I. The final rates problems in the Singapore Math, Primary Mathematics 4B Workbook:
II. The final rates problem in the Everyday Math, Student Math Journal, Volume 2:
I. The final rates problems in the Singapore Math, Primary Mathematics 4B Workbook:
This morning, I finally got to this week's NY Times Sunday Review. And there in the Gray Matter column, lo and behold, is yet another reference to the highly influential studies that Judith Rich Harris cites in The Nurture Assumption. The Gray Matter columnist is Nancy L. Segal, a professor of developmental psychology at California State University and the author of “Born Together — Reared Apart: The Landmark Minnesota Twin Study." Segal reaffirms that:
Identical twins raised apart match closely on genetically influenced traits such as intelligence and personality. In fact, identical twins raised apart are as similar in personality as are identical twins raised together. In contrast, biologically unrelated siblings (i.e., when one or both are adopted) show little behavioral resemblance despite their shared rearing.This, in a nutshell, is what leads to Harris' conclusion in The Nurture Assumption that:
Children would develop into the same sort of adults if we left them in their homes, their schools, their neighborhoods, and their cultural or subcultural groups, but switched all the parents around.As some commenters have pointed out, this is not as strong a statement as it first appears. Harris is:
A commenter on my recent post on The Nurture Assumption suggests that book's conclusions about the low influence that parents have on their kids also applies to schools. Here, again, is the book's nutshell conclusion:
Children would develop into the same sort of adults if we left them in their homes, their schools, their neighborhoods, and their cultural or subcultural groups, but switched all the parents around.The analogous point about schools would be:
Children would develop into the same sort of adults if we left them in their homes, their families, their neighborhoods, and their cultural or subcultural groups, but switched all the schools around.While this may well be a reasonable corollary to Harris' argument, it's important to keep a couple of things in mind.
Your children are not your children.So opens Judith Rich's 1998 book, The Nurture Assumption. Assailing trends still current in the Parenting Advice Industry, this book revolutionized my thinking about parenting over 15 years ago. Recently, I've found myself thinking about it anew.
...You may give them your love but not your thoughts,--Kahlil Gibran
For they have their own thoughts.
You may house their bodies but not their souls,
For their souls dwell in the house of tomorrow,
which you cannot visit, not even in your dreams.
You may strive to be like them,
but seek not to make them like you.
Children would develop into the same sort of adults if we left them in their homes, their schools, their neighborhoods, and their cultural or subcultural groups, but switched all the parents around.I've looked around at some of the critical reviews of Harris' book, and, so far as I can tell, no one has cited facts that refute this finding.
I. The final problems of the 5th grade Everyday Math Student Math Journal Volume 2:
“I found it, hiding between bed and chest.”
Not a particularly remarkable statement, except that it came from a child with a neurological condition that supposedly makes this sort of language inaccessible.
The child would be my son J; his neurological condition, mild-moderate autism; and the language in question, the non-literal language he used to depict his bathrobe as an entity that hides itself.
Personification is a subtype of metaphor in which a non-human entity is compared to a person. Metaphors are often claimed to be an area in which autistic children struggle—and there are even experiments that back up this claim. The problem, supposedly, is inherent to autism: autistic individuals are said to have difficulty with non-literal language in general.
But J not only gets metaphor: he revels in it. Is he really that unusual among his peers? Or might it be that a great many more people on the autistic spectrum have a much greater capacity for figurative language than we give them credit?
To see what it takes to grasp the figurative sense of “hiding,” it’s useful to divide instances of metaphor (and personification in particular) into three categories:
1. Frozen metaphors: phrases
2. Generalized, conventionalized metaphors
3. Novel, creative metaphors
Frozen metaphors are metaphors expressed in fixed expressions—“music to my ears”; “boiling mad”--that are used so often that they’re practically idiomatic. Indeed, some of what has been claimed to be examples of metaphor (see here and here) might just as easily be classified as idioms or idiomatic expressions: “an angel,” “a snake,” “a pig,” “a rock,” “bubbly,” or “sweet” (said of a person); “warm,” “cold,” “dark,” “light,” “up,” “down,” “blue,” or “sour” (said of a mood); “heated” (said of an argument); “spinning” (said of someone’s head); “loud” (said of colors); “reeks of” (as in “suggests”); “fishing for” (as in “seeking”), “a melting pot,” “a broken heart,” “the light of my life,” “the apple of my eye,” and “emotional rollercoaster.” Like idiomatic expressions, these phrases/usages can be learned one by one just as individual vocabulary words are.
Generalized, conventionalized metaphors are the ones that linguist George Lakoff and philosopher Mark Johnson talk about in Metaphors We Live By: standard, general metaphors that form the basis for more particular ones. Here are two sets of generalized metaphors from Lakoff and Johnson (courtesy TheLiteraryLink):
ARGUMENT IS WAR
Your claims are indefensible.
He attacked every weak point in my argument.
His criticisms were right on target.
I demolished his argument.
I've never won an argument with him.
You disagree? Okay, shoot!
If you use that strategy, he'll wipe you out.
He shot down all of my arguments.
TIME IS MONEY
You're wasting my time.
This gadget will save you hours. I don't have the time to give you.
How do you spend your time these days? That flat tire cost me an hour.
I've invested a lot of time in her.
1 don't have enough time to spare for that.You're running out of time.
You need to budget your time.
Put aside aside some time for ping pong.
Is that worth your while?
Do you have much time left?
He's living on I borrowed time.
You don't use your time, profitably.
I lost a lot of time when I got sick.
Thank you for your time.
And here’s my elaboration of their personification metaphor:
ENTITIES ARE HUMAN/SENTIENT/PURPOSEFUL
The engine is tired.
My browser is confused.
The sunlight is dancing on the waves.
The river is trying to break through the dam.
The wind is wailing and howling.
My bathrobe is hiding between the bed and the chest.
While these instances of metaphor far outnumber their frozen counterparts (all sorts of specific mappings and wordings being possible), they can be mastered much more systematically. You simply learn the general metaphors that underlie them: e.g., argument is war, time is money, and entities are human, and, together with a little analogic reasoning, you have the tools you need to interpret them or to create them yourself.
Novel, creative metaphors are a different story. These include things like (courtesy TheEnglishClub and LiteraryDevices) “The moon was a ghostly galleon tossed upon cloudy seas” (from Alfred Noyes, The Highwayman ) or “She is all states, and all princes, I.” (from John Donne, “The Sun Rising”). Making sense of these involves a level of reading comprehension and sensitivity to speaker intention that autistic children may lack. Luckily, these more challenging, creative metaphors are mostly restricted to the sort of literary texts that are already difficult for autistic children for many other reasons.
Indeed, the overwhelming majority of metaphors that we hear, read, and utter in everyday life belong to the first two categories: frozen metaphors than can be memorized, and generalized conventional metaphors whose underlying principles can likewise be committed to memory--and applied using general analogic reasoning skills. A mild/moderately autistic, high functioning child who is stumped by these instances of metaphor, I suspect, is a child who’s simply lacking in exposure and instruction.
When it comes to autism, where so much of the world is tuned out, and so much must be taught explicitly, it’s easy to confound teaching issues with deeper, conceptual issues. And sometimes what looks like a failure of imagination by children on the autistic spectrum may instead be a failure of imagination by those who assess and/or teach them.
Why do so few kids read for pleasure these days? Why do even fewer
read with deep and lasting comprehension? We can find causes at home, school
and in the environment, but it sure wouldn’t hurt if there were a little
more peace and quiet.
Inventing your own math problems, as it turns out, isn't just a 21st century skill...
I. From Hamilton's Essentials of Arithmetic:
According to a report by Common Sense Media:
53% of 9-year-olds vs. 17% of 17-year-olds are daily readers.
The proportion who "never" or "hardly ever" read tripled since 1984. A third of 13-year-olds and 45% of 17-year-olds say they've read for pleasure one to two times a year, if that.Then there's reading quality:
Not all time spent reading is fully focused. Even before electronic books, some children “media multitasked” while reading — in other words, used some other medium at the same time they were reading, such as having music or television on in the background.
The Kaiser Foundation’s study of media multitasking (Foehr, 2006), using data from 2003–2004, found that 28% of seventh through twelfth graders used another medium “most of the time” when they were reading, and another 30% said they did so “some” of the time they read.I'm guessing that listening--especially extended, full-focused listening--is also down, though this is harder to measure. Current trends in education, however, are highly suggestive.
One thing that complicated our recent (and lively) discussion of reading comprehension challenges is the question of what makes a text challenging. Challenges come in two distinct forms: language and content.
It's this first sense of challenging that the Lexile scale attempts to measure. A big part of it is vocabulary and sentence structure (the Lexile scale actually measures sentence length, but structure is far more important, with sentences that have lots of certain types of subordinate clauses being much more difficult to process than sentences that are merely long). Other language-based challenges (not captured by the Lexile scale) are how densely packed a text is with content words--especially nouns and verbs--and how interconnected it is with pronouns and other anaphoric devices ("that strategy," "this issue") that have antecedents elsewhere in the text. Some of these language-based challenges are dictated by the content: some ideas, themselves complex, are most clearly expressed in sophisticated vocabulary and complex, dense, interconnected prose (and, as I've discussed earlier, simplifying that prose actually makes the ideas harder to follow). Other language-based challenges are gratuitous: poorly defined terms, ambiguous antecedents, modifiers that are ambiguous in what they modify, and sentences that are generally hard to follow (certain German philosophers come to mind).
Then there's content. Some content is inherently difficult (quantum mechanics comes to mind); some content is difficult because it presupposes lots of background knowledge (part of what makes texts written by specialists for specialists about specialized topics hard to read).
Hard content sometimes requires hard language, but not always. And when we worry about boring kids with "easy" texts, it's important to keep this in mind. One route to boredom are texts whose content is easy or familiar--even if their language is hard. Another route to boredom (or tedium) is language that is too hard to follow. But what about a text whose language is easy while its content is hard--or at least new and interesting. Does the low reading level of the text make it boring? If so, then I'm boring myself every time I read the newspaper.
In fact, it takes a really great writer to describe complex ideas in simple prose. Steven Pinker ("How the Mind Works"); Isaac Asimov ("On Chemistry"); Richard Dawkins ("Climbing Mount Improbable"): I'm sure all these books are below my reading level, whatever that is, but who cares?
What I find most tedious--if not boring--are texts (or movies, for that matter) that some critics praise as "difficult," but that turn out to be difficult only because they are hard to follow; not because they contain complex and interesting characters, emotions, settings, themes, composition, and/or ideas. Call me pedestrian, but I'll always pick the "easy" piece that inspires me over a "hard" one that offers little in return for all that effort.
I. The first equivalent fractions problem set in 4th grade Singapore Math (Primary Mathematics 4A Workbook, Unit 3, p. 67) [click to enlarge]:
How often have you found yourself arguing with someone who misrepresents what you say and shoots down the resulting straw men, presents “I disagree with you” and various vacuous appeals to authority as arguments, and/or simply restates his or her position along with his or her original arguments—all this, instead of actually engaging with your actual arguments?
It may be harder to teach adults new tricks, but at least we should be able to do better by our children. My impression, however, is that the prevailing approach to research and opinion-writing in today’s K12 schools is on the wrong track. The typical assignment seems to be: identify your opinion on such and such an issue, and then track down x pieces of evidence (typically over the Internet) that support your opinion (where x typically equals 3). Problems with this approach include:
1. Opinions preceding evidence.
2. Supporting your opinion crowding out curiosity as the driving force behind research
3. Internet data mining turning up “evidence” in support of anything
4. The cutting and pasting of that “evidence” substituting for argumentation.
Decent argumentation involves accurately characterizing all known counter-arguments (drawing the line somewhere out on the crackpot end of the spectrum) and anticipating other possible (non-crackpot) counter-arguments, and then actually engaging with the substance of those arguments— without distorting it into straw. It means making a convincing case either that a given counter-argument is faulty, or that its merits vs. the merits of your opposing point(s) boil down to matters of opinion.
Of course, this should be a life skill as well as a classroom skill. But how often does life hold us accountable? As for today's children, how often do their grades reflect their ability to argue--in the best, most timeless sense of the term?
On a recent post, a commenter took issue with my notion of “full comprehension” as a measure of reading skill. In the age of post-post-modernism, how can anyone still think that full comprehension is possible?
So let me draw a distinction between linguistic comprehension—which is mostly (though not entirely) about a text’s literal meaning—and literary interpretation, which goes far beyond it, and leaves plenty of room for—well—interpretation.
Linguistic comprehension involves basic things like: knowing what the words mean in the given context, knowing what the pronouns (“it”, “they”, “this”, etc.) as well as generic terms like “that idea” and “the statesman” refer to in the context of the text and its assumed background knowledge; identifying where an interrupted clause picks up after an appositive; knowing which parts of a sentence its various modifiers modify; and having the basic background knowledge to make connections elements of the text that, literally, seem disconnected. Without this, the text will remain incoherent, and comprehension will be faulty or incomplete.
Consider this passage from "Gulliver’s Travels to Lilliput," a rather tough passage that I helped my daughter work her way through just this past week. It occurs just after a section describing how, in service of the emperor of Lilliput, Gulliver, a.k.a. "Man-Mountain," dragged fifty men-of-war from the fleet of the empire of Belfuscu over to the shores of the empire of Lilluput.
His Majesty desired that I would take some opportunity of bringing all the rest of his enemy's ships into his ports. And so unmeasurable is the ambition of princes, that he semed to think of nothing less than reducing the whole empire of Blefuscu into a province and governing it by a viceroy; of destroying the Big-endian exiles and compelling that people break the smaller end of their eggs, by which he would remain the sole monarch of the world. But I endeavored to divert him from his designs by many arguments drawn from the topics of policy as well as justice, and plainly protested that I would never been an instrument of bringing a free and brave people into slavery. And, when the matter was debated in council, the wisest part of the ministry were of my opinion. This bold declaration was so opposite to the schemes and politics of his Imperial Majesty, that he could never forgive me. And from this time began an intrigue between his Majesty and a junto of ministers maliciously bent against me, which broke out in less than two months and has like to have ended in my utter destruction.
About three weeks after this exploit there arrived a solemn embassy from Blefuscu, with humble offers of peace; which was soon concluded on conditions very advantageous to our Emperor. There were six ambassadors with a train of about five hundred persons. When their treaty was finished, wherein I did them several good offices by the credit I now had at court, their Excellencies invited me to that kingdom in the Emperor, their master’s name. I desired they would present my most humble respects to the Emperor, whose royal person I resolved to attend before I returned to my own country. Accordingly, the next time I had the honour to see our Emperor, I desired his license to wait on the Blefuscudian monarchy, which he was pleased to grant me, as I could plainly perceive, in a very cold manner but could not guess the reason, till I had a whisper from a certain person that Flimnap, the High-Treasurer, and Bolgolam, the Admiral, had represented my intercourse with those ambassadors as a mark of disaffection from which I am sure my heart was wholly free.Here are the questions I would ask to assess whether a child completely understands the text, in the linguistic sense of understanding:
I. The final volume problems in the Singapore Math Primary Mathematics Workbook (Volume 2, pp. 40-42):