really concrete like 'the price of the bat', or more abstract like 'the difference between the price of the bat and ball'. If you're able to rapidly 'just see' the answer to the bat and ball question, how do you do it? 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. Age 14 to 16 Short Challenge Level. 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've written about close reading recently, and this has been something like a close reading of the bat and ball problem. He was having considerable difficulty with his algebra, so a tutor would come. Do the math, and you will see. 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. The language correctly reflects there are two things we should consider. In (1), 5 and 10 are both similarly small compared to 100 and 110. But probably I'd learn something from going through the process, and it's something that could maybe happen in the future. This is an equation. 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. However, for the other two I 'just see' the correct answer. The bat and ball still gets me, though. Or something else. Three experiments explored the effects of word problem cueing on debiasing versions of the bat-and-ball problem. this (apparently unpublished) ‘extremely rough draft’, suggested early on by Kahneman and Frederick. People just really like the answer 'ten cents', it seems. This showed a clear difference: 94% of 'five cent' respondents could recall the correct question, but only 61% of 'ten cent' respondents. Probably nowhere much for a while, as I have other priorities. I suspect that person would have been corrected by saying "the bat costs precisely one dollar more than the ball", The bat and ball problem I answer in what I'll call one conscious time-step with the correct “five cents”, but it happens too fast for me to verify how (beyond the usual trouble with verifying internal reflection). Whew, I was thinking to write a separate post on this, but now I don't have to! 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. The widget problem presents a process which doesn’t change how the machines and widgets relate to each other in its solution. You can find more short problems, arranged by curriculum topic, in our short problems collection. 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. But I'm not sure how general this is. My favourite thing I found was this (apparently unpublished) ‘extremely rough draft’ by Meyer, Spunt and Frederick from 2013, revisiting the bat and ball problem. I think the solution to bat and ball of "10cents, oh no, that doesn't work. Ultimately you would think that the ball costs 10 cents because ($1+$0.10=$1.10). The correct answer is 5¢. If you wanted to solve the bat and ball problem without having to 'do it by algebra', how would you go about it? 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. 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). Abstract. In the original problem, the 110 units and 100 units both refer to something abstract, the sum and difference of the bat and ball. Or it could improve ability to come up with an 'unintuitive' solution, like solving the corresponding simultaneous equations by a rote method. The correct answer is 5p. Instead they seem to focus mainly on the first condition (adding up to $1.10) and just use the second one as a vague check at best ('the bat would still cost more than $1'). This is quite interesting in itself, but I was most excited by the comments section. 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. Click on the link to see learn about why the batter's grip doesn't matter during the baseball-bat collision, including a more detailed discussion of the Todd Frazier "no-hands" home run. But especially in the context of expecting a trick question, I second-guess it and come up with the correct answer fairly quickly. 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. We all used some variant of the method suggested by Marlo Eugene in the comments above. The text is 100 columns wide. yup that's better" is all done on system 1. The jump in success rate could be down to better trained intuition. David Chapman recommended Formal Languages in Logic by Dutilh Novaes for more background on this. This is wrong because if the ball costs 10 cents then 1 dollar more then 10 cents would be $1.10. I created the machine.minute unit (equivalent to the kWh unit) that allowed me to understand that a widget is made in 5 machine.minutes. 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. 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. Only a few build up the necessary repertoire of tricks to solve the problem quickly by insight. In ordinary language, "that costs $1.00 more than the other one" is not incorrect if the difference is $1.01. Plenty of people noped out of mathematics long before they got to simultaneous equations, so they won't be able to solve it this way. But as be KNOW A+B=110 the only number for B on it’s own is 5. In (3), 24 is small compared to 48, but 47 isn't. My suspicion is still that very few people solve this problem with a fast intuitive response, in the way that I rapidly see the correct answer to the lilypad question. Finally some examples of how the problem is solved in the wild! 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. 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). The Ferrari costs $100,000 more than the Ford. For the second one though, I was too careful: I immediately started transcribing the problem in a pertinent format, i.e. (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.). “Bat & Ball” Cognitive Reflection Test. If this test was really testing my propensity for effortful thought over spontaneous intuition, I ought to score zero. There's a kind of natural inertia to this kind of puzzles. How many seats have I booked? How much does a bat cost? The most common and reactive answer is probably what you just said aloud or in your mind. she was about 50% likely to give the right answer. 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. If there are 'System 1' ways to get the correct answer, the whole thing gets much more muddled, and it's hard to disentangle natural propensity to reflection from prior exposure to the right mathematical concepts. However, I haven't found any great evidence either way for this guess. The first sentence presents two things, a bat and a ball. If the ball were to … Careful reflection definitely seems to improve the chance of a correct answer in general, but many of the responses don't really fit the neat 'fast vs slow' division of the original setup. 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. Same numbers, same mathematical formula to reach the solution. Thus if the ball equals to x, the bat equals to x plus 1... ', Correct answer, incorrect start. It correctly sounds like a + b; two things. 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. I hear your pain. The ‘instant’ answer that comes to mind is that the ball costs 10p. 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. This lack of a rich background in puzzling out the answer to specific concrete problems means most of us lean hard on formal rules in this domain, even if we're relatively mathematically sophisticated. 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. So, here is the problem: A bat and ball cost $1.10 The bat costs one dollar more than the ball How much does the ball cost ? 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 … Example given: 'I see. For the bat to cost $1 more than the ball, the ball has to cost 5 cents and the bat $1.05. Performance on it has been shown to correlate with intertemporal choice, risky choice, moral reasoning, strategic behavior, and belief in god. This problem is taken from the UKMT Mathematical Challenges. I have a vague suspicion that Frederick trawled through something like 'The Bumper Book of Annoying Riddles' to find some brainteasers that don't require too much in the way of mathematical prerequisites. 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 framing is important for interpreting the CRT. Nevertheless, the ball is hit for a home run, demonstrating in dramatic fashion that the batter's grip plays no role in the ball-bat collision. Imagining “100” afterwards feels wrong, but less immediately so than “ten cents” did. 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’. (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.). endstream endobj 2416 0 obj <. Third was 47 On the 48th day it was full so on the 47th it was half there cause each day if halves. I just saw the answer to the bat and ball problem within a few seconds. Anyway, this is a separate rant.). Is an easier question getting substituted? ", "I'm trying to find out what x is, like in 2x + 7 = 15. �Loڀ47��Uؙ�� %�]�2����dl�r���0����|0K�H/���D -��Pv The surprisingly high rate of errors in this easy problem illustrates how lightly System 2 monitors the output of System 1: people are not accustomed to thinking hard, and are often content to trust a plausible judgment that quickly comes to mind. But I also share your sense that the answer to (3) is 'wildly off', whereas the answer to (1) is 'close enough'. 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. The work of seeing the equations as relating to something concrete has mostly been done for you. To what extent might reasoners be led to modify their judgement, and, more specifically, is it possible to facilitate 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… Ah yeah, I meant to make this bit clearer and forgot. Notice that this is missing the 'more than the ball' clause at the end, turning the question into a much simpler arithmetic problem. A baseball bat and a ball cost $1.10 together, and the bat costs $1.00 more than the ball, how much does the ball cost? Meyer, Spunt and Frederick tested this hypothesis by getting respondents to recall the problem from memory. 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 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. Problem 18: During a baseball game, a baseball is struck at ground level by a batter.

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