A presentation given at Oregon's Teacher of Teachers of Mathematics (TOTOM) Conference - an AMTE Affiliate.

1. Using Rich Assessments and the
STEM Pedagogical Content Knowledge Rubric
to Examine Teacher Learning
Nicole Miller Rigelman Teachers of Teachers of Mathematics
Conference
Portland State University Monmouth, Oregon
rigelman@pdx.edu September 2014

2. • Students were given the following task:
It takes 3/4 liter of paint to cover 3/5
m2. How much paint is needed to paint
1m2? Explain your reasoning and
justify your answer.
What are some strategies, both correct and incorrect, that
you think students will use to solve this task? Show the
specific ways you are thinking will come forward.
Painting Task

3. • Doing the math
• Framing our assessment work
• Defining Pedagogical Content Knowledge (PCK)
• Eliciting and assessing PCK
• A sample assessment pairing
• Sample teacher responses
• What we are learning
• Considering implications for your own practice
Session Overview

4. “… assessment is a powerful tool
for raising the quality of teaching
and learning. It should be used
diagnostically and interactively,
not as a form of autopsy.”
- Shulman, 2009, p. 237

5. Supports for
implementation
of effective
STEM
practices
Effective
teacher
professional
development
experiences
Effective
practices for
STEM learning
environments
STEM Common
Measurement System
Effective
student learning
environments
in STEM
- Saxton, et al., 2014

6. Supports for
implementation
of effective
STEM
practices
Effective
teacher
professional
development
experiences
Effective
practices for
STEM learning
environments
Effective
student learning
environments
in STEM
Academic
Identity
Supportive
Teacher-Student
Relationships
Teacher
Self-Efficacy
Transformational
Leadership
Collective
Teacher Efficacy
Motivational
Resilience
Higher-Order
Cognitive Skills
Application of
Conceptual
Knowledge
Instructional Practices
Pedagogical Content Knowledge
- Saxton, et al., 2014

7. • A teacher’s PCK is an important outcome in describing
the quality of teaching because it impacts every part of a
teacher’s professional practice from planning,
implementation, assessment, reflection, and revision for
the future.
- Park, et al., 2011; Shulman, 1987
Developing PCK

8. Specialized
Content
Knowledge
(SCK)
Knowledge
of Content
and Students
(KCS)
Knowledge
of Content
and Teaching
(KCT)
Pedagogical Content
Knowledge
Knowledge
of Content
Curriculum
Subject Matter
Knowledge
Common
Content
Knowledge
(CCK)
Horizon
Content
Knowledge
Mathematical Knowledge for
Teaching (MKT)
- Ball, Thames, & Phelps, 2008

9. • There are relatively few studies, in narrow content areas,
examining how PD focused on PCK influences classroom
performance and student learning.
• PCK is not well-defined; KCS and KCT are
underspecified and the relationships to student
achievement are undemonstrated.
- Hill, Ball, & Schilling, 2008
Dilemmas regarding PCK

10. 1. Teachers’ knowledge of student thinking about
specific STEM topics including prior knowledge,
misconceptions, learning progressions, common
difficulties, and developmentally appropriate levels of
understanding.
2. Teachers’ understanding and use of the effective
strategies for specific STEM topics including strategies
to engage students in inquiry, represent STEM
phenomena, and guide discourse about the STEM topic.
Defining PCK

11. STEM Pedagogical Content Knowledge (PCK) Rubric D3.0 V2
Knowledge of student thinking about specific STEM topics including prior
knowledge, misconceptions, learning progressions, common difficulties, and
developmentally appropriate levels of understanding.
In planning, implementation, &/or reflection the teacher demonstrates:
Understanding and use of the effective strategies for specific STEM topics
including strategies to engage students in inquiry, represent STEM phenomena, and
guide discourse about the STEM topic.
In planning, implementation, &/or reflection the teacher demonstrates:
4
a) Sophisticated understanding of student prior knowledge including misconceptions
and common learning difficulties; this level of understanding includes student
thinking from multiple experiences in and out of school as the basis of learning.
b) Careful consideration of the specific STEM topic based on the grade level of
students, learning progressions, and developmentally appropriate levels of
understanding, which reveals an understanding that the specific idea is
challenging depending on grade level of students.
c) Many, diverse strategies including inquiry strategies to challenge student
thinking or resolve learning difficulties about the specific STEM topic.
d) Sophisticated understanding of representations that are specific to the STEM
topic, pedagogically effective, accurate, and strongly-linked to students’ prior
knowledge and experience.
e) Language use around the specific STEM topic includes multiple modes of
communication, an emphasis on students communicating reasoning with many
questions clearly designed to probe for student understanding of the specific
STEM topic.
5/14 Draft 3. Based on Park et al., 2011; Lee et al., 2007; Schneider & Plasman, 2011. PMSP, Office of Research and Assessment. Contact Emily Saxton, esaxton@pdx.edu
4
3
a) Adequate understanding of student prior knowledge including misconceptions
and common learning difficulties; this level of understanding includes specific
examples of possible student thinking as well as ways to look for that student
thinking by listening to students, reading students’ work, etc.
b) Some consideration of the specific STEM topic based on the grade level of
students and developmentally appropriate levels of understanding, which reveals
an understanding that the specific idea is challenging to students of a certain age
or when they don’t have enough background knowledge to understand.
c) Some inquiry strategies to challenge student thinking or resolve learning
difficulties about the specific STEM topic.
d) Adequate understanding of representations that are specific to the STEM topic,
pedagogically effective, accurate, and well-linked to students’ prior
knowledge.
e) Language use around the specific STEM topic includes some variety in terms
of modes of communication (i.e. discussion (possibly in groups of various
sizes) and writing (e.g., journals) for students to describe their ideas and
explain their thoughts about the specific STEM concept) with some questions
designed to probe for student understanding of the specific STEM topic.
3
2
a) Narrow understanding of student prior knowledge including misconceptions and
common learning difficulties; this level of understanding includes primarily
consideration of wrong ideas/misconceptions or vague assumptions about student
thinking.
b) Limited consideration of the specific STEM topic based on the grade level of
students or developmentally appropriate levels of understanding, which reveals a
general understanding of the specific idea being challenging because of
vocabulary or abstractness.
c) Few inquiry strategies to challenge student thinking or resolve learning
difficulties about the specific STEM topic; those strategies planned are not
diverse strategies.
d) Narrow understanding of representations that are specific to the STEM topic;
planned or implemented representations are pedagogically limited,
underdeveloped, and/or weakly linked to students’ prior knowledge.
e) Language use around the specific STEM topic is somewhat limited including
whole class discussion, reading textbooks, or writing reports or summaries
with few questions designed to probe for student understanding of the specific
STEM topic.
2
1
a) No understanding of student prior knowledge including misconceptions and
common learning difficulties, the assumption that students have no prior beliefs
beyond what was taught in school, or an awareness of prior knowledge but no
clear incorporation into lessons.
b) No consideration of the specific STEM topic based on the grade level of students
or developmentally appropriate levels of understanding, which reveals an
assumption about the specific idea not being challenging for students or
challenges being the same challenges teachers themselves had as students.
c) Limited strategies and no inquiry strategies to challenge student thinking or
resolve learning difficulties about the specific STEM topic.
d) Limited or no understanding of representations (i.e. illustrations, examples,
models, analogies, and demonstrations) that are specific to the STEM topic;
planned or implemented representations are ineffective, inaccurate, and/or not
linked to students’ prior knowledge.
e) Language use around the specific STEM topic is limited to teacher directed
instruction and student–student talk is not included or no questions designed
to probe for student understanding of the specific STEM topic.
1
0 Evidence either missing or to insufficient to score. Evidence either missing or to insufficient to score. 0

12. • Solve the task in multiple ways.
• Examine student work and list observations about each.
Student What I notice… What I wonder…
A
B
C
• Suggest potential next steps with these students that
would move their thinking forward.
Anticipating, Examining, & Using Student
Thinking

13. • Examine the teacher work samples.
• Which aspects of pedagogical content knowledge do you
see evidence of in the teacher response?
• What edits would you suggest to the prompts in order to
elicit more from the teacher?
Anticipating, Examining, & Using Student
Thinking

14. • Identify/adapt a low floor high ceiling task to implement
in your classroom next year (Boaler, 2014).
• Create a discourse plan, anticipating student thinking, and
strategizing how you will orchestrate a whole group
discussion about students’ ideas on the task (Smith &
Stein, 2011).
• Plan the launch, explore, and summarize portions of the
lesson including specific questions you will ask or moves
you will make to encourage students’ own mathematical
thinking and problem solving (Rigelman, 2011).
A Project as a Post-Assessment

15. • Teachers have a difficult time anticipating students’ strategies
on tasks, particularly when the task is new to them.
• When content is unfamiliar/challenging to teachers, their
planned pedagogical moves tend to be generic.
• There are differences between teachers’ performance with a
task that is outside their curriculum versus a task selected from
inside their curriculum. This may be due to several factors
including: teacher choice, resources available, and time.
• To better elicit teacher knowledge through their planning, it
may be necessary to call for annotations on the lesson plan
such that the reasoning behind the teacher actions/decisions is
made clear.
What we are learning…

16. • What thoughts/ideas have emerged for you regarding how
you design assessments in your courses for teachers?
• What are potential implications for your practice?
Implications

17. Boaler, J. (2014). Low floor high ceiling tasks. Available at: http://youcubed.org/teachers/2014/low-floor-high-ceiling-
tasks/
Olson, T. A., & Olson, M. (2013). The importance of context when presenting fraction problems to help
students formulate models and representations as solution strategies. Journal of Mathematics Education
Leadership 14(2), 38-47.
Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing
and measuring teachers’ topic-specific knowledge of students. Journal of Research in Mathematics
Education, 39(4), 372-400.
Park, S., Jang, J., Chen, Y., & Jung, J. (2011). Is pedagogical content knowledge (PCK) necessary for reformed
science teaching? Evidence from an empirical study. Research in Science Education, 41, 245–260.
Rigelman, N. R. (2011). Bring•do•leave: Nurturing reasoning and sense making. Teaching Children
Mathematics, 18(3), 190-197.
Saxton, E., Burns, R., Holveck, S., Kelley, S., Prince, D., Rigelman, N., & Skinner, E. A. (2014). A common
measurement system for K-12 STEM education: Adopting and evaluation methodology that elevates
theoretical foundations and systems thinking. Studies in Educational Evaluation, 40(1), 18-35. DOI:
http://dx.doi.org/10.1016/j.stueduc.2013.11.005.
Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational
Review, 57(1), 1–21.
Shulman, L. S. (2009). Assessment of teaching or assessment for teaching? Reflections on the invitational
conference. In G. H. Gitomer (Ed.), Measurement issues and assessment for teaching quality. Thousand
Oaks, CA: Sage Publications.
Smith, M. S. & Stein, M. K. (2011). Five practices for orchestrating productive mathematics discussions.
Reston, VA: National Council of Teachers of Mathematics.
References