Classroom Challenges for Formative Assessment: Supporting the Common Core State Standards for Mathematics Mathematics Assessment Project (MAP) Mary Bouck / Hugh Burkhardt-October 30 2012

Mathematics Assessment Project (MAP) Research and development project out of University of California, Berkeley Purpose—develop assessments that support the Common Core State Standards for Mathematics: --Content --Eight Practices

MAP Materials Well-engineered tools to support those facing new challenges with CCSSM Tools for Formative Assessment Lessons, “Classroom Challenges” Professional Development Modules Test Tasks Prototype Summative Tests

Formative Assessment Lessons Assess and develop student’s integrated understanding Provide diagnosis and treatment Use rich tasks that assess reasoning, linking content and practice standards Help teachers and students move reasoning forward, A “protein supplement” to any curriculum diet

FALs: “Classroom Challenges” Concept development lessons To reveal and develop students’ interpretations of significant mathematical ideas and how these connect to their other knowledge. “Proficient students expect mathematics to make sense” Problem solving lessons To assess and develop students’ capacity to apply their math flexibly to non-routine, unstructured problems, both from pure math and from the real world. “They take an active stance in solving mathematical problems”

“Model” Materials MAPs FAL are Lessons, they are opportunities for students to learn as well as opportunities of teachers and students to note what students can do on a topic and with the practices and what they are struggling with. MAPs FAL’s—are examples/models of how one can do formative assessment and instruction at the same time in a lesson.

“Repeating Decimals” Classroom Challenge: a formative assessment lesson for “concept development”

FAL Task: Repeating Decimals Read through the task and answer the questions the way you think your “average” student would do so. Directions to students: Read through the opening text ‘How to write repeating decimals’ on the handout,. Answer as much as you can. Show your work so I (the teacher) can understand your reasoning. Don’t worry if you can’t do everything. There will be a lesson on this material that will help you improve your work.

FAL Task: Repeating Decimals Collect students’ responses to the task. Make some notes on what their work reveals about their current levels of understanding and their problem solving strategies. We suggest that teachers do not score students’ work. The research shows that this will be counterproductive.

Fraction Conversion Challenge On your own, take about 5 minutes and change these fractions into decimals. Do as many as you can. You likely will be able to do some mentally and others may need to do some working out on paper. You might see some patterns that will help you!

Fraction Conversion Challenge Use a calculator and try to finish the sheet. Note any patterns you see in the table and write them down on the back of your paper.

Whole Class Discussion: Patterns Look at the pattern in the thirds-- 0.3333333... Do the threes in this pattern go on forever? How can you be sure? What does the pattern for two thirds and three thirds look like? Look at the pattern in the sixths-- 0.16666... (or 0.16666....7, if the calculator rounds the number) Is a sixth a repeating decimal? Why? How can we write this using “a line” in the notation?

Whole Class Discussion: Patterns Look at sevenths. Do you see any patterns? 0.142857142...; 0.285714285...; 0.428571428...; ... How can we write these using our notation? Look at the ninths: 0.111111...; 0.222222...; 0.333333...; 0.444444… How does this pattern continue? What happens with nine ninths?

Direct Teaching: Converting a repeating decimal to a fraction If needed, teach the students a method for figuring out the fractional value of a repeating decimal (shown on the first individual task). x = 0.17 100x = 17.7 100x - 10x = 17.7 – 0.17 90x = 16 x = 16/90 x = 8/45

Collaborative Activity: Continuing with the same work and Ideas. Please get into groups of 2 or 3. Each group will be given three sets of cards: Set A—Decimals, Set B—Equations, and Set C—Fractions (along with large sheet of paper) Take turns matching cards. the expressions and the words. Place the matches next to each other on the paper. Each time you make a match explain to your partner(s) why. If you think there is no suitable card that matches, write one of your own. There are some blank cards for doing this. . 15

Collaborative Activity: Teacher’s role Please get into groups of 2 or 3 (self selected). Each group will be given three sets of cards: Set A—Decimals, Set B—Equations, and Set C—Fractions Take turns matching cards. the expressions and the words. Place the matches next to each other on the paper. Each time you make a match explain to your partner(s) why. If you think there is no suitable card that matches, write one of your own. There are some blank cards for doing this. Teachers are to: Listen and watch students carefully. Note different student approaches to the task. Notice any difficulties that students encounter, and the ways they justify and explain to each other. Try not to make suggestions that move students towards a particular approach to the task. Instead, ask questions that help students to clarify their thinking. 16

Whole Class Discussion Post completed posters and compare solutions It is suggested teachers: Select a set of cards that most groups matched correctly as this may encourage good explanations Then select one or two matches that most groups found difficult. Ask students to share their thinking with the whole

FAL Task: Repeating Decimals: revisited Directions to students: Look at your original responses and the questions (on the board/written on your script.) Think about what you have learned. Look at the new task sheet, Repeating Decimals (revisited). Use what you have learned to answer these questions.

Reflection Turn to a couple of people near you and talk about: What mathematics could your students learn or deepen their understanding of with this lesson? What would they struggle with in this lesson? How is this lesson like what goes on in your district’s mathematics classrooms? How is it different? What do you think might be the benefits of using a lesson like this? What could teachers learn about their students from using lessons like this? What could students learn about themselves from lessons like this?

What Makes Assessment Formative? Items: The tasks used with the students look like classwork and homework implied by standards (content and practices in the CCSS). Feedback: The task provides information to both the teacher and the students as to his/her mathematical understanding/skills. The feedback has content (mathematical) not just value judgment (grade) Revisions: There is an opportunity for students to revise their work. Timing: The assessment is available and used by teachers while instruction is still going on for the topic, while there is still time to adapt the teaching to meet the needs of the students.

Assessment Task: Design Recommend these be given to students one or more days before lesson Suggest students answer the questions without assistance from the teacher Collect work, but do not give a score, as the purpose is to help teachers see what sense students are making, so as to prepare for the lesson Suggest written comments be given; “help students to make further progress by summarizing their difficulties as a list of questions.”

Constructive Feedback Recomendations Recommendation: Research shows that students benefit most from feedback that: Focuses on the task, not on grades or scores. Is detailed rather than general. Explains why something is right or wrong. Makes clear what has been achieved and what has not. Suggests what the student may do next Offers specific strategies for improvement

Common Issues/Question

Lesson Structure Before the lesson Students work on an assessment task. The teacher reviews the work, noting mistakes and misconceptions. Whole-class introduction Teacher asks questions or provides students with guidance on the context of the lesson. Collaborative activity Students then work in small groups, producing a poster of their work. Whole-class discussion Students engage in an interactive discussion where they justify their solutions. Review individual solutions to the assessment task Students use comments from the teacher regarding their work and what they have learned in the lesson to try to improve their solution.

Collaborative Activity: Teacher’s role Please get into groups of 2 or 3 (self selected). Each group will be given three sets of cards: Set A—Decimals, Set B—Equations, and Set C—Fractions Take turns matching cards. the expressions and the words. Place the matches next to each other on the paper. Each time you make a match explain to your partner(s) why. If you think there is no suitable card that matches, write one of your own. There are some blank cards for doing this. Teachers are to: Listen and watch students carefully. Note different student approaches to the task. Notice any difficulties that students encounter, and the ways they justify and explain to each other. Try not to make suggestions that move students towards a particular approach to the task. Instead, ask questions that help students to clarify their thinking. 25

Reviewing Work Students revisit their answers to the assessment task and revise their work. They either use a different colored pen when reviewing their work or complete a fresh copy or slightly different version of the task sheet.

Lesson Structure Before the lesson Students work on an assessment task. The teacher reviews the work, noting mistakes and misconceptions. Whole-class introduction Teacher asks questions or provides students with guidance on the context of the lesson. Collaborative activity Students then work in small groups, producing a poster of their work. Whole-class discussion Students engage in an interactive discussion where they justify their solutions. Review individual solutions to the assessment task Students use comments from the teacher regarding their work and what they have learned in the lesson to try to improve their solution.

Summary Points of FALs Good tasks—CCSS content and opportunities to engage in one or more practices Collect information (intro task), inform teacher and lesson plan Given written feedback and not scores During “collaborative” work, teacher is to work hard at listening to what students can do and what they struggle with Opportunity for revising work (understanding, thinking)

Teacher Guides: Components Math goals Standards addressed: content and practice Introduction/overview Materials needed Time needed   Lesson outline --Before lesson (individual task and analysis) --Launch, Explore, Summarize Solutions Student sheets

Links to Common Core Standards Mathematical Content Number 8.NS: Know that there are numbers that are not rational, and approximate them by rational numbers. Equations and Expressions 8.EE: Analyze and solve linear equations. Mathematical Practices 1.Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 6. Attend to precision.

“Gold Rush: Maximizing Area” Classroom Challenge: a formative assessment lesson for “problem solving”

FAL Task: Gold Rush Read through the task and think about how a “typical” middle school student in your district would answer the questions. Directions to students: Read through the questions carefully and try to answer them as well as you can. Show all your work so that I (the teacher) can understand your reasoning. Try to show your findings in an organized way. Don’t worry if you can’t do everything. There will be a lesson on this material that will help you improve your work.

FAL Task: Sample Teacher Feedback What does the length of the rope given to a prospector measure? How could you measure the amount of land enclosed by the rope? Now investigate if combining ropes affects how much land each prospector gets. How can you now organize your work? How do you know for sure your answer is the best option?

Collaborative Activity: Producing a Joint solution Please get into groups of 2 or 3 (self selected). Take turns to explain your method and how you think your work could be improved. Listen carefully to each other. – Ask questions if you don’t understand. Once everyone in the group has explained their method, plan a joint method that is better than each of your separate ideas. Make sure that everyone in the group can explain the reasons for your chosen method. On poster paper, write a brief outline of your group’s method. State any assumptions your group has made. 34

Collaborative Activity: Teacher’s role Please get into groups of 2 or 3 (self selected). Take turns to explain your method and how you think your work could be improved. Listen carefully to each other. – Ask questions if you don’t understand. Once everyone in the group has explained their method, plan a joint method that is better than each of your separate ideas. Make sure that everyone in the group can explain the reasons for your chosen method. On poster paper, write a brief outline of your group’s method. State any assumptions your group has made. Teachers are to: Listen and watch students carefully. Note different approaches to the task and what assumptions students make. 35

Whole Class Discussion Display posters Hold a class discussion that has groups explaining different approaches and/or different assumptions.

Analyzing Sample Student Responses Imagine you are the teacher and have to assess the student work. Individually review each piece of work. Note the mathematics the student has used and if they have made any errors. Explain your findings to the rest of the group. Listen carefully to explanations by others in your group. Ask questions if you don't understand. Once everyone agrees on what the student has done, complete the questions below the students’ work. (Make sure the student who writes the answers is not the student who explained them.)

Sample Response: Ann

Sample Response: Jake

Sample Response: Kodie

Sample Response: Mark

Whole Class Discussion of Samples Hold a class discussion that has groups explaining the different approaches assumptions in the samples. Which method did you like best? Why? Which method did you find most difficult to understand? Why? How could the student (Ann, Mark, . . . ) improve his/her answer?

Reflecting: How Did You Work

Lesson Structure Before the lesson Students work on an assessment task The teacher reviews the work, noting mistakes and misconceptions, and writes questions that will push students to improve their solutions. Whole-class introduction Provides students with feedback and they work to improve solution Collaborative activity Students work in small groups to provide a joint solution. Analyzing Student Work Students discuss sample work and give written analysis. Whole-Class Discussion Organize a discussion about what has been learned and some of the different approaches. Review or Reflect on work and learning Students reflect on their individual and group solutions and work on an improved solution or they reflect on their work and learning.

Links to Common Core Standards Mathematical Content: Geometry: 6.G. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. 7.G. Draw, construct, and describe geometrical figures and describe the relationships between them. Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Links to Common Core Standards Mathematical Practices Make sense of problems and persevere in solving them. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics.

Questions, Comments, Concerns After looking at two lessons, what questions, comments or concerns do you have : About the formative assessment lessons and or MAP Lessons? Implementation of those lessons in classrooms??

Mathematics Assessment Project (MAP) Open source: Classroom Challenges for Formative Assessment: Supporting the CCSS for Mathematics Mathematics Assessment Project (MAP) Open source: http://map.mathshell.org.uk/materials/ Mary Bouck