Comments on: A little experiment IV

By: Eric

Eric — Sat, 14 Aug 2010 02:00:17 +0000

Well I thought of it in the following manner:
Ok, we have x^4. Now there’s a 2. Hmmm here is a larger expression, lets expand so I have 4x^3, ** then right here there was a very sudden switch, maybe because there were only x-terms to think of, so I started thinking of numbers instead of all those confusions x’s.

1 4 0 0 2, then the next terms make this work, I have 1 4 4 4 2, now 1 4 6 4 2, and finally 1 4 6 4 1 so the answer is clear.

Don’t now why I thought of it this way, but made it really fast and no backtracking required, whole thing took 10 seconds or so.

By: palibacsi

palibacsi — Wed, 09 Jun 2010 22:52:05 +0000

At the beginning, I found it even harder than the third experiment. I had to restart twice after half the expression. I then noticed that the expression contains only powers of x. I tried to memorize them as (a, b, c,…) again, where (a, b, c,…) means a+bx+cx^2…
I mistook the -1 for +1 at the end and went through the expression again as I could not immediately simplify it. After getting it right, I almost immediately factorized to (x+1)^4.

By: esmeyny

esmeyny — Wed, 09 Jun 2010 22:05:32 +0000

This was much, much easier than the rest.

By: rgrig

rgrig — Thu, 03 Jun 2010 15:37:18 +0000

Now that I read what I just wrote, step 2 is obviously wrong, if only because the square magically disappeared.

By: rgrig

rgrig — Thu, 03 Jun 2010 15:32:57 +0000

1. I skipped the parenthesis and saw that what’s outside is (x^2+1)^2.
2. I looked in parenthesis and noticed (x^2+1) again so I split it out giving (4x+1)(x^2+1)+4x^2.
3. Again I wasn’t sure at all that I didn’t make mistakes and I decided to give up, especially because I was a bit frustrated/annoyed by episode III that made my eyes hurt.

The Prezi version feels much easier than the garbage version, and I suspect I would have persevered if wouldn’t be already irritated by my failure in episode III.

By: Jason Dyer

Jason Dyer — Thu, 13 May 2010 03:17:20 +0000

I finally had time for an honest try. A couple observations:

1. Despite your warning, I had a very strong mental tendency to want to read that ‘x’ next to the parenthesis as multiplying.

2. I found myself mentally “slotting” the various exponents (so the x^4 term was “farther to the left”, the x^2 was “in the middle”, etc.)

3. I was unable to get anywhere past having a polynomial until I wrote it out and recognized the binomial theorem expansion. Again, any sort of “pattern matching” shut down when I was struggling to merely store the parts in my head.

By: Vania

Vania — Tue, 11 May 2010 05:57:31 +0000

I agree with Erik on the short memory effects, though from my experience (being hard of hearing) I can tell you that my “aural” short memory is much worse off than my visual one (I can hardly ever benefit from someone spelling a phone number or a word to me really fast, while I can take in written symbols with so much more effectiveness). Maybe this suggestion about audio input only highlights how much our short term abilities matter in our approach to math symbols (just as if it were an actual language).

By: Erik Nelson

Erik Nelson — Tue, 11 May 2010 05:27:20 +0000

I read through the first version in little experiment III but didn’t try to simplify it. Scrolling down the pdf file I was surprised to find that I could now guess the next symbol on each page and I got a little ways toward simplifying it. I think if i scrolled through the pdf again I could probably close my eyes and see the formula.

Perhaps the short term memory effects have something to do with having to re-assemble, or simplify the original image. I wonder if the experiment would work out differently if you used a voice recording of the formula being read-off like on wiktionary entries.

By: Pete

Pete — Sat, 08 May 2010 17:26:31 +0000

This felt much much easier than the previous version: I guessed that there would be nothing like (x+y)(x-1), expanded everything out as I went along, and recognised that this was clearly going to be (x+1)^4 before seeing the final -1. I think I’d have had a lot more trouble if there had been some complicated brackets as above: I don’t think I can at all easily deal with the need for separate ‘stores’. Of course, normally I’d write down such an intermediate calculation rather than memorise it.

I wonder how much of this is really to do with it being hard to do things like store multiple complicated expressions mentally and how much is to do with never having to practice it. Perhaps it would be interesting to ask someone who for whatever reason cannot read / write (e.g. someone blind or with muscle problems) and who therefore has had to practice?

By: Micha

Micha — Sat, 08 May 2010 17:01:15 +0000

Well, as the aim was to simplify, I directly took the party, seeing x^4 and 4x^3, that we were trying to get (x+1)^4. The rest of the parsing was more about checking than guessing, then (actually, I was doing something like “What is (x+1)^4 minus the expression I’m reading. I wonder what I would have done if the “4x” were *after* the (x^2+1+x).).

By: gowers

gowers — Sat, 08 May 2010 16:14:20 +0000

Ah, I’m glad you wrote that. There was a typo in what I wrote, but it should be OK now.

By: Kristal Cantwell

Kristal Cantwell — Sat, 08 May 2010 16:03:49 +0000

I kept going back and forth over it I found I could simplify using the expression (x+1)^4
I ended up with
(x+1)^4-2x^2+2x
and I factored the final term
(x+1)^4-2(x-1)x

By: Vania

Vania — Sat, 08 May 2010 13:54:06 +0000

This one seems to be more of a psychological experiment on short term memory than math, though pattern recognition should be essential still. For people with little short term memory it can be a hassle to keep in mind an expression you get to see symbol by symbol (which is not realistic in math practice, of course).

In my case I started suspecting that it might be all about the binomial expansion of (x+1)^4 when I got to 4x(x^2, and then the suspicion was confirmed when finally the -1 came at last. So again, without a good guess and a terribly strong short-term memory I would consider this test quite a bit harder than the equation-based we’ve seen, and I might have needed to keep track of the expression on paper as it unfolded.

This suggests that our impact of formulas on us, despite our famous care for individual symbols and details, inherits a lot from our well-known custom of taking in words as a whole, so that for example even a text with words neatly separated but with the letters within the words well-scrambled would still be intelligible to us.

By: ogerard

ogerard — Sat, 08 May 2010 13:02:05 +0000

The animation is smooth. There are sometimes glitches with a parenthesis dangling before a “+” sign. It feels a like a simulation of what it would be to do algebra for a visually deficient student, for instance ambliopic, using a magnifying glass installation to read a textbook.

The two first experiments where much more a test on the interaction between the algebra training and the personal style of a selected public (the readers of this blog), always dealing on known territory, all methods allowed. They are more a creativity and variety contest than a human species cognitive test.

The third experiment is more radical as it tries to understand if we can become extempore lexical and syntactic mathematical parsers and expression reducer with one byte buffers.

Concerning these two (III and IV) experiments in general, if mathematicians had adopted in daily use a general style of written notation for expressions close to the Reverse Polish Notation, it would have changed our ability and allowed more of us to be confortable with such a practice of expression manipulation.

For the expression used in experiment IV which is

x^4 + 2 + 4*x * (x^2 + 1 + x) + 2*x – 1

RPN would give something like

x , 4, ^, 2, +, 4, x, *, x, 2, ^, 1, +, x, +, *, 2, x, *, +, 1, – ;

with all operators of arity 2.