, we will have th S: When hIQ: is greater than 0, shift h units to the left, we will thehparabola = a(x0, − shift h units relationships? Li? Review that the values of a, h, and k have on the parabolas. What are the corresponding S: When h is greater than 0, shift h units to the left, we willh)have the parabola y = a(x − h)2 . 2. h)2. S: When h is greater than 0, shift h units to the left, we will have the parabola y = a(x − that the valuesLi? ofIQ: a, h, and k have on the What the corresponding relationships? Review thatparabolas. the values of a,are h, and k have on the parabolas. What are the correspondin T: What question did I ask you? FQ: Redirecting (Refocusing) What question did I ask you? FQ: Redirecting (Refocusing) h)2T: . S:T:What relationships? Li? IQ: Review question did I ask you? When h is greater than 0, shift h units to the left, we will Li? haveIQ: theReview parabola y = a(x − relationships? S: [silent] S: [silent] T: What question ask you? than FQ: Redirecting (Refocusing) 2. didhIis S:[silent] When greater 0, shift h units left,h we will have the0,parabola y = a(x − left, we will have the parabola y = a(x h) S: to the When is greater than shift h is, units the S: T: T: the What asked thetothe direction of opening, vertex, and the axi What I asked is, the direction of opening, vertex, and axisIof symmetry, S: [silent]T: What 2 h)2. question did I ask you? FQ: Redirecting h) vertex, . (Refocusing) T: What I asked is, the direction of opening, and the axis of symmetry, the relationship relationship between them and the values of a, h, and k. FQ: Re relationship between them and the values of a, h, and k. FQ: Redirecting (Cueing) T: What S: IT:asked is, question the direction vertex, and the axis oo sommetro, the What did I of askopening, you? FQ: Redirecting [silent] T: What (Refocusing) question did I ask you? FQ: Redirecting (Refocusing) S: The open directions [of the three parabolas] are all related to the S: The open directions [of the three parabolas] are all related to the value of a. relationship between them values h,and and k. k.vertex, FQ: Redirecting (Cueing) [silent] T:S:between What Ithem asked is,and thethe direction of opening, and the axis of symmetry, the and the values ofof a,a, h,S: [silent] T: Yes, good. T: Yes, good. S: The open [of between theis, three parabolas] are all related to theand value ofRedirecting a. of symmetry, T: directions What I asked thethem directionthe of opening, vertex, thethe axis the vertex, and the axis of symmetry, t relationship values of a, h, and k. FQ: (Cueing) T: What I asked ofof opening, S: good. open directions [of the and three parabolas] are all related todirection the value a. S: is, Then their…hum… S:TheThen their…hum… T: Yes, relationship between and the values of a, all h, and k. FQ: Redirecting S: The open directions [ofthem the three parabolas] are related to the value of the a.(Cueing) relationship between them and values of a,Redirecting h, and k. FQ: Redirecting (Cueing T: What about the vertex? FQ: (Cueing) T: their…hum… good. T:Yes,What about the vertex? FQ: Redirecting (Cueing) S: Then Thegood. open directions [of the three parabolas] all directions related to [of the the value of a. T:S: Yes, S: Theare open three parabolas] are all related to the value of a. S: vertices The vertex, their vertices S:Then The vertex, to the value of h. The are shifted h unitsare related to the value of h. The vertice S: S: their . . .their hum . . . are related T: What about the vertex? FQ: vertices Redirecting (Cueing) T: Then Yes, good. their…hum… T: Yes, good. to the left or right. to the left or right. S: The vertex, their arevertex? relatedFQ: to the valueS:of h.(Cueing) The vertices are shifted h units Thenvertices their…hum… T:S:What What about the Redirecting Then their…hum… T: about the vertex? T: Does it have to be left or right? FQ: Probing (Clarification) T: orDoes it have to be left or right? FQ: Probing (Clarification) to the S: left right. T: The What about thevertices vertex? are FQ:related Redirecting (Cueing) vertex, their to value about of h. The h units(Cueing) T:theWhat thevertices vertex? are FQ:shifted Redirecting Or, vertices upwardsare or downwards. S: itS:have TheOr, vertex, their vertices are related to the value of S:h. The shifted h units to the left upwards or downwards. T: Does to be left or right? FQ: Probing (Clarification) S: toThe vertex, their vertices are related to the value of h.their The vertices are shifted htounits the left or right. S: The vertex, vertices are related the value of h.That The vertices are shifted uni the vertex, it has is an ordered pair.hWh T:or right. So,orthe vertex, it has coordinates. That is an orderedT:pair.So, What you have saidcoordinates. is S: Or, upwards downwards. to the left or right. T: Does it have to be left or right? FQ: Probing (Clarification) to the left or right. Now, what if we look at its coordinate about its coordinates. position. Now, what ifordered we look at What its coordinates? FQ: T: So, theS:T:vertex, it it has That is an pair. youabout have its saidposition. is Inviting Does haveor todownwards. be left or right? FQ: Probing Or, upwards T: Does(Clarification) it have to be left or right? FQ: Probing (Clarification) T: Does it have to be left or right? comments (Supplement) comments (Supplement) about T:S: its position. Now, if we look That at itsis coordinates? FQ: Inviting Or,the upwards oritwhat downwards. So, vertex, has coordinates. an upwards ordered pair. What you have said is S: Or, or downwards. S: Y-coordinate. Y-coordinate. S: S: Or, upwards or downwards. comments (Supplement) T: about So, theitsvertex, it has coordinates. is pair. What youFQ: have saidisisan ordered pair. What you have said position. Now, what if That we atordered its coordinates? Inviting T: look So,anthe vertex, it has coordinates. That T: Okay. Please be seated. Let’s have look at this. Okay. Please seated. Let’s have look at this. S: Y-coordinate. T: T:So,comments the vertex, itbehas coordinates. That islook an ordered pair. What you have said about its aposition. about its (Supplement) position. Now, what ifa we atitsits coordinates? FQ: Inviting io we lookisat its coordinates? about oosition. Now, what be seated. Let’s have a look at this. T: Okay.S:Please Incomments summary, thewere initiation questions commentsthe (Supplement) In Y-coordinate. summary, initiation questions asked in the (multiple) sequences reported to takeasked in the IRF (multiple) seq Inviting (Supplement) Now, what if we look at its coordinates? FQ:IRF up a regular proportion of the total number up a regular proportion of the total number of questions in each lesson of the whole unit. Here it was of questions in each lesson o S: Y-coordinate. T: the Okay. Pleasequestions be seated.asked Let’s in have look at this. sequences were reported to take S: (multiple) Y-coordinate. In summary, initiation the aIRF S:found Y-coordinate. found that the breakdown of the initiation questions in IRF (multip that the breakdown of the initiation questions in IRF (multiple) sequences demonstrated T: Okay. Please be seated. Let’s have a look at this. T:in each Okay. Please be seated. Let’sHere haveita was look at this. up a regular proportion of thethe total number of questions of the whole unit. In summary, initiation questions asked in thelesson IRF (multiple) sequences were reported to takeunit. In other words, the teach various characteristics across the whole T: Okay. Please be seated. Let’s have a look at this. various characteristics across the whole unit. In other words, the teachers tended to use a regular found thatup the breakdown ofthethe initiation questions IRF sequences In summary, initiation questions in the(multiple) IRF sequences were reported to take a regular proportion of the total numberasked ofin questions in (multiple) each lesson ofquestions thedemonstrated whole unit. Here it was In summary, the initiation asked in the IRF (multiple) sequences were repo proportion all the questions in each lesson as initiation questions to st proportion of all the questions in each lessonwords, as initiation questions to of start IRF (multiple) sequences, various characteristics across the whole unit. In other the teachers tended to use a regular up aIn regular proportion of the total number of questions in each lesson of the whole unit. Here it was found that the breakdown of the initiation questions in IRF (multiple) sequences demonstrated up a regular proportion of the total number of questions in each lesson of the whole unit. summary, the initiation questions asked in the IRF (multiple) sequences were reported to take but the purposes of starting IRF (multiple) sequences were different, whi but the purposes of starting IRF (multiple) sequences were different, which might reflect the teachers’ proportionvarious of all the questions in each lesson as initiation initiation to start IRF (multiple) sequences, found that the breakdown of questions in IRFthe (multiple) demonstrated characteristics across thethe whole unit. In other teachers tended towhole use a unit. regular foundquestions that thewords, breakdown of thesequences initiation questions in IRFit(multiple) sequences dem up a regular proportion of the total number of questions in each lesson of the Here was stable and flexible strategies in using IRF (multiple) sequences. stable and flexible strategies in using IRF (multiple) sequences. but the purposes of starting IRF questions (multiple) sequences were different, which might reflect the teachers’ various characteristics across in the whole unit. other words, the teachers tended to use a regular proportion of all the each lesson as In initiation questions to start (multiple) various characteristics across theIRF whole unit. Insequences, other words, the teachers tended to us that the breakdown of (multiple) the initiation questions in IRF (multiple) sequences demonstrated various stable andfound flexible strategies in using IRF sequences. proportion of allof the questions in each lesson as initiation questions to start IRF (multiple) sequences, but the purposes starting IRF (multiple) sequences were different, which might reflect the teachers’ proportion of all the questions in each lesson as initiation questions to start IRF (multiple) 5. Discussion 5. Discussion across the whole unit. In other words, the characteristics teachers tended use athe regular proportion but the purposes starting IRF (multiple) sequences wereof different, mightto reflect teachers’ stable and flexibleof strategies in using IRF but (multiple) sequences. the purposes starting which IRF (multiple) sequences were different, which might reflect th 5. Discussion ofstable all the questions in each lesson as initiation questions to start IRF (multiple) sequences, but the and flexible strategies in using in IRF (multiple) sequences. stable and flexible strategies in using IRF (multiple) sequences. 5.1. Adjustment of Teacher Questioning in the Sequence of Consecutive Lesson 5.1. Adjustment of Teacher Questioning the Sequence of Consecutive Lessons purposes of starting IRF (multiple) sequences were different, which might reflect the teachers’ stable 5. Discussion 5.1. Adjustment ofThis Teacher Questioning in the of Consecutive Lessons This how studythecontributes in demonstrating and exemplifying how study contributes in Sequence demonstrating and exemplifying teachers adjusted and 5. Discussion 5. Discussion and flexible strategies in using IRF (multiple) sequences. regulated their questioning strategies in different lessons of a teachin regulated their questioning strategies in different lessons of a teaching unit. It was shown in the 5.1. Adjustment of Teacher Questioning in theexemplifying Sequence of Consecutive This study contributes in demonstrating and how the Lessons teachers adjusted and previous studies (e.g., Reference [21]) that one teacher tended to change previous studies (e.g., Reference [21]) that one teacher tended to change the ways of asking questions 5.1. Adjustment of Teacher Questioning in the Sequence of Consecutive Lessons 5.1. Adjustment of Teacher in theinSequence of Consecutive Lessons Discussion regulated 5. their questioning strategies inindifferent lessons of a exemplifying teaching unit.Questioning It was shown the This study contributes demonstrating and how the teachers adjusted and students’ needs. This study r in a lesson in order to scaffold different in a lesson in order to scaffold different students’ needs. This study reports that one teacher also previous studies (e.g.,their Reference [21]) that teacher tended to change ways of asking This study contributes inone demonstrating and exemplifying how the teachers adjusted and regulated questioning strategies in different lessons ofthe a teaching unit. Itquestions was shown the This study contributes in demonstrating and in exemplifying how theThe teachers ad changed the ways of asking questions in different lessons. two parti changed the ways of askingQuestioning questions inin different lessons. The two participating teachers both asked Adjustment of Teacher the Sequence of Consecutive Lessons in a lesson5.1. in order to scaffold different students’ needs. This study reports that one teacher also regulatedstudies their questioning strategies inregulated different lessons oftoachange teaching unit. It was shown in theof a teaching unit. It was sh previous (e.g., Reference [21]) that one teacher tended the ways of asking questions their questioning strategies in different lessons a substantially different number of questions a ways substantially different number of questions inThe different lessons. In addition, the question types used in different lessons. In addi changed the of studies asking questions in different lessons. two participating teachers both asked (e.g., [21]) that one teacher tended toReference change the ways ofone asking questions inprevious a lesson in order to Reference scaffold different students’ needs. This study reports that teacher also also previous studies (e.g., [21]) teacher tended to change ways of askin This study contributes in demonstrating and exemplifying how the teachers adjusted and the in the initiation stages of IRF sequences changed from one lesson in the initiation stages of IRF sequences also changed from one lesson to another over the lesson a substantially different number of questions in different lessons. Inorder addition, the question types usedteacher in a lesson in order to scaffold different students’ needs. This study reports that one also changed the ways of asking questions in different lessons. The two participating teachers both asked in a lesson in to scaffold different students’ needs. This study reports that one t regulated their questioning strategies in different lessons of a teaching unit. It was shown in the sequence. sequence. in the initiation stages IRFof sequences also changed from oneways lesson totwo another over lessontypes theof ways asking questions in changed different lessons. The participating teachers both asked achanged substantially different number of questions in different lessons. In addition, thethe question used the of asking questions in different lessons. The two participating the unit of lessons, the pedagogical goal ofteachers one less In the unit of consecutive lessons, the pedagogical goal ofIn one lesson is consecutive related to but different sequence. ina substantially of questions inchanged differentfrom lessons. In addition, the question types used the initiationdifferent stages ofnumber IRF sequences one lesson to another over the lesson aalso substantially different number of questions in different lessons. In addition, the question from another lesson. In order to fulfill the changing pedagogic fromofanother lesson. In order to fulfill goal the of changing pedagogical goals,different the teachers can In thesequence. unit consecutive lessons, thesequences pedagogical one from lesson isofrelated to in the initiation stages of IRF one lesson to but another over the lesson inalso the changed initiation stages IRF sequences changed from one lesson to another over approximately plan howalso many approximately plan how many questions he or she can ask and determine when tocan startquestions the IRF he or she can ask and deter from another lesson. In order to fulfill the changing pedagogical goals, the teachers sequence. In the unit of consecutive lessons, the pedagogical goal of one lesson is related to but different sequence. sequences in order to direct students inthe the ways aligned with the teach sequenceshow in order to questions direct students thecan ways aligned with the teachers’ expectations to fulfill approximately hetoorin she and togoals, the IRF In the unitmany of consecutive lessons, the pedagogical goal of onewhen lesson isstart related to but different Inask the unitdetermine ofpedagogical consecutive lessons, the goal of of the onechanges lesson isinrelated to b fromplan another lesson. In order fulfill the changing pedagogical thepedagogical teachers can goals. Therefore, the evidence the numbe pedagogical goals. Therefore, the evidence of with the changes in theexpectations number of questions and in the use sequences in orderanother to direct students in order the ways thecan teachers’ fulfill from lesson. to aligned fulfill changing the teachers can pedagogical goals, the te from lesson. order goals, toto fulfill the changing approximately plan howInmany questions he the oranother she ask pedagogical andIndetermine when tothe start the IRF


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previous studies (e.g., Reference [21]) that one teacher tended to change the ways of asking questions in a lesson in order to scaffold different students’ needs. This study reports that one teacher also changed the ways of asking questions in different lessons. The two participating teachers both asked a substantially different number of questions in different lessons. In addition, the question types used in the initiation stages of IRF sequences also changed from one lesson to another over the lesson sequence. In the unit of consecutive lessons, the pedagogical goal of one lesson is related to but different from another lesson. In order to fulfill the changing pedagogical goals, the teachers can approximately plan how many questions he or she can ask and determine when to start the IRF sequences in order to direct students in the ways aligned with the teachers’ expectations to fulfill the pedagogical goals. Therefore, the evidence of the changes in the number of questions and in the use of initiation questions across the unit of consecutive lessons might reflect the teachers’ changes in his or her questioning practices in order to satisfy the different pedagogical goals of different lessons. For example, in the case of the teacher CHN1, the earlier lessons involved the investigation of the methods to solve right triangles whereas the later lessons asked students to solve mathematics tasks by applying the methods of solving right triangles. In this regard, the later lessons in the unit were less investigation-based but more practice-based than the earlier lessons. To scaffold the students’ investigations of the new methods, the teacher CHN1 asked a larger number of questions in the earlier lessons. However, she reduced the number of questions in the later lessons of the unit so that the students could have more time practicing the methods of solving right triangles. The changes of questioning practices across consecutive lessons extend our understanding of how teachers employ questioning strategies to cope with different pedagogical goals in different lessons. The findings in this study lead us to the argument that teacher questioning can be seen as the implementation of the teacher’s pedagogical strategies to cope with adjustments to the teacher’s instructional goals in order to accommodate changing circumstances as the instruction proceeds in a teaching unit. 5.2. Consistencies of Teacher Questioning Practices across the Lessons This study reported the consistencies regarding teacher questioning practices over the lesson sequences. Each participating teacher was inclined to use a consistent proportion of all the questions in each lesson to initiate IRF (multiple) sequences. For each initiation question in the IRF (multiple) sequences, each participating teacher tended to ask about two follow-up questions. These consistencies might be interpreted as the routinized and habitual aspects of teaching practices in the participating teacher’s classes. According to some researchers (e.g., Reference [1]), the teaching profession required teachers to continually make decisions on a daily basis in classrooms about how to interact with students, and such decision-making processes, regardless of teachers’ awareness, become routinized over time. Although the teachers could plan how to initiate a questioning sequence, it is impossible for teachers to accurately prepare for what students could respond to teachers’ initiation. Therefore, the teachers might have to make instant decisions to determine when to elicit students’ longer responses and how to build up on students’ responses. In this regard, the consistencies in the use of follow-up questions in IRF sequences could be interpreted as the teachers’ routine and habitus developed in the teaching experiences. 5.3. The Complexity of Teaching and Classroom Observation Research In this study, the act of asking questions in classroom interaction not only includes the teachers’ conscious planning so as to accomplish pedagogical goals, but also involves the teachers’ unconscious routine in how to build up on students’ thinking. The above results added evidence regarding the complicated process of how teachers’ classroom practices could be shaped by various potential factors arising from teaching planning and implementation. By presenting both of these variations and consistencies evident in the questioning


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practices across the consecutive lessons, this study supports other researcher’s arguments about the complexity of teaching [30–32]. The complexity documented in one teacher’s teaching unit in this study suggests that it is also hard to represent one teacher’s instructional practices by observing a limited number of his or her lessons. Therefore, the results in this study support the claim that there is a further complexity with regard to the reliability of classroom observations for generalizing results [33–35]. Schlesinger and Jentsch argued that it is challenging to determine the number of lessons in order to obtain enough information for one teacher’s instructional quality [35]. This study reminds us that it is very easy to draw inaccurate or even misleading conclusions by examining a teacher’s practices in one or two lessons. Instead, a unit of consecutive lessons might be a better lens through which the complexity of one teacher’s teaching practices could be investigated and understood in better ways. 6. Conclusions Teacher questioning is one the most frequently used and significant ways of orchestrating students’ thinking and reasoning [7]. A better documentation and fine-grained analysis of teacher questioning practices could help to understand how classroom interaction could be initiated and sustained, which thereby contributes to the construction of the desirable environment for constructive and rich classroom talk and to the cultivation of positive learning attitudes [36]. This study contributes to the research community by utilizing the IRF framework to examine two Chinese teachers’ questioning practices in a unit of consecutive lessons. Both participating teachers’ questioning practices showed variations and consistencies over the lesson sequence. It implies the potential influences that the changing instructional objectives and teaching routines might exert on teacher questioning practices. The explicit observation of the complicated nature of teacher questioning practices was enabled by adopting the IRF framework and the lens of consecutive lessons. It can be argued that the IRF framework and the context of consecutive lessons made the functions of teacher questioning more visible and explicit. Therefore, the challenges of determining the number of selected lessons when conducting classroom observation research might be partially resolved by choosing the lesson sequence as the analytical unit. There are some limitations in this study. First, this study examined teacher questioning practices in one single culture, and therefore this study could not provide information about the potential influences that cultural differences might exert on teacher questioning practices. Second, the unit topics in the selected classrooms in two Chinese classrooms were rather limited to algebra and functions. We did not videotape classroom teaching of other topics such as statistics and probability. Therefore, this study could not conclude whether the unit topic could have significant impacts on the teachers' employment of questioning practices. Author Contributions: Conceptualization, L.D., D.C., and W.T.S.; Methodology, L.D., D.C., and W.T.S.; Formal Analysis, L.D., L.W., and Y.C.; Investigation, L.D. and L.W.; Resources, Y.C. and D.C.; Writing-Original Draft Preparation, L.D.; Writing-Review & Editing, L.D., L.W., and Y.C.; Supervision, D.C., Y.C., and W.T.S.; Project Administration, W.T.S.; Funding Acquisition, Y.C., and D.C. Funding: This research was funded by China National Education Sciences Grant (2018): Middle School Students’ Cognitive interaction and Social interaction in Collaborative Problem Solving (Grant No. BHA180157). Acknowledgments: This study was supported by Beijing Normal University and The University of Melbourne. The video data in this study was from the database co-established by Yiming Cao at Beijing Normal University and David Clarke at The University of Melbourne. Conflicts of Interest: The authors declare no conflict of interest.

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A continuous record of temperature evolution over a sequence of ...

A continuous record of temperature evolution over a sequence of ...

questioning the ethics of using opaque web practices ...

BEST PRACTICES FOR TEACHER EFFECTIVENESS

Changing demographics, changing practices: teacher

BEST PRACTICES FOR TEACHER EFFECTIVENESS

QUESTIONING

Over a Decade of recA and tly Gene Sequence Typing of the ... - MDPI

Learning Experiences and Practices of Elementary Teacher

Improving teacher questioning in science ... - Wiley Online Library

Teacher questioning in Chinese Language classrooms - NIE Digital