For an overabundance of ideas on the concept of zero, Charles Seife wrote a wonderfully engaging book titled "Zero: the biography of a dangerous idea." If you are looking for a book to help translate Math into English, Ben Orlin's newest book, "Math for English Majors," is equally fantastic as all his others. (My favorite of his books is "Change is the Only Constant," as it so clearly illustrated to me the beauty of the philosophy of calculus, tho I don't claim to be able to use calculus for any purpose.)
I could go on about math books and math education (but not actually about math) for a significant time, but what I came here to say is that a numeral is a figure, symbol, or group of figures or symbols denoting a number. No numeral is either positive or negative: they are all symbolic. The base-10 system necessarily uses 10 numerals to denote its numbers. When "0" is used in conjunction with "1" in the proper order, the numerals together denote the number ten.
The 10-bead-per-bar style abacus, popular with young learners in the west, uses beads, not numerals, to represent the way we group our numbers in tens for counting. The beads (or stones or marks in the dust) may also represent anything else you choose: seeing as this style of abacus is most efficiently used for mere tallying, there is no need to use the middleman of "number" when keeping track of how many games of Sorry I've played in a row on the abacus. I can just say, "I'll play one row of Sorry games today and no more." If you know I'm using an abacus with 10 beads per row and you know your numbers, you'll know I mean 10 games. Otherwise, you'll know when we're done because all the beads are moved. (Tho if we're playing Sorry, you know your base-10, so that was a poor example.)
The number of beads on the bars of this poor abacus you seem to find so offensive (your offense is likely the only reason I am defending it, as I agree there are more efficient styles) aligns nicely with the number of fingers on our two hands. You have kindly failed to point out that every child counting on their fingers has, likewise, been born with the wrong number of fingers to count in a base 10 system, and that for this reason we all ought to be counting base 11. Wait, whaaat? Yes, thank you; most people would find that sort of imposed mathematical conversion traumatizing.
And speaking of imposed mathematical conversion, I will conclude with a story of one of my favorite people: my mom. The household abacus with 10 beads per bar was missing, and all my dad could find was the silly little abacus with 5 beads per bar. "I can't count my exercises on this; there's not enough beads," he said. She replied, "Just count in base 5 and you'll be fine."
My 10-fingered children play with many formats of abaci.
Re: ten fingers: ha! You're entirely right that, to be consistent, I should look askance on our manual digits. Thanks for helping me wear my anti-ten-abacus vitriol a little more ironically. And I've just put a hold on those books. Thanks!
A very deep truth in education is that people don't stumble so much because things are hard, but because things are abstract. The bit that catches people out in math is not logical thinking or manipulating numbers, but the level of abstraction. Many adults struggle on the Wason selection task, until you phrase it in a specific context (see the wiki page) and suddenly everyone gets it. Someone once did a study where they gave the task to, I think 6-year olds, using stuffed toys instead of cards, and in a context with a nice story around it, and they almost all got it right.
I've seen kids who were supposed to solve problems like 5 ?? 2 x 3 where the ?? needs to be filled with the symbol for greater, equals or less-than. Quite apart from getting the symbol the wrong way round ("the crocodile's mouth opens towards the larger number"), they'd think something like "well 5 is bigger than 2, and 5 is bigger than 3, so it's 5". But ask them whether they'd rather have a stack of five $1 bills (or coins) or two stacks of 3, and suddenly they not only pick the right answer but can even explain "well if you have two stacks of 3, you put them on top of each other you have 6, so that's like the 5 and then you have one left over".
Also, everyone ever who runs away screaming at "x + 2 = 5" because the x summons demons or something, but "[ ] + 2 = 5, what goes in the box?" they can do without a problem.
So yes, kids need to count pebbles, blocks of wood (Cuisenaire rods! ❤), and other things that they can not only visualise, but also manipulate with their hands and feel them as physical objects as they're moving them around or counting them - that's where the somatic part comes in. Before you learn to count, you learn to count objects. And then you learn to do it on the abacus, still moving beads around and still using your hands, but the beads now have superpowers.
I've recently been reading H. Wu's "Basic Skills versus Conceptual Understanding" (https://bioscience.tripod.com/readings/basicskillvsconcpt.pdf) which I like in many ways - it defends teaching longhand addition, digit by digit and carry the ones, not only because that means you can use it to solve subproblems in more interesting questions later on, but also because the whole thing actually makes sense and you can teach it that way. Wu's suggestion is, I think, the philosophic to the abacus' somatic - an explanation of the algorithm in terms of properties of addition like that a+b is the same as b+a or that you can do a+b+c+... in any order you like, which is one of the things that makes both the abacus and the algorithm work. But I'd personally teach the abacus first, and then when you do the paper version someone or other will go "It's just like the abacus! But on paper!"
About the tenth bead - I think it's supposed to be to carry 1, so you can add numbers and then adjust at the end to make say 2,10,4 become 304. Or to do the "borrow" thingy when you're subtracting. I would make the tenth bead a different color! (EDIT: https://journals.sagepub.com/doi/full/10.1177/10323732221132005 - yes, it's a "carry one" thing. The idea is that for 8+3, you set 8 beads on the units, then slide 3 over one by one, if you run out of beads because all 10 are over already, you reset the row, add one on the tens, and move on. If you end up with all 10 over at the end of the sum, you have to adjust upwards.)
But you can also shift the carry over immediately when you need to. Otherwise you get stuck when you have to carry more than one, when you're multiplying. There is another argument for extra beads in some cases, like on the Chinese suanpan: https://web.archive.org/web/20110927085436/http://webhome.idirect.com/~totton/suanpan/ but you can also work with other methods just fine that don't need the extra beads. If you're trying to get kids to learn place value in the decimal system, as you said, you need 9 beads. Not more, and not less.
What a beautiful response — I stand corrected that the tenth bead is a literal mistake (even if I think it bad), and will in the future be less likely to vandalize my local Ikea with a pliers (though possibly more likely to with a jar of paint).
Re: the demons summoned up by using letters in algebra: when I was a math tutor, I found that random emoji were weirdly useful for this. E.g. "x + 4 = 11" was hard for some kids, but "🦕 + 4 = 11" was child's play. (Although this short TED talk is also wonderful: https://www.ted.com/talks/terry_moore_why_is_x_the_unknown )
We like the abacus from RightStart math, which has ten beads per row, but each side labeled in a different way so that you can be literal and use it up to 100 or abstract and use it for higher numbers.
We've used MathUSee's "Decimal Street" model, where each "house" can only hold 9. We've also loved James Tanton’s Exploding Dots for teaching the concept of bases (Thanks for the previous recommendation!)
Coloring the last bead sounds like the best of both worlds, but I did find one 9-bead abacus currently available on Amazon:
For an overabundance of ideas on the concept of zero, Charles Seife wrote a wonderfully engaging book titled "Zero: the biography of a dangerous idea." If you are looking for a book to help translate Math into English, Ben Orlin's newest book, "Math for English Majors," is equally fantastic as all his others. (My favorite of his books is "Change is the Only Constant," as it so clearly illustrated to me the beauty of the philosophy of calculus, tho I don't claim to be able to use calculus for any purpose.)
I could go on about math books and math education (but not actually about math) for a significant time, but what I came here to say is that a numeral is a figure, symbol, or group of figures or symbols denoting a number. No numeral is either positive or negative: they are all symbolic. The base-10 system necessarily uses 10 numerals to denote its numbers. When "0" is used in conjunction with "1" in the proper order, the numerals together denote the number ten.
The 10-bead-per-bar style abacus, popular with young learners in the west, uses beads, not numerals, to represent the way we group our numbers in tens for counting. The beads (or stones or marks in the dust) may also represent anything else you choose: seeing as this style of abacus is most efficiently used for mere tallying, there is no need to use the middleman of "number" when keeping track of how many games of Sorry I've played in a row on the abacus. I can just say, "I'll play one row of Sorry games today and no more." If you know I'm using an abacus with 10 beads per row and you know your numbers, you'll know I mean 10 games. Otherwise, you'll know when we're done because all the beads are moved. (Tho if we're playing Sorry, you know your base-10, so that was a poor example.)
The number of beads on the bars of this poor abacus you seem to find so offensive (your offense is likely the only reason I am defending it, as I agree there are more efficient styles) aligns nicely with the number of fingers on our two hands. You have kindly failed to point out that every child counting on their fingers has, likewise, been born with the wrong number of fingers to count in a base 10 system, and that for this reason we all ought to be counting base 11. Wait, whaaat? Yes, thank you; most people would find that sort of imposed mathematical conversion traumatizing.
And speaking of imposed mathematical conversion, I will conclude with a story of one of my favorite people: my mom. The household abacus with 10 beads per bar was missing, and all my dad could find was the silly little abacus with 5 beads per bar. "I can't count my exercises on this; there's not enough beads," he said. She replied, "Just count in base 5 and you'll be fine."
My 10-fingered children play with many formats of abaci.
Re: ten fingers: ha! You're entirely right that, to be consistent, I should look askance on our manual digits. Thanks for helping me wear my anti-ten-abacus vitriol a little more ironically. And I've just put a hold on those books. Thanks!
So much to love here!
A very deep truth in education is that people don't stumble so much because things are hard, but because things are abstract. The bit that catches people out in math is not logical thinking or manipulating numbers, but the level of abstraction. Many adults struggle on the Wason selection task, until you phrase it in a specific context (see the wiki page) and suddenly everyone gets it. Someone once did a study where they gave the task to, I think 6-year olds, using stuffed toys instead of cards, and in a context with a nice story around it, and they almost all got it right.
I've seen kids who were supposed to solve problems like 5 ?? 2 x 3 where the ?? needs to be filled with the symbol for greater, equals or less-than. Quite apart from getting the symbol the wrong way round ("the crocodile's mouth opens towards the larger number"), they'd think something like "well 5 is bigger than 2, and 5 is bigger than 3, so it's 5". But ask them whether they'd rather have a stack of five $1 bills (or coins) or two stacks of 3, and suddenly they not only pick the right answer but can even explain "well if you have two stacks of 3, you put them on top of each other you have 6, so that's like the 5 and then you have one left over".
Also, everyone ever who runs away screaming at "x + 2 = 5" because the x summons demons or something, but "[ ] + 2 = 5, what goes in the box?" they can do without a problem.
So yes, kids need to count pebbles, blocks of wood (Cuisenaire rods! ❤), and other things that they can not only visualise, but also manipulate with their hands and feel them as physical objects as they're moving them around or counting them - that's where the somatic part comes in. Before you learn to count, you learn to count objects. And then you learn to do it on the abacus, still moving beads around and still using your hands, but the beads now have superpowers.
I've recently been reading H. Wu's "Basic Skills versus Conceptual Understanding" (https://bioscience.tripod.com/readings/basicskillvsconcpt.pdf) which I like in many ways - it defends teaching longhand addition, digit by digit and carry the ones, not only because that means you can use it to solve subproblems in more interesting questions later on, but also because the whole thing actually makes sense and you can teach it that way. Wu's suggestion is, I think, the philosophic to the abacus' somatic - an explanation of the algorithm in terms of properties of addition like that a+b is the same as b+a or that you can do a+b+c+... in any order you like, which is one of the things that makes both the abacus and the algorithm work. But I'd personally teach the abacus first, and then when you do the paper version someone or other will go "It's just like the abacus! But on paper!"
About the tenth bead - I think it's supposed to be to carry 1, so you can add numbers and then adjust at the end to make say 2,10,4 become 304. Or to do the "borrow" thingy when you're subtracting. I would make the tenth bead a different color! (EDIT: https://journals.sagepub.com/doi/full/10.1177/10323732221132005 - yes, it's a "carry one" thing. The idea is that for 8+3, you set 8 beads on the units, then slide 3 over one by one, if you run out of beads because all 10 are over already, you reset the row, add one on the tens, and move on. If you end up with all 10 over at the end of the sum, you have to adjust upwards.)
But you can also shift the carry over immediately when you need to. Otherwise you get stuck when you have to carry more than one, when you're multiplying. There is another argument for extra beads in some cases, like on the Chinese suanpan: https://web.archive.org/web/20110927085436/http://webhome.idirect.com/~totton/suanpan/ but you can also work with other methods just fine that don't need the extra beads. If you're trying to get kids to learn place value in the decimal system, as you said, you need 9 beads. Not more, and not less.
What a beautiful response — I stand corrected that the tenth bead is a literal mistake (even if I think it bad), and will in the future be less likely to vandalize my local Ikea with a pliers (though possibly more likely to with a jar of paint).
Re: the demons summoned up by using letters in algebra: when I was a math tutor, I found that random emoji were weirdly useful for this. E.g. "x + 4 = 11" was hard for some kids, but "🦕 + 4 = 11" was child's play. (Although this short TED talk is also wonderful: https://www.ted.com/talks/terry_moore_why_is_x_the_unknown )
We like the abacus from RightStart math, which has ten beads per row, but each side labeled in a different way so that you can be literal and use it up to 100 or abstract and use it for higher numbers.
We've used MathUSee's "Decimal Street" model, where each "house" can only hold 9. We've also loved James Tanton’s Exploding Dots for teaching the concept of bases (Thanks for the previous recommendation!)
Coloring the last bead sounds like the best of both worlds, but I did find one 9-bead abacus currently available on Amazon:
https://www.amazon.com/ZIJIA-9-Bead-Column-Counting-Education/dp/B07L3H59FT
J. W. Van Cleve has an article on nine-bead abacuses, with thoughts on construction:
https://docslib.org/doc/2776946/nonus-the-nine-bead-abacus-its-theory-and-derivation
I don't know if he still makes them, but his contact information is at the end of this blog article:
https://joevancleave.blogspot.com/2009/07/bead-frame-madness.html?m=1