Mathematical argument in the elementary grades : what and why? -- Elementary students as mathematicians -- The teaching model -- Using the lesson sequences : what the teacher does -- Mathematical argument in the elementary classroom : impact on students and teachers.

Based on extensive research conducted by the authors, Reasoning and Sense Making in the Mathematics Classroom, Pre-K-Grade 2, is designed to help classroom teachers understand, monitor, and guide the development of students' reasoning and sense making about core ideas in elementary school mathematics. It describes and illustrates the nature of these skills using classroom vignettes and actual student work in conjunction with instructional tasks and learning progressions to show how reasoning and sense making develop and how instruction (...) can support students in that development. Students who can make sense of mathematical ideas can apply those ideas in problem solving, even in unfamiliar situations, and can use them as a foundation for future learning. Without them, students are reduced to rote learning, often experiencing frustration and failure. But what do reasoning and sense making during learning and teaching look like? Each chapter of Reasoning and Sense Making in the Mathematics Classroom, Pre-K-Grade 2 explores a different topic that young children encounter in mathematics, demonstrating with actual student work and classroom dialogue how their mathematical knowledge and reasoning ability move through "levels of sophistication" or learning progressions: After opening with a discussion of the nature of reasoning and sense making and their critical importance in developing mathematical thinking, chapter 1 examines how young students attempt to make sense of the concepts of place value and length measurement. Chapter 2 focuses on how early childhood instruction can engage students in mathematical reasoning while helping them construct a rich sense of number and operations. Chapter 3 identifies core algebraic ideas and shows how students can engage with these ideas in ways that not only deepen their understanding of arithmetic but also lays the foundation for the future study of algebra. Children's reasoning and sense making as they decompose and compose geometric shapes--including reasoning about area--is examined in chapter 4. The use of learning progressions to understand students' reasoning and to guide their sense making with appropriate teaching is also discussed. Not just a theoretical discussion, the book also provides specific suggestions for related instructional activities for each topic. Supplementary online resources can be accessed at NCTM's More4U website. Reasoning and Sense Making in the Mathematics Classroom, Pre-K-Grade 2 will be a valuable and practical addition to your professional library. (shrink)

Examining the nature of teachers’ emotions and how they are managed and regulated in the act of teaching is crucial to assess the quality of teacher’s instructions. Despite the essential role emotions play in teachers’ lives and instruction, research on teachers’ emotions has not paid much attention on teachers’ state emotions in the context of daily teaching. Significant portion of literature has described teachers’ emotions by foregrounding trait emotions through deductive methodological approaches. This paper explored elementary teachers’ (...) state and trait emotions while preparing for teaching and during teaching, reasons that underlie these emotions, and the relationship between their emotions and the quality of their mathematics instruction. Participants were seven elementary teachers working in the U.S. who participated in Holistic Individualized Coaching (HIC) professional development that consisted of 5 cycles of coaching over an year. For each coaching cycle pre-coaching conversation and post-coaching conversation data were collected regarding state emotions teachers felt in anticipation of teaching and during teaching retrospectively. Also, teachers responded adapted Teaching Emotion Scale (TES) that was aimed to measure trait emotions. In order to compare teachers’ emotions with instructional quality, coaching session were video recorded and analyzed to determine the quality of instruction. Findings of this study showed that teachers reported 6 categories of emotions (positive, negative, neutral, mixed-positive, mixed-negative, and mixed-complex), and often in non-typical ways (e.g., “not nervous”, “anxious but in a positive way”), and mixed emotions were the most dominant category. Also, teachers had more positive emotions anticipating teaching, than actually teaching the lesson, and there was no noticeable change of emotions across the 5 coaching cycles. The reason teachers felt mixed emotions reflected complex and context-specific nature of teaching, which was not articulated in the literature. There were no clear relationships between emotional experiences and instructional quality. This study allowed participants to freely describe their authentic, complex, overlapping, and ambiguous emotions in the context of active teaching, which contributes opening up the possibilities of diversifying teacher emotion research and shows the significance and usefulness of understanding teachers’ state emotions. (shrink)

This book offers an innovative introduction to the psychological basis of mathematics and the nature of mathematical thinking and learning, using an approach that empowers students by fostering their own construction of mathematical structures. Through accessible and engaging writing, award-winning mathematician and educator Anderson Norton reframes mathematics as something that exists first in the minds of students, rather than something that exists first in a textbook. By exploring the psychological basis for mathematics at every level - including (...) geometry, algebra, calculus, complex analysis and more - Norton unlocks students' personal power to construct mathematical objects based on their own mental activity, and illustrates the power of mathematics in organizing the world as we know it. Including reflections and activities designed to inspire awareness of the mental actions and processes coordinated in practicing mathematics, the book is geared towards current and future secondary and elementary mathematics teachers who will empower the next generation of mathematicians and STEM majors. Those interested in the history and philosophy that underpins mathematics will also benefit from this book, as well as those informed and curious minds attentive to the human experience more generally. Anderson Norton is Professor in the Department of Mathematics at Virginia Tech, USA, where he has been teaching mathematics courses for future teachers for 15 years. He is the editor of Constructing Number, and the co-author of Developing Fractions Knowledge and Numeracy for All Learners. (shrink)

In this paper, we discuss a proposal for reform in the teaching of Euclidean geometry that reveals the symbiotic relationship between axiomatics and pedagogy. We examine the role of intuition in this kind of reform, as expressed by Mario Pieri, a prominent member of the Schools of Peano and Segre at the University of Turin. We are well aware of the centuries of attention paid to the notion of intuition by mathematicians, mathematics educators, philosophers, psychologists, historians, and others. (...) To set a context for Pieri’s proposal, we only seek to open a small window on views of the pedagogical role of intuition, from primary education to university study that may have informed early 20th century efforts to improve the teaching of geometry at the secondary school level. Pieri addressed the topic of intuition in many of his axiomatizations, including those in projective geometry which was his main area of concentration in foundations. We focus here primarily on his axiom systems for elementary geometry, which embraced the transformational approach of Felix Klein’s vision for the subject. Our goal is to convey Pieri’s thoughts on how to integrate two types of intuition, denoted as sensible and rational, in his endeavors to improve the teaching of the geometry of Euclid. We show how Pieri’s views on geometric intuition and pedagogical reform were either ignored or misrepresented in several notable publications at the turn of the 20th century. In particular, we give Pieri a voice in response to specific comments made in the early 1900s by Federigo Enriques, Ugo Amaldi, and Florian Cajori in widely circulated publications inspired by Klein. (shrink)

The pendulum has had immense scientific, cultural, social and philosophical impact. Historical, methodological and philosophical studies of pendulum motion can assist teachers to improve science education by developing enriched curricular material, and by showing connections between pendulum studies and other parts of the school programme, especially mathematics, social studies, technology and music. The pendulum is a universal topic in high-school science programmes and some elementary science courses; an enriched approach to its study can result in deepened science (...) literacy across the whole educational spectrum. Such literacy will be manifest in a better appreciation of the part played by science in the development of society and culture. Such history, philosophy and science (HPS)-informed teaching and study of pendulum motion can serve as an exemplar of the benefits of HPS-informed teaching across the science curriculum. (This chapter draws on material in Matthews (1998, 2000, 2001, 2004), and on contributions to Matthews et al. (2005)). (shrink)

Context: Technology has not only changed the way we teach mathematical concepts but also the nature of knowledge, and thus what is possible to learn. While geometric transformations are recognized to be foundational to the formation of students’ geometric conceptions, little research has focused on how these notions can be introduced in elementary schooling. Problem: This project addressed the need for development of students’ reasoning about and with geometric transformations in elementary school. We investigated the nature of students’ (...) understandings of translations, rotations, scaling, and stretching in two dimensions in the context of use of the software application Graphs ’n Glyphs. More specifically, we explored the question “What is the nature of elementary students’ reasoning of geometric transformations when instruction exploits the technological tool Graphs ’n Glyphs?” Method: Using a design research perspective, we present the conduct of a teaching experiment with one pair of fourth-graders, studying translation and rotation. The project investigated how and to what extent activity using Graphs ’n Glyphs can elicit students’ reasoning about geometric transformations, and explored the constraints and affordances of Graphs ’n Glyphs for thinking-in-change about geometric transformations. Results: The students proved adept using the software with carefully designed tasks to explore the behavior of two-dimensional shapes, distinguish among transformations, and develop predictions. In relation to varied conditions of transformations, they formed generalizations about the way a shape behaves, including the use of referent points in predicting outcomes of translations, and recognizing the role of the center of rotation. Implications: The generalizations that the students developed are foundational for developing an understanding of the properties of transformations in the later years of schooling. Dynamic technological approaches to geometry may prove as valuable to elementary students’ understanding of geometry as it is for older students. Constructivist content: This study contributes to ongoing constructivism/constructionism conversations by concentrating on the transformation of ideas when engaging learners in activity through particular contexts and tools. Key Words: Geometry, transformations, constructionist technologies. (shrink)

Creativity is an essential factor in ensuring the sustainable development of a society. Improving students’ creativity has gained much attention in education, especially in Science, Technology, Engineering, Arts, and Mathematics education. In a quasi-experimental design, this study examines the effectiveness of a project-based STEAM program on the development of creativity in Chinese elementary school science education. We selected two fourth-graders classes. One received a project-based STEAM program, and the other received a conventional science teaching over 6 (...) weeks. Students’ creativity was assessed before and after the intervention using a multi-method approach, including a test of divergent thinking, a story completion through the Consensus Assessment Technique, a creative self-efficacy measure, and a group-based creative project. Moreover, all students received a test of their science knowledge after the intervention. The results showed that compared with the control group, the creativity of the experimental group students improved significantly for 6 weeks at both individual and group level, even though their knowledge in science were comparable. This result confirmed the effectiveness of a project-based STEAM educational program improving elementary school students’ creativity. Implications are discussed. (shrink)

This paper is part of a larger study that contributes to theory and method related to the study of how racism permeates teaching practice. I combine concepts from critical realism and critical race theory to develop a theory to better describe how local social interactions that occur in a mathematics classroom can disrupt common patterns of interactions that lead to the reproduction of the racial structure that permeates contemporary U. S. society. Drawing primarily on the concept (...) of norm circles, I discuss how specific mathematical teaching practices supported the creation of a conflictive normative space inside of a classroom in which local disruption of racism is more likely to occur. With data collected from an elementary mathematics laboratory classroom, I refine and improve the theory. I also discuss some methodological considerations that include the need to capture unexpected disruption of (racist) patterns and confirmation of such a disruption being normative in the classroom. I illustrate such work by reporting on how keeping the focus on mathematics supported the creation of intersectional normative spaces in which Black children were more likely to engage in doing mathematics and to expect and be expected to do so. In these spaces, they were also less likely to be disciplined or have their thinking immediately evaluated and corrected. (shrink)

In their examination of elementary science classrooms, Amerine and Bilmes (1988) found that following instructions requires students to understand the relationship between the projected outcome and the corresponding course of actions. One of the most important resources for instructions is the lesson plan, which prescribes the sequence of teaching. However, there is often a gap between what is planned and what actually happens in the classroom. This raises the question of how teachers come to terms with contingent variants (...) and unexpected outcomes that real-time interactions occasion and how lesson plans are configured into these processes. This study examines a teacher education program that uses lesson plans as a central resource for teaching mathematics. The results suggest that classroom teachers use lesson plans as communicative resources to identify problems, specify assumptions about their teaching and act on the evolving contingency of classroom interaction. The interactional contingency is the locus of teaching practices, not an obstacle to the application of procedures in lesson plans. (shrink)

The nature of moral values has occupied philosophers and educationalists for centuries and a variety of claims have been made about their origin and status. One tradition suggests they may be thoughts in the mind of God; another that they are eternal truths to be reached by rational reflection (much like the truths of mathematics) or alternatively through intuition; another that they are social conventions; and another (from the logical positivists) that they are not verifiable facts but simply the (...) expression of emotional likes and dislikes. Standard introductory texts (e.g., Bowie 2004; Vardy and Grosch 1999) on the subject of ethics rarely mention Darwin or Darwinism (Mepham 2005 is a useful exception) possibly mindful of the fact that the relationship of evolutionary biology to moral questions has had a troublesome history. The effect of this has been that whole generations of moral philosophers have given the biological sciences a wide berth and consequently often remain poorly informed about recent advances in evolutionary thought and the neurosciences. On the other hand, scientists have developed interesting models of the evolution of the moral sentiments and are using new imaging techniques to explore the centres of the brain associated with emotion and motivation, but many have been fearful of committing the naturalistic fallacy and so have steered clear of extrapolating their findings to ethical questions. No one after all wants to be seen to be committing an elementary logical blunder. But in the last 20 years, evolutionary biologists have regained the confidence to explore the implications of evolution for the study of ethics (de Waal 1996; Wilson 1998; Wright 1994; Greene 2003). This paper is designed to encourage those entrusted with the teaching of ethics to be open to the potential of Darwinism as a source of ideas on the origins and status of ethical thought and behaviour. It is also hoped that it will illustrate for science educators the enormous explanatory power inherent in Darwinian thought. (shrink)

By means of data from the most comprehensive source of teacher data in the nation, Schools and Public School Teacher Staffing Survey (SASS), we designeda follow-up quantitative study to test the effects of two decades of national policy mandates on instructional time allotments for core academic subjects. We used data from the SASS data from National Center for Educational Statistics (NCES) (1993/1994, 1999/2000, 2003/2004, 2007/2008) to examine national trends of continued marginalization of social studies by exploring the influence of (...) recent educational policy on defining elementary curricula in the 21st Century. With the reauthorization of NCLB in 2007 and newly mandated science testing in grades 3 through 5, we sought to understand statistically how this policy change has affected instructional decisions regarding time allocated to core academic subjects (ELA, mathematics, science and social studies) in elementary schools. Findings provide evidence of the national trend of the declining role of social studies in an era in which testing is associated with importance. Moreover, grade level disparities unique to social studies intensified the effects of 2007 NCLB policy mandates of science testing for 3rd through 5th grade students. Results provide a broader and nationally generalizable understanding of the declining role of social studies in elementary schools and the reduction of time practitioners spend teaching social studies. Thus, the growing imbalance in instructional time distributions for social studies and the tipping of the time distribution scales in favor of science document teacher responses to the expansion of testing mandates and the continued absence of studies from the national testing landscape. (shrink)

The intersection between art, poetry, philosophy and science was the leitmotif which guided the lives and careers of romantic natural philosophers including that of the Danish natural philosopher, H. C. Ørsted. A simple model of Ørsted’s career would be one in which it was framed by two periods of philosophical speculation: the youth’s curious and idealistic interest in new attractive thoughts and the experienced man’s mature reflections at the end of his life. We suggest that a closer look at the (...) epistemological aspects of his works on the theory of beauty reveals a connection between this late work and his early philosophical work including experimental philosophy, but also with the work in teaching and textbook writing, that lies in between. The latter includes Ørsted’s view on the application of mathematics in natural philosophy as well as his failed attempt at a genetic presentation of elementary geometry.Keywords: H. C. Ørsted; Anschauung; Naturphilosophie; Aesthetics; Genetic epistemology; Geometry. (shrink)

Tasks in Primary Mathematics Teacher Education is intended to advance relevant research and innovative international practices in the preparation and professional development of mathematics teachers. Emerging from discussion at the ICMI study on teacher professional development, this volume, focused on primary and elementary teachers, culls a richness that can only be found by gathering wisdom from varied experiences around the world. The choice of tasks, and the associated pedagogies, is a key aspect of teaching and (...) learning mathematics. Arguing that what students learn is largely defined by the tasks they are given, several major themes are presented. One such major strand, the form, function and focus of tasks, is discussed throughout several chapters, offering analysis, discussion of implementation, and exemplars of a broader category of illustrative techniques for developing critical understanding. (shrink)

By the end of the 19th century, mathematics teaching underwent a crisis of similar proportion in several European countries. Constrained by the values of classical Antiquity, which dominated in secondary education, mathematics was taught as a formal discipline enhancing elementary geometry and intended as some gymnastics of the mind. Moreover, a specific feature in Germany was that the country was split up into numerous independent states each having a separated educational system of its own. Many of (...) these conferred an only marginal status to mathematics instruction. This complex situation is analyzed for the traditional school system for boys, as well as for the emerging system for girls, and for primary schooling.The study describes the key initiatives by Felix Klein, who forged a dynamic movement of teachers for modernising secondary school teaching. Their culmination in the famous “Meran” reform program is depicted, a program which became the starting point for the first international reform movement in mathematics teaching. (shrink)

By the end of the 19th century, mathematics teaching underwent a crisis of similar proportion in several European countries. Constrained by the values of classical Antiquity, which dominated in secondary education, mathematics was taught as a formal discipline enhancing elementary geometry and intended as some gymnastics of the mind. Moreover, a specific feature in Germany was that the country was split up into numerous independent states each having a separated educational system of its own. Many of (...) these conferred an only marginal status to mathematics instruction. This complex situation is analyzed for the traditional school system for boys, as well as for the emerging system for girls, and for primary schooling.The study describes the key initiatives by Felix Klein, who forged a dynamic movement of teachers for modernising secondary school teaching. Their culmination in the famous “Meran” reform program is depicted, a program which became the starting point for the first international reform movement in mathematics teaching. (shrink)

The purpose of this exploratory action research study was to examine how the modeling by a collaborative team of instructors regarding technology integration and information literacy would affect the quality of the lessons that elementary teacher-education students designed and taught in their field placements. The research was conducted over two distinct years with two different cohorts of methods students placed at a local elementary school that had received new interactive whiteboards, SMART boards, in every classroom at the (...) beginning of the previous school year. Based upon field-supervisor/instructor observations, reflections, and oral and written feedback from host-teachers and students, an analysis was conducted to determine the effectiveness of the interventions. Findings suggest that teacher education students' level of engagement increased regarding the integration of technology, information literacy, ethical information use, and children were more engaged and actively involved during the teaching of methods students' mathematics and science lessons. (shrink)

This is a critical edition of a set of particularly detailed and carefully prepared lecture notes, taken down in 1804 during a course on mathematical and experimental physics given at the University of Tübingen by Christoph Friedrich von Pfleiderer. Since Pfleiderer had been appointed to the chair of mathematics and physics in 1782, and had previously held a similar post at the Warsaw Military Academy, when he delivered these lectures he had been teaching the subject for nearly forty (...) years. Besides providing these public courses on “theoretical physics,” he also held private classes in elementary and advanced mathematics and in experimental physics. The lectures were based on a general textbook, which was supplemented with information from a wide range of other sources. Up until the summer term of 1792, they followed the layout of the first edition of a standard work on the subject, Anfangsgründe der Naturlehre, by Wenceslaus Johann Gustav Karsten, professor of the mathematical sciences at Halle. Pfleiderer’s personal copy of this book, annotated in his own hand, is still preserved in the University Library at Tübingen, and throws a lot of light on the precise manner in which he elaborated on the basic text. From 1792 until 1815 he made use of a similar but more up-to-date work, with the same title, by Georg Simon Klügel, and it is therefore this book which forms the basis of the lecture notes now published. At this stage in his career, however, Pfleiderer had got into a well established routine, and the similarities between these 1804 lectures and the manner in which he annotated Karsten, indicate that the notes also give a pretty accurate account of the way in which “theoretical physics” was being taught at Tübingen a decade or so earlier. Like the whole of Pfleiderer’s private collection of books, the manuscript of the notes is preserved in the University Library at Tübingen, which acquired it in 1879. The note taker was Gottlieb Friedrich Harttmann, who studied at Tübingen from 1799 until 1804, subsequently took holy orders and taught at Maulbronn, and eventually emigrated to America. (shrink)