developing teachers’ and teaching researchers’ professional competence in mathematics through...

ORIGINAL ARTICLE

Developing teachers’ and teaching researchers’ professionalcompetence in mathematics through Chinese Lesson Study

Rongjin Huang • Hongyu Su • Shihong Xu

Accepted: 20 October 2013

� FIZ Karlsruhe 2013

Abstract This study examines co-learning of mathematics

practicing teachers and mathematics teaching researchers

through parallel lesson study in China. Two cases are illus-

trated and compared to highlight what practicing teachers

and teaching researchers learned. The practicing teachers

developed their competence in identifying instructional

objectives, improving instructional process, selecting and

sequencing mathematical tasks, and developing professional

vision. The mathematics teaching researchers developed

their professional competence in effectively carrying out

teaching research activities, effectively mentoring teachers,

and deepening the understanding of teaching.

Keywords Professional development �Mathematics teaching researcher � Chinese lesson

study � Parallel lesson study � Practicing teachers

1 Introduction

In China there is a well-established, multi-tiered teaching

research system through which teachers and teaching

researchers work together to design, deliver, and revise les-

sons to promote a high quality of student learning (Huang

et al. 2010; Ma 1999). Based on the observation that there is

‘‘a much greater alignment in China between the interests of

teachers and the interests of researchers,’’ Stigler and his

colleagues (2012) suggested ‘‘the culture of teaching as

public activity, and the infrastructure of teaching research

educational system at all levels support the interactions

between teachers and mathematics education researchers’’

(p. 231). Researchers have documented how Chinese teach-

ers could improve their teaching and develop their expertise

through developing public lessons (Han and Paine 2010) and

exemplary lessons (Huang and Li 2009; Huang et al. 2011)

with the support of teaching researchers in various ways.

Huang and his colleagues (2012), investigating the

characteristics and roles of mathematics teaching

researchers (MTRs) who have worked with and helped

teachers improve their teaching, noted their crucial roles in

mentoring and assessing teaching, conducting teaching

research activities, and supporting implementation of new

curricula. Similar to mathematics coaches and lead teach-

ers in Western countries, the school-based MTRs in China

have proven to be valuable for teachers’ professional

development, building a professional learning community,

and promoting students’ learning (Campbell and Malkus

2011; Huang et al. 2012). However, little is known about

how the MTRs develop their own professional competence

when working with teachers in differing professional

development programs. Thus, an examination of how

Chinese MTRs and practicing teachers co-learn through

participating in a specific teaching research activity, par-

allel lesson study, can provide valuable insight into the

professional development of both school-based mathe-

matics teacher educators and practicing teachers. More-

over, these findings may also have implications for

university-based mathematics teacher educators who are

primarily responsible for teacher preparation but also care

about practicing teacher professional development. This

paper addresses the following research questions: (1) How

R. Huang (&)

Middle Tennessee State University, Murfreesboro, USA

e-mail: [email protected]

H. Su

South China Normal University, Guangzhou, China

S. Xu

Guangdong Academy of Education, Guangzhou, China

123

ZDM Mathematics Education

DOI 10.1007/s11858-013-0557-8

do MTRs develop their professional competence through

mentoring teachers in parallel lesson study? (2) How do

practicing teachers develop their competence in teaching

through participating in parallel lesson study?

2 Background and theoretical framework

2.1 Overview of the teacher education system in China

A three-stage teacher education system has evolved over

the past decade: primary school teachers are trained in 3- or

4-year teacher colleges; junior and senior high school

teachers are trained in 4-year teacher colleges or compre-

hensive universities; and some senior high school teachers

are required to attain postgraduate-level degrees. Candi-

dates from non-teacher education universities can become

teachers if they pass some required examinations, usually

general pedagogy and subject didactics, within a 2-year

probation period. Middle and high school mathematics

teachers typically are graduates of mathematics programs

(pure mathematics or mathematics education). Mathemat-

ics teacher preparation programs focus on profound

mathematics knowledge and advanced mathematics liter-

acy, review and study of elementary mathematics from an

advanced perspective, with a limited teaching practicum

experience (6 weeks) (Li et al. 2008).

A coherent and institutionalized in-service teacher

education program has been in place for decades (Stewart

2006). Within this system, teachers have developed their

expertise in teaching through participating in various pro-

grams such as apprenticeship practices, school-based

teaching research activities (Huang et al. 2010), and public

lesson development (Han and Paine 2010; Huang and Li

2009). A professional ranking and promotion system,

established in 1993, has evolved for supporting teachers’

professional development (Ministry of Education 1990,

2001). According to China’s secondary school teachers’

professional position system, the positions of secondary

teachers include senior (Gao Ji), intermediate (Zhong Ji),

and primary (Chu Ji). For each level, political, moral, and

academic qualifications are specified. Recently, alternative

professional titles such as exceptional teacher and

advanced teacher, equivalent to university professor status,

have been proposed. This system not only specifies com-

ponents of teacher professional expertise, but also provides

incentives and a culturally supported mechanism for tea-

cher professional development (see Li et al. 2011).

2.2 Teaching research system and teaching researchers

Another infrastructure for supporting teaching research

activities is the teaching research system (Yang and Ricks

2012). Teaching research (Jiaoyan) is a special term that

refers to various activities of professional development at

different levels (school, district, city, or national levels),

and is organized by teaching research institutes (Jiaoyan

Jigou). Teaching research institutes, initially established in

1956 (Wang 2009), have evolved into a hierarchical system

(central, provincial, municipal, and district) hosted in dif-

ferent departments including educational bureaus, educa-

tional science research academies, and curriculum

development centers at both national and local levels. They

are responsible for guiding teaching research, overseeing

teaching administration in schools on behalf of educational

bureaus, providing consultation for educational authorities,

mentoring the implementation and revision of new curric-

ula, building the bridge between modern educational the-

ories and teaching experiences, and promoting high-quality

classroom instruction.

There are more than 100,000 teaching researchers

(inclusive of other disciplines) working in teaching research

institutes (Wang 2009). They play multiple roles, including:

(1) putting forward opinions regarding the implementation

of teaching plans, syllabi, and materials based on local

contexts; (2) providing evidence and suggestions on deci-

sion making for local education authorities; (3) organizing a

variety of teaching research activities at different levels;

and (4) helping teachers study teaching materials, imple-

ment teaching schedules, and improve their teaching effi-

ciency. With the reform of the teacher education system

(Ministry of Education 2001), teaching researchers have

shifted their focus to research on teaching, and guidance of

teaching research and public service for teaching research.

Specific requirements for recruiting teaching researchers

have been set by the Ministry of Education and are further

specified by local education authorities (Huang et al. 2012).

For example, the educational bureau in a big Southern city

requires teaching researcher candidates: (1) be dedicated to

their work, and excellent in teamwork, mentoring, and

administration; (2) be familiar with and have understanding

of the teaching culture, be respected by students, and have

ample experience in preparing exams with an excellent

record of students’ achievement; and (3) have strong

teaching research ability and good writing skills. In sum-

mary, a teaching researcher must be an excellent teacher

with good teaching research ability and leadership.

2.3 An enriched Chinese Lesson Study: parallel lessons

Improving teachers’ professional competence by studying

lessons is the fundamental feature underlying all teaching

research activities in China. Similar to Japanese Lesson

Study (e.g., Lewis 2002), the Chinese Lesson Study refers to

a model of professional development that includes cycles of

collaboration of lesson plans, delivering lessons and

R. Huang et al.

123

classroom observation, post-lesson debriefing and reflection,

and revision (Huang and Bao 2006; Yang and Ricks 2012). In

addition to similarities shared with the Japanese Lesson

Study in terms of their activity structures, the Chinese model

focuses ‘‘on both content and pedagogical knowledge and

skills, and an open, learner-centered implementation com-

ponent’’ (Lerman and Zehetmeier 2008, p. 139).

Recently, Parallel Lesson Study (PLS), an enriched

Chinese Lesson Study, has become a very popular model

embedded in the notion of exemplary teaching (Klafki

2000) and responds to the call of new curricula which

require teachers to creatively and innovatively use their

textbooks in their classrooms to provide differentiated

instruction for their students (Li 2009). To conduct a PLS,

typically a core topic is selected based on extensive dis-

cussions among teachers and MTRs. A lesson study group

usually consists of a MTR from a district educational

bureau, a master teacher, a demonstrating teacher (who

takes the main responsibility for developing and teaching

the selected content), and other mathematics teachers.

Through the process of Chinese Lesson Study, at least two

independent lesson study groups develop exemplary les-

sons of teaching the selected content. Then, a teaching

research activity at the cross-district level is organized,

inviting teachers from different study groups to demon-

strate their respective lessons. A post-lesson meeting

focuses on comparing and contrasting the public lessons.

2.4 Co-learning of teachers and teacher educators

Based on document analysis and a case study, Huang et al.

(2012) concluded that MTRs could help teachers develop

their professional competence through giving public lec-

tures, organizing teaching research activities, and individ-

ual supervision of developing public lessons. However, the

mechanism and process by which MTRs learn is largely

unknown.

To examine teachers’ and teacher educators’ learning,

Jaworski (2001, 2003, 2008) developed a theory of co-

learning between teachers and teacher educators in pro-

moting classroom inquiry whereby teachers and teacher

educators learned from operating with and reflecting on

three levels of activities:

1. Mathematics power: mathematics and provision of

classroom mathematical activities for students’ effec-

tive learning of mathematics;

2. Pedagogical power: mathematics teaching and ways in

which teachers think about developing their

approaches to teaching;

3. Educative power: the roles and activities of teacher

educators in contributing to developments in levels one

and two (Jaworski 2001).

Specifically, Zaslavsky (2008) proposed a model that

provides insight into the role of teacher educators as

designers and orchestrators of tasks that foster teacher

learning, while highlighting the dynamic nature of teacher-

educators’ practice and development. In Zaslavsky’s model

the crucial mediating objects are mathematical tasks. By

extending tasks to lessons, we revised the model as shown

in Fig. 1. In this model, the object examined is lessons; the

learners (practicing teachers) construct their knowledge

through developing lessons and reflecting on their work.

Meanwhile, the facilitator (MTR) develops their knowl-

edge through mentoring lesson development and reflecting

on learners’ work.

Constantly reflecting on-action and in-action is a fun-

damental feature of mathematics teacher educators’ learn-

ing (Cochran-Smith 2003; Jaworski 2008). Teacher

educators could develop their capacities through ‘‘reflec-

tion, inquiry, research and writing, and mentoring or co-

mentoring’’ (Jaworski 2008, p. 355).

In this study, co-learning of teachers and teacher edu-

cators in a community of inquiry that pursues excellence in

teaching through a lesson study approach is the overarch-

ing framework. We examine how teachers and teacher

educators co-learn through working on the mediating

object of exemplary lesson development.

3 Methods

3.1 Setting and participants

Two MTRs from different districts voluntarily participated

in this study. Mr. Zhu, a teaching researcher in charge of

teaching research activities at the middle school level in

district A, formed a lesson study group A. Mr. Wu is the

demonstrating teacher in group A. Mr. Hu, a teaching

researcher in charge of teaching activities at both middle

and high schools in district B, formed another lesson study

group B. Miss Han is the demonstrating teacher in group B.

According to a unified teaching research activity schedule

Facilitator constructs knowledge

Learners construct knowledge

Learners engage inlesson study

Learners try to develop the lesson

Learners reflect on their work

Facilitator reflects on learners’ work

Facilitator supports the lesson development

Fig. 1 Facilitator–learner mechanism of construction of knowledge

through lesson study (adapted from Zaslavsky 2008)

Developing teachers’ and teaching researchers’ professional competence

123

in a Southern city, these two lesson study groups focused

on teaching an activity-based lesson (AL) in seventh grade

on exploring patterns embedded in calendars using alge-

braic expressions. An activity-based lesson, a newly rec-

ommended instruction model in the new curriculum

(Ministry of Education 2011), aims to get students engaged

in mathematics activities that include the process of

observation, experimentation, conjecture making, justifi-

cation, and communication based on their existing

knowledge and experience. These two lesson study groups

had a quite similar teaching research schedule, but each

group developed their AL independently.

3.2 Instruments

3.2.1 Teacher interview protocol

Based on teachers’ lesson plans from initial to final stages,

and teachers’ reflection reports, which identified the major

changes and provided the rationale for these changes as

required by the MTRs, the first author made a list of sub-

stantial changes made by each of the teachers. For example,

in the lessons taught by Miss Han the changes included: (1)

adding a review of basic knowledge related to a calendar

year; (2) designing a game motivating students and leading to

the main topics of the lesson; (3) designing follow-up prac-

tice questions after the first activity; (4) designing various

frames such as diagonal and other combination models as

scaffoldings; (5) creating a challenging application problem;

and (6) improving the summary of the lesson. Surrounding

these changes, the teacher interview protocol focused on the

following: (1) How did you make those major changes? (2)

What major roles did the teaching researcher play? (3) What

have you learned from the process of exemplary lesson

development? (4) Are there any ways to improve the lesson if

there was an opportunity to reteach?

3.2.2 MTR interview protocol

The purposes of the MTRs’ interview include under-

standing the process of PLS, how MTRs supervise teach-

ers, and what they have learned from mentoring the PLS.

Thus, the MTRs’ interview protocol included the follow-

ing: (1) What is the main process of conducting the PLS?

(2) How do you help the teacher(s) develop the lessons

during the process? (3) What specific suggestions have you

provided for making the major changes the teacher made?

(4) What have you learned from mentoring the PLS?

3.2.3 Data collection

Five types of data were collected: (1) lesson plans of var-

ious versions created during the process; (2) teachers’

reflection reports on the learning from participating in the

PLS; (3) videotapes of the final public lessons; (4) teacher

interviews; and (5) MTR interviews.

The third author was responsible for supervising the

implementation of this PLS. This PLS is part of teach-

ing research activities scheduled by the municipal

teaching research institute in which the third author

served as a supervisor (a MTR at city level). The two

MTRs collected and provided a set of data including

lesson plans, demonstrating teachers’ reflection reports,

and videos of the final public lessons. With the help of

the second author, the first author, who did not know

the teachers and MTRs, conducted an interview with

each of the teachers and MTRs via Skype. The inter-

views lasted approximately 40 min, and were audio

recorded. The first author transcribed the records, and

sent transcripts to relevant interviewees for corrections

and confirmations. All interviewees returned their

revised transcripts, which were double-checked by the

first author prior to analysis.

3.2.4 Data analysis

With regard to lesson plans and videos, attention was given

to the final lesson plans associated with the videos to

provide an accurate description of the two exemplary

lessons.

A grounded theory approach (Corbin and Strauss 2008)

was used to analyze the interview data. Regarding MTRs’

interview data, the first author read the transcripts of the

two interviews carefully to make sense of the interviews;

then, through constant comparison with a focus on how the

MTRs learned as facilitators and reflective researchers

through mentoring lesson development, a tentative code

table was developed for each MTR’s interview. Through

comparing the two code tables and consulting with original

transcripts when necessary, a final category was developed.

The MTRs’ gains mainly included learning how to orga-

nize and implement a PLS; learning how to facilitate

teachers; and developing a better understanding of how to

design and teach ALs.

Similarly, teachers’ gains were identified in five cate-

gories: instructional objectives; instructional process;

mathematical tasks; classroom interaction; and profes-

sional vision. In addition, what teachers learned from the

supervision of MTRs was identified as: conception of

instructional design and implementation; and techniques of

instructional design and implementation. To develop tri-

angulation, the first author discussed these categories with

the two MTRs, who confirmed these findings. Then, the

first author developed illustrations of each category by

citing and translating relevant excerpts from Chinese

transcripts.

R. Huang et al.

123

4 Results

The results are presented in three sections. The first

describes the co-learning of lesson study group A; the

second reports the co-learning of lesson study group B; the

third presents how teachers learned from observing

exemplary lessons and collaborating with MTRs.

4.1 Co-learning of Mr. Hu and Miss Han

Mr. Hu, the novice MTR, was an excellent senior teacher

with a bachelor’s degree in mathematics. He had 3 years of

experience in serving as a MTR. Miss Han, the experienced

demonstrating teacher, had a bachelor’s degree in mathe-

matics, with an intermediate professional rank and 6 years

of teaching experience. Miss Han worked collaboratively

with colleagues to develop the lesson, with the support of

Mr. Hu. In this case study a description of the final public

lesson is provided, followed by an overview of the tea-

cher’s learning from participating in the PLS, and the

MTR’s learning from mentoring the teacher.

4.1.1 The lesson

The topic of this lesson is ‘‘mathematics in monthly cal-

endars.’’ Prior to the lesson, students were formally intro-

duced to the use of letters to represent numbers, the

concept of algebraic expressions, and addition and sub-

traction of algebraic expressions. The purposes of the les-

son were to develop students’ ability in using letters to

represent numbers and using algebraic expressions to

express quantitative relationships embedded in life situa-

tions, and to experience the process of discovering and

expressing patterns embedded in monthly calendars.

Through cycles of revisions of lesson plans and trial

teachings, group A developed an exemplary lesson. In an

auditorium, the final public lesson was presented to 53

seventh graders and more than 200 middle school mathe-

matics teachers. The lesson lasted 45 min, with four major

phases: introduction, exploring the main activities, appli-

cation with feedback, and summary.

Introducing The lesson started with questions concern-

ing basic facts about calendars, for example what date is

next Friday (if today is Friday). This line of questions led

to the topic of the lesson: mathematics in calendars. The

teacher then launched four interconnected tasks for stu-

dents to explore and share. The teacher showed a calendar

and asked students, ‘‘What patterns can you find regarding

the consecutive numbers in a row and column?’’

Students reported their observations that numbers

increase by 1 in a row and 7 in a column. The teacher then

organized a game where students had to add three con-

secutive numbers in a row, and tell the teacher the sum. In

figuring out the three numbers immediately, the teacher’s

magic motivated students to explore the strategy to find the

three hidden numbers. Based on the game, the teacher

explicitly raised the question: given the sum of three con-

secutive numbers in a row [in a monthly calendar], find the

three numbers. The teacher then summarized the two pat-

terns based on students’ responses.

The students were then asked to explore extended

problems using the same calendar: first, when observing

three consecutive numbers in a column, what pattern can

you find? Afterwards, students were asked to solve a

similar question with five consecutive numbers.

Self-exploratory task Use the following frames (Fig. 2)

to cover numbers in a monthly calendar and analyze the

patterns of the covered numbers; if the frame is moved, do

the results you discovered remain? Please explain your

results using appropriate methods.

Students presented and explained their patterns (see

Fig. 2a, b) (typically, students raised their hands, and then

one of them was called on to respond while the teacher

wrote the answers on the board; occasionally, a student was

called to the board to explain their answers).

Challenging task Students were asked: When a 3 9 3

grid is moved, the numbers covered by the 3 9 3 grid

frames change correspondingly; are there any patterns

regardless of where it is located? What do you find? How

would you express your findings? Based on individual

exploration and peer discussion, students were invited to

present their findings on the board: (1) if we are given one

of three numbers, we can figure out the other two numbers

in a row or a column (filling out the table as shown in

Fig. 2c); and (2) the sum of all nine numbers equals nine

times the number in the center of the grid.

Application tasks If all odd numbers are displayed in the

following style, can you answer the following questions?

1 3 5 7 9 11

13 15 17 19 21 23

25 27 29 31 33 …

(1) What are the patterns among the numbers in a row

and in a column?

(2) If the sum of two consecutive numbers in a column is

34 in the given arrangement, find the two numbers.

(a) (b) (c)

Fig. 2 a Diagonal model, b plum-flower model, c 3 9 3 grid


123

(3) If you cover nine numbers using a 3 9 3 grid frame,

what patterns can you find?

After individual trials and peer discussions students

solved the first two questions. The third question was left

for after-class activities. The class teacher ended with a

short discussion about the most impressive moments and a

summary of the key points of the lesson.

4.1.2 The learning of Miss Han from participating

What Miss Han learned from participating in PLS is cat-

egorized in five aspects: instructional objectives, instruc-

tional process, mathematical tasks, classroom interaction,

and professional vision.

Changing of instructional objectives Miss Han noted her

shift from ‘‘focusing on finding patterns’’ to ‘‘focusing on

the methods of discovering and expressing patterns, and

mathematical methods underlying these processes of dis-

coveries’’ which guided the whole process of teaching.

Improving instructional process Developing an effective

instructional process is a common concern. Miss Han

mentioned different aspects of instructional procedures: (1)

building connections between the new knowledge and

previous knowledge and experience, connecting mathe-

matics to daily life, and motivating students through

games; (2) giving explicit instruction about activities; (3)

summarization of class/activity; (4) variation practice/

problems; and (5) transition between activities.

For example, Miss Han indicated that the lesson pro-

ceeded more smoothly by ‘‘adding daily introductory

questions such as how many days in a week, and adding

some transition exercises between activity 1 and activity

2.’’ She also realized that using variation problems helped

students develop flexibility by stating: ‘‘In addition to

various illustrative examples, I created a challenging var-

iation problem to explore relevant patterns in an arrange-

ment of odd numbers.’’ Regarding the summary, Miss Han

mentioned her shift from focusing on solving problems to

ways of discovering patterns and relevant mathematical

thinking methods:

I previously asked an open-ended question: What we

have learned today? There are many patterns in cal-

endar problems, what method can be used to help

discover patterns?… Later on, other observation

teachers and the MTR believed that students may

raise several key points by answering the open-ended

question, but it is important to connect these impor-

tant points and emphasize the understanding of the

general methods and the process of discovering pat-

terns. Thus, in the final lesson, I added a slide sum-

marizing the major methods and steps of discovering

patterns.

Selecting and using mathematical tasks Miss Han paid

particular attention to selecting appropriate tasks. She was

not satisfied with the design of the self-exploratory activ-

ities in the exemplary lesson, and expressed her desire for

further improvement in her reflection report as follows:

I will let students design diagrams and explore rele-

vant mathematical relationships in groups. Then, I

will ask them to exchange diagrams among groups so

that each group can explore other patterns. Thus,

students will be deeply involved in activities. But

there is a risk that [responses to] the activities may be

too open to be converged when discussing. But

anyway, the current design is too limited [regarding

the space for students to explore] and I have to revise.

Classroom interaction Although Miss Han learned how

to design an activity, she regretted guiding and talking too

much in the final public lessons by saying: ‘‘In this vid-

eotaped lesson, the teacher heavily led students to complete

the task; due to the teacher’s constraint or some objective

environment constraint, the students were not really

involved. Although they were thinking of patterns, they

were lacking exchanges. There were no group activities

and collaboration.’’

Developing professional vision Miss Han changed her

overall views about teaching, learning, and good lessons,

which has had a long-term impact on her professional

development. She believed that the biggest achievement

she made was:

To understand that the essential goal of mathematics

instruction may not be content points … but letting

students understand how to realize the importance of

using letters to represent numbers through solving

daily life situation problems, and how to apply

algebra to solve real-world problems. In other words,

helping students understand that mathematics comes

from life, for life, may be the essential goal.

4.1.3 The learning of Mr. Hu from mentoring

Based on the interviews with Mr. Hu, the novice teacher

researcher, his learning is related to how to carry out a PLS,

how to mentor a teacher’s teaching, and how to teach an

AL.

How to carry out a PLS Reflecting on the entire pro-

cess, Mr. Hu learned how to effectively carry out a PLS

regarding the selection of demonstrating teachers, the

selection of content, the collection of data during the

process, and the organization of post-lesson meetings.

First, the selection of demonstrating teachers is of utmost

importance. For example, a previous teacher was not

aware of the importance of teaching ALs, and was not

R. Huang et al.

123

willing to accept other teachers’ suggestions, resulting in

replacement of the teacher. Teachers who are not com-

fortable with taking lead roles in developing an exemplary

lesson can participate and benefit from observation and

discussion. As reflected in a statement by the MTR: ‘‘In

the future, we should select young teachers who are

willing to accept different opinions as demonstrating

teachers. So the bidding method will be used to select

demonstrating teachers [who are willing to embrace the

challenges but not necessarily experienced teachers] as the

other district did.’’

Second, it is important to select a topic in which stu-

dents have the appropriate pre-existing knowledge. The

MTR expressed concern about students’ readiness in

algebra that impacted the implementation of the AL as

follows:

The teaching schedule should be considered care-

fully. It is necessary that the students have prior

knowledge preparation in order to carry out an AL

effectively. For example, in this study the students’

knowledge in algebra expression is not strong enough

(based on students’ feedback in the classroom)

because the exploration of these activities requires

students to have the connection of different types of

mathematics knowledge.

Third, it is important to record critical events such as

post-lesson meetings, teachers’ reflection journal, and

teaching researcher journal during the process of carrying

out a PLS. The MTR expressed the need to conduct

teaching research activity more systematically:

The teaching research activity should be imple-

mented systematically and scientifically. It is a pity

that we did not record much important resources in a

timely manner. We only focused on product (e.g.,

final videotaped public lesson), but ignored the pro-

cess of how the teacher improves. In fact, it will be

useful to let teachers know how tortuous it is to carry

out an AL. It will help other teachers to understand

and learn. In the future, we should emphasize both

product and process.

Fourth, the MTR realized the importance of organizing

post-lesson meetings productively. In this activity, the

MTR required teachers to comment regarding the

strengths, weaknesses, and suggestions for improvement

that had been learned from the other district. The MTR also

thought there was a need to specify a framework for

evaluating lessons so that ‘‘the discussions focus on pre-

determined important aspects and freely generated ideas as

well.’’

How to supervise teachers The MTR noted several

challenges involved in the supervision of teachers’

teaching of an AL. When reflecting on the ways to improve

teacher facilitation, the MTR indicated that it would be

helpful for MTRs to teach the AL prior to critiquing

teachers. It is important to facilitate teachers from their

perspective, as stated by Mr. Hu:

I believe that when we realize the weaknesses of

teachers, we should not rashly tell them how to

improve. We should analyze the problems based on

the teachers’ situation, and help them step-by-step.

We cannot be too rushed or too demanding. Some-

times even teachers understand my ideas, but they

may not be able to implement them in their class…Similarly, as a teaching researcher, it would be

helpful to teach the content, get an experience in how

to teach the content before mentoring teachers. Thus,

our supervision will fit teachers’ needs more. We will

understand our teachers more and show less blame.

Regarding teaching an AL, he gave more detail:

Teaching an AL is different from teaching normal

lessons in terms of their focus [ALs usually focus on

developing students’ ability in applying learned

knowledge to solve contextual problems; normal

lessons refer to lessons of introduction of new con-

cepts or skills, application or exercise, and review].

So the supervision method could be different…Teachers could not only teach students how to do it,

rather than help students to get experiences in

abstracting mathematics thinking methods, and doing

mathematical activities, and help students to get

insight into mathematical thinking and methods

embedded in the mathematical activities. In fact,

many teachers adopted traditional teaching methods

in teaching an AL, rather than treating it as another

specific lesson…

How to teach an AL The MTR summarized his devel-

opment of understanding of teaching an AL: the meaning

of an activity, the characteristics of an AL, and the purpose

of teaching an AL. He further explained:

Through this teaching research activity, I have

developed the understanding of a mathematics AL. I

believe that the purpose of an AL is to promote stu-

dents’ development of mathematical thinking. An

activity is the tool and form. Mastering mathematical

thinking methods, accumulating experiences in doing

mathematics activities are the goals. Activities could

include multiple forms such as hands-on, thinking in

your head, oral expressions, writing, etc. Activities

include different levels such as discovering, posing,

analyzing and solving problems, and activity pro-

cesses and cognitive strategies.


123

4.2 Co-learning of Mr. Zhu and Mr. Wu

Mr. Zhu, the experienced teaching researcher, with a

bachelor’s degree in mathematics and a senior professional

rank, had 15 years of teaching experience in middle school

and had served as a MTR for 7 years. He was an excellent

teacher, teaching researcher, and practicing teacher devel-

oper. Mr. Wu, the novice demonstrating teacher, with a

master’s degree in mathematics and statistics, had just

started his first year of teaching. In this second case a

description of the final public lesson is provided, followed

by an overview of the teacher’s learning from participating

in PLS and the MTR’s learning from mentoring the

teacher.

4.2.1 The lesson

Through cycles of revision of lesson plans and trial

teachings, group B developed an exemplary lesson. The

lesson covered the same topic taught by Miss Han, but the

design and implementation were quite different.

The lesson began with an observation of a monthly

calendar, and the teacher asked students to fill out numbers

in a row, a column, and a diagonal diagram (two or three

consecutive numbers) when one number was given. After

students found the patterns based on this activity, they were

asked to express these patterns algebraically and were

encouraged to conjecture the relationships between the sum

of the three numbers and the number in the middle in a

row, and then prove the conjecture using algebraic

expressions. After doing some simple application prob-

lems, students were asked to explore the patterns among

numbers in 3 9 3 grid frames in a monthly calendar. Two

concrete examples were explored, and then a general pat-

tern about the relationships between the sum of these nine

numbers and the number in the center of the grid was

revealed. Students were led to give an algebraic justifica-

tion of this pattern. The teacher ended the lesson by sum-

marizing the main ideas, and assigned two variation

problems for students to explore after class.

4.2.2 Mr. Wu’s learning from participating

Similar to Miss Han, Mr. Wu’s gains from participating in

the PLS fall into the following categories: instructional

objectives, instructional process, classroom interaction,

mathematical tasks, and professional vision.

Changes of instructional objectives Mr. Wu developed

comprehensive instructional objectives with the support of

Mr. Zhu:

At the beginning, I focused on how to solve problems

themselves, how to solve similar problems. Later on,

I gradually developed a better understanding of the

purposes of the lesson. We should focus on how to

find the patterns, and mathematics thinking methods

underlying the patterns. We should emphasize

strengths and usefulness of expressing quantitative

relationships using letters to represent numbers.

Improving instructional process Mr. Wu improved his

instructional process through: (1) building connections

between the new knowledge and previous knowledge and

experience, connecting mathematics to daily life; (2) giv-

ing explicit instruction about activities; (3) summarization

of class/activity; (4) variation practice/problems; and (5)

transition between activities.

For example, Mr. Wu improved his transitions during

the lesson by adding scaffolding questions:

In a trial lesson in my school, colleagues believed

that directly exploring the sum of numbers and the

number in the center of a 3 9 3 grid may be difficult

for low-achieving students. Adding some exploration

with concrete numbers in a row, column, and diag-

onal will help students overcome the learning diffi-

culty. I adopted this suggestion.

Furthermore, he implemented variation by applying

‘‘the principle of practicing with variation problems to

develop students’ logical thinking ability.’’ For example,

after discussing the 3 9 3 grid, he created a ‘‘cross’’ frame

and 4 9 4 frame for students’ further exploration.

Selecting and using mathematical tasks Mr. Wu detailed

how he constructed appropriate scaffolding tasks and de-

constructed inappropriate scaffolding problems. For

example, he explained the process of deconstructing

unnecessary tasks:

In our earlier design of the lesson, students explored

the patterns in the first two tasks adopting the

inductive method; we designed the third task (pattern

in 3 9 3 grid) adopting the same method. However,

in one trial teaching, students directly presented the

patterns using an algebraic expression; thus, we felt it

is unnecessary to explore the concrete cases, partic-

ularly for high-achieving students. Thus, we deleted

these specific cases in the final lesson and encouraged

proving the pattern deductively.

Classroom interaction Mr. Wu made improvements in

balancing between teacher’s guidance and students’ self-

exploration. For example, he designed some problems to be

competitive and ‘‘gave students more time to think and

exchange, thus the class atmosphere became more exciting.’’

Developing professional vision Mr. Wu deepened his

views about teaching, learning, and classroom instruction.

He stated this in his reflection:

R. Huang et al.

123

Mathematics lessons should focus on the process of

exploring and acquiring, rather than understanding

the knowledge itself. The process not only can lead

students to better understanding of knowledge, but

also can lead students to think mathematically when

doing activities, experience the value of knowledge,

increase the awareness of application of mathematics

knowledge, and experience the connections between

mathematics and daily life.

He further illustrated his three major shifts: (1) teachers

have transferred their roles from being knowledge trans-

mitters to being organizers, guides, collaborators, and co-

learners of students’ learning by listening to students; (2)

students become active learners and learn how to learn,

master textbook knowledge, and discover knowledge

through the acquisition of mathematics; (3) the lesson

includes ‘‘fluency, openness, collaboration, and guidance’’

through frequent conversations and peer collaborations

among students and teachers about solving problems.

4.2.3 Mr. Zhu’s learning from mentoring

In his interview, Mr. Zhu summarized his learning in the

following aspects: how to carry out a PLS; and deepening

understanding of how to teach ALs.

How to carry out a PLS Mr. Zhu expressed that it is

necessary to help practicing teachers conduct an AL

because teachers’ quality in such teaching is low. His plan

for improvement includes ‘‘improving the design procedure

of carrying out an AL, and enhancing the understanding of

an AL itself.’’ Specifically, teachers need to learn ‘‘how to

select mathematics activities, how to explore activities, and

how to present activity results.’’ In addition, it will be

helpful ‘‘to provide teachers with systematic training about

how to conduct an AL because teachers often teach an AL

as a transmission of skills rather than focusing on the

development of reasoning ability.’’

Deepening understanding of teaching an AL Mr. Zhu

gained valuable insight about preparing for, implementing,

and reflecting on the PLS. Prior to supervising teachers

carrying out an AL, Mr. Zhu chose to do some preparation:

searching for relevant literature and lesson cases. In par-

ticular, he conducted an AL teaching of congruent trian-

gles, identified several plausible factors related to effective

implementation of an AL, and published his findings.

During the implementation of the PLS his responsibilities

included observing public lessons, reflecting on lessons,

and supervising teachers. After the PLS, he reflected upon

the problems that occurred and sought ways to improve a

PLS.

Second, Mr. Zhu learned how to help teachers under-

stand and implement these ideas in their classrooms. As

indicated in Mr. Wu’s interview and reflection, Mr. Zhu

asked him (and his colleagues) to read some articles to

understand the characteristics of ALs, and gave them some

general suggestions on the selection of problems, explo-

ration of solutions, and evaluation of student work. How-

ever, when observing the AL, Mr. Zhu reflected on the

differences between what was intended and what really

happened. After watching Mr. Wu’s trial teaching, he

reflected in his blog:

The teacher [acts] likes a tour guide, leading students

to visit Grand View Garden regarding knowledge.

The students greatly appreciated the teacher’s strong

foundation. But, from a students’ perspective, they

maybe, like me, became more and more confused. At

the very beginning, the teacher led students to explore

the patterns of three consecutive numbers in a row…and then in a diagonal, and then in a 3 9 3 grid

diagram and so on. My confusions include: why do

we need to study the patterns with three, or five or

seven consecutive numbers, rather than the patterns

with two, four and six consecutive numbers?… They

only know to explore the issues that the teacher

designed by following the methods that the teacher

demonstrated mechanically. It is only to do some

imitation, not favorable for high-order thinking abil-

ity development.

The confusion led Mr. Zhu to realize that the essential

issue is:

How to explore patterns. In this lesson, basically, the

students follow the phases designed by the teacher

step-by-step. The freedom for students’ exploration is

not great enough. The openness of exploratory

problems is not great enough. If students do not know

the directions of exploration, and the methods of

exploration, how can we promote students’ thinking

ability?

Third, Mr. Zhu tried to locate the factors that affect

teachers’ practice in teaching ALs, and develop his theory

of teaching. As discussed previously, the purpose of the AL

should focus on how to explore and discover patterns,

rather than just finding patterns. Mr. Zhu wondered if it

was necessary to purposefully train teachers how to con-

duct an AL.

Fourth, Mr. Zhu raised questions for further exploration.

Why do many teachers not like to teach ALs? How can

ALs really promote students’ thinking and ability? To

address the first question, Mr. Zhu has to convince teachers

that teaching ALs is valuable. In China, teaching is exam-

oriented; therefore teachers feel justified in not teaching

ALs because they do not see the relevance of such teach-

ing. However, based on Mr. Zhu’s experience with grading


123

high school entry examinations, he realized that many

students were failing to solve the challenging problems

because of their weakness in mathematical thinking and

exploration ability. ALs are expected to address such

weaknesses. To address the second question, Mr. Zhu is

learning about improved ways to support teachers to con-

duct an AL.

4.3 Teachers’ learning from watching parallel lessons

In a cross-district teaching research activity, Miss Han and

Mr. Wu delivered their lessons in the same school to two

different classes. They had an opportunity to observe each

other’s lesson, and both teachers noted how valuable this

was. However, the nature of learning differed. Miss Han

commented on other parallel lessons more critically and

developed a new design of the lesson based on these

reflections.

Miss Han gave positive comments on the lesson taught

by Mr. Wu with respect to a clear instructional objective

and reasonable procedures but criticized the appropriate-

ness of the manipulative activity. She gained insight into

the teaching of pattern recognition by watching two other

lessons on the same topic. By comparing the three lessons,

she redesigned her lesson to address two critical issues:

discovering and expressing patterns and developing flexi-

bility in solving problems.

Mr. Wu felt that Miss Han’s lesson was excellent. He

learned many things from her lesson such as how to launch

a task, engage students using games, and use multiple

teaching strategies to differentiate instruction. He further

realized that it is critical to create a relaxed classroom

environment: engaging, encouraging, and appreciating

students during mathematically rich tasks.

4.4 Teachers’ learning from the supervision of MTRs

The teachers specifically expressed what they learned from

the supervision of MTRs: the conceptions of instructional

design and implementation, and the techniques of instruc-

tional design and implementation.

The conceptions of instructional design and implemen-

tation. Miss Han learned how to develop and implement

instructional objectives:

He often asks me what my instructional objectives

are. Did I stick to these objectives? Finally, I speci-

fied overall instructional goals: the process objectives

became more important while the knowledge objec-

tive become less important. Mr. Hu believes that it is

not enough letting students master using letters to

represent numbers and seeking patterns. The objec-

tives should include the experience in the connections

among different concepts, and the thinking methods

of discovering patterns.

Moreover, she noted the critical relationship between

design and implementation as one of the most important

things learned from Mr. Hu:

Teaching designs are only ‘‘intended designs,’’ and

you can improve them endlessly. Others can always

critique your design. You can purposefully adopt

different ideas to develop a reasonable instructional

design. However, when you implement your design,

no matter how you formed your design (reflecting on

your lessons, or adopting others’ comments), you

have to seriously consider students’ learning, asking

questions such as how can I implement for these

students, how can I guide them to learn effectively in

the class.

The techniques of instructional design and implemen-

tation Miss Han also received some specific suggestions

from Mr. Hu, including: (1) specifying the details of the

game design; (2) deconstructing unnecessary scaffolding in

the 3 9 3 grid activity; (3) designing the final challenge

problem; and (4) summarizing key points.

Likewise, Mr. Wu mentioned several helpful sugges-

tions given by Mr. Zhu. For example, in designing the

exploration activities, the MTR suggested providing at

least three cases to explore to make use of inductive rea-

soning. Additionally, in his reflection, Mr. Wu indicated

that he received the following suggestions from Mr. Zhu:

(1) understanding the essence of the problems and their

contexts; (2) summarizing activities and evaluating stu-

dents’ work during and after each activity; (3) identifying

the characteristics of a problem; (4) synthesizing different

methods and integrating numerical and geometrical repre-

sentations; and (5) specifying instructions for activities.

5 Discussion

5.1 Teachers’ learning from participating

The teachers improved their teaching and developed their

professional competences simultaneously. The teachers

made significant improvements to their respective lessons

including setting appropriate instructional objectives,

optimizing instructional procedures, and selecting and

sequencing mathematical tasks.

First, the teachers made substantial changes in their

instructional objectives: from mathematical knowledge and

skills to mathematical knowledge, mathematical thinking,

and mathematical activity experience. For example, Miss

Han shifted her instruction objectives from mastering

R. Huang et al.

123

‘‘seeking patterns’’ to ‘‘thinking methods of seeking pat-

terns.’’ This shift meets the call of new mathematics

standards (Ministry of Education, P. R. China 2011). It is

stated that mathematics instruction should help students to

‘‘understand and master basic mathematics knowledge and

skills, experience and use mathematics thinking and

methods, and obtain basic mathematical activity experi-

ence’’ (p. 42).

Second, the teachers improved their instructional pro-

cedures: introducing, exploring new topics, exploring new

knowledge and skills, practicing with variation, and sum-

marizing. They used different strategies to build connec-

tions between students’ previous knowledge/experience

and new topics. They each provided cognitively highly

demanding mathematical tasks and improved the

sequencing of the selected tasks progressively. Variation

problems were provided to help develop students’ flexi-

bility in applying knowledge. Summarization of the lesson

was emphasized to balance students’ individual reflection

on their learning and teacher-led summaries of key points.

Third, the selection and sequence of mathematics tasks

have been improved significantly. In Miss Han’s lesson, the

game motivates students to explore mathematical patterns

in calendars. Then several deliberately designed explor-

atory tasks paved the way for exploring patterns (3 9 3

grid). The challenging problem involving the odd number

series was provided to develop students’ ability to apply

methods and strategies to new situations. In Mr. Wu’s

lesson introductory tasks and transitions were improved to

lead to discovering patterns, and developing students’

reasoning ability. These findings further support the

observations made by others (Han and Paine 2010; Huang

et al. 2011).

In addition to the specific improvements in teaching the

AL, the teachers have developed their professional com-

petence in instruction that includes their views about

mathematics instruction and teaching ALs, and their

awareness of improving teaching. For example, Miss Han

realized that the ultimate goals of mathematics instruction

should not only include mathematics concepts and skills,

but also more importantly, include mathematical thinking

and methods to help students understand that ‘‘mathematics

comes from life and for life.’’

Through comparing their own teaching with others, each

teacher developed an awareness of improving their teach-

ing and found directions for improvement. The experienced

teacher tried to develop an innovative teaching design

through reflecting on parallel lessons and the novice tea-

cher realized his weaknesses and found areas to improve.

Their increased awareness of strategies of designing and

implementing an AL and the relationship between design

and implementation had a substantial impact on their

professional growth.

5.2 MTRs’ learning from mentoring

The two case studies illustrate how the MTRs learned from

mentoring PLS. By comparing the characteristics of the

two MTRs’ learning one can ascertain that there are more

similarities than differences. Both MTRs realized that their

gains in carrying out a PLS activity involved developing a

better understanding of how to design and deliver an AL.

Since the MTRs had different years of experience in that

role, it would be interesting to notice the subtle differences

between them.

5.2.1 Learn how to conduct a PLS

Both MTRs acknowledged their gains in carrying out a

PLS, but they noted different aspects. Mr. Hu’s (novice

MTR) learning focused on the administrative aspects such

as selecting demonstrating teachers, setting schedules,

organizing post-lesson meetings, and recording data related

to the process of developing the lesson. However, Mr.

Zhu’s (experienced MTR) learning focused on essential

aspects of providing teachers with systematic training

pertinent to conducting high-quality ALs. Mr. Zhu had a

training plan in mind, which included steps on how to: shift

teachers’ tendency of transmitting skills to developing

reasoning ability; design lessons based on a deep under-

standing of ALs; and select, explore, and present mathe-

matically rich ALs. Based on his self-study and

experimentation prior to conducting this PLS, Mr. Zhu

hypothesized how to conduct ALs. He wrote articles dis-

seminating his theories, and planned to conduct systematic

training programs to further test and develop theories about

conducting ALs. This method is similar to conducting

design research to develop theories (Cobb et al. 2003). The

differences between the novice and the experienced MTRs

suggest that novice MTRs developed knowledge-in- prac-

tice while the experienced MTRs intentionally developed

knowledge-of- practice (Cochran-Smith and Lytle 1999).

This finding suggests that although both MTRs learned

from mentoring PLS, it is ideal to shift the pattern of

learning from developing knowledge-in-practice to devel-

oping knowledge-of-practice, which is more transferable.

5.2.2 Deepening understanding of teaching ALs

Teaching ALs is a new endeavor in classroom instruction

to meet the new curricula. Although both MTRs were

excellent teachers with knowledge and skills in teaching

traditional concept and review lessons, they did not have

experience in teaching ALs. Through this PLS activity they

developed a better understanding of teaching ALs: the

purpose should focus on the ways and process of discov-

ering patterns. Teachers should strike a balance between


123

guidance and student self-exploration. The MTRs realized

that it is important to provide pertinent training to teachers

regarding how to select, explore, and present activity

results. In addition, Mr. Zhu raised new questions for fur-

ther investigation. Thus, the experienced MTR intended to

pose and solve problems with a theoretical reflection, while

the novice MTR intended to solve problems with a prac-

tical consideration.

6 Conclusion

This study provided evidence of how practicing teachers

could develop their professional competence through par-

ticipating in PLS. It is encouraging that the participating

teachers developed specific instructional skills such as

setting comprehensive instructional objectives, selecting

and sequencing mathematics tasks, effective transition and

summary, and long-term professional visions such as

beliefs about good lessons.

Meanwhile, MTRs could develop their professional

competence in carrying out Chinese Lesson Study, men-

toring teachers, and deepening understanding of teaching.

Building on their strong knowledge and skills in school

mathematics, teaching mathematics, and conducting

teaching research activities (Huang et al. 2012), it appeared

that the mentoring of PLS provided a promising way for

MTRs to continuously develop their professional compe-

tence as educators. The features of Chinese MTRs’ learn-

ing through mentoring teaching research activities provide

a vivid description of learning through reflection, inquiry,

research, writing, and mentoring (Jaworski 2008).

Huang et al. (2011) argued that repeated teaching of the

same lesson and obtaining immediate feedback from

knowledgeable sources is a form of deliberate practice,

which enhances participant teachers’ instructional exper-

tise (Ericsson et al. 1993). This study further showed how

MTRs could also benefit from mentoring a PLS. That is,

the PLS demonstrates its merits in forming a co-inquiry

learning community (Jaworski 2008) which promotes

teachers’ and teaching researchers’ professional develop-

ment. This professional learning community is rooted in

‘‘the culture of teaching as public activity’’ (Stigler et al.

2012; Li et al. 2011), motivated and supported by two

infrastructures of teaching research and professional rank-

ing systems (Huang et al. 2010).

Studying Chinese teaching researchers and teachers’

learning is an underdeveloped but worthy effort. Although

this study provides a preliminary exploration of a rich

description of co-learning of practicing teachers and MTRs

through PLS, there are certainly some limitations. The data

of this study relies on instructional products and partici-

pants’ reflection and interviews about the lessons, with

limited reflection about the learning process when devel-

oping the lessons. Methodologically, an ethnographical

approach focusing on the process of learning and interac-

tion among participants would provide further insight into

the learning within this distinct, co-constructed profes-

sional community.

Acknowledgments We thank editors, Dr. Gabriele Kaiser, Dr.

Barbara Jaworski, and anonymous reviewers for their invaluable

feedback on earlier versions of the paper. We are grateful to Dr.

Glenda Anthony from Massey University, New Zealand, for her

helpful comments on the improvement of the paper. We appreciate

Mr. Kyle Prince and Mrs. Teresa Schmidt from Middle Tennessee

State University for their help in editing the manuscript. Our thanks

goes to participating teachers and teaching researchers for their

commitment to the Lesson Study and support of data collection.

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