Uncovering Student Thinking about Mathematics in the Common Core, Grades 6-8: 25 Formative Assessment Probes

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Cheryl Rose Tobey & Carolyn B. Arline

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    Acknowledgements

    To my grandmother, Hazel Gangi, for inspiring in me the love of teaching and my former teacher, Helen Cogswell, for igniting my love of mathematics.

    —Cheryl

    To those who are faithful in using their passion for mathematics, love of learning, and hearts of gold to serve kids of all ages, and to my dear lifelong friend Darwin Ziegelmann who never fails to check in on me when I need it most.

    —Carolyn

    Preface: Mathematics Assessment Probes

    Overview

    Formative assessment informs instruction and supports learning through varying methods and strategies aimed at determining students' prior knowledge of a learning target and using that information to drive instruction that supports each student in moving toward understanding of the learning target. Questioning, observation, and student self-assessment are examples of instructional strategies educators can incorporate to gain insight into student understanding. These instructional strategies become formative assessments if the results are used to plan and implement learning activities designed specifically to address the specific needs of the students.

    This book focuses on using short sets of diagnostic questions, called Mathematics Assessment Probes (Probes). The Probes are designed to elicit prior understandings and commonly held misunderstandings and misconceptions. This elicitation allows the educator to make sound instructional choices, targeted at a specific mathematics concept and responsive to the specific needs of a particular group of students.

    Diagnostic assessment is as important to teaching as a physical exam is to prescribing an appropriate medical regimen. At the outset of any unit of study, certain students are likely to have already mastered some of the skills that the teacher is about to introduce, and others may already understand key concepts. Some students are likely to be deficient in prerequisite skills or harbor misconceptions. Armed with this diagnostic information, a teacher gains greater insight into what to teach. (McTighe & O'Connor, 2005)

    The Mathematics Assessment Probes provided here are tools that enable school teachers in Grades 6 through 8 to gather important insights in a practical way, and that provide immediate information for planning purposes.

    Audience

    The first collection of Mathematics Assessment Probes and the accompanying Teacher Notes were written for the busy classroom teacher eager for thoughtful, research-based diagnostic assessments focused on learning difficulties and aimed at enhancing the effectiveness of mathematics instruction. Since the publication of the first three Uncovering Student Thinking in Mathematics Resources (Rose & Arline, 2009; Rose, Minton, & Arline, 2007; Rose Tobey & Minton, 2011), we have continually received requests for additional Probes. Both teachers and education leaders have communicated the need for a collection of research-based Probes that focus on narrower grade spans. As well as additional Probes for each grade span, educators were eager for an alignment of the Probes to the Common Core Mathematics Standards (Council of Chief State School Officers [CCSSO], 2010). In response to these requests, we set to work writing, piloting, and field-testing a more extensive set of Probes for middle school teachers with a focus on targeting mathematics concepts within the new standards. This book is one in a series of Uncovering resources for the K–2, 3–5, 6–8, and 9–12 grade spans.

    Organization

    This book is organized to provide readers with the purpose, structure, and development of the Mathematics Assessment Probes as well as to support the use of applicable research and instructional strategies in mathematics classrooms.

    Chapter 1 provides in-depth information about the process and design of the Mathematics Assessment Probes, and develops an action-research structure we refer to as a QUEST Cycle. Chapters 2 through 6 contain the collection of Probes categorized by concept strands with accompanying Teacher Notes to provide the specific research and instructional strategies for addressing students' challenges with mathematics. Chapter 7 highlights instructional considerations and images from practice to illuminate how easily and in how many varied ways the Probes can be used in mathematics classrooms. This chapter also highlights how use of the Probes can support students' proficiency with the Common Core's Mathematical Practices.

    Acknowledgments

    We would like to thank the many mathematics educators who during attendance at various professional development sessions gave valuable feedback about features of the Probes, including structures, concepts to target, and purposes of use.

    We especially would like to acknowledge the contributions of the following educators who provided ideas and field-tested Probes, gave feedback on Teacher Notes, scheduled classroom visits, and/or opened their classrooms to us to try Probes or interview students: John Aresta, Lisa Beede, Joelle Drake, Michael Dyer, Judy Forrest, Tracey Harnett, Renee Henry, Tracy Kinney, Tom Light, Guy Meader, Judy Morgan, Sandra Roderick, Pamela Rodriquez, Joan Savage, Steven Shaw, Kristin Spangler, Jessica West, Ed Worcester, and the math teachers from Maine's RSU 2.

    A very special thanks to Nancy Philbrick and Ruth Wilson for their ongoing support in the piloting of the ideas from this book in their classrooms.

    We would like to thank our Corwin editor, Jessica Allan, for her continued support and flexibility, and Page Keeley, our science colleague, who designed the process for developing diagnostic assessment Probes and who tirelessly promotes the use of these assessments for formative assessment purposes, helping to disseminate our work in her travels.

    Mostly, we are grateful for the continued support, sacrifice, and patience shown by our families—Corey, Grandad, Carly, Jimmy, Bobby, Samantha, and Jack; and Liz, Kate, Adam, Sophie, and Gram—throughout the writing of this book.

    Publisher's Acknowledgments

    Corwin gratefully acknowledges the contributions of the following reviewers:

    Frederika Reisman Professor and Director, Drexel-Torrance Center for Creativity and Innovation Goodwin College, Drexel University Philadelphia, PA

    Debra A. Scarpelli Middle School Mathematics Teacher North Smithfield, RI

    About the Authors

    Cheryl Rose Tobey is a senior mathematics associate at Education Development Center (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 National Science Foundation (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 co–principal 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 co–principal 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 three prior books in the Uncovering Student Thinking series (2007, 2009, 2011), 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.

    Carolyn B. Arline is a secondary mathematics educator, currently teaching high school students in Maine. Carolyn also works as a teacher leader in the areas of mathematics professional development, learning communities, assessment, systematic school reform, standards-based teaching, learning and grading, student-centered classrooms, and technology. She has previously worked as a mathematics specialist at the Maine Mathematics and Science Alliance (MMSA) and continues her work with them as a consultant. Carolyn is a fellow of the second cohort group of the Governor's Academy for Science and Mathematics Educators and serves as a mentor teacher with the current cohort. She participated as a mathematics mentor in the NSF-funded Northern New England Co-Mentoring Network (NNECN) and continues her role as a mentor teacher. She serves as a board member of the Association of Teachers of Mathematics in Maine (ATOMIM) and on local curriculum committees. Carolyn received her BS in secondary mathematics education from the University of Maine.

  • Appendix A: Information on the Standards for Mathematical Practice

    The Standards for Mathematical Practice are not a checklist of teacher to-dos but rather support an environment in which the CCSS for mathematics content standards are enacted and are framed by specific expertise that you can use to help students develop their understanding and application of mathematics. (Larson, Lott Adams, Fennell, Dixon, & Kanold, 2012, p. 26)

    Formative assessment begins by identifying a learning goal, such as a grade level expectation from the CCSSM. The Common Core State Standards for Mathematics define what students should understand and be able to do in K-12 mathematics. Since the grade-level expectations in the CCSS define what students should “understand” or “be able to do,” it is important for teachers to find out what students know and can do both conceptually or procedurally in relation to the expectation for learning. In addition to these content standards, an important feature of the CCSSM is the Standards for Mathematical Practices. These practices describe a variety of processes, proficiencies, and dispositions that teachers at all grade levels should seek to develop in their students. Since the CCSS do not define the methods and strategies used to determine the readiness and prior knowledge necessary to achieve the standards, the Mathematics Assessment Probes in this book complement CCSS's eight Standards for Mathematical Practices and their link to mathematical content (adapted from Keeley & Rose Tobey, 2011, p. 30).

    Figure A.1 The Progression Project's Structure of the Mathematics Standards

    Source:McCallum (2011).

    Structuring the Mathematical Practice Standards

    The Institute for Mathematics and Education's Progression Project is organizing the writing of final versions of the progressions documents for the K–12 CCSS. The work is being done by members of the original team along with mathematicians and educators not involved in the initial writing of the standards (Institute for Mathematics and Education [IME], 2012). The Progression Project created the diagram in Figure A.1 to provide some higher order structure to the practice standards, in the way that the clusters and domains provide higher order structure to the content standards.

    The remaining part of this appendix will address each of the practice clusters using language from a variety of resources including the Common Core document, the Common Core Learning Progressions documents, and the Unpacking documents created by North Carolina's Department of Public Instruction. In Chapter 7, we describe how the Probes can be used in relationship to the ideas of each cluster.

    Reasoning and Explaining Practice Cluster (Practices 2 and 3)

    Each of the Probes includes a selected answer response and an explanation prompt. These explanation prompts are the key to the practices within this cluster.

    Mathematical Practice 2. Reason Abstractly and Quantitatively

    Students demonstrate proficiency with this practice when they make sense of quantities and relationships while solving tasks. This involves both decontexualizing and contextualizing. When decontextualizing, students need to translate a situation into a numeric or algebraic sentence that models the situation. They represent a wide variety of real world contexts through the use of real numbers and variables in mathematical expressions, equations, and inequalities. When contextualizing, students need to pull from a task information to determine the mathematics required to solve the problem. For example, after a line is fit through data, students interpret the data by interpreting the slope as a rate of change in the context of the problem (CCSSO, 2010; Common Core Standards Writing Team, 2011–2012; North Carolina Department of Public Instruction, 2012a, 2012b, 2012c). Students who reason abstractly and quantitatively are able to

    • move from context to abstraction and back to context;
    • make sense of quantities and their relationships in problem situations;
    • use quantitative reasoning that includes creating a coherent representation of the problem at hand;
    • consider the units involved;
    • attend to the meaning of quantities (not just how to compute with them);
    • know and flexibly use different properties of operations and objects; and
    • use abstract reasoning when measuring and comparing the lengths of objects.
    Mathematical Practice 3. Construct Viable Arguments and Critique the Reasoning of others

    Students demonstrate proficiency with this practice when they accurately use mathematical terms to construct arguments, engage in discussions about problem-solving strategies, examine a variety of problem-solving strategies, and begin to recognize the reasonableness of them, as well as similarities and differences among them. Middle school students should construct arguments using oral or written explanations accompanied by expressions, equations, inequalities, models; and graphs, tables, and other data displays including box plots, dot plots, and histograms (CCSSO, 2010; Common Core Standards Writing Team, 2011–2012; North Carolina Department of Public Instruction, 2012a, 2012b, 2012c).

    Students who construct viable arguments and critique the reasoning of others are able to

    • make conjectures and build a logical progression of statements to explore the truth of their conjectures;
    • recognize and use counterexamples;
    • justify their conclusions, communicate them to others, and respond to the arguments of others;
    • distinguish correct logic or reasoning from that which is flawed and, if there is a flaw in an argument, explain what it is;
    • construct arguments using concrete referents such as objects, drawings, diagrams, and actions; and
    • listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve arguments, including “How did you get that?” “Is that always true?” and “Why does that work?”
    Seeing Structure and Generalizing Practice Cluster (Practices 7 and 8)

    Adolescents make sense of their world by looking for patterns and structure and routines. They learn by integrating new information into cognitive structures they have already developed.

    Mathematical Practice 7. Look for and Make use of Structure

    Students demonstrate proficiency with this practice when they look for patterns and structures in the number system and other areas of mathematics such as modeling problems involving properties of operations. Students examine patterns in tables and graphs to generate equations and describe relationships. For instance, students recognize proportional relationships that exist in ratio tables, double numbers, graphs, and equations describing the multiplicative properties. Students apply properties to generate equivalent expressions (e.g., distributive property and properties of exponents) and solve equations. In geometry, students compose and decompose two- and three-dimensional figures to solve real-world problems involving area and volume and explore the effects of transformations and describe them in terms of congruence and similarity (CCSSO, 2010; Common Core Standards Writing Team, 2011–2012; North Carolina Department of Public Instruction, 2012a, 2012b, 2012c).

    Students who look for and make use of structure are able to

    • identify patterns in various displays of distributions of univariate and bivariate data;
    • describe the multiplicative properties of proportional relationships;
    • measure the attributes of three-dimensional shapes, allowing them to apply area formulas to solve surface area and volume problems; and
    • categorize shapes according to properties and characteristics.
    Mathematical Practice 8. Look for and Express Regularity in Repeated Reasoning

    Students demonstrate proficiency with this practice when they look for regularity in problem structures when problem solving, notice if calculations are repeated, and look for both general methods and shortcuts; use repeated reasoning to understand algorithms; and make generalizations about patterns. For example, they make connections between covariance, rates, and representations showing the relationships between quantities. They connect place value and prior work with operations to understand algorithms to fluently divide multi-digit numbers and perform all operations with multi-digit decimals. They use iterative processes to determine more-precise rational approximations for irrational numbers (CCSSO, 2010; Common Core Standards Writing Team, 2011–2012; North Carolina Department of Public Instruction, 2012a, 2012b, 2012c).

    Students who “look for and express regularity in repeated reasoning” are able to

    • notice if processes are repeated;
    • look for both general methods and shortcuts;
    • continually evaluate the reasonableness of their intermediate results; and
    • repeat the process of statistical reasoning in a variety of contexts.
    Modeling and Using Tools Practice Cluster (Practices 4 and 5)

    Students use multiple different tools (e.g., rulers, counters, base-ten blocks, calculators) in the primary mathematics classroom. How the tools are used depends on the mathematics topic of focus, and the same tool might be used in a variety of contexts. When given a problem, students need to be able to determine what tool would be appropriate, how the tool could be used in solving the problem, and how to communicate about their process. In the early grades, students often use tools to model a problem. It is also important for students to be able to communicate about the modeling process by representing the process using numbers and symbols.

    Mathematical Practice 4. Model with Mathematics

    Students demonstrate proficiency with this practice when they model real-life mathematical situations with a number or algebraic sentence or equation and check to make sure that their equation accurately matches the problem context. For example, students form expressions, equations, or inequalities from real-world contexts and connect symbolic and graphical representations. They begin to explore covariance and represent two quantities simultaneously. They use number lines to compare numbers and represent inequalities. They use measures of center and variability and data displays (e.g., box plots and histograms) to draw inferences about and make comparisons between data sets. Students use scatterplots to represent data and describe associations between variables. Students need many opportunities to connect and explain the connections between the different representations. They should be able to use all of these representations as appropriate to a problem context (CCSSO, 2010; Common Core Standards Writing Team, 2011–2012; North Carolina Department of Public Instruction, 2012a, 2012b, 2012c).

    Students who model with mathematics are able to

    • apply what they know to make approximations;
    • identify important quantities in a problem situation;
    • analyze relationships between quantities; and
    • reflect on whether the results make sense.
    Mathematical Practice 5. Use Appropriate Tools Strategically

    Students demonstrate proficiency with this practice when they access and use tools appropriately. Students consider available tools (including estimation and technology) when solving a mathematical problem and decide when certain tools might be helpful. For example, students might draw pictures, use applets, or write equations to show the relationships between the angles created by a transversal. They may decide to represent figures on the coordinate plane to calculate area. Number lines are used to understand division and to create dot plots, histograms, and box plots to visually compare the center and variability of the data. Additionally, students might use physical objects or applets to construct nets and calculate the surface area of three-dimensional figures (CCSSO, 2010; Common Core Standards Writing Team, 2011–2012; North Carolina Department of Public Instruction, 2012a, 2012b, 2012c).

    Students who use appropriate tools strategically are able to

    • consider available tools when solving a mathematical problem;
    • make sound decisions about when each of these tools might be helpful;
    • explain their choice of a particular tool for a given problem; and
    • detect possible errors by strategically using estimations.
    Overarching Habits of Mind of Productive Thinkers Practice Cluster (Practices 1 and 6)

    Productive disposition refers to the tendency to see sense in mathematics, to perceive it as both useful and worthwhile, to believe that steady effort in learning mathematics pays off, and to see oneself as an effective learner and doer of mathematics. Developing a productive disposition requires frequent opportunities to make sense of mathematics, to recognize the benefits of perseverance, and to experience the rewards of sense making in mathematics. (NRC, 2001, p. 131)

    Mathematical Practice 1. Make Sense and Persevere in Solving Problems

    Students demonstrate proficiency with this practice when they make sense of the meaning of the task and find an entry point or a way to start the task. Students solve real-world problems involving ratio, rate, area, and statistics through the application of algebraic and geometric concepts. Students use concrete manipulative, pictorial, and symbolic representations as well as mental mathematics. Students also are expected to persevere while solving tasks; that is, if students reach a point in which they are stuck, they can think about the task in a different way and continue working toward a solution. They may check their thinking by asking themselves, “What is the most efficient way to solve the problem?” “Does this make sense?” and “Can I solve the problem in a different way?” They develop visualization skills connected to their mathematical concepts as they recognize the existence of, and visualize, components of three-dimensional shapes that are not visible from a given viewpoint (CCSSO, 2010; Common Core Standards Writing Team, 2011–2012; North Carolina Department of Public Instruction, 2012a, 2012b, 2012c).

    Students who use appropriate tools strategically are able to

    • start by explaining to themselves the meaning of a problem and looking for entry points to its solution;
    • make conjectures about a solution;
    • plan a solution pathway rather than simply jumping into a solution attempt;
    • monitor and evaluate their progress and change course if necessary;
    • rely on using concrete objects or representations to help conceptualize and solve a problem;
    • check their answers to problems using a different method;
    • continually ask themselves, “Does this make sense?”; and
    • make sense of the problem-solving approaches of others, noticing similarities and differences among approaches.
    Mathematical Practice 6. Attend to Precision

    Students demonstrate proficiency with this practice when they are precise in their communication, calculations, and measurements. Students continue to refine their mathematical communication skills by using clear and precise language in their discussions with others and in their own reasoning. For example, students become more precise when attending to attributes, such as describing a triangle, square, or rectangle, as being closed figures with straight sides. They state precisely the meaning of variables they use when setting up equations, including specifying whether the variable refers to a specific number, or to all numbers in some range. During tasks involving number sense, students consider if their answer is reasonable and check their work to ensure the accuracy of solutions. When measuring or using measurement data, students attend to the unit (CCSSO, 2010; Common Core Standards Writing Team, 2011–2012; North Carolina Department of Public Instruction, 2012a, 2012b, 2012c).

    Students who attend to precision are able to

    • communicate precisely to others;
    • use clear definitions in discussion with others and in their own reasoning;
    • state the meaning of the symbols they choose, including using the equal sign consistently and appropriately; and
    • specify units of measure to clarify the correspondence with quantities in a problem.

    Learn more about how the Probes support teachers in assessing ideas related to the mathematical practices in Chapter 7.

    Appendix B: Developing Assessment Probes

    Developing an assessment Probe is different from creating appropriate questions for comprehensive diagnostic assessments and summative measures of understanding. The Probes in this book were developed using a process similar to that described in Mathematics Curriculum Topic Study: Bridging the Gap Between Standards and Practice (Keeley & Rose, 2006; Mundry, Keeley, Tobey, & Carroll, 2012). The process is summarized as follows:

    • Use national standards to examine concepts and specific ideas related to a topic. The national standards used to develop the Probes for this book are Common Core State Standards for Mathematics (CCSSO, 2010). The Common Core Standards for Mathematics (referred to as CCSSM) define what students should understand and be able to do in K-12 mathematics.
    • Within a CCSSM grade-level expectation, select the specific concepts or ideas you plan to address, and identify the relevant research findings. The sources for research findings include the Research Companion to Principles and Standards for School Mathematics (NCTM, 2003), Elementary and Middle School Mathematics: Teaching Developmentally (Van de Walle et al., 2013), articles from NCTM's Journal for Research in Mathematics Education, Second Handbook of Research on Mathematics Teaching and Learning (NCTM, 2007), and additional supplemental articles related to the topic.
    • Focus on a concept or a specific idea you plan to address with the Probe, and identify the related research findings. Keep the targeted concept small enough to assess with a few items because Probes are meant to be administered in a short amount of time. Rather than trying to target as much information about a topic as possible it is better to be more narrow and focused.
    • Choose the type of Probe format that lends itself to the situation (see more information on Probe format in Chapter 1's “What Is the Structure of a Probe?” beginning on page 13). Develop the stem (the prompt), key (correct response), and distractors (incorrect responses derived from research findings) that match the developmental level of your students.
    • Share your assessment Probe(s) with colleagues for constructive feedback, pilot with students, and modify as needed.

    Feedback on the assessment Probes developed for this resource was collected from 6–8 educators across multiple states, and the Probes were piloted with students across multiple grade levels. The feedback and student work were used to revise the Probes and to support the development of the accompanying Teacher Notes.

    Appendix C: Action Research Reflection Template

    QUEST Cycle
    Q uestions to Consider About the Key Mathematical Concepts

    What is the concept you wish to target? Is the concept at grade level or is it a prerequisite?

    Uncovering Student Understanding About the Key Concepts

    How will you collect information from students (e.g., paper and pencil, interview, student response system, etc.)? What form will you use (e.g., one-page Probe, card sort, etc.)? Are there adaptations you plan to make? Review the summary of typical student responses.

    Exploring Excerpts from Educational Resources and Related Research

    Review the quotes from research about common difficulties related to the Probe. What do you predict to be common understandings and/or misunderstandings for your students?

    Surveying the Prompts and Selected Responses in the Probe

    Sort by selected responses and then re-sort by trends in thinking. What common understandings/misunderstandings did the Probe elicit? How do these elicited understanding/misunderstandings compare to those listed in the Teacher Notes?

    Teaching Implications and Considerations

    Review the bulleted list and decide how you will take action. What actions did you take? How did you assess the impact of those actions? What are your next steps?

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