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 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)

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

The Mathematics For Topic X: What is it? How do you do 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, …

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

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

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) Perhaps this combines with misconceptions …

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

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) Note the word trigonometry only appears in high school, but the ideas of trigonometry occur K-12

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

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]

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)