"chunk decomposition" - the matchstick problems again

Belatedly, I'm posting this abstract from the matchstick math study. It relates to the Why is SAT geometry hard? post I began writing the other day; it also relates to the two different meanings of the word "chunking" that cropped up in the matchstick comments.

When we're talking about memory, "chunking" means chunking several bits of information together into one larger chunk, allowing working memory to hold more than the 3 or 4 separate items it is capable of holding at one time. So, for instance, instead of remembering 2 - 0 - 3 as three separate numbers, you come to remember 203 as just one chunk.

When we're talking about perception, "chunking" means something closer to an automatic and entirely unconscious perceptual bias towards seeing -- visually seeing -- 'wholes' or 'chunks' instead of the parts that make up the chunk. "Visual chunking" happens instantly and naturally, whereas memory chunking requires practice over time. Crucially, visual chunking is extremely difficult to resist or to undo.

I've mentioned in a couple of comments threads, I think, that I believe autistic people (and children and animals) much more readily perceive parts instead of wholes -- something Temple Grandin absolutely believes. Temple told me once that the hidden figures in hidden figures puzzles always 'pop' at her, and I believe it. After 9/11 she and I used to talk about using high-functioning autistic people to man the carry-on scanners at the airport. 

Here's the abstract:

Constraint relaxation and chunk decomposition in insight problem solving.
By Knoblich, Günther; Ohlsson, Stellan; Haider, Hilde; Rhenius, Detlef
Journal of Experimental Psychology: Learning, Memory, and Cognition, Vol 25(6), Nov 1999, 1534-1555.
Abstract
Insight problem solving is characterized by impasses, states of mind in which the thinker does not know what to do next. The authors hypothesized that impasses are broken by changing the problem representation, and 2 hypothetical mechanisms for representational change are described: the relaxation of constraints on the solution and the decomposition of perceptual chunks. These 2 mechanisms generate specific predictions about the relative difficulty of individual problems and about differential transfer effects. The predictions were tested in 4 experiments using matchstick arithmetic problems. The results were consistent with the predictions. Representational change is a more powerful explanation for insight than alternative hypotheses, if the hypothesized change processes are specified in detail. Overcoming impasses in insight is a special case of the general need to override the imperatives of past experience in the face of novel conditions.

This study of Perceptual contributions to problem solving: Chunk decomposition of Chinese characters looks interesting.

culture clash

passage from Battle Hymn of the Tiger Mother:

In one study of 50 Western American mothers and 48 Chinese immigrant mothers, almost 70% of the Western mothers said either that "stressing academic success is not good for children" or that "parents need to foster the idea that learning is fun." By contrast,roughly 0% of the Chinese mothers felt the same way. Instead, the vast majority of the Chinese mothers said that they believe their children can be "the best" students, that "academic achievement reflects successful parenting," and that if children did not excel at school then there was "a problem" and parents "were not doing their job." Other studies indicate that compared to Western parents, Chinese parents spend approximately ten times as long every day drilling academic activities with their children. By contrast, Western kids are more likely to participate in sports teams.
p. 14

change

Superintendent's Resignation Linked to Performance Review

Agreement and Mutual Release

irreconcilable differences

Algebra Review for Pre-Calculus (follow-up)

I recently wrote about Algebra Review for Pre-Calculus and discussed some problems that I typically assign in the College Algebra courses I teach. Here is a link to some examples of those problems including algebraic simplification, simplifying rational expressions and solving equations involving rational expressions.

I'll write next about the Rational Roots Theorem and polynomial factorization. Here's a problem I found from a Japanese University Entrance Exam from 1990 that makes interesting use of these concepts. The document I took this from is here - the problem is from the first sample test on page 4.

Suppose the polynomial P(x) with integer coefficients satisfies the following conditions:

(A) If P(x) is divided by x^2-4x+3, the remainder is 65x-68

(B) If P(x) is divided by x^2+6x-7, the remainder is -5x+a

Then we know that a={?}.

Let us find the remainder bx+c when P(x) is divided by x^2+4x-21.
Condition (A) implies that {?}b+c={?}.
Condition (B) implies that {?}b+c={?}.
It follows that b={?} and c={?}

I've changed the notation of the answers a little in the hope of making it less confusing - you can see the original by clicking through the link.

I got pretty tangled up in solving this problem because I had never seen the Remainder Theorem used in quite this way before. The answers for this are on page 5 of the original.

housekeeping

Turns out I have a Spam Box at Blogger, which contains 170 comments Blogger thought might be spam.

Am going through them now.

AP calculus scores & stereotype threat

As I understand it, stereotype threat is essentially a self-fulfilling prophecy that takes place below the level of consciousness -- although when I experienced stereotype threat while competing on a television game show my thoughts were conscious, intense, and pretty close to crippling.

I had won a spot on Sale of the Century, and when the big moment arrived I was to play against two men: one white and one black.

I was gripped by stage fright. Sitting in the studio with the group of aspiring contestants who had made the first cut, waiting to see whether I would be called, I felt so terrified I wanted to bolt from the room.

The only thing stopping me bolting from the room was the stage fright I felt at the prospect of people watching me bolt from the room. Which is worse? Having a roomful of strangers watch you play a game show? Or having a roomful of strangers watch you run screaming from the room instead of playing a game show?

I chose door number one and stayed put in my chair; and when I was called to play, two men were called to play, too. I was led to the seat between them.

Suddenly, entirely unbidden, my Betrayer Self began to think: "I can't win against men. I can't win against men. I can't win against men." Over and over again. "I can't win against men."

I was aghast.

I had an Ivy League degree and a Ph.D.; I was a committed feminist; I thought the idea that men were my betters was hooey. Plus I knew boatloads of random factoids and trivia, and Sale of the Century was a random factoids and trivia contest. Yet there I was thinking -- thinking very loudly -- "I can't win against men." Until that moment, I had had no idea I felt that way.

So when it comes to stereotype threat, I'm a believer.

I won that game, but the reason I won was that the black contestant, who was a better player than I, appeared to be suffering an even worse case of stereotype threat than the one threatening to derail me. I won't tell that part of the story here. Suffice it to say that he made an on-camera allusion to what he was feeling just before choosing the wrong square on the board and giving the wrong answer: and losing the game. It was a painful moment. More than painful; it was excruciating. I recall a murmur of what sounded like distress running through the audience.

I managed to defeat the white male contestant mostly because a) he appeared to be just as panicked as I was and b) I knew a lot more random factoids and trivia than he did. And heaven only knows what kind of stereotype threat might have been hammering his brain. Stereotype threat isn't just for for blacks and women; it's for everyone. In fact, you can lower the math performance of white male students attending Stanford University if you remind them, before they take the test, that they aren't Asian.

After I won the game, I calmed down and won two more games, then retired with cash instead of gambling my certain winnings to play on in hopes of making it to much bigger winnings at the top: a classic example of Kahneman and Tversky's concept of loss aversion.

Back to stereotype threat. Sian Beilock discusses stereotype threat at length in Choke. Turns out there is a (disputed) study of stereotype threat and AP calculus scores finding that when test-takers fill out the personal info form, which includes gender, after they've finished the test instead of before, girls score better. (The dispute has to do with statistic methodologies and significance.)

From the 'pro' authors:

Pragmatically speaking, the “trivial” differences carefully dismissed in Stricker and Ward (2004) can translate into very large practical effects, with real theoretical meaning. The inquiry manipulation reduces the gender difference to less than one third its original size. Instead of a ratio of about 6 girls receiving AP credit for every 9 boys who obtain credit, the new manipulation generates a ratio of about 8 girls receiving AP credit for every 9 boys.

How would this manipulation affect females at the population level of all students taking AP Calculus AB? Stricker and Ward (2004) told us that 52,465 boys and 47,275 girls took the test in 1995 (p. 669). ... [C]hanging the way the tests are administered would increase the number of girls receiving AP Calculus credit from 15,081 to 17,870 in a year—an increase of 2,789 young women starting college each year with Calculus credit.

This size number should not be below the radar. The number of people taking the AP Calculus AB test is increasing. In 2004, there were 88,809 boys and 81,521 girls who took the exam (College Board, 2004), which represents an increase of 70.8% since Stricker and Ward collected data in 1995. All other things being equal, we estimate that 4,763 more women would receive AP Calculus AB credit if the timing were changed. We are convinced that stereotype threat in real-world testing situations can have a significant effect on test takers, and Stricker and Ward’s (2004) data support this conclusion.

Stereotype Threat, Inquiring About Test Takers' Ethnicity and Gender, and Standardized Test Performance
Lawrence J. Stricker, William C. Ward
Journal of Applied Social Psychology
Volume 34, Issue 4, pages 665–693, April 2004


Stereotype Threat in Applied Settings Re-Examined
Kelly Danaher and Christian S. Crandall
Journal of Applied Social Psychology
Volume 38, Issue 6, pages 1639–1655, June 2008
[figures drawn from Danaher and Crandall]


Stereotype Threat in Applied Settings Re-Examined: A Reply†
Lawrence J. Stricker, William C. Ward
Journal of Applied Social Psychology
Volume 38, Issue 6, pages 1656–1663, June 2008

SAT equivalent scores

Convert individual and mean scores from the original scale to the recentered scale

In April 1995, the College Board recentered the score scales for all tests in the SAT Program to reflect the contemporary test-taking population. Recentering reestablished the average score for a study group of 1990 seniors at about 500—the midpoint of the 200-to-800 scale—allowing students, schools, and colleges to more easily interpret their scores in relation to those of a similar group of college-bound seniors.

A 700 on critical reading today was a 640 prior to 1995.

A 700 on math today was a 710 prior to 1995, and the math test today covers more material.

test prep

Debbie sent me a link to a story that contains this fabulous passage:

José Solís, 36, shut down his application process for a master’s in business and environmental science because of noncompetitive GMAT results. After taking the test in 2009, he knew grad schools were “going to blow right past me in the application process.” While working for an architecture firm in Houston, he started taking algebra, statistics and accounting classes at a community college, which was less expensive than commercial prep courses. “I really learned the fundamentals and not just tricks,” he says. “Taking all these courses kind of got me back in the mode of studying again.”

Almost a year later, he retook the GMAT. “When I saw my score, which you can see instantly, I almost fell out of my chair. It was exactly the average of the schools I was applying to.” As an afterthought, he took the G.R.E. and scored just short of a perfect 800 on the math. He has accepted a fellowship at the University of Michigan.
A Little Rusty?
By KATHLEEN McELROY
Published: July 22, 2011

sleep

What have I learned taking timed SAT math tests?

Apart from how to answer tricky SAT math questions fast?

I have learned once and for all that sleep loss is catastrophic.

I had always been vaguely aware that I "don't function well" on too-little sleep, but it wasn't until I started taking -- and more importantly scoring -- SAT math sections that I realized exactly how not well.

From Permission to Sleep In by Christopher Shea | July 7, 2011, 12:02 PM ET:

Researchers [at Stanford University] measured shooting percentages, sprint and reaction times, and subjective mood for 11 members of the men’s basketball team, for two-to-four weeks during the 2005 and 2008 seasons, as the players followed their ordinary sleep schedules. (Average sleep, according to a motion-detection bracelet: 6.7 hours.)

Then for five to seven weeks the players boosted their sleep time to 10 hours. They were encouraged to take daytime naps when they couldn’t meet that goal.

With the additional pillow time, players dropped their average time on a routine sprinting drill (from the baseline to half court and back, then to the opposite baseline and back) to 15.5 seconds, from 16.2 seconds. Performance on free throws improved to 88%, from 79%. Three-point shots made jumped to 77% from 68%.

Basic reactions, gauged by having players push a button after spotting a stimulus on a screen, improved by 12%, and players were happier.
Source: “The Effects of Sleep Extension on the Athletic Performance of Collegiate Basketball Players,” Cheri D. Mah, Kenneth E. Mah, Eric J. Kezirian and William C. Dement, Sleep (July)

These are young, healthy varsity athletes.

When I take an SAT math section on a good night's sleep (or maybe a good week of good nights' sleep), I miss 0 to 1 questions. Two questions at most.

The other day I took a PSAT math section in a hot room after several nights of going to bed late and getting up early, and I missed 5 questions out of 18, all of them problems I normally get right.

A five-out-of-fifteen miss rate at this stage of the game is, for me, a collapse. Projecting the same percent correct across the 3 math sections of an SAT I, that's a drop from a score in the 700 to 800 range to a score at the bottom of the 600s.

One hundred points. At least.

I'm asking myself what this translates to in terms of lost writing productivity over the years.