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Formative Re-Engaging Lessons

Formative Re-engaging Lessons involve a cycle of inquiry, instruction, assessment, analysis, selection, and re-engagement around a mathematical concept. Here, you can see how multiple teachers have approached designing lessons to formatively re-engage 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 field-tested in multiple settings and refined over time.

Understanding Re-engagement

Hillary Lewis-Wolfsen
5th Grade Math - Proportions & Ratios 

In this clip, Linda Fisher, Carolyn Dobson, and Hillary Lewis-Wolfsen discuss the idea of “re-engagement” 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.

Re-teaching VS. Re-engaging

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, minute-to-minute and day-by-day. 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 in-depth understanding of mathematical concepts and effective strategies for instruction. Without these in-depth 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 Re-engaging Lessons?

    Re-engagement is not re-teaching. Re-teaching presents the same material again to a group of students. Re-engagement involves students in thinking about mathematical concepts in a new way. Formative re-engaging 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 follow-up or re-engagement lessons featured here model strategies for designing lessons using the formative re-engagement 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 well-designed re-engagement lesson addresses students' learning needs across this continuum.

  • Strategies for Formative Re-engagment

    One way to get started with formative re-engagement 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 Re-engaging Lessons

1st Grade Math - Base Ten Menu

Liz O'Neill leads a re-engagement 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 post-assessments.

3rd Grade Math - Interpreting Multiplication & Division

Mia Buljan leads a re-engagement 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 re-engagement lesson on fractions using symbolic notation, area models, measurement (number lines), sets, and fractional situations (word problems).