Counting to Three to Count to Two
My eldest daughter is two and some months old. She is at an age where she is quite taken with counting and reciting the alphabet. She does not do either with perfect consistency or fidelity, but rather varying degrees of approximation to the physical and vocal patterns that are embodies when counting or singing the ABC’s.
Of course, I like to ask her “how many” of various things there are. It’s not a quiz, it’s an opportunity for her to demonstrate he prowess. She loves it.
Something interesting — nay, crucial — about her learning revealed itself as I watched her count various things.
She was particularly apt at counting to three: “Wan, doo, FWEE!”. So apt was she, that if I gave her two items, and asked her how many, she would be sure to count one of them twice, so that she could count to three. If I gave her three items, she would count each once, also counting to three.
It took a bit of practice to show her that with two items, if we count each once, we STOP at TWO. She got it, but it was uncomfortable for her at first. She wanted to complete the pattern, to enact the entire temporal pattern as she understood it. For her counting to three was not some action build up of parts, but an inseparable whole.
Logically, counting to three necessitates counting to two. It’s Peano arithmetic: 3 is nothing but 2+1. Yet, for Sig, she has to extract the sub-pattern of “one-two” from the larger pattern of “one-two-three”.
This is not a one-off or a fluke. I’ve written elsewhere about how for organisms, unlike machines, it is the whole that begets its parts. The organs of the body did not get “put together” to make the organism.
The same consideration applies to the life of the mind. Wholes beget parts. Once begotten, those parts may develop a wholeness of their own, but it would be a mistake to assume that all mental patterns are constructed from smaller parts, rather than being extracted from larger patterns.
This is no small thing. This is one of the mysteries that lies at the so-called foundation crisis of formal mathematics. Our formal theories operate a ruthless deductive manner: axioms and production rules lead to theorems. Everything is built up mechanically in that way. Yet mathematicians seem to be operating in a different way, which is why they may venture conjectures that only become proven via a chain of deduction later, or demand a novel axiom be introduced to a system. From where did the conjecture arise? From where did the axiom emerge?
These are indeed active mysteries. But the recognition that we do not seem to behave as formal machines, at least not the ones we commit to paper and code, is crucial for how and how we create opportunities for learning. First: patterns and gut. Then: abstraction and head.
The working assumption that we are ourselves machines doesn’t only impact pedagogy, but the entire structure of the industrial ‘education system’ we have societally constructed.
We are at the beginning of a great educational awakening. It’s time to treat ourselves and our children as something more than machines again.
I think it works both ways in the real world and human minds: Big to small and small to big. Reality is immense.
It is perhaps the smaller mind that fragments, and the entire social structure has been built up by fragments, and so we see fragments and think in fragments. We’re blind to the unifying singular order, and doomed to disorderly repetition in a vain attempt to create the very order our fragmented thinking has in fact destroyed. We learn to count up, starting from one. Perhaps we should learn to count down, starting from infinity preferably.