You could introduce your child to CircleMath, invented by Dr Stephen Taylor. It is a very simple process for using 2 bases at once, eg 9 & 10, 10 & 11, 9 & 11 is best because it has many more patterns and involves 'Greek 9s'. The method relies on estimation, which I suggest is the starting point for all calculations. This works in any base... and, until metrics and decimals and calculators took over, everyone once used multiple bases every day... pints/gallons, inches/feet, days/weeks, etc.

When I first met Dr Taylor in 1985 he was tutoring a half dozen 5-year olds in his lounge using a whiteboard. The children could do 'sums' like 37x58 in their heads... in under 10 seconds... and tell me, a complete stranger - how they did it. As I was at Teachers' College at the time, preparing to teach high school math, I realised I knew very little... So, Dr Taylor and I became friends, I have most of his books, and still teach CircleMath when I get a chance. I recently wrote a PowerPoint intro - for adults - if anyone would like to see it.

I frequently get amazed looks as people realise how intuitively obvious it is, but it has been overgrown by dull modern mono-base thinking... and there is always a real buzz about this 'new thing'. It's the buzz all teachers and parents love...

As an HoD Math I put this to the test at one of our regional math assoc meetings, and talked about 15 capable, qualified math leaders through the basics, building up to this one: 114622/514. Following the very simple rules they got the answer correct in about 20 seconds... then looked at me and each other in silence... then said 'how the hell does that work?' Tada! CircleMath :)

It had taken only 45 mins, and they all got it correct, but few really got it at all.

Sad to say, for years afterwards some of them joked about that session as the workings of a crazed mind... not at all open for people whose job it was to open students' minds.

Edit for clarification: there was a link but has lapsed since Dr Taylor died in the last couple of years. I'll contact his son who was also brilliant at this stuff.

Edit2 for clarification: Dr Taylor wrote a book Theory of Mind, in which he identified and promoted the human brain's 'intrinsic base', meaning if things are couched in the right way then they resonate with the brain's OS, as it were, thus obviating the need for us to try to engender or pander to the child's interest. Things put in the right way will be interesting to the child. It is a provocative book about what is intrinsic and extrinsic to mind, learning and understanding.

2Perhaps keep on experimenting with (positive) whole numbers. For example, can he see how the sum of two odd numbers is always even? – Joel Reyes Noche – 2015-12-09T13:10:56.680

1One thing I could recommend, from the way I was taught by my parents, is to show that an "advanced" concept exists (ie. multiplication) and explain a possible derivation. My parents explained multiplication when I was about 6 (amazing I remember!), and then I remember writing tons of numbers down in a notebook and discovered simple "rules" of multiplication. Try powers of 2 to be simple maybe. But I was also an odd child who wanted to prove everything to myself. – aidan.plenert.macdonald – 2015-12-09T17:04:38.963

2I remember being interested in the problem of making two lines of equal length using blocks of unequal length (i.e. least common multiple). – Rhymoid – 2015-12-09T17:47:20.887

6

"We talked about the fact that there's an infinite number of even numbers but that if you add in the odd numbers you'll get twice as many numbers. That's crazy." The real crazy part is that you can match up the elements of the combined infinite set of odds/evens with the elements of just one. https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel (It might be too early to teach him about the cardinalities of infinity, though.)

– JAB – 2015-12-09T19:04:32.7373As JAB almost said, I don't think it's quite true to say you get twice as many numbers when you add in the odd numbers. You get the same number of numbers. Infinity + infinity = infinity. – bdsl – 2015-12-10T00:39:07.773

1I read the title as if you were trying to create some sort of mathematical super power in a child – Aequitas – 2015-12-10T02:25:53.693

2What does it mean if I still find 3 of those 4 things awesome? – Cort Ammon – 2015-12-11T03:43:31.847

Thanks for all the great feedback! There's a lot here to keep us busy. – Mathdad – 2015-12-11T09:55:16.267

A bit early for a 6yo, but I really enjoyed reading The Number Devil at about age 10. And it helped me understand a large part of high school math.

– Kevin – 2015-12-11T14:49:05.437I'm still on the lookout for a good prime-number-decomposition game (preferably a physical one rather than a tablet one, both are good though, maybe the tablet one for further ages?). – Jack Maddington – 2015-12-11T19:08:11.317

Just an anecdote: I once got an eight year old to think math was cool enough to talk to his friends about it by explaining to him that it is impossible to write down the number googolplex because it has more zeros ($10^{100}$) than there are atoms in the universe (about $10^{35}$). – Todd Wilcox – 2015-12-11T22:23:20.317

" there's an infinite number of even numbers but that if you add in the odd numbers you'll get twice as many numbers. That's crazy." -- No, you get the same number of numbers, which some think is crazier. You can see that there are as many whole numbers as even whole numbers because they can be matched up -- each whole number to its double, each even number to its half. If you want to see how much fun and fascinating math can be, check out Vi Hart's youtube videos. – Jim Balter – 2015-12-12T11:02:02.797

I know my dad taught me negative numbers sometime around grade 0. Got in trouble with the teacher at school who gave me 2 pencils and asked how it is possible to take 3 away. Shame on you, teacher! – Richard Le Mesurier – 2015-12-13T12:43:48.737