Secondary school mathematics teachers’ conceptions of proof

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Many consider proof to be central to the discipline of mathematics and the practice of mathematicians. In fact, Ross (1998) contended that "the essence of mathematics lies in proofs" (p. 254). Yet, surprisingly, the role of proof in secondary school mathematics has traditionally been peripheral at best, usually limited to the domain of Euclidean geometry. According to Wu (1996), however, the scarcity of proof outside of geometry is a misrepresentation of the nature of proof in mathematics. He argued that this absence is a glaring defect in the present-day mathematics education in high school, namely, the fact that outside geometry there are essentially no proofs. Even as anomalies in education go, this is certainly more anomalous than others inasmuch as it presents a totally falsified picture of mathematics itself (p. 228). National Council of Teachers of Mathematics Similarly, Schoenfeld (1994) maintained that "proof is not a thing separable from mathematics, as it appears to be in our curricula; it is an essential component of doing, communicating, and recording mathematics. And I believe it can be embedded in our curricula, at all levels" (p. THEORETICAL PERSPECTIVES Authors have suggested various roles that proof plays in mathematics: to verify that a statement is true, to explain why a statement is true, to communicate math-1 My use of the term conceptions includes both subject matter knowledge and beliefs. Although separating teachers' knowledge and beliefs serves as a useful heuristic for thinking about and studying factors influencing their instructional practices, the separation is less distinct in reality than it is in theor

Symmetrized models of last passage percolation and non-intersecting lattice paths

It has been shown that the last passage time in certain symmetrized models of directed percolation can be written in terms of averages over random matrices from the classical groups $U(l)$, $Sp(2l)$ and $O(l)$. We present a theory of such results based on non-intersecting lattice paths, and integration techniques familiar from the theory of random matrices. Detailed derivations of probabilities relating to two further symmetrizations are also given.Comment: 21 pages, 5 figure

Ten New and Updated Multi-planet Systems, and a Survey of Exoplanetary Systems

We present the latest velocities for 10 multi-planet systems, including a re-analysis of archival Keck and Lick data, resulting in improved velocities that supersede our previously published measurements. We derive updated orbital fits for ten Lick and Keck systems, including two systems (HD 11964, HD 183263) for which we provide confirmation of second planets only tentatively identified elsewhere, and two others (HD 187123, and HD 217107) for which we provide a major revision of the outer planet's orbit. We compile orbital elements from the literature to generate a catalog of the 28 published multiple-planet systems around stars within 200 pc. From this catalog we find several intriguing patterns emerging: - Including those systems with long-term radial velocity trends, at least 28% of known planetary systems appear to contain multiple planets. - Planets in multiple-planet systems have somewhat smaller eccentricities than single planets. - The distribution of orbital distances of planets in multi-planet systems and single planets are inconsistent: single-planet systems show a pile-up at P ~ 3 days and a jump near 1 AU, while multi-planet systems show a more uniform distribution in log-period. In addition, among all planetary systems we find: - There may be an emerging, positive correlation between stellar mass and giant-planet semi-major axis. - Exoplanets more massive than Jupiter have eccentricities broadly distributed across 0 < e < 0.5, while lower-mass exoplanets exhibit a distribution peaked near e = 0.Comment: ApJ accepted. v.2 makes minor corrections to author list, citations, and provides a stable set of orbital parameters for HD 183263 in Table 1. v.3 makes preprint consistent with ApJ version, minor changes to wording, orbital parameters of HD 217107 and HD 187123c in Table

Collagen type X is essential for successful mesenchymal stem cell-mediated cartilage formation and subsequent endochondral ossification

n tissue engineering, endochondral ossification (EO) is often replicated by chondrogenically differentiating mesenchymal stromal cells (MSCs) in vitro and achieving bone formation through in vivo implantation. The resulting marrow-containing bone constructs are promising as a treatment for bone defects. However, limited bone formation capacity has prevented them from reaching their full potential. This is further complicated since it is not fully understood how this bone formation is achieved. Acellular grafts derived from chondrogenically differentiated MSCs can initiate bone formation; however, which component within these decellularised matrices contribute to bone formation has yet to be determined. Collagen type X (COLX), a hypertrophy-associated collagen found within these constructs, is involved in matrix organisation, calcium binding and matrix vesicle compartmentalisation. However, the importance of COLX during tissue-engineered chondrogenesis and subsequent bone formation is unknown. The present study investigated the importance of COLX by shRNA-mediated gene silencing in primary MSCs. A significant knock-down of COLX disrupted the production of extracellular matrix key components and the secretion profile of chondrogenically differentiated MSCs. Following in vivo implantation, disrupted bone formation in knock-down constructs was observed. The importance of COLX was confirmed during both chondrogenic differentiation and subsequent EO in this tissue engineered setting

The Scientific Investigation of Unidentified Aerial Phenomena (UAP) Using Multimodal Ground-based Observatories

Peer reviewedPublisher PD

Isolating Pediatric Mesenchymal Stem Cells with Enhanced Expansion and Differentiation Capabilities

Mesenchymal stem cells/marrow stromal cells (MSCs) are attractive for applications ranging from research and development to use in clinical therapeutics. However, the most commonly studied MSCs, adult bone marrow MSCs (A-MSCs), are limited by significant donor variation resulting in inconsistent expansion rates and multilineage differentiation capabilities. We have recently obtained permission to isolate pediatric MSCs (P-MSCs) from surplus iliac crest bone chips. Here, we developed a simple and easily replicable isolation protocol yielding P-MSCs, which adhere to MSC defining guidelines. After confirming immunophenotypic marker expression, we compared expansion rates, senescence, morphology, and trilineage differentiation of P-MSCs to A-MSCs for multiple donors. We found P-MSCs have faster in vitro replication, consistently show significantly lower senescence, and are capable of more reproducible multilineage differentiation than A-MSCs. We, therefore, believe P-MSCs are a promising candidate for use in research applications and potentially as part of an allogeneic therapeutic treatment

Classification of current anticancer immunotherapies

During the past decades, anticancer immunotherapy has evolved from a promising therapeutic option to a robust clinical reality. Many immunotherapeutic regimens are now approved by the US Food and Drug Administration and the European Medicines Agency for use in cancer patients, and many others are being investigated as standalone therapeutic interventions or combined with conventional treatments in clinical studies. Immunotherapies may be subdivided into “passive” and “active” based on their ability to engage the host immune system against cancer. Since the anticancer activity of most passive immunotherapeutics (including tumor-targeting monoclonal antibodies) also relies on the host immune system, this classification does not properly reflect the complexity of the drug-host-tumor interaction. Alternatively, anticancer immunotherapeutics can be classified according to their antigen specificity. While some immunotherapies specifically target one (or a few) defined tumor-associated antigen(s), others operate in a relatively non-specific manner and boost natural or therapy-elicited anticancer immune responses of unknown and often broad specificity. Here, we propose a critical, integrated classification of anticancer immunotherapies and discuss the clinical relevance of these approaches

Bridging Psychology and Mathematics Education: Reflections on Boundary Crossing

Collaborations between psychology and mathematics education have the potential to yield progress on critical questions about the teaching and learning of mathematics. In this paper, we describe our experience of collaborating at this boundary. We have found that collaboration has many benefits: it strengthens the research, it is professionally enriching, and it brings novel perspectives to disciplinary communities. However, collaboration is also challenging, because different views about the nature of knowledge and the aims of inquiry can be difficult to bridge. Collaboration can also raise difficult questions about professional identity. We consider several factors that are critical to success in interdisciplinary collaboration, including methodological openness, a broad view of what constitutes “basic” and “applied” research, and an appreciation for diverse perspectives and varying levels of analysis. We close by offering some advice for others who wish to collaborate at the boundary of psychology and mathematics education

Proof: Examples and beyond.

Asking middle school students to verify the math they do requires them to think about proof. By doing so, students construct arguments in the middle school and are more ready for proof in high school