One of the key ideas of the modern-day elementary mathematics teacher education deals with mediating learning by visual thinking to enable transition from seeing and acting on concrete objects to describing the visual and the physical through culturally accepted symbolic representations. This paper shares mathematical activities designed originally for teacher candidates and used with students in upper elementary classrooms at a school in Upstate New York with minority student enrollment 97%. Because successful use of conceptual thinking by young students does have positive impact on their future teachers, connection of work in the school to a master’s level elementary mathematics education course taught by the authors is discussed. It is shown how using a spreadsheet and Wolfram Alpha allows for the research-like extension of the activities to the secondary level of mathematics education.

During the COVID-19 pandemic, online learning has become an important and widely used form of education. Many studies have pointed out that interaction is key to online learning. The Interaction Hierarchy Theory categorizes interactions in remote teaching into three types: operational, informational, and conceptual. Operational interaction serves as the foundation for all types of interactions and refers to the interface interactions that learners engage in at the behavioral level through the use of media features and tools in online learning. However, should we simply encourage higher intensity operational interaction? Specifically, live teaching, as a form of remote teaching, has a higher sense of immediacy and synchronicity compared to asynchronous learning. Should we encourage and guide students to engage in more operational interaction during live teaching? How would it affect learners' informational and conceptual interactions? In this study, 137 students from 21 live classes were grouped according to class size and operational interaction intensity, and their levels of informational and conceptual interaction were explored. The results showed that the conceptual interaction intensity of learners in live teaching was higher than the informational interaction intensity, and operational interaction intensity and class size both had an impact on informational interaction, but a weaker impact on conceptual interaction. Operational interaction can affect conceptual interaction through informational interaction, especially through the mediation of student-resource informational interaction. The contribution of this study lies in verifying the establishment of the interaction hierarchy tower in the live teaching scene, that is, there are three different levels of interactive influence chains from operational interaction, informational interaction and conceptual interaction. Operational interaction and class size have a strong influence on information interaction directly and conceptual interaction indirectly. In online learning aiming at high-level interaction such as conceptual interaction, designers should not blindly promote operational interaction, but should pay attention to the promoting effect of operational interaction on informational interaction, and the operational interaction without effect on learners' informational interaction is invalid. In addition to enhancing operational interaction, controlling class size is also a way to facilitate informational interaction.

The educational landscape in North and West Africa stands to gain significant advantages through collaborative government-entrepreneurial investments in Educational Technology (EdTech). The establishment of public-private partnerships plays a pivotal role in effectively infusing technology into education systems, thereby enhancing student accessibility, improving educational quality, and fostering better learning outcomes. Beyond its immediate impact on education, EdTech investments also hold the potential to stimulate economic growth. Moreover, these investments can nurture a pool of indigenous entrepreneurs armed with expertise in both business and information technology, thereby contributing to the expansion of the EdTech sector. This burgeoning sector, prevalent in emerging regions, becomes a magnet for foreign investments, effectively bolstering national economies and creating ripple effects across diverse industries. Such synergistic collaborations within the EdTech sphere underscore the political commitment of governments toward the advancement of education. This dedication resonates on a global scale, attracting the attention of international organizations and donor nations, as investing in education for sustainable development fosters not only improved socio-economic conditions but also favorable diplomatic relationships. This research investigates the impact of collaborative EdTech investments between governments and entrepreneurial entities on the education. By examining specific case studies within North and West Africa, namely Ghana, Nigeria, Morocco, and Algeria, it becomes evident that technology-driven educational reforms can lead to profound improvements in accessibility, educational quality, economic growth, and even political alliances. Through a comprehensive exploration of the interplay between government initiatives and entrepreneurial endeavors, this review paper delves into the multifaceted outcomes of EdTech investments across these nations.

By advocating a “non-semantic” perspective of words as conditioned acoustic symbols rather than fixed representations, this book challenges the mainstream philosophy of language. It critiques the dualism and literalism that underpin linguistic theory, arguing that meaning emerges from usage, not semantics. In seven chapters, original frameworks aim to provide more accurate analyses of contextual human symbol use, which resolve enduring puzzles in the philosophy of language, mathematics, and epistemology. It is acknowledged in the book that discussing language using language itself is inherently difficult. This unified theory rejecting semantic assumptions could significantly transform the understanding of language acquisition, communication across cultures, and biases embedded in linguistic symbols if empirically validated.  This seminal contribution provides valuable new insights into the emergence of meaning in human thought and communication by emphasizing conditioned usage and over-representation. In this review, we summarize the philosophy and significance of the book, and discuss its contribution to the field of study.

This book provides a gradient model of intersubjectivity and social cognition in language. Through an interdisciplinary synthesis, the author proposes three levels of linguistic acts: those aimed at the speaker’s benefit alone, those expressing concern for an interlocutor, and those indicating concern for societal reactions. There is a continuum of egocentric to extended intersubjectivity, which includes co-actionality, immediate intersubjectivity, and collective constructions. The model is applied to study children’s mastery of intersubjective polysemy and verbal organization, and to determine “social mind” capabilities in autism spectrum disorders. While non-linguistic topics and further empirical validation are outside the scope, this innovative interdisciplinary approach meaningfully extends understanding of the intertwined nature of language, cognition, and social engagement.