This session was presented at the annual National Council of Teachers of Mathematics (NCTM) Annual Conference & Exposition held in San Franciso, CA from April 13-16, 2016.

1. I will demonstrating the use of one
of my favorite video tools. If you
don’t wish to be on camera, just
turn your head or look down
please. Thanks!

2. Goals for Today’s Session
 Learn two frameworks that can be used to inform
instructional practice
 Problem Solving Cycle (Borko et. al, 2015)
 Productive Practices for Mathematical Reasoning (Heng, 2015)
 Relate frameworks to high leverage effective teaching
practices in mathematics
 Experience how frameworks combined with video reflection
can have a powerful effect on how teachers’ consider
instruction
 Preparing for video use in a school setting

3. Background
 Stockton received state level Math Science Partnership
grant targeting common core math and instructional
practice of elementary & MS teachers
 Used 2 frameworks as foundations for our work
 Problem Solving Cycle (PSC)- overall steps taken w/teachers
 Productive Practices/Noticing- provides deeper dive into
steps of PSC & links to high leverage effective teaching
practices
 4 PD sessions over course of year with instructional
coaching between sessions (f2f & virtual via EdThena)
 Video blended into PD for training then shifted to video of
teacher practices in their classrooms

4. 0 We are going to begin by watching a 2nd grade teacher
instructing a small group of students.
0 Write down what you notice. We will refer back to
your notes later!

5. Mathematical Knowledge for Teaching or MKT is “the
professional knowledge that mathematics teachers need to
effectively carry out the mathematical work of teaching”
(Borko et al, 2015)
The focus of our grant was on the development of MKT.

6. Knowing
Solve the following problem…
Julie has 38 boxes of oranges in her
delivery truck. Each box holds 12
oranges. How many oranges does Julie
have in her truck?

7. Knowing for teaching
You will be shown an example of student work
for the same problem as before. Be ready to
consider….
How was the answer produced?
What might lead a student to make this error?
What methods could you use to teach this
concept beyond the algorithm to help students
see the error in their ways?
Sample work from:
http://mathmistakes.org/multiplication-strategies-my-students-are-starting-with/

10. The Mathematical Knowledge of
Teaching (MKT)
Mathematical
Knowledge for
Teaching
Mathematical
Content
Knowledge
Common
Knowledge of
Mathematics
Specialized
knowledge of
Math
Pedagogical
Content
Knowledge
Knowledge of
Content &
Teaching
Knowledge of
Content &
Students
(Ball, Thames, & Phelps, 2008)

11. Mathematical Content
Knowledge
Common
knowledgeBasic understanding of math
skills, procedures, and concepts
acquired by a well-educated
adult.
You can….
-calculate an answer correctly
-use terms and notations
accurately
-recognize a wrong answer….
Specialized knowledge
Deeper, more nuanced
understanding of
mathematics.
You can….
-Respond to “why” questions
-Modify tasks to make it
easier or harder
-Evaluate plausibility of a
student’s claim….

12. Pedagogical Content
Knowledge
Knowledge of
Content & Teaching
Knowing about the content of
mathematics and methods of
teaching it in a way that is
accessible to learners.
You can….
- Sequence mathematical
content
- Select appropriate ways to
illustrate representations of
content
Knowledge of
Content & Students
Knowing about students and
how they make sense of, learn,
and understand mathematics.
You can….
- Anticipate what students are
thinking
- Predict what students will
find interesting & motivating
- Anticipate what a student will
find difficult

13. Problem Solving Cycle
Framework
Is a professional development cycle focusing on problem
solving that supports the development of MKT by
 Providing opportunities for you to develop math and
pedagogical knowledge through the lens of problem solving
in your classroom
 Offering relevance by looking at what happens in your own
classroom during the cycle
 Building awareness of instructional moves and practices of
others through conversation, collaboration, and focused
workshops
(Borko et. al, 2015)

14. The Problem Solving Cycle
Solve
Problem and
Develop
Lesson Plan
Teach and
Video-
record
Problem
Video
Analysis of
Student
Thinking (&
Instruction)
Video
Analysis of
Instruction
(& Student
Thinking)
+ student artifacts

15. “Problems” of the
Problem Solving Cycle
 Address multiple mathematical concepts and skills
 Are accessible to learners with different levels of
knowledge
 Have multiple entry and exit points
 Have an imaginable context
 Provide a foundation for productive mathematical
communication
 Are both challenging for teachers and appropriate for
students
Solve
Problem &
Develop
Lesson
Teach &
Video
Analyze
Student
Thinking
Analyze
Instruction

16. Which is a true “problem”?

19. Preparing the lesson…..
 Explore & identify “true” problem
 Work collaboratively on the problems
 Get solution
 Consider potential strategies to solve
 Develop a unique lesson plan for the
classroom
 Set learning goal
 Select problem
 Predict solution strategies
 Structure procedure of lesson to get at student
thinking*
 Key questions
 Organizing students
Solve
Problem &
Develop
Lesson
Teach &
Video
Analyze
Student
Thinking
Analyze
Instruction

20. Getting a sense of the PSC
 Prior to teaching lessons, we spent some time looking
at exemplars of practice from NCTM’s Principles &
Actions toolkit (http://www.nctm.org/PtA/)
 We discussed & focused in on specific effective
mathematics teaching practices
From: NCTM’s Principles to Actions toolkit (see references)

21. Effective Mathematics Teaching Practices
1. Establish mathematics goals to focus learning.
2. Implement tasks that promote reasoning and problem
solving.
3. Use and connect mathematical representations.
4. Facilitate meaningful mathematical discourse.
5. Pose purposeful questions.
6. Build procedural fluency from conceptual
understanding.
7. Support productive struggle in learning mathematics.
8. Elicit and use evidence of student thinking.
National Council of Teachers of Mathematics. (2014). Principles to actions:
Ensuring mathematical success for all. Reston, VA: Author.

22. Take a moment to review the task below. Consider how students might
approach the problem and what they might struggle with.

23. Watch video
Use the back of your half task sheet to jot down what
you see. Make two columns. One for teacher actions and
one for student actions. Be ready to discuss

24. 1. Establish mathematics goals to focus
learning.
2. Implement tasks that promote
reasoning and problem solving.
3. Use and connect mathematical
representations.
4. Facilitate meaningful mathematical
discourse.
5. Pose purposeful questions.
6. Build procedural fluency from
conceptual understanding.
7. Support productive struggle in
learning mathematics.
8. Elicit and use evidence of student
thinking.
What were the
teacher actions?
Student actions?
What effective
practices were
integrated?

25. 5 Steps for Orchestrating
Productive Mathematics
Discussions
We found using this book helped teachers to
organize their lessons and focus on what to do with
students
5 practices….
1. Anticipating likely responses
2. Monitoring students actual responses
3. Selecting particular students to present
4. Sequencing the student responses that will be
shared
5. Connection different students responses and
to key mathematical ideas

26. The Case of Mr. Harris and
the Band Concert Task

27. How does each representation match the story
situation and the structure of multiplication?
Jasmine Kenneth
Teresa
Consider Lines 52-57.
Why did Mr. Harris select and sequence the work of these
three students and how did that support student learning?

28. Learning from teaching to improve
teaching requires teachers to develop
the eyes to see, the ears to hear and the
mind to think.
Mathematics Teacher Noticing
Heng, 2015

29. Productive Classroom Practice
Design tasks that reveal student
thinking
Listening to and responding to
student thinking
Reflecting about student thinking

30. Solve
Problem
&
Develop
Lesson
Teach &
Video
Analyze
Student
Thinking
Analyze
Instruction
Problem Solving Cycle

31. Heng, 2015

32. Attending to Student
Thinking
• Attending to students thinking means
noticing students’ thinking.
• Attending to student thinking helps the
teacher determine the extent to which
students are reaching the learning goals.
• When you attend to student thinking what
you notice should be used to make
instructional decisions during the lesson
and to prepare for subsequent lessons.

33. Let’s try “noticing”
We are going to watch the half problem again. This time
focus on:
-what students say and how they describe the math
-where there is confusion or points of understanding
Use the chart to track your thoughts….
Who Viewing Analyzing Refining

34. Teacher “Noticing”
 Let’s watch the video from the very beginning again.
For this one, focus on “noticing”. Specifically notice
student responses. Listen carefully to the math talk
going on.
 You are watching a 2nd grade teacher instructing a
small group of students.
 Jot down observations on the Analysis chart you are
given.

35. Steps to develop instructional
skills using PSC model….
0 Learned & discussed research proven high leverage practices in
math
0 Use Principles to Actions book & web-based resources at NCTM’s
website
0 Spent time on finding or developing quality problems/tasks to
use
0 Good places to get started… Inside Mathematics POW, Illustrative
Math POW, & NCTM Problem of Month
0 Taught how to orchestrate productive mathematics discussion
0 Great framework in the 5 Practices NCTM book
0 Practiced productive noticing with focus on “attending” (ie.
focusing on student conversation & thinking vs teacher action)

36. Why video?
0 Teaching is complex with many actions going on at same
time.
0 You can view the same video many times to look at things
from different perspectives.
0 You can examine students’ thinking and learning.
0 Makes conversations about teaching relevant because it is
about your own classroom.
0 Allows you to share methods with colleagues.
Solve
Problem
&
Develop
Lesson
Teach &
Video
Analyze
Student
Thinking
Analyze
Instructio
n

37. A look at the impact of PSC &
why we chose it….
 2 year PSC project
 13 teachers & 5 teacher leaders videorecorded
 One in beginning
 One at end
 Two times- “typical” lesson & PSC specific
 Used the Mathematical Quality Instrument (MQI)
observation scale (Learning Mathematics for Teaching
Project, 2011)
(Borko et al., 2015)

38. Impact of a sustained PSC

40. What we’ve learned so far
 Teachers are more apt to consider change when it
focuses on the students
 Video is hard at first, but soon becomes a tool for
conversation. The focus quickly shifts from judging
their teaching to listening/attending to student
thinking
 Centering PD around real classroom practice in their
classrooms & schools helps build authority &
ownership over the work
 It’s essential to offer foundational structures to help
analyze instruction and focus on effective
instructional practices

41. Challenges
 Having access to the technology tools needed to
capture and view video
 Comfort watching yourself teaching
 Timing the PDs so you can review video then provide
time to implement
 Virtual vs face-to-face
 Teacher buy in and acclimating to a new way of
approaching PD. It takes time!

42. Considerations as you get started
 Determine school policy regarding video in classrooms
 Do your homework on the value of video-based
reflection. There is a great deal of evidence on why it’s
valuable. (See my references)
 Establish “by in” with administration & teachers
 Value as a PD tool
 Only used in-house
 Non-evaluative for “coaching”
 Focus on student vs teacher
 Have the right equipment and tools including recording
& accessing video We used….
 Recording…Microsoft Surface Pros, iPad/iPod w/Swivl
 Sharing…. EdThena (*can be costly), Swivl Cloud Pro (cheaper)
I highly recommend El et. al article in Mathematics Teacher Educator!

43. Set ground rules for video use as PD
NORMS FOR WATCHING VIDEO
Video clips are examples, not exemplars.
 To spur discussion, not criticism
Video clips are for investigation of teaching and learning,
not evaluation of the teacher.
 To spur inquiry, not judgment
Video clips are snapshots of learning, not an entire lesson.
 To focus attention on a particular moment, not what came
before or after
Video clips are for an examination of a particular
interaction.
 To provide evidence for claims by citing specific examples
(Borko et. al, 2015, p.45)

45. References
 Borko, H., Jacobs, J., Koellner, K., Swackhamer, L. (2015).
Mathematical professional development: Improving teaching using the
problem-solving cycle and leadership preparation models. NY:
Teachers College Press & NCTM.
 Es, E., Stockero, S., Sherin, M., VanZoest, L., & Dyer, E. Making the
most of teacher self-captured video. Mathematics Teacher Educator,
4(1), 6-19.
 Heng, C. (2015). The FOCUS framework: Snapshots of mathematical
teacher noticing. MME Staff & Graduate Student Colloquim
presentation. Singapore: National Institute of Education.
 National Council of Teachers of Mathematics (2014). Principles to
Actions: Ensuring mathematical success for all. Reston, VA: NCTM.
Additional info at: http://www.nctm.org/PtA/
 Smith, M., Stein, M. (2011). Five practices for orchestrating productive
mathematics discussions. Reston, VA: NCTM.
Check NCTM Bookstore for these books!