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Deep Pedagogical Content Knowledge

Deep Pedagogical Content Knowledge. Some current related work …. PCK – Shulman 1986 Framework for content knowledge and pedagogical content knowledge – Ball, Thames, and Phelps 2008 Measuring Teachers’ Mathematical Knowledge – Heritage and Vendlinski 2006

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Deep Pedagogical Content Knowledge

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  1. Deep PedagogicalContent Knowledge

  2. Some current related work … • PCK – Shulman 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)

  3. Bottom Line of DPCK What is the mathematics? And how can I help my students understand it deeply?

  4. The Mathematics For Topic X: • What is it? • Deeply, simply, essentially … • How do you do it? • Compute it, operate on it, with it … • What’s it connected to? • Interconnected web of mathematical ideas, concepts, methods, representations … • What’s it good for? • Applications, contexts, models, …

  5. Content-Specific Pedagogy 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

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

  7. 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)

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

  9. 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)

  10. 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

  11. 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]

  12. 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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