A. From the last page of exercises of volume 1 (of 2 volumes) of Everyday Mathematics Student Math Journal, p. 163
4 children share 12 slices of pizza equally. How many slices does each child get? Draw a picture.
Each child gets ____ slices.
5 + 6 + 23 = _____
_____ = 3 + 3 + 12
4 + 3 + 17 = _____
____ = 9 + 2 + 9
Count by 2s.
70, 72, 74, ___, ___, ___, ___, ___, ___,___, ___, ___
B. From the last page of exercises of volume 1 (of 2 volumes) of Singapore Math Primary Mathematics Workbook (2A), p.175
312 boys and 195 girls took part in a swimming test. How many more boys than girls were there?
There were _____ more boys than girls.
There are 292 men, 149 women and 68 children on a train. How many people are there on the the train?
There are _____ people on the train.
Meihua and her sister saved $502 altogether. Meihua saved $348. How much did her sister save?
Her sister saved $______
C. Extra Credit:
Which problem set do you think the University of Chicago Math Department, as opposed to the University of Chicago Math Project, would prefer to see 2nd graders doing half-way through 2nd grade math?
Friday, November 28, 2008
A. From the last page of exercises of volume 1 (of 2 volumes) of Everyday Mathematics Student Math Journal, p. 163
Tuesday, November 25, 2008
When he insisted it was all a dream, I asked J whether it was also all just a dream for Daddy, whom I had called after it had happened so that J could hear me communicate to the other domestic authority figure the consequence of his Friday night behavior--"never, never going to the Chinese restaurant again." J dropped the subject and hasn't revisited it since.
A week later, this past Friday, I appeared unexpectedly in his doorway while he was sprawled on the floor doing something on his computer that he didn't want me to see (either trying to download games for free, or sending emails to people asking to play with their wiis and ceiling fans). He reflexively slammed the door with his foot and unintentionally hit me, hard, in the temple.
He immediately apologized, and, seeing how upset I was, ran down two flights of stairs on his own initiative and brought up an ice pack, which he'd jury-rigged from a sandwich bag around an ice cube, handed it to me, and repeatedly asked me whether I was OK.
Then, suddenly, I could see a thought flit across his eyes.
"Did you lose your memory?" He asked, ever so slightly hopefully.
Sunday, November 23, 2008
J. has been addicted to fans since he was 6 months old, but no fans enthrall him more than the ceiling fans at the local Chinese restaurant. I can practically hear them wincing at the other end of the line when I call up to order J's favorite dish, squid with spicy salt, with "please just the squid with nothing else on it--no peppers, no garlic." They most assuredly recognize who the request is coming from, and hope, hope, that that crazy blond boy won't be accompanying his mother into the restaurant when she comes to pick up the order.
Problem is, every time I bring him in he charges past the takeout counter into the dining room, and, if the fans aren't all on fast, climbs onto the first booth to reach across tea cups and torsos to grope at the fan controls. So, naturally, it had been a while since I'd taken him there with me. The last time was about 6 months ago, way back in summer when the fans--whew!--were already on high.
But last week when my husband was away and J had had a good enough week to earn his Friday squid--squid being the one meat he eats, and one he craves it almost as much as fans-on-high--I had no choice but to bring him. He'd matured a lot since summer; surely I could talk him into controlling himself.
The whole way over there, all the rewards for self-restraint and all the penalties for putting fans-on-fast were gone over ad nauseam. Yes, yes, he repeated: he understood. No sooner do we enter the restaurant, than he wrenches his hand out of mine, hurtles his way into the dining room, and throws himself across the booth, nearly knocking the teacups out of the hands of the horrified customers, while I yell "stop" over and over again, tugging back at his jacket until it practically bursts at the seams. It was the loudest, most frenzied scene the two of us had made in years.
"You are never, never, ever, in your entire life, ever going into that restaurant ever again," I repeat, repeatedly, as we head back home. "Never, ever, ever. Do you understand?"
"Yes," he repeated, contritely.
End of story? Three days later, J explains to me that on Thursday night I had fallen into a deep sleep from which I didn't awake until Saturday. He'd had to go to school by himself, come home by himself, and fend for himself for supper. Everything that I might have thought had happened on Friday, he assured me, was nothing more than a dream.
Friday, November 21, 2008
A. From "Finding Fractions of a Whole," a 5th grade Investigations (TERC) homework sheet:
1. In a school election, 141 fifth graders voted. One-third voted for Shira and two-thirds voted for Bree.
a. How many votes did Shira get? ____
b. How many votes did Bree get? ____
2. Bob, Liz, and Eli drove from Chicago to Denver. (1,050 miles).
Bob drove 1/10 of the distance.
Liz drove 4/10 of the distance.
Eli drove 1/2 of the istance.
How many miles did each person drive?
Check to make sure the total is 1,050 miles.
a. Bob: _____ miles b. Liz: _____ miles c. Eli: _____ miles
3. Carlos and Rick paid $8.75 for a present. Carlos paid 2/5 of the total amount and Rick paid 3/5 of the total.
a. How much did Carlos pay? _____
b. How much did Rick pay? ______
4. A pizza costs $12.00, including tax. Scott paid 1/4 of the total cost. Trung paid 1/3 of the total cost. Pritish paid 1/6. Bill paid the rest. How much did each person pay?
a. Scott: $____ b. Trung: $____ c. Pritish $_____ d. Bill: $____
B. From 5th grade Singapore Math Primary Mathematics 5a, fractions word problems:
1. A tank is 4/5 full of water. If 40 gal more water are needed to fill the tank completely, find the capacity of the tank.
2. There are 1400 students in a school. 1/4 of the students wear eyeglasses. 2/7 of those who wear eyeglasses are boys. How many boys in the school wear eyeglasses?
3. Larry spent 1/2 of his money on a camera and another 1/8 on a radio. The camera cost $120 more than the radio. How much money did he have at first?
4. Mrs. Ricci had $580. She used 2/5 of it to buy an electric fan. She also bought a tea set for $60. How much money did she have left?
Which problem set involves more "inquiry," "critical thinking," and "argumentation"--all those critical 21st Century Skills we keep hearing about?
Wednesday, November 19, 2008
Events in my son's school life since yesterday inspire three additions:
1. Keep all written homework directions sketchy enough that bored kids who don't pay attention won't remember the key details you only give orally.
2. Base tests not on textbooks or other written resources, but only on the notes that students take in class--another way to disfavor bored/distracted kids, who tend not to take notes, as well as disorganized kids, who lose track of their notes, along with the penmanship impaired, whose notes are often minimal and/or impossible even for him/her to decipher.
3. Never, never teach penmanship! That way only certain students (the penmanship "naturals") will end up writing fluently and clearly enough to take good notes and meet today's high standards for neatness.
Tuesday, November 18, 2008
...and make sure not too many go to the math buffs and other analytical children that traditional teachers considered "smart."
1. Don't collect homework; leave it up to kids to hand things in. Many smart but disorganized children will lose points to this.
2. Grade via inflexible rubrics that contain at least one "visual" dimension (color, creativity, neatness) that disfavors the artistically impaired. That way, no matter how well a smart, non-artistic kid does on the more academic components, he'll still fall short of a top grade. All the better if you design the entire assignment to strike both smart kids and their parents as inane. That way many of these parents won't bother to make sure their unmotivated children fulfill all the requirements.
3. Don't hand things back until the end of the marking period. This keeps parents from knowing that their kids are losing points for trivial reasons--e.g., for not showing multiple solutions or explaining their answers or using enough colors--and intervening accordingly.
4. Assure kids--and their parents--that the second tier grade that most of them are getting (a.k.a., "B", "3", or "proficient") is a very good grade.
5. Keep the math problems easy and define "exceeding the standard" to include explaining your answers as verbosely as possible, drawing neat, colorful visual representations, and providing multiple "creative" strategies by solving simple problems over and over again. The easier the student finds the math, the less motivated s/he will be to comply, and the more points s/he will lose.
6. Do the same with science, language arts, and foreign language: water down the academic, analytical components, and up the "creative," showy visual requirements.
Sunday, November 16, 2008
Somehow, though, my son is getting A's in his reading and writing assignments.
But then there's his "All About Me" shield:
|In not wanting to "privilege" the written word, we're shortchanging those with art disabilities.|
...And those whose parents aren't motivated to oversee the more questionable assignments.
Thursday, November 13, 2008
1. From the beginning of the 4th grade investigations curriculum
2. From the beginning of 4th grade Singapore Math, Primary Mathematics 4A
Wednesday, November 12, 2008
The more papers I read in cross-cultural pragmatics, the more convinced I am that foreign language classes are overly marginalizing grammar and vocabulary, even in light of their new emphasis on "communicative competence."
I just finished re-reading, for a class I'm about to teach, a 2001 paper by Thom Hudson entitled "Indicators for pragmatic instruction: some quantitative tools." Focusing on speech acts like apologies, requests, and refusals, Hudson notes that intermediate-level ESL students have little trouble figuring out when a particular speech act is called for, or observing the appropriate linguistic conventions in formulating the act in question.
What trips them up, instead, are the more language-intensive aspects of effective apologies, requests, and refusals--e.g., providing appropriate amounts of accompanying explanation. In apologizing for missing a meeting with your boss, or asking a neighbor to lower the volume of his stereo at night, or turning down an invitation for lunch, it turns out, it helps to have a large enough vocabulary and repertoire of syntactic structures to effectively explain why you missed the meeting, or are bothered by the music, or aren't available at lunch time.
However reluctant today's foreign language teachers are to teach the more left-brained nuts and bolts of language content, it turns out that these are key to successfully executing the more social, right-brained aspects of language use.
Monday, November 10, 2008
And this child isn't even one of the kids we're corrupting on the Continental Math League.
Rather, it was his mother who showed him "how I learned to solve this kind of problem." And he found her way more efficient than the "splitting" that his teacher prefers. But then his teacher caught him stacking, and advised him that when you split instead of stack it's much easier to explain your answer (key to getting full credit).
Meanwhile, at Continental Math League practice today, I reminded kids that they can solve the multi-digit calculations using any method they want. The first few times they opted for stacking, I asked why.
"It's much faster," said one. "With splitting you have to keep splitting and splitting, and then you lose track of what you're doing and forget what's going on."
Saturday, November 8, 2008
Allison's post yesterday at kitchentablemath on how difficult it is to convince fellow parents of Reform Math's shortcomings coincides with my receipt of our school's latest Home and School Association report.
According to this report, 89% of the 70+ parents who attended the school math consultant's parent presentation rated it "helpful" to "extremely helpful." This, despite the fact that she made an evidence-free argument against having children solve problems by "stacking" (arranging numbers vertically and using the standard algorithms), asserted that we should be teaching our children the multiplication tables at home because teachers don't have time to in school, and spent about half of her presentation soliciting our multiple strategies for multiplying 85 by 4 and dividing 120 by 3, thus demonstrating how it is that time is used up in the classroom.
Like Allison, I feel that no one takes me seriously when I carry on about the dire effects of Reform Math.
No one, that is, except my friends. But these people, for all the concerns they've been expressing, appear to have been written off by the HSA as "friends of lefty."
Here, for what it's worth, is the latest from one of these friends, the parent of a 1st grade mental math whizz:
Just as we expected, J has been struggling with the “show your work” portion of his math homework. He got an “incomplete” one week for failing to illustrate how he added 2 and 2, or some such thing. This week, on a “3 + 5” problem, he had a solution: he drew a stick figure with a thought bubble, put “3 + 5” in it, and labeled it “me”!
Thursday, November 6, 2008
1. This week's 2nd grade Investigations homework:
2. From a similar point in the 2nd grade Singapore Math workbook (Primary Mathematics 2A):
Doubles vs. multiples. (Apologies for the poor resolution; I'm still learning how to scan).
Wednesday, November 5, 2008
Over the last few weeks I've been trying, and thus far failing, to acquire some basic programming software for my autistic son that I can run on a computer at home. BASIC? FORTRAN? Even C?
The more time I spend on this, the more I wonder how many people still have opportunities to learn the programming fundamentals that to me seem not just essential, but fun: recursion, nested loops, flow charts, logic.
Back when I was in high school, students could write basic programs directly onto their primitive home computers. What about today's high school students? How many of them are nesting loops or creating self-calling functions--as opposed to writing html tags, cutting and pasting from someone else's code, or "programming" games using high-level programs like Director?
In the course of these musings, I came across a recent blog by a Penn graduate that questions whether even college-level courses are now teaching such fundamentals as pointers and recursion.
The more today's programming limits itself to "cut and paste" and high-level parameter setting, the more we lose the kinds of flexibility and creativity crucial for novel software applications--whether for entertainment, research, or education.
All the more reason for me to track down a programming package for my autistic son, who may one day find his most promising niche in what may already be a dying vocation! Any pointers?
Monday, November 3, 2008
Psychologists say we're biased towards the perception that other people are more biased than we are.--Today's Phildelphia Inquirer Science Section.
Perhaps that explains my reaction to the latest Grand Theory of Autism that's crossed my desk: the Binaural Phase Shift Theory. This theory, which, so far as I know, has one acolyte (whom I'll refer to as "BPST") holds that the language deficit in autism is caused by out of sync processing of auditory information from right and left ears. It's like seeing double, except that it's hearing double.
When BPST emailed me about his theory two days ago, it was my first time being directly contacted by a Grand Theorist. And, since I have a deaf, autistic son with a cochlear implant who hears only out of one ear, I was tempted to reply:
As I understand it, your theory is based on phrase disynchronicity inThis, of course, led to a more extensive exchange, which I excerpt here:
*binaural* hearing. This assumes that binaural hearing exists in a given
child in the first place.
My favorite theory of autism, because it fits *my* son so well, is the
Theory of Mind/Extreme Maleness theory. It explains why he's so unusually
good at math, so good at visual reasoning, and so unusually physically
But, since autism is such an extremely heterogeneous condition, it would
be in appropriate for me to assume that this theory applies to autistic
children in general.
BPST: And exactly how profound is profoundly deaf. Most autistic children are not profoundly deaf.Arguably, BPST isn't worth two seconds of my time. But he claims to be an MD, and claims to be surgically inserting "unilateral large tympanostomy tubes" (ear tubes) to reduce hearing in one ear as a "treatment" for autism.
Lefty: Yes, indeed: no surprise there. Most autistic children aren't deaf at all.
BPST: Many autistic children will be missing or reduced certain segments of their hearing bandwith like the lower frequencies
Lefty: Actually, it's typically the higher frequencies that are missing in deafness. That's why many deaf people use low-frequency doorbells, for example.
BPST: I believe that lower frequencies with longer bandwidths are the easiest to synchronize and are heard in the womb by the placenta [sic].
Lefty: "I believe" isn't an argument.
BPST: I know a profoundly autistic adult with right ear has normal hearing at all frequencies but her left ear has lost all frequencies except the midrange. As a newborn she never was able to synchronize low range and gradually progress to synchronizing the mid range. At her cortex the left midrange frequencies appear out of phase with right midrange frequencies. The left midrange frequencies reach the cortex out of phase with the right midrange. Voices that have midrange are profoundly blurred and she has no functional use of speech.
Lefty: Nor is one anecdote.
BPST: Are you saying that before the unilateral implant the BAER were totally flat? Did you see the BAER? I bet you did not.
Lefty: That's rather presumptuous of you. Yes, I did see the BAER. No evidence of hearing at any level.
BPST: I am no implant expert but I sincerely doubt that an implant can produce anywhere near the complex signal discrimination and resolution of a single normal auditory system. I would not expect a unilateral cochlear implant to provide the resolution of ordinary unilateral hearing.
Lefty: He can discriminate all speech sounds, as can most kids implanted before 2 years of age.
BPST: Unless a unilateral side is completely wiped out with no BAER signal at any frequency, I would assume that binaural traces from the weak ear still reach cortex and could interfere with auditory resolution from the strong ear.
Lefty: Such an assumption isn't warranted in my son's case.
BPST: I am no expert on the theory of the mind although I read and forgot it briefly. However autistics need simultaneous attention of same abstract issue to develop vocabulary [nouns] of abstract things.
Lefty: My son has an excellent vocabulary of nouns (not to mention verbs, prepositions, and function words), and of abstract things in general. (What's the connection between "abstract things" and binaural phase shift???)
BPST: It is difficult for autistics to follow the rapid verbal thought processing of the nonautistic and for them to precisely follow what abstract thing is being named.
Lefty: My son has no trouble with abstraction. Many people underestimate the capacity of autistic children for abstraction.
BPST: As for " extreme maleness" the median age that boys learn speech occurs after girls. Your son does not have more testosterone than other boys unless he has a hormone problem.
Lefty: Do you have actual evidence against Baron-Cohen's findings on the testosterone connection?
BPST: The "extreme male mind theory" has its roots in cultural bias that accepts delayed verbal and enchanced math skills in males vs females.
Lefty: Do you have actual evidence against Baron-Cohen's findings on this? Remember, Baron-Cohen's findings relate to statistical tendencies, not absolutes.
BPST: If [your son's] cortex is so excellent at conscious reasoning [association patterns] then why can't he form association pattens for auditory input? Ask your self why his auditory association patterns fail to meet the performance standard of the visual association areas. [there are many questions you can't approach]
Lefty: What makes you think he can't form association patterns for auditory input? In fact, he does so all the time. You don't know him, and you shouldn't be diagnosing him at a distance.
BPST: [The Binaural Phase Shift Theory] likely applies to a large subset of autistic children but not all.
Lefty: "likely?" "large subset"? I would not presume this of my pet theory.
BPST: You fail to understand the significance of simultaneous cortical synchronization.
Lefty: Peer reviewed articles?
BPST: I can sense your bias against it, and to accept it would fundamentally change your belief system. Your mind is controlled by what you heard in the past. There is simply too much programming in your head to change. But I suspect I was right about your son and his ability puzzles. A lot of people thought their were weapons of mass destruction in Iraq in March 2003. You have believed a lot a things that were wrong. So would it even matter if you were wrong about this?
Lefty: And you? And the vaccine theorists, the Diet Theorists, the Interactive Metronome Theorists, the Mozart Effect theorists, the Floor Time theorists? I've been in this business for over 10 years, and when it comes to "causes" and "cures," I've heard it all. But I'm always open to actual evidence, as opposed to beliefs, assumptions, anecdotes, and wishful thinking.
And he's actively recruiting young children as subjects.
And I worry that he'll find a way divert funding towards his practices and away from (what I consider to be) more promising areas of research and treatment.