Sunshine Greene and Carmel Schettino Mathematics Department, Emma Willard School

1. “There was one time during class that I put a problem up at the board and got the entire thing correct. I was actually, in a way, disappointed because I feel like I learn better from my mistakes.” “Whenever I have a question about a problem I ask the question, why?” “This course makes you think for yourself and use what you know to solve new material. I love this approach.” “I am now more confident in my abilities in math, and I am now more open to sharing my ideas about different ways to get the same answer.” “When I was presenting a problem that we did in class where I could talk with my classmates and figure it out as a group, it really helped me gain confidence because it showed that I knew what I was doing.” “I like how it is focused on yourself figuring out the problem – though that was hard for me to adjust to – however it’s made me much more independent math-wise.” “I think that presenting my solution is most useful because I am forced to participate and learn from other people.” “Every time I put a problem up on the board I felt confident because I knew that even if I got it wrong, the correct solution would be explained so that I would do it correctly next time.” “You can see that everyone here wants to figure out how they got a certain problem. There’s more of an interest than just getting something right, instead of just getting an A. It’s something you know that they want to understand how. “ “My best description is that of a light bulb. I could see where the rest of the problem might lead and the concept that it covered.” “I was presenting a problem that I wasn’t quite sure I had answered right, and with a little push from the class, the concept suddenly clicked in my mind. I was able to finish the rest of the problem quickly and felt good about my success.” “I hated it at first because I couldn’t do what I had always done and do well. But by the beginning of the second semester, I had started to figure out what it was all about. And this year in my other classes, like English and history, I’m THINKING better, I’m analyzing differently because of that Geometry class.” Sunshine Greene and Carmel Schettino Mathematics Department, Emma Willard School

2. Creating a Geometry Classroom Based on the Relational Learning of Girls Sunshine Greene and Carmel Schettino Mathematics Department, Emma Willard School

3. Comparison of Traditional and Relational Classroom *Belenky,et al, 1986, +Boaler, 2008

4. The Theory of Girls’ Learning • Preference for Collaboration • Active Learning • Connected • Prior Knowledge • People • Current Knowledge • Need to feel valued, listened to • Questioning, Curious for real understanding

5. The Pedagogy • Feminist Mathematics Pedagogy • Dissolution of Hierarchy • Empowerment and Agency • Inclusion of all Voices • Ownership in Learning • A Pedagogy of Relation • Relational Equity • Relational Authority Solar, 1995, Anderson, 2005, Bingham, 2004, Boaler, 2008

6. Teacher Behaviors Schettino, 2009

7. The Curriculum • Problem-Based • Discourse-Driven • Spiraled and Parallel Topics • Built on Prior Knowledge • Student Construction of New Knowledge • Teacher Facilitation and Scaffolding - Key • Multiple Representations • Assessment Variations • Ownership of New Knowledge via Metacognitive Journaling

8. Problem Purposes • reviewing material from past courses • triggering prior knowledge for an upcoming problem • inspiring construction of new knowledge • introducing new terminology • practicing a new skill • challenging the more able students (differentiated instruction) • seeing the same new idea from a different representation

9. Slope – Multiple Perspectives The vertices A, B and C are collinear. Find the dimension n.

10. Slope – Multiple Perspectives An airplane is flying at 36000 feet directly above Lincoln, Nebraska. A little later the plane is flying at 28000 feet directly above Des Moines, Iowa, which is 160 miles from Lincoln. Assuming a constant rate of descent, predict how far from Des Moines the airplane will be when it lands.

11. Slope – Multiple Perspectives • The diagram at right shows the graph of 3x + 4y = 12. The shaded figure is a square, three of whose vertices are on the coordinate axes. The fourth vertex is on the line. Find the lengths of the sides of the square.

12. Right Angles – Multiple Perspectives Let A = (3, 2), B = (1, 5), and P = (x, y). Find x- and y-values that make ABP a right angle. Describe the configuration of all such points P.

13. Right Angles – Multiple Perspectives Find coordinates for the vertices of a lattice rectangle that is three times as long as it is wide with none of the sides horizontal.

14. References • Anderson, D. L. (2005). A portrait of a feminist mathematics classroom: What adolescent girls say about mathematics, themselves, and their experiences in a "unique" learning environment. Feminist Teacher, 15(3), 175-193. • Belenky, M., Clinchy, B., Goldberger, N. and Tarule, J. (1986). Women’s Ways of Knowing: The Development of Self, Voice and Mind. Basic Books:. • Biesta, G. (2004). Mind the gap. In C. Bingham & A. M. Sidorkin (Eds.), No education without relation (pp. 11-22). New York: Peter Lang. • Bingham, C. (2004). Let's treat authority relationally. In C. Bingham & A. M. Sidorkin (Eds.), No education without relation (pp. 23-38). New York: Peter Lang. • Boaler, J. (2008). Promoting 'relational equity' and high mathematics achievement through an innovative mixed-ability approach. British Educational Research Journal, 34(2), 167-194. • Fisher, B., M. (2001). No Angel in the Classroom: Teaching Through Feminist Discourse. Lanham, MD: Rowman & Littlefield Pulishers, Inc. • Hmelo-Silver, C., & Barrows, H. (2006). Goals and strategies of a Problem-Based Learning facilitator. The Interdisciplinary Journal of Problem-Based Learning, 1(1), 21-39. • Hmelo-Silver, C., & Barrows, H. (2008). Facilitating collaborative knowledge building. Cognition and Instruction, 26, 48-94.

15. References • Solar, C. (1995). An inclusive pedagogy in mathematics education. Educational Studies in Mathematics, 28(4), 311-333. • Taylor, C., & Robinson, C. (2009). Student voice: Theorising power and participation. Pedagogy, Culture and Society, 17(2), 161-175. • Thayer-Bacon, B., J. (2004). Personal and social relations in education. In C. Bingham & A. M. Sidorkin (Eds.), No education without relation. New York: Peter Lang. • Our Curriculum can be accessed at my website at a geometry course link http://community.emmawillard.org/Math/Schettino/index.htm or directly by using this urlhttp://tinyurl.com/2djncvb