I thought I would share one student's response to chapter's 1-4, and my response to her. I am curious about comments from anyone who happens to read this. Simply, I wonder what impact her present take-aways from the book may be, and to what effect my response may elicit further consideration.
In the first Chapter of Young Children Reinvent Arithmetic, a key concept that stood out for me were three kinds of knowledge and how each of them can be expanded through math, just through different means. The physical knowledge is expanded just by observing objects and the social knowledge is expanded by showing students what has been constructed for them to know. The most important and effective knowledge for math,of course, is logico-mathematical knowledge allows for students to be the constructors of their own understanding. This is achievable by the teacher creating an experience for students so that they can be the authority in their mathematical thinking as we talked about in Friday's class. I appreciated how in class we were able to both be a part of and observe this type of set-up when we constructed our own conventions about consecutive numbers. We were provided with the launch to think about at home and try to create expressions for the 1-2-3-4 puzzle. In class, we were then able to discuss our expressions with peers, either agreeing with or disagreeing with one another. Then we were prompted to explore some patterns we saw that connected these problems together and express a convention to share. I can definitely see how this style (launch-explore) is much more beneficial than merely showing expressions.The second chapter talked about representation and a major take-away for me here was the use of manipulatives as being a symbol for numbers. Kamii talked about how kids prefer to use pictures to represent numbers when counting on. Chapter 3 was all about how social interaction is vital to understanding math and logic. Cooperation is necessary because it both mutually benefits the learners in the group when they can decenter and constructively criticize one anothers work and explain the "why" to one another. I appreciate how Kamii relates this to being beneficial to moral development as well. This was demonstrated in our class session, like I stated before when we compared our expressions and either agreed/disagreed with each other. Finally, chapter 4 was about allowing fro autonomy rather than promoting heteronomy. A key takeaway was allowing for children to make decisions for themselves by giving them choices not just in the intellectual realm but also in the moral. The title for the Kamii book is appropriate because it is all about creating opportunities that allow students to invent ways to connect math to the realities they experience. This is the only way they will take ownership and have a positive experience while learning math.
You have made an important mistake in trying to understand how we know people learn. The purpose for naming three types of knowledge is more about what the knowledge is of -- the physical world or the social world, or the knowledge that is built upon previous knowledge through the brain's inventions (constructions) of relationships between those other forms of knowledge. One human cannot transfer that knowledge (mathematical or otherwise) to anyone else--ALL knowledge is constructed. So the work of the teacher is to teach (math) "indirectly" -- because there is no other way. Even when "telling" - the student's mind must take in the perceptions (teacher's voice, images, etc.) and construct knowledge from that.
When children think (all people), we work with images in our mind. When we put those images on paper (for example) -- within that representation of our mental are all the ideas and connections we associate with that. Then we can operate on the picture, reducing the taxation on our mind to hold all of those thoughts while operating on them. As a learner, we invent our own manipulatives -- only then can a manipulative be that representation of our mental images. And thus, usually the (forced) use of manipulatives in a classroom just become another physical or social knowledge for children to take in, rather than to invent the relationships among children's current knowledge that the manipulative is intended to invoke.
Your summary is quite excellent. Your response reflects a *core* misunderstanding of learning theory, one that plays itself out in classrooms across the country, leading to the poor learning opportunities you have observed. Children will not "invent ways to connect math to the realities they experience" first because math is not something external to them, ever. Math is external to no one--everyone of us has our own mathematics. And the "realities they experience" also may not be the most precise use of words for the modern learning theorist -- again, reality (while it may exist out there) is not so much experienced as it is invented by us as a biological organism living in this "world." And since no one can actually know a "true" reality (assuming there is one), we have no way of ever knowing if our knowledge is correct or not. We only have viability in the world as we experience it--that we remain alive, functioning, happy, and we have other humans who seem to confirm our knowing of this reality...