# Dimension, On the Shoulders of Giants

I’ve spent forty years teaching mathematics to undergraduate and graduate students in mathematics education. Through all these years, it was geometry that I most enjoyed teaching. I still have a passion for geometry, and it is this passion and some of the things I’ve learned and continue to learn that I hope to share in many of my blog posts. In this post I want to begin by telling you about a book and an essay in the book that have tremendously influenced my thinking about geometry and about how and when it should be taught.

The book is On the Shoulders of Giants, and the essay, written by the mathematician Thomas Banchoff, is titled simply, “Dimension.” Among other things, some of which I hope to share in future posts, this essay introduced me to Friedrich Froebel, the inventor of the Kindergarten in Germany more than 150 years ago.

As Froebel established kindergartens in different towns, he developed curricular materials and accompanying manipulatives that were based on his philosophy and understanding of child development and learning.

Froebel referred to the activities included in his curricular materials as “occupations” and the manipulatives he called “gifts.” Geometric concepts were the focus of nearly twenty of these occupations and gifts.

The following video clip shows a few of the gifts that were manufactured by the Milton Bradley Company in the US in the late 1800s, and also a few that are currently being manufactured in the US by Froebel USA.

Thomas Banchoff describes Froebel’s philosophy as follows: “If children could be stimulated to observe geometric objects from the earliest stage of their education, these ideas would come back to them again and again during the course of their schooling, deepening with each new level of sophistication.”

Froebel believed that “a correct comprehension of external materials things is a preliminary to a just comprehension of intellectual relations. A correct perception is a preparation for correct knowing and thinking.”

Banchoff not only affirmed Froebel’s philosophy and curriculum, but also suggests that it be expanded upward through the grades. He states, “objects should always be nearby. Awareness of space and volume should be a continuing part of mathematical experience in school at all levels.”

This is the philosophy that will drive not only my posts on this blog, but is the philosophy that we at AIMS embrace.

We’ll continue in the next post with a wonderful testimonial to the effectiveness of Froebel’s philosophy by Frank Lloyd Wright, who experienced the Froebel materials as a young child.