Something is happening with this famous example that makes people fail so reliably. Essentially, it is using a “magician’s force.”
If the ball cost problem was presented mathematically like:
x + (x + 1.00) = 1.10
I doubt that the failure rate of University students would still be 50%. Would it be 5% or 0.5% or lower? Probably.
The wording of the problem is intended to engage a verbal attempt at solving it. Remember all the parsing needed to pull meaning from the scenario.
“A bat and ball cost $1.10”
(Two objects are presented in a specific order, and equated with a number also separated in order by two pieces by the decimal. The first object is bat and the first number segment is a dollar; the second object is ball and matches ten cents. Even if the problem is read aloud, “dollar ten” is still two ordered items. This results in:
{bat, dollar}, {ball, ten cents}
Next is the direct sentence further suggesting the cost of the bat is a dollar.
“The bat cost $1...”
Of course, left alone it would be a direct lie, but it continues “more than the ball.”
Since a dollar is more than 10 cents the bat would certainly cost more than the ball. A true statement. The lack of a comma means we are supposed to parse “more” in the sense of an inequality and some people recognize that and restructure the scenario mathematically. But most people will not jump off the initial word base association that was “forced” onto them.
“A Pickle and an Onion when purchased together cost 90 cents. The Onion costs twice as much as the Pickle. How much does each cost?”
How many of your friends would fail simple math problems when presented without the magician’s force?
I and many of my peers, when struggling with an academic topic, would receive the confusing injunction to apply ourselves. I'm not sure anyone who said it could articulate what applying one's self is. They might have had something shaped like system three in mind, however vaguely. Applying one's self likely also involves something else. But system three is a start.
"Apply yourself", "think", "be rational"—these injunctions confused me when I was a student, because I had no idea how to interpret them as concrete actions inside my head.
I'm more used of thinking of improving my intuition with the concept of 'muscle memory'. it's like any physical task, at first you have to actively think through it then you can just feel your way through it.
>I discovered in ninth grade that the only way to get around that was to verify after every three lines that what I was writing still made sense and that I really believed it.
In my experience, its normal to have a term thats one line or longer, and to just simplify it for half a page or so, without those intermediary steps clearly corresponding to anything. There are often ways to make the proof with fewer and/or smaller algebra blackboxes, but those are a lot more work to find, and are valued accordingly. I dont think you mean that you always go for these, so I guess the visualisations for those intermediary steps dont always connect to the larger-scale objective?
>In practical terms, here’s what that means. When my intuition tells me A and rationality tells me B, I put myself in the position of a referee....
This is what I would have said as well, it just doesnt seem as central to me. I mean, for me to have an intuition about fact, that I disprove, and that doesnt go away in the process of finding the disproof, thats pretty rare. Usually you see the way it *could* be wrong, before you fully prove it.
Also, many proofs dont proceed by seeing something is true, and then finding a way to express it. They really are more like "find" tasks, where you might not know the result until quite late, but be quite sure you were productive so far. These are more so about seeing good plays at each step, and thats "intuition" of some sort, but IMO quite different from the ones that some statement is true. Both are used, including within each other.
And these intuitions about the way forward are often triggered by the representation. For example, to show that a binomial model converges to poisson as time divisions get finer, I would write out the expression we want to limit, and notice right away the option for a compound-interest-e, and this is clear even if I didnt know what result I want to get.
>The most incredible thing was that everyone spoke to me about “doing the calculations,” as if there were any need for that, as if it weren’t visually evident that the right answer was 5¢.
I feel like I do the calculation here, though at a speed that may get the same reaction from your friend. (I encountered the topic in reading, so it seemed in line with his theory to me.) Interestingly, the cognitive reflection test has three questions originally, which seem useful in this context also: The one with the machines can be done "explicitly" without doing arithmetic at all, and the one with the lilypads I expect many people in technical fields to get right intuitively, because this is a calculation as basic as 2+2=4 for exponentials.
>I have these pictures in my head because, in my life, I have made a lot of calculation errors. Instead of concluding that I was terrible at math
It seems strange that someone would conclude that. I think the difference between the "losing a sign" error and the "I dont know what to do, lets guess an operation to apply to the numbers found in the text problem" error is a very obvious one, and looking at ones classmates its clearly the second thats important to how good you are at math.
>Because I have terrible handwriting... I have a tendency to make mistakes in calculations.
I found significant improvements on this front from writing with a finer pen. It doesnt look better, arguably the opposite, but its less ambiguous.
This is a great post. You can indeed program your intuition, and that is a major (and underappreciated) part of what school should be for. I find frequent, rapid in-class estimation plus metacognition to be a very good way of developing this (very rare) habit with normal students (that is, those who are mostly not future math PhDs). Eg "4672 / 3 is <,>,= 1500? Quick!"
This is very in line with the work of Gary Klein. A lot of people think his work is about training "intuition," but that is pretty misleading as the experts we study (disclaimer: Gary is my boss) do what you are calling System 3. Though we typically call it Sensemaking, which I think of as a dynamic reciprocal relationship between realizing what is relevant, and framing it up.
> In a sense, it picks up the classic opposition between left brain and right brain, but in a modern version, without the anatomical nonsense.
It's argubably still non-sense. The systems don't exist, and Kahneman recognizes them as mere metaphors. The reason the distinction is still used is merely because, as a field, we haven't coalecesed around a less false way of talking about it.
However, here is how I think about it: when you solve 47x83, you break it down into a series of system 1 steps. So why evoke System 2 at all? System 2 is not a different system, but instead about bringing structure (a frame) to a series of System 1 steps to bound it in a productive way. That's what we see in your example of the bat and the ball, as well. You found a representation (a frame) which constrained your system 1 pattern matching to something you could mentally handle. Why should we understand that to be a any different that breaking down 47x83 into a series of similar pattern matching steps?
This is why one of the initial and most important findings of Naturalistic Decision-Making (the field Gary founded) is the following: "The ways in which individuals made sense of situations often exerts greater influence on their actions than deliberation over a set of predefined options." Because the way you represent and frame a problem is the most important part of reasoning, as everything follows from that representation.
My most recent Substack post gets a little more into this, as well
(btw, your interview on EconTalk was great, and I definitely plan on reading the book)
Thanks for your comment. I do like "sensemaking", it's a great word.
I agree that the System 2 doesn't really exist in a biological sense—but it makes sense as a pragmatic description of script-based procedural information processing, which is an undeniable facet of mathematical reasoning.
I remember reading a while back about how often there are flaws in mathatical proofs, and how that hasn't led to a replication crisis. I've looked for that reference for a long time without being to find it. Good to finally find something that makes a similar argument.
We both agree on the benefits (even if I may be less platonist). The thing I want to add to the conversation is just that this process of finding a frame which bounds System 1 into a procedure isnt any different than what we mean by the words "common sense" or "expertise." It also helps us to avoid the word "intuition" which is often too ambiguous.
"Expertise" has some extra implications in that experts can more easily pattern match to a good representation through experience and by noticing cues and anomalies. Thats how you get firefighters who are able to sense a building is about to collapse, or who just know what to do without any conscious deliberation.
But in the actual cognitive process being used, expertise is not different than common sense. Perspicuity is a property of the framing of a problem, and experts are just better at framing it up (consciously and unconsciously).
In Kahneman and Klein's adversarial collaboration, they identified the conditions for intuitive expertise. I think what was missing in that paper was a greater clarification of the nature of intuition - pattern matching to a frame, and pattern matching within that framing.
A fabulous post, and one that articulates thoughts I've had in the past. The development of System 3, both in myself and in my students, is a central goal for me as a learner and as an educator.
I gave specific recommendations in specific contexts, because once in a while it's worth immersing oneself into an excellent math book. For the general public, I have no blanket recommandations, except watching 3blue1brown's videos.
I visualise the bat/ball problem in exactly the same way as you. At first the ‘ball’ is also ball shaped, then it morphs into a shorter rod shape when I realise it costs less than the bat, just before it duplicates and achieves a specific length. As a small child I was taught basic arithmetic using Cuisenaire Rods (is that still common?) which I would definitely recommend though I suspect people with aphantasia would still struggle to apply this system.
I passed this along to my AP Calculus students. No matter how many times we say, “When the position is increasing, the velocity is positive”, there will be a velocity graph and you’ll ask when the position is increasing and they’ll identify intervals upon which the velocity is increasing. System 1! Next semester I’m going to think about being more deliberate with system 1 vs. system 2 vs. system 3.
My name for this is Mad Maths! Borrowed from the oeuvre of a certain actor.
In philosophy we call intuition an heuristic, a mental short-cut. It's the default way of thinking. The light is green so it's safe to go. This checkout line looks quicker. I usually get that wrong! For important decisions, reason can help but it's not infallible. My father who had a senior job with an investment company, told me: never make decisions about money when you're in a good mood, and never make decisions about people when you're in a bad mood.
System 3 looks like a good solution. Can I live with the rational decision?
Something is happening with this famous example that makes people fail so reliably. Essentially, it is using a “magician’s force.”
If the ball cost problem was presented mathematically like:
x + (x + 1.00) = 1.10
I doubt that the failure rate of University students would still be 50%. Would it be 5% or 0.5% or lower? Probably.
The wording of the problem is intended to engage a verbal attempt at solving it. Remember all the parsing needed to pull meaning from the scenario.
“A bat and ball cost $1.10”
(Two objects are presented in a specific order, and equated with a number also separated in order by two pieces by the decimal. The first object is bat and the first number segment is a dollar; the second object is ball and matches ten cents. Even if the problem is read aloud, “dollar ten” is still two ordered items. This results in:
{bat, dollar}, {ball, ten cents}
Next is the direct sentence further suggesting the cost of the bat is a dollar.
“The bat cost $1...”
Of course, left alone it would be a direct lie, but it continues “more than the ball.”
Since a dollar is more than 10 cents the bat would certainly cost more than the ball. A true statement. The lack of a comma means we are supposed to parse “more” in the sense of an inequality and some people recognize that and restructure the scenario mathematically. But most people will not jump off the initial word base association that was “forced” onto them.
“A Pickle and an Onion when purchased together cost 90 cents. The Onion costs twice as much as the Pickle. How much does each cost?”
How many of your friends would fail simple math problems when presented without the magician’s force?
Great point! I never considered how much difference the lack of a comma makes in tricking your intuition into latching on to an answer too early.
I and many of my peers, when struggling with an academic topic, would receive the confusing injunction to apply ourselves. I'm not sure anyone who said it could articulate what applying one's self is. They might have had something shaped like system three in mind, however vaguely. Applying one's self likely also involves something else. But system three is a start.
"Apply yourself", "think", "be rational"—these injunctions confused me when I was a student, because I had no idea how to interpret them as concrete actions inside my head.
When they tell you to snap out of it, click your fingers and say, "Done."
I'm more used of thinking of improving my intuition with the concept of 'muscle memory'. it's like any physical task, at first you have to actively think through it then you can just feel your way through it.
>I discovered in ninth grade that the only way to get around that was to verify after every three lines that what I was writing still made sense and that I really believed it.
In my experience, its normal to have a term thats one line or longer, and to just simplify it for half a page or so, without those intermediary steps clearly corresponding to anything. There are often ways to make the proof with fewer and/or smaller algebra blackboxes, but those are a lot more work to find, and are valued accordingly. I dont think you mean that you always go for these, so I guess the visualisations for those intermediary steps dont always connect to the larger-scale objective?
>In practical terms, here’s what that means. When my intuition tells me A and rationality tells me B, I put myself in the position of a referee....
This is what I would have said as well, it just doesnt seem as central to me. I mean, for me to have an intuition about fact, that I disprove, and that doesnt go away in the process of finding the disproof, thats pretty rare. Usually you see the way it *could* be wrong, before you fully prove it.
Also, many proofs dont proceed by seeing something is true, and then finding a way to express it. They really are more like "find" tasks, where you might not know the result until quite late, but be quite sure you were productive so far. These are more so about seeing good plays at each step, and thats "intuition" of some sort, but IMO quite different from the ones that some statement is true. Both are used, including within each other.
And these intuitions about the way forward are often triggered by the representation. For example, to show that a binomial model converges to poisson as time divisions get finer, I would write out the expression we want to limit, and notice right away the option for a compound-interest-e, and this is clear even if I didnt know what result I want to get.
>The most incredible thing was that everyone spoke to me about “doing the calculations,” as if there were any need for that, as if it weren’t visually evident that the right answer was 5¢.
I feel like I do the calculation here, though at a speed that may get the same reaction from your friend. (I encountered the topic in reading, so it seemed in line with his theory to me.) Interestingly, the cognitive reflection test has three questions originally, which seem useful in this context also: The one with the machines can be done "explicitly" without doing arithmetic at all, and the one with the lilypads I expect many people in technical fields to get right intuitively, because this is a calculation as basic as 2+2=4 for exponentials.
https://en.wikipedia.org/wiki/Cognitive_reflection_test#Test_questions_and_answers
>I have these pictures in my head because, in my life, I have made a lot of calculation errors. Instead of concluding that I was terrible at math
It seems strange that someone would conclude that. I think the difference between the "losing a sign" error and the "I dont know what to do, lets guess an operation to apply to the numbers found in the text problem" error is a very obvious one, and looking at ones classmates its clearly the second thats important to how good you are at math.
>Because I have terrible handwriting... I have a tendency to make mistakes in calculations.
I found significant improvements on this front from writing with a finer pen. It doesnt look better, arguably the opposite, but its less ambiguous.
This is a great post. You can indeed program your intuition, and that is a major (and underappreciated) part of what school should be for. I find frequent, rapid in-class estimation plus metacognition to be a very good way of developing this (very rare) habit with normal students (that is, those who are mostly not future math PhDs). Eg "4672 / 3 is <,>,= 1500? Quick!"
This is very in line with the work of Gary Klein. A lot of people think his work is about training "intuition," but that is pretty misleading as the experts we study (disclaimer: Gary is my boss) do what you are calling System 3. Though we typically call it Sensemaking, which I think of as a dynamic reciprocal relationship between realizing what is relevant, and framing it up.
> In a sense, it picks up the classic opposition between left brain and right brain, but in a modern version, without the anatomical nonsense.
It's argubably still non-sense. The systems don't exist, and Kahneman recognizes them as mere metaphors. The reason the distinction is still used is merely because, as a field, we haven't coalecesed around a less false way of talking about it.
However, here is how I think about it: when you solve 47x83, you break it down into a series of system 1 steps. So why evoke System 2 at all? System 2 is not a different system, but instead about bringing structure (a frame) to a series of System 1 steps to bound it in a productive way. That's what we see in your example of the bat and the ball, as well. You found a representation (a frame) which constrained your system 1 pattern matching to something you could mentally handle. Why should we understand that to be a any different that breaking down 47x83 into a series of similar pattern matching steps?
This is why one of the initial and most important findings of Naturalistic Decision-Making (the field Gary founded) is the following: "The ways in which individuals made sense of situations often exerts greater influence on their actions than deliberation over a set of predefined options." Because the way you represent and frame a problem is the most important part of reasoning, as everything follows from that representation.
My most recent Substack post gets a little more into this, as well
(btw, your interview on EconTalk was great, and I definitely plan on reading the book)
Thanks for your comment. I do like "sensemaking", it's a great word.
I agree that the System 2 doesn't really exist in a biological sense—but it makes sense as a pragmatic description of script-based procedural information processing, which is an undeniable facet of mathematical reasoning.
My interpretation (see https://davidbessis.substack.com/p/weve-been-wrong-about-math-for-2300 and https://davidbessis.substack.com/p/the-curious-case-of-broken-theorems) is that this idealized imaginery mechanical process has actual neuroplastic benefits.
I remember reading a while back about how often there are flaws in mathatical proofs, and how that hasn't led to a replication crisis. I've looked for that reference for a long time without being to find it. Good to finally find something that makes a similar argument.
We both agree on the benefits (even if I may be less platonist). The thing I want to add to the conversation is just that this process of finding a frame which bounds System 1 into a procedure isnt any different than what we mean by the words "common sense" or "expertise." It also helps us to avoid the word "intuition" which is often too ambiguous.
"Expertise" has some extra implications in that experts can more easily pattern match to a good representation through experience and by noticing cues and anomalies. Thats how you get firefighters who are able to sense a building is about to collapse, or who just know what to do without any conscious deliberation.
But in the actual cognitive process being used, expertise is not different than common sense. Perspicuity is a property of the framing of a problem, and experts are just better at framing it up (consciously and unconsciously).
In Kahneman and Klein's adversarial collaboration, they identified the conditions for intuitive expertise. I think what was missing in that paper was a greater clarification of the nature of intuition - pattern matching to a frame, and pattern matching within that framing.
A fabulous post, and one that articulates thoughts I've had in the past. The development of System 3, both in myself and in my students, is a central goal for me as a learner and as an educator.
I know you said math books are not meant to be read, but have you ever given math book recommendations?
I gave specific recommendations in specific contexts, because once in a while it's worth immersing oneself into an excellent math book. For the general public, I have no blanket recommandations, except watching 3blue1brown's videos.
I visualise the bat/ball problem in exactly the same way as you. At first the ‘ball’ is also ball shaped, then it morphs into a shorter rod shape when I realise it costs less than the bat, just before it duplicates and achieves a specific length. As a small child I was taught basic arithmetic using Cuisenaire Rods (is that still common?) which I would definitely recommend though I suspect people with aphantasia would still struggle to apply this system.
Amazing post! My instinctive reaction to the A vs B question was also "sounds like I'm going to have to mull over it"
I passed this along to my AP Calculus students. No matter how many times we say, “When the position is increasing, the velocity is positive”, there will be a velocity graph and you’ll ask when the position is increasing and they’ll identify intervals upon which the velocity is increasing. System 1! Next semester I’m going to think about being more deliberate with system 1 vs. system 2 vs. system 3.
My name for this is Mad Maths! Borrowed from the oeuvre of a certain actor.
In philosophy we call intuition an heuristic, a mental short-cut. It's the default way of thinking. The light is green so it's safe to go. This checkout line looks quicker. I usually get that wrong! For important decisions, reason can help but it's not infallible. My father who had a senior job with an investment company, told me: never make decisions about money when you're in a good mood, and never make decisions about people when you're in a bad mood.
System 3 looks like a good solution. Can I live with the rational decision?