Action-property duality in embodied design

For those working on embodied design it is a challenge to create tasks that enable students to develop abstract mathematical concepts. We approach this issue from the perspective of Sfard's notions of saming, encapsulation, and reification. We discuss a duality of properties and actions, and use this duality to review saming, encapsulation and reification from an action-and perceptionbased perspective. To illustrate the power of this theoretical contribution we discuss one new embodied task design and two from literature: MIT-P for proportion and a design for the gradient of a plane using the Augmented Reality Sandbox

Supporting students to compress mathematical knowledge while problem solving

Compression of mathematical objects, procedures and statements could play a major role in successful problem solving. We report on a study in which we aim to design an instrument, named heuristic tree, to stimulate the process of compression in students while working in a digital problem-solving environment. In particular, we report on the evidence a second pilot study provided that improvements in the design helped to overcome issues with help-seeking that presented themselves in a first pilot

Embodied approaches to functional thinking using digital technology: A bibliometrics-guided review

Digital technology offers many opportunities for embodied approaches to mathematics education. To investigate what is known from literature about such approaches for the case of Functional Thinking, we carried out a systematic literature review, followed by a bibliometric and an expert content analysis. We included 36 peer-reviewed articles from 1986 to 2020 in the study. As a result, we identified five research themes in the field, which are further merged into three categories labelled Embodiment not central, Pseudo embodiment and Embodiment

Heuristic Trees as a Digital Tool to Foster Compression and Decompression in Problem‑Solving

This design-based study addresses the issue of how to digitally support students’ problem-solving by providing heuristics, in the absence of the teacher. The problem is that, so far, digital tutoring systems lack the ability to diagnose students’ needs in open problem situations. Our approach is based on students’ ability to self-diagnose and find help. To this purpose, we introduce a new type of digital, interactive, help-seeking tool called a heuristic tree. Students’ use of this tool is supported by a help-seeking flowchart. The design of heuristic trees is based on our reinterpretation of the notion of heuristic in terms of terms of compression. Our research question is: How do heuristic trees and the help-seeking flowchart influence students’ problem-solving behaviour? This question was studied in the context of a number theory course for in-service mathematics teachers. During five weeks, fifty students worked on fifty-five problems supported by heuristic trees. Our data consists of video observations of two small groups of students, a teacher log, interviews with these two groups, and a survey filled in by twenty-three students. The main results are that the support by heuristic trees and the help-seeking flowchart allows students to work in the absence of a teacher and to engage strongly with problems, maintaining ownership of the solution methods. Moreover, as intended by the tree structure, students learned to focus not just on the small steps of the solutions, but also on the general heuristic techniques, theorems, and concepts that should be learned in the process of finding those solutions

Geometric quantization of Hamiltonian actions of Lie algebroids and Lie groupoids

We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a prequantization line bundle. Next, we discuss a version of K\"{a}hler quantization suitable for this setting. We proceed by defining a Marsden-Weinstein quotient for our setting and prove a ``quantization commutes with reduction'' theorem. We explain how our geometric quantization procedure relates to a possible orbit method for Lie groupoids. Our theory encompasses the geometric quantization of symplectic manifolds, Hamiltonian Lie algebra actions, actions of families of Lie groups, foliations, as well as some general constructions from differential geometry.Comment: 40 pages, corrected version 11-01-200

Activate students? Let them fold! Mathematical paper folding in secondary education in France and Germany

Though the potential of mathematical paper folding for learning mathematics is known, this practice is still rare in mathematics classes. To inform the design of mathematics lessons enriched with folding, we investigated via a literature review what mathematical paper folding is done in secondary education and why. The research question that we explore in this paper is: What reasons do teachers from France and Germany report for implementing mathematical folding activities in authentic classroom situations? Through a grounded theory approach, we found reasons like: ‘to activate students by letting them manipulate paper’ and ‘to visualise mathematics’. Teachers state that folding allows for dynamic representations that support the transition from informal to formal mathematics and the practice of skills. We summarise these findings in two major categories of reasons: 'to activate students by providing folding tasks’ and ‘to grow mathematical understanding by folding’

Hypertension with hidden causes:the cognitive and behavioral profile of an adult female with chronic stress and 16p11.2 microdeletion

This case report aims to alert physicians to neuropsychological features and chromosomal variants that may underly resistant hypertension. We present a 35-year-old female patient with hypertensive crisis (BP 260/160 mmHg), initially treated with a combination of calcium antagonists, beta blockers, diuretics and angiotensin-converting enzyme (ACE)-inhibitors, though with little improvement. Cushing's syndrome, Conn's syndrome, and glucocorticoid receptor deficiency were ruled out. Multidisciplinary examination of medical history and (hetero)anamneses including psychosocial factors revealed mild dysmorphic body features, developmental delay, early diagnosis of autism spectrum disorder, a history of being bullied at school, little peer contact, learning disabilities, and special education. Neuropsychological assessment demonstrated below average to low average intelligence quotient, cognitive impairments, and psychopathology. Parallel genetic analyses revealed a rare 16p11.2 microdeletion syndrome. These concurrent examinations explained the patient's life-long high stress levels. After psychological treatment, with additional support at home, her blood pressure lowered to normal levels and antihypertensive drugs were no longer needed.</p