Mathematics Formative Assessment, Volume 2

50 More Practical Strategies for Linking Assessment, Instruction, and Learning

Page D. Keeley, Cheryl Rose Tobey

• Publisher: Corwin
• Year: 2016
• ISBN:
  9781506336640
• Edition: 1
• Topic: Éducation
• Subtopic: Enseignement des mathématiques

This one-of-a-kind resource helps you build a bridge between your students' initial ideas and correct mathematical thinking. Includes an annotated reference guide.

Chapter 1 An Introduction to 50 More Formative Assessment Classroom Techniques (FACTs)

Classroom Snapshot of Formative Assessment in Practice

Sixth-grade students are learning to multiply and divide fractions and mixed numbers. As an exit ticket several days into the unit, they individually complete a Slide Sort in which they are asked to use estimation to determine whether the quotient of two numbers is less than or more than 1. The teacher collects the exit tickets and looks at the students’ responses and explanations prior to planning. As she examines the students’ reasoning, she notices many are relying on rounding or calculating rather than reasoning about the size of the numbers. She realizes she needs to give them experiences with visual models using a series of problems that allow students to see patterns (i.e., dividing a larger fraction by a smaller fraction, dividing a smaller fraction by a larger fraction, dividing two fractions close in size). She creates the following Learning Intention and Success Indicators to focus students on what they will learn and how they will show they learned it.

• Learning Intention: Reasoning about the size of fractions is helpful in determining estimates for the sum of two fractions
• Success Indicator 1: I can use a strategy to determine whether a quotient is larger or smaller then a given benchmark.
• Success Indicator 2: I can explain the strategies I use to estimate quotients of fractions and mixed numbers.

She then plans a series of scaffolded activities that provide opportunities for students to develop a conceptual meaning of division and reason about how the sizes of the dividend and divisor affect the size of the quotient, including building from whole numbers, thinking about division as how many __ are in __? (e.g., How many

lbs are in 4 lbs?), and using models to find approximate and exact answers. Between activities, the teacher helps students Take Stock of their learning by revisiting the Success Indicators. During these discussions, she uses Talk Moves to probe further and assess students’ understanding of various strategies for estimating the size of the quotient, being sure to include questions for different sizes of dividends and divisors. For example, after investigating the results of dividing a smaller number by a larger number, a student claims that when dividing a smaller number by a larger number the quotient will always be less than 1. After asking students to restate and support the claim in pairs and having several pairs share with the full group, she moves on to the next activity in the lesson.

The teacher concludes the lesson by referring students to the Learning Intention that was posted at the start of the lesson: “Reasoning about the size of fractions is helpful in determining estimates for the sum of two fractions.” She uses the Thumbs Up, Down, and Sideways technique—a self-assessment for students to indicate the extent to which they feel they’ve met the three Success Indicators listed at the beginning of the lesson—as evidence of meeting the Learning Intention and then has the students do a quick write on Success Indicator 2 by asking students to explain at least two strategies for estimating the quotient of two fractions.

This brief classroom snapshot is an example of the inextricable link between formative assessment, good instruction, and learning. Formative assessment is frequently referred to as assessment for learning rather than assessment of learning, which is summative assessment. The preposition makes a difference as formative assessment’s primary purpose is to inform instructional decisions and simultaneously support learning through continuous feedback to the learner. However, a third preposition can also be added: assessment as learning. You can see from the snapshot provided that purposeful formative assessment classroom techniques (FACTs) can become learning opportunities.

The FACTs described in this snapshot are just a few of the ways teachers can utilize various strategies to elicit students’ ideas, monitor changes in their thinking, provide feedback, engage students in self-monitoring, and support reflection on learning. Throughout the process, the teacher is taking into account how well students are moving toward a learning target and what needs to be done to bridge the gap between where students are in their understanding and where they need to be. The 50 FACTs in this book, combined with the 75 FACTs in Volume 1 (Keeley & Tobey, 2011) will help you build an extensive repertoire of strategies that will inform instruction and promote learning—through a process called formative assessment. While you may be tempted to skip ahead and go directly to Chapter 3 to choose FACTs you can use in your classroom, you are encouraged to read the rest of this chapter and Chapter 2 so you can make effective use of the FACTs and strengthen your knowledge of formative assessment in mathematics instruction.

Why 50 More FACTs?

Formative assessment is a process that informs instruction and supports learning, with instructional decisions made by the teacher or learning decisions made by the students being at the heart of the process. Dylan Wiliam (2011) describes the central idea of formative assessment as follows: “Evidence about learning is used to adjust instruction to better meet students needs—in other words, teaching is adaptive to the learner’s needs” (p. 46). This overarching idea is broken down into five key strategies (Leahy, Lyon, Thompson, & Wiliam, 2005):

1. Learning Intentions and Success Criteria
2. Designing and facilitating productive classroom discussions, activities, and tasks that elicit evidence of learning
3. Providing feedback that moves learning forward
4. Activating learners as instructional resources for one another
5. Activating learners as the owners of their own learning

This book includes 50 new techniques that will help teachers and students utilize these five key strategies. In addition to the 75 FACTs published in the first volume of this series (Keeley & Tobey, 2011) and several of the FACTs in the science versions (Keeley, 2015, 2016) that are not repeated in the mathematics versions, teachers and teacher educators now have a total of 163 FACTs to embed throughout a cycle of instruction. Table 1.1 at the end of this chapter lists the combined collection of FACTs across all the current books in this series. A rich repertoire of FACTs helps learners interact with assessment in a variety of ways—writing, drawing, speaking, listening, questioning, investigating, modeling, and more. Furthermore, these FACTs provide mathematics-specific examples that are often lacking in general formative assessment resources.

Misunderstandings are likely to develop as a normal part of learning mathematics. These misunderstandings can be classified as conceptual misunderstandings, overgeneralizations, preconceptions, partial conceptions, and common errors. Misconceptions are a problem in mathematics for two reasons. First, when students use them to interpret or apply them to new mathematics experiences, misconceptions interfere with their learning. Second, because students have often actively constructed their misconceptions, they are emotionally and intellectually attached to them. Even when students recognize that a misconception affects their learning, they are reluctant to let go of it (Tobey & Arline, 2014a, 2014b). For this reason it is important for teachers to have an expansive repertoire of effective techniques, such as the ones provided in this book, for uncovering, monitoring, and providing feedback on student thinking.

Another feature of the 50 new FACTs included in this book that is important in navigating today’s mathematics education landscape is the connections to mathematics standards that include mathematics content, processes, and practices. Whether your state has its own mathematics standards or whether your state adopted the Common Core, the connection between the formative assessment classroom technique (FACT) and mathematics standards is included for each FACT.

The first volume of this book, Mathematics Formative Assessment: 75 Practical Strategies for Linking Assessment, Instruction, and Learning (Keeley & Tobey, 2011) includes important background information on formative assessment in mathematics. You are encouraged to obtain a copy of the first book as a companion to this volume to read and learn more about the following:

• Types of assessment and purposes for using formative assessment
• The research that supports formative assessment
• Classroom environments that support formative assessment
• The connection between teaching and learning
• Making the shift to a formative assessment-centered classroom
• Integrating assessment and instruction
• Metacognition
• The MAIL (Mathematics Assessment, Instruction, and Learning) Cycle
• Suggestions for selecting FACTs
• Suggestions for Implementing FACTs
• Using data from FACTs

Two purposes of formative assessment that are emphasized in this collection of 50 FACTs are elicitation of student thinking related to the learning goal and supporting productive math discourse, both of which relate directly to two of the National Council of Teachers of Mathematics’ (NCTM) eight mathematical practices (NCTM, 2014). Each of these purposes has special nuances in mathematics that often are not explicitly addressed in general formative assessment strategies. While there are other purposes for which the FACTs are used in this book, it is important to understand these two purposes, because these are central to assessment for learning in mathematics.

Elicitation FACTs

“Effective teaching involves finding the mathematics in students’ comments and actions, considering what students appear to know in light of the intended learning goals and progression, and determining how to give the best response and support to students on the basis of their current understandings” (NCTM, 2014, p. 56). Elicitation FACTs are techniques that can be used to bring this Principle to Action to life in the classroom by supporting teachers in eliciting ideas both prior to and during the instructional cycle. Many of the Elicitation FACTs in this book are designed to draw out students’ existing ideas, especially those that use faulty mathematics reasoning, so that responsive action can be taken. For example, many students struggle with the concept of fractions and decimals. Often the difficulty lies in overgeneralizations from their work with whole numbers. Students learning to compare fractions often treat the numerator and denominators as separate whole numbers, incorrectly identifying the fraction with the larger number in the denominator as the larger fraction (i.e., incorrectly reasoning that

is larger than

). When comparing decimals, students often overgeneralize the rule used when comparing whole numbers, “a number with more digits is larger” (i.e., incorrectly reasoning that 0.235 is larger than 0.43). This is why elicitation of preexisting ideas is an important part of mathematics formative assessment.

Examples of Elicitation FACTs in this book include but are not limited to Conjecture Cards, Ranking Tasks, and Slide Sort. Elicitation FACTs that target preexisting ideas are used at the beginning of a unit, cluster of lessons, or a single lesson, as well as mid-instruction, to provide an opportunity for students to surface their initial ideas and give the teacher a sense of students’ thinking prior to instruction. They are used to challenge students’ existing ideas or conceptual models, uncover common errors in terminology use or interpretation of representations, and expose faulty explanations of common or familiar mathematics concepts. As a formative assessment that informs instruction, it helps the teacher gauge initial student thinking, plan for enacting or modifying the lesson to follow, gauge progress midstream to determine a responsive action, or choose a new lesson that better addresses where students are in their mathematical understanding. Using a FACT for elicitation promotes learning by engaging students, stimulating further thinking, and setting the stage for the activities and/or discussion that will follow.

When an Elicitation FACT is selected, it should be designed so that every student can have an answer or opinion, regardless of whether they are correct. The intent of using an Elicitation FACT is that every student will have an opportunity to share their thinking either verbally or through writing. The data from Elicitation FACTs help the teacher set learning goals for activities designed to address students’ ideas, as well as eliminate activities that may not be necessary if students demonstrate conceptual understanding.

Early in the school year as you enact formative assessment, some students may feel uncomfortable sharing their initial ideas in a whole-class setting. This will eventually change as you work to establish a classroom culture that makes it safe to discuss and evaluate peer ideas. Your goal should be to eventually move students toward public sharing and critique of their thinking. In the meantime, you might consider using an anonymous elicitation strategy such as Fingers Under Chin or Extended Sticky Bars. One way you can anonymously share students’ thinking as they are writing their responses to a FACT is to say, “As I walked around the room, I noticed several of you wrote . . .”; or, after you have collected students’ written responses to a FACT and scanned through them, you might say, “I noticed several of you think . . .”; or you might list students’ ideas or solutions on a chart as an initial record of the class thinking without critique at this point. These anonymous techniques provide a way for students to see that not everyone has the right answer and that in a mathematics community, students often have alternative explanations and ways of thinking. The goal is to work toward a common, accepted understanding by considering the ideas of others and gaining new information that can be used to construct mathematical explanations. These Elicitation FACTs also show students that you, the teacher, value their ideas regardless of whether they are right or wrong. Eventually students transition toward publicly sharing their thinking because they experience a learning environment where it is safe and interesting to share different ideas and ways of thinking.

First and foremost, remember the goal of elicitation is to promote student thinking, mentally commit to an answer, and participate in discussions (either immediately after an elicitation question is posed and/or in follow-up discussions) that reveal students’ existing ideas. The biggest challenge for teachers right after students respond to an Elicitation FACT or during follow-up elicitation discussions, is to refrain from giving the answer and/or passing judgment on students’ thinking. Let students grapple with ideas while you guide them toward understanding and a consensus of the best ideas or solutions.

Provide an opportunity for students to talk in pairs or in small groups before facilitating a whole-class discussion following use of an Elicitation FACT. Circulate and listen as students discuss their ideas and defend their arguments. When pairs or small groups are talking, the teacher’s role should be as a facilitator who draws out the students’ ideas without i...