supporting and extending children’s mathematical thinking

Supporting and Extending Children’s Mathematical Thinking

Vicki Jacobs San Diego State University

Funded by NSF (ESI 0455785) TDG, February 2011

The Challenge

  Vision of a responsive style of instruction

  Eliciting and building on children’s thinking   Meshing research on children’s thinking with knowledge of an

individual child’s thinking in the moment

  Examine a particular slice: one-on-one conversations between a teacher and child   Reduce some of the complexity of classrooms   Important part of instruction

Goals for Today

  Engage with a framework we have found useful for talking about one-on-one teacher-student interactions   Range of expertise

(in terms of engaging with children’s mathematical thinking)   Special attention to responsive interactions

(toolbox for supporting children when they are stuck & extending their thinking when they have been successful)

  Consider data about teachers’ development of expertise   Put it all together and consider implications (So what?)

Collaborators University of California-Davis

Rebecca Ambrose Heather Martin

San Diego State University Randy Philipp

Lisa Lamb Jessica Pierson

Bonnie Schappelle Candy Cabral John Siegfried

Major sources of videos & ideas   About a decade of informal work analyzing K–3 video with

Rebecca Ambrose (across multiple PD settings)

  STEP (Studying Teachers’ Evolving Perspectives)— Randy Philipp & I co-direct this NSF project to study the effects of sustained professional development focused on children’s mathematical thinking. Participants were129 teachers. Each teacher

  interviewed 3 children;   posed 4 (or more) problems per child (problems provided

but could be adjusted)’   was given a goal for the interview: Discover how each child

thinks (correct answers not stressed).

Framework of Engagement With Children’s Thinking

Exploration of Children’s Thinking

Minimal Active

Imposition of

Teachers’ Thinking

Active

Minimal

Watch Erica (Grade 3)

Framework of Engagement with Children’s Thinking

Exploration of Children’s Thinking

Minimal Active

Imposition of

Teachers’ Thinking

Active

Minimal

Framework of Engagement with Children’s Thinking

Exploration of Children’s Thinking

Minimal Active

Imposition of

Teachers’ Thinking

Active

Interrupted Exploring of Children’s

Thinking (Erica)

Minimal

Watch Gavin (Grade 3)

Framework of Engagement with Children’s Thinking

Exploration of Children’s Thinking

Minimal Active

Imposition of

Teachers’ Thinking

Active

Interrupted Exploring of Children’s

Thinking (Erica)

Minimal

Framework of Engagement with Children’s Thinking

Exploration of Children’s Thinking

Minimal Active

Imposition of

Teachers’ Thinking

Active Imposing

Teachers’ Thinking (Gavin)

Interrupted Exploring of Children’s

Thinking (Erica)

Minimal

Watch Matthew (Grade 2)

Framework of Engagement with Children’s Thinking

Exploration of Children’s Thinking

Minimal Active

Imposition of

Teachers’ Thinking

Active Imposing

Teachers’ Thinking (Gavin)

Interrupted Exploring of Children’s

Thinking (Erica)

Minimal

Framework of Engagement with Children’s Thinking

Exploration of Children’s Thinking

Minimal Active

Imposition of

Teachers’ Thinking

Active Imposing

Teachers’ Thinking (Gavin)

Interrupted Exploring of Children’s

Thinking (Erica)

Minimal Limited

Interacting (Matthew)

Watch Matai (Grade 2)

Framework of Engagement with Children’s Thinking

Exploration of Children’s Thinking

Minimal Active

Imposition of

Teachers’ Thinking

Active Imposing

Teachers’ Thinking (Gavin)

Interrupted Exploring of Children’s

Thinking (Erica)

Minimal Limited

Interacting (Matthew)

Framework of Engagement with Children’s Thinking

Exploration of Children’s Thinking

Minimal Active

Imposition of

Teachers’ Thinking

Active Imposing

Teachers’ Thinking (Gavin)

Interrupted Exploring of Children’s

Thinking (Erica)

Minimal Limited

Interacting (Matthew)

Exploring Children’s Thinking (Matai)

So What?

By characterizing one-on-one teacher-student interactions with this framework,

 what do professional developers gain?

 Should this information be shared with teachers? If so, how and when? (What else might they need to know first?)

STEP Participant Groups (N = 129, 30+ per group) Emerging Teacher Leaders At least 4 years of sustained

professional development and some leadership activities

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

K–3

Tea

cher

s

*Average of 14–16 years of teaching per group; range 4–33 years

Professional Development   Goal—help teachers learn about the research on 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

  Discussion of classroom artifacts (video and written student work) and the underlying mathematics

  Problems to try in teachers’ own classrooms between meetings

STEP Participant Groups (N = 129, 30+ per group) Emerging Teacher Leaders At least 4 years of sustained

professional development and some leadership activities

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

K–3

Tea

cher

s

*Average of 14–16 years of teaching per group; range 4–33 years

Cautions….what do we worry about?

  “Labeling” teachers

Framework of Engagement With Children’s Thinking

Exploration of Children’s Thinking

Minimal Active

Imposition of

Teachers’ Thinking

Active

Minimal

Framework of Engagement with Children’s Thinking

Exploration of Children’s Thinking

Minimal Active

Imposition of

Teachers’ Thinking

Active

ETL: 3% 16%

Minimal

3% 78%

Framework of Engagement with Children’s Thinking

Exploration of Children’s Thinking

Minimal Active

Imposition of

Teachers’ Thinking

Active

ETL: 3%

PST: 57%

16%

17%

Minimal

3%

20%

78%

6%

Framework of Engagement with Children’s Thinking

Exploration of Children’s Thinking

Minimal Active

Imposition of

Teachers’ Thinking

Active

ETL: 3%

IP: 10% PST: 57%

16%

16% 17%

Minimal

3%

45% 20%

78%

29% 6%

Framework of Engagement with Children’s Thinking

Exploration of Children’s Thinking

Minimal Active

Imposition of

Teachers’ Thinking

Active

ETL: 3% AP: 26% IP: 10% PST: 57%

16% 16% 16% 17%

Minimal

3% 26% 45% 20%

78% 32% 29% 6%

What have we learned about the development of expertise?

  Not something everyone can automatically do   Can be learned

  Not learned from teaching experience alone   Takes time   Development is not linear

Let’s look more closely at the Exploring Children’s Thinking cell….

Framework of Engagement With Children’s Thinking

Exploration of Children’s Thinking

Minimal Active

Imposition of

Teachers’ Thinking

Active Imposing

Teachers’ Thinking (Gavin)

Interrupted Exploring of Children’s

Thinking (Erica)

Minimal Limited

Interacting (Matthew)

Exploring Children’s Thinking (Matai)

Toolbox for this type of interaction   For each problem — consider the interaction in 2 parts:

  Supporting conversation before a correct answer   Extending conversation after a correct answer

  Categories of supporting and extending teacher moves   No “best” move (many productive moves)   Moves need not lead to a correct answer (or the desired

effect) to be considered productive.   Productive moves are well-selected, well-timed, and well-

implemented in response to a child’s ideas in a particular situation.

  Productive moves are consistent with the research on children’s thinking.

Cautions….what do we worry about?

  “Labeling” teachers

  Over-emphasizing a correct answer

  Talking about teacher moves in isolation

Extending Toolbox Potentially productive moves to extend a child's thinking after a correct answer is given are to

 promote reflection on the strategy the child just completed,

 encourage the child to explore multiple strategies and their connections,

 connect the child's thinking to symbolic notation, and

 generate follow-up problems linked to the problem the child just completed

Revisit Matai (Grade 2)

Watch Sam (Grade 1)

Watch Daniella (Grade 1)

Watch Juventino (Grade 1)

Pretend you are the teacher. How would you respond?

Supporting Toolbox

Potentially productive moves to support a child's thinking before a correct answer is given are to

 ensure that the child understands the problem,

 change the mathematics in the problem to match the child's level of understanding,

 explore what the child has already done, and

 remind the child to use other strategies.

Revisit Matai (Grade 2)

Watch Leilani (Grade 1)

Pretend that you are the teacher. How would you respond?

So What?

  What do professional developers gain by considering these toolboxes for supporting and extending?

  Should this information be shared with teachers? If so, how and when? (What else might they need to know first?)