Mathematics learning encompasses a broad range of processes and skills that change over time. Magnitude and equivalence are two fundamental mathematical ideas that students encounter early and often in their mathematics learning. Numerical magnitude knowledge is knowledge of the relative sizes of numbers, including whole numbers, fractions, and negative numbers, within a given scale. Understanding mathematical equivalence means understanding that two or more specific quantities with the same value can be represented in a variety of ways and remain equal and interchangeable. A major area of research on equivalence is knowledge of the equal sign. Both equal sign knowledge and magnitude knowledge are foundational in that they predict later learning in mathematics, including algebra. Implications for practice include the use of number lines and more variation in the way that arithmetic problems are formatted.
Article
Key Mathematical Competencies From Arithmetic to Algebra
Kristie J. Newton, Christina A. Barbieri, and Julie L. Booth
Article
Mentoring Epistemologies Beyond Western Modalities
Carol A. Mullen
Commitment to mentorship, while necessary to benefit mentoring parties, is insufficient to work with the complexities of contemporary educational settings, especially in pursuit of engagement and learning for all. Mentoring that makes a profound difference for all participants, worldwide, is oriented at the outset to call into question such organizational constraints as hegemony, hierarchy, and culture. Traditional versus alternative approaches to mentoring is a critical binary that can be differentiated in the abstract. However, context and culture are existing organizational realities for which mentoring forms, enactments, and activities (such as mentoring circles) either perpetuate the status quo or produce significant change. Thus, alternative mentoring approaches work within both the traditional view of mentoring and any alternative to it.
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Literacy, Basic Processes, and Interventions for Struggling Readers
Robert Savage
Literacy is a gateway to education, and yet universal literacy remains an aspiration rather than a reality. The science of reading has, however, made significant progress in understanding the key factors that impact development. Five relevant factors can be identified. The first factor is the developmental focus of models. Here the richness and dynamic nature of development is central. Models must clearly explain change and phenomena such as bi- and multilingualism. A second factor concerns bioecological influences on development. Stronger models include understandings of the complexity of gene–environment interactions in development. A third pertinent factor concerns the precise nature of the learning task facing the beginner reader, and in particular the influence of distinct orthographies. A fourth factor concerns the coherent exposition of the cognitive processes involved in “word-level” and “text-level” reading processes. Finally, contextual effects on literacy are profound. Historical and politicoeconomic forces are often linked to wide country- and region-based differences in literacy.
A detailed treatment of what is known about effective interventions for struggling readers can be built on the basis of this theorizing. Here, evidence from meta-analysis suggests that both the word-level decoding and text-level comprehension aspects of reading development can be measurably improved through evidence-based interventions. For word-level interventions studies focusing on phonics currently furnish the most secure evidence of impact. For text-level comprehension, interventions focusing on oral language development and text-based meta-cognitive strategy appear the most efficacious. Measure of treatment effects for such interventions show modest but reliable impacts on development and form the basis of ongoing efforts to optimize interventions.
Article
Problem Solving
Richard E. Mayer
Problem solving refers to cognitive processing directed at achieving a goal when the problem solver does not initially know a solution method. A problem exists when someone has a goal but does not know how to achieve it. Problems can be classified as routine or non-routine, and as well-defined or ill-defined. The major cognitive processes in problem solving are representing, planning, executing, and monitoring. The major kinds of knowledge required for problem solving are facts, concepts, procedures, strategies, and beliefs. The theoretical approaches that have developed over the history of research on problem are associationism, Gestalt, and information processing. Each of these approaches involves fundamental issues in problem solving such as the nature of transfer, insight, and goal-directed heuristics, respectively. Some current research topics in problem solving include decision making, intelligence and creativity, teaching of thinking skills, expert problem solving, analogical reasoning, mathematical and scientific thinking, everyday thinking, and the cognitive neuroscience of problem solving. Common theme concerns the domain specificity of problem solving and a focus on problem solving in authentic contexts.
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Critical Realism for Ethnography
Grant Banfield
While specific applications of critical realism to ethnography are few, theoretical developments are promising and await more widespread development. This is especially the case for progressive and critical forms of ethnography that strive to be, in critical realist terms, an “emancipatory science.” However, the history of ethnography reveals that both the field and its emancipatory potential are limited by methodological tendencies toward “naïve realism” and “relativism.” This is the antimony of ethnography. The conceptual and methodological origins of ethnography are grounded in the historical tensions between anti-naturalist Kantian idealism and hyper-naturalist Humean realism. The resolution of these tensions can be found in the conceptual resources of critical realism. Working from, and building upon, the work of British philosopher Roy Bhaskar, critical realism is a movement in the philosophy of science that transcends the limits of Kantian idealism and Humean realism via an emancipatory anti-positivist naturalism. Critical realism emerged as part of the post-positivist movement of the late 1960s and early 1970s. From its Marxian origins, critical realism insists that all science, including the social sciences, must be emancipatory. At its essence, this requires taking ontology seriously. The call of critical realism to ethnographers, like all social scientists, is that while they must hold to epistemological caution this does not warrant ontological shyness. Furthermore, critical realism’s return to ontology implies that ethnographers must be ethically serious. Ethnography, if it is to hold to its progressive inclinations, must be about something. Critical realism for ethnography pushes the field to see itself as more than a sociological practice. Rather, it is to be understood as a social practice for something: the universalizing of human freedom.
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Theories of Generative Change in Teacher Education
Arnetha F. Ball
In 1950, Erik Erickson introduced the concept of generativity in psychosocial development when referring to an individual’s desire to produce new knowledge that contributes to the guidance of the next generation. Nearly fifty years later, Epstein built on the term generativity in his research when referring to the generation of new or novel behavior in problem-solving. According to Epstein, generativity theory is a formal, predictive, empirically based theory of ongoing behavior in novel environments. Because it can be used to predict generative behavior and engineer new performances, it is also predictive of creativity and offers important contributions to the study of the transformative processes needed by teachers who desire to work effectively with students in culturally and linguistically complex classrooms. The evolution of theories of generativity can be traced from their use in studies of psychosocial development, to their use in studies of education, teacher education, and the preparation of teachers who work effectively in complex, 21st century classrooms. It should be noted that the theme that runs throughout the research literature on generativity over the last seventy years is a focus on using the term generativity theory to refer to a formal, predictive theory of creative behavior in individuals. When applied to education and the development of teachers to teach in culturally and linguistically complex classrooms, it is important to note that oftentimes teachers—many of whom have never worked with diverse student populations before—must develop the ability to translate their desire to teach into a conscious concern to serve the next generation—into a generative commitment to teach all students. They must make decisions to establish goals for generative behavior and then turn those decisions into generative actions and the creation of effective pedagogical solutions that meet the needs of their diverse students. One meaning of generative behavior is to generate things and people, to be creative, productive, and fruitful, to “give birth” to creative pedagogical problem-solving both figuratively and literally. The scholarship on generativity theory emphasizes the notion that generativity, unlike simple altruism or general prosocial behavior, involves the creation of a product or legacy. The qualities emphasized in generativity theory are the qualities needed by teachers who hope to be effective in their work with diverse populations. Generative behavior involves the conservation, restoration, preservation, cultivation, nurturance, or maintenance of that which is deemed worthy of such behavior, as in nurturing children and adapting traditions that link generations and assure continuity over time—through generative concern, action, and narration. Reflection is not enough. Rather, generative action that stems directly from teachers’ commitment, enhanced belief, and stimulated by concern, inner desire and cultural demand is needed. Generative action—which includes the behaviors of creating, maintaining, and offering to others—is the ultimate result of generativity. Narrations of generativity and the use of writing as a pedagogical tool for deep thinking are two means by which the complex relations among demand, desire, concern, belief, internalization, commitment, and action can be captured and analyzed.
Article
Status, Content, and Evaluation of Lesson Study in Japan on Teacher Professional Development
Takashi Nagashima
In Japan, various styles of Lesson Study (LS) have been born over 140 years. The first issue is what should be the focus of observation in the live lesson. There are two trends with regard to the target of observation. One is teacher- and lesson-plan-centered observation since the Meiji era (1870s), and the other is child-centered observation since the Taisho era (1910s). The former is closely related to administrative-led teacher training. The latter is more complex and can be further divided into five types. The second issue is which activities are given priority in the LS processes: observation of the live lesson itself, preparation before the lesson, or reflection after the lesson. Furthermore, each activity can be designed as a personal or a collaborative process. Thus, there are roughly six types of LS in Japan related to this issue. Which type is adopted depends on the period, lesson-study frequency, and school type. In addition, it is noteworthy that the type of LS implemented is closely related to which of demonstration teacher or observers are regarded as the central learners. The third issue is whether to regard LS as scientific research or as literary research. Teachers and researchers in 1960s Japan had strong interest in making lessons and lesson studies more scientific. On the other hand, as teachers attempt to become more scientific, they cannot but deny their daily practice: making improvised decisions on complicated situations without objective evidence.
Although lesson studies have been revised in various forms and permutations over the last 140, formalization and ceremonialization of lesson studies has become such that many find lesson studies increasingly meaningless and burdensome. What has become clear through the discussions on the three issues, the factors that impede teacher learning in LS are summarized in the following four points; the bureaucracy controlled technical expert model, exclusion of things that are not considered scientific, the view of the individualistic learning model, and the school culture of totalitarian products.
To overcome obstruction of teachers’ education in LS and the school crisis around the 1980s, the “innovative LS Cases” has begun in the 1990s. The innovative LS aims not for as many teachers as possible but for every teacher to learn at high quality. In the innovative LS Case, what teachers are trying to learn through methods of new LS is more important than methods of new LS itself. Although paradoxical, in order to assist every single teacher to engage in high quality learning inside school, LS is inadequate. It is essential that LS address not only how to actualize every single teacher to learn with high quality in LS but also through LS how to improve collegiality which enhances daily informal collaborative learning in teachers room. Furthermore, LS cannot be established as LS alone, and the school reform for designing a professional learning community is indispensable. Finally, the concept of “the lesson study of lesson study (LSLS)” for sustainable teacher professional development is proposed through organizing another professional learning communities among managers and researchers.
Article
Complexity Theory as a Guide to Qualitative Methodology in Teacher Education
Fiona Ell, Marilyn Cochran-Smith, Mary Hill, Mavis Haigh, Lexie Grudnoff, and Larry Ludlow
Qualitative teacher education research is concerned with understanding the processes and outcomes of teacher preparation in ways that are useful for practitioners, policymakers, and the teaching profession. Complexity theory has its origin in the biological and physical sciences, where it applies to phenomena that are more than the sum of their parts—where the “higher order” form cannot be created by just putting together the pieces that it is made from. Complexity theory has moved to social science, and to education, because many social phenomena also seem to have this property. These phenomena are termed “complex systems.” Complexity theory is also a theory of learning and change, so it is concerned with how complex systems are learning and changing. This means that methods to investigate complex systems must be able to identify changes in the system, termed “emergence,” and must also account for change over time and the history of the complex system. Longitudinal designs that involve the collection of rich data from multiple sources can support understanding of how a complex system, such as teacher education, is learning and changing. Comparative analysis, narrative analysis, extended case studies, mapping of networks and interactions, and practitioner research studies have all been used to try to bring complexity theory to empirical work in teacher education. Adopting a complexity theory approach to research in teacher education is difficult because it calls into question many taken-for-granted assumptions about the nature of research and what is possible to find. Linear, process-product thinking cannot be sustained in a complexity framework, and ideas like “cause,” “outcome,” “change,” and “intervention” all have to be re-thought. A growing body of work uses complexity thinking to inform research in teacher education.
Article
Teachers’ Knowledge for the Digital Age
Margaret L. Niess
The 21st-century entrance of digital media into education has required serious reconsideration of the knowledge teachers need for guiding students’ learning with the enhanced technological affordances. Technological Pedagogical Content Knowledge (TPCK or TPACK) describes the interaction of the overlapping regions of technological knowledge, pedagogical knowledge, and content knowledge that also creates four additional regions (technological pedagogical knowledge, technological content knowledge, pedagogical content knowledge, and technological pedagogical content knowledge). These knowledge regions are situated within a contextual knowledge domain that contains macro, meso, and micro levels for describing the dynamic equilibrium of the reformed teacher knowledge labeled TPCK/TPACK. Teacher educators, researchers, and scholars have been and continue to be challenged with identifying appropriate experiences and programs that develop, assess, and transform teachers’ knowledge for integrating information and communication technologies (ICT) that are also spurring advancements in artificial intelligence (AI) as learning tools in today’s reformed educational environments. Two questions guide this literature review for engaging the active, international scholarship and research directed toward understanding the nature of TPCK/TPACK and efforts guiding the transformation of the teacher’s knowledge called TPCK/TPACK. The first question considers the nature of a teacher’s knowledge for the digital age and how it differs from prior descriptions. Three distinct views of the nature of TPCK/TPACK are explained: the integrative view; the transformative view; and a distinctive view that directs how the primary domains of pedagogy, content, and technology enhance the teacher’s knowledge. The second question explores the research and scholarship recommending strategies for the redesign of teacher education towards developing, assessing, and transforming teachers’ TPCK/TPACK. These strategies recognize the importance of (1) using teacher educators as role models, (2) reflecting on the role of ICT in education, (3) learning how to use technology by design, (4) scaffolding authentic technology experiences, (5) collaborating with peers, and (6) providing continuous feedback. This research further characterizes teacher educators with strong ICT attributes as the gatekeepers for redesigning teacher education programs so that today’s teachers are better prepared to engage in the strategic thinking of when, where, and how to guide students’ learning given the rapid advancements of digital technologies. These cumulative scholarly efforts provide a launchpad for future research toward transforming teachers’ knowledge for teaching with the technological advancements of the digital age.
Article
Academic Optimism
Martinette V. Horner, Derrick D. Jordan, and Kathleen M. Brown
Academic optimism was developed in 2006 as a latent concept that provides insight into the improvement of student outcomes especially for those who, because of socioeconomic status, ethnicity, and other demographics, have historically been labeled as underperforming. The three main components of academic optimism (academic emphasis, collective emphasis, and faculty trust) underscore the reality that the teachers, parents, and students all play a critical role in the education arena when it comes to ensuring that students fully grow and stretch to the fullest extent possible. High academic optimism in a school suggests that academic achievement is valued and supported; the faculty has the capacity to help students achieve; and students and parents can be trusted as partners of the school for student achievement. Each of these can be controlled by the actions and decisions of school leaders and faculty so that schools can overcome the effects of poverty on student achievement.
