Basic Math in Monkeys and College Students (Cantlon and Brannan)
Human and multiple non-human species have demonstrated a variable capacity for intuitive number sense (approximate quantitative representation, counting, and simple addition). Mathematics has a significant link with verbal and visual symbolic representations (ie, the arabic number symbols) that have been used by humans to extend mathematical analysis far beyond the most basic intuitive quantitative analysis. Humans and a much more limited number of non-human species demonstrate any capacity for "symbolic sense" for even basic mathematics - the (variable) ability to associate verbal or visual symbols with the species' (variable) number sense. Major limiting factors on the results and comparison of results among previous studies have included small sample sizes and methodological differences (particularly when compared to human experiments). Although results should be taken with requisite skepticism, some of the differences found are still clearly in the domain of basic mathematical operations; multiple species can track quantity in a +1 addition experiment, but few non-human species seem to exhibit an intuitive concept of set addition. Furthermore, humans, although more capable than many other species, are clearly limited in their intuitive capacity for set addition. In the study above, adult human (n = 14) demonstrations of intuitive set addition was similar to that of adult female rhesus macaque monkeys (n = 2).
The study used the same method to compare the species, but sample size continues to be an issue. Results, though, do seem plausible: that humans performed a bit more accurately and quickly, and could more accurately add larger sets. The limitations found in human capacity also seem plausible, showing a declining accuracy as set size increased and also as set size and similarity in set quantity increased. If my assumption is correct that symbolic sense in mathematics is constructed upon numerical sense, results would support the inference that learning the rules of symbolic manipulation for mathematics doesn't necessarily mean that there is a depth of understanding in the meaning of results; in other words, it's likely that there is a significant difference between the ability to do symbolic math and the ability to understand the meaning of both the process and the result.
There are profound educational implications of the relationship between intuitive mathematics and symbolic mathematics. For one example, I've encountered this many times in teaching about the history of life on Earth and the concept of biological evolution. Current estimates put the age of our planet at about 4.5 billion years, and evidence of life has been found in fossils nearly 4 billion years old. It's easy to look over what I've just done because it's so commonplace: I represented the astronomically incredible age of our planet with 10 squiggly marks on the screen, and the age of life on our planet with only 8 squiggly marks! To really understand evolution, we need to understand huge quantities of time. I have heard a variety of analogies used to help students understand large quantities. However, many of these analogies are rooted in symbolic math, and, unsurprisingly, still involve thinking about huge quantities: for example, a mole, about 6.023 x 10^23, is huge.
To increase the intuitive understanding for the enormity of "billions" of years, I have students do an activity that specifically builds symbolic math upon a simple but powerful physical experience. The student prepares with a blank piece of paper and a writing implement, and for five minutes draws lines - "hash marks" - on a page. (Of course, students can take a break if their hand cramps a bit or if they are uncomfortable at all - I try to cheer them on a bit like a sport coach...) After five minutes I have them count up their marks, and then help them to think about each of them as representing one full Earth year. We then work through a calculation of how much time it would take them - with no breaks for eating, sleeping, etc - to make 4 billion marks. The usual result is usually around 40 years, so I ask them to think about what they'll be like when forty years from now -- it's enough to get the gears moving in a much more powerful way than just saying and/or writing 4 billion.
I think there are some interesting philosophical implications of these studies, too, though I'm admittedly superficial and short in my ability to deal with them. But I think they come down to the remaining "open" question of whether or not mathematical logic is created or discovered - ie: is math a property of the universe or a property of the human mind? Given that the human mind is based in the human brain, and the human brain shares at least some degree of common ancestry with the brains of all species tested (and certainly a great deal with the monkey), perhaps the question on the "universality" of mathematical logic is refined to whether it is a property of the universe or of the vertebrate mind.