Ah, thank you! I’m excited for this to be out in the world!

There’s lots more to say here, but I’ll stop where our paper does: learning haptic 1:1 procedures for counting may not transfer as easily to the problem of set-matching as it does visually, but it’s clear that procedural knowledge is important for expressing an understanding of Hume's principle.

Also, if kids didn’t use 1:1 correspondence when blindfolded first, they were less likely to do so when tested w/ visual access second. Together, these results suggest that challenges to implementing 1:1 correspondence were not conceptual, but a problem of executing an effective procedure.

So, in Study 2, we only tested the kids who had some visual sensitivity. We tested them twice: once w/ visual access (as usual) and once blindfolded, counterbalanced. We found two things: kids’ performance dropped when blindfolded, resembling those who completely lacked visual inputs.

Even more surprisingly - blind individuals who had some small gradient of visual sensitivity (e.g., a little peripheral vision) successfully matched large sets like numerate, sighted adults. But those who didn’t have any visual sensitivity did not. Further scaffolding didn’t help the latter group.

This isn’t what we found though. Instead, blind adults and kids who had a verbal counting system AND used 1:1 correspondence to count & construct sets didn’t extend this procedure to set-matching, instead opting for alternate strategies (e.g., approximation, matching continuous extent).

But for blind people, an approximation strategy should be just as laborious as using 1:1 strategy since it can’t be done in “a glance”. We thought that this might even lead to an advantage for blind kids who may be more likely to use 1:1 correspondence w/o access to quicker alternatives.

With this in mind, we tested blind adults and kids, reasoning that if sighted kids learn to set-match from an analogy to counting, then we might expect that blind individuals who have learned a haptic counting procedure to also analogically extend this procedure to haptic set-matching.

One reason for “failures” on the task could be that alternate strategies can be implemented more quickly than 1:1. People may be more inclined to approximate sets or match line extent because it can be done in a single shot. But 1:1 correspondence requires attending to each object in a set.

Still with me? Now i’m finally talking about our study! Here, we ran with Schneider et al.’s idea, exploring why this procedural understanding may be hard to arrive at.

Schneider et al. suggested that children who had learned to count might have a concept of exact number, but that matching large sets required some procedural knowledge, which might come from experience with counting sets (which features a similar 1:1 relation between objects and number words).

Past work in our lab led by Rose Schneider explored this question in US kids. Schneider et al. (2022) found that children who could count to large numbers (CP-knowers) were more likely to match large sets than children who couldn’t count (subset-knowers), but many still didn’t use a 1:1 strategy.

Yet many innumerate groups approximate instead. Of course, lots has been said about such failures to use 1:1 correspondence in this task. Many have pointed to flaws in task instructions, participant motivation, differences in cultural context, as potential explanations beyond the role of language.

In one test of knowledge, participants are asked to construct a set that matches the experimenter’s set in number. The idea being that, even those who don’t have a counting system may recognize that two sets are equal if they are in 1:1 correspondence.

Thus far, the debate in this literature has been centred around the claim that people who lack linguistic symbols for large numbers cannot reason about Hume’s principle — that 2 sets placed in 1:1 correspondence are exactly equal.

@tpcanoe.bsky.social had us create geotags for sites and map migratory patterns on google maps. This was years ago (so I don’t remember specifics) but I remember it being a fun assignment!