Click on the title to read each story. You’ll find the stories available online as a web page and as a PDF for download. (last updated 02/13/18)

#### Fact Fluency

Lily’s “Invisible Fingers” Strategy for Solving Subtraction Fact Problems

Moving On From Counting On: Developing Addition Fact Fluency

Working on Fact Fluency with Addition and the Importance of Derived Fact Strategies

Cultivating Addition Fact Fluency through Discussion of Strategies

#### Multidigit Word Problems

Bailey and Emma Use Direct Modeling to Solve an Addition Word Problem

Avery Explains That You Cannot Take Eight from Seven on a Subtraction Word Problem

Comparing Jeovani, Desean, and Cesar’s Strategies for Solving a Multidigit Subtraction Word Problem

Mya’s Solution with Cubes Leads to a Strategy Involving Relational Thinking

Students Learning to See Ten as a Unit to Solve Multidigit Addition Word Problems

Catherine’s Strategy and the Move from Concrete Models to Abstract Thinking in Addition Problems

#### Place Value Concepts

Using Maxwell’s Counting by Tens Strategy to Solve a Groups of Ten Word Problem

Students Moving from Direct Modeling with Ones to Direct Modeling with Tens

Breanna Counts by Tens to Solve a Word Problem Involving Groups of Tens

Student Use of Direct Modeling Strategies on Groups of Ten Word Problems

#### Fractions

Students Using Anticipatory Strategies on Equal Sharing Word Problems

Jessiah Uses Repeated Halving Strategies on Equal Sharing Problems

Students Making Sense of Equal Sharing Problems

Using Jack’s Work to Introduce Notation for Multiplication on Fractions

### Background

*What’s Next?* is a collection of stories documenting professional development experiences shared by elementary teachers working to advance student understanding in mathematics. Each story recounts the experience of practicing teachers engaging in collaborative inquiry focused on attending to the details in student thinking and using what they learn about student thinking to inform instructional decisions.

The stories provide narrative cases of elementary teachers engaging with real students in a school setting. The teachers perform an assessment of student thinking, identify a goal for student learning, formulate a plan to help students move toward that goal, carry out the plan, assess the result, and reflect on the experience. In other words, the teachers in these stories are learning through engagement in the problem-solving process as it applies to teaching and learning. At the same time, the stories depict examples of teachers engaging in the processes of formative assessment and lesson study.

The teachers in each story start by posing mathematics problems to individual students and paying close attention to how these individual students think when they attempt to solve the problems. They use this freshly gathered knowledge on that very same day to set near-term learning goals for these students and develop a lesson plan tailored to these specific students on that specific day. Next, one of the teachers implements the planned lesson while the others observe the lesson. The teachers then gather to discuss and reflect on their observations and insights into teaching and learning. Occurring during the 2015–16 and 2016–17 school years, each of the individual stories in the *What’s Next?* collection typically describes events that took place over a period of a single school day.

Although the stories in the *What’s Next?* collection could be interpreted as though teachers were self-guided, they were not. In every case, the group of teachers was guided by a workshop leader as part of a classroom-embedded workshop day. These workshop days occurred as part of the academic-year component of the *Cognitively Guided Instruction* or *Extending Children’s Mathematics* professional development programs created by Teachers Development Group under the direction of Linda Levi.

One important technique people use to develop and improve their ability to perform complex tasks—such as teaching—is to break a task down into fundamental components and practice those components very slowly and deliberately. All accomplished athletes and musicians use this technique extensively as part of their practice as a primary way to improve their ability to perform their crafts. Practicing slowly and deliberately with abundant self-observation and reflection throughout the process supports deep learning. It allows practitioners to concentrate on fine details of the practice in a way they cannot while they are performing the more complex task at full speed.

The design of the classroom-embedded days involves a deliberate slowing down of the practice of teaching so that teachers can think deeply about how the various critical elements involved in the practice of teaching interrelate. Some of these elements include assessing student understanding, setting learning goals for students based on assessment of students’ current understanding, planning lessons, implementing classroom experiences designed to advance student understanding, and reflecting on these experiences to inform next steps.

The purpose of the *What’s Next?* collection is to document and share ideas about teaching and learning mathematics. The stories document some of the experiences teachers had on these classroom-embedded days as a way to archive the learning opportunities and share with a broader group of interested educators.

Each story contains a lesson plan or description of a lesson as it was implemented, but these stories do not purport to offer examples of perfect lessons that are guaranteed to work in every classroom. If another teacher takes a problem and/or lesson plan from one of these stories and poses the same problem(s) or teaches the same lesson to other students, the story is unlikely to unfold just as it did in the story. In fact, several stories in this collection start with the same initial set of problems and unfold very differently. The stories diverge, because student thinking informs teachers’ instructional decisions, each student is a unique person with a unique background, and every classroom has a different constellation of people in it. Moreover, each group of teachers had different ideas about how to advance the students’ understanding. Nonetheless, we wrote these stories to create opportunities for teachers who were not a part of the classroom-embedded workshop days to learn from these teachers’ experiences.

Although some simplification was necessary to condense a full day of conversations into one- or two-dozen pages of text, the stories do not offer a simplification or idealization of the process of teaching and learning. They do provide opportunities for the reader to consider the situations and ideas encountered by the teachers in the workshop. We expect each reader to create a personal interpretation and conclusions, and we think that readers who put forth the effort to study and reflect on these stories will be rewarded with new insight into teaching and learning.

The stories involve real teachers and real students. The names of school sites and teachers are omitted to maintain confidentiality. We are extremely grateful to the school districts, principals, teachers, instructional coaches, and students for their participation and support for the teacher professional-development workshops.

The writers of these stories have had many great conversations based on their experiences observing and describing the events in these stories. We have learned a great deal from observing teachers learning together, and we hope the readers of these stories can benefit similarly.

*Robert C. Schoen, Ph.D., is the Associate Director of LSI’s Florida Center for Research in Science, Technology, Engineering, and Mathematics (FCR–STEM), as well as the founder and director of Teaching is Problem Solving. His research involves mathematical cognition, the mathematical education of teachers, the development and validation of educational and psychological measurement instruments, and evaluation of the effectiveness of educational interventions.*