## A little experiment III

This one is a little bit different, and the associated poll questions are rather vague. I am curious to know what effect it would have on our ability to do routine manipulations if we could look at only one symbol at a time. Maybe at some point after the experiment I will say what my motivation is for this, but for now I want to influence what happens as little as possible.

After the fold, you will find a mathematical expression that can be simplified. Usually, one would look at the expression and take in chunks of it at a time, but I have embedded it in a lot of junk, so that that will not be possible. I want it to be easy to find the individual characters that make up the expression, but not easy to look at more than one at a time, so the non-junk characters are quite widely separated, but they are signalled by being enclosed in dollar signs. For example, to convey the expression $e^x$ I would write something like

asdf\$e\$oijgeqwrfnoifwefhppsd\$^\$pofiewqt;jklnrewfppas\$x\$eiorfoi4

Your task, if you feel like participating in the experiment, is to simplify the expression as much as you can in your head. (If you write it down, then obviously it makes it completely pointless for me to have written it in this strange form.) I would then like to hear, in as much detail as you can remember, what thoughts went through your head, and in what order. I am particularly interested in what your eyes were doing and how they interacted with these thoughts. It may not be easy to remember all that, but if you do the best you can then I’ll be happy.

Here, then, is the expression to be extracted and simplified.

qweiuorh23\$y\$5%98haoiuhf0987a\$^\$&fsdoifpoiuq%hq9fw9hf78&hf

sdofs&gi\$2\$ouyewr9%98u82923hisdfhoeq&w0\$+\$9fhewiur%h1291

ww\$(\$oi%&q2034og0jp&oigu2\$x\$4309jge%sfoiune@aw\$+\$f05432

123098jap\$y\$oijra0erj12o3je12-09ij\$)\$a;soijdfpoqae\$^\$wrhoiof99e

aw\$4\$oenf9923oiaof\$+\$ijawioejf0e2piuh01209381kj4u\$3\$23kjufaih

sertp98%uq23\$x\$4309gsdf&0sliuhqr3%0\$^\$q0h230hrf02%hfjrj&02

as\$2\$df0j12on12399oi\$+\$j)ojsd&oiwefoij%oijasdfoij@oi\$y\$jwe%f@

asdf9jqweoi\$^\$jeqwr9gjoeqwroij1\$2\$30aewroijh(oijewr0)%ija\$+\$s

sdg\$5\$9fj%joijesr)joijear\$x\$99OIjewroij2jOORWjoweroaspdjkfkhkk

asdoifjw23r@\$y\$09iuoweroijD\$+\$Foijwer9Oweroouodfgyosdf&&

weonf\$2\$wpkfejwjjjwefpok23408wlskdjf%oijwef&Jsdvojwgjjjjoj\$+\$afoij@

ooj%%oi\$y\$jsdfgoijolllcdl&@ojs\$^\$dfoijsdf(oijweroFJSOoijgh\$2\$fa

SOJF\$+\$HOWerkjoafhovnononon\$x\$oj%ojf@ojwefojsdvqowhf\$y\$joxcv

Now for a little spoiler space.
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Let me briefly explain my choice of expression. If you do write it down, then it sort of jumps out at you that you have a sum of two homogeneous polynomials in $x$ and $y$, one quartic and one quadratic, together with a constant term. The natural thing to do then is collect together the terms of the quadratic, which are rather scattered, in order to see what it is. At this stage, one sort of forgets about the quartic and constant terms. Then one establishes a simplification of the quadratic, after which the problem becomes fairly easy.

Now my guess is that something like this process happens even when the expression is presented in the peculiar way I have presented it, but it is harder to keep in mind what is going on. With the usual way of writing something down, one can always glance at another part of the expression for a quick reminder. Here, one has to make a conscious effort to store it in one’s mind. I am also guessing that most people will at some stage resolve to ignore everything except the quadratic part of the polynomial, which will feel like the part that “we do not yet understand”. Then when one does come to understand it, it will have the flavour of a “technical lemma”, in that after proving it one can hold just the answer in one’s head and consciously ignore the quadratic part (though since it is split up character by character that is harder than usual).

I am particularly interested in how much people hold in their heads. At one extreme, you might have somebody who went through the entire expression several times with the aim of learning it off by heart, so that they could get access to its parts in something approximating the usual way. (This is almost like ignoring my request not to write anything down — such a person would be writing it down on “mental paper”, so to speak.) At the other end would be somebody who tried very hard to concentrate on only one thing at a time. For instance, such a person, while counting how many $y^2$ there are, would try very hard not to be distracted by anything that was not a $y^2$, with the thought that once they have finished counting the $y^2$ they can remember the answer, forget the “working” and then think about other parts of the expression. In this poll I invite you to say where you are on a scale of 1 to 4, where 1 means that you did everything you could to minimize any effort you might need to make to memorize parts of the expression, and 4 meant that you made a huge effort of memorization before even starting to think about simplifying the expression. Options 2 and 3 are like options 1 and 4 but less extreme: you might go for option 3 if you have a good memory and decided that even though you couldn’t remember the whole expression, you did make a conscious effort to commit parts of it to memory in order to help you along, and you might go for option 2 if, although you didn’t want to tax your memory any more than you had to and had a general tendency to focus on just one part of the expression at a time, you didn’t put in too much effort to think of ways of avoiding using your memory. As I said at the beginning of this post, these are rather vague categories, so I will be particularly interested in narrative accounts of how the process felt to people.

### 46 Responses to “A little experiment III”

1. Jitse Niesen Says:

I found it surprisingly hard. I went through the whole mess once to get a general feel for the expression and found the homogeneous quartic, some quadratic terms and lower order terms (it was only at the very end that I found out that the lower order terms are in fact only one constant term). Then I tried summing the homogeneous quadratic part, but I found it very hard to keep the three coefficients in my head and be confident in the result, so in the end I had to concentrate on summing the y^2 terms first. Then I summed the x^2 terms, and I “summed” the xy term. I added the result and saw that the result factorizes (as I had expected all along); I went again through the expression to check. Then I went to look for the non-quadratic terms, found out that there is only one term of order less than quadratic and at that point it became possible to keep the whole expression in my head.

2. ogerard Says:

I clearly started reading the first time with the intent of doing 4 and ended doing mostly 2.

There was a marked difference between the first “scan” of the expression and all the others, complete or partial. During the first reading of the expression, I was all the time trying to guess the next term, in fact to guess the whole expression, often wishing it to end so that I could start simplifying it. I was several times disappointed (for instance, the quartic exponent came as a surprise for me, also I was expecting less terms, even if the quartic exponent should have told me that it was not likely hereafter.) It is perhaps an automatic strategy which can in some cases ease the finding and certification of the next character and certainly is part of memorizing the expression so far, because one expects it to be coherent.

Also it seems to me than when visualizing a traditionnal printed expression, one can assess the whole size fairly automatically and so that one does not expect or wish so much about the expression in case.

In the process of doing this little experiment, I ended asking me conciously questions about the expression, its length and number of terms, the order of this and that, much more systematically than what I remember in dealing with similar algebraic expressions in the mix of visual, mental, oral and gestual thought and perception that typically accompany this kind of manipulation.

I sometimes feared to be in a closed loop trying to find my bearings before safely eliminating terms.

I have answered 2 to the poll because it matches the way I acted for the longest cumulated duration of working out this exercise, but my sensation is that depending on the phase of my treatment of the exercise the focus on larger scale memorization versus local work focus was oscillating between 1 and 4.

I find this “1-byte buffer” experiment really enlightening. Thanks for having shared it with us. I have now just written a little program to automatically “encode” expressions in a similar way to play with the idea without having only one example.

sidenote: I noticed when scanning that I usually found first visually the rightmost \$ sign than the one bracketing the character on the left. Did others’ experience differ ?

3. Tom Ellis Says:

I got half way through the equation, reading aloud the terms as I came across them. Then I though “I’m never going to fit this in my head” so I stopped.

4. Uwe Stroinski Says:

This was hard.

I go through the text. (x+y)^4? There must be a trick. I try to add up the coefficients of x^2, y^2 and xy. To my own surprise I have problems. I go through the text again and again. After a couple of times, reading it becomes easier. At least. I realize 1+1+1=3, save that for later use, but have problems with the 5. There is another xy at the end that gives a 6. I factor and get the (x+y)-expression. Is this the result? I am very insecure. I memorize the expression and try to factor it. “Somehow” the 3 becomes a 2 and I get (1+(x+y)^2)^2 instead of (1+(x+y)^2)(2+(x+y)^2). I have no confidence in my last step and stop.

My vote was 3.

5. Alejandro Zuleta Says:

I normally like to make mathematics in my head, in order to exercise my memory, which is usually very bad, so I could say I am somewhat used to memorize complex expressions (It is never something easy, though).

I had to read this problem twice. The first time I tried to find a pattern to simplify the expression as I was reading it. When I reached the “xy” part, however, it was something I wasn’t expecting, so it distracted me and thus the whole expression started to become fuzzy in my head. When I reached the “2”, I was expecting it to be factor of x or y, but it wasn’t, so, again, the whole expression started to look extremely fuzzy in my head. So I had to start again.

The second time, I already knew that I was going to find some factors of y, some factors of x, of xy, so it all went more smooth than in the first time, as my memory was prepared. Thus, I summed the like terms as I was reading, making the whole expression a lot easier to keep in my head. Finally, I arranged the whole expression in a familiar fashion (the quartic first, etc), and made the final simplification without much effort.

6. Steve Flammia Says:

I went through the whole thing once, and realized that it contained a only quartic, quadratic and constant terms. Then I went back through again to try to learn the coefficient on each monomial. I realized that the quartic and constant term could be safely “ignored” for the purpose of simplifying the quadratic. I went through it one more time to make sure I remembered the quadratic part correctly. Then once I collected all the quadratic monomials, I simplified them. I had no confidence that I wasn’t missing something, though.

It was surprisingly difficult. A few times, I found that I had missed something. For example, on my first pass, I parsed the symbols “(” and “+” as being next to each other, which doesn’t make sense. So I had to backtrack a few times.

7. Greg Martin Says:

I did pretty much what you described in your post. Scan through to see what’s there; uh oh, lots of terms; scan through again, this time with an eye towards counting how many terms there are (as an aid to storing in “mental paper”); realize that most terms are quadratic; scan through again picking out just the quadratic terms; simplify and factor the entire quartic; then spend a few moments looking in vain for further simplifcations of the individual factors.

On the multiple scans I described, my eyes were literally scanning the text line by line, left to right. At some point my goal was to memorize the actual expression on mental paper; but I found my brain trying to simplify (the quadratic part, for example) before I would have agreed that the expression was wholly “uploaded”.

8. Jason Dyer Says:

I accidentally missed some of the \$ terms and the whole thing was nonsense (or at least, harder than I was sure you intended). Then I tried it again and I did it again and I decided to stop.

Maybe a video would be better? I know the extra garbage interfered wildly with my thought processes and even if I could simplify anything it wouldn’t be an “authentic” mental measure.

• gowers Says:

If I knew how to produce such a thing, I’d have a box on the screen, and if you pressed the right arrow it would “move to the right” (meaning that large characters would appear on the right-hand side of the box, move to the left, and disappear), and if you pressed the left arrow it would “move to the left”. Hiding the characters in the garbage was the best low-tech approximation I could think of.

• Jason Dyer Says:

Options I know of for you:

1. The low level programming language known as BLOG.

2. Prezi (presentation software that allows text to zoom around however you like).

http://prezi.com/

3. Powerpoint (which can embed in a blog in multiple ways, but I don’t have a recommendation offhand).

• ogerard Says:

I created a pdf version of the experiment, with a page for each symbol and mailed it to you. It should be readable inside the browser of most readers.

9. Phil Says:

I didn’t realise we were “allowed” to take multiple passes, which obviously skewed my approach. I made a massive effort at memorisation, holding a visual and “audio” image of the current state of the polynomial in my head.

Something of the form \$x\$junk\$^\$junk\$2\$, say, would first add an x to the end of the polynomial, and then when I discovered the hat, I’d skim forward to find the exponent and replace the x with x^2. Whenever I hit a + sign I’d try to simplify the polynomial I had thus far in my head, and then commit the polynomial in simplified form to memory again before continuing.

I essentially just did it all on “mental paper”, but paper that I couldn’t glance at (and that is far less reliable than actual paper!). Interestingly (well, maybe that’s too strong a word), every time I accessed or modified the polynomial I physically looked up as I worked with the mental image and so had to hunt for where I was in the text when I looked down.

When memorising, I would mentally repeat the polynomial to myself as well as imagining the image, and when reading the polynomial I’d have a clear image of one part of the polynomial with a vague sound-based memory of the bits I wasn’t mentally “looking” at. Apologies if that’s a bit confused; it’s hard to explain or recall exactly what your mind’s doing!

10. Anonymous Says:

It was too hard, I gave up half way through.

11. Richard Seguin Says:

I took one look at this and didn’t proceed. At age 58 my short term memory is no longer quite efficient and vast enough to separate the content and structure of the real information from all that noisy junk without resorting to pencil and paper. I think that in general the brain is much more adept at “seeing” visual images obscured by heavy noise than doing the rough equivalent with text (processing text is too recent of an important skill to have been finely tuned by evolution). And, I suppose, I could be lazier than I used to be when I was young and would sometimes do things like factor integers in my head during otherwise idle moments.

12. Terence Tao Says:

This was remarkably difficult for me; I think my way of doing mathematics is optimised for having a decent amount of visual assistance (e.g. a page of scratch paper, or a blackboard) available, and not to rely on “registers” to do these sorts of manipulations. I got as far as being able to collect the quadratic terms (after multiple passes) before giving up.

Afterwards, I tried writing down the expression, and indeed one could then easily simplify and factorise it in one’s head. So there is definitely a noticeable time-memory tradeoff here.

13. Kristal Cantwell Says:

I ended up memorizing the entire expression. The main reason I did this was I kept going over it to figure out the quadratic expression was I wanted to make sure I got it right. In particular I had problems with the repeated y^2. I kept going over it to make sure I had it right In the process I ended up memorizing the entire expression. I simplified to (x+y)^4 + 3(x+y)^2 +2 and that simplified to ((x+y)^2+1)((x+y)^2+2).

14. Rahul Says:

Midway through, while sorting the quadratic term out, I decided it was too hard. But my unconscious was apparently still interested in the problem – it made me take a few more glances, and sorted the thing out. I was working so hard on the deciphering problem itself that I didn’t really have the same consciousness of my thought processes as in your other two experiments.

Because my handwriting is very poor, I tend not to use visual aids, and do as much as I can in my head. This is very limiting, but also a motivation to find the simplest possible approach.

15. Guy Srinivasan Says:

I saw y ^ 2 and thought “y … to the … 2 oh okay y squared”. This pattern persisted, whereas usually I would see the term as squared up front. For example, (x+y)^4 I thought “weird I wonder why it’s in parentheses” before getting to the ^.

Seeing the quartic I thought “screw that” and continued with “if something looks NICE I’ll go back and expand that quartic otherwise it’s simplest as-is”. Seeing the next y^2 I remembered “wasn’t the first term y^2? yep okay 2y^2”. When I got to the third “+1 then, 3y^2”.

The last xy went “oh I know I’ve seen xy before right? Let’s check… yes there it is and huh, I missed the 5 the first time around.” Somehow I didn’t see the coefficient of the 5xy term on the first pass.

Finally I thought “am I done? 3y^2, a quartic, something xy, and a constant… sounds like it”.

It wasn’t until I wrote the above and skimmed the comments that I realized there was an x^2 term. I’m sure I saw it while reading, but it was completely gone from memory by the end and during writing this.

16. Andy Says:

I missed the plus sign between the y^2 and the (x+y)^4 at first. Thus seeing a y^2(x+y)^4 term which I kept in the back of my mind, but mostly ignored. I saw that most of the rest were at most quadratic and added most of them, arriving at 3x^2 + 6xy + ?y^2 + 2. I continued to ignore the first part and started in the middle again, achieving 3x^2 + 6xy + 2y^2 + 2. This didn’t seem so great when combined with the putative y^2(x+y)^4, so I started at the beginning, saw the plus sign, and was essentially done.

17. Graham R. Shutt Says:

In the preliminary example, you use dollar signs, a familiar TeX
convention, to mark up the three elements of the expression \$e^x\$:

asdf\$e\$oijgeqwrfnoifwefhppsd\$^\$pofiewqt;jklnrewfppas\$x\$eiorfoi4

\$e\$ \$^\$ \$x\$

When I saw the test problem, I realized that it uses the same TeX
convention as the example. However, there are lots of random
characters to scan while searching for the dollar signs. By the time I
reached the opening parenthesis my short term memory was overflowing
and I ground to a halt.

I confess that even when I used a TeX buffer in Emacs as an aid I
failed to parse the equation correctly my first time through. (I am
not a professional mathematician, and so I am not as familiar with the
quartic and quadratic patterns as a professional mathematician would
be).

An interesting experiment!

18. Anonymous Says:

I think this is too hard for people with a bad memory, like me.
I guess I could have tried harder, but I’ve always hated any kind of conscious memorization and rote learning, so I gave up quite soon.
I kept reading once and again to see if there was some obvious simplification but it is just some random expression with no global structure, and it’s pretty hard to remember all terms, let alone keep track of how the terms in (x+y)^4 interact with the rest. I ended up writing the expression on paper as I couldn’t hold everything needed in my head.
I voted for 4 because memorization was very hard for me, but I would have liked to keep the amount of memorization low had it been possible.

19. david Says:

I think this is too hard for people with a bad memory, like me.
I guess I could have tried harder, but I gave up pretty soon as I’ve always hated any kind of memorization and rote learning. I ended up writing the expression on paper.
I tried to keep reading to see if there was some global structure that allowed simplification, but there is none, it’s just a random expression and it’s pretty hard to remember all the terms, let alone keep track of how the terms in the expansion of (x + y)^4 interact with the rest.
I voted for 4 because memorization was hard for me, but I would have liked to minimize the amount thereof had it been possible.

20. DC Says:

I missed the “3” in front of the “3x^2” when reading it (something I must have done 20 times) and got nowhere. If it does factor when you remove the 3, I couldn’t figure out how.

I started by memorizing little bits, reading left to right from the beginning. Once I had a part in my mind’s eye, I only kept a trace of it (e.g. “sum of squares in x, y, and x+y” for the first few terms) in my head, and moved on to the next part. When I got to the end, I had a feel for what the whole expression “looked like”, so and I went back and reread it trying to put the whole thing in memory. So on the first pass I was reading the “feel” of it, and once I had that, went for the whole thing. I didn’t think I stood any chance of memorizing the whole thing until I knew its “feel.” On my second pass I did end up memorizing it, but then I couldn’t factor it (because of the missing 3).

I felt that this format forced me to work left to right. In a visual representation, answers to questions like “what is the highest power appearing in this expression?” or “does this expression have variables besides x and y in it?” are essentially immediate, and their answers allow me to identify the most “essential” parts of the expression almost instantly. I seem to need to know the “gist” of an expression before I begin seriously thinking about how to manipulate it. Here I had no way to get the gist without reading the whole thing— and I seemed to have to do this left-to-right, as any other way would add effort. (If you read from right to left, or just plop randomly down in the middle, and see a “2”, you don’t whether it’s an exponent or not until you read more.)

It’s interesting that we often think of symbolic manipulation as purely “mechanical”, in contrast to, say, “geometrical” things that must be “visualized”. What I learned here is that the way I do algebra depends essentially on having a visual representation of what I’m manipulating. Without it, it’s almost like I don’t know algebra anymore.

21. Sune Kristian Jakobsen Says:

This made me wonder: Do there exist any blind mathematicians? I guess that blind people are much better at tasks like remembering and manipulating strings in their heads, but still it seems to be a huge disadvantage.

22. Jax Says:

Sune: Wasn’t Euler blind, or nearly so, at the later stages of his life?

I found this very hard. I went for Approach 1, but it would’ve taken ages to even get to the point where I had a handle on it so I gave up.

I’m sure somebody here would program your alternative approach mentioned above. I’ll give it a shot if nobody else wants to.

• gowers Says:

I had a look at Jason Dyer’s suggestion of Prezi and made some progress with it, but not yet enough to produce a nice embeddable equation presented in that form. It would be quite interesting though.

23. Mark Bennet Says:

Interesting – I read one of the ‘x’ symbols as ‘multiply’ first time through.

24. Gavin Says:

Examining the expression when it is presented as a sequence of symbols like this is very different to how you would normally do it, because you have to put more conscious effort into building up the structure of it. In an expression like x^2 + 2xy + 2y^2, you see this as 5 chunks. Each term is seen as an expression and each operator is seen as a plus sign. We immediately recognize a pattern of alternating products of variables and operators as a valid mathematical expression, and furthermore when the operators are plusses we recognize it as a sum. This example suggests that we see mathematical notation in a hierarchical manner, divided into chunks and sub-chunks. We associate patterns in what we see with concepts that we have already learned, some more specific than others. (This is true of objects outside of mathematics as well.) (Another example of a concept is that of a polynomial in a single variable. As soon as we see something like 2x^3 + 3x^2 – 7x + 13 it immediately has the “flavour” of this concept.)

In order to solve a problem like this, first of all we have a desire to solve the problem. This motivates us to take actions that may lead to doing so. Some of these actions are learned procedures, where we know how to get from one step to the next. If we don’t have an exact mental algorithm to go through, we can put our minds in certain directions that we feel that may be productive, which is informed by a semi-subconscious mathematical intuition. (I believe that much of the mystery in how people do mathematics is not knowing exactly where these feelings come from.) There is a parallel with human action outside of mathematics here. If we want to achieve some goal, we carry out actions that we believe are likely to reach it, and intermediate goals acquire the appeal of the final goal.

25. Todd Trimble Says:

Sune: there’s a mathematician named Bernard Morin who has been blind since age 6.

To all: I am completely baffled. Am I the only one who sees random-looking gobbledygook? I’m going to copy and paste what I saw on my screen and see if it looks any different:

qweiuorh23\$y\$5%98haoiuhf0987a\$^\$&fsdoifpoiuq%hq9fw9hf78&hf

sdofs&gi\$2\$ouyewr9%98u82923hisdfhoeq&w0\$+\$9fhewiur%h1291

ww\$(\$oi%&q2034og0jp&oigu2\$x\$4309jge%sfoiune@aw\$+\$f05432

123098jap\$y\$oijra0erj12o3je12-09ij\$)\$a;soijdfpoqae\$^\$wrhoiof99e

aw\$4\$oenf9923oiaof\$+\$ijawioejf0e2piuh01209381kj4u\$3\$23kjufaih

sertp98%uq23\$x\$4309gsdf&0sliuhqr3%0\$^\$q0h230hrf02%hfjrj&02

as\$2\$df0j12on12399oi\$+\$j)ojsd&oiwefoij%oijasdfoij@oi\$y\$jwe%f@

asdf9jqweoi\$^\$jeqwr9gjoeqwroij1\$2\$30aewroijh(oijewr0)%ija\$+\$s

sdg\$5\$9fj%joijesr)joijear\$x\$99OIjewroij2jOORWjoweroaspdjkfkhkk

asdoifjw23r@\$y\$09iuoweroijD\$+\$Foijwer9Oweroouodfgyosdf&&

weonf\$2\$wpkfejwjjjwefpok23408wlskdjf%oijwef&Jsdvojwgjjjjoj\$+\$afoij@

ooj%%oi\$y\$jsdfgoijolllcdl&@ojs\$^\$dfoijsdf(oijweroFJSOoijgh\$2\$fa

SOJF\$+\$HOWerkjoafhovnononon\$x\$oj%ojf@ojwefojsdvqowhf\$y\$joxcv

26. Steve Flammia Says:

I was just reminded of something that this experiment reminds me of…
When learning a foreign language, there is a period where you have quite a large vocabulary, but are still not very comfortable with long sentences. Since you basically still haven’t progressed beyond the ability to interpret the individual words, the sentences don’t make any collective sense until you read it multiple times. In particular, by the time you finish reading the sentence you’ve forgotten all of the words at the beginning, and their context in the long sentence. You have to build up a “global” understanding by identifying clauses, prepositional phrases, etc. piece-by-piece and then later add them up just like a series of lemmas, as you suggest. (Well, this has been my personal experience with foreign languages, anyway.)

This also makes me think this phenomenon is probably well studied in the psychology and linguistics literature. As a math problem though, probably not. But I bet it is worth exploring connections with published experiments.

• esmeyny Says:

With regard to the comment on learning a foreign language, I have a different take.

As a student, I spent a few weeks in another country and was asked to talk to students who were beginners in my native language. They would say “Slower, slower” all the time, until I finally spoke so slowly that I, too, forgot the beginning of my sentences while speaking.

So, you have to have a minimum speed in a language to be able to handle long sentences automatically, native language or no.

27. Daniel Moskovich Says:

I scanned it once to get the general idea, then again (misread it), then a third time I just summed similar terms and got (x+y)^4+3x^2+6xy+y^2+2. While thinking about how to factor that expression, I forgot it. So I repeated the first steps once… same story… and once again, and saw the factorization once (x+y)^4+3(x+y)^2+2. Once more I recalculated to reconfirm, then I factored as (x+y)^2(3+(x+y)^2)+2, and this is as far as I got.

28. Pietro Says:

I overestimated my own memory and tried to remember everything. I’m usually not good with paper and write down very little when I work, except to summarize. But I found that the extra work of filtering out garbage nearly wiped out my short-term memory.

So I went back several times to tally up the different order terms, but I found it quite hard even to remember small bits of information (like the coefficients of x^2 and y^2 while I tallied xy) while filtering out noise. Eventually I grouped everything and then it was easy to somewhat simplify the expression, because the 1-2-1 pattern in 3x^2 + 6xy + 3y^2 jumped out at me.

29. GM Hurley Says:

I started going through it both saying the expression to myself and trying to visualise it mentally, which was OK up to the quartic expression, then I had to go back and start again, and get a bit further, then go back and start again, and before I reached the end had just thought, “well, that’s why we write things down.” I think I was comfortable holding about 8 pieces of information at one time (in the sense that (x+y)^4 is 7 characters, but I’d count as 3 pieces of information). Once something more went in on the end the beginning was lost.

A comparison would be with trying to follow a geometric construction when the completed diagram is on one page and the actual construction is (infuriatingly!) on another page. I can manage by flipping backwards and forwards to track the build-up of a diagram with 6 or 7 construction lines: beyond that I have to draw or photocopy the diagram to put it alongside the narrative. How could one count the “pieces of information” involved?

(On further consideration, the problem with working through a construction against a completed diagram is remembering which lines you’ve already got. If they’re on the same page, it seems easier mentally to “bold” the lines you’ve got and “grey” the rest, or even just use fingers to track them, while the page-flipping and consequent reorientation seems to introduce enough time delay that you risk losing it and having to go back to the start. Practice would no doubt make a lot of difference, both to your exercise and the geometrical one.)

30. Michalis N Says:

When I carefully went through this mess the first time, I got the feeling that there were a lot of second order terms (I remembered several y^2 ‘s and xy’s).
Going through it the second time I collected every x^2, y^2 and xy and noticed that I could simplify them to 3(x+y)^2. At this point I remembered the whole expression, but I went through it a third time to check if my memory was correct. Then I simplified it to ((x+y)^2+1)((x+y)^2+2).

31. Ricardo Sandoval Says:

I choose option 4

On my first read I memorized the first two terms, after that I just got the gist of the rest.

I read it again making sure of the first ones.

Getting somewhat frustated about having to remember it all, since nothing obviously jumps at my face.

I notice there are some repeated terms. Read some stuff again and again until confident I am not mixing things (I was affraid of mixing how many x’s or y’s)

On my third read more or less I was collecting the y^2, x^2 and xy (as a bonus) got happy in the end after collecting 3y^2 matching 3x^2 and even 6xy… Perfect!! After that all was fast and straightforward.

Thinking better I should have marked 2 or 3.

32. Richard Elwes Says:

First I went through reading each term out loud, and then reached the point where I realised totally memorising it was going to require more effort than I had invested (or perhaps more ability than I have), so I continued to the end in the same manner, a little quicker to get the general picture. Then went back to where I remembered the quadratic terms fitted into the skeleton of the expression, using the brackets and “^ 4” as landmarks to navigate around. I analysed the quadratic terms in a mental ‘rolling tally’ form, remembering the previous total at each stage adding up as I went along.

I rememberd the quartic expression (x+y)^4 without having to reread it, but nearly tripped up by forgetting the constant term, only remembering it on a final quick scan though to ‘check’.

Then I wanted to factorise the whole expression, and I did visualise the terms written down, and manipulate them on ‘virtual paper’ by making the substitution z=(x+y)^2 (even to the extent of waving an imaginary pen around in the air).

I voted 3 as it reflected my experience, and where my effort was focussed (aside from that involved in filtering out the junk, of course). Thinking about it though, I’m not quite clear how I could have significantly reduced the amount of memorization required.

An interesting experiment, thank you.

33. rgrig Says:

The general strategy was to keep in mind the shortest possible form that is equivalent to what I read so far.

For the first attempt, I got confused after the first set of parentheses because I bumped into what seemed like mismatching parenthesis and my eyes became hazy. I stopped, made the fonts a lot bigger, and started again. This time I decided to skip the outermost parentheses in the first pass. It turned out that there weren’t any other (unless I made a mistake). Applying the strategy of keeping to simplify after each step, I got something like (x+y)^4+3(x^2+y^2)+2xy+2. Then I realized I’m very uncertain about the correctness of this and thought about double checking, but I gave up, because I though that if I repeat the process I might forget the expression I’m supposed to double check. I decided to give up. (While writing this I think that if I would have went for double-checking I would have planned a bit in advance and decide to focus on x and y and xy in turn, so there’s less strain on memory.)

Now, let me write it down to see what it was. 🙂
[2min later] OK, now I give up because my eyes hurt too much. 😦 (But it looks like what I did mentally was completely off.)

34. palibacsi Says:

I found it really hard and I was constantly afraid of failing to notice some of the \$ signs. I went through the expression and wanted to remember everything. That did not work at all. Moreover, I lost track of which line I was reading. Fortunately, this happened only once, but after it I did not feel comfortable any longer and wanted to write everything down that matters. Of course, I did not do so. I went through it, stored the (x+y)^4 term, realized that there were a lot of x^2 and y^2 and stored ax^2+by^2 as (a, b). During this process, I noticed that there was not that much left any more. I stored (x+y)^4+3x^2+3y^2 and went through it once again to collect the rest. I noticed that 6xy fits in and simplified to (x+y)^4+3(x+y)^2+2. I tried (x+y)^2(3+(x+y)^2)+2 but did not like it. Then I gave up without trying to factorise the expression.

35. esmeyny Says:

Quite hard and a disagreeable experience.

I went through it once to see what kind of terms I got and interestingly thought at the end that (x+y) was squared. After the second swipe, I decided to count the coefficients of the quadratic as three numbers in one go, but made errors because quickly scanning for the quadratic terms I missed some coefficients and was disappointed that the result was not symmetric in x and y.
I tried again, it was symmetric and I could finish.

It made me think of a student of mine who could only read little bits of the exam paper with a huge lens.

36. deden Says:

wooww.. it’s hard…

37. dinov Says:

good experiment 🙂

38. Elijah Laws Says:

In the 1st pass, I wanted to get an idea of what the expression contained while committing very little to memory.

In the 2nd pass, my goal was to reduce as much ‘memory space’ as possible by combining like terms. Afterward, the string became much easier to remember in whole.

Overall, I thought it was challenging.

39. Zavosh Says:

I went through the lines several times, writing on ‘mental paper’, sometimes going back to check if I had missed any symbols (which I had a few times.) I verbalized the terms as I spotted them as aid, and found that it was easier to hold the terms in my head ‘suspended in limbo’ as opposed to linearly arranged on mental paper. My eyes learned where to look for the symbols and finding them got easy by the end. Sometime before I had it all ‘written,’ I realized there are three different x^2’s which match up with the 3y^2, then checked for the xy terms to find that indeed the quadratic terms make up 3(x+y)^2. At this point I wrote ‘(x+y)^4+3(x+y)^2+2’ on mental paper, factored A^2+3A+2=(A+2)(A+1) on another ‘sheet’ of mental paper, and finally got ((x+y)^2+2)((x+y)^2+1). Wondered whether it simplified further and considered writing as (B+1/2)(B-/12) but realized that’s pointless.