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This study aims to explore teacher noticing differences of the sampled US and Chinese elementary teachers from cross-cultural mathematics videos. A total of 34 expert teachers commented on 25 video clips online. We coded what and how teachers noticed from the videos both quantitatively and qualitatively. Findings reveal teachers' strong interests and profound reflections, especially in the teaching domain including representations, communication, and teacher questioning/guide. Cross-cultural differences in teacher noticing were identified, which were discussed based on possible cultural influences. Implications in research and practice, teacher support, and methodology are discussed.

This exploratory study examined whether instruction aligned with the Institute of Education Sciences recommendations (i.e., use of worked examples, representations, deep questions) predicts student learning of early algebra, particularly inverse relations, in elementary classrooms. Instructional quality was determined through an opportunity-propensity analysis of cross-cultural data (N = 569) from the United States and China, which showed that teaching played a stronger role in student learning than previously reported. Results from structural equation modeling indicated that teaching quality--especially teachers' use of deep questions--had both direct and indirect effects (through prior knowledge) on student learning, which together demonstrated greater predictive power than what was reported in previous studies. Specifically, direct path results showed that instructional quality, rather than teacher characteristics, played a more important role in predicting student learning. Indirect effects showed that instructional quality, mediated through students' prior knowledge, predicted the most about students' mathematics achievement. Taken together, the results of the analysis provided strong evidence that implementation of worked examples, representations, and deep questions were positively related to student learning in early algebra. Hierarchical linear regression confirmed these findings but indicated that teaching quality in the current data explained more additional variance in "correctness" than "inverse understanding" for student learning. These findings suggest that the kind of instruction examined in prior opportunity-propensity studies had a weaker effect because it did not align with the Institute of Education Sciences recommendations. Findings also shed light on the extent to which classroom instructions are better at developing students' algebraic readiness.

In this survey paper we focus on mathematics learning in Chinese contexts, as a way to contribute to broader discussions about mathematical learning. We first review the features of Chinese students' mathematical learning depicted in the literature, followed by a review of student mathematical learning in recent Chinese research journals. This leads to an introduction of the papers on Chinese students' learning in this issue. For Chinese students' learning contexts, we discuss four aspects, namely, classroom instruction, teachers' professional learning, curriculum materials, and learning outside of school. For each context, we review the literature findings on the identified features, introduce emerged practices and most recent policies under the reformed era, and discuss the relevant papers in this special issue. Whenever possible, we connect findings on Chinese students' learning with the associated contexts and relate these findings in the Chinese contexts to findings in the broader world context. We conclude this survey paper with possible lessons learned from Chinese students' learning features and from the varied Chinese contexts. In particular, we discuss these aspects from culturally contextualized and semantically decontextualized dimensions, which is expected to facilitate broad international discourse centering on the three questions proposed at the end of this paper.

Collaborative learning has been advocated for the last several decades. However, classroom practice has indicated that this method often comes with challenges and low efficiency, suggesting the need for better support for teachers. The aim of this study is to explore teachers' learning about collaborative learning in a professional development (PD) environment. In particular, we focused on teacher perceptions of the two aspects of collaborative learning, namely, tasks and collaboration. Teachers learned that tasks for collaborative learning should be open-ended, challenging, and feature a low floor but a high ceiling. Such tasks call for true collaboration among group members and provide a space for interactive engagement. In addition, teachers learned that true collaboration should occur in two layers, including both peer collaboration and collaboration between the PD provider (researcher) and teachers. For peer collaboration, there should be 'deep participation' in which every group member actively contributes ideas and engages with others' thinking. For the researcher-teacher collaboration, the PD provider serves as the teachers' learning partner through careful task design and facilitation of learning. Based on their direct experience with collaborative learning in this PD workshop, teachers learned about the roles of tasks and collaboration in their own classrooms. Implications of the findings and future directions for research and practice are discussed.

In this editorial, we review 400 articles and reviews published in the International Journal of STEM Education during its first decade (2014–2023). Using bibliometric analysis, we examine these publications to assess the journal's major contributions to STEM education research and identify emerging trends over the years. The results present a dynamic picture of the growth of STEM education, highlighting key topics, such as STEM integration, equity, and emerging technologies. These findings also reveal evolving "hot topics" that reflect the shifting interests of researchers in the field. This review suggests that many areas of STEM education research are still in the growth phase. We encourage readers to use these insights as a foundation for developing future research agendas and advancing STEM education globally. [ABSTRACT FROM AUTHOR]

Instructional principles gleaned from cognitive science play a critical role in improving classroom teaching. This study examines how three cognitive instructional principles including worked examples, representations, and deep questions are used in eight experienced elementary teachers' early algebra lessons in the U.S. Based on the analysis of 32 videotaped lessons of inverse relations, we found that most teachers spent sufficient class time on worked examples; however, some lessons included repetitive examples that also included irrelevant practice problems. Most teachers also situated new teaching in concrete contexts, which were faded into abstract representations. However, connections between concrete and abstract were not always made. The largest challenge was rooted in teachers' inability to ask deep questions that elicited students' deep explanations. Some teachers focused on key words and provided students with direct explanations. Implications are discussed.

This study examines how sampled Chinese and U.S. third and fourth grade students (N[subscript China] = 167, N[subscript US] = 97) understand the commutative, associative, and distributive properties. These students took both pre- and post-tests conducted at the beginning and end of a school year. Comparisons between students' pre- and post-tests within and across countries indicate different learning patterns. Overall, Chinese students demonstrate a much better understanding than their U.S. counterparts. Among these properties, the associative and distributive properties appear to be most challenging, especially for the U.S. students. By the end of grade 4, some Chinese students demonstrate explicit understanding of the associative and distributive properties across tasks; almost no U.S. students achieve a comparable level of understanding on these properties. Student understanding in different contexts also reveals cross-cultural differences. Chinese students tend to reason upon concrete contexts for sense-making, which is rare with U.S. students. Finally, the Chinese student sample shows clear growth of understanding across grades, but this is not seen in the U.S. sample. This understanding gap between the two countries is found to dramatically increase over time. Implications are discussed.

In this study we investigate the teaching of the associative property in a natural classroom setting through observation of classroom video of several elementary math classes in a large urban school district. Findings indicate that the associative property was often conflated with the commutative property during teaching. The role of the associative property in many computational tasks remained fully implicit, even after the property had been formally introduced. Most classrooms did not present the associative property with substantial justification in terms of concrete representations, especially those in which the abstract property was formally introduced -- while a few classrooms did situate the property in rich concrete contexts, the property remained implicit in these classes, indicating a lack of linking between the concrete and the abstract when teaching the property. Instruction also did little to develop the notion of the associative property as a property of an operation conceptualized as a mental object, rather than as a rule governing the outcome of a procedure. Much of the instruction displayed a significant focus on computational strategies, which aggravated the challenge of providing a clear explanation of the nature and meaning of the associative property.

Linear quantity models such as pre-tapes, tape diagrams, and number line diagrams have drawn increasing attention in mathematics education around the world. However, we still know relatively little about how teachers actually use these models in the classroom. This study explores how exemplary US and Chinese elementary teachers use linear quantity models during mathematics instruction. Based on an examination of 64 videotaped lessons on inverse relations, we identified 110 "diagram episodes." An analysis of these episodes reveals that linear quantity models, especially tape diagrams, were used more frequently in US classrooms than in Chinese classrooms. However, Chinese lessons used these models for the sole purpose of modeling the underlying quantitative relationships, whereas US lessons mainly used them to aid in computation. In addition, while US teachers rarely involved students in discussion of linear quantity models, Chinese teachers spent significant time engaging students in co-constructing, comparing, and explaining these models.

This study reports how an expert Chinese teacher implements mathematics textbook lessons in enacted instruction. Our video analysis indicates that both textbook and enacted teaching included only one worked example; however, the teacher engaged students in unpacking the example in great depth. Both the textbook and the enacted teaching showed "concreteness fading" in students' use of representations. However, the Chinese teacher incorporated students' self-generated representations and facilitated students' active modeling of quantitative relationships. Finally, the Chinese teacher asked a greater number of deep questions than were suggested by the textbook. These deep questions often occurred as clusters of follow-up questions that were either concept-specific or promoted comparisons which facilitated connection-making between multiple representations and solutions.

Tape diagrams, also known as strip diagrams, bar models, fraction strips, or length models, are linear representations that can be used to effectively model quantitative relationships during problem solving. Tape diagrams are a powerful model recommended by the Common Core, but they are relatively new to teachers and students in the United States. This study aims to support teachers who are less familiar with tape diagrams in learning to use this model productively, as exemplified by two Chinese teachers.

This study explores how preservice teachers (PSTs) transfer the intended specialized content knowledge (SCK) to elementary classrooms. Focusing on the case of the associative property of multiplication, we compared three PSTs' SCK during enacted lessons in fourth grade classrooms with their own learning in professional development (PD) settings. Findings revealed the PSTs' successes and challenges in unpacking an example task, especially in areas of making connections between concrete and abstract representations and asking deep questions that target quantitative interactions. Factors that may have supported or hindered PSTs' SCK transfer include the complex nature of teacher knowledge, the PD effort and the outside factors such as the support from textbooks and cooperating teachers. Implications for teacher education and directions for future research are discussed.

Prior studies show that elementary school children generally "lack" formal understanding of inverse relations. This study goes beyond lack to explore what children might "have" in their existing conception. A total of 281 students, kindergarten to third grade, were recruited to respond to a questionnaire that involved both contextual and non-contextual tasks on inverse relations, requiring both computational and explanatory skills. Results showed that children demonstrated better performance in computation than explanation. However, many students' explanations indicated that they did not necessarily utilize inverse relations for computation. Rather, they appeared to possess partial understanding, as evidenced by their use of part-whole structure, which is a key to understanding inverse relations. A close inspection of children's solution strategies further revealed that the sophistication of children's conception of part-whole structure varied in representation use and unknown quantity recognition, which suggests rich opportunities to develop students' understanding of inverse relations in lower elementary classrooms.

Researchers have long debated the meaning of mathematical understanding and ways to achieve mathematical understanding. This study investigated experienced Chinese mathematics teachers' views about mathematical understanding. It was found that these mathematics teachers embrace the view that understanding is a web of connections, which is a result of continuous connection making. However, in contrast to the popular view which separates understanding into conceptual and procedural, Chinese teachers prefer to view understanding in terms of concepts and procedures. They place more stress on the process of concept development, which is viewed as a source of students' failures in transfer. To achieve mathematical understanding, the Chinese teachers emphasize strategies such as reinventing a concept, verbalizing a concept, and using examples and comparisons for analogical reasoning. These findings draw on the perspective of classroom practitioners to inform the long-debated issue of the meaning of mathematical understanding and ways to achieve mathematical understanding.

This study, focusing on inverse relations, examines how representative U.S. and Chinese elementary textbooks may provide opportunities to learn fundamental mathematical ideas. Findings from this study indicate that both of the U.S. textbook series (grades K-6) in comparison to the Chinese textbook samples (grades 1-6), presented more instances of inverse relations, while also containing more unique types of problems; yet, the Chinese textbooks provided more opportunities supporting meaningful and explicit learning. In particular, before presenting corresponding practice problems, Chinese textbooks contextualized worked examples of inverse relations in real-world situations to aid in sense making of computational or checking procedures. The Chinese worked examples also differed in representation uses especially through concreteness fading. Finally, the Chinese textbooks spaced learning over time, systematically stressing structural relations including the inverse quantities relationships. These findings shed light on ways to support students' meaningful and explicit learning of fundamental mathematical ideas in elementary school.

Background: Specialized content knowledge (SCK) is a type of mathematical content knowledge specifically needed for teaching. This type of knowledge, although serving as a critical component for preservice teacher education, is often challenging to develop with preservice elementary teachers (PTs). The purpose of this study is to investigate PTs' development of SCK for teaching fundamental mathematical ideas in a university methods course. Focusing on the case of the associative property (AP) of multiplication, the author as the course instructor identified three instructional opportunities (a formal introduction and two delayed revisits) to stress two SCK components, representations, and explanations. PTs' learning progresses were assessed through three diagnostic tests (a pretest, a mid-term exam, and a final exam) and two prompts, which informed the upcoming lesson design. Meanwhile, the course instructor conducted ongoing reflections on PTs' learning, which also informed the corresponding lessons. Results: It was found that PTs initially generated abstract number sentences without reasoning about the contexts of word problems. This representational sequence indicates a symbol precedence view. When prompted for explanations, PTs focused on individual numbers rather than quantitative relationships, and they could not consistently apply the basic meaning of multiplication for reasoning. The methods course, when designed to address these issues, promoted PTs' SCK development. At the end of the course, the majority of PTs were able to generate number sentences based on the word problem structures and provided reasonable explanations; however, the methods course also faced dilemmas due to PTs' robust symbol precedence view and the tension between PTs' learning and children's learning. Conclusions: Very few studies have explored ways to support PTs' knowledge growth in SCK, especially for teaching fundamental mathematical ideas. This study, by carefully documenting the successes and challenges in developing PTs' SCK, contributes to the existing literature. Based on our findings, this study highlights the importance of stressing basic meanings so as to develop PTs' explanation skills. Meanwhile, to develop PTs' representation skills, university instructors should be aware of the tension between PTs' and children's learning as manifested by PTs' symbol precedence view. Finally, to support PTs' SCK growth, it is also important to emphasize the role of elementary textbooks.

Through examining a representative Chinese textbook series' presentation of the distributive property, this study explores how mathematics curriculum may structure representations in ways that facilitate the transition from concrete to abstract so as to support students' learning of mathematical principles. A total of 319 instances of the distributive property were identified. The representational transition among these instances was analyzed at three tiers: within one worked example, from the worked example to practice problems within one topic, and across multiple topics over grades. Findings revealed four features that facilitate the transition process in the Chinese textbook series. First, it situates initial learning in a word problem context, which serves as a starting point of the transition process. Second, it sets up abstract representations as an ultimate goal of the multi-tier transition process. Third, it incorporates problem variations with connections in carefully designed tasks that embody the same targeted principles. Fourth, it engages students in constant sense making of the transition process through various pedagogical supports. Implementations and future research directions are also discussed.

This study explores, from both constructivist and cognitive perspectives, teacher guidance in student-centered classrooms when addressing a common learning difficulty with equivalent fractions--lines or pieces--based on number line models. Findings from three contrasting cases reveal differences in teachers' facilitating and direct guidance in terms of anticipating and responding to student difficulties, which leads to differences in students' exploration opportunity and quality. These findings demonstrate the plausibility and benefit of integrating facilitating and direct guidance in student-centered classrooms. Findings also suggest two key components of effective teacher guidance including (a) using pretraining through worked examples and (b) focusing on the relevant information and explanations of concepts. Implementations are discussed.

Lesson study (LS), a teacher-oriented, student-focused professional development approach that originated in Asia, has spread globally. However, findings related to the effect of LS on teacher learning and growth have varied widely. This suggests a need to review current research on LS in the field of mathematics education. This paper focuses on mathematics teachers' learning and professional growth through LS using Clarke and Hollingworth's interconnected model of professional growth (IMPG) that contains three components: domains of change, mediating mechanisms, and change environment. With regard to domains of change, we found that although previous LS studies—often small scaled, qualitative studies—attended to all domains of change, they have more frequently reported changes in the personal (or collective) domain (e.g., teacher knowledge) but have relatively less frequently reported changes in other domains. In fact, there is a lack of research on systematic changes across domains in the IMPG. Regarding the mediating mechanisms, research has well explored both enactment (planning and teaching) and reflection (debriefing). For enactment, the planning process is less appreciated, and there is an inconsistent understanding of the role of the knowledgeable others. For reflection, there is a lack of focus on student thinking and constructive feedback during this process. Finally, research has suggested that LS has received support from the change environment but that tensions (e.g., time limits) have remained. These challenges call for systemic support and cross-cultural collaborations to develop sustainable and large-scale LS. We suggest future directions for continued research and practice. [ABSTRACT FROM AUTHOR]

This study explored a group of elementary teachers' ("n" = 35) learning to construct high-quality lesson plans that foster student understanding of fundamental mathematical ideas. The conceptual framework for this study was gleaned from the recently released Institute of Education Sciences (IES) recommendations, including (a) interweaving worked examples and practice problems, (b) connecting concrete and abstract representations, and (c) asking deep questions to elicit student self-explanations. Comparisons between teachers' pre- and postsurveys, and among teachers' initial, revised, and end-of-course lesson plans, indicated teachers' growth in using worked examples, representations, and deep questions during their lesson planning. Issues related to teachers' learning as they constructed lesson plans that aligned with the IES recommendations were also revealed. (Contains 3 tables and 5 figures.)

This study examines preservice elementary teachers' (PTs) knowledge for teaching the associative property (AP) of multiplication. Results reveal that PTs hold a common misconception between the AP and commutative property (CP). Most PTs in our sample were unable to use concrete contexts (e.g., pictorial representations and word problems) to illustrate AP of multiplication conceptually, particularly due to a fragile understanding of the meaning of multiplication. The study also revealed that the textbooks used by PTs at both the university and elementary levels do not provide conceptual support for teaching AP of multiplication. Implications of findings are discussed. (Contains 4 tables and 7 figures.)

This study qualitatively explored teachers' responses to doubling errors (e.g., 3/4 x 2 = 6/8) that typically reflect students' difficulties in understanding the "rule" for finding equivalent fractions (e.g., 3/4 x 2/2 = 6/8). Although all teachers claimed to teach for understanding in interviews, their responses varied in terms of effectiveness and types. One teacher grasped the doubling error to explain the reasons underlying the rule using symbolic representations. The other two used concrete representations to explicate the reasons but tended to either prevent or ignore the doubling error. Although these teaching actions appeared different in form, all teachers essentially held the same perception-based perspectives; that is, they either assumed students could see what they saw or students could directly comprehend mathematical ideas through visualizing pictures only. Such perspectives overlook students' available mathematical concepts and might be at the root of the difficulties with those particular teaching strategies.

This study took a broad view of generalization, exploring how classroom instruction within opportunities of implicit generalization may be related to students' abilities to generalize. From an integrated perspective of generalization and problem solving, we investigated twelve lessons of two middle level teachers based on three pattern-based problems. We found teachers' limited awareness and knowledge of problem situations, especially problem structures, were obstacles hindering their ability to support students' understanding of implicit generalization. Teachers tended to focus on particular "results" for individual cases rather than the "process" of obtaining those results, resulting in missed opportunities to develop students' sense of general patterns. Without attention to implicit generalization, explicit generalization rarely took place in a meaningful way during problem solving. (Contains 5 figures and 3 tables.)

A steady stream of research has shown that many elementary school teachers have weak mathematical knowledge in some areas, including place value and fractions. Since a teacher's mathematical knowledge affects their students' performance, improving elementary school teachers' knowledge is critical. A better understanding of the mathematical misconceptions held by future (pre-service) elementary school teachers will help break this cycle. This article contributes to the understanding of this problem. As suggested by the Conference Board of the Mathematical Sciences (CBMS) report entitled, "The Mathematics Education of Teachers," many colleges now require several mathematics courses for pre-service elementary school teachers. But, because of a shortage of college mathematics educators these courses are often taught by mathematicians or mathematics educators (or sometimes graduate students) with little or no experience at the K-6 level. College instructors may find themselves under-prepared to anticipate the ways that their students, pre-service elementary school teachers, think of and engage the material. In this article, the authors present an analysis of the errors pre-service elementary school teachers tend to make as they solve mathematics problems. (Contains 8 figures.)

In this study, we investigated Chinese teachers' attributions and coping strategies for classroom misbehaviour across grade levels. A total of 244 teachers (Grades 1-12) from the Chinese mainland participated in this survey. Results indicated that Chinese teachers first attributed misbehaviour to student characteristics, such as being "lazy, not making enough effort", and second to "bad learning habits". Looking across grade levels, elementary teachers first blamed student learning habits while secondary teachers blamed student effort. With regard to coping strategies, inconsistencies were found across grade levels and between teachers' perceptions and actions. The majority of sampled elementary teachers tended to choose "praising good students" as the most effective and often-used strategy, while secondary teachers believed in "talking after class". In fact, "talking after class" was viewed to be more effective as grade level increased. However, teachers reported that they did not actually use the strategy of "talking after class" very often when coping with misbehaviours. (Contains 4 tables.)

This study examines presentations of the distributive property (DP) in two widely used U.S. elementary text series and one main Chinese text series along three dimensions: problem contexts, typical problem types within each problem context, and variability in using the DP. In general, the two U.S. texts were found to resemble each other but to differ considerably from the Chinese text series. Both U.S. texts are computation-dominated, presenting various strategies centering on "breaking apart a factor to perform multiplication." These strategies limit the use of the DP mainly with whole numbers and in a regular direction. The underlying principle of these strategies is seldom made explicit. In contrast, the Chinese text approaches focus on the underlying principle and are well aligned with cognitive research suggestions. Multiple-step word problems with particular structures are used in a systematic and hierarchic manner across grades to help students learn and transfer the DP. The Chinese texts also tend to ask students to "compute in convenient ways" involving various numbers (e.g., whole numbers, decimals, fractions, and percents) and using the DP in both regular and opposite directions. The introduction of repeated variables is a timely application of the DP, which provides an entry to algebra (e.g., expressions and equations with repeated variables). The Chinese approaches (e.g., contextual interferences, spaced practice, and encoding variability) suggest alternative insights into developing U.S. students' understanding of the DP and readying them for algebra. (Contains 3 figures, 9 tables and 2 footnotes.)

This study focuses on Chinese teachers' perceptions of students' classroom misbehaviour. A questionnaire was designed to assess teachers' general concerns about classroom management, teachers' perceptions of the most frequent and troublesome types of misbehaviour, and teachers' perceived needs for help with improving classroom management. A total of 244 responses were collected, from lower elementary to high school teachers in two provinces in mainland China. In general, the majority of Chinese teachers (65.6%) do not think that classroom management is a great concern. They perceive "daydreaming" to be the most frequent and troublesome misbehaviour. This contrasts with many prior studies in Western settings where "talking out of turn" has been reported as the biggest concern. In addition, Chinese teachers are eager to understand the psychological reasons behind student misbehaviour. They also suggest that the Chinese system of "teaching for testing" should be reformed. (Contains 7 tables and 1 figure.)

This study reports findings from comparative samples of sixth-grade Chinese and U.S. students' interpretations of the equal sign. Ninety-eight percent of the Chinese sample correctly answered 4 items indicating conceptions of equality and provided conceptually accurate explanations. In contrast, only 28% of the U.S. sample performed at this level. We examine how teacher preparation materials, students' textbooks and teachers' guidebooks treat equality in each country. U.S. teacher preparation textbooks rarely interpreted the equal sign as equivalence. On the contrary, Chinese textbooks typically introduced the equal sign in a context of relationships and interpreted the sign as "balance," "sameness," or "equivalence" and only then embedded the sign with operations on numbers. (Contains 4 tables.)

The authors examined the extent to which teacher interventions focused on students' mathematical thinking in naturalistic cooperative-learning mathematics classroom settings. The authors also observed 6 videotapes about the same teaching content using similar curriculum from 2 states. They created 2 instruments for coding the quality of teacher intervention length, choice and frequency, and intervention. The results show the differences of teacher interventions to improve students' cognitive performance. The authors explained how to balance peer resource and students' independent thinking and how to use peer resource to improve students' thinking. Finally, the authors suggest detailed techniques to address students' thinking, such as identify, diversify, and deepen their thinking. (Contains 7 tables and 3 figures.)

Teachers and researchers have long recognized that students tend to misunderstand the equal sign as an operator; that is, a signal for "doing something" rather than a relational symbol of equivalence or quantity sameness. Students' equal sign misconception has been researched for more than thirty years (Weaver, 1971, 1973) with little refinement in the theory. It was popularly believed that younger students were not developmentally ready to work variations of open numbers sentences, such as missing addend problems (Thompson & Babcock, 1978). In fact, misconceptions about the equal sign were identified in kindergarten students even before formal instruction (Falkner, Levi, & Carpenter, 1999). However, it is clear that with specific instructional guidance, elementary students can understand that the equal sign expresses a relation (Baroody & Ginsburg, 1983; Carpenter, Levi, & Farnsworth, 2000; Saenz-Ludlow & Walgamuth, 1998). In this article, the authors examine variables that could contribute to student's equivalence misconception and whether the equal sign misconception is still manifest in a U.S. sample and present in a Chinese sample. Findings indicate that misconceptions are still manifest in the U.S., and textbooks do little to mitigate the problem in the United States, while in China students are able to interpret the equal sign as a relational symbol of equivalence. The authors also found that the inclusion of multiple representations for equivalence in textbooks and guidebooks in China make a difference in assisting students to correctly interpret the equal sign.