One fun one is that people used counting boards for all of their complicated calculations (literally "calculi" = "pebbles", i.e. counters) for many thousands of years, starting we don't know precisely when but maybe sometime before 3000 BC in Mesopotamia, and at least in Europe continuing up until only a few centuries ago (in some places until the 18th century or after) and now almost no one has even heard of them, let alone has any idea how to use one.
(For what it's worth: I think a counting board is still the best way to get small kids doing some basic calculations and understanding a positional number system: moving buttons or pennies around on a piece of paper with some lines drawn on it takes much less manual dexterity than writing, and the representation is much more direct and concrete than written symbols.)
The abacus is a standard part of elementary education in several Asian countries, for precisely the reasons you mentioned regarding numerical intuition. In American education, a student might only learn about the abacus from a brief paragraph in their history textbook.
Even today, there are average people in the Chinese countryside who know how to calculate the solution to a set of linear equations with counting sticks (a technique known as fāngchéng - 方程). My point being that usage of mechanical calculation assistance is indeed a useful skill, and would probably be beneficial in American/western education as well.
A sliding-bead suanpan or soroban is a practical and very portable tool for doing basic sums and differences, but after working with my own kids I don't think it's as good as a teaching tool as a counting board is, and I expect it's probably not as effective for doing more advanced calculations either, compared to a flat counting board where counters can be positioned arbitrarily, and where it's easy to add as many additional counts as you want by just making some more lines on a new piece of paper.
The real advantages of a counting board are (1) it needs no special equipment beyond a pile of pebbles, pennies, buttons, or other tokens; (2) it can be easily modified to apply to different number systems or specific calculations (though it's perhaps not as conveniently flexible as symbolic writing); and (3) there are many different representations of any number, and the game of calculation is about starting the problem off immediately with one version of "the right answer" already on the board and then performing various meaning-preserving operations to simplify the representation until arriving at one which is convenient to interpret or compare. This seems quite different psychologically from the use of a soroban (disclaimer: I'm not an expert) which is more about performing a sequence of steps in a pre-determined algorithm to obtain a correct answer, with intermediate steps not showing a representation of the same number because the soroban has only one unique way to represent any particular string of digits. I think the more flexible and representation-agnostic tool better promotes an essential skill which only increase in use as people get to higher levels of mathematics and other technical subjects. The soroban might be better for an accounting tool but the inflexibility is a deficiency for a teaching/thinking tool.
I have a toddler, do you have any recommendations of things you did with a counting board with your kids for teaching basic math at a young age?
We use a variant of Steve Stephenson's counting board, which we call "button arithmetic" as an activity. Stephenson (since deceased) was a retired engineer turned high school teacher who got very interested in counting boards in the 2000s. He made some YouTube videos here:
https://www.youtube.com/playlist?list=PL545ABCC6BA8D6F44
and some web pages:
https://ethw.org/Ancient_Computers
https://web.archive.org/web/20170903104702/http://sks23cu.ne...
Some of Stephenson's historical speculations are somewhat implausible, but it's fun to think about, or try to invent your own alternative ideas, and overall I think ancient calculation methods are underestimated by many modern scholars.
With my kids (now 9 and 6), we haven't bothered with Stephenson's floating-point-with-exponents system, but we do base ten arithmetic using horizontal lines for powers of ten and a vertical line to separate positive/negative. The space between two lines represents (as in medieval Europe) five times the previous power of ten.
I went to a fabric store and examined every type of button they had in bulk, then bought a bunch of my favorite type: some round metal ones, somewhat smaller than pennies, symmetrical on top/bottom, with a slightly domed shape that makes them much easier to pick up than coins. But pennies also work okay, as do carefully chosen beach pebbles.
I think counting boards are quite helpful for kids, a powerful and flexible tool that they can grow into. They can get started with it at age 3–4, before having the manual dexterity to write numerals.
Thank you!
> The abacus is a standard part of elementary education in several Asian countries, for precisely the reasons you mentioned regarding numerical intuition. In American education, a student might only learn about the abacus from a brief paragraph in their history textbook.
IIRC, Montessori schools use them, or something like them.
The abacus is awesome, and fun to learn. My parents bought me a miniature one on a trip to San Francisco when I was 8 years old (first time visiting Chinatown). It came with an illustrated pamphlet and I started practicing with it and figured out how to use it for basic math. I'd recommend it for any kid.
American schools do use "manipulatives" to introduce counting and numbers, addition, subtraction, etc. They might use checkers, or popsicle sticks, or anything small and easy to hold/move.
The use of various kinds of mathematical manipulatives and concrete materials is great (including base ten blocks, cuisenaire rods, ten frames, number lines, dice, balance scales with weights, geoboards, pattern blocks, multi-link cubes, etc.). I'm a fan of all of them. But I think the counting board, per se, is a sadly neglected tool, not least because it gives a nice connection to the past.