";s:4:"text";s:30694:"I've written about close reading recently, and this has been something like a close reading of the bat and ball problem. One conjecture for what people are doing when they get this question wrong is the attribute substitution hypothesis. Dual Process Theory (System 1 & System 2), Thinking, inherently fast and inherently slow, No, seriously, the answer isn't ten cents. The thread with anders in particular goes into lots of other examples of how we think through solving various problems, and is well worth reading in full. The ‘obvious wrong answers’ for 2. and 3. are completely unappealing to me (I had to look up 3. to check what the obvious answer was supposed to be). It could also be due to better access to formal methods. The closest I found was this paper by Szaszi et al, who did carry out these sort of interview, but it doesn't include any examples of individual responses. The later paper by Meyer, Spunt and Frederick is much more interesting to me, because it really starts to pick apart the specifics of the bat and ball problem. h�b```��,@������b�@�1AXP��A�P@P@�A��M�L�~,�~W��=�ڦ�1l����~L߫�O�#N����C�G� This simple problem does have 'ten cents' as the answer, so it's very plausible that people are getting confused by it. But some options are: Testing theories? This framing is important for interpreting the CRT. she was about 50% likely to give the right answer. If B=6 then the solution would be 100+6+6 and A+B would be 112. And I have to stare at it for a minute or so to work it out, slowed down dramatically by the fact that Obvious Wrong Answer is jumping up and down trying to distract me. My original post on the problem was a pretty quick, throwaway job, but over time it picked up some truly excellent comments by anders and Kyzentun, which really start to dig into the structure of the problem and suggest ways to 'just see' the answer. The explicit moral is that we are too willing to lean on System 1, and this gets us into trouble: The bat-and-ball problem is our first encounter with an observation that will be a recurrent theme of this book: many people are overconfident, prone to place too much faith in their intuitions. You can find more short problems, arranged by curriculum topic, in our short problems collection. When you first read the problem and hear that the bat is a dollar more than the ball, and the bat and the ball cost a dollar and ten cents, your brain assumes that the ball is automatically 10 cents. Connect more specifically to Stanovich's idea of cognitive decoupling. I don't really see it as good evidence for my guess either way. Some of Anders' variant questions might fit the bill, how close in magnitude the intuitive-but-wrong answer is, as in TheManxLoiner's comment. As it is, I score two out of three, because I've trained my intuitions nicely for ratios and exponential growth. There’s no object level mirror trick in the other two problems, they’re straight forward maths mapping an object level visual representation. ", "I'm trying to find out what x is, like in 2x + 7 = 15. (It feels like a similar emotion to noticing I've gotten the wrong amount of change, in fact.). How many fence posts are there? Frederick's original paper on the Cognitive Reflection Test is in that generic social science style where you define a new metric and then see how it correlates with a bunch of other macroscale factors (either big social categories like gender or education level, or the results of other statistical tests that try to measure factors like time preference or risk preference). 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. I haven't found exactly what I wanted, but I did turn up a few interesting studies on the way. Do the math, and you will see. Problem 18: During a baseball game, a baseball is struck at ground level by a batter. It also has the biggest jump in success rate when comparing university students with non-students. (One exception is the sort of puzzle sheets that are often given to young kids, where the unknowns are just empty boxes to be filled in. I learned the most from the individual responses, though. This showed a clear difference: 94% of 'five cent' respondents could recall the correct question, but only 61% of 'ten cent' respondents. Frederick's original paper has been cited nearly 3000 times, and dredging through that for the good bits is a lot more work than I'm willing to put in. some effort was made to reconsider the answer given, even if it was ultimately incorrect. I was allowed to sit in a corner while the tutor would try to teach my cousin algebra. Repeating it with numbers, but avoiding currency symbols, the original problem specifies that the bat and the ball cost 1.10 together, and the bat costs 1 more than the ball. Bat-and-ball games (or safe haven games) are field games played by two opposing teams, in which the action starts when the defending team throws a ball at a dedicated player of the attacking team, who tries to hit it with a bat and run between various safe areas in the field to score points, while the defending team can use the ball in various ways against the attacking … Printing both versions was slightly more successful, bumping up the correct response to 35%, but it was still a small effect. The three items on the CRT are “easy” in the sense that their solution is easily understood when explained, yet reaching the correct I'd kind of assumed that there'd be some kind of serious-business Test Creation Methodology, but for the CRT at least it looks like people just noticed they got surprising answers for the bat and ball question and looked around for similar questions. Split the difference evenly for 5 cents? However, if you do the math, you see that the difference between a $1 and 10 cents is actually 90 cents, not $1. I have to consciously remind myself to apply some extra effort and get the correct answer. With the total cost of bat & ball at $1.10, and the difference between the two being $1, the ball couldn’t cost 10¢ because that’d make the bat cost $1.10, which would bring the combined price to $1.20. A problem which I would volunteer for a CRT is the snail climbing out of a well. I also need to divide by 2. 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. I do the bat and the ball problem with low effort, and the exponential growth one with no effort, but I find the machines one a bit confusing. Certainly I seem to have better intuition there, without having to resort to rote calculation. This is wrong because if the ball costs 10 cents then 1 dollar more then 10 cents would be $1.10. 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’. 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. They apparently find cognitive effort at least mildly unpleasant and avoid it as much as possible. As I remember, my thought process was something like: Could it be 10 cents? Three balls and two bats cost £210 in total. People just really like the answer 'ten cents', it seems. 2415 0 obj
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My gut hasn't internalised anything useful, and it's super keen on shouting out the wrong answer in a distracting way. A ball and a bat cost £90 in total. I wanted to see if others had raised the same objection, so I started doing some research into the CRT. We all used some variant of the method suggested by Marlo Eugene in the comments above. with the “bat and ball” problem to form a simple, three-item “Cognitive Reflection Test” (CRT), shown in Figure 1. The correct answer to this problem is that the ball costs 5 cents and the bat costs — at a dollar more — $1.05 for a grand total of $1.10. This is instead hived off into its own separate subject, called 'algebra', and the rules are taught much later in a much more formalised style, without much attempt to build up intuition first. (When we think of 'solving simultaneous equations' we imagine people pulling the answer out, rather than pushing the solution in and seeing if it fits - solving versus checking as it were.). (4b) There is a 20m fence in which the fence posts are 2m apart. The first sentence presents two things, a bat and a ball. Let X the price of baseball bat and Y the price of the ball… This is exactly what bothers me and resulted in me wanting to look up the question online. The Ferrari costs $100,000 more than the Ford. I think that after reasoning my way through all these perspectives, I'm finally at the point where I have a quick, 'intuitive' understanding of the problem. However, it's pretty obvious that a lot of people won't have access to this method. �Loڀ47��Uؙ��
%�]�2����dl�r���0����|0K�H/���D -��Pv Whew, I was thinking to write a separate post on this, but now I don't have to! There's a variant 'Ford and Ferrari' problem that is somewhat related:> A Ferrari and a Ford together cost $190,000. Every day, the patch doubles in size. If you instead remove the same quantity b you get 100 cents. 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. I think the solution to bat and ball of "10cents, oh no, that doesn't work. I just wanted to say this was a really fun read. On the quiz the other 2 questions were definitive. Imagining “24” afterwards feels some intermediate level of wrong between “ten cents” and “100”; my mental graph of the growth curve puts the expected value 24 at “way too low” intuitively before I can compute the actual exponent. Imagining “100” afterwards feels wrong, but less immediately so than “ten cents” did. I'd still appreciate more detailed transcripts, including the time taken to solve the problem. It could improve participants' ability to intuitively see the answer. The correct answer is 5¢. 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. It correctly sounds like a + b; two things. The cognitive reflection test (CRT) is a task designed to measure a person's tendency to override an incorrect "gut" response and engage in further reflection to find a correct answer.It was first described in 2005 by psychologist Shane Frederick.The CRT has a moderate positive correlation with measures of intelligence, such as the Intelligence Quotient test, and it correlates highly … This is not quite true even for the examples he gives — faceblind people and calculating prodigies exist. I did however find this meta-analysis of 118 CRT studies, which shows that the bat and ball question is the most difficult on average – only 32% of all participants get it right, compared with 40% for the widgets and 48% for the lilypads. But this gives a good example of an idea of "training mathematical intuitions" I hadn't thought about before. First they tried bolding the words more than the ball to make this clause more salient. 2428 0 obj
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At this point they completely gave up and just flat out added “HINT: 10 cents is not the answer.” This worked reasonably well, though there was still a hard core of 13% who persisted in writing down 'ten cents'. Or something else. One of the striking early examples in Kahneman's Thinking, Fast and Slow is the following problem: (1) A bat and a ball cost $1.10 in total. The correct answer is 5p. So why do so many people answer incorrectly? I'll only summarise the bat-and-ball-related parts of the comments here. I'm going to continue reading Dutilh Novaes and some ethnomethodology. _____ minutes, (3) In a lake, there is a patch of lily pads. Our brains don’t have to mash up the pond and the lilies to separate the visual presentation to an abstract level. In the case of the OP, system 1 has been trained to really understand exponential growth and ratios. If the ball were to … The wrong answer is just incredibly compelling. In this post, I'm going to assume you've come across the Cognitive Reflection Test before and know the answers. Many people will immediately think the answer is … They also demonstrate one way to recover from an incorrect solution (think about the answer you blurted out and see if it actually works). I must have missed this comment before, sorry. I suspect that this is less true the other two problems - ratios and exponential growth are topics that a mathematical or scientific education is more likely to build intuition for. If I recall correctly, the third question was easier than the second question, which was easier than bat & ball: I think I generated the correct answer as a suggestion for 2 and 3 pretty much immediately (alongside the supposedly obvious answers), and I just had to check them. The first sentence is ‘this plus that equals $1.10’. Here are questions which might be similar to (I): (4a) I booked seats J23 to J29 in a cinema. A baseball bat and a ball total cost is $8.50. They apparently find cognitive effort at least mildly unpleasant and avoid it as much as possible.” This is a really interesting point. However, I haven't found any great evidence either way for this guess. “Bat & Ball” Cognitive Reflection Test. X + Y = Equation 1 X = Y + The baseball bat is $3.00 more than the ball. For all three questions, the wrong answer comes to my mind first*. “The bat-and-ball problem is our first encounter with an observation that will be a recurrent theme of this book: many people are overconfident, prone to place too much faith in their intuitions. For the bat to cost $1 more than the ball, the ball has to cost 5 cents and the bat $1.05. It was apparent to me that simply phrasing the problem in terms of concrete objects was activating something like visualization which made the problems easy, and just phrasing it as abstract numbers was failing to activate this switch. 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). Can they all work on one widget simultaneously, speeding that one up? The lilypads one has a family resemblance to the classic grains-of-wheat-on-a-chessboard puzzle, for instance.). However, getting people to actually answer the question correctly was a much more difficult problem. These demonstrate one way to reason your way to the correct answer (solve the simultaneous equations) and one way to be wrong (just blurt out the answer). I don't really share drossbucket's intuition - for me the 100 widget question feels counterintuitive the same way as the ball and bat question, but neither feels really aversive, so it was hard for me to appreciate the feelings that generated this post. This is where you really get to see the variety of ways that people tackle the problem. Then I looked at the question, and computed that you’d need 500 machine.minutes to make 100 widgets, so 5 minutes with 100 machines. ", "Yeah, but you did it by arithmetic. Can participants reproduce the correct question from memory? Three experiments explored the effects of word problem cueing on debiasing versions of the bat-and-ball problem. How many seats have I booked? First of all the bat cost $1 more then the ball. If you haven't, it's only three quick questions, go and do it now. Still, the examples given for their response categories give a few clues. 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. For the first one the bat had to be one dollar MORE than the ball so, if the ball was 10 cents the bat had to be $1.10cents that plus another 10 cents is $1.20. Interesting: these could cover a couple of misunderstandings, one is that B>=100, the other that "The bat costs $1.00 more than the ball" does not mean B-b=100, but that B-b>=100. yup that's better" is all done on system 1. h�bbd``b`�$ׂm7Dd �Pa⺂X� �D���2��Sg&F�@#:��G�� ~4 I can also imagine (c) I'm leaping to the “wrong” answer, then trying to verify it, noticing it's wrong, and correcting it, all in the same subconscious flash, but that feels off. Closeness to correct answer. The paper just states the following: Motivated by this result [the answers to the bat and ball question], two other problems found to yield impulsive erroneous responses were included with the “bat and ball” problem to form a simple, three-item “Cognitive Reflection Test” (CRT), shown in Figure 1. I'm not really sure what to make of that statement you put in italics. When I see 1., however, I always think ‘oh it’s that bastard bat and ball question again, I know the correct answer but cannot see it’. If together they cost €1.10, and the bat costs €1 more than the ball... the solution should be 10 cents. For example, writing B for the bat and b for the ball, we get the two equations. This made surprisingly little impact: 29% of respondents solved it, compared with 24% for the original problem. Ultimately you would think that the ball costs 10 cents because ($1+$0.10=$1.10). I haven't thought about this much. I'm interested in any comments on the post, but here are a few specific things I'd like to get your answers to: My rapid, intuitive answer for the bat and ball question is wrong (at least until I retrained it by thinking about the problem way too much). 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. (Btw, the bit about them adding a hint and there still being people who wrote 10 cents made me laugh out loud, that's hilarious.). The answer is .05.’ And then I checked my answer by doing 1.05 + .05 and 1.05 - .05. Whenever possible, the bat should be captured and sent to a laboratory for rabies testing. This is an equation. Marlo Eugene's solution, for instance, is a mixed solution of writing the equations down in a formal way, but then finding a clever way of just seeing the answer rather than solving them by rote. I think this reliance on formal methods might be somewhat less true for exponential growth and ratios, the subjects underpinning the lilypad and widget questions. Only a few build up the necessary repertoire of tricks to solve the problem quickly by insight. and she immediately started answering the questions 100% correctly, very rapidly too. In Kyzentun's version these become much more concrete objects, the width of the text and the total width of the margins. Anders came up with a load of similar problems in the comments. Sometimes you get 2+3=□, sometimes it's 2+□=5, but either way you go about the same process of using your wits to figure out the answer. Is an easier question getting substituted? You could do this in excel by doing each of the calculuations in a cell, but that would simply be superfluous, it's quite easy to do this mentally, or on paper, as you can see. That leaves five for the ball. this (apparently unpublished) ‘extremely rough draft’, suggested early on by Kahneman and Frederick. 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. I'm not sure where I'm going to take this next. This is quite interesting in itself, but I was most excited by the comments section. I found this to actually be the best coverage of the whole test, and it's analysis of people's reasoning to be a significant step up from what I've seen in other coverages of the test. The impact between bat and ball is a collision between two objects, and in its simplest analysis the collision may be taken to occur in one-dimension. Third was 47 On the 48th day it was full so on the 47th it was half there cause each day if halves. Frederick's original Cognitive Reflection Test paper describes the System 1/System 2 divide in the following way: Recognizing that the face of the person entering the classroom belongs to your math teacher involves System 1 processes — it occurs instantly and effortlessly and is unaffected by intellect, alertness, motivation or the difficulty of the math problem being attempted at the time. which we could then solve in various standard ways, e.g. How do people go about designing tests like these? So there's an extra 10 cents--oh, of course, the difference between $1 and $1.10 has to be distributed evenly between both items, so the answer is 5 cents. Obviously the machine-widget ratio hasn’t changed, and obviously exponential growth works like exponential growth. Also, there is a kind of problems like "one wallet contains ten coins, another one contains twice more, and the total is twenty; explain" that get asked much earlier than kids learn algebra, if I remember right. Did you arrive at 10 cents? The distinctive mark of this easy puzzle is that it evokes an answer that is intuitive, appealing, and wrong. The first sentence presents two things, a bat and a ball. She was very frustrated by it, and if I verbally asked "What's seven minus five?" And psych research sounds like a gigantic minefield even if you are knowledgeable, so I'd probably end up wasting my time. So when they gave me the residual my next thought was “now all the objects are the same, so whatever I do to one I do to all of them”. 2434 0 obj
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<. So as you say, for more tricky arithmetic problems, it may be the case that what mental circuits are "activated automatically" determine the first answer you arrive at, and you can exploit that effect with edge cases like this. This makes me think of ordinary real life contexts where I would say “costs $1.00 (or $20 or $100) more than.” It seems possible it might be clear to both me and my listener I meant ‘at least x more than,’ ‘as much as x more than,’ or ‘approximately x more than.’ I wonder if changing the wording to “The bat costs exactly $1.00 more than the ball” would help any. In (1), 5 and 10 are both similarly small compared to 100 and 110. We generally learn arithmetic as young children in a fairly concrete way, with the formal numerical problems supplemented with lots of specific examples of adding up apples and bananas and so forth. These are designed to be cognitively unpleasant in the same way as the bat and ball, so I keep putting them off. Impact of feedback on the bat-and-ball problem. The correct answer to the bat & ball question is 5¢. The widget problem presents a process which doesn’t change how the machines and widgets relate to each other in its solution. (Aside: I'd like to know how these other two problems were chosen. I just saw the answer to the bat and ball problem within a few seconds. I think that for me both "quickly check that your answer is right" and "try something vaguely sensible and see what happens" are both ingrained as general principles that I don't have to exert effort to apply them to simple problems. Yes, the most common answer for that for what the ball costs is 10 cents. really concrete like 'the price of the bat', or more abstract like 'the difference between the price of the bat and ball'. This question first turns up informally in a paper by Kahneman and Frederick, who find that most people get it wrong: Almost everyone we ask reports an initial tendency to answer “10 cents” because the sum $1.10 separates naturally into $1 and 10 cents, and 10 cents is about the right magnitude. Request PDF | The bat-and-ball problem: a word-problem debiasing approach | Three experiments explored the effects of word problem cueing on debiasing versions of the bat-and-ball problem… However, after more than a little head scratching I’ve gained an understanding of this puzzle. The problem as presented is non-intuitive because the objects visualization it suggests doesn’t reflect the shape of the formal solution. X + Y = $1.10. It could have been .04 0r.03 and the bat would still cost more than $1. Conversely, finding √19163 to two decimal places without a calculator involves System 2 processes — mental operations requiring effort, motivation, concentration, and the execution of learned rules. That sounds a bit abstract, so let's look at some responses (I'll paste all these straight in, so any typos are in the original). Here’s the solution: Although $1.00 + $0.10 does equal $1.10, if you take $1.00 – $0.10 you get $0.90, but the problem requires that the bat costs $1 more than the ball. *In the third question, the actual answer "24" does not come to mind first, but the general sense of "half that number" does. The widgets problem I do a noticeable double-take on, but it's rapidly corrected within one conscious time-step; the “100” is a momentary flicker before my brain settles on the correct answer. The categories are: Correct answer, correct start. How much does the Ford cost?So here we have correct answer: 45000, incorrect answer: 90000Here the incorrect answer feels somewhat wrong, as the Ford is improbably close in price to the Ferrari. It's possible that there is a different common cause of both the 'ten cent' response and misremembering the question, but it at least gives some support for the substitution hypothesis. One is that 5 cents and 10 cents both just register as 'some small change', whereas 24 days and 47 days feel meaningfully different. University students overwhelmingly tend to provide the biased answer to this problem. It was around the same for me: I knew I had to be careful for the first problem (that was accentuated by the fact that we were warned about the failure rate). I've taken a course that covered simultaneous equations, but my memory of it is hazy enough that I'm sure that method would've taken me much longer. I hear your pain. Three studies in which we used a range of second guess elicitation methods show that biased reasoners predominantly give second guesses that are smaller than the intuitively cued heuristic response ("10 cents"). Kahneman's examples of system 1 thinking include (I think) a Chess Grandmaster seeing a good chess move, so he includes the possibility of training your system 1 to be able to do more things. The bat costs $1.00. Anyway, this is a separate rant.). So far I've read maybe a third of it. So, what are people doing when they solve this problem? There are a couple of variants of this explained in the comments. I learned algebra, fortunately, not by going to school, but by finding my aunt's old schoolbook in the attic, and understanding that the whole idea was to find out what x is - it doesn't make any difference how you do it. This was suggested early on by Kahneman and Frederick, and is a fancy way of saying that they are instead solving the following simpler problem: (1) A bat and a ball cost $1.10 in total. Here there's an obvious but wrong answer but I think if you realise that it's wrong then the correct answer isn't too hard to figure out. There are a couple of possible reasons for this. You have to do it by algebra.". Although in some ways it's the simplest of the three problems, solving it in a 'fast', 'intuitive' way relies on seeing the problem in a way that most people's education won't have provided. I'm done.'. I would speculate, in decreasing order of intuitive probability, that in order to get the answer, either (a) I've seen an exactly analogous “trick” problem before and am pattern-matching on that or (b) I'm doing the algebra quickly using my seemingly well-developed mathematical intuition. How much does a bat cost? I feel as though psychologists and psychiatrists get together every now and then to prove how stoopid I am. We have a bat, B. Abstract. What might be less obvious, at least if you mostly live in a high-maths-ability bubble, is that these people may also be missing the sort of tacit mathematical background that would even allow them to frame the problem in a useful form in the first place. Now that I'm thinking of the problem in this way, I directly see the equations as being 'about a bat that's halfway between 100 and 110 cents', and the answer is incredibly obvious. The conceptual emphasis seems to lie within the word MORE. What is the price of the baseball bat and the price of the ball? _____ days. But also, it could be to do with relative size compared to the other numbers that appear in the problem setup. I still finished what I was doing, then looked at it as a whole, and saw the "you need more widgets but have more machines so you need the same time" intuitively.And for the last one, it was very obvious of course. 307.7KB. The bat costs $1.00 more than the ball. The language correctly reflects there are two things we should consider. Earlier today I set you six ‘bat and ball’ puzzles, meaning puzzles that require you to overrule a wrong ‘gut’ answer. The lily pads question takes me a conscious time-step longer to answer than either of the other two; the initial flash is “inconclusive”, and then I see myself rechecking the part where the quantity doubles every step before answering “47”. I have the same experience as you, drossbucket: my rapid answer to (1) was the common incorrect answer, but for (2) and (3) my intuition is well-honed. ";s:7:"keyword";s:20:"bat and ball problem";s:5:"links";s:847:"Annual Flowers Australia,
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