kitchen table math, the sequel: 7/17/11 - 7/24/11

Friday, July 22, 2011

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.

Battle Hymn of the Tiger Mother

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

Thursday, July 21, 2011

when smart is dumb


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


Wednesday, July 20, 2011

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.

Choke: What the Secrets of the Brain Reveal About Getting It Right When You Have To

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


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

Choke: What the Secrets of the Brain Reveal About Getting It Right When You Have To

a company labels its products...


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.

Tuesday, July 19, 2011

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)

Monday, July 18, 2011

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.

 Dr. John Chung's SAT Math

Outsmarting the SAT

Sunday, July 17, 2011

Algebra Review: Laying the Groundwork for Pre-Calculus

I've been using Michael Sullivan's Algebra and Trigonometry textbook for the last few years to teach College Algebra/Pre-Calculus/Trigonometry on the quarter system, but we're switching to John Coburn's Algebra and Trigonometry next year. They're both pretty good textbooks.

In building Pre-Calculus curriculum I've drawn mainly from four textbooks:

Mary Dolciani - Modern Introductory Analysis (original copyright 1964, my edition is from 1986)

Richard Brown/David Robbins - Advanced Mathematics (this copyright is from 1984, a newer edition of this book is here)

Paul Foerster - Pre-Calculus with Trigonometry (copyright 1987)

Max Sobel/Norbert Lerner - Pre-Calculus Mathematics (copyright 1995)

I had thought about using the Sobel/Lerner book for the Pre-Calculus course I developed for Clatsop Community College, and even e-mailed Max Sobel asking him about a new edition of that text, but he very kindly replied that he had retired (I also have the Harper & Row Algebra I and II textbooks from his series with Evan Maletsky and like these as well).

One of the things I liked about the Sobel/Lerner textbook was the coverage of rational expressions that several of the other books I had considered didn't have. I like rational expressions because this topic requires a firm grasp of many of the most important concepts from algebra – factoring and multiplying of bi- and trinomials, simplifying complex expressions, and combining like terms in the context of manipulating algebraic fractions. If students understand numerical fractions it helps a lot.

When I teach College Algebra courses at Clatsop CC, I often begin with exercises like (x+7)(2x-3) – (x+1)(x+5) to address these topics and prepare the students for when they see this again in working with rational expressions. Another good example is (x+6)(3x+1) – (x+2)^2, to get them used to seeing the squared binomial. ( I make the squared binomial a regular visitor in most of my algebra classes). These expressions often appear as the numerator of a combined fraction in a problem like (x+7)/(x+1) – (x+5)/(2x-3).

Here is a link to a collection of problems I often assign for this topic.

I recently began to reacquaint myself with the College Board Math II Subject Test which I had taken after taking Pre-Calculus in the spring of 1982. The first question on the sample test I looked at was intriguing, and an understanding of rational expressions is really useful in finding a quick solution:

If 3x+6=(k/4)(x+2), then k=

a) ¼

b) 3

c) 4

d) 12

e) 24

In this problem, dividing through by (x+2) so that 3=k/4 (and 12=k) gives the almost instantaneous answer we need for a timed test. This is an interesting problem because it really gets at the concepts involved in working with factors in an equation.

In the Sobel/Lerner textbook, Sections 1.7, 1.8 and 1.9 cover the algebra review (multiplying polynomials, combining like terms, factoring polynomials and rational expressions) necessary to move on. This is where it is important to illustrate these topics with problems, because when I say “combining like terms,” I don't mean 2x+5x. While this type of problem could be appropriate when first teaching the concept, in the context of review for Pre-Calculus, a problem from the Sobel/Lerner text like (x^3-2x+1)(2x)+(x^2-2)(3x^2-2) is better practice for using these skills together.

This is something that I consider extremely important and that textbooks and assessments often don't include enough of – using the skills together. Learning skills in isolation is useful to grasp each skill individually, but to really DO MATH, a student must be able to make decisions about what to do and when. Something that I like about the Pre-Calculus curriculum is that it lends itself well to the type of problem in which the tools of algebra must be applied in a variety of situations.

The Brown/Robbins text covers complex numbers and the quadratic formula in Chapter 1 (1-4, 1-5) and then the solution of equations involving rational expressions in Section 2-2. Chapter 2 goes on to examine the graphing of quadratic and polynomial curves and finishes with material on finding rational roots.

The Sullivan book covers polynomials and algebra in sections R.4, R.5 and R.7, Coburn covers this in sections R.3, R.4 and R.5.

The Foerster and Dolciani texts don't really cover much algebra review at all, but, as a result, they explore a number of topics the other books don't. I'll probably follow a path similar to the Brown/Robbins book and talk about rational roots next.