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  1. Many teacher education models involve reflecting on teaching practice for the sake of improving it. Such reflection must be carefully structured to help practitioners identify and act upon significant opportunities for improvement (SOIs). Learning from SOIs requires the cognitive activities of noticing students’ mathematical thinking and its connection to instructional practice, along with an affective disposition to view sub-optimal teaching practices as learning opportunities. We draw upon existing literature and theory related to the notion of developing positive error cultures to identify design principles for helping teachers learn from their own sub-optimal practices rather than becoming discouraged by them. The design principles include experience-based learning, low-stakes settings, collaboration, process reflection, and exploration of disagreements. We then describe a mathematics teacher education environment incorporating the design principles. Examples of pre-service teachers’ work within the environment are analysed for possible patterns of learning from SOIs within a positive error culture. Based on these examples, a four-quadrant model to characterise teachers’ learning from SOIs is proposed. The four quadrants describe various outcomes related to recognising and resolving SOIs.
  2. In this article, we discuss how rising fourth graders' understandings of fraction equivalence developed over the course of a ten-week summer program. We found that all students steadily improved in their abilities to give conceptual explanations during the study, yet most struggled with equivalent fractions that could not be formed from a given fraction by halving or doubling the numerator and denominator. We found that the use of discrete manipulatives helped students with these difficulties.
  3. Undergraduate research is increasingly prevalent in many fields of study, but it is not yet widespread in mathematics education. We argue that expanding undergraduate research opportunities in mathematics education would be beneficial to the field. Such opportunities can be impactful as either extracurricular or course-embedded experiences. To help readers envision directions for undergraduate research experiences in mathematics education with prospective teachers, we describe a model built on a design-based research paradigm. The model engages pairs of prospective teachers in working with faculty mentors to design instructional sequences and test the extent to which they support children’s learning. Undergraduates learn about the nature of systematic mathematics education research and how careful analyses of classroom data can guide practice. Mentors gain opportunities to pursue their personal research interests while guiding undergraduate pairs. We explain how implementing the core cycle of the model, whether on a small or large scale, can help teachers make instructional decisions that are based on rich, qualitative classroom data.