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A Habits of Mind Framework Supports Knowledge for Teaching Geometry

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### A Habits of Mind Framework Supports Knowledge for Teaching Geometry

Geometric Habits of Mind (G-HOMs)Workshop Overview

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

- 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 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)

- 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 (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 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)

- Seeking Relationships
- Checking Effects of Transformations
- Generalizing Geometric Ideas
- Balancing Exploration with Deduction

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

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-HOMs)

- Seeking Relationships
- Checking Effects of Transformations
- Generalizing Geometric Ideas
- Balancing Exploration with Deduction

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 rhombus?

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-HOMs)

- Seeking Relationships
- Checking Effects of Transformations
- Generalizing Geometric Ideas
- Balancing Exploration with Deduction

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

Connecting the midpoints of this quadrilateral created a parallelogram. What other quadrilaterals will this work for?

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?"

- Seeking Relationships
- Checking Effects of Transformations
- Generalizing Geometric Ideas
- 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?

Internal Questions:

“What happens if…?”

“What did that action tell me?”

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
- How FGT field test addresses impact of G-HOM-based materials on 2 types of Knowledge for Teaching Geometry
- Questions

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 Overview

- Geometric Habits of Mind framework (G-HOMs)
- 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

- 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…”

Balancing Exploration with Deduction

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

C

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

“Why is this happening?”

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

- 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.

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 Overview

- Geometric Habits of Mind framework (G-HOMs)
- Knowledge for Teaching Geometry
- Questions

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

- Geometric Habits of Mind framework (G-HOMs)
- Knowledge for Teaching Geometry
- Questions

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

Rachel Wing

rwing@edc.org

Mark Driscoll

mdriscoll@edc.org

Fostering Geometric Thinking website

www.geometric-thinking.org

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