Learning to Do the Mathematical Work of Teaching Deborah Loewenberg Ball Hyman Bass, Tim Boerst, Yaa Cole, Judith Jacobs, Yeon Kim, Jennifer Lewis, Laurie Sleep, Kara Suzuka, Mark Thames, and Deborah Zopf University of Michigan New England Comprehensive Center Mathematics Leadership Network Webinar • November 25, 2008
Overview of session • A larger perspective: why are we doing this, and what will it take? • Mathematical knowledge for teaching (MKT) • Learning to do the mathematical work of teaching • Tasks for developing teachers’ MKT and their skill with the mathematical work of teaching • Discussion: Issues for teacher development
The problem How can we improve students’ learning? . . . Teachers’ mathematical knowledge is a key factor shaping what they are able do. What mathematical knowledge do teachers need?
Different questions From teacher knowledge to knowledge for teaching: • What mathematics do teachers need to know? • What mathematics do teachers know? • What mathematics do teachers use? • What mathematics does teaching entail?
Elements of our “practice-based” approach • Study instruction and identify the mathematical work of teaching • Analyze what mathematical knowledge is entailed by the work (MKT) • Test the working hypotheses based on these analyses by developing measures of MKT, validating teacher scores against practice and against student achievement gains • Develop and evaluate approaches to helping teachers learn mathematical knowledge for teaching
Mathematical task, grade 4-5 • What fraction of the big rectangle is shaded blue? • What fraction of the big rectangle is shaded green? • What fraction of the big rectangle is shaded altogether?
Video clip • Discussion of warm up problem • Focused attention on equal parts • Developing working ideas about fractions • Identify the whole • Make equal parts • Count how many equal parts out of the whole
Viewing foci • What is the work of teaching in this clip? • What mathematical “knowledge” is entailed by that work?
Examples of work of teaching • Selecting/designing tasks • Identifying and working toward the mathematical goal of the lesson • Listening to and interpreting students’ responses • Teaching students what counts as “mathematics” and mathematical practice • Making error a fruitful site for mathematical work • Attending to ambiguity of “big rectangle” • Deciding what to clarify, what to make more precise, what to leave in student’s own language
Mathematical knowledge for teaching Subject Matter Knowledge Pedagogical Content Knowledge Knowledge of Content and Students (KCS) Common Content Knowledge (CCK) Specialized Content Knowledge (SCK) Knowledge of curriculum Knowledge at the mathematical horizon Knowledge of Content and Teaching (KCT)
Common content knowledge (CCK) • Knowing the names of the shapes (rectangle, triangle) • Identifying what fraction of the whole is shaded green, blue, together
Specialized content knowledge (SCK) • Choosing or making this figure and understanding what its key mathematical point is • Why might someone think that 1/2 of the rectangle is shaded green? Or that 1/6 of the rectangle is shaded blue?
Knowledge of students and content (KCS) What might be difficult for students about this task? What are the different answers students might give for this problem? How do students’ experiences with fractions lead them to think that the blue triangle is 1/6 of the whole? Knowledge of teaching and content (KCT) What questions would you pose about this figure? How could you help students understand that the green rectangle and the blue triangle are each 1/8 of the whole? In a whole-class discussion, what answers to this task would you want presented, and in what order? What drawing would you present to students next, and why?
Other tasks of teaching mathematics • Analyzing mathematical treatments in textbooks • Identifying mathematical distortions or inaccuracies in textbooks and modifying them • Responding to students’ “why” questions • Unpacking and decomposing mathematical ideas • Explaining and guiding explanation • Using mathematical language and notation • Generating examples • Sequencing ideas • Choosing and using representations • Analyzing errors • Interpreting and evaluating alternative solutions and thinking • Making mathematical practices explicit • Attending to issues of equity (e.g., language, contexts, mathematical practices)
The challenge • How can MKT be developed? How can opportunities for learning SCK be provided? • Easy to work on mathematics, or on students; less easy to create opportunities to develop the specialized knowledge of mathematics needed for teaching • Still harder––to develop teachers’ capacities with doing the mathematical work of teaching
Our approach to the problem • Find/develop tasks that create opportunities for learning mathematical knowledge for teaching • Situate teachers’ opportunities to learn in the contexts of use • Provide opportunities to practice the kinds of mathematical thinking, reasoning, and communicating used in teaching • Enact tasks in ways that maintain the focus on developing MKT and the ability to use it in teaching
Designing tasks to develop MKT What is the difference between a good mathematics task and one that is good for developing mathematical knowledge for teaching? What are tasks that afford opportunities to learn to do the mathematical work of teaching?
Mathematical knowledge for teaching (MKT) • Frame: knowledge used in practice • “knowledge entailed by the work of teaching” • What do we mean by “knowledge”? • Mathematical knowledge, skill, habits of mind • What do we mean by the “work of teaching”? • The activities in which teachers engage, and the responsibilities they have, to teach mathematics, both inside and outside of the classroom
Designing MKT tasks • What distinguishes a good mathematics task from a good MKT task? • What are tasks that afford opportunities to learn to do the mathematical work of teaching?
Features of tasks designed to develop MKT • Unpacks, makes explicit, and develops a flexible understanding of mathematical ideas that are central to the school curriculum • Opens opportunities to build connections among mathematical ideas • Provokes a stumble due to a superficial “understanding” of an idea • Lends itself to alternative/multiple representations and solution methods • Provides opportunities to engage in mathematical practices central to teaching (explaining, representing, using mathematical language, analyzing equivalences, proving, proof analysis, posing questions, writing on the board)
Sample MKT task:Writing division story problems Write as many different stories as you can that correspond to this division expression and that represent different interpretations of the meaning of division or what it means in specific situations. 38÷4
Sample MKT task:Analyzing errors For each problem: • Do the problem correctly and explain the procedure you use and why it works. • What mathematical steps likely produced each incorrect answer? • 3.4 • x 2.4 • 136 • 680 • 81.6 • 42 • x 83 • 326 • 24 • x 53 • 72 • 120 • 192 • 283 • x 4 • 2062
MKT task #1:Analyzing solutions What fraction of the rectangle is shaded? What reasoning could produce each of these answers?
MKT task #2:Analyzing textbook treatments Examine three different textbooks to see how fractions are defined. How are “equal parts” defined? How is the notion of the “whole” addressed? How is the difference between counting and areas dealt with?
MKT task #3:Choosing examples to assess student understanding • Make up three examples to assess whether or not students can correctly attend to the whole and to “equal parts” in identifying fractions • Consider different contexts for naming fractions in choosing examples: sets of discrete objects, number line, areas • Provide clear mathematical and pedagogical justifications for your choices
Questions to consider when examining tasks • What opportunities to learn MKT does each task provide? • How well does each task provide practice with learning to do the mathematical work of teaching? • What are the challenges of enacting MKT tasks with teachers?
Tasks as enacted by teacher and students Tasks as they appear in curricular materials Tasks as set up by teachers Student learning The Mathematical Task Framework (MTF) Stein, Grover & Henningsen (1996), Smith & Stein (1998); Stein, Smith, Henningsen & Silver (2000)
Tasks as enacted by teacher educator & teachers Tasks as they appear originally Tasks as set up by teacher educators Teacher learning The Mathematical Task Framework adapted to teacher education
Challenges of teaching MKT • Staying focused on the mathematics, and not on how to teach the math • Keeping the problems focused on MKT and not just “M” • Unpacking the mathematics sufficiently and convincingly helping them see what there is to learn and do • Making visible the connections to the kinds of mathematical thinking, judgment, reasoning one has to do in teaching • Actually practicing the mathematical reasoning and problem solving needed in practice 34 34
Enactment:What are key questions and moves that can be used to provide opportunities to learn to do the mathematical work of teaching? • Asking teachers to explain their solutions to the others • Having teachers explain what is/was confusing them • Asking teachers to figure out what might be confusing/difficult for someone else about the problem • Having teachers ask questions to become more clear about their colleagues’ solutions • Asking teachers to make correspondences between solutions and/or representations • Asking teachers to explain someone else’s thinking • Providing opportunities to “talk mathematics” and write on the board • Provoking a common error • Narrating how something a teacher does/says relates to or is a skill used in teaching
Discussion • What are the challenges of shifting the focus in professional development to “the mathematical work of teaching”? • Finding and developing good tasks • Developing teacher developers’ own skills • Keeping the focus on practice, and not merely knowledge or belief • . . . ?