Noticing Children’s Mathematical Thinking During Instruction
Transcript of Noticing Children’s Mathematical Thinking During Instruction
Noticing Children’s Mathematical Thinking
During Instruction
Vicki Jacobs San Diego State University
SDSU Funded by NSF (ESI 0455785) February 14, 2009
High-Leverage Practices in Mathematics Teaching
Important to teaching and the subject matter Happen frequently Generative - help teachers continue to learn
Are there hidden high-leverage practices in mathematics teaching?
What happens in classrooms?
Child says or does something
and then the teacher responds.
What is going on behind the scenes before the teacher responds?
What would we like to have going on?
Reform Recommendations
“sizing up students’ ideas and responding” as a core activity of mathematics teaching (Ball, Lubienski, & Mewborn, 2001)
Focus on children’s mathematical thinking (Lester, 2007; NCTM, 2000, NRC, 2001)
Children think about mathematics differently than adults
Instruction can be improved when teachers attend to children’s mathematical thinking • Student achievement • Generative teacher learning
Hidden Practices
What does it mean to size up students’ ideas and respond in the midst of instruction? Professional Noticing of Children’s Mathematical Thinking Attending to children’s strategies
Interpreting children’s understandings
Deciding how to respond on the basis of children’s understandings
Professional Noticing
Draw on the work of others studying noticing (Goodwin, 1994,Mason, 2002, Stevens & Hall, 1998; van Es & Sherin, 2002, 2006, & 2008)
Not everyday noticing Our focus
• Hidden practices before a teacher responds to a child’s action, question, or comment
• In-the-moment decision making • Children’s mathematical thinking (lens)
Kindergarten teacher asked Rex to solve these problems in June
Rex had 13 cookies. He ate 6 of them. How many cookies does Rex have left?
Today is June 5 and your birthday is June 19. How many days away is your birthday?
Rex had 15 tadpoles. He put 3 tadpoles in each jar. How many jars did Rex put tadpoles in?
Please React to the Prompt
After the teacher posed the tadpole problem, Rex commented, “I don’t even know that one. That’s hard.” Describe some ways you might respond to Rex, and explain why you chose those responses.
Decision Making
How did you decide how you would respond to Rex? (What types of things did you consider?)
Consider the 4 responses. What are the similarities and differences?
Describe some ways you might respond to Rex, and explain why you chose those responses.
We categorized responses by the extent to which teachers considered Rex's mathematical thinking. Was Rex’s thinking on the past 2 problems explicitly referenced in the rationale for the proposed interaction?
Was there space for Rex's thinking (not just the teacher’s thinking) in the proposed interaction?
Responding Categories
USE OF CHILDREN’S THINKING Robust Responding on the Basis of Rex’s
thinking
Limited Responding on the Basis of Rex’s thinking
LACK OF USE OF CHILDREN’S THINKING General comments
Dominance of teacher’s thinking
Responding Categories
USE OF CHILDREN’S THINKING Robust Responding on the Basis of Rex’s
thinking (Response 2) Limited Responding on the Basis of Rex’s
thinking (Response 3) LACK OF USE OF CHILDREN’S THINKING General comments (Response 4) Dominance of teacher’s thinking (Response 1)
So What?
How could these categories help you when designing professional development? When facilitating professional development? When coaching in classrooms?
Studying Teachers Evolving Perspectives (STEP) Project
5-year NSF-funded-project
Project Personnel Randy Philipp, Vicki Jacobs, Lisa Lamb, Jessica Pierson, Bonnie Schappelle, Candy Cabral, John Siegfried, Chris Macias-Papierniak, Courtney White
STEP Overview
Evolution of teachers involved in sustained professional development focused on children’s mathematical thinking
Study the perspectives of 4 groups of teachers • Beliefs • Mathematical content knowledge • Responsiveness to children’s thinking in
1-on-1 interviews • Professional Noticing of Children’s
Mathematical Thinking
Practicing Teachers (average of 14-16 years of teaching experience per group)
Emerging Teacher Leaders: At least 4 years of sustained professional development
Advancing Participants: 2 years of sustained professional development
Initial Participants: 0 years of sustained professional development
Prospective Teachers: Undergraduates enrolled in a first mathematics-for-teachers content course
Teacher GroupsN=131 (30+ per group)
STEP Professional Development
Help teachers learn about children’s mathematical thinking and how to use this knowledge to inform their instruction
Drew heavily from the Cognitively Guided Instruction (CGI) project (Carpenter et al., 1999, 2003)
5 full-day meetings per year Problems to try in teachers’ own classrooms
between meetings Discussion of classroom artifacts (video &
written student work) and the underlying mathematics
Components of Professional Noticing of Children’s Mathematical Thinking
Attending to children’s strategies
Interpreting children’s understandings
Deciding how to respond on the basis of children’s understandings
Components of Professional Noticing of Children’s Mathematical Thinking
Attending to children’s strategies
Interpreting children’s understandings
Deciding how to respond on the basis of children’s understandings
Deciding How to Respond on the Basis of Rex’s Thinking
Prospective Teachers
Initial Participants
Advancing Participants
Emerging Teacher Leaders
Robust Use Limited Use
General Comments Teacher’s Thinking
Deciding How to Respond on the Basis of Rex’s Thinking
Prospective Teachers
Initial Participants
Advancing Participants
Emerging Teacher Leaders
Robust Use Limited Use
General Comments Teacher’s Thinking
Deciding How to Respond on the Basis of Rex’s Thinking
Prospective Teachers
Initial Participants
Advancing Participants
Emerging Teacher Leaders
Robust Use Limited Use
General Comments Teacher’s Thinking
Deciding How to Respond on the Basis of Rex’s Thinking
Prospective Teachers
Initial Participants
Advancing Participants
Emerging Teacher Leaders
Robust Use Limited Use
General Comments Teacher’s Thinking
Deciding How to Respond on the Basis of Rex’s Thinking
Prospective Teachers
Initial Participants
Advancing Participants
Emerging Teacher Leaders
Robust Use Limited Use
General Comments Teacher’s Thinking
Deciding How to Respond on the Basis of Rex’s Thinking
Prospective Teachers
Initial Participants
Advancing Participants
Emerging Teacher Leaders
Robust Use 0% 0% 19% 42%
Limited Use 0% 19% 23% 24%
General Comments 47% 42% 10% 18%
Teacher’s Thinking 53% 39% 48% 15%
Deciding How to Respond on the Basis of Rex’s Thinking
Prospective Teachers
Initial Participants
Advancing Participants
Emerging Teacher Leaders
Robust Use 0% 0% 19% 42%
Limited Use 0% 19% 23% 24%
Generic Comment 47% 42% 10% 18%
Teacher’s Thinking 53% 39% 48% 15%
Deciding How to Respond on the Basis of Rex’s Thinking
Prospective Teachers
Initial Participants
Advancing Participants
Emerging Teacher Leaders
Robust Use 0% 0% 19% 42%
Limited Use 0% 19% 23% 24%
Generic Comment 47% 42% 10% 18%
Teacher’s Thinking 53% 39% 48% 15%
Deciding How to Respond on the Basis of Rex’s Thinking
Prospective Teachers
Initial Participants
Advancing Participants
Emerging Teacher Leaders
Robust Use 0% 0% 19% 42%
Limited Use 0% 19% 23% 24%
Generic Comment 47% 42% 10% 18%
Teacher’s Thinking 53% 39% 48% 15%
So What?
Using children’s thinking to decide how to respond is challenging
Professional development matters – teachers can learn how to use children’s thinking to decide how to respond
Categories help us understand teachers’ perspectives & how to customize PD to help them grow
Components of Professional Noticing of Children’s Mathematical Thinking
Attending to children’s strategies • Can teachers describe the details of strategies?
Interpreting children’s understandings
Deciding how to respond on the basis of children’s understandings
Please React to the Prompt
Please describe in detail what Rex said and did in response to this tadpole problem. (We recognize that you had the opportunity to view this video only one time, so please just do the best you can.)
Attending to Rex’s Strategy
How would you characterize the 2 responses?
Assessing Teachers’ Attending
Two Categories: Most Details & Few Details
Possible Mathematical Details Direct modeling strategy - built groups of
3 cubes up to 15 & then counted number of groups
Counted by 3s up to 9 then by 1s to 15 Answer confusion - number of jars vs.
number of tadpoles
Attending to Rex’s Strategy
Prospective Teachers
Initial Participants
Advancing Participants
Emerging Teacher Leaders
Most Details
Few Details
Attending to Rex’s Strategy
Prospective Teachers
Initial Participants
Advancing Participants
Emerging Teacher Leaders
Most Details 19% 35% 77% 88%
Few Details 81% 65% 23% 12%
Attending to Rex’s Strategy
Prospective Teachers
Initial Participants
Advancing Participants
Emerging Teacher Leaders
Most Details 19% 35% 77% 88%
Few Details 81% 65% 23% 12%
Attending to Rex’s Strategy
Prospective Teachers
Initial Participants
Advancing Participants
Emerging Teacher Leaders
Most Details 19% 35% 77% 88%
Few Details 81% 65% 23% 12%
Relationships
How are the 2 skills related?
If you decide how to respond on the basis of children’s understandings, do you also attend to the details of children’s strategies?
If you attend to the details of children’s strategies, do you also decide how to respond on the basis of children’s understandings?
Relationship Between Attending and Deciding How to Respond
20 teachers showed Robust Use of Rex’s Thinking in deciding how to respond 19 of 20 were also able to describe most details of Rex’s strategy
=>Teachers who decided how to respond on the basis of Rex’s thinking could also describe most strategy details
Relationship Between Attending and Deciding How to Respond
20 teachers showed Robust Use of Rex’s Thinking in deciding how to respond 19 of 20 were also able to describe most details of Rex’s strategy
=>Teachers who decided how to respond on the basis of Rex’s thinking could also describe most strategy details
71 teachers described most details of Rex’s strategy More than 10 of 71 were in each of the 4 categories for deciding how to respond
=>Teachers who described most strategy details did not necessarily decide how to respond on the basis of Rex’s thinking
STEP Assessments for Professional Noticing of Children’s Mathematical Thinking
Individual Interview (Rex)
Classroom Video (Lunch Count)
Written Student Work (M&Ms)
Lunch Count Prompts
Attending to children’s strategies • Please describe in detail what the children said and
did in response to this problem. • (Pair 1, Pair 2, Sunny)
Interpreting children’s understandings • Please explain what you learned about these
children’s understandings.
Deciding how to respond on the basis of children’s understandings • Pretend that you are the teacher of these children.
What problem or problems might you pose next? (Problem or Problems, Rationale)
So What?
How could the construct of Professional Noticing of Children’s Mathematical Thinking help you?
• Attending to children’s strategies • Interpreting children’s understandings • Deciding how to respond on the basis of children’s
understandings How could our results about the developmental trajectory of professional noticing skills help you? Is Professional Noticing of Children’s Mathematical Thinking a hidden high-leverage practice?