A Meditation on Ratio
What do you think a ratio is? I emphasize “think” because I’d like you to be interested only in the fragmentary pictures conjured by the question—you want to be aware of what you think a ratio is, not what you think you know about ratios, which are two different thinks.
So, perhaps try now to answer the question, but don’t censor your first “thinks” here. Pay close attention to what thoughts arise, but don’t try to change them into anything else. And if very few thoughts arise, simply notice that too. Don’t try to manufacture thoughts about this. The idea is to simply be aware of your initial impressions about a particular concept, not to judge what’s coming.
Only after you have noticed what you have noticed about your thoughts about ‘ratio’ can you then set this bubble of incomplete thoughts, bits of pictures, and perhaps even some emotional reactions in front of you for criticism, editing, and analysis.
Good. So Now We Share Our Noticings.
I’d like to share with you one noticing of mine about the concept of ratio that has come out of something like this ‘math meditation’ described above, and it is my hope—and my sense—that you will be able to relate to it:
There is a ‘twoness’ about ratio that shouldn’t be there.
That is, my impressions—my “first thinks”—about ratio are of two parts, two quantities, and I have to work a little to see that ‘ratio’ has a meaning and identity as a single object. Incidentally, by contrast, ‘sum’ and ‘product’ each have immediate meaning to me as individual things. One is the result of addition and the other is the result of multiplication. They are each single values, and the work involved is in the other direction: I have to work a little for ‘sum’ and a little more for ‘product’ to see these as being decomposed into two or more parts. This is as it should be. When I think mathematically, my primary mental access to these concepts should be as coherent units, not as collections of parts. Analogously, if my first access to ‘cat’ is “see ay tee” or “whiskers, claws, tail” and I have to work to identify ‘cat’ as a single thing, my cognition about this animal will be impaired—a deficit that will become more obvious the more complex is my work with cat concepts.
This seems to be the situation we’re in with regard to multiplicative reasoning in particular in schooling, beginning possibly with the concept of ratio (but likely even “before” that with the concept of multiplication). The primary psychological relationship we allow students to have with ratio is one in which a ratio is two things rather than one. If you doubt this, perhaps you can imagine giving students (or even adults) a simple prompt to write 5 ratios. How many do you think would write a whole number (not written in fraction form) as one of their responses? I would expect close to none. But perhaps another good test is to watch the little video I made here and notice whether something about it goes against your grain. That feeling—I would suggest—is likely the result of the collision of the two notions of ratio: the one we have primary access to, and the one that would allow for a more productive relationship with this concept.
I could be totally wrong, of course. You should just imagine that written on a sign and slung over every post I write here.