Understanding and Growing Together: Mathematics Teachers’ Reciprocal Learning Between China and Canada

An Analytical Framework of Teacher Noticing

van Es and Sherin (2006) proposed a framework for analyzing teacher noticing, comprising four dimensions: agent, topic, stance, and focus. These analytical dimensions provide a foundation for further comprehensive inquiry. Teachers can effectively promote students’ mathematical thinking and their own professional development by attending to these dimensions. Teacher noticing is an effective strategy for fostering students’ mathematical thinking and for supporting teachers’ professional development (Didis et al., 2016). Subsequently, the interrelationship between teacher noticing and teachers’ knowledge construction has been extensively examined from a comparative perspective (Roller, 2016). These analytical dimensions provide a foundation for further comprehensive inquiry.

In this study, teacher noticing themes were examined from the communication between the teachers, then noticing stance and pre-decision-making were analyzed to study how teachers learn mathematics teaching with noticing perspectives. The framework (Table 1) was constructed by Xie (2020), based on van Es and Sherin (2006) and van Es's (2011) framework.

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Table 1.

An Analytical Framework for Teacher Noticing.

Dimension	Subdimension	Details
Themes (What did the teacher notice?)	Agent/topic	Whom (such as the student, the teacher, or others) did they notice as being involved in reciprocal learning?
		What topics (such as mathematical thinking, pedagogy, climate, and management) had teachers noticed as being pertinent to reciprocal learning?
Stance (How did the teacher analyze the events they noticed? How deep was their analysis? How did they offer other alternative strategies?)	Type of teacher noticing representation	Description: The statements that recounted the events that unfolded. Evaluation: The statements that were judgmental in nature, in which the teachers commented on what was good or bad, or could or should have been done differently. The evaluation was more concerned with how. Interpretation: The statements in which the teachers made inferences about what they noticed, with the intent of explaining what happened and why.
	Pre-decision-making	Teachers could express the plans or strategies related to a similar situation.

Participants

The Mathematics Education Team was a collaborative research team within the Canada–China Reciprocal Learning project, comprising four universities and over 20 researchers. Among the four universities, the University of Toronto and Northeastern Normal University joined together in 2013 in a reciprocal learning partnership and focused on primary mathematics education, while the other two universities focused on secondary mathematics education. The reciprocal learning partnership network was established between Changchun and Toronto, comprising nine schools and 20 teachers and administrators. There were also five graduate assistants as part of the university research team. These students supported the research in many ways: conducting literature reviews, interviewing participants, translating interactions, creating slide presentations, conducting online meetings, and conducting interview recordings. They were so important to the data collection, data analysis, and report writing.

Relying on university–school partnerships, the math reciprocal learning team invited schools and teacher participants from Canada and China with respect to their willingness to participate in the project. The teacher participants were selected according to the following criteria: a willingness to invest in a reciprocal learning partnership over an extended period, an interest in the development of mathematics as a major field of study, exemplary teaching ability, and a willingness to utilize electronic communication as a primary means of communication. Additionally, the teacher participants from China were initially selected based on their willingness to join, and subsequently, the principals determined the final participants from the list of volunteers. This study involved four sister-schools and two pairs of teacher reciprocal learning partnerships, with each set including one teacher from China and one from Canada (Table 2). The paired teachers were assigned codes: Ch-G and Ca-K were one pair, and Ch-W and Ca-L were the other pair.

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Table 2.

Participants of This Study.

Pair	Country-primary School	Teacher Participants	Involved Date
1	China-F	Ch-G: Male, 11 years of teaching experience, a fifth-grade math teacher.	September 2013
	Canada-G	Ca-K: Female, 15 years of teaching experience, a sixth-grade generalist teacher (excluding French and drama), with 14 years of experience teaching mathematics.	March 2014
2	China-J	Ch-W: Female, 17 years of mathematics teaching experience, a fifth-grade math teacher, and grade subject leader.	December 2013
	Canada-B	Ca-L: Female, 16 years of teaching experience, sixth-grade teacher of IT, math, language arts, and social studies, team leader of the grade and literacy, with 6 years as a math teacher.	December 2013

The data were collected from interviews and conducted with a view to exploring teacher noticing patterns in reciprocal learning contexts. For this study, an interview protocol was prepared and used by all reciprocal learning researchers conducting semistructured interviews with the participating teachers. The semistructured interview protocol consists of a list of questions that were designed at the beginning of the project and modified based on the interactions of the teachers. This study was guided by Connelly and Xu's (2015, May) narrative vision for the partnership project, which allowed the collection of a multitude of data types ranging from observations, interviews (formal and informal), and document analysis (Stake, 1995; Yazan, 2015).

In the initial phase, the pairs of teachers used email to communicate with each other. The researchers posed questions that sought to gather information about the teaching practices of the teachers. For example, some questions asked were: “What constitutes an effective classroom?” “What are your current explorations of pedagogical reform in mathematics?” and “What recent developments have occurred in teaching content and activities?” The goals of the interviews were to encourage the teachers to share their current practices of teaching mathematics and to help set a foundation for identifying changes that might occur in teaching practices while interacting with a teacher from another country.

In the second phase, the university researchers invited the teachers to identify an area of their teaching practice that they would like to improve. Following their discussions, the teachers recorded their classroom instructions and shared them with the other teacher for peer review. The teachers agreed to watch each other's classroom teaching videos and write down their reflections. Subsequently, the teachers attended an online session hosted by the university researchers to discuss “What did you notice while watching the videos and why?” The topics of videos are listed in Table 3. During this stage, the focus of the teachers gradually shifted toward noticing the similarities and differences in teaching practice.

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Table 3.

Content of Mathematics Instruction Recording.

Country	Instructional Content	Grade	Semester
China	Multiplication of three-digit numbers by two-digit numbers	Grade 4	Semester I
	Division of decimals	Grade 4	Semester II
	Area of a circle	Grade 6	Semester I
Canada	Multiplication of three-digit numbers by two-digit numbers	Grade 6	Semester I
	Probability	Grade 6	Semester II
	Time	Grade 6	Semester II

In Ontario, Canada, the mathematics curriculum standards for Grades 1–8 require that, by the end of Grade 5, students can use a variety of strategies to solve multiplication problems involving two-digit numbers, can summarize algorithms—especially the standard algorithm—and can use estimation to judge and verify results when solving problems. By the end of Grade 6, students should be able to calculate the multiplication of four-digit numbers by two-digit numbers. Although the textbook includes content on the multiplication of three-digit numbers by two-digit numbers in Grade 6, the curriculum standards do not have explicit requirements for it, allowing teachers more flexibility in handling it. Ca-K, based on the students’ learning progress, arranged this lesson at the end of Grade 5, completing it in 3.5 lessons (each lesson is approximately 2 h long).

In the third phase, teachers in the reciprocal partnership shared more about their daily teaching practices and exchanged ideas on WeChat. They visited each other's schools and taught in each other's classes. All these activities were aimed to facilitate a more profound comprehension of mathematics teaching and learning in different cultural contexts.

Furthermore, observation was employed in this study, aiming to collect data on the interaction processes and teaching activities among reciprocal partners. A grounded theory approach was employed for inductive analysis, and a teacher growth transformation model was constructed from the perspective of teacher noticing.

Ethical Issue

The study had been subjected to a rigorous review process by the External Research Review Board of the University of Toronto Research and Ethics Office and the City School Board and had been granted approval. To guarantee the confidentiality of the data, the teachers and schools involved in the study were assigned pseudonyms in accordance with the established processes at these institutions. All faculty members who participated in the study signed a formal consent form and a consent letter on behalf of all the children involved in the student interaction and their guardians.

Research Credibility

The data collected for the study comprise interviews, classroom observations, video analyses, and meeting documents. The interview data were analyzed by coding, utilizing a codebook developed from interview responses and literature reviews. Some codes were clustered into themes. The formation of theory and mutual interpretation of evidence were conducted through triangulation, which involved the integration of data from the following: “the researcher–the research participant–the research materials,” and “observation–interview–stuff collection.” The researchers assumed the role of observers and recorders in the teacher reciprocal learning process, responsible for coordination without intervening or deliberately guiding the topics of the teachers’ discussions.

Phase I (June 2013–July 2014): Familiarization With Educational Contexts and School Profiles

In this phase, online communication facilitated by university researchers enabled cross-cultural dialogues, with a focus on personal and professional exchanges. Teachers from Canada and China engaged in online (Skype) discussions; however, Skype does not have translation software, and there were challenges with Internet connections. The university researchers maintained a noninterventionist stance, allowing discussions to flow freely and respectfully. Personal introductions, teaching experience, educational background, and current roles were shared, as well as classroom materials and photos of students’ activities, which reflect their daily school life. As they inquired about each other's interests, the conversation naturally progressed from daily routines to broader topics in primary education and mathematics teaching, which built on mutual interests.

To learn more about a general overview of primary education in their respective countries, Ca-L, based in Toronto, Canada, and Ch-W, based in Changchun, China, discussed their daily work routines through Skype. Ch-W stated:

I arrive at school at 7:30 AM, teach two to three math classes a day, and spend the rest of my time preparing lessons and grading homework. I also exercise in the gym at noon or talk to students about their learning. Officially, I finish work at 3 PM, but I usually leave school around 5 PM or even later.

In addition to daily routines, online discussions also included specific aspects of daily teaching, such as teaching approaches, design ideas, instructional facilities, and the students’ learning process. Based on this exchange of information, teachers gained a general understanding of each other's school days. However, the information exchanged between the two sides remained at a factual surface level, focusing primarily on the details about “what” was happening. For instance, during a Skype discussion on students’ academic evaluation between Ca-K and Ch-G, Ch-G asked, “I’d like to know if Canadian primary school graduates take exams before entering a particular secondary school, and if the scores are important?” Ca-K responded, “Students take the Ontario provincial exams in Grades 3 and 6, and there is also another test called the CAT4.” Ch-G inquired, “What is the CAT4?” Ca-K explained:

It is for recent graduates, mainly assessing students’ knowledge and skills in literature and mathematics. According to the school enrollment district system, the school a student attends is determined by the residential area in which they live. However, students can also choose schools based on a combination of district assignment and exam scores.

Ch-G commented, “This is similar to the school district system we have in China now.” Ca-K asked, “What is the relationship between exam scores and the school district system? Could you explain?” Ch-G responded, “In China, the transition from primary to secondary school is typically determined by school districts, but some top schools conduct entrance exams to select outstanding students. Although such practices are nearly banned, they sometimes occur covertly.”

Regarding mathematics education, the conversations encompassed teaching methodologies, curricular content, learning strategies, and students’ achievement evaluation. For example, Ca-K inquired, “How many instructional hours are allocated to mathematics classes per academic year? And how long is a period class?” Ch-G responded, “Approximately 80 instructional hours, with each class lasting 40 min.” Ca-K further probed, “Do students prefer classes that are taught step by step, or do they prefer classes that are diverse, incorporating inquiry, teamwork, presentations, and so on?”

Ch-G indicated, “They prefer to use various approaches to conquer the problem.” Ca-K asked, “What do you think is the most significant challenge your students encounter when learning mathematics?” Ch-G said, “How to use existing knowledge to stimulate new knowledge, learn to think mathematically, and solve problems logically.” At this stage, the topics related to mathematics education raised by the teachers in the reciprocal learning partnership remained at a general level, often manifesting in a straightforward question-and-answer format.

According to the communication among reciprocal teachers, it shows evidence that their noticing is primarily centered on the teachers. The topics of teacher noticing mainly involve general education overview of the countries (such as education system, evaluation methods, curricula, etc.), an overview of curriculum reform (mainly about primary school mathematics curriculum reform, such as curriculum aims, design, content, teaching methods, student learning, assessment, etc.), an overview of schools (school educational philosophy, student demographics, class size, regular and special activities, support for teacher professional development, management system, home-school cooperation, etc.), and the routine of mathematics teaching (lesson plans, textbook editions, teaching methods).

These noticing themes are often raised in the form of general “what” questions, thus manifesting as statements of specific facts or recollections of events. These identified themes arise mainly due to individuals’ gaps in background knowledge about each other's culture and educational practices. Additionally, some topics are raised by linking personal current experiences and interactions. At this stage, there are no significant national differences in the topics of teacher noticing between the two countries.

Phase II (August 2014–October 2015): Focused Discussion on Subject-Specific Teaching (“Why,” “How” Question Style)

As the interactive communication within the project progressed over a year, the reciprocal teachers pieced together their understanding of each other's primary education, curriculum reform, and mathematics teaching profiles in their respective countries. This understanding did not result from a systematic, externally imposed, comprehensive introduction. The teachers set their own discussion topics and focused on similarities and differences between their teaching practices and school culture.

In the second phase, the reciprocal learning teacher pairs agreed to expand their focused discussions toward subject-specific themes. Teacher noticing shifted toward the analysis and discussion of teaching particular math content. To overcome space-time barriers—it was unrealistic to live-stream classroom teaching online due to the time difference—the Canadian and Chinese teachers proposed to record and exchange classroom teaching videos and then conduct discussions and analyses based on specific teachings after watching the recordings. Each teacher provided three classroom teaching recordings. The teaching discussions were conducted around the specific topics that interested the teachers and facilitated the conversations.

During this phase of discussion, the mathematics teacher team had its own advantages compared to the other partnership teams. The teachers could understand the subject content and teaching activities without translation due to the common understanding of mathematical symbols. The graduate students who were part of the university research team made translations of the conversations to assist in understanding teaching designs and thoughts when language barriers occurred.

The teachers wrote down comments and questions while reviewing the videos. Questions such as “Why is this context introduced at the beginning of the class?” were raised. Some topics for discussion emerged from the videos, such as “How can we evaluate and ensure that students have ultimately learned?” “What is the intention behind designing this particular content in the textbook?” and “What is the teaching philosophy?” Although reciprocal teachers knew something about each other's educational backgrounds and cultural customs from the first phase, the lack of systematic, continuous personal experience led to differences or even biases in understanding on some issues. Comparisons and criticisms arose, shifting the dialogue from “what” to “why/why not?” and “how can they …?” (Table 4).

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Table 4.

Observation Notes.

China	Canada
“How can such a simple problem be covered in five or six classes with such a slow pace? Will they be able to finish by the end of the term?” “Do they not teach the standard algorithm for vertical multiplication?” (Ch-G's questions after observing Ca-K's video on “Multiplication of three-digit numbers by two-digit numbers”).	“Can students absorb information in such a short time?” “Will pointing out students’ mistakes so directly in class cause them embarrassment?” (Exclamations from Ca-K after observing the video.)
“These are simply not feasible in my classroom. I only have 45 min, and I do not have the authority, time, or space to integrate content from other subjects, even though these themes would be attractive to students.” (Ch-W commented after observing teaching videos and reviewing some teaching materials shared in the schedule.)	“How can China allow such large class sizes for teaching?” “Goodness! How is it possible to teach in such conditions? How can you address individual differences?” (Ca-L's exclamations upon viewing the observation video.)

In this phase, teacher noticing expanded from the individual reciprocal teachers to the mathematics teacher community, with a growing focus on the students. Since everyone independently identified focus topics from each other's videos, raised questions or comments, and then attempted to respond to each other or exchange opinions, nationally focused differences were shown in terms of teacher noticing. The analysis that follows is based on individual observations of three videos, personal note-taking, and three collective discussions, focusing on the identification of teacher noticing response points and frequency statistics (Figures 1 and 2).

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Figure 1.

Statistics of Chinese mathematics teacher noticing topics.

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Figure 2.

Statistics of Canadian mathematics teacher noticing topics.

Both Chinese and Canadian teachers demonstrate their teacher noticing on the overall design and section design. Chinese teachers place greater emphasis on section design compared to Canadian teachers. Both teacher pairs noticed various aspects, including content design and selection (considering appropriateness, arrangement, and alignment with student recognition levels), activity design, and specific teaching section design (considering timing, methods, rationality, clarity, fluency, coherence, appropriateness, gradient, etc.). We noted that Chinese teachers focus on textbook reorganization, while Canadian teachers focus on content and material selection due to the lack of a unified textbook version. Chinese teachers focus on the ability to design activities, while Canadian teachers focus on the format of activity design. Chinese teachers focus on goal-oriented activity design, while Canadian teachers focus on task-based learning for students.

Chinese and Canadian teachers share a focus on the teaching environment and student engagement, with nuanced differences in their noticing of specific aspects and teaching effectiveness. In terms of teacher noticing on the teaching process, both Chinese and Canadian teachers focus on the teaching environment, student engagement, learning methods, classroom creation and adjustment, teaching methods, and support for student learning. Canadian teachers exhibit a greater frequency of teacher noticing in the first four aspects. Student engagement is the central focus among all the teacher noticing. They all value the state, depth, and breadth of student engagement, emphasizing whether students are positively engaged, focused, and interested in the learning content.

There are some similarities and differences in what the teachers focused their attention on in their teaching practices. The Chinese teachers focus on creating appropriate mathematical learning contexts, while Canadian teachers value the psychological environment construction. In terms of collaborative learning, Chinese teachers emphasize the facilitation of teamwork, while Canadian teachers focus on the internal mechanisms of teamwork. Increasing efficiency is a key concern for Chinese teachers, while Canadian teachers focus on the zone of proximal development. Lastly, Chinese teachers focus on achieving learning outcomes and overcoming challenges, while Canadian teachers focus on learning differences.

In terms of noticing teaching effectiveness, both Chinese and Canadian teachers value increasing students’ competencies, especially their critical thinking skills and problem awareness. Chinese teachers focus on the achievement of teaching objectives and students’ academic outcomes, while Canadian teachers focus on students’ abilities in problem-solving and sustainable development.

Analysis of the Chinese and Canadian teacher noticing stances reveals shared common perspectives: Deep student engagement in the classroom is one measure to evaluate teaching effectiveness; teaching should be constructed as a “teaching triangle” among teachers, students, and teaching content; the learning process should be the students’ understanding process of the subject matter structure; creation and adjustment in the classroom are manifestations of “student-centered” approaches and teaching wisdom; and mathematical thinking is formed and activity experience accumulated through deep thinking and dialogue.

However, differences are noted: The Chinese teachers believe that “teaching design should highlight the nature of the discipline and teaching philosophy.” This includes how knowledge is created and accepted, the structure of existing knowledge, and the different ways of understanding knowledge within the field. Meanwhile, the Canadian teachers assert that “teaching design should have ‘blanks’ and ‘flexibility’”; while the Chinese teachers advocate for creating a “relaxed, democratic, yet orderly classroom culture,” whereas Canadians aim to foster a “harmonious yet challenging classroom culture”; Canadian teachers view “students’ deep understanding as the essence of an effective lesson”; Chinese teachers consider “achieving all teaching objectives as an important indicator to evaluate an effective lesson.”

Phase III (November 2015–June 2019): In-Depth Understanding and Learning Based on Teaching and School Exchange Experiences

During this phase, with the increasing frequency and depth of interaction, most discussions focused on a very specific teaching content and delved deeply into the intricacies of mathematics teaching and learning. Additionally, they engaged in reciprocal school visits and teaching in each other's classrooms, a beneficial opportunity to acquire vivid, firsthand experiences. Consequently, teachers from both countries became increasingly adept at appreciating and understanding each other's culture, education, pedagogy, and context. They had shifted from a stance of criticizing and comparing to one of understanding and respecting. Meanwhile, they managed to select ideas from their reciprocal communication to transform, collaborate, and apply within their own teaching domains.

For instance, one Chinese teacher said to the researcher:

At the beginning, when I communicated with them, I usually compared unconsciously and looked for gaps. But now I understand that reciprocal learning should be mutual, learning from each other's strengths, and growing together. (Ch-G)

Another Chinese teacher felt that there were many differences in education:

Considering the cultural and geographical differences between China and Canada, there are significant differences in education. Some of my previous doubts now seem rather narrow-minded and biased. (Ch-W)

A Canadian teacher identified cultural background as contributing to their understanding: “The weakness I thought at that moment might not be a real weakness; I should interpret it based on their cultural background.” (Ca-K).

Teachers mentioned that they are actively considering how to draw upon each other's effective approaches and apply them to their teaching and learning (Ch-G, Ch-W, Ca-L, Ca-K).

The Chinese and Canadian reciprocal teaching partners collaborated closely, sharing experiences and enriching their teaching practices. Ch-G sent several teaching designs and small activity examples (such as learning about symmetrical figures through Chinese paper-cutting) to Ca-L, which were enthusiastically received and beloved by students upon implementation in Ca-L's classroom. Ch-W solicited curriculum integration resources from Ca-K (e.g., How many pairs of jeans are there in your family? —a synthesis of environmental protection, mathematical statistics, and art design). This led Ch-W to explore interdisciplinary thematic learning in conjunction with teachers from other disciplines.

Although interdisciplinary thematic learning was not explicitly articulated in the Chinese compulsory curriculum reform plan at that juncture, these instances have enabled Ch-W to acquire effective strategies, thereby endowing her with foresight and sagacity in navigating subsequent reforms. After class, both reciprocal partners analyzed how the integrated curriculum was brought about in terms of knowledge integration, experience transfer, and problem-solving in real-world contexts. However, due to practical limitations, such courses were not widely implemented in China, even though Ch-W found them very meaningful and valuable.

After several activities created by the university researchers designed to help the teachers get to know each other, Ca-K and Ch-G gradually began to use a chat app to communicate. Ca-K shared her thoughts on “How to calculate the perimeter of a triangle by using a square,” and then Ch-G shared “The basic properties of fractions.” Ca-K commented, “These methods are very creative. If my students have difficulty learning this knowledge, I will use your intuitive models to help them learn.” Ca-L shared some classroom blackboard writings and Ch-G uploaded one of his lesson studies for sharing and discussion.

In Ca-L's classroom, two geckos were invited to live in the class for 5 months and participate in the students’ Science, Technology, Engineering, and Mathematics courses. Ca-L used a math class to calculate and compare the costs of keeping geckos as pets. “This is a particularly innovative teaching method that greatly promotes the multisensory development of students. If conditions permit in the future, we can do it as well,” Ch-G remarked. Canadian teachers realized that the pronunciation of the Chinese language plays a role in Chinese students’ learning of mathematics, such as the semantics of “place value” in two-digit multiplication with three-digit numbers, which is very conducive to students’ understanding of arithmetic principles (e.g., in Chinese, “20” is expressed as “二十” (èr shí), which means “two tens,” and “40” is “四十” (sì shí), meaning “four tens”).

From 2016 to 2019, the teachers visited their reciprocal learning teacher partner's school. These visits provided immersive insights into educational practices and cultural contexts. During these visits, teachers conducted observations in the classroom and campus, reviewed various school documents (including policies, lesson plans, textbooks, assessment portfolios, students’ homework, and students’ creations). They engaged in conversations with superintendents, principals, teachers, and students, which facilitated a deeper understanding of mathematics teaching within specific cultural and comprehensive educational settings, as well as the school environment and routine activities. These visits also allowed teachers to address previously held questions and doubts. Throughout this phase of reciprocal learning, the themes of educational practices, curriculum reform, and teaching implementation, which had been professionally noticed in the first two phases, were contextualized within the culture. This cultural embedding led to a more nuanced understanding and interpretation of these themes through concrete teaching scenarios, reducing biases and fostering greater understanding and respect.

During their school-based visits, teachers observed distinct educational approaches and challenges in Chinese and Canadian schools. Teachers noted that Chinese schools typically offer higher standards of education and attract higher-quality students, while Canadian sister-schools face more diverse and complex student demographics, exhibiting considerable variation in students’ learning backgrounds. In Canada, primary school teachers manage a small class (about 15–25 students), while in China, one subject teacher (normally a Chinese language teacher) oversees a larger class (about 35–45 students), with other subject teachers rotating in. Canadian teachers are allowed to select textbooks on their own, as well as have flexible teaching schedules and examination requirements.

In mathematics teaching, Canadian teachers independently rearrange content and teaching steps based on curriculum standards and individual teaching plans, and teaching time is flexible. All the above was interpreted as “Canadian teaching is more flexible and autonomous, some teaching conducted in an integrated way that maximizes the value of curriculum and is particularly conducive to fostering students’ core competencies.” In contrast, Chinese teachers are specialized by subject, and teaching is conducted independently in large classes, using a single version of textbooks issued by the authority. They have collaborative lesson plans and follow the same teaching pace, schedules, materials, and even the same exercises and evaluations. The typical Chinese mathematics teaching includes reviewing previous knowledge, introducing new concepts, independent learning based on tasks, group discussion and presentation, exercises, and summary. Canadian teachers viewed this as “Chinese teachers construct a strong foundation with collective lesson planning, which helps them to teach professionally and efficiently. To some extent, single-subject teaching has its own advantages. They maintain a tight rhythm in their teaching, which may guarantee teaching quality and effectiveness.”

In this phase, the teacher's noticing of agency expanded its focus from concentrating primarily on teachers to include both teachers and students, with the noticed agents shifting from being solely related to teachers to connecting with how to support students’ learning and development. The teacher noticing topics shifted from the specific content of teaching in Phase II to subject matter knowledge and approaches within the learning area, as well as the context (such as grades, schools, district curriculum, educational reforms, etc.) associated with it in Phase III. All the teacher noticing points were gradually connecting and constructing a systematic framework. They could provide objective and impartial descriptions, interpretations, and evaluations based on their teaching noticing framework. Sometimes, they would say, “If I stand in her shoes, I would …,” which showed pre-decision-making or experience transformation.

In general, the characteristics of Chinese teacher noticing are manifested as follows: macrosystemic teacher noticing, efficiency evaluated by “objective-oriented” achievement, enhancement of subject-specific competencies, and differentiated learning tailored to group differences. Conversely, the characteristics of Canadian teacher noticing are evident in the following aspects: microspecific teacher noticing, flexible teaching guided by “student-centered” approaches, sustainable development for lifelong learning, and differentiated learning due to individual differences.

Reconstruction of Teacher Knowledge Through Teacher Noticing During Cross-Cultural Reciprocal Learning

During the three phases of teacher reciprocal learning, various types of teacher knowledge are identified through teacher noticing perspective, which include original teacher knowledge (K₀) as a system, general education knowledge (K_Ge), general educational context knowledge (K_Gec), general mathematics education knowledge (K_Gme), subject matter knowledge and related pedagogical content knowledge (K_S&Pck), practical knowledge (K_p), and cross-cultural education knowledge (K_Cr−cE). All this knowledge has been shifted and reconstructed from phase to phase (Figure 3).

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Figure 3.

Transformation pattern of teacher knowledge within reciprocal learning.

Reconstruction of teacher knowledge through knowing and criticizing. Voluntary participation in the project fosters the emergence of general education knowledge (K_1−Ge) and begins the process of knowledge reconstruction through teacher noticing. Since all teachers joined the project willingly, they were encouraged to show curiosity and a desire to exchange information about an unfamiliar country and educational environment. The various and abundant information satisfies their mutual interests and meets their expectations. General education knowledge (K_1−Ge) emerges and is enriched through communication. K_1−Ge contains educational reform, educational system, curriculum, and school-based education (such as general activities, daily routines, etc.). Meanwhile, it also involves general mathematics education knowledge (K_−Gme), although K_−Gme is present in small amounts and scattered. This is the reason that specific mathematics teaching and learning knowledge do not appear in the knowledge map above. Consequently, the knowledge framework does not sufficiently empower or apply in teaching. All this knowledge is identified through teacher noticing, which relies on comparing with each other's experience and interests, so that growing new knowledge can be connected to the original teacher knowledge (K₀) system, but has not yet been integrated into K₀.

In the second phase, the shift toward content-specific communication leads to a deeper examination of mathematics education knowledge (K_2−Gme) and the identification of cross-cultural disparities. During the second phase, communication is highly related to content teaching, and despite the interests, the K_1−Ge knowledge shifts to K_2−Gme, which concentrates more on mathematics. Meanwhile, some part of K_2−Gme is growing based on general educational context knowledge (K_2−Gec). Although it is not sufficient, it still helps teachers to place what they notice in a specific context to better understand mathematics education.

During Phase II, although cross-cultural education knowledge (K_2−Cr−cE) serves as a foundation to support reciprocal learning, it is still not fully integrated. The more specific their noticing becomes, the more comparisons they make. All these comparisons direct them to identify disparities or gaps between their national mathematics teaching and learning practices. They begin to evaluate or criticize from their own stance, using their personal criteria, as K_1−Ge achieved in Phase I, which is not effectively applied to explain what they notice. Subject matter knowledge and related pedagogical content knowledge (K_2–S&Pck) are noticed based on the period of teaching they reviewed from video recordings, which remains isolated from the new knowledge framework.

Reconstruction of teacher knowledge through understanding and learning. Through in-depth visits and communication interactions in Phase III, reciprocal teachers enrich their direct experiences, fostering understanding and respect based on frequent and deep interactions. In their attempt to integrate new knowledge based on cross-cultural education knowledge (K_3−Cr−cE), which served as a cultural background to understand each other's education, reciprocal teachers enrich their practice. Subject matter knowledge and related pedagogical content knowledge (K_3–S&Pck) are developed based on teacher noticing and improved based on the connection with general mathematics education knowledge (K_3−Gme) and general educational context knowledge (K_3−Gec). This integration helps teachers understand the nature of mathematics teaching. Practical knowledge (K_3−P) is achieved by integrating K_3–S&Pck, K_3−Gme, and K_3−Gec.

Reciprocal teachers reconstructed their knowledge framework, which enables the interpretation of professional noticing topics, correcting many of the previous misunderstandings or doubts. When teachers notice new events, although they sometimes compare them with their own education, they also prefer to interpret these events within a specific educational context and culture of each other. This approach is supported by research from the National Academies of Sciences, Engineering, and Medicine (NASEM, 2018), which shows that cultural values may have a significant impact on learners’ ways of thinking and objectives, highlighting the need for understanding and respect in education.

Reciprocal teachers share lessons with specific purposes in mind, such as exploring essential themes, addressing difficulties in teaching and learning, and reforming educational practices. While the lessons shared may not be perfect, this does not detract from the value of reflective practices. In fact, discussions often transcend the specific lessons, leading to abundant conversations and the generation of more research questions. Ultimately, within the cross-cultural professional learning community, professional pedagogical knowledge is formed and enriched.

Reconstruction of teacher knowledge through cocreation within reciprocal learning. In the post-Phase III of reciprocal learning, an inclusive knowledge framework has been thoroughly integrated into the existing knowledge framework (K₀), forming a new knowledge framework (new-K₀). This new framework breaks away from interpreting other cultures based on single-personal values, enabling an understanding of each other's educational teaching experiences and identifying the similarities and differences within cultural contexts. It strives to unbiasedly understand reciprocal teachers’ thinking and activities, even though indirectly. New knowledge is continuously created by selectively applying knowledge and transforming experiences. Studies show that all learning processes are influenced by a large cultural system (Nasir & Hand, 2006; Tomasello, 2016). In fact, to disrupt the equilibrium of existing cognition, one must perceive the significance of new knowledge. Learners want to embrace new knowledge, not to prove correctness, but to prove its value (Whitehead, 2012).

Under sustainable collaboration, teachers continuously share their daily teaching practices, instructional designs, and students’ learning outcomes. They exchange thoughts behind their teaching through think-aloud methods, which has evolved into a sustainable, conscious, and autonomous dialogue. Even after the project ended, the reciprocal learning teaching partners maintained contact and engaged in intermittent exchanges. Teachers collectively achieve the recreation of knowledge, applying and transferring knowledge creatively, and continually constructing and enriching their educational concepts and teaching methods to overcome new mathematics teaching challenges.

Within reciprocal cross-cultural learning contexts, teachers have a chance to reflect on themselves, understand themselves, and develop themselves through mutual inquiries and collaboration. This process enables them to transcend culture-specific problem-solving and coconstruct pedagogical knowledge for universal educational challenges, thereby creating knowledge and sharing experiences.

Understanding Based on Real Contexts Is Key to Forming Stable Achievements in Teacher Reciprocal Learning

The formation of international understanding and respect in reciprocal learning is a gradual process, stemming from indirect learning, direct experience, and positive construction. In this process, the more important aspects are walking into authentic contexts and engaging in deep interaction. Context refers to the deep structure of culture, the web of connections with unique cultural activities within a certain culture (Sun, 2004, p. 5). Context encompasses multiple dimensions such as politics, economy, culture, systems, and communities (Hallinger, 2018). Teachers achieve continuous learning and professional development by participating in communities and interacting with others and their environment (Darling-Hammond et al., 2017; Opfer & Pedder, 2011). This kind of context-based interaction within cultures forms an understanding and the respect for cross-cultural perspectives based on cultural identity.

In the context of cross-cultural reciprocal learning, teachers engage in the sharing of practical experiences through deeper learning and collaboration. They transcend borders not only in geographical terms but also in achieving a profound understanding of teaching practices and philosophies. This cross-border collaborative teaching promotes teachers’ professional growth and provides students with a diversified learning experience. Through face-to-face interactions both online and in schools, teachers can directly observe and learn from mathematics education rooted in different cultural contexts; thereby, teaching strategies in practice are continuously created and enhanced. This sustainable, creative process of reciprocal learning strengthens the connections among teachers and provides a continuous impetus for innovation and development in educational practices.

Cross-cultural reciprocal learning allows participants to immerse themselves in cross-cultural educational settings, engaging in long-term, in-depth, and interactive processes. This enables participants to appreciate cultural differences, respect diverse perspectives, and engage in equitable collaborative learning across cultural contexts. Through interaction, biases are eliminated, understanding is deepened, and empathy and respect are formed based on specific cultural contexts; meanwhile, participants foster a collaborative learning environment. Everyone's life trajectory is influenced by the cultural, social, cognitive, and biological contexts. Based on understanding the multiple dimensions of the learner, such as development, culture, context, and history, the core of human learning can be understood (NASEM, 2018, p. 2). All learners grow and learn within culture-based contexts and in ways determined by culture (NASEM, 2018, p. 225). This reciprocal learning project has propelled participants to share knowledge and experiences within a cross-cultural teacher professional development community.

The Mathematics Education Reveals Unique Cultural Characteristics in Reciprocal Learning

Mathematics education in cross-cultural reciprocal learning showcases distinct linguistic models and cultural influences on teaching practices. Mathematical language is characterized by its high degree of abstraction, conciseness, and precision (Shao & Liu, 2005). When presented in the form of symbols, numbers, and diagrams, the language of mathematics permits teachers from different countries to share a common language. Although language barriers pose a challenge to effective communication between reciprocal teachers, the language of mathematics simplifies their communication, offering the possibility for direct exchange in collaborative teaching endeavors.

Within the context of cross-cultural reciprocal learning, three distinct linguistic models are identified: the scientific language of mathematical symbols, the professional language of pedagogy, and the communicative language of social interaction. The universality of mathematical symbols and educational terminology serves as constructive elements in the process of communication and enhances mutual understanding. The characteristic of professional pedagogy language is that it has special and shared expressions within the profession. Combined with translation, these expressions can be understood. It has been shown that mathematics education in cross-cultural reciprocal learning has subject advantages.

Mathematics teachers from different cultural backgrounds exhibit significant differences in teaching methods, preferences, and influences on students’ learning. Different countries with varying cultural backgrounds impact teachers’ professional learning and action. In reciprocal learning, teachers identify characteristics of teaching practices from a teacher noticing perspective. Canadian mathematics teaching is student-centered, emphasizes the accumulation and transfer of student experiences, encourages deep learning, is more flexible in instruction, and focuses on students’ core competencies and critical thinking skills. On the other hand, Chinese mathematics teachers exhibit a strong sense of order and procedure in their teaching, maintain a consistent pace within grades, focus on cultivating subject-based foundational abilities, grasp the essence of the discipline, and emphasize the effectiveness of the curriculum under goal-oriented instruction.

As Vistro-Yu (2013) noted, enhancing collaboration and partnerships further improves our understanding of the complexity of mathematics teaching and deepens our understanding of a country's uniqueness and individuality. Thus, when learning from international mathematics education cases, cultural relevance must be considered as a factor influencing reciprocal mathematics teaching and learning, and even teacher development. All these considerations would provide a higher quality and excellent mathematics education for all within a global vision.

The Role of Teachers Influences the Perspectives of Teacher Noticing and the Transformation of Experience

Teachers’ roles influence teacher noticing perspectives and experience transformation, with differences between Chinese and Canadian teachers in several aspects. The Chinese teachers were selected by their principals due to their roles as subject leaders in mathematics teaching. When identifying events from the perspective of teacher noticing, they exhibit a more holistic and foresight-oriented approach, closely related to current educational reforms. Chinese teachers also have a mission to promote mathematics teaching supervision at both the school and district levels, as well as to establish long-term collaboration within reciprocal partnerships. Consequently, teacher noticing is highly related to current teaching reform practices (such as interdisciplinary thematic learning, homework designed for different levels, etc.). They are more active and leading in reciprocal learning, making progress with the knowledge gained, and then transferring the experience into their reform practices both in schools and districts.

Canadian teachers are invited entirely based on their willingness to participate and have no additional school roles. The events they notice are specific and highly related to their own personal teaching. They apply some gained knowledge to their practical mathematics teaching. They are interested in the high effectiveness of Chinese mathematics teaching, especially considering PISA test conclusions, focusing on understanding the successful experiences of Chinese mathematics education. Thus, they attempt to understand how Chinese children learn mathematics, which is deemed a strategy to decode. Then, they try to help Chinese children in their classes learn mathematics better.