Consider the numeral 3. In isolation, without context, 3 represents this: * * *. Three items. In some systems, * * * had a dedicated symbol. In the Greek Ionic system, for instance, γ’ was 3, λ’ was 30, τ’ was 300, ,γ was 3000, ,λ was 30000, and ,τ was 300000. This allowed for numbers up to 999999, and didn’t require any sort of place holder. So 3,030 would be ,γλ’.
The Ionic system, like the Acrophonic system (adapted by the Romans), and our current system, was decimal. When we adapted the Indian placeholder system, we modified the numeral set (from Arabic numerals, which came from Indian numerals). Since then, we have been consistently using a set of ten single-character numerals, or digits (likely from Proto-Italic *digitos, “pointer”, and connected to the ten pointers we have on our hands).
Like the other numerals, the symbol 3 still has a consistent meaning in mathematical notation, but in a placeholder system, it has a more tenuous relationship to language. 3 means * * * of SOME unit, but it doesn’t necessarily mean “three”, and “three” isn’t necessarily written as 3.
Consider the number 35.43. This contains two copies of 3, but neither of those copies represents three items. The first one represents thirty items (* * * tens) and the second one represents three hundredths of an item.
Now consider the number 11 in binary, base two. This contains zero copies of 3, but it DOES represent three items (one pair of items, plus another item, that is, * * + *). Likewise, 10 in base three represents three items (as does 111 in the rarely used base one).
This is part of a general trend, as mathematical notation has evolved, of distancing that notation (and its underlying language) from natural language. Advantage: It streamlines the notation. Disadvantage: It makes the notation more difficult to learn, often (as in this case) in ways that are difficult to see for the adept mathematics user.
If our education is effective, we internalize that the 3 in 365 represents three hundred and we stop thinking about it. But for students who are struggling with this concept, that’s a big hill to climb. The early grade materials that I see spend a long time on place value, and then, at some point, we assume they’ve got it and we mostly stop mentioning it.
It also feels like we’ve largely abandoned discussions of base. I went to school during the computer revolution; I entered high school with a basic four-function calculator and left it with a personal computer. Learning binary and hexadecimal, and hence the concept of base systems, was a STEM priority then.
I mention bases because, in any base other than base ten, 30 is not properly read “thirty”. In base four, it is generally read “three zero” and represents the value twelve; it still represents * * * of some unit, but that unit is “fours”. We often refer to base ten as base 10, out of some mix of sloppiness and convenience, but ALL bases are base 10 (which is read as “base one-zero” is all bases except base ten).
If that last paragraph made your head hurt, I’m there with you. Fluent readers of basic mathematical symbols have generally internalized the linguistic connection of 3 to “three” (or its translation in your preferred language), 30 to “thirty”, and 13 to “thirteen”, that we struggle to detach that connection, just as students who have not reached that fluency struggle to attach it. And while true dyscalculia may be rare, that can add another layer to the struggle.