extra help

C. has completed all but one of his make-up math homework assignments & was in the midst of doing the "POW" (From 1994 through 1997, the cost per mile of owning and operating a car increased by 2.2 cents per year, etc.) when a kid in his class called for help.

A gratifying moment. Possibly a milestone!

I could hear the boy on the other end jabbering away, firing off one question after another. Chris' answers were precise and correct. (I think.)

The other boy didn't sound convinced.

"Are you sure that's right?"

"Yeah, that's the answer."

"Are you sure that's right?"

"That's the answer I got."

"Are you sure that's right?" The kid seemed to be needing a lot of reassurance.

Finally C. said, "It's right, my mom checked it," and I thought yikes. The blind leading the blind.

When he hung up, C. said "That was fun, getting called for help with math." I agreed that it was, and after we'd basked in glory for a bit I said, "Why did he call you?"

"He said he tried everyone else and no one was answering their phone."


Happy New Year!

time travel

I have spent most of today thinking tomorrow was New Year's Eve.

It's not.

New Year's Eve is today.

Which means the brisket should have gone into the oven quite some time ago.

Maybe we'll just skip the food and go straight to the champagne.

white paper lollapalooza

For awhile there, after it became impossible for C. to share a room with Andrew, C. was sleeping on the floor of my office.^{i, ii} This arrangement led to a new bedtime ritual: each night, after Ed read C. a story, the two of them would make a bet on how many files would be open on my (Mac) Desktop. I think the count once topped out in the 50s.

(Opening up way too many files is one of the behaviors I may be attempting to correct in the New Year. see: Book Club)

My point is: I am supposed to be writing a chapter on pigs.

Which means my Next Action is finding all my stuff about pigs, and that means my First Next Action is drilling down through the open documents on my computer screen to find all my stuff on pigs.

That's what I was doing when I found this passage in a 2006 Brookings report:

Although the 2006 edition of the CWI [Child and Youth Well-Being Index] indicates that overall well-being increased somewhat in 2005, once again children’s performance in the education domain was flat. This outcome for 2005 continues a trend that has now lasted for three decades. The lack of significant improvement in educational achievement is especially remarkable because national, state, and local policy has focused on improved educational performance almost continuously since the launch of Sputnik in 1957. The nation has been alerted to achievement problems by a host of national reports, and per-pupil spending has more than doubled since 1970. Moreover, schools have undergone wave after wave of educational reform. Yet the student achievement flatline persists. To make matters worse, the gap in performance between poor and minority students on the one hand and middle class students on the other, has narrowed only slightly and is still very large.

Sauve qui peut.


ⁱ He was sleeping on the floor because the room was too small to hold a desk and a bed at the same time.

ⁱⁱ An example of the difficulty of "breaking set." It took months for us to figure out that C. should keep the large-ish bedroom the two of them had shared, Andrew should move into my tiny office (I think -- don't know -- autistic kids sometimes fair better in smaller spaces), and I should set up shop downstairs in the dining room. Of course once we'd made the cognitive shift from "dining room" to "home office," the solution seemed obvious.

the tennis ball problem

A tennis ball can with radius r holds a certain number of tennis balls also with the same radius. The amount of space in the tennis ball can that is not occupied by the tennis balls equals at most the volume of one tennis ball. How many tennis balls does the can hold?

Barry sent me this problem months ago & I've been avoiding it because geometry scares me.

I finally shamed myself into attempting it just now & got an answer of 2. Unfortunately, I typed up my solution, loaded it to flickr, but flickr is on the blink so I can't post.

I solved it (assuming I did solve it) algebraically, then resorted to "logic and reasoning" to check.

Unfortunately, I'm confused by logic and reasoning at the moment.

I was thinking that because a sphere is 2/3 of a cylinder of same radius, with each ball you put inside a same-radius cylinder you end up with 1/3 of a ball's worth of empty space....which now implies to me that the answer should be 3 balls, not 2.

sigh


update (1):

I'm mixing things up

The 1/3 that's left over isn't 1/3 of a tennis ball. It's 1/3 of a cylinder with the same radius as the tennis ball.

I better forget the logic and reasoning & stick to algebra.

Assuming I didn't screw up the algebra, that is.


update (2):

OK, so in between dealing with screaming autistic youths, loading the dishwasher, & microwaving a taco for Jimmy, I realized that I don't need to know "how much of a tennis ball-sized volume is left over."

I just need to know how much empty volume is left over, period, then figure out how many multiples of that empty space add up to the volume of 1 tennis ball.

volume of tennis ball with radius r: 4/3πr^3
height of cylinder that fits just one tennis ball: 2r
volume of cylinder w/height of 2r: πr^2h = 2πr^3

vol. of cylinder - vol. of 1 tennis ball = vol. of empty space

2πr^3 - 4/3πr^3 = 2/3πr^3 empty space left over when 1 tennis ball is in cylinder

2 tennis balls leaves 2 empty spaces, each 2/3πr^3 in volume:

2/3πr^3 + 2/3πr^3 = 4/3πr^3, which is the volume of 1 tennis ball

so: 2 tennis balls

update (3):

Barry says the problem comes from Dolciani's Algebra 2! (I include the exclamation point because I'm happy to discover I am able to solve a problem from that book. cool.)

original wording:

A tennis ball can in the shape of a cylinder with a flat top and bottom of the same radius as the tennis balls is designed so that the space inside the can that is NOT occupied by the balls has volume at most equal to the volume of one ball. What is the largest nubmer of balls the can will contain?

wild goose chases

Found the quote -- it's from what is apparently a classic book on Mathematical Problem Solving by Alan Schoenfeld.

The Wild Goose Chase in Problem Solving

When solving standard mathematics problems, students normally recall and apply learned procedures in a straightforward way. However, if the problem is unfamiliar, some students simply pick a method and keep persistently on the same track for a long time without getting anywhere. Schoenfeld (1987) described this behaviour as chasing the wild mathematical goose.

That's not the meaning I remember.

I probably got it wrong.

knock on wood
true confession
wild goose chase

true confession

I found the wild-goose-chase problem, the one that took me many steps not to do while C. solved it at once using "logic and reasoning."

Find the value of r so the line that passes through each pair of points has the given slope.

44. (-2, 7), (r, 3), m = 4/3 ^*

C.'s solution:
7 - 3 = 4/3 (-2 - r)
4 = (-8 - 4r)/3
12 = -8 - 4r
20 = -4r
r = -5

My non-solution:
y = 4/3x + b
7 = 4/3(-2) + b
7 = -8/3 + b
21/3 + 8/3 = b
29/3 = b

I seem simply to have stopped at this point. I don't know why. Looking at my solution page now is a bit like examining a single-vehicle accident scene trying to determine how the driver flipped his car twice in broad daylight & nice weather.

Alternatively, I didn't stop with 29/3 = b (I remember having done more steps) but the rest of my perambulations are recorded on some other piece of paper, not this one.

We'll never know.

knock on wood
true confession
wild goose chase


^* source: Glencoe Algebra, p. 261

Elmore on the politics of education reform

"Hence the knowledge that is enacted in curriculum and pedagogy becomes a
byproduct of the political incentives that operate on teachers-discrete bits of information, emphasis on coverage rather than depth, diffuse and hard-to-understand expectations for student learning, little convergence between the hard day-to-day decisions about what to teach and the largely content-free tests used to assess student performance, and a view of pedagogy as a function of the personal tastes and aptitudes of teachers rather than as a function of external professional norms. Students who do well in such a system recognize that they are being judged largely on their command of the rules of the game, which reward aptitude rather than sustained effort in the pursuit of clear expectations. All systems have a code; the job of the student is to break it. Some do, some don't."

Richard Elmore
The Politics of Education Reform

knock on wood

C. is having a good math year.

I've hesitated to say so; I think all of us, the teacher included, are wondering whether this is real. But I think we're past the point at which it makes sense to wonder whether this is a fluke.

The year started badly - dreadfully, in fact - with a D- on the first test.

Things picked up a bit when I pitched in with multiple-houred sessions of parent reteaching, but I figured I was looking at another school year of emergency preteaching, reteaching, and all the rest of the fol de rol.

I wasn't.

Those days are behind us (knock on wood!) I don't preteach, I do very little reteaching, and -- this is the interesting part -- when I do need to reteach, C. picks up the concept rapidly.

This vacation I'm doing a fair amount of reteaching because C. missed 4 days of the last 9 full days of school. He could go in for Extra Help and ask his teacher to reteach the material, but since I want him to return to school after break having completed all 4 of the missed homework assignments, I also need to teach the lessons. In other words, this is something I want to do, not something I absolutely have to do.

Going into the break, I was a bit worried. I had found the homework assignments challenging myself. Somehow the point-slope formula was mixing me up, and I'd never done a word problem involving linear functions.

So I had a time of it figuring the concepts out myself. How was C going to handle it?

He handled it fine.

Yesterday he correctly and efficiently did a point-slope problem I had hosed entirely, then explained his success by saying, "I used logic and reasoning."

Yeah, well, I used logic and reason, too, but I went on a wild goose chase instead of seeing the simple and should-have-been obvious solution. ^* C. looked at the problem and saw the solution immediately.

C. also missed the classroom lesson on finding the equation of a line parallel to another line. That seemed like a fairly big concept to me.

No problem. He instantly saw that a parallel line would have to have the same slope in order to be parallel -- and that it would have to have a different y-intercept if it wasn't going to be the same line. Looking at a line on the coordinate plane while thinking all this over, he had one of his "Oh, yeah" moments. After that he knew how to find the equation without further instruction.

We owe all of this to C's teachers. He has two math teachers this year because 8th graders have "Math Lab" 3 days a week. Both are terrific as far as I can tell. They're also experienced. The main teacher has been at the school for close to 10 years (I think); I'm not sure how long the other teacher has been here but I do know that he taught in NYC schools prior to coming to Irvington. He may also be in the 10-year category or close to.

They've done a fantastic job.


Math A archived exams
Math A Regents prep

knock on wood
true confession
wild goose chase


^* A few years back I read a study that distinguished between great math students and OK math students that I probably can't find again. The difference between the two groups was that the great students produced efficient and elegant solutions while the OK math students went on wild goose chases. They'd solve the problem, but it wasn't pretty.

Mmm... Tasty Brains...

Just what we've been searching for - a scientific explanation for all of the recipe posts at KTM: cooking may have been a significant driver of human evolution. On the other hand, this theory seems only to deepen the mystery of how some tribes managed to flourish without ever having learned to cook.

I tend to think of the advent of cooking as having a huge impact on the quality of the diet. In fact, I can't think of any increase in the quality of diet in the history of life that is bigger. And repeatedly we have evidence in biology of increases in dietary quality affecting bodies. The food was softer, easier to eat, with a higher density of calories—so this led to smaller guts, and, since the food was providing more energy, we see more evidence of energy use by the body. There's only one time it could have happened on that basis; that is, with the evolution of Homo erectus somewhere between 1.6 [million] and 1.8 million years ago. [/SNIP]

...Homo erectus is the species that has the biggest drop in tooth size in human evolution, from the previous species, which in that case was Homo habilis. There wasn't any drop in tooth size as large as that at any later point in human evolution. We don't know exactly about the gut, but the normal argument is that if you reconstruct the ribs, you have reduced flaring of the ribs. Up until this point you have ribs that went out to apparently hold a big belly, which is what chimps and gorillas are like, and then at this point [when Homo erectus arose] the ribs go flat, meaning you've got now a flatter belly and, therefore, smaller guts. And then you have more energy being used; people interpret the locomotor skeleton as meaning that the distances traveled every day are much farther. And the brain has one of its larger rises in size.

*I originally intended to title this post with a pithy reference to Brillat-Savarin, but George Romero is really much more my style.

Jeanne d'Arc

We went to a party on the Jeanne d'Arc helicopter carrier last night.

Men in uniform.

Woo hoo!

today's factoid 2

...somewhere between 97 and 98 percent of American voters have never looked at a blog in their lives...
source:
Foggy Bloggum by David Frum

Does that make us outliers?

in Pakistan

John Moore, photographer, describes the day.

I'm going to show this to C.

Siouxsie and the Banshees - Il Est Né le Divin Enfant

I'd never heard of Siouxsie, but I like this version....

Il est né le divin enfant

Benazir Bhutto, rip

Here's Hitchens:

The sternest critic of Benazir Bhutto would not have been able to deny that she possessed an extraordinary degree of physical courage.

long division in the time of computers

"with apologies to Gabriel Garcia Marquez" (pdf file)

"When we assert that “this is the factorization of a number into primes,” the Fundamental Theorem of Arithmetic is lurking in the background."

One of these days I will be a person who knows what that means.

[pause]

Well, I have now skimmed the entire pdf file and I have no idea what it's talking about, or what the author's views on the place of long division in the curriculum is or is not.

So that was enlightening.


the long version (pdf file)

from anonymous...

Thanks for reminding me why we homeschool.


chuckle

edu-jargon

EDUCATIONAL JARGON GENERATOR

I'm very amused by this.

Educational Jargon Generator.

Singapore Math v. Everyday Math -- You Be The Critic

I recently had a fantastic discussion with an administrator at my local public middle school. This person is new to the Town and school. My impression is that she is less pedagogically dogmatic than most I have met. She does not have an direct involvement with EM. In Middle School she see the pre-algebra and algebra math of 7th and 8th grade. She hears from many other teachers and administrators that EM is fantastic and wonderful and perfect for our school. However, she, and most other educators in town notice that this "wonderful" elementary math program isn't connecting well to middle school and high school. Kids aren't doing all that great on those high school courses.

The reigning wisdom has been that the problem can't be EM. It must be the middle school math program (with are traditional pre-algebra and algebra courses).

There is a failure to analyze their underlying assumptions. Nobody is willing to consider that EM might not be the best preparation for advancing in math.

But this administrator has shown an interest. I gave her my opinion on the matter at a forum a couple weeks ago and she was interested in what I had to say. I advised that she put EM and Singapore Math side by side and compare to cut through all the rhetoric. She could make up her own mind about it. I finished by saying if there was one thing I could convince this district to do, it would be to teach bar models as a means of problem solving.

She had just read an article about bar models. She wants to know more.

So here is your chance, everyone. If you had limited time available with an interested administrator, and you had 1st through 6th grade Singapore and Everyday Math at your disposal -- where would you start? What pages, links, sections would you highlight or focus on?

Chocolate Pecan Pie & Fractions

I have been baking chocolate pecan pie almost every Christmas for over 20 years. Tonight my 5th grade daughter helped me with this recipe:

CHOCOLATE PECAN PIE
1 pie shell, unbaked
Filling:
1/4 cup (1/2 stick) unsalted butter
2 ounces unsweetened chocolate
3 large eggs
1 cup sugar
3/4 cup dark corn syrup or sugar cane syrup
1/2 teaspoon pure vanilla extract
3 tablespoons bourbon or rum (optional)
1/4 teaspoon salt
1 1/2 cups pecan halves
Preheat the oven to 350 degrees F.
To make the filling: melt the butter and chocolate in a small saucepan over medium-low heat, remove from heat and let cool. Beat the eggs in a large mixing bowl until frothy and then blend in the sugar. Stir in the syrup, vanilla, bourbon, salt, and the melted butter mixture until well blended.
Arrange the pecans on the bottom of the pie crust and carefully pour the egg mixture over them. Bake until the filling is set and slightly puffed, about 45-50 minutes. Test for doneness by sticking a thin knife in the center of the pie, if it comes out pretty clean, you're good to go. Transfer the pie to rack and cool completely before cutting.

We made two pies because we’re having 15 guests for dinner on Christmas Day. My daughter instantly converted all the fractions to the quantities needed for two pies. She was faster than I was. It may not seem like such a great achievement to some, but to me it was wonderful.

Thank you, Kumon!

where does tofu come from?

Kitchen Chemistry at MIT

MIT physics lectures on the web

Walter H. G. Lewin, 71, a physics professor, has long had a cult following at M.I.T. And he has now emerged as an international Internet guru, thanks to the global classroom the institute created to spread knowledge through cyberspace.

Professor Lewin’s videotaped physics lectures, free online on the OpenCourseWare of the Massachusetts Institute of Technology, have won him devotees across the country and beyond who stuff his e-mail in-box with praise.

“Through your inspiring video lectures i have managed to see just how BEAUTIFUL Physics is, both astounding and simple,” a 17-year-old from India e-mailed recently.

Steve Boigon, 62, a florist from San Diego, wrote, “I walk with a new spring in my step and I look at life through physics-colored eyes.”

Professor Lewin delivers his lectures with the panache of Julia Child bringing French cooking to amateurs and the zany theatricality of YouTube’s greatest hits. He is part of a new generation of academic stars who hold forth in cyberspace on their college Web sites and even, without charge, on iTunes U, which went up in May on Apple’s iTunes Store.

At 71, Physics Professor is a Web Star

Or, if physics doesn't interest you, you can go audit the fancy-shmancy Yale course on death.


Chronicle of Higher Education on Yale online courses
Yale Offers Free Online Courses
Open Learning Initiative at Carnegie Mellon
MIT Open Courseware

the key to life

beautiful

show your kids

Intercepts reports that Will Smith and his wife homeschool their children.

a question most 8th graders miss

Maria has $5.00 more than Joseph. Together they have $37.50. Which of these equations would you use to find the amount of money Joseph has?

A. j + (5 x j) = $37.50

B. j + ( j ÷ 5) = $37.50

C. 5 x j = $37.50 + j

D. 2 x ( j + 5) = $37.50

E. j + j +5 = $37.50


This item ... is missed by the majority of eighth-grade students in the NWEA norm group.

The Proficiency Illusion
John Cronin, et al
October 2007

I'm going to guess C. would get this right.

We'll see.

Of course, this is a classic bar model problem in Singapore Math. Third grade level, I believe, although you don't use variables. Still, the bar model is as abstract as a letter variable, or close to.

today's factoid

How many teachers in the U.S.?

3,000,000 according to Gadfly

Direct Instruction post at Fordham

check yourself before you wreck yourself

Fordham draws the familiar analogy between scripting in medicine and scripting in education: do you want your physician to be inventing new concepts in hand-washing on the spot or do you want him to do it the way the Best Practices checklist says to do it?

What often goes unremarked in these discussions is the fact that master teachers and professors often write their own scripts. These scripts will be revised, edited, and polished over the years, but they are scripts nonetheless. Teachers of older students and professors who teach seminars write questions and create discussion structures they repeatedly use. Here, for instance, is a terrific set of videos of a professional development session in which the presenter explains how to write one's own DI questions. Again, these questions and structures are revised, edited, and polished from one semester or school year to the next, but the fact is that superb teachers and professors don't wing it. Nor do they reinvent each course from scratch each year.

Creative people use scripts.

Another point: creativity comes in many forms. There is no reason to assume that a 1st grade teacher following a script could be replaced by a robot following the same script. Here is Ken on the question of scripting: (you may have to hit refresh to bring the page up)

Inevitably, whenever Direct Instruction (DI) is discussed the subject of “scripting” is raised. One frequent objection is that the scripts stifle teacher creativity. Nothing could be further from the truth.

Before we jump in, let’s first look at some sample scripts. Here’s a sample script on Writing Fractions. Here’s one on Subtraction. And, here’s another. (pdf files)

In DI, teachers use pre-designed scripts when teaching. The scripts are based on extensive research regarding student retention, and every aspect of every script is based upon results that were demonstrated through research. The great advantage of this approach is that every teacher using the script becomes the beneficiary of that research and will probably teach much more effectively than if left to her own devices.

DI designers test the programs carefully before publishing them and each DI program is extensively revised based on specific student error data from the field test. Scripting the lessons allows sharing of these “polished stones” across teachers. Also scripting helps reduce the amount of teacher talk. Children learn by working through the sequence of tasks with carefully timed comments from the teacher. Children learn little from straight teacher talk. Too much teacher talk decreases pupil-motivation, draws out the lesson length unnecessarily, and often causes confusion by changing the focus of the tasks, disrupting the development of the larger generalization, of which a teacher the first time through is usually unaware.

Also, without guidance, teachers may use language that students do not understand or that distracts students’ attention from examples. Scripts also allow aides, parents, and other paraprofessionals to assume teaching responsibilities, resulting in increased quality instructional time.

Moreover, even though the DI programs are carefully tested and scripted, successful use of them requires training in the special techniques of delivery. Teachers must make many decisions in response to the children's performance. Some of the most important decisions involve placing each child appropriately and moving the children through the lessons at a pace that maximizes their learning potential.

Lastly, the scripted presentations do not comprise the whole lesson, and the lessons do not comprise the whole school day. There are opportunities for group and independent work. A good DI teacher also creates additional activities that allow students to make use of their learning in various situations. So, there is a great deal of teacher creativity involved in the interpretation and presentation of the script, in attending to the needs and progress of all students and in designing expansion activities.

why is scripting used in Direct Instruction?
big concepts in DI
Explicit Direct Instruction professional development videos (brief - very worthwhile)
Carol Gambill in a nutshell

seamless (w)holes

from instructivist:

[Carolyn once said that math was "a seamless whole" inside her head,...]

I don't know if this ties in with the idea of a seamless whole, but it has occurred to me that discrete skills are needed first before one can appreciate the connectedness of math. Without these concrete skills, math is more like a seamless black hole.

This became apparent to me again when teaching a group of seventh and eighth graders brought up on EM and currently using CMP who are a tabula rasa when it comes to the simplest bits of math knowledge. They can't do any operations with fractions (e.g. change mixed numbers to improper fractions let alone addition and division), can't divide decimals, don't have knowledge of even rudimentary geometry... One wonders what they have been doing for seven and eight years.

The seventh graders are currently in the CMP stretching and shrinking stage. Their homework consisted of finding the scale factor of two rectangles the width of which goes from 1.5 cm to 3 cm. So the idea was to divide 3 by 1.5 (they can't do it because they can't divide decimals). When I tried to show an alternative way of division using fractions to demonstrate the connectedness of math (seamless whole), I ran into trouble, too. They don't have the discrete skills of seeing 1.5 as 1 1/2, then changing this mixed number to 3/2 and dividing 3 by 3/2 (they absolutely can't divide fractions and moreover don't see 3 as 3/1. It would have been spectacular to make them experience with understanding that the more complicated decimal division problem 3/1.5 virtually solves itself when you divide the respective fractions (3 divided by 3/2). Invert and multiply but they have never heard of reciprocals and how they work. The 3 cancels and 2 is left standing without much ado!

So the upshot is: they use Connected Mathematics but can't see the connectedness of math because they don't have discrete skills (skills they could have learned through drill and kill but haven't). So to them, math is a seamless black hole from which not even light can escape.

This one's going in the Greatest Hits file. (on the sidebar)


wholes, not parts
top down teaching
whole math taught wholly

book club

We may need to form a book club for interested folks. I say this selfishly because I need people to help me brainstorm my way through Karen Pryor's Don't Shoot the Dog, especially now that I'm hyper-aware of how difficult it is to generalize newly-acquired knowledge.

Pryor's book is brilliant. Reading it I see that I've been barking up the wrong tree thinking cognitive science has the answers. (barking?) Cognitive science does have answers; it's a valuable and riveting field. But when it comes to education the behaviorists are way out in front. Direct Instruction, Precision Teaching, Don't Shoot the Dog -- they're so far out in front they're disappeared from view.

No wonder their work is no longer taught in ed schools. (scroll down)


life-altering factoid number 1

Chickens procrastinate.

Using either fixed or variable schedules, extremely long sequences of behavior can be trained. A baby chick can be induced to peck a button a hundred times or more for each grain of corn. For humans there are many examples of delayed gratification. One psychologist jokes that the longest schedule of unreinforced behavior in human existence is graduate school. [ed.: followed closely by middle school]

[snip]

Another phenomenon occurs on very long schedules: slow starts. The chick pecks away at a steady rate once it gets started, because each peck brings it nearer to reinforcement, but researchers have noted that a chick tends to "put off" starting for longer periods as the schedule of reinforcement gets longer.

This is sometimes called delayed start of long-duration behavior, and it's a very familiar aspect of human life. On any long task, from doing the income taxes to cleaning out the garage, one can think of endless reasons for not starting now. Writing, even sometimes just the writing of a letter, is a long-duration behavior. Once it gets started, things usually roll along fairly well, but, oh! it's so hard to make oneself sit down and begin, James Thurber found it so difficult to start an article that he sometimes fooled his wife (who was understandably anxious for him to write articles since that was how the rent got paid) by lying on a couch in his study all morning reading a book in one hand while tapping the typewriter keys at random with the other.

Don't Shoot the Dog, Revised Edition, p. 25

palisdesk on change in ed school curriculum
The Misbehavior of Organisms
Marian Breland Bailey: many lives (pdf file)
book club
a heroine I didn't know I had

Rote Learning in Core Knowledge, An Example

Last week, my son came home from school with a study sheet for the last big test for the year. The test covered some basic US geography, including the names of the Great Lakes, some facts about the Mississippi River, some facts about the US Flag, and locating some major geographical features. In addition, the students were supposed to be able to name at least 25 of the states, given a list showing only the first letter of each state.

Over the next one-and-a-half weeks, A. and I studied for his test about 15 minutes each night. This studying was the exactly the sort of studying that ed. schools teach as being the most harmful: pure rote memorization.

Early in this process, A. objected to the continued practice of the entire list on the basis that, "I only have to know 25 states, not all the states." My sympathy was notably limited; we studied the whole list. 8-)

By a couple of days before the test, A. was pretty reliably naming all the states starting with a given letter when prompted with that letter. (BTW, "M" is particularly annoying.) By this point, he was starting to think that getting the entire list right was pretty cool.

The day before the test, the teacher announced to the class that any child who could name all the states would get extra credit and a small prize.

Of the 26 kids in the class, 8 named all the states on the test the next day. None of those kids needed to have his or her self-confidence artificially boosted after the test, and they all now have a much better understanding of the value of hard work.

fight the power

Math is hard.

Didn't Barbie say that?

Mr. Percent

I'm having way too much fun.

how to pronounce ISEE

I - see

Now you know.

On the other hand, if you go around calling the ISEE the I-S-E-E, people do know what you're talking about.

personal narrative

from Tracy W:

Okay, I love my calculator. Sharp EL-5120. It's on my desk at the moment. It's not much to look at, but its functionality means that it rocks my world. In terms of calculator-adoration I am probably in the top 1% of the world's population. My calculator has literally travelled around the world with me (there's no way I'd trust it to any removal company). I'm not a poet, but if I was I would write love poems to my calculator. The only reason I do not sleep with my calculator is that I fear it will disappear down the end of the bed and I will never see it again. When it comes to using calculators, I strongly suspect I am not normal. However, despite my deep and undying affection for my calculator I am sometimes without it, and on those occasions it is useful to be able to do basic arithmetic such as long division with pencil and paper or in my head. This may not be normal, but why should we educate kids merely to be normal people anyway?

Priceless!

SAT problem

YouTube rules.

I love this guy

But I still need a percent chart.

help desk - percent chart

Is there an all-around mode of charting the values in percent problems similar to the charting taught for distance problems?

this one's fun

During a sale, a bookstore sold 1/2 of all its books in stock. On the following day, the bookstore sold 4,000 more books. Now, only 1/10 of the books in stock before the sale are remaining in the store. How many books were in stock before the sale?

source:
SSAT & ISEE 2007 Edition Kaplan
p. 161

C. came pretty close to doing this one on his own.

Of course, close doesn't cut it on a standardized test.

I realize that.

This problem can't be done.

Right?

Not enough information

Or way too much information, as the case may be.

Two trains are loaded with equal amounts of rock salt and ball bearings. Train A leaves Frogboro at 10:00 A.M. carrying 62 passengers. Train B leaves Toadville at 11:30 A.M. carrying 104 passengers. If Train A is raveling at a speed of 5 mph and makes four stops, and Train B is traveling at an average speed of 86 mph and makes three stops, and the trains both arrive at Lizard Hollow at 4:30 P.M., what is the average weight of the passengers on Train B?

source:
Kaplan SSAT & ISEE 2007 edition
p. 155

Center for Environmental Therapeutics

A psychiatrist friend of mine recommends this web site. She says it has a self-administered diagnostic test for seasonal affective disorder that tells you exactly when to use light therapy.

We have enormous, aging light boxes all over the basement, but we've never known when to use them. We got them because John (Ratey) told me he'd visited the NIMH and all the shrink researchers there had light boxes on their desks. That was enough for me. We used to train one on Jimmy every day.

Today's Times has an article. (sorry - this is a paid subscription link, I believe - can't get the Times link generator pages to open.)

In 2001, Dr. Thomas A. Wehr and Dr. Norman E. Rosenthal, psychiatrists at the National Institute of Mental Health, ran an intriguing experiment. They studied two patient groups for 24 hours in winter and summer, one group with seasonal depression and one without.

A major biological signal tracking seasonal sunlight changes is melatonin, a brain chemical turned on by darkness and off by light. Dr. Wehr and Dr. Rosenthal found that the patients with seasonal depression had a longer duration of nocturnal melatonin secretion in the winter than in the summer, just as with other mammals with seasonal behavior.

Why did the normal patients show no seasonal change in melatonin secretion? One possibility is exposure to industrial light, which can suppress melatonin. Perhaps by keeping artificial light constant during the year, we can suppress the “natural” variation in melatonin experienced by SAD patients.

There might have been a survival advantage, a few hundred thousand years back, to slowing down and conserving energy — sleeping and eating more — in winter. Could people with seasonal depression be the unlucky descendants of those well-adapted hominids?

Regardless, no one with SAD has to wait for spring and summer to feel better. “Bright light in the early morning is a powerful, fast and effective treatment for seasonal depression,” said Dr. Rosenthal, now a professor of clinical psychiatry at the Georgetown Medical School and author of “Winter Blues” (Guilford, 1998). “Light is a nutrient of sorts for these patients.”

The timing of phototherapy is critical. “To determine the best time for light therapy, you need to know about a person’s individual circadian rhythm,” said Michael Terman, director of the Center for Light Treatment and Biological Rhythms at the Columbia University Medical Center.

People are most responsive to light therapy early in the morning, just when melatonin secretion begins to wane, about eight to nine hours after the nighttime surge begins.

How can the average person figure that out without a blood test? By a simple questionnaire that assesses “morningness” or “eveningness” and that strongly correlates with plasma melatonin levels, according to Dr. Terman.

The nonprofit Center for Environmental Therapeutics has a questionnaire on its Web site (www.cet.org).

Once you know the optimal time, the standard course is 30 minutes of fluorescent soft-white light at 10,000 lux a day.

and....

It may sound suspiciously close to snake oil, but the newest promising therapy for SAD is negative air ionization. Dr. Terman found it serendipitously when he used a negative ion generator as a placebo control for bright light, only to discover that high-flow negative ions had positive effects on mood.

Now that is exciting. I've been interested in negative ions forever. Negatives ions probably explain why it's impossible to be depressed on the beach.

Santa may be bringing me a negative ionizer for Christmas.


source:
Brought on by Darkness, Disorder Needs Light
December 18, 2007
Brought on by Darkness, Disorder Needs Light
By RICHARD A. FRIEDMAN, M.D.

Shadow Syndromes

Enhancing Academic Motivation

I've just ordered Enhancing Academic Motivation from Research Press thanks to a friend of mine whose child is seeing Dr. Brier. Every word out of Dr. Brier's mouth so far has rung true.

When I discovered that Dr. Brier had published with Research Press, I was sold. Research Press published the two books that shepherded Ed and me through our first years with Jimmy: Gerald Patterson's Living with Children and Wesley Becker's Parents Are Teachers. Both are classics.

Wes Becker worked with Engelmann on Project Follow-Through:

During the Project’s third year, we found out that Carl was leaving to go to Canada and become an investigator for the Ontario Institute for Studies and Education and a professor at the University of Toronto. He invited Valerie and me to join him. Valerie accepted; I tentatively declined.

Carl’s impending departure presented serious problems to the preschool project. The reason was that I was not qualified to head the project. The only degree I had was a BA in philosophy, and the position I held then was Senior Educational Specialist, which did not allow me to administer projects. Neither Jean nor Cookie could assume directorship of the project because they also lacked formal credentials.

The rumors were that the Institute for Research on Exceptional Children would take over the project and change it as soon as Carl left. I later found out that Jean and Carl met with Wesley Becker, a gifted professor in the Department of Psychology. Their goal was to seek his help in preserving the project. I had heard a lot about Wes Becker from my sister-in-law, Geraldine Piorkowski, who earned her PhD at the University of Illinois. Wes was her advisor; from her descriptions of him I assumed he could even run on water. Among other achievements, he had set the all-time track record at Stanford for attaining a PhD, entering as a freshman and taking only six years to earn a PhD in clinical psychology and statistics.

At the time he advised Geraldine, Wes was a cognitivist, but shortly after she received her PhD, he became an energetic exponent of Skinner’s behaviorism, which is based on evidence that behavior may be changed by manipulating positive or negative consequences that follow responses. Wes abandoned his earlier orientation because it lacked data of effectiveness, a signature characteristic of Wes. The professional articles that Wes wrote in the '60s show his change in orientation from 1961 to '67: “Measurement of Severity of Disorder in Schizophrenia by Means of the Holtzman Inkblot Test” (1961); “A Circumflex Model for Social Behavior in Children” (1964); “The Parent Attitude Research Instrument” (1965); “How We Encourage Cheating” (1966); and “The Contingent Use of Teacher Attention and Praise in Reducing Classroom Behavior Problems” (1967).

I had met Becker only once. He had presented to our project staff and graduate students. He summarized his current research, which involved working with teachers in failed classrooms and teaching them techniques for using positive reinforcement with their students. His data showed that even though most teachers had to be instructed in how to give praise, and even though the praise some of them issued sounded contrived and unnatural, it changed students’ behavior. The basic thrust of Wes’s training was, “Catch kids in the act of being good.” His studies were among the first applications of Skinner’s version of behaviorism to humans and school settings.

After the meeting I told him about some of the observations we had made in the preschools. He listened, then asked, “Where’s the data?”

I told him I didn’t have any formal data related to the observations. He smiled and shrugged. The message this gesture conveyed was that if I wanted to demonstrate the validity of my assertions, I needed data.

Jean and Carl had set up their meeting to ask Wes if he would assume the role of director of our project. They didn’t have a chance to ask him, however. As they entered his office, he greeted them, and said, “I know why you’re here, and the answer is yes.”

I count this as one of the more amazing commitments a person could make. The project was embroiled in controversy. The work was demanding. By saying “yes,” Wes made an official break with the fortress of higher learning and moved to the trenches, the gritty realities of working with teachers and kids.

Wes brought some of his graduate and undergraduate students with him. Thirty years later, I still work with three of them: Doug Carnine, a shy undergraduate who had already authored articles that appeared in professional journals; Linda McRoberts, an adventurous and outspoken graduate student who later would become Linda Carnine; and Susan Stearns (now Susan Hanner), only nineteen years old but very smart and industrious.

[snip]

Wes devoted some of his “free time” to writing the book, Applied Psychology for Teachers, a very ambitious work that covered everything related to effective practices and background information—from behavioral principles to the theoretical underpinnings of effective instruction and how to interpret data on student performance. I believe that Wes considered this book his ultimate achievement, an opus that positioned effective teaching and the analysis of learning in a framework that could be comprehended by undergraduates and that would establish DI as at least a contender in the field of education. The work, positioned as a textbook for undergraduates, was published by SRA in 1986. It was a colossal work—472 pages, in 8 1/2” x 11” format, with 275 references (over 11 pages).

It was another false hope. The book did not sell, was not adopted by more
than a handful of the faithful, and after only a few years, was discontinued by SRA. No other publisher was interested in it. I recently bought a copy of it online. It was in very good condition and cost $4.50.

[snip]

Goodbye to a Good Guy

Wes and Julie were divorced in1980. Wes continued in his role as associate
dean until 1992, when he became involved in political wars with the College of Education and quit the University. After retiring, he refused to talk about education. On three or four occasions, I tried to discuss the book we had started. I got the same response each time. He said that he would talk about golf or other sports and the stock market, but that was all. He declined to talk about the Association for Direct Instruction, or about anything else related to education. He told me, “That is something from a past life. It’s dead and I have no interest in it.”

In 1993, Wes sold his shares in Engelmann-Becker Corp. and moved from
Eugene to Sedona, Arizona. There was no going-away party or celebration
because he didn’t want one. Just before he left, I asked if there was anything I could do for him. He asked if I would give him a painting I had done of a lion. Yes.

I called him several times in Arizona to see how things were going. Not well. I called him once around noon and he sounded as if he’d been drinking. His leg had gone bad so he couldn’t play golf, and the stock market had not been kind to him. His son David lived with him for a while but left. Wes never remarried and lived alone. A couple of times I asked him when he was coming back to visit us in Eugene. He seemed to entertain the idea but it apparently didn’t make the seriousplanning list. I never saw Wes again after he moved to Arizona.

[snip]

Wes’ death came as a great shock. I hadn’t been in touch with him for months.
I knew he was getting frequent tests, but I had no idea that he would die at 73. We felt we should do something to honor him and decided to hold a memorial service for him in Eugene. We put a notice in the paper, made many calls, and arranged to hold the service in the church that Wes had attended (the Unitarian Church). A lot of people showed up for the service, including Don Bushell, Wes’ daughter Jill (who is a professor of biopsychology at the University of Michigan), his son David, and his ex wife, Julie (who lived in Florida). We took turns telling Wes stories and feeling sad.

I said, “Those who worked with him were routinely amazed, not only by his
skill, but the speed with which he could do things. Perhaps his most impressive quality, however, was the strength of his will. In the face of terrible setbacks and impossible deadlines, Wes prevailed. If he promised to get something done by a particular time, it was not only done on schedule, but done very well.”

Several others echoed this observation. One researcher who studied under
Wes said something that I had observed many times, the amazing speed at which Wes could identify glitches in raw data or elaborate calculations. About the time I was looking at the first few numbers on a spreadsheet of data, Wes would point to a set of scores in the middle of the display and say something like, “It’s impossible for them to have a correlation of point 9 with these data. These scores account for no more than 5 percent of the variance.” Possibly a minute later, I would see what he meant, but if I’d figured it out on my own, it would probably have taken closer to an hour.

I pointed out that even with the incredible number of things he had to do,
Wes was a good dad (a lot better than I was during the Follow Through years). Wes’ daughter Jill expanded on this theme. She told about some of the nice things he had done and indicated that the only time he lied to her was a couple of months before he died. She had visited him in a hospital in California. The last thing she said before leaving was, “Now, you take care of yourself. I’ll be back in three months.”

He said, “I’ll be fine.”

The clinical causes of Wes’ death had to do with his liver, kidneys and blood
pressure. One of the contributing causes was that he probably drank too much. These may have been the measurable causes, but the psychological cause was that he killed himself. When the establishment rejected Wes and his beliefs in data, he rejected education. To do that, he had to reject a huge part of himself. The image of himself that he had to maintain afterwards was one with many amputated parts, the hollow core that could survive on what had been peripheral interests. When his physical health failed, he had nothing.

The sad part of this equation was that Wes had to reject himself not because
he did anything reprehensible but because the establishment made a mockery of his beliefs and accomplishments. Jill believes that someday he will be recognized for his singular contribution to Follow Through. I hope she’s right, but I can sympathize with Wes. It is not very comforting to know that you can help thousands of kids and teachers, but you lack credibility and have no access to these victims. It hurts to see your professional beliefs trampled by educators who cling desperately to myth and folklore.

A colleague recently showed me a picture from the ‘70s, taken at a “Zignic” (a
picnic at the Veneta property for all our trainers and friends). Six people, including Wes, Bob, and I, were wearing t-shirts with the motto, “Show me the data.” For Wes, it was a way of life.

How is Wes remembered? In 2003, the College of Education at the University of Oregon launched a fund-raising campaign to support construction of a mega-building to house the College. Part of what the planners did was to make up a price list for “dedications.” If you want an office named after somebody, donate $25,000, and the plaque goes up. For a decent-sized classroom, the ticket is about $100,000.

Shortly after the list came out, Doug called me about raising enough money to have a classroom named after Wes. I told him that we shouldn’t have to pay anything. My feeling was that the College should have dedicated an entire wing to Wes, with no donation required. The College didn’t see it that way. Doug is currently trying to negotiate the price of a plaque for both Wes and Bob at the entrance to the Clinical Services Building, which was one of Bob’s projects.

Siegfried Engelmann 2007

Parents Are Teachers table of contents
Living with Children table of contents

Rye Country Day

wow----

School says that since 1996, 97% of kids taking the Advanced Placement BC Calculus course received perfect scores on the AP exam.

How the Schools Stack Up (pdf file)
WSJ

still can't do fractions?

Tom Loveless is back:

The 2007 NAEP test results showed small but statistically significant gains in both math and reading. Mathematics scores at fourth and eighth grade continued the steady progress registered since the main NAEP test was first administered in 1990. Both grade levels notched 2 point gains in scale scores. Table 1-1 reports the magnitude of the math gains in scale score points and years of learning. Figure 1-1 illustrates the upward trajectory of the scores. The gains indicate that fourth and eighth graders in 2007 knew more than two additional years of mathematics compared to fourth and eighth graders in 1990. On the face of it, this is an amazing accomplishment. Previous Brown Center Reports have raised questions about such gains. The primary question concerns the content of the NAEP math tests. Students are clearly making progress, but at learning what kind of mathematics? Suffice it to say that students are making tremendous progress on the mathematics that NAEP assesses, in particular, problem solving with whole numbers, elementary data analysis and statistics, basic geometry, and recognizing patterns. NAEP pays scant attention to computation skills, knowledge and use of fractions, decimals, and percents, or algebra beyond the rudimentary topics that are found in the first chapter of a good algebra text. In sum, we know that students are getting better at some aspects of math. But we do not know how American students are doing on other critical topics, including topics that mathematicians and others believe lay the foundation for the study of advanced mathematics. Thus, the years of learning gain must be taken with a grain of salt.

The 2007 Brown Center Report on American Education:
How Well Are American Students Learning

plus ça change (scroll down)

Spellbound

wonderful

perfect

especially for Christmas

or the Fourth of July!

Spellbound

daily lit for Christmas

My siblings don't know it, but they're about to receive free novels via email for Christmas.

I think.

I'm getting one for me, too. Maybe I'll finally be able to finish The Prince.

middle school math teacher preparation

William Schmidt is working on an international report. I like Schmidt. He's the coherent curriculum guy.

snippets:

• The mathematics knowledge of US future middle school mathematics teachers generally is very weak compared to future teachers in Taiwan and Korea. It is also weak compared to German future teachers in all areas except statistics.
  

• Taiwanese and Korean future teachers were the top performers in all five areas of mathematics knowledge – including algebra, functions, number, geometry and statistics.

• School algebra (which includes functions) is the topic that, across almost fifty countries studied in TIMSS, was the major focus of instruction at seventh and eighth grade.
  

• On the algebra and functions tests, US future teachers performed at or near the bottom among the six countries—over a full standard deviation below the performance of future Taiwanesse teachers.
  

• The results for the statistics test were the only bright spot in future US teachers’ performance. The US future teachers scored enar the mean of the six countries.
  

• Future middle school teachers prepared by a secondary program performed somewhere between one-half to three-fourths of a standard deviation higher in algebra, functions, geometry and number compared to those prepared in either of the other two programs. The difference was slightly less in statistics.

COURSE TAKING

• On average, the Taiwanese and Korean future teachers reported taking courses that covered around eighty percent or more of the advanced mathematics topics typically covered in undergraduate mathematics programs.
  

• For analysis (the study of functions) Taiwanese future teachers covered virtually all of the topics (ninety-six percent) while, in Korea, the coverage was seventy-nine percent.
  

• In the algebra and analysis courses which provide the mathematical background for middle school algebra, the Taiwanese, Korean and Bulgarian future teachers all covered around eighty percent or more of the possible topics while Germany covered around 60 to 70 percent.
  

• Mexican and US future teachers covered less than half of the analysis topics. The same was true for Mexico on the algebra topics, but US future teachers covered on average 56 percent of those topics.

etc.

Bill would ban military slot machines - CNN.com

Bill would ban military slot machines - CNN.com

Why not, everything else is banned in the Military.

The perfect military troop: Non-smoking, non-drinking, church going, Habitat for Humanity voluteering, non-extreme sporting, anti-gambling, feminish supporting, college educated, safety oriented killer.

why is there 1?

Cleared away one of my floor stacks today. I'm glad I did because one of the objects in the stack was the Algebra 2 notebook in which I'd written an anecdote about C. & math. The notebook went missing a few months ago and now I've found it.

This is C. last summer at the picnic table outside our kitchen. We were probably working on percent (scroll down).

I don’t like math.

Why is there 1!?

1 + 1 equals 2!

Why is that?

I don't like math.

Words make sense. If I read “the dog,” I know what “the” means.

Math doesn't make sense.

- CHB summer 2007, 12 years old

I know you will all be impressed by the fact that I did not say, "You know what 'the' means?"

Myrtle on procedural and conceptual knowledge

Maybe I can make a convincing argument that a student who only thinks of multiplication as iterative addition and can't multiply 24X86 has neither procedural nor conceptual knowledge.

There is more to the "concept" of multiplication than iterative addition. (Try applying iterative addition to 1/8 x 2/5.) Perhaps iterative addition is appropriate for 2nd and 3rd graders learning their multiplication tables (or is it 3 and 4th graders these days?) But "the" concept of multiplication includes the fact that it distributes over addition (and that it's associative as well). The multiplication algorithm invisibly makes use of the distributive "concept," and does not employ an iterative "concept." Perhaps I'm overdoing the disdain quotes but I've been lied to too many times by people telling me that something is the "concept" of a procedure or rule and it turns out not to be.

A child with a conceptual knowledge of multiplication, and a lot of time on his hands, could successfully multiply two digits numbers without the multiplication algorithm:

24 X 86 means that
(20 + 4)(80 + 6) which means/implies that...

Etc. You see where I am going with this. One of the benefits of Singapore is that the kid does end up with a conceptual understanding of multiplication, and can apply his knowledge of concepts to come up with correct answers.

Notwithstanding operations on super hairy numbers, he is capable of doing the algorithm on paper when he needs to and can resort to "concepts" when he needs to do mental calculations.


the multiplication algorithm invisibly makes use of the distributive "concept"

I love that!

I love the whole Comment, in fact. People like me -- people who value liberal arts education in general and mathematics education in particular but who aren't expert in mathematics and probably never will be, have no way to get at these things.

I intuitively grasp the notion that there is some kind of "starter understanding" a person can have without being fluent in procedures. Seeing that 6x4 is the repeated addition of 6 4s or 4 6s as the case may be (I've spent quite a bit of time muddled over that one!) strikes me as superior to not seeing it. (I had no idea multiplication could be called repeated addition until I started reteaching myself math, and then I noticed it on my own.)

But at the same time I am gripped -- and gripped is the correct word -- by the conviction that a starter understanding is not a real understanding.

And yet because I lack a real understanding I have no way to express this and thus no means of combating the forces of reform math when they threaten to overrun my son's education.

I'm logging this post under Greatest Hits so I'll know where it is when I need it.

compare and contrast

presidential primary candidate:






professor of psychology and education:

WHAT DO WE WANT CHILDREN TO LEARN?

Beyond basic literacy and numeracy, it has become next to impossible to predict what kinds of knowledge people will need to thrive in the mid-21st century....[T]he only defensible answer to the question of what we want schools to accomplish is that they should teach students to use their minds well, in school and beyond (Kuhn, 2005). The two broad sets of skills I identify as best serving this purpose are the skills of inquiry and the skills of argument. These skills are education for life, not simply for more school (Anderson et al., 2000). They are essential preparation to equip a new generation to address the problems of the day.

Deanna Kuhn, Professor of Psychology and Education Teachers College
Is Direct Instruction an Answer to the Right Question?
a response to Why Minimally Guided Instruction Does Not Work (pdf file)
EDUCATIONAL PSYCHOLOGIST, 42(2), 109–113



Apparently Mike Huckabee has not set foot inside an ed school any time within recent memory.

bonus observation: I might actually be willing to pay more taxes to stop the extraordinary professional development and ongoing education teachers "require."

Starting with the workshops on writing to learn in math and science. I would pay to have my district's science and math teachers not attend another one of these things.

Blind Father Needs Help

This question was just posted on another list I follow.

Hello all. I am totally blind, but my wife and children are sighted.

My son is nine years old and in the fourth grade, and he is having a little bit of difficulty with long division--Especially when dividing a double-digit number into another number (e.g. 5128 divided by 47).

Can any one give me some pointers on how I might explain and illustrate the concepts of how to perform these types of problems, with emphasis on how to explain how to estimate?

I hope that this question is clear enough and that someone may have some ideas that will help me.

Thank you for your assistance.

This request was posted on a list where most of the members are adult blind mathematicians who are unlikely to know what is currently going on with grade school math and to what extent the son's problem is likely to be related to the educational environment.

I don't think this Blogger interface is very accessible to persons who use screenreaders. However, if anyone has any advice or suggestions, I'm happy forward them to this father.

hyperspecificity in Math A

Another classic example of hyperspecificity:

C. did not recognize this expression as a case of the distributive property:

2x(x+1) + 3(x+1)

Can't say I blame him.


update

from Barry:

This isn't real obvious when you first come across it. So do this. Let's let x + 1 = T.

Now substitute T in the expression and you get:

2x(T) + 3(T)

Can you factor out the T?

Yes. You get T(2x + 3)

Now substitute the x + 1 back in. You get:

(x+1)(2x+3)

I think it was Ron Aharoni who referred to seeing expressions such as (x+1) as single entities as "chunking". To help students do such "chunking" it helps to do what I did above so they can see that x + 1 represents a number, and as such it can be factored.

I'm amazed by how difficult it is to see that "expression X" is the same as "expression Y." This is an ongoing source of pain in my mental life these days (not to put too fine a point on it), because I came to feel, shortly after Animals in Translation was completed, that Temple's & my thesis concerning hyperspecificity in animals and autistic people is wrong in some important way -- either wrong or perhaps right for the wrong reasons.

We argued that autistic people, children, and animals are hyperspecific compared to typical adults. Autistic people, children, and animals are splitters; nonautistic adults are lumpers; etc. (I know I've said all this before, but feel I must repeat in case newcomers stop by.)

The classic hyperspecificity story re: autistic children is the little boy who was painstakingly taught to spread butter on bread and then had no clue how to spread peanut butter on bread.

Until I began to reteach myself math, this kind of thing seemed to me incontrovertible evidence of the otherness of the autistic brain. But now that I'm factoring trinomials I've discovered I have something in common with that little boy. That's probably why God or the universe decided I should take up math. I needed an object lesson.

Still, the observations Temple has spent a lifetime making of animals' (and autistic people's) hyperspecificity aren't wrong. Normal adult humans aren't hyperspecific in the same way animals and autistic people are hyperspecific.

Sometimes I wonder whether the issue is simply that non-autistic adults pass through the hyperspecific stage of knowledge more quickly or more frequently than autistic people do. When a "typical" adult (typical being the preferred term these days) encounters brand-new material he, too, is hyperspecific, as I am with math. Everyone starts out a splitter.

But I don't think that's quite it, either.

I'm getting the feeling that animals may not be hyperspecific across the board, but perhaps only in certain realms. Maybe animals are more hyperspecific than adult humans when it comes to sensory data? e.g.: To a horse a saddle feels completely different at a walk, a trot, and a canter -- so different that he will buck his rider off when he moves from a trot to a canter if he hasn't been carefully trained to tolerate the saddle at all 3 gates individually.

Better story: Temple's black hat horse.

This was a horse who was terrified of people wearing black hats. He wasn't terrified of people wearing white hats or red hats. Just black hats.

I'm thinking, this morning, that humans may be relatively oblivious to "sensory data," that we're lost in words -- so perhaps words are the place where you'll see us being hyperspecific ? (There's evidence that language masks sensory data, but I don't know it/remember it well enough to summarize.)

Temple complains about this all the time. She'll give a talk and her audience will take away a too-specific meaning from her words; then they'll go out and apply her advice all wrong & bollocks things up. "People get hung up on the specific words," she'll say. (I'll write down the next example of this that crops up - can't think of one offhand.) These conversations have gotten to be quite funny because, after years of reading countless articles on autistic people being literal-minded and "concrete," I am now spending my time listening to an autistic person complain that normal people are literal-minded.

Well, she's right. Looking at an expression like 2x(x+1) + 3(x+1), I'm like the horse with the saddle and so is my 13-year old son.

"x+1" next to 2x is completely different from "x+1" next to 3.

Different enough to make us start pitching our riders into the haystack.


percent troubles
Robert Slavin on transfer of knowledge
rightwingprof on what students don't know
Inflexible Knowledge: The First Step to Expertise

what understanding without procedural knowledge looks like

Extremely useful comment left by palisdesk:


on the 5th grade students she assessed:

I assessed a whole [5th grade] cohort in math (only one was a special education student, and he was no worse than the rest), with similar results.

They scored well in understanding "concepts." Whoop-de-doo. But none could reliably do computation with regrouping, do mental computation of any sort, measure accurately, use a number line, name fractions, or do any operations with fractions or decimals. Surprisingly they could not even count money correctly!! They could not figure out elapsed time, nor read non-digital clocks.

What good is all this great "conceptual understanding" if you can't count coins under $3.00, measure the length of a board, find the perimeter of a triangle or determine whether to add or subtract to compare two numbers? The one thing most were good at was reading graphs -- pictographs, bar graphs and simple tables.

Some of these students were quite bright and had good reasoning ability but what they ALL lacked was knowledge of number facts, facility with algorithms, precise vocabulary (perpendicular, acute angle, numerator, range), an organized approach (if guessing didn't work they were stuck), in short MASTERY at any level. Like [instructivist's] students, all have had nothing but fuzzy blah-blah since entering school. Few to none can afford Kumon and most don't have computers at home or access to them, so even that kind of practice is not available to them.

A recent study of teacher competence in our district found that most teachers in 5-8 grade mathematics did not themselves show mastery of the subject at that level. I think we may have come full circle. The system is now being run by people who are its products, and many are quasi-literate and numerate, however bright and caring they may be. Also, the lack of any kind of intellectual rigor or scientific and statistical training in their preparation has left them vulnerable to every fad that comes down the pipeline.

conceptual understanding w/o procedural knowledge:

The best way to illuminate my point would be to contrast my findings with what might have occurred in "the old days." Time was -- the 50's? 60's? when you might often find students who could proficiently, or at least adequately, perform basic operations -- including, in many cases, operations with fractions and decimals -- but would have been at a loss to explain what they were doing, or why they were (for instance) regrouping in subtraction, or inverting fractions to divide them. It was not considered necessary for students to have a "deep" understanding of the number system per se. Most of us (I remember this myself) "got it" in the process of learning the algorithms and how to apply them, and did (eventually) understand why you had to regroup, what you were doing when inverting fractions (I would have wanted to show how it works, even now it would be hard to explain succinctly in words, but it's easy enough to demonstrate).

The children I was assessing -- with a detailed, widely-used norm-referenced diagnostic math test and also with a locally developed "performance assessment" -- showed the opposite pattern. They understood about place value (haven't played with Base 10 blocks for years for nothing), knew that multiplication is repeated addition, that fractions can name parts of an object, or members of a group, and so on. They could show you (with the ever-present manipulatives), or draw a diagram and explain. They could tell you why you have to rename the ones as tens, why you have to keep the decimal points lined up, what the value of various coins etc. is, what the "big hand" and the "little hand" on a clock indicate, and so forth.

What they could NOT do was reliably apply a procedure to come up with an answer. Given a problem like 24X86, they knew this means you make 24 groups of 86 (or 86 groups of 24), but would get lost trying to build them with blocks or count out the tally marks. If they tried to use the algorithm, typically they got directionality, order of steps, etc. all mixed up, and they didn't know the number facts. The brighter ones would figure out the answer had to be something around 2000, but many did not even get that far. When counting coins, they lacked a strategy such as, counting the quarters first, then the dimes, then the nickels, etc. They randomly counted each one separately and continually lost count. They didn't know how to "count up" to find a difference (or an interval between numbers or clock times).

This is so helpful. It makes sense to me that a student could have some degree of conceptual understanding without procedural knowledge, and yet when I try to think of how that would work I come up with a blank. It's clear to me, for instance, that my "conceptual understanding" of unfamiliar subjects -- economics, say -- is thin at best. God is in the details.



hyperspecificity for conceptual knowledge -- ?

One thing I think I see happening with palisdesk's 5th graders is "hyperspecificity" for conceptual understanding. I'm used to seeing hyperspecificity for concrete knowledge. I hadn't really thought about hyperspecificity for concepts such as the meaning of multiplication. It makes sense, though. I'm pretty sure it happens to me all the time, teaching myself algebra 2. I'll have a basic conceptual understanding of a concept -- logarithms, say -- that doesn't immediately transfer to a problem type I haven't done before. I'll try to come up with an example to post.

As usual, I'm hamstrung by a lack of terminology. Our sturdy workhorse words -- procedural, conceptual -- are failing to give me the distinctions I need within the category of "conceptual understanding."

These 5th graders have for arithmetic what I have for logarithms: some kind of start-up understanding of the concept that won't take them very far when confronted with an actual logarithm problem in the flesh.



thank you palisdesk, pissed-off teacher, instructivist, redkudu, dy/dan, nyc educator, exo, smartest tractor (I'm sure I'm leaving others off ....)

For parents and the broader public schools are a black box. Mike Schmoker says that's by design; the official term for management in schools is "loose-coupling," which means, I gather, that the goal of management is to protect the core functions of the organization from outside scrutiny. (more later...much, much later)

Teachers who share their experiences with outsiders are functioning as the education reporters our country needs but does not have.


Robert Slavin on transfer of knowledge
hyperspecificity posts
loose coupling & instrutional leadership

tales from the front, part 2

from pissed off teacher --

A zillion years ago when my son was in first grade and computers were still a novelty, he decided to write his spelling words on the computer. The little twerp was using copy and paste to write them the required ten times. When I made him stop, he told me "but my teacher said it was okay". As I said, computers were new and she did not know how he was using it.

Kids have always cheated. We just have to be smarter than them to stop them in their tracks.

The middle school's Civil War Museum exhibit last spring put the kids' papers on display. Ed looked at every paper & saw a huge amount of internet text. Same thing the year before at the Jason Project exhibit. Many papers included downloaded text. A couple of papers were the same paper.

Of course most people didn't notice because they were looking at the crafts projects, not the papers.

Crafts projects are to a middle school what misdirection is to a magician.

Possibly.