There are some interesting ideas in this post. I teach binary numeral notation, and then arithmetic with that notation, to third graders each year as part of the math classes I teach in my town. One of my favorite resources is the book Algebra by Gelfand and Shen,
which includes problems in representing numbers in binary notation and doing arithmetic with binary notation that are very approachable to young learners. (The problems are also very good review for undergraduate math majors
in which the implicit base is base sixty. The link shown here mentions speculation from ancient Greece that that base was chosen because it has many different prime factors. That Babylonian system of numerals, whatever its origin, appears to be related to historical relics such as counting sixty minutes in an hour or 360 degrees of arc in a circle.
A slight variation that I read years ago using only the socratic method (the teacher only asked probing questions) to teach third graders binary arithmetic. Interesting how he weaved aliens into it.
For the tens you're counting lines of blocks, and for the ones you're counting blocks, so it doesn't match up. Maybe if the ones were lined horizontally so you're still counting vertical lines, they just happen to be 1 block tall instead of 10 blocks tall.
yes. this confused the heck out of me. the units need to be horizontal, or (when using physical objects) the multiples could be piled on top of each other. as it is, it's just misleading.
There was a cool computer game called the Zoombinis that we used to play at home. One of the puzzles taught binary arithmetic using strange characters that had two expressions. There's a picture here:
The addition went up to 15 I think. You had to aim the pinballs at the group that would 'overflow' and jump in the river.
There were some very challenging puzzles in the Zoombinis titles - but they were teaching the conceptual foundations and encouraging intuition, not focussing on the terminology and modern applications.
Edit: a more useful link, with educational context and a larger picture, is this, although it's in French:
Man, Zoombinis, those were the days. A few friends and I all "played" it in our elementary days; however I am convinced that the lessons I learned there about binary numbers, problem solving, and spacial and mathematical efficiency stick with me to this day.
As was written in this article, many students apart from CS never fully understand number bases, which is unfortunate. I applaud this attempt and others (Zoombinis, etc) to teach some of these basic concepts at an early age.
He could have also mentioned the old babylonians and their base 12 system. We still have an extra word for 12 (a dozen) and it explains our strange counting of time (212 hours in a day, 512 in an hour or minute).
http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773
which includes problems in representing numbers in binary notation and doing arithmetic with binary notation that are very approachable to young learners. (The problems are also very good review for undergraduate math majors
http://www.ocf.berkeley.edu/~abhishek/chicmath.htm
and help adults think more deeply about mathematics, which is why I like teaching with this book as a source of lesson topics.)
Edit after seeing other comment: I also mention to the children in my classes the Babylonian numerals,
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babyl...
in which the implicit base is base sixty. The link shown here mentions speculation from ancient Greece that that base was chosen because it has many different prime factors. That Babylonian system of numerals, whatever its origin, appears to be related to historical relics such as counting sixty minutes in an hour or 360 degrees of arc in a circle.