Lessons Learned from a Lesson Study Approach to High School Mathematics

1. Lessons Learned from aLesson Study Approach toHigh School Mathematics Signature Pedagogies Unpack/Repack Process Deep Pedagogical Content Knowledge

2. Presentation Outline • Overview of lesson study project • Introduction, with examples, of: • Signature pedagogies • Unpack/Repack process • Deep Pedagogical Content Knowledge

3. ImportantMathematicsandPowerfulPedagogy

4. IMAPP ProjectImportant Mathematicsand Powerful Pedagogy • Iowa high school mathematics lesson study project • Math Science Partnership Program Grant • Iowa Board of Regents and Iowa Department of Education • University of Iowa, Maharishi University of Management, Great Prairie AEA, Local School Districts

5. Project Vision …

7. Project Overview • “Important Mathematics” June Professional Development Institute • “Powerful Pedagogy” July Professional Development Institute • Teaching and Learning Mathematics in the Classroom – Lesson Study Approach Academic Year

8. IMAPP Lesson Study Model • Plan – Unpack/Repack • Teach – Signature Pedagogies • Observe – Intentional yet Flexible • Debrief – Math, Teaching, Learning • Revise – Refocus on Deep Understanding of Important Mathematics

9. Planning the Lesson • Unpack/Repack (see later) • For topic taught, consider: • What is it?(deep conceptual knowledge) • How do you do it, operate with it/on it?(deep procedural knowledge) • What’s it good for?(apply) • What’s it connected to?(connected and coherent)

10. Planning the Lesson (cont.) • Focus Question • Misconceptions, trouble spots • Pivotal points • Questions that probe and deepen student understanding • Opportunites for critical reflection • Summarize

11. Teaching the Lesson • Signature Pedagogies (see later) • Formative Assessment – re: Dylan Wiliam: “formative instruction”

12. Observation Guidelines • Intentional (not random, not transcript) • Math, teaching, learning (separately or all together) • What I saw; What I heard • Time stamps (partly so you can reference the video later) • Could do assigned observing: • Questioning • What’s happening on problem x • Technology use • Group work • Teaching strategies • Stationary or roving • Follow one student, one group • Take a few minutes to reflect on your observation notes, decide what you observed that was most relevant, important, salient • Take notes flexibly, as most helpful to you • Goal: Get observation data to help deepen knowledge of math, teaching, learning; and create a more effective lesson

13. Debrief the Lesson:Math, Teaching, Learning • Observer notes • Discussion • Commentary • Based on what I saw, what I heard

14. Some Revision Guidelines • Sharpen the focus on targeted, student-learning-appropriate, important mathematics • Design instruction to ensure students engage in critical reflection • Revise based on student understanding/learning – examine data from observation and debrief (see checklist) • Good questions, good questions, good questions • Design and maintain connected and coherent richness • Design and maintain high level of cognitive demand

15. IMAPP Lesson Study Model •  Plan – Unpack/Repack •  Teach – Signature Pedagogies •  Observe – Intentional yet Flexible •  Debrief – Math, Teaching, Learning •  Revise – Refocus on Deep Understanding of Important Mathematics

16. Academic Year Activities and Data • Checklist …

17. SignaturePedagogies

18. Signature Pedagogies • Potent teaching approaches that warrant common application • Global to local • Phrase borrowed and idea adapted from Shulman (2005)

19. Shulman – Signature Pedagogies • In the professions of law, medicine, engineering, the clergy, with implications for teacher education • NRC workshop, Feb 2005 (MSP resource) • A characteristic: Habitual, Routine • Rules of engagement always the same • Compare to: Must have variety in teaching • Not boring, novelty comes from applying to different subject matter, not from changing the pedagogy

20. Signature Pedagogies:Global Examples • Investigative approach (e.g., Baroody 2003) • Teaching through problem solving(e.g., Schoen 2003, Kilpatrick 2001, Grouws 2000) • Teaching focused on reasoning and sense making (e.g., NCTM 2009, Focus in High School Math: Reasoning and Sense Making)

21. Signature Pedagogies:More Global Examples • Questioning(e.g., Redfield and Rousseau 1981) • Multiple, connected representations(e.g., NCTM 2000) • Connected and coherent(e.g., NCTM 2000) • In context (e.g., RME, Freudenthal Institute)

22. Signature Pedagogies:Strand Examples • Algebra • Functions approach to teaching algebra • Geometry • Conjecturing approach to teaching geometry(re: Polya: “First believe it, then prove it.”) • Statistics and Probability • Focus on the big idea of variability • Simulation • Real data

23. Signature Pedagogies:Topic Examples • Functions • Include a recursive view of functions • Recursion • Include use of pedagogically powerful informal notation, like NEXT and NOW • Vertex-Edge Graphs • Discrete mathematical modeling

24. Signature Pedagogies • Potent teaching approaches that warrant common application • Global to local

25. Unpack/RepackProcess

26. Planning the Lesson • Unpack the Math What is the connected and coherent web of mathematical knowledge that comprises and surrounds this topic? • Unpack Teaching and Learning What theory and practice related to teaching and learning are germane to this topic? • Repack into an Effective Lesson How will you take everything you have unpacked, and repack into a lesson that will help students achieve deep understanding of important mathematics? See: Unpack/Repack Diagram, Unpack/Repack Organizer …

28. Unpack/Repack Organizer See attached …

29. Example – unpack • Fractions lesson • IMAPP teacher team, last week … • For more, go to their ICTM presentation

30. Deep PedagogicalContent Knowledge

31. Some current related work … • PCK – Shulman (again) 1986 • Framework for content knowledge and pedagogical content knowledge – Ball, Thames, and Phelps 2008 • Measuring Teachers’ Mathematical Knowledge – Heritage and Vendlinski 2006 • Effects of Teachers’ Mathematical Content Knowledge for Teaching on Student Achievement – Hill, Rowan, Ball 2005 • Mathematics for Teaching – Stylianides and Stylianides 2010 • High school level, Counting – Gilbert and Coomes 2010 • re: conceptual and procedural knowledge (e.g., Lesh 1990?) and deep (e.g., Star 2006)

32. Some Components of Deep Pedagogical Content Knowledge Understand, with strategies for addressing: • Misconceptions • Student Content Difficulties • Learning Progressions • Task Choice and Design • High School Mathematics from an Advanced Perspective • Questioning • Pedagogical Mathematical Language

33. Misconceptions • Anticipate • Identify • Resolve Example: • Modeling circular motion with trig – doubling the angle will double the height?

34. Student Content Difficulties • Anticipate • Identify • Resolve Example: • Counting – The issue of “order” implicit in the Multiplication Principle of Counting (sequence of tasks) versus the issue of order in permutations (choosing from a collection: AB counted as a different possibility than BA) (also see: Gilbert and Coomes 2010)

35. Task Choice and Design • Focus and depth (targeted important mathematics) • Sequence • Questioning • Scaffolding (“goldilocks”, ZPD) • Pivotal Points (identify, facilitate) Examples: • Recursion lesson begins with “pay it forward” (exp, hom) or “handshake problem” (quad, non-hom)? • Slope of perpendicular lines – pattern in data and/or nature of a 90° rotation

36. Learning Progressions • Develop • Analyze • Implement Examples: • Trigonometry, K-12 (large grain), 6-12 (with details) • (Note NCTM discrete mathematics K-12 learning progressions for Counting, Recursion, Vertex-Edge Graphs)

37. School Mathematics from an Advanced Perspective • Direct connections • Inform HS curriculum and instruction (perhaps indirectly) Examples: • Linear – HS algebra vs. linear algebra (e.g., KAT, MSU 2003, used in IMAPP) • Factoring – Factor Theorem, Fund. Thm. of Alg., prime versus irreducible • Independence in probability – trials, outcomes, events, random variables

38. Questioning • General questions and taxonomies of questions are helpful • Content-specific questions are crucial (e.g., Zweng 1980, Hart 1990, Ball 2009) • Provide effective instruction, formative assessment, differentiation Example: • HS teacher: “This table [showing y = 2x] shows constant rate of change.” Questions: What is the constant? [2] How is the change constant? [It goes up by 2 at each step.] How does it go up, by what operation? [multiply by 2] How is “rate of change” defined? [change in y over change in x] And how is the “change in y” computed, what operation? [oh, subtraction, right, so I guess it isn’t constant rate of change] How about this table for y = 2x. The y’s are going up by 2. Is this constant rate of change? [yeah, it goes up by adding 2 each time] So subtraction? [yeah] How does this relate to the features of arithmetic and geometric sequences? … [constant difference versus constant ratio]

39. Pedagogical Mathematical Language • Mathematically accurate • Pedagogically powerful (e.g., bridging, meaning-laden) • Benefits and limitations Example: NEXT/NOW for recursion • Captures essence of recursion used to describe processes of sequential change • Helps make idea accessible to all students • Promotes “semantic learning” as opposed to just “syntactic learning” (a danger when going too fast to subscript notation) • Limitations – very useful for linear and exponential, less for quadratic (hom versus non-hom)

40. Some Components of Deep Pedagogical Content Knowledge Understand, with strategies for addressing: • Misconceptions • Student Content Difficulties • Learning Progressions • Task Choice and Design • High School Mathematics from an Advanced Perspective • Questioning • Pedagogical Mathematical Language

41. Lessons Learned from aLesson Study Approach toHigh School Mathematics Signature Pedagogies Unpack/Repack Process Deep Pedagogical Content Knowledge

43. Sharpen the focus on central mathematical ideas • Make it explicit for teacher and student (mathematical residue) • Make this the focus for the launch, explore, summarize, homework, and assessment • Focus question(s)

44. Critical Reflection • Key to learning, where learning really happens • Can be developed as a student skill, habit, practice • Japanese PS model • Understand the problem • Solve the problem (engage in solving) • Critical reflection • Summarize • Critical reflection woven throughout PS, launch, explore, summarize

45. Maintain connected & coherent • Locally (in individual lessons) and globally (in sets of lessons, units, courses, curricula) • How? Levels of implementation: • Mention connections, prior and future (within strand, across strands, across disciplines, to world outside the classroom, etc.) • Include tasks, problems, questions in the lesson that focus on connecting • Organizational strategy for unit, course, curriculum

46. Maintain cognitive complexity • Stein, et al., QUASAR, 2000 • Problem-Based Instructional Tasks (ESC) • Authentic Intellectual Work • Rigor and Relevance • Bloom