Enhancing students’ mathematical reasoning in the classroom: teacher actions facilitating generalization and justification

A proof is a connected sequence of assertions that includes a set of accepted statements, forms of reasoning and modes of representing arguments. Assuming reasoning to be central to proving and aiming to develop knowledge about how teacher actions may promote students’ mathematical reasoning, we conduct design research where whole-class mathematical discussions triggered by exploratory tasks play a key role. We take mathematical reasoning as making justified inferences and we consider generalizing and justifying central reasoning processes. Regarding teacher actions, we consider inviting, informing/suggesting, supporting/guiding and challenging actions can be identified in whole-class discussions. This paper presents design principles for an intervention geared to tackle such reasoning processes and focuses on a whole-class discussion on a grade 7 lesson about linear equations and functions. Data analysis concerns teacher actions in relation to design principles and to the sought mathematical reasoning processes. The conclusions highlight teacher actions that lead students to generalize and justify. Generalizations may arise from a central challenging action or from several guiding actions. Regarding justifications, a main challenging action seems to be essential, while follow-up guiding actions may promote a further development of this reasoning process. Thus, this paper provides a set of design principles and a characterization of teacher actions which enhance students’ mathematical reasoning processes such as generalization and justification.     

The dynamics of connecting universities,non-governmental organizations and community members by means of academic projects directed at people in need

In this article we discuss an action research experience that took place from 2002 to 2015. We analyze the inception and progress of several project-based learning-centered academic courses that were aimed at developing the project management skills of graduate and undergraduate students. The experience involved approximately 1800 students from two universities and 40 community partners. The students developed real-life projects on behalf of institutions and non-governmental organizations located in the area that help people in need; we refer to these agencies as ‘community partners.’ We analyze the development of the course from a systemic perspective, making use of causal loop diagrams in order to understand the main driving dynamics. We discuss the results achieved by these projects and the benefits brought to the community partners. We also present testimony from the community partners themselves. We conclude by presenting the lessons we learned over the years we have been running this course.     

Immigrant girls as caregivers to younger siblings: a transnational feminist analysis

This article employs a transnational feminist lens to examine the experiences of racialised immigrant girls who provide care for their younger siblings. The article draws on findings from a participatory action research study in Victoria, British Columbia, Canada to examine the role that immigrant girls’ caregiving practices play in transnational families’ social, economic, and cultural survival. It argues that greater attention should be given to the diverse and complex contexts, consequences, costs, and contributions of immigrant girls’ caregiving labour.     

Early career challenges in secondary school music teaching

The article reports an Economic and Social Research Council‐funded study of the early career experiences of secondary school music teachers in England, set within a wider national picture of decreasing age‐related pupil engagement with school music, career perceptions of music teaching, variable patterns of teacher recruitment and possible mismatches between the musical biographies of young people and intending music teachers. Qualitative and quantitative data were collected from a short‐term longitudinal survey (first questionnaire: n = 74, second questionnaire: n = 29), supplemented by case studies (n = 6) and open‐ended, written questions (n = 20). Analyses suggest that only a half of the newly qualified participants chose to teach full‐time in a mainstream, state‐funded school music classroom. Of these, the majority were faced with a range of early career challenges stemming from curricular, extra‐curricular and non‐curricular school expectations. These included the need to balance their existing musical performer identity with that of being a new teacher.     

Novel model reference adaptive control laws for improved transient dynamics and guaranteed saturation constraints

In classical model reference adaptive control (MRAC), the adaptive rates must be tuned to meet multiple competing objectives. Large adaptive rates guarantee rapid convergence of the trajectory tracking error to zero. However, large adaptive rates may also induce saturation of the actuators and excessive overshoots of the closed-loop system’s trajectory tracking error. Conversely, low adaptive rates may produce unsatisfactory trajectory tracking performances. To overcome these limitations, in the classical MRAC framework, the adaptive rates must be tuned through an iterative process. Alternative approaches require to modify the plant’s reference model or the reference command input. This paper presents the first MRAC laws for nonlinear dynamical systems affected by matched and parametric uncertainties that constrain both the closed-loop system’s trajectory tracking error and the control input at all times within user-defined bounds, and enforce a user-defined rate of convergence on the trajectory tracking error. By applying the proposed MRAC laws, the adaptive rates can be set arbitrarily large and both the plant’s reference model and the reference command input can be chosen arbitrarily. The user-defined rate of convergence of the closed-loop plant’s trajectory is enforced by introducing a user-defined auxiliary reference model, which converges to the trajectory tracking error obtained by applying the classical MRAC laws before its transient dynamics has decayed, and steering the trajectory tracking error to the auxiliary reference model at a rate of convergence that is higher than the rate of convergence of the plant’s reference model. The ability of the proposed MRAC laws to prescribe the performance of the closed-loop system’s trajectory tracking error and control input is guaranteed by barrier Lyapunov functions. Numerical simulations illustrate both the applicability of our theoretical results and their effectiveness compared to other techniques such as prescribed performance control, which allows to constrain both the rate of convergence and the maximum overshoot on the trajectory tracking error of uncertain systems.     

Inspiratory muscle warm-up and inspiratory muscle training: separate and combined effects on intermittent running to exhaustion

In the present study, we examined the independent and combined effects of an inspiratory muscle warm-up and inspiratory muscle training on intermittent running to exhaustion. Twelve males were recruited to undertake four experimental trials. Two trials (Trials 1 and 2) preceded either a 4-week training period of 1 × 30 breaths twice daily at 50% (experimental group) or 15% (control group) maximal inspiratory mouth pressure (PImax). A further two trials (Trials 3 and 4) were performed after the 4 weeks. Trials 2 and 4 were preceded by a warm-up: 2 × 30 breaths at 40% PImax. Pre-training PImax and distance covered increased (P < 0.05) similarly between groups after the warm-up (~11% and ~5-7% PImax and distance covered, respectively). After training, PImax increased by 20 ± 6.1% (P < 0.01; d = 3.6) and 26.7 ± 6.3% (P < 0.01; d = 3.1) when training and warm-up were combined in the experimental group. Distance covered increased after training in the experimental group by 12 ± 4.9% (P < 0.01; d = 3.6) and 14.9 ± 4.5% (P < 0.01; d = 2.3) when training and warm-up interventions were combined. In conclusion, inspiratory muscle training and inspiratory muscle warm-up can both increase running distance independently, but the greatest increase is observed when they are combined.     

How full is your luggage? Background knowledge of zoo visitors regarding sharks

For the general population, sharks have a reputation that does not really fit with their biological and ecological nature. Informal surveys often classify sharks as dangerous, aggressive and/or man-eaters. This apparent common knowledge seems difficult to detach from the conscience of many worldwide zoo visitors, even with the help of shark-focused educational programmes. As so, how can zoos and aquariums contribute to a change in this paradigm? Are the education and conservation strategies effective to the average zoo visitor? In this study, young visitors’ perceptions regarding sharks were assessed with the goal of identifying a valid Learning Progression, thus helping to update and develop different and more effective educational strategies and methodologies. The study took place at Zoomarine, an oceanographic park located in southern Portugal, with visitors aged between 8 and 16. A case study was conducted using a qualitative research approach (drawings and interviews) resulting in a Learning Progression Hypothesis. The proposed Learning Progression is composed of three knowledge levels for the participants’ biological perceptions regarding sharks. The results of this study showed a clear predominance of anthropomorphic and anthropocentric visions, as well as a utilitarian view of sharks and ecosystem, thus conditioning the visitors’ epistemological knowledge.