But is speaking “simply speaking” and is adding “simply adding”?
Imagine a very young child who is beginning to say what its parents recognize as the word “mummy.” The parents probably respond, enthusiastically:
- “Oh, you said mummy!”
By repeating “mummy,” the parents naturally assume they support the child to say mummy again. Their focus is entirely on “mummy”: on the child’s saying of “mummy” and on their repetitions of “mummy.” By encouraging the child to say “mummy” again (and more clearly), they are teaching the child to speak.
No doubt their encouraging repetitions do support the child. However, the parents didn’t merely repeat “mummy.” They also said:
- “Oh, you said mummy!”
From the very first words a child utters, parents respond not only by repeating what the child says, but also by speaking about speaking:
- “Say daddy!”
- “Do you want to speak to mummy?”
- “You said you wanted cookies”
- “Which cookie did you mean?”
- “What’s your name?”
- “What you said isn’t true”
- “Don’t use that word!”
Parents’ natural attitude is that they teach the child simply to speak. But, more spontaneously, without intending or noticing it, they initiate the child into the notions of speaking. One might call this neglected dimension of teaching: the reflexive dimension. When we teach the child X, we simultaneously initiate it into the reflexive notions of X: into the concept of what it learns.
This should apply also to learning addition, and I assume to just about anything we learn. There is an easily neglected initiation into a reflexive dimension of what is learned.
I suppose one reason why the reflexive dimension is neglected is that it is what enables talk about what the child learns. Reflexivity draws our attention away from itself, and thus from the fact that the child not simply learns what learns, but also the concept of what it learns.
If you want to read more about reflexive practices – how they are acquired, how they practically contribute to making language what it is (said to be); how they tend to be intellectually sublimated as theories of language – I want to recommend the writings of Talbot J. Taylor.
One article by Taylor that especially clearly demonstrates the early onset of reflexive language use in children is:
Taylor’s work on reflexivity challenges me to reconsider the nature of philosophy. For philosophy seems to be concerned with the kind of notions we fail to notice we initiate children into, when we say, “You said mummy!”
Philosophy is “about” what we don’t notice we learn as children.