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Abstract

There has been a call for more theory-driven research that investigates how teachers learn from participating in Lesson Study. This study responds to that call by using the commognitive theory for investigating teachers' learning in and from Lesson Study. Learning is regarded as a change in discourse, and the study investigates teachers' discourse on teaching. From analysis of an empirical example, three characteristics of the teachers' discourse on teaching are identified. Firstly, students' learning is described as static conditions. Secondly, assumptions are made about prerequisites for developing understanding of students in these static conditions. Thirdly, dialogue between "weak" and "able" students are described as important for students' learning. These three characteristics become interesting when studying teacher learning in Lesson Study-in terms of change in discourse.
Teacher learning in Lesson Study: Identifying characteristics in
teachers’ discourse on teaching
Anita Tyskerud
University of Stavanger, Faculty of Arts and Education, Norway; anita.tyskerud@uis.no
There has been a call for more theory-driven research that investigates how teachers learn from
participating in Lesson Study. This study responds to that call by using the commognitive theory for
investigating teachers’ learning in and from Lesson Study. Learning is regarded as a change in
discourse, and the study investigates teachers’ discourse on teaching. From analysis of an
empirical example, three characteristics of the teachers’ discourse on teaching are identified.
Firstly, students’ learning is described as static conditions. Secondly, assumptions are made about
prerequisites for developing understanding of students in these static conditions. Thirdly, dialogue
between “weak” and “able” students are described as important for students’ learning. These three
characteristics become interesting when studying teacher learning in Lesson Study - in terms of
change in discourse.
Keywords: Lesson Study, mathematics teachers, professional development, discourse.
Introduction
Research on the development of teachers’ professional teaching practice in mathematics has
increased during the past decades, and recent studies have investigated the potential of practice-
based approaches to contribute to this development (Thames & Van Zoest, 2013). Dudley (2015)
and others suggest that Lesson Study (hereafter LS) should be implemented in schools as part of
continued professional development. While the Japanese school system has applied LS as a
sustainable form of teacher driven professional development for more than a century (Ronda, 2013;
Saito & Atencio, 2013), researchers from other countries have become interested in LS the past two
decades (Dudley, 2013, 2015; Lewis, 2002). Much of the interest in LS in the western world arose
after Stigler and Hiebert (1999) wrote “The Teaching Gap”.
Research question(s), prediction and observation are three important aspects of the LS process.
Teachers investigate their own teaching practice (Olson, White & Sparrow, 2011), they plan,
conduct and evaluate a research lesson in order to answer their own research question(s) (Chokshi
& Fernandez, 2004). The entire process requires that the teachers are open minded and eager to
better understand student learning or uncover new ideas of a particular aspect of the teaching of
mathematics (Lewis & Hurd, 2011). When planning the research lesson, prediction and observation
are crucial to help the teachers understand how students learn (Bjuland & Mosvold, 2015), and to
develop their own instructional interactions in their teaching practice (Lewis & Hurd, 2011).
Previous research on teacher learning in LS has focused on what teachers learn from planning
meetings (e.g. Cajkler, Wood, Norton & Pedder, 2014), teachers’ reflection as an important part of
mathematics teachers’ professional development (Ricks, 2011), and how observation of students
influences teacher learning (e.g. Warwick et al., 2016). Xu and Pedder (2015) call for more research
on how LS teachers learn and develop practice through participation in LS, within a clear
theoretical framework. This study aims at contributing to this strand of research, by using the
commognitive theory (Sfard, 2008) as a theoretical and analytical framework for investigating
teacher learning in LS.
The data presented in this paper is taken from a larger ongoing study in a lower secondary school in
Norway. The study regards knowledge as shared and collective rather than individual. Learning is
considered to develop through social negotiation (Radford, 2008), and is visible as a change in
discourse (Sfard, 2008). In terms of teacher learning in LS, a distinction can be made between
discourse on teaching and discourse of teaching. The former refers to when teachers’ talk about
(their own) teaching practice and student learning. The latter, discourse of teaching, refers to the
discourse and routines that the teachers use in the classroom. This study investigates change in
teachers’ discourse on teaching. A prerequisite for investigating change in discourse is to identify
key characteristics of the discourse. The aim of this study is thus to identify key characteristics in
the teachers’ discourse on teaching from planning meetings in the first of three LS-cycles. The
following research question is approached:
What are some characteristics of teachers’ discourse on teaching that might be relevant to
investigate in terms of teacher learning in LS?
The characteristics of discourse identified in this particular LS group are intended to serve as
exemplars of characteristics that might be relevant to focus on in studies of teacher learning in LS.
Theoretical and analytical framework
Sfard (2008) considers thinking as communication and she has developed the term commognition: a
combination of communication and cognition, which she claims are two processes of the same
phenomenon. A discourse is defined as “different types of communication (and thus of
commognition) that draw some individuals together while excluding some others” (Sfard, 2008, p.
91). A mathematical discourse is characterized by four critical properties: word use, visual
mediators, routines, and endorsed narratives. Word use refers to how the user defines the meaning
of words, and “is responsible for what the user is able to say about the world” (Sfard, 2008, p. 133).
Sfard (2008) describes development of word use in four stages: passive use, routine-driven use,
phrase-driven use and object-driven use. Passive use refers to hearing the word, without actively
using it. Routine-driven use refers to using the word in a concrete situation. Phrase-driven use
relates to being able to use the word in similar situations. Finally, object-driven use refers to “the
users’ awareness of the availability and contextual appropriateness of different realizations of the
word” (Sfard, 2008, p. 182). Visual mediators are visible objects, either iconic, concrete or
symbolic. Narratives are defined as any sequence of utterances framed as a description of a
mathematical object, and endorsed narratives are often by the discursants (participator in the
mathematical discourse) labeled as true. Routines are repetitive patterns characteristic of the given
discourse, and divided into three types: explorations, rituals and deeds. The first type of routine is a
how routine, meaning you can recall, sustain and construct narratives. Rituals are when routines,
referring to when it is appropriate to use the different narratives. Deeds are to consider as practical
actions that result in a physical change. Sfard (2008) defines learning as a permanent change in
discourse. The change can take place on two levels. She distinguishes between object-level learning
and meta-level learning. On the object-level, the change in discourse (learning) expands by
developing new routines, new objects or endorsed narratives. In contrast, meta-level learning
involves a change in metarules, which can only occur if there has been a commognitive conflict
(e.g. that two individuals use the same word, but with different meanings).
Objectification is important in discourse development (Sfard, 2008). It is a process where discourse
on human behavior and actions develops into an impersonal discourse on objects. This process
consists of two closely related but not inseparable sub-processes: reification and alienation.
Reification is the first step in this process and refers to the process of turning a discourse into an
object (Sfard, 2008). For instance, instead of saying, “A pupil has solved many of the tasks perfectly
in the test”, one can state, “The pupil has developed a mathematical understanding of the subject”.
To make this statement an alienation, the utterance must release the subject, then “mathematical
understanding” is a way to simplify a long story about the students’ skills and activities.
Subjectifying is an accompanying term which “refers to a special case of the activity of
objectifying, the one that takes a discursive focus shift from actions and their objects to the
performers of the action” (Sfard, 2008, p. 290). One trap of objectification of a person’s former
actions and subjectification, is that it might affect as constrain to the persons’ abilities and
motivation. As Sfard states, “Words that make references to action-outlasting factors have the power
to make one’s future in the image of one’s past” (Sfard, 2008, p. 56).
Methods
The LS-group consists of four mathematics teachers, one participant from the school administration
(the group leader), and one external expert (the author of this paper). The first LS-cycle took place
in the spring of 2016. The main data sources are video-recorded observations from the group’s
meetings. Three meetings were conducted before the research lesson. The first meeting was an
introduction to LS followed by two planning meetings. The presented examples are from the first
planning meeting (see Table 1 below).
Table 1
An overview of the data collection
Part of the first cycle Video-recordings
Introduction to LS
Planning meeting 1
73 min.
154 min.
Planning meeting 2 162 min.
In the first step of the analysis process, video-recordings were transcribed (by the author of this
paper). In the second step, a data reduction was made. In this process, two particular aspects of
discourse on teaching were isolated: 1) teachers’ narratives on students and student learning, and 2)
teachers’ narratives on teaching practice. The third step was to identity characteristics in the
teachers’ discourse, within these two core aspects. The theoretical concepts that informed this third
step of analysis were Sfard’s (2008) four properties of mathematical discourse: word use, visual
mediators, narratives, and routines, and the metaphors of objectification and subjectification.
Empirical example
The mathematical theme of the research lesson is the concept of volume. The teachers want the
students to understand volume as the relation between the base area and height, not only to calculate
the answer of some three-dimensional shapes (using formulas). The discussion arises in the first
planning meeting, in which the transcript presented below is taken from. The discussion continues
in the second planning meeting. The tasks for the lesson have not yet been selected, and the teachers
have not yet decided how to organize the students. Early in the conversation, the teachers have two
focus areas: how to differentiate and how to pair the students in groups. They stress that it is
important to differentiate, because there is a significant gap in the students’ mathematical
understanding. The following dialogue takes place in this discussion1:
1 Teacher 4: There are only students at the top and at the bottom in this class?
2 Teacher 1, 2 and 3: Yes (In unison).
3 Teacher 1: But that is okay, it is like that in some classes.
4 Teacher 4: And then it is the bottom there, it is enough just to do the calculation.
5 Teacher 3: It is like that in class C as well. It is the top and the bottom. In this class,
students achieve all grades, except grade one.
6 Teacher 1: But the differences, it is not in the same way.
7 Teacher 1: I think the groups should be mixed. Slightly different levels, but not too big a
gap. In addition, I think it would be better if we do not put all the weak
students in the same group.
In the continuing discussion on how to pair the students, the teachers ponder whether the students
should choose their own groups based on what task they want to elaborate upon, given tasks with
different shapes, or if the teachers should set the groups beforehand. In the latter case, they have to
consider whether they should group students homogeneously or mixed (7). One argument that the
teachers present in support of “mixing students” is that, when a student explains something to a
fellow student, both the explainer and the listener learn from the dialogue. They want the students to
explain to each other how they got their answer – not only to exchange their answer, but to argue
mathematically. The teachers assume that it is more difficult to find the volume of a shape with a
complex base area than for instance a plain rectangular prism. They agree that when calculating the
volume of a prism with different base areas, a rectangular base is easier than a triangular base; a
cylinder is even more difficult. The teachers predict that the “weak students” need a shape with
single base area, while the more able students can be given a shape with more complex shapes, for
instance a shape with two or three different base areas, like a swimming pool with different depths.
One of the teachers would like to hand out a concrete three-dimensional figure to each group, as a
visual mediator. He proposes a task in which the students calculate the volume of the figure on the
hand-out, first individually, then in groups, discussing their answers. To assess if the students have
understood the relation between base area and height, the teachers want to study the students’
discourse. The discussion proceeds as follows when the teachers plan on how to facilitate and
observe student dialogue:
1 The transcripts have been translated from Norwegian by the author of this paper.
8 Teacher 3: Do they understand how to calculate the volume?
9 Teacher 1: Mm, and do they catch the connection between the base areas multiplied the
height. We can check if they got it right, if we give the groups complex
shapes.
10 Teacher 4: I feel it is most appropriate to take “the house-task”2.
11 Teacher 1: Yes, but at the same time, they can be too caught up in that task.
12 Teacher 4: Yes, they can.
13 Teacher 1: So, I do not think we give them “the house-task”, we can rather find other
geometric shapes.
14 Teacher 3: So, is there a correlation between base area and volume. (Sitting and writing,
reading what she has written)
15 Teacher 1: Mm
16 Teacher 3: We are wondering whether they can explain what they are doing in their
calculation or not. Then they must be able to show their understanding,
explaining to each other how they have done it.
17 Teacher 1: Mm
18 Teacher 4: I think it is a good idea that they can explain to each other.
19 Teacher 1: Yes, I think so too.
20 Teacher 3: I can write, “They must explain the procedures”.
21 Teacher 4: Most likely, one of the group members is able to solve the task and explain
how.
22 Teacher 1 and 3: Mm
Analysis and discussion
Analysis of the teachers’ discourse identify three potentially relevant categories of the teachers’
discourse on teaching: 1) narratives on students, 2) narratives on students’ learning and 3)
narratives on teaching practice. The first relates to subjectification, whereas the two latter relate to
teachers’ different expectations of the students’ routines, and creating dialogues.
Narratives on students
As the teachers predict how students will respond to the given task, they are concerned about the
significant gap in students’ mathematical skills and understanding. Teacher 4 (1, 4) and Teacher 3
(5) refer to the students as “students at the top” and “students at the bottom”. The dialogue (1–7)
illustrates how the teachers categorize the students based on their grades. This kind of statement of
the students’ understanding, describing and putting their skills as something (or someplace) the
students are, on the behalf of their former actions, is by Sfard’s (2008) term referred to as
subjectifying. Another example of this kind of subjectifying is given by the teacher talking about
“weak students” (7). Talking about “students at the bottom” or “weak students” is problematic as it
might tend to function as a self-fulfilling prophecy (Sfard, 2008). If you are initially labelled as a
student “at the bottom”, it is hard to motivate the student for further development.
Narratives on student learning
2 “The house-task” is a practical task, which the students have elaborated on in an earlier project. They are supposed to
build a model of a house, including mathematical calculations, in order to complete the task.
As visual mediators, one teacher wants to hand out different three-dimensional figures to each
group of students. Teachers’ prediction indicates that “weak students” choose rectangular prism,
whereas “able students” choose complex shapes. In addition, the teachers have different
expectations of the students “at the top” and “at the bottom”. Firstly, because the teachers are
pleased if students “at the bottom” recall previously endorsed narratives (working on familiar
shapes, e.g. a shape with rectangular base area). Secondly, following Sfard (2008), an interpretation
can be made of the utterance by Teacher 4: “And then it is the bottom there, it is enough just to do
the calculation” (4). “Just to do the calculation” (4) can be seen as a deed. If the students know the
formula, they are able to calculate the volume without necessarily understanding the relation
between base areas and height.
What does it mean “to understand the concept of volume”, and how are the teachers going to find
out whether the students have learned something or not? The first two lines in the second transcript
(8, 9) indicate that in the teachers view, if the students calculate the volume correctly, they know the
relation between volume, base area and height of the shape. These two first lines, viewed as
separate utterances, one could recall as a deed (cf. Sfard, 2008). However, Teacher 3 (16, 20) and
Teacher 4 (18) later stress that the students should explain their procedures to each other, and they
expect students to use endorsed narratives. In this way, they want to observe students’ utterances
and evaluate their reasoning. The teachers thus, have different expectations to theirs students’
routines. Routines for the “weak” students can be seen as a deed, and to recall narratives (4), in
contrast to the “able” students that supposed to sustain and construct narratives as in an explorative
routine (21).
Narratives on teaching practice
In the conversations from the planning meetings, narratives indicate that the teachers consider
learning as participating in an activity. In the “house-task” (10, 13), students with practical skills
were as much participants in solving the task (building the model) as the students who did the
mathematical calculation. The teachers want students to explain to each other their mathematical
thinking and understanding. Also, they claim that by listening to fellow students, it is easier to
construct, substantiate or recall endorsed narratives. If some students do not understand the task, the
teachers stress that in the learning process, the students’ own mathematical language can be more
helpful for fellow students than the explanations provided by themselves as teachers. A main focus
in their discourse on teaching practice is to facilitate dialogue among students, where students help
other students to develop new endorsed narratives (16, 18, and 20). Narratives from the reflection
meeting, held after the lesson was conducted, is an account of the teachers’ observations. The
observations revealed that the students only focused on what the right answer was, not why it was
correct, and mathematical conversations between students did not occur. This was one of the main
goals of the teachers, they wanted to create dialogues that invite the students into explorative
routines.
Concluding discussion
From analysis of these three categories of teachers’ narratives on: students, student learning and
teaching practice, three interrelated characteristics can be identified in the teachers’ discourse on
teaching. Firstly, there is the issue of how teachers talk about students’ skills as something static – a
condition – and categorize the students as being “at the top” or “at the bottom”. There are different
expectations about what the students are able to achieve. According to Sfard (2008), this kind of
subjectification might have a negative impact on student learning, because it tends to function as a
self-fulfilling prophecy and affect students’ identity (see e.g., Mosvold, 2015; Mosvold & Ohnstad,
2016). Secondly, the subjectification of students influence the teachers’ expectations of the students’
performances (routines). The teachers predict that students “at the bottom” only understand plain
shapes such as rectangular prisms and calculate the volume without understanding the relation
between the base area and the height of the shape. Together these two aspects affect the teachers’
teaching practice, for instance in the way the teachers organize the students to create student
dialogues, and how they decide to differentiate. My interpretation of their way of differentiating, is
that the teachers want to facilitate all students’ opportunity to construct endorsed narratives (cf.
Sfard, 2008), students need different three-dimensional shapes to work on, based upon their already
known narratives. Thirdly, the teachers assume that learning develops through conversations
between “able” and “weak” students, in which students use their mathematical language and
explore their mathematical thinking and understanding. Warwick et al. (2016) support this kind of
thinking on learning through dialogue. In their study, they accentuate how LS contributes to making
a dialogical space amongst teachers in order to improve future teaching intentions. They advocate
that inter-thinking thinking out loud together creates a good learning environment for the
teachers. Analysis of the teachers’ discourse in this paper indicates that the teachers desire this kind
of learning environment for their students. This study claims that the findings reported on, might be
of interests in further studies of teacher learning in LS. The three characteristics are examples of
how teachers’ discourse on teaching can be identified. If LS processes contribute to change these
characteristics, thus change the teachers’ discourse on teaching, interpretations of teacher learning
in LS can be made (i.e. learning, cf. Sfard, 2008).
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Article
Full-text available
A characterization of recent Nordic research in didactics of mathematics is presented based on the 32 research reports from the Nordic and Baltic countries in the proceedings from NORMA17 (The eighth Nordic conference on didactics of mathematics). Recent Nordic research in didactics of mathematics is observed from several aspects such as choice of problem, theory, method, result, and target for the message. The analyses of the papers on different levels and from a manifold of perspectives build up an image of what Nordic research in DM (didactics of mathematics) contains and represents currently. The closeness and cooperation between researchers in the Nordic countries is characteristic as well as the breadth and variation in the choice of questions, theories, and methods. Research activity seems to flourish most in Norway. Small-scale studies as well as larger projects are visible. A variety of messages about mathematics teaching and learning for all age groups are directed to students, mathematics teachers, teacher educators, and policymakers. Quantitative and qualitative empirical studies dominate. Conceptual or theoretical investigations are rare. Studies of outcomes of interventions, including teaching approaches and experiments are most common, followed by studies of learning and cognition, including problem-solving. There is a need for surveys and overviews as so many new results are exposed.
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