"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.

working memory in children

[T]he findings from this study indicate that the three main components of the Baddeley and Hitch (1974) model of working memory are in place by 6 years of age. The capacity of each component increases linearly from age 4 to early adolescence.


The Structure of Working Memory From 4 to 15 Years of Age
Susan E. Gathercole
University of Durham Susan J. Pickering, Benjamin Ambridge,
and Hannah Wearing
University of Bristol
Developmental Psychology 2004, Vol. 40, No. 2, 177–190

Children have lower working memory than adults, and lower working memory has ramifications for language learning and some cases of problem solving.

Tiger Mom

Debbie's been telling me for ages how wonderful Battle Hymn of the Tiger Mom is, and I finally bought the book.

She's right. It's incredible.

Here is the first page:

This is a story about a mother, two daughters, and two dogs. It’s also about Mozart and Mendelssohn, the piano and the violin, and how we made it to Carnegie Hall.

This was supposed to be a story of how Chinese parents are better at raising kids than Western ones.

But instead, it’s about a bitter clash of cultures, a fleeting taste of glory, and how I was humbled by a thirteen-year-old.

A lot of parents whose kids are heading off to college will find the book moving, especially parents who've been part of kitchen table math. All the things you wanted to do, and tried to do, and failed to do because your kids had other ideas....your kids and your schools and your culture: all the things you didn't manage to do because no one thought it was a really good idea to spend 4 years of your child's life reteaching math (and spelling) at home so he could be on par with his peers in Europe and Asia.

Hard to sort it all out.

That's what the book is about, though for Amy Chua's kids the contested territory was music, not math.

Battle Hymn of the Tiger Mother is the memoir of a Chinese afterschooler.

Are Schools Preparing Students Well for the SAT?

The College Board's website says in many different places that the best way to prepare for the test is to do well in school:

Keep in mind that the foundation of a student's SAT and college preparation is a rigorous curriculum of English, mathematics, science, history, and other academic subjects. Students should read extensively and develop good writing skills.

I want that for my children.

My question is, are schools really teaching this rigorous academic curriculum that the college board says is the best prep for the test ?

I took my son's 10th grade PSAT the other day, and found myself aghast (again) at how darn hard this test is. I'm not opposed to rigor, by the way; just wondering if our schools are on the same page.

There were passages dealing with Descartes, dualism, genomes, and neuroscience. Students had to compare two passages that were extremely sophisticated, with both authors agreeing on the main point, but from different perspectives, and their distinctions were subtle.

Not easy......

Cross posted on Perfect Score Project

when smart is dumb

Directions:

Make each statement true by moving just one matchstick.

In Choke, Sian Bielock discusses a study in which more than 90% of adults with normal working memory correctly answered the first problem. Roughly the same number of people with damaged working memories also got it right.

Only 43% of normal adults got the answer to the second problem, while 82% of patients with damage to the prefrontal cortex figured it out.

I believe high-functioning people with autism (or a healthy loading of autism genes) will also have a high rate of success on problem number 2, but that's just me.

Better without (lateral) frontal cortex? Insight problems solved by frontal patients
Carlo Reverberi,1,2 Alessio Toraldo,3 Serena D’Agostini4 and Miran Skrap4
Brain (2005), 128, 2882–2890

working memory

Just came across this textbook chapter on working memory and thought I'd share it. Don't know who wrote it.

Bielock writes that "working-memory differences across people account for between 50 percent to 70 percent of individual differences in abstract reasoning ability or fluid intelligence."

Working memory also makes you dumber in some situations.

I'll get to that later.

adults and children - language learning

Age Effect Problems in F.L. Acquisition

• Language is better learned at an earlier age.

• Despite numerous methods of explicit language instruction, older children and adult learners do not reach a native level of language proficiency.

• Adults generally learn the word order and semantic aspects of language more quickly than children but usually never master the grammatical aspects.

The Application of the Less is More Hypothesis in Foreign Language Learning (Powerpoint)
(Simone L. Chin, Alan W. Kersten)
Cognitive Science Program
Kwon Ahram

article:

The Application of the Less is More Hypothesis in Foreign Language Learning
Simone L. Chin (schin2@fau.edu)
Florida Atlantic University, Psychology Department
777 Glades Road, Boca Raton, FL 33431
Alan W. Kersten (akersten@fau.edu)
Florida Atlantic University, Psychology Department
777 Glades Road, Boca Raton, FL 33431

second language learning - the "less is more" hypothesis

I'm reading Choke by Sian Beilock, a terrific book. Mostly it's about why and how people choke under pressure, but I've been surprised at the number of ancillary topics that turn out to be related to choking -- including foreign language learning by grown-ups.

Beilock says that the reason children learn language better than adults is that children have less working memory (pdf file). Less is more.

She has fascinating things to say about math and problem solving, too.

Statement of Research Interests (pdf file)
Alan W. Kersten
This research has been testing one hypothesis for why adults have so much difficulty successfully acquiring a second language, namely the “Less is More” hypothesis of Elissa Newport (1990). According to this hypothesis, the reduced working memory capacity of children relative to adults actually results in better language learningby forcing children to focus on small chunks of language. Adults, on the other hand, can remember larger chunks of language, allowing them to memorize useful expressions in a foreign language (e.g., “Where is the bathroom?”), but making it difficult for them to extract the lower-level meaning elements from which those expressions are constructed. Adults are thus limited to the set of phrases that they have acquired, and are unable to recombine the lower level elements from which those phrases are constructed to express novel meanings. If this hypothesis is correct, one may predict that adults will learn a language better if they are forced to focus on small chunks of language rather than being allowed to learn entire phrases. We have tested this prediction using a miniature artificial language learning paradigm (see Kersten & Earles, 2001). One group of adults was presented immediately with complete “sentences” from this language, whereas a second group was presented initially only with individual words from the language. This second group was subsequently presented with incrementally longer chunks of language until ultimately they were hearing the same sentences that the other group heard all along. The group that was initially forced to focus on small chunks of language showed better ultimate learning of the word meanings and morphology of that language, consistent with the “Less is More” hypothesis. We are currently investigating whether starting small benefits the acquisition of a natural language with more complex grammar, namely French (Chin & Kersten, in press).

update: Less Is Less in Language Acquisition

a company labels its products...

AA2

A company labels its product with a three-character code. Each code consists of two letters (not necessarily different) from the 26 letters of the English alphabet, followed by one digit, as shown above. What is the total number of such codes that are available for labeling the company's product?

I'll post the answer in the comments thread later.

My answer is wrong, and I don't understand why.

David Sedaris on language lessons

Ah, grammar. How dry. How boring. Right up there with those tedious times tables and those soul-sapping algorithms of arithmetic.

But then here's David Sedaris in the latest New Yorker reflecting on the Pimsleur language program:

Thanks to Japanese I and II, I’m able to buy train tickets, count to nine hundred and ninety-nine thousand, and say, whenever someone is giving me change, “Now you are giving me change.” I can manage in a restaurant, take a cab, and even make small talk with the driver. “Do you have children?” I ask. “Will you take a vacation this year?” “Where to?” When he turns it around, as Japanese cabdrivers are inclined to do, I tell him that I have three children, a big boy and two little girls. If Pimsleur included “I am a middle-aged homosexual and thus make do with a niece I never see and a very small godson,” I’d say that. In the meantime, I work with what I have.

The problem is that Pimsleur is all about mimicry:

Pimsleur's a big help when it comes to pronunciation. The actors are native speakers, and they don't slow down for your benefit. The drawbacks are that they never explain anything or teach you to think for yourself. Instead of being provided with building-blocks which would allow you to construct a sentence of your own, you're left using the hundreds or thousands of sentences you have memorized. That means waiting for a particular situation to arise in order to comment on it; either that or becoming one of those weird non-sequitur people, the kind who, when asked a question about paint color, answer, "There is a bank in front of the train station,"or, "Mrs. Yamada Ito has been playing tennis for fifteen years."

What are those "building-blocks which would allow you to construct a sentence of your own," and, equally importantly, the rules that tell you how to put those building blocks together? That dry, tedious, soul-sapping entity known (or sort of known) as Grammar.

However tedious and soul-sapping it is to do so, mastering a language's grammar rules is the only way to move beyond mimicry and use the language creatively: the only way to move from  "I have three children, a big boy and two little girls" to “I am a middle-aged homosexual and thus make do with a niece I never see and a very small godson.”

Pimsleur isn't alone in presuming that you can master a language without learning its grammar; the biggest seller of this fiction is Rosetta Stone (whose slogan, ironically, is "More than Words. Understanding.") Other grammar-denialists (as I discuss here) are k12 foreign language curriculum developers, as well as (as I discuss here) autism therapists and the general American public. It's a vicious cycle that worsens with each succeeding generation of mis-educated students, more of whom need to spend time attempting to converse with Japanese cabbies before foisting their language lessons on the rest of us. Thank you, David Sedaris, for yours!

(Cross-posted at Out in Left Field)

The Learning Part is Easy; It's the Remembering That's Hard

A commenter pointed me to this Wired article about Piotr Wozniak and SuperMemo. Has anyone tried SuperMemo?

These quotes from the article describe how I feel, to a tee.

Learning things is easy. But remembering them — this is where a certain hopelessness sets in.

Wozniak felt that his ability to rationally control his life was slipping away. "There were 80 phone calls per day to handle. There was no time for learning, no time for programming, no time for sleep...."

Our capacity to learn is amazingly large. But optimal learning demands a kind of rational control over ourselves that does not come easily. Even the basic demand for regularity can be daunting.

(cross posted on Perfect Score Project)

how the SAT changed in 2006

I'm tossing old papers, or trying to, and in my rummaging came across this Times article from 2005.

I was tickled to see that one of the problem types added to the revised test was the absolute value inequality word problem, a category I had never seen or imagined until I encountered one in Dr. Chung's SAT Math:

For pumpkin carving, Mr. Sephera will not use pumpkins that weigh less than 2 pounds or more than 10 pounds. If x represents the weight of a pumpkin, in pounds, he will not use, which of the following inequalities represents all possible values of x?
a. | x - 2 | > 10
b. | x - 4 | > 6
c. | x - 5 | > 5
d. | x - 6 | > 4
e. | x - 10 | > 4

Pop Quiz; New and (Maybe) Improved
Published: November 7, 2004

Talk about inflexible knowledge. Somehow I had concluded that absolute value inequality calculations were just that: calculations. Arithmetic. I was stunned to discover that you could have an absolute value inequality word problem.

Wonders never cease.

Elizabeth King's explanation of these problems is excellent.