Bringing Math Students Into the Formative Assessment Equation: Tools and Strategies for the Middle Grades

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Susan Janssen Creighton, Cheryl Rose Tobey, Eric Karnowski & Emily R. Fagan

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    Acknowledgements

    To Joan Ferrini-Mundy and Richard Schwab, two very influential teachers and mentors who helped set me on this path, and shaped my approach to the teaching profession in so many valuable ways: with appreciation and gratitude. —SJC

    To my late mother, who encouraged me to do what I loved and to be who I am. —EK

    To the FACETS teachers, whose energy, dedication, and thoughtfulness provided rich experiences for us to learn from. —SJC, EF, EK, CT

    Preface

    In 2008, we saw an increase in attention to formative assessment yielding interesting work: excellent books being published, such as those by Black and Wiliam, Lee, and Bright and Joyner; informative articles such as those by Wylie and Heritage for the Council of Chief State School Officers (CCSSO); and district and statewide initiatives being built, such as those in Iowa and Syracuse, New York. At the same time, with only a few exceptions (such as Bright and Joyner’s, 2004, Dynamic Classroom Assessment), these materials and initiatives cut across school content, using examples from science, language arts, and social studies as well as mathematics. Our experiences with professional development for teachers led us to believe that while these cross-content descriptions are invaluable to the field, a focus on what formative assessment looks like in the mathematics classroom is needed for teachers of math. We often hear frustration from cross-content professional development, as teachers say, “I see how this works in the example subject, but I don’t see how it fits in mine.” (This is true whether, for example, the teacher’s subject is math and the example is language arts, or vice versa.) As you’ll see in this book, effective implementation of formative assessment is deeply tied to the subject content, so providing examples and suggestions specifically for mathematics teachers is paramount.

    To respond to this need, we have written this book for middle grades (5–8) teachers of mathematics, including special education teachers. While much of the information within is transferable to other grades, and even to other content, we maintain our focus on mathematics at the Grade 5–8 level to make this book as relevant to this audience as possible.

    This book provides

    • an extensive description of formative assessment, including examples from middle grades mathematics classes to focus support for math teachers in implementing formative assessment in ways that a subject-general book cannot, and
    • concrete tools, strategies, and resources, informed by research and field-tested by teachers, to
      • support your implementation of various formative assessment practices: setting and using learning goals and criteria for success, eliciting and interpreting evidence, and acting on the information; and
      • make explicit to students their role in the formative assessment process: understanding the learning goal; self-monitoring, self-assessment and reflection; and using peers as resources to provide and act on feedback.

    Through the professional development work and associated research that helped us to produce this book (see “Our Work”), the ideas in this book have been tested. We are proud to have included voices of teachers throughout the book, as we have learned much from the teachers who have participated in and helped to improve and refine our work by using in it their classrooms. We hope that hearing from them helps you to bring the ideas into practice.

    This book is rich with information and resources that you will find useful for many years. We hope you will not read it once and put it back down—rather, we hope that you will return to it again and again, as you deepen your understanding and move forward to integrate formative assessment into your daily practices.

    Our Work

    With a grant from the National Science Foundation (Grant # DRL-0918438), we began work to develop Formative Assessment in the Mathematics Classroom: Engaging Teachers and Students (or FACETS), a program and resources to introduce middle school mathematics teachers to formative assessment and to support them as they began to implement formative assessment practices in their classrooms. We based our initial work on that of many educational researchers and teacher educators and on our own prior work in teacher professional development, formative assessment, and mathematics curriculum development and teacher training. After a year of initial development, we worked with one group of teachers for 2 years, learning about how they made sense of the information we were providing.

    As we implemented our initial program with our first cohort of teachers, the FACETS research team gathered and interpreted data to study teacher’s learning of formative assessment. We worked with a second cohort of teachers for another 2 years, again learning even more about what worked and what didn’t.

    The researchers provided significant input to help us shape the program, guided by the questions:

    • How do middle grades teachers make sense of and learn to implement formative assessment practices?
    • What barriers and challenges do they face in implementing formative assessment?
    • What are supports that can help them move through these challenges?

    For example, we learned that

    • envisioning students as the primary consumers of formative assessment information in teacher’s conception of the formative assessment process early made even the initial work more meaningful and powerful.
    • teacher learning of formative assessment takes time. As you’ll see, formative assessment has many pieces to it; teachers need time to get to know each piece individually and then more time to put those pieces together.

    Figure 0.1 illustrates, in a simplified way, how teachers were able to approach learning about formative assessment. As they were becoming familiar with formative assessment and considering how to manage the different pieces, they were mostly focused on their own actions, though the student role in formative assessment did get some attention. Later, they were able to switch their focus to be more on the student, as they worked toward full integration of the pieces within their teaching practices.

    This book, and the companion resource website (Resources.Corwin.com/CreightonMathFormativeAssessment), are the results of that work.

    How to Use this Book

    We’ve written this book to be read through from beginning to end, though we recognize that different people will want different things from a resource like this. If you decide not to read each chapter in order, we highly recommend at least reading Chapters 1 and 2 first. (Depending on how you plan to bring this information into your classroom, Chapter 8 might also be a good one to read after that—see the last paragraph in this section.) Given our emphasis on student involvement and self-regulation, Chapter 5 may seem like an important chapter to read early, however, we believe it will make most sense after you have read Chapters 1 through 4. Those chapters include some previews of what you will read in Chapter 5, so you will still be getting some idea of the student role in the formative assessment process throughout the early chapters.

    Figure 0.1 A Simplied View of Teacher Learning of Formative Assessment

    Resources, available at Resources.Corwin.com/CreightonMathFormativeAssessment, are highlighted throughout the book. When you see a reference to a resource, you may want to stop reading to look—we support that desire, but with a few exceptions, this isn’t expected. For those exceptions, such as a couple of places in Chapter 1, we explicitly request that you take a moment to look over or work with a particular resource before continuing. In those cases, we are trying to re-create some popular and effective professional development activities that we think will help you develop some initial understanding of formative assessment. Of course, you may choose to continue reading rather than stop. You know how you learn better than we do!

    Two of the following icons accompany each resource, both in this book and on the companion resource website. One icon represents what type of resource it is (interactive, poster, etc.), and the other represents how it’s meant to be used.

    Finally, most chapters include several recommendations to help you integrate formative assessment into your classroom practices; we suggest you give yourself time to integrate them gradually—choose a few to focus on until they become familiar, and then you can choose another small group when you’re ready to move on. Chapter 8 provides some specific recommendations on how to do this gradual implementation, so if you prefer to begin implementing before you’ve completed the whole book, we recommend looking at that chapter—and then return to Chapter 8 often as you progress.

    Acknowledgments

    We would like to acknowledge many people who have helped and supported us in creating this book.

    First, a very grateful thank you to the teachers who worked with us, giving us their time and attention to provide feedback on their experiences: Nancy Adamoyurka, Sara Allegretti, Bekah Aucoin, Lisa Beede, Maryanna Biedermann, Laurie Boosahda, Susan Brass, Nancy Cachat, Debra Chamberlain, Jan Chamberlain, Sara Churchill, Terri Clark, Laura DeSimone, Jessica Dineen, Nancy Dobbins, Joelle Drake, Michelle Eastman, Eli Edinson, Kerri Falcone, Dory Fish, Carolyn Flanders, Sheri Flecca, Judy Forrest, Josh Fox, Maura Ghio, Linda Gloski, Megan Goff, Angie Goldberg, Meredith Gonzalez, Michele Greco, Maureen Griffin, Tracey Hartnett, Renée Henry, Pat Henyan, Sharon Johnson, Dan Johnston, Patrick Kelcourse, Tracy Kinney, E. J. Kluge, Marguerite Lackard, Tami LaFleur, Kathleen Lam, Jesse Latimer, Jane Lewandowski, Errol Libby, Tom Light, Jessica Lloyd, Megan Maguire, Mandy Marrella, Laura McDuff, Guy Meader, Judy Morgan, Jonathan Newman, Michele O’Connor, Lauren O’Malley, Patricia Paul, Nancy Philbrick, Joan Savage, Michelle Schechter, Ricki Scheeder, Nicole Simkins, Diana Smith, Mike Soucy, Wendy Stebbins, Michele Torkomian, Bridget Wade, Nichole Walden, Alison Washington, Mae Waugh, Adam West, Jessica West, Ruth Wilson, Ed Worcester, Yan Yii, and Steve Zakon-Anderson. We also would like to thank the following educators who embraced many of these ideas during a schoolwide effort to implement formative assessment: Amanda Bolanda, Karla Bracy, Amy Bru, Laura Cummings, Peggy Dorf, Terri Eckes, Laura Foley, Georgina Grenier, Melissa Guerrette, Kelly Langbehn, Lauren LaPointe, Susan Leunig, Cathleen Maxfield, Nathan Merrill, Stephanie Pacanza, Roberta Polland, Cynthia Powers, Kim Ramharter, Sara Roderick, Joann Smith, Gal Stetson, Lisa Stevens, and Kimberly Tucker.

    We are most grateful for the support, sacrifice, and patience shown by our families throughout the work involved in bringing this book to fruition: Doug Creighton, Evan Janssen, Alison Janssen; Mark Saperstein; Corey Tobey, Robert Spadea, Carly Rose, Jimmy Rose, Bobby Rose, Samantha Tobey, Jack Tobey; Sean Fagan, Nellie Fagan, and Seamus Fagan.

    Next, our colleagues at Education Development Center (EDC): Lynn Goldsmith and Sophia Mansori formed the research team for FACETS, and Cynthia Char (of Char Associates) served as our external evaluator. We would have been lost without our administrative assistants, Mari Halladay, Carlene Kaler, and Michelle Raymond. Our former colleague, Fred Gross, was instrumental in assembling the team. And we enjoyed and appreciated the support of many others in our EDC community.

    We also want to thank the advisors to the FACETS project: Colleen Anderson, David Baumgold, Steve Benson, Paul Black, Mark Driscoll, Margaret Heritage, Karen King, Kit Norris, Paola Sztajn, and Nancy Zarach.

    As we drafted this book, we received wonderful feedback from several reviewers: Carolyn Arline, George Bright, Peg Brown, Nicole Clark, Deb Cook, Heather Daigle, Michelle Eastman, Matthew Lunt, Diana Smith, Joan Taczli, and Ruth Wilson.

    We want to extend a special thank you to the folks at Corwin who worked diligently to help us bring you this book, and the companion resource website: Desirée Bartlett, Arnis Burvikovs, Robin Najar, Andrew Olson, Michelle Ponce, Ariel Price, Maura Sullivan, and Veronica Stapleton Hooper.

    Finally, as coauthors, we want to acknowledge each other. Working together over the last 5 years to create and evolve the ideas, tools, and strategies that are the foundation of this resource has been a truly collaborative experience.

    Publisher’s Acknowledgments

    Corwin would like to thank the following individuals for taking the time to provide their editorial insight and advice:

    • Lyneille Meza

      Coordinator of Data & Assessment

      Denton ISD

      Denton, TX

    • Dr. Marc Simmons

      Principal

      Ilwaco Middle School

      Long Beach, WA

    • Rita Tellez

      Math Coordinator

      Ysleta Independent School District

      El Paso, TX

    • Morris White

      Secondary Math Teacher

      Alamosa High School

      Alamosa, CO

    About the Authors

    Susan Janssen Creighton is a senior mathematics associate at Education Development Center (EDC) in Massachusetts. She has worked in mathematics education for 30 years, both in schools and at EDC, where her work has focused largely on K–12 mathematics curriculum development and mathematics teacher professional development. Currently, her work focuses on helping mathematics teachers adopt and successfully implement formative assessment practices and on supporting teachers’ understanding and use of the CCSS Standards for Mathematical Practice. As a member of the National Science Foundation (NSF)–funded project, Formative Assessment in Mathematics Classrooms: Engaging Teachers and Students (FACETS), she was a lead facilitator for several of the participating districts. Creighton has written print and online materials for numerous clients, including the international Department of Defense schools, the Columbus, Ohio, public schools, the National Board of Professional Teaching Standards, Everyday Learning publishers, the PBS TeacherLine project, the Massachusetts Department of Education, and the E-Learning for Educators project funded by the U.S. Department of Education. She has also served as the director of the MathScape Curriculum Center, a national center that supported the implementation of the NSF-funded mathematics curriculum MathScape developed at EDC, for which she was also one of the writers, and has led numerous teacher professional development opportunities for middle and high school teachers on the teaching and learning of mathematics. Prior to coming to EDC, she taught middle school and high school mathematics for a number of years in Portland and Saco, Maine and in Brookline, Massachusetts. She received a BA in mathematics and a MEd in Secondary Education, with a concentration in curriculum, both from the University of New Hampshire. She currently lives in western Massachusetts with her husband, her two teenagers, and the world’s softest dog.

    Cheryl Rose Tobey is a senior mathematics associate at EDC in Massachusetts. She is the project director for Formative Assessment in the Mathematics Classroom: Engaging Teachers and Students (FACETS) and a mathematics specialist for Differentiated Professional Development: Building Mathematics Knowledge for Teaching Struggling Students (DPD); both projects are funded by the NSF. She also serves as a director of development for an Institute for Educational Science (IES) project, Eliciting Mathematics Misconceptions (EM2). Her work is primarily in the areas of formative assessment and professional development. Prior to joining EDC, Tobey was the senior program director for mathematics at the Maine Mathematics and Science Alliance (MMSA), where she served as the coprincipal investigator of the mathematics section of the NSF-funded Curriculum Topic Study, and principal investigator and project director of two Title IIa state Mathematics and Science Partnership projects. Prior to working on these projects, Tobey was the coprincipal investigator and project director for MMSA’s NSF-funded Local Systemic Change Initiative, Broadening Educational Access to Mathematics in Maine (BEAMM), and she was a fellow in Cohort 4 of the National Academy for Science and Mathematics Education Leadership. She is the coauthor of six published Corwin books, including seven books in the Uncovering Student Thinking series (2007, 2009, 2011, 2013, 2014), two Mathematics Curriculum Topic Study resources (2006, 2012), and Mathematics Formative Assessment: 75 Practical Strategies for Linking Assessment, Instruction, and Learning (2011). Before joining MMSA in 2001 to begin working with teachers, Tobey was a high school and middle school mathematics educator for 10 years. She received her BS in secondary mathematics education from the University of Maine at Farmington and her MEd from City University in Seattle. She currently lives in Maine with her husband and blended family of five children.

    Eric Karnowski is a senior mathematics associate EDC in Massachusetts. He has worked in mathematics education for 25 years, initially as a teacher, then as a textbook editor, and finally as a curriculum developer and teacher professional development provider. Since joining EDC, he has directed the development of the K–5 program Think Math! and written numerous activities for the award-winning Problems with a Point website. He directed projects to develop several online teacher professional development courses for PBS TeacherLine, Louisiana Algebra 1 Online Professional Development, and most recently, the National Board of Professional Teaching Standards in both mathematics and science. In addition, he was a contributing author on Ways to Think about Mathematics and the MathScape curriculum. Prior to joining EDC, Karnowski had the distinct privilege to edit influential secondary textbooks for Janson Publications and Everyday Learning, including Contemporary Mathematics in Context by the Core-Plus Mathematics Project, Contemporary Calculus by the North Carolina School of Science and Mathematics, and Impact Mathematics by EDC. He received a BS in Liberal Arts (honors mathematics) and an MS in Mathematics, both from the University of Tennessee, Knoxville. He currently lives in Boston with his husband, Mark, and two large cats, Endora and Tabitha.

    Emily R. Fagan is a senior curriculum design associate at EDC in Massachusetts where she has developed print and online curricula as well as professional development and assessment materials in mathematics for 14 years. She was director of the MathScape Curriculum Center, a project funded by the NSF to support schools, districts, and teachers in curriculum implementation, and she directed the revision of MathScape: Seeing and Thinking Mathematically (McGraw-Hill, 2005). She was a developer and facilitator of three NSF-funded projects, Addressing Accessibility in Mathematics and Differentiated Professional Development: Building Mathematics Knowledge for Teaching Struggling Students (DPD) aimed at supporting struggling math learners, particularly those with learning disabilities, and Formative Assessment in the Mathematics Classroom: Engaging Teachers and Students (FACETS) the inspiration for this book. Fagan is the coauthor of two books: Uncovering Student Thinking About Mathematics in the Common Core, Grades K–2 (2013) and its companion for Grades 3 through 5, as well as book chapters and articles about curriculum implementation and instruction. Prior to joining EDC, Emily taught high school and middle school mathematics in Philadelphia and in Salem and Brookline, Massachusetts. She was a mentor teacher, math coach, and member of the Massachusetts faculty of the Coalition of Essential Schools. She has long been interested in accessibility in mathematics education and improving opportunities for all students to learn and love math. While mathematics has been her focus for the last 2 decades, she has also taught science, social studies, and Spanish. Fagan holds an AB cum laude from Harvard University. She lives in Sudbury, Massachusetts, with her husband and their two children.

  • Appendix A: Resources

    The following sections present some resources that can help you implement the recommendations found in this book. Each of these resources is referenced in relevant chapters, but here we provide a consolidated list. All resources, plus additional videos, can be found at Resources.Corwin.com/CreightonMathFormativeAssessment.

    Learning Resources

    These resources support your learning about the critical and supporting aspects of formative assessment.

    Guidance Documents
    • Reflecting on Your Implementation provides indicators by which you can assess the extent to which you are incorporating the principles into your instructional practices and consider next steps as you work gradually toward full implementation.
    Web-Based Interactives
    • Formative Assessment Overview provides an approach to considering different characteristics of formative assessment. The interactive web page provides several statements about formative assessment, by selected notable authors, which you can sort to look for connections among the ideas.
    • Lesson Purposes, Practice Interactive provides practice considering purposes for lessons when writing learning intentions and success criteria; it also helps you learn how to use the “Lesson Purposes, Planning Interactive” listed in the planning resources.
    • Responsive Actions—During provides practice deciding which responsive action is reasonable, for individual students, small groups, or a whole class.
    • Responsive Actions—After provides practice deciding which responsive action is reasonable after the lesson is concluded.
    • Feedback Characteristics—Content provides practice deciding whether feedback examples meet the characteristics related to content.
    • Feedback Characteristics—Usability provides practice deciding whether feedback examples meet the characteristics related to student usability.
    Images From Practice
    • Examples of Learning Intentions and Success Criteria. Although writing learning intentions and success criteria is dependent on many factors including curricula, unit goals, and data from previous lesson, many teachers have found reviewing examples helpful. This searchable database includes the examples used throughout this book as well as additional examples.
    • Images of Posted Learning Intentions and Success Criteria shows various ways teachers have posted learning intentions and success criteria for reference and use during a lesson.
    • Sample Unit Progressions we created collaboratively with teachers in our professional development program.
    • Modeling Reasoning and Approach Example Video portrays a teacher modeling how to share your reasoning and approach to a mathematics problem.
    External Resources
    • Books and Articles List is a bibliography of recommended sources for additional information, for yourself or others.
    Reference Resources

    These resources summarize key ideas about formative assessment practices presented in this book.

    Summary Cards
    • Chapter 1 Summary Card is an index-sized Summary Card that provides a summary of the critical and supporting aspects of formative assessment.
    • Chapter 2 Summary Cards provides index-sized Summary Cards for the following:
      • Characteristics of Learning Intentions and Success Criteria summarizes the characteristics described in this chapter.
      • Learning Intentions and Success Criteria Starter Statements includes a list of sentence starters based on using understand as the basis of the learning intention.
      • Lesson Purposes lists the lesson purposes described more fully in the “Overarching Purposes and Guidelines for Writing LI and SC.”
      • Framework for Using LI and SC With Students summarizes ways to help students understand and use learning intentions and success criteria during a lesson. The framework is described in detail in the classroom resource listed next.
      • Recommendations for Using Mathematics Learning Intentions and Success Criteria includes all the recommendations from Chapter 2.
    • Chapter 3 Summary Cards provides index-sized Summary Cards for the following:
      • Planning for Eliciting Evidence summarizes considerations when planning for eliciting evidence.
      • Process for Eliciting and Interpreting Evidence summarizes evidence related steps from planning for the lesson to enacting the lesson to determining next steps after the lesson.
      • Responsive Actions summarizes four responsive actions.
      • Recommendations for Gathering, Interpreting, and Acting on Evidence includes all the recommendations from Chapter 3.
    • Chapter 4 Summary Cards provides index-sized Summary Cards for the following:
      • Characteristics of Formative Feedback summarizes the characteristics described in this chapter.
      • Recommendations for Providing and Using Formative Feedback includes all the the recommendations from Chapter 4.
    • Chapter 5 Summary Card is an index-sized Summary Card that includes all the recommendations described in Chapter 5.
    • Chapter 6 Summary Cards provides index-sized Summary Cards for the following:
      • Building a Unit Progression summarizes the steps involved in building a unit progression.
      • Using a Unit Progression to Write LI and SC summarizes the steps described in the guideline document in Planning Resources.
      • Recommendations for Using Mathematics Learning Progressions includes all the recommendations from Chapter 6.
    • Chapter 7 Summary Cards provides index-sized Summary Cards for the following:
      • Considerations for Environment includes brief descriptions of each of the elements described in Chapter 7.
      • Recommendations for Establishing a Classroom Environment includes all the recommendations from Chapter 7.
    • Chapter 8 Summary Card is an index-sized Summary Card that includes all the recommendations from Chapter 8.
    Guiding Documents
    • Formative Assessment Recommendations is a printable PDF that includes all the recommendations from this book.
    • Implementation Principles for Formative Assessment provides a full list of the principles.
    • Implementation Indicators for Formative Assessment includes the full list of the principles and associated indicators (also in Appendix B).
    • Correlation of the Principles, Recommendations, and Resources provides the principles correlated with associated recommendations and resources.
    Formative Assessment Cycle
    • Formative Assessment Cycle is a color version of the full Formative Assessment Cycle that we build throughout this book.
    • Formative Assessment Cycle Video is an audiovisual tour of the Formative Assessment Cycle that we build throughout this book.
    Planning Resources

    These planning tools are resources to support your lesson planning when writing learning intentions and success criteria for your mathematics lessons.

    Unit-Level Planning
    • Building a Unit Progression includes a step-by-step process for building a unit progression.
    • Unit Progression Builder is an interactive web page that uses the process from the “Building a Unit Progression” resource to help you build a progression for any unit you are planning.
    Lesson-Level Planning
    • Guiding Questions for Formative Assessment includes guiding questions to help you plan for formative assessment practices.
    • Formative Assessment Planner Templates can be used for creating a formative assessment plan for a lesson.
    • Guidelines for Writing LI and SC includes a step-by-step process for writing learning intentions and success criteria after you have created a unit level progression.
    • Evaluating and Refining LI and SC provides questions designed to help you evaluate and refine your learning intentions and success criteria. These questions are also included in the “Guidelines for Writing LI and SC” resource.
    • Lesson Purposes, Planning Interactive provides a structure (similar to the “Lesson Purposes, Practice Interactive”) for selecting purposes for lessons within a unit.
    • Overarching Purposes and Guidelines for Writing LI and SC is a printable guide for considering purposes for lessons when writing learning intentions and success criteria. It can also be used as a reference when using the “Lesson Purposes, Planning Interactive.”
    • Using Exit Tickets as Evidence describes a way to sort exit tickets to inform your work for the next day.
    Classroom Resources

    These resources illustrate various classroom routines that you can use during instruction. Each routine provides a structure that you can use or adapt to routinize your practice around the use of learning intentions and success criteria.

    Strategies and Techniques
    • Framework for Using LI and SC With Students describes a way to help students understand and use learning intentions (LI) and success criteria (SC) during a lesson. It describes actions for both teacher and students to take at the start of the lesson, at midway points during the lesson, and at the end of the lesson.
    • Sharing Strategies includes several strategies for use when introducing students to LI and SC for the first time.
    • Revisiting Strategies includes several strategies for use when revisiting LI and SC throughout the lesson.
    • Wrapping-Up Strategies includes several strategies for use referring to LI and SC to wrap up a lesson.
    • Teacher Summary Cards for the Strategies is a set of large-sized cards that provides step-by-step summaries of teacher moves for the strategies described earlier.
    • Elicitation Strategies includes a description of elicitation techniques.
    • Feedback Techniques lists formative assessment classroom techniques (FACTS) related to feedback that can be found in the book Mathematics Formative Assessment: 75 Practical Strategies for Linking Assessment, Instruction, and Learning by Keeley and Tobey (2011).
    • Feedback Response Strategy describes a strategy that you can give your students to help them respond to feedback.
    • Self-Assessment Techniques lists FACTS for student self-assessment that can be found in the book Mathematics Formative Assessment: 75 Practical Strategies for Linking Assessment, Instruction, and Learning by Keeley and Tobey (2011).
    Lessons to Teach Formative Assessment Practices
    • Introducing Students to Learning Intentions and Success Criteria is an introductory lesson to help students understand the purpose of learning intentions and success criteria.
    • Introduction to Formative Feedback (for Students) is a lesson plan for introducing the characteristics of formative feedback to students.
    • Peer Feedback is a lesson plan for introducing and practicing how to give formative feedback to peers.
    Classroom Materials

    These resources can be copied or re-created to be used in your classroom.

    Student Summary Cards
    • Student Summary Cards for the Strategies is a set of large-sized cards that provides step-by-step summaries of student moves for the strategies described with the Classroom Resources. These cards can be distributed to each student or pairs of students for use when learning about a strategy.
    • Student Summary Card for Participating as a Listener is a large-sized card that provides a summary for students for the listening strategy in Chapter 7.
    Templates
    • Feedback Templates for using the feedback techniques described above.
    Posters
    • Self-Regulation Poster can be posted in your classroom to encourage students to think like a self-regulating learner.
    • Self-Assessment Poster can be posted in your classroom to provide prompts that encourage students to provide evidence from their work in their self-assessment.
    • Feedback Poster can be posted in your classroom for ongoing reference after introducing students to the characteristics of feedback through the “Introduction to Formative Feedback (for Students)” lesson. The file can be printed poster size though your local office supply or copy center.

    Appendix B: Implementation Indicators for Formative Assessment

    Learning Progressions

    Implementation Principle A: Be clear about what’s mathematically important or significant in your content (the concepts as well as the procedures) and how that content connects with other mathematics and grows both within a unit and across units.

    • I use learning progressions to help inform my thinking about the mathematics content I will teach.
    Learning Intentions and Success Criteria

    Implementation Principle B: Learning intentions help your students focus on what they should be learning, and success criteria help them gauge the extent to which they are learning it.

    • My learning intentions make clear what the central focus of learning is for the lesson.
    • My learning intentions are about conceptual understanding.
    • For each learning intention, I include success criteria that focus on students demonstrating their understanding through explanations, justifications, or other higher-order thinking activities, as well as those that focus on completing procedures correctly.
    • My success criteria focus on evidence that students can tangibly demonstrate.
    • My success criteria for a learning intention, taken collectively, provide enough evidence for my students and me to be confident whether the learning intention has been reached.
    • I write learning intentions and success criteria whose language and math content are appropriately accessible to students.
    • The lesson activities provide experiences that enable students to make progress toward the learning intention.

    Implementation Principle C: You and your students both need access to the learning intention and success criteria throughout the lesson.

    • I post the learning intentions and success criteria, or provide them on a handout, so that my students and I can refer to them throughout the lesson.
    Eliciting and Interpreting Evidence

    Implementation Principle D: Students need opportunities and encouragement to articulate the reasoning behind their work.

    • I provide opportunities for students to articulate the reasoning behind their work, as it relates to the success criteria, for correct as well as incorrect work.

    Implementation Principle E: Students need opportunities to provide evidence of each of the success criteria.

    • When I plan my lessons, I think about the kind of evidence I need, with regard to my learning intention and success criteria.
    • My lessons include opportunities for students to provide evidence for each of the success criteria.

    Implementation Principle F: You need variety in how, when, and from whom you gather evidence, to get a full picture of the class’s understanding, as a whole.

    • I collect evidence from either a few individuals, small groups, or the whole class, as appropriate at given points in a lesson
    • I collect evidence at key points throughout the lesson (not just at the end).
    • I collect evidence through student explanations, justifications, and so on, as well as through their completion of procedures and problems.

    Implementation Principle G: All gathering and interpretation of evidence needs to be in terms of the success criteria.

    • I focus on the success criteria when I assess the current status of students’ progress.
    • I choose and carry out a responsive action during instruction, based on evidence I have gathered during the lesson (e.g., use evidence to group students for a follow-up activity or to back up to provide more instruction).
    • I use evidence gathered from a lesson to plan a responsive action for the next day’s lesson.
    Formative Feedback

    Implementation Principle H: Formative feedback communicates to the student how he or she has met, as well as not yet met, the success criteria, and it provides hints, cues, or models for next steps.

    • I provide feedback that includes what has been met (including what in the work indicates this to me).
    • I provide feedback that includes what hasn’t been met (and why not, when appropriate).
    • I provide feedback that includes hints, cues, or models—not solutions or answers—to help students meet the success criteria.

    Implementation Principle I: Formative feedback should be focused and specific.

    • I focus my feedback on the skills and concepts described in the success criteria.
    • My feedback to students focuses on no more than one or two things at a time, so the students can digest it and act on it.

    Implementation Principle J: You need to prepare yourself to provide formative feedback and to plan opportunities for students to respond to it.

    • During my lesson planning, in order to help me plan for providing feedback, I articulate for myself what a student would say or do if he or she is meeting the success criteria.
    • I provide structures and time for students to act on feedback.
    • I follow-up with students to see if they acted on feedback (during the lesson or the next day).
    Student Ownership and Involvement

    Implementation Principle K: Students need to understand what the learning intention and success criteria mean.

    • I discuss learning intentions and success criteria and make efforts to clarify their meanings.
    • I share and discuss sample work against success criteria, as a way to build understanding of the success criteria and what it looks like to meet them.
    • I refer to learning intention or success criteria (verbally or by pointing to a displayed version) to refocus students or help them solidify their understanding.
    • I return to the learning intention and success criteria as part of lesson closure.

    Implementation Principle L: Students need to use the success criteria to guide their self-assessment and self-monitoring.

    • I provide opportunities for my students to self-assess according to the success criteria.
    • I provide structures and strategies designed to help students monitor their own learning.
    • My students refer to the success criteria to help focus and guide their work.
    • When giving feedback to a peer, my students refer to the learning intention or success criteria.

    Implementation Principle M: Students need to learn how to provide and use evidence of their mathematical reasoning and understanding in their self-assessment, and they need encouragement to do so.

    • My students supply evidence of their work, with a comparison to the success criteria, in their self-assessments.
    • My students ask questions that are focused on moving their learning forward (i.e., are more specific than “I don’t get it” and focus on a particular element of what they are learning).

    Implementation Principle N: Students need to understand what feedback is, why it’s helpful, and how they should use it.

    • I spend time in class teaching students what feedback is and how they will use it.
    • I build in opportunities to check that students understand the feedback they are receiving (for example, summarizing it in their own words or telling me what they think they need to do next).
    • My students use teacher and peer feedback to improve their work. They may do so with prompting from me but also sometimes without prompting from me.

    Implementation Principle O: Students need opportunities to provide and receive peer feedback, as a way to learn to provide feedback to themselves about their own work.

    • I provide opportunities for students to practice giving peer feedback.
    • I help students connect the process of peer feedback with the process of self-assessment.

    Implementation Principle P: Students understand a variety of ways to act on the results of their self-monitoring to take the next steps to move their learning forward.

    • My students access resources (including peers, print resources, and the teacher) to help them make progress toward meeting the success criteria. They may do so with prompting from me but also sometimes without prompting from me.
    Classroom Environment

    Implementation Principle Q: To support formative assessment practices, the social/cultural environment of the class promotes intellectual safety and curiosity.

    • I establish clear guidelines for my students about everyone communicating their ideas safely, and I use structures during the lesson to ensure this happens.
    • I provide appropriate think time after a question or comment before continuing.
    • I ask questions about students’ thinking, both for correct thinking as well as incorrect thinking.

    Implementation Principle R: To support formative assessment practices, the instructional environment optimizes learning and encourages and makes visible students’ thinking.

    • When I plan lessons, I think about framing my math activities in a way that invites my students to do some mathematical thinking and discussion.
    • I try to provide just enough support and information so that students can do the majority of the work themselves.

    Implementation Principle S: To support formative assessment practices, the physical environment allows easy access to various resources.

    • I have a predictable location for making available my learning intentions and success criteria, so students always know where to look.
    • I arrange my classroom to give students ready access to the formative assessment tools they need as well as each other.

    References

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