a habits of mind framework supports knowledge for teaching geometry daniel heck horizon research,...

A Habits of Mind Framework Supports Knowledge for

Teaching Geometry

A Habits of Mind Framework Supports Knowledge for

Teaching Geometry

Daniel Heck

Horizon Research, Inc.

Chapel Hill, North Carolina

Rachel Wing & Mark Driscoll

Education Development Center

Newton, Massachusetts

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

Workshop OverviewWorkshop 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 OverviewWorkshop 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 (FGT)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 OverviewWorkshop Overview Geometric Habits of Mind framework (G-HOMs)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 RelationshipSeeking 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

Which two make the best pair?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?”

Which two make the best pair?Which two make the best pair?

Seeking Relationships

Checking Effects of Transformations

Generalizing Geometric Ideas

Balancing Exploration with Deduction

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

Checking Effects of TransformationsChecking 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-A square’s diagonals always intersect at 90-degree angles. Is this true of a rhombus?degree angles. Is this true of a rhombus?

Internal Questions:

"What changes? Why?"

"What stays the same? Why?”

A square’s diagonals always intersect at 90-A square’s diagonals always intersect at 90-degree angles. Is this true of a rhombus?degree angles. Is this true of a rhombus?

Seeking Relationships

Checking Effects of Transformations

Generalizing Geometric Ideas

Balancing Exploration with Deduction

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

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

Connecting the midpoints of this quadrilateral Connecting the midpoints of this quadrilateral created a parallelogram. What other quadrilaterals created a parallelogram. What other quadrilaterals

will this work for?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?"

Connecting the midpoints of this quadrilateral Connecting the midpoints of this quadrilateral created a parallelogram. What other quadrilaterals created a parallelogram. What other quadrilaterals

will this work for?will this work for?

Seeking Relationships

Checking Effects of Transformations

Generalizing Geometric Ideas

Balancing Exploration with Deduction

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

Balancing Exploration with DeductionBalancing 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 Is it possible to draw a quadrilateral that has exactly 2 right angles and no parallel lines? exactly 2 right angles and no parallel lines?

Internal Questions:

“What happens if…?”

“What did that action tell me?”

Workshop OverviewWorkshop 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 GeometryKnowledge 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

Knowledge for Teaching GeometryKnowledge for Teaching Geometry

Knowledge to Assess Students’ Geometric Thinking

Knowledge to Advance Students’ Geometric Thinking

Knowledge for Teaching GeometryKnowledge for Teaching Geometry

Knowledge to Assess Students’ Geometric Thinking

Knowledge of productive ways of thinking in geometry.

Knowledge of the development of geometric thinking.

Knowledge of common difficulties students encounter in geometry.

Knowledge of how to elicit and analyze students’ thinking.

Knowledge to Advance Students’ Geometric Thinking

Knowledge for Teaching GeometryKnowledge for Teaching Geometry

Knowledge to Assess Students’ Geometric Thinking

Knowledge of productive ways of thinking in geometry.

Knowledge of the development of geometric thinking.

Knowledge of common difficulties students encounter in geometry.

Knowledge of how to elicit and analyze students’ thinking.

Knowledge to Advance Students’ Geometric Thinking

Knowledge of how to promote productive, and potentially productive, ways of geometric thinking in students.

Knowledge of how to encourage students to reconceptualize areas of difficulty or misunderstanding.

Workshop OverviewWorkshop 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 Knowledge of productive ways of thinking in geometryof 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 Knowledge of productive ways of thinking in geometryof 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 BPerimeter of A = Perimeter of B

-20 -15 -10

A B

Perimeter of A = Perimeter of B = Perimeter of CPerimeter 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 for Teaching GeometryKnowledge for Teaching Geometry

Knowledge to Assess Students’ Geometric Thinking

Knowledge of productive ways of thinking in geometry.

Knowledge of the development of geometric thinking.

Knowledge of common difficulties students encounter in geometry.

Knowledge of how to elicit and analyze students’ thinking.

Knowledge to Advance Students’ Geometric Thinking

Knowledge of how to promote productive, and potentially productive, ways of geometric thinking in students.

Knowledge of how to encourage students to reconceptualize areas of difficulty or misunderstanding.

Knowledge of how to promote productive, Knowledge of how to promote productive, and potentially productive, ways of and potentially productive, ways of geometric thinking in students.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 RelationshipsTangrams and Seeking Relationships

1. 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.

2. T: So, would you say then that they’re all 1/7th of the shape?

3. 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.

4. T: So they’re not the same size pieces, but there are 7 pieces.

5. S: Yeah. And they’re all 1/7th of this thing, even though some are smaller than the others.

6. T: So this (small triangle) is 1/7th and this (large triangle) is 1/7th?

7. 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, Knowledge of how to promote productive, and potentially productive, ways of and potentially productive, ways of geometric thinking in students.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

““B’s area is bigger”B’s area is bigger”

-20 -15 -10

A B

“Area of A = x 3 x 3; Area of B = 2 x 6 x 6”

-20 -15 -10

A B

But…“B’s area is bigger”But…“B’s area is bigger”

-20 -15 -10

A B

Workshop OverviewWorkshop 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 Knowledge for Teaching Geometry in the Context of FGT Materialsin 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 OverviewWorkshop 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 TestFGT 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

Questions?

Contact InformationContact Information

Rachel WingRachel Wingrwing@edc.orgrwing@edc.org

Mark DriscollMark Driscollmdriscoll@edc.orgmdriscoll@edc.org

Fostering Geometric Thinking websiteFostering Geometric Thinking websitewww.geometric-thinking.orgwww.geometric-thinking.org