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.     

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.     

Cost economies in the provision of higher education for international students: Australian evidence

In the past few decades, the additional revenues available via higher education exports (through both relatively higher prices and increased enrolments) have attracted the attention of providers in many developed countries, not least in Anglophone countries like the USA, the UK, Canada and Australia. However, while the revenue case is strong, the institutional cost structures underlying the provision of higher education services for international students remain relatively unknown at the sector level. Accordingly, we offer a comprehensive analysis of the cost economies underlying higher education provision for international students using a sample of 37 Australian public universities over the period from 2003 to 2012. The findings suggest that it is appealing to enrol additional overseas students given their lower average and marginal costs and the significant economies of scale prevailing in higher education generally. Further, while we find evidence of economies of scope for overseas students only in smaller institutions, there is no evidence of diseconomies of scope, implying the current number of overseas students and their joint production with domestic students at the least does not lead to unnecessarily higher overall costs.     

Dynamics of undergraduate student generic problem-solving skills captured by a campus-wide study

The ability to effectively problem solve is a highly valued competency expected of university graduates, independent of their area of study. Evaluation of problem-solving skill (PSS) development is hindered by a shortage of available tools for monitoring student progress and by lack of defined instructional strategies for development of these skills. Our research is aimed at addressing these problems. We have developed an evaluation tool, which we applied to study the dynamics of undergraduate student PSS. We tested first- and upper-year students from 26 different courses (total enrollment of 2229 students). Overall improvement of PSS was detected for the first-year students over their first term of study. There were no significant differences between the PSS of first- and upper-year students, and no improvement was detected over a single term by measuring PSS in upper-year students. Only three courses were effective at facilitating PSS. Our data indicate that most of the standard lecture approaches do not develop undergraduate student PSS and that universities and individual instructors must take active steps to advance this critical skill set in university students.     

Academic careers and the valuation of academics. A discursive perspective on status categories and academic salaries in France as compared to the U.S., Germany and Great Britain

This article explores the changing dynamics between gender, cultural capital and the state in the context of higher education expansion in contemporary China. With a particular focus on the one-child generation and women’s opportunities and aspirations, I draw upon empirical evidence from a first-hand survey study and in-depth semi-structured interviews with female undergraduates from one-child families in 2007. The findings from the survey study suggest that singleton status might mediate the impact of socioeconomic status and cultural capital on students’ academic performance and elite opportunities. The qualitative interview data provide further evidence on how singleton women’s aspirations are related to their socioeconomic and cultural backgrounds. The most significant finding is concerned with singleton girls’ strategy of applying for Chinese Communist Party membership as a way to minimize their social and gender disadvantages. I argue that there emerges a bottom-up approach of women empowerment through qualifications and political selection during China’s transition. Political selection is dressed up in seemingly meritocratic selection, thus becoming more appealing to female undergraduates who, in turn, take advantage of party membership to add a silver lining of political loyalty to higher education qualifications.     

“I want to speak like the other people”: Second language learning as a virtuous spiral for migrant women?

This article contributes to scholarship on migrant women’s second language (L2) education in North America and Europe. Questioning reductionist understandings of the relationship between female migrants, their receiving communities and L2 education, the authors consider existing literature as well as their own qualitative work to investigate the challenges, opportunities and agency of migrant women. Weaving together and thematically presenting previous scholarship and qualitative data from interviews, participant observations and classroom recordings from a mixed-gender L2 adult migrant classroom in Austria and an all-women L2 migrant classroom in the United States, they trouble conceptualisations which position women primarily as passive recipients of education and in need of emancipation, while simultaneously elevating communities as agentic providers of these. Specifically, the authors emphasise that (1) L2 proficiency is not a guarantee for migrant women’s social inclusion or socioeconomic advancement; (2) migrant women’s complex challenges and agency need to be recognised and addressed; and (3) all involved in L2 education of migrant women do well to become learners of their own experiences of oppression, including their complicity in it.