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A Habits of Mind Framework Supports Knowledge for Teaching Geometry. Rachel Wing & Mark Driscoll Education Development Center Newton, Massachusetts. Daniel Heck Horizon Research, Inc. Chapel Hill, North Carolina.

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a habits of mind framework supports knowledge for teaching geometry

A Habits of Mind Framework Supports Knowledge for Teaching Geometry

Rachel Wing & Mark Driscoll

Education Development Center

Newton, Massachusetts

Daniel Heck

Horizon Research, Inc.

Chapel Hill, North Carolina

Research reported here is support by the National Science Foundation under Grant ESI-0353409

workshop overview
Workshop Overview
  • Geometric Habits of Mind framework (G-HOMs)
  • Knowledge for Teaching Geometry
  • How G-HOMs and Fostering Geometric Thinking (FGT) materials support 2 types of Knowledge for Teaching Geometry
  • Engage in FGT activities and consider the extent to which the G-HOMs framework supports the 2 types of Knowledge for Teaching Geometry
  • How FGT field test addresses impact of G-HOMs-based materials on 2 types of Knowledge for Teaching Geometry
  • Questions
workshop overview1
Workshop Overview

Fostering Geometric Thinking

  • Geometric Habits of Mind framework (G-HOMs)
  • Knowledge for Teaching Geometry
  • How G-HOMs and Fostering Geometric Thinking materials support 2 types of Knowledge for Teaching Geometry
  • Engage in FGT activities and consider the extent to which the G-HOMs framework supports the 2 types of Knowledge for Teaching Geometry
  • How FGT field test addresses impact of G-HOM-based materials on 2 types of Knowledge for Teaching Geometry
  • Questions
fostering geometric thinking fgt
Fostering Geometric Thinking (FGT)
  • Identify productive ways of thinking in geometry (G-HOMs)
  • Create professional development materials based on G-HOMs
    • 40 hours  20 two-hour sessions
    • Group-study materials
    • Structured Exploration Process (Kelemanik et al. 1997)
      • Stage 1: Doing mathematics
      • Stage 2:Reflecting on the mathematics
      • Stage 3: Collecting student work
      • Stage 4: Analyzing student work
      • Stage 5: Reflecting on students’ thinking
    • G-HOMs framework is a lens for analysis

FGT is support by the National Science Foundation under Grant ESI-0353409

fostering geometric thinking fgt1
Fostering Geometric Thinking (FGT)
  • Field Test

Research Questions: In what ways does the use of professional development materials that target students’ geometric ways of thinking…

      • Q1: …increase teachers’ content knowledge and understanding of student thinking in geometry and measurement?
      • Q2: …affect instructional practice in geometry?

 Knowledge for Teaching Geometry

workshop overview2
Workshop Overview
  • Geometric Habits of Mind framework (G-HOMs)
  • Knowledge for Teaching Geometry
  • How G-HOMs and Fostering Geometric Thinking materials support 2 types of Knowledge for Teaching Geometry
  • Engage in FGT activities and consider the extent to which the G-HOMs framework supports the 2 types of Knowledge for Teaching Geometry
  • How FGT field test addresses impact of G-HOMs-based materials on 2 types of Knowledge for Teaching Geometry
  • Questions
geometric habits of mind g homs
Geometric Habits of Mind (G-HOMs)
  • Seeking Relationships
  • Checking Effects of Transformations
  • Generalizing Geometric Ideas
  • Balancing Exploration with Deduction
seeking relationship
Seeking Relationship
  • Actively looking for relationships within and between geometric figures, in one, two, and three dimensions.
  • Relationships can be between/among:
  • figures
  • whole figures and their parts
  • concepts
slide10

Which two make the best pair?

Internal Questions:

“How are these figures

alike?”

“In how many ways are

they alike?”

“How are these figures

different?”

“What would I have to do to this object to

make it like that object?”

geometric habits of mind g homs1
Geometric Habits of Mind (G-HOMs)
  • Seeking Relationships
  • Checking Effects of Transformations
  • Generalizing Geometric Ideas
  • Balancing Exploration with Deduction
checking effects of transformations
Checking Effects of Transformations
  • Analyzing which attributes of a figure remain invariant and which change when the figure is transformed in some way.
  • Attributes of the figure that may be affected by transformations include:
  • Orientation
  • Location
  • Area, perimeter and volume
  • Side lengths and ratio of side lengths
  • Angles
a square s diagonals always intersect at 90 degree angles is this true of a rhombus1
A square’s diagonals always intersect at 90-degree angles. Is this true of a rhombus?

Internal Questions:

"What changes? Why?"

"What stays the same? Why?”

geometric habits of mind g homs2
Geometric Habits of Mind (G-HOMs)
  • Seeking Relationships
  • Checking Effects of Transformations
  • Generalizing Geometric Ideas
  • Balancing Exploration with Deduction
generalizing geometric ideas
Generalizing Geometric Ideas
  • Wanting to understand the "always" and the "every" related to geometric concepts and procedures.
  • Generalizing progresses through stages:
  • Conjecturing about the “always” & “every”
  • Testing the conjecture
  • Drawing a conclusion about the conjecture
  • Making a convincing argument
slide17
Connecting the midpoints of this quadrilateral created a parallelogram. What other quadrilaterals will this work for?
slide18
Connecting the midpoints of this quadrilateral created a parallelogram. What other quadrilaterals will this work for?

Internal Questions:

"Does this happen in every case?"

"Why would this happen in every case?"

"Can I think of examples when this is not true?"

"Would this apply in other dimensions?"

geometric habits of mind g homs3
Geometric Habits of Mind (G-HOMs)
  • Seeking Relationships
  • Checking Effects of Transformations
  • Generalizing Geometric Ideas
  • Balancing Exploration with Deduction
balancing exploration with deduction
Balancing Exploration with Deduction
  • An iterative and cumulative process that alternates between:
  • exploring structured by one or more explicit limitation/restriction
  • and
  • taking stock of what is being learned through the exploration
is it possible to draw a quadrilateral that has exactly 2 right angles and no parallel lines
Is it possible to draw a quadrilateral that has exactly 2 right angles and no parallel lines?

Internal Questions:

“What happens if…?”

“What did that action tell me?”

workshop overview3
Workshop Overview
  • Geometric Habits of Mind framework (G-HOMs)
  • Knowledge for Teaching Geometry
  • How G-HOMs and Fostering Geometric Thinking materials support 2 types of Knowledge for Teaching Geometry
  • Engage in FGT activities and consider the extent to which the G-HOMs framework supports the 2 types of Knowledge for Teaching Geometry
  • How FGT field test addresses impact of G-HOM-based materials on 2 types of Knowledge for Teaching Geometry
  • Questions
knowledge for teaching geometry
Knowledge for Teaching Geometry

Influenced by…

  • Our work developing the FGT materials and measures for the field test
  • Previous work on Connected Geometry and Fostering Algebraic Thinking Toolkit
  • Van Hiele (1959; 1986) model of cognitive development in geometry
  • Ball et al.’s (2001; 2003) work on Content Knowledge for Teaching Mathematics
workshop overview4
Workshop Overview
  • Geometric Habits of Mind framework (G-HOMs)
  • Knowledge for Teaching Geometry
  • How G-HOMs and Fostering Geometric Thinking materials support 2 types of Knowledge for Teaching Geometry
  • Engage in FGT activities and consider the extent to which the G-HOMs framework supports the 2 types of Knowledge for Teaching Geometry
  • How FGT field test addresses impact of G-HOMs-based materials on 2 types of Knowledge for Teaching Geometry
  • Questions
knowledge of productive ways of thinking in geometry
Knowledge of productive ways of thinking in geometry
  • FGT  G-HOMs framework
    • Language for thinking and talking about geometric thinking
if we could find the area of all the pieces and then take the square root
“If we could find the area of all the pieces and then take the square root…”

Balancing Exploration with Deduction

knowledge of productive ways of thinking in geometry1
Knowledge of productive ways of thinking in geometry
  • FGT  G-HOMs framework
    • Language for thinking and talking about geometric thinking
    • Increase use of G-HOMs in problem solving
perimeter of a perimeter of b perimeter of c
Perimeter of A = Perimeter of B = Perimeter of C

C

“Will the perimeter always be the same no matter how many mounds?”

“Why is this happening?”

slide34
Knowledge of how to promote productive, and potentially productive, ways of geometric thinking in students.
  • FGT  G-HOMs internal questions and emphasis on potential
    • Question students in a way that encourages G-HOMS
tangrams and seeking relationships
Tangrams and Seeking Relationships
  • S: If you take the whole thing…there are 7 different parts of this (rectangle they’ve built). Some of them may be different shapes, different sizes, but they’re all equal to 1 part of this shape.
  • T: So, would you say then that they’re all 1/7th of the shape?
  • S: Yeah…well, no…kind of ‘cause like this (points to large triangle) could be 1/7th or like if you had a greater denominator it could be greater than 1/7th ‘cause it’s bigger than everything else.
  • T: So they’re not the same size pieces, but there are 7 pieces.
  • S: Yeah. And they’re all 1/7th of this thing, even though some are smaller than the others.
  • T: So this (small triangle) is 1/7th and this (large triangle) is 1/7th?
  • S: Yeah, because like this (small triangle) has to be at least 1/7th because you don’t really go into decimals and that stuff.
slide36
Knowledge of how to promote productive, and potentially productive, ways of geometric thinking in students.
  • FGT  G-HOMs internal questions and emphasis on potential
    • Question students in a way that encourages G-HOMS
    • Capitalize on potential
workshop overview5
Workshop Overview
  • Geometric Habits of Mind framework (G-HOMs)
  • Knowledge for Teaching Geometry
  • How G-HOMs and Fostering Geometric Thinking materials support 2 types of Knowledge for Teaching Geometry
  • Engage in FGT activities and consider the extent to which the G-HOMs framework supports the 2 types of Knowledge for Teaching Geometry
  • How FGT field test addresses impact of G-HOMs-based materials on 2 types of Knowledge for Teaching Geometry
  • Questions
knowledge for teaching geometry in the context of fgt materials
Knowledge for Teaching Geometry in the Context of FGT Materials
  • Explore FGT problem: Dissecting Shapes
  • Reflect on G-HOMs’ ability to capture productive ways of thinking in geometry
  • Video of students working on Dissecting Shapes
  • Discuss how the G-HOMs framework affected how you analyzed students’ thinking in the video
  • How FGT would use this artifact with teachers to promote productive, and potentially productive, ways of geometric thinking
workshop overview6
Workshop Overview
  • Geometric Habits of Mind framework (G-HOMs)
  • Knowledge for Teaching Geometry
  • How G-HOMs and Fostering Geometric Thinking materials support 2 types of Knowledge for Teaching Geometry
  • Engage in FGT activities and consider the extent to which the G-HOMs framework supports the 2 types of Knowledge for Teaching Geometry
  • How FGT field test addresses impact of G-HOMs-based materials on 2 types of Knowledge for Teaching Geometry
  • Questions
fgt field test
FGT Field Test
  • Groups randomly assigned to 2 conditions: Treatment (N=137), Wait-Listed Control (N=140)
  • Measures
    • Geometry Survey
      • Multiple choice geometry problems
      • Open-ended questions about approaches to a problem
      • Open-ended questions about analyzing student work
    • Observations
      • Extent to which lessons promoted G-HOMs
      • Extent to which teachers promoted G-HOMs
      • Extent to which students employed G-HOMs
contact information
Contact Information

Rachel Wing

rwing@edc.org

Mark Driscoll

mdriscoll@edc.org

Fostering Geometric Thinking website

www.geometric-thinking.org