A friend asked me the other day my philosophical reasons for intending to homeschool my six-month-old son, and I began by telling her this story. As a mathematician, I like this story because it controls many variables. . .
One year when I was working as a freelance math and drama teacher in the Portland, OR homeschooling communities, I taught two geometry classes in two locations: a homeschooling center and a charter school for homeschoolers.
The homeschooling center was a privately-run non-profit where I had virtually complete control of my geometry class. Students (in some cases influenced by their parents) took my class by choice.
The charter school was government-funded, so it needed to prove that its students were learning. So, I was required to give grades. In order to give accurate grades, I gave my students tests. It was possible to receive a diploma from this school, so the school had requirements (including some number of years of math) and there was a sense of needing (not choosing) to take the class.
The homeschooling center and the charter school served overlapping populations. Quite a few students attended classes at both.
So, I was the same teacher teaching the same course to the same students, but. . .
At the homeschooling center, my geometry students and I had a blast: there was a sense of energy and fun in the room. I focused on the idea of building up a mathematical theory from first postulates and they grasped it. If they found a certain series of steps in a proof irksome (because it appeared so often and seemed silly to have to write out so many steps) we would prove it as a class lemma, name it (We came up with some awesome names!) and from then on any student could use it as a single step in any proof. We came up with our own acronyms and symbols for common (or uncommon!) theorems to save us time in writing.
They were curious about all sorts of aspects of geometry, and I often launched into historical tangents. We spent some time looking at Euclid's Elements and his wonderful definitions (such as a line is a "breadthless length"). At the end of each chapter, I assigned a set of about twelve problems which I corrected at home (all other homework problems solutions were provided in the book or in class) and I spent time going over every concept that a student had struggled with, not moving on until I felt that the class had mastery of the material. I built in an extra day at the end of each chapter for hands-on projects: building all of the Platonic solids and a truncated icosahedron, measuring the height of trees using similar triangles, drawing a flower of life using a compass. . .
I was able to be a guide and a mentor to my students. They knew me as an individual, enjoyed my company, and appreciated my knowledge of math and teaching ability. At the end of the year, they had increased appreciation for, understanding of, and skills in geometry.
On the other hand. . .
At the charter school, students were more confused about the material, never really understood how to write proofs, complained about test scores but seldom asked me for help understanding the concepts behind the problems they had gotten wrong, and at the end of the year were confirmed in their opinion that math is confusing and not much fun. I struggled to connect with each student; they thought of me as an authority figure doling out grades and I had to work very hard to combat that impression.
I was a better teacher at the homeschooling center than at the charter school. The constraints of more traditional schooling made it harder for me to teach well, establish rapport with individual students, create an energized and fun classroom culture, and help each student to enjoy learning and achieve mastery of geometry. It is the (more) traditional school system itself that impedes learning and stifles curiosity, joy and positive relationships.
And keep in mind, the "more traditional school" in my story is a charter school for homeschoolers with more flexibility and less structure than a public school.
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