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Lesson Study and the development of professional teaching practice in

mathematics

Anita Tyskerud

University of Stavanger, Faculty of Arts and Education, Norway; anita.tyskerud@uis.no

To investigate what teachers learn from Lesson Study (LS), focusing on teaching practice and student

learning in mathematics, this study uses a coordination of activity theory and the commognitive

theory as theoretical and analytical framework. LS and teaching practice are considered as object-

oriented activities, and learning is regarded as a change in discourse. Analysis of the group members’

discourse, provide an indication of what the teachers have learned during LS processes. This paper

presents some of the findings from the study, which investigates the discourse from the first cycle.

The findings elaborated upon here, provide a foundation for further work by identifying the discourse

in the first cycle. This makes it possible to compare the discourses in the latter work – in order to

investigate if the discourse have changed.

Keywords: Lesson Study, mathematics teachers, professional development, teaching practice.

Introduction

Research on 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 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 only the past two decades become interested in LS. Much of the interest in LS in the

western world arose after Stigler and Hiebert (1999) wrote “The Teaching Gap”.

There are two main issues in the process of LS. Firstly, the teachers are conducting research on their

own teaching practice. Together, teachers plan, conduct and evaluate a research lesson in order to

answer their own research question(s). The entire process requires that the teachers are open minded

and want to better understand student learning or uncover new ideas of a particular aspect of the

teaching of mathematics. Secondly, there are two important elements in LS: prediction and

observation. These are crucial in helping the teachers to understand how students learn, and in

addition to develop their own instructional interactions in their teaching practice.

One important finding from Dudley (2013), is that by the opportunity to rehearse aloud, to consider

the effect of the exchanges as if they happened in class, teachers can utilize the tacit knowledge in

their response to improve their practice. The Lesson Study teachers developed other forms of

pedagogical content knowledge; they reported how they were later able to apply and to use new

practice or pedagogical content knowledge in subsequent teaching long after the process was over

(Dudley, 2013). According to Dudley (2013), the role of verbal interaction in the learning process is

essential because new knowledge develops through talk in social interaction.

The data presented in this paper, is taken from a larger ongoing study in a lower secondary school in

Norway. The study – in a sociocultural stance – regard knowledge as shared and collective rather

than individual, which develops through social negotiation (Radford, 2008), and learning as a change

in discourse (Sfard, 2008). Research on human development and learning have become the study of

development of discourse – focusing on what teachers learn with particular attention to teaching

practice and student learning. A coordination of two sociocultural theories – activity theory (Leontiev,

1978) and the commognitive theory (Sfard, 2008) – are used as a theoretical and analytical

framework. Activity theory is used as a grand theory; Lesson Study and teaching practice are seen as

activities. Sfard’s (2008) theory of thinking as communicating is used as a local theory, then learning

becomes a permanent change in discourse. To be able to discover if and in what way the discourse

has changed, identification of the discourse is a prerequisite. The findings elaborated upon here,

explain how the study identifies utterances from the planning stage in the first LS cycle, when a

discussion emerges on how students learn through conversations and when the teachers predict the

students’ mathematical understanding of volume. The following research question is posed:

What characterizes teachers’ discourse on: 1) students’ mathematical understanding of

volume and 2) their perspectives on teaching volume as a mathematical concept?

How the discourse on students’ understanding and teaching practice are identified in the action “plan

the lesson”, serve as a foundation of comparing the discourse on students learning and teaching

practice in the other actions as well as the second LS-cycle.

Theoretical and analytical framework

To Leontiev (1978), all human activities are oriented towards an object with a certain motive. The

activities consist of three dynamic levels: object-motive, actions-goals and operations (Leontiev,

1978). LS and teaching practice is considered as an object-oriented activity. Mosvold and Bjuland

(2016) describe LS as an object-oriented activity by defining the object-motive as “investigating your

own teaching practice to improve students’ learning” (Mosvold & Bjuland, 2016, p. 188, my

translation). They divide the activity into four actions which constitute the four stages in a LS cycle:

1) plan the lesson, 2) conduct the lesson, 3) observe the lesson and 4) evaluate the lesson. Each of

these actions have their own specific goals and operations (Mosvold & Bjuland, 2016). Table 1 gives

an overview of the five operations in the action “plan the lesson”, from which the findings in this

paper are collected.

Plan the lesson (Action) Prepare teaching (Goal)

Study other

textbooks,

curriculum, etc.

(Operation a)

Formulate lessons

goals, research

question

(Operation b)

Select artefacts, design

worksheets, group

work, differentiation

(Operation c)

Predicting

student

responses

(Operation d)

Planning

observation

(Operation e)

Table 1. The goal and operations of the action “Plan a lesson” in the activity Lesson Study

(translated and adapted from Mosvold & Bjuland, 2016, p. 188).

Sfard (2008) considers communication as part of commognition, and different types of commognition

are defined as different discourses. She defines learning as a permanent change in the discourse

(Sfard, 2008), and this change can take place on two levels. Sfard (2008) distinguishes between

object-level learning and meta-level learning. On the object-level, the change in discourse (learning)

expands endogenously. In contrast, meta-level learning involves a change in metarules and relates to

exogenous change in discourse. The latter can only occur if there has been a commognitive conflict

e.g. that two individuals use the same word, but with different meanings.

A mathematical discourse is characterized by four critical, discursive properties: word use, visual

mediators, routines and endorsed narratives. Word use, indicates 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 terms of individualization. This is a process

divided into four stages: passive use, routine-driven, phrase-driven and object-driven. Passive use

refers to hearing the word, without using it one-self, routine-driven refers to using the word in a

concrete situation, and phrase-driven relates to be able to use the word in similar situations. The

object-driven stage 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,

narratives are defined as any sequence of utterances framed as a description of the object, and

endorsed narratives are often labeled as true. Routines are repetitive patterns characteristic of the

given discourse, and divided into three types: explanation, rituals and deeds. The first type of routine

is a “how” routine, rituals are a “when” routine, and deeds considered as a practical action, that is an

action resulting in a physical change.

Objectification is, according to Sfard (2008), an important metaphor in discourse development. It is

a process where discourse on human behaviour and actions develop 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 and data

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

spring 2016. The main data sources are video-recorded observations from the teachers’ meetings. The

utterances on teaching practice and student learning were in the larger study at first categorized into

the four different actions (plan-, conduct-, observe-, and evaluate the lesson), secondly into each

operations. The data presented below provide some examples from the action “plan the lesson” in the

first cycle. The utterances are from one of the meetings before the first research lesson, which was

selected because they accentuate the most prominent focus in the way the teachers talk about students’

learning and teaching practice. The whole conversation takes place in operation c, d and e, the given

examples are all from the second planning meeting. The findings are a part of the second step of the

analysis, when discourse is identified by using the properties of mathematical discourse as described

above. In a third step of the analysis – not yet conducted – an attempt is made to compare the

utterances from the action “plan the lesson” in both cycles, in order to discover a change in discourse

i.e. learning (Sfard, 2008).

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 figures (using formulas). The discussion arises in the operation

“select artefacts, design worksheets, group work and differentiation” (operation c). The transcript

presented below is taken from the operation “prediction on students’ response” (operation d), and the

discussion continues in the operation “planning observation” (operation e). The tasks for the lesson

are not yet decided, neither is whether the students are supposed to work in groups or individually.

Early in the conversation, the teachers have two focus areas: how to differentiate and how to pair the

students in groups (from operation c). They stress that it is important to differentiate, because there

is a significant gap in the students’ mathematical understanding. The following dialog takes place in

this discussion

1

:

1 Teacher 3: There are only students at the top and at the bottom in this class?

2 Teacher 1, 2 and 4: Yes (In unison).

3 Teacher 1: But that is okay, it is like that in some classes.

4 Teacher 3: And then it is the bottom there, it is enough just to do the calculation.

5 Teacher 4: 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.

In the 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. If the latter, they have to consider whether they

should group students homogeneously or mixed. One teacher suggests: 7 “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.” One argument in favour of “mixing the groups”

is that the teachers stress 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 figure with a complex base area than

1

The transcripts have been translated from Norwegian by the author of this paper.

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 students “on bottom” need a figure with single base area, while

the more able student can be given a figure with more complex shapes, for instance a figure 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 handed 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 how the students talk to each other, by observing their

conversations. The discussion proceeds as follows:

8 Teacher 4: 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 3: 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 3: 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 4: So, is there a correlation between base area and volume. (Sitting and writing,

reading what she has written)

15 Teacher 1: Mm

16 Teacher 4: 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 3: I think it is a good idea that they can explain to each other.

19 Teacher 1: Yes, I think so too.

20 Teacher 4: I can write, “They must explain the procedures”.

21 Teacher 3: Most likely, one of the group members is able to solve the task and explain

how.

22 Teacher 1 and 4: Mm

Analysis and discussion

The teachers’ main goal is for the students to understand the connection between the base area and

the height, in addition to calculate the volume of some geometric figures. The example is taken from

the two first steps of the analyses, and provides an impression on how the teachers are planning their

teaching practice. What characterizes teachers’ discourse on students and students’ mathematical

understanding of volume indicates a conjunction with how the teachers think about differentiating,

which in addition, correlate with their view on teaching practice.

Students and students’ 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 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 3 and 4 (turn 1, 4 and 5)

refer to the students’ skills as “students at the top” and “students at the bottom”. Turn number 5

illustrates how the teachers categorize the students based on their grades. This kind of statement of

the pupils understanding, describing and putting the pupils’ skills as something (or someplace) the

pupils are, on the behalf of their former actions, is by Sfard’s term called subjectifying. Another

example of this kind of subjectifying is given by the teacher talking about “weak students” (turn 7).

Talking about “students on 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.

Differentiation

As visual mediators, the teachers want to hand out different figures to each group of students. My

interpretation of this way of differentiating, is that the teachers want to facilitate all students’

opportunity to construct endorsed narratives, students need different three dimensional shapes to work

on, based upon their already known narratives. In addition, the teachers have different expectations

of the students “on the top” and “on the bottom”. The utterances of Teachers 3 “And then it is the

bottom there, it is enough just to do the calculation” (turn 4), I interpret in terms of Sfard (2008) in

two ways. Firstly, “just to do the calculation” can be seen as a deed. If they know the formula, they

are able to calculate the volume without necessarily understanding the relation between base areas

and height. Secondly, the teachers are pleased if students “at the bottom” recall previously endorsed

narratives.

The word use of understanding

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 (turn

8 and 9) indicate that if the students calculate the volume correctly, they know the relation between

volume, base area and height of the figure. These two first lines viewed as separate utterances, one

could recall as a deed. However, Teacher 3 (turn 18) and Teacher 4 (turn 16 and 20) 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.

Teaching practice

In a holistic point of view of the transcripts from the first cycle, the word use in the conversations

from the action “plan a lesson” indicates that the teachers consider learning as participating in an

activity. In the “house-task”-project (turn 10 and 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 teachers’ explanations. One of the main goals for their teaching practice is to

contribute to a dialog among students, where students help other students to develop new endorsed

narratives. Findings from the reflection meeting later in the first cycle after the lesson was conducted,

confirms that there are no mathematical conversations between students. The students only focused

on what the right answer was, not why it was correct. This is crucial to the teachers. It is one of their

main goals. This challenge is further elaborated on in the second LS cycle.

Concluding discussion

The characteristics of the discourse on students’ mathematical understanding of volume and how to

teach this mathematical concept, can be summed up in three interrelated perspectives. Firstly, there

is the issue of how teachers talk about students’ skills as something static, a condition, and categorize

the students understanding “on top” or “on bottom” based upon their grades and correlate the different

expectations on what the students are able to achieve. According to Sfard (2008), this 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 teachers

predict that students “on 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

figure. Thirdly, the teachers assume that learning develops through conversations between “able” and

“weaker” 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

dialog. 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. The teachers presented in

this study desire this kind of learning environment for their students.

The aim of this paper was to illuminate some examples from how the study identifies the teachers’

discourse on student understanding of volume, and how to teach this mathematical concept by

studying the transcript from the action “plan the lesson” in one LS cycle. In the study, learning is

regarded as a change in discourse. Further analysis of the discourse from the other actions in LS is

required in order to investigate if permanent change in discourse or learning (Sfard, 2008) occurs.

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