What Matters Most: A Comparison of Expert and Novice Teachers' Noticing of Mathematics Classroom Events
Abstract
In this study, we examined 10 expert and 10 novice teachers' noticing of classroom events in China. It was found that both expert and novice teachers, who were selected from two cities in China, highly attended to developing students' mathematics knowledge coherently and developing students' mathematical thinking and ability; they also paid attention to students' self-exploratory learning, students' participation, and teachers' instructional skills. Furthermore, compared with novice teachers, expert teachers paid greater attention to developing mathematical and high-order thinking, and developing mathematics knowledge coherently, but paid less attention to teachers' guidance. Moreover, we further illustrated the qualitative differences and similarities in their noticing of classroom events. Finally, we discussed the findings and relevant implications.
Background
Researchers have explored how to enhance teachers' learning through developing their noticing of mathematics classroom events (e.g., Sherin, 2007; Sherin & van Es, 2009; van Es & Sherin, 2008). In a series of studies, Sherin and her colleagues found that teachers can change their noticing of classroom events and improve their ability of reflecting on their own practices through watching and discussing videotaped lessons or lesson clips (Sherin & Han, 2004; van Es & Sherin, 2008). Eventually, teachers' learning that has occurred in the video clubs can be extended to their classroom instruction (Sherin & van Es, 2009). For example, Sherin and Han (2004) described nicely how middle school mathematics teachers can shift their paradigm of noticing classroom events from a primary focus on the teacher to increased attention to students' actions and ideas; further from simple restatements of students' ideas to detailed analyses of students' thinking. Finally, the teachers can reframe their discussions of pedagogical issues in terms of the students' thinking. Findings from these existing studies suggest that helping teachers shift their focus of noticing to critical aspects of classroom instruction is important for them to reflect upon and improve their own practice.
In fact, some researchers argued that improving teachers' ability to notice should be an explicit focus of teacher education by providing appropriate opportunities and a deliberative framework for participants to develop their ability to notice and further develop their professional vision (Jacobs, Lamb, & Philipp, 2010; Santagata, Zannoni, & Stigler, 2007; Sherin & van Es, 2005, 2009; Star & Strickland, 2008). However, existing frameworks used for guiding teachers' noticing are mainly derived from researchers' intentions and perspectives (e.g., Jacobs et al., 2010; Star & Strickland, 2008). Because teachers play a key role in enhancing students' learning, their own beliefs and awareness of effective teaching have directly impacted their attention to certain aspects of classroom teaching. It can be assumed that teachers' views about what constitutes important and effective teaching practices needs to be examined and understood before we will know how to best help them. Yet, little attention has been given to teachers' voices about what they are really concerned about effective teaching (Cai, Perry, Wong, & Wang, 2009).
Thus, this study was designed to examine teachers' noticing of important classroom events when viewing videotaped lessons. Different from the aforementioned studies on what teachers notice in classroom instruction, we aimed to understand what aspects of classroom instruction that teachers may care about the most. To get a deeper understanding of teachers' predilections toward noticing of classroom events, we took an expert–novice teacher's comparative approach in the case of examining Chinese mathematics teachers' noticing. The selection of Chinese mathematics teachers allowed us to build upon our own knowledge and experience in mathematics teaching and teacher education in China. In the current context of learning more about mathematics teachers and teaching in high-achieving education systems in East Asia (e.g., Li & Shimizu, 2009; Ma, 1999), an examination of Chinese expert and novice teachers should also provide an important case for mathematics educators in other parts of the world to reflect on teachers' practice and find effective ways to improve their teaching.
While mathematics teaching and teacher education in China share many similarities with other education systems, there are also some unique features in Chinese mathematics teaching and teacher education. For example, Chinese mathematics classroom instruction is typically presented as teachers' lecture-dominated, large-size, whole-class teaching. However, researchers found that Chinese mathematics classroom instruction is actually coherent, and polished (Chen & Li, 2010; Huang & Li, 2009b; Leung, 1995, 2005; Stevenson & Lee, 1997), with teachers given considerable considerations about students and their learning (e.g., Huang & Leung, 2004; Mok, 2006). Consistently, China has a coherent in-service teacher education system (Huang, Peng, Wang, & Li, 2010; Stewart, 2006) that provides mathematics teachers with plentiful classroom observation opportunities. It can be assumed that frequent classroom observation should help teachers develop a certain ability to notice culturally valued aspects of mathematics classroom teaching.
Furthermore, China has a systematic and official teacher promotion and ranking system (Li, Huang, Bao, & Fan, 2011) that enables us to select expert and novice teachers more objectively. Therefore, a comparison of mathematics expert and novice teachers' noticing of classroom events should help us better understand possible differences and similarities in teachers' noticing of mathematics classroom events. The result should provide mathematics educators and researchers in other education systems an important reference to reflect on culturally valued noticing of classroom events in their own systems. In particular, this study aimed to answer the following questions:
- 1
What important classroom events did Chinese secondary mathematics expert and novice teachers notice in general?
- 2
What are possible similarities and differences in noticing of important classroom events between Chinese expert and novice teachers?
Theoretical Considerations
We first clarify and define the meanings of teachers' noticing of classroom events. Then, we review relevant studies on contrasting expert and novice teachers' noticing of classroom events and indicate its limitations in terms of research methods. Lastly, we justify and describe our research framework.
What Does It Mean to Notice?
Developing teachers' noticing of classroom events has been recognized as an important issue in teacher education (Frederikson, Sipusic, Sherin, & Wolfe, 1998; Sherin & Han, 2004; Sherin & van Es, 2009; Star & Strickland, 2008; van Es & Sherin, 2002, 2008). The definition proposed by van Es and Sherin (2008) is a commonly used description of what noticing of classroom events means. According to van Es and Sherin, noticing should include three aspects: (1) identifying what is important or noteworthy about a classroom situation; (2) making connections between the specifics of classroom interactions and the broader principles of teaching and learning that they represent; and (3) using what one knows about the context to reason about classroom interactions.
In order to guide teachers' attention to noticing, different frameworks were proposed. For example, Frederikson and his colleagues (1998) developed hierarchical categories under four general criteria: pedagogy, climate, mathematical thinking, and management. Similarly, Star and Strickland (2008) adopted the following five categories: classroom environment, classroom management, task, mathematics thinking, and communication. Some researchers only focus on a few elements such as (1) goal(s) of the lesson; (2) student learning; and (3) teaching alternatives (Santagata et al., 2007). However, these mentioned frameworks mainly reflect researchers' intentions to examine teachers' notices, with less attention to practicing teachers' own perspectives.
Expert and Novice Teachers' Notices of Classroom Events
There are some studies comparing experts' and novices' comments on classroom teaching. In their study, Sabers, Cushing, and Berliner (1991) asked seven experts and five novices to view three television monitors simultaneously, with each focusing on one working group of a junior high science class, and to express their thoughts in terms of classroom management and instruction. It was found that expert teachers frequently made sense of events they saw and made evaluative judgments and interpretations about them, while novice teachers often gave descriptive details of what they were seeing.
In the study by Sato, Akita, and Iwakawa (1993), five expert and five novice teachers were asked to watch a videotaped fifth-grade Japanese poetry lesson without stopping it and to express their perceptions and feelings aloud. It was found that expert teachers' thinking contained several characteristics, including (1) thinking in the process of reaction, (2) utilizing multiple points of view and a broad perspective, (3) incorporating active, sensitive, and deliberative involvement in a situation observed, and (4) relating to content, cognition, and context. In addition, by investigating five expert and five novice teachers' comments on a videotaped seventh-grade Estonian grammar lesson, Krull, Oras, and Sisask (2007) found that expert teachers typically noticed more classroom events than the novice teachers, and they were more reflective and talkative in their comments. Expert teachers were more sensitive toward certain instructional events, such as teacher presentation of learning information, teacher guidance to pupils, and enhancing the retention and transfer of learning. In general, they were more concerned with the general teaching strategy and classroom atmosphere.
Several studies about expert and novice teachers documented the differences between these two groups of teachers on mathematics subject content knowledge (e.g., Leinhardt & Smith, 1985) and mathematics teaching performance (e.g., Borko & Livingston, 1989; Leinhardt, 1989). Berliner (2001) summarized and highlighted the following differences: (1) Expert teachers excel mainly in their own domain and in particular contexts; (2) expert teachers are more opportunistic and flexible in their teaching than are novices; and (3) expert teachers are more sensitive to the task demands and social situations surrounding them when solving problems.
In summary, comparative studies on expert and novice teachers suggest that expert teachers are able to: (1) effectively recognize meaningful patterns and make sense of multiple events, (2) selectively attend important events, (3) flexibly utilize multiple perspectives in accordance with different contents, contexts, and cognitive situations, and (4) hypothesize reasons for observed behaviors and offer solution strategies for identified problems. These differences in thinking and actions of expert and novice teachers can be accounted for by assuming that novices' cognitive schemata are less elaborate, less interconnected, and less accessible than that of the experts, and that the novices' pedagogical reasoning skills are less developed (Borko & Livingston, 1989).
However, some issues with respect to expert–novice teachers' comparison arise. First, the different criteria of selecting expert teachers in different studies may have an influence on the research results. Second, the different subject contents being focused on in different studies may also impact teachers' thinking and responses. As far as mathematics education is concerned, it is necessary to compare expert–novice teachers' comments on mathematics lessons with rigorous and objective standards to identify expert teachers.
Focusing on Chinese Expert and Novice Teachers' Noticing of Classroom Events
Comparing experts and novices is a common approach to exploring complex cognitive phenomena such as teaching and teachers' thinking (e.g., Borko & Livingston, 1989; Leinhardt, 1989). While the identification of novice teachers is relatively easy, the identification and selection of expert teachers vary from study to study on the basis of years of teaching experience, students' test scores, or recommendation by superintendents. There is a call for identifying expert teachers based on a systematic and rigorous teacher credential mechanism (Berliner, 2001).
Coincidently, China has a systematic and nationwide teacher ranking and promotion system that positions teachers at different levels according to their educational background, teaching performance, and students' performance and teaching research ability. With regard to secondary teachers, they can be promoted from primary (Chuji), to intermediate (Zhongji), and then to senior (Gaoji). For each teacher rank, there are specific requirements in political, moral, and academic aspects. For example, the conditions for being a senior secondary teacher includes five years or more of serving as a secondary school teacher at the intermediate level or having a Ph.D., and demonstrating the ability to take the responsibility of being a senior secondary teacher. Moreover, the candidates should have (1) either systematic and sound fundamental theory and subject content knowledge and plenty of teaching experience and good teaching effectiveness, or specializing in political and moral education, classroom management, achieving high performance, and acquiring rich experiences; as well as (2) engaging in the education research on secondary education and teaching, writing a study summary, reflection report, and research paper which are of integration of theory and practice at a certain academic level, or making remarkable contributions to the improvement of other teachers' academic level and teaching ability. In addition, there is one honorary rank called “teachers of the exceptional class,” which is provided to senior-ranked teachers who are exceptional in certain aspects (Li, Huang, Bao, & Fan, 2011). Expert teachers in this study refer to those with a senior or exceptional rank. So they are officially conferred and publicly recognized in the system.
Building upon existing studies on expert and novice teachers' noticing of classroom events, we conducted this study through video-stimulated interviews. Specifically, we asked expert and novice teachers to watch two contrasting videotaped lessons. Then, participants were invited to write responses to semi-structured questions with regards to the three aspects of noticing (van Es & Sherin, 2008). By comparing expert and novice teachers, we aimed to explore emerging patterns in teachers' noticing of classroom events in a particular context.
Methodology
Participants
Ten experts and 10 novice teachers were conveniently selected from two east coast cities in China as participants in this study. Expert teachers were defined as teachers who had a senior position (advanced and above position) with more than 10 years of teaching experience. Novice teachers were defined as those who had a junior position with less than three years of teaching experience. All 10 expert teachers were experienced teachers with at least 14 years of teaching experience. Nine of them held a bachelor's degree in mathematics while the other held a master's degree in mathematics education. They either taught at a secondary school or supervised secondary school teachers (hereafter, these expert teachers are denoted as ET1∼ET10). All 10 novice teachers had a junior position with an average of 1.8 years of teaching experience. Five of them had a master's degree in mathematics education and the remaining five had a bachelor's degree in mathematics (hereafter, these novice teachers are denoted as NT1∼NT10). Local mathematics educators recruited all of these volunteer subjects.
Videotaped Lessons
Two videotaped lessons were deliberately selected for this interview. One lesson won the first prize in a nationwide classroom instruction competition in 2002 (Li & Li, 2009) due to its success in reflecting reform-oriented teaching notion to a certain extent (denoted as lesson one [L1]). The other, given by a locally respected expert teacher, attempted to adopt a reform-oriented teaching approach, but eventually demonstrated many traditional features of teaching (denoted as lesson two [L2]). The contrasting features of these two lessons were intended to stimulate and broaden interviewees' opinions on effective teaching. The general processes of these two lessons are described as follows.
L1 consists of the following phases: (1) establishing a contextual learning situation, (2) working on the situational problems through questioning and discussion, (3) exploring new knowledge, (4) consolidation and familiarity of new knowledge through exercise with variation, and (5) summary and assignment. At the very beginning of the lesson, the teacher showed a set of dynamic pictures related to an annular eclipse by using the computer and then posed the following questions to the students: “After seeing these pictures, do you have any questions related to mathematics?” and “Can you state any relation between the positions of the circular shapes of the moon and the sun?” Through class discussions between the teacher and students, students realized that the relationship between the moon and sun could be represented by the positions of the two circles with the use of two different-sized circular plate models. After that, the students were asked to use two prepared circular models placed on a transparent paper to explore the position relationships of two circles by moving them toward or away from each other along the line that passes through the centers of the two circles. Furthermore, students were asked to explore the quantitative relations in line with the position relations of the two circles. Then, the teacher assigned the class a set of exercise problems related to the position relations and relevant quantitative relations of two circles. Finally, the students were invited to share their learning and gains in the lesson, and some homework was assigned to the students at the end. In this lesson, the teacher was successful in adopting several reform-oriented teaching styles (Ministry of Education, P. R. China, 2001), including introducing new topics through exploring a contextual situation with the support of multiple media, developing new knowledge further through manipulative activities in groups and sharing in public, and finally students' learning reflection through discussion and sharing.
L2 includes the following stages: (1) reviewing and introduction, (2) exploring and proving the properties of a midline, (3) application of the properties, and (4) summary and assignment. The teacher first asked students about the properties and definition of the median and then introduced the new topic through presenting an instructional task with the support of the Sketchpad (Key Curriculum Press, Emeryville, California, USA). Next, students were grouped together to explore the properties of the midline of a triangle by doing another instructional task: They drew a right triangle, an obtuse-angled triangle, or an acute-angled triangle and measured lengths of sides and midlines, then made conjectures on the relationships among different sides. Then, students conjectured that the midline is parallel to the third side and equal to half of the length of the third side of the triangle, namely the triangle midline theorem. One proof of the conjecture was introduced and formally written on the chalkboard with the help of Sketchpad software. After that, the teacher presented a set of varying exercises based on the same diagram from the textbook. Finally, the teacher summarized the lesson and, as homework, asked students to create a set of questions according to the last problem they had discussed in the last part of the lesson. Although the teacher adopted hands-on activities organized in groups and made use of multiple media appropriately, he still demonstrated many traditional features of mathematics teaching. For example, he gave extensive explanations, wrote clear and structural proof procedures on the blackboard with the inputs from students, and summarized key points in time (Huang & Li, 2009b).
Interview
These two videotaped lessons were first sent to interviewees. After watching the videotaped lessons, they were invited to write their responses to the following questions: (1) What are the characteristics of these lessons? (2) Why do you think they are unique or impressive? (3) What are your suggestions for improving these lessons, if any? (4) In your opinion, how should mathematics be best taught and learned? And (5) in your opinion, what dimensions are important to consider in evaluating the effectiveness of mathematics instruction? The first question aimed to draw participants' attention to important classroom events. The second question asked participants to explain why they are important. The third question aimed to interpret classroom events for improving classroom instruction. These three questions are intended to draw participants' attention to the three aspects defined by van Es and Sherin (2008) correspondingly. The last two questions aimed to explore participants' views of effective mathematics learning and teaching. There was no time limitation to complete the responses, but interviewees were explicitly required to provide their own opinions honestly. Because these research participants were recruited on a one-by-one basis from different schools through local coordinators, the recruitment process helped ensure that these participants were not aware of others' voluntary participation in this study and would not have an exchange about their comments on these questions in advance.
Data Analysis
In this article, we aimed to investigate teachers' noticing of classroom instruction events by addressing our research questions through analyzing participants' responses to the first three interview questions. All 20 participants' responses to the first three questions were analyzed by adopting a mixed method. The data analysis was based on the Chinese transcripts. First, we read through each participant's responses to the questions. Through making constant comparisons within and across individual responses (Corbin & Strauss, 2008), we developed a coding system shown as the first two columns (category and item) in Table 1. The coding system includes 20 items grouped in four categories. The categories include (1) instructional objectives, (2) instructional design, (3) instructional process, and (4) teacher quality and personality. Instructional objectives include mastering knowledge and skills, developing mathematical thinking and ability, and fostering appreciation of mathematics. Instructional designs include coherent development of knowledge, treatment of difficult and important content points, arrangement of classroom exercise and homework, selection of teaching methods, use of teaching aid tools, and consideration of students' preparation. Instructional processes refer to the process of delivering a lesson. It includes the following aspects: classroom atmosphere, students' interest and motivation, students' participation, students' articulation (e.g., students' problem posing and expressing their opinions), students' self-exploratory learning, teachers' listening to students and giving them feedback, teachers' effective guidance, students' participation, and students' higher order thinking. Teacher quality and personality includes teachers' image, instructional language, and board writing; teachers' mathematical foundation; and teachers' enthusiasm and passion.
| Category | Item | Expert | Novice |
|---|---|---|---|
| Frequency (No. of Teachers) | Frequency (No. of Teachers) | ||
| Instructional objectives | Mastering mathematics knowledge and skills | 4 (3) | 4 (4) |
| Developing students' mathematical thinking and ability | 81 (10) | 25 (8) | |
| Fostering students' appreciation of mathematics | 8 (3) | 3 (2) | |
| Instructional design | Coherent development of knowledge | 93 (10) | 57 (10) |
| Classroom exercise and homework | 14 (5) | 11 (7) | |
| Selection of proper teaching methods | 7 (4) | 4 (2) | |
| Adequate use of teaching aid tools | 16 (8) | 21 (9) | |
| Adequate treatment of difficult and important content points | 7 (4) | 6 (4) | |
| Consideration of students' preparation | 5 (4) | 4 (3) | |
| Instructional process | Classroom atmosphere | 8 (6) | 16 (6) |
| Students' interest and motivation | 13 (7) | 7 (6) | |
| Students' participation | 28 (9) | 11 (5) | |
| Students' articulation | 18 (8) | 10 (5) | |
| Listening to students | 6 (4) | 5 (3) | |
| Students' self-exploration | 21 (8) | 15 (8) | |
| High-order thinking | 24 (8) | 9 (6) | |
| Teachers' guidance | 1 (1) | 11 (6) | |
| Teachers' quality and personality | Teachers' mathematics knowledge | 10 (6) | 6 (4) |
| Teaching language and board writing | 13 (4) | 15 (9) | |
| Teachers' passion and humor | 6 (6) | 1 (1) |
- Note. Frequency refers to the total numbers of occurrence of that item. The number in bracket refers to the number of teachers who stated the corresponding item.
Because the majority of these items are self-explanatory, we only discuss several items here for illustration. For example, the following statement by ET5, “the effectiveness of a lesson depends on the gain of students' mathematics thinking, mathematics concepts, and mathematical methods,” was coded as “knowledge and skill” and “developing mathematical thinking and ability.” In another example, the excerpt from ET1, “students participated in all classroom activities. They play an active role in mathematics learning” was coded as “students' participation” and “students' self-exploration and learning.” Two mathematics educators coded all of the Chinese transcriptions independently. All disagreements among the educators were resolved through extensive discussions.
The frequency of each item produced by the expert and novice teachers were counted and entered into SPSS 15 (SPSS Inc., Chicago, IL, USA) for analysis. We first identified and analyzed the most and least frequently noticed items. After that, we did a Mann–Whitney U-test to examine whether there are significant differences between expert and novice teachers in various items. In addition, the qualitative differences between expert and novice teachers were further illustrated.
Findings
In the sections that follow, we first presented an overall teachers' predilection toward the noticing of classroom events. Then, we focused on the noticing differences between expert and novice teachers. Finally, we described the differences and similarities qualitatively.
Overall Predilections of Expert and Novice Teachers' Noticing
The first column of Table 1 includes four overarching categories. The second column includes all the items noticed by the teachers. The third column presents the frequency (i.e., the number of times that the teacher(s) commented on the item when responding to the first three questions; it often happened that a teacher commented on an item more than one time) and the numbers of expert teachers who attended relevant items. The fourth column presents the frequency and the numbers of novice teachers who attended relevant items. For example, under the category of instructional objectives, the item of developing students' mathematical thinking and ability was mentioned by 10 experts 81 times while it was mentioned by 8 novices 25 times.
There are 383 occurrences of the 20 events mentioned by the 10 expert teachers. Based on the percent of each event with regard to the total occurrences (i.e., the times of occurrence of a particular event divided by the total occurrences of 383), we listed the most frequently noticed six aspects by the 10 expert teachers as follows: (1) developing knowledge coherently (93 times, 24%), (2) developing mathematical thinking and ability (81 times, 21%), (3) students' participation (28 times, 8%), (4) developing high-order thinking (24 times, 6%), (5) students' self-exploratory learning (21 times, 5%), and (6) use of teaching aid tools (16 times, 4%) in turn. Nevertheless, the least-mentioned aspects are: listening to students and giving their feedback (six times, 2%), teachers' enthusiasm and passion (six times, 2%), readiness of students' preparation (five times, 1%), students' mastering knowledge and skills (four times, 1%), and teachers' effective guidance (one time) in turn.
However, for the 10 novice teachers, there are 241 occurrences of the 20 events. The most frequently mentioned six aspects are: (1) developing knowledge coherently (57 times, 24%), (2) developing mathematical thinking and ability (25 times, 10%), (3) use of teaching aid tools (21 times, 9%), (4) classroom atmosphere (16 times, 7%), (5) students' self-exploratory learning (15 times, 6%), and (6) teachers' image, instructional language, and board writing (15 times, 6%) in turn. On the other hand, they are least aware of the following aspects: listening to students and giving their feedback (five times, 2%), selection of teaching method (four times, 2%), readiness of students' preparation (four times), students' mastering of knowledge and skills (four times), developing students' appreciation of mathematics (three times, 2%), and teachers' enthusiasm and passion (one times).
If we treat all these teachers as a whole, then, the first 12 highly noticed aspects include: (1) developing knowledge coherently (24%), (2) developing mathematical thinking and ability (17%), (3) students' participation (6%), (4) use of teaching aid tools (6%), (5) students' self-exploratory learning (6%), (7) developing high-order thinking with appropriate exploration (5%), (8) classroom atmosphere (4%), (9) teachers' image, instructional language, and board writing (4%), (10) students' problem posing and expressing own opinion (4%), (11) exercise and homework 4%), and (12) students' motivation and interest (3%). In addition, teachers' enthusiasm and passion, readiness of students' preparation, and students' mastering knowledge and skills were attended less. In the following parts, we will present the differences and similarities in noticing of classroom events between experts and novices.
Quantitative Differences in Noticing Between Expert and Novice Teachers
Based on the Mann–Whitney test, it was found that there are significant differences in six aspects between expert and novice teachers as shown in Table 2.
| Expert MR | Novice MR | U | r | |
|---|---|---|---|---|
| Developing mathematical thinking and ability | 14.80 | 6.20 | 7.0*** | −.73 |
| Developing knowledge coherently | 13.75 | 7.25 | 17.5** | −.55 |
| Teachers' enthusiasm and passion | 13.00 | 8.00 | 25.0** | −.51 |
| Teachers' effective guidance | 8.35 | 12.65 | 28.5** | −.45 |
| Developing high-order thinking | 12.85 | 8.15 | 26.5* | −.41 |
| Students' participation | 12.65 | 8.35 | 28.5* | −.37 |
- Note. MR = Mean Rank; r = effect size.
- * p < .10. ** p < .05. *** p < .01.
The table shows that expert teachers paid significantly greater attention to developing mathematical thinking and ability (Mean Rank [MR] = 14.8) than novice teachers (MR = 6.2), U = 7.0, p < .001, r = −.73. Similarly, expert teachers paid significantly greater attention to developing knowledge coherently (MR = 13) than novice teachers (MR = 8), U = 17.5, p < .014, r = −.55, and expert teachers paid significantly greater attention to teachers' enthusiasm and passion (MR = 13) than novice teachers (MR = 8), U = 25, p < .022, r = −.51. Moreover, expert teachers also paid moderately greater attention to developing high-order thinking (MR = 12.85) and students' participation (MR = 12.65) than novice teachers (MR = 8.15 and MR = 8.35 correspondently) (p < .10).
However, expert teachers paid significantly less attention to teachers' effective guidance (MR = 8.35) than novice teachers (MR = 12.65), U = 28.5, p < .045, r = −.45.
Qualitative Descriptions of Difference and Similarities in Teachers' Noticing
In this part, we describe the qualitative differences or similarities in those 12 highly noticed aspects between novice and expert teachers. Some subtle qualitative differences were found in explaining attentions to the following aspects: developing knowledge coherently, developing mathematical thinking and ability, use of teaching aid tools, and selecting teaching methods. However, both experts and novices showed similar attention to the following aspects: students' participation; students' self-exploration; classroom atmosphere; teachers' image, instructional language, and board writing; and students' motivation and interest.
Qualitative Differences in Teachers' Noticing
Developing knowledge coherently. How to develop mathematics knowledge coherently is the most frequently noticed aspect by all the expert (93 times) and novice teachers (57 times). They all noticed the instructional design and flow which were described as “reasonable, smooth, and progressive.” For example, one expert teacher appreciated the teacher in L1 who implemented the instructional principles of “development progressively, and integration of theory and practice” (ET9). However, some subtle differences between expert and novice teachers were found. For the expert teachers (10 teachers), they also noticed if mathematics methods and ability had been developed while coherently developing mathematics knowledge and skills. For example, one expert teacher (ET9) wrote:
This lesson consists of four phases: introduction to the topic, exploration of the property proposition, proof of the proposition, and application of the property theorem. Each stage reflects the principles of developing students' mathematical thinking and methods, and enhancing students' learning ability.
Relatively, the novice teachers tended to pay attention to mathematics knowledge development as explained in the following:
This lesson has a clear instructional design: to start with the introduction, and then move to students' exploration and summary of key points of learned knowledge, and finally to application of knowledge (NT8).
Developing mathematical thinking and ability. Both expert teachers (10 teachers, 81 times) and novice teachers (8 teachers, 25 times) noticed how to develop mathematics thinking and methods, and logical reasoning ability. Basically, they paid attention to developing the following mathematics thinking and methods: classification, analogy, induction, deduction, equation method, modeling, and transformation.
In addition to classification and analogy as noticed by novice teachers, the expert teachers (six teachers) also noticed more broad and general dimensions of mathematical thinking methods such as movement, transformation, equation method, and scientific method. In particular, one expert teacher also attended to general scientific methods as follows:
In addition, discussing the relationship between two circles in terms of their radiuses was based on analysis of simple and specific situations. These discussions reflect the scientific thinking methods such as the generality and specificity, quantitative changes and qualitative changes. We should understand the methods from scientific methodology and philosophical aspects (ET7).
Use of teaching aid tools. Both expert and novice teachers attended to the use of teaching aid tools. The novice teachers mentioned it more frequently (9 teachers, 21 times) than did expert teachers (8 teachers, 16 times). In general, these teachers appreciated the use of teaching aid tools in the following aspects: using computer demonstrations to motivate students and using multiple media course software to visualize concepts.
However, only expert teachers (five teachers) noticed the appropriateness of adopting multiple media as follows:
When extensively using multiple media, the students have few opportunities to write and draw (for example, searching for the relationship among the position relations of two circles and the numerical relation between centers and radius). Thus, the plentiful use of media is not necessarily better for student learning. (ET10)
Selection of teaching methods. Both expert teachers (four teachers, seven times) and novice teachers (two teachers, four times) recognized group learning and manipulative activities as innovative teaching methods. In general, these teachers appreciated the positive effects of innovative teaching methods because collaborative efforts can help overcome students' learning difficulties (ET1), and sharing and discussion in groups motivates students' learning (NT9). However, one expert teacher challenged the legitimacy of discovering the triangle midline theorem through measuring and making a conjecture, as shown later:
In the midline property lesson, using measurement to calculate the doubling length of the midline is equal to the length of the third side. It is helpful for proving the theorem. However, measurement by ruler will produce 1.7 and 1.8 as 2; thus, the conjecture will not be true. It is not a mathematical method. Moreover, spending too much time on measuring the lengths of segments is not wise (ET5).
Qualitative Similarities in Teachers' Noticing
Expert and novice teachers shared more qualitative similarities than differences in the following aspects: students' participation; students' self-exploratory learning; classroom atmosphere; developing high-order thinking; teachers' image, instructional language, and board writing; and students' motivation and interest.
Students' participation. Expert teachers paid more attention to students' participation (9 teachers, 28 times) than novice teachers (5 teachers, 11 times). Expert teachers appreciated students' active learning under the teachers' guidance. For example, an expert teacher stated that “a lot of knowledge does not need a teacher's explanation; students can learn and understand by themselves. Innovative classroom instruction should include students' active and independent learning to achieve instructional objectives rather than teacher-led teaching alone” (ET1). Novice teachers noticed students' active participation in general aspects such as “motivating students' enthusiasm through participation, and encouraging them to demonstrate in public” (NT7).
Students' self-exploratory learning. Expert teachers more frequently mentioned this event (8 teachers, 21 times) than novice teachers (8 teachers, 15 times). Both experts and novices appreciated students' self-exploratory learning through doing experiments. For instance, an expert said:
In exploration activities, students were asked to measure the lengths of all drawn midlines and sides, and then make conjectures on midline property. Based on students' experiments and conjectures, it will benefit students' scientific sprit of respecting objective fact and seeking truth from facts (ET7).
Developing high-order thinking. Experts (8 teachers, 24 times) and novice teachers (6 teachers, 9 times) indicated that teachers should leave appropriate times for students to explore and develop their high-order thinking. For instance, an expert teacher emphasized creative mathematics thinking as follows:
It is not necessary to have high climaxes through a whole lesson, but it should include some highlighting episodes. Creative mathematics learning mainly is manifested by creative thinking. A lesson that goes through smoothly does not mean a good one. It is necessary to include some kind of cognitive conflicts in order to result in some innovative ideas and thinking methods (ET1).
While, a novice teacher suggested to give students much space for them to explore:
When discussing the distances of the centers of two circles and two radiuses with regard to different position relationships of two circles, the teacher lets students observe the dynamitic demonstration and get the results. Is it possible for the students to explore and find the results first, and then the teacher play the dynamitic movie? (NT6)
Students' problem posing and expressing their own opinions. The expert teachers (8 teachers, 18 times) noticed this event slightly more than novice teachers (5 teachers, 10 times). In general, these teachers noticed the importance of encouraging students to express what they know. For instance, an expert noticed that “The students expressed their opinions and found three relationships between two circles: separation, tangency and intersection. It was expressed by students rather than the teacher” (ET2). A novice appreciated that “There are many opportunities for students to explain their findings. This method not only motivates students, but also develops students' ability of language expression” (NT1).
Classroom atmosphere. Both expert teachers (6 teachers, 8 times) and novice teachers (6 teachers, 16 times) appreciated a harmonious classroom atmosphere. For example, one expert teacher stated, “During the lesson, students do hands-on activities, and demonstrate their findings in public. They actively participate in the classroom activities through manipulation, thinking, writing and expression. The classroom atmosphere is very vivid” (ET9). Similarly, novices appreciated a good learning environment, which was described as “leaving enough time for students to manipulate and discuss, and letting students explore and solve problems through activities” (NT3).
Teachers' image, instructional language, and board writing. Both expert teachers (4 teachers, 13 times) and novice teachers (9 teachers, 15 times) paid attention to teachers' image, instructional language, and board writing. Expert teachers stated that “the teacher's [in L2] elegant mathematical language, sound mathematics foundation, soulful eye contact with students and fluent and graceful board writing, and appropriate enlightening gave me a deep impression” (ET9). Similarly, novice teachers also praised the teacher's [in L1]“nice and natural teacher's image” (NT2) and “clear and well-structured board writing, and rhythmic language, and very nice teacher's image” (NT5).
Students' motivation and interest. Both expert (7 teachers, 13 times) and novice teachers (6 teachers, 7 times) appreciated the efforts to motivate students and get them engaged in learning. For example, expert teachers mentioned, “the teacher [in L1] fully encourages students to actively participate in the learning” (ET5) and “it is good to motivate students' learning enthusiasm” (ET9). However, novice teachers noticed specific activities for motivating students and provided the following statement: “introducing new topics through annular eclipse is novel, and can motivate students” (NT6).
Summary
The most frequently attended five aspects of the sampled teachers include the following: (1) developing mathematical knowledge coherently, (2) developing mathematical thinking and ability, (3) students' participation, (4) use of teaching aid tools, and (5) students' self-exploratory learning. Moreover, some differences in noticing between expert and novice teachers are identified. Compared with the novice teachers, the expert teachers paid significant and greater attention to developing mathematical thinking and ability, developing knowledge coherently, and developing high-order thinking; they also paid significant and greater attention to teachers' enthusiasm and passion, and students' participation. However, the expert teachers paid significant and less attention to teachers' guidance.
In addition, both expert and novice teachers explained similar reasons for them to attend the following aspects: students' participation, students' self-exploratory learning, classroom atmosphere, developing high-order thinking, and so on. However, they explained some subtle different reasons for their attention to the following aspects, including developing knowledge coherently, developing mathematical thinking and ability, use of teaching aid tools, and selecting teaching methods. Expert teachers focused on developing mathematics thinking and methods while learning basic mathematics knowledge than did novice teachers. Although expert and novice teachers appreciated the roles of using multiple media, only expert teachers noticed the side product of the abuse of multiple media. Both expert and novice teachers realized the importance of group learning and manipulative activities, but only expert teachers recognized the appropriateness and necessity of using manipulative activities. Overall, compared with novice teachers, expert teachers seemed to notice more mathematically essential aspects (e.g., developing mathematical thinking and methods, mathematical language) and contextual aspects (e.g., appropriateness of using multiple media and using manipulative activities).
Conclusion and Discussion
Is Teachers' Noticing of Mathematics Classroom Events Universal or Cultural Specific?
This study revealed that both expert and novice teachers noticed the following overarching aspects when watching videotaped lessons: instructional objectives, instructional design, instructional process, and teachers' quality and personality. However, the commonly noticed aspects paint a complicated picture about these Chinese teachers' predilections toward attending mathematics classroom events. On one hand, these teachers did pay attention to some traditionally valued aspects of Chinese math classroom instruction, such as emphasizing connection and development of knowledge, mathematical thinking and reasoning, teachers' guidance, and so on (Huang & Li, 2009b), while diverting their attention from some other aspects, such as emphasizing important content points, dealing with difficult content points, and emphasizing students' mastering knowledge and skills. On the other hand, the teachers did notice some reform-oriented instruction elements, such as students' participation, students' self-exploration, students' articulation, students' interest and motivation, and teachers' listening to students' opinions. It may be a positive effect of the new curriculum reform (Liu & Li, 2010). The new curriculum advocates innovative instructional ideas including (1) using multiple teaching methods and measures such as self-exploration, cooperation, and exchange to guide students' active learning through mathematical activities, and (2) understanding and mastering basic knowledge, skills, and underlying mathematical ideas and methods.
With regard to the observation of less attention given to some traditionally valued aspects such as “dealing with important and difficult content points” and emphasizing “students' mastering knowledge and skills,” we should interpret it cautiously. Mastering basic knowledge and skills through appropriately dealing with difficult and important content points is one of the unique features of mathematics classroom instruction in China (Huang & Li, 2009b; Li & Li, 2009; Zhang, Li, & Tang, 2004). Does this observation reflect the negative effect of the new curriculum reform? To interpret this phenomenon, we would like to explain the following two points. First, these two videotaped lessons are basically good lessons (one is the nationwide mathematics classroom instruction competition winner and the other is given by a master teacher). In these two lessons, the teachers did a good job in this regard, as pointed out by ET9 that “[In L2] the content is rich, the explanation is appropriate and the important contents are highlighted” and NT3 that “the teacher [L2] appropriately explained the important points.” Second, in their views of effective mathematics instruction, these participating teachers valued students' mastering of mathematics knowledge and skills (Huang, Li, & He, 2010). Thus, it could be assumed that these teachers' ignorance of dealing with important and difficult content is most likely due to the specification of these lessons.
The observation that these Chinese experts paid less attention to teachers' guidance is in line with experts' views of effective mathematics instruction (e.g., Huang & Li, 2009a; Zhao & Ma, 2007). One possible interpretation is that teachers' guidance is a basic teaching skill, which all teachers should have in traditional classrooms, whereas the innovative ideas such as students' self-exploratory learning should be attended to in the curriculum reform context. The expert teachers, who are skilled in classroom management and students' guidance, are more sensitive to the student-centered teaching events, while the novice teachers, who may be struggling with effective guidance of students, paid close attention to that aspect.
In views of Chinese culture, it is worth noticing that while it has learned from the West, its own tradition has still remained in essence. As argued by Zheng (2006), the evolution of modern education in China “is mainly a process of assimilation. That is to say, rather than being alienated by foreign factors, the foreign values were absorbed and assimilated into the Chinese culture” (p. 382). Instead of going to the extremes, Chinese mathematics educators prefer to get a balance between various conceptions in education, which is the central idea of “Zhong yong,” a spirit of Chinese culture. Thus, it may be possible to develop some unique features of mathematics classroom instruction by adopting some innovative notions from Western theories while keeping some good Chinese traditions alive.
How Do Expert and Novice Teachers Differ?
This study revealed that the sampled expert teachers were most likely to be aware of developing students' mathematics thinking and ability when helping students develop mathematics knowledge. They were sensitive to properly selecting teaching aid tools and teaching methods in terms of the specifications of students and contents. These findings are consistent with other findings on teachers' noticing of classroom events such as evaluative judgment (Sabers et al., 1991), selectivity and reflection (Krull et al., 2007), and contextual relevance (Sato et al., 1993). Moreover, this study suggests that noticing of fundamental mathematics substance, such as mathematical thinking methods and ability, and the appropriateness of selecting teaching methods to develop mathematical contents in classroom instruction are keys to differentiating experts from novices. Thus, in addition to previous general features, focusing on mathematical essence, rather than superficial aspects of guidance and management, should be a critical element of mathematics instruction that expert teachers may notice.
Implications
This study not only contributes to a better understanding of expert–novice similarities and differences in their noticing of classroom instruction events, but also provides insights for teacher professional development.
The emerging pattern in the selected Chinese mathematics teachers' noticing of classroom events may provide some complementary components for Western educators to construct their frameworks for guiding teachers' attention to classroom events in teacher education programs. On one hand, the Chinese pattern shares some elements with Western literature, including instructional goals, teaching alternatives (Santagata et al., 2007), and mathematical thinking (Star & Strickland, 2008). On the other hand, it demonstrates some unique features such as developing mathematical thinking methods and ability while mastering knowledge and skills, placing an emphasis on knowledge development and instructional coherence, and balancing teachers' guidance and students' self-exploratory learning. However, less attention is paid to students' learning and classroom management. Grounded in teachers' own noticing of classroom events, this pattern may reflect Chinese mathematics teachers' practice. As argued by some researchers (Leung, 2005; Zhao, 2005), it is crucial to get a contextualized understanding of other cultural practices (particularly effective ones) from social, cultural, and systematic contexts before deciding what we can learn from them. The findings of this study provide a kind of catalyst for mathematics educators to reflect on ways to improve teacher education programs. It sheds light on making balances between developing mathematical thinking methods and developing fluency in basic knowledge and skills, and between teachers' guidance and students' self-exploratory learning. Hopefully, the Chinese pattern of noticing of mathematics classroom events may help mathematics educators in other cultures to develop their own ways of enhancing teachers' noticing of classroom events in hopes of improving teacher education and classroom instruction. For example, U.S. mathematics teachers may need to focus on developing students' knowledge coherently, developing students' mathematical reasoning, and mastering mathematics knowledge.
In addition, the differences in noticing between expert and novice teachers help identify those aspects that novice teachers really need to improve. Practically, the following aspects should be the focus when developing novice teachers' ability to notice: how to focus novice teachers' attention on developing students' mathematics thinking and ability while developing basic mathematics knowledge, and how to select teaching methods and teaching aid tools in terms of context. Thus, when developing teacher professional development programs, it will be crucial to guide participants to focus on mathematical essence (developing concept understanding, mathematical thinking methods) and flexibility in selecting teaching methods and aid tools in terms of students' learning.
Limitations and Suggestions
Although many of this study's findings confirm the features of Chinese teachers' beliefs about effective mathematics teaching (e.g., Cai et al., 2009) and classroom instruction (e.g., Huang & Li, 2009b; Leung, 2005), these results are not guaranteed to apply to other teachers due to the small sample size and the limitation inherent in the sampling procedure. Although the written responses to the interview questions make it impossible to probe interviewees' opinions in depth, they still provide rich and valuable information for researchers to capture their noticing of classroom events. In addition, analyzing and quantifying teachers' responses in terms of coded frequencies of items can help highlight selected similarities and differences in teachers' views, but these methods carry the risk of oversimplifying or even misinterpreting teachers' rich descriptions. In addition, the specific geometrical contents of the videotaped lessons may also impact teachers' noticing to classroom events. It should be helpful to examine teachers' noticing when watching videos of teaching algebraic contents. As a result, future studies will be needed to investigate Chinese teachers' noticing of mathematics classroom events with the rich data that can be obtained through face-to-face interviews and analyzed in ways to reveal teachers professional vision in detail. Moreover, a cross-national examination of teachers' noticing to classroom events when teaching the same content will help us understand cultural differences in teachers' noticing and further provide insight into developing teachers' noticing ability. Finally, it is very important to investigate how teacher education programs with a focus on changing ways of noticing classroom events work and how the changing of noticing of classroom events may impact on teachers' teaching practices and students' learning outcome.
