I am genuinely stumped about how to explain why 3 is smaller than 10 in a way that isn’t either “because it is” or requires early university math. And the moment we go to university maths all the comments about ambiguity are true and it’s unsolvable.

You gotta remember, numerals are just arbitrary symbols that we assign meaning to. To small children, you could rearrange the order of numbers and teach them to count to ‘10’ like “7, 4, 5, 1, 8, 3, 9, 2, 6, 70!” And to them it would make perfect sense.

The symbols dont have meaning til we assign them
meaning. The teacher probably implemented some way of tying meaning to the symbols, such as using tally marks. The teacher repeats the exercise many times, and then gets the kid to repeat the exercise on paper. The answer to the question is probably “3 is smaller than 10 because ‘III’ has less than ‘IIIII IIIII’”

The number 10 was probably chosen because it contains the number 1, which is less than 3, and requires an understanding of base-10 numeral system.

Its more of an abstraction and repetition question than a math question. Its hard for us to understand why a child might struggle with this, but I do remember being corrected lots of times for writing my numbers and skipping 10. Id jump straight from 9 to 11. I felt that ‘10’ didnt make sense. I insisted that 10 doesn’t exist. That was one of the hardest years of college for me.

Yes for sure. But it didn’t have to be that way. Other cultures used base 12 (egyptians) or base 60 (babylonians).

You can count to 12 on one hand using your finger segments, and can count to 60 with both hands, using your fingers as tally markers indicating 12, 24, 36, 48 and 60.

I think my problem is that if you inversely asked why those are true, my answer would be “because 3 is smaller than 10”.

I think I’d just write something like “1+1+1=3, 1+1+1+1+1+1+1+1+1+1=10, there are clearly more 1s in 10”. But that also just feels like a “because i am defining 3 and 10 this way” = “because it is”. Though now that I think about it, that’s kinda just the simple version of the university level answer, it works i guess.

In reality I would hope anything somewhat sensible would be correct here anyway, since it’s more about making the child think about their answer than anything.

Honestly why is 3 smaller than 10 could be a philosophical question.

But I think to a 5 year old we just kinda want them to know that numbers can be added together to get bigger. Or maybe they really are supposed to be able to decode the structure of the universe wtf do I know I want to public school in the US

I am genuinely stumped about how to explain why 3 is smaller than 10 in a way that isn’t either “because it is” or requires early university math. And the moment we go to university maths all the comments about ambiguity are true and it’s unsolvable.

You gotta remember, numerals are just arbitrary symbols that we assign meaning to. To small children, you could rearrange the order of numbers and teach them to count to ‘10’ like “7, 4, 5, 1, 8, 3, 9, 2, 6, 70!” And to them it would make perfect sense.

The symbols dont have meaning til we assign them meaning. The teacher probably implemented some way of tying meaning to the symbols, such as using tally marks. The teacher repeats the exercise many times, and then gets the kid to repeat the exercise on paper. The answer to the question is probably “3 is smaller than 10 because ‘III’ has less than ‘IIIII IIIII’”

The number 10 was probably chosen because it contains the number 1, which is less than 3, and requires an understanding of base-10 numeral system.

Its more of an abstraction and repetition question than a math question. Its hard for us to understand why a child might struggle with this, but I do remember being corrected lots of times for writing my numbers and skipping 10. Id jump straight from 9 to 11. I felt that ‘10’ didnt make sense. I insisted that 10 doesn’t exist. That was one of the hardest years of college for me.

Do you think we use base 10 because we have 10 fingers?

Yes for sure. But it didn’t have to be that way. Other cultures used base 12 (egyptians) or base 60 (babylonians).

You can count to 12 on one hand using your finger segments, and can count to 60 with both hands, using your fingers as tally markers indicating 12, 24, 36, 48 and 60.

Ten fingers gives you either base six (one hand for zero magnitude, other hand for one magnitude) or base eleven (both hands for zero magnitude).

A number line, or any approximation, would work.

The problem sheet, or the one prior, likely have an example of the answer format as well.

solution 1:

numbers 1 to 10 on the first row

arrow under 3 with text “smaller”

arrow under 10 with text “bigger”

solution 2:

“10 has more numbers/digits”

Both would be - imo - thinkable and sufficient solutions at quite young age.

(keep in mind that this is before they get asked “is 5 divisible by 2” and “No” or “Only with remainder 1” are the expected answers.)

The “more digits” answer is the one I’d expect from a kindergarten kid

Hm, how about “you can fit 3 into 10” or “if you take 3 away from 10, you still have something left over”?

I think my problem is that if you inversely asked why those are true, my answer would be “because 3 is smaller than 10”.

I think I’d just write something like “1+1+1=3, 1+1+1+1+1+1+1+1+1+1=10, there are clearly more 1s in 10”. But that also just feels like a “because i am defining 3 and 10 this way” = “because it is”. Though now that I think about it, that’s kinda just the simple version of the university level answer, it works i guess.

In reality I would hope anything somewhat sensible would be correct here anyway, since it’s more about making the child think about their answer than anything.

Honestly why is 3 smaller than 10 could be a philosophical question.

But I think to a 5 year old we just kinda want them to know that numbers can be added together to get bigger. Or maybe they really are supposed to be able to decode the structure of the universe wtf do I know I want to public school in the US

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