How do you raise a child as a rationalist? I can't say that that was exactly what I had in mind but it seems to make for a fitting title here. A more precise title could have been: "How to deeply educate a child such that it fun and natural".
Today I'd like to tell you about the lullabies I sung and to what that led.
When my firstborn was very young I adapted a classic German lullaby "Schlaf kindlein schlaf" to numbers. It started with with only a few verses but grew over time (in part by the need to cover longer times until he slept).
I did sing it in German but I tried to translate it here to give you a better idea. It goes to the melody of "Schlaf Kindlein Schlaf which you may not know but can google easily (note that in German there are nicer rhymes for 100, million, googol):
Sleep, baby, sleep!
Thy father counts the sheep,
One, two, three and four
Little baby sleep some more
Sleep, baby, sleep!
The refrain repeats and the verses are replaced as follows:
Five, six, seven, eight - Tired at the dreamland gate.
Nine, ten and eleven - Sleeping in the number heaven.
Twelve, thirteen and fourteen - Sleeping babies have you seen.
Fifteen up to twentyone - Dreaming baby sleep is done.
Twentytwo to hundredtwo - Baby I will care for you.
Hundredthree to thousendfive - Caring for you all your life.
Thousandsix to millionthree - Of your dream you can break free.
Millionfour to one googol - Dreaming of a giant ball.
Googol-one to googolplex - Steaming in the dreamland tracks.
Googolplex to infinity - I will always care for thee.
I get slower during the song and very slow with infinity - mostly they slept then.
I have dreams of this song where sheep accumulate to larger and larger blocks until the block number thats raising in blocks and everything ends in white noise.
I do no longer sing it to my older sons but they accompany me sometimes when singing it to my youngest (two years old). And they do know what googol means already.
I also have a bed ritual where I let them give the number of times I put the blanket on their face (they like it). When they give too large numbers I use blocks. These tended to get high too.
One time I asked for lower numbers (that was when my second oldest already knew halfs and quarters) which led to gaming for unusual fractions and ultimately to his insight that "There is no larger fraction than one half that can divide one" (by a seven year himself).
It seems to have put numbers so deeply in their mind and interest that my seven year old can do simple fractions, exponentials and roots in his head. I tried hard to avoid too much arithmetic before school lest they bore of math in school and that worked for his older brother (who nonetheless tops his class in math) but he just asks and asks and I have just given up and keep just answering his questions and posing comparable return questions at his Zone Of Proximal Development.
There are dialogs that run like this (contracted):
He: "In school we had to give tasks to get 50. I was allowed to give 5*10"
Me: "Can you give some other examples?"
He: "2*20+10" thinking a bit "20 time 2 and a half equals 50"
Me: "What about division?"
He: "100 divided by 2 obviously. Or 50/1."
Me: "How long is the side of a cube containing one litre?"
He: "10?" (omitting centimeters)
Me: "How do you know that?"
He: "You have told me." (*I* can't remember when; must be month's)
Me: "And how long is the side of a cube with 27 liters?"
He (dividing 27 then adding or something like that): "18,5?"
Me: "No. How did you get there?"
He: "There must be some number multiplied to get 27" (or something like that)
Me. "Yes, the side time the side times the side." (expecting him to try some numbers)
He: "What is the root of 27?" (he has picked up that root is the reverse of times the same number)
Me: "Good idea. Here whe have three times or a number to the 3rd power - so we need the 3rd root."
He: "And what is the third root or 27?"
Me: "Try it."
He: "2*2*2 equals 16 no 8" (he seems to remember a few powers of 2)
Me: "Yes. That is too small"
He: "5^3? 5*5*5?"
Me: "That is 125 - too large"
He: "What is 10*10*10*10*10?"
Me: "You mean 10 to the 5th power? That's hundred thousand"
He: "What 10*10*... [lots of 10s)?"
Me: "You mean 10 to the 30th power? Thats nonillion."
He: "What is 10 to the 100th power?"
Me: "That is called googol. A 1 with 100 zeros."
He: "What is 10^100^100^100?"
Me: "Do you mean 10^100 and that to the 100th power or 10 to the 100^100th power?"
He: Somehwat confused asks differnt questions, dialog levels off.
(note that in German "to the xth power" is simply "hoch" thus much easier to concatenate)
I have to say that I am quite proud of my children and wouldn't be surprised when you called me overly so. I have to add that we, my wife and I, invest significant time into our children, so just singing this song may not be enough. And it also may be that I was lucky that they are (partly) gifted with math (like me). But I have to emphasize that we did no rote memoization or repeated training whatsoever (and left that to school).
There are other things we do for 'rationalist training' which I will try to post some time soon.
- Bed time stories with complex patterns (endless stories, simply nested stories, parallel stories, forking stories).
- Everyday Experiments for young children.