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R E PRO DUCI B LE

Common Core Mathematics in a PLC at WorkTM, Leader’s Guide © 2012 Solution Tree Press • solution-tree.comVisit go.solution-tree.com/commoncore to download this page.

Figure 1.5: High-Leverage Unit-By-Unit Actions of

Mathematics Collaborative Teams

Teaching and Learning1. The team designs and implements agreed-on prior knowledge skills to be assessed and taught during

each lesson of the unit. The collaborative teacher team reaches agreement for teaching and learning in the lessons and unit.

2. The team designs and implements agreed-on lesson-design elements that ensure active student engagement with the mathematics. Students experience some aspect of the CCSS Mathematical Practices, such as Construct viable arguments and critique the reasoning of others or Attend to precision, within the daily lessons of every unit or chapter.

3. The team designs and implements agreed-on lesson-design elements that allow for student-led summaries and demonstrations of learning the daily lesson.

4. The team designs and implements agreed-on lesson-design elements that include the strategic use of tools—including technology—for developing student understanding.

Assessment Instruments and Tools1. The team designs and implements agreed-on common assessment instruments based on high-

quality exam designs. The collaborative team designs all unit exams, unit quizzes, final exams, writing assignments, and projects for the course.

2. The team designs and implements agreed-on common assessment instrument scoring rubrics for each assessment in advance of the exam.

3. The team designs and implements agreed-on common scoring and grading feedback (level of specificity to the feedback) of the assessment instruments to students.

Formative Assessment Feedback1. The team designs and implements agreed-on adjustments to instruction and intentional student

support based on both the results of daily formative classroom assessments and the results of student performance on unit or chapter assessment instruments such as quizzes and tests.

2. The team designs and implements agreed-on levels of rigor for daily in-class prompts and common high-cognitive-demand tasks used to assess student understanding of various mathematical concepts and skills. This also applies to variance in rigor and task selection for homework assignments and expectations for make-up work. This applies to depth, quality, and timeliness of teacher descriptive formative feedback on all student work.

3. The team designs and implements agreed-on methods to teach students to self-assess and set goals. Self-assessment includes students using teacher feedback, feedback from other students, or their own self-assessments to identify what they need to work on and to set goals for future learning.

R E PRO DUCI B LE

Common Core Mathematics in a PLC at WorkTM, Leader’s Guide © 2012 Solution Tree Press • solution-tree.comVisit go.solution-tree.com/commoncore to download this page.

Table 1.1: Seven Stages of Teacher Collaboration

Stage Questions That Define This Stage

Stage one: Filling the time What exactly are we supposed to do as a team?

Stage two: Sharing personal practice What is everyone doing in his or her classroom for instruction, lesson planning, and assessment?

Stage three: Planning, planning, planning What should we be teaching during this unit, and how do we lighten the load for each other?

Stage four: Developing common assessments How will we know if students learned the standards? What does mastery look like for the standards in this unit?

Stage five: Analyzing student learning Are students learning what they are supposed to be learning? Do we agree on student evidence of learning?

Stage six: Adapting instruction to student needs How can we adjust instruction to help those students struggling and those exceeding expectations?

Stage seven: Reflecting on instruction Which lesson-design practices are most effective with our students?

Source: Graham & Ferriter, 2008.

The powerful collaboration that characterizes

professional learning communities is a systematic

process in which teachers work together to analyze

and improve their classroom practice. Teachers

work in teams, engaging in an ongoing cycle of

questions that promote deep team learning. This

process, in turn, leads to higher levels of student

achievement.

The Critical Issues for Team Consideration guide

the collective inquiry and action research of each

collaborative team in a professional learning com-

munity. This plan book explores these issues in

greater detail at strategic intervals. You and your

teammates will be challenged to “build shared

knowledge”—to learn together—about each issue

and ultimately generate a product as a result of

your collective inquiry and action research.

10 Professional Learning Communities at Work Plan Book © 2006 Solution Tree • www.solution-tree.com

REPRODUCIBLE

1. ___ We have identified team norms and proto-cols to guide us in working together.

2. ___ We have analyzed student achievement dataand have established SMART goals that weare working interdependently to achieve.

3. ___ Each member of our team is clear on theessential learnings of our course in general aswell as the essential learnings of each unit.

4. ___ We have aligned the essential learnings withstate and district standards and the high-stakes exams required of our students.

5. ___ We have identified course content and/ortopics that can be eliminated so we candevote more time to essential curriculum.

6. ___ We have agreed on how to best sequencethe content of the course and have estab-lished pacing guides to help students achievethe intended essential learnings.

7. ___ We have identified the prerequisite knowl-edge and skills students need in order tomaster the essential learnings of our coursesand each unit of these courses.

8. ___ We have identified strategies and createdinstruments to assess whether students havethe prerequisite knowledge and skills.

9. ___ We have developed strategies and systemsto assist students in acquiring prerequisiteknowledge and skills when they are lackingin those areas.

10. ___ We have developed frequent common form-ative assessments that help us to determineeach student’s mastery of essential learnings.

11. ___ We have established the proficiency stan-dard we want each student to achieve oneach skill and concept examined with ourcommon assessments.

12. ___ We have developed common summativeassessments that help us assess the strengthsand weaknesses of our program.

13. ___ We have established the proficiency stan-dard we want each student to achieve oneach skill and concept examined with oursummative assessments.

14. ___ We have agreed on the criteria we will use injudging the quality of student work relatedto the essential learnings of our course, andwe practice applying those criteria to ensureconsistency.

15. ___ We have taught students the criteria we willuse in judging the quality of their work andhave provided them with examples.

16. ___ We evaluate our adherence to and the effec-tiveness of our team norms at least twiceeach year.

17. ___ We use the results of our common assessmentsto assist each other in building on strengthsand addressing weaknesses as part of aprocess of continuous improvement designedto help students achieve at higher levels.

18. ___ We use the results of our common assess-ments to identify students who need addi-tional time and support to master essentiallearnings, and we work within the systemsand processes of the school to ensure theyreceive that support.

1 2 3 4 5 6 7 8 9 10

Not True of Our Team Our Team Is Addressing True of Our Team

Critical Issues for Team Consideration

Team Name:__________________________________________________________________________________________

Team Members:__________________________________________________________________________________

Use the scale below to indicate the extent to which each of the following statements is true

of your team.

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Comparing Two Mathematical Tasks1 Martha’s Carpeting Task Martha was re-carpeting her bedroom which was 15 feet long and 10 feet wide. How many square feet of carpeting will she need to purchase? The Fencing Task Ms. Brown’s class will raise rabbits for their spring science fair. They have 24 feet of fencing with which to build a rectangular rabbit pen in which to keep the rabbits.

If Ms. Brown's students want their rabbits to have as much room as possible, how long would each of the sides of the pen be?

How long would each of the sides of the pen be if they had only 16 feet of fencing?

How would you go about determining the pen with the most room for any amount of fencing? Organize your work so that someone else who reads it will understand it.

1 Source: Adapted from Smith, Stein, Arbaugh, Brown, and Mossgrove, Characterizing the Cognitive Demand of Mathematical Tasks: A Task Sorting Activity, Professional Development Guidebook for Perspectives on the Teaching of Mathematics, NCTM, 2004.

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Analyzing Mathematical Tasks1

Categorize each of the elementary school tasks below as High Level Cognitive Demand or Low Level Cognitive Demand. Then, develop a list of criteria that describe the tasks in each category. Task A Manipulatives or Tools Available: One triangle pattern block Using the edge of a triangle pattern block as the unit of measure, determine the perimeter of the following pattern-block trains.

Task B Manipulatives or Tools Available: Grid paper, interlocking cubes 1. The first grade class is going to watch a play in the cafeteria in the morning. There are 25 students.

In what different ways could the chairs for them be arranged so that all the rows are equal?

2. The two third grade classes are going to watch the play in the cafeteria in the afternoon. There are 48 students. In what different ways could the chairs for them be arranged so that all the rows are equal?

3. What do you notice about your solutions for problem 1 and problem 2

Task C Manipulatives or Tools Available: None

About how big is 4/5 of this rectangle? Show your answer by shading in the rectangle.

What other fractions are near 4/5 in size?

1 Adapted from Smith, Stein, Arbaugh, Brown, and Mossgrove, Characterizing the Cognitive Demand of Mathematical Tasks: A Task Sorting Activity, Professional Development Guidebook for Perspectives on the Teaching of Mathematics, NCTM, 2004.

Task D Manipulatives or Tools Available: None Use the graph to answer the questions. 1. How many students picked red as

their favorite color? 2. Did more students like blue or

purple?

3. Which color did only one student choose as a favorite color?

Task E Manipulatives or Tools Available: None Identify the place value for each of the underlined digits. 1. 351 2. 76 3. 4,789 4. 1.2 Task F Manipulatives or Tools Available: Grid paper, interlocking cubes Solve the two sets of problems shown below. You can use interlocking cubes or grid paper to make arrays to help you solve these problems. Try to solve the last problem in each set by thinking about the other problems in the set. You can add problems to the set that help you solve the final problem better. After you solve the problems, write about how you solved the last problem in each set.

10 x 6 3 x 6 6 x 6 13 x 6

4 x 6 4 x 10 4 x 12 4 x 30 4 x 36

Task G Manipulatives or Tools Available: None Do the brownies below have ½ of the brownie shaded? Explain your thinking.

Adapted from K. Chval, J. Lannin & D. Jones (2013) Putting Essential Understanding of Fractions to Practice, 3-5. NCTM: Reston, VA Task H Manipulatives or Tools Available: Color tiles (a) The shaded region below is ¼ of the whole. What does one whole look like?

(b) The shaded region below is 2/3 of the whole. What does one whole look like?

(c) The shaded region below is 3/2 of the whole. What does one whole look like?

Adapted from K. Chval, J. Lannin & D. Jones (2013) Putting Essential Understanding of Fractions to Practice, 3-5. NCTM: Reston, VA

Task I (Grades K-2) Manipulatives or Tools Available: Interlocking cubes (optional)

1. How many legs are on the people in our classroom? Show how you figured it out. 2. How could you tell someone how to figure out the number of legs for any number of people? Can

you think of a rule? For example, how would you finish this sentence: “To find the number of legs in a group of people, you . . . “

3. What if there were 26 legs? How many people would there be?

Task J Manipulatives or Tools Available: base-ten blocks, tens and ones mat

Write the tens and ones. Circle the number that is greater. The first one is started for you.

1. 56 __5_ tens _6__ones

61 __6_ tens _1__ones

2. 74 ____ tens ____ones

89 ____ tens ____ones

3. 31 ____ tens ____ones

19 ____ tens ____ones

Task K (Grades K, 1) Manipulatives or Tools Available: red and blue crayons or cubes; colored pencils; crayons or markers

I want to put 8 crayons in my box. I want to have a mix of red and blue crayons. How many of each color could I have? How many blues? How many reds? Find as many different ways as you can.

Task L (Grades 1-2) Manipulatives or Tools Available: Counters, interlocking cubes

Write and solve a number story for 26 8.

Show how you solved the problem using words, counters, tallies, or pictures.

Classification of Tasks

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Leaves and Caterpillars Vignette1

Students in David Crane’s fourth-grade class were solving the following problem: “A fourth-grade class needs five leaves each day to feed its 2 caterpillars. How many leaves would they need each day for 12 caterpillars?” Mr. Crane told his students that they could solve the problem any way they wanted, but emphasized that they needed to be able to explain how they got their answer and why it worked. As students worked in pairs to solve the problem, Mr. Crane walked around the room making sure that students were on task and making progress on the problem. He was pleased to see that students were using lots of different approaches to the problem – making tables, drawing pictures, and, in some cases, writing explanations. He noticed that two pairs of students had gotten wrong answers as shown below. Darnell and Marcus Missy and Kate

Mr. Crane wasn’t too concerned about the incorrect responses, however, since he felt that once these students saw several correct solution strategies presented, they would see what they did wrong and have new strategies for solving similar problems in the future. When most students were finished, Mr. Crane called the class together to discuss the problem. He began the discussion by asking for a volunteer to share their solution and strategy, being careful to avoid calling on the students with incorrect solutions. Over the course of the next 15 minutes, first Kyra, then Jason, Jamal, Melissa, Martin and Janine volunteered to present the solutions to the task that they and their partner had created. Their solutions are shown on the back. During each presentation, Mr. Crane made sure to ask each presenter questions that helped them to clarify and justify their work. He concluded the class by telling students that the problem could be solved in many different ways and now, when they solved a problem like this, they could pick the way they liked best because they all gave the same answer. _______________________________________________ 1From: Smith, M.S., & Stein, M.K. (2011) 5 practices for orchestrating productive mathematics discussions. Reston, VA: National Council of Teachers of Mathematics.

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What Principals Need to Know About Teaching and Learning Mathematics © 2012 Solution Tree Press • solution-tree.comVisit go.solution-tree.com/leadership to download this page.

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TRU Math: Teaching for Robust Understanding in Mathematics Scoring Rubric Release Version Alpha | December 20, 2013

This work is a product of The Algebra Study (NSF Grant DRL--‐0909815 to PIs Alan Schoenfeld, U.C. Berkeley, and Robert Floden, Michigan State University), and of The Mathematics Assessment Project (Bill and Melinda Gates Foundation Grant OPP53342 to PIs Alan Schoenfeld, U. C Berkeley, and Hugh Burkhardt and Malcolm Swan, The University of Nottingham).

TRU Math addresses five general dimensions of mathematical classroom activity, and one dimension that is algebra-specific (which is not included here). Each of these dimensions is coded separately during whole class discussions, small group work, student presentations, and individual student work.

Level Mathematical Focus,

Coherence and Accuracy

Cognitive Demand Access Agency: Authority and

Accountability Uses of Assessment

1

Classroom activities are purely rote, OR disconnected or unfocused, OR consequential mistakes are left unaddressed.

Classroom activities are structured so that students mostly apply familiar procedures or memorized facts.

Classroom management is problematic to the point where the lesson is disrupted, OR a significant number of students appear disengaged and there are no overt mechanisms to support engagement.

The teacher initiates conversations. Students’ speech turns are short (one sentence or less) and shaped by what the teacher says or does.

The teacher may note student answers or work, but student reasoning is not surfaced or pursued. Teacher actions are limited to corrective feedback or encouragement.

2

The mathematics discussed is relatively clear and correct, BUT connections between procedures, concepts and contexts (where appropriate) are either cursory or lacking.

Classroom activities offer possibilities of conceptual richness or problem solving challenge, but teaching interactions tend to “scaffold away” the challenges and mostly limit students to providing short responses to teacher prompts.

The class is engaged in mathematical activity, but there is uneven participation and the teacher does not provide structured support for many students to participate in meaningful ways.

Students have a chance to say or explain things, but “the student proposes, the teacher disposes”; in class discussions, student ideas are not explored or built upon.

The teacher refers to student thinking, perhaps even to common mistakes, but specific student ideas are not build on (when potentially valuable) or used to address challenges (when problematic).

3

The mathematics discussed is relatively clear and correct, AND connections between procedures, concepts and contexts (where appropriate) are addressed and explained

The teacher’s hints or scaffolds support students in “productive struggle” in building understandings and engaging in mathematical practices.

The teacher actively supports (and to some degree achieves) broad and meaningful participation, OR what appear to be established participation structures result in such participation.

Students put forth and defend their ideas. The teacher may ascribe ownership for students’ ideas in exposition, AND/OR students respond to and build on each other’s ideas.

The teacher solicits student thinking and subsequent instruction responds to those ideas, by building on productive beginnings or addressing emerging misunderstandings.

Detailed instruction regarding the use of the rubric are provided in The TRU Math Scoring Guide available at <http://ats.berkeley.edu/tools.html>

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Copyright © 2007 by The National Council of Teachers of Mathematics, 1906 Association Drive, Reston, VA 20191-1502, Tel: (703) 620-9840, Fax: (703) 476-2690, www.nctm.org.

Assessment

Research Brief

Five “Key Strategies” for Eff ective Formative Assessment

IN ORDER to build a comprehensive framework for forma-tive assessment, Wiliam and Thompson (2007) proposed that three processes were central:

1. Establishing where learners are in their learning

2. Establishing where they are going

3. Establishing how to get there

By considering separately the roles of the teacher and the students themselves, they proposed that formative assess-ment could be built up from fi ve “key strategies.”

1. Clarifying, sharing, and understanding goals for learning and criteria for success with learners

There are a number of ways teachers can begin the process of clarifying and sharing learning goals and success criteria. Many teachers specify the learning goals for the lesson at the beginning of the lesson, but in doing so, many teachers fail to distinguish between the learning goals and the activities that will lead to the required learning. When teachers start from what it is they want students to know and design their instruc-tion backward from that goal, then instruction is far more likely to be effective (Wiggins and McTighe 2000).

Wiggins and McTighe also advocate a two-stage process of fi rst clarifying the learning goals themselves (what is wor-thy and requiring understanding?), which is then followed by establishing success criteria (what would count as evidence of understanding?). Only then should the teacher move on to ex-ploring activities that will lead to the required understanding.

However, it is important that students also come to under-stand these goals and success criteria, as Royce Sadler (1989, p. 121) notes:

The indispensable conditions for improvement are that the stu-dent comes to hold a concept of quality roughly similar to that held by the teacher, is continuously able to monitor the qual-ity of what is being produced during the act of production it-self, and has a repertoire of alternative moves or strategies from which to draw at any given point.

Indeed, there is evidence that discrepancies in beliefs about what it is that counts as learning in mathematics class-rooms may be a signifi cant factor in the achievement gaps ob-

served in mathematics classrooms. In a study of 72 students between the ages of seven and thirteen, Gray and Tall (1994) found that the reasoning of the higher-achieving students was qualitatively different from that of the lower-achieving stu-dents. In particular, the higher-achieving students were able to work successfully despite unresolved ambiguities about whether mathematical entities were concepts or procedures. Lower-achieving students were unable to accept such ambi-guities and could not work past them. By refusing to accept the ambiguities inherent in mathematics, the lower-achieving students were, in fact, attempting a far more diffi cult form of mathematics, with a far greater cognitive demand.

A simple example may be illustrative here. When we write 6 1

2, the mathematical operation between the 6 and the 1

2 is ac-tually addition, but when we write 6x, the implied operation between the 6 and the x is multiplication, and the relationship between the 6 and the 1 in 61 is different again. And yet, very few people who are successful in mathematics are aware of these inconsistencies or differences in mathematical notation. In a very real sense, being successful in mathematics requires knowing what to worry about and what not to worry about. Students who do not understand what is important and what is not important will be at a very real disadvantage.

In a study of twelve seventh-grade science classrooms, White and Frederiksen (1998) found that giving students time to talk about what would count as quality work, and how their work was likely to be evaluated, reduced the achievement gap between the highest- and lowest-achieving students in half and increased the average performance of the classes to such an extent that the weakest students in the experimental group were outperforming all but the very strongest students in the control group.

This is why using a variety of examples of students’ work from other classes can be extremely powerful in helping stu-dents come to understand what counts as quality work. Many teachers have found that students are better at spotting errors in the work of other students than they are at seeing them in their own work. By giving students examples of work at different standards, students can begin to explore the differ-ences between superior and inferior work, and these emer-gent understandings can be discussed with the whole class.

1

As a result of such processes, students will develop a “nose for quality” (Claxton 1995) that they will then be able to use in monitoring the quality of their own work.

2. Engineering eff ective classroom discussions, questions, activities, and tasks that elicit evidence of students’ learning

Once we know what it is that we want our students to learn, then it is important to collect the right sort of evidence about the extent of students’ progress toward these goals, but few teachers plan the kinds of tasks, activities, and questions that they use with their students specifi cally to elicit the right kind of evidence of students’ learning. As an example, con-sider the question shown in fi gure 1 below.

Fig. 1. Diagnostic item on elementary fractions

Diagram A is the obvious answer, but B is also correct. However, some students do not believe that one-quarter of B is shaded because of a belief that the shaded parts have to be adjoining. Students who believe that one-quarter of C is shaded have not understood that one region shaded out of four is not necessarily a quarter. Diagram D is perhaps the most interesting here. One-quarter of this diagram is shaded, although the pieces are not all equal; students who rely too literally on the “equal areas” defi nition of fractions will say that D is not a correct response. By crafting questions that ex-plicitly build in the undergeneralizations and overgeneraliza-tions that students are known to make (Bransford, Brown, and Cocking 2000), we can get far more useful information about what to do next. Furthermore, by equipping each student in the class with a set of four cards bearing the letters A, B, C, and D and by requiring all students to respond simultaneous-ly with their answers, the teacher can generate a very solid evidence base for deciding whether the class is ready to move on (Leahy et al. 2005). If every student responds with A, B, and D, then the teacher can move on with confi dence that the students have understood. If everyone simply responds with A, then the teacher may choose to reteach some part of the topic. The most likely response, however, is for some students to respond correctly and for others to respond incorrectly, or incompletely. This provides the teacher with an opportunity

to conduct a classroom discussion in which students with dif-ferent views can be asked to justify their selections.

Of course planning such questions takes time, but by in-vesting the time before the lesson, the teacher is able to ad-dress students’ confusion during the lesson, with the students still in front of him or her. Teachers who do not plan such questions are forced to put children’s thinking back on track through grading, thus dealing with the students one at a time, after they have gone away.

3. Providing feedback that moves learning forwardThe research on feedback shows that much of the feed-

back that students receive has, at best, no impact on learn-ing and can actually be counterproductive. Kluger and DeNi-si (1996) reviewed more than three thousand research reports on the effects of feedback in schools, colleges, and workplaces and found that only 131 studies were scientifi cally rigorous. In 50 of these studies, feedback actually made people’s per-formance worse than it would have been without feedback. The principal feature of these studies was that feedback was, in the psychological jargon, “ego-involving.” In other words, the feedback focused attention on the person rather than on the quality of the work——for example, by giving scores, grades, or other forms of report that encouraged comparison with others. The studies where feedback was most effective were those in which the feedback told participants not just what to do to improve but also how to go about it.

Given the emphasis on grading in U.S. schools, teach-ers may be tempted to offer comments alongside scores or grades. However, a number of studies (e.g., Butler 1987, 1988) have shown that when comments are accompanied by grades or scores, students focus fi rst on their own grade or score and then on those of their neighbors, so that grades with comments are no more effective than grades alone, and much less effective than comments alone. The crucial requirement of feedback is that it should force the student to engage cog-nitively in the work.

Such feedback could be given orally, as in this example from Saphier (2005, p. 92):

Teacher: What part don’t you understand?

Student: I just don’t get it.

Teacher: Well, the fi rst part is just like the last problem you did. Then we add one more variable. See if you can fi nd out what it is, and I’ll come back in a few minutes.

Written feedback can support students in fi nding errors for themselves:

• There are 5 answers here that are incorrect. Find them and fi x them.

Five “Key Strategies”for Effective Formative Assessment

2

A B C D

In which of the following diagrams is one quarter of the area shaded?

Copyright © 2007 by The National Council of Teachers of Mathematics, 1906 Association Drive, Reston, VA 20191-1502, Tel: (703) 620-9840, Fax: (703) 476-2690, www.nctm.org.2

• The answer to this question is … Can you fi nd a way to work it out?

It can also identify where students might use and extend their existing knowledge:

• You’ve used substitution to solve all these simulta-neous equations. Can you use elimination?

Other approaches (Hodgen and Wiliam 2006) include en-couraging pupils to refl ect:

• You used two different methods to solve these problems. What are the advantages and disadvan-tages of each?

• You have understood … well. Can you make up your own more diffi cult problems?

Another suggestion is to have students discuss their ideas with others:

• You seem to be confusing sine and cosine. Talk to Katie about how to work out the difference.

• Compare your work with Ali and write some ad-vice to another student tackling this topic for the fi rst time.

The important point in all this is that as well as “putting the ball back in the students’ court,” the teacher also needs to set aside time for students to read, respond to, and act on feedback.

4. Activating students as owners of their own learning

When teachers are told they are responsible for making sure that their students do well, the quality of their teach-ing deteriorates, as does their students’ learning (Deci et al. 1982). In contrast, when students take an active part in mon-itoring and regulating their learning, then the rate of their learning is dramatically increased. Indeed, it is common to fi nd studies in which the rate of students’ learning is doubled, so that students learn in six months what students in control groups take a year to learn (Fontana and Fernandes 1994; Me-varech and Kramarski 1997).

In an attempt to integrate research on motivation, meta-cognition, self-esteem, self-effi cacy, and attribution theory, Monique Boekaerts has proposed a dual-processing theory of student motivation and engagement (Boekaerts 2006). When presented with a task, the student evaluates the task accord-ing to its interest, diffi culty, cost of engagement, and so on. If the evaluation is positive, the student is likely to seek to increase competence by engaging in the task. If the evalua-tion is negative, a range of possible outcomes is possible. The

student may engage in the task but focus on getting a good grade from the teacher instead of mastering the relevant ma-terial (e.g., by cheating) or the student may disengage from the task on the grounds that “it is better to be thought lazy than dumb.” The important point for teachers is that to maxi-mize learning, the focus needs to be on personal growth rath-er than on a comparison with others.

Practical techniques for getting students started include “traffi c lights,” where students fl ash green, yellow, or red cards to indicate their level of understanding of a concept. Many teachers have reported that initially, students who are focusing on well-being, rather than growth, display green, in-dicating full understanding, even though they know they are confused. However, when the teacher asks students who have shown green cards to explain concepts to those who have shown yellow or red, students have a strong incentive to be honest!

5. Activating students as learning resources for one another

Slavin, Hurley, and Chamberlain (2003) have shown that activating students as learning resources for one another pro-duces some of the largest gains seen in any educational inter-ventions, provided two conditions are met. The fi rst is that the learning environment must provide for group goals, so that students are working as a group instead of just working in a group. The second condition is individual accountability, so that each student is responsible for his or her contribution to the group, so there can be no “passengers.”

With regard to assessment, then, a crucial feature is that the assessment encourages collaboration among students while they are learning. To achieve this collaboration, the learning goals and success criteria must be accessible to the students (see above), and the teacher must support the students as they learn how to help one another improve their work. One par-ticularly successful format for doing this has been the idea of “two stars and a wish.” The idea is that when students are commenting on the work of one another, they do not give evaluative feedback but instead have to identify two positive features of the work (two “stars”) and one feature that they believe merits further attention (the “wish”). Teachers who have used this technique with students as young as fi ve years old have been astonished to see how appropriate the com-ments are, and because the feedback comes from a peer rath-er than someone in authority over them, the recipient of the feedback appears to be more able to accept the feedback (in other words, they focus on growth rather than on preserving their well-being). In fact, teachers have told us that the feed-back that students give to one another, although accurate, is far more hard-hitting and direct than they themselves would

Five “Key Strategies”for Effective Formative Assessment

3Copyright © 2007 by The National Council of Teachers of Mathematics, 1906 Association Drive, Reston, VA 20191-1502,

Tel: (703) 620-9840, Fax: (703) 476-2690, www.nctm.org. 3

have given. Furthermore, the research shows that the person providing the feedback benefi ts just as much as the recipi-ent because she or he is forced to internalize the learning in-tentions and success criteria in the context of someone else’s work, which is less emotionally charged than doing it in the context of one’s own work.

ConclusionThe available research evidence suggests that consider-

able enhancements in student achievement are possible when teachers use assessment, minute-by-minute and day-by-day, to adjust their instruction to meet their students’ learning needs. However, it is also clear that making such changes is much more than just adding a few routines to one’s normal practice. It involves a change of focus from what the teacher is putting into the process and to what the learner is getting out of it, and the radical nature of the changes means that the support of colleagues is essential. Nevertheless, our experi-ences to date suggest that the investment of effort in these changes is amply rewarded. Students are more engaged in class, achieve higher standards, and teachers fi nd their work more professionally fulfi lling. As one teacher said, “I’m not babysitting any more.”

By Dylan WiliamJudith Reed, Series Editor

REFERENCESBoekaerts, Monique. “Self-Regulation and Effort Investment.”

In Handbook of Child Psychology, Vol. 4: Child Psychology in Practice, 6th ed., edited by K. Ann Renninger and Irving E. Sigel, pp. 345–77). Hoboken, N.J.: John Wiley & Sons, 2006.

Bransford, John D., Ann L. Brown, and Rodney R. Cocking. How People Learn: Brain, Mind, Experience, and School. Washington, D.C.: National Academies Press, 2000.

Butler, Ruth. “Task-Involving and Ego-Involving Properties of Evaluation: Effects of Different Feedback Conditions on Motivational Perceptions, Interest and Performance.” Journal of Educational Psychology 79, no. 4 (1987): 474–82.

———. “Enhancing and Undermining Intrinsic Motivation: The Effects of Task-Involving and Ego-Involving Evaluation on Interest and Performance.” British Journal of Educational Psychology 58 (1988): 1–14.

Claxton, G. L. “What Kind of Learning Does Self-Assessment Drive? Developing a ‘Nose’ for Quality: Comments on Klenowski.” Assessment in Education: Principles, Policy and Practice 2, no. 3 (1995): 339–43.

Deci, Edward L., N. H. Speigel, R. M. Ryan, R. Koestner, and M. Kauffman. “The Effects of Performance Standards on Teaching Styles: The Behavior of Controlling Teachers.” Journal of Educational Psychology 74 (1982): 852–59.

Fontana, David., and M. Fernandes. “Improvements in Mathematics Performance as a Consequence of Self-Assessment in Portuguese Primary School Pupils.” British Journal of Educational Psychology 64, no. 4 (1994): 407–17.

Gray, Eddie M., and David O. Tall. “Duality, Ambiguity, and Flexibility: A ‘Proceptual’ View of Simple Arithmetic.” Journal for Research in Mathematics Education 25 (March 1994): 116–40.

Hodgen, Jeremy, and Dylan Wiliam. Mathematics inside the Black Box: Assessment for Learning in the Mathematics Classroom. London: NFER-Nelson, 2006.

Kluger, Avraham N., and Angelo DeNisi. “The Effects of Feedback Interventions on Performance: A Historical Review, a Meta-analysis, and a Preliminary Feedback Intervention Theory.” Psychological Bulletin 119, no. 2 (1996): 254–84.

Leahy, Siobhan, Christine Lyon, Marnie Thompson, and Dylan Wiliam. (2005). “Classroom Assessment: Minute-by-Minute and Day-by-Day.” Educational Leadership 63, no. 3 (2005): 18–24.

Mevarech, Zemira R.., and Bracha Kramarski. “IMPROVE: A Multidimensional Method for Teaching Mathematics in Heterogeneous Classrooms.” American Educational Research Journal 34, no. 2 (1997): 365–94.

Sadler, D. Royce. “Formative Assessment and the Design of Instructional Systems.” Instructional Science 18, no. 2 (1989): 119–44.

Saphier, Jonathon. “Masters of Motivation.” In On Common Ground: The Power of Professional Learning Communities, edit-ed by Richard DuFour, Robert Eaker, and Rebecca DuFour, pp. 85–113. Bloomington, Ill.: National Education Service, 2005.

Slavin, Robert E., Eric A. Hurley, and Anne M. Chamberlain. “Cooperative Learning and Achievement.” In Handbook of Psychology, Vol. 7: Educational Psychology, edited by W. M. Reynolds and G. J. Miller, pp. 177–98. Hoboken, N.J.: John Wiley & Sons, 2003.

White, Barbara Y., and John R. Frederiksen. “Inquiry, Modeling, and Metacognition: Making Science Accessible to All Students.” Cognition and Instruction 16, no. 1 (1998): 3–118.

Wiggins, Grant, and Jay McTighe. Understanding by Design. New York: Prentice Hall, 2000.

Wiliam, Dylan, and Marnie Thompson. “Integrating Assessment with Instruction: What Will It Take to Make It Work?” In The Future of Assessment: Shaping Teaching and Learning, edited by C. A. Dwyer. Mahwah, N.J.: Lawrence Erlbaum Associates, 2007.

Five “Key Strategies”for Effective Formative Assessment

4Copyright © 2007 by The National Council of Teachers of Mathematics, 1906 Association Drive, Reston, VA 20191-1502,

Tel: (703) 620-9840, Fax: (703) 476-2690, www.nctm.org.4

Grade 2 Problems

1. Hannah has 42 rocks. Aviva has 17 rocks. Who has more? How many more?

2. Samuel has 53 rocks. Paolo takes 26 of them and hides them. How many does Samuel have now?

3. Sasha started a rock collection. Avi gives her 18 rocks for her collection. Now she has 43. How many did she have before Avi’s gift?

4. Austin has 16 rocks in her collection. She wants to have 43 rocks. How many does she need to find to reach her goal?

Copyright © 2007 by Mathematics Assessment The Baker Resource Service. All rights reserved.

The Baker This problem gives you the chance to: • choose and perform number operations in a practical context The baker uses boxes of different sizes to carry her goods.

1. On Monday she baked 24 of everything. How many boxes did she need? Fill in the empty spaces. cookie boxes ____________ donut boxes ____________

muffin boxes ____________ bagel boxes ____________

2. On Tuesday she baked just bagels. She filled 7 boxes. How many bagels did she make? ____________ Show your calculations. 3. On Wednesday she baked 42 cookies. How many boxes did she fill? ____________ How many cookies were left over? ____________ Explain how you figured this out.

__________________________________________________________________

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4. On Thursday she baked 32 of just one item and she filled 8 boxes. What did she bake on Thursday? _______________________________ Show how you figured this out.

Cookie boxes hold 12 cookies.

Donut boxes hold 4 donuts. Muffin boxes hold 2 muffins.

Bagel boxes hold 6 bagels.

Bagel boxes hold 4 bagels.

10

Copyright © 2005 by Mathematics Assessment Page 8 Fractions Test 5Resource Service. All rights reserved.

FractionsThis problem gives you the chance to:• show the position of fractions on a number line• compare the sizes of fractions

Here is a number line.

1. Mark the position of the two fractions and on the number line.

2. Explain how you decided where to place and on the number line.

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

3. Which of the two fractions, or , is nearer to ? _______________

Explain how you figured it out.

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

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6

23

25

0 112

23

25

12

23

25

MARS Task Anticipation Sheet Task Name:__________________ Grade:_____ Year:______ Tot Pts. _____ Core Pts._____ In anticipating the student work where will students show success? What parts of the task will students be successful?

In terms of knowing and doing mathematics what does this indicate?

In anticipating the student work where will students struggle? What parts of the task will students be unsuccessful?

In terms of knowing and doing mathematics what does this indicate? What understandings or skills do the students need to learn?

Considering strengths and weaknesses from students, what are plans for future teaching? What are the implications for future instruction?

What specific instruction or lesson experiences will you design for students?

MARS Task Analysis Sheet Task Name:_________________ Grade:__ Year:____ Tot Pts. ___ Core Pts.___ Core____% In analyzing the student work where did students show success? What parts of the task did students demonstrate success?

In terms of knowing and doing mathematics what does this indicate?

In analyzing the student work where did students struggle? What parts of the task were students being unsuccessful?

In terms of knowing and doing mathematics what does this indicate? What understandings or skills do the students need to learn?

Considering strengths and weaknesses from students, what are plans for future teaching? What are the implications for future instruction?

What specific instruction or lesson experiences will you design for students?

Action Plan Worksheet Page 1 of 2

Activity Action Plan Worksheet

Activity ___________________________________________________________________________________________________________________

Team Members: _____________________________________________________________________________________________________________ Goals for this Activity:

Action Steps Individuals Responsible Time Frame Measure of Achievement

Action Plan Worksheet Page 2 of 2

Activity: ____________________________________________________________________________________________________________________ Resources Needed: (ex. people to recruit, materials, space, financial)

Item

Other Considerations

Fractions This problem gives you the chance to:

• show the position of fractions on a number line • compare the sizes of fractions

L

Fractions This problem gives you the chance to:

• show the position of fractions on a number line • compare the sizes of fractions

N