The Tale of Two Teachers’ Use of Prompts in Mathematical Discussions

Facilitating mathematical discussions has consistently been identified as beneficial to students’ mathematical learning, with teachers’ use of questioning a primary identifier of appropriate facilitation. Although many teachers report familiarity with appropriate questioning techniques, we hypothesized that some teachers may not work in contexts where they can implement what they understand as best practices in their classroom. To explore this potential interaction, two primary teachers with similar dispositions towards mathematics pedagogy, but dissimilar institutional obligations were observed over a 10-week period. The types and frequencies of teachers’ questioning and their students’ responses during whole class mathematical discussions were observed. Despite both teachers holding similar conceptions of and dispositions towards facilitating mathematical discussion, the effectiveness of teachers’ various prompts in eliciting students’ mathematical descriptions was substantially different. Findings suggest that differences in the respective teachers’ institutional obligations may have affected the effectiveness of one teacher’s probing questions.


INTRODUCTION
Students who engage more frequently in talking about, explaining, or discussing mathematics tend to have higher mathematics achievement scores than those students who engage in mathematical discussion less frequently [11,22].However, such positive effects depend greatly on the teacher and school students are enrolled [23].Therefore, factors that influence the effectiveness of teachers' facilitation of mathematical discussion and talk are of great importance.Various authors have supplied recommendations to improve the quality of whole class discussion for teachers to implement which recommend teachers: establish certain ground rules for conversation [28]; facilitate dialogic discourse [36], and use appropriate questioning strategies [30].Within mathematics education research in elementary settings, such strategies have been observed to relate with deeper conceptualizations on the part of students [7,15].However, the quality of teachers' questioning in mathematical discussions has been found to relate with teachers' level of mathematical knowledge for teaching (MKT) [13], teachers' descriptions of what counts as effective discussion [12,24], and teacher's dispositions towards supporting student autonomy [21].Yet, there is also evidence that curricular demands influence how teachers facilitate mathematical discussion through usage of certain questioning types [1].
Mathematics teachers' pedagogical decisions are influenced by individual resources they bring to such scenarios (knowledge, beliefs, experiences, etc.), but obligations to the teaching profession also influence pedagogical decisions [10].Within the context of facilitating mathematical discussion, and more particularly regarding teachers' use of questioning in such contexts, it is worth considering to what degree certain professional obligations influence individual teachers' decisions.The present study examines this issue in the case of two teachers with relatively higher demonstrated MKT, and generally positive mathematics pedagogy dispositions, but who taught in different districts with different curricular demands placed.Thus, the purpose of this study is to compare the mathematical questioning practices, and students' responses, of two elementary teachers with similar beliefs, knowledge and dispositions but who taught with different curricular expectations and demands.

TEACHER QUESTIONING
A primary means for teachers to facilitate mathematical discussion is through the purposeful use of effective questioning practices.Observations of upper elementary teachers in the U.S. has revealed a connection between a press for meaning via questions that solicited explanation and justification with deepening students' mathematical understandings [19].Supporting these observations, researchers have found that students whose teachers elicited more explanation and justification via questioning had higher mathematics achievement [11].Such questions which solicit explanation and justification are also referred to as probing questions [1].Although probing questions are generally encouraged by findings from observational and empirical studies, another form of questioning is much more prevalent.Referred to as gathering information questions, such prompts solicit factual/answer-only responses, recalled/memorized procedures, and similarly simplistic mathematical statements [1].Gathering information prompts are the dominant form of mathematical questioning by teachers in the U.S., and although such prompts can sometimes stimulate discussion, probing questions have been observed as more consistent in eliciting deeper descriptions of mathematics [34].
Although the questioning that facilitates effective mathematical discussion has been well documented in the literature, the effectiveness of such questioning varies significantly across schools and classrooms [23].Several research efforts have sought to examine the potential causes of such variance.One description of press for meaning in teachers' facilitation stemmed from observations of four teachers [19]."All four teachers encouraged their students to describe how they solved the problems and circulated in the room during small-group activity to talk to students about their work" [19].Yet, teachers who pressed students for meaning sought more than mere descriptions of procedures.They pressed students for explanations through argumentation.Observations of New Zealand teachers' facilitating argumentation [16] support the findings of this prior research [19].
Studying teacher and researcher agreement for teaching logs of instruction, researchers have found that many elementary teachers had different interpretations of justification and proof than did researchers [12].This difference may be due to elementary teachers' access to mathematical language, since they typically do not have strong backgrounds in mathematics [12].However, researcher examining high school teachers' depictions of instruction for facilitating mathematical argumentation has found that teachers' interpretations of what counts as argumentation typically lacks an inclusion of probing questions or solicitation of mathematical explanation [24].When examining for relationships with college-level mathematics coursework, teachers with greater exposure to college-level mathematics were less likely to depict probing questions as a means of scaffolding argumentation [24].Yet, various research has observed relationships with teachers' specialized content knowledge for teaching mathematics [12,24] suggesting that particular kinds of mathematical knowledge may relate to which questions teachers pose and how they do so.
Much of the research on teachers' mathematical questioning has identified effective questioning practices [7,11].To a lesser degree, the literature also includes study of teachers' conceptions of mathematical questions and facilitation of discussion [12,24] as well as how teacher knowledge influences such questioning practices [2,13,20,21].Yet, factors beyond the individual teachers' control may provide an additional reason for why questioning practices vary in their effectiveness.Comparing the questioning practices of teachers at schools with different curriculum, researchers have observed that teachers with more reform-oriented curriculum posed more probing questions and fewer gathering information questions [1].A similar finding has been observed in the context of questions on written assessments [6].Specifically, questions drafted by mathematics teachers can be "affected by the requirements of institutions they work" [6] as well as knowledge and experience, but not necessarily by the curriculum that was mandated.Rather, it was how the particular institutions mandated such curriculum that may have affected such assessment questions.Unfortunately, much of the literature examining teachers' use of questioning in mathematical discussions have focused either on the individual-based resources teachers bring to such scenarios and/or their students' interaction in mathematical discourse as a result of such questioning.The theory of practical rationality affords a means of explaining how individual and institutional factors influence teachers' pedagogy of mathematical questioning.

PRACTICAL RATIONALITY OF TEACHER QUESTIONING
There is a triadic relationship between different stakeholders of mathematics instruction (the teacher, the student(s), and the mathematical content), which is commonly referred to as the instructional triangle [4].Among other things, the concept of the instructional triangle allows for explaining how teacherstudent interactions can often be mediated by the mathematics content at-hand.From a perspective of explaining how mathematical questioning facilitates discussion, the instructional triangle is useful in characterizing both the relationships between a teachers' questioning and students' response to such questioning, as well as how the mathematics at-hand can mediate this studentteacher interaction.
The literature base suggests certain key resources that aid teachers' posing of appropriate questions in mathematical discussions.These include, but are not limited to, supporting students' autonomy and dialogic discourse [25,26,28,28], possessing a higher degree of MKT [2,8,20,21], and, related to MKT, possessing a developed understanding of appropriate questions for discussion [13,24].Although such resources are not necessarily predictive of good questioning practices, the preceding paragraphs describe studies that infer strong relationships with such practices.Yet, as has previously been alluded to, there may be other influences beyond the content, teacher, and particular students in a classroom that influence the use and effectiveness of mathematical questioning.Herbst and Chazan expanded the original The Tale of Two Teachers' Use of Prompts in Mathematical Discussions conception of the instructional triangle [4] to include the influence of the institutional environment (or instructional situation) as well as the influence of other various systems that influence students and teachers both in and out of the classroom [10].The rationale for this inclusion is that teaching includes the activity of instruction, but also activities which influence instruction.Some of these influences stem from particular obligations teachers have to the teaching profession.Four such professional obligations described by researchers [9] include: the disciplinary obligation, or obligation to teach mathematics as a valid representation of the discipline; the individual obligation, or obligation to attend to students as individuals with particular needs and ways of being; the interpersonal obligation, or obligation to the various social dynamics that exist in the classroom context; and the institutional obligation, or obligation to "the department (e.g., textbook choices, curriculum coverage), to the school (e.g., calendar, bell schedules), to the district where the school is (e.g., assessment instruments and goals), to professional associations and unions" [9] and so forth.Although teachers are influenced by each professional obligation [37], the present study focuses specifically on the institutional obligation.
Using alternate language, various studies of mathematics instruction lend support to the claim that teachers' decision-making is influenced by the institutional obligation [9].For example, organizational mandates alter middle school mathematics teachers' intended actions in the classroom [29].Similarly, researchers have observed teachers who used an approach they believed was pedagogically unsound because of institutional demands from where they were teaching [18].Focusing on science teachers, researchers have found that early career teachers in schools with more restrictive curriculum were less likely to teach using reform-oriented pedagogy, which included appropriate questioning practices [17].Particular to mathematics, teachers' questioning patterns may be influenced by curriculum [1].Unfortunately, little study has examined how teachers' professional obligations to the institution affect their questioning practices.Yet if, as various research suggests, both individual resources and institutional demands affect the quality of teachers' prompts to facilitate class discussions, it is important to understand the nature of this interaction to improve teaching reform.It is with this focus on such interactions that the present study asks the following research question:

Participants and Context
Data were collected from two primary grade teachers in a Midwestern U.S. state in the 2012 -2013 academic year.Mary was a grade 3 teacher with a Bachelors and Master's degree in education.Susan was a grade 1 teacher with a Bachelors in psychology and Master's degree in education.Each teacher had taught for three or more years, indicating they were no longer 'novice' teachers [3].Mary and Susan previously participated as part of a larger sample in a survey-based study on teachers' perceptions, dispositions and knowledge for facilitating mathematical discussions [21].Mary and Susan were purposefully selected given the similarity of their self-reported perspectives.Each teacher was observed for 10 mathematics lessons.For the present study, we report on data from both teachers' survey responses and classroom observations.
Prior to being asked to participate in the present study, Mary and Susan completed a survey packet that included a measure of their MKT [14], their disposition for supporting student autonomy [31], their disposition for supporting dialogic discourse in mathematics, and several items regarding their perceived frequency of posing certain types of mathematical questions during discussion [36].Regarding the MKT score, both Mary and Susan had scores representing higher than average ability 1 .The statistic is relevant because prior research has previously observed that teachers with higher MKT generally used questioning effectively to facilitate discussion [13].Regarding dialogic disposition, both Mary and Susan's scores represent positive dispositions towards dialogic discourse.Finally, both teachers were found to have scores representing support for students' autonomy (as opposed to a controlling classroom).Three additional items assessing teachers' perceived frequency of using certain prompts during discussion were also examined: asking students to explain their procedures; to justify their solution strategies; and to provide a rationale following an explanation.Both teachers reported doing so every or almost every discussion for the first two questioning types and about half discussions for the latter questioning type. 1 Note that the LMT measures are designed to assess the effectiveness of teacher education and professional development efforts, but not individual teachers' teaching ability.Our use of MKT scores should be interpreted in concert with these aims.Thus, MKT provides a proxy for the effectiveness of teachers' prior professional development regarding mathematics content.Specific scores for teachers are not provided to protect the nature of the assessment, and the individual teachers.Both teachers' MKT scores were significantly above the average indicator (0.00).

The Tale of Two Teachers' Use of Prompts in Mathematical Discussions
The institutional obligation states that "the teacher is obligated to observe various aspects of the schooling regime.These include attending to school policies, calendars, schedules, examinations, curriculum, extracurricular activities, and so on" [10].Mary and Susan indicated similar institutional demands in a number of regards.For example, both teachers had similar amounts of time devoted to mathematics instruction (30 to 40 minutes per day) and often negotiated this allotted time with scheduled assemblies and special events in the school calendar.During the timeframe of observations, each teacher had a shortened mathematics lesson due to a scheduled assembly in their school.
Central to the purpose of the present study are the differences in institutional demands each teacher faced.The different contexts which Mary and Susan taught conveyed different institutional obligations, particularly regarding how each school districts' adopted curriculum was to be interpreted into classroom pedagogy, and different expectations due to the grade levels each teacher taught.Prior to observation, the first author met with each teacher to discuss the teachers' participation, assess their background and disposition, school and district context, and so forth.Mary's school district had adopted the Saxon mathematics curriculum and had adopted a policy that mandated teachers were to have students complete a curriculum-provided worksheet each day to reinforce mathematics skills as part of the lesson.This mandate applied to all primary grade levels in the district.Mary indicated concern that these curricular expectations negatively affected her ability to foster effective mathematical discussions -particularly regarding time demands that would not always allow for more in depth discussions.Susan's school district had adopted the Investigations mathematics curriculum.Whereas Mary had described specific curricular expectations from her school district, Susan stated she was expected to teach the required content, but was allowed a degree of professional discretion on how to do so.
The U.S. state that Mary and Susan taught required a standardized test be taken in mathematics every year beginning in third grade (Mary's grade level).Students' scores in a teacher's classroom was classified as an indicator of whether a teacher was performing well.Therefore, preparing students for the state mandated test was an additional institutional obligation Mary faced that Susan did not.

Data and Analysis
The present study incorporated a correlational embedded mixed method design.Specifically, correlational embedded designs collect qualitative data that is used to help explain relationships and trends in correlational or comparison studies [5].To fulfill the purpose of this study, Mary and Susan's questioning prompts and their students' responses to such prompts were examined and classified with two coding rubrics.The frequencies of each code allow for a non-parametric comparison of different teacher questions and student responses for each class.A supplementary qualitative analysis of how mathematical discussions were carried out in these classrooms was also conducted.The latter analysis provides a means of explaining observed non-parametric relationships between the two teachers, and provide a richer interpretation of such findings.

Quantitative Data and Analysis
Observational data was video recorded and transcribed.Transcripts were coded alongside viewing the video for teachers' question types and students' response types.To do this, data was parsed into exchanges, which included the opening and closing of discourse regarding particular discursive objects.Following is an example excerpt from Mary's fifth observation (teachers' names are emphasized in bold for readability).In this brief example, the beginning and ending statements serve to open and close the discourse for students to contribute to how the mathematics is conveyed for 53 +  = 100.Within each exchange, we initially looked for whether teachers incorporated sequences of questions or if their mathematical prompts were isolated.prompts (see Table 1 for descriptions of each code and descriptive statistics) [1].Both authors coded exchanges independently before coming together to compare and reconcile coding.Prior to reconciling codes, a Kappa statistic of .81 was calculated, suggesting strong interrater reliability for the 583 central prompts coded across both classrooms.Following coding, we categorized students' responses into six primary codes (shown with descriptive statistics in Table 2).Of particular interest for the present study are three variations of mathematical description: minimal, procedural and rationalized.Minimal descriptions conveyed some form of relationships, but provided relatively little more in the way of mathematical

Inserting Terminology
Solicits appropriate use of mathematical language.

Exploring Mathematical Meanings and/or Relationships
Emphasize or target particular mathematical relationships or meanings.

Probing
Solicits elaboration and description from students.

Generating Discussion
Solicits contributions from various students.

Linking and Applying
Used to make connections to the real-world or to other mathematical topics.

Extending Thinking
Connects to similar contexts or tasks.

Orienting and Focusing
Focuses on particular elements of a problem or task.

Establishing Context
Connects to topics outside of mathematics.details (e.g., in response to a word problem: "one person sits in the middle, one person sits on the other side, and one person sits on the other.").Procedural descriptions provided descriptions of mathematical procedurals, but did not provide rationales for these procedures (e.g., "The four plus five equals nine, and then three plus seven equals ten").Rationalized descriptions provided some form of justification or rationale either accompanying procedures or as standalone statements (e.g., describing parallel lines: "because even though they're not the same length, they're still-they can still go on forever and they're not gonna cross").A total of 854 student responses nested in 583 exchanges were coded.Student responses were coded independently by both authors and then reconciled.Interrater reliability of this process indicated strong reliability (Kappa = .91).

QUALITATIVE DATA AND ANALYSIS
A microethnographic approach was used to examine how the teachers and students interacted during mathematical discussions.Specifically, micrethnography involves examining how certain macro-cultural characteristics manifest themselves in the subtleties of face to face interactions [33].Microethnography examines such interactions sequentially, meaning that actions inform actions which in turn inform other actions, and so on.Spoken and non-spoken actions of individuals in interactions are of interest, as well as how such actions build off of one another [27].A central feature of microethnography is the examination of specific moments in a discourse, as situated in the larger discourse at hand.In this particular study, exchanges (identified in the quantitative data and analysis section) were considered specific moments in larger, whole class mathematical discussions.However, these class discussions were also considered in the context of the teacher's classroom in their particular grade level and school environment.Emergent themes from this analysis were used to make inferences regarding results for the quantitative analysis.

Quantitative Results
Table 3 provides a contingency table for Susan and Mary's observed central questions in analyzed exchanges, alongside counts which would be expected by chance.Because certain question types occurred infrequently, they were excluded from quantitative analysis to comply with requirements for calculating the Chi-Square statistic [32].A comparison between frequencies of prompt types found that these differences were independent from chance ((df = 4) = 10.12,p = .04).Specifically, Mary was observed to ask probing questions and explore mathematical meaning questions comparatively more often than Susan.However, Susan was observed to ask insert terminology and generate discussion questions comparatively more often than Mary.By comparatively more often, we mean to suggest that the observed frequencies were comparatively larger than expected by chance alone.
Next, student responses between teachers were examined (see Table 4).Student responses in each class were observed to be statistically significant from chance ((df = 5) = 40.60,p < .001).Post hoc analysis of Z-scores for observed and expected differences found that students in Susan's class provided more minimal, procedural and rationalized descriptions than would be expected by chance, while students in Mary's class provided fewer such descriptions than would be expected by chance.No observable differences were found for answer-only, affirmation or student solicits information responses.Next, we compared particular student response types with each teacher's questioning prompts (see Table 5) in order to better understand the observed prevalence of various mathematical descriptions in Susan's classroom compared to Mary's.The ratio between student descriptive response and each teacher question type is displayed in Table 5 (i.e., of Susan's six probing questions, there were seven descriptive responses).Interestingly, 20.3% of Susan's gathering information prompts elicited some form of descriptive response from her students, whereas only 3.7% of Mary's gathering information prompts.Additionally, all of Susan's probing prompts elicited some form of description from her students (100.0%)whereas only 26.3% of Mary's probing  questions did the same.Although Mary asked a higher proportion of probing questions, Susan's probing questions were found to yield a higher proportion of descriptive responses.

QUALITATIVE FINDINGS
Both Susan and Mary were observed to solicit and emphasize descriptions from their students, as well as redirecting students to make connections between each other's mathematical comments and strategies.However, particular differences emerged in examining how both teachers facilitated mathematical discussions.Across observations, Susan generally allowed 1-2 more seconds of wait time than Mary.Maintaining 3 to 5 seconds of wait time significantly improves the quality of discussions [35].Although Mary generally provided 3 to 5 seconds of wait time, her pace of instruction was quicker than Susan's.Another observation was that Susan's class used manipulatives and other mathematical representations much more frequently, and with more time devoted to these representations.Both of these observed general differences appear to be related to Mary's allocation of mathematics lesson time for students to complete the mandated worksheet during each observed lesson.Specifically, handouts associated with the Saxon curriculum were often integrated within instruction and took at least 10 minutes of allocated math time, on average.A useful comparison of how these handouts affected mathematical discussions lay in a strategy both Mary and Susan used.Both Mary and Susan would often pause dialogic discourse at a key prompt, but rather than having students answer right away, each would have students work in partners or small groups to investigate the question for a period of time.This partners-as-waittime strategy often generated better descriptions and more accurate responses from students.Susan's use of this strategy often incorporated teacher-created worksheets for students to record thoughts and ideas while they worked with manipulatives related to the task at-hand.Mary also used worksheets, provided by the mandated curriculum, but these often limited mathematical explorations within the small groups to symbolic manipulation.It is important to note that Mary did use manipulatives and dynamic mathematical representations, but such usage was observed less frequently and with less allocated time than in Susan's lessons.In fact, the manner in which Susan and Mary were observed to engage their students in mathematical discussion often appeared quite similar in the level of detail they were able to elicit from students.
During her third observation, Susan was working with students on identifying properties of shapes.Students were discussing an activity where they attempted to fill various shapes on a worksheet with pattern blocks.One student brought up a word they suggested as another name for a hexagon (i.e., polygon).less class time.Mary did press both Tim and Caleb for more clarity in their descriptions.This resulted in a description of Caleb's actions that conveyed an understanding related to subtraction, which Mary recognized.Comparing both excerpts, Susan's class spent more time exploring one student's conjecture (i.e., Preston's conjecture about vertices on a circle) than was done in Mary's class (i.e., Caleb's conjecture of 8 ÷ 2 as taking away one group of four).

DISCUSSION
Susan and Mary faced very different institutional obligations involving curriculum and testing expectations.However, both teachers were similar in many ways.Mary and Susan both had positive dispositions towards supporting student autonomy, engaging students in dialogic mathematical discourse, and had higher levels of MKT.Although Mary and Susan had similar dispositions and beliefs regarding mathematical discussion, quantitative results from the present study suggest that Susan's prompts were more effective in eliciting descriptive responses from students.Susan elicited 4.5 times more descriptive responses than Mary.Although a relatively higher proportion of Mary's prompts were probing questions than Susan's, Susan had a higher percentage of student descriptive responses to such prompts than Mary (100% versus 26.3%).Qualitative findings strongly suggest that the context in which teachers taught mattered significantly.One primary reason for this difference may be that Susan allocated more mathematics lesson time to specific exchanges within class discussions than did Mary.Mary and Susan had similar lengths of time for their mathematics lessons, but Mary spent as much as a quarter of her mathematics lesson time having her students complete a mandated worksheet from the district's chosen curriculum.Prior to observation, Mary explicitly described the worksheet as something she felt she had to do because the district had told her to, although she was concerned that following this protocol lessened the quality of her class's mathematical discussions.Another key difference in how time was allocated within discursive exchanges was the larger length wait time provided by Susan compared to Mary.Providing an appropriate amount of wait time is a critical component of effective mathematical discussions [35].Although wait time is often considered an instructional decision made explicitly by the teacher, findings from the present study suggest that a teacher's decision to include wait time may be influenced by contextual factors related to a teacher's institution.
Describing the theory of practical rationality, researchers suggest that enactment of or deviation from particular actions in the milieu of mathematical instruction can be justified via particular norms and obligations [10].For the present study, differences in Mary and Susan's obligation towards the institution was observed to have a substantial effect on their practical rationality for questioning in mathematical discussions.This was evident in Mary and Susan's descriptions of their instructional context, and the observed differences in how time was managed in whole class discussions in mathematics lessons.Further, the findings presented here suggest that particular institutional demands may affect not only what teachers do, or do not do, but students' actions as well.Prior research findings have suggested that students in classrooms where descriptive responses are more prevalent also have higher mathematics achievement scores [11].Therefore, an important implication of the present study is that teachers who are otherwise knowledgeable in effective pedagogy, but perceive they must adhere to certain institutional obligations, may not teach to their potential.Similar observations regarding other professional obligations have been discussed elsewhere [37].However, the findings from the present study have important implications specifically regarding the current environment of high stakes testing and its influence on teacher accountability.Rather, in many states, teachers like Mary are evaluated using score-based metrics (i.e., valueadded measures).Although we stress that the findings from the present study are not generalizable, and need further investigation, we consider the findings to be highly significant with important implications regarding comparisons of teachers.
The findings presented here are useful in continuing the work of understanding teachers' questioning strategies to facilitate mathematical discussion, as well as factors that influence the enactment of such strategies.Further research comparing teachers of similar conceptions for facilitating mathematical discussion under varying sets of institutional obligations is needed to help illuminate how such institutional demands interact with facilitation of mathematical discussions.By further investigating such interactions, we believe the field will be better informed to improve teacher education and professional development efforts.

:
Fifty-three plus what equals one-hundred?What do you think now?Steven?Steven: Fifty-seven?Mary: You're close.Jenny?Jenny: Forty-seven.Mary: Forty seven… A lot of kids are gonna say fifty-seven.You have to keep in mind what we noticed over here.The digits in the ten's place have to add up to make nine The above example yields two student responses with three different prompts, but only one of these prompts organized the exchange: "fifty-three plus what equals 100?" Therefore, coding of prompts in an exchange focused on the central prompt, or prompt directing the nature of the exchange.To classify these central prompts, we used Boaler and Brodie's classifications for mathematical The Tale of Two Teachers' Use of Prompts in Mathematical Discussions 119

Table 1 :
Descriptive Statistics for Mary and Susan's Questioning Across 10 Observations.

Table 2 :
Descriptive Statistics for Student Responses across All Observations.

Table 3 :
Contingency table for teachers and observed question types.
Note: Observed counts are in normal text, expected counts are italic.

Table 4 :
Contingency table for students' response types by teachers.Note: Observed counts are in normal text, expected counts are parenthesis.

Table 5 :
Comparison of Teachers' Prompts with Students' Descriptive Responses.