news to me

instructivist looked up the top 100 teacher salaries in the state of Illinois:


Top 100 Teacher Salaries
2006 search results.

Bode Jeanne E $441,612 CARY CCSD 26

Morrison Donna M $439,803 BELLWOOD SD 88

Schackmann Deborah K $329,404 SOUTH EASTERN SP ED PROGRAM

Marszalek Christine D $174,403 LOCKPORT TWP HSD 205

Hall James W $170,172 LOCKPORT TWP HSD 205

Gonsiorowski Marvin $165,370 LEMONT TWP HSD 210

Vallicelli Richard E $164,232 LEYDEN CHSD 212

Swords Robert G $163,327 LEYDEN CHSD 212

Martinez Lino $163,184 ADLAI E STEVENSON HSD 125

Wolter James H $163,144 NORTHFIELD TWP HSD 225

Sincora Craig D $162,432 ADLAI E STEVENSON HSD 125

Weber Peter W $161,622 ADLAI E STEVENSON HSD 125

Vanderschoot Gerard $161,502 LEMONT TWP HSD 210

Dezurko Steve $160,729 LEYDEN CHSD 212

..................................

I wonder if these folks grade on a curve.

SAT lollapalooza

right here

lots of good stuff

Question of the day

If you wanted to make a real impact on improving K-12 education, and someone gave you one million dollars to do it with* - what would you do?

And I'm not talking here about what you would do as a school or district official: I'm talking about KTMII participants. Informed parents. Outside agitators.

Would you create your own school to show how it should be done? Would you launch a public engagement campaign in your district to empower other parents? Would you buy ads across the country to let people know about Singapore Math and Direct Instruction? Would you underwrite a voucher campaign? Would you spend it lobbying public officials to make changes you think should be made? Or what?

A million dollars isn't what it used to be - you have to pick your battles and leverage that investment to the extent possible.

So what would you do - and why?

* No, I do not have a million dollars to give you - this is just a thought exercise to see where there greatest points of leverage in school improvement might be.

who am I, college edition

I love Michele Hernandez. Even if she can't add.

You'll notice that most colleges have you start by filling in your parents' college backgrounds and professions. I know that when I filled in these boxes, I was convinced that the admissions committee would be impressed by my parents' backgrounds and jobs. I thought I would get "extra points" because I came from a family that valued education. In fact, I thought the committee would see that I had a good chance of succeeding since I came from a successful family. I could not have been more wrong.

The main reason colleges ask this question is to see if you qualify as a "legacy" at their college, meaning that either your mother or father graduated from their school (see chapter 12 for more details). Another reason is to see if you have come from a much less sophisticated background and therefore would have lower standardized test scores and perhaps a less polished application. What these questions are not there for is to impress the reader, some of whom have been known to harbor grudges against kids who "have it easy." It's a matter of preconceived notions and expectations, as well as personal biases.

If an officer is reading the application of a student from Groton whose father went to Harvard Medical School and is the chief neurosurgeon at a major hospital and whose mom has a Ph.D. in clinical psychology and is in private practice, he starts out by expecting a lot of the student. Clearly, the student comes from a family where there has always been enough money to put a child in the finest nursery schools, kindergarten, after-school activities, and so on. Undoubtedly, they have been able to afford private tutors for both standardized tests and high school subjects. If the officer then sees that this applicant has mid-600 SAT scores, a B high school average, and has been involved in extracurricular activities such as lacrosse, sailing, and horseback riding, he is bound not to be very impressed. The officer's thoughts would run as follows:

He has had every possible advantage and still only managed mediocre [by Ivy League standards, that is] grades and scores and has never really gone beyond the classroom to seek additional academic challenges.

Now let's look at a student with similar achievements from a totally different background. The first thing the officer sees about Susan is that neither parent went to college; her father is an auto mechanic and her mom is a postal clerk. She goes to a poor inner-city high school where only 15 percent of the population goes on to four-year colleges. Before the officer even starts reading her application, he mentally adjusts his expectations and attitude. No need to feel threatened or jealous of the advantages she might have had. Despite her background, and the fact that she could not afford SAT prep courses, she was always a reader from early childhood. She scores 660 on the verbal, 650 on math, and ends up ranked tenth in her class of 110 students (let's say she earned mostly A's, with a few B's), taking the hardest course load available to her. She is captain of two athletic teams, has done significant community-service work, and teachers say she adds that extra spark to class discussions. Susan would get the vote of any highly selective admissions committee because she rose to the top with limited resources and managed to stand out. The readers can feel good, thinking that they are helping out someone less fortunate by giving her that ticket to an Ivy League school so she can succeed in life.

Some of you may take my advice too much to heart and be tempted to lie or invent. I am not suggesting that path. I am a firm believer that the cream always rises to the top and the best kids shine, no matter where they are from. What I am saying is that you don't need to be overly specific. You have to state where your parents were educated, but when it comes time to put down their jobs, you might want to be somewhat vague. For example, if your father is the president and CEO of a big-name investment bank, the committee is going to be expecting quite an amazing applicant, one who has gone beyond his comfy lifestyle to make himself known. You might just write down "banker" for occupation. It's not a lie, but at the same time, it doesn't create such a high expectation in terms of wealth and privilege. Rather than saying "chief neurosurgeon," why not just M.D.? Rather than "chief partner in a major law firm," just put "lawyer." I'm not suggesting you deceive the readers; rather, I'm proposing that you be modest and exercise a level of humility in both your personal part of the application and the description of your parents' jobs.

A is for Admission
Chapter 1

So . . . is author out? And if author is out, then what?

I'm pretty sure housewife-blogger, while admirably non-hoity-toity, doesn't lower expectations in quite the way postal clerk might. Housewife-blogger, in fact, could be just as annoying in its way as neurosurgeon dad.

Maybe I'll have C. put down math warrior when the time comes.

inflexible knowledge redux

Another example of inflexible knowledge.

C. and his friend J. were assigned a "digit word problem" for homework. Example:

The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased by 27. Find the number.

source: purplemath

Neither of them had a clue how to do it. (I think digit problems had been taught in class that day for the first time. They'd both seen 1 or 2 worked examples.)

I tried to teach it by writing the number 23 and then, below that, writing a variable for each digit, then asking them something wildly imprecise like, "What would you have to do to these variables to get them to equal 23?"

Actually, as I think about it now, I don't know what words a person educated in mathematics would use to express this, so maybe you all can fill me in.

In any case, they did get the idea they were to let x equal 2 and y equal 3, and they were to write an equation using these values for x and y that would equal 23.



Looking at this arrangement of numerals and variables didn't help.

J. said, "x + y?"

"That's 5," I said.

"x times y?"

"That's 6."

They were stumped.

This is a classic case of inflexible thinking, for two reasons.

First of all, they've seen "plus" and "times" more often than they've seen 10x + y. Confronting a novel problem, their brains went straight to the more familiar instead of the less familiar.

Of course, that's what a brain ought to do: When you hear hoof beats on the bridge don't think of zebras.

The problem was that, pace Willingham, once they figured out the hoof beats weren't made by horses they didn't think of zebras next.

Both boys know how to set up word problems using letter variables; both boys are proficient (or close to) at solving number and consecutive integer problems. They can set up and solve coin problems, distance problems, and age problems; they can also set up and solve simple linear function problems.

They've been writing algebraic expressions for at least a couple of years.

And they were stopped cold by a simple digit problem.

I wish I'd taken notes on what I did next. I think I said something like, "What is the 2?" (That's imprecise, too, right? What's the correct language there? "What value does 2 represent"?)

I don't think I had to go as far as to remind them of the base-10 place system, but, on the other hand, maybe I did and I'm repressing it.

In any case, the moment it dawned on them that the key was recognizing the 2 as 20 they could both set up and solve digit problems rapidly and efficiently.

Knowledge transfer and generalization are core, unsolved problems in education.

update: Looking at this now, I think I should have had them start with the "3" and write an equation using y alone, then had them write an equation using x and 2 (or 20). If I wasn't simply going to show them a worked example myself (should I have? I don't know) I needed to break this down into the smallest, simplest possible steps. And I should have started with the ones digit, not the tens.

I think.


cumulative practice and problem solving

Speaking of which, I keep promising to write a post about one of the three most valuable papers I've ever read on the subject of teaching math:

The Effects of Cumulative Practice on Mathematics Problem Solving
Kristin H. Mayfield and Philip N. Chase

Here is the introduction:

Over 35 years of international comparisons of mathematics achievement have indicated problems with the performance of students from the United States. According to the latest international study, the average score of U.S. students was below the international average, and the top 10% of U.S. students performed at the level of the average student in Singapore, the world leader (Wingert, 1996). In addition, recent tests administered by the U.S. National Assessment of Educational Progress revealed that 70% of fourth graders could not do arithmetic with whole numbers and solve problems that required one manipulation. Moreover, 79% of eighth graders and 40% of 12th graders could not compute with decimals, fractions,
and percentages, could not recognize geometric figures, and could not solve simple equations; and 93% of 12th graders failed to perform basic algebra manipulations and solve problems that required multiple manipulations (Campbell, Voelkl, & Donahue, 1997).

These statistics reveal students’ deficits in the fundamental skills of mathematics as well as mathematical reasoning and problem solving. Indeed, poor problem-solving skills have been targeted by the National Council of Teachers of Mathematics (Carpenter, Corbitt, Kepner, Lindquist, & Reys, 1980; Carpenter, Kepner, Corbitt, Lindquist, & Reys, 1980; Kouba et al., 1988; National Council of Teachers of Mathematics, 1989, 2000). Thus, it seems appropriate that current behavior-analytic research in mathematics education should address problem-solving skills as well as basic mathematics skills (e.g., Wood, Frank, & Wacker, 1998).

I'm hoping I can get to this later on today.

But first -- must write chickens chapter!

SAT question of the day

SAT question of the day

Vicky S just reminded me of these questions; I've been subscribing to them for awhile now, too.

So today I'm going to subscribe for C.

I'm sure he'll be pleased.

Michelle Hernandez on admissions & SATs

The most obvious result of recentering is the fact that what used to be considered an unusually high score on the SAT I is now just an average score. To be exact, for years a combined score of 1,400 was considered the dividing line between normal scores and extremely high scores for highly selective colleges. This si no longer the case. The average combined score for the admitted class of 2000 at Dartmouth was over 1,410. In fact, for the class of 2000, the average scores of all 11, 400 applicants who applied to Dartmouth (this includes all the weakest applicants in the pol) was 662V, 677M, almost 1,340 combined. For the class of 2001, the average was even higher: 664V, 683M.

While it used to be extremely rare to see students with verbal scores over 700 (remember, only 1 percent of all students taking the unrecentered test scored over a 700, and those students could not have missed more than three or four questions on the whole test to achieve that score), now it is much more common. In fact, half of the members of the class of 2000 scored over 710 on the verbal section, about the same number who scored over 710 on the math section.

Here is an SAT I breakdown of the acceptance rates at Dartmouth for the class of 20000:

Verbal SAT Math SAT
399 or less 0%
400-499 1.6%
450-499 4.1%
500-549 5.4%
550-599 7.5%
600-649 10.1%
650-699 14.8%
700-749 28.2%
750-800 52.2%

Math SAT
399 or less 0%
400-499 2.4%
450-499 6.3%
500-549 5.4%
550-599 7.1%
600-649 9.5%
650-699 15.1%
700-749 24.4%
750-800 44.0%

All these statistics taken together mean that parents, students, and counselors need to readjust their standard for excellence dramatically in terms of scores. When a college counselor or a parent calls an admissions officer to ask whether or not a student has a chance of being admitted, he usually starts out by announcing that the student has very high SAT I scores. The problem is, when the officer asks what he means by "very high," he usually says, "Over fourteen hundred," which, as we have just seen, is not only not "very high"; it is actually below the class average.

[snip]

It is still, as it has always been, more impressive to see high verbal scores than high math scores, since most students at the highly selective colleges will be doing much more writing and reading than math. Verbal ability is still a good indicator of how strong a reader the student is. The ability to read well will ultimately have a bigger impact on most college students than the ability to do SA I math very well, especially since the level of SAT math is not particularly high. There are many students who do terribly on the SAT I math and yet who manage to get the highest score of 5 on the AP calculus exam. If any math is useful at the college level, it is calculus, not the basic math covered on the SAT I. Therefore, SAT I scores of 750 V, 630M would be much more impressive for most highly selective colleges than a 640V, 780M, even though the latter score has a higher combined total by forty points.

The final point I want to make about interpreting SAT I recentered scores is that because most admissions officers are not gifted in math, there is still the tendency to use the 700 cut-off as the magic number between good and excellent scores, even though, as we have seen, the averag score is over 1,420 combined....Remember probably 99 percent of all current admissions officers took the old SAT I, the nonrecentered test. In their own personal histories, 1,400 has always been a high score. It is extremely difficult to consider applicants under a totally different system from the one you had in high school. [Hernandez' book was published in 1997, so this is 10 years ago.]

[snip]

Oddly enough, and I saw this during my last two years in admissions, officers still respond to any scores over 700 by saying that they are strong, whereas if the scores are in the mid to high 600s, they tend to refer to them as average. This is simply not accurate. In Ivy applicant pools, a 700V, 700M is now an average, even below-average, score, although some officers will not be able to draw the correct conclusion when analyzing student scores.

Despite this inability on the officers' part to be 100 percent accurate in interpreting your scores 100 percent of the time, they will undoubtedly improve as the years go on and they see years' worth of recentered scores. In the meantime, all applicants should be aware of the reality of SAT I, which is that if you want to stand out in the Ivy applicant pool, you should aim for a combined score of about 1,490 or higher, or at least a verbal score over 730 or so, which would allow you to have a modest 650 math score and still look strong in the applicant pool

source:
A Is for Admissions: The Insider's Guide to Getting into the Ivy League and Other Top Colleges
by Michele A. Hernandez
pp. 46-49

Is she right about SAT-level math?

That strikes me as wrong -- at least based in the fact that I constantly see algebra called the "gatekeeper to advanced courses in math," etc.

Hernandez was an assistant director of admissions at Dartmouth. She's a high end college counselor now. Mathew says she's fantastic, and her books are terrific.

Garrison Keillor and NCLB

Of all the people who could be expected to defend NCLB, I never would have included Garrison Keillor on that list. Yet here he is, on Salon.com.

The heart of it:

Face it, the schools are not run by Republican oligarchs in top hats and spats but by perfectly nice, caring, sharing people, with a smattering of yoga/raga/tofu/mojo/mantra folks like my old confreres. Nice people are failing these kids, but when they are called on it, they get very huffy. When the grand poobah Ph.D.s of education stand up and blow, they speak with great confidence about theories of teaching, and considering the test results, the bums ought to be thrown out.

There is much evidence that teaching phonics really works, especially with kids with learning disabilities, a growing constituency. But because phonics is associated with behaviorism and with conservatives, and because the Current Occupant has spoken on the subject, my fellow liberals are opposed.

Liberal dogma says that each child is inherently gifted and will read if only he is read to. This was true of my grandson; it is demonstrably not true of many kids, including my sandy-haired, gap-toothed daughter. The No Child Left Behind initiative has plenty of flaws, but the Democrats who are trashing it should take another look at the Reading First program. It is morally disgusting if Democrats throw out Republican programs that are good for children. Life is not a scrimmage. Grown-ups who stick with dogma even though it condemns children to second-class lives should be put on buses and sent to North Dakota to hoe wheat for a year.

Who’d a thunk it?

Hat tip to Alexander Russo at This Week in Education.

not your father's SAT

I probably won't be able to find his comment now, but I believe lrg has raised the question of whether student performance and school quality have in fact declined over the years or whether performance has remained the same while expectations have risen.

The answer is that performance has declined at all levels, including the top, although -- and this will strike most of us around here as ironic -- I think it's possible that performance in math may actually have risen in students who take the SAT. There is no question that performance on reading tests has declined across the board.

I spent quite a long time wondering about this issue myself. Had American schools really declined or did people just think they had? I became especially curious about this question after reading an edu-blogger's post saying that American schools "serve rich white kids well." (The original post seems to have been deleted from the blogger's archives.)

That struck me as wrong, but I didn't know.

For quite awhile I was on the lookout for evidence concerning advantaged white students, a phrase I prefer to "rich white kids." Eventually I learned that this question had been asked and answered. The decline is real. It started in 1969 and ended in the early 1980s.

Best sources for a quick trip through what has taken place:

Verbal, Math and Combined scores 1962-2001

SAT scores were "recentered" in 1995. Today's SAT-V scores are roughly 70 to 80 points higher than the same score prior to 1995: SAT I Individual Score Equivalents.

Best primer on decline in student achievement: Waiting for Utopia


decline at the top

The recentering of SAT scores is a major factor in suburban parents' perception that their children are attending high-quality schools. No one in the wider public knows the scores have changed -- and school districts don't make a habit of filling parents in on the real value of a 600 on the SAT-V.

A 600 today was a 520 in 1994. Same test, same raw score, different conversion formula.

No one knows. Parents "read" their kids' scores through the filter of their own scores back in the day.



more on SAT-M later

also see:
The Seeds of Growth

drive-by

The nonprofit National Council on Teacher Quality (NCTQ) reports that, despite many calls for performance pay coming from state capitals, only 14 states require school systems to evaluate their public school teachers at least once a year, while some are much more lax than that. Tennessee, for example, requires evaluations of tenured teachers only twice a decade.

[snip]

Teachers union contracts dictate the professional requirements for teachers in most school districts. But the NCTQ study found that only two-thirds of them require teachers to be evaluated at least once a year and a quarter of them require evaluations only every three years.

The evaluations themselves are typically of little value—a single, fleeting classroom visit by a principal or other building administrator untrained in evaluation wielding a checklist of classroom conditions and teacher behaviors that often don’t even focus directly on the quality of teacher instruction. “It’s typically a couple of dozen items on a list: ‘Is presentably dressed,’ ‘Starts on time,’ ‘Room is safe,’ ‘The lesson occupies students,’ ” says Michigan State University Professor Mary Kennedy, author of Inside Teaching: How Classroom Life Undermines Reform, who has studied teacher evaluation extensively. “In most instances, it’s nothing more than marking ‘satisfactory’ or ‘unsatisfactory.’”

It’s easy for teachers to earn high marks under these capricious rating systems, often called “drive-bys,” regardless of whether their students learn.

source: Rush to Judgment: Teacher Evaluation in Public Education
by Thomas Toch and Robert Rothman

My favorite was the department chair who told me that so-and-so was "an excellent teacher" because Department Chair had visited so-and-so's classroom and "all the students understand."

We were in the midst of a conversation about the fact that my own child, a member of so-and-so's class, manifestly did not understand.

That didn't affect Department Chair's evaluation because "he's the only one"^* and "you're the only parent complaining."^**

Principal concurred. "So and so is a fine young teacher. I've visited so-and-so's class many times."

That was it.

I've visited the classroom. They didn't even look at the freaking bulletin boards, for God's sake.


^* He wasn't.
^** I wasn't.

food fight!

Is School Choice Enough?

from Hirsch's take:

The choice movement is a structural approach. It relies on markets to improve outcomes, not venturing to offer guidance on precisely what the schools should be teaching. Such guidance would go against the “genius of the market” approach, which is to refrain from top-down interference with curriculum. Stern shows—rightly, I believe—that this is a fundamental failing of the choice movement.

But market-based “choice” is not the only structural reform of the recent past that has ignored substance. The government-funded “whole-school-reform” project was another metastructure that essentially said, “Let a thousand flowers bloom,” not concerning itself with what kinds of flowers. President Bush’s No Child Left Behind law uses carrots and sticks to induce gains in reading and math. But it, too, is a structural approach, grandly leaving to states the details about what is taught and learned. The one area where the law does opine on substance—regarding the teaching of phonics—is the one area where genuine improvement has taken place in the earliest grades.

Let’s not forget the structural approach of the state-standards movement, either. Though state standards are, in my view, potentially the most promising reform project, they decline to offer guidance on the substance of what is taught and learned in the language arts, the most important area of early schooling.

[snip]

And then there’s the structural orientation of the dominant theories within the educational establishment—that activities, rather than “mere facts” or a “rote-learned” academic curriculum, will induce academic progress, and that children’s natural development will allow them to gain or “construct” the needed knowledge.

All of these ideas leave the grade-by-grade specifics of the curriculum to be determined by some quasi-divine agency—the magic of the market, the wisdom of the locality, the nature of the child—in short, not mere policy makers, but somebody or something else.

Yet the grade-by-grade core substance of the curriculum is what schooling is.

I alternate between the Sol Stern camp and the wild-eyed radical John Taylor Gatto give the money to the homeschoolers and let's be done with the whole thing camp.

Let a thousand homeschools bloom.

That's what I say.

Depending on the day and the hour, of course.

Kindergarten Math in the Twilight Zone

My kindergartener received her report card yesterday. So, what do they grade a five year old on in math these days? Just take a look at the expectation for January:

Applies Understanding of Number Concepts and Basic Operations
Uses one to one correspondence to count objects
Identifies numerals 0-20
Writes numeral 0-20
Counts 0-70
Counts backward 10-0

Applies Mathematical Thinking and Reasoning
Sorts and classifies objects
Recognizes, reproduces and extends patterns
Understands estimation
Performs data collection and creates simple graphs

This is NOT the expectation for January:

• Counts 0-115
• Counts backward 20-0
• Understands the meaning of addition
• Understand the meaning of subtraction
• Recognizes penny, nickel, dime and quarter
• Tells time to the hour

Students aren't expected to know that until June.

This is life in the world of Everyday Math. It's like another plane of reality where it's more important for a five year old to learn how to estimate, collect data, and create graphs than it is to learn how to add, subtract, count coins, tell time or count backward. Actually, the expectation isn't even that these kindergarteners will enter first grade actually adding and subtracting, it's enough that they "understand the meaning" of adding and subtracting.

Yes folks, the problem with Everyday Math begins as early as kindergarten and will haunt your children each step of the way.

Then they wonder why parents get upset.

Cross posted at Mindless Math Mutterings.

down for the count, possibly

Came home from Fordham Prep yesterday (much to report) to find that our internet router (or whatever it's called....) is kaput so my internet access is iffy. At the moment I'm pirating my neighbor's network, so we'll see.

Back for sure sometime Wednesday.

"Orton-Gillingham for Math": "Making Math Real" -- Introductory Post

Making Math Real

http://www.makingmathreal.org/

I just finished the 2-day, 14-hour introduction:

http://www.makingmathreal.org/index.php?option=com_content&task=view&id=38&Itemid=57

My brain is either fried or overloaded. Lots more sympathy for kids whose learning styles aren't supported in the classroom.

[Aside: hush!--already about the "learning styles". I agree that lots of the edu-babble about learning styles is content-free. However -- I sat, listened, and wrote for 120-minute blocks, about a subject that I am deeply committed to. I'm an adult with a well-honed capacity for taking in new information by listening, and retaining new information by writing. By minute 90, I was overloaded. You go shadow your kid throughout her school day and see how well you could keep up. /Aside]

Making Math Real (MMR) isn't a math curriculum, it is a specific, structured approach to math instruction.

A good analogy is:

Orton-Gillingham approaches to reading = MMR approach to arithmetic and mathematics.

The training is expensive.

Question: Is it worth it?
Answer: If your district is using Everyday Math (or other purely constructivist math curricula), it is definitely worth it to offset the fog of confusion EDM engenders in many kids, with or without LDs.

It may also be less expensive in the long run than putting your kids in Kumon, Sylvan, or other after-school remediation/tutoring programs.

Here's a description from the MMR website:

Multisensory structured methodologies deliver all instruction via the three processing modalities: visual, auditory and kinesthetic-motoric. Students who are struggling experience processing difficulties in either one or more of these processing modalities. Best instructional practices require linking all incoming information across the three channels to maximize successful processing.

Structured curriculum means starting with the simplest elemental foundation and building developmentally in an incremental and systematic progression from the concrete to the abstract. The most powerful aspect of a multisensory structured program is that each current activity and lesson builds the essential developmental tools for success at the next level thereby reaching the full diversity of learning styles and educational needs in all classrooms.

Schwablearning.com was a great source of discussion and information -sharing for non-standard kids. Here are links to previous discussions of Making Math Real on the parent-to-parent board (listed in chronological order):

• http://www.schwablearning.org/message_boards/view_d iscussion.aspx?thread=3601
• http://www.schwablearning.org/message_boards/view_d iscussion.aspx?thread=4982
• http://www.schwablearning.org/message_boards/view_d iscussion.aspx?thread=15320
• http://www.schwablearning.org/message_boards/view_discussion.aspx?thread=18313
• http://www.schwablearning.org/message_boards/view_d iscussion.aspx?thread=19498
• http://www.schwablearning.org/message_boards/view_d iscussion.aspx?thread=24091
  
• http://www.schwablearning.org/message_boards/view_d iscussion.aspx?thread=25871

BTW #1: , as part of the class, I am now "bombproof" on my 13 multiplication facts, thanks to the "nine lines". I'd lay it all out for you now...but I am totally used up, cognitively.

And actually, I'm kind of appreciative of that experience -- because, for some of our kids, they are "totally used up, cognitively" before the end of the school day. That doesn't happen so often for us as adults.

BTW #2 -- being "cognitively tapped-out" is why I think homework in k-3 is stupid a waste of the (a) teacher's, (b) student's (c) parent's time and energy.