NoOneToWaste < Kitchen

23 Oct 2006 - 21:39

no one to waste

Now that I've been involved in "remediating" at least two people (not counting a couple of the kids in Singapore Math &, I guess, not counting me, either) I've realized that: I am interested in the question of remediation.

It's entirely possible that we have some real research avilable to us in the area of remediation, as opposed to your standard edu-research, which is not research but log-rolling.

If so, this would jibe with my feeling that when it comes to teachers & principals, you want the teacher or principal who's taught special ed.

Christopher's ELA teacher, the one who gave them the SWB and so assignment, has taught special ed.

So has the new principal; special ed is his background, I believe.

And then there's Siegfried Engelmann.

Teachers and administrators who've worked in special ed have spent a great deal of time endeavoring to teach the kids who need teachers, a large point in their favor. They haven't been able to coast on the gifted kids who "breathe it in"; nor have they been able to "teach to the middle."

They've been teaching to the bottom, by definition. I don't mean this statement to hurt feelings; I know special ed kids can be brainy — I've got some brainy special ed types around here. But when a special ed teacher is teaching a high-end special ed kid with strong native intelligence, she's still teaching a kid who has many gaps in his knowledge and probably a roaring case of edu-shame and demoralization along with. Special ed teachers are working with the most challenging population, regardless of how naturally intelligent any given special ed kid may be.

Beyond this, special ed teachers have spent a great deal of time — countless hours of time, in fact, and I am in a position to know — sitting in face to face, legally mandated meetings with parents hashing out what will be taught in the upcoming year. These teachers experience strong external legal, moral, and emotional demands to get content inside their students' heads.

Obviously any good teacher feels internally pressured to get content inside students' heads. Depending upon the school, she may be feeling heat from NCLB & state standards, too.

But special ed teachers live with intense outside pressure from law and from parents.

They've been shaped by different forces.

And then there's the fact that people who choose to go into special ed are the salt of the earth.

I was drafted. They volunteered.

So I've come to feel optimistic the minute I hear that a teacher Christopher is going to have has a background in special ed. My reaction may be common; my neighbor said "Good" the minute she heard that the new middle school principal had a background in special ed.

Anyway, I realized this afternoon that this principle may also apply to education research. It's possible that research into remediation is more serious and less ideology-ridden than research into "literacy" and "mathematical reasoning" and the like.

So I'm going to start paying closer attention to the field.

no one to waste

I've been prospecting for remediation factoids for awhile now.

Here are a couple:

The study found that students who are successfully remediated become productively employed. Almost 16% become professionals; 53.7% obtain mid-level, white-collar or technical positions; 19.8% become high-skill, blue-collar workers; and only 9.2% remain in unskilled or in low-skill jobs.

source:
No One To Waste: A Report to Public Decision-Makers and Community College Leaders

Students who require remediation in reading are at a greater disadvantage than those with a math deficiency (McCabe, 2000).

source:
Remediation PostSec

...about half of the students entering college aren't prepared for credit-bearing college-level work and have to take remedial courses, according to the Department of Education (2001). Overall, more than one million entering college students take remedial courses, says R. H. McCabe in "No One To Waste" (2000). McCabe cites studies that show 20 percent of entering students are underprepared in reading, 25 percent in writing, and 34 percent in math. Students who take remedial courses are much less likely to graduate.
source:
ACTIVITY

It's always interesting seeing the same study characterized as optimistic ("students who are successfully remediated become productively employed") and pessimistic ("Students who take remedial courses are much less likely to graduate") both.

McCabe's study doesn't seem to be available online, but I've found an interview with him.

Remedial students range from those with a minor deficiency in only one area to people who are deeply deficient in all three. There are roughly one million students a year who begin in college and are assigned to remedial courses.
PC: Is that number for all institutions or just community colleges?
RM: In all of them. It’s 29 percent of all of the entering college students. In community colleges it’s about 40 percent, and I do not know of an urban community college where it is not at least half.

[snip]
...the average student takes a little over seven semester credits for remediation, approximately one-fourth of a college year.
PC: What is known about whether, or how well, remediation works? Your report talks about the perception, especially among some state legislators, that we simply are paying for failure one more time?
RM: There is that perception, but more often than not it’s based on faulty premises. First, some of these legislators see continuity between high school and college that doesn’t exist. They think that if students graduate from high school, they should be ready for standard college work. But the criteria for high school graduation do not match the competency requirements to begin standard college courses....
Second, our definitions of success may be too narrow. A variety of studies, including my own, show that between 40 and 50 percent of students who begin in community colleges are successfully remediated. But, the fact is that a high percentage is not going on to bachelor’s degrees, even though remedial programs are really geared for that purpose. Twice as many earn occupational associate degrees or certificates.
If you look at remedial students nine years later, 90 percent are employed in work above an entry-level type of job. Less than two percent are out of work. Remediation in the community colleges is much more occupational and vocational than we have thought, and measures of its success should recognize this.
For the seriously deficient, the program is basically a total failure. Only 20 percent of those students complete remediation, and very few go on to anything after that. Programs for the seriously deficient need to be redesigned altogether.
PC: How do you define “seriously deficient”? Out of the million students mentioned earlier, how many are there?
RM: Seriously deficient students are those who are deficient in reading, writing and math, and are required to take at least one course prior to the standard remedial courses. For example, they simply could not take one remedial math course and be ready—they would have to take a math course prior to that math course. As for numbers, I lack data for all institutions, but in the community colleges I looked at, about one-fourth were seriously deficient.
[snip]
Actually, remediation is the most productive education program we have. With just one percent of the budget, it salvages the lives of a half-million people, and enables them to become positive, con-tributing individuals in our society. It is a particularly good investment because of the country’s changing demography and economy. The part of our population that is growing the most is also the part that is least prepared for college and skilled jobs. We believe in opportunity and access, and we can’t have either without remedial education.
...you assert that remediation is essential for quality, not a detriment to it. Is this true?
RM: It is absolutely true. In regular courses, faculty must expect that students are prepared, and only remedial testing and placement can assure that this expectation is met. If it is not met, then quality will suffer because faculty will be forced to lower their expectations to meet the competencies of the unprepared, or too many will fail.
[ed.: this reminds me of my hypothesis that when you refuse to accelerate gifted children you also refuse to accelerate regular children]
...it is essential to America, morally, socially and economically, that a high percentage of young Americans have education or training beyond high school—as high as 80 percent, in my opinion.
[ed.: I think it's Robert Murnane (not sure) who has shown that the only reason it's essential for 80% of young Americans to have training beyond high school is that high school isn't high school any more. It's junior high. Will attempt to track down the source at some point.... yes, it's Murnane^*
[snip]
...teaching less-prepared students is hard and often frustrating work, for they need more personal attention and support. A belief that their job is helping human beings to develop, and that academic work is, in fact, an instrument to that end, is not typical of college faculty.
[snip]
One of the most disappointing things to me is that typically community colleges, where most academically deficient students enroll, do not seem to use the resources that they are given for remedial programs. In fact, they often use mostly part-time instructors without additional and necessary support. They give remediation low priority and run cheap programs.
[snip]
PC: If all of the K–12 improvement programs that the country has been working so hard on are successful, can’t we look to the day when remediation will not be around in higher education?
...I simply do not see reform wiping out the need for remediation in the fore-seeable future. We have to dig in for the long run.
[ed: amen]
[snip]
Testing should recognize a continuum; secondary school testing should be closely related to tests used to place people in college. With regard to under-prepared students, experience together with educational research has produced substantial knowledge of effective learning practices. Typically community colleges do not use that knowledge. They must.
[snip]
Most programs simply identify someone who is deficient in math or reading or writing, and then assign students to subject classes based on test results. That is a great waste of resources and student time; nothing is done to identify the differences in deficiencies within and across subject fields, or to relate deficiencies to the learning program.
We have the capacity to produce diag-nostic assessments and to align each student’s program to the results.
[snip]
A final word, if I may. I was surprised by our data on just how educationally far behind ethnic minorities are. [ed.: No kidding.] There is a monumental problem here that, in my opinion, needs to be addressed with mega effort and resources, not just by schools, but also by states and by communities.

what skills does one need to hold a middle-class job?

RICHARD MURNANE, Co-Author, "Teaching the New Basic Skills:" To qualify for a job that will pay a middle class wage as minimum the graduate needs to be able to read well enough to understand training manuals, basically ninth grade, able to do the mathematics that’s typically included in training manuals, fractions and decimals and line graphs, mastery of that, the ability to problem solve, to take a problem and find what will work, to shape it, to design a solution towards it, and two kinds of what we call soft skills, the ability to communicate effectively both orally and in writing, the ability to work productively with people from different backgrounds, and enough familiarity with computers to have the self-confidence and the knowledge to learn to use new software. You might say these skills are extremely modest, and they are in one sense, and there are lots of jobs that require, that pay good wages that require a lot more than these skills, but there are almost none, outside of professional sports, that do not require at least these new basic skills and also, remember, in terms of whether this is a challenge for schools to provide this, roughly half of American high school seniors are graduating without these new basic skills.
DAVID GERGEN: And if they graduate without them and never get them, they’re condemned to live in very low wages? RICHARD MURNANE: They can find work in most cases but these are jobs that pay six and seven dollars an hour, not enough to support children.
DAVID GERGEN: And if they can get the skills?
RICHARD MURNANE: If they can get the skills, they have a chance at acquiring middle-class jobs and have access to subsequent training when they need it. And these aren’t jobs that will be jobs for--that one holds for twenty-five or thirty years. To a large extent, those jobs have disappeared from the economy, but it will be the opportunity to move from job to job and to earn enough to support kids.

If you don't get these skills in high school, and black and Hispanic students routinely do not, you've got to get them somewhere else.

That somewhere else is going to be our community colleges.

Our community colleges & John Saxon.

^* Murnane and coauthor Levey do not appear to hold the view that public school decline has made it necessary for high school graduates to pick up what once were K-12 skills in college:

FRANK LEVY: That’s right. And that’s what makes the problem so hard to diagnose. Schools today are a little better than they were 15 years ago, but the job market skills have just escalated much faster than that. I mean, we have an example in the book, "Looking At a Modern Automobile Plant," and about half of today’s high school graduates couldn’t make the cut-off to be a production worker at a modern automobile plant.

DAVID GERGEN: Say it one more time. Half of the 17-year-olds--

FRANK LEVY: Half of 17-year-olds don’t have the skills necessary to make the cut-off of the production worker at a modern automobile plant today.

-- CatherineJohnson - 23 Oct 2006

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I disagree with Murnane and Levy. Public schools have declined. They may be marginally better than they were 15 years ago, but what about compared to 30 or 40 years ago? My dad took "dummy math" in the '50s, and his math skills are still better than those of most high school graduates today.

Schools are only quick to "adapt" to changes when it means dumbing down the curriculum. Adapting upward to meet changes in skill requirements is beyond the school system's ability.

-- BrendaM - 24 Oct 2006

"I've been prospecting for remediation factoids for awhile now."

How about the primary factoid:

"An ounce of prevention is worth a pound of cure."

Remediation, by definition, implies previous failure.

-- SteveH - 24 Oct 2006

How about continuous remediation? I call it teaching.

This is when the teacher does his/her job and sees right away when something is going wrong. When I taught, it starts in the classroom with looking at their eyes and body language to determine if another approach or more explanation is needed. The next level is homework. After that it's tests. If teachers (or schools) don't deal with the problem (and they sure as heck know there is a problem) and send the kids off to the next grade, the teachers and schools need remediation.

-- SteveH - 24 Oct 2006

"Actually, remediation is the most productive education program we have. With just one percent of the budget, it salvages the lives of a half-million people,..."

In our public schools, they use a phonics-based approach for remediation. This doesn't mean that remediation is good. It means that phonics is good. If your point is that this is lost on schools, then I would have to agree. But, then again, they probably think that it is not a preferred method - only a last resort.

You can argue that remediation is wrong, but you really can't argue that effective remediation techniques need to be used in place of the primary techniques. You could argue that these techniques need to be included in a teacher's arsenal so that remediation can be avoided. They just might find that these techniques work better than their primary ones.

-- SteveH - 24 Oct 2006

"Schools today are a little better than they were 15 years ago, but the job market skills have just escalated much faster than that. I mean, we have an example in the book, "Looking At a Modern Automobile Plant," and about half of today’s high school graduates couldn’t make the cut-off to be a production worker at a modern automobile plant."

Don't tell me that the average Game Cube, Nintendo, Ipod, cell phone, computer-junkie, MySpace^?, internet surfing 17-year old doesn't have the basic common sense to work at a "Modern Automobile Plant". Communication? Team work? Problem solving? I hate all of this garbage about "21st Century Skills"! This doesn't mean (obviously) that I think schools are doing a great job. I just think that this line of argument is always used in one self-serving way or another.

-- SteveH - 24 Oct 2006

15 years ago American schools were well into the NCTM sponsored decline of fuzzy math. That was 1991, two years after NCTM put out their manifesto Curriculum and Evaluation Standards. I don't think things have changed much for the better since then.

I agree with SteveH^? in principle here. I think it is irrelevant whether or not a hs graduate could make the cut-off to be a production worker at a modern automobile plant. When's the last time GM opened a new plant? We have lots of unemployed auto workers ready to fill the plant, should one open.

Unfortunately, the huge political debates over the H1b visa program expansion indicate one important thing. We don't seem to be graduating kids out of HS that are ready to go on to be engineers and computer techies with a collegiate degree. There's little doubt in anyone's mind (I think) that today's game-cube, yada yada yada, nerds are not making the grade in college level math and science. We have to import workers for that.

Remediation isn't going to help, because that sets the bar to low. But, of course, we have to do a better job of teaching real math everyday.

-- LynnGuelzow - 24 Oct 2006

As one of the tutors in my tutoring office says, "By the time I get them, it's too late." She tutors language arts.

But it is absolutely true in mathematics.

And I have to disagree about Special Education teachers. The students that I tutor who have been in a special education program all the way through high school are hopelessly behind. I believe that since elementary school, they have been stuck in a math class with a calculator and told to use it. Then they come to community college and test into the most basic math class where.....calculators are not allowed.

Then they have to try to cover all of elementary and middle school math in one semester.

College is not a good place to remediate. First of all, even at community college, you usually cannot pick your instructor. Some instructors go much faster than others: I am tutoring students in the same class. One student is on sec 4.4, one student is on sec 3.1. Secondly, in a college class, most teachers do not do homework problems unless specifically asked about a problem. If asked, they will go over a problem, but never seem to ask what the student got wrong. The teachers do not go over any problems on the test. And they never actually watch a student do a problem. (There may be exceptions but this is what I've seen). Finally, a college class assumes no gaps in knowledge. If a student cannot do a certain topic, there is no mechanism for assessing the reason and closing the gap.

So a remedial college class does not actually remediate. The class simply reteaches the same material at a faster pace.

In the college algebra class that I take notes in, approximately 50% of the class had pre calc or AP calculus in high school. But if your SAT score is low enough (I think it is 600 in math), then you have to take a placement test. There have been two tests in this class and we have lost 1/3 of the class. One student actually scored a 14 out of 100.

-- AnneDwyer - 24 Oct 2006

How about the primary factoid:

"An ounce of prevention is worth a pound of cure."

I love it!

damn straight

-- CatherineJohnson - 24 Oct 2006

The students that I tutor who have been in a special education program all the way through high school are hopelessly behind.

That's exactly Christian's situation, along with his friend.

They don't qualify for the pre-pre-algebra course.

-- CatherineJohnson - 24 Oct 2006

Public schools have declined. They may be marginally better than they were 15 years ago, but what about compared to 30 or 40 years ago?

I'm with you.

My own dad, who is in his 80s, has far more math knowledge and comprehension than I did at the end of high school.

-- CatherineJohnson - 24 Oct 2006

When I taught, it starts in the classroom with looking at their eyes and body language to determine if another approach or more explanation is needed. The next level is homework. After that it's tests. If teachers (or schools) don't deal with the problem (and they sure as heck know there is a problem) and send the kids off to the next grade, the teachers and schools need remediation.

That is what we have EVERYWHERE.

This is why I've "enlarged the problem" here in Irvington.

It's not just about the math.

The whole K-12 system here needs reform.

For me, the central fact about my district's failures remains the fact that when Christopher failed 1/3 of his entire 4th grade math course nobody notified us. Nor did do anything to remediate his learning.

They just let the teacher go and we all moved on.

I'm now hearing from parents whose kids have suffered all kinds of damage to their math learning, self confidence, and liking for the subject.

These were kids who were in Phase 4 & were dropped down.

I'll post that story tomorrow, probably. I've actually written a coherent narrative of what happened - though in the past couple of days another piece of the puzzle has fallen in place.

-- CatherineJohnson - 25 Oct 2006

My district doesn't blink when the regular-ed kids fall by the wayside.

If they fall, they fall.

They're still above average & that's what matters.

"In Irvington, everyone is above average."

That's a quote.

School Board president wrapping up the presentation on the 8th grade ELA scores.

-- CatherineJohnson - 25 Oct 2006

I surmise that these two simply don't want to get into the public-schools-stink mode.

I think they have an ethical duty (I'm big on ethical duties these days, aren't I? I'm going to have to get off that-) to do so.

They don't need to bash K-12, but they should figure out an honest position and state it freely.

The problem with our public schools isn't that Hispanic kids keep coming across the border.

The problem with our public schools is that they take zero responsibility for finding out whether kids have learned the content.

Obviously Christian could learn arithmetic perfectly well as a child - and he did learn quite a bit of arithmetic.

The only reason he didn't learn what he should have learned was that his schools didn't take responsibility.

ACTUALLY, tonight he's saying that the Mamaroneck schools were good.

I don't think I agree.

He moved to Yonkers in 7th grade; if Mamaroneck had been doing its job he should have fallen off the math cliff at the end of 6th, not back in 3rd.

-- CatherineJohnson - 25 Oct 2006

It's entirely possible, however, that Mamaroneck did a good job moving his reading ahead. He had the advantages of a white kid in that realm, since his mother and aunt (don't know about Dad) were readers. Maybe his grandmother, too.

He came from a reading family & so was probably a good match for Mamaroneck schools in that realm.

The amazing thing, though, is that his friend, who dropped out in 10th grade, is also a good reader. He says she's super-literate, meaning she reads all the time.

I should find out whether he knew her in Yonkers or Mamaroneck.

-- CatherineJohnson - 25 Oct 2006

As one of the tutors in my tutoring office says, "By the time I get them, it's too late." She tutors language arts.

But it is absolutely true in mathematics.

You know, I think it's actually exactly the reverse.

I read some fascinating research on adult reading programs (don't know if I can find the citation) and the most progress they can get is 1 to 1.5 years.

Even if adults are in the program for much longer than that, they only pick up 1 to 1.5 years (as I recall - I'll try to find the study).

Reading, basically, is IQ. (This is what all the IQ researchers say; the SAT measures IQ.)

IQ has its "range of reaction"; you can move someone to the top of their range, or press them down to the bottom.

But I'm not sure how easy it is to move someone up his range when he's an adult.

Math is a subject matter.

That's what I've realized in the wake of doing a lot of reading about standardized tests, etc.

Reading tests measure verbal IQ; math tests measure math.

You can pick up a new subject matter any time.

-- CatherineJohnson - 25 Oct 2006

So a remedial college class does not actually remediate. The class simply reteaches the same material at a faster pace.

That isn't remediation.

I don't know why on earth anyone would call it remediation.

That's review.

-- CatherineJohnson - 25 Oct 2006

That's horrifying.

I wish I had John Saxon textbooks for college.

Of course, what I really wish is that I had a John Saxon statistics textbook.

Preferably a John Saxon textbook on edu-data mining.

-- CatherineJohnson - 25 Oct 2006

Lynn, I see the auto plant story as an illustration of how work has changed (although while GM may not be opening any new plants, Honda certainly is). His point is the "unskilled"-level jobs that included the possibility of advancing into a "middle-class job" don't exist today, in the auto industry or anywhere else.

I started to write a long screed in response to the rest of your post, but boiling it down:

Murnane is right. Most jobs, including many that purportedly "require" college degrees, really don't need a "college education." I know men of my father's generation who finished their careers as high corporate executives, and never went to college (but they did finish high school!). But...
Elementary and secondary school education has deteriorated. Today's high school diplomas are meaningless as a measure of education or skill level.
We are graduating a lot of high school students who aren't ready for college-level math and science. In fact, they don't even have 9th grade skills. Tragically, they (and their parents) don't find this out until it's too late. (Read KDeRosa^?'s comment on this thread.) The result...
Companies hang the "college degree" requirement on jobs that only require a 9th grade education, in hope that the "degreed" applicants will be as qualified as the high school graduates of two generations ago.

But I see the "H1b" argument as a smokescreen. I'd postulate that a ton of qualified students who could have become "engineers and computer techies with college degrees" looked at the job market and the working conditions in those industries, then made the sensible (for them) economic decision: Better to be a lawyer (after all, there aren't any H1b lawyers competing against you), or an MBA, or speculate in real estate (or start their own company). Why spend 4-8 years grinding through hard science courses only to wind up competing with H1b visa holders for depressed wages at companies known for "death march" development, disconnected management, and "outsizing" (or "offshoring") the moment the company's executives decide that the quarter's figures aren't up to snuff?

We "have to import workers" because the imported workers are willing to work for minimal wages in practically indentured positions, knowing that they can go back to their low-cost-of-living countries and live like millionares off the proceeds, after their employer is done with them. Americans generally want to stay in America, and generally have American expectations.

-- OldGrouch - 25 Oct 2006

I see the auto plant story as an illustration of how work has changed

Right

That's his point; he's saying that an entry-level auto plant job isn't what it used to be. It requires higher skills.

-- CatherineJohnson - 25 Oct 2006

We are graduating a lot of high school students who aren't ready for college-level math and science.

Right, and in fact we have figures for this....let me find them (from ACT's latest)...Nearly Half of ACT-Tested Graduates Aren’t Ready for College Reading

Of the nearly 1.2 million 2005 high school graduates who took the ACT® test, only 51 percent met the college readiness benchmark score of 21 on the Reading Test. Students who reach or exceed that score are likely ready to handle the reading requirements for typical credit-bearing, first-year college social science courses.

The percentage of graduates meeting or exceeding the reading benchmark in 2005 was the lowest in more than a decade. It peaked at 55 percent in 1999 and has declined since.

Experience reading complex texts in high school is the key to developing college-level reading skills, according to Reading Between the Lines. The report calls for major changes in high school reading standards and instruction. ACT data clearly reveal factors that make a difference in a student’s ability to read at college level. The report suggests that many high school teachers are either not teaching reading skills or not exposing students to the types of texts they will encounter in college and the workforce.

ACT’s findings suggest the ability to read complex texts is the best differentiator between students who are more likely to be ready for college-level reading and those who are less likely to be ready. Across the nation, however, states are virtually ignoring the complexity of texts in their high school learning standards in reading. Most states don’t define the types of reading materials high school students in each grade should be exposed to, and not a single state defines complex texts.

Their math remediation figure is elsewhere; shouldn't take much time to find it.

-- CatherineJohnson - 25 Oct 2006

"The problem with our public schools is that they take zero responsibility for finding out whether kids have learned the content."

I can't quite figure it out. Most teachers care. Perhaps they think the kids will eventually learn the material. This is the point of spiraling the curriculum. However, our public schools have a curriculum that states in which year mastery must be achieved for each. It doesn't get done. Perhaps they try, but what if students are struggling with material from two years ago? How is the teacher supposed to deal with them? What are the other kids doing in the meantime?

Social promotion and curriculum spiraling seem to have the very damaging effect of removing all responsibility. Everybody thinks it's someone else's job. Wait long enough and you can blame it on the parents or external causes. Before social promotion and spiraling, all teachers had to make the pass/flunk/summer school decision. Teachers made sure that parents knew what was going on because there might be a good chance that little Johnnie isn't going to be moving on to fourth grade with his friends. When I was growing up, summer school was an unthinkable horror. Flunking was not an option. Nowadays, this leverage and "Real World" motivation is gone.

Don't teachers see what's going on? There is absolutely no reason why kids should be getting to fifth grade without mastering their times table. This is very easy to test. Failure = Summer School! They don't do it. Perhaps they've convinced themselves that it isn't necessary. Perhaps the upper grade teachers find it easier to complain about getting kids ready for testing than yelling at the administration for passing along unprepared kids.

Has anyone done any research into when social promotion got to be so common? Why, exactly, do they do it? The only things I have ever heard are 1) it's better for the kids to stay with their peer group, and 2) kids who are held back have a higher rate of dropping out. The second reason is absurd and the first is just ignoring the problem and postponing the inevitable.

-- SteveH - 25 Oct 2006

This is a horrifying statistic.

Reading is IT, the big kahuna.

Math is content; reading is IQ.

Our schools are in all likelihood graduating kids at the low end of their IQ "range of reaction."

-- CatherineJohnson - 25 Oct 2006

From 2003, this is pretty bad:

Only a fourth (26 percent) of 2003 graduates earned a score of 24 or higher on the ACT Science Test, while just four in ten earned a score of 22 or higher on the ACT Math Test. ACT research has shown that students who attain these college readiness benchmarks are more likely than others to be ready for college biology and algebra courses, respectively. They also have a high probability of completing these first-year courses with a grade of C or higher.

In comparison, the large majority (67 percent) of students scored 18 or higher on the ACT English Test, indicating they likely have the skills necessary to be ready for college English composition classes.

This could be a temporary drop, but a decline from 67% of ACT test takers (assuming the test population is more or less stable) to 50% in 3 years sounds bad to me.

-- CatherineJohnson - 25 Oct 2006

So on ACT we've got:

51% in 2006 ready for college level reading

40% in 2003 ready for college level math

25% in 2003 ready for college level science

-- CatherineJohnson - 25 Oct 2006

Most teachers care. Perhaps they think the kids will eventually learn the material. This is the point of spiraling the curriculum.

I've got to get to bed, but tomorrow I'll quote a passage from my book on teaching to mastery.

It's simply assumed that all classes will be mixed ability — very mixed ability.

The authors say that one of the main edu-objections to teaching to mastery is that if you taught to mastery you'd have everyone waiting on the slow kids.

This is a direct, unironic statement of fact; the authors simply take it for granted that all classes, everywhere, have 3 distinct levels of students learning at 3 distinctly different speeds.

They say that it is standard practice to teach to the middle — and they explicitly define the "middle" as too fast for the bottom 3rd & too slow for the top.

(I think it's a 3rd.)

In one passage they indicate that our entire public school system is deliberately not teaching anyone to mastery because it would mean everyone moving at the slowest possible pace.

-- CatherineJohnson - 25 Oct 2006

Spiraling lets you move at a good clip.

-- CatherineJohnson - 25 Oct 2006

It's another scheduling problem.

Like what we've got here at every level of the district.

-- CatherineJohnson - 25 Oct 2006

A group of parents fought 8 years for a foreign language program in the lower grades.

The district fought back for 8 years on grounds that foreign languages would be too hard to schedule.

-- CatherineJohnson - 25 Oct 2006

I'd postulate that a ton of qualified students who could have become "engineers and computer techies with college degrees" looked at the job market and the working conditions in those industries, then made the sensible (for them) economic decision

I suspect that's true myself (based in part on conversations my brother-in-law has had with his rich-Indian Silicon Valley entrepreneur brother-in-law.

The guy says Americans will never do the hard work of an 8-year Ph.D.

They can get out of school and make good money for less work.

I'm sure there's some truth to that.

-- CatherineJohnson - 25 Oct 2006

[J]ust four in ten earned a score of 22 or higher on the ACT Math Test. ACT research has shown that students who attain these college readiness benchmarks are more likely than others to be ready for college biology and algebra courses[.]

Even scarier? The "college algebra" that a student is ready for with an ACT score of 22? This is the material taught in your typical high school Algebra 1 and Algebra 2 classes. I think that the ACT cut-off to be ready to start calculus is 26.

-- RudbeckiaHirta - 25 Oct 2006

The "college algebra" that a student is ready for with an ACT score of 22? This is the material taught in your typical high school Algebra 1 and Algebra 2 classes. I think that the ACT cut-off to be ready to start calculus is 26.

What's the highest score on the ACT?

Is it 28?

I have to check.

I don't need to check to know you're right, though. After I finished Saxon's Algebra 1 I got an 800 (or close to) on a sample SAT test, and this was doing the problems in my head.

Interestingly, I printed out the NY Regents A exam & did quite a bit of it last night; there were a couple of problems I had to think about. (There was a word problem with 4 variables that I needed to be sitting at a desk to do, as opposed to lying down in bed in dim light using a pencil...)

-- CatherineJohnson - 25 Oct 2006

Again, you've put your finger on my dilemma.

Unless Christopher gets through AP calculus here, I have zero confidence he'll be able to take any kind of math apart from elementary statistics in college.

I suspect, now that I've worked with Saxon's high school books, that he'd be prepared for college math if he worked through Saxon's high school books, too.

-- CatherineJohnson - 25 Oct 2006

High school student follows up top SAT score with perfect ACT score

While most students hope to improve their academic performance from one year to the next, it will be hard for William Park Cram to outdo his junior year.

The rising senior at Richland Northeast High School in Columbia learned this week that he had scored a perfect 36 on the ACT. According to ACT officials, Cram was the only South Carolina student to accomplish the feat and one of only 60 nationwide.

Amazing, this was Cram’s second round of college-prep perfection. He had already scored a 1600 on the SAT, the highest possible mark.

In addition to a superlative academic record, Park was a member of the school’s 13-time international champion Model United Nations Team and is currently a member of the Junior Academy of Science program and the Mu Alpha Theta math club. He is also on the Richland Northeast High School SAT Team (AAAA state champions in 2000-01) and is a volunteer with the nationally recognized "Souperbowl of Caring."

Cram’s double play marked the second consecutive year that a student from Richland District 2 achieved top scores on both the ACT and SAT. State officials can’t recall any other instance where that has occurred.

According to the College Board, which adminsters the SAT, only one student in 5,000 earned a top score on that exam last year. Likewise, ACT, Inc. reports that only one in 6,700 made perfect marks on the ACT.

Thursday, June 20, 2002

-- CatherineJohnson - 25 Oct 2006

"The authors say that one of the main edu-objections to teaching to mastery is that if you taught to mastery you'd have everyone waiting on the slow kids."

They use this as a justification for not teaching ANYTHING to mastery for ALL kids? When did mastery get thrown out of K-8?

My impression is that, compared to when I was in public schools, there is a much wider spread of abilities in each classroom. In the old days, the idea was main-streaming the lower ability kids, but they still had to meet the same yearly grade level pass/fail criteria (at least get close). Now, with full-inclusion, there seems to be no yearly pass/fail criteria, but this is eliminated for all kids.

"They say that it is standard practice to teach to the middle — and they explicitly define the "middle" as too fast for the bottom 3rd & too slow for the top."

This has always been the case, but now these ranges are larger.

I guess my question is when did schools start dropping yearly pass/fail expectations? What is their justification for this? They don't think mastery is important enough? Do they think it will just happen?

"In one passage they indicate that our entire public school system is deliberately not teaching anyone to mastery because it would mean everyone moving at the slowest possible pace."

This is a cop-out. There has always been multiple levels of abilities in schools. One could argue that schools never guaranteed mastery for kids moving on to the next grade, even in the old days. They tried, however, and if your mastery was not good enough, you failed. Now they seem to be using this as an excuse for full-inclusion.

The problem is not that they can't do it. The problem is that they don't want to do it. They don't even try. Why?

Everyday Math offers very little practice in their Home Links. This is not even enough for the top third of the class to achieve mastery. They could say that they will try for mastery, but it's not a yearly pass/fail criteria. They don't even do this. They just don't like mastery. They don't like repetitive practice. It's drill and kill. Somehow they think that hard work is not necessary for mastery.

In my meeting with my son's fifth grade EM math teacher last Friday, we talked about the potential for changing to a better curriculum (note that the ability level of the students is not a concern in this discussion). She mentioned that in one of their conversations, a few teachers commented about "rote" knowledge related to doing practice of basic arithmetic skills. They are mixing up mastery with rote knowledge and drill and kill.

They are NOT doing this just to keep the advanced kids from waiting for the slower kids. They are damaging a lot of the advanced and average kids.

" ...deliberately not teaching anyone to mastery"

"Deliberately"? Perhaps, but that is NOT the reason. If they moved at a faster pace, then the ability spread would get larger and they would have a real problem on their hands. Besides, what advantage is there for moving at a faster pace when there is no mastery? Most parents who move their kids to other schools do so because the pace is so slow and the expectations are too low.

What is the real reason?

-- SteveH - 25 Oct 2006

"Unless Christopher gets through AP calculus here, I have zero confidence he'll be able to take any kind of math apart from elementary statistics in college."

AP Calculus or calculus of any form in high school is not very important for college. The key is to be in that track, taking those courses, and getting good grades. The danger is that a student ends up in one of the lower, go-nowhere, math tracks.

-- SteveH - 25 Oct 2006

AP Calculus or calculus of any form in high school is not very important for college. The key is to be in that track, taking those courses, and getting good grades.

The reason I assume he needs AP Calculus - and obviously I don't know this - is that will "keep his hand in," presumably expose him to the very best math teacher they've got in the high school, and give him an extra year of practice.

Unfortunately, our accelerated math track, in the middle school, may be quite damaging.

I wouldn't be at all surprised to find that the average track kids learn more than the accelerated kids - and this isn't the Ms. K problem so much as it's the lousy course construction problem.

-- CatherineJohnson - 25 Oct 2006

They use this as a justification for not teaching ANYTHING to mastery for ALL kids?

According to these authors, who are in the heart of the field, yes.

-- CatherineJohnson - 25 Oct 2006

I guess my question is when did schools start dropping yearly pass/fail expectations?

I think that happened many years ago.

"social promotion"

-- CatherineJohnson - 25 Oct 2006

Everyday Math offers very little practice in their Home Links. This is not even enough for the top third of the class to achieve mastery.

Exactly.

I would bet money that the only kids in Christopher's class last year who achieved anything close to mastery of the majority of concepts are the handful of mathematically gifted kids.

I would also bet money that when it comes to mastery Christopher has more than (and certainly not less than) any of the regular smart kids in the class, entirely because I forced him to do extra practice, drill, etc.

And think about it: Christopher, I'm pretty sure, still shows the effects of having flunked 1/3 of 4th grade.

He's coming from WAY behind the pack, and yet he almost certainly has more mastery than all but the truly gifted kids.

(Scratch that. There's another kid whose dad is a math teacher; he'd probably wipe Christopher's bu**. But it's only kids in anomalous positions like Christopher & that kid who have anything even resembling mastery.)

There are a few topics all of the kis probably reached some level of mastery on, because the skills were re-practiced in every unit.

Integers is one.

They learned integers at the very beginning of the year — huge challenge for Christopher — and then they used integers in nearly every unit thereafter.

Ed was in despair over integers.

He'd tell me dramatically, "Christopher doesn't understand integers at all. He can't do integers at all."

Well, every unit had integers, and Ed kept reteaching them and recomplaining about them, until finally Christopher could do them.

-- CatherineJohnson - 25 Oct 2006

" ... expose him to the very best math teacher they've got in the high school"

This should be the case if he is in the honors math track, even though he is one year behind the top students. Doing well in this track is much more important than getting to AP Calculus.

-- SteveH - 25 Oct 2006

At the state university where I work, math and English placement is based on SAT and ACT scores.

We don't let students into Calc I unless they have an ACT math score of 29 or SAT math score of 650. I would estimate that about 70% of students who took pre-calc or calculus in HS don't test into Calc I. When their parents hear this, they usually freak out (as they should).

I remember in 1993, my elementary school in Idaho wanted to promote a 4th grade boy who couldn't read and wasn't placed in special ed. His mother worked 3 jobs to support her 3 kids and lazy bum husband. When she found out they were going to promote him, she livid! She stormed into the principal's office and declared, "You WILL NOT pass my child. You WILL teach him to read before he goes into 5th grade." The principal and teacher did the "he'll be biggger and older than the other kids" song and dance, but she refused to back down. She won, and they DID teach him to read that next year.

-- AndyJoy - 25 Oct 2006

"social promotion"

I guess that is what I am trying to find out. When and why this happened. How has it changed.

They had it, at some level, when I was growing up. Some borderline kids, who really should have been held back, were allowed to go on to the next grade.

Before, it was expedient. You can't keep a kid in third grade forever. Now, it philosophical and pedagogical and the root cause for so many problems. Actually, I recall someone once telling me that the reason for social promotion is that parents won't allow it.

-- SteveH - 25 Oct 2006

This should be the case if he is in the honors math track, even though he is one year behind the top students. Doing well in this track is much more important than getting to AP Calculus.

Absolutely. I agree.

Unless Ed and I manage to make some changes around here, the chances of Christopher getting into an honors course are only 1 in 4 (assuming it's the same formula used to limit enrollment in Regents Earth science in 8th grade).

Our honors courses are strictly rationed.

The Irvington goal is always to cut kids from the high-end track.

-- CatherineJohnson - 25 Oct 2006

I would estimate that about 70% of students who took pre-calc or calculus in HS don't test into Calc I. When their parents hear this, they usually freak out (as they should).

wow

factoid of the day

-- CatherineJohnson - 25 Oct 2006

Actually, I recall someone once telling me that the reason for social promotion is that parents won't allow it.

well, there's usually a parent element in these things

however, my current perception of public schools is that the parent "position" is invariably a logical reaction to lose-lose situation

for instance, in the case of social promotion, the school didn't teach the kid anything in 3rd grade, so how likely is it they're going to teach him anything the second time around?

I know that in Andy's story the school did, but I've seen this over and over again.

Parents' seemingly irrational or unaware-of-the-research views & preferences are frequently a case of going for the least-worst option.

-- CatherineJohnson - 25 Oct 2006

His point is the "unskilled"-level jobs that included the possibility of advancing into a "middle-class job" don't exist today, in the auto industry or anywhere else. I started to write a long screed in response to the rest of your post

Oh gosh, I hope not. I think (in my inartful way) that I agree with you. I guess I didn't make that clear.

Unskilled jobs don't really exist that can launch anyone into middle class. Agreed. Lots of entry service level jobs exist, with no hope of advancement, and HS grads appear to be pretty well prepared for those. But I wouldn't call that a success.

lots of other good points.

But I see the "H1b" argument as a smokescreen. I'd postulate that a ton of qualified students who could have become "engineers and computer techies with college degrees" looked at the job market and the working conditions in those industries, then made the sensible (for them) economic decision: Better to be a lawyer (after all, there aren't any H1b lawyers competing against you), or an MBA, or speculate in real estate (or start their own company).

Really? See, I'm unsure on this. I don't see a ton of qualified students who simply made other choices. My anecdotal experience is that many kids sail along through elem and then ms, maybe even hs, doing low expectation math. The first time they are really given real math, they crumple. They've never failed at anything and their confidence is shot. They give up, partly because they don't know how to overcome their weaknesses. That and they don't want anyone to know that they aren't as smart as they thought. So they take up history (nothing against history) because they "love it" and that's that. I know too many people that couldn't survive college engineering and math so they switched majors. It isn't long term look at future job prospects.

And then there's the law degree. The media makes it seem like every lawyer is a Wall Street lawyer and they all start at gazillion dollars a year. Many, many lawyers would step up in salary to H1B visa rates. The job market isn't great for lawyers.

What are the chances any kid in college really has any good idea about the realistic earning potential of career decisions?

I still believe we don't produce enough highly skilled math and science grads because they are too ill prepared to make the grade when they get to college. Our K-12 schools have failed them.

-- LynnGuelzow - 26 Oct 2006

WSJ had an op-ed on the lawyer issue; there are ZILLIONS of them; the market is saturated, and even the "invented" market (the market invented by legislator-lawyers) is saturated.

-- CatherineJohnson - 26 Oct 2006

hmmm.....found the article; not sure I've characterized it correctly

here's part:

On the surface, the legal profession appears to be booming. Although growth has slowed since the 1960s and '70s, each year 40,000 new lawyers join a field that now totals one million, about the same size as the nation's state prison population. Salaries have climbed steadily, and lawyers at the top firms can expect to make about $160,000 upon graduation from law school. But look beneath the statistics and a few facts jump out. First, large law firms, those employing more than 500 lawyers, lose nearly 40% of their associates within four years of hiring them. After six years, the ratio climbs to 60%.

Some might suggest that the fault here lies with the firms' policies regarding advancement. A number of recent articles have bemoaned the lack of female partners (only 17% of the partners at major law firms are women, while women compose nearly half of all law-school graduates). The number of males who don't stick around long enough to make partner, however, is only a few percentage points lower. Thus, while it may not be easy to be a woman in law, the guys aren't doing much better. In fact, it could be argued that women are leaving in slightly higher numbers because they can while many men, trapped by their gender-typed "provider" roles, have fewer options.

The attrition numbers are even worse in other parts of the profession. According to a recent study by the National Association for Law Placement Foundation, 42% of lawyers in small firms (and 50% in solo practices) have changed jobs within three years of graduation, and two-thirds of them have switched two or more times. One way to interpret the numbers is to conclude that such lawyers have plentiful opportunities and are moving to better jobs. The same group, however, tends to have less stellar credentials and to have graduated lower in their class than their colleagues at big firms, leaving them fewer options, and suggesting that these attorneys are even more dissatisfied than their big-firm contemporaries.

What happens to the recently departed? While many go to other law firms, or into other legal jobs, such as in-house counsel at corporations, anecdotal evidence shows that a significant percentage drop out of the legal profession entirely. This doesn't surprise me: Among my own law school classmates, for example, only one of my friends is still practicing at the firm he joined upon graduation. The rest have moved on or dropped out of the profession.

[snip]

The legal profession is really two professions: the elite lawyers and everyone else. Most of the former start out at big law firms. Many of the latter never find gainful legal employment. Instead, they work at jobs that might be characterized as "quasi-legal": paralegals, clerks, administrators, doing work for which they probably never needed a J.D. Although hard data about the nature of these jobs is difficult to come by (and relies on self-reporting, which is inherently unreliable), the mean salary for graduates of top 10 law schools is $135,000 while it is $60,000 for "tier three" schools. It's certainly possible that tier-three graduates tend to gravitate toward lower-paying public-interest and government jobs, but this lower salary may also reflect the nonlegal nature of many of these jobs and the fact that these graduates are settling for anything that will pay the bills. At $38,000 a year for law school, plus living expenses, law-school graduates certainly have a lot of debt ($60,000 on average, upon graduation). For this price, college students and their parents should be thinking harder about their choices. When I went to law school, nearly everyone tried to convince me that doing so would "keep my options open." All this really means is: "You can still be a lawyer."

If I wanted to be a screenwriter, waiting tables would have kept my options open, too. In fact, many wannabe screenwriters find themselves going to law school, misled by adults into thinking that it will help them get into the movie business. It won't. [ed.: having spent a number of years on the edges of the film industry, I question this. Sure, you can be a talent agent or a movie producer with a law degree, but you can be one without a degree, too. yes. This is certainly true. Most of the skills you learn in law school (and legal practice) won't help you make a movie, and the few that will may not be worth the cost (more than $120,000, including tuition, living expenses, as well as three years of forgone experience and salary). Rather than keeping options open, the crushing debt of law school often slams doors shut, pushing law students to find the highest-paying job they can and forever deferring dreams of anything else.

It's time those of us inside the profession did a better job of telling others outside the profession that most of us don't earn $160,000 a year, that we can't afford expensive suits, flashy cars, sexy apartments. We don't lunch with rock stars or produce movies. Every year I'm surprised by the number of my students who think a J.D. degree is a ticket to fame, fortune and the envy of one's peers -- a sure ticket to the upper middle class. Even for the select few for whom it is, not many last long enough at their law firms to really enjoy it.

Mr. Stracher is the publisher of the New York Law School Law Review and the author of "Double Billing: A Young Lawyer's Tale of Greed, Sex, Lies and the Pursuit of a Swivel Chair."

-- CatherineJohnson - 26 Oct 2006

My husband was a little older when he went to law school and I think that made a difference. He wasn't messing around. He took a course on taking the LSAT which helped him to clock in a high score. He made straight A's in the school all three years, so when the big Chicago firms started looking for summer associates he was on their lists. From there it was far easier to move right into a large firm (and their nice salaries. Starting salaries in Chicago for associates, I think, are at or near $100,000.)

I remember him telling me that a lot of the kids were straight out of college. Between that and Dad footing the bill, they really might not have realized that they still had to compete.

-- SusanS - 26 Oct 2006

older is a big help in that way

I'm almost goofily serious about Algebra 2

I would really like to know whether I'm the only middle-aged woman on the planet sitting around teaching herself Algebra 2

I don't see how that can possibly be the case, but otoh I don't seem to stumble across other folks doing the same

did I tell you I've hatched a plan to take the AP test in calculus?

after I teach myself calculus, that is?

well I have

-- CatherineJohnson - 26 Oct 2006

I was wondering whether I'd have to beg and plead my school district to let me take the AP exam - along with begging and pleading them to allow my child to take an Honors course or two - but it turns out College Board will give the tests to anyone

so we'll see

-- CatherineJohnson - 26 Oct 2006

I'm getting ahead of myself

I finished my Regents Math A exam last night

why don't I go score that?

I passed the CAHSEE (CA high school exit exam) with flying colors

-- CatherineJohnson - 26 Oct 2006

I know NOTHING about how many combinations of 5 you can get out of a set of 9

-- CatherineJohnson - 26 Oct 2006

5! (as in "5 factorial: 5x4x3x2x1") Wrong! Vlorbik corrects me in the next post, which serves me right for trying to show off without checking my references! - OG The proof is left as an exercise. Can you develop a general formula for combinations of n taken from a set of m? ;-)

-o-

I know too many people that couldn't survive college engineering and math so they switched majors.

Agreed. That was exactly Ken DeRosa^?'s point in the thread I linked. Those students believed they were qualified. Everybody told them they were qualified. They had accumulated the necessary credentials that labeled them "qualified." And the university that admitted them must have assessed them as qualified. Turned out they weren't. Terrible waste of time and potential, and needs to be addressed.

But that says nothing about the smart, qualified people who take a look at the field and say, "No thanks, not for me." You'll never see them, and it's my opinion that there are a lot more of them out there than you think.

-- OldGrouch - 26 Oct 2006

the number of "combinations" of 5 objects
from a set of 5 isn't 5! (!), but rather,
C(9,5) (in texts, usually 9C5 with the 9 & 5
subscripted; in journals [and university maths],
usually a 9 above a 5 within a pair of parens).
i pronounce it "9 choose 5" wherever i see it.

the formula C(n,r) = n!/[r!(n-r)!]
is "easily" derived (it can be clearly presented
in a single lecture to algebra-unready students).
somebody please stop me before i try it here.

anyhow (exercise) C(9,5) works out to 126.
you can also work this out by writing down
nine rows of "pascal's triangle" and counting
out to the fifth entry (starting with zero).
these numbers — C(n,r) — are called
"binomial coefficients" because they appear
as coefficients in the expansion of (x+y)^n
(the caret ["^"]denotes exponentiation).

thanks for your kind attention and good day.

-- VlorbikDotCom - 26 Oct 2006

"the formula C(n,r) = n!/[r!(n-r)!] is "easily" derived (it can be clearly presented in a single lecture to algebra-unready students). somebody please stop me before i try it here."

I won't stop you, but I do have a request. I have worked with the formulas for combination and permutations, athough it's not my "thing". The problem is that there are so many darned variations of these and some problem statements are extremely confusing. On top of that, these are some of the favorite problems to give kids for discovery. What I would like, however, it a source or link that explains how to translate all sorts of different problem statements into one of these simple formulas. I have Schaum's guide to Probability and Statistics that gives some of the formula variations, but it requires more study than I am willing to give it.

-- SteveH - 26 Oct 2006

If someone has such a link, I hope they'll share it. I confess to having trouble figuring out exactly when is a problem a permutation and when is it a combination. I guess the trouble must be that I never learned the subject to mastery and have forgotten more than I ever knew. Each time something comes up, I am like Catherine at the U.S. Open -- I have to rethink everything from the very beginning.

-- LynnGuelzow - 26 Oct 2006

You're right, missed the divisor. Well, it's only been 40 years...

-- OldGrouch - 26 Oct 2006

Time for me to recommend The Cartoon Guide to Statistics again. This is more of a "guide to probability with a little statistics" than it is a "guide to statistics", but IIRC it does have illustrations of permutations and combinations.

Also, I'm sure that I have at home some math fun books geared toward pre-adults that introduce permutations and combinations, but I don't remember the titles off the top of my head.

Permutation: the order is important, so there will be more of these (than combinations).

Combination: order is not important, so there will be fewer of these (than permutations). Fewer is your cue to divide by something.

If the question just boils down to "how many ways are there to choose m objects from n objects?" then you want the number of combinations, because order is not important. If the question implies or states order, then you want number of permutations.

Nobody's stopping us, so...

Think of 9 marbles: abcdefghi. I want to choose 5 of them.

For the first one, I can pick a, or b, or c, or ..., or i. That's 9 different choices for the first one, or 9 ways to choose 1.
For the second one, I can pick any of the 8 remaining, so there are 9 x 8 ways to choose 2 in a particular order.
For the third one, I can pick any of the 7 remaining, so there are 9 x 8 x 7 ways to choose 3 in a particular order.

From here you can see that there is a pattern of 9 x ... x (9-m+1) ways to choose m marbles in a particular order. But this is just: [9 x 8 x ... x (9-m+1) x (9-m) x ... x 2 x 1] / [(9-m) x ... x 2 x 1] = 9! / (9-m)! Choosing objects in a particular order gives us the number of permutations, so the number of permutations is 9! / (9-m)!

If we don't care in which order we choose the m marbles, but just want an unordered collection of m marbles, then we need to eliminate the effect of the permutations that are formed from the same marbles. That is, abc is the same as acb is the same as cab. The number of ways you can combine m marbles is m!, which you can derive in the same manner as the above derivation, or prove by induction.

Therefore, if we want to choose m marbles from 9, we combine the two pieces of knowledge above to arrive at the number of combinations = 9! / m!(9-m)! .

Generalize to _nC_m = n! / m!(n-m)!

-- GoogleMaster - 26 Oct 2006

the cartoon guide to statistics:
heck, yes! larry gonick is the man!
probably the most effective teacher (of academic subjects;
spiritual teachers are somewhat hard to compare)
working in english today (in the sense of having
taught the most stuff to the most people;
individual annie-sullivan-style miracle workers
are also hard to compare and anyway usuallly anonymous).

a new volume of the cartoon history is due soon.
you can darn well bet i'll get it (and study it).

-- VlorbikDotCom - 26 Oct 2006

Those students believed they were qualified. Everybody told them they were qualified. They had accumulated the necessary credentials that labeled them "qualified." And the university that admitted them must have assessed them as qualified.

right, and what I'm seeing is that this is happening everywhere in k-12, in part because kids are given standardized tests that mean nothing

-- CatherineJohnson - 26 Oct 2006

well I wish I'd talked to you guys about factorials (factorials?) before I took the test

"logic" quote-unquote may be beyond me

-- CatherineJohnson - 26 Oct 2006

ok, I'm sold on Larry Gonick!

factorials larrygonick regentsexam

-- CatherineJohnson - 26 Oct 2006

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