Rational or Irrational Numbers

How does the title of the example "Using Irrational Numbers" jibe with the question "To the nearest ten meters, how much farther is ..."?

I get that this is a pre-algebra textbook, but it seems to me that a discussion of irrational numbers should deal with irrational numbers, or at least with answers rounded to the nearest hundredth or something. 

Maybe the start of a discussion of significant figures and tolerance and error?

Algebra for all in the 8th grade is disastrous

from Joanne:

Stop teaching dumbed-down algebra to unprepared eighth graders and we can solve America’s math problem argues Jacob Vigdor in an American Enterprise Institute report. Unprepared students don’t benefit from Algebra Lite and prepared students are turned off to math, he argues.

Charlotte-Mecklenburg schools pushed most eighth graders into algebra classes, he writes. Students scored much lower on the end-of-course exam than those allowed to take algebra in ninth grade — and accelerated students did worse in geometry. The district abandoned the experiment after two years.
Our math problems are largely “self-inflicted,” Vigdor writes. In order to bring low performers up to the standard, schools have lowered standards.

Closing the achievement gap by improving the performance of struggling students is hard; closing the gap by reducing the quality of education offered to high performers—for example, by eliminating tracking and promoting universal access to “rigorous” courses while reducing the definition of rigor—is easy.

The first step to improving math performance is to concede that students differ in abilities, he concludes.

I have an idea. Let's try this everywhere.

Algebra is worthless - to a PolySci professor.

So Andrew Hacker got on the NYTimes and asked "Is Algebra Necessary?"

"This debate matters. Making mathematics mandatory prevents us from discovering and developing young talent. In the interest of maintaining rigor, we’re actually depleting our pool of brainpower."

From a cognitive sense, brain-power depletion is a myth and he's barking up the wrong tree to start with it. His points get worse as he continues.

"The toll mathematics takes begins early. To our nation’s shame, one in four ninth graders fail to finish high school. Most of the educators I’ve talked with cite algebra as the major academic reason."

Really? Every school I know of has some program in place for those who cannot pass algebra - consumer math, business math, basic algebra - they can get their three math credits and move on. Then, too, the educators who responded to this man's anecdotal survey probably had no idea of the real reasons kids dropped out - math was just the convenient scapegoat to help make his argument.

States such as California who make Algebra I a requirement for 8th grade are forgetting that not everyone will be a STEM major and that there are plenty of adults in the world who can't "do math" yet who are doing just fine and consider themselves successful. Mandatory 8th grade algebra is bad policy. I have never been a proponent of 8th grade algebra except for the 15% or so who are ready for it. Making it a requirement for all 8th grade students is educational abuse in my book.

There are certainly students who will never do well in a strictly abstract, mathematically intense field ... but that's okay. The world needs artists, too. And property management, and construction, and politicians, and entrepreneurs, and salesmen, and fast food, and so on. Where Hacker goes astray is in taking the previous paragraph and then running out of the algebra classroom, Pied-Piper-style, taking the students with him.

Why? All high school students should climb the mathematical ladder to the greatest height they can manage. 9th graders are not in much of a position to know their strengths and weaknesses and should be pushed, cajoled, tutored, helped, or cheered as necessary. It's not until the summer jobs between 10th and 11th grade that students start getting a sense of "this is necessary" or "this is useful" and decide to apply themselves a little more.

"If you can't swim the first time
you get in the water, you should never try
to learn." seems to be his message.

Why not let them experience algebra first? Many students just don't realize what they are good at. Many parents and elementary school teachers (the two biggest influences in students' lives so far) are rarely good at math and pass on that limitation to the kids.  Give me some time to reverse that before you write off this generation as math losers.

The depressing conclusion of a faculty report: “failing math at all levels affects retention more than any other academic factor.” A national sample of transcripts found mathematics had twice as many F’s and D’s compared as other subjects. (Which means that math teachers are grading for knowledge and other disciplines for development? Hardly an argument for changing the curriculum.  C.)

If students fail in the abstract path, they should be diverted to a parallel ladder of courses that are less abstract (more vo-tech, perhaps) while still teaching them math.

Hacker is also fairly sloppy about mixing in topics from pre-calculus and calculus and using somewhat obscure terminology to scare the reader ... "vectorial angles and discontinuous functions," two topics that wouldn't be a graduation requirement anywhere and I wouldn't expect very many of the NYT readers to know (I'm not sure Hacker knows either - I think he just picked up a math book).

I'll throw in my favorite, "But there’s no evidence that being able to prove (x² + y²)² = (x² - y²)² + (2xy)² leads to more credible political opinions or social analysis." Well, Andrew, no one ever claimed that it would.

It’s true that students in Finland, South Korea and Canada score better on mathematics tests. But it’s their perseverance, not their classroom algebra, that fits them for demanding jobs.

Actually, it's their perseverance that helps them score better on the tests. It's that same perseverance that dictates whether they will succeed in college and in life. Removing algebra doesn't change that.

But a definitive analysis by the Georgetown Center on Education and the Workforce forecasts that in the decade ahead a mere 5 percent of entry-level workers will need to be proficient in algebra or above.

Not to press the point, but entry-level jobs rarely require much of anything. They won't ask the entry-level intern to write a report, but grammar and writing skills are necessary.  An entry-level construction worker isn't interpreting designs, and the entry-level brewery go-fer isn't using his biology knowledge. It's at the next levels, where the decisions are made, that companies require the writing, programming, algebra, science, and tech skills.

I'll end with this:

Go ahead and deny the algebra classes to anyone who seems unlikely to use the skills. Tell them that they aren't going to take algebra, geometry, calculus. Remove them from all the purely theoretical classes you want.

If you aren't sued for discrimination and neglect, keep on doing it. My students will continue to return and thank me for what we did in class and have an easier time of it because your students won't even be in their rear-view mirror.

I’ll grant that with an outpouring of resources, we could reclaim many dropouts and help them get through quadratic equations. But that would misuse teaching talent and student effort. It would be far better to reduce, not expand, the mathematics we ask young people to imbibe.

Works for me.

Algebra in the Third Grade?

I got the NASCO 2011 catalog in my mailbox this morning. If you haven't dealt with them, they are a pretty typical school supply catalog - prices are double what you can find if you search diligently or ten times the scrounger's price. What it is really good for is as a wish list, a source for ideas. "How does that toy work? I could do that in the garage for $2 in materials."

My problem with them is that sometimes they really miss the mark. Here's the cover:

Really? Third grade and up? I rail all the time about my students arriving in 9th grade without a clear grasp of fractions and clueless about operations with fractions (sans calculator), limited fluency with percents and decimals, and an unsteady grasp of their multiplication tables and other basic stuff. I had usually blamed the 3rd through 6th grade teachers for not really understanding arithmetic and passing on any math phobias, but this seems like a major problem right here.

If the expectation is that this kind of thing is possible for 3rd grade and up, maybe it shouldn't be a surprise that fractions and arithmetic aren't getting as much attention.

My questions:

• Is this typical? Do third grade teachers really do this?
• Is it just a stupid cover by a catalog desperate to sell overpriced AlgeBlocks to a gullible school system?
• Shouldn't we be solidifying knowledge to the point of automaticity instead of spreading algebraic materials ever lower?
• I'm pretty sure that a few third graders could get this but is it appropriate for that level? 
• Is it possible without manipulatives at this age?

 Then, there's the other debate:

• Are manipulatives appropriate? 
• Does the use of something tangible and obviously fixed in size get in the way of learning an abstract idea about a variable?

Cattle Puzzle

Farmer Jones sold a pair of cows for $210. On one he made 10 percent and on the other he lost 10 per cent, cleaning up just 5 per cent. What did the cows originally cost him?

Sam Loyd, Cyclopedia of Puzzles, 1914

answer here.

Checkbook Puzzle

Try this very simple checkbook-balancing problem.

Beginning balance for the month $54.00

Check #0221 $20.00 Check #0222 $20.00 Check #0223 $10.00 Check #0224 $ 4.00 Total $54.00

Balance $34.00 Balance $14.00 Balance $ 4.00 Balance $ 0.00 Total $52.00

You naturally added everything very carefully so you should be able to tell where the missing two dollars have gone.

Math in the Crosshairs again, this time Maryland

"... many graduates do not have a grasp of the basics."
"... schools have deemphasized drilling students."
"... taught too early to rely on calculators."

A calculator is a tool. It should be used as a tool. As soon as it replaces thought, it should itself be replaced.

I have decided to make a new slogan, signifying my reluctance to rely on the thrilling new technology of calculators because of the very real effects on the kids' development.

"Thrill and Kill."

"... ninety-eight percent had to pay for remedial classes." Okay, it's a community college and you expect that many of the attendees would be looking to improve their math skills. But 98% ??

"Across the nation, slightly more than one-third enroll in remedial classes." That's bad, people.

The report gets specific but, in my view, misses the mark. " ... particularly critical of the Algebra I standards, saying that they are watered down because educators must teach material for the High School Assessments, which includes data analysis. It is not what any mathematician would consider an algebra course."

No, I think the algebra I course is watered down because, (A) it is taught to eighth graders and they had to water it down so they could pass more easily and (B) mainstreaming and the refusal to place students in an appropriate class means that every room has kids who slow down the group. This insistence on placing kids in a course based on emotion, faulty pedagogy, self-delusion and parental desire instead of mathematical ability will ruin your classrooms every time.

Anyway, the article that started this train of thought appears below the fold.

Double Dose of Algebra and some Ramblings

Over at Curriculum Matters, they're asking Does 'Double-Dose' Algebra Work?

Across the country, one of the strategies schools are trying to help struggling students in algebra is essentially doubling the amount of time spent on that course. It's a popular tactic in other areas of math, and in reading, too.

A new study, however, says that double-dose courses produced mixed results in Chicago schools. On the one hand, the 9th graders studied saw their test scores rise. But the policy did not appear to result in fewer students failing the course, as school officials had hoped, the authors report. The grades of some struggling students increased, after the double-dosing, though the weakest students did not see their grades rise.

Ignoring the grammatical and structural problems in those paragraphs, I was struck by the apparent surprise. This result seems very obvious to me. With more time devoted to drilling and practicing for the test, you'd expect the test scores to converge on the mean. Look no further than KIPP.

The grades in the course, on the other hand, are based on different criteria. Teachers routinely count "participation," homework, classwork, and behavior. Extra credit and test corrections and grade-adjustment skew the scores. Parents routinely ask for "make-up work," code for "just give him more points without making him do anything or we'll raise hell."

Should advocates of double-dose math courses be pleased with these results? After all, one could argue that student learning—if test scores accurately reflect that—increased. Or should persistently high failure rates raise red flags?

Sure. But Red flags signifying which issue? That the class is different from the test? DUH. Would you be upset if the class averages were in the 90s and the test scores were dismal? Yes, you should. But you'd be worried about the wrong things. Do class averages measure learning? Does the State test measure learning?

When the State scores are low, the usual response is to bitch about the content of the course or the quality of the teacher. This is occasionally the problem. A few people complain about the motivation of the student, which is a big problem. No one ever talks about the idea that the teacher might be teaching material that the State isn't testing, like the algebra the kids need instead of the geometry that the State assumes they should be ready for at age 15.

When the scores don't correlate, too many people spout off about "test-taking strategies" and "test anxiety" and "some kids don't test well" and "only a snapshot."

What is wrong with a "snapshot," anyway? I can't tell you how many times I've had people tell me that the (3-hr, well-constructed and extensively peer-reviewed before and after the fact) State Tests are bad because they're only one day's measurement and thus should be eliminated. These same folks then turn around and push for a final exam (1.5hr, 'constructed' by the teacher at 12 midnight and never reviewed) to be 20% of the kid's grade.

No one ever complains about the most common problem and the biggest reason why grades and State Scores rarely correlate: the teacher has too much control over the way the course is graded and his tests are scored.

Even the newest, most inexperienced teachers are routinely given control over grading, curriculum and student progress.

I have had brand-new teachers tell me that they are going to teach chapters 1, 6, 8, 7, 2, 4, in that order. Why? They thought it was better that way. Ch3 and ch5 are ignored because why? The answer usually has more to do with the teacher's lack of understanding than the stated reason, "Those chapters on matrices and blahblah are not current and blah blah blah." No account is given for the idea that the book was written and vetted by people far smarter (and with a lot more time to consider these things) to be done in a particular order. No thought is spared for the linearity of the material and the fact that most of the problems in ch4 assume that you've done (or at least understand) ch 1-3. Going out of order confuses even the best students; they are constantly asking "Should I have known that really important fact or idea? Am I stupid or did we just not cover it?"

The standards-based movement exacerbates this problem. Instead of following a curriculum, the teacher is supposed to pick and choose from the Standards and individualize the curriculum for each kid based on test scores. The grades coming out of this train-wreck usually have more to do with some vague idea of "effort" than of any assessment of ability.

Beyond the curriculum matters, the teachers also mess up grading. Do you count homework or grade it? Some have participation and brown-nosing scores as high as 10%. Some have tests, quizzes and homework roughly equal. In my school, we even have control over the relative weight of the terms if you can believe it. Some faculty count the two terms and the final as 40-40-20 while others are 45-45-10. Others don't give a final.

Is it any wonder that grades aren't changing in lockstep with the State scores?

Is it surprising that we don't achieve in the same way as the schools in other countries do? You know, the ones that have a national curriculum that aligns with the TIMMS and PISA tests so that those countries will do well on the TIMSS and PISA tests? That have long (by comparison) teacher mentorships so that the new teachers are in line with the older ones and not spinning off like some subatomic particle in their own personal cloud-chamber? Have we learned nothing from the reality of Stand and Deliver or were we just seduced by the feel-good fiction?

I'm not saying this is the ideal but if we wish to repeat their successes, we'll need to at least partially repeat their processes. We also have to consider that learning might not equal scores or grades.

Curriculum Matters continues with another obvious point:

Chicago is, of course, coping with many of the same challenges in algebra that other districts are. The new study follows another one, released last month, which found that Chicago's failure rates increased when the district mandated that students take algebra in 9th grade.

I nominate Curriculum Matters for Poor Elijah's Emperor Awards 2009 (2008 here) "Archimedes Eureka Honorarium" which spotlights the imaginative world of education research. Congratulations to all.

Math Tales from the Spring: The Avoider

Math Tales from the Spring: The Avoider.

Take some comfort: It's not just you, Mrs. H. These clowns are all over the place.

And their unfortunate kids, too.

Algebra expansion - $3.1 Billion

WTF?

Jill Tucker, San Francisco Chronicle Staff Writer with lots of reader comments
August 13, 2008
SACRAMENTO -- California's schools will need an additional $3.1 billion annually - $2,100 more for every middle school student - to implement the governor's new eighth-grade algebra testing requirement, California State Superintendent of Public Instruction Jack O'Connell said Tuesday.

What a concept. First, it's silly to assume that 100% of eighth graders are ready for anything, not to mention algebra. From the Newsweek article on Teach For America,

Locke High School in Watts. At Locke, a school hemmed in by competing gangs, 2 percent of ninth graders are proficient in algebra; 11 percent read at grade level. Too many can't read at all.

so we're now going to accomplish state-wide algebra proficiency at the 8th grade level instead?

Second, spending that much only to realize that a large percentage of the students are not ready for algebra seems somewhat counter-productive.

Thirdly, that's a heck of a lot of money and I doubt that much of it is going to any teachers. I think it more likely that some testing company is scoring a big, fat juicy contract here (assuming the number isn't wildly inflated to score PR victory).

Most troublesome from my perspective, the unintended consequence will be that algebra 1 will have to be watered down to the level of the test that 90% of the kids COULD pass.

Of course, they could always set the passing score to 30 out of 90. That seems to be the way of the world these days.

California to require every 8th-grader to take Algebra I class

"The Road to Hell" and all that. It's easy and simplistic. "We Have Spoken: Students Will Achieve."

I give it 3 years. In the first year, they'll test and realize 7 months later that it didn't work. They'll panic and re-jigger things, and it still won't work. They'll cheer themselves when a dozen schools manage to get all of their kids to pass, and quietly return to normal at the end of the 3rd year. And those schools who got 95% passing? I'd be looking VERRRY carefully at test administration.



By Dana Hull / San Jose (CA) Mercury News
7/10/2008

California education officials set a high bar Wednesday: All eighth-grade students will be tested in algebra.

While the 8-1 vote from the state Board of Education was immediately applauded by several business groups, some educators warned it set unrealistic expectations since half the state's students are still struggling to master basic math skills.

The decision means that an intense ramp-up period is about to begin, as the state scrambles to hire and train more algebra teachers, align curriculum and get young students ready for more rigorous course work. The program could launch in as little as three years.

The move makes California the first state in the nation to require algebra at such an early level, signaling the state's determination to set high standards for its 6.3 million K-12 public school students.

"Algebra is the key that unlocks the world of science, innovation, engineering and technology," said Gov. Arnold Schwarzenegger in a statement. "This is California's future."

By adopting an Algebra I test as the sole federal math assessment for eighth-graders, state education officials expect that all eighth-graders will ultimately take algebra classes. If they fail the test, students can still advance to the next grade.

The latest round of math wars arose earlier this year when the federal Department of Education found California out of compliance with the No Child Left Behind law when it came to testing students. Some eighth-grade students were tested in algebra, while others were tested in a lower-level general math.

The federal government told California to enroll all students in Algebra 1 within three years, or develop an alternate test that would include some Algebra 1 concepts.

State schools Superintendent Jack O'Connell and his staff chose to develop a new eighth-grade math test that included some algebra. But many business leaders, education advocates and the governor objected, calling it "algebra lite." The state education board rejected O'Connell's plan.

"Algebra I is now the sole test of record for eighth-grade math," said state Board of Education President Ted Mitchell. "There is one set of standards, no matter what a student's ZIP code, race, ethnicity or income level. We're committed to creating a system in which they master skills for the 21st century."

Some critics say that pushing students into math courses they are not ready for could exacerbate California's dropout rate. And many question the time frame under which the new rules are being put into place.

"You need to lay the groundwork if you're going to make this kind of policy shift," said Santa Clara County schools chief Chuck Weis. "We need to invest in the teaching of mathematics at the lower grade levels."

Traditionally, high school students have taken Algebra I in the ninth grade. But in recent years, there has been a growing effort to shift Algebra I to the eighth grade. About 52 percent of all California eighth-graders now take Algebra I courses, up from only 16 percent at the beginning of the decade, according to EdVoice, a statewide education advocacy group.

But while more students are taking algebra earlier, not all of them are passing it.

"There's a big void at the middle-school level," said Pioneer High School math teacher Stephen McMahon. "We have a lot of students who take Algebra I in the eighth grade, and then they repeat it in ninth grade. A lot of students aren't ready for that level of math."

See also
How to screw up with data and Once more into Eighth Grade Algebra.

How to screw up with data

A recent principal had no idea of how math worked or should be taught, she just knew that our "scores weren't good enough." She bounced into an end-of-year faculty meeting with the graph at right (taken from a State publication). Essentially, she said, "The math teachers aren't pushing our kids enough. We can improve our scores by pushing the kids into more rigorous courses. I've talked with Guidance and we're scheduling next year's freshmen. We're 'STEPPING them up! Students who were recommended for pre-algebra will be in intro-algebra instead. Intro-Algebra is now pushed into College prep algebra. Those in intro-algebra this year can move into algebra II if they pass. Our scores will rise because this graph shows that students in more rigorous classes do better on the exam."

Wow. The graph shows the NEAP math scores attained by tenth-graders, sorted according to the math course they were taking. The idea our fearless leader was pushing was that artificially placing the kids up a course would raise their scores.

My comment that we should push them all the way into pre-calculus and do even better was not appreciated. All of us math teachers began trying to explain that it was rather natural that 10th graders in pre-calculus would do better than 10th graders in pre-algebra. We were told to "Get with the program." "Why are you obstructing progress?" "This chart is from the State DOE. It's based on real data." "We need to raise our scores."

The plan happened. Over the summer, the principal took the lists of freshmen and made placements based on a 1.5 year-old test instead of teacher recommendations. She put them into classes of 20 and decreed that these classes would stay together all day for all of their courses. Imagine that: 20 kids are well-placed together for math, science, English, history, and PE.

The eighth grade teachers were furious because they had done a lot of work placing these kids, holding meetings to best group them, examining tests across the board, and so on. They had done a typically bang-up job. ( We HS teachers were rarely surprised by an out-of-place student. Year after year, we knew what we were getting and the kids knew they were going into classes they were ready for. The parents knew the placements were good -- this was a small town and everyone had an older sibling who had Mrs. W., etc.) The MS teachers told the kids their classes for next year, everything was A-ok.

The kids showed up at HS in August and found they were STEPped. Lines of students formed at guidance trying to change schedules. They were all refused. Parents were told of the new plan, which mollified them somewhat. They thought their kids were really going to do better.

What really happened was that one math teacher had all of the freshmen classes and had lowered the expectations across the board. Hand in homework and be nice were the two primary criteria for passing. College prep algebra was reduced to the level of an introductory algebra; intro algebra had some algebra in it but not much. The school-wide algebra tests and exams were used but the scoring was wildly curved. (Sort of like the NY regents: 30 out of 95 was a typical passing score.) Grades went "up". Parents were happy.

Then these kids moved into 10th grade. Almost all of them did poorly. Their new teachers were accused of making the classes too difficult and of not teaching properly. We were told that maybe we should change to block scheduling (what?) or perhaps an integrated math program. We were told to "individualize your classes. Take each child from wherever she or he is and help them make improvement. We are not standardized here." It didn't matter if the kid didn't really know much algebra 1 because his next teacher was supposed to start where he was and build.

There were kids in Geometry who had never dealt with square roots except by punching a calculator. Algebra II kids who didn't know how to factor. Algebra I kids who couldn't reduce a simple fraction. We'd had all these problems before, of course, but never in these numbers.

Kids and parents were also not happy. "I had an A last year, now I'm getting a D" was the most common complaint. A few admitted that "Mr. C. didn't teach us much." "We never did 'graphing lines' or word problems. We went outside and calculated the slope of a mountain we were walking up. Actually, T. held up a protractor and we all used that number. We wrote about it in our journals." Notice that slope was 22 degrees, not (y2-y1)/(x2-x1). This was totally new to most. Not like the usual "Oh, I remember" but more of a "Huh? We never did that."

What data was used to see if the program worked? "The freshmen's grades this year were higher than last year's so it must be a good change. Their grades went down as sophomores so the other math teachers did a bad job."

Don't you just love education? Where else can a perfect storm of incompetence screw up so many kids?

Just to wrap up, a look at the cast of characters:

The principal who thought that all students could have finished Geometry by the end of the ninth grade and that 50% of the students could therefore take calculus as seniors. The principal who couldn't average 100 and 92 without a calculator.

The freshmen math teacher who is the head of the union and had been installed as department chair by that principal because he agreed with her. He's not allowed to teach Geometry, Pre-Calculus, or Calculus because the State says he doesn't have the credentials or ability. We didn't even like him teaching Algebra II because he did such a terrible job. Kids with math talent hated getting placed in his class.

The curriculum coordinator (a math phobe) who quit to become a principal at another school. This man felt that all kids can succeed in any course if the teacher individualizes the class. Apparently, when you put a "B" in algebra on a transcript, colleges magically know that you individualized the class at a basic math level.

The guidance counselor who had to quit because she needed to renew her credentials, which required that she pass the Praxis I math test. She couldn't. (You know the one: Which of the following is equal to a quarter of a million? (A) 40,000 (B) 250,000 (C) 2,500,000 (D) 1/4, 000, 000 (E) 4/1, 000, 000 ). The other two guidance counselors also left for better jobs where they didn't have to defend decisions they didn't make or agree with.

The other 5 math teachers: 3 quit, listing politics and bad administration as reasons. They're teaching at nearby schools. The remaining two stayed for other reasons, but were "this close" to quitting anyway.

How about that?

Once more into Eighth Grade Algebra

To California we go: the LA Times here and copied here (the Times archives it's articles).

Some years ago, California decided that all 8th graders should take algebra. While a wonderful concept theoretically, did anyone stop to think that not everyone is at the same level at the same time? Math is not like Literature. You need to understand the previous work before moving on to the following work. How has everyone forgotten this?

My theory is that the majority of teachers and education reformers are liberal arts majors. They just don't know what they're talking about when it comes to teaching math or developing appropriate math education policy.

Elementary teachers, by and large, "don't do math" and have very little sense of the topic. The middle school teachers, for the most part, are general education type folks with only a small percentage having a STEM background. Narrow your list down to MS math teachers and you still don't have just math people. You can find English majors who have covered a basic math course for a year or two. or three. You have those who can't teach Geometry or Algebra II in the high school and have drifted down to the level they can teach.

The Amsco salesman told me a joke some years ago: "I know the first name of almost every middle school math and science teacher on the East Coast," he said. When I looked at him askance, he laughed, "Coach."

The high school teachers are competent in their fields, but only a tenth of them teach math. I'm not even going to try and pretend that all HS math teachers are competent. The upshot is that virtually no one understands the teaching of math and yet everyone has opinions. Only a tiny portion of the total teaching cohort has ever taught a HS math course. Unfortunately for the math students, schools tend to be a democracy and those who know get out-voted by their principals, school boards, guidance personnel, do-gooders, loud and pushy parents, fellow teachers, and the like.

To be fair, math teachers should probably push back far harder than they do. We do tend to be quiet and just "go along to get along" sometimes.

From the LA Times: Testing for Algebra

As California debates college-prep math testing, educators must remember to put learning first.

July 9, 2008

California has made great strides toward teaching algebra in the eighth grade. Six years ago, fewer than a third of eighth-graders took the course that's considered the linchpin to college-prep mathematics. Now, more than half do.

But the state is caught between the rigid mandates of the No Child Left Behind Act and its own lofty academic ambitions. The federal act requires the state's proficiency exams to test whatever the standard is, and the state'sstandard is for all eight-graders to take algebra. What to do about the 48% who take lower-level math?

The answer to be considered by the state Board of Education this week is to give those students a new, tougher test with a sprinkling of algebra questions, while algebra students would continue to take the full algebra test. That makes the feds happy. That makes the state Department of Education happy. The only problem is that this is unsound education.

Though we support standardized testing as a way to measure the progress of schools, testing students on material they haven't learned is the educational tail wagging the dog. Teachers will throw a few simple algebra concepts into a curriculum in which they make no sense in the hopes of a better score; it's the proverbial "teaching to the test" in its worst form.

The state already has inducements in place to prod schools toward eighth-grade algebra, which has led to the progress so far. But the eighth-grade algebra standard is not a requirement, and though there's a movement afoot to make it one, that would be a mistake. Math demands progressively sophisticated skills. Students have to master Step 1 before they can successfully attempt Step 2, and the public schools have long allowed students to move on to the next step while they're still shaky on the previous one. That's why algebra remains the single biggest obstacle to high school graduation.

The problem begins long before middle school; in fact, one of the major factors is failure to master multiplication tables. The state needs to think out its curriculum before it starts testing students on it.

California has adopted materials for an algebra-readiness course for middle schools, but it will be years before the curriculum is fully in place. Let's not lose sight of the real goal here, which is to ensure that students learn math -- not just take it, or pass it, but actually learn it. Better for lagging students to be prepared properly for algebra in ninth grade than for them to take it early, only to fail, and fail again.