Integer Play and Playing with Integers (Chapter Two of Exploring the Integer Addition and Subtraction Landscape: Perspectives on Integer Thinking)

This chapter describes instances of play within a teaching episode on integer addition and subtraction. Specifically, this chapter makes the theoretical distinction between integer play and playing with integers. Describing instances of integer play and playing with integers is important for facilitating this type of intellectual play in the future. The playful curiosities arising out of integer addition and subtraction tended to be concepts that we think of prerequisite knowledge (e.g., magnitude or order, sign of zero) or knowledge that is more nuanced for integer addition and subtraction (e.g., how negative and positive integers can “balance” each other). Instances of integer play and playing with integers are connected to the work of mathematicians, highlighting the importance of play in school mathematics

Preservice Teachers\u27 Temperature Stories for Integer Addition and Subtraction

Ninety-eight elementary and middle school preservice teachers posed eight stories for integer addition and subtraction number sentences. Stories that were posed about temperature were analysed using a modified Marthe‘s (1979) framework for integer problem types. This framework was modified based on the stories provided by the preservice teachers. This paper reports on the problem types utilized by the preservice teachers. Results highlight that preservice teachers do not frequently use some problem types. Also, results may indicate that some number sentence types (e.g., -23 – -5=☐) support different problem types (e.g., State-State-Translation)

Conceptual Models for Integer Addition and Subtraction

In this article, we report the findings of a study conducted with 6 Grade 8 students in the United States. The students posed stories for open number sentences involving addition and subtraction of integers. We analysed the stories posed by the students to build models that describe the conceptual structures behind these posed stories – the conceptual models for integer addition and subtraction. These four conceptual models for thinking about and using integer addition and subtraction include Bookkeeping, Counterbalance, Relativity, and Translation, and are generated from the students’ posed stories. We also provide profiles of conceptual model use for two of the 6 students that describe how the students posed stories to accommodate conceptual model use, such as posing unconventional or unrealistic stories or changing the structure of the number sentences. The conceptual models and descriptions of how the students used them provide perspective into student thinking about integers and contexts, highlighting the mathematics of the students, and calling for a re-examination of contexts used in school mathematics

Preservice teachers’ pictorial strategies for a multistep multiplicative fraction problem

Previous research has documented that preservice teachers (PSTs) struggle with under- standing fraction concepts and operations, and misconceptions often stem from their understanding of the referent whole. This study expands research on PSTs’ understanding of wholes by investigating pictorial strategies that 85 PSTs constructed for a multistep fraction task in a multiplicative context. The results show that many PSTs were able to construct valid pictorial strategies, and the strategies were widely diverse with respect to how they made sense of an unknown referent whole of a fraction in multiple steps, how they represented the wholes in their drawings, in which order they did multiple steps, and which type of model they used (area or set). Based on their wide range of pictorial strategies, we discuss potential benefits of PSTs’ construction of their own representations for a word problem in developing problem solving skills

Mathematics Education Communities: Crossing Virtual Boundaries

The growth of social media has yielded a range of virtual communities focused on issues related to education (Carpenter & Krutka, 2014; Hur & Brush, 2009). These communities, which operate across a range of different platforms, create an evolving landscape for users to navigate. Moreover, interactions within and across virtual communities has become a norm within society at large as well as within mathematics education. The Math Twitter Blog-o-Sphere (MTBoS), Mathematics Stack Exchange, specialized Facebook groups, and myNCTM are just a few examples of communities that are currently popular with mathematics teachers and educators in North America. Similarly, students of mathematics use virtual communities to make records of information that, in earlier times, would have been available through more informal channels. For example, solution clearinghouse sites (e.g., Chegg. com) allow students to request or post answers to problems sets and teacher-rating sites (e.g., RateMyProfessor.com) offer a platform where students can trade information about their instructors. With the ubiquity of internet-enabled devices, negotiating virtual communities has become a norm within mathematics teaching and learning. Consequently, educators, both new and old, who participate in these communities are encountering issues and ideas that they likely have limited experience with. This raises a number of questions for mathematics teacher educators seeking to help themselves, preservice teachers (PSTs), and current teachers understand these virtual communities. For example: How can the differences, similarities, and affordances of communities be highlighted? How can the boundaries that define and separate these communities be made clear? Within this chapter, we seek to address these and related questions by providing a framework for understanding these communities. We then use this framework to examine several communities currently popular within North America

What Does It Take to Be a Fox? New Horizons for Communities of Practice

In this theoretical research report we reflect on the challenges of becoming more fox-like in mathematics education work. Using a communities of practice motivating theoretical lens, we compare and discuss the differences in defining, creating, and accessing knowledge between virtual and scholarly communities of practice in mathematics education. We present four claims that virtual communities of practice in mathematics education are inherently foxy work. As part of our claims, we discuss how scholarly communities of practices are inherently hedgehog work. We conclude with a list of recommendations of those within the scholarly communities of practice in mathematics education. These recommendations include looking toward the successful fox-like attributes of the virtual communities in mathematics education

Variation in the cortical area map of C57BL/6J and DBA/2J inbred mice predicts strain identity

BACKGROUND: Recent discoveries suggest that arealization of the mammalian cortical sheet develops in a manner consonant with principles established for embryonic patterning of the body. Signaling centers release morphogens that determine regional growth and tissue identity by regulating regional expression of transcription factors. Research on mouse cortex has identified several candidate morphogens that affect anteroposterior or mediolateral cortical regionalization as well as mitogenesis. Inbred strains of laboratory mice can be exploited to study cortical area map formation if there are significant phenotypic differences with which to correlate gene polymorphism or expression data. Here we describe differences in the cortical area map of two commonly used inbred strains of laboratory mice, C57BL/6J and DBA/2J. Complete cortical hemispheres from adult mice were dissected and stained for the cytochrome oxidase enzyme in order to measure histochemically defined cortical areas. RESULTS: C57BL/6J has the larger neocortex, relatively larger primary visual cortex (V1), but relatively smaller posterior medial barrel subfield of the primary somatosensory cortex (PMBSF). The sample of C57BL/6J and DBA/2J mice can be discriminated with 90% accuracy on the basis of these three size dimensions. CONCLUSION: C57BL/6J and DBA/2J have markedly different cortical area maps, suggesting that inbred strains harbor enough phenotypic variation to encourage a forward genetic approach to understanding cortical development, complementing other approaches

Fibromatosis of the Plantar Fascia: Diagnosis and Indications For Surgical Treatment

Plantar fibromatosis is a rare, benign lesion involving the plantar aponeurosis. Eleven patients (13 feet) underwent 24 operations, including local excision, wide excision, or complete plantar fasciectomy. Clinical results were evaluated retrospectively. There were no differences among the subgroups in postoperative complications. Two primary fasciectomies did not recur. Three of six revised fasciectomies, seven of nine wide excisions, and six of seven local excisions recurred. Our results indicate that recurrence of plantar fibromatosis after surgical resection can be reduced by aggressive initial surgical resection

Alveolar soft-part sarcoma responding to interferon alpha-2b

A 23-year-old woman with an alveolar soft-part sarcoma of her calf with pulmonary metastases unresponsive to chemotherapy is described. Interferon (IFN) alpha-2b induced an impressive tumour response still ongoing after IFN treatment had to be stopped because of a psychosis. An explanation of this effect is still speculativ