Formative ReEngaging Lessons
Formative Reengaging Lessons involve a cycle of inquiry, instruction, assessment, analysis, selection, and reengagement around a mathematical concept. Here, you can see how multiple teachers have approached designing lessons to formatively reengage their learners. These lessons were developed by the Silicon Valley Mathematics Initiative and are taught by practicing teachers and professional developers. These lessons have been extensively fieldtested in multiple settings and refined over time.
Understanding Reengagement
Hillary LewisWolfsen
5th Grade Math  Proportions & Ratios
In this clip, Linda Fisher, Carolyn Dobson, and Hillary LewisWolfsen discuss the idea of “reengagement” and what they hope the fifth grade students and observing teachers will get out of the demonstration lesson on proportions and ratios.
To prepare, they picked out some interesting work on the task “Candies,” a fifth grade assessment, to show teachers a variety of strategies and models used by students to make sense of the problem as well as to present common misconceptions.
Reteaching  VS.  Reengaging 

teaching the unit again addressing missing basic skills do the same problems over more practice; learn procedures focus mostly on underachievers cognitive load usually lower 
revisiting student thinking addressing conceptual understanding examine the task from different perspectives critique approaches, make connections engage entire class in mathematics cognitive load usually higher 

Why Formative Assessment?
The information that is most valuable for teaching must focus on student thinking. Dylan Wiliam states that, "The central idea of formative assessment, or assessment for learning, is that evidence of student learning is used to adjust instruction to better meet student learning needs." He describes formative assessment practice as students and teachers using evidence of learning to adapt teaching and learning to meet immediate learning needs, minutetominute and daybyday. Most teachers don't actively use these practices, in part because few teachers are trained to use formative assessment and have no apprenticeship implementing its use in classrooms.
Quality math performance assessments, coupled with effective professional development for classroom teachers and leaders, can support improved instruction and student achievement. Teachers need indepth understanding of mathematical concepts and effective strategies for instruction. Without these indepth understandings, it is challenging to design instructional experiences that drive significant student achievement. Teachers can improve instructional effectiveness by using a cycle of formative assessment practice. As they examine the student thinking revealed in the assessments and consider each student's current knowledge and misconceptions, teachers also clarify and strengthen their own understanding of mathematical concepts.

What are Formative Reengaging Lessons?
Reengagement is not reteaching. Reteaching presents the same material again to a group of students. Reengagement involves students in thinking about mathematical concepts in a new way. Formative reengaging lessons are directly tied to the results of formative assessments. They reengage students in the core mathematical ideas of the assessment task in order to deepen their understanding of the core math and build a better conceptual foundation to learn further mathematics. The followup or reengagement lessons featured here model strategies for designing lessons using the formative reengagement process.
To do so, teachers engage in a process of examining student work. It's common for test results in a class to range from students indicating little success to those students who successfully complete the task. A welldesigned reengagement lesson addresses students' learning needs across this continuum.

Strategies for Formative Reengagment
One way to get started with formative reengagement is to use student work from the class or the MARS tasks. The work is transcribed to assure anonymity. Students are asked to examine the work, determine if it is mathematically sound and either to justify the findings or show where the work lacked mathematical accuracy or logic. The challenge of critiquing and explaining other students' thinking and misconceptions requires and develops high cognitive skills.
Asking the class to explore the student thinking in unique or mathematically interesting approaches, or in intuitive and logical approaches which contain mathematical flaws, are productive ways to deepen student understanding. Comparing alternative approaches is also valuable. The teacher may select a few different approaches and have the class examine and compare the methods to make connections between ideas and representations. The elaborated level of the task might also be explored through the examination of other students' thinking.
Formative Reengaging Lessons
1st Grade Math  Base Ten Menu
Liz O'Neill leads a reengagement lesson on composing and decomposing numbers within twenty. Students engage in a variety of activities to think of ten in as many ways as possible, to compose and decompose numbers in a variety of ways, and to justify their thinking. The lesson includes pre and postassessments.
3rd Grade Math  Interpreting Multiplication & Division
Mia Buljan leads a reengagement lesson on the relationship between multiplication and division, using different representations of math stories/contextual word problems.
4th Grade Math  Understanding Fractions
Michelle Makinson leads three days of instruction building students’ understanding of unit fractions using area and set models, verbal representations, numerical representations, number line representations, and contextualized representations or word problems.
5th Grade Math  Interpreting Fractions
Michelle Kious leads a reengagement lesson on fractions using symbolic notation, area models, measurement (number lines), sets, and fractional situations (word problems).