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Student and teacher interventions: a frameworkfor analysing mathematical discourse in the classroom
Ove Gunnar Drageset
� Springer Science+Business Media Dordrecht 2014
Abstract Mathematical discourse in the classroom has been conceptualised in several ways,
from relatively general patterns such as initiation–response–evaluation (Cazden in classroom
discourse: the language of teaching and learning, Heinemann, London, 1988; Mehan in learning
lessons: social organization in the classroom. Cambridge, MA: Harvard University Press, 1979)
to concepts for more fine-grained description such as the ‘Advancing Children’s Mathematics’
framework (Fraivillig et al. in J Res Math Educ 30(2):148, 1999). This article suggests a
framework to be used for detailed studies of mathematical discourse on a turn-by-turn basis.
This framework was used to study how single turns affect each other to form patterns in one
teacher’s practice. The method used belongs to conversation analysis: studying single turns and
characterise these according to their role in the conversation. Two main repeating patterns were
identified: one between student explanations and the teacher’s focusing actions, and the other
between the teacher’s progressing actions and students’ teacher-led responses. The findings
also included other connections that demonstrate how various student interventions (expla-
nations, teacher-led responses, unexplained answers, partial answers, and initiatives) are fol-
lowed by different types of teacher actions. One implication is that, by developing concepts
capable of describing qualities of a discourse on a turn-by-turn basis, it then becomes possible to
analyse when mathematical talk fosters delivery of facts and when it fosters mathematical
argumentation, debate, and critique.
Keywords Communication � Discourse � Interaction � Conversation � IRE �Focusing � Funnelling
Introduction
Multiple frameworks and concepts have been developed in order to characterise classroom
communication in general, such as initiation–response–evaluation (IRE) (Cazden 1988;
O. G. Drageset (&)UIT, The Arctic University of Norway, Tromsø, Norwaye-mail: [email protected]
123
J Math Teacher EducDOI 10.1007/s10857-014-9280-9
Mehan 1979), and the four levels of communication suggested by Brendefur and Frykholm
(2000). Within mathematics education specifically, several such frameworks have been
developed, for example the Advancing Children’s Thinking framework (Fraivillig et al.
1999) and a framework of eight communicative features (Alrø and Skovsmose 2002).
Common to most existing frameworks for describing mathematics communication in the
classroom is that they describe situations and practices, rather than conversations on a turn-
by-turn basis. As a result of this, we know little about how different types of turns affect
each other and how they form patterns in mathematical classroom discourses. To clarify,
even though we do know that individual turns are thoroughly affected by previous turns
and that they set the premise for subsequent turns (Linell 1998; Sidnell 2010), we none-
theless know little of how turns affect each other and form patterns. In order to describe
mathematical discourses in the classroom on a turn-by-turn basis, there is a need for
detailed frameworks with categories and concepts able to describe turns on an individual
basis.
The main question this study deals with is: How do turns affect each other to form
patterns?
In order to answer this question, we need a framework, able to describe qualitative
differences, that distinguishes between different types of turns. These different types of
turns might then form patterns.
Conceptualising classroom discourse
Scholars have developed several concepts for characterising communication in teaching
practice. One of the most cited frameworks is IRE, which describes a discourse pattern
where the teacher initiates the questions, the students respond to them, and the teacher
evaluates the responses (Franke et al. 2007). IRE is described by Cazden (1988) as the
‘default option’, which is normally used unless the teacher makes a deliberate decision not
to. In this pattern, the students are normally engaged in a procedure-bound discourse, such
as calculating answers and memorising procedures, with little emphasis on ‘students
explaining their thinking, working publicly through an incorrect idea, making a conjecture,
or coming to consensus about a mathematical idea’ (Franke et al. 2007, p. 231). It is easy to
recognise this pattern, but it has limited value as a tool for characterising communication as
it lacks detail and has a dichotomous nature. Either a practice is described as IRE, or it is
not. This suggests that IRE is too coarse a concept to describe different qualities at a
detailed level, but still viable for describing classroom cultures. In what could be seen as an
improvement on describing classroom discourses simply as IRE or not, Brendefur and
Frykholm (2000) suggest four levels of communication: unidirectional, contributive,
reflective, and instructive. The first two levels both fit into the classic definition of IRE and
thus provide more detail. In reflective communication, the objective is to share ideas and
for the student to participate in the evaluation rather than just respond to teacher initiatives.
In instructive communication, the student also has to initiate questions. This means that the
last two levels of Brendefur and Frykholm’s model give the student responsibility for more
than just the ‘R’ of IRE and, by the same token, identifies two distinct patterns that are ‘not
IRE’. As with IRE, the four-level communication model suggested by Brendefur and
Frykholm (2000) is a viable tool for describing the culture of communication in the
classroom in general. Another such tool is offered by Wood et al. (2006), who describe
four different classroom cultures: conventional textbook culture, where the main interac-
tion pattern is dominated by IRE; conventional problem-solving culture, where the main
O. G. Drageset
123
interaction pattern is the teacher giving hints; strategy-reporting classroom culture, where
the students report strategies; and enquiry/argument classroom culture, where questions for
further clarification, challenge, and disagreement aim to train the student in justification
and assessment, in order to develop more robust mathematical arguments and reasoning.
There are many similar observations between the frameworks of Brendefur and Frykholm
(2000) and of Wood et al. (2006), especially as both differentiate between patterns based
on how students contribute to the discourse.
The above examples of conceptualising classroom discourse use dialogue to describe
classroom cultures in a more general sense. But there is also room within IRE for con-
siderable variation, and Wells (1993) illustrates how and why qualitatively different ini-
tiatives, evaluations, and responses all could be characterised as IRE. This illustrates that
there is more to learn from studying dialogue at a more detailed level. For example, using
conversation analysis involves studying interaction on its own terms, rather than as a
window through which we can view other processes (Hutchby and Wooffitt 1998; Sidnell
2010). There are examples of studies within mathematics education, which describe and
conceptualise classroom dialogue, where the findings for the interaction itself are more
important than using interaction as a window to observe something else. For example,
Fraivillig et al. (1999), and Cengiz et al. (2011), describe studies of how teachers actively
use student ideas to lead them towards more powerful, efficient, and accurate mathematical
thinking, and in which situations this process occurs. Fraivillig et al. (1999) present a
framework titled ‘Advancing Children’s Thinking’ (ACT), based on an in-depth analysis of
one skilful first-grade teacher. The framework has three components: eliciting children’s
solution methods, supporting children’s conceptual understanding, and extending chil-
dren’s mathematical thinking. Each of these components is defined by several categories of
instructional techniques, for example ‘encourage elaboration’, ‘remind student of con-
ceptually similar situations’, and ‘demonstrate teacher-selected solution methods’. Elicit-
ing, supporting, and extending can also function as tools for teachers to use when
orchestrating classroom discourse. Alrø and Skovsmose (2002) also offer similar tools in
their description of eight communicative features that present both in student–student and
in teacher–student interaction: getting in contact, locating, identifying, advocating,
thinking aloud, reformulating, challenging, and evaluating. Fraivillig et al. (1999), Cengiz
et al. (2011), and Alrø and Skovsmose (2002) describe communication in such detail that it
is possible to consider single interventions and smaller episodes, in order to describe or
understand the dialogue.
A major challenge for a teacher is what to attend to and what not during a discourse.
Such decision-making seems to be connected to a complex balance between goals for the
lesson, time constraints, beliefs, and several other factors. In addition to this, recognising
important mathematical moments and using them productively are difficult for both novice
teachers (Peterson and Leatham 2009) and experienced teachers (Chamberlin 2005). But
noticing whether or not teachers respond to student thinking only yields limited infor-
mation. Stockero and Van Zoest (2013) move on from this by developing five types of
what they call ‘pivotal teaching moments’ (PTM), from which it becomes possible to
analyse which types of student thinking a teacher is able and unable to recognise. In
addition, three types of productive teacher responses to PTMs are observed (extend,
pursue, and emphasise) and two that are less productive (ignore/dismiss, acknowledge but
continue as planned). By developing such concepts, Stockero and Van Zoest (2013) have
enabled us to describe both (a) different types of important opportunities for exploring,
discussing, and learning mathematics, which arise during a discourse (PTM), and (b) if and
how the teacher uses these opportunities. One finding worth mentioning is that teachers’
Student and teacher interventions
123
inability to notice PTMs had a negative impact on student learning. This illustrates how the
development of such concepts enables researchers to inspect and understand mathematical
discourse in more detail.
Frameworks such as ACT (Fraivillig et al. 1999), the eight communicative features
model (Alrø and Skovsmose 2002), and PTM (Stockero and Van Zoest 2013) offer con-
cepts that enable us to describe classroom discourse in a much more detailed manner than
is possible using IRE and the four communication levels model (Brendefur and Frykholm
2000), or the four classroom cultures model (Wood et al. 2006). For example, concepts
such as ‘unidirectional communication’ (Brendefur and Frykholm 2000) and ‘strategy-
reporting classrooms’ (Wood et al. 2006) are feasible tools to describe an entire practice or
classroom culture, but are typically unhelpful when studying single interventions and their
effects in a discourse. In this case, concepts such as ‘remind student of conceptually similar
situation’ (Fraivillig et al. 1999) and ‘challenging’ (Alrø and Skovsmose 2002) would be
more helpful, as they enable us to describe single-teacher interventions at a level of detail
that is not possible using more generalised concepts. However, looking at single inter-
ventions yields a very limited scope. According to Sacks et al. (1974), turns are the most
fundamental feature of conversation. Yet, even though people take turns in speaking,
sequentially and one at a time, it is not possible to characterise a conversation as a series of
individual actions. Instead, conversations are social practices where each turn is thoroughly
dependent on previous turns, and individual turns cannot be understood in isolation from
each other (Linell 1998). When responding, it is normally possible to give different types
of responses, but usually one or more responses are preferred to others. Sidnell (2010)
exemplifies this by saying that the preferred response to a dinner invitation is to accept. If
accepted, there is no need for an explanation, but is the invitation is rejected, this then
requires an accompanying explanation. Linell (1998) explains a similar concept called
relevance, stating that some responses are more relevant than others. The concept of
appropriation (Newman 1990) highlights the need to look at sequences as well as single
interventions. Appropriation describes a process of teachers’ interactive support for stu-
dents, where feedback given on students’ work helps students to learn the overall structure
and purpose of the activities assigned them. However, there is still a need to develop
concepts that describe single interventions and how they act together on a turn-by-turn
basis, in order to understand such processes as appropriation.
This does not mean that concepts at a detailed level are always used to describe
interaction on its own terms in mathematics education. There are several concepts and
frameworks developed in order to study interaction in detail, as a window through which
one endeavours to view other processes. One such example is ‘The Knowledge Quartet’
(Rowland et al. 2005; Rowland and Turner 2009; Rowland et al. 2009; Turner and
Rowland 2011), a framework that comprehensively describes situations where teachers’
subject content knowledge can be identified from their interaction with students in the
classroom. Another example is the video coding tool called ‘Mathematical Quality of
Instruction’ (MQI) (Learning Mathematics for Teaching project 2006, 2011). In order to
study the quality of instruction in mathematics, the lessons are divided into five-minute
segments and the classroom discourse is studied and quantified using numerous codes
divided into six scales. An example of a detailed study of classroom interaction conducted
in order to compare the quality of instruction in mathematics across different cultures is the
first TIMSS video study (Stigler and Hiebert 1999). While the TIMSS video study
developed codes in order to describe the entire complexity of the classrooms, O’Keefe and
Clarke (2006) report a study where the use of Kikan-Shido in eighteen classrooms across
five countries is compared (as part of The Learners’ Perspective Study). Kikan-Shido is a
O. G. Drageset
123
Japanese term meaning ‘between-desk instruction’, where the teacher walks around the
classroom while monitoring and guiding students. This is an example of picking a pattern
or specific part of mathematics classroom practice and comparing its methodologies in
different classrooms and/or cultures. Whether one wants to capture the whole picture or
focus on specific parts of a practice, there are many challenges during the construction of
classroom analysis schemes for empirical use. Schoenfeld (2013) describes that the pur-
pose of constructing such schemes is about capturing the classroom practices that we
believe leads to students’ robust understandings and study whether or how these are related
to student performance.
Both the first TIMSS video study (Stigler and Hiebert 1999) and the Learner’s Per-
spective Study (LPS) (O’Keefe and Clarke 2006) developed their codes from their own
data. The Mathematical Quality of Instruction instrument (MQI) (Learning Mathematics
for Teaching project 2006) and the TRU math observation scheme (Schoenfeld 2013) used
another approach, as these projects have developed their codes based mainly on the lit-
erature studies, using a combination of established theoretical concepts and codes origi-
nally developed by other studies. A major challenge for coding schemes of all types is that
it can be difficult to see the superordinate connection between the codes. Without any
pointed superordinate connection, the codes may appear to have been put together by some
degree of coincidence, or at least, to be based on concepts and codes available in the
research literature, and not by what is needed to explain the data.
Whether or not there is a clear superordinate connection between the codes is not the
only problem; it is even more important to consider how the choice of codes and categories
affects the result. In spite of many school reforms in this direction, ‘researchers continue to
report that principal modes of instruction (lecturing, recitation, demonstration, seat work)
continue to dominate despite the increasing number of options that are being constructed’
(Klette 2010, p. 1006). This means that two practices with the same amount of lecturing
may seem similar, while it is possible that substantial differences in in-class lecturing are
hidden in these general categories. Differences and changes are therefore hidden by the
choice of categories in such studies. Klette (2010) argues that it is necessary to move from
looking at what is done to a more detailed inspection of how it is done. This also points out
a need for more detailed concepts and frameworks in order to describe qualitative dif-
ferences within mathematical discourses in the classroom.
It seems to be an established truth across subjects that effective instructional practices
demand student verbal participation, especially those that engage students actively in
reflective discussions (Mercer and Howe 2012; Walshaw and Anthony 2008). This means that
just talking is not sufficient. A skilful orchestration of classroom discourse must be facilitated
in order to foster argumentation, debate, and critique concerning important mathematical
ideas. An alternative way of putting it is that in order to encourage classroom practice
efficacy, it is necessary to change the discourse from what Mortimer and Scott (2003) ‘call
authoritative talk’, to a more real dialogue with active student participation. NCTM (2007)
suggests that this can be achieved through the teacher orchestrating a discourse by (a) lis-
tening carefully to students’ ideas and deciding what to pursue in depth, (b) posing questions
to elicit and challenge, and (c) asking students to clarify and justify. A similar way to
orchestrate more productive mathematical discussions is suggested by Stein et al. (2008),
who present five practices to move beyond the ‘show and tell’ phase of classroom discourse.
The suggestions of both NCTM (2007) and Stein et al. (2008) illustrate that in order to engage
students in reflective discussions, the teacher should not withdraw to a passive role, but
instead should actively facilitate the discourse to foster argumentation, debate, and critique.
In other words, promoting student talk is not sufficient; its usefulness depends on the content
Student and teacher interventions
123
and structure of the talk. The content should focus on important mathematical ideas, and the
structure should encourage the fostering of mathematical argumentation as opposed to merely
the delivery of facts and procedures.
As the examples above illustrate, categories at a detailed level are not found only in
theory-generating studies, but also in explorative, descriptive, and comparative studies, and
in tools developed for studying mathematics teaching. The main parts of the categories and
concepts are developed in order to use the resulting discourse as a window into other
related processes, but few studies describe mathematical discourse on a turn-by-turn basis
in order to develop concepts to describe the discourse itself. A related observation is that
most frameworks and coding tools are based on a study of episodes (sequences, situations,
etc.) rather than on single interventions and their role in classroom discourse.
There is a concern that without enough detail in the framework applied, important
qualitative differences can be missed, as illustrated by Wells (1993). At the same time,
categories and concepts that are developed from practice in such a way that a superordinate
connection is established will reduce the chances of missing important observations or
letting the codes and categories decide what is possible to see and not. Without such a
focus, the description of the category system often becomes very complex, may seem to be
put together by coincidence or to be based on what is available in the research literature;
additionally, the results become less generalisable because the choice of categories appears
to be more subjective and connected to the specific researcher.
Method
During the project ‘Mathematics in Northern Norway’, 356 teachers individually completed a
test related to their mathematical knowledge for teaching and a questionnaire related to their
beliefs. From these data, two knowledge constructs (‘common content knowledge’ and
‘specialised content knowledge’) and two belief constructs (‘rules’ and ‘reasoning’) were
established (Drageset 2009, 2010). Twelve teachers with diverse profiles were invited to
participate in a video study in order to describe their practices. Five of these twelve teachers
accepted and gave the researcher access to film in the classroom. These five teachers achieved
either high or average related to the knowledge constructs and emphasised either rules,
reasoning, or both belief constructs. Generally, the researcher and the teachers did not know
each other in advance. The agreement with the five teachers was that, in return for giving the
researcher access to the classroom, they would all receive a short course in mathematics
education, which also included a discussion about preliminary findings from their class-
rooms. Also, all five were invited a year later to a presentation of the findings. They could
withdraw their participation at any point, and their consent was free and informed. The project
adapted to the five main ethical issues suggested by Hammersley and Atkinson (2007):
informed consent, privacy, harm, exploitation, and consequences for future research.
The five teachers all taught in upper primary school (grades five to seven, students aged
ten to thirteen) and were all educated as general teachers. Typically, teachers teaching
these grades follow one or two student groups and teach most subjects for these groups.
Almost all teachers in Norwegian primary and lower secondary schools are educated
generalist teachers, graduates of a 4-year college programme implemented in 1992.
All mathematics teaching for 1 week was filmed for each of these teachers, from the
start of the topic of fractions. This varied between four and six lessons of 45 min each. The
camera followed the teacher, who also wore a microphone that captured all the dialogues in
which the teacher participated.
O. G. Drageset
123
In order to get a first and general overview, all the lessons were divided into segments
that lasted from two to ten minutes, typically including a section where one task was
solved, one student or group was helped, or a new method was introduced.
Analysis was conducted separately for teacher and student interventions, but never in
isolation, as every intervention was analysed as part of the discourse in order to see from
where it emerged and what effect it had on the next turn. In regard to the teacher inter-
ventions, the analysis was conducted in three main phases. The first phase started with a
rather coincidental observation that several teacher interventions in a segment were a
combination of a confirmation and a question (see ‘correcting questions’ in the next
section). While looking at similar patterns in other segments, the teacher interventions
were analysed one at a time, describing them as part of the dialogue. Similar interventions
were put together and characterised as groups, which were later developed into categories.
The categories were developed independently for each segment studied, and after ten
segments, the number of categories was growing so large that it was difficult to retain an
overview. At the same time, several categories were obviously closely related. During the
second phase, all the categories from the various segments were put together and com-
pared; several were merged and others refined. In the third phase, all the remaining seg-
ments were coded using the categories developed during the first two phases. During this
process, two new categories were developed, while several of the existing categories were
adjusted and refined. In total, more than 1,800 teacher interventions were used in the
development, resulting in thirteen categories organised in three superordinate groups.
The student interventions were analysed in a similar way, through two phases rather
than three. In the first phase, the student interventions were analysed one at a time, as part
of the dialogue as opposed to in isolation. Also, in this phase, similar interventions were
put together and were gradually developed into categories. Every time, a new intervention
was added to a group, and this description of the group was looked over and quite often
altered slightly, which sometimes led to a division into two groups or a merge with another
group. The result of the first phase was 26 categories describing a wide range of student
interventions. Even though some of these categories were interesting, many of them were
quite similar, and all too often interventions could fit into more than one category. As a
consequence, it was decided that the second phase would look for and determine more
general categories. In the second phase, the 26 initial categories were grouped together in
several different ways, in order to look for superordinate categories. The result was five
superordinate categories that collectively included all the initial 26 categories.
Part of the research design involved bringing together particular interventions, groups of
similar interventions, and initial category descriptions, into discussions in a local research
group, which consisted of five to ten researchers and teacher educators within the field of
mathematics education. These discussions were an important part of the process, and
feedback and disagreement resulted in changed names, changed or sharper definitions, and
merging and splitting of certain categories. Also, ideas for initial categories were suggested
to external groups of researchers, and substantial discussions related to these were
important for the development of the framework.
In order to answer the research question, the practice of one of the original five teachers
is analysed below. This teacher, who has been given the name Hanna, has more than
20 years of experience in teaching mathematics and several other subjects in primary
school (grades one to seven; students aged five to thirteen). Like most teachers in Nor-
wegian primary school, Hanna was educated as a general teacher. To qualify as a general
teacher in Norway, one must undertake a 4-year education programme, typically including
Student and teacher interventions
123
one semester devoted to mathematics education. Hanna is also regarded among colleagues
as a skilful teacher of mathematics.
A framework describing teacher and student interventions
Teacher actions
The development of categories to describe qualitatively different teacher interventions has
resulted in thirteen categories grouped according to three different actions: redirecting,
progressing, and focusing actions (Drageset 2014).
Redirecting actions are used by teachers in order to change student approaches. This is
done either by putting the intervention aside, advising a new strategy, or asking a cor-
recting question. Progressing actions are used by teachers in order to move the progress
forward. This is done either (a) by the teacher demonstrating a solution, (b) through
simplification where the teacher adds information or alters the task to make it simpler,
(c) by requesting closed process details, or (d) by initiating an open progress. Focusing
actions are used by teachers to stop progress, in order to look into details or reasons behind
an answer or approach. This is done either by requesting students to do something, or by
the teacher pointing out. Requests are made by either asking students to enlighten details,
requesting justification, asking them to apply themselves to a similar task, or asking other
students to make an assessment. Pointing out is done either by telling the students to notice
something that the teacher finds important during task solving, or by recapitulating the
steps taken to reach a solution (Table 1).
Progressing actions are used by teachers to move the progress forward. On average,
more than half of the five teachers’ actions recorded in the study are progressing actions,
and ‘closed progress details’ is the type of teacher intervention used most frequently by all
of them. The following is an example from Hanna’s practice:
T: How much is one of… one-fifth then of… of twenty-five?
S: Five
T: It is five, yes. How much is two-fifths?
S: … ten
T: Then it becomes ten. How much is three-fifths?
S: Fifteen
T: How much is four fifths?
Several students: Twenty
T: And how much is five fifths:
Several students: Twenty-five
S: One whole
T: One whole, yes. Yes, good. Great
(Excerpt 1)
It is typical of ‘closed progress details’ that the teacher asks for one detail at a time,
moving along one step at a time. Instead of asking for the final answer to a problem, the
teacher splits it into several smaller tasks and asks for answers to each of these. One aim of
this strategy might be to ensure, by leading them through each important step, that every
student is able to follow the line of thought. Regardless, the result is that the teacher takes
control of the process and most likely reduces the complexity of the task for the students,
as they do not need to see the whole picture. The large proportion of these interventions
O. G. Drageset
123
indicates that it is characteristic of Hanna’s practice that she, in her role as the teacher,
controls the process, splits multistep tasks into simpler individual steps, and asks quite
simple one-step questions. A similar pattern is described by Lithner (2008) as ‘guided
algorithmic reasoning’. In guided algorithmic reasoning, questions typically have only one
correct or desired response, which is quite often easy to find. ‘Simplification’ denotes
another way to reduce the complexity of tasks for the student, typically by adding infor-
mation and giving hints. Sometimes, this results in the students solving a completely
different task [similar to the Topaze effect described by (Brousseau and Balacheff 1997)].
‘Demonstration’ describes teacher interventions where the teacher simply solves the entire
task, either as an example or to help students who are not able to solve it on their own. The
‘open progress initiatives’ category describes interventions where the teacher seeks pro-
gress but leaves it to the student to choose his or her own method.
As mentioned previously, focusing actions are used by teachers to stop progress, in
order to look into details or reasons behind an answer or approach. Focusing actions are
also used frequently, especially in the categories of ‘enlighten detail’ and ‘notice’. One
example of the use of ‘notice’ occurs after Hanna has asked how students can be certain
that five sixths is larger than one-half:
S: Because three… is one half and…T: Yes. Because three-sixths is one half (writes 3/6 = 1/2). Right? When we have the
half in the numerator of the denominator, then we have one half. Then, we can find all
that have more than the half in the numerator
(Excerpt 2)
The student’s answer seems to be both correct and interesting, but is not very well
articulated, so the teacher chooses to repeat and clarify this point. ‘Notice’ typically
involves interventions aimed at emphasising or pointing out important elements during a
dialogue. The teacher often changes the original statement slightly, adds new information
to make the point clearer, or reminds them of information or findings on which they have
agreed earlier in the solution process. The purpose appears to be to help students by
pointing out important elements that they should use in their solution process, or by
pointing out important aspects that they may need to understand or use in future mathe-
matical studies. ‘Recap’ is also used relatively frequently, and while ‘notice’ describes the
teacher pointing out important aspects during a solution process to help students
Table 1 Redirecting, progressing, and focusing actions (for further details and examples, see Drageset2014)
Redirecting actions Progressing actions Focusing actions
Put aside Demonstration Request
Advising a new strategy Simplification Enlighten detail
Correcting question Closed progress details Justification
Open progress initiatives Apply to similar problems
Request assessmentfrom other students
Point out
Notice
Recap
Student and teacher interventions
123
understand, ‘recap’ refers to when the teacher points out important aspects from the
solution process after reaching a solution.
An example of ‘enlighten detail’ can be seen in the following excerpt:
T: How many do I have altogether when these three chips are one-fifth?
S: Fifteen
T: It is fifteen. How can… how did you find the answer?
(Excerpt 3)
Instead of just accepting the answer and continuing, Hanna halts progress and requests the
student to tell how he or she was able to find the answer. Interventions in this category
typically request that the students stop and explain what something means or how some-
thing happens. Such detailed explanations might be necessary for other students to follow
the line of thought, for the teacher to understand how the student thinks, or to check
whether or not the student really knows or understands. According to Franke et al. (2007),
making details explicit is one of the most powerful pedagogical moves a teacher can make.
‘Justification’ also involves making details explicit, but while ‘enlighten detail’ refers to
when the teacher asks a student to explain how an answer is found and what has been done
to find it, ‘justification’ involves the teacher asking the student why an answer or method is
correct.
Redirecting actions are the least frequently used types of action in these five practices.
Redirecting actions are interventions where the purpose seems to be to change the student’s
approach. This is typically done without asking the student their reasons for choosing the
current approach. The most frequent type of redirecting action noted in Hanna’s practice is
the ‘correcting question’. One example is when Hanna draws three dots on the blackboard
with a circle around them and states that this is one-fifth of all the dots. The task is to find
out how many dots there are altogether when three dots equal one-fifth. She then draws one
more circle with three dots and asks how many they have now:
S: You have… er… six
T: Yes but how many five… fifths do I have now? (draws a line from both circles to the
same place at the blackboard and waits)
(Excerpt 4)
This example is typical for ‘correcting questions’, as it includes a confirmation followed by
a question from the teacher in order to redirect the student towards another approach. Such
questions therefore also act as corrections.
Redirecting, progressing, and focusing actions allow one to comprehensively describe
teacher practice in ways that highlights many different qualities. However, this method is
limited to describing teacher interventions, and as will be illustrated later, different types of
teacher interventions are often related to particular types of student interventions.
Student responses
The development of categories describing qualitatively different student interventions has
resulted in five superordinate categories: explanations, initiatives, partial answers, teacher-
led responses, and unexplained answers.
The ‘explanations’ category emerged from three groups of different types of explana-
tions: explaining concept, explaining reason (why), and explaining action (what or how).
Typically, interventions categorised as explanations were requested by the teacher and
gave explicit details about concepts, reasons, and actions. Such explanations seem to serve
O. G. Drageset
123
both as a control of a student’s understanding and as a way to make details explicit in order
to share knowledge.
The ‘initiatives’ category represents a break in the flow. The filmed practices of the
five teachers were generally controlled by the teacher asking questions and assessing
student responses, thus creating a flow of questions and (mostly correct) answers.
However, sometimes student initiatives surfaced, through student interventions such as
(a) asking what or how to do something, (b) making suggestions, and (c) pointing out
important details. Typically, these interventions involved a student stopping the flow of
discourse, either to ask (why, how or what), to suggest, or to point out or make a
correction. A study of such cases might give information about how a teacher copes with
decision-making.
‘Partial answers’ is a category developed from incomplete responses or interventions.
Some of these responses are clearly insufficient, while others are closer to being correct.
Typically, the partial answers are neither entirely correct nor entirely false, but somewhere
in between.
The ‘teacher-led responses’ are the most frequent category of student interventions seen
in this study. These interventions were grouped together because they all seemed to involve
answers that to some degree were provided by the teacher, usually by reducing complexity
or by leading the students towards the answer. The two main ways the teacher did this were
by dividing a task into smaller steps and by adding information that resulted in the sim-
plification of the task. Typically, the student interventions in this category are correct
responses to basic tasks. An example from Hanna’s practice arises when she sets the task of
writing down how large a fraction of boys are in a group:
T: How many are there altogether in Bjørg’s group?
S: Four
T: Four. And then how many are boys?
S: Two
T: Yes, and then it becomes?
S: Half
This example illustrates how a teacher’s use of ‘closed progress detail’ interacts with a
student’s ‘teacher-led response’. The teacher divides the task into single steps that are so
simple for a student in upper primary school that the answer is almost given in the process.
The exception may be the step before the final answer, where there is a hidden step in the
student’s process from two-fourths to one-half, but even this is quite a basic process at this
level. ‘Teacher-led responses’ are typically student answers to overly basic tasks. This
often occurs as a response to ‘closed progress detail’ interventions, as demonstrated in the
example above, or as a response to ‘simplification’ interventions. These findings indicate
that it is the characteristic of this practice for the teacher to control the process and
dominate the discourse, often leaving the students to answer basic one-step tasks.
The interventions categorised as ‘unexplained answers’ are correct or incorrect
answers without any obvious reason or information as to how the student deduced or
reasoned to arrive at this answer. Responses where the student was unable to answer are
also included. To summarise, interventions in this category are typically answers without
explanations, which seem to come from nowhere, and where there are hidden steps
between the question and the answer. Such hidden steps mean that there is often potential
to enlighten interesting details or create discussions about how and why. One such
example from Hanna’s practice is:
Student and teacher interventions
123
S: One-fourth of sixty is fifteen
T: How did you think then?
As this student does not explain the reasoning behind the answer, subsequently asking
them to explain might be necessary both to enlighten the process for other students to
understand, and to show the teacher that it was not just lucky guessing. ‘Unexplained
answers’ are typically correct or incorrect answers without any obvious reason or infor-
mation about how the student thought. The answers seem to come from nowhere, and there
are some hidden steps between the questions and the answer. As the example above
illustrates, there is often a potential for enlighten interesting details or creating discussions
about how and why.
Both ‘teacher-led responses’ and ‘unexplained answers’ typically come without any
explanation or information as to how the student was thinking or why the student thought
the answer correct. Initially, these two categories were one, but this larger category
included student interventions very different from each other, and in sorting out these
different types of interventions, the term ‘distance’ proved to have explanatory power. The
interventions categorised as ‘unexplained answers’ have a tangible distance between the
question and the answer, meaning that normally there are several smaller steps to be taken
in order to reach an answer. Often, there are several ways to solve the problem or explain
the result. The term ‘distance’ is, in this context, either an expression of the work that the
student has to do to arrive at an answer, the distance that has to be covered between a task,
or the distance between a question and its answer. The larger the distance, the more work is
left to the student. It is typical of these interventions that they lack information about how
this distance is travelled, that is, the information regarding the smaller steps, or indeed any
relevant explanation, is not expressed. On the other hand, the interventions categorised as
‘teacher-led responses’ are so basic that normally there is no need to explain how the
problems are completed or why each answer is correct.
The main challenge involved in using the idea of distance to separate the categories of
teacher-led responses and unexplained answers is the subjective determination of which
problems would not normally need several steps or an explanation for students at these
grade levels. Yet, even with such challenges, the difference is principally interesting. The
term ‘distance’ developed when trying to differ between two observed types of student
interventions, both lacking information about reasoning and thinking. It also opens up the
possibility to see whether teachers treat unexplained answers with hidden steps differently
from more basic and teacher-led responses where there is little to explain, discuss, or
enlighten.
More examples will follow in the next section. For further details and examples, see
Drageset (2013, 2014).
Results—student interventions and their subsequent teacher interventions
By combining the categories of teacher and student interventions, a framework is
developed. This framework is able to describe qualities in a discourse on a turn-by-turn
basis. Using the concepts of the framework, it becomes possible to describe the role of
single interventions, how different types of interventions are typically or frequently
related, and the qualities of longer sequences or practices. In this section, student
interventions and their subsequent teacher interventions will be inspected predominantly
(Table 2).
O. G. Drageset
123
The ‘teacher-led responses’ are the most frequent student interventions noted in Han-
na’s practice. These are typically followed up with progressing actions. ‘Closed progress
details’ are the dominating type of progressing actions used to follow teacher-led responses
by Hanna (55 of 70 %). This is not surprising, as closed progress details and teacher-led
responses often come in repeating patterns like this one from Hanna’s practice:
T: Yes. So if I have thirty chips here and then divide them into six equal piles, then how
many are there in each pile then?
S1: There are five (hold up five fingers)
T: Five. But how much is two-sixths of thirty, then?
S2: Ten
T: Ten. How much is three-sixths?
S2: Fifteen
T: And four sixths?
S2: Twenty-five
S1: Twenty, twenty
T: And f… six sixths?
S2: Thirty
T: Yes. And… six sixths, how much do I have then?
S2: One whole
T: One whole. And then, this time the entire quantity was?
S2: Thirty
T: Thirty yes
(Excerpt 5)
Here, every ‘closed progress detail’ intervention from the teacher is followed by a ‘teacher-
led response’ from the student, which again leads to a ‘closed progress detail’ intervention
from the teacher, and so on. However, this example is longer than most such patterns, as
two or three repetitions seem to be more normal.
‘Teacher-led responses’ are also followed up relatively frequently by focusing actions.
One example of this is when Hanna has 22 beads, six of them blue:
T: How can you write this?
S: Six of twenty-two
T: Yes. What does twenty-two mean?
(Excerpt 6)
In this case, the student answers an easy, one-step question about how to write the fraction.
Instead of just accepting the answer and continuing, the teacher stops the progress and asks
what the number twenty-two means in the fraction. This is a typical example of the teacher
intervention category of ‘enlightening detail’.
‘Unexplained answers’ are the other major type of student interventions in Hanna’s
practice. The most frequent response to ‘unexplained answers’ interventions is to use
focusing actions. The main types of focusing action used are ‘justification’ (18 of 46 %),
‘enlightening detail’ (10 of 46 %), and ‘notice’ (10 of 46 %). An example of how Hanna
uses justification as a response to ‘unexplained answers’ is this one:
S1: One-sixth of eighteen is three
T: Mm (confirming). What is three-sixths of eighteen?
S1: Er… don’t know
T: But if one-sixth is three
Student and teacher interventions
123
S2: Nine
T: Yes, but I asked S1. S1, if one-sixth of eighteen is three
S1: Nine
T: Yes, but why?
(Excerpt 7)
When the first student does not know the answer, the teacher tells the student to notice that
we already know that one-sixth is three. Another student then answers correctly, but the
teacher makes a point of letting the first student answer. It is not possible to know whether
the first student knows the answer or is just repeating the answer of the second student. The
student’s answer is an ‘unexplained answer’ because no reason is given as to how the
student reached the answer or why the student thought this answer to be correct. There is
also a distance between the task and the answer here, as it is not immediately obvious that
the answer must be nine. When the teacher asks why the answer is nine, it is a request for
justification, which might be a way to check whether the first student knows the answer or
has just repeated what the second student said.
Almost as frequently, ‘unexplained answers’ are followed up by progressing actions.
The main types of progressing actions used are ‘closed progress detail’ (22 of 38 %) and
‘simplification’ (10 of 38 %). An example of how Hanna uses closed progress details as a
response to ‘unexplained answers’ is:
S1: One-sixth… no, one-seventh of fourteen is two
T: Yes, and how much is four sevenths… of fourteen?
S1: Of fourteen four sevenths is… seven
T: Yes, but you said that one was two, and two…S1: Two. Six… eight
T: Yes. Do you follow, everybody? S2? If one-seventh is two, then two-sevenths
becomes four. Three becomes…S2: Six
T: And four becomes…S2: Eight
(Excerpt 8)
Table 2 Student interventions and their subsequent teacher interventions—how does Hanna respond todifferent types of student interventions? Each row adds up to 100 %
Redirecting actions(11 % of Hanna’sinterventions)
Progressing actions(54 % of Hanna’sinterventions)
Focusing actions(35 % of Hanna’sinterventions)
Explanations (12 % of all studentinterventions)
2 36 62
Teacher-led responses(45 % of all student interventions)
3 70 27
Unexplained answers(27 % of all student interventions)
17 38 46
Initiatives (7 % of all studentinterventions)
23 40 37
Partial answers (4 % of allstudent interventions)
5 68 26
O. G. Drageset
123
This case is quite similar to the example of ‘justification’ as a response to ‘unexplained
answers’ (see excerpt 7). In both cases, the students are unable to answer correctly; the
teacher subsequently tells them to notice how much one part of the fraction is (one-sixth in
the first example; one-seventh in the second). In both cases, this results in a correct answer,
but in this latter example the teacher does not ask why, but instead asks one-step questions
(closed progress details) until the student arrives at the correct answer. It seems that one
purpose of using ‘closed progress details’ is to highlight the steps that are hidden between
the task and the answer; another is to lead the student through the steps needed to arrive
safely at a correct answer.
When Hanna redirects ‘unexplained answers’, they are nearly always incorrect. And
when Hanna uses focusing actions, the answers are nearly always correct. This shows that
this teacher has a tendency towards making details explicit only when the ‘unexplained
answers’ are correct, and not when they are incorrect.
As a response to student ‘explanations’, Hanna normally employs focusing actions.
Hanna’s most frequent interventions as a response to ‘explanations’ are ‘notice’ (27 of
62 %), ‘apply on a similar problem’ (13 of 62 %), and ‘recap’ (11 of 62 %). One example
of Hanna’s use of ‘notice’ as a response to ‘explanations’ is:
S: One-third of twenty-four and one-fourth of twenty. One-third of twenty-four is largest
T: Is it largest? Why is it largest?
S: Because one-third is larger than one-fourth
T: Yes but you have, you do not have the same quantity to take it from
(Excerpt 9)
This student explanation results directly from the teacher’s request for justification. Hanna
is not satisfied with the student’s initial explanation. Her response is to tell the student to
notice that it is not enough to just look at the denominator fraction, but also to consider the
value of the whole.
When Hanna follows up students’ ‘explanations’ with progressing actions, she most
frequently employs ‘closed progress details’ (24 of 36 %). One such example is when the
teacher draws four dots on the blackboard with a circle around them, states that this is one-
third of the total, and asks how much the total quantity is:
S: Twelve
T: It becomes twelve. Why do you think so?
S: Because… I just… don’t know… saw it at once because… four that is… four
multiplied by three is twelve and then four becomes one-third of twelve
(Excerpt 10)
The teacher then clarifies the student’s explanation, draws two more circles with four dots,
and starts to ask ‘closed progress details’ questions like ‘How many circles do I have?’ and
‘How much is two circles?’ These requests for ‘closed progress details’ seem to be used to
keep the students’ attention when the teacher is clarifying something.
Hanna frequently uses all three types of actions when responding to student initiatives.
The following example illustrates a rather common response to student initiatives. In this
scenario, the student has forty chips and is working on dividing them into groups:
S: It is not possible to divide three by forty
T: Forty divided by three. Divide them in equal groups. What was it they should be
divided by?
S: They shall be divided by three
Student and teacher interventions
123
T: Yes divide them by three then there is some… then some will be… (interrupted)
S: But we do not know how many we have to put in each
T: Yes, then you have to try. Start with four. Start with four. In three groups. That is four
in three groups, then we add two more to each group
(Excerpt 11)
The student first claims that the calculation is impossible to do (he confuses the numbers,
but based on visual impressions from the video, it seems that he means forty by three). This
is classed as an initiative, as the student comes up with it himself without being asked to do
it. Later, it becomes obvious that the student has problems with dividing into equal piles
without knowing how many chips should be in each pile. The teacher tells what has to be
done, step by step. The effect is that the task is simplified and the student just counts and
does what the teacher tells him to do. Together with ‘simplification’, closed progress
details were the main redirecting actions used as a response to student initiatives. The main
focusing action used was ‘notice’, which means that the teacher either shared the idea with
other students or pointed something out for the student as correct or important to
remember. It is also worth noticing that even though the redirecting actions are the least
frequently used overall in Hanna’s practice, redirecting actions follows student initiatives
more often than any other type of student intervention (mainly advising a new strategy).
When responding to partial answers, Hanna most frequently uses progressing actions.
S: They eat five slices of pizza, altogether
T: Yes. And one slice is one eight. Then five slices becomes?
S: Five slices…T: Five such ones… What is one such slice?
S: Hmm, one eight [sic]?
T: Yes, and five slices becomes?
S: Five eights [sic]
(Excerpt 12)
Partial answers range from almost correct to insufficient. In this example, the student
answer is clearly insufficient as it gives no new information (line three). The teacher
responds by asking one question for each step, or each consideration needed, in order to
find the correct answer. Such responses are typical for the category of ‘closed progress
details’. Almost every second response to partial answers can be categorised as closed
progress details. The two main types of focusing actions are (requests to) ‘enlighten detail’
and ‘notice’. ‘Enlighten detail’ might be used to find out the reason for the student’s partial
intervention (for example, by enquiring how the student worked out the answer to a
problem). ‘Notice’ might be used to point out important parts of the student intervention in
order to help the student.
Conclusion
The most frequent student interventions—teacher-led responses—are normally followed
by a progressing action by the teacher (mainly ‘closed progress details’). It is typical, as
several excerpts have illustrated, for ‘teacher-led responses’ and ‘closed progress details’
to occur in sequences, thus forming a circular pattern between teacher-led responses and
progressing actions. In such a sequence, the teacher controls the process and the student
responds to basic tasks that look like mere control questions. In Hanna’s practice, the
O. G. Drageset
123
mathematical work the students are doing in such sequences is at a lower level than could
be expected at this grade; it typically involves counting and one-digit computations.
Following ‘teacher-led responses’, the teacher rarely redirects as the students normally
answer these basic questions correctly, but sometimes the teacher uses focusing actions.
Looking at the second most frequent student intervention, ‘unexplained answers’, a
different picture is drawn. The most frequent teacher interventions here are focusing
actions, meaning that the teacher either requests more information from the student about
how the answer is reached (i.e., ‘enlighten detail) and/or why the student thinks it is correct
(i.e., ‘justification’), or she points out important elements (‘notice’ or ‘recap’). This means
that the teacher uses the potential of the hidden steps in ‘unexplained answers’ to explore
student reasoning or to point out important elements, which she appears to do nearly every
second time the opportunity rises. Almost as often, the teacher continues without exploring
student thinking. A probable explanation is that this is likely a natural procedure in most
classrooms, owing to the need to progress and the teacher’s ability to assess the ‘unex-
plained answer’ without needing to explore further or point out anything in the answer
given by the student.
Following student ‘explanations’, Hanna usually uses focusing actions. Students nor-
mally explain following a request to do so (i.e. when Hanna requests either ‘enlighten
details’ or ‘justification’). This indicates a circular pattern of focusing actions and expla-
nations, and illustrates both how the student’s turn is affected by the teacher’s request and
also how the teacher’s next turn is affected by the type of student answer.
The two circular, or repeating, patterns of ‘teacher-led response—progressing actions’
and ‘student explanations—focusing actions’ illustrate how a turn is thoroughly affected by
previous turns and that an individual turn can never be understood in isolation from others
(Linell 1998). It seems that a student explanation often triggers a focusing action from the
teacher, which again seems to trigger a new explanation from the student, and so on. In the
same way, it seems that progressing actions (especially closed progress details) triggers
teacher-led responses, which in turn trigger new progressing actions, and so on. This
illustrates the point that some responses are preferred to or more relevant than others
(Linell 1998; Sidnell 2010). It also illustrates how one turn affects the next to form
patterns.
When a teacher-led response does not initiate a pattern, or the pattern breaks, typically
this occurs when the teacher either asks the student to explain his thinking (‘enlighten
details’), or points out an important detail during the solution process (‘notice’). Similarly,
when a student explanation does not instigate a pattern, or the pattern breaks, typically this
is because the teacher has ended the discussion by asking closed questions in order to move
forward (closed progress details).
Following student’s unexplained answers and student initiatives, the subsequent teacher
interventions are nearly equally divided between focusing and progressing actions. Both
unexplained answers and student initiatives give the teacher opportunities to focus, either
on the hidden thinking involved (unexplained answers) or on an idea broached by a student
(student initiatives). However, no teacher can fruitfully use every opportunity to ask for
details or justification, or to point out important details. To do so would probably result in
no progress within the classroom at all, and the process of reaching solutions is also an
important part of mathematics. However, the balance between focusing and progressing
actions when following unexplained answers and student initiatives might be interesting to
look further into, possibly by comparing teachers who demonstrate very different
approaches to such student interventions.
Student and teacher interventions
123
The key to describing patterns was the development of detailed categories to describe
interventions on a turn-by-turn basis (Drageset 2013, 2014). The development of the
framework is also an answer to the call for using more detailed categories from which to
report findings (Klette 2010), and adds to the examples of how new concepts enable us to
better understand and differentiate between qualities of mathematical communication
(Alrø and Skovsmose 2002; Fraivillig et al. 1999; Stockero and Van Zoest 2013).
Several scholars have emphasised that student talk is not enough to facilitate student
learning; its effectiveness depends on both the content and the structure of the discourse
(Mortimer and Scott 2003; NCTM 2007; Stein et al. 2008; Walshaw and Anthony 2008).
By using this framework, it becomes possible to characterise the talk down to the basic
elements, that is, each individual intervention and how they build patterns. It then becomes
possible to analyse on a turn-by-turn basis if the content is related to important mathe-
matical ideas and if the structure is one that fosters mathematical argumentation, debate,
and critique, instead of merely the delivery of facts.
It is possible to study classroom communication in mathematics as a window into such
things as teacher knowledge, student learning, or level of mathematical content. For
example, the Knowledge Quartet (Rowland and Turner 2009; Turner and Rowland 2011) is
designed to use classroom communication as a window into teacher knowledge, while the
Mathematical Quality of Instruction tool (Learning Mathematics for Teaching project
2006, 2011) is designed to use classroom communication as a window into the quality of
instruction. This article also studies classroom communication, but describes the com-
munication on its own terms, rather than as an observational tool for other aspects of
mathematical learning. The aim is to develop concepts and use them to identify the patterns
of communication. Such concepts and patterns might later be used to provide insight into
student learning, level of mathematical content, or teacher knowledge.
Some studies have attempted to research and describe entire practices, such as the
Mathematical Quality of Instruction coding tool (Learning Mathematics for Teaching
project 2006, 2011) and the first TIMSS video studies (Hiebert et al. 2003; Stigler and
Hiebert 1999). Other investigations report findings from a study of selected elements, such
as the study of Kikan-Shido (between-desk instruction) in eighteen classrooms across five
countries (O’Keefe and Clarke 2006). The study in this article is closer to describing an
entire practice than selected elements, with the important difference that student conver-
sation that does not include the teacher is not registered. Also, the categories of teacher and
student interventions are developed from the data, in a similar way to the first TIMSS video
study (Stigler and Hiebert 1999) and the Learner’s Perspective Study (O’Keefe and Clarke
2006). Others, such as the Mathematical Quality of Instruction tool (Learning Mathematics
for Teaching project 2006, 2011) and the TRU math observation scheme (Schoenfeld
2013) were developed from the literature studies, using existing theoretical concepts and
codes originally developed by other studies. Both approaches face serious challenges. By
developing the concepts or codes from the data, it is difficult, but still important, to connect
the findings with established knowledge in order to progress as a research field. It is easy to
envisage that if everybody developed their own concepts and codes themselves, this would
result in a jungle of similar and overlapping concepts that could hinder the establishing of
connections, general findings, and progress. By using existing concepts and codes, though,
there is the danger of reporting within established categories (Klette 2010), consequently
risking missing important details related to, for example, the relevant specific practice or
culture.
Further studies based on the framework presented here might follow a quantitative or a
qualitative path. Having established a framework with new concepts and categories opens
O. G. Drageset
123
up the possibility to map the terrain using quantitative measures: filming teaching at a
larger scale, counting single interventions, counting specific patterns, and possibly use this
for comparative studies or to connect it to student achievement. It would be natural to
expect some challenges in reaching an acceptable level of intercoder reliability using these
exact categories, but even with an adjustment made for this reason, the framework pre-
sented in this article still has the potential to give new insight using quantitative studies. A
qualitative study could go deeper into relations between categories, for instance, by
describing patterns more explicitly. This article ascertains that teacher-led responses and
progressing actions seem to be connected in circular patterns, and that a similar connection
exists between student explanations and focusing actions. It would be interesting to explore
the shifts in discourse, as seen in excerpt 6, when the discourse moves between a ‘teacher-
led response’—‘progressing actions’ pattern and a ‘student explanation’—‘focusing
actions’ pattern; other areas to investigate include how a pattern typically ends, and how it
typically moves from one to another.
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