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Papert explains Piaget's work and provides case studies of how the programming language, LOGO, can help. He provides a wonderful contrasting explanation of the weaknesses of how math and physics are usually taught in schools.
I learned quite a few things from this that I did not know before. People are very good at developing theories about why things work the way they do. I knew that these theories are almost always wrong. What I did not realize is that if you give the person a way to test their theory, the person will keep devising new theories until they hit on one that works. What is usually missing in education is the means to allow that testing to occur.
An especially imaginative part of this book were the discussions of how to create theory testing solutions that are much simpler and easier to apply than any school problem you ever saw in these subjects. Papert works from a very fundamental and deep understanding of math and physics to reach the heart of the most useful thought processes for applying these subjects. It is thrilling to read about what you have known for many years, and to suddenly see it in a totally different and improved perspective.
Another benefit I got from this book were plenty of ideas for how to help my teenage daughter with her math. She is very verbal, and Papert points out that math seldom teaches a vocabulary for talking about math. As a result, she memorizes a lot and gets dissociated from the subject. I got a lot of ideas for how to encourage her to personalize the concepts and problems by moving her own body. From that I realized that I often solve the same kinds of problems by recalling physical situations I have been in. But I have failed to help her make that connection because I was unaware of it on a conscious level.
If you want to improve as a learner, help others learn better and faster, or simply want to understand more about different ways to think, this is a great book. I hope that all teachers get a chance to read and apply it.
Enjoy learning more!