supporting and extending children’s mathematical thinking
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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?)
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