Is math by hand better than math by keyboard?

Sorting books this afternoon, I came across Tahir Yaqoob's What Can I Do to Help My Child with Math When I Don't Know Any Myself? and found this passage:

The actual process of using your muscles to write something is a powerful long-term memory aid. The more that you write out things (and in different ways), the more your long-term memory will be etched out. It is not good enough simply to read and think (although this is important for reviewing large amounts of material shortly before taking an exam, but only if you have done the long-term ground work). Writing out full solutions to problems in math is especially important compared to other subjects, whether it is part of reviewing for exams or whether you are learning new material.

Writing things out can also help you to understand difficult problems. For example, if you see a fully worked solution to a problem in a textbook, but don't understand one or more of the steps, try simply writing out the solution yourself. You may be surprised that while you are doing that, you suddenly understand something that you didn't before. Sometimes the brain has a strange way of working. Despite its enormous capacity , the. brain can really benefit from an external "scratch pad." When you come across something that you don't understand, sometimes just writing out the steps in a brief form can make a great deal of difference.

What Can I Do to Help My Child with Math When I Don't Know Any Myself? Paperback – February 7, 2011 by Tahir Yaqoob - p133

I've always found this to be true, both for C. and for me. I don't know why. One of these days I'll get around to reading The Hand: How Its Use Shapes the Brain, Language, and Human Culture, which I hope will explain the phenomenon.

The OECD report on students and technology (Students, Computers and Learning: Making the Connection) found that using the computer for drill was associated with reduced achievement:

The decline in performance associated with greater frequency of certain activities, such as chatting on line at school and practicing and drilling, is particularly large (Figure 6.6). Students who frequently engage in these activities may be missing out on other more effective learning activities. Students who never or only very rarely engage in these activities have the highest performance.

Given my experience, the "other more effective learning activities" these students are missing may be drilling by hand.

Apparently students who attend Ivy League schools are good memorizers

Yale students seem to have spent so much time memorizing history facts and dates that Professor Joanne Freeman feels compelled to warn students against the Revolutionary War "fact bubble":

Tip number one is don't get lost in the dreaded Revolutionary War fact bubble, which I have to say it makes me think of the first time that I taught this course. I was actually a brand new professor and I had just come to Yale and it was my first course and it was my first lecture in my first course and I'm [sound cuts out] It actually was in Connecticut Hall, which, for those of you who don't know, dates back to the period when this course is talking about and was Nathan Hale's -- essentially his dorm. So there I am. I'm a brand new professor to Yale and I'm teaching a course about the Revolution and it's in a building that dates to the Revolution, so I'm having sort of a "wow" Yale moment as it is, and I'm off, I'm giving my lectures, and I'm really excited. I give about three of them and someone raises their hand after about three lectures and they have a kind of a puzzled expression on their face. I said, "Yes?" And he says, "Excuse me, Professor Freeman. What are we supposed to be memorizing? Where are the facts and dates?" [laughs] So as a new professor my first impulse was: Darn! I forgot the facts and dates. [laughter] I got it wrong. [laughs] But actually, the fact of the matter is, they're not the star of the show. Certainly, dates are not the star of the show. There are dates you're going to have to remember so don't think Easy Street; there's not a date I have to know. There will be some dates, but this isn't a story about dates. It's obviously something a lot more interesting and a lot broader than that. Okay. Avoid fact bubble.

Joanne Freeman "American Revolution" | Lecture 1 Introduction: Freeman's Top Five Tips for Studying the Revolution

Obviously, excessive memorization didn't keep these students out of Yale.

I'm sure there's a reason for that, the reason being that excessive memorization actually helped.

And here is Daniel Willingham:  prior knowledge & working memory in 1 paragraph

Prior knowledge gets around working memory limits

[I]t’s well known that extensive background knowledge allows one to circumvent the limitation of working memory. To take an obvious example, if I ask you to hold six letters in mind for one minute, it will be much easier to do with B-R-A-K-E-S than with X-P-W-M-Q-R. Although both are a string of six letters, the first forms a word, so you can treat it like a single unit. It’s like holding one thing in working memory, not six. Naturally, this saving of space in working memory only works if you know the word “brakes.” The same phenomenon is observed in many other domains. The chess expert looking at a board does not see 16 white pieces—she sees several clusters of pieces, each cluster defined by the relationship of the pieces to one another and to opposing pieces. Whether it’s chess pieces or letters in a word, the compacting of many things into one thing in working memory is based on prior knowledge.
Have Technology and Multitasking Rewired How Students Learn? by Daniel T. Willingham

Why students need to memorize, Common Core edition

"...anything that occupies your working memory reduces your ability to think."
- Daniel Kahneman | Thinking Fast and Slow

The only way to clear space in working memory is to store knowledge in long-term memory.

AND SEE:
Why students have to memorize things
#whystudentsneedtomemorize

automated essay grading

In the Times today:

Essay-Grading Software Offers Professors a Break By JOHN MARKOFF
Published: April 4, 2013 | New York Times

I'm actually in favor of essay grading software, in theory. I've been interested in automated essay scoring ever since reading Richard Hudson's paper Measuring Maturity in Writing (which I need to re-read, so nothing more on that at the moment):

Abstract
The chapter reviews the anglophone research literature on the 'formal' differences (identifiable in terms of grammatical or lexical patterns) between relatively mature and relatively immature writing (where maturity can be defined in terms of independent characteristics including the writer's age and examiners' gradings of quality). The measures involve aspects of vocabulary as well as both broad and detailed patterns of syntax. In vocabulary, maturity correlates not only with familiar measures of lexical diversity, sophistication and density, but also with 'nouniness' (not to be confused with 'nominality'), the proportion of word tokens that are nouns. In syntax, it correlates not only with broad measures such as T-unit length and subordination (versus coordination), but also with the use of more specific patterns such as apposition. At present these measures are empirically grounded but have no satisfactory theoretical explanation, but we can be sure that the eventual explanation will involve mental growth in at least two areas: working memory capacity and knowledge of language.

Maturity of writing, in this sense, can be measured by software, and I would be using automated scoring software myself if I could buy essay-scoring software on Amazon. EdX says it's giving software away free to 'institutions' (does that leave out individuals?) so I'll have to see if my department might throw its hat in the ring.

That said, a lot of this is nonsense:

Anant Agarwal, an electrical engineer who is president of EdX, predicted that the instant-grading software would be a useful pedagogical tool, enabling students to take tests and write essays over and over and improve the quality of their answers. He said the technology would offer distinct advantages over the traditional classroom system, where students often wait days or weeks for grades.

[snip]

“It allows students to get immediate feedback on their work, so that learning turns into a game, with students naturally gravitating toward resubmitting the work until they get it right,” said Daphne Koller, a computer scientist and a founder of Coursera.

[snip]

“One of our focuses is to help kids learn how to think critically,” said Victor Vuchic, a program officer at the Hewlett Foundation. “It’s probably impossible to do that with multiple-choice tests. The challenge is that this requires human graders, and so they cost a lot more and they take a lot more time.”

None of these things is going to happen. Students aren't going to write essay responses "over and over again;" if they do write essay responses over and over again it's not going to feel like a fun game; and nobody's going to learn to think critically from automated essay scoring software.

Oy.

We need a Writing Renaissance, not a "Writing Renaissance"

Here is is! (Cross-posted from Out in Left Field.)

According to an article in this past week's Edweek, K12 writing instruction is undergoing a renaissance:

Teachers are focusing on writing instruction like never before. More and more, they're asking students to write about what they read, helping them think through and craft their work, and using such exercises as tools not only to build better writers, but to help students understand what they're studying.

This renaissance, the article claims, includes a shift towards explicit instruction:

The shift is still nascent, but people in the field are taking notice. It marks a departure from recent practice, which often includes little or no explicit writing instruction and only a modest amount of writing, typically in the form of stories, short summaries, or personal reflections, rather than essays or research projects on topics being studied.

In fact there appears to be little or no increase in explicit instruction, but simply a shift in quantity and genres. For example, rather than taking inspiration from Dr. Seuss to write their own whimsical stories:

First graders in South Strafford, Vt., are reading Dr. Seuss' The Lorax, for fun, then for greater understanding, and then to hunt for evidence. They look for events in the plot that illustrate how the whimsical protagonist tries to protect the Earth and assemble examples into a simple paragraph to support the theme of the story.

In keeping with the new Common Core standards for English and Language Arts, the article notes, "these kinds of projects are unusual for the way they connect writing and reading." But the ultimate goal seems not to be to improve writing, but reading:

"Now we're seeing a lot more attention to the idea that writing about a text can improve reading about that text," said literacy expert Timothy Shanahan, the chairman of the department of curriculum and instruction at the University of Illinois at Chicago.

The article cites a 2010 study:

a meta-analysis of 93 studies of writing interventions, which found that writing had consistently positive effects on students' reading skills and comprehension. Writing about what they read was particularly helpful to students' comprehension, but so were taking notes on what they read, answering questions about it, and simply writing more often.

In other words, despite the fact that ever since No Child Left Behind, as the article itself notes, concerns about reading comprehension have eclipsed concerns about writing, the ultimate goal of the "Writing Renaissance" continues to be reading.

Worse, this emphasis on writing for comprehension has become yet another excuse to water down math class:

A math teacher in Brighton, Mich., found that writing had a powerful effect on helping her 6th grade students understand algebra concepts. Julie Mallia and a colleague from the English department, Don Pawloski, teamed up in spring 2009 to have students write 10-page "how to" books for the next fall's 6th graders. Drawing both on math and on writing instruction, students had to explain concepts such as solving a problem with x.

What goals the article does mention that pertain specifically to writing are about quantity and argumentation rather than technique. Students should be writing more, and they should be writing pieces that shift from:

"opinion untethered to evidence" and "decontextualized" writing—writing not based on the reading of a text—in favor of writing that requires students to read, comprehend, and respond to text, grounding their interpretations in evidence found there.

As for writing strategies, we find nothing here about corrections and revisions. Instead of rewriting, there's rereading:

They read a text again and again, first to make sense of it and note their questions, as the teacher works the room to help,... A second round of annotating focuses on looking for elements of the genre and how it works. They read again to spot structural decisions the writer made to create meaning, she said. The students then use what they learned in their own writing.

There's something to be said for "spotting structural decisions" and trying to emulate these. Indeed, the one instance of direct writing instruction the article cite pertains to organization:

When Ms. Leddy teaches The Lorax, she walks through the text repeatedly with students, discussing it from a different angle each time. When they're through, students learn to write short "hand paragraphs," with the thumb as the topic sentence—the Lorax cares for the Earth—followed by three examples of how he does that and a "pinky sentence" restating the interpretation.

But none of this addresses a much more fundamental problem that affects all types of writing--no matter whether it's fiction, nonfiction, personal writing, summaries, or more involved, reading-connected writing assignments. This problem, which has become the talk of professors at campuses all around the country, is the problem that growing numbers of students have with the basic building blocks of all writing: phrases and sentences.

In none of the many Edweek articles on English and Language Arts do we find any mention of the steep decline in students' ability to write well-formed sentences. But this is arguably the greatest problem with today's writing, which, even at the college and graduate levels, is riddled with comma splices, dangling modifiers, subject-verb agreement problems, and the kind of garbling that results from a dearth of direct instruction and feedback from teachers and a failure by students to review and revise. Here are just a few examples from my growing collection:

1. Comma between subject and verb: Children who experience the world in a more rigid and narrow manner, will have difficulty with social inferences.

2. Comma splice: Generalization is a tough skill for ASD students to learn, teachers are sometimes baffled that they act a certain way in one subject and completely different in other.

3. Failed subject verb agreement: Two of the defining characteristics of autism includes impairments in social interactions and communication.

4. Failed preposition agreement: There are three types of aphasia to which a child can be diagnosed.

5. Dangling modifiers: In thinking about students transitioning from high school to college, the issues of accepting the disability and self-advocacy are crucial.

6. Wordiness: By providing direct instruction, this assists the students with improving their ability to give examples.

7. Awkwardness (and wordiness): Because of these results, it suggests that “object and subject relative sentences” need different amounts of working memory to be understood by the reader.

8. Displaced modifier: I first inquired about this young man’s high school experience, who I will call RC.

9. Incoherence (and wordiness): Due to the fact that these children with autism are unable to proper engage in social situations eliminates the knowledge base that they would normally acquire.

None of the above-described elements of the so-called "Writing Renaissance" will solve these problems. For this, we need direct, sentence-focused writing instruction. Yet, for all the empirical support there is for this kind of instruction, the tide shifted away from it long ago, and it will take a true Writing Renaissance to bring it back.

why there are no child prodigies in literature, history, or philosophy

Abstract: Writing skills typically develop over a course of more than two decades as a child matures and learns the craft of composition through late adolescence and into early adulthood. The novice writer progresses from a stage of knowledge-telling to a stage of knowledge-transforming characteristic of adult writers. Professional writers advance further to an expert stage of knowledge-crafting in which representations of the author's planned content, the text itself, and the prospective reader's interpretation of the text are routinely manipulated in working memory. Knowledge-transforming, and especially knowledge-crafting, arguably occur only when sufficient executive attention is available to provide a high degree of cognitive control over the maintenance of multiple representations of the text as well as planning conceptual content, generating text, and reviewing content and text. Because executive attention is limited in capacity, such control depends on reducing the working memory demands of these writing processes through maturation and learning. It is suggested that students might best learn writing skills through cognitive apprenticeship training programs that emphasize deliberate practice.
Kellogg, R.T. (2008). Training writing skills: A cognitive developmental perspective. Journal of writing research, 1(1), 1-26

Knowledge of fractions & division predict success in algebra

When I first started writing kitchen table math, with Carolyn Johnston, Carolyn told me that fractions are the math cliff.

Yesterday, Glen left a link to a new study in Psychological Science confirming the critical importance of fractions -- and long division -- to a child's future success in algebra:

Our main hypothesis was that knowledge of fractions at age 10 would predict algebra knowledge and overall mathematics achievement in high school, above and beyond the effects of general intellectual ability, other mathematical knowledge, and family background. The data supported this hypothesis.

and:

Early knowledge of whole-number division also was consistently related to later mathematics proficiency.

and:

The greater predictive power of knowledge of fractions and knowledge of division was not due to their generally predicting intellectual outcomes more accurately.

More from the article:

ABSTRACT
Identifying the types of mathematics content knowledge that are most predictive of students’ long-term learning is essential for improving both theories of mathematical development and mathematics education. To identify these types of knowledge, we examined long-term predictors of high school students’ knowledge of algebra and overall mathematics achievement. Analyses of large, nationally representative, longitudinal data sets from the United States and the United Kingdom revealed that elementary school students’ knowledge of fractions and of division uniquely predicts those students’ knowledge of algebra and overall mathematics achievement in high school, 5 or 6 years later, even after statistically controlling for other types of mathematical knowledge, general intellectual ability, working memory, and family income and education. Implications of these findings for understanding and improving mathematics learning are discussed.

[snip]

Marked individual and social-class differences in mathemat- ical knowledge are present even in preschool and kindergarten (Case & Okamoto, 1996; Starkey, Klein, & Wakeley, 2004). These differences are stable at least from kindergarten through fifth grade; children who start ahead in mathematics generally stay ahead, and children who start behind generally stay behind (Duncan et al., 2007; Stevenson & Newman, 1986). There are substantial correlations between early and later knowledge in other academic subjects as well, but differences in children’s mathematics knowledge are even more stable than differences in their reading and other capabilities (Case, Griffin, & Kelly, 1999; Duncan et al., 2007).

These findings suggest a new type of research that can con- tribute both to theoretical understanding of mathematical development and to improving mathematics education. If researchers can identify specific areas of mathematics that consistently predict later mathematics proficiency, after controlling for other types of mathematical knowledge, general intellectual ability, and family background variables, they can then determine why those types of knowledge are uniquely predictive, and society can increase efforts to improve instruction and learning in those areas. The educational payoff is likely to be strongest for areas that are strongly predictive of later achievement and in which many children’s understanding is poor.

In the present study, we examined sources of continuity in mathematical knowledge from fifth grade through high school. We were particularly interested in testing the hypothesis that early knowledge of fractions is uniquely predictive of later knowledge of algebra and overall mathematics achievement.

One source of this hypothesis was Siegler, Thompson, and Schneider’s (2011) integrated theory of numerical development. This theory proposes that numerical development is a process of progressively broadening the class of numbers that are understood to possess magnitudes and of learning the functions that connect those numbers to their magnitudes. In other words, numerical development involves coming to understand that all real numbers have magnitudes that can be assigned specific locations on number lines. This idea resembles Case and Okamoto’s (1996) proposal that during mathematics learning, the central conceptual structure for whole numbers, a mental number line, is eventually extended to rational numbers. The integrated theory of numerical development also proposes that a complementary, and equally crucial, part of numerical development is learning that many properties of whole numbers (e.g., having unique successors, being countable, including a finite number of entities within any given interval, never decreasing with addition and multiplication) are not true of numbers in general.

One implication of this theory is that acquisition of fractions knowledge is crucial to numerical development. For most children, fractions provide the first opportunity to learn that several salient and invariant properties of whole numbers are not true of all numbers (e.g., that multiplication does not necessarily pro- duce answers greater than the multiplicands). This understanding does not come easily; although children receive repeated instruction on fractions starting in third or fourth grade (National Council of Teachers of Mathematics, 2006), even high school and community-college students often confuse properties of fractions and whole numbers (Schneider & Siegler, 2010; Vosniadou, Vamvakoussi, & Skopeliti, 2008).

This view of fractions as occupying a central position within mathematical development differs substantially from other theories in the area, which focus on whole numbers and relegate fractions to secondary status. To the extent that such theories address development of understanding of fractions at all, it is usually to document ways in which learning about them is hindered by whole-number knowledge (e.g., Gelman & Williams, 1998; Wynn, 1995). Nothing in these theories suggests that early knowledge of fractions would uniquely predict later mathematics proficiency.

Consider some reasons, however, why elementary school students’ knowledge of fractions might be crucial for later mathematics—for example, algebra. If students do not under- stand fractions, they cannot estimate answers even to simple algebraic equations. For example, students who do not under- stand fractions will not know that in the equation 1/3X = 2/3Y, X must be twice as large as Y, or that for the equation 3/4X = 6, the value of X must be somewhat, but not greatly, larger than 6. Students who do not understand fraction magnitudes also would not be able to reject flawed equations by reasoning that the answers they yield are impossible. Consistent with this analysis, studies have shown that accurate estimation of fraction magnitudes is closely related to correct use of fractions arithmetic procedures (Hecht & Vagi, 2010; Siegler et al., 2011). Thus, we hypothesized that 10-year-olds’ knowledge of fractions would predict their algebra knowledge and overall mathematics achievement at age 16, even after we statistically controlled for other mathematical knowledge, information-processing skills, general intellectual ability, and family income and education.

Early Predictors of High School Mathematics AchievementRobert S. Siegler1, Greg J. Duncan2, Pamela E. Davis-Kean3,4, Kathryn Duckworth5, Amy Claessens6, Mimi Engel7, Maria Ines Susperreguy3,4, and Meichu Chen4Psychological Science 23(7) 691–697

I'm back!

Wow.

That was intense. Eight-hour classes during the day, 3-hour reading assignments at night, tests each morning, no family, no dogs, no kitchen, AND a whole new group of classmates to get to know --- Working memory blowout!

By the end of the Week 2, I was having mini-blackouts in class. I would be sitting in my Learning Position, wearing my Learning Expression and Tracking the Speaker with my eyes, and....I would suddenly come to and have no idea how much time had passed since the last time I actually heard something the speaker said. It was like SAT reading, only for listening.

Plus try jumping rope 100 times inside a hotel room. (I hit 100 in June.)

All worth it.

I've just spent two weeks of my life witnessing what is probably the best teaching on earth.

Many notes to share.

letter to Andrew Rosenthal

re: Texas Republicans and "Knowledge-Based Education," I've sent this email to an address that I hope belongs to Andrew Rosenthal:

Hi -

I am a writer (Animals in Translation; Animals Make Us Human) and an instructor of freshman composition.

My class blog is here.

My husband, Ed Berenson, is Director of the Institute of French Studies at NYU (his new book is The Statue of Liberty: A Transatlantic Story).

Both of us strongly support “knowledge-based education,” and we are likely in the majority of parents, including liberal parents living in New York.

Although it’s not obvious from the platform’s wording, knowledge – not critical thinking per se – is the issue the Texas Republican Party has taken a position on. The phrase “critical thinking” means something quite different inside public education than out, and I’m hoping you’ll consider writing a follow-up to clarify.

Boiling it down, there are two fundamental issues in the ‘education wars,’ one involving values, the other involving empirical research on the brain.

In terms of values, a majority of parents (and taxpayers and liberal arts professors) want schools to transmit to students knowledge of the liberal arts disciplines.

The K-12 establishment disagrees. Education professors [tend to] believe knowledge is changing so quickly that material taught today will be obsolete tomorrow, so content doesn’t matter. Instead of teaching knowledge, schools should teach students to ‘think critically’ and to ‘learn how to learn.’

(If you're interested, I compare my own district's ‘content doesn’t matter’ 7th grade reading program to the Core Knowledge reading sequence here. My district spends $29K per pupil.)

In terms of research on the brain, the K-12 establishment believes that ‘knowing’ and ‘thinking’ are separate functions. In the age of the internet, they argue, there is no reason for students to 'memorize' and 'regurgitate' knowledge because you can find any information you need on Google.

That sounds logical, but cognitive science has shown that it’s wrong. In reality, it's not possible to think about content stored on Google. While you are thinking, content must be stored inside 'working memory,' and working memory for “external,” unlearned content is tiny -- while working memory for knowledge stored in long-term memory is much larger.

In short, “knowledge” stored in the brain is biologically different from “knowledge” stored outside the brain, and the difference matters to the quality of thought. Thinking depends on knowing.

Cognitive scientist Daniel Willingham’s article for teachers is worth reading:
Critical Thinking: Why Is It So Hard to Teach?

In closing, I’ll mention that Ed headed the California History/Social Science Project in the ‘90s. CHSSP was a state-wide effort by the superintendent of schools to remove professional development from education schools and put it in the hands of disciplinary specialists – in other words, to make professional development “knowledge-based.”

I’m sure Ed would be happy to talk to you if you’re interested.
Hoping you’ll look into this further and consider writing a follow-up –

Catherine Johnson

Of course, I've omitted the question of direct instruction in values...

group work, IQ, and underperformance

re: group work lowers IQ

I've skimmed the article (free online).

Assuming the findings are confirmed in other studies (I suspect they will be), this is bad news.

Set-up

• Subjects had the same IQ: 126.
• They were put in small groups of 5 and introduced to each other.
• They took an IQ test with no feedback as to how they did.
• Then they took a second computer-administered IQ test.
• After each question, they were told whether they got the answer right or wrong.
• At the same time, they were also given their rank inside the group (rank determined by each person's # of correct answers).
• They were also given their rank vis a vis 1 particular member of the group, chosen "pseudorandomly."
• 2 people had brain scans during the test-with-feedback condition.

Results

• Everyone did worse in the beginning. Across the board. Everyone. Everyone did worse than his/her measured ability.
• As the test went on, some people recovered. Their performance went back up to the level predicted by their IQ scores. 
• The others never improved. They started low, and they stayed low. 
• Females were more likely to start low and stay low than males. 
• For high performers, brain scans showed activity in the amygdala (likely fear), decreasing over time. (That is, they were likely feeling less fear, perhaps growing more confident.)
• For high performers, activity in the lateral PFC increases over time. Activity in lateral PFC is associated with IQ tasks, with working memory tasks, and with increased task difficulty. 

So. All told, membership in a small group produced a net decrease in number of correct answers compared to what you would predict for the same 5 people working alone.

Everyone's performance dropped at the outset.

Some people recovered, others didn't.

The people who recovered simply went back up to where they had been going in, before experimenters assigned them to a small group.

Questions

• How small is small? Would group of 20 students in whole class instruction show the same pattern? 25 students? 30? 
• If so ... yikes.
• In the wake of this study, mixed-ability groups strike me as an even worse idea than I've thought in the past. Lower ability children in a mixed-ability group are going to be getting constant negative feedback about their status vis a vis the higher ability children. On the other hand, the study did not include a condition that manipulated feedback in this manner. That would be interesting.
• Assuming this study picked up on a personality difference (which we don't know, of course), what would happen if you grouped the 'nervous' kids together, putting the 'confident' kids in their own group? Would two groups still separate out in this way?
• Would same-sex groups change the results?
• What does this tell us about grades and grading? 

The blue bar represents the low performers, the red bar high performers. All have the same measured IQ (126), and at the beginning of the study all are performing well below the level their IQ would predict. The higher performers then recover, and their performance increases to a "126" level. The the low performers do not recover, and their performance remains suppressed.

Implicit signals in small group settings and their impact on the expression of cognitive capacity and associated brain responses

David Brooks has a really bad idea

"When you visit The New American Academy, an elementary school serving poor minority kids in Crown Heights, Brooklyn, you see big open rooms with 60 students and four teachers. The students are generally in three clumps in different areas working on different activities. The teachers, especially the master teacher who is floating between the clumps, are on the move, hovering over one student, then the next. It is less like a factory for learning and more like a postindustrial workshop, or even an extended family compound.

The teachers are not solitary. They are constantly interacting as an ensemble. Students can see them working together and learning from each other. The students are controlled less by uniform rules than by the constant informal nudges from the teachers all around.

[snip]

[The principal] revitalized one of the most violent junior high schools in the South Bronx and with the strong backing of both Klein and Randi Weingarten, the president of the teachers’ union, he was able to found his brainchild, The New American Academy.

[snip]

He has a grand theory to transform American education, which he developed with others at the Harvard School of Education. The American education model, he says, was actually copied from the 18th-century Prussian model^* designed to create docile subjects and factory workers. He wants schools to operate more like the networked collaborative world of today.

He talks fervently like a guerrilla leader up in the mountains with plans to take over the whole country."
The Relationship School By DAVID BROOKS
Published: March 22, 2012

Yes, I just bet he does. Other people's children are always a tempting target of world conquest, it seems.

I wish David Brooks would stop writing about public schools. He doesn't seem to know anything about education per se, and what he thinks he knows about "the social animal" appears to have blinded him to seemingly every aspect of school apart from its social and moral elements. Learning, memory, knowledge, curriculum, the importance of deliberate practice: I've never seen David Brooks venture an opinion about the actual knowledge that does or does not get transmitted to the next generation inside a public school, or about the processes by which students do or do not acquire that knowledge.

As to classroom management, which Mary Damer once told me was the single most critical challenge facing any new teacher even if he's a Marine just coming out of the service (I agree), here is Brooks:

The New American Academy takes a different approach than the other exciting new education model, the “No Excuses” schools like Kipp Academy. New American is less structured. That was a problem at first, but Waronker says the academy has learned to get better control over students, and, on the day I visited, the school was well disciplined through the use of a bunch of subtle tricks.

For example, even though students move from one open area to the next, they line up single file, walk through an imaginary doorway, and greet the teacher before entering her domain.

That, my friends, is what you call a red flag.

They put the kids in an open classroom (remember those?), chaos ensued, so they invented imaginary doorways inside the big open classroom: imaginary doorways the kids had to be trained to imagine and use. Instructional time was taken away from reading, writing, and arithmetic and redirected to teaching disadvantaged kids to pretend to walk through a doorway that isn't there -- and all because the folks at Harvard School of Education couldn't be bothered to read up on the history of the open classroom or on anything behaviorists have figured out in the past 50 years.

And Brooks approves.

The New American Academy has two big advantages as a reform model. First, instead of running against the education establishment, it grows out of it and is being embraced by the teachers’ unions and the education schools. If it works, it can spread faster.

Second, it does a tremendous job of nurturing relationships. Since people learn from people they love, education is fundamentally about the relationship between a teacher and student.

Union-slash-Teachers College charter public schools with imaginary doorways and no curriculum! Yes, that should spread like wildfire. Parents have been clamoring for no curriculum and no doorways  for years.

Brooks's own children have all attended Jewish day schools in the D.C. area.

and see:
the founder, chair, and CEO of Netflix has a really bad idea
Larry Summers has a really bad idea
Wash U professor on Reed Hastings' really bad idea

David Brooks has a really bad idea
David Brooks has a really bad idea, part 2
David Brooks has a really good idea

a difficult passage - & terrific advice

[Terri] LeClercq offers a very helpful technique to check for coherence in a multi-paragraph text: As you edit your rough draft, separate each topic sentence from your text and examine each one to make sure it is a strong introduction to the main idea of that paragraph. Then examine the coherence of the topic sentences as they relate to the overall thesis set-up by seeing whether the topic sentences form a coherent paragraph.
“Writing Good Paragraphs with Topic Sentences.” Legal Writing Tips 1.8 (2005). Print. Web. 14 March 2012.

My students had trouble with this passage today. They understood the first and second sentences (the 2 independent clauses linked by the colon), but they stumbled over the last one.

I'm not crazy about the last sentence myself: reading it, you have to keep too much information lit up in working memory until you finally get to the most important bit, which is at the very end. The end is where the most important bit usually should be, but still. There's an awful lot to hang on to until you get there.

I don't mean to sound harsh. That last sentence is perfectly serviceable, and impressive in its way. It's nicely linked to its partner sentence, the one that comes just before it, and it manages to pack a great deal of information into a small space, which is not easy. You'd have to be an experienced writer to write it.

Nevertheless, if it were my sentence, I would keep on writing it before I stopped. I would revise.

Teaching basic composition, I've come to realize how important it is for college students to be able to read prose I wouldn't advise any of them actually to write. A fair portion of academic and professional prose is not very good, and some of it is god-awful. But students have to read it.

Of course everyone knows this, but I hadn't thought about the implications until now. College students have to be able to read bad writing. Not just difficult writing, not just sophisticated writing, not just writing with a lot of big words. College students have to be able to read all those things, but they also have to be able to read difficult, sophisticated writing with a lot of big words that is bad.

So how do they acquire this skill?

Composition textbooks come stocked with dozens of heavily copy-edited essays written by journalists and originally published in popular books and magazines: these are works that have been professionally engineered to be maximally swift, cohesive, and clear. They make sense as models for writing, but they're useless for reading. They're so well written they practically read themselves.^*

Where are the composition texts featuring 100s of pages of dense and mystifying academic prose, I ask?

And how does one teach students to read badly-written prose?

Is it different from teaching students to read well-written prose?

I'm thinking it might be.

Last but not least, speaking of bad and good prose, I think LeClercq's advice is brilliant. It's an amazing fit for William Kerrigan's X-1-2-3s.

X-1-2-3

^* I don't remotely believe that well-written prose practically reads itself. Certain genres, however, are written to be effortlessly readable, and that's the writing that appears in college composition texts.

gobsmacked

I'd read chemprof's and EM's descriptions of very bright students with serious reading problems, and I'd been appropriately horrified. But until this morning I'm not sure I really, truly grasped what they were talking about. Which is: very bright, very talented students who have major problems reading and who haven't been diagnosed with dyslexia.

This morning, one of the smartest students I've ever had reluctantly showed me the thesis statement he'd just written. He didn't want me to look at it because the spelling was bad, and I won't post it without permission. I'll just say that he had misspellings on the order of 'charitturs" for characters; "sirve" for serve; and "morle" for moral. Of 19 words, 9 are misspelled, and the words spelled correctly include only one noun.

His spelling is so bad, he said, that spellcheck doesn't work. At some point his laptop decided he might be writing in Spanish, so half the time spellcheck serves up a menu of Spanish options. When Microsoft Word does present him with a selection of words in English, my student often has no idea which one to pick.

I had him read out loud a difficult paragraph, written by Maria Tatar, on fairy tales. He could more or less do it (he's very sharp), but he kept missing the short words: prepositions and short verbs, too, I think. He kept missing the short words because he automatically filled in whatever word he thought would come next instead of reading the word that actually did come next. (I guess somebody taught him to 'make predictions.')

Here's an example of what I mean (I don't remember whether he misread this section):

Like many fairy tales, the Grimms’ narrative begins by framing a prohibition

He would likely trip over 'by,' reading it as 'with' instead (or whatever word seemed most likely to appear). Although he can spell all the prepositions -- prepositions are the main words he can spell -- he often misreads them.

Another problem: he doesn't read left-to-right. Instead, he jumps around in a sentence looking for words he can base his reading of the sentence in: he's looking for a kind of anchor word, I think. Once he's found an anchor word, he goes back to the beginning of the sentence and starts over. Then, if he has no luck on the second go-round, he'll jump forward again and look for a second anchor word that might help.

He couldn't read the word "prohibition" at all, but he demonstrated for me the method he would use to tackle it. He would start with the syllable "pro," which he could read; then he would hit "hibition," which he couldn't read at all. Then he would skip to "tion" at the end (which he saw as a separate syllable - that's good). Then he would try to guess "prohibition" on the basis of "pro" and "tion."

And he would fail, in spite of the fact that he does indeed know the word 'prohibition' and recognized it the moment I said it.

He's in his mid-20s, he's extremely intelligent, and he cannot read "prohibition" using phonics and syllables.

I know very little about dyslexia, so I don't know whether that's an issue. I do know a little about phonics, and it seems clear that he doesn't read phonetically. At least, not fluently. I'm sure he was taught to read using whole language or balanced literacy. As a child he memorized all the words he was taught and then, via high IQ, high energy, and a scrappy personality, figured out how to reverse-engineer paragraphs in order to wrest some meaning from them.

Interestingly, his problems have led him to a theory of paragraph development I've been mulling over myself: he believes paragraphs typically have a concluding sentence that sums everything up. He reads that sentence first, then goes back to the beginning of the paragraph to try and decipher the whole thing.

I've been wondering whether paragraphs have conclusions and have been operating under the theory that most of them do not. Now I wonder.

He can't really write at all (he says -- he's so verbal, I find that hard to believe). But he has somehow figured out how to think in whole paragraphs -- maybe even in whole 5-paragraph essays or perhaps beyond. Today he actually came up with a sophisticated thesis sentence AND three coherent supporting topic sentences almost entirely in his head. He says he got through high school on oral presentations, and by oral presentation he doesn't mean Powerpoint. He means thesis, topic sentence, elaboration, and specific support. He produces more content talking than I have the working memory to deal with: he's got to figure out how to write if only so other people will be able to follow what he's saying.

This is a guy trading in complex ideas entirely inside the oral register. You can't do that! (Well, you can, but most of us can't 'hear' it ---- )

He told me a few weeks back that when he was a kid his school introduced a new reading program that was so terrible, and left him with such profound spelling deficits, that his parents had wanted to sue the school. No surprise there.

Of course, as we know, a parent can't sue a school for failing to teach their very bright child to spell. Educational malpractice doesn't exist.

Question. What is the way forward here? (If he wants a way forward, that is. He's not a kid, and he's figured out work-arounds that are serving him reasonably well.)

It strikes me that he needs to look into voice recognition software ---- although to use voice recognition software for writing anything more serious than a short email you have to be able to read what you've dictated. So I'm sure about that.....

He says he's often thought he needs to take ESL classes in English.

That doesn't strike me as a bad idea -- I've begun to rely upon lessons in English as a second language myself -- but I think what he really needs is phonics. Phonics and a lot of practice reading left-to-right. His jump-around habits are ingrained; he'd need to practice until he developed a new ingrained left-to-right habit.

Is there a phonics program anyone out there would recommend for this student?

Other thoughts or suggestions?

Why students have to memorize things

re: Larry Summers' claim that "in a world where the entire Library of Congress will soon be accessible on a mobile device..., factual mastery will become less and less important":

Larry Summers is wrong.

Factual mastery has not and will not become less important, for the simple reason that it is not possible to think about something stored on Google.

While you are thinking about something, that something has to be lodged inside working memory, not Google.

Biology does not work the way Larry Summers thinks it works.

Working memory

If I ask you to multiply 36 by 3 inside your head, working memory is what you use to do it.

Working memory (WM) does three things:

1. Holds the problem -- "multiply 36 by 3" -- in consciousness 
2. Retrieves the relevant knowledge from long-term memory (the times tables, in this case)
3. Performs the calculation

Boiling it down, working memory is:

1. a form of storage
2. a search engine 
3. a "computer" or thinker

"Critical thinking" is accomplished by working memory.

3 to 5

The fact that we can think only about things stored inside working memory leads directly to the need for "factual mastery."

Factual mastery—knowledge stored inside long-term memory—is essential because although long-term memory is vast, working memory is tiny:

...cognitive tasks can be completed only with sufficient ability to hold information as it is processed. The ability to repeat information [you have just heard or read] depends on task [difficulty]... but can be distinguished from a more constant, underlying mechanism: a central memory store limited to 3 to 5 meaningful items in young adults.

The Magical Mystery Four: How Is Working Memory Capacity Limited, and Why? by Nelson Cowan

Working memory can hold three to five items at once. That's it. That's the limit.

Three to five.

I hit this limit all the time trying to write about new topics. The basal ganglia, for instance. For well over a year, I have been endlessly working and re-working a project on the basal ganglia, a subject I knew essentially nothing about going in. Where the basal ganglia were concerned, my long-term memory was a blank slate.

The upshot: I was not able to write about the basal ganglia until I actually learned about the basal ganglia: learned as in committed the material to memory. It didn't matter how many times I looked up basal ganglia on the internet. I looked up the basal ganglia on the internet a lot, as a matter of fact; then I forgot whatever it was I had looked up while I was looking up something else to do with the basal ganglia, after which I'd have to go back and re-look up the first thing all over again.

Try it if you don't believe me.

Here are some terms related to the basal ganglia:

Dorsal striatum
Ventral striatum
Putamen
Nucleus accumbens
Ventral tegmental area
Orbital frontal cortex
Dopamine
Two pathways
OCD
Addiction
Habit
Impulsive
Compulsive
Intuition
Probabilistic learning
Associative learning
Statistical learning
Serotonin
Orbitofrontal cortex
Cortico-striatal circuit

Now supposing I handed you a laptop and asked you to look up each term on Wikipedia, then write a coherent, reasoned 5-paragraph essay on the basal ganglia: what it is and what it does. Just a quick summary organized into 5 coherent paragraphs.

You couldn't do it.

You couldn't do it because every time you wrote about the ventral striatum, the dorsal striatum, and the orbitofrontal cortex, you would forget the VTA and the putamen—and you would forget the VTA and the putamen because your working memory will hold only 3 to 5 things at once. Something has to go.

That's what happened to me when I took the SAT with a calculator I didn't know how to use. Each time I swapped the steps for using the calculator into working memory, my brain swapped the information for the problem I was doing back out of working memory. Then, when I tried to cram the information for the problem back into working memory, the calculator steps got squeezed out again.

I could remember the problem, or I could remember the calculator, but I couldn't remember both at the same time. Too much information, literally.

My calculator fiasco illustrates the reason you need to practice until you learn content and skills to the point of 'automaticity.' (Automaticity is another basal ganglia term, by the way. The basal ganglia are the part of the brain that underpins automaticity.) Once you've learned something so well you don't have to think about it, you free up space in working memory to hold other things.

Thus if you know the times tables "by heart," you don't need to pull "3x6=18" into working memory. Working memory can locate "3x6=18" inside long-term memory and use it without displacing "36x3."

Knowledge stored inside the brain is different from knowledge stored outside the brain

Experts always possess factual mastery of their fields. Always.

The reason experts always possess factual mastery of their fields is that knowledge stored in long-term memory is different from knowledge stored on Google.

Knowledge stored in long-term memory is (or becomes) biologically connected, or "chunked." Thus to an expert on the basal ganglia, ten facts about the basal ganglia are just one or two big facts about the basal ganglia.

Chunking is the magic, because working memory doesn't care about chunk size. Working memory can hold 3 to 5 small and simple items or 3 to 5 large and complex items. Either will do. Chunking gets around the limits on working memory.

Dan Willingham's demonstration of working memory

For a demonstration of the chunking principle, read the list below, then look away and try to remember what you've read:

CN

NFB

ICB

SCI

ANC

AA

How many letters did you recall?

To find out how many letters you would have recalled via prior chunking inside long-term memory, see Daniel Willingham's explanation in "How Knowledge Helps" (American Educator | Spring 2006).

(The answer is all of them.)

You can't Google knowledge chunks

Knowledge chunks can be created only inside the brain, via learning. You can't Google someone else's complex knowledge chunks and swap them into your own working memory. It doesn't work that way. Your own brain has to do the work of chunking, and your brain does that work through the process of learning, bit by bit and step by step.

Which means that the process of storing content in long-term memory is not a simple matter of "memorizing facts" so you can "regurgitate" them later.

Over time, memorization creates the complex knowledge chunks that allow knowledgeable people to engage in complex thought.

Experts think better than novices because experts have factual mastery

To a gratifying degree, I can now think about nearly all 19 items on the basal ganglia list at the same time. I'm still struggling with "putamen" and "ventral tegmental area," but the other 17 are stored in memory: my memory, not Google's. So, for me, those 17 items are no longer 17 separate items, but closer to 2 or 3. When I think about 1 item on the list, I'm thinking about the others.

I reached this point by committing these terms and concepts to memory. As the terms entered my long-term memory, they became biologically connected and chunked. Now that I can think about them at the same time, which means I can write about them, too.

What makes experts expert, to a large degree, is factual mastery of their fields. Factual mastery allows experts to think deeply and well because the content they are thinking about has been biologically connected and chunked inside their brains, and there is no obvious limit to the amount of chunked content working memory can manage so long as knowledge has been chunked into no more than 3 to 5 separate entities.

Factual mastery is required for complex thought.

Which brings me back to Larry Summers.

If our schools are going to ask students to 'think' about material they haven't learned, students are going to be thinking about 3 to 5 small, not-well-elaborated items at a time. Period. Their thinking will be superficial, and the conclusions they reach will be superficial, too.

Which is exactly what we see in Larry Summers' op-ed about education, a field in which he is neither expert nor learned.

AND SEE: 
Superior Memory of Experts and Long-Term Working Memory (LTWM)
Extremely fast learning & extended working memory
The Number and Quality of Representations in Working Memory by Weiwei Zhang and Steven J. Luck
How Knowledge Helps by Daniel T. Willingham American Educator Spring 2006
#whystudentsneedtomemorize

Larry Summers has a really bad idea

In today's Times, Larry Summers weighs in on the question of what college students ought to learn in college.

Larry's answer: not too much, because the entire Library of Congress will soon be accessible on a mobile device with search procedures that are vastly better than any card catalog!

Larry bases his novel and highly original thesis (to wit: "factual mastery will become less and less important") on "what we now understand about how people learn."

(Does Harvard have node chairs, I wonder? Sounds like no.)

OK, I'm going to go look up calculus on the internet. I've always been interested in calculus, so now that I've received a mobile device for Christmas, I'm going to look it up. Then I'm going to collaborate with some friends who also looked up calculus on the internet to figure out what to do about the 21st century global world meltdown.

I'm going to do this because I've noticed that economists use calculus in their collaborative group papers.

[pause]

There is a reason why students must commit content to memory as opposed to looking it up on a mobile device with search procedures that are vastly better than any card catalog.

That reason has to do with working memory.

More anon.

What You (Really) Need to Know by Lawrence A. Summers

update: Why students have to memorize things
and see: Extremely fast learning & extended working memory

AND SEE:
The founder, chair, and CEO of Netflix has a really bad idea
Larry Summers has a really bad idea
Wash U professor on Reed Hastings' really bad idea
Barry Eichengreen has a really bad idea
President Obama has a really bad idea

David Brooks has a really bad idea
David Brooks has a really bad idea, part 2
David Brooks has a really good idea

The Daily has a really bad idea

anonymous on lefties taking the SAT

Good advice:

Lefties should be particularly vigilant about the chairs/desks. I've heard stories about them having to take long tests on right-handed flip-up deskettes; a true nightmare. I know the registration forms for my grad comps asked lefties to identify themselves, because almost all of the seats in the auditorium had right-hand deskettes. They brought in as many extra lefty ones as they needed. Is there a similar question on SAT registrations?

I have got to find time to write a quick post on working memory.

I believe that the "transmission mechanism" from right-hand desks for left-hand test-takers to reduced performance is working memory blowout.

And see: death by calculator. Death by calculator is a case of working memory blowout.

(The real-world term for working memory blowout problems is cognitive load theory.)

Definitions, Precision, Coherence

What's missing from today's school math, and particularly, from middle school math?

Definitions, precision, and coherence.

Without a proper introduction to definitions, students don't get clarity in what a math statement is and what it is not. They don't get a sense of the abstractness of it. Without having actual definitions, they don't learn to work with definitions, so they can never learn how to use them to derive new things that are true of "all" of a given set. Without definitions, they cannot learn to REASON about mathematics because they have no basis for reasoning.

Without precision, students cannot make clear, unambiguous statements. They are not able to properly manipulate the symbols they are given, and they can't even say what their manipulations refer to and what they do not refer to. (As Wu says, In the same way that we do not ask “Is he six feet tall?” without saying who “he” is, we do not write down xyzrstuvw = a + 2b + 3cdefghijklmn
without first specifying what a, b, . . . , z stand for either (this is
an equality between what? Two random collections of symbols??
What does it mean??))

Without coherence, math is a series of unrelated facts designed merely to trick students. Without coherence, math is just a test of how big your working memory can be when you can never integrate any understanding together. There is no notion that the results you get follow from other results. Reasoning can't exist without coherence.

The reason SAT math feels so "tricky" to so many students is because they were never taught math in a coherent, reasoned way with definitions and precision. So the test seems to bejust a set of tricks designed to "catch" you, as opposed to a test of how what you know relates to other things you already know. That is why it seems to just test working memory. That is why it seems to have so much associative interference. The reason is because you were never taught that math was reasoned, with every piece of it following from the other pieces, a seamless whole that reinforced the same truths from a zillion different directions. And you were never taught what it meant.

rat psych - "careless errors" in reading the SAT

During my year of living dangerously, doing SAT math prep off and on with C., I was chronically amazed stunned by the number and type of "careless errors" he and I both made taking timed sections of the test. In particular, I made repeated errors of "simple" reading, particularly when I was tired or the room was hot. I made so many reading errors that when I finally took the real test, I had no way to predict my math score at all: no way to estimate how many reading errors I had -- or had not -- made.

I eventually came up with a theory of careless errors, the details of which I've forgotten at the moment. I do recall that it had to do with working memory. Arguably the SAT tests working memory above all: all 10 sections put you into working memory blowout. I experienced working memory blowout so often that I began to notice a connection. As far as I can tell, you make more careless errors when your working memory is overtaxed (and you hit the limits of working memory much more quickly when you're sleep-deprived or overheated).

I've just come across a new study that I think confirms my subjective experience:

This study resolves two long-standing debates in the field. Does our working memory function like slots, and after our four slots [emphasis added] are filled with objects we cannot take in any more; or does it function like a pool that can accept more than four objects, but as the pool fills the information about each object gets thinner? And is the capacity limit a failure of perception, or of memory? [emphasis added]

“Our study shows that both the slot and pool models are true,” says Miller. “The two hemispheres of the visual brain work like slots, but within each slot, it’s a pool. We also found that the bottleneck is not in the remembering, it is in the perceiving.” [emphasis added] That is, when the capacity for each slot is exceeded, the information does not get encoded very well. The neural recordings showed information about the objects being lost even as the monkeys were viewing them, not later as they were remembering what they had seen.
Picower: 1 Skull + 2 Brains = 4 Objects in Mind

Failures of working memory are failures of perception!

Subjectively, that's what I experienced taking practice sections; that's what it felt like. Once I hit a certain level of tiredness, or heat, or working memory blow-out, I stopped being able to read.

The same thing happens on the reading and writing sections, too. The reading and writing sections are so taxing that you reach points where you simply cannot take in what the sentence or paragraph before you says. ^* I'm not talking about losing the ability to answer questions about the sentence or paragraph.

I'm talking about losing the ability just to read the words on the page.

I'm a 10
rat psych: what to do about SAT math (part 1)
rat psych: what to do about SAT math (part 2)
rat psych: what to do about SAT math (part 3)
rat psych: careless reading errors on the SAT

^* I say "you" because I know I am not alone in this.

rat psych - what to do about SAT math (part 3)

Your typical high school student, I presume, has spent several years setting up equations and solving for x. At least, let's hope so. I certainly did.

The SAT uses this fact to elicit many wrong answers from test-takers who have worked a problem correctly. The student gets the solution right but the answer wrong because the answer isn't x. The answer is 3x, say, or xy. I seem to recall a problem or two where the answer was -x, for god's sake, but I might be making that up.

Other times the test will give you a value for x + y, say, and you're supposed to see that you should simply insert that value some place else in the problem, et voilà: the answer they're looking for pops up.

Here's a typical problem, medium difficulty (according to the College Board):

If 4(x + y)(x - y) = 40 and (x - y) = 20, what is the value of x + y?

A kid who's had no test prep at all will likely miss this question -- either miss it outright or take too much time spotting the solution, thus leaving him too little time to finish the test and increasing the likelihood he'll make "careless errors" on the questions he does get to because now he's working too fast trying to make up for the time he lost on the x + y problem.

For what it's worth, I think using x + y as the value, instead of x or y alone, is an interesting and instructive way to write a problem. (I'm curious what math people think). It seems to me that writing problems in which x + y is the salient unit may be a way of teaching what Ron Aharoni calls the fifth fundamental operation of arithmetic:

In addition to the four classical operations, there is a fifth one that is even more fundamental and important. That is, forming a unit, taking a part of the world and declaring it to be the “whole.” This operation is at the base of much of the mathematics of elementary school. First of all, in counting, when you have another such unit you say you have “two,” and so on. The operation of multiplication is based on taking a set, declaring that this is the unit, and repeating it. The concept of a fraction starts from having a whole, from which parts are taken. The decimal system is based on gathering tens of objects into one unit called a “10,” then recursively repeating it.

The forming of a unit, and the assigning of a name to it, is something that has to be learned and stressed explicitly. I met children who, in fifth grade, knew how to find a quarter of a class of 20, but had difficulty understanding how to find “three-quarters” of the class, having missed the stage of the corresponding process of repeating a unit in multiplication. What I Learned in Elementary School by Ron Aharoni

Maybe I'm wrong, but it seems to me that the x+y questions test math as opposed to obedience under pressure, which is what the Find xy questions test.

Still, there is no doubt in my mind that these questions elicit wrong answers from test takers who know the math involved, can do the math involved, and have a reasonable understanding of the math involved. Students who have spent years of their lives solving for x aren't going to break the Solve for x habit for the first time ever when they're working at breakneck speed and their eyes are bleeding from the Ella Baker passage.

Which brings me back to extinction learning. Test prep for SAT math involves spending a fair amount of time building new habits that conflict with ingrained old habits. You've been conditioned to solve for x; now you have to condition yourself not to solve for x. Also, you have to build as much speed as possible at not solving for x because you are never going to forget solve-for-x. The two impulses are inside your head, competing with each other, and the competition takes time (and probably eats up some precious working memory resources to boot).

Funny thing: during the time we spent doing SAT math prep around here, I overlearned don't solve for x to the degree that a couple of weeks before taking the real test I came across a practice problem that did ask the test-taker to solve for x. I was so surprised that I wasted several seconds reading and re-reading and re-reading again to make sure I hadn't misunderstood. You can't win.

For parents: your child needs to spend enough time not solving for x that he or she gets to be really, really fast at not solving for x.

Then he should be on the lookout for problems that say Solve for -x.


I'm a 10
rat psych: what to do about SAT math (part 1)
rat psych: what to do about SAT math (part 2)
rat psych: what to do about SAT math (part 3)
rat psych: careless reading errors on the SAT