There’s a fascinating article on BBC News today, about a really interesting study that proposes that an internal mechanism in the Arabidopsis thaliana plant (which is used widely in scientific experiments as a model organism) regulates starch consumption in the absence of sunlight in a way that requires the plants to be able to mathematically “divide” the numbers of two different types of cells. Now I’m not a botanist and I can’t say whether the result is correct, but I do take issue with the claim that “They’re actually doing maths in a simple, chemical way”. The last quote from the article is more accurate: “This is not evidence for plant intelligence. It simply suggests that plants have a mechanism designed to automatically regulate how fast they burn carbohydrates at night. Plants don’t do maths voluntarily and with a purpose in mind like we do.”
All sorts of natural processes can be modelled using mathematics – so, for instance, Fibonacci patterns appear in a variety of plants in the operation of phyllotaxis (the arrangement of leaves on stems). We don’t say that these plants ‘do math’. And the same principle applies above to the new finding above. It’s incredibly cool that these mathematical patterns emerge, and it’s a very interesting question why they emerge biochemically. But that raises an even more interesting question: what do we mean when we say that humans ‘do math’?
Humans are organisms and thus part of the physical world, and so lots of the things they do unconsciously or without explicit reflection can thus be modelled mathematically. But this is not the same as saying that all humans do mathematics. This seems to be what is being suggested in the last quotation: that ‘doing math’ involves conscious, explicit, purposeful reflection on the mathematical aspects of reality. Being able to throw a curveball is not ‘doing mathematics’; being able to model the trajectory of a curveball is. And the overlap between the sets of humans able to do each task is minimal.
Let me give another example related to the plant study above. A child has a pile of 23 candies and wants to divide it among some gathered group of five kids including herself. She starts to her right giving one candy to each friend, continuing to pass them out until they’re all gone. When the process is complete, each child will have 4 candies and the three to the right of the distributor will have 5 each. We could, if we wished to, define ‘division’ as ‘the process of dividing up a group of objects among another group’ and then say ‘thus, the kids are dividing 23 by 5 and getting 4 with a remainder of 3’. But I think most of us would be reluctant to argue that the first child understands division, or knows how to divide. Even though distributing the candy is a conscious decision, and even though it requires some general process (one candy to one child), it does not require that the child be able to do mathematics.
For the same reason, I sometimes have some skepticism when my colleagues in ethnomathematics describe the mathematics of some human activity in terms of fractal geometry or the Fibonacci series. It is, of course, possible that people have some awareness of the processes behind their activities, and ethnographically, when they can talk about that, it is very interesting. For instance, if the child above says “Well, I know I have 23 candies and so they won’t go evenly, so there are going to be some left over at the end,” then we do indeed know that the child has some explicit knowledge of division. I worry, in fact, that because so many natural processes result in such sequences, that we confuse the result with the conscious awareness of the process. In doing so, we fail to investigate the explicit mathematical knowledge that humans do actually encode in all sorts of things they do, and we falsely attribute a sort of explicit consciousness to activities that have no explicitness underlying them (in humans, animals, plants, and even in nonliving things).
Glossographia 
Would you say that the the weaver of a piece of Tellem fabric or of a Chitimatcha basket were “doing math,” even if they didn’t call it such?
It’s a really interesting question. I’d argue that we don’t know the answer automatically just from the fact that we can discern a pattern in it. What we need to do is figure out what the makers know about the patterns they are creating – what kinds of knowledge are involved – before we could make those kinds of judgements. It’s not about what they call it – obviously ‘mathematics’ as a category is Western-centered – but about the forms of explicit knowledge and ways to manipulate the world that are entailed.
To clarify a bit more: basically my point in the last paragraph of the original post is that if you just look and say “Ooh, a Fibonacci sequence!” you’re actually not saying very much, since these sorts of patterns can emerge in completely non-sentient entities including non-living things. So to do good ethnomathematics, you need to be especially attentive to what people actually know and say (i.e., the cultural knowledge about the patterns and their creation), not the patterns themselves.