1. Pedagogical Content Knowledge
Secondary Mathematics
NextGen STEM Teacher Preparation in WA State
2019 PCK Working Group

2. Essential Questions
What is pedagogical content knowledge (PCK) as it relates to secondary mathematics?
What knowledge should secondary math students acquire?
What professional development is needed for teaching pedagogical content knowledge (PCK) in secondary mathematics?
Essential Questions for this presentation

3. PCK and Secondary Math
Subject matter knowledge: teacher’s understanding of big ideas (Hauk et al., 2014).
Pedagogical Content Knowledge: how the understanding informs thinking and builds knowledge about content & student; content & teaching; and content of curriculum (Hauk et al., 2014).
Subject matter knowledge for secondary and post-secondary is different from K-8
To build a PCK model, one must first unpack ideas:
Address ways of understanding and thinking about mathematics to teach
Be mindful of the cultural heterogeneity in the classroom

4. Common Core State Standards in Mathematics
What knowledge should secondary math students acquire?
In WA State, this is defined by the CCSS. This image illustrates the content strands, broken up into K-5, 6-8, and high school. It is important for teachers to know the content knowledge that is expected of high school students, as well as the content knowledge that students will come into high school classrooms with. Not present in this image is the “modeling” strand, which exists in all the HS content clusters.

5. What Does the Research Say?
Ball, Thames, & Phelps, 2008
Hauk et al., 2014
How do teachers need to know the content required to teach in their own classrooms?
Extended Model of PCK- MKT model plus Knowledge of Discourse
Analyzing teaching practice and developing instruments to test the ideas
How do language and culture shape teaching and learning?
Content Knowledge → CCK and Specialized Content Knowledge
Pedagogical Content Knowledge → knowledge of content & students and knowledge of content & teaching
Curricular Thinking, Anticipatory Thinking, & Implementation Thinking
Ball et al. - proposed the egg diagram with suggestion that more research is needed to refine and continue the the work of definition how exactly teachers need to know the content in their own classrooms to teach it.
Hauk et al. - proposed the extended model to include Mathematical Knowledge for Teaching (MKT)/

6. Ball et al. (2008) - Guiding Questions
What are the recurrent tasks and problems of teaching mathematics? What do teachers do as they teach mathematics?
What mathematical knowledge, skills, and sensibilities are required to manage these tasks?
Teaching includes:
interactive work of teaching lessons planning
evaluating student work writing and grading assessments
explaining classwork to parents making and managing homework
attending to equity dealing with administration
The researchers decided that rather than focus on the what, they wanted to focus on the HOW. The what has an “obvious” answer of - the content that teachers are required to teach in their specific classrooms.

7. Further Research Suggested by Ball et al.
Are there aspects of teachers’ content knowledge that predict student achievement?
Do different approaches to teacher development impact particular aspects of teachers’ PCK?
Is it possible to get a clearer sense of the actual content teachers need to teach to clarify the content curriculum?
As we move forward, Ball et al. (2008) suggest that we look into these three questions to guide our future work.
Related to third question, final CCSS standards weren’t announced until June these national standards could be used to guide the work to answer the third question, but also bring to light the question of “what do we do if CCSS goes away?”

8. More suitable for K-8 Secondary and Post
Hauk et al. extended the existing model which they felt was geared more toward the K-8 to include language. Research shows that language and illustrations can be used to support/enhance conversations in the classroom.
MKT seem to be more linear in that KCS + KCT + KCC = PCK, whereas a more complex model beyond the linear assumption may be a better fit for the dynamics of secondary mathematics and higher education.

9. Knowledge of Discourse
Knowledge about the culturally embedded nature of discourse
Inquiry and forms of communication in mathematics
Teacher knows of normative and non-standard mathematical vocabularies, representations, and artifacts
Curricular thinking
How one might approach topics, procedures and concepts
Anticipatory thinking
How learners may engage in the content, processes and concepts
Implementation thinking
How to enact teaching intentions in the classroom
Curricular thinking
Connects knowledge of discourse with knowledge of curriculum (curricular thinking)
Thinking about how to approach mathematical topics, procedures and concepts as well as their relationships
Conventions for reading, writing, and speaking them that’s in the curriculum
This contributes to a profound understanding of mathematics (Ma, 1999)
Anticipatory thinking
Connects knowledge of discourse with knowledge of content and students
Thinking about how learners may engage in the content, processes, and concepts
Includes awareness of and responsiveness to student thinking
Piaget’s “decentering”- shifting from ego-centric to an ego-relative view for seeing or communicating about an idea or way of thinking from the perspective of another
Perspective - shifting - deeply connected to discourse through awareness of ‘other’ as different from ‘self’
May be connected to intercultural orientation
Implementation thinking
Connects knowledge of discourse with knowledge of content and teaching
Thinking about how to enact teaching intentions in the classroom
How to adapt teaching according to the content and socio-cultural context
How to act on decision shaped by intercultural orientation

10. What does Knowledge of Discourse look like in the classroom?
From Vignette 2 (Hauk et al., 2014, p. A30:
Curricular Thinking
Questioning provides cognitive conflicts about central conflicts (temporal ordering).
Scaffolds the use of key mathematical practices (visual organization of information (table), parsing arithmetical relationships and reasoning).
Opportunity to reflect on what they think.
Anticipatory Thinking
Questioning to elicit student thinking.
Teacher anticipates prior knowledge.
Students are asked to share out their rationale as part of working through the problem.
In this scenario, keep in mind this is the early stage of building Knowledge of Discourse.
Implementation Thinking
Teacher elicits and connects student thinking to procedures and concepts.
Students are able to listen and make sense of each other’s ideas.
The teacher emphasizes reasoning rather than the product.
Multiple modes of discourse: visual reference on the board, student generated statements, table, confirming questions.
Students are asked to clarify their terminology and language.
Curricular Thinking
Questioning provides cognitive conflicts about central conflicts (temporal ordering)
Scaffolds the use of key mathematical practices (visual organization of information (table), parsing arithmetical relationships and reasoning)
Opportunity to reflect on what they think
Anticipatory Thinking
Questioning to elicit student thinking
Teacher anticipates prior knowledge
Students are asked to share out their rationale as part of working through the problem
In this scenario, keep in mind this is the early stage of building Knowledge of Discourse
Implementation Thinking
Teacher elicits and connects student thinking to procedures and concepts
Students are able to listen and make sense of each other’s ideas
The teacher emphasizes reasoning rather than the product
Multiple modes of discourse: visual reference on the board, student generated statements, table, confirming questions
Students are asked to clarify their terminology and language
Hauk et al., 2014, p. A30

11. Next Steps - Essential Questions to consider
How is PCK taught to novice teachers in Washington State Teacher Programs?
What resources are currently available for teaching PCK to Washington State Teacher Programs?