Deep PedagogicalContent Knowledge

Some current related work … PCK – Shulman 1986PCK – Shulman 1986 Framework for content knowledge and pedagogical content Framework for content knowledge and pedagogical content

knowledge – Ball, Thames, and Phelps 2008 knowledge – Ball, Thames, and Phelps 2008 Measuring Teachers’ Mathematical Knowledge – Heritage Measuring Teachers’ Mathematical Knowledge – Heritage

and Vendlinski 2006and Vendlinski 2006 Effects of Teachers’ Mathematical Content Knowledge for Effects of Teachers’ Mathematical Content Knowledge for

Teaching on Student Achievement – Hill, Rowan, Ball Teaching on Student Achievement – Hill, Rowan, Ball 20052005

Mathematics for Teaching – Stylianides and Stylianides Mathematics for Teaching – Stylianides and Stylianides 20102010

High school level, Counting – Gilbert and Coomes 2010High school level, Counting – Gilbert and Coomes 2010 re: conceptual and procedural knowledge (e.g., Lesh re: conceptual and procedural knowledge (e.g., Lesh

1990?) and deep (e.g., Star 2006)1990?) and deep (e.g., Star 2006)

Bottom Line of DPCK

What is the mathematics?What is the mathematics?And how can I help my students understand And how can I help my students understand it deeply?it deeply?

The Mathematics

For Topic X:For Topic X: What is it?What is it?

Deeply, simply, essentially … Deeply, simply, essentially …

How do you do it?How do you do it? Compute it, operate on it, with it …Compute it, operate on it, with it …

What’s it connected to?What’s it connected to? Interconnected web of mathematical ideas, concepts, Interconnected web of mathematical ideas, concepts,

methods, representations …methods, representations …

What’s it good for?What’s it good for? Applications, contexts, models, …Applications, contexts, models, …

Content-Specific Pedagogy

Understand, with strategies for addressing:Understand, with strategies for addressing: MisconceptionsMisconceptions Student Content DifficultiesStudent Content Difficulties Learning ProgressionsLearning Progressions Task Choice and DesignTask Choice and Design High School Mathematics from an High School Mathematics from an

Advanced PerspectiveAdvanced Perspective QuestioningQuestioning Pedagogical Mathematical LanguagePedagogical Mathematical Language

Misconceptions

AnticipateAnticipate IdentifyIdentify ResolveResolve

Example:Example: Modeling circular motion with trig – doubling Modeling circular motion with trig – doubling

the angle will double the height?the angle will double the height?

Student Content Difficulties

AnticipateAnticipate IdentifyIdentify ResolveResolveExample:Example:

Counting – The issue of “order” implicit in the Counting – The issue of “order” implicit in the Multiplication Principle of Counting (sequence of Multiplication Principle of Counting (sequence of tasks) versus the issue of order in permutations tasks) versus the issue of order in permutations (choosing from a collection: AB counted as a different (choosing from a collection: AB counted as a different possibility than BA)possibility than BA) (also see: Gilbert and Coomes 2010) (also see: Gilbert and Coomes 2010)

Task Choice and Design Focus and depth (targeted important mathematics)Focus and depth (targeted important mathematics) SequenceSequence QuestioningQuestioning Scaffolding (“goldilocks”, ZPD)Scaffolding (“goldilocks”, ZPD) Pivotal Points (identify, facilitate)Pivotal Points (identify, facilitate)Examples:Examples:

Sequence – Recursion lesson begins with “pay it Sequence – Recursion lesson begins with “pay it forward” (exp, hom) or “handshake problem” (quad, forward” (exp, hom) or “handshake problem” (quad, non-hom)?non-hom)?

Focus and depth – Slope of perpendicular lines: pattern Focus and depth – Slope of perpendicular lines: pattern in data and/or nature of a 90° rotationin data and/or nature of a 90° rotation

Learning Progressions

DevelopDevelop AnalyzeAnalyze ImplementImplement

Examples:Examples: Trigonometry, K-12 (large grain), 6-12 (with details)Trigonometry, K-12 (large grain), 6-12 (with details) (Note NCTM discrete mathematics K-12 learning (Note NCTM discrete mathematics K-12 learning

progressions for Counting, Recursion, Vertex-Edge progressions for Counting, Recursion, Vertex-Edge Graphs)Graphs)

School Mathematics from an Advanced Perspective Direct connectionsDirect connections Inform HS curriculum and instruction (perhaps Inform HS curriculum and instruction (perhaps

indirectly)indirectly)

Examples:Examples: Linear – HS algebra vs. linear algebra (e.g., KAT, MSU Linear – HS algebra vs. linear algebra (e.g., KAT, MSU

2003, used in IMAPP)2003, used in IMAPP) Factoring – Factor Theorem, Fund. Thm. of Alg., prime Factoring – Factor Theorem, Fund. Thm. of Alg., prime

versus irreducibleversus irreducible Independence in probability – trials, outcomes, events, Independence in probability – trials, outcomes, events,

random variablesrandom variables

Questioning General questions and taxonomies of questions are helpfulGeneral questions and taxonomies of questions are helpful Content-specific questions are crucialContent-specific questions are crucial (e.g., Zweng 1980, Hart (e.g., Zweng 1980, Hart

1990, Ball 2009)1990, Ball 2009) Provide effective instruction, formative assessment, differentiationProvide effective instruction, formative assessment, differentiationExample:Example:

HS teacher: “This table [showing y = 2HS teacher: “This table [showing y = 2xx] shows constant rate of ] shows constant rate of change.” Questions: What is the constant? [2] How is the change 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 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? 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” [change in y over change in x] And how is the “change in y” computed, what operation? [oh, subtraction, right, so I guess it computed, what operation? [oh, subtraction, right, so I guess it isn’t constant rate of change] How about this table for y = 2x. The 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 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 goes up by adding 2 each time] So subtraction? [yeah] How does this relate to the features of arithmetic and geometric sequences? this relate to the features of arithmetic and geometric sequences? … [constant difference versus constant ratio]… [constant difference versus constant ratio]

Pedagogical Mathematical Language Mathematically accurateMathematically accurate Pedagogically powerful (e.g., bridging, meaning-laden)Pedagogically powerful (e.g., bridging, meaning-laden) Benefits and limitationsBenefits and limitationsExample: NEXT/NOW for recursionExample: NEXT/NOW for recursion

Captures essence of recursion used to describe Captures essence of recursion used to describe processes of sequential changeprocesses of sequential change

Helps make idea accessible to all studentsHelps make idea accessible to all students Promotes “semantic learning” as opposed to just Promotes “semantic learning” as opposed to just

“syntactic learning” (a danger when going too fast to “syntactic learning” (a danger when going too fast to subscript notation)subscript notation)

Limitations – very useful for linear and exponential, Limitations – very useful for linear and exponential, less for quadratic (hom versus non-hom)less for quadratic (hom versus non-hom)