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Before this summer, I hadn't realized how important it was to actually do math with other people. I remember enjoying that as a student, but didn't realize how important it would be for me as an adult, moreover, as a teacher. Working together is not only more fun, but we also get more out of it!

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I loved becoming a constructivist learner in this course. I feel that now I truly am one. I am no longer solely answer focused. I now approach tasks with the intent to learn. If I have solved and "answered" but not learned, then I'm not done. And, even if I have learned by doing a task, I'm still not done. Exploration never has to be over.

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I'm thinking about the power of questions in the problem solving process. Through my experiences at Bank Street, including the tangram puzzle, I have consistently asked many more questions than I have gotten answers. While I have learning a lot of mathematics through this process, I sometimes feel that I don't make as much progress in problem solving as I'd like to. I wonder how I can ask questions through my process while also challenging myself to stay focused on the guiding question that I start with. I think there are times when it can be helpful to stay focused on one question, while there are other times that make sense to explore questions as they come up.

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Thinking linearly was my default approach to any mathematical problem. However, this course has forced me to change my habit of thinking from looking for a constant rate as a predictable pattern to using other mathematical considerations.

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I have learned that communicating and reflecting, while seeming very cliché, have been crucial in elevating and developing my own understanding of mathematics in general.  Having to work in partnerships to share my thoughts and my voice has helped me to better understand the power of communicating my ideas.  I have gotten better at being able to clarify my ideas via learning from someone else’s way of interpreting a given scenario.  My repertoire of methodologies to use for problem solving has expanded and I know it is currently much greater than before starting this program.  While unlearning the “be quiet and think to yourself” mentality of my early childhood learning will not change overnight, I am slowly but surely beginning to open up and not be afraid to speak for fear of being wrong, feeling inadequate, and quite frankly, feeling stupid. Finally, I have learned that I have much more to learn about myself as a mathematician and having more to learn and discover is part of the art of and beauty of learning.

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I have learned to challenge my thinking and examine what I really know about mathematics. Using manipulatives and visuals helped to deepen my understand and realize why procedures and certain formulas work.(triangular numbers) .I also like the Math Congress because it gives us a platform to hear each other thinking and see if there are any connections. Overall, I learned the power of manipulatives and communication. You need it to deepen your understanding.

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I have learned about the importance of open-ended exploration in building my understanding of math ideas. I have learned that collaborating is crucial to building understanding in math and clarity is so important when communicating about mathematical ideas. When I learned about some of the real-world applications of the seemingly abstract math that I learned years ago, I realized how making these connections makes the math more engaging for me.

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Collaboration has been the most useful experience I am learning to embrace during this process. The shift of moving from receiving help being seen as cheating to help being a way to learn and grow together is not easy to release but I believe may be tremendously beneficial to both my students and me. I am grappling with how much help to allow in order to keep my personal enthusiasm and ownership of my learning. I am sure some of my students will have the same concern but this is the beauty of learning more each day.

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I have shifted by perspective about the role of collaboration in learning. Previously collaboration felt like it wasn't time well spent and/or that if you needed to collaborate it was an indication that you're not capable. Now, I see collaboration as essential to my learning process and as a result I have been able to access tasks that I most likely would not be able to alone.

I have also experienced first hand the value of using specific, guiding questions to support students in developing their own conceptual understanding.

I have learned that I am capable doing challenging mathematics.

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The value of collaboration! Doing this work, in conjunction with working with peers with mathematical knowledge more sophisticated than my own, has been an enlightening process.I have learned to ask other's to slow down the thought process they are describing to construct meaning of it for myself. I have also learned how valuable the skill of listening for understanding is.

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I am learning that it is ok to seek help or support from outside resources when my learning has “hit a wall.” Aside from acknowledging that there are gaps in my knowledge, I need to act and do something about it. There will be moments when I feel stuck or frustrated and I will need to take a break, or seek help from others whether that be colleagues or the internet. In the end, it is ok to take the found information as long as I am making sense of it for myself.

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This was a quite different experience for me. I was always taught math with a purposeful agenda. Lesson objective, followed by a mini lesson and practice. During class I felt very helpless during most of our explorations. I am a little uncomfortable with this approach but I am starting to see that if you are proficient in the content and child developmental you can craft lessons with a purposeful end. And if I am understanding the course design it means it is ok not to end with mastery of the objective.  This inherently  no different  in both a progressive or traditional approach.

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Working together in class made my groups' collaboration easier and work more smoothly since we already knew how we needed to work.  We each brought a new idea to the table and each of us thought of this problem in a different way. We accomplished more together than I would have been able to do alone.

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Throughout this course I learned to appreciate math as a collaborative process. In the past I have always been a solution oriented independent worker during math learning. It took me a while, but I am starting to appreciate hearing others thinking. My biggest fear was that hearing from someone else would ruin my experience and diminish my interest in the problem or task. I have learned to value others thoughts and view them as tools to deepen my own understanding. I look forward to establishing a collaborative learning community with my students and promoting this idea of doing math together.

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I am learning that through open tasks, students are allowed to address many different concepts, understand concepts deeply, and make many more connections. As a learner, the takeaways are much more impactful when we develop the math on our own. Jumping to the answer does not do much for me as a learner. Collaborative learning and open tasks allow for teachers to have more access to getting to know their students thinking about math and to facilitate learning. I have also learned responding to student questions with another question can help clarify a student's thought and continue their learning in a positive direction.

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I'm realizing that you don't have to discover everything on your own in order to make sense of things in math. Sometimes when we don't have access to the math, we need to do some learning about it and then have the opportunity to make sense from there. There is no way I could have made progress throughout this exploration if I had worked in isolation the entire time. I needed Millie and Ayanna to share with me and push my thinking in new ways in order for me to move forward.

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When learning and doing mathematics, it is so essential that when you are working with others that norms are developed and practiced. Reflecting back on my experience with working with others, trust and respect needs to be developed when working with one another so that everyone is comfortable contributing ideas and sharing their thinking. Additionally.....its so essential to ask learner questions and not teacher questions and to organize your thinking to help move forward in a mathematical problem.

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Collaboration is not about finding the right answer together it's about finding the next questions together

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I have learned that I am tenacious when interested in the mathematics. The course provided an opportunity to engage in ideas that I had long forgotten about and made connections between these ideas in a way that supported sense-making and ownership.  Thinking of learning as a social construct, I really benefited from the conversations and thinking of others.  We not only explored what it means to do math together but also learned how to frame new questions together.

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Collaborating with those who have a don't have the same content knowledge background as you is helpful to see the problem from a different perspective and provides a whole host of observations and connections that couldn't be made with a more homogenous pairing.