I have another experiment that may not add all that much to the previous one, but I’m posting it anyway because I don’t want to waste the (admittedly not huge) effort it has just taken me to follow Jason Dyer’s suggestion and create a presentation using Prezi. If you follow this link, you will be taken to it. If you hover near the bottom of the box, a left arrow and a right arrow will appear, which will allow you to move right or left in an expression that, again, needs to be simplified. I’ve made it a bit easier than the last one, and you also get to see slightly more than one character at a time. There is also a button that allows you to shrink the entire expression so that it fits into the box: obviously if you use that then it counts as giving up on the experiment, but it may be interesting to do the simplification in your head according to the strict rules first and then see what happens to the information in your head when you then click the pan-out button.

Incidentally, I’m not sure that Prezi supports mathematical symbols, so I’ll save you some potential irritation by pointing out in advance that every x is an x rather than a “times”. (There’s one that looks a bit like a “times” because of the unusual spacing.)

I haven’t bothered with a poll this time. If anyone has anything interesting to report, then by all means let me know. Otherwise, think of it as a weird form of entertainment.

If you haven’t done the previous experiment (or even if you have but just want to see the same idea in a different format), you might like to look at the following two pdf files, kindly sent to me by Olivier Gerard, both of which present one chunk per page. In the first file the definition of “chunk” is the same as it was for me, so would have e on one page, ^ on the next and x on the next. In the second, exponents are attached to the previous symbol, so would be on a single page, and would be on five pages of which the fifth would be .

Fully separated version

Version with exponents not separated

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May 8, 2010 at 2:02 pm |

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.

May 8, 2010 at 2:54 pm |

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.

May 8, 2010 at 5:03 pm |

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

May 8, 2010 at 5:14 pm

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

May 8, 2010 at 6:01 pm |

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).).

May 8, 2010 at 6:26 pm |

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?

May 11, 2010 at 6:27 am |

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.

May 11, 2010 at 6:57 am |

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).

May 13, 2010 at 4:17 am |

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.

June 3, 2010 at 4:32 pm |

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.

June 3, 2010 at 4:37 pm |

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

June 9, 2010 at 11:05 pm |

This was much, much easier than the rest.

June 9, 2010 at 11:52 pm |

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.

August 14, 2010 at 3:00 am |

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.