Search

This is the result of a collaborative and creative experiment, a process in which six scholars—Garcia Lazo, Locke, Rupcich, O’Connor, Yoon and Longley—agreed to co-author an article through written and/or visual pieces of work. The emphasis taken in the article is to perform a methodology of fragmentation. The article is to be read as a series of ‘folds’, where collages or fragments of text are contained and while each fold is numbered, the reader is invited to read any fold in any order. Using fragmentation as method in writing also inspired our interpretation of a body of work made up of multiple ‘folds’ from the collaborative parlour game played by the surrealist movement. Known as cadavre exquis, surrealists enjoyed the strange juxtaposition created by each artist’s contribution as they drew part of a body onto a folded piece of paper without seeing the previous drawings in each fold. While often referred to in the context of psychoanalysis, we instead emphasise the fragmentation and hybridity of this form of collaboration in order to critique of universalising narratives about education (and writing) that we consider to be at odds with the post-capitalist realities of contemporary education and life.

Tuberous sclerosis complex (TSC) is a rare autosomal dominant neurodevelopmental disorder characterized by variable expressivity. TSC results from inactivating variants within the TSC1 or TSC2 genes, leading to constitutive activation of mechanistic target of rapamycin complex 1 signaling. Using a mouse model of TSC (Tsc2-RG) in which the Tsc2 gene is deleted in radial glial precursors and their neuronal and glial descendants, we observed increased ornithine decarboxylase (ODC) enzymatic activity and concentration of its product, putrescine. To test if increased ODC activity and dysregulated polyamine metabolism contribute to the neurodevelopmental defects of Tsc2-RG mice, we used pharmacologic and genetic approaches to reduce ODC activity in Tsc2-RG mice, followed by histologic assessment of brain development. We observed that decreasing ODC activity and putrescine levels in Tsc2-RG mice worsened many of the neurodevelopmental phenotypes, including brain growth and neuronal migration defects, astrogliosis and oxidative stress. These data suggest a protective effect of increased ODC activity and elevated putrescine that modify the phenotype in this developmental Tsc2-RG model.

This chapter examines the notion of personal models of modeling, and presents examples of the use of Reflection Tools (RT). It begins with a line of reasoning to suggest why RTs can be valuable both for the learner in developing modeling competences and for the researcher in understanding that development. The efforts with RTs are promising but preliminary, involving assumptions and approaches that require far more testing and refinement. The chapter aims to encourage modeling-eliciting activities research and practice communities to consider the further development and systematic inclusion or adoption of RT-like activities. It summarizes research directions that might help determine where and how RTs or similar mechanisms might be most useful. The Individual Roles RT is designed to attune problem-solvers both to the specific roles they take and to identifiable transitions in group modeling activities.

Our project seeks to draw attention to the rich mathematical thinking that is generated when students work on contextual mathematising tasks. We use a design-based research approach to create ways of reporting that raise the visibility of this rich mathematical thinking while retaining and respecting its complexity. These reports will be aimed for three classroom stakeholders: (1) students, who wish to reflect on and enhance their mathematical learning; (2) teachers, who wish to integrate contextual mathematising tasks into their teaching practice and (3) researchers, who seek rich tasks for generating observable instances of mathematical thinking and learning. We anticipate that these reports and the underlying theoretical framework for creating them will contribute to greater awareness of and appreciation for the mathematical value of contextual mathematising tasks in learning, teaching and research.

Enacting the work of diagnostic teaching is challenging and demands that teachers pay attention to mathematical details when designing tasks, orchestrating discussions and reflecting on their lessons. This chapter presents the FOCUS Framework on teacher noticing, which can be used to characterise teachers’ efforts to notice productively during all three phases of diagnostic teaching: lesson planning, delivery and review. Using the two key components of the framework, the focus and its focusing, we provide snapshots of a teacher’s mathematical noticing in each of the phases. The findings from this research suggest that productive noticing in all the three phases is highly consequential, and illustrates how the FOCUS Framework can be used to analyse a teacher’s mathematical noticing.

Students often hold on to algorithms that are inappropriate for the problem at hand, despite being presented with evidence of their inappropriateness during testing. Past research has tended to focus on the rejection and replacement of an algorithm in its entirety, rather than small refinements to an otherwise intact algorithm. In this study, we take a fine-grained view, focusing on the iterative refinement and augmentation of an algorithm, rather than its wholesale replacement. We ask: how do students decide what to change and what to keep when revising their algorithm upon testing? We analyze the collaborative work of three students on a graph theory task which invited the students to develop an algorithm for a contextualized problem. Two terms, localized considerations and patching, are introduced to describe how the group revised and validated their algorithms while retaining one specified attribute of their initial algorithm throughout.

With the proliferation of new affordable recording technologies, many universities have begun offering students recordings of live lectures as a part of the course resources. We conducted a survey to investigate why some students choose to attend lectures in person rather than simply watching the recordings online, and how students view the two types of lectures. Students attending live lectures in five large undergraduate mathematics lecture streams were invited to respond to the survey. A significant number of respondents viewed recorded lecture as superfluous to their needs which were met upon attending live lecture. Surprisingly, however, an equally large number of students described compelling reasons for watching both live and recorded lectures. A number of factors were identified as determining students’ perceptions of live and recorded lectures as competing or complementary: personal learning styles, study habits, esteem for the lecturer, and the possibility of interaction in the lecture.

This paper describes part of an international project considering graphical construction of antiderivative functions in the secondary mathematics classroom. We use Schoenfeld’s resources, orientations, and goals (ROGs) framework to analyse the decisions made by a teacher, Adam, during a lesson on graphical antiderivatives. We present details of Adam’s ROG and see how this is related to resolution of the conflict between his competing goals and the decisions he makes. The results suggest that a beneficial professional development strategy might be to assist teachers to become more aware of their ROG and its influence on in-the-moment classroom decisions.

Cadaveric biomechanical study.To determine the degree of segmental correction that can be achieved through lateral transpsoas approach by varying cage angle and adding anterior longitudinal ligament (ALL) release and posterior element resection.Lordotic cage insertion through the lateral transpsoas approach is being used increasingly for restoration of sagittal alignment. However, the degree of correction achieved by varying cage angle and ALL release and posterior element resection is not well defined.Thirteen lumbar motion segments between L1 and L5 were dissected into single motion segments. Segmental angles and disk heights were measured under both 50 N and 500 N compressive loads under the following conditions: intact specimen, discectomy (collapsed disk simulation), insertion of parallel cage, 10° cage, 30° cage with ALL release, 30° cage with ALL release and spinous process (SP) resection, 30° cage with ALL release, SP resection, facetectomy, and compression with pedicle screws.Segmental lordosis was not increased by either parallel or 10° cages as compared with intact disks, and contributed small amounts of lordosis when compared with the collapsed disk condition. Placement of 30° cages with ALL release increased segmental lordosis by 10.5°. Adding SP resection increased lordosis to 12.4°. Facetectomy and compression with pedicle screws further increased lordosis to approximately 26°. No interventions resulted in a decrease in either anterior or posterior disk height.Insertion of a parallel or 10° cage has little effect on lordosis. A 30° cage insertion with ALL release resulted in a modest increase in lordosis (10.5°). The addition of SP resection and facetectomy was needed to obtain a larger amount of correction (26°). None of the cages, including the 30° lordotic cage, caused a decrease in posterior disk height suggesting hyperlordotic cages do not cause foraminal stenosis.N/A.

Diagrams and physical manipulatives are often recommended as useful semiotic resources for visualising area and volume problems in which nonlinear reasoning is appropriate. However, the mere presence of diagrams and physical manipulatives does not guarantee students will recognise the appropriateness of nonlinear reasoning.

Visualisation is often suggested as a useful heuristic for generating new ideas when one is stuck on a problem. Yet generating ideas is just one aspect of mathematical activity. Visualisation can also help students generalise mathematical discoveries and communicate mathematical ideas.

What role do gestures play in advanced mathematical thinking? We argue that the role of gestures goes beyond merely communicating thought and supporting understanding � in some cases, gestures can help generate new mathematical insights. Gestures feature prominently in a case study of two participants working on a sequence of calculus activities. One participant uses gestures to clarify the relationships between a function, its derivative and its antiderivative. We show how these gestures help create a virtual mathematical construct, which in turn leads to a new problem-solving strategy. These results suggest that gestures are a productive, but potentially undertapped resource for generating new insights in advanced levels of mathematics.

Students often play a passive role in large-scale lectures in undergraduate mathematics courses: they observe the lecturer demonstrate mathematical procedures, but they rarely engage in authentic mathematical activity themselves. This study uses semi-structured interviews of undergraduate students to investigate the implicit and explicit social norms and expectations that influence students to maintain their passive roles during lectures. Students were aware that their passivity was influenced by social norms, but perceived these norms as necessary for allowing the lecturer to get through the content in the allotted lecture time, while enabling students to avoid being publicly embarrassed in the lecture. However, the students appreciated opportunities to work on examples in small groups during lectures. We argue that the success of small group interactions during large-scale lectures depends on students and lecturers establishing supportive social norms, and adjusting their lecture goals from �covering the content� to �developing mathematical understanding�

This paper examines how a person’s gesture space can become endowed with mathematical meaning associated with mathematical spaces and how the resulting mathematical gesture space can be used to communicate and interpret mathematical features of gestures. We use the theory of grounded blends to analyse a case study of two teachers who used gestures to construct a graphical anti-derivative while working on a professional development task in a calculus modelling activity. Results indicate that mathematical gesture spaces can encourage mathematical experimentation, lighten the cognitive load for students and can be limited by a person’s physical constraints.

Tablet PCs have been increasingly used in undergraduate courses to create recorded lectures that are close copies of the live lectures. Research has shown that students are largely positive about the availability of tablet PC recorded lectures. However, there is some concern that the availability of such faithful recordings may encourage students to miss live lectures, which may in turn lead to lower achievement in the course. In this study, we surveyed students on their use of recorded lectures in two large undergraduate mathematics courses. We investigated patterns in their use of recorded lectures and live lecture attendance, how and why they used recorded lectures and how this use was associated with their final grade. The results suggest that the practice of missing live lectures intentionally because the recordings were available was not associated with final grade. However, those respondents who intended to watch more recorded lectures than they actually did achieved significantly lower grades.

Two complementary processes involved in mathematical modelling are mathematising a realistic situation and applying a mathematical technique to a given realistic situation. We present and analyse work from two undergraduate students and two secondary school teachers who engaged in both processes during a mathematical modelling task that required them to find a graphical representation of an anti-derivative of a function. When determining the value of the anti-derivative as a measure of height, they mathematised the situation to develop a mathematical model, and attempted to apply their knowledge of integration that they had previously learned in class. However, the participants favoured their more primitive mathematised knowledge over the formal knowledge they tried to apply. We use these results to argue for calculus instruction to include more modelling activities that promote mathematising rather than the application of knowledge.

Using traditional educational research methods, it is difficult to assess students’ understanding of mathematical concepts, even though qualitative methods such as task observation and interviews provide some useful information. It has now become possible to use functional magnetic resonance imaging (fMRI) to observe brain activity whilst students think about mathematics, although much of this work has concentrated on number. In this study, we used fMRI to examine brain activity whilst ten university students translated between graphical and algebraic formats of both linear and quadratic mathematical functions. Consistent with previous studies on the representation of number, this task elicited activity in the intra-parietal sulcus, as well as in the inferior frontal gyrus. We also analysed qualitative data on participants’ introspection of strategies employed when reasoning about function. Expert participants focused more on key properties of functions when translating between formats than did novices. Implications for the teaching and learning of functions are discussed, including the relationship of function properties to difficulties in conversion from algebraic to graphical representation systems and vice versa, the desirability of teachers focusing attention on function properties, and the importance of integrating graphical and algebraic function instruction.

Mathematical leaps of insight—those Aha! moments that seem so unpredictable, magical even—are often the result of a change in perception. A stubborn problem can yield a surprisingly simple solution when one changes the way one looks at it. In mathematics, these changes in perception are usually structural: new insights develop as one notices new mathematical objects, attributes, relationships and operations that are relevant to the problem at hand. This paper describes a novel analytical approach for studying these insights visually using “mathematical SPOT diagrams” (SPOT: Structures Perceived Over Time), which display evidence of the mathematical structures students perceive as they work on problems. SPOT diagrams are used to compare the conceptual development of two pairs of participants, who investigate whether a gradient (derivative) graph yields information about the relative heights of points on its antiderivative; one participant pair experiences a leap of insight, whereas the other does not. Each pair’s SPOT diagrams reveal key differences in the structural features they attend to, which can account for the disparate outcomes in their conceptual development.

Objectives: The purpose of this study was to examine the predictive value of auditory steady-state response (ASSR) evoked potential thresholds and predicted behavioral thresholds in a group of children with steeply sloping sensorineural hearing loss (HL). Study Design: Case series. Methods: Twenty-nine children with sloping sensorineural HL underwent behavioral audiometric evaluation, impedance testing, distortion product otoacoustic emissions, and steady-state response testing. A t test was performed to compare the means of ASSR predicted behavioral thresholds and behavioral responses. Pearson correlation coefficients were calculated at each tested frequency, 500 Hz, 1,000 Hz, 2,000 Hz, and 4,000 Hz, using the same data. Results: Bracketed thresholds were obtained at 500 Hz, 1,000 Hz, 2,000 Hz, and 4,000 Hz. Nineteen ears were used in this evaluation. A comparison of threshold difference as a function of bracketing revealed that the means were statistically different (P < .05). The mean threshold differences were calculated, and Pearson r values were determined between the behavioral thresholds and the predicted thresholds using the Rance 95 algorithm. The results revealed no difference of means at 500 Hz between predicted and measured behavioral thresholds. Linear regression analysis revealed strong correlation at 500 Hz, 1,000 Hz, and 2,000 Hz. Conclusions: The GSI Audera appears to predict the configuration of HL in children with steeply sloping sensorineural HLs and over-predicts the severity of the loss by 15 to 20 dB above 500 Hz at each test frequency (1,000, 2,000, and 4,000 Hz). Correlation coefficients display a strong correlation at 500 Hz, 1,000 Hz, and 2,000 Hz.

The authors observed a group of 4 eighthgrade students who conducted a series of inquiry-based science investigations over 6 weeks. The authors examined evidence of students' motivational beliefs, thoughtfulness, nature of their conceptual understanding, and changes in that understanding. The authors also used pre- and posttests of conceptual understanding. Marked variability in terms of the types of students' motivations was found. The variations were associated with differences in the quality of their thoughtfulness and in the development of their conceptual understanding. Evidence of student mastery-goal orientation appeared most strongly related to increased understanding.

If a curriculum developer's goal is to create a single linear sequence of tasks that lead to the development of some important mathematical concept, then some researchers have suggested that these sequences should follow progressions similar to stages of development that have been identified in Piaget-like research on the relevant concept(s). These research-based sequences are referred to as learning trajectories. Other researchers emphasize that conceptual development can involve interactions among ideas expressed using a variety of representational media and can occur along a variety of "dimensions" such as concrete-abstract, simple-complex, or situated-decontextualized. Therefore, different paths can be appropriate for different students, and trying to funnel development along any single developmental path can be inappropriate for some students. These researchers often envision trajectories to be specific paths within a branching tree diagram that portrays the space of possibilities. This article empha...

This chapter describes an evolving new class of collaborative research methods called multitiered design experiments. After a brief introduction to the general concept of such studies, the chapter is divided into four sections. The first compares and contrasts the origins of the term design research in cognitive science versus mathematics and science education. The second describes how design research requires a reconceptualization of many traditional notions about researchers’ roles, subjects, relevant theories, and the nature of results. The third gives a concrete example of a multitiered design experiment that investigated the development of knowledge for three interacting types of subjectsstudents, teachers, and experts who know about the kinds of mathematical knowledge needed for success beyond school. The fourth describes two common design flaws in design experiments; then, several quality assurance principles are described which should apply to the ways that such studies are conducted.

IntroductionLordotic cage insertion through lateral transpsoas approach is being used increasingly for restoration of sagittal alignment. However, the degree of correction achieved by varying cage ...

What is the nature of typical problem-solving situations where elementary-but-powerful mathematical constructs and conceptual systems are needed for success beyond school in a technology-based age of information? What kind of “mathematical thinking” is emphasized in these situations? What does it mean to “understand” the most important ideas and abilities that are needed in the preceding situations? How do these competencies develop? What can be done to facilitate development? How can we document and assess the most important achievements that are needed: (i) for informed citizenship, or (ii) for successful participation in the increasingly wide range of professions that are becoming heavy users of mathematics, science, and technology?

Background The widespread use of broad-spectrum antibiotics has resulted in an increase in the prevalence of methicillin-resistant Staphylococcus aureus (MRSA). Its presence in sinonasal cultures in patients with sinusitis suggests its pathogenicity. However, the efficacy and safety of treatment modalities for MRSA sinusitis remain incompletely described. Methods A retrospective chart review of six patients treated for MRSA sinusitis with outpatient intravenous (i.v.) antibiotics was performed for patient demographics, history of antibiotic use, history of prior sinus surgery, and treatment-related complications. A quality-of-life survey and endoscopically guided cultures before and after therapy were used to measure treatment outcomes. Results The cohort consisted of five women and one man with a mean age of 50.8 years. All patients had undergone multiple sinus procedures with a mean number of 2.7 procedures per patient (range, 1–6 procedures). Five patients (83.3%) experienced negative cultures after outpatient i.v. antibiotics. The single patient who had persistent cultures experienced clinical and endoscopic improvement in her symptoms. The quality-of-life scores improved in five of the six patients (83.3%) after therapy. Four patients (66.7%) experienced five adverse events including allergic reaction (four events) and neutropenia (one event), all of which resolved with a change in medication. Conclusion Outpatient i.v. antibiotics may be an effective therapy for the treatment of MRSA sinusitis. The occurrence of adverse events requires a dedicated protocol to therapy. Future studies are required to investigate long-term efficacy.

This chapter illustrates how Applications and Modelling promote the learning of mathematics by developing the student’s mathematical language and her use of tools, and by developing the learner’s capacity to ask and answer questions in, with, and about mathematics.

Summary Objective Plexiform neurofibromas of the head and neck in neurofibromatosis type 1 (NF 1) carry a significant morbidity with substantial loss of function as well as significant cosmetic problems. We describe our experience with early aggressive surgical intervention in such patients in order to avert these problems. Methods Retrospective review of four consecutive pediatric patients with massive head and neck plexiform neurofibromas who underwent single stage near total or sub-total tumor resections. Results All four patients were referred for obstructive airway symptoms. Each patient experienced complete relief of symptoms and return of function without additional neurological deficits. There were two minor complications and no major complications of surgical resection. There have been no recurrences to date, with follow-up ranging from 15 months to 5 years. Conclusions Early surgical intervention of NF 1 patients with plexiform neurofibromas of the head and neck with a goal of near total resection avoids the loss of function associated with these tumors, such as tracheostomy dependence, swallowing difficulty, and speech problems, and prevents the inexorable progression of substantial cosmetic deformity. Successful management of these complex lesions requires detailed preoperative planning, advanced surgical techniques, and vigilant postoperative care.