Transcript LessonsLearnedLSA

Lessons Learned from a
Lesson Study Approach to
High School Mathematics
Signature Pedagogies
Unpack/Repack Process
Deep Pedagogical Content Knowledge
Presentation Outline
Overview of lesson study project
 Introduction, with examples, of:
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Signature pedagogies
 Unpack/Repack process
 Deep Pedagogical Content Knowledge
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Important
Mathematics
and
Powerful
Pedagogy
IMAPP Project
Important Mathematics
and Powerful Pedagogy
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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
Project Vision …
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
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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
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Planning the Lesson
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Unpack/Repack (see later)
For topic taught, consider:
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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)
Planning the Lesson (cont.)
Focus Question
 Misconceptions, trouble spots
 Pivotal points
 Questions that probe and deepen student
understanding
 Opportunites for critical reflection
 Summarize
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Teaching the Lesson
Signature Pedagogies (see later)
 Formative Assessment – re: Dylan
Wiliam: “formative instruction”
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Observation Guidelines
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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:
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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
Debrief the Lesson:
Math, Teaching, Learning
Observer notes
 Discussion
 Commentary
 Based on what I saw, what I heard
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Some Revision Guidelines
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Sharpen the focus on targeted, student-learningappropriate, 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
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
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Academic Year
Activities and Data
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Checklist …
Signature
Pedagogies
Signature Pedagogies
Potent teaching approaches that warrant
common application
 Global to local
 Phrase borrowed and idea adapted from
Shulman (2005)
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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
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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
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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)
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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)
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Signature Pedagogies:
Strand Examples
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Algebra
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Geometry
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Functions approach to teaching algebra
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
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Signature Pedagogies:
Topic Examples
 Functions
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Recursion
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Include a recursive view of functions
Include use of pedagogically powerful
informal notation, like NEXT and NOW
Vertex-Edge Graphs
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Discrete mathematical modeling
Signature Pedagogies
Potent teaching approaches that warrant
common application
 Global to local
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Unpack/Repack
Process
Planning the Lesson
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Unpack the Math
What is the connected and coherent web of mathematical
knowledge that comprises and surrounds this topic?
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Unpack Teaching and Learning
What theory and practice related to teaching and learning
are germane to this topic?
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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 …
Unpack/Repack Organizer
See attached …
Example – unpack
Fractions lesson
 IMAPP teacher team, last week …
 For more, go to their ICTM presentation
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Deep Pedagogical
Content Knowledge
Some current related work …
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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)
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
Misconceptions
Anticipate
 Identify
 Resolve
Example:
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Modeling circular motion with trig – doubling
the angle will double the height?
Student Content Difficulties
Anticipate
 Identify
 Resolve
Example:
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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)
Task Choice and Design
Focus and depth (targeted important mathematics)
 Sequence
 Questioning
 Scaffolding (“goldilocks”, ZPD)
 Pivotal Points (identify, facilitate)
Examples:
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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
Learning Progressions
Develop
 Analyze
 Implement
Examples:
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Trigonometry, K-12 (large grain), 6-12 (with details)
(Note NCTM discrete mathematics K-12 learning
progressions for Counting, Recursion, Vertex-Edge
Graphs)
School Mathematics from an
Advanced Perspective
Direct connections
 Inform HS curriculum and instruction (perhaps
indirectly)
Examples:
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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
Questioning
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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]
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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)
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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
Lessons Learned from a
Lesson Study Approach to
High School Mathematics
Signature Pedagogies
Unpack/Repack Process
Deep Pedagogical Content Knowledge
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)
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Critical Reflection
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Key to learning, where learning really happens
Can be developed as a student skill, habit, practice
Japanese PS model
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Understand the problem
Solve the problem (engage in solving)
Critical reflection
Summarize
Critical reflection woven throughout PS, launch,
explore, summarize
Maintain connected & coherent
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Locally (in individual lessons) and globally (in sets of
lessons, units, courses, curricula)
How? Levels of implementation:
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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
Maintain cognitive complexity
Stein, et al., QUASAR, 2000
 Problem-Based Instructional Tasks (ESC)
 Authentic Intellectual Work
 Rigor and Relevance
 Bloom
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