Posing purposeful questions is one of the most effective teaching practices (NCTM, 2014). Smith and Stein (2011) highlighted the importance of teachers’ questioning practices that “make students’ thinking visible … while they are exploring the task” in orchestrating productive mathematics discussions (p. 10). Research on teacher questioning has a long history in education (Verschaffel et al., 2020). Teacher questioning can be employed for broad purposes, including (a) drawing attention to relevant information, (b) exploring meanings, (c) probing thinking, (d) generating reasoning and discussion, (e) establishing context, and (f) supporting mathematical modeling and problem solving (Boaler & Brodie, 2004; Maxfield, 2011; Rittle-Johnson & Koedinger, 2005).

In a mathematical problem-solving activity, teachers engage students in activities or tasks that might be constructed in different forms (e.g., computation questions, academic problems, contextual word problems, or authentic tasks) depending on the curricular content and learning goals (Herrington et al., 2009). Scholars have concluded that, due to their linguistic and numerical complexity, the commonly used word problems in K-12 mathematics classrooms are the most intricate problem types for elementary students to solve in school mathematics (Daroczy et al., 2015; Jonassen, 2003). Nevertheless, a problem-solving activity is considered to result in meaningful learning when students are able to (a) demonstrate their understanding of the task, (b) perform the strategies that make sense to them, (c) actively construct new mathematical knowledge, and (d) see the application of the learned knowledge in a real-life context (Carpenter et al., 2015; NCTM, 2000). The role of contextual knowledge in supporting students’ mathematical problem solving has been highlighted with the implementation of “situating problems in story contexts” (Rittle-Johnson & Koedinger, 2005, p. 316). Moreover, Jackson et al. (2013) advocated that supporting students in developing a common language to describe contextual features and mathematical relationships specific to the task on which they are working is critical for providing high-quality learning opportunities.

Because the process of solving mathematical problems is complex, it can be challenging for teachers to support this effort effectively (Nilsson, 2009). Researchers have investigated the effects of contextualization on students’ mathematical word problem solving (WPS) (Clarke & Roche, 2018; Yee & Bostic, 2014) and discussed the process of decontextualizing, contextualizing, and recontextualizing in students’ development of mathematical proficiency (Moynihan, 2015; NGA Center & CCSSO, 2010). However, research on how teachers enact questioning, the most-employed instructional technique, to trigger and guide the contextualization process is rather scarce. Thoroughly analyzing the teacher–student interactions initiated by teachers’ questions would enrich the existing knowledge of the types, rationales, timing, and quality of questions (Moyer & Milewicz, 2002; Schwartz, 2015; van Zee & Minstrell, 1997). For example, Sahin and Kulm (2008) found that inservice teachers asked more factual questions and less guiding questions, while novice teachers asked more probing questions than experienced teachers. Franke et al. (2009) concluded that asking probing sequences of specific questions could better elicit further elaboration from the student than enacting general, singular, and leading questions. Furthermore, additional analysis of such teacher questioning would extend our understanding of how inexperienced teachers’ questions (a) focus on students’ process or product in solving mathematical word problems (Crespo & Nicol, 2003), (b) elicit students’ mathematical thinking and process (Colonnese et al, 2022), and (c) further support or hinder students’ contextualization of mathematics problems (Dominguez, 2016).

This study aimed to investigate elementary preservice teachers’ questions with different functions (e.g., helping students understand the word problem or inquiring about students’ existing solution strategies) and how those questions affected students’ contextualization in WPS. It is hoped that mathematics teacher educators can utilize these findings to help teachers increase their effectiveness and efficiency in questioning.

Learning to enact questioning in mathematical field experience

Researchers have compared experienced and novice teachers’ questioning performance and found that novice teachers, including preservice teachers (PSTs), usually perform less efficiently and fluently in their questioning than their more senior counterparts (Jacobs et al., 2011; Tienken et al., 2009). Examining novice teachers’ questioning performance has illuminated the dilemmas and challenges that inexperienced teachers encounter in their practices (Moyer & Milewicz, 2002; Schwartz, 2015; Webel & Conner, 2017; Weiland et al., 2014). These studies agree that effectively enacting questioning is a challenging practice and that certain constraints complicate questioning, especially for PSTs. Accordingly, it is imperative to scrutinize PSTs’ questioning practice, including how their questions function and what effects those questions have on their interactions with students (Crespo & Nicol, 2003; Johar et al., 2017). Only once these questions are answered can teacher educators provide appropriate assistance in teacher education and professional development programs to help novice teachers learn to enact effective questioning.

Since the initiation of teacher education programs, there have been substantial discussions and debates regarding the role of field experiences in teacher preparation (Mewborn & Stinson, 2007; Scherff & Singer, 2012; Shaughnessy et al., 2015, 2019; Zeichner, 2010). Although PSTs may enter their teacher education programs with apprenticeships or observation experience (Lortie, 2002), Shaughnessy and Boerst (2018) concluded that “preservice teachers made use of some moves that were less productive for learning about student thinking” (p. 51), such as forcing students to solve a problem via a different strategy or moving them through problem solving in a specific way (Weiland et al., 2014; Wood, 1998). Studies have shown that field experiences provide PSTs with opportunities to apply questioning strategies to understand students’ mathematical thinking (Colonnese et al., 2022; Mewborn & Stinson, 2007) and to analyze their questioning strategies with appropriate assistance (Crespo & Nicol, 2003; Moyer & Milewicz, 2002). For example, PSTs in Chamberlin and Chamberlin’s (2010) study mentioned learning about “questioning the students to stimulate their thinking, to refocus them on the problem at hand, to understand the students’ thinking, or to challenge the students in their thinking” (p. 402) in their field experiences.

In general, PSTs have relatively little experience working with elementary students, particularly on mathematical tasks. Therefore, field-based activities in practicum settings can serve as great resources and opportunities to expose PSTs to an environment in which they can gain knowledge of children’s thinking in mathematical problem solving. Although Zeichner (2010) warned that there exists a “disconnection between what students are taught in campus courses and their opportunities for learning to enact these practices in their school placements” (p. 91), Feiman-Nemser (2001) advocated that “[o]bservation, apprenticeship, guided practice, knowledge application, and inquiry all have a place in field-based learning” (p. 1024). Moreover, Mewborn and Stinson (2007) contended that “[f]ield experiences provide a rich ground for questioning why we do the things we do and how we might do them differently if we are serving the goal of creating opportunities for PSTs to engage in assisted performance” (pp. 1482–1483). These findings support the idea that developing questioning proficiency relies on using carefully designed tasks in an assisted learning environment rather than on PSTs’ self-evolving development over time.

Only a few studies have examined PSTs’ questioning practices exclusively in mathematical field experience. Moyer and Milewicz (2002) investigated PSTs in one-on-one mathematics interviews with elementary students and stated that “using the diagnostic interview format allowed them to recognize and reflect on effective questioning techniques” (p. 293). They further concluded that having PSTs focus on “the skill of questioning in a one-on-one diagnostic interview may be an effective starting point for developing the mathematics questioning skills they will use as future classroom teachers” (p. 297). Weiland et al. (2014) concluded that the field experience approach provided rich opportunities for PSTs not only to develop the core practice of questioning but also to “adapt their questioning practice to offer more competent questions in their interactions with students” (p. 349). Colonnese et al. (2022) employed predetermined question types (NCTM, 2014) to examine PSTs’ questioning to elicit student thinking in a practice-based field experience and found that PSTs’ questions focused more on eliciting process than “probing understanding of mathematical ideas and encouraging reflection” (p. 208). In summary, these studies revealed the following:

  1. 1.

    PSTs’ questioning techniques can be developed within the context of face-to-face interaction with small numbers of students.

  2. 2.

    The function of a question is related to the students’ responses and the environment.

  3. 3.

    There exist difficulties and dilemmas in teachers’ questioning practices that can serve as starting points for teacher educators to intervene.

The aforementioned studies have revealed the strengths and areas for improvement in elementary PSTs’ questioning practice in their mathematical field experience, but a detailed construct of question–answer interactions and the effects of the interactions stemming from teacher questioning remains limited in mathematics teacher education.

Functional moves in mathematical word problem solving (WPS)

In elementary grades, word problems are part of the school mathematics curriculum and are still predominantly used in mathematics classrooms (Alghamdi et al., 2020; Daroczy et al., 2015; Hoogland et al., 2018; Mandal & Naskar, 2019). Working on word problems provides students with opportunities to engage in mathematical modeling, apply problem-solving skills, and associate the learned knowledge with reality (Chapman, 2006). Jonassen (2003) explained that word or story problems “typically present a quantitative solution problem embedded within a shallow story context” (p. 267). The context of the word problem normally resonates with problem solvers’ past experience so that “mathematical practices arising from experiential situations can be compared with story problems that gloss the experiential situations” (Roth, 1996, p. 490).

Nonetheless, the effectiveness of using arithmetic word problems in schools is debated (Chapman, 2006; Verschaffel, 2002). For example, Verschaffel et al. (1994) warned about such problems promoting “a strong tendency to exclude real-world knowledge and realistic considerations” (p. 273) in students, echoing the notion of the “suspension of sense-making,” in which students concentrate on structural features and prioritize mechanical calculations while solving word problems (Chapman, 2006; de Corte et al., 2000). Because most teachers cannot avoid employing word problems in mathematics instruction, it is necessary to examine and analyze not only the types and characteristics of such problems but also the factors that prevent students from contextualizing them. This study focused on mathematical word problems with contextual features, most of which were adapted from the word problems demonstrated in Cognitively Guided Instruction (CGI) programs (Carpenter et al., 2015; Empson & Levi, 2011).

A typical problem-solving framework proposed by Polya (1957) contains four distinct stages: (a) understanding the problem, (b) devising a plan, (c) carrying out the plan, and (d) looking back. Solving a word problem requires students to proceed through these stages, explicitly or implicitly, although “most often, students use a procedural approach to their solution, directly translating story values into solvable algorithms” (Jonassen, 2003, p. 267). To elicit students’ mathematical thinking related to their problem-solving performance, it is inevitable that teachers must ask follow-up questions, which usually have varying purposes corresponding to the four stages of problem solving. In this study, every question was assigned a specific function based on the goal that the question intended to achieve. For example, the question “Can you explain what the problem is about to me?” was asked to inquire about students’ understanding of the task, so it has the function of helping students understand the task. The conceptual framework section will provide more details about these questioning moves with varying functions (henceforth referred to as functional moves).

Research questions

Three research questions guided this study:

  1. 1.

    What was the function of questions asked by the PSTs when the students were solving word problems?

  2. 2.

    How did the functional moves in the teacher-initiated questioning influence the students’ contextualization?

  3. 3.

    What were the successes and difficulties when PSTs enacted questioning to support the students’ contextualization?

The purpose of this study is to provide mathematics teacher educators with insight into the role of teacher questioning in students’ contextualization of WPS. It is hoped that such insight will inform the development of interventions to improve the effectiveness and efficiency of teacher questioning.

Conceptual frameworks

Jacobs and Empson (2016) proposed a framework that addresses the intention of mathematics teachers’ instructional moves and found that experienced teachers demonstrate a sequence of professional techniques while working with elementary students on story problems by (a) ensuring that the child is making sense of the story problem, (b) exploring details of the child’s existing strategies, (c) connecting the child’s thinking to symbolic notation, and (d) encouraging the child to consider other strategies. Because the participants in the present study were all PSTs with very little mathematics teaching experience, the four themes from Jacobs and Empson’s (2016) framework were employed as general guidance for analyzing PSTs’ questioning performance, not as comprehensive categories. For example, teachers might ask a question with the intention of ensuring that the student is making sense of the story problem, but the question could result in different functions, such as clarifying terms for the student or seeking interpretations from the student.

In addition to describing the function of teachers’ questions, this study goes further to investigate how these functional moves influenced students’ contextualization of word problems. A questioning move can be an interrogative expression that always ends syntactically with a question mark (?) when captured in print, while other formats include directive and informative initiations (Mehan, 1979b). A questioning move should be conducted when the questioner is trying to seek information from the listener for an intended purpose; moreover, this move may initiate a reaction or a series of conversation exchanges, such as a dialogue. According to Wood (1998), this dialogic function of communication plays a role in mathematics classrooms because it enhances the likelihood of “a thinking device to act as a generator of new meanings for the receiver” (p. 168). Since a questioning move may elicit different levels of responses, Hogan et al.’s (1999) framework of interaction patternsFootnote 1 was adopted in this study to guide the characterization of the responses initiated by a teacher’s question—specifically, here, in one-on-one interviews—in the attempt to answer the second research question. In a one-on-one interview, the student could engage in off-task conversations like greetings (i.e., a nonconceptual reaction, Hogan et al., 1999) or follow the teacher’s directions and silently produce written expressions on a paper. These reactions were coded as “nonresponse” in this current study because they did little to enrich and move the teacher–student interaction forward. Similarly, the “consensual response” coding implied that the student simply agreed with, accepted, or repeated the previous statement in the interaction (Hogan et al., 1999), which often led to a stagnation of the teacher–student dialogue. Therefore, the “nonresponse and consensual reactions” were combined to form the NC pattern, and the relatively rich “responsive and elaborative reactions” were merged into the RE pattern to better characterize the teacher–student interactions captured in this study. Although some questions were initiated by students, the discussion of these student-initiated turns is beyond the scope of this paper. Table 1 shows the two categories of interactive patterns that were employed to describe the identified functional moves: (1) teacher-initiated nonresponse or consensual reactions (TNC) and (2) teacher-initiated responsive or elaborative reactions (TRE).

Table 1 The Framework of Interactive Patterns, adapted from Hogan et al. (1999)
Full size table

Methods

Participants and settings

This research was conducted in the teacher education program for early childhood majors (certification Pre-K–5) at a large public university in the southeastern USA. The author was the researcher and conducted observations and data collection in the class. The participants were six pairs of PSTs and elementary school students, who were selected through convenience sampling (Patton, 2002). The six PST participants were students in their first mathematics methods course and were invited to participate in this study investigating PSTs’ questioning skills in their field experience. In this mathematics methods course, all PSTs conducted a single-student mathematical interview with the same student once a week for eight weeks in a public elementary school. Three of the teacher participants worked with first graders and three with fourth graders. The methods course instructor provided interview demonstration videos and weekly interview protocols composed of whole-number word problems adapted from CGI (Carpenter et al., 1989), but PSTs were allowed to adapt or redesign the given word problems to fit students’ problem-solving ability. The mathematics methods course covered the discussion of talk moves (Ginsburg, 1997), which included questioning tips such as “look beneath the answer to understand the thought that produced it” and “ask the fundamental question” without directly providing them with questions suggested by the instructor. In addition, the class watched several CGI videos with a focus on teacher questioning (Carpenter et al., 1989), and the importance of asking follow-up questions was emphasized before and after every interview. Table 2 shows the components of an interview protocol along with two sample word problems from different weeks.

Table 2 Example interview protocol
Full size table

Data collection and analysis

Each interview session was video-recorded and lasted for approximately 35 to 45 min. The interview protocols, PSTs’ field notes, and students’ written work were collected after the interviews were completed. In addition, supplementary materials used in the analysis included the debriefing forms, course assignments, and the final portfolios that required PSTs to identify their strengths and areas of improvement in questioning and reflect on questions like “What question did you ask the most during the interview?” and “How were they useful?” All video clips were transcribed verbatim, and nonverbal actions were included to enrich the interview records. Every turn that occurred in the teacher–student interactions was defined as an interactive turn, which referred to either the teacher’s question or the student’s response. Most teacher-initiated interactive turns consisted of a questioning move, conceptualized as a verbal expression that prompted the student to provide a verbal response or nonverbal action. In the analysis, several interactive turns were initially grouped together and examined on (a) the first turn’s function based on the Jacobs and Empson (2016) framework and (b) the interaction pattern of the grouped turns described in the Hogan et al. (1999) framework. The analysis was then followed by a microanalysis of the subtheme according to the principles of thematic analysis. For example, the teacher participant might offer to “read the word problem again” for the student. This interactive turn has the intention to “ensure the student is making sense of the word problem,” even if the student declined this offer.

The data were analyzed by applying thematic analysis (Braun & Clarle, 2006), defined as a method “for identifying, analyzing and reporting patterns (themes) within data” (p. 79), and then conducting microanalysis to identify emergent subthemes based on the teaching move framework and the framework of interaction patterns. The method contained five steps:

  1. 1.

    Familiarization means that the author delved deeper into the data by repeatedly reading the transcripts, along with watching the corresponding videos.

  2. 2.

    Coding and indexing indicate that the data were coded by highlighting the characteristics of interactive turns. First, questioning moves were distinguished from responding moves. Then, whether the initial interactive turn was teacher-initiated or student-initiated was determined. Although all interactive turns were analyzed in this study, the focus of this paper was the teacher-initiated turns; the discussion of student-initiated turns is beyond the scope of this paper. Every teacher-initiated turn was classified based on the main themes of the teaching moves, and the subsequent responses and follow-ups were analyzed based on the interaction pattern. Different characteristics within the same theme were labeled and indexed. The interactive turns employed for the same purpose were then grouped as a functional move, and irrelevant turns (e.g., moves such as “Do you think this chair is comfy?” and “I like your dress!”) were detached.

  3. 3.

    Thematizing emphasizes that potential themes and grouped coded data were distinguished under broad themes. The author read through the coded interactive turns several times to discern potential themes and subthemes.

  4. 4.

    Reexamining means that the themes were thoroughly checked to confirm the patterns, and the classification of functional moves was reassessed if necessary. After the main theme and subtheme of these grouped interactive turns were identified, the function of each group was then determined, and a functional move was assigned to the grouped interactive moves that were employed to achieve a particular goal in the WPS (see Fig. 1). To check for reliability, this coding process on partial data was repeated by the researcher every two weeks, and all questions were coded separately at least three times, achieving 92% intracoder agreement across several time points.

    Fig. 1
    figure 1

    A flowchart for analyzing interactive turns and determining a functional move

    Full size image
  5. 5.

    Interpreting the subthemes formed the fifth stage in the thematic analysis. This stage revealed the seven emergent subthemes with the meanings assigned by producing thick descriptions to interpret the functional moves. One purpose of this study is to identify the detailed function of teachers’ questioning moves.

Figure 2 shows the correspondences among the four main strands of teaching moves and the seven categories of functional moves. Task clarification (TC) was used to ensure that the student was making sense of the posed problem, so this move could occur either when the teacher clarified or when the student sought the given information in the original task. The category of plan elicitation (PE) moves was not present in Jacobs and Empson’s (2016) framework, so this category was added herein to further analyze teachers’ questioning moves in students’ mathematical WPS. To explore the details of the students’ existing strategies, the teachers centered their questions on three aspects: the students’ procedural understanding (PU), their ability to make connections (MC), and the rationale behind their strategies (RA). The math terminology (MT) moves corresponded to connecting the student’s thinking to symbolic notation, and the alternative strategy (AS) moves accorded with the teaching moves “encouraging the child to consider other strategies.”

Fig. 2
figure 2

Correspondences among the four categories of teaching moves and the seven categories of functional moves

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Results

To reveal the function of questioning moves and how they affected students’ contextualization in WPS, the frequency and percentage of functional moves are presented by category to provide an overview of the quantitative outcomes in the data. Figure 3 provides a breakdown of the functional moves employed by teacher participants with the frequency at different grade levels (the first and fourth grades). These data help answer the first research question: How did elementary PSTs’ questioning moves function? This quantitative data is then followed by the analysis of the interaction patterns, including the identification of the initiator and the respondent’s contributions to the conversation, to address their impact on students’ contextualization in mathematical WPS and the successes and difficulties in PSTs’ questioning while interacting with students in solving these problems.

Fig. 3
figure 3

Frequency of functional moves in Grade 1 and Grade 4 by category

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The most frequently used moves in this study were: (a) procedural understanding (PU) moves (23% in the fourth grade and 17% in the first grade), (b) task clarification (TC) moves (10% in the fourth grade and 9% in the first grade), and (c) alternative strategy (AS) moves (8% in the fourth grade and 5% in the first grade). The results suggest that questions were predominantly enacted to elicit the procedure of solving the mathematical word problems posed to students by the teacher participants. To better illustrate the use and effect of different questioning moves, the following sections focus on how teacher participants orchestrated the relatively popular functional moves to (a) promote task understanding (the TC moves), (b) disclose existing strategy (the PU, MC, and RA moves), and (c) incorporate an alternative strategy (the AS moves).

The use of task clarification (TC) moves

Figure 4 contains a quantitative comparison of the number of teacher-initiated TC moves by grade. A total of 16 moves were used in the first grade and 12 in the fourth grade. This dataset also shows that the fourth graders acquiesced to their teachers’ offerings more often than the first graders did (8:5, fourth graders to first graders), but the first graders demonstrated higher verbal engagement with the teacher-initiated TC moves (4:11, fourth graders to first graders).

Fig. 4
figure 4

Frequency of teacher-initiated TC moves by grade level and pattern

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Regarding the format of teacher-initiated TC moves, the teachers either repeated the entire task verbatim or merely highlighted a particular part of the word problem for students. Although most moves appeared to facilitate students’ understanding of the task, some teacher-initiated TC moves were either rejected or malfunctioned. In the following sections, three selected scenarios are presented to illustrate the outcomes in situations where these teacher-initiated TC moves did not function well.

Scenario 1. Clay animal problem. Interview conducted by Alison with first-grade student Caesar.

Task: Our class made some clay animals. The following day, our class made five more clay animals. Now there are nine clay animals. How many clay animals were there before our class made any more?

S:

[Drawing five circles, and then freezing]

Fig. A The student’s solution (recreated by the author)View full size image

T:

Do you want me to read the question again?

S:

[Shaking head]

T:

No? Okay

S:

[Drawing nine circles after the 5 circles. Numbering circles from 1 to 5. Writing down \(5+9=14\).] (see Fig. A)

(Grade 1, Week 1, Task #3)

In this interview, Alison spontaneously enacted a TC move (in italics and highlighted) after noticing an inaccurate written representation and a long pause from Caesar, but Caesar declined this offer and continued to solve this task based on his perception of the contextual features in this word problem. Accordingly, this task was concluded with the answer 11, which was not the anticipated solution to the original word problem.

Scenario 2. Cookie problem. Interview conducted by Alison with first-grade student Caesar.

Task: Eric made eight cookies. Three were chocolate chip, and the rest were oatmeal raisin. How many oatmeal raisin cookies did Eric make?

S:

[Drawing 8 circles, and then three more circles, and another set of three, and another set of three, writing down \(8+3\), then pausing] (see Fig. B)

Fig. B The student’s solution (recreated by the author)View full size image

…

T:

So, I said that he made three chocolate chip cookies, right? And the rest were oatmeal raisin, so how many oatmeal raisin cookies did he make?

S:

Four?

T:

So eight…there are eight total cookies, right?

S:

[Nodding]

T:

And he made three chocolate chip cookies, so how many oatmeal raisin cookies did he make? You wanna do it with cubes? Maybe it’d be easier to look at?

S:

[Pulling out eight cubes, grabbing three more in his hand] …8, and, made 3…

T:

Okay, so what do you think the answer is? It’s 11, right?

S:

[Nodding]

T:

Yeah? Okay, all right

(Grade 1, Week 1, Task #10)

In Scenario 2, Alison tried to help Caesar understand the word problem by restating the given information “eight total cookies and three chocolate chip cookies” and even suggested using cubes to make the inclusion relationship “easier to look at.” Nevertheless, as reflected by his drawings of additive images representing 8 + 3 on the paper, Caesar was unable to process this task as a part-part-whole; part unknown word problem (CGI, Carpenter et al., 1989). Although Caesar tried to adopt different math representations (e.g., cubes, numerals, drawings) to solve this task, he remained stuck in his misinterpretation of the mathematical relationship between the “whole eight cookies” and the subset entity “three chocolate chip cookies.” This situation highlights that when the problem solver does not construct a mathematical relationship that matches the word problem, the PST enacting the TC move by merely restating the given information and suggesting the use of manipulatives might not be sufficient for clarifying a mathematical task.

The aforementioned scenarios show that the enacted TC moves could fail if students consciously reject or misinterpret the TC moves provided by the teacher participants. This phenomenon reveals that PSTs’ skills in attending, interpreting, and decision making while enacting questioning (Jacobs et al., 2010) need attention. More questioning techniques and opportunities should be provided to equip PSTs with effective questioning moves to support students’ WPS. The next scenario involves, Gina, the PST, successfully redirecting Nicole’s attention to the correct mathematical relationships by requiring her to explain why the given word problem had to be worked from the beginning. This questioning move not only helped Nicole understand the context of this word problem but also interrupted her incorrect computation.

Scenario 3. Dinosaur problem. Interview conducted by Gina with fourth-grade student Nicole.

Task: 49 dinosaurs are coming, and they have 8 caves. If they spread the family out as equally as possible, how many dinosaurs will need to sleep in each cave?

In this scenario, Gina attempted to use a TC move to help Nicole clarify the task after she noticed that Nicole was using an incorrect number in her computation

T:

Okay, well, fifty-…so what does this number mean? This number down here? We just added up all these [numbers] right?

S:

[Adding all numbers in her drawing up by conducting \(7+7+7+7+8+7+7+7=28+29\) but resulting in only 47] It’s supposed to be 57, but I am checking my answer’cause we did it so fast

T:

Well, is that how many dinosaurs were coming, in the beginning?

S:

I am checking my answer to see if we did something. Um, 7, that would be…8…38; I’m off by 10…But where?

T:

Wait so, okay, so repeat the problem to me, what were we trying to do in the beginning. I wanna hear from the very beginning. What is–what are we trying to figure out in this problem?

S:

How many can fit in 8 caves if we have 49 dinosaurs?

(Grade 4, Week 6, Task #4)

In this study, when the PSTs initiated a TC move to clarify the contextual features and mathematical relationships of word problems, they could encounter difficulties in which the students (a) completely ignored the move; (b) only provided partial or superficial interpretation in their responses (e.g., merely addressed the referent unit without noticing the mathematical relationships); or (c) ineffectively processed the story context and the relationships among numbers. In contrast, if a functional move did not result in the outcomes mentioned above, it was considered a successful TC functional move. For example, after Nicole completely ignored Gina’s first TC move in Scenario 3 due to focusing on her computation, Gina attempted to enact another TC move differently by asking Nicole to repeat the problem “from the very beginning,” which successfully redirected her to the original context.

In comparing teachers’ questioning practices at different grade levels, the author noticed that the TC moves enacted by the teachers working with first-grade students functioned in the following ways: (a) clarifying students’ uncertainty, (b) ensuring that students were informed, (c) helping students correctly represent the task in their solution (e.g., completing a number sentence or select the correct number of cubes), or (d) correcting students’ misunderstanding while noticing a mistake in first graders’ WPS. For example, one teacher participant in this study requested her first-grade student to interpret the mathematical relationship in her own words based on the given information; only when the student could not do so did the teacher step in to provide needed information. In contrast, the teachers working with the fourth-grade students often enacted TC moves to complement their students’ incomplete statements about the task, which generally occurred in the early stages of the student’s WPS. Because this field experience required PSTs to work with students on solving word problems with a specific focus on understanding students’ mathematical thinking, the learning objectives were not distinguished for different grade levels. Therefore, the observed phenomena can be attributed to the fact that students who had a better comprehension of the text outperformed students with poor reading comprehension (Vilenius-Tuohimaa et al., 2008). Accordingly, the PSTs in this study may have assumed that the first graders needed more assistance in understanding the word problems while believing that the fourth graders should be capable of understanding the tasks without substantial teacher assistance.

Most of the teacher-initiated TC moves were identified as valid moves because they elicited one of the three reactions—consensual, responsive, and elaborative—whereas only a few TC moves resulted in a nonresponse reaction in the interactions. Most of the word problems conducted in the fourth grade contained up to three valid TC moves, except for one task that consisted of five valid TC moves. This case occurred in Alice’s interview with a fourth grader, Sandy, when Sandy was required to solve a multistep problem with the mathematical structure “\(5\times 6+4-3=\_\_\times 6+\_\_\)” (Grade 4, Week 2, Task #5). In the following sequence, Alice attempted to clarify one condition at a time alongside Sandy’s step-by-step computational procedure by initially posing the task with the first TC move highlighting “six peanuts in each bag” to help the student write the correct numerical sentence: “\(5\times 6=30\).” Then, she provided a second TC move by saying “Four extra peanuts not in a bag and three peanuts were given away. How many left?” Sandy added 4 to 30 and then subtracted 3 to obtain 31 as her final answer, so Alice then employed the third TC move by asking “how many complete bags of six will she have left?” and the fourth TC move by saying “6 peanuts in each bag and you wanna know how many bags you can make.” Sandy then performed the division by using the long-division method to divide 31 by 6 and concluded that the answer was 5 with a remainder of 1. Accordingly, Alice employed the fifth TC move by asking “so how many complete bags?” and switched to another functional move after receiving the answer “five bags” from the student.

In this case, the word problem was broken down into small chunks of information, and all five TC moves were effectively enacted with the respective provision of each information chunk during the problem-solving process. In this case, the teacher lessened the student’s working memory load by decomposing this multistep word problem, thereby enhancing the likelihood of the student constructing the mathematical relationship embedded in this task. This result echoes the notion that students’ ability to solve word problems is influenced by their working memory capacity and that cognitive strategies should be employed to lighten their working memory load to increase their understanding of mathematics (Cook et al., 2012; Swanson, 2014). Moreover, these TC moves were precisely enacted in alignment with their corresponding procedural steps, so they functioned as step-by-step directions in the problem-solving task. Although the process meant that the teacher had to provide fragmented information as directions, which might also limit the student’s creativity in solving this task, the student nevertheless successfully solved this task and could explain the meaning of each number and operation she produced in this process.

In sum, the teacher-initiated TC moves could be enacted in the following ways: the PSTs asked the students to rephrase the task and reinforced the contextual features, or they broke down information for the students to lighten their working memory load. TC moves can be employed in different formats, and how they are enacted may affect students’ performance in WPS, so it is important to equip PSTs with diverse ways to enact TC moves.

The use of PU, MC, and RA moves

To explore the students’ mathematical thinking related to their work in progress, the PST participants employed three different types of functional moves: procedural understanding (PU), making connections (MC), and clarifying the rationale behind a strategy (RA). The quantitative data show that the PSTs working with the fourth-grade students used more PU, MC, and RA moves (total 33 moves) than those working with the first-grade students (total 24 moves).

After the PU moves were initiated by the teachers, the students in both first and fourth grade provided responsive or elaborative reactions, which suggests that the students were able and willing to enrich the conversation related to their procedural steps. Compared by grade level, the making connections (MC) and rationale behind a strategy (RA) moves had similar numbers of moves: There were six MC moves in the first grade and seven in the fourth grade, as well as 13 RA moves in the first grade and 14 RA moves in the fourth grade. In addition, when it was the teachers who initiated MC and RA moves, they were more likely to receive responsive reactions from the students (TRE moves), except for the use of MC moves in the first grade. This finding implies that, when employing MC and RA moves, the PSTs were inclined to ask open-ended questions to invite students’ elaboration on the mathematical relationship among the numbers and the rationale behind their strategies (Fig. 5).

Fig. 5
figure 5

Frequency of teacher-initiated PU, MC, and RA moves by grade level and pattern

Full size image

In this study, enacting MC moves resulted in the following three outcomes:

  1. 1.

    A complete contextualized connection in student work was produced,

  2. 2.

    The contextual features were maintained but the key mathematical relationship was diminished, or

  3. 3.

    The mathematical relationship was ignored.

The contextualized connection

While enacting MC moves, the teacher participants who worked with the fourth-grade students paid more attention to the accuracy of the description of the referent unit to help students draw connections between the solutions and contextual features.

Scenario 4. Marker problem. Interview conducted by Tina with fourth-grade student Alia.

Task: Julie had some markers. She gave four markers to her brother. Now she only has two markers. How many markers did Julie start with?

After Tina posed the task, Alia came up with her solution strategy (see Fig. C) and explained “because 2 plus 4 is 6, and you add those [numbers] up. I did an inverse [\(6-2\)] so I got 4. I also knew 4 plus 2 was 6, so if you added 2 plus 4, it’d be 6 markers. So it’s six markers.” Tina then initiated the MC move as shown in the following exchange:

T:

So the 2 is what in the problem?

Fig. C The student’s solution to \(\_\_\_-4=2\) (recreated by the author)

S:

The 2 is how many markers she gave to her brother, and the 4 is how many markers she had left

T:

And what’s the 6?

S:

The 6 is how many she had before she gave 2 to her brother, and before she had four left

(Grade 4, Week 1, Task #1)

Scenario 5. Diamond ring problem. Interview conducted by Tina with fourth-grade student Alia.

Task: Rosa had 69 diamond rings, and then she bought 316 more. How many diamond rings does Rosa have now?

After Tina posed the task, Alia then fluently performed a computational procedure by explaining “So, if she had 69 rings and she bought 316. Six plus 9 is 15, so you add the one [in the tens place] up here. Six plus 1 is 7, plus 1 more is 8, so you have 385” with a vertical form of the addition \(69+316=385\). Tina then prompted with an MC move as shown in the following exchange:

T:

So the 69 is what?

S:

Sixty-nine is how many rings she has, and 316 is how many she bought

T:

And then 385 is what?

S:

How many she has in all

(Grade 4, Week 1, Task #2)

Based on the observations, the fourth-grade students demonstrated proficiency in making contextualized connections on their own even when the teachers’ questions lacked specific indicators to help them relate to the original task. In both Scenarios 4 and 5, Alia not only knew the unit of the given numbers but also specified the structure of the problem, which involved an action related to the given numbers. However, different performance was found in the first-grade students.

The diminished mathematical relationship

In this study, the first-grade students were observed to maintain the contextual features (e.g., a superficial impression such as the unit of the number) while diminishing the key mathematical relationship in a task. Thus, the correct mathematical structure was not completely carried over in the student work, and this omission hindered the possibility of solving this task as a separate start unknown word problem, one of the problem types described in CGI.

Scenario 6. Playground problem. Interview conducted by Alison with first-grade student Caesar.

Task: There are some kids on the playground. After six kids went home, 10 kids were still left on the playground. How many kids did they start off with on the playground?

After Alison posed the task by reading it once, Caesar immediately represented and solved this problem by drawing 10 circles in total, crossing out 6, then writing down \(10-6=4\) (see Fig. D). Caesar then explained as follows: “First, I did 10. Second, I x-ed out 6. Third, I counted, and then the number was 4…when I counted them. Only 4 were left.” Alison then initiated an MC move as shown in the following excerpt:

T:

Okay, what does the 10 and the 6 and the 4 stand for? What would be the word problem you give them?

Fig. D The student’s solution (recreated by the author)View full size image

S:

[long pausing]

T:

So if they ask you what does 10 minus 6 equal 4 mean? What would you say then?

S:

Equals…

T:

So what is… what does the number 10 mean? Is it the number of kids on the playground? Or is it the number that was left?

S:

The number that left… think… okay, the playground?

T:

Okay, so it’s the kids on the playground, so then what does 6 represent?

S:

The kids that went home?

T:

The kids that went home. And so what does 4 mean, then?

S:

Kids that were still on the playground

T:

Okay, good!

(Grade 1, Week 1, Task #9)

In Scenario 6, Caesar mentioned some contextual features from the task stem, such as “on the playground,” “went home,” and “left,” but Alison did not leverage the contextual features Caesar retained to help him reconstruct the correct mathematical relationship to succeed in solving this word problem. Start unknown problems have been proven to be more challenging for children to solve (Peterson et al., 1989). Caesar’s performance showed that he did not comprehend the key mathematical relationship in this separate start unknown word problem, and his misinterpretation of the contextual features was reflected in his solution strategy: “Ten on the playground, six went home, and four were still on the playground.” This difficulty caused by lack of comprehension is supported by previous studies, in which researchers have asserted that mathematical relationships and contextual features both play a critical role in solving word problems (Jackson et al., 2013). Instead of allowing the student to misinterpret the contextual features of this word problem, Alison could have asked questions to better scaffold Caesar’s problem-solving process, such as “I found that you got the answer that 4 kids were left after 6 kids went home, but the problem said after six kids went home, 10 kids were still left on the playground. How can we have both 4 kids left and 10 kids left?” or “The problem said there are some kids on the playground, but it did not say how many. Do you think we know how many?”.

Ignoring the contextual features

When students were immersed in working on computational procedures, they sometimes failed to consider the contextual features of the task.

Scenario 7. Cookie problem. Interview conducted by Tina with fourth-grade student Alia.

Task: Eric brought 11 boxes of cookies to school. There are 12 cookies in each box. How many cookies did Eric bring to school?

While solving this task, Alia drew 11 circles with 12 dots in each circle and explained “So 12 plus 12 is 24 [repeating five times] and there’s gonna be one extra of 12. So this is 48, this is 48, and this [48 plus 48] would be 96, and you add 96 and 24” (see Fig. E). Tina then initiated an MC move, as shown in the following exchange:

T:

Here you had 24…you did 96, I am just making sure I am following, so 96 is this [pointing two sets of 48] number, right?

Fig. E The student’s solution to \(11\times 12=\_\_\_\) (recreated by the author)View full size image

S:

Yes

T:

And that represents…

S:

Eight plus 8 is 16, but that, you can’t do, so you have to add 1 to, um, each of…well, that is 8. So you have to add 1 to 8, which is 9,

T:

Okay

S:

And you put the 6, is 96

T:

And the 96 represents what?

S:

Um, how many I added up, for…so like, I added those up, which is 24,

T:

Okay

S:

And I add it 4 up, which is 48,

T:

Okay

S:

And then, I added those up, and I got 96

T:

So 96 represents…

S:

How many this [circling written representations, 8 groups of 12 dots] is

T:

Okay! gotcha!

(Grade 4, Week 1, Task #4)

In Scenario 7, Tina enacted MC moves three times, but Alia continued replying with her computational steps without associating them to the context of this task. The same tendency was present in other scenarios in which such decontextualization occurred. Enacting MC moves offered the teacher participants the opportunity to assess students’ understanding of the contextual features and mathematical relationships and ascertain their capacity to make connections between the given information and the solution. In Scenario 7, Alia was fluent in demonstrating and articulating all computational steps by providing written representations and verbal expressions, and she was able to devise a solution strategy for the task, carry out the plan, and obtain a satisfactory result. Although Alia was capable of performing complicated computations and obtaining the correct answer, the MC moves seemed mainly to elicit procedural explanations of the computation rather than conceptual explanations of the way that the numbers and operations corresponded to the story in the task. That is, Tina made multiple attempts to elicit contextual explanations using MC moves, but Alia interpreted them as requests for explanations about the procedure.

The use of AS moves

For the teacher-initiated AS moves, the students in both first and fourth grades demonstrated a responsive tendency in the interactive patterns, and there was no considerable difference between the number of AS moves used in both grade levels (13 in the first grade and 18 in the fourth grade) (Fig. 6).

Fig. 6
figure 6

Frequency of teacher-initiated AS moves by grade level and pattern

Full size image

The strategies the students carried out in this study included drawings, number sentences, equations that correspond to the mathematical relationship in either horizontal or vertical form, or manipulatives such as cubes. The student participants used an alternative strategy or representation when (a) the teacher encouraged more strategies (i.e., an open-ended move) after the initial attempt produced the anticipated solution, (b) the students wanted to try another strategy because the first attempt did not work well, or (c) the teachers directly reminded the students about a particular strategy (i.e., a closed move) regardless of the implementation of the initial strategy. In addition, the questions “How can you check your solution by using another way?” or “Is there another way you can write it?” elicited alternative strategies (including representation) in students’ solutions. Open-ended AS moves usually left more freedom for students to employ their desired strategies to solve or examine their existing solutions. However, enacting open-ended AS moves also resulted in some consequences that are worth attention, which will be illustrated by the following scenarios from the first grade.

Scenario 8. Clay animal problem. Interview conducted by Alison with first-grade student Caesar.

Task: Our class made some clay animals. The following day, our class made five more clay animals. Now there are nine clay animals. How many clay animals were there before our class made any more?

Caesar initially solved the task by drawing circles and writing an equation corresponding to his misunderstanding. After Caesar concluded the task with the wrong solution 14, Alison employed an AS move to suggest using cubes (the highlighted part)

S:

[Drawing nine circles after the five circles. Numbering circles from 1 to 5. Writing down \(5+9=14\).] (see Fig. A)

T:

So what’s the answer?

S:

14

T:

Do you think, um…using the cubes or anything would help you with the problem? Do you wanna try that?

S:

[Nodding]

T:

Yeah? You can do that

S:

[Grabbing the bag of cubes, taking some blocks out]

(Grade 1, Week 1, Task #3)

After receiving the suggestion to use cubes instead of written representations, Caesar placed five cubes on the table and held one cube in his hand with a long pause, and admitted that this task was too hard for him. Therefore, Alison terminated this task. Although Caesar accepted the suggestion of incorporating cubes, the mathematical relationship of this task had been misunderstood, so Alison’s suggestion to use a second strategy resulted in a duplication of Caesar’s initial false solution strategy. In the next scenario, AS moves were used to elicit computational manipulation instead of a new problem-solving strategy.

Scenario 9. Cookie problem. Interview conducted by Amy with first-grade student Bonnie.

Task: A teacher brings 24 cookies for her class. Amy’s mom brings 12 more cookies. How many cookies does the class have in all?

Bonnie initially tried to solve this task by deriving from known facts including \(12+15=27\) and \(10+15=25\) but was interrupted by Amy’s first AS move that suggested breaking down the numbers

T:

Can you make it break down somehow?

S:

Yeah! Like that other problem that we did

T:

Mm-hmm

S:

24 [Writes 24 and 12 vertically]

…

S:

36! Cause I know that 4 + 2 = 6. So it’s gonna be 36

T:

Is that another way you can write that maybe?

S:

[Writes down \(12+24=36\)] 36. There!

T:

Is there any way you can break it down and write it? Do you know?

S:

If I wanted to do a minus one, then it would be 36 minus 12, or 24

T:

36 minus what?

S:

36 minus 12 equals 24

T:

How do you know that all [sentences are] equal to each other? How do you know that you can flip it around and that it can do that?

S:

Because, um, I know that 24 plus 12 equals 36, so if I did 36 minus 12, it equals 24

T:

Good job

(Grade 1, Week 8, Task #5)

In Scenario 9, Amy employed AS moves three times, in response to which Bonnie created, respectively, (1) a vertical form of the equation \(24+12=36\), (2) an equation \(12+24=36\) based on the commutative property, and (3) an inverse equation \(36-12=24\). Amy then inquired into Bonnie’s solutions by asking “How do you know that all equal each other?” and Bonnie replied, “Because I know \(24+12\) equals 36, so if I did \(36-12\), it equals 24.” Amy praised Bonnie and moved to the next task. The application of arithmetic properties became the focus in Scenarios 9, in which Bonnie produced her second equation based on the commutative property when Amy asked for “another way you can write it.” That is, after creating the first equation via the “break down” method, the AS move elicited Bonnie’s knowledge about the addition and subtraction facts instead of the contextual features and mathematical relationship of the task. Once the number sentence was constructed, the focus of the conversation could easily be shifted to the manipulation of the written equation or the numerical relationship rather than using the contextual features of the task to approach the problem with an alternative strategy.

Discussions and implications

The purpose of this study was to examine the function and effects of PSTs’ questions on students’ contextualization along with the successes and difficulties they encountered while enacting questioning. The findings not only enrich the existing knowledge of what questions PSTs asked and how PSTs enacted questioning but also extend our understanding of how the identified functional moves impact the students’ contextualization of solving mathematical word problems. Learning to enact questioning can be “cognitively demanding,” as it “requires considerable pedagogical content knowledge (PCK) and necessitates that teachers know their students well” (Boaler & Brodie, 2004, p. 773). Although PSTs’ unfamiliarity with the assigned student’s academic performance and mathematical disposition could have impacted their PCK and questioning performance, Walsh and Sattes (2016) advocated that “[q]uality questioning is an organic and dynamic process…from continual teacher learning through reflection on daily classroom practice” (p. 2). Hence, purposeful and continuous practice of enacting questioning in mathematics teaching is indispensable to PSTs, especially since previous studies have confirmed that reflecting on enacted questioning moves is necessary and beneficial to enhance questioning effectiveness (Crespo & Nicol, 2003; Moyer & Milewicz, 2002; Weiland et al., 2014). Furthermore, PSTs’ increased knowledge of instructional strategies and understanding of students through enacting questioning effectively would also promote PCK development (Berry et al., 2016; Şahin et al., 2016). Accordingly, mathematics teacher educators should facilitate PSTs’ reflection on enacting questioning through the proposed framework of functional moves in a practice-based field experience. Moreover, general teacher educators need to be aware that teacher candidates who are new to enacting questioning in any subject need time and experience to acquaint themselves with its practicality and versatility in terms of (re)contextualizing the learned materials (Colonnese et al, 2022; Dominguez, 2016).

To promote task understanding, most PSTs habitually repeated the word problem verbatim, and this lack of flexibility in the delivery of intended tasks failed to help the students, especially those in first grade, rectify misunderstandings. However, when the PSTs required the students to rephrase mathematical relationships in their own words, the task clarification (TC) moves worked as desired. The observed difficulties and successes have instructional and curriculum implications for field-based activity design and implementation. Teacher educators should consider providing more assistance to help PSTs select, pose, and clarify tasks with complete contextual features in which the learning goals could be clarified efficiently (Schwartz, 2015). For example, after the tasks are provided to PSTs, teacher educators could invite them to consider appropriate questions that they could ask before giving the tasks to the students and perhaps even try them out in an informal setting such as homeschooling or tutoring children, like the mirror and window proposed by Dominguez (2016). In conclusion, PSTs need to enact TC moves with intention rather than as box-checking questions, as illustrated in several studies (Moyer & Milewicz, 2002; Weiland et al., 2014). With the given task protocols, teacher educators could openly invite PSTs to discuss their intended questioning moves before the student interview as well as prompt their reflections on the successes and difficulties of questioning after completing the interview. This idea echoes the notion Stein et al. (2009) has advocated: Teacher educators should provide a scaffold for teachers to “allow them to do something that they would otherwise not be able to do” (p. 135).

After initiating PU moves, the PSTs seemed either to repeat what the students said or merely to comment with a brief acknowledgment. Such lack of follow-up inquiry was most evident in the fourth-grade pairs, where the PSTs granted mathematical agency to their students in WPS (Gresalfi et al., 2009). A functional move was considered effective when it helped students advance the justification of their mathematical thinking (NCTM, 2014). Based on the observations in this study, as the PSTs enacted their PU moves, their students remained immersed in the computational procedures in an unbalanced manner (Colonnese et al., 2022), without paying attention to the contextual features. Furthermore, some teacher participants did not enact questioning to help students identify the errors in their reasoning when they received flawed explanations from students. Although it is important to grant students their learning agency, teachers should also be aware of the necessity of building a learning community in which every member, including the teacher, can act to ensure their action contributes to the interaction in which they are participating. Therefore, when students present invalid reasoning or overconcentrate on numerical expressions and equations that are decontextualized from the word problems, PSTs should step in by confidently enacting professional inquiry and probing questioning.

With regard to the MC moves, most of the complete contextualized connections occurred in the fourth-grade interviews, and a plausible interpretation could be that the fourth-grade students were more likely to share a common basis of mathematics language and knowledge with peers and teachers than the first-grade students (Cobb et al., 1992). Hence, they were more capable of drawing connections among the operational mathematical elements, the solutions they produced, and their meaning in the context. The difficulties usually occurred when the mathematical relationship was diminished (e.g., Scenario 6) or ignored in the student’s explanations (e.g., Scenario 7), and the PSTs in this study seemed unable to resolve this difficulty through enacting questioning. Such difficulty can be attributed to the fact that the PSTs were not encouraged to correct the students’ mistakes, so they maintained a conservative standpoint in their questioning even when students were making irrelevant or wrong connections. Moreover, excessively associating MC moves with the written representations could have prompted students to deviate from the contextual features and mathematical relationships of the word problem they were trying to solve. These considerations need to be incorporated into components like mathematical task selection and field experience design in teacher education programs.

In the practice-based field experience, PSTs were encouraged to invite students to use alternative strategies/representations (AS), and they accomplished this either through allowing any possible strategies or by directly assigning one. In this study, the AS moves promisingly served as an incentive for students to create more ways to solve or represent the posed word problem. However, these AS moves sometimes resulted in decontextualization when the students only paid attention to the number sentence or the computational process they created to solve the problem. For example, the last AS move employed in Scenario 9 elicited the equation \(36-12=24\), which neither corresponded to the problem structure “join result unknown” nor matched the student’s existing strategy “adding 20 and 10 first and then adding the ones.” This case prompted a practical question: In what way and to what extent can PSTs’ questions align better with their students’ solution strategies and the contextual features? Based on the findings in this study, a potential solution could be a thorough examination of the effects of enacting these AS moves after the interviews, which is needed in future studies.

Overall, the single-student mathematical field experience focusing on questioning practice is considered a successful activity because, on the one hand, teacher participants recognized questioning as a powerful technique in mathematical instruction. On the other hand, they determined to enhance their teaching repertoire based on the knowledge they gained from this questioning experience after struggling with enacting questioning effectively. Therefore, it could be helpful for future teachers to refine mathematics teaching practices by experiencing similar struggles in their early field experiences and learning to analyze functional moves in questioning systematically to be more responsive to students’ thinking and enhance the contextualization of the implemented mathematical tasks.

Conclusion and limitations

In this study, functional moves including clarifying the task (TC), inquiring about the procedure (PU), encouraging alternative strategies (AS), making connections (MC), and eliciting the rationale behind a strategy (RA) were identified through fine-grained qualitative analyses. The timing and modes of enacting these functional moves had different effects on students’ contextualization of mathematical WPS. Thus, implications were discussed to provide mathematics teacher educators and researchers with insight into the role of teacher questioning in support of students’ contextualization. The findings confirmed that it is feasible to help PSTs reflect on or even improve their questioning practice through developing theory-based knowledge and practice-based approaches (Cohen et al., 2003; Moyer & Milewicz, 2002; Shaughnessy et al., 2015, 2019). Furthermore, the proposed framework of functional moves can serve as an analytical tool for examining practically the relationship between teacher questioning and student contextualization.

In terms of limitations, the captured moment-to-moment teacher–student interactions were analyzed to illustrate how various functional moves influenced students’ contextualization, but without interviewing teacher participants, their intention to enact a particular move type is unknowable. The lack of teacher intention limits our understanding of the relationships among word problem context, student contextualization process, and the use of the enacted functional moves. That is, an interview with teacher participants in future studies may help clarify whether particular functional moves enacted by teachers are triggered by students’ reactions or preconceived based on teachers’ interpretation of the word problem. In addition, while the small sample size of the qualitative analyses still revealed the successes and difficulties of students’ contextualization in the dynamic teacher–student interactions, quantitative data with a larger sample size would unveil the trends of enacting functional moves to support students’ mathematical problem solving and increase the reliability while generalizing the data. Future research with a focus on teacher questioning may incorporate quantitative measures so that mathematics teacher educators can make more confident generalizations in terms of PSTs’ use of functional moves in mathematical settings.

Regardless of the aforementioned limitations, the current study provides a thorough investigation of the types, quality, and enactment of functional moves. Its findings may be useful to extend mathematics teacher educators’ understanding of how teacher questioning could be enacted to support students’ contextualization. This study is also expected to contribute to mathematics teacher education programs in terms of preparing teachers for using a variety of functional moves on questioning to enhance the effectiveness of mathematical instruction.