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



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






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%






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?



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






Most Details

Few Details






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?

noticing children’s mathematical thinking during instruction

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