This study examined preservice elementary teachers' (PETs') epistemological beliefs (EB) and their views about mathematics (VM) and the impact on these parameters by their prior experience in learning mathematics. A total of 541 PETs from an American university completed EB and VM questionnaires. It was found that participants’ EB and the relationship to their VM are different compared to those of other college students or middle/high school students. Participants hold strong beliefs about “how to study” but not “what knowledge is.” Furthermore, the more participants believe that knowledge is constructed by authority and a collection of facts, the more they make efforts to improve their knowledge of mathematics. Concerning their prior mathematics experience, the more mathematics courses they have taken during secondary education, the more developed EB and VM they have, but their prior experience in college mathematics education did not have any significant correlation with their EB and VM.
The aim of this study is to investigate task design for differentiated instruction in mathematics in professional learning communities. Based on the cultural-historical activity theory, we conceptualize a professional learning community as an activity system and use the analytical construct of contradictions to give an account of structures that bring forward the teachers’ work. Eight Swedish upper secondary teachers, engaged in designing tasks for differentiated instruction in mixed-ability mathematics classrooms, are studied. The analysis outlines three contradictions, manifested as three dilemmas, and shows how the teachers, by noticing a dilemma and making it an explicit object of inquiry, came to address a diversity of issues related to differentiated instruction in mathematics. For example, the teachers addressed students’ different learning needs, problem-solving activities for a mixed-ability classroom, and design of tasks that are challenging for all students. With input from our findings on the three dilemmas, we then discuss the design of a task analysis guide as a means for facilitating the development of a professional learning community that is inquiry-oriented with a strong content focus on designing tasks for differentiated instruction in mixed-ability mathematics classrooms.
In this paper we explore how participation in a constructivist-oriented, classroom aligned mathematics intervention program advanced the learning and positive dispositions of Grade 1 students who were failing to thrive when learning mathematics. Intervention programs are an approach to differentiated instruction that some schools adopt, but there are questions about whether such interventions advance equity and inclusion for students. To provide insight about these issues, we draw on data from the Extending Mathematical Understanding for All study in NSW, Australia (2016-2020) to address two research aims. The first aim was to evaluate the effectiveness of the EMU intervention approach for progressing the mathematics learning of students who were failing to thrive mathematically, and for closing the performance gap with their peers. The second aim was to consider the impact of participating in EMU intervention on students’ dispositions for learning mathematics. The findings confirm the potential of intensive constructivist-based intervention approaches for closing the performance gap for students who were not thriving in mathematics, and for improving the learning dispositions of these students. However, two years after the intervention, the performance gap was again apparent for many students. It is likely that a one-off intervention program is not sufficient for enabling all students to thrive mathematically, and that well-focused classroom-based differentiated instruction and further intervention support may be warranted for some students.
There is widespread belief that addressing differentiation within classroom practice is critical for school improvement in mathematics. School mathematics leaders and consultants are faced with the challenge of interpreting the various models of differentiation that are presented throughout professional learning programs and aligning these with existing beliefs and cultures within schools. This paper reports on a self-study of a learning consultant who supported primary mathematics teachers to plan lessons for implementing differentiation in their classrooms and aims to highlight the Mathematical Knowledge for Teaching (MKT) required to lead such planning sessions. Over three weeks, a mathematics consultant assisted teaching teams across three different schools by facilitating planning meetings. The consultant recorded self-reflections after each session and the qualitative data were coded to identify the categories of MKT the consultant relied on during her interactions with teachers. Overall, the results highlight the dominance of three categories, Knowledge of content and students (KCS), Knowledge of content and teaching (KCT) and Specialised content knowledge (SCK) emphasised by the consultant, as well as the influence different contexts had in developing a shared understanding of differentiated learning in mathematics. The findings have implications for ways in which schools and facilitators develop strategies that comprehensively support teachers to plan suitable differentiated lessons for their students. Furthermore, this study highlights the need to develop greater understanding around models of planning that adequately support teachers to plan effectively for differentiating instruction in mathematics.
Pre-symbolic algebra has been advocated for as a mathematics topic elementary students should experience to better prepare them for middle and high school algebra. However, most elementary pre-service teachers have little to no experience with pre-symbolic algebra. The study reported here analysed the struggles that ten elementary pre-service teachers experienced when learning about pre-symbolic algebra in a mathematics content course. Three types of struggles emerged, struggles with changes in the artifacts of algebra, the objects of algebra, and pre-service teachers’ role while doing algebra. This study could inform efforts to better support elementary PSTs in preparing to teach pre-symbolic algebra.
Both in New Zealand and internationally, there has been a focus on the use of differentiation in mathematics instruction to raise achievement levels and provide equitable outcomes. New Zealand has a long history of the use of ability grouping to provide differentiation. Recently, this practice has been challenged in a large scale professional learning and development (PLD) initiative entitled Developing Mathematical Inquiry Communities (DMIC) which focuses on shifting to strength-based, capability focused heterogeneous grouping practices. This article draws on a case study of five teachers involved in the DMIC PLD initiative to examine the enabling factors and barriers to shifting teacher beliefs in relation to mathematical ability grouping. The findings indicate the persistence of teacher beliefs related to fixed ability levels in mathematics. Assessment practices focusing on narrow domains in regard to success in mathematics appeared to act as a barrier to changing teacher beliefs. In contrast, a focus on strengths that students brought to a mathematical task and collaborative group-work appeared to be a key enabling factor to change in teacher beliefs and practices. A key implication of the article is that changes to assessment practices are required alongside changes to pedagogical practices to support teachers to move from gap gazing to recognising multiple strengths in the mathematics classroom.
In this article we propose the notion of differentiation from an advanced standpoint as a teaching strategy, particularly valuable for working with students with low prior attainment. The notion arose from an enactivist analysis of the work of three teachers, engaged in action research in their own classrooms. All three teachers chose to teach their students (who were aged 15-16) topics that are usually only offered to those with relatively high prior attainment in mathematics. No intermediate or bridging topics were offered, instead, these teachers found ways to differentiate work for their classes, from this advanced standpoint. There is tentative evidence of students experiencing their relationship to mathematics in new ways, recognising they were doing “A-grade” work, and of gains in their attainment. There is also evidence of the teachers’ own surprise at what their students could achieve.
Ability grouping is a common practice in elementary mathematics instruction, but some research suggests that grouping by ability can exacerbate existing inequities, and there is evidence that alternatives to grouping can improve learning experiences for all students. In this paper, we describe an effort to support teachers in using equitable teaching practices that was part of an Elementary Mathematics Specialist (EMS) certification program at a public university in the United States. We employ multiple perspectives in our collaborative action research approach: the first author was the director of the EMS program, the second author was a graduate researcher working to support the program, and the third, fourth, fifth, and sixth authors were all teachers in the program. We start by introducing efforts to address equity from the perspective of program designers, including specific activities used during a Summer Institute to prompt consideration of alternatives to ability grouping. Then we share several experiences of reducing ability grouping from the perspectives of four teachers who were participants in the program, noting successes and challenges. We conclude with implications for research and practice.
This report examines one teacher’s attempts to differentiate instruction every day, in a dynamic teaching and learning environment. The report is part of a larger study that utilised a network of theories approach, coordinating the documentational approach to didactics (DAD) and Thompson and Harel’s theory of meanings, to examine teachers’ understandings as they engaged in and discussed their attempts to differentiate instruction in remote and hybrid learning environments. Analyses involved building models of teachers’ understandings of differentiated instruction and the resources they utilised to support differentiation and exploring how these models persisted or changed throughout the study. The report’s findings highlight the importance of teachers’ meanings in their attempts to differentiate instruction and the role digital resources play in supporting or hindering such practices. Finally, this report adds to a growing body of research on teacher’s work with and on digital resources as a means to support differentiated instruction, particularly as remote and hybrid teaching and learning continues throughout the U.S. and across the globe.
In order to transform mathematics classrooms for students with disabilities, teachers must design based on learners at the margins. Universal Design for Learning (UDL) is a framework built on these principles (Meyer et al., 2014), but lacks a robust process through which educators can reimagine mathematics. We theorize that when educators merge the principles of Universal Design for Learning in mathematics with the process of Design Thinking, more accessible designs for curriculum, routines, tools, and spaces would result. Using design research methodology, we analysed a 6-week summer course for mathematics educators on UDL and Design Thinking. Data sources included class video recordings and transcripts, class artefacts, as well as pre- and post-participant surveys. Findings indicate shifts in participants’ knowledge of UDL from a passive to an active role, empathy as a central guiding principle, and the potential of using Design Thinking as a process to enact UDL.“UDL is the What, Design Thinking is the How:” Designing for Differentiation in Mathematics
Professional organisations have provided recommendations for prospective elementary teachers (PTs) to engage in mathematics coursework designed for teachers. In this study, 37 syllabi obtained from instructors of these courses – termed Mathematics Content for Elementary Teachers (MCfET) courses in this paper – were analysed through the lens of messages communicated about the nature of mathematics teaching and learning. Findings indicate that the syllabi in this study communicated messages about mathematical content as well as messages related to mathematical disposition and the role of collaboration in the mathematics classroom. Syllabi presented mathematical disposition messages in two forms: mathematical (i.e., seeing mathematics in the world, mathematics as a sense-making activity) and personal (i.e., belief in oneself, self-assessment), while other syllabi presented potential to promote mathematical disposition. The role of collaboration was described in general and mathematical terms. Careful construction of these messages can help PTs develop a productive mathematical disposition and consider the role of collaboration as they prepare to teach mathematics.
This paper describes a professional development (PD) programme design integrating just-in-time learning (JITL) to support teachers to learn about and enact inquiry teaching. Through JITL, teachers are provided with support that is responsive and applicable to their needs. This case study reports on the professional journeys of three teachers who received JITL via online resources, face-to-face meetings, opportunities for community building within and across schools, and reflections on pupils’ reactions to learning mathematics through inquiry. Findings indicate that JITL embedded within PD facilitated teacher learning about inquiry enactment, since the PD was responsive to teachers’ immediate and contextual needs. We suggest that explicit attention to JITL is given in the design of teacher PD, providing support that is made readily available for teachers to access and utilise.
A study was conducted to explore the beliefs and practices of 49 New South Wales (NSW) primary school teachers regarding their beliefs and practices concerning the use of concrete materials in the learning and teaching of Number and Algebra. This paper reports on elements of the study regarding why and how teachers use concrete materials. Not only do teacher beliefs influence their classroom practice, Buehl and Beck (2014) propose that teacher practices may impact teacher beliefs suggesting they are interrelated. This paper sought to situate teacher beliefs and practices upon a conceptualisation of this interrelationship based on aspects of classroom practice involving concrete materials. The study employed survey methodology utilising a questionnaire and semi-structured interviews to gather data that were analysed to identify key themes regarding teachers’ prevailing beliefs about the use of concrete materials. These themes included a belief in a constructivist approach to learning and teaching mathematics, a cognitive dimension and engagement. On the basis of these findings an overview is suggested positioning teacher beliefs and practices relating to the use of concrete materials.
This study sought to investigate what dimensions of didactic-mathematical knowledge preservice teachers discussed when engaged in lesson study and how lesson study supported the preservice teachers in implementing reform-minded teaching of mathematics. We employed an interpretive case study methodology with the intent of conceptualising the lesson study experience of the preservice teachers. The present lesson study took place in methods of teaching mathematics course at a state university in Turkey. We studied the experiences of two cohorts of preservice teachers during two consecutive years. The current study found that the lesson study experience engaged preservice teachers in discussions about the didactical dimension of teacher knowledge and supported their learning to teach from a reform-minded teaching approach.
The current pool of highly qualified secondary mathematics teachers is woefully inadequate to address the needs of schools across the United States and other countries internationally. In STEM (Science, Technology, Engineering, Mathematics) areas, providing quality instruction in a changing world requires continuous change and innovation as programs prepare and train teachers. University teacher preparation programs wrestle with ways to provide wider professional experiences (WPE) within social learning environments called communities of practice (CoP). This qualitative study examines a university-led undergraduate scholarship program, aimed at recruiting, training, and retaining highly qualified secondary preservice mathematics teacher candidates. With increased exposure to mathematics content, mathematical teaching pedagogy, and community outreach beyond traditional preparation requirements, the goal of the study is to determine the immediate and potential value participants, undergraduate students, found engaging in a unique, CoP-based program. Findings reveal that participants concurrently reported both immediate and potential value in teaching experiences and ideas even when engaging in more mathematics or indirect teaching environments. Further, while mentoring is a key feature of the program, participants rarely identified mentoring or faculty support as an immediate or potential value although mentors were often the conduit for participants’ engagement in WPE.
The pre-service teacher followed in this paper aimed to act as an agent of change during a time of reform in Irish post primary mathematics education. A teacher-researcher led action research project was conducted to depict the reality of the situation regarding the implementation of mathematical problem solving from a variety of perspectives – namely those of the pre-service teacher/teacher-researcher, the second-level students, and the other mathematics teachers in the school. A problem-based approach to teaching and learning was implemented by the teacher-researcher using Van de Walle’s teaching through problem solving framework. Following this intervention, participating students showed an improvement in engagement with mathematical problem solving and expressed positive opinions regarding the problem based approach to teaching and learning. Meanwhile, other mathematics teachers in the school expressed concerns relating to the reality of implementing problem solving in their classrooms, however these concerns were not issues faced by the author when implementing problem solving in the same setting. When these concerns were indirectly challenged by the pre-service teacher a change in conversation occurred.
