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But how are metacognitive feelings relevant to performance on theory of mind tasks like Spelke et al’s famous habituation task?

Spelke, Breinlinger, Macomber, & Jacobson (1992, p. figure 6 (part))

feeling of surprise

There is a feeling of surprise which has features characteristic metacognitive feelings.

‘the intensity of felt surprise is [...] influenced by [...]
the degree of the event’s interference with ongoing mental activity’

Reisenzein et al, 2000 p. 271; cf. Touroutoglou & Efklides, 2010

In particular,
‘the intensity of felt surprise is not only influenced by the unexpectedness of the surprising event, but also by the degree of the event’s interference with ongoing mental activity, [...] the effect of unexpectedness on surprise is [...] partly mediated by mental interference’ (Reisenzein, 2000, p. 271)
That is, the feeling of surprise is a sensational consequence of mental interference. (This can be tested by increasing cognitive load: this intensifies feelings of surprise without, of course, making the events themselves more suprirsing. But see (Reisenzein, Horstmann, & Schützwohl, 2017) for an alternative interpretation of such findings.)
So whereas the feelings of agency and familiarity are both consequences of unexpected fluency of processing, the feeling of surprise is supposed to be the opposite: it is a consequence of unexpected interference in processes.
\footnote{ An alterantive is proposed by Foster & Keane (2015, p. 79): ‘the MEB theory of surprise posits that: Experienced surprise is a metacognitive assessment of the cognitive work carried out to explain an outcome. Very surprising events are those that are difficult to explain, while less surprising events are those which are easier to explain.’ Foster & Keane (2015) is about reactions to reading about something unexpected, whereas Reisenzein (2000) measures how people experience unexpected events (changes to stimuli while solving a problem). The latter is much closer to what I’m after. }

synchronic

early-developing,
automatic process

???metacognitive feeling
 

looking duration
(v-of-e or dishabituation)

early-developing,
automatic process

???
 
metacognitive feeling
+ learnt associations

verbal response
(or manual search, ...)

So here’s my thought.
Earlier I asked, How does the early-developing, automatic belief-tracking process measurably influence looking duration?
I conjecture that its influence goes via a metacognitive feeling, perhaps surprise.
\textbf{Metacognitive feelings are important in development because they can create create stop-and-think moments on the basis of mental interference in the operation of early-developing, automatic processes.}
Earlier I also asked [second question], Why does the early-developing, automatic belief-tracking process not measurably influence verbal responses?
Importantly for me, metacognitive feelings will not get you from an early-developing, automatic process to a verbal response. This is because \textbf{metacognitive feelings are intentional isolators}. They have no intentional objects (or none that are related to what they are usually taken them to be about).% \footnote{ Of course it is also true that in most experiments where verbal responses are measured, there is typically no mental interference in any early-developing, automatic belief-tracking process. There are exceptions, however (Wimmer & Mayringer, 1998, p. e.g.). } If I have a feeling of surprise (say) and then you ask me a question. How should I answer? The feeling doesn’t tell me. At most it might cause me to stop and think, but it cannot tell me what’s happened, nor what I should think.
\textbf{A metacognitive feeling might leave you both surprised and not knowing what has surprised you. It may take further effort to identify which event was surprising.}
This is not to say that metacognitive feelings never influence verbal judgements: of course we know that they can because lots of experiments with adults involve questions about the subjects’ feelings. But for them to do this, there must have been some learning about what the feeling means. A metacognitive feeling could in principle inform a verbal response only if there were some way of interpreting it, such as a learnt association between the feeling and a feature like familiarity or surprise. (And interpretation may not carry us that far. As far as we know metacognitive feelings are not very specific: there may be no ’metacognitive feeling of false belief’ but only a feeling of surprise.)
\textbf{Metacognitive feelings are important in development for a second reason: they allow for the intentional (conceptual/representational) isolation of two or more distinct kinds of process.}
I mentioned earlier Koriat’s proposal that

metacognitive feelings ... allow a transition from the implicit-automatic mode to the explicit-controlled mode of operation.’

(Koriat, 2000, p. 150)

Koriat, 2000 p. 150

Although he wasn’t talking about development, I think we can see that there’s something to this.
Metacognitive feelings have a dual role. By triggering ‘stop-and-think’ responses to events which interfere with automatic processing, they may create opportunities for learning.
But because they are intentional isolators, they also serve to keep the later-developing, less automatic processes separate from the more automatic, early-developing processes.
So they both ‘allow a transition’ of one kind and prevent a transition of another kind.
So let me finish by offering you a conjecture about how which distinct processes interact ...

Conjecture:

Metacognitive feelings


connect developmentally unchanging, fast processes
for tracking objects and minds
to slow processes.

I think this is a nice story. But of course it’s quite a wild idea to invoke metacognitive feelings in elaborating a theory about development. What are the predictions of conjecture about metacognitive feelings?
I have to admit that it is difficult to generate predictions because relatively little is know about the development of metacognitive feelings.% \footnote{ *TODO: evidence of metacognitive feelings early in development (I predict there are). } For now, I have just one prediction which is not very specific:

Predictions:

Manipulations which affect
metacognitive feelings of surprise in adults
will also have task-irrelevant effects on
infants’ performance in violation-of-expectation tasks.