Let them discover $\;\ell=2\pi R\;$ by their own, at least the invariance of $\ell/R$
Do you think that the following method works well in a mathematics class of elementary school? What do you think about the idea that math education system would remove the dictation of the math formulas, instead, it would encourage the students to observe some part or the whole formula themeselves? In the following post I mainly focuse on "The perimeter of circle" in elementary schools.
As a generalization of this question, at the last lines of this post, I will extend this question to high school geometry, for example sphere area.
Do you have any ideas or similar suggestions or some situations you already experienced? When you were student, how did you feel about memorizing and be dictated of mathematical formula?
Here is my suggestion about discovering processess of circle perimeter formula in elementary school:
"Let's distribute a few circles, with different radii, among some different groups of students of elementary school. (Circles that are made of flexible wires and easily form a line segment. Or we can use some pulley and thread or-string. The radius of pulley is easilly measurable. The perimeter of circle is the length of thread ) Then let the students calculate the ratio of the perimeter of the circle to the length of the radius and they compare the result of each other. We do this in a situation where they are not yet familiar with the formula for the circumference of a circle. But they would observe that the ratio of perimeter by radius is independent of our chosen circle.They would observe that the ratio is the same for measurments of all groups. Perhaps they would be surprised.
We do not expect them to discover the number$\pi$ but hopefully they observe and discover that $\ell/R$ does not depend on datas so is a universal constant."
Generalization to high school geometric formulas: How can one extend this idea to high school problems for example observing that the surface area of sphere divided by $R^2$ is independent of chosen sphere?
Your answers and comments are very appreciated.