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.     

Changing classroom culture,curricula, and instruction for proof and proving: how amenable to scaling up,practicable for curricular integration,and capable of producing long-lasting effects are current interventions?

This paper is a commentary on the classroom interventions on the teaching and learning of proof reported in the seven empirical papers in this special issue. The seven papers show potential to enhance student learning in an area of mathematics that is not only notoriously difficult for students to learn and for teachers to teach, but also critically important to knowing and doing mathematics. Although the seven papers, and the intervention studies they report, vary in many ways—student population, content domain, goals and duration of the intervention, and theoretical perspectives, to name a few—they all provide valuable insight into ways in which classroom experiences might be designed to positively influence students’ learning to prove. In our commentary, we highlight the contributions and promise of the interventions in terms of whether and how they present capacity to change the classroom culture, the curriculum, or instruction. In doing so, we distinguish between works that aim to enhance students’ preparedness for, and competence in, proof and proving and works that explicitly foster appreciation for the need and importance of proof and proving. Finally, we also discuss briefly the interventions along three dimensions: how amenable to scaling up, how practicable for curricular integration, and how capable of producing long-lasting effects these interventions are.     

Teaching multidigit multiplication: combining multiple frameworks to analyse a class episode

This paper provides an analysis of a teaching episode of the multidigit algorithm for multiplication, with a focus on the influence of the teacher’s mathematical knowledge on their teaching. The theoretical framework uses Mathematical Knowledge for Teaching, mathematical pertinence of the teacher and structuration of the milieu in a descending and ascending a priori analysis and an a posteriori analysis. This analysis shows a development of different didactical situations and some links between mathematical knowledge and pertinence. In the conclusion, the contribution of the frameworks from both French and Anglo-American origins is briefly addressed.     

The perceived impact of external evaluation: the system,organisation and individual levels—Estonian case

The purpose of this article is to provide an overview of how the employees of higher education institutions perceive the impact of external evaluations. The study was conducted using the concurrent mixed method and involved 361 employees from Estonian universities and professional higher education institutions. The results indicated that evaluation is mostly deemed necessary at the levels of the organisation and the system; from the personal viewpoint of an employee, positive effects of external evaluations are perceived to a lesser degree. The negative influences mentioned were increased bureaucracy, extra work and stress. Top managers of institutions most often saw the positive influences from external evaluations since they help to implement reorganisations and development activities within an organisation. Managers perceived the positive effects of external evaluations on their everyday work as well, more than other employees did. Researchers felt the positive effects of external evaluations least. Nearly a quarter of all employees of higher education institutions do not perceive the effects of external evaluations on their everyday work at all. Employees of professional higher education institutions perceive the positive effects of external evaluations to a greater extent than employees of universities do, indicating that external evaluation has contributed to the development of their organisations. The differences in opinions of the employees of higher education institutions regarding perceived positive and negative impacts of external evaluations—whether on individual, organisation or system points of view—indicate that employees do not identify closely enough with the organisation and its strategic goals.     

Student perceptions of the fairness of grading procedures: a multilevel investigation of the role of the academic environment

The purpose of this study is to examine the effects of assessment method (essays vs. examinations) and instruction method (seminars vs. lectures) on student perceptions of the fairness of the assessment process. Department-specific combinations of these factors give a unique profile to the assessment process and to the way students interact with faculty. It is argued that the conditions thus created place students in some departments in a more advantageous position when it comes to meeting justice-related expectations. The variables of interest are procedural justice (PJ) and informational justice (IJ). For PJ, aspects regarding the amount of control students can exert on the grading process (PJ-C) are distinguished from aspects regarding the perceived validity of grading procedures (PJ-V). The sample consists of 1549 students from 48 departments of a German university. Analysis is done via multilevel mixed effects models. Models also check for cross-level interactions between effects of the academic environment and student socioeconomic status (SES). Results show that PJ-C and PJ-V are significantly affected by the assessment method. Higher proportions of essays relative to examinations in a department lead to higher ratings of PJ-C, while they decrease ratings of PJ-V. Ratings of IJ are higher as well if assessment is more essay-based, although this only affects low-SES students. Regarding the instruction method, a higher proportion of seminars was found to significantly increase PJ-C and IJ. Again, effects on IJ are moderated by parental SES. Policy implications for reducing feelings of injustice are discussed.     

How experiential learning in an informal setting promotes class equity and social and economic justice for children from “communities at promise”: An Australian perspective

Educational research often portrays culturally, linguistically and economically disenfranchised (CLED) children’s disengagement from school learning as individual behaviour, ignoring the contribution of race, gender, socio-cultural, ethnic and social class factors. This paper analyses a specific community engagement programme in Australia which uses experiential learning in an informal setting. The programme, which has been running for seven years, partners pre-service teachers, volunteer high school students and volunteers from a national bank with primary schools where many pupils are experiencing learning difficulties and school engagement problems as a result of their socio-economic status, their poverty, and their ethnic and cultural diversity. Drawing on the perspectives of the children and volunteers participating in the pilot study, and privileging their voices, this paper illustrates how community partnerships may be developed and sustained. The programme’s conceptual framework of Connecting-Owning-Responding-Empowering (CORE) pedagogy is explored for its potential to enhance student engagement, achievement and empowerment through focused community involvement. The findings show that when students feel connected to and involved in their community, all participants are empowered in their learning and teaching.     

Interfaith education: A new model for today’s interfaith families

With societal changes rapidly transforming cultures that had been largely homogenous, today’s multi-cultural – and in particular interfaith – families need new educational strategies to help them understand their cultural roots and identify and clarify what aspects of their heritages they wish to nurture and transmit to their children. This paper focuses on a new model for religious education, namely non-doctrinaire “dual-faith” education, which the principal author has helped to develop in the United States (US) through the Interfaith Community (IFC), a small, independent non-profit organisation created and led by dual-faith Jewish/Christian families. The model is premised on the notion that families can have two different faiths in one household and that – with respect and education – families can be harmonious, religion can be transmitted, and tolerance broadly nurtured. While the model is particular to the US and to families with Jewish and Christian heritages, its premises and structure have significant potential to be adaptable to other religious combinations and other cultures and countries. After reviewing relevant literature and situating the IFC model in the global and US contexts, the paper sets out to clarify the importance of the concept, describe its elements, and discuss its implications for religious education in this time of changing ethos and demography.