Many of us mathematicians feel that there is something special about mathematics and we try to show it to others. The most common way this is done is by speaking of the 'beauty' of mathematics and/or its applicability.
Let's take an example: Summing a geometric progression. (1/2 + 1/4 + 1/8 + ... + 1/2^n -- multiply it by 2, subtract, etc... Wow isn't that beautiful?! it all just works out!)
Try to explain this to someone. They don't need to have much mathematical background, other than fractions. I tried with many people and hardly anyone seems to think this is nice.
Children tend to ask questions that adults think. When children study mathematics (and probably all other subjects) there are two questions that come up a lot: "Who cares?" and "How could someone come up with this?"
"Who cares?" - The answer usually given to them is either "it's on the test" or at best "you'll see it used later". In fact, most don't see it used later, and if they do it will be in some other mathematical notion, and if they ask "Who cares?" there they will be given the same answer until some of them perhaps study physics or mathematics later on, and actually, the same answer is given in university, where the end-goal is implicitly partaking in active scientific research. So by giving this answer we're basically saying that all students of mathematics are on their way to being professors.
"How could someone come up with this?" - Usually the answer is something like "the person who came up with this had some incredible insight". By giving this answer we're basically telling students that the only way they can truly discover mathematics is by having some incredible insight. If they don't they are only left to admire what these mathematicians were doing, and at best imitate or superficially use their results (as most homework and exam problems in mathematics are now). This forces people studying to memorize mathematics (there's no other way when presented like this). [Note well-known claim: 'stand on the shoulders of giants']
To sum up this, we are not allowing children to make mathematics their own -- it is always someone else's and at best they can somehow relate to it. We are also teaching them that mathematics is a tool. But those of us who continued our study of mathematics into university usually don't see mathematics as a tool. Many of us see it as an experience. And by teaching mathematics in this way we are not allowing the students to experience mathematics.
And, what happens is that since it is all based on memorization, they forget. Usually right after the exam.
Speak about Erdos's "The Book" and in general our role models in mathematics (Fields medal until age 40, people who's work cannot be comprehended, "geniuses")
[Present the naive discovery by halving the distances each time - as a discussion where only questions are asked]
This way the student is empowered, feels self-confident, understands, doesn't need to memorize (and really will be able to reconstruct it when needed), will be able to show it to others and they will (with higher chance) enjoy it, and have a true mathematical experience. He will be able to collaborate with others or similar questions because they will also have a feeling for it.
The Pythagoreans did not write. Maybe this is one of the reasons. This sort of discussion just presented fits much better as an actual discussion than something written. Different people will take slightly different routes and each can be allowed, whereas a written text is static to a certain mode of thought.
So if this is so nice, why isn't this discovery method usually presented? Because it's longer. Mathematics is affected by a very strong force to brevity. As long as we adhere to that force, we will lose naiveness and always have "mystical" presentations.
How can one find such "naive discoveries" of mathematical results?
Be very critical, thinking like a child, clearing your mind of prior conceptions and always asking "why?" and "how?"
Find people to talk with about these things and try to discuss it together.
Read good books and websites [insert links]
History - in many cases, since the force of brevity brings us to mystical presentations, the original discoverers had a much more naive route. If you try to find it (and in many cases several mathematicians discovered the same fact -- find all of their methods) you may find a naive method. But take care!
Sometimes the historical story may be naive but still very long. Try to find shorter (but still naive) possible discoveries, using history as a partial guide
Sometimes the discovery was truly a work of inspiration. In these cases many times someone after the original discoverer has a different presentations, and in many cases that other presentation is what people studied from, not the original discovery.
Reversal of applications - Look for applications of the issue at hand (wikipedia, books, etc) and try to reverse them -- See if you can state the application as a question and use the attempt to answer that question to lead to the discovery of this issue
Meditate. Not in the "sit cross-legged" sense but just sit and stare and let your ideas drift. Since we're trying to be more like children here, try to clear your mind so that your mind is closer to a child's.
Some more issues:
Multiple naive discoveries - In many cases there can be several different approaches to discover the same mathematical notion. They are all valid, and represent different ways of thought that might allow for different people to feel comfortable with. Try to find many discoveries for the same issue.
Sometimes a complete naive discovery might end up very very long. So it could be nice to find "less mystical" proofs or "not completely naive" discoveries as something in between. We can even have a "scale" between brevity/mysticality and length/naiveness and have many different presentations for different steps in between.
Using heavyweight tools - If trying to compute the area of a circle, one can use the integral and it turns out somewhat simple. But in order to do that one must understand what an integral is, coordinate geometry and much more. So it's not that simple after all. For this, it would be nice to have two different naive discoveries - one if you already have some knowledge of the integral and one if you do not (the way Euclid and Archimedes did it). Both are valid and should be written.
Just like Thales introduced the notion of a proof, Pythagoras spreaded it, and Euclid made it commonplace upto the point that today all mathematics is proven, we can start a process like this - start by presenting possible discoveries of basic mathematics results, create a community that tries to discuss and discover these and perhaps reach a point where all of mathematics has a form of naive discovery. If this works, then someone trying to discover a new result might adhere to inspiration to find it, but will then sit and try to find a naive discovery of the same result.
[Insert some more examples - numbers, prime numbers, negative numbers, quadratic formula?, combinatorics? -- things we've been talking about at the school]
We invite you all to join the society for the naive discovery of mathematics.