Supporting Secondary Mathematics Teachers’ Purposeful and Powerful Discourse Michelle Cirillo, Beth Herbel-Eisenmann, & Mike Steele MDISC, PIs

Thank you RAs: Kate Johnson, Jen Nimtz, Sam Otten, Shannon Sweeny, Alex Theakston, & Rachael Todd (Heather Bosman, Lorraine Males, Faith Muirhead) NSF AB members: Ryota Matsuura, David Pimm, Mary Schleppegrell, Ed Silver, Peg Smith, Randy Phillipp Evaluation team: Horizon Classroom teachers who have allowed us to use their classroom observations as the basis of these materials (NSF, Grant # , Herbel-Eisenmann, PI)

Mathematics Discourse in Secondary Classrooms (MDISC): A Case-Based PD Curriculum How can teachers’ and students’ purposeful attention to discourse contribute to productive and powerful mathematical experiences for students? ▫Productive: How does discourse in the classroom help construe students’ understandings of mathematics? ▫Powerful: How does discourse in the classroom help with developing students’ identities as people who can know and do mathematics?

The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.

MDISC Timeline Project Timeline Design & External Review Field Testing Large-Scale Publication & Internal& Revision& Revision Pilot Dissemination Review & Revision Phase IPhase IIPhase III Phase IV Phase V

Overview of materials Setting the stage: building community, exploring beliefs, and noticing beginning aspects of classroom discourse Constellation 1: Evidence, Explanations & Tacit Expectations (focus on students) Constellation 2: Teacher Discourse Moves and opening up classroom discourse (focus on teachers) Constellation 3: Planning for productive and powerful discourse Constellation 4: (Focus: launch & explore stages) Constellation 5: (Focus: summarize & assessment) Capstone and cycles of action research

Overview of Presentation Key discourse constructs shaping the materials (Beth) Overview of a subset of materials (Mike) Observations from small groups (Michelle) Comments & discussion (Ryota)

Key constructs: Language and language learning Beth Herbel-Eisenmann Michigan State University

Beginning point: “Talk Moves” (Chapin, O’Connor & Anderson, 2003) using appropriate wait time [Wait Time]; Revoicing, asking students to explain someone else’s reasoning [Restating], prompting students to participate in substantive mathematical conversations [Prompt]; and having students apply their own reasoning to other people’s reasoning [Apply]

Teacher Discourse Moves (TDMs) Inviting student participation Waiting Revoicing Asking students to revoice Probing a student’s thinking Creating opportunities to engage with another’s reasoning

Why not focus on just these TDMs? Previous work focused on elementary Are somewhat pedagogically generic Provide a tool for focusing attention Secondary teachers wanted more mathematical substance to the discourse readings Don’t focus attention on increasing sophistication of mathematical discourse Don’t necessarily help with considering when, why, or how to use these TDMs

Mathematically proficient students… ▫… justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. ▫…able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. ▫…try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose... (CCSS)

Considering why, when & how: Systemic Functional Linguistics (Halliday & colleagues) Not a ‘transformational’ view of language, which suggests that children are born with a genetic blueprint for language learning Learners learn to use language for a range of purposes--how adept they become at controlling language and understanding the purposes (consciously or not) is dependent on the contexts in which learners find themselves

Register The set of grammatical and lexical features that realize a specific context (Halliday & Hasan, 1989) Mathematics register: ▫The meanings that belong to the language of mathematics (the mathematical use of natural language, that is; not mathematics itself) and that a language must express if it is being used for mathematical purposes. (Halliday, 1978, p. 195) ▫Not only about how mathematical terms are used but also characteristic phrases and certain modes that are acceptable (Pimm, 1987)

Developing Math Register: Mode Continuum (Gibbons, 2003, 2004, 2006, 2008, 2009) Can serve as a mechanism for teachers and teacher educators to analyze mathematical talk and support the deliberate development of the mathematics register Focuses on gradations of communication between spoken and written mode of communication Movement from context-dependent language to more abstract and discipline-based use of language

Imagine how language might be used as… Students engage with trying to generalize the division of exponential expressions that have the same base

Consider how language changes as… a small group of students work at their desks to try to figure out how to generalize the division of exponential expressions that have the same base; one student from that group is asked to describe the solution to other students after the groups worked on the task; a student might write up a formal explanation; and textbook explanation

Supplementing Gibbons: How language relates to mathematical representations Classroom Generated Language (“slantiness”) Bridging Languages ▫Contextual Language (“dollars per mile”) ▫Transitional Mathematical Language (“the steepness,” “what it goes up by”) Official Mathematical Language (“slope”) (Herbel- Eisenmann, 2002) Not a linear progression, rather teachers & students need to move back and forth across the mode continuum. The decisions about when, how, and why to move back and forth must be informed by what the teacher knows about the students with whom she works…

Attending to mathematics register alone is not sufficient! Interpersonal function of language is also important… Productive… and Powerful!

Positioning …the ways in which people use action and speech to arrange social structures… recognizes that there can be multiple kinds of conversation happening in any mathematics classroom, each of which assigns fluid roles to the participants. (Wagner & Herbel-Eisenmann, 2009) People can position themselves &/or others Not necessarily intentional Who is considered knowledgeable in my classroom? About what (e.g., procedures? concepts?)? Who is considered a struggling learner? What does it mean to know mathematics in this classroom? What is emphasized, thinking processes or doing processes? Do we generate mathematics collaboratively or is it something that is mostly done individually?

What: ▫TDMs: a place to focus teacher’s attention When, how, & why: ▫Mathematics register & mode continuum: productive ▫Positioning: powerful

Applying the constructs in our work Mike Steele Michigan State University

Why not focus on just these TDMs? Previous work focused on elementary Are somewhat pedagogically generic Provide a tool for focusing attention Secondary teachers wanted more mathematical substance to the discourse readings Don’t focus attention on increasing sophistication of mathematical discourse Don’t necessarily help with considering when, why, or how to use these talk moves

Goals of Constellation 1 Explore criteria & assumptions related to evaluating student work Examine student written solutions and other contributions with nuance associated with the Mode Continuum Use student written work and other contributions to think about what students know and what we are not sure students know Examine student discourse with respect to positioning and mathematical dispositions Identify factors that open up & close down student discourse

Constellation 1 Activities 1.Reflective Questions 2.Solve Triangle Area and Perimeter Task 3.Create an Imagined Transcript 4.Examine Student Work 5.Compare Two Classroom Cases 6.Analyze Classroom Video 7.Looking Back/Looking Forward

Constellation 1 Activities 1.Reflective Questions 2.Solve Triangle Area and Perimeter Task 3.Create an Imagined Transcript 4.Examine Student Work 5.Compare Two Classroom Cases 6.Analyze Classroom Video 7.Looking Back/Looking Forward

…because they don’t make sense without the work session activities. If you’re interested in piloting the materials, contact Beth Mike or Michelle

Observations from small groups Michelle Cirillo University of Delaware

Comments & Discussion Ryota Matsuura St. Olaf’s College

Discussion Questions In what ways do you see these activities helping to achieve the goals we described? How might these activities help spur change (through field experiences or action research) to secondary mathematics classroom discourse? What do you see as the challenges of engaging teachers in reflecting on their classroom discourse? Looking ahead, what kind of evidence would convince you that teachers changed their thinking and classroom discourse practices?