The concept of proof has attracted considerable research attention over the pastdecades in part due to its indisputable importance to the discipline of mathematics and tostudents' learning of ...mathematics. Yet, the teaching and learning of proof is an instructionallyarduous territory, with proof being recognized as a hard-to-teach and hard-to-learn concept atall levels of education. Prior research has examined, documented, and cast light on theprocesses underpinning different problems of classroom practice in the area of proof, buthas paid less emphasis on acting upon such problems to generate possible solutions throughresearch-based interventions in mathematics classrooms. In this Editorial, we first situate thecontributions in this Special Issue in a brief chronological account of scholarly work on research-based interventions in the area of proof, and we conclude with a proposal for somehigh-leverage directions for future research.
In this paper, we investigate using vignettes in educational research, particularly for eliciting value-laden constructs such as teacher beliefs and understandings and how these influence teacher ...practices. Drawing on research where vignettes have been used as the central instrument for data collection, we argue that methodological consistency is achieved when the research aims, methodologies and vignette methods are aligned, and this helps satisfy internal validity and supports the findings. To assist with achieving methodological consistency we introduce and discuss a vignette framework that identifies three key elements for vignette construction (conception, design, and administration), supplemented by characteristics for each element and descriptions. We then illustrate the framework by using an empirical study from mathematics education where two purposefully formulated vignettes were used to elicit diverse teacher beliefs and practices for promoting student engagement. This illustration shows that using vignettes in educational research can be particularly effective for gaining insights into interpretations and concerns that teachers may have about particular phenomena, such as student engagement in mathematics. We propose that carefully formulated vignettes aligned with the phenomena being investigated can help capture participants' beliefs leading to a more nuanced understanding of the phenomena.
Although students of all levels of education face serious difficulties with proof, there is limited research knowledge about how instruction can help students overcome these difficulties. In this ...article, we discuss the theoretical foundation and implementation of an instructional sequence that aimed to help students begin to realize the limitations of empirical arguments as methods for validating mathematical generalizations and see an intellectual need to learn about secure methods for validation (i.e., proofs). The development of the instructional sequence was part of a 4-year design experiment that we conducted in an undergraduate mathematics course, prerequisite for admission to an elementary (Grades K-6) teaching certification program. We focus on the implementation of the instructional sequence in the last of 5 research cycles of our design experiment to exemplify our theoretical framework (in which cognitive conflict played a major role) and to discuss the promise of the sequence to support the intended learning goals.
Assumptions play a fundamental role in disciplinary mathematical practice, especially concerning the relativity of truth. However, much is still unclear about ways to help students recognize key ...aspects of this role. In this paper, we propose a set of principles for task design to introduce students to the role of assumptions in mathematical activity, with particular attention to the following two learning goals: recognize that (1) a conclusion depends on the assumption(s) underlying the argument that led to it; and (2) making the underlying assumption(s) explicit is crucial to reaching consensus on the conclusion. In the context of a 3-year design research study, we first used existing literature to construct an initial version of task design principles which we then empirically tested and refined by designing and implementing two tasks in Japanese school classrooms. One of the tasks was in the area of functions at the secondary level and the other in the area of geometry at the elementary level. We analyze two classroom episodes to discuss the promise and evolution of our proposed task design principles. In addition, our analysis sheds light on the role of the teacher's instructional moves and the students' mathematical knowledge during the implementation of the designed tasks.
Research on classroom-based interventions in mathematics education has two core aims: (a) to improve classroom practice by engineering ways to act upon problems of practice; and (b) to deepen ...theoretical understanding of classroom phenomena that relate to these problems. Although there are notable examples of classroom-based intervention studies in mathematics education research since at least the 1930s, the number of such studies is small and acutely disproportionate to the number of studies that have documented problems of classroom practice for which solutions are sorely needed. In this paper we first make a case for the importance of research on classroom-based interventions and identify three important features of this research, which we then use to review the papers in this special issue. We also consider the issue of ‘scaling up’ promising classroom-based interventions in mathematics education, and we discuss a major obstacle that most such interventions find on the way to scaling up. This obstacle relates to their long duration, which means that possible adoption of these interventions would require practitioners to do major reorganizations of the mathematics curricula they follow in order to accommodate the time demands of the interventions. We argue that it is important, and conjecture that it is possible, to design interventions of short duration in mathematics education to alleviate major problems of classroom practice. Such interventions would be more amenable to scaling up, for they would allow more control over confounding variables and would make more practicable their incorporation into existing curriculum structures.
Ambitious teaching is a form of teaching that requires a high level of teacher responsiveness to what students do as they actively engage with the subject matter. Thus, a teacher enacting ambitious ...teaching is often confronted with uncertainties about how to advance students' learning while also building on students' contributions. In this article we propose a framework that aims to deepen understanding about the role of instructional engineering in helping reduce the uncertainties of ambitious teaching, particularly with regard to the design and implementation of task sequences that target academically important but difficult-to-achieve learning goals. To illustrate the framework, we consider how instructional engineering helped reduce the uncertainties in enacting ambitious teaching to advance university and secondary students' understanding of what counts as "proof" in mathematics.
Promoting engagement is crucial for encouraging student participation, interest, and learning in mathematics. Student engagement has been conceptualized as interrelated types comprising behavioural, ...emotional, and cognitive characteristics.
Cognitive engagement
, our focus in this paper, relates to students’ psychological investment in learning and practices used to enhance learning, such as self-regulatory strategies and metacognitive processes. Although crucial for students’ learning, research suggests that teachers’ practices for promoting students’ cognitive engagement are not well understood. In this qualitative study, we investigated the beliefs of 40 secondary mathematics teachers across eight English schools concerning promoting cognitive engagement in mathematics classrooms, and whether teachers with different cognitive engagement beliefs differ in the features of classroom practice they attend to in relation to promoting student self-regulation and metacognition. We developed a Cognitive Engagement Framework (CEF) for the following purposes: (1) to develop vignettes that described the practices of two contrasting teachers (Teacher A and Teacher B), who differed in their use of specific self-regulation and metacognitive processes; and (2) to use as a tool for analysis. 17 participants identified with Teacher A who favoured a controlling style towards student strategy use such as activating knowledge, planning, and enacting and regulating strategies, and a passive approach towards students’ use of self-reflection. 14 participants identified with Teacher B who favoured promoting student autonomy for planning and enacting and regulating strategies, self-reflection, and acknowledged affective elements. In addition to its findings, the paper makes a methodological contribution by using ‘vignettes’ as a new way of investigating teachers’ beliefs, and a theoretical contribution through the development of the CEF.
In this paper, we argue that posing new researchable questions in educational research is a dynamic process that reflects the field’s growing understanding of the web of potentially influential ...factors surrounding the examination of a particular phenomenon of interest. We illustrate this thesis by drawing on a strand of mathematics education research related to students’ justification schemes that has evolved rapidly during the past few decades. Also, we reflect on the possible boundaries of the domain of application of the thesis, and we hypothesize that it would apply equally to other strands of educational research. To support this hypothesis, we briefly consider how the thesis would be applicable in two additional research strands. We conclude by elaborating on three important implications of the thesis: (1) as new potentially influential factors about the phenomenon of interest are identified, findings from past research that had not accounted for those factors might prove to be insufficient or be put into question; (2) there are increased methodological challenges for researchers as they seek to design new studies that pay due regard to research advances about all relevant and potentially influential factors surrounding the phenomenon of interest; and (3) as a wider range of potentially influential factors get discovered and considered about a particular phenomenon, research knowledge becomes not only more refined, and presumably more accurate, but possibly more fragmented too.
Despite widespread agreement that the activity of reasoning-and-proving should be central to all students' mathematical experiences, many students face serious difficulties with this activity. ...Mathematics textbooks can play an important role in students' opportunities to engage in reasoning-and-proving: research suggests that many decisions that teachers make about what tasks to implement in their classrooms and when and how to implement them are mediated by the textbooks they use. Yet, little is known about how reasoning-and-proving is promoted in school mathematics textbooks. In this article, I present an analytic/methodological approach for the examination of the opportunities designed in mathematics textbooks for students to engage in reasoning-and-proving. In addition, I exemplify the utility of the approach in an examination of a strategically selected American mathematics textbook series. I use the findings from this examination as a context to discuss issues of textbook design in the domain of reasoning-and-proving that pertain to any textbook series.
Self-efficacy in mathematics is related to engagement, persistence, and academic performance. Prior research focused mostly on examining changes to students’ self-efficacy across large time intervals ...(months or years), and paid less attention to changes at the level of lesson sequences. Knowledge of how self-efficacy changes during a sequence of lessons is important as it can help teachers better support students’ self-efficacy in their everyday work. In this paper, we expanded previous studies by investigating changes in students’ self-efficacy across a sequence of 3–4 lessons when students were learning a new topic in mathematics (n
Students
= 170, n
Time-points
= 596). Nine classes of Norwegian grade 6 (
n
= 77) and grade 10 students (
n
= 93) reported their self-efficacy for easy, medium difficulty, and hard tasks. Using multilevel models for change, we found (a) change of students’ self-efficacy across lesson sequences, (b) differences in the starting point and change of students’ self-efficacy according to perceived task difficulty and grade, (c) more individual variation of self-efficacy starting point and change in association with harder tasks, and (d) students in classes who were taught a new topic in geometry had stronger self-efficacy at the beginning of the first lesson as compared to those who were taught a new topic in algebra (grade 10), and students in classes who were taught a new topic in fractions had steeper growth across the lesson sequence as compared to those who were taught a new topic in measurement (grade 6). Implications for both research and practice on how new mathematics topics are introduced to students are discussed.
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