Here is an excerpt from an email I wrote:
The general idea of developing the hierarchy of a subject or concept is critical. Take even something simple like handwriting. The breakdown of skills Montessori has for teaching the skill of handwriting is ingenious. I remember learning handwriting in school, by non-Montessori methods. It was hard to learn. All of a sudden I had to hold this big damn pencil in my hand, when I had no preparation for anything like that. (But I think another factor toward the difficulty was bad fine motor skills.) The experience was traumatic — maybe mildly, but still traumatic. Instead of getting encouragement, or, even better, a slow breakdown and build up of the skill, I got berated. At least, that is what I remember. Montessori, in contrast, has the young student do sensorial activities such as button a shirt, tie laces, make things with beads — to learn eye-hand coordination and muscular control and to develop muscle strength, as the student learns useful life skills; has the student trace letters on sand paper and felt — to learn the motor skills needed for tracing letters as he/she trains herself to make distinctions in touch; has the student put knobbed cylinders of various dimensions into the correct hole in a wood block — to learn the motor skills needed for holding a pencil (by grasping the knob) as he learns about volumes.
There should, of course, also be a hierarchical development for the core subjects. Algebra and geometry, as taught today, usually hit students over the head like a wood rolling pin. (“Why do we use letters in math???”) Montessori has students learn about and work with geometric shapes (in 2-D and 3-D) from an early age. Students reach a point in arithmetic where they study squaring and cubing — following the pattern laid out by the ancient Greeks — as geometric ideas. When you square a number you make a square that has sides of length, for example, 6
and area of 36. When you cube a number you make a cube that has sides of length, for example, 6 and volume of 216. Students learn to do a bit more geometry than that with their manipulatives, too. I think this kind of foundation is important, even for kids who could pick things up fast. (Just look at the [bureaucrats] who run and ruin our economy to see how fast and intelligent does not mean grounded and objective!!)
I think the problem too many students in most schools have in geometry is two-fold: they have not learned enough about shapes and the interrelationships amongst shapes, and they have not learned logic. Arguments and progressions of ideas are alien to many students and to modern education. Hell, I had coworkers when I taught in public high schools who could not understand a freshman text in logic!! No wonder students can’t learn it — when teachers don’t understand it. The coworkers were generally intimidated [by] logic, and did not even try to get it. But in regard to shapes, the ancients knew quite a bit before they actually made geometry into a science and before they grasped what methods they needed in proofs. And it took some time and thought to go from Thales and Pythagorus to Euclid. We should, too.
The NCTM (National Council of Teachers of Mathematics) advocates some ideas like this. But as the NCTM comes from the perspective of Dewey — and Marx — they are outside the realm of reason and rational pedagogy.