How do you define geometric thinking?

1. How do you define geometric thinking?

2. FGT Staff • Mark Driscoll, Principal Investigator • Rachel Wing DiMatteo, Research Associate • Johannah Nikula, Research Associate • Mike Egan, Curriculum Specialist

3. Session Agenda • Why Geometry? • FGT’s Approach to Professional Development • Reflect on Geometric Thinking Through Problem Solving • Analyze Student Work • Preliminary Findings of the Project • Conclusion and Questions

4. Why Geometry? “Broadly speaking I want to suggest that geometry is that part of mathematics in which visual thought is dominant whereas algebra is that part in which sequential thought is dominant. This dichotomy is perhaps better conveyed by the words ‘insight’ versus ‘rigour’ and both play an essential role in real mathematical problems. The educational implications of this are clear. We should aim to cultivate and develop both modes of thought. It is a mistake to overemphasize one at the expense of the other and I suspect that geometry has been suffering in recent years” (p. 29). From Sir Michael Atiyah’s 1982 essay “What is geometry?”, reprinted in Pritchard, C. (2003). The changing shape of geometry: Celebrating a century of geometry and geometry teaching. Cambridge: Cambridge University Press.

5. Merging Algebraic and Geometric Thinking: Excerpts from Wu’s Curricular Proposals* On Algebra Curriculum: “Students need to be totally at ease in moving between the geometric data of a straight line and the algebraic data of a linear equation. This cannot happen if they are never taught similar triangles before embarking on the study of linear equations and their graphs, and have never been exposed to the explanation of why the equation of a line is linear and why the graph of a linear equation is a line” (pp. 3-4). On Geometry Curriculum: “A dilation with center at the origin O and scale factor r (r ≠ 0) is a transformation of the plane that sends a point (a, b) to the point (ra, rb)...Once [the students] buy into this concept of dilation, they are ready for the definition two figures to be similar if one figure is congruent to a dilated version of the other” (p. 6). * From Wu, H. (2005). Key mathematical ideas in grades 5-8. Paper presented at the Annual Meeting of the National Council of Teachers of Mathematics, Anaheim. Available: http://math.berkeley.edu/~wu.

6. Are We Neglecting Geometry in Schools? • U.S. 8th graders’ weakest performance areas in the Trends in International Mathematics and Science Study (TIMSS) were geometry and measurement1 • U.S. 12th graders posted the lowest geometry scores of any participating country on the TIMSS assessment2 • Data from the Programme of International Student Assessment (PISA) study also reveals that U.S. students are weakest in geometry3 • TIMMS data shows that U.S. 8th graders receive proportionally less geometry instructional time than students in most comparison countries3 1Mullis, I.V.S., et al. (2001). Mathematics benchmarking report – TIMSS 1999 8th grade. Chestnut Hill, MA: International Study Center, Boston College. 2Mullis, I.V.S., et al. (1998). Mathematics and science achievement in the final year of secondary school: IEA’s Third International Mathematics and Science Study. Chestnut Hill, MA: International Study Center, Boston College. 3Ginsburg, A., et al. (2005). Reassessing U.S. international mathematics performance: New findings from the 2003 TIMSS and Pisa. Washington DC: American Institutes for Research.

7. Should We Increase Middle School Students’ Exposure to Geometry? “Since geometry correlated highly with [important higher-order thinking skills], geometry may be something of a gateway skill to the teaching of higher order mathematics thinking skills...These findings suggest that the curriculum in the United States should put more emphasis on teaching geometry, because geometry may enable teaching of important mathematical thinking skills....Surprisingly, algebra did not correlate with these mathematical thinking skills but was related to computational skills. Thus, educators concerned with designing effective mathematics curricula might ask: Is the emphasis on algebra in current U.S. mathematics curricula sufficient to effectively teach logical reasoning and higher level judgmental skills to this age group [of 8th graders]?” (pp. 922-923)* *Tatsuoka, K.K., et al. (2004). Patterns of diagnosed mathematical content and process skills in TIMMS-R across a sample of 20 countries. American Educational Research Journal, 41(4), 901-926.

8. How Can We Prepare Middle Grade Teachers to Effectively Teach Geometry?

9. FGT’s Approach to Professional Development – Overview • Written professional development materials • * Developed with funding from the National Science Foundation • (NSF Grant ESI-0353409) • * For use by a facilitator with a group of teachers • * Intended for grades 5-10 teachers • * 40 hours of PD  broken into 20 two-hour sessions. • * Currently in second year of field testing

10. FGT’s Approach to Professional Development – Guiding Structures • Structured Exploration Process (experience today) • Geometric Habits of Mind (G-HOMs) Framework (examples today) • Additional Supports (Questioning, Language, Cognitive Demand)

11. FGT’s Approach to Professional Development – SE Process • 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

12. FGT’s Approach to Professional Development – The G-HOMs • Generalizing Geometric Ideas • Reasoning with Relationships • Investigating Invariants • Balancing Exploration and Reflection

13. Generalizing Geometric Ideas • Wanting to understand and describe the "always" and the "every" related to geometric phenomena. • "Does this happen in every case?" • "Have I found all the ones that fit this description?" • "Can I think of examples when this is not true, and, if so, should I then revise my generalization?”

14. Reasoning with Relationships • Actively looking for and trying to use relationships (e.g., congruence, similarity, parallelism, etc.), within and between geometric figures. • "How are these figures alike?" • "In how many ways are they alike?"   • "How are these figures different?"

15. Investigating Invariants • An invariant is something about a situation that stays the same, even as parts of the situation vary. • "What changes? Why?" • "What stays the same? Why?"

16. Balancing Exploration and Reflection • Trying various ways to approach a problem and regularly stepping back to take stock. • "What happens if I (draw a picture, add to/take apart this picture, work backwards from the ending place, etc.….)?" • "What did I learn from trying that?"

17. FGT’s Approach to Professional Development – The G-HOMs • Generalizing Geometric Ideas • Reasoning with Relationships • Investigating Invariants • Balancing Exploration and Reflection

18. Reflect on Geometric Thinking Through Problem Solving • Take a few minutes to play with the problem yourself. • Continue working on the problem with others at your table. • Goals: • Explore the problem • Prepare to analyze students’ thinking about the problem • Pay attention to how you are working on the problem

19. Reflect on Geometric Thinking Through Problem Solving • Two vertices of a triangle are located at (0,6) and (0,12). • The area of the triangle is 12 units2.

20. Reflect on Geometric Thinking Through Problem Solving • a) What are all possible positions for the third vertex? • b) Explain how you know these vertices create triangles with an area of 12 units2. • c) How do you know there aren’t any more?

21. Reflect on Geometric Thinking Through Problem Solving • Take a few minutes to play with the problem yourself. • Continue working on the problem with others at your table. • Goals: • Explore the problem • Prepare to analyze students’ thinking about the problem • Pay attention to how you are working on the problem

22. Geometric Habits of Mind elicited by Finding Area in Different Ways • Generalizing Geometric Ideas • "Does this happen in every case?" • "Have I found all the ones that fit this description?" • "Can I think of examples when this is not true - should I then revise my generalization?” • Reasoning with Relationships • "How are these figures alike?" • "In how many ways are they alike?"   • "How are these figures different?" • Investigating Invariants • "What changes? Why?" • "What stays the same? Why?" • Balancing Exploration and Reflection • "What happens if I (draw a picture, take apart this picture, work backwards, etc.….)?" • "What did I learn from trying that?"

23. Analyze Student Work • What relationships are the students paying attention to as they try to find triangles with an area of 12 units2?. • What steps are the students taking toward generalizing their ideas?

24. Preliminary Findings • Increase in geometric content knowledge, especially in the area of measurement • Increase in attention to students’ mathematical thinking

25. Transcript Question: What stood out for you in terms of the work the students and teachers were doing in this activity? Before FGT: “…the teacher was leading the students by the questions she was asking. I also liked the way the kids had a chance to think about the problem and come back to answer questions.” After FGT: “It was clear that they had worked on the volume of a rectangular prism and they had filled a rectangular prism with layers of cubes to find the volume of the prism. They actually took this thought and applied it to the cylinder…”

26. Transcript Question: What stood out for you in terms of the work the students and teachers were doing in this activity? Before FGT: “(The) teacher was not giving any answers, formulas or even strategies (and) was simply working through a discussion with the class to see how they solved it and what direction it went. Each class discussion could be totally different.” After FGT: “ --Analyzing cylinders and boxes --volume of prisms versus cylinders --understanding of height and radius --dimensions of shapes”

27. Importance of Geometry • U.S. 8th graders’ weakest performance areas in TIMSS were geometry and measurement • “… geometry correlated highly with [important higher-order thinking skills], geometry may be something of a gateway skill to the teaching of higher order mathematics thinking skills • TIMMS data shows that U.S. 8th graders receive proportionally less geometry instructional time than students in most comparison countries

28. Will be published by Heinemann in 2008 To find out more about the project: www.geometric-thinking.org Mark Driscoll (mdriscoll@edc.org) Mike Egan (megan@edc.org) Johannah Nikula (jnikula@edc.org) Rachel Wing DiMatteo (rwing@edc.org)