Guiding reinvention of conventional tools of mathematical logic: students’ reasoning about mathematical disjunctions

Motivated by the observation that formal logic answers questions students have not yet asked, we conducted exploratory teaching experiments with undergraduate students intended to guide their reinvention of truth-functional definitions for basic logical connectives. We intend to reframe the relationship between reasoning and logic by showing how logic emerges within students’ mathematical activity. This activity entails reflecting on and systematizing their own language use across diverse semantic content. We present categories of students’ untrained strategies for assessing the truth-values for mathematical disjunctions. Students’ initial reasoning heavily reflected content-specific and pragmatic factors in ways inconsistent with the norms and conventions of mathematical logic. Despite this, all student groups reinvented the standard truth-functional definition for simple disjunctions. We demonstrate how this learning depended upon particular forms of reasoning about logic. We also contrast various strategies for assessing quantified disjunctions and their different affordances in students’ mathematical activity.     

Visualization and its role in students’ assessment of scientific explanations

It is argued that an important first stage in students’ assessment of scientific explanations is that of visualization. A test of visualization is developed and the results for students in their first year of university education are presented. Certain characteristics of defective visualization and the circumstances in which blocks to satisfactory explanation assessment are created emerge rather clearly.

     

An investigation of relationships between students’ mathematical problem-posing abilities and their mathematical content knowledge

The importance of students’ problem-posing abilities in mathematics has been emphasized in the K-12 curricula in the USA and China. There are claims that problem-posing activities are helpful in developing creative approaches to mathematics. At the same time, there are also claims that students’ mathematical content knowledge could be highly related to creativity in mathematics, too. This paper reports on a study that investigated USA and Chinese high school students’ mathematical content knowledge, their abilities in mathematical problem posing, and the relationships between students’ mathematical content knowledge and their problem-posing abilities in mathematics.     

An investigation of factors impacting on mainstream teachers’ beliefs about teaching students with learning difficulties

The relationship between teacher experience, further professional development training, and beliefs and attributions about teaching students with additional learning support needs was studied in a sample of 199 mainstream general class primary school teachers. Using multiple regression, it was found that none of the teacher experience or professional training variables were significant predictors of locus of causality, stability, or controllability attributions, or of teacher self‐efficacy with students with difficulties in learning. Self‐efficacy, however, was a positive predictor of attributions, and sympathy a negative predictor. Multivariate analysis of variance found no relationship between teacher experience, further professional development training, and dependent outcome variables: self‐efficacy with learning difficulties, coping with learning difficulties, interactions with people with disabilities, or general optimism. Implications for continued professional development are discussed.     

Walking the walk: Rasch analysis of an exploratory survey of secondary teachers’ attitudes and understanding of students with learning difficulties

Abstract

This paper describes the use of low stakes Rasch analysis to examine the findings of a 2004 survey focused on secondary teachers’ attitudes and understanding of students with learning difficulties. The responses of 280 secondary teachers from across Queensland have been interpreted in complementary ways. Two sets of quantitative data were subjected to analysis using the Rasch model for rating scale data (Andrich, 1988; Bond & Fox, 2001). Attitudinal data was collected by Liken scale responses while data obtained from statements about student characteristics was analysed for understanding. These results were linked to demographic indicators and to qualitative data of selected teacher groups. This exploratory study revealed a uniformity of teacher attitude and understanding regardless of demographic indicator. This has significant implications for teacher educators, pedagogy, school organisation and culture as well as for the large numbers of students with learning difficulties enrolled in secondary schools.     

Managing reading and related literacy difficulties: University students’ perspectives

This study explored the experiences of university students who identified as always having had reading difficulties. Thirty-three participants, all from one Australian university, were surveyed and 10 were also interviewed. Findings revealed that many shared academic challenges related to studying, along with a belief that their difficulties were not well understood by staff or peers. While participants disclosed many similar personal struggles related to completing their studies, positive attributes such as determination and time management strategies were frequently mentioned as instrumental in allowing them to reach university. Experiences were more variable regarding awareness of, access to, and satisfaction with university-based support and academic adjustments. To facilitate optimal inclusive educational practices, this study supports the need for greater awareness among staff about challenges faced by these university students along with a more individualised and targeted approach to providing them with academic supports and accommodations.     

Learning about friction: group dynamics in engineering students’ work with free body diagrams

In educational research, it is well-known that collaborative work on core conceptual issues in physics leads to significant improvements in students’ conceptual understanding. In this paper, we explore collaborative learning in action, adding to previous research in engineering education with a specific focus on the students’ use of free body diagrams in interaction. By looking at details in interaction among a group of three engineering students, we illustrate how they collectively construct a free body diagram together when learning introductory mechanics. In doing so, we have focused on both learning possibilities and the dynamic processes that take place in the learning activity. These findings have a number of implications for educational practice.     

‘Human nature’: Chemical engineering students’ ideas about human relationships with the natural world

While importance of environmental ethics, as a component of sustainable development, in preparing engineers is widely acknowledged, little research has addressed chemical engineers’ environmental concerns. This study aimed to address this void by exploring chemical engineering students’ values regarding human–nature relationships. The study was conducted with 247 3rd–4th year chemical engineering students in Israeli Universities. It employed the New Ecological Paradigm (NEP)-questionnaire to which students added written explanations. Quantitative analysis of NEP-scale results shows that the students demonstrated moderately ecocentric orientation. Explanations to the NEP-items reveal diverse, ambivalent ideas regarding the notions embodied in the NEP, strong scientific orientation and reliance on technology for addressing environmental challenges. Endorsing sustainability implies that today's engineers be equipped with an ecological perspective. The capacity of Higher Education to enable engineers to develop dispositions about human–nature interrelationships requires adaptation of curricula towards multidisciplinary, integrative learning addressing social–political–economic–ethical perspectives, and implementing critical-thinking within the socio-scientific issues pedagogical approach.     

Teachers’ estimates of their students’ motivation and engagement: being in synch with students

Being aware of, monitoring and responding constructively to students’ signals of motivation and to students’ signals of engagement represent two important teaching skills. We hypothesised, however, that teachers would better estimate their students’ engagement than they would estimate their students’ motivation. To test this hypothesis, Korean high-school teachers rated three aspects of motivation and four aspects of engagement for each student in their class, while students completed questionnaires to provide referent self-reports of these same aspects of their motivation and engagement. Multi-level analyses showed that, after statistically controlling for the potentially confounding information of student achievement, teachers’ engagement estimates corresponded significantly to their students’ self-reports while their motivation estimate did not. These findings validate teachers’ skill in inferring their students’ classroom engagement and lead to the recommendation that teachers monitor classroom engagement to be in synch with their students during instruction.     

Fostering mathematical modelling competency of South African engineering students: which influence does the teaching design have?

Educational Studies in Mathematics - This paper reports on empirical results about the influence of two different teaching designs on the development of tertiary students’ modelling...     

The impact of engineering curriculum design principles on elementary students’ engineering and science learning

The Framework for K-12 Science Education and the Next Generation Science Standards propose that students learn core ideas and practices related to engineering as well as science. To do so, students will need high-quality curricular materials designed to meet these goals. We report an efficacy study of an elementary engineering curriculum, Engineering is Elementary (EiE) that includes a set of hypothesized critical components designed to encourage student engagement in practices, connect engineering and science learning, and reach diverse students. To measure the impact of the curriculum, we conducted a cluster randomized controlled trial in 604 classrooms in 152 schools in three states. Schools were randomly assigned to either the treatment curriculum or to a comparison curriculum that addressed the same learning goals but did not include several critical components. Results show that students who used the treatment curriculum (EiE) regardless of demographic characteristics outperformed students in the comparison group on outcome measures of both engineering and science content learning. The results show that curriculum design affects student-learning outcomes.     

“It was smart when:” Supporting prospective teachers’ noticing of students’ mathematical strengths

Learning to name and notice students’ mathematical strengths is a challenging process requiring time and multiple iterations of practice for prospective teachers (PTs) to adopt. Mathematics teacher educators (MTEs) can approximate and decompose the complex practice of naming and noticing students’ mathematical strengths so PTs learn to teach mathematics while emphasizing what students know and can do. This study uses two tools MTEs can use to support PTs as they learn to name and notice students’ mathematical strengths: A LessonSketch experience, a digital platform with comic-based storyboards showing children engaged in a mathematics task, and a strengths-based sentence frame. Our study presents the findings from the 111 noticing statements from 18 PTs as they engaged in the LessonSketch digital experience and practiced making noticing statements about what children know about mathematics. The study found that after a sentence-frame intervention, the PTs are more likely to use strengths-based language and more likely to identify mathematical evidence in their noticing statements. Uncommitted language (statements that do not align with a strength- or deficit-based coding scheme), suggests a fruitful, yet complex space for supporting more PTs as they learn to name and notice students’ mathematical strengths. The paper concludes with implications for future research in teacher education.

     

The construction of a double-entry table: a study of primary and secondary school students’ difficulties

Tables are a presentation format that is commonly used to organize information, and they are widely present in many scenarios of our students’ everyday activities; however, there is a scarcity of studies devoted to their analysis. Some of these studies point out that the organization of data into a double-entry table presents difficulties for primary and secondary school students. The present study analyzes the following: (1) the level of competency of primary and secondary school students in constructing a double-entry table from a set of data and (2) the main difficulties encountered by these students during the task. Our findings showed that the percentage of middle-school students who succeeded in contructing a conventional table was relatively low, and the number did not significantly increase over four school years. A set of difficulties is identified and discussed in terms of cognitive and graphical processes.     

Connecting with the horizon: developing teachers’ appreciation of mathematical structure

A professional learning program for teachers of junior secondary mathematics regarding the content and pedagogy of senior secondary mathematics is the context for this study of teachers’ mathematical and pedagogical knowledge. The analysis of teachers’ reflections on their learning explored teachers’ understanding of mathematical connections and their appreciation of mathematical structure. The findings indicate that a professional learning program about senior secondary mathematics can enable practicing teachers to deepen and broaden their knowledge for teaching junior secondary mathematics and develop their practice to support their students’ present and future learning of mathematics. Further research is needed about professional learning approaches and tasks that may enable teachers to imbed and develop awareness of structure in their practice.     

A longitudinal study of engineering students’ approaches to their studies

This longitudinal study draws on data from a larger project and examines how students’ perceptions of their opportunities to influence their study environment may be enacted in approaches aimed at influencing their studies, and whether this changes during the course of their studies. Ten students from a 4.5‐year Master’s programme in Engineering were studied throughout their education by means of semi‐structured in‐depth interviews, which were analysed thematically. The results indicate that students’ perceptions of their study environment were enacted in three approaches aimed at influencing their study environment: (i) to adapt to the environment and to study alone; (ii) to try to change the programme, to create an individual curriculum and to interact with teachers; and (iii) to cooperate with their peers. The thematic analysis suggests that students’ perceptions of their study environment were enacted in the different approaches and that these changed along with external demands in the programme.     

Prediction assessments: Using video-based predictions to assess prospective teachers’ knowledge of students’ mathematical thinking

In order to evaluate the effectiveness of an experimental elementary mathematics field experience course, we have designed a new assessment instrument. These video-based prediction assessments engage prospective teachers in a video analysis of a child solving mathematical tasks. The prospective teachers build a model of that child’s mathematics and then use that model to predict how the child will respond to a subsequent task. In this paper, we share data concerning the evolution and effectiveness of the instrument. Results from implementation indicate moderate to high degrees of inter-rater reliability in using the rubric to assess prospective teachers’ models and predictions. They also indicate strong correlation between participation in the experimental course and prospective teachers’ performances on the video-based prediction assessments. Such findings suggest that prediction assessments effectively evaluate the pedagogical content knowledge that we are seeking to foster among the prospective teachers.     

Characterizing a highly accomplished teacher’s noticing of third-grade students’ mathematical thinking

This study investigated a highly accomplished third-grade teacher’s noticing of students’ mathematical thinking as she taught multiplication and division. Through an innovative method, which allowed for documenting in-the-moment teacher noticing, the author was able to explore teacher noticing and reflective practices in the context of classroom teaching as opposed to professional development environments. Noticing was conceptualized as both attending to different elements of classroom instruction and making sense of classroom events. The teacher paid most attention to student thinking and was able to offer a variety of rich interpretations of student thinking which were presented in an emergent framework. The results also indicated how the teacher’s noticing might influence her instructional decisions. Implications for both research methods in studying noticing and teacher learning and practices are discussed.     

Using I poems to illuminate the complexity of students’ mathematical identities

In this article, we demonstrate how I poems, part of the Listening Guide method, can help researchers to better understand the multiple voices that might comprise a student’s mathematical identity. We provide an analysis of one student’s interview as an example of this method. We argue that the use of the Listening Guide method illuminates the complexity of students’ mathematical identities, especially their fluid, negotiated, and sometimes contested character. In the example shared here, we show how others’ discourses become part of a student’s developing mathematical identity, as well as highlight how the addition of drawings enhances the Listening Guide method.     

Forms of mathematical interaction in different social settings: examples from students’, teachers’ and teacher–students’ communication about mathematics

The study presented in this article investigates forms of mathematical interaction in different social settings. One major interest is to better understand mathematics teachers’ joint professional discourse while observing and analysing young students mathematical interaction followed by teacher’s intervention. The teachers’ joint professional discourse is about a combined learning and talking between two students before an intervention by their teacher (setting 1) and then it is about the students learning together with the teacher during their mathematical work (setting 2). The joint professional teachers’ discourse constitutes setting 3. This combination of social settings 1 and 2 is taken as an opportunity for mathematics teachers’ professionalisation process when interpreting the students’ mathematical interactions in a more and more professional and sensible way. The epistemological analysis of mathematical sign-systems in communication and interaction in these three settings gives evidence of different types of mathematical talk, which are explained depending on the according social setting. Whereas the interaction between students or between teachers is affected by phases of a process-oriented and investigated talk, the interaction between students and teachers is mainly closed and structured by the ideas of the teacher and by the expectations of the students.

Self-efficacy,mathematics’ anxiety and perceived importance: an empirical study with Portuguese engineering students

The accomplishment in mathematics has gained attention from educators and arises as an emerging field of study, including in engineering education. However, in Portugal, there is still incipient research in the area; so it is high time to explore factors that might enlighten the gap in the study of the relationship between Portuguese engineering students and the learning of mathematics. The main purpose of this study is to explore three factors identified in the literature as influencing the learning of mathematical concepts – self-efficacy, anxiety towards mathematics and perceived importance of mathematics – and search for differences by gender and by type of engineering course, a dimension not much reported in the literature but which was revealed as important in the team's previous research. Based on a sample of 140 undergraduate students of different engineering courses from University of Minho, results only identify differences in the type of course and not in gender. These results constitute a contribution and open new paths for future research in the engineering education.