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•# ScienceDirect Procedia

Social and Behavioral Sciences

Procedia Social and Behavioral Sciences 8 (2010) 41-49

International Conference on Mathematics Education Research 2010 (ICMER 2010)

What Mathematics Mentor Teachers Decide: Intervening in Practice

Teachers' Teaching

Chih-Yeuan, Wang*

General Educatoin Centre, Lan Yang Institute of Technology

Abstract

In this paper, we mainly investigate, through the pedagogical critical incidents of practice (CIPs), the ways mathematics mentor teachers intervene in the teaching of practice teachers, and the reasons and underlying values for their interventions, based on case studies of a group of 8 mentor-practice teachers and their student classes in secondary schools. When watching practice teachers' teaching, due to lack of professional knowledge and experiences, mentors may get the feeling that their students are confused about the teaching, or situations of classrooms are not under their control, so that they must deal with the situations at the moment. One general technique mentors might have used is to "intervene-in-action" of practice teachers, when the CIPs appear. According to our raw data, we distinguish the forms of mentor teachers' on-the-spot interventions into three major categories: "active", "passive", and "no" intervention. Two subcategories "direct" and "indirect" intervention are also salient within "active intervention". The preliminary results show that (1) the reasons and ways of mentor teachers' interventions were varied; and (2) mentor teachers developed frameworks of mentoring decision-making closely related to the specific modes of intervention that they chose. © 2010 Elsevier Ltd. All rights reserved.

Keywords: Critical incidents of practice; Mentoring decision-making; Teaching intervention

1. Introduction

Student teachers of secondary mathematics in Taiwan study both mathematical and educational courses in the university, followed by a paid placement of teaching practice at a junior or senior high school as practice teachers. Some experienced school teachers are assigned to be their mentor teachers. This new internship of addressing in-school teaching practice and mentoring plays an important role in Taiwanese teacher preparation programs. It was reported that a novice mathematics mentor switched his role in the one-year mentoring process from 'mentor' to 'co-mentor' and then to 'inner-mentor' (Huang & Chin, 2003). Mentor teachers thus may play different roles to foster the professional development of practice teachers in different periods, for example model, coach, supervisor, helper, guide, supporter, facilitator, observer, evaluator, critical friend, etc. (Furlong & Maynard, 1995; Jaworski & Watson, 1994; Tomlinson, 1995). They may offer practice teachers every opportunity to learn, including designing material, planning lesson, grading, observing mentors or other teachers' teaching, teaching in the mentors' classes, to improve their "mathematical power" and "pedagogical power" (Cooney, 1994). Thus the pedagogy of teachers

ELSEVIER

* Corresponding author.

E-mail address: wcyeuan@gmail.com

1877-0428 © 2010 Published by Elsevier Ltd. doi:10.1016/j.sbspro.2010.12.006

should be at the heart of promoting the professional growth of teachers (Clarke & Hollingsworth, 2002). What most mentors usually do is to organize and offer practice teachers opportunities to teach in a few pre-selected topics, and to discuss the collected critical incidents of practice (CIPs) (Skott, 2001) with them later. We are then interested in understanding the ways and principles of mentors' decisions on intervening in such CIPs.

Shulman (1986) distinguished teachers' professional knowledge into three major categories: subject matter content knowledge (MK), pedagogical content knowledge (PCK), and curricular knowledge. Wilson, Cooney & Stinson (2005) suggested that teachers' perspectives on good teaching includes requiring prerequisite knowledge, promoting mathematical understanding, engaging and motivating students, and organizing effective classroom. Bishop & Whitfield (1972) also suggested that good or effective teachers are those who are aware of the variables they can control, aware of the likely effects of manipulating these variables in different ways, and able to manipulate them so that they can achieve what they regards as effective learning. As novices in the profession, the practice teachers might be unable to understand fully what and how students think, to represent accurately what subject content they know, to manage the classroom situations effectively, as Ponte, Oliveira & Varandas (2002) observed that "it is not enough for pre-service mathematics teachers to have knowledge of mathematics, educational theories, and mathematics education" (p. 96). As can be foreseen that the majority of practice teachers are deficient in the professional knowledge required and they are not yet good or effective enough in teaching, so school mentors are mainly responsible for this. Nilssen (2003) also agreed that mentors should endeavour to develop student teachers' understandings of child-centred approaches to teaching and pupil learning in the subject.

In general, school mentors possess two different identities. On the one hand, they are mentors of teaching for the practice teachers; on the other hand, they still are teachers for the students. Although they offer the practice teachers opportunities to teach in their classrooms, but at the same time, they must consider the learning of his or her students. When watching practice teachers' teaching, due to lack of professional knowledge and experiences, mentors might get the feeling that the students are confused about or ignorant of what the teaching is going on, or classroom situations are not under teacher's control, so that they must deal with the situations at the critical moment. Bishop (1976) considered critical incidents (CIs) as the teaching events where the pupil(s) indicated that "they don't understand something, by making an error in their work or in their discussion with the teacher, or by not being able to answer a teacher's question, or by asking a question themselves" (p. 42). Lerman (1994) described CIs as "ones that can provide insight into classroom learning and the role of the teacher, ones that in fact challenge our opinions and beliefs and our notions of what learning and teaching mathematics are about" (p. 53), and "critical incidents are those that offer a kind of shock or surprise to the observer or participant" (p. 55). Skott (2001) further addressed that CIP possesses the feathers of offering potential challenges, requiring decision making, and revealing conflicts. In the light of this, CIPs can be conceived from both teaching and mentoring aspects, because the incidents invoke the conflicts and challenges of mentors' beliefs and values, as well as thinking about their roles or identities from both a teacher and a mentor's stand for making the best on-the-spot decisions on the teaching-mentoring process.

It is likely that when teachers become more experienced in their teaching, then a kind of decision schema or criteria develops (Bishop, 1976). The teacher's value structure also monitors and mediates the on-going teaching situation, connecting choices with criteria for evaluating them, and then they carry out the decisions in a consistent manner (Bishop, 2001). Gudmundsdottir's (1990) research indicated that teachers' PCK has been reorganized to take into considering students, classrooms and curriculum revolving around their personal values, in other words, the values decided what teaching methods are important for students' learning the teachers believe. Decision-making is therefore an activity at the heart of the teaching process (Bishop, 1976). We then consider mentors as decision makers in mentoring, paralleled to the view of teachers as decision makers in teaching, in this case, a decision-making system of mathematics teaching both informs and is informed by a decision-making system of mentoring. When mentoring CIPs occur and mentors encounter the conflicts and challenges of their beliefs and values, whether the factors underlying these incidents occur are due to lack of practice teachers' professional knowledge or capacity of managing classrooms, the value judgments must be activated (Goldthwait, 1996) and decide how they should do at the moment. One general technique a mentor might use is to "intervene-in-action" of practice teachers when such CIPs appear. Our interest is to describe the values underlying decision-making for mentoring.

2. Methodology

The case study method, including classroom observations, pre and post-lesson interviews, and mentor-tutor conferences, was used as the major approach of inquiry to investigate the values of mathematics mentors. The systematic induction process and the constant comparisons method based on the grounded theory (Strauss & Corbin, 1998) were used to processing data and confirming evidence characterized the method of our study. Eight mentors (Mi, i=1—8) and their practice teachers (Ai, i=1—8) and students (S) were participated in the 2005 academic year as the first of this 3-year longitudinal case studies on the development of mentors' "educative power" (Jaworski, 2001). Mi are all mathematics teachers with at least 4 years of teaching experiences, but most of them might have insufficient experiences in mentoring Ai. We as both the researchers and tutors visited every Ai twice during the academic year, one in the first semester and the other in the second semester, observed Ai's classroom teaching with Mi and interviewed Mi in the later mentor-tutor conferences. The classroom observations were focused on collecting CIPs of teaching and how Mi would react when the CIPs occurred, and what ways Ai interact with Mi and S. The post-lesson interviews helped us clarify and consolidate our observations, and we could explore the principles and underlying values of the Mi' decision-making. And all classroom observations and post-lesson interviews were tape recorded and later transcribed.

3. Findings

All mentor teachers in our research appoint their practice teachers to teach in their classes. Through classroom observations and interviews, first we find that mentor teachers asked practice teachers to design material, plan lesson or carry out pre-lesson micro teaching, and then they mutually discussed and modified the contents before and after the lessons. Secondly, some mentor teachers intervene in practice teachers' in-lesson teaching, either intentionally or unintentionally, through self interventions or pushing some students to ask questions for him/her. Generally speaking, we conceive the above occasions as the kinds of mentor teachers' intervention in practice teachers' classroom teaching. Whatever tactics mentor teachers adopt to monitor practice teachers' classroom behaviours, they must make decisions regardless of pre, in or post lesson. At the same time, the values or principles behind these on-the-spot mentoring decisions could also be activated and become salient.

Because our research focuses primarily on the forms that mentor teachers intervene in the in-lesson teaching of practice teachers, thus according to the data collected through classroom observations and interviews in mentor-tutor conferences, we distinguish initially the manners of mentor teachers' in-lesson teaching interventions into three major categories: active intervention, passive intervention, and no intervention. Two subcategories direct intervention and indirect intervention are also salient within active intervention category. Outline of these categories and mentor teachers' major concerns are given in table 1. We will describe in detail the transcripts and interpretations of some representative CIPs and mentor teacher interviews related to the active and passive interventions.

Table 1. Categories and concerns of eight mentor teachers' on-the-spot interventions

Categories Active intervention Passive No intervention

Direct Indirect intervention

Cases M3,M5 M5 M6 M1,M2,M4,M7,M8

Major principles of intervention for the mentor teachers Caring about students Concerning teacher self-esteem Supporting teacher authority Caring about students Caring about students Considering professional identity Concerning teacher self-esteem Supporting teacher authority Solving problems Accumulating experience

After introducing the concept of 'the equation of circle', A3 asked students to do the exercise: 'Find out the shortest and longest distance between point P(-3,5) and circle: x2+y2-2x-4y-4=0, and the coordinates of these points'. A series of teacher-student dialogues were then developed as follows:

A3: Given an equation of circle and a coordinate of point, what is the shortest and longest distance between point P(-3,5) and circle: x2+y2-2x-4y-4=0, and the coordinates of these points? (A3 drew a circle on the blackboard)

A3: Where can we find the nearest point? (A3 drew the point P outside the circle)

S1: (The first student's response) Teacher, why the point P is outside the circle?

A3: It must be outside the circle according to the meaning of the question.

S2: Teacher, if the point P is inside the circle, how would it be?

A3: We can't do it if the point P is inside the circle.

S3: Why not?

A3: Maybe we can do it, but... (A3 was thinking)

M3: We can do it either the point P is outside or inside the circle, but just the answers will be different.

A3: Right, we can do it regardless where point P is. (A3 continued the lesson)

In CIP1, we found that A3 was too urgent in solving the problem through the action of "drawing the point P outside the circle". He didn't consider that students might trouble in seeing the exact position. Consequently, one student came up with "why the point P is outside the circle?" and the other was then asking "if the point P is inside the circle, how would it be?" We think that he was lack of understanding students' mathematical experiences while learning the topic. His subject matter content knowledge was also questionable of saying "we can't do it if the point P is inside the circle" and "maybe we can do it, but.". When A3 was thinking the students' questionings, M3 stated directly that "we can do it either the point P is outside or inside the circle, but just the answers will be different". M3 considered the content A3 taught might let the students confuse or misunderstand, even influence their future learning, so he had to clarify it immediately. We viewed CIP1 as a teaching CIP, because it resulted in challenging A3's teaching and leading M3's active intervention directly.

In the post-lesson interview, we asked M3 what problems there were in the CIP1. He said that "A3' trouble was that he sometimes thinks students all understand the contents; so, he drew point P outside the circle directly today. And his subject matter content knowledge was more or less problematic; it might then have embarrassed the students". When we asked the principles of his sudden intervention, he indicated that "if practice teachers let students confuse due to their faults or misleading, and then might further influence the learning of students, I will intervene in their teaching immediately". In the interview, he described his underlying belief for this intervention as "the most important thing what teachers must consider in teaching is the learning of students". We asked why he had to intervene in A3' teaching actively and immediately, and whether it would attack A3 self-esteem, teaching authority and the students' feelings about him. He mentioned that "when students having the reflection and question, if I didn't deal with and clarified it at the moment, maybe they would forget it after some days and the misunderstanding would still remain in the mind of students" and "the students' feelings about him were not so bad, I was just addressing problem and I didn't intend to take the lead". In this case, when mentor teachers think that the subject matter content knowledge of practice teachers was problematic and it could let the students confuse, then they may actively intervene in the teaching directly; and the most important focus for them is on student's learning.

A5 was lecturing the topic of 'the equation of circle'. After introducing the standard form and the general form of circle, she prepared to interpret how to transfer the general form to the standard form by squaring and the discriminant of circle. A series of teacher-student dialogues were developed as follows:

A5: We can transfer the general form of circle ' x2 + y2 + dx + ey + f = 0 ' to the equation

( d ^2 ( e ^2 d 2 e2

' 1 x + d) +1 y + ~2 ) = +'- f ' by squaring. But we are still unable to ascertain whether it is a

d2 + e 2 - 4 f

circle or not. We can discover the term-— is critical, because it is the 'radius'' in terms of the

standard form of circle. The radius must be larger than zero, and then the equation above can represent a

circle. We call d2 + e2 - 4 f the discriminant of circle.

(M5 encouraged the student (S1) near her to ask A5 some question)

d2 + e2 - 4 f

S1: Teacher, the term---— is the radius or the square of radius?

A5: Oh, yes. It is the square of radius. Hey, very good. All of you don't be fooled, and it is the square of radius in fact. (The whole class was laughing)

A5: So, when d2 + e2 - 4f > 0 , then x2 + y2 + dx + ey + f = 0 is a circle. Can you tell me where the position of centre of the circle is? - d - e 2 , 2 A5: And the radius?

S : Vd2+e2 -4f i: 2 .

(A5 continued the lesson)

In CIP2, we found that A5's subject matter content knowledge was little questionable by the state of "we can

d2 + e2 - 4 f

discover the term---— is critical, because it is the radius". M5 thought that A5 made a serious mistake

which would confuse the students' correct conceptions and influence their future learning. However, we observed that M5 didn't intervene in her teaching directly, but she encouraged a student near her to question. We felt that M5 intended to correct A5's mistake and clarify students' misconception through 'the agency of question'' indirectly. We could appreciate M5's intention according to A5's reflection of "Oh, yes. It is the square of radius. Hey, very good.

Jd2 +e2 -4f

All of you don't be fooled, and it is the square of radius in fact", and Si's response of "----". We

considered CIP2 as a teaching CIP, because it resulted in challenging A5's teaching and leading M5's active intervention indirectly.

In the post-lesson interview, we asked M5 why you encourage student to question rather intervene in A5's teaching by yourself. She indicated that "I once intervene in her teaching of some topic directly, but I reflect whether she is hurt or not. So, I decide not to intervene in her teaching of the topic". And then she said that "but she shows the term d2 + e2 - 4 f

---— is radius, students will think it is correct if not to remedy it in time. The misconception will influence

the future learning of students". We further clarified the question above, and she answered that "I encourage student to question based on these reasons above, but I don't do it until the problem is enough serious to influence students' learning; that is, her subject matter content knowledge is problematic". We asked her that "you directly talked to her before, but you designate students as your agency now". She mentioned that "yes, I designate students to do it, and it is such thing that must be clarified. At the same time, I hope students to understand they can also ask questions when practice teachers are teaching". In the case, although mentor teachers think that the subject matter content knowledge of practice teachers was problematic and students are confused, but they actively intervene in the teaching indirectly. Although the most important focus for them is the student's learning, but they also concern practice teachers' self-esteem and support their authorities.

A6 was lecturing the topic of 'the formulary solution of the system of linear equations'. She illustrated the operation of determinant expansion in 'Cramer's rule'. When she introduced 'normal vector' and 'vector product' with determinant expansion and suddenly got a feeling that the content of teaching was out of her control. A series of mentor-practice teacher dialogues were then developed as follows:

A6: Mentor, do I speak far away from the topic? I connect it with the meaning of geometry. (A6 was looking at M6)

M6: You can't go back to the beginning now. (The whole class was laughing) You can ask them, and then

you would perhaps understand their problem through their facial expressions. A6: I need help (from M6).

M6: Let me take it over. (The whole class was laughing and clapping again)

M6: (To the whole class) A6 is lack of teaching experience that you all understand, isn't it? (M6 took over the teaching and finished the lesson)

In CIP3, we found that A6's subject matter content knowledge was alright, but she was just unable to adopt a more accessible way of introducing the concept. We thought that A6 and M6 were aware of the condition by "Do I speak far away from the topic? I connect it with the meaning of geometry" and "You can't go back to the beginning now". Although M6 was aware of some students' confusions, but he didn't intend to intervene in A6's teaching in the beginning; he intervened until A6 asking for help. M6 conceived that A6's problem was about pedagogical content knowledge and teaching experience rather subject matter content knowledge, so that he was just observing how A6 would do with the situation till A6 conveying the signal for help, so he was forced to intervene in A6's teaching. We considered CIP3 as a teaching CIP, because it just challenged A6's teaching and led M6 to passive intervention.

In mentor-tutor conference, M6 confessed that he would not have taken A6's teaching if she did not ask for immediate help by saying that "no, I just observe how she deals with the condition; I play the role of an observer". We asked if A6 encountered difficulties in teaching but didn't ask for help, then whether he would help her or not? He answered that "I would certainly not intervene in her teaching, since my roles are observer and mentor, not a teacher, at the moment, and I have no reason to intervene instantly when time is sufficient for me to lead the students to re-visit the concept later". We then asked M6 "if you consider the students' learning at that moment". He then indicated that "she is just a bit lack of pedagogical content knowledge and teaching experiences, her subject matter content knowledge is alright" and "she just uses a more complicated method to illustrate the subject, if she is unaware of using a simpler method then I will correct it next lesson". But M6 took the lead to lecture the content finally, he said to us "in such situation, the teaching process couldn't be gone on well, so I was forced to intervene in her teaching at that critical moment". Therefore, if mentor teachers think that the subject matter content knowledge of practice teachers is unproblematic and is just lack of general teaching experiences, they are not necessarily intervening in teaching on the spot, and may just talk to practice teachers in after lesson or correct later by themselves. Sometimes the mentor teachers are forced to intervene in the teaching of practice teachers due to their expectations and invitations (for help).

Most of our mentor teacher cases didn't intervene immediately in the in-lesson teaching of practice teachers except the three mentor teachers in CIP1~3. In post-lesson interviews or mentor-tutor conferences we checked with them about the reasons or principles of not intervening at the moment although Ai's teaching seemed to be inappropriate or problematic. For example, we asked M1 "why you don't intervene in A1's teaching", and she answered "I will talk about the teaching with her post lesson, and I wish that she revises the content of teaching by herself" and "I think that I should concern her self-esteem and support her authority". When we asked M4 about why he didn't intervene in A4's teaching even though A4 needed help from him, he indicated that "I just remind him how to do in private, he must confront and deal with these questions by himself" and "I think that he will encounter some difficult in the future and he must face and solve it of his own, and this event is a valuable experience for him to learn and feel". And then we asked M4 that whether he would intervene in the teaching of A4 on the spot in the future. He said that "It is not good to intervene in practice teachers' teaching immediately, as they would feel frustrated. I could make it up in the following lessons, and it is better not to intervene on the spot". We asked M7 that whether he would intervene in A7's teaching when some troubles are apparent. He mentioned that "I would let him finish the lesson by himself, and correct the mistakes or solve the deficiencies in next lesson". As we observed that M8 assisted the teaching of A8 when students were doing exercises, and then we asked him "why?" in the post-lesson interview. He described that "It is no good to intervene in her teaching on the spot I would remind her either before or after the lesson, and usually she did very well".

In terms of the above exemplary CIPs, we find that the teaching CIPs of practice teachers appear when their professional knowledge is not properly used or their teaching decisions are moving toward an inappropriate direction. At the same time, mentor teacher views these CIPs as mentoring CIPs and uses them as the opportunities to guide practice teacher' professional development. Thus, mentor teachers may adopt a variety of ways and strategies based on their value priorities to intervene in practice teachers' teaching. In our research, we find that mentor teachers adopt different forms of intervention including active, passive or no intervention to intervene in the in-lesson teaching of practice teachers, and at the moment, they also reveal the varied underlying values about

mathematics mentoring. We think that different forms of intervention should reflect different underlying values which include caring about students, concerning teachers' self-esteem, supporting teachers' authority, considering teachers' professional identity, accumulating experience, solving problems and so on. Because mentor teachers possess two different professional identities, they must activate their value judgments when they make their own decisions of intervention or not on the spot. Although major concern of the mentor teachers was "caring about students" no matter active or passive intervention, but still others considered "the self-esteem, authority or identity of practice teachers" as the first priority, in this case, they adopted the forms of indirectly active, passive or no intervention to intervene in their teaching. Moreover, mentor teachers may change their forms of intervention in different periods through reflection on her action of intervention (Schon, 1983), M5 modified the form of intervention in separate periods of one-year mentoring process. Does this case suggest that mentor teachers could convey their value priorities when they make decisions of mentoring strategies about teaching interventions?

4. Conclusion

4.1 Understanding the varied principles for and forms of teaching intervention

The forms of, and enacted principles for mentor teachers' interventions in classroom teaching practice are varied depending on the values upheld, and different intervention forms entails different underlying values. We find that what the mentor teachers indicate most frequently is about the shortage of practice teachers' professional knowledge, teaching experiences and management capabilities; and what they concern most is the learning of students. But there were mentor teachers who did not intervene in the CIPs where practice teachers were teaching, even if the occasions that they had professed were appeared eventually. We understand that mentor teachers' active or passive interventions are informed by some underlying values; however, there appear to be other values being activated when they decide not to intervene at the critical moment in the teaching of practice teachers. We also find that some teachers' mentoring strategies were changing in the forms of intervention at different tutoring periods due to self reflection-on-mentoring, indicating their values priorities about concerns in the mentoring process. So, the affective dimension of mathematics mentoring seems to be salient for the enactment of the mentoring principles which in turn influences mentors' approaches of classroom teaching intervention. What other aspects would mentor teachers concern? Are there other underlying values and specific enacted principles for mathematics mentoring except those we have discovered?

Our previous proposal of 'distinguishing the forms of on-the-spot intervention into active intervention including direct and indirect format, passive intervention and no intervention' is perhaps oversimplified and needs to be further examined. There might be other forms of intervention in mentoring. For example, first, mentor teachers may invite other mentor teachers to observe practice teachers' teaching in their own classes together, and then open the teaching of practice teachers to other mentors' comments. Secondly, mentor teachers could encourage practice teachers to observe other mentor or mathematics teachers' teaching, and then share and interchange substantial ideas of and about teaching mathematics with practice teachers. Thirdly, mentor teachers could arrange practice teachers to teach other mentor or mathematics teachers' classes, and then involve other school teachers in a form like "co-mentoring" (Jaworski & Watson, 1994). Fourthly, mentor teachers may invite university tutors to engage in mentoring jointly. Finally, we may extend the forms of intervention to include any formal or informal forms of interventions before and after lesson.

4.2 Developing the framework of decision-making for mentoring

When mentor teachers decide whether they should intervene in the teaching of practice teachers or not, as if they make decisions about mentoring, their underlying values and beliefs about mentoring are likely to be activated at that moment. The hidden mentoring decision-making systems then may play similar role, forcing mathematics mentor teachers to judge and decide while mentoring, as the teaching making-decision systems for mathematics teachers. In this case, mentor teachers enact their value structures about mentoring through the relevant knowledge, beliefs and experiences, the structures monitor and mediate the on-going teaching-mentoring situations. Facing to the selected teaching CIPs, they make choices in terms of certain intervention criteria and then carry out the resulting decisions in mentoring; and the criteria and choices may reorganize mentor teachers' values structure, it

might reveal other values priority in the next intervention. Some mentor teachers' decision-making systems are fairly stable, so their forms of on-the-spot intervention would not change radically; but other mentor teachers may transfer their decisions according to some specific values judgments in different periods, so their intervention forms are modifiable. What are the components or structures of the expected mathematics mentoring decisions framework? Is it possible that the underlying values of teaching and mentoring decision-making framework are the same?

4.3 Learning-to-see through teaching CIP

The meaning of mentor teachers' teaching interventions is not only for correcting the practices teachers' faults and caring the students learning; the major purpose for the interventions is for education which means to foster the practice teachers' mathematical and pedagogical powers through teaching interventions while mentoring. At the same time, we can view CIPs of teaching interventions as the catalysts to advance mentor teachers' educative power. In other words, mentor teachers learn to understand how to mentor and practice teachers learn to develop the sense of how to teach through those teaching-mentoring CIPs. As Ma (1999) argued that teachers must be able to reorganize what they know in response to a specific context. To do this, Ball & Bass (2000) proposed several ideas asking mathematics teachers to de-compress their own apprehensions of mathematical knowledge, to de-compose mathematical contents by considering the diverse possible trajectories of enactment and engagement, and to unpack their own highly compressed understandings that are the mark of expert knowledge. Thus, in sharing these three aspects with the practice teachers, mentor teachers should also assist them to engender the processes of decomposition, decompression and unpacking for teaching mathematics. At the same time, mentor teachers should learn to de-compress their apprehensions about mentoring knowledge, to de-compose their mentoring strategies and to unpack their own highly compressed understanding of mentoring knowledge, to improve the professional knowledge of practice teachers, paralleled to the perspectives of Ball & Bass.

But most of the mentor teachers we studied were still beginners in mentoring, so, 'how to discover and effectively use these CIPs of mentoring?' is a question worthy to be re-examined. For example, in post-lesson interview M6 indicated that "I can see A6's apparent trouble in and struggling with the interaction of the students, but I don't know how to let her know about this?" We expect that practice teachers learn to develop their mathematical and pedagogical powers, and meanwhile mentor teachers learn to develop their educative power through their own CIPs; that is, mentor teachers and practice teachers can both "learn-to-see in mentoring" (Furlong & Maynard, 1995), and engender their own professional growth through "the co-learning cycle of teaching and mentoring" (Huang & Chin, 2003). What are the de-compressing, de-composing and unpacking processes of mentor teachers' professional teaching-mentoring knowledge? How and in what ways mentor teachers help practice teachers to de-compress, decompose and unpack their professional knowledge of teaching?

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