Asking that question was one of the dumber things I've done on this list. Apologies to all
Thanks for reminding me about pre-concepts. I've been thinking about something similar and wondering if this is part of what makes doing mathematics 'mathematical.' Historically, by the way, mathematics grew out of manipulating such material objects; however, there are indications that, at some point (and it may have happened more than once), there was sort of a leap.
Mathematics is considered a science; for instance, of patterns or, as Hegel puts it, quantity. I agree for a mathematician symbols of various sorts are effectively 'things'.
In the 80s some mathematicians (School Mathematics Study Group) in the US put together a formal curriculum - my aunt used it - which was a disaster (and a real pain for the kids involved). Indications are children learned little.
So to add a little to a discussion that possibly has continued far longer than it should. Mathematics may have a few characteristics that may distinguish it from other disciplines such as
1. A student has the ability, in principle, to be able to independently of teachers or peers verify a grade appropriate mathematics statement (not a definition although definitions admit, in a sense, a sort of empirical verification).
2. Solutions to problems are, in general, not subject to social conventions (which probably is included in the above). Amusingly, I believe in the US a state legislature once tried to set the value of pi to 3.1417
However, I'm not sure how such would fit together into a useful unit of analysis.