1. 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

2. 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

3. 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

4. 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

5. 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

6. Workshop 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

7. Geometric Habits of Mind (G-HOMs) Seeking Relationships Checking Effects of Transformations Generalizing Geometric Ideas Balancing Exploration with Deduction

8. 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

9. Which two make the best pair?

10. 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?

11. Seeking Relationships Checking Effects of Transformations Generalizing Geometric Ideas Balancing Exploration with Deduction Geometric Habits of Mind (G-HOMs)

12. 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

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

14. Internal Questions: "What changes? Why?" "What stays the same? Why?” A square’s diagonals always intersect at 90- degree angles. Is this true of a rhombus?

15. Seeking Relationships Checking Effects of Transformations Generalizing Geometric Ideas Balancing Exploration with Deduction Geometric Habits of Mind (G-HOMs)

16. 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

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

18. 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 created a parallelogram. What other quadrilaterals will this work for?

19. Seeking Relationships Checking Effects of Transformations Generalizing Geometric Ideas Balancing Exploration with Deduction Geometric Habits of Mind (G-HOMs)

20. 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

21. 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?”

22. 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

23. 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

24. Knowledge for Teaching Geometry Knowledge to Assess Students’ Geometric Thinking Knowledge to Advance Students’ Geometric Thinking

25. Knowledge 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

26. Knowledge 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.

27. 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

28. Knowledge of productive ways of thinking in geometry FGT  G-HOMs framework Language for thinking and talking about geometric thinking

29. “If we could find the area of all the pieces and then take the square root…” Balancing Exploration with Deduction

30. 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

31. Perimeter of A = Perimeter of B A B

32. 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?”

33. Knowledge 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.

34. 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

35. Tangrams 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.

36. 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

37. “B’s area is bigger” A B

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

39. But…“B’s area is bigger” A B

40. 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

41. 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

42. 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

43. 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

44. Questions?

45. Contact Information Rachel Wing rwing@edc.org Mark Driscoll mdriscoll@edc.org Fostering Geometric Thinking website www.geometric-thinking.org