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.

]]>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.

]]>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.

]]>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.)

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.

]]>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.

]]>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). ]]>

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.)

]]>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.

]]>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.

]]>Oh, sorry, yes. I understand now. Thanks.

]]>That looks as it is supposed to look. Almost all of it is indeed random gobbledygook. For an explanation of how to find what isn’t gobbledygook, see the second paragraph of the post.

]]>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

]]>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.

]]>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.

]]>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.

]]>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.

]]>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.

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