Supporting Students in Agency, Identity and Authority David Foster Silicon Valley Mathematics Initiative CMC South Palm Springs November 6, 2015

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TRU Math Dr. Alan Schoenfeld UC Berkeley

Agency, Authority, and Identity Core Question: What opportunities do students have to explain their own and respond to each other’s mathematical ideas? Many students have negative beliefs about themselves and mathematics, for example, that they are “bad at math,” or that math is just a bunch of facts and formulas that they’re supposed to memorize. Our goal is to support all students—especially those who have not been successful with mathematics in the past—to develop a sense of mathematical agency and authority. We want students to come to see themselves as mathematically capable and competent—not by giving them easy successes, but by engaging them as sense-makers, problem solvers, and creators of mathematical ideas. Agency, Authority, and Identity Pre-observationReflecting After a LessonPlanning Next Steps What opportunities exist in the lesson for students to explain their own and respond to each other’s mathematical ideas? What opportunities did students have to explain their own and respond to each other’s mathematical ideas? What opportunities can we create in future lessons for more students to explain their own and respond to each other’s mathematical ideas? Think about: o Who generates the mathematical ideas that get discussed. o Who evaluates and/or responds to others' ideas. o How deeply students get to explain their ideas. o How the teacher responds to student ideas (evaluating, questioning, probing, soliciting responses from other students, etc.). o How norms around students' and teachers' roles in generating mathematical ideas are developing. o How norms around what counts as mathematics (justifying, experimenting, practicing, etc.) are developing. o Which students get to explain their own and respond to others' ideas in a meaningful way.

Another Look at a Classroom

Making Sense Accentuate the Negative: Introducing Addition of Integers Explore operations on integers. Look for patterns and repeated reasoning, to determine a general method to add two integers.

Insert Video Lily Integer Task

Agency, Authority, and Identity

Stage 1: Denial (those aren’t real students, this must have been staged, the camera made them smart, etc.) Stage 2: Low Expectations (those aren’t like my students, my students can’t talk like that, those students come from rich schools, they must have had good math teachers in the younger grades, etc.) Stage 3: How Is This Done? (why don’t I know how to teach like that, why haven’t I been shown?, how do you get students to express themselves and listen to others to learn? etc.) Viewing Model Classrooms Videos

What do we mean by classroom discourse? The National Council of Teachers of Mathematics (NCTM) in their 1991 professional standards describes discourse as ways of representing, thinking, talking, agreeing, and disagreeing; the way ideas are exchanged and what the ideas entail; and as being shaped by the tasks in which students engage as well as by the nature of the learning environment.

The Mathematics Teaching Rubric Worthwhile Tasks The Learning Environment Teachers Role in Discourse Students Role in Discourse Tools for Enhancing Discourse Teaching and Learning Analysis The Silicon Valley Mathematics Initiative

The teacher self- assesses their teaching using the rubric and the coach uses the findings to engage in a coaching conversation. 4 – Exemplary Teaching3– Successful Teaching2– Improving Teaching1– Limited Teaching The teacher orchestrates discourse in the class. The teacher poses questions and tasks that elicit, engage, and challenge each student’s thinking. The teacher listens carefully to the students’ ideas and discerns mathematical meaning and relevancy from student responses. Students are asked to clarify and justify their ideas orally and in writing. The teacher decides what to pursue in depth from among the ideas that students bring up during a discussion. Care is given by the teacher to develop concepts thoroughly and insure students’ ownership and understanding. The teacher decides when and how to attach mathematical notation and language to students’ ideas. The teacher has full understanding of the mathematical goals of the lesson and decides when to provide information, when to clarify an issue, when to model, when to lead, and when to let a student struggle with difficulty. These decisions are consistent with the goal and pace of the lesson. The teacher monitors students’ participation in discussions and decides when and how to encourage each student to participate. These decisions are predicated on insuring all students will learn and be successful in mathematics. The teacher often orchestrates discourse in the class. The teacher often poses questions and tasks that elicit, engage, and challenge each student’s thinking. Often the teacher listens to the students’ ideas and makes sense of their responses. Often students are asked to clarify and justify their ideas orally and in writing. The teacher may decide what to pursue in depth from ideas that students bring up during a discussion. Some care is given by the teacher to develop concepts and to encourage students’ ownership and understanding. The teacher attempts to attach mathematical notation and language to students’ ideas. The teacher has good understanding of the mathematical goals of the lesson and is often successful in determining when to provide information, when to clarify an issue, when to model, when to lead, and when to let a student struggle with difficulty. These decisions are often consistent with the goal and pace of the lesson. The teacher attempts to monitor students’ participation in discussions and decides when and how to encourage students to participate. Attempts are made to reach all students. The teacher directs the class and attempts to foster discourse. Sometimes the teacher poses questions and tasks that elicit, engage, and challenge students’ thinking. The teacher listens to some of the students’ ideas. The teacher may have difficulty following students’ math thinking. Sometimes students are asked to clarify and justify their ideas orally and in writing. The teacher rarely deviates from lesson plan to pursue ideas that students bring up during a discussion. Some care is given by the teacher to develop concepts and to encourage students’ ownership. The teacher follows the textbook in deciding when and how to introduce mathematical notation and language. The teacher follows the curricula math goals of the lesson. The teacher is improving their questioning strategies. The teacher is developing the pacing and generating good questioning. The instructional decisions may be aligned with lesson goals. The teacher attempts include all students. The teacher directs the class. The teacher poses questions and tasks that focus students’ work. The teacher may listen to some of the students’ responses. The teacher may have difficulty following students’ math thinking and usually ignores those responses. Rarely are students asked to clarify and justify their ideas orally and in writing. The teacher rarely deviates from the textbook to pursue ideas that students bring up during a discussion. The teacher usually follows the textbook in deciding when and how to attach mathematical notation and language to students’ ideas. The teacher follows the curricula math goals of the lesson. The instructional decisions may be aligned with state standards and with the goal of covering the topics on state test. The teacher modifies expectation from student to student. Teacher’s Role in Discourse Silicon Valley Mathematics Initiative Mathematics Teaching Rubric How is the instrument used?

Experiences Designed To Establish Classroom Discourse Find someone nearby and give one and get one. “One way I develop classroom discourse in math at the beginning of the year is ____________.” Repeat with another colleague.

Five Practices for Orchestrating Productive Discussions Anticipating likely student responses Monitoring students’ actual responses Selecting particular students to present their mathematical work during the whole class discussion Sequencing the student responses Connecting different students’ responses—to each other and to key mathematical ideas. Smith & Stein, 2011

Five Talk Moves Using wait time Revoicing a student’s contribution Asking students to restate another student’s contribution Prompting students for further participation Asking students to apply their own reasoning to someone else’s reasoning Chapin, O’Connor, & Anderson (2009)

Diane Briars – President National Council of Teachers of Mathematics

Principles to Actions, p. 35

Using Math Talk Moves to Deepen Conceptual Understanding

Number Talk When you find one solution path, put up our “thumbs up” signal. When you find a second way to solve it, put up two fingers If you can find three ways, put up three fingers.

Mia’s Second Grade Number Talk

How is the teacher using number talks to promote classroom discourse? How are students using math talk moves to deepen their conceptual understanding of mathematics?

Agency, Identity and Authority

Agency The capacity, condition, or state of acting or of exerting power : operation A person or thing through which power is exerted or an end is achieved: instrumentality “Students are owners of their own learning, they take responsibility for themselves and their peers. They have a productive disposition - habitual inclination to see mathematics as sensible, useful, and worthwhile.”

Identity The distinguishing character or personality of an individual. The relation established by psychological identification. “ Student has a growth mindset about learning and effort, with a strong belief in diligence and one’s own efficacy.”

Authority An individual cited or appealed to as an expert Power to influence or command thought, opinion, or behavior Freedom granted by one in authority Persons in command “ Where does the authority reside in math class – teacher acknowledging correctness, answers in back of the book, or in the logic of the mathematics?”

How should teachers establish a community of learners? What are most essential characteristics of that community?

Work Together in Groups of Four 1.Each group member gets 1 clue with a white dot. 2.The other two clues may be used if there are more than four group members. 3.Each group member may read her/his clue to the group as often as needed. 4.No clue may be physically share with another group member. 5.Everyone should have opportunities to use and build with the cubes. 6.Each group must verify that they have constructed a figure that meets the constraints of all four white dot clues. 7.Respectfully work together sharing air time, working as partners and collaborate solve the puzzle.

Building Group Dynamics: Get It Together

Reflect on the Video How effective was the group work? What is your evidence? How did the math activity foster students’ discourse and Agency, Identity and Authority? What was common among the groups? Where there some issues that arose with the groups? If so, what where they and how might they be addressed? What do we think the teacher did to make the lesson as successful as it was?

CCSS Mathematical Practices OVERARCHING HABITS OF MIND 1. Make sense of problems and persevere in solving them 6. Attend to precision REASONING AND EXPLAINING 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others MODELING AND USING TOOLS 4. Model with mathematics 5. Use appropriate tools strategically SEEING STRUCTURE AND GENERALIZING 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning

What Mathematical Practices are Students Engaged in? Explain How does student discourse deepen conceptual understanding? Explain

Mia’s Second Grade Students Reasoning Quantitatively, Using Structure and Repeated Reasoning

What Mathematical Practices are Students Engaged in? Explain How does student discourse deepen conceptual understanding? Explain

Socio-Mathematical Norms Errors are gifts…they promote discussion and learning. The making sense is important…not only the answer. Ask questions…until it makes sense. Think with language…use language to think. Use multiple strategies…multiple representations.

A Look at Agency, Authority, and Identity in the Classroom

Making it Precise: Distance/Time Graphs 8 th Grade

Agency, Authority, and Identity

Five Talk Moves Example: Using wait time.....

Five Talk Moves Example: Revoicing a student – S: “I minused the 60 first.“ – T: "So you're saying that you broke apart the 64 into and used the 6 tens first? Is that right?"

Five Talk Moves Example: Asking students to restate one another – T: “Can you restate what he just said in your own words?” – T: “John, what did you hear Danica say?”

Five Talk Moves Example: Prompting for further participation – T: “Would someone like to add on or share another method?” – T: “Who found a different way to solve it?” – T: “Did we find all the possible combinations?”

Five Talk Moves Example: Asking students to apply their reasoning to someone else’s – T: “Do you agree or disagree and why?” – T: “How is Mary's thinking similar to or different than Juan's?” – T: “Do you think Angelique’s method will always work?”

Talk Moves from our Subtraction Discussion Which moves did you notice? What was their apparent function? Did you see a place where a talk move could have been used? Wait Time Revoicing Restating Prompting Applying

Reflect on Talk Moves Why should we use these talk moves? How do they support classrooms discourse? How do they affect Students’ Agency, Identity and Authority?

Functions of the Talk Moves Using wait time – Allows students time to think – Minimizes students’ tendencies to reason hastily – Increases opportunities for equitable participation Back

Functions of the Talk Moves Revoicing a student’s contribution – Amplification – Elaboration, increase clarity of reasoning – Bridge to more mathematical language – Set up alignments and oppositions – Demonstrate attention and concern for student thinking and voice Back

Functions of the Talk Moves Asking a student to restate – Build a community of active listeners – Provide another phrasing of reasoning for students to engage with – Formative assessment Back

Functions of the Talk Moves Prompting for further participation – Increase opportunities for participation from a variety of students – Get multiple solutions/ideas on the table – Push to deeper levels of mathematical thinking Back

Functions of the Talk Moves Asking students to apply their own reasoning to someone else’s reasoning – Encourage students to engage with one another’s ideas – Direct attention to reasoning rather than answers – Make mathematical connections – Promote community argumentation and justification Back

Connect Classroom Discourse, Talk Moves and Agency, Identity and Authority. Make a poster.

SVMI Resources and Tools: MAC/MARS performance tasks K - 12, Problems of the Month, Formative Assessment Lessons, Formative Re-Engagement Lessons, Coaching Tools, etc. Professional Development Services: Summer Math Institutes, Year-long Professional Development Workshops, Webinars, MAC/MARS Scoring Training, Leadership Training, etc. Now Funded By

Inside Mathematics Website Silicon Valley Mathematics Initiative Mathematics Assessment Project UC Berkeley & Shell Centre for Mathematical Education

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