For a long time, a side interest of mine has been how people think when they are doing mathematics. Two difficulties in investigating this general question are (i) that it is quite hard to examine one’s own thought processes reliably (since very often what one remembers of these processes after solving a problem is a very tidied up version of what actually happened) and (ii) that in any case I am just one mathematician with my own particular style and my own little bag of tricks.
I would therefore be grateful to anyone who was prepared to spend about 90 seconds contributing to a little experiment. What you have to do is solve a simple equation that appears just after the fold and pay close attention to your thought processes as you do so. Once you have done that, you can look at further instructions about how to record the result of your participation.
I stress once again that the whole thing should be quick and easy. But please don’t look at the equation until you are ready to start thinking about it and remembering the sequence of your thoughts, since otherwise there is a danger that the tidying-up process will take over and it will be impossible to get a reliable result.
The equation is this.
*
*S
*P
*O
*I
*L
*E
*R
*
*S
*P
*A
*C
*E
*
*
*
*
*
Now there are at least three approaches to this equation.
1. The blindingly obvious approach: cancel 
2. Rewrite the left-hand side as 
3. Rewrite the equation as 
Now once you’ve thought of 1, anything else seems silly, and once you’ve thought of 2, 3 seems a bit silly. But there’s no disgrace in having silly thoughts — it’s an important part of doing mathematics — and the whole point of this experiment is to get some idea of the extent to which we have them.
To record what happened to you, I have a poll. I have represented the various possibilities by numbers or sequences of numbers. Each number corresponds to the numbers above, where I listed the approaches, and a sequence of numbers such as 31 would indicate that your first thought was to use 3 but then you abandoned that thought after noticing 1. I have listed only decreasing sequences, since my expectation is that people are unlikely to abandon an approach for another approach that is (as I see it anyway) strictly sillier. But if you have had an interesting experience that is not covered by the options in the poll, then there is always the possibility to leave a comment.
Please try hard to remember exactly what happened. If you very briefly thought of doing 2 but almost immediately realized that you could do 1 and that 1 was better, then your answer should be 21 rather than 1, even if you didn’t at any stage feel a commitment to 2. It’s your fleeting thoughts that interest me.
Gowers's Weblog 
April 22, 2010 at 6:30 pm |
I voted for 1.
A bit more explicitly, in case it’s of any interest: I simply read the equation, from left to right, thought “it’s a quadratic”, then immediately thought “something’s odd, oh, right, same coefficient of the squared terms on both sides, I can cancel, leaving a linear equation, etc”.
The whole process felt subjectively almost instantaneous. Also, although I’ve transcribed the process verbally, that gives an inaccurate account of what’s going on. When doing mathematics like this I’m often only vaguely aware of any verbal content to my thought.
April 22, 2010 at 6:41 pm |
Yep, as Michael says, I originally interpreted it as a quadratic, but subconsciously cancelled the terms as I read it leaving me with a linear equation.
Rearranging that was done again almost subconsciously – I didn’t manually think about how to do it.
April 22, 2010 at 6:41 pm |
My line of thought was exactly like Michael Nielsen had described, although I voted 21.
April 22, 2010 at 7:09 pm
The idea of doing 2 never occurred to me, perhaps because by the time it would have, I’d already noticed that something was a bit odd.
There’s a subtlety here that I find really interesting. I’m guessing – and perhaps I’m wrong, Gabriel, please correct me if so – that Gabriel immediately had the idea of solving the quadratic, while reading the equation. But while I noticed it was a quadratic, I held back from the next obvious idea (solving it) until after I was done reading the equation. By that point I’d noticed something was odd.
(Take all this with a grain of salt, obviously. Confabulation in this kind of analysis must be almost impossible to avoid.)
April 22, 2010 at 6:43 pm |
I also read from left to right. I therefore saw
before reading on. By the time I reached the end I was already at $(x+1) \neq x$ so $x+1 = -x$. At this point I looked back and said “oops …”
April 22, 2010 at 6:44 pm |
I voted for 1. I’ve taught enough algebra to instantly pattern-recognize matching things that can cancel. The closest way I can describe my mental process is spotting a matching pair of tiles in a video game.
April 22, 2010 at 7:04 pm |
As the equation was rather of small size, I had a global view at first sight and pattern matched the canceling terms. Then linear equation and stuff. A longer equation would have probably led me to 21, though.
April 22, 2010 at 7:08 pm |
I would highly recommend reading Douglas Hofstadter’s book Fluid concepts and creative analogies in which he and his team of researchers attempt to answer very much this sort of question by writing programs to solve very simple questions in the way that we appear to go about answering them. On studying this sort of puzzle one can in principle learn a great deal about the way the human mind functions.
April 22, 2010 at 7:08 pm |
I had basically the same experience as Micha: this could just be training/indoctrinating, but I always try to look at the whole equation and “pattern match” to see possible cancellation/simplifications.
April 22, 2010 at 7:09 pm |
I did 1 immediately, and then went back to 2 to make sure I didn’t lose a solution in dropping a degree.
April 22, 2010 at 7:13 pm |
My first thought was ‘there’s a square on each side’, before pretty much immediately seeing ‘that’s linear pretending to be quadratic’ and then going back just to check out what went wrong with the other ‘solution’ [1=0] obtained by taking the square root of both sides, just for interest. So I put 21.
April 22, 2010 at 9:59 pm
When I was young the formulae for squares of sums and differences were the first really elegant and neat ones I learned – and I guess that my first instinct with a quadratic is to find a way of taking a square root.
For me this pulls against the ‘get it all on the same side’ instinct, because ‘it is in a neat form I might be able to do something with’.
Mind you the difference of two squares is elegant too – I think the evaluation of 3 as more primitive than 2 is not entirely justified. Method 3 shows that there is one solution, contains an implicit understanding of why (because one factor is constant) and avoids tangling with the sign of the square root.
April 22, 2010 at 7:31 pm |
I think because the right-hand side was not “=0”, I saw the x^2 that could be simplified and thus did it this way (so 1).
I wonder if this is due to very old — 30 years ago — training in school doing hundreds of exercises involving finding common factors, expanding and simplifying algebraic expressions in 2 or 3 variables (our teacher had the peculiar insistence that the exercise only got full points if any common factor that was used was underlined in our write-up before being used — something which apparently was not common practice, but which seems to make very good sense).
(Incidentally, although I’m sure most readers here know it, it may be worth recalling that the operation of going from A+B=A+C to B=C, called “al-muqabala”, was one of the two reduction steps emphasized by Arabic scholars in solving equations ; the other one, going from A+B-C=D to A+B=C+D, was “al-jabr” and led to the name “Algebra”; hence if Al Khwarizmi had exchanged them in the title of his book, we would probably be speaking of “Almucabalic geometry” instead of “Algebraic geometry”…)
April 22, 2010 at 7:35 pm |
My experience was similar to the other commenters’. Although I voted 1, I did have the left-hand side factored as (x+1)^2 before I got to the right-hand side. I must have been expecting a second variable there, as I did a bit of a double-take that caused me to look at the equation globally; then I saw and carried out 1.
April 22, 2010 at 7:37 pm |
I did 2, then checked it by doing 1. This is probably because x^2 + 2x + 1 just screams out “factor me!”, to the point where I almost read it as if it were (x+1)^2.
April 22, 2010 at 8:01 pm |
My thought process (which boils down to 21), as I tried to observe it:
Look at LHS and convert it to
.
and picture the graphs (two parabolas).
.
is the only possibility, so that something funny must be going on.
terms cancel, so that the equation is really linear. More or less simultaneously, solve 
(by visualization rather than calculation).
.
See
Infer
Wonder whether that’s actually possible, and realise that it is, but that
Notice that the
Check that my answer really does satisfy the simpler equation
April 22, 2010 at 8:09 pm |
I immediately solved it with option 1, but then (since I suspected this was a trick, even though it wasn’t) I checked my answer. I had a moment of doubt when I saw I had only one answer to a quadratic equation, but then quickly realized that this was actually just a linear equation written in a confusing way – in canonical form there is no second order term.
April 22, 2010 at 8:14 pm |
I vote 1.
My thought process was (as far as I could tell) the following: I did recognize (x+1)^2. However, then I saw x^2, remembered that I saw it on the left-hand side, knew I could cancel it out and that what I was left was a linear equation. Subsequently I derived the equation, solving it led me to the right answer.
By the way, if you want to scientifically investigate this issue you might want to control for various factors (e.g. level of mathematical instruction).
April 22, 2010 at 8:15 pm |
Also, this is very interesting! It says a lot about how we were all taught to do simple algebra, and what patterns we see first – do we first think of polynomials as sums of monomials or as products of simpler polynomials?
Could you perhaps post a followup problem which might take a little longer to solve (maybe 5 minutes)? I think so many of us learned how to do the current problem by rote; it’d be interesting look at the more creative aspects of problem solving which come into play with a slightly longer problem.
April 22, 2010 at 8:15 pm |
My thought process:
SPOI… *oh, that’s not the equation*
x^2 + … *quadratic: will have two solutions*
… = x^2 *huh, funny quadratic, with only one solution.
x^2 + … = x^2 *cancel the x^2s*
2x+1=0. *no solutions*
2x+1=0. *wait, wrong, there are solutions, because there’s still an x there.*
x = -1/2. *Ok, time to read the instructions.*
April 22, 2010 at 8:15 pm |
I saw 1 only: the repetition of the
factor was more or less the first thing that came into view. (I tend to read sentences from their most prominent features first, rather than from left to right.)
April 23, 2010 at 6:40 pm
“I tend to read sentences from their most prominent features first, rather than from left to right”
OMG, this is how I felt the first time that superman revealed that he can fly around the world real fast to turn back time
April 22, 2010 at 8:18 pm |
Since A-Level Mathematics I’ve always collected the terms of a polynomial on one side before attempting to factorize, so my process was 1.
April 22, 2010 at 8:18 pm |
Last comment: The comments on Terry Tao’s “Blue-eyed Islander problem” page are fascinated in this regard as well. It’s a harder problem and lots of people are convinced of the wrong answer:
April 22, 2010 at 8:33 pm |
I voted 1, but after I solved the linear equation, I also noticed that you have (x+1)^2 in the lhs and that it could be also solved that way (i.e. 2).
All that was after I used 1 to solve (and before I scrolled further down).
April 22, 2010 at 8:57 pm |
x^2 + 2x + 1 ==> “That factors.”
… = x^2 “oh wait cancel it”
“2x = -1”
“x = -1/2”
Recorded as 21.
April 22, 2010 at 10:08 pm |
I find it interesting that many people factored before they carried on. Whilst I see “x^2 + 2x + 1” and think “It can be factored”, the actual factorization doesn’t occur to me. Similarly, if I were to see 6*7, I’d merely think “I can evaluate that” rather than see 42.
So for me, whilst upon glancing at it I knew the left hand side could be factored, I wanted to avoid it (as that involves actually obtaining the factorization, even though it is a trivial one) and so did the obvious thing, 1. I also didn’t want to factorize as the next step wasn’t obvious (given that I didn’t ‘know’ at that point that it was a perfect square).
In fact, even when reading three-digits numbers I merely think “I can read this number” rather than actually reading it until I need to know it.
I’m not really sure if there is anything for you to learn from my post, but I guess it is that I have a distinction between “I can do X” and actually doing X, even for very simple operations that are instant for other people.
(I wonder if this has anything to do with dyslexia? Upon reading something I hear it in my mind, but I cannot store it (even in short term memory) without extra effort. Ask me what 132 + 164 is and I will just forget the numbers and so cannot do it. Which appears to others as pretty poor for somebody with a degree from Cambridge!)
April 22, 2010 at 10:37 pm
I actually think that this “dyslexic” phenomenon — the very small amount of short-term memory we possess — is extremely important and explains a huge amount about the way we do mathematics. I can manage 132+164 in my head, but I have an almost physical sense of discomfort that arises from pushing close to my storage limit. And it’s the same with proofs: if I am reading a paper and it states a rather complicated and unmotivated lemma, then I simply cannot understand the paper. Or rather, I can but I have to do some work reorganizing the argument so that I can get by with holding less in my head.
April 22, 2010 at 10:40 pm |
1 for me. I saw the whole line of text, rather than read it, and canceled the repeated terms, then read the linear part properly and solved it.
I find that I often scan over things faster than I read them properly, and am drawn to the patterns which stand out. Here I saw that something was repeated before actually seeing that is was a square.
April 22, 2010 at 10:44 pm |
I’m not a professional mathematician. So I saw on the left a square, linear and constant term, and, ignoring the coefficients, thought “That’s a quadratic, which I can solve if it’s in standard form with a zero on the right, what puts a zero on the right? oh, it’s linear, move the constant over and divide.” Then I thought: this is too easy, what’s the catch?
Presumably the “catch” is that people who do this all the time and see lots of these things would see the “squareness” of the LHS faster than they’d see that it’s an instance of a quadratic, whereas people who only see equations on special occasions default to using a general technique. If the question had just presented the LHS, and said “factorise this”, I could have done it without thinking, because it’s one of the ones I recognise: but the setup was “solve a simple equation”, and to me a “simple equation” has zero on one side, and anything that doesn’t is going to need work. So I’d focus on doing the work, whereas perhaps more experienced people just start start at the left and, in a programming sense, “interpret” the symbols as they come.
There’s one of Stephen Jay Gould’s essays where he talks about doing palaeontological fieldwork in the African Rift Valley with people who were looking for hominid fossils. He didn’t see any of those at all, but “boy, did I see a lot of snails”, which nobody had previously noticed in that fauna. Because of course by training he was a snail man, not a mammal man, and accordingly he spotted the things he had the right mental template to spot.
April 23, 2010 at 12:41 am |
I think the cancellation is too apparent. Maybe you should devise a slightly trickier test.
April 23, 2010 at 1:09 am |
This could be my unreliable memory, but I think I did 1 and 3 simultaneously. Math contests train you to spot even partial factorizations quickly; I almost won $1000 once because of such a factorization.
April 23, 2010 at 1:13 am |
It would be interesting to run two tests simultaneously, with people randomly selected to see slightly different versions of the problem. E.g., you could vary the font size in the exponent of x^2, to make the exponent more or less obvious. This sounds silly, but I wouldn’t be surprised if the font size changed the results significantly. I’d certainly bet swapping the two sides of the equation, so it became x^2 = x^2+2x+1, would change the results.
In general, studying how the presentation of problems affects people’s solutions seems like quite an interesting subject for study.
April 23, 2010 at 5:41 pm
I voted 21, but I bet if the equation had been written x^2=x^2+2x+1 my vote would have been 1.
April 23, 2010 at 1:57 am |
1
I saw the equation, I just copied it without any thinking about it on a paper (I was expecting something a little bit more complicated because of what you wrote that do not read the rest until you are ready to think about the problem). Then I just crossed the x^2 from both sides and saw that the answer is -frac{1}{2}.
April 23, 2010 at 1:59 am |
Despite the fact 1 is most obvious, there are other solutions which depending on your education I think are natural. For example, students who have not done algebraic manipulations might try guess and check, using a calculator like TI-93 (unfortunately), or plot the curves y = (x+1)^2 and y = x^2 and find their intersection.
April 23, 2010 at 2:01 am |
ps: before writing the comment I put -frac{1}{2} in the original equation (in my mind) to make sure I have not made a mistake.
April 23, 2010 at 3:19 am |
My left eye picked up the first x^{2}, my right the second – there was a little, quick moment where i felt my eyes starting to cross to check the opposite sides of the equation, followed by a moment of slight dis-belief at the extra x^{2}.
April 23, 2010 at 3:52 am |
Sir,
After option 1 (which as you say is blindingly obvious), I read options 2 and 3.
I found option 3 to be more comfortable than option 2, because one side of the equation is zero and hence it was naturally ‘more’ intuitive to solve for (2x+1) = 0
They only way I could resolve (x+1) = (+/-)x was (x+1) + (-/+)x = 0. This is essentially option 3.
I voted 31 because before getting to x = (-)1/2 via option 1, I contemplated on solving the equation using what later turned out to be option 3.
April 23, 2010 at 4:19 am |
Spotting the cancellation instantly, I also saw 1 only. I’ve also taught elementary algebra, but I doubt that’s the real reason — 1 is the only natural approach if you initially see the equation as a whole. Terry’s comment about reading sentences from their most prominent features first is interesting. I think that I also do that somewhat, and I wonder if it explains why I’m not a particularly fast reader. I’ve always had a feeling that I read differently than others. Hmmm. Now that I think about it, that’s how I view visual scenes as well (photography has been one of my hobbies: http://gallery.me.com/richardseguin#100037 ).
This was interesting. Maybe we should try more of these. I rarely hear people actually discuss how they reason at the micro level.
April 23, 2010 at 7:53 am |
I read the equation from left to right, so had a fleeting moment of thinking of the left-hand side as (x + 1)^2 before reaching the right-hand side and immediately realising that I could cancel and so get a linear equation. Despite noticing that factorisation immediately, I still cancelled rather that noticing that I had two squares equal to each other. I voted 21.
Vicky
April 23, 2010 at 7:56 am |
My thoughts and emotions were:
tension … x^2 … quadratic …. relief … 2x +1 … binom … I factor … = x^2 … that cancels … linear … disappointment … solve …
That is remarkably inefficient. I guess I factored before I checked the right-hand side since in my experience the right-hand side is mostly zero anyway. When I realized that it is not zero I did not finish strategy 2 because … actually I have no idea … probably because the problem was to easy to solve once recognized as linear.
April 23, 2010 at 9:19 am |
Let me say a tiny bit more about the motives behind this experiment. One of the interesting questions about how we operate when we do mathematics is the extent to which we are simply running algorithms. It seems that every level of our thinking has algorithmic and non-algorithmic aspects to it. I don’t mean “non-algorithmic” in a Penrose sense — I think ultimately the brain is purely algorithmic — but rather in the sense that the correct description of what is going on at the high level is non-algorithmic. Just to illustrate what I mean, if you ask your pocket calculator to multiply together two positive integers, then it will run the same procedure whatever those two integers are. But if you ask a fellow mathematician to do it, then they will have different approaches to the following six examples:
. (You could still say that they are running “the same” algorithm, but that would be an artificial description, since the first step of this algorithm would be, “Hmm, what subalgorithm shall I go for?” It would be a bit like a “unified” proof of Fermat’s last theorem and the Poincaré conjecture that simply proved FLT then PC and then had FLT&PC as a corollary.)
A better description of what we are doing seems to be that we have lots of little procedures in our brains that compete with each other. (This idea can be found in the book of Hofstadter recommended by Jonathan above, which I have.) When we are solving equations, for instance, we have one little bot that says, “Aha, I can cancel!” and another that says, “Aha,
— I recognise that!” A simple algorithm might have a rule like “always cancel if you can”, but that is not always a good thing to do: sometimes it destroys a pattern like 
that would have been helpful. So in fact it seems that there is some rather more interesting architecture that determines which bot wins out at any given time. (Again, this is something that is discussed a lot in the Hofstadter book.) The fact that a significant minority of people voted for 21, and several people have reported at least noticing the factorization even if they opted for 1, bears this out, though exactly what is going on is still a bit mysterious.
April 23, 2010 at 8:38 pm
>if you ask your pocket calculator to multiply together two positive integers, then it >will run the same procedure whatever those two integers are.
That may be true of pocket calculators, but certainly the best mathematical software does may very different algorithms depending on what the input is (e.g., for factoring integers, most sophisticated algorithms are not better than trial-and-error division by small primes unless the integer to factor is of a certain size). Even for “simple” multiplication, I think that different algorithms may be used for different arguments.
April 23, 2010 at 8:40 pm
(I typed “enter” a bit too early on the previous comment; the garbled sentence “does may very different algorithms” is supposed to be “may use very different algorithms”…)
April 23, 2010 at 9:01 pm
That’s interesting. But the tendency to use a variety of procedures would be far more pronounced in humans. For instance, it wouldn’t be worth programming a computer to spot that it could work out
by thinking of it as 
times 
. But that is exactly what an experienced human mathematician would do to get the answer 
.
April 25, 2010 at 2:32 pm
In the case of the six multiplications you propose, I presume you mean something like
1 x 2 : 1 is neutral, remains 2.
100 x 123 : adding 00.
102 x 102 : (a+2)^2 or 100*b + 2*b
64 x 64 : convert to log2 then use memory, 2^12 = 4096
48 x 52 : (a-b)(a+b) = a^2 – b^2 a= 50
47 x 73 : (a-b)(a+b) = a^2 – b^2 a= 60
I would have added this multiplication to the other 6 ones:
99 x 199
In any case, like for symbolic computation systems as recalled by Emmanuel, there is a tradeoff between the number of patterns you can check and the time you gain when choosing the best algorithms for that pattern. If you know (and test) too many different patterns, and if you further need a decision procedure when several different patterns match, you may be less efficient for very usual tasks.
April 25, 2010 at 2:52 pm
I agree with that. But for humans the assessment of costs and benefits is different. For us the main constraint, if we are doing mental arithmetic, is that we can hold only a very small number of distinct pieces of information in our heads at once. (People often talk of “the law of seven”.) And we have a distinct preference for minimizing the number of pieces even when we can manage more. So for us it pays to have quite a few tricks hard-wired into our brains in order to reduce the burden on our short-term memory. A symbolic computation system doesn’t have to worry about this.
April 25, 2010 at 4:39 pm
There is some interesting pattern matching going on, as well as a difference between those who see it ‘all at once’ and those who read left to right.
The patterns you match depend on the patterns you know. I could do 147 x 73 very easily [start with 137 x 73]. So that’s how I attacked the last one.
April 23, 2010 at 9:29 am |
I put down “1”, but perhaps I was more of a “21”. The process was something like:
x^2 + 2x + 1 …
“Well there’s some sort of quadratic, so I’ll definitely be able to use the quadratic formula later if I need to…”
… = x^2
“…Wait, I won’t need to, cos that’s gonna cancel.”
So my first thought was a “2”-ish thought, but with the quadratic formula in place of factorisation. I guess I only processed that is was *some* quadratic, my mind hadn’t really twigged which particular one yet.
April 23, 2010 at 9:54 am |
First thing I saw were the two
on either side of an 
sign and cancelled without thought. Then saw the 
. Solved it out of habit. Though it happened almost immediately due to years of school practice, it happened ‘visually’… in the sense the +1 ‘flew’ to the other side to become ‘-1’ etc.
I hope it’s alright to offer my two bits to this interesting discussion. I am just a lowly grad student, but, I have spent a bit of time (mostly during moments of self-doubt) thinking about how people’s mind ‘move’ when they do mathematics. So, while in final year undergrad, I tried to do something similar as this post. Since, I had also wanted to try the experiment on acquaintances from the humanities department, many of whom had an aversion to math, the question in my experiment had, at least superficially, no high school algebra or geometry etc.
In case anybody is interested, the question i asked was this:
Say you have a machine where when you plug in a number (people assume you mean natural number), if the number is odd (but not one), the machine adds one to it and outputs the result, if the number is even, the machine halves it and outputs the result. The machine stops if you input 1. Suppose you continue feeding the output back to the machine, what do you think will be the ultimate behaviour of the machine? Does it go on forever for some/all number or does it stop for some/all number?
One thing I learned from the large pool of 14 people I asked was that the first thing most people (10 if i remember correctly) tried was to reduce it to a form that they could then handle mechanically, using whatever level of math they knew. This is interesting because the first thing the other 4 tried was experimenting with small integers.
April 23, 2010 at 12:03 pm |
My process went something like this:
“A equation, let’s write it down”
“So here is some quadratic on the LHS and on the rig– wait a minute I have seen the x^2 term before”
Cancel and solve the equation. I had no thought of the factorization until I read further.
April 23, 2010 at 1:18 pm |
I think more options are missing. I guess what you wrote makes me
look silly, but what I did was exactly the opposite to 21: considered 1. for a moment, but then I thought “hey, this is (x + 1)^2, maybe this is faster” and went with 2. So my answer should be 12, but this option is not available.
April 23, 2010 at 2:05 pm
I should stress that my main reason for leaving out options was not that I judged them to be crazy but rather that I didn’t want a poll with fifteen options, so I tried to guess which ones were unlikely to be chosen. I’m interested to discover that I got it wrong, and thanks for the extra data point.
April 23, 2010 at 4:20 pm |
A Level student here! Being taught to read the whole question before attempting it I did method 1 straight away. I think this question proves how important it is to read the question thoroughly to use the most effective method to solve it.
April 23, 2010 at 9:32 pm |
Purely 1 for me. First thing I see about the equation: it is a sum of simple things. One of those things occurs twice, so move it left to simplify, it becomes zero, so vanishes. The result is 2x + 1 = 0; that’s the classic “x-intercept or factor” sort of linear equation; form 1/2 from the coefficients, flip the sign (because stuff is implicitly moving across the equals sign) and you’ve got x = -1/2.
For context, I’m in the US but was homeschooled and didn’t get any sort of formal algebra instruction (but lots of hobby programming experience, including mathematical work for simulations/games/graphics), then jumped into college (CS undergrad now), starting the mathematical portion with a “precalculus” course (lots of algebra/trig/basic analytic geometry). Because of this, I only learned to factor quadratics two years ago, so it is *not* the first thing that comes to mind. Also, I don’t read linearly first (I barely managed to not completely read the equation before the instructions — feed readers don’t see “folds”), so I saw the overall structure of the equation before I would ever notice the convenient values of the coefficients.
My current exposure to physics and linear algebra may also have some slight influence in looking for sums (or rather, linear combinations) of objects as the first thing.
April 23, 2010 at 11:58 pm |
Well i vote 1. I actually saw the quadratic equation, started rearranging terms to put it into canonical form; in doing so
canceled so I ended up with the linear equation and solved it.
April 24, 2010 at 12:05 am |
My thought process was the same as many other commenters.
1. x^2 + … (“It’s a quadratic.”)
2. x^2 + 2x + 1 (“Ah, it’s a perfect square.”)
3. x^2 + 2x + 1 = x^2 (“Oh, I can just cancel the x^2 terms, so it’s linear.”)
4. 2x + 1 = 0
5. x = -1/2
(The first 3 steps occured in only about 1 second, as I was reading the equation from left to right.)
Then I thought: “Wait, that was too easy. Should there be a second solution that I missed by canceling the x^2?” This instinct probably comes from many experiences where simplifying a problem subtly removes a solution … for example, if you cancel a common factor, forgetting that this factor might actually be zero. This must have been drilled into me in high school to the point where it became instinctive. So, then I checked by doing approach 2″
6. (x+1)^2 = x^2
7. x+1 = \pm x (“Yup, looks like there’s going to be a second solution.”)
8. 2x + 1 = 0 or 1 = 0 (“Ah, so that’s why there was only one solution!”)
April 24, 2010 at 2:46 am |
This reminds me the famous John von Neumann storie, when asked about the two trains and fly problem
http://mathworld.wolfram.com/TwoTrainsPuzzle.html
he choose the correct answer. Faling to notice the “inteligent” way to handle the problem, he prefer to sum a series instead of solving a linear equation!
Concerning the experiment proposed here, I regret to admit I choose 1; I would prefered to be on van Newman side (I’m sure he would multiply both sides by x and solve the cubic…). Anyway, I don’t think this particular problem serves the purpose it was intended because it is very elementary. My thoughts could be more or less describe like this:
A quadratic equation? ok, lets look at it. Failing to notice the “pattern” on the left, I saw x^2 on both sides. Which I new was my enemie here. The big guy. But I had to look again: it can not be true. it would be to easy. I check the left side again. It is there. This is silly… 90 seconds for this? Ok, maybe there is a trick… Did someone said it was real numbers? They thought I could be that naive. Ok, I don’t care, lets finish this. now look out with the minus sign. be carefull there. divide by 2.
April 24, 2010 at 11:27 am |
Ok, my thought process was the following, reading from left to right:
1. “x²+2x+1” : oh it’s quadratic, I will have to do some computations (but at this stage, I didn’t notice that it can be factored… oh well)
2. “…=x²” : nice! a cancellation, I won’t have to do some terrific computations for this experiment!
3. Now 2x+1=0 is equivalent to 2x=1 (I know, I did’nt manage to properly solve this equation… I was not concentrated enough!), so x=1/2.
4. I read the solutions and felt kind of ashamed!
I answered 1, because even though I first thought it was a quadratic equation, I saw the cancellation before to notice the left-hand side could be factored.
April 25, 2010 at 2:13 am |
I voted 1, since after thinking “how do I solve this” I spotted x^2 on both sides of the ‘=’ sign. Then I cancelled both x^2, and the rest followed. But, the experience became more interesting upon reflection and rereading of the previous paragraphs.
Although Dr. Gowers did mean x^2, x, +, 2, and 1 as real or complex number or integers, he doesn’t state this anywhere before the equation appears. So, we’ve all assumed this.
To show this consider what would have happened had we assumed x^2, x, +, 2, and 1 in the sense of interval arithmetic… as in 1 corresponds to the 1 element in the set of interval numbers, 2 does something similar. In short, 2 corresponds to [2, 2], 1 corresponds to [1, 1], + corresponds to [a, b]+[c, d]=[a+c, b+d]. Letting x=[y, z], x^2 corresponds to x*x=[y, z]*[y, z]=[y^2, z^2] where x just has positive values (i. e. any closed interval [a, b] where a is greater than 0, and b comes as a real number). – for [a, b] and [c,d] then corresponds to [a, b]-[c, d]=[a-c, b-d] since -[a, b]=[-b, -a]. Then x^2-x^2=[y, z]^2-[y, z]^2=[y^2, z^2]-[y^2, z^2]=[y^2-z^2, z^2-y^2] which does not equal 0*=[0, 0], but only contains 0* as a point. A similar sort of thing happens in fuzzy arithmetic.
As a simple example [1, 4]^2=[1, 16], while -[1, 16]=[-16, -1], so [1, 4]^2-[1, 4]^2=[-15, 15] which does not equal [0, 0].
This may seem a bit strange from the viewpoint of an interval as meaning that a point x comes as greater or equal to its infimum and less than or equal to its supremum. But, from the viewpoint of interval arithmetic one often takes intervals themselves and does computations, at least for the purposes of showing things about the structure of calculations with uncertain quantities. E. G. that such calculations satisfy the axioms for a commutative monoid (but not for a group).
This all goes to show what Dr. Gowers originally said “Two difficulties in investigating this general question are (i) that it is quite hard to examine one’s own thought processes reliably (since very often what one remembers of these processes after solving a problem is a very tidied up version of what actually happened).”
April 25, 2010 at 3:53 am |
“[a, b]-[c, d]=[a-c, b-d]”
Sorry, I meant [a, b]-[c, d]=[a-d, b-c]
April 25, 2010 at 1:06 pm |
I didn’t even notice the factorization initially, instead I just chunked the whole thing as “quadratic so group powers of x and apply the formula”.
April 25, 2010 at 2:47 pm |
In view of several of the comments to your simple experiment, one separate series of questions you might have included in your poll would have been:
– did you use paper to think about this problem?
– If yes did you start solving or thinking about the problem before writing it completely ?
– did you bother to write down the final solution ?
– did you pause, fearing a trick because it looked too simple ?
– did you use one other method to check ?
This could be correlated to the train of thought used.
April 25, 2010 at 11:22 pm |
I saw and cancelled x^2 without reading the rest of the equation.
To me it looked like x^2 + blah = x^2.
A couple of seconds later I noticed that the
left hand side was also a square.
April 26, 2010 at 4:09 am |
After looking at the equation and being confused by the asterisks followed by letters below the equation, I decided to read the instructions. Revisiting the equation. I solved it by process 1 with no hesitation.
April 26, 2010 at 6:21 am |
As a number of other people have said, I saw it as a quadratic first but didn’t factor the LHS since I planned to put it in standard form, but then it became obvious that it was actually just linear. I guess that’s 1.
Out of curiosity, are you planning on doing a similar experiment with a slightly more difficult problem? My only concern is that so many people learned to solve quadratics by rote, that I wonder whether my thought process is more a function of my education than anything else.
April 26, 2010 at 10:50 pm |
I only saw 1. The first thing that I noticed was that you could cancel the x^2
term, leaving a linear equation. The process took me a second or less, and I
must confess that 2 and 3 never even crossed my mind until I read further.
April 28, 2010 at 12:33 am |
I immediately thought of 1 and then didn’t bother solving for x once I noticed it was linear. But then I thought:
“I only did this over C. What about characteristic \not= 0? This won’t work for characteristic 2.”
Then I actually computed the result to be -1/2.
I didn’t think of solutions 2 or 3 until I read them, but thought they were cute when I did =)
April 30, 2010 at 2:33 am |
I put 21. My first thought was “Oh. Factor this”, but then I saw the right hand side of the equation.
April 30, 2010 at 6:42 am |
Like few others my process was 12. I saw 1, solved, saw the perfect square and thought, wow maybe there’s a fancier way to solve it and did 2 (not so fancy anyway…)
Maybe this is due to a way of reading. I tend to be a fast reader, which means that I do not scroll text from left to right but rather focus on big blocks of text altogether. Seeing the equation as a whole the cancellation of x^2 first comes to your mind, scrolling it from left to right the perfect square first hits your brain.
April 30, 2010 at 10:09 am |
I think there were two things for me — conscious processing and subconscious processing. I don’t think I consciously thought “there are some quadratics here”, but I am sure that I was subconsciously aware of something like it — this was a setup I felt familiar with. My first conscious thought was “there is a symbol to the right of the ‘=’ that is the same as a symbol to the left of the ‘=’, which means these can be cancelled/ignored”. I think the algorithm that did this would have acted the same whether the the symbol was x^2 or x^3 or whatever, but probably beyond a certain level of complexity I would not have spotted it as the first thing.
After this I thought “the result is a linear equation — does that count as having solved it? I know I can certainly solve it mechanically from here”.
Then I solved it mechanically by moving the symbols around in the obvious way.
May 1, 2010 at 2:22 am |
“I stress once again that the whole thing should be quick and easy…”
I wonder if the preamble above may have biased most to see only 1. As I recall my thoughts, the appearance of the “x^2” on the left side led me to look at the right side of the equation and to already assume something will simplify.
It would be interesting to repeat the experiment with various elementary problems in random order to see what the results would be.
May 1, 2010 at 2:56 am |
2
My brain immediately pattern-matches and I saw the equation as . Then I thought x can’t equal x+1…no, wait, maybe 0 or 1…no, wait, two squares are equal when their magnitudes are equal–when does a number have the same magnitude as one more than than number? Ah, -1/2.
May 2, 2010 at 4:19 pm |
I am struck that so many commentors, as they describe both what they voted and what they thought, did not in fact follow the precise instructions by the author of the experiment as to how to describe one’s thought.
Particularly not respected is:
“If you very briefly thought of doing 2 but almost immediately realized that you could do 1 and that 1 was better, then your answer should be 21 rather than 1, even if you didn’t at any stage feel a commitment to 2.”
Many participants should have put 12 or 21 according to their explanations while they chose 1 or 2 as their answer to the poll and they rarely express regret of having misused the codification. They should have known better, we could hope, having been trained in mathematics, and they should have been able to follow instructions for the duration of a little game.
So the poll results are probably meaningless, despite the qualified audience because we cannot estimate the error measure for the most numerous options.
May 3, 2010 at 8:46 pm
Criminals, all of them. Off with their heads.
May 15, 2010 at 2:20 pm |
I am not a trained mathematician, though I have been reading some maths of late, and was somewhat lazy about this -I factorised (almsot subliminally) and saw that the solution implied x^2=(x+1)^2 and realised the solution was was either + or -1/2 then read on without checking which. I post this merely to note that one doesn’t have to be a trained mathematician to find the factorisation more obvious than the cancelling. Perhaps the professional fluency helps one to see things more simply; or perhaps the professional is more used to categorising equations as ‘linear’ or ‘quadratic’ and is used to reducing the order of polynomials wherever possible where the amateur just sees an equation. I think also my mind latched onto the apparent implication ‘x=x+1’ and got bothered about that so that kept me busy (perhaps I was hoping to prove 1=0).
There are some quite interesting analogies here with chess, where some players will find they have analysed a few moves ahead before absorbing the whole position and noticing it’s checkmate in one – the analytical machine gets going as soon as any patterns are noticed.
Many of the same phenomena work in the arts – a word or two sets of multiple chains of possibilities – though perhaps there’s less premium on finding the simplest path there.
By the way, I think we overlapped at school – compliments on your very short introduction which I greatly enjoyed.
May 15, 2010 at 3:42 pm
When I saw your name, I thought, “I wonder if that’s the same John Mackinnon I remember from school,” and then got to your final sentence …
June 3, 2010 at 3:43 pm |
1. I wrote it down, without thinking.
2. “What’s this thing doing on the right? I should move it to the left.”
3. “Oh, it cancels, I don’t actually have to move it.”
4. “Now it’s linear with solution -1/2.”
5. “Was I just tricked into a wrong solution?”
6. “Nope, I don’t think so.”
July 5, 2010 at 8:41 am |
1. I glanced at the problem.
2. I thought of few strategies and selected the strategy that seemed to be the simplest to implement. I started with complex strategies and continuously moved towards 1.
July 5, 2010 at 11:03 am |
used 1 as many have described
process – looks like a quadratic —
oh x^2 on both sides … well i could cancel … leaves a simple linear equation
vague feeling of unease – have i lost a solution in doing that
July 6, 2010 at 11:33 am |
After seeing the equation, and before I could consciously decide on a strategy, I recognized that $x^2$ occurs on both sides. The recognition triggered an automatic and compulsive process of visual simplification, whereby I saw the $x^2$ terms disappear, 1 move to the right side, followed by 2 which arrived in a denominator. I then saw x=-1/2 and
realized that’s it.
Sometime after I saw the $x^2$’s disappear, and before the other steps, I had a thought confirming that I was engaged in the correct process, along the lines of “Yes, that’s the correct way.” I think that it was verbal but internally unvoiced.
As I was scrolling down to see your further instructions, I glanced at the equation again and saw that it said $(x+1)^2=x^2$. I then felt a bit repulsed, thought something like “that’s the bad way to go,” and immediately abandoned that line of thought.
July 6, 2010 at 7:58 pm |
My option isn’t on the poll: I essentially went 2 1.
Here’s the record I took of my thoughts. Afterward, I have an explanation.
“we subtract x^2 from both sides
actually wait we have
(x+1)^2=x^2
so x+1=x which is bad
so x+1=-x
so 2x+1=0
so obviously x=-1/2
done”
Now for the explanation. My first inclination was to do what in many cases is the most obvious and in a sense the most brute force thing you can do: subtract something that you see on both sides. Then I saw what I would call something “nice,” something “out of the ordinary”: The left hand side was the square of a simple expression! There was a kind of pattern, something you had to recognize. Whenever you see a nice pattern, something nice and nontrivial, it’s a general problem solving rule that you go for it! Subtracting something from both sides is obvious – noticing that pattern is subtler. You should also go for the trickier thing.
So I went for it, and that’s how I solved it. Maybe in this case it was easier to subtract from both sides, but I think a general rule is that noticing interesting and nontrivial patterns is a very important way of moving forward in math, and in other problems it may be more fruitful. In fact, why do we sometimes actually add something to both sides to that we can factor one of the sides? This is part of my point.
September 10, 2010 at 3:13 am |
I was on one immediately, tried to remove anything to make it simpler so canceling was there. Yet the equation being left justified might have influenced me. It could be nice to make the poll with the equation being centered and if there are any significant variations.
June 7, 2011 at 3:45 pm |
Thought of doing ‘1’ instantly, thought “that seems too straightforward” so then did ‘2’ for some reason and then asked myself why I bothered second guessing myself in the first place.
July 21, 2011 at 9:18 am |
Here is an experiment I though could be interesting. Ask each person in the experiment one of the following two questions (chosen randomly):
One and a half kilo of something costs 180$. What does one kilo cost?
One and a half year of some service costs 180$. What does one year cost?
Now compare the time it takes to answer the two questions. Is it faster to do the second calculation because it is about years and one and a half year is 18 months?