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This mixed methods study explored the development of elementary and middle school preservice teachers' (PSTs) learning to assess student thinking through analysis of students' written work. Six sections of the first mathematics content course for teaching implemented two writing assignments focused on analysis of student work aligned to the mathematics topics covered in the PSTs' classes. Using quantitative methodology, we examined the PSTs' (n = 99) change in their noticing and assessment skills of attending to and interpreting students' thinking. We also documented their mathematical content knowledge (MKT) as measured by the Learning for Mathematics Teaching in the domains of specialized content knowledge and knowledge of content and students. Findings show that the PSTs increased in their assessment skills and in their MKT. Using a qualitative methodology on interview data of a subset of eight PSTs, we found they acknowledged that having a firm grasp of mathematical content is essential to understanding and explaining student thinking. Findings indicate that engaging PSTs in the real work of teaching such as noticing and assessing students' thinking through written work at the early stages of their teacher preparation may motivate PSTs to learn mathematics more deeply with an eye towards strengthening their assessment skills.

This study examined the development of prospective teachers' (PTs) understanding of productive struggle using video episodes which PTs analyzed through the lens of professional teacher noticing. Our qualitative study included 66 PTs in four sections of a semester-long mathematics content course for prospective elementary and middle school teachers that focused on numbers and operations. The goal was to give PTs opportunities to observe students struggling with the course content the PTs were studying and to enact their specialized content knowledge and pedagogical knowledge for teaching simultaneously. Findings suggest that the PTs develop the ability to attend to and interpret the mathematics underlying the student struggles through video analyses. They also begin to identify teaching strategies and practices that appear potentially useful for supporting productive struggle. The use of a productive struggle framework helped the PTs develop a language for discussing productive struggle. Findings also suggest that PTs who had very little knowledge of productive struggle became more aware of what it looks like in practice, discuss ways to support students' struggle and suggest its potential value in supporting students' understanding of mathematics. One implication for this study is to carefully weave opportunities to develop teaching practices such as support of productive struggle into the content course for teaching. The PTs may better bridge the stance between being a student and becoming a teacher in what productive struggle looks like in learning mathematics.

Making math more engaging for students is a challenge that every teacher faces on a daily basis. These authors write that they are constantly searching for rich problem-solving tasks that cover the necessary content, develop critical-thinking skills, and engage student interest. The Mystery Fraction activity provided here focuses on a key number theory concept of divisibility and connects it to many important concepts in middle school mathematics, including place value, fraction sense, scale factors, reasoning and proof, and counting. The main focus of this activity is to explore possible strategies to determine a mystery fraction that must satisfy certain conditions and is equivalent to a given number.

An episode in which students encounter difficulty while working on a challenging task can be viewed as an opportunity for them to grapple with important mathematical ideas. Teachers can use these instances to acknowledge struggle as a natural part of learning while providing appropriate guidance and support to maintain the mathematical goals and cognitive demand of the task (Smith 2000). Research suggests that a range of teacher support and responses are possible in episodes of student struggles to advance students productively toward a resolution of understanding (Warshauer 2014). This article outlines teaching strategies that remind students of the positive aspects of struggle and student actions that indicate productive engagement toward a resolution. It concludes that by incorporating instructional approaches that acknowledge student struggles and effectively support and guide the students' thinking toward a productive resolution, students are given opportunities to strengthen their disposition toward engaging in challenging tasks. Over time, they will persist through their effort to make sense of and understand important mathematics.