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The Bat and Ball Problem Revisited

I haven't thought about the bat and ball question specifically very much since writing this post, but I did get a lot of interesting comments and suggestions that have sort of been rolling around my head in background mode ever since. Here's a few I wanted to highlight:

Is the bat and ball question really different to the others? First off, it was interesting to see how much agreement there was with my intuition that the bat and ball question was interestingly different to the other two questions in the CRT. Reading through the comments I count four other people who explicitly agree with this (1, 2, 3, 4) and three who either explicitly disagree or point out that they find the widget problem hardest (5, 6, 7). I'd be intrigued to know if other people also disagree that the bat and ball feels different to them.

Concrete vs abstract quantities. Out of the people who agreed with that the bat and ball is different, this comment from @awbery does a particularly good job of giving a potential explanation for why:

The problem is a ‘two things’ problem. The first sentence presents two things, a bat and a ball. The language correctly reflects there are two things we should consider. The first sentence is ‘this plus that equals $1.10’. It correctly sounds like a + b; two things. The first sentence presents the state of affairs, not the problem itself. The second sentence presents the problem. The language of the second sentence reinforces the two things idea because there’s still the bat and the ball and they’re compared against each other: ‘there’s this one and it’s more than that one’. The trickiness is that it is a two things problem, but the two things we need to consider are not the most object level single units, but the bat, and the bat-plus-ball. Our brains are pulled toward the object level division of things by the language and the visual nature of the problem. We have to think really hard to understand that the abstract construct of the problem is the same shape as the state of affairs – there are two things to consider in relation to each other – but while the bat and the ball are still involved, they’re reconfigured by a non-intuitive/non-object-like division.

There’s no object level mirror trick in the other two problems, they’re straight forward maths mapping an object level visual representation. The widget problem presents a process which doesn’t change how the machines and widgets relate to each other in its solution. Our brains don’t have to mash up the pond and the lilies to separate the visual presentation to an abstract level. We can see that the pond is the same pond, half covered with lilies then fully covered with lilies at the next step. We don’t suddenly have some new abstract unreal configuration of lilies and pond to contend with.

I think this is why Kyzentun and Ander’s methods help get at the bat and ball problem intuitively – because they bypass the conflict between object level and abstract and translate it into the formal algebra realm. The problem as presented is non-intuitive because the objects visualization it suggests doesn’t reflect the shape of the formal solution.

So I think this is a particular type of problem, one in which visual shape and language of the presentation collude to obfuscate the visualization of the solution at an abstract/formal level. It’s a different type of problem to the other two in this sense, because the objects they present can be used as given in the solution.

Closeness to correct answer. Another interesting possibility is in TheManxLoiner's comment - that the bat and ball problem is difficult because the incorrect answer is 'close to the real one', whereas for the other two problems the incorrect answer is 'wildly off'. I've written a comment in response but I need to think about this more.

Ethnomethodology. David Chapman pointed out that these introspective accounts of what people are thinking when they solve maths problems are very unreliable, and that I'd probably be better concentrating strictly on what people do, as in ethnomethodology:

Yes, the fundamental principle of ethnomethodological methodology is “look at what people say and do, and don’t ever speculate about what’s happening in their head, because we can’t know.” At first that seems like a straitjacket, and highly unintuitive; but it forces you to really look, and then you see what is going on.

This sounds promising. I'm only just getting round to reading some ethnomethodology, and I haven't got my bearings yet.

Cognitive decoupling. There's a link with cognitive decoupling (in Stanovich's original sense) that could be worth exploring further. Success in the bat and ball problem seems to involve decoupling from the noisy wrong answer. David Chapman recommended Formal Languages in Logic by Dutilh Novaes for more background on this. So far I've read maybe a third of it. I've also written a bit more about cognitive decoupling and the history of the term here.

Next steps. I'm not sure where I'm going to take this next. Probably nowhere much for a while, as I have other priorities. But some options are:

  • Anders came up with a load of similar problems in the comments. These are designed to be cognitively unpleasant in the same way as the bat and ball, so I keep putting them off. I should actually go through them!
  • I'm going to continue reading Dutilh Novaes and some ethnomethodology.
  • Connect more specifically to Stanovich's idea of cognitive decoupling.

Testing theories? Further out, it could be interesting to actually test some theories by trying alternative, disguised versions of the question, on Mechanical Turk or something. Right now I've barely considered this, because I haven't thought through what I'd want carefully enough yet, but it might be interesting to test variations in:

  • how concrete the things the quantities refer to are (e.g. really concrete like 'the price of the bat', or more abstract like 'the difference between the price of the bat and ball'. Some of Anders' variant questions might fit the bill
  • how close in magnitude the intuitive-but-wrong answer is, as in TheManxLoiner's comment

I'm very ignorant about experiment design, so to do this I'd to get help from someone more knowledgeable. And psych research sounds like a gigantic minefield even if you are knowledgeable, so I'd probably end up wasting my time. But probably I'd learn something from going through the process, and it's something that could maybe happen in the future.

The Bat and Ball Problem Revisited
A possible reason for this is that the intuitive but incorrect answer in (1) is a decent approximation to the correct answer, whereas the common incorrect answers in (2) and (3) are wildly off the correct answer. For (1) I have to explicitly do a calculation to verify the incorrectness of the rapid answer, whereas in (2) and (3) my understanding of the situation immediately rules out the incorrect answers.

I must have missed this comment before, sorry. This is a really interesting point. Just to write it out explicitly,

(1) correct answer: 5, incorrect answer: 10
(2) correct answer: 5, incorrect answer: 100
(3) correct answer: 47, incorrect answers: 24

Now, for both (1) and (3) the wrong answer is off by roughly a factor of two. But I also share your sense that the answer to (3) is 'wildly off', whereas the answer to (1) is 'close enough'.

There are a couple of possible reasons for this. One is that 5 cents and 10 cents both just register as 'some small change', whereas 24 days and 47 days feel meaningfully different.

But also, it could be to do with relative size compared to the other numbers that appear in the problem setup. In (1), 5 and 10 are both similarly small compared to 100 and 110. In (3), 24 is small compared to 48, but 47 isn't.

Or something else. I haven't thought about this much.

There's a variant 'Ford and Ferrari' problem that is somewhat related:

> A Ferrari and a Ford together cost $190,000. The Ferrari costs $100,000 more than the Ford. How much does the Ford cost?

So here we have correct answer: 45000, incorrect answer: 90000

Here the incorrect answer feels somewhat wrong, as the Ford is improbably close in price to the Ferrari. People appeared to do better on this modified problem than the bat and ball, but I haven't looked into the details.

For the metaphors

Not quite knitting, but close - you may like this piece by Sarah Perry explaining a spinning metaphor of Wittgenstein's:

And we extend our concept of number as in spinning a thread we twist fibre on fibre. And the strength of the thread does not reside in the fact that some one fibre runs through its whole length, but in the overlapping of many fibres.
MLU: New Blog!

Just curious, are you planning to migrate blog comments too? I didn't know about Netlify, but it looks very promising for what I want - a mostly static site with some support for storing form submissions - so I'm going to investigate it a bit now.

Why exactly is the song 'Baby Shark' so catchy?

There was a Durham University study running from 2010 to 2013 where they asked the public to record their earworms (I contributed a few).

They suggest a few features that go into a particularly persistent earworm. A couple that stood out to me:

- Simple exposure. Songs that are currently popular tend to predominate. (There is a lot of Lady Gaga in their corpus.)

- A melody in the 'sweet spot' where it's generic enough to be easy to remember and sing but also has some kind of distinctive 'hook' like an unusual interval.

The popularity feature definitely fits Baby Shark. I think the melodic-sweet-spot feature does too: it's overall an extremely generic and repetitive tune, but also has the distinctive, painfully memorable 'doo doo do doo doo' bit.

The paper they published is here (pdf link). From a quick skim I'm not convinced that the stats are going to be all that great, but you'll have to read it more closely to judge for yourself. At the least it might give you some useful hints on other references and some terminology to google.

(And if you find anything interesting, let us know! I'm extremely prone to getting songs stuck in my head and would also like to know more about earworms.)

Book Review: The Eureka Factor

Oops, I fixed that in my blog version and then accidentally posted the old draft here. Edited now, thank you!

The Bat and Ball Problem Revisited

Ooh, I'd forgotten about that test, and how the beer version was much easier - that would be another good one to read up on.

How did academia ensure papers were correct in the early 20th Century?

Not a full answer, but I would expect most of this kind of debate to be in more informal channels rather than journals (as in LiorSuchoy's answer).

Einstein, for example, was a prolific letter writer, and corresponded with many of the great physicists and mathematicians of the day, e.g. Born, Cartan and Schrödinger (from a quick google it looks like the Schrödinger letters are still not published as a collection, so I haven't linked them).

I read the Cartan letters, some time ago. I don't have access to a copy now, but IIRC they get much more into picking at disagreements/clearing up confusions than anything you'd find in journals. For example, I opened up the Google Books preview, and immediately found the following from Einstein (on page 13):

I am sending to you my articles on the subject, published so far by the Academy. The second, on the approximate field equations, suffers, however, from the drawback that, with the choice made there for the Hamiltonian, a spherically symmetric electric field is impossible...

Then as well as letters, there'd be conversations at conferences, gossip over lunch and in department common rooms, question sessions after lectures. This stuff is mostly lost, though, whereas the letters can still be read now, so that's where I'd look.

All of this still goes on between researchers now, of course, and that's still how news travels in individual research areas. If you want to know what's wrong with published papers you're much better off talking people in that field than trying to find retractions in the published literature. But academia was so much smaller then that informal networks of correspondence might plausibly cover large areas of science rather than a small research speciality.

The Bat and Ball Problem Revisited

Strangely, it can sometimes also go the other way!

One of my most eye-opening teaching experiences occurred when I was helping a six-year-old who was struggling with basic addition – or so it appeared. She was trying to work through a book that helped her to the concept of addition via various examples such as “If Nellie has three apples and is then given two more, how many apples does she have?” The poor little girl didn’t have a clue.
However, after spending a short time with her I discovered that she could do 3+2 with no problem whatsoever. In fact, she had no trouble with addition. She just couldn’t get her head around all these wretched apples, cakes, monkeys etc that were being used to “explain” the concept of addition to her. She needed to work through the book almost “backwards” – I had to help her understand that adding up apples was just an example of an abstract addition she could do perfectly well! Her problem was that all the books for six-year-olds went the other way round.

I think this is unusual though.

The Bat and Ball Problem Revisited

Ah yeah, I meant to make this bit clearer and forgot.

I'm not really sure what to make of that statement you put in italics. The jump in success rate could be down to better trained intuition. It could also be due to better access to formal methods. I don't really see it as good evidence for my guess either way.

If I get more time later I'll edit the post.

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