Author Archives: Coach

Giving Thanks For Turkey And Nuts


Giving Thanks For Turkey And Nuts
DEBRA J. SAUNDERS
Tuesday, November 25, 1997

THIS THANKSGIVING, I raise my glass to the nuts. I toast Americans who care enough to fight and push and demand quality in the world around them. I give thanks to parent activists who have worked to restore phonics to reading instruction, sanity to math curricula and rigor to dumbed-down schools.

I can’t name them all, there are many. Suffice it to say that when I think of the great nuts, I think of people like the godlike — watch her thunder — Marion Joseph, the Phonics Queen recently appointed by Governor Wilson to the state board of ed. Molecular biologist Mike McKeown and statistician Paul Clopton and all the other nuts at Mathematically Correct in San Diego, who are pushing for strong math programs. Stanford’s Hoover Institution prof. Bill Evers and the dissident parents of Palo Alto. Wisconsin mother Leah Vukmir who founded Parents Raising Educational Standards in Schools. Bay Area math teacher Monica Brown and colleagues who have begged for challenging books and programs.

For their trouble, they’ve been called every name in the book — evil, mean, rigid, anti-education, anti-public school, pinheads who don’t want children to think for themselves, rote nazis, tools of the Religious Right, extremists. Nuts.

It’s true. They are nut cases. Many could afford to send their kids to private schools that cleave more closely to their idea of a solid education; instead they work to change the public schools. A thankless task. They don’t just think of helping their own children, they also want to help other people’s children — and that makes them nuts.

They make waves. They think their own thoughts. They rely on their common sense instead of common dogma. They believe in knowledge, not process- mongering. They must be crazy.

When they could be dining out, they are pouring through research on various curricula. They don’t blindly trust in the experts. Unlike many of the reporters who write about proposed standards and many board members who vote on them, these loons actually read drafts in their entirety. Folly. Worse: work!

Then they go to board meetings where they are told that their informed opinions don’t count and aren’t welcome. Good parents aren’t supposed to question educational orthodoxy. Good parents should sit up straight with their hands folded. Good parents should behave like battered wives — love the system, even when it hurts the kids.

Sometimes, these parents tell me I am brave. Sorry, I get paid to write a column. I know that what I produce will be taken seriously by someone. These parents put in months on commissions, or attending groups for parents who aren’t welcome on commissions, often only to be shunned. They pour their hearts into projects — all the while knowing that they are battling an establishment more interested in proving itself not to have been wrong than in exploring how it could improve. They labor, then they’re laughed at.

They question school authority, then fear their children may be punished for it. They stick to their principles.

This week, the PBS series “Liberty! The American Revolution” has shone the spotlight on America’s unlikely patriots — dreamers who risked all for the hope of a better world.

Modern Americans have elevated achieving personal comfort to an art form. Life is so easy that standing up for an ideal is something only quacks do. In an age when eating in all week is considered a hardship, I sometimes think such giants could not be found today. But then I think of Marion Joseph, Mike McKeown, Bill Evers, Leah Vukmir and Monica Brown.

You can read Debra J. Saunders on The Gate at www.sfgate.com.

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URL: http://sfgate.com/cgi-bin/article.cgi?file=/chronicle/archive/1997/11/25/ED25276.DTL


©2005 San Francisco Chronicle

The Fuzziest of Disciplines

The Fuzziest of Disciplines

by Alexander Nazaryan and Alexander Wilson

In the early 1990s, a group of public school districts clustered in the American West — California, Arizona, Texas, Iowa — selected a set of Math curricula, for children from kindergarten through their last year in high school, that promised new approaches to learning and a more enticing mathematical climate. It was hoped these changes would increase students’ interest in and capability with Math and ultimately help close the lauded mathematical gap between lagging American students and those in the rest of the industrialized world.

These school districts received about $50 million in Federal aid from the National Science Foundation to implement these curricula, collectively (and colloquially) known as `The New New Math.’

Five years on, these programs are a universal educational disaster. In one California school district, the average standardized test score moved from the 86th percentile to the 55th in only the first two years after the New New Math curricula was implemented. The test scores quickly rose back to the 77th percentile after the school’s old, traditional Math curricula were reinstated.

This is consistent with other results, and though the National Science Foundation has announced it will withdraw funding from any school district which abandons the New New Math curricula, many of those school districts who initially adopted these curricula are reverting to more traditional (proofs and rote) methods of learning mathematics.

The academic tail, however, may well still be wagging the educational dog. Whatever the level of its popular rejection, the New New Math is still extensively taught in American graduate schools. Though New New Math was only developed in 1989, it is estimated that fully half of American Math teachers are trained in Fuzzy Math, the central component of the New New Math.

The impetus for the New New Math curricula is political and, as its name suggests, it is a political antecedent to the `New Math’ movement of the 1960s.

In the 1960s, Math teachers, overwhelmed by notions of cultural subjectivism and inclined, as was the contemporary zeitgeist, to question rigid theories of learning as potentially autocratic, decided that the Arabic number system’s (the number system we use) reliance on the number `10′ as a base for arithmetic, geometric, and algebraic learning was arbitrary and probably Euro-centric. Instead, the New Math devoted extensive time to learning alternate number bases, such as 7. (In a base seven system, the number `6′ would still be written `6′, but `12′ would be written as `15′ and `16′ would be written as ’22’).

The idea was not to develop a better basis for understanding higher math (since base 7 is pretty useless in, for example, calculus) but to convince students of the fallacy of the objective reality of numbers. `12′, in the New Math, is not necessarily ’12,’ since in a base 7 system it would be ’15.’ Numbers, for New Mathematicians, are nothing other than `numerals,’ signifiers. Though the concept of ’15 elephants’ is real, it would be insufficient to abstract that to ’15,’ since `15′ is unnecessarily filial to the base ten number system. Instead, to answer the question, “How many elephants are there?” a student would have to answer something along the lines of “The amount of elephants that, in a base ten number system, we would usually express using the numeral `15′.” The New Math also focused extensively on things like set theory and `congruence arithmetic’ rather than multiplication tables and long division.

This system, the `New Math’ of the 1960s, was, predictably, a resounding failure which did little more than confuse students. Test scores plummeted, students turned away from Math, and before long the New Math was dropped.

Thirty years later, enter the New New Math.

The New New Math, originally conceived as a curricular notion in 1989 by a report of the National Council of Teachers of Mathematics, takes the same impulse, the rejection of the objective abstract reality of numbers, and combines it with a series of voguish concepts in Mathematical education — `fuzzy math’, `ethnomathematics’, and `constructivism’.

Since numbers are not really `real,’ in the conception of New New Mathematicians, focusing the study of mathematics on finding a `correct’ answer is pretty trivial. The emphasis thus belongs to the process of learning math, recognizing patterns in the numeric symbols – not numbers – and on making “shrewd guesses” from visual models. There is a lot of emphasis, in the New New Math, on using two and three-dimensional models, which can show the reality (`three blocks’) from which numbers stem, with little emphasis on numeric manipulation itself. (Blocks and tiles are in, pens and pencils are out).

This branch of the New New Math (which holds that the teachers should discourage abstract numerical manipulation in search of an absolute answer and encourage shrewd, intuitive guesses, regardless of whether they are right or not, based on recognized visual patterns in objects like tiles or blocks) is called `Fuzzy Math.’

Martin Gardner, writing in The New York Review of Books, gives a good example of what Fuzzy Math means: “Teachers traditionally introduced the Pythagorean theorem by drawing a right triangle on the blackboard, adding squares on its sides, and then explaining, perhaps even proving, that the area of the largest square exactly equals the combined areas of the two smaller squares. According to Fuzzy Math, this is a terrible way to teach the theorem… [The students] cut from graph paper squares with sides ranging from two to fifteen units. Then they play the following `game.’ Using the edges of the squares, they form triangles of various shapes. The `winner’ is the first to discover that if the area of one square exactly equals the combined area of the other two squares, the triangle must have a right angle with the largest square on the hypotenuse. For example, a triangle of sides 3,4,5.”

Students who never discover the theorem are said to have `lost’ the game. Thus, with no help from teacher, the children are supposed to discover that with right triangles a(squared) + b(squared) = c (squared).” (This is `constructivism,’ perhaps best defined as the method of teaching New New Math, which stresses that students should figure out answers for themselves – elsewhere called `experiential learning’ – instead of being fed formulas and theorems by the teacher).

`Ethnomathematics,’ which stems from a whole range of multicultural impulses, is a secondary component of the New New Math curricula, though it is more sociological than mathematical in structure. It has two effective curricular influences.

The first is to encourage the study of the way primitive tribes counted and added. There is a popular text, released only in 1997 but already in full use, called Africa Counts: Pattern and Number in African Culture by a woman named Claudia Zaslavsky which gives extensive examples of different methods of dealing with Math and numbers. (As Carl Sagan and others have pointed out, the arguments for the mathematical and scientific success of ancient tribes are, in large part, fabrications).

The fact, of course, is that Math, like Physics or Chemistry, is a progressive effort. We don’t study Aristotle’s physics because we have Einstein’s. The effect of ethnomathematics is to reduce science from a way of understanding the world to a way of understanding culture.

The second effect of ethnomathematics is to suffuse the texts of the discipline with multicultural “examples” that have at best a dubious relation to mathematics. One popular textbook quotes a Maya Angelou poem in whole (there is an accompanying photograph of Ms. Angelou with President Clinton) and claims the parallelism in the poem is weighty evidence, contributing to an enhanced understanding of parallel lines and geometric structures.

The same text asks, “Is the time it takes to read an Alice Walker novel always a function of the number of pages?” John Leo, the US News & World Report columnist, examined several New New Math texts and found numerous “photos of President Clinton, and Mali wood carvings, lectures on what environmental sinners we all are and photos of students with names such as Taktuk and Esteban who … offer their thoughts on life.”

Since fuzzy math and constructivism stress the formation of subjective and personal mathematical algorithms and ethnomathematics stresses sociology and Maya Angelou, the bulk of responsibility for ensuring that students do well on traditional measures of mathematical acuity, like standardized tests, falls squarely on the shoulders of calculators. Calculators are a necessary part of the New New Math curricula, and are hence now being introduced as early as the first grade. The upshot is that students no longer need to know multiplication tables or how to do long division — they can simply press a few buttons.

Some states and standardized testing agencies have caught on to this effort to cheat the evaluation system, and so (California is notable here) have banned the use of calculators on tests.

Parents and educational policy-makers alike have been consistent in their opposition to the New New Math, and the associated dependence on the calculator.

Nevertheless, the development and persistence of contemporary trends in American schools of education seems to indicate that the New New Math may be here to stay.

It is ultimately teachers that determine the way students are taught, and the number of teachers who are trained in and teaching fuzzy math, constructivism, ethnomathematics, and the rest of it, is rising.

What will fix public education? A teacher, a chalkboard and a roomful of willing students

My Turn: Forget the Fads—The Old Way Works Best

What will fix public education? A teacher, a chalkboard and a roomful of willing students

By Evan Keliher
NEWSWEEK

Sept. 30 issue — I’ve never claimed to have psychic powers, but I did predict that the $500 million that philanthropist Walter Annenberg poured into various school systems around the country, beginning in 1993, would fail to make any difference in the quality of public education. Regrettably, I was right.

BY APRIL 1998, it was clear that the much-ballyhooed effort had collapsed on itself. A Los Angeles Times editorial said, “All hopes have diminished. The promised improvements have not been realized.” The program had become so bogged down by politics and bureaucracy that it had failed to create any significant change.
How did I know this would be the result of Annenberg’s well-intentioned efforts? Easy. There has never been an innovation or reform that has helped children learn any better, faster or easier than they did prior to the 20th century. I believe a case could be made that real learning was better served then than now.
Let me quote Theodore Sizer, the former dean of the Harvard Graduate School of Education and the director of the Annenberg Institute for School Reform, which received some of the grant money. A few years ago a reporter asked him if he could name a single reform in the last 15 years that had been successful. Sizer replied, “I don’t think there is one.”
I taught in the Detroit public-school system for 30 years. While I was there, I participated in team-teaching, supervised peer-tutoring programs and tussled with block scheduling plans. None of it ever made a discernible difference in my students’ performance. The biggest failure of all was the decentralization scheme introduced by a new superintendent in the early 1970s. His idea was to break our school system into eight smaller districts—each with its own board of education—so that parents would get more involved and educators would be more responsive to our students’ needs. Though both of those things happened, by the time I retired in 1986 the number of students who graduated each year still hadn’t risen to more than half the class. Two thirds of those who did graduate failed the exit exam and received a lesser diploma. We had changed everything but the level of student performance.
What baffles me is not that educators implement new policies intended to help kids perform better, it’s that they don’t learn from others’ mistakes. A few years ago I read about administrators at a middle school in San Diego, where I now live, who wanted a fresh teaching plan for their new charter school and chose the team-teaching model. Meanwhile, a few miles away, another middle school was in the process of abandoning that same model because it hadn’t had any effect on students’ grades.

The plain truth is we need to return to the method that’s most effective: a teacher in front of a chalkboard and a roomful of willing students. The old way is the best way. We have it from no less a figure than Euclid himself. When Ptolemy I, the king of Egypt, said he wanted to learn geometry, Euclid explained that he would have to study long hours and memorize the contents of a fat math book. The pharaoh complained that that would be unseemly and demanded a shortcut. Euclid replied, “There is no royal road to geometry.”

There wasn’t a shortcut to the learning process then and there still isn’t. Reform movements like new math and whole language have left millions of damaged kids in their wake. We’ve wasted billions of taxpayer dollars and forced our teachers to spend countless hours in workshops learning to implement the latest fads. Every minute teachers have spent on misguided educational strategies (like building kids’ self-esteem by acting as “facilitators” who oversee group projects) is time they could have been teaching academics.
The only way to truly foster confidence in our students is to give them real skills—in reading, writing and arithmetic—that they can be proud of. One model that incorporates this idea is direct instruction, a program that promotes rigorous, highly scripted interaction between teacher and students.
The physicist Stephen Hawking says we can be sure time travel is impossible because we never see any visitors from the future. We can apply that same logic to the subject of school reforms: we know they have not succeeded because we haven’t seen positive results. But knowing that isn’t enough. We should stop using students as lab rats and return to a more traditional method of teaching. If it was good enough for Euclid, it is good enough for us.

Keliher is the author of “Guerrilla Warfare for Teachers: A Survival Guide.”

© 2002 Newsweek, Inc.

Flunking the Tests


Flunking the Tests
DEBRA J. SAUNDERS
Sunday, February 15, 1998

THE HOUSE killed funding for President Clinton’s proposed national education tests this month by a 242-to-174 vote. While Education Secretary Richard Riley denounced the vote as a “partisan attack” on “voluntary national tests,” the issue isn’t as simple as test supporters make it out to be.

The plus of the tests is clear: Give every fourth grader a reading test and every eighth grader a math test, as the administration has proposed, and parents and teachers should know which children need remedial attention. Schools then should provide remedial education. Done right, a national test could prevent the social promotion of illiterate students who otherwise might be doomed to spend their school careers in a haze of half understanding.

But can parents trust federal educrats to do the test right?

I certainly wouldn’t trust federal edu-swamis with math. Consider the saga of National Science Foundation grant-monger Luther Williams. Williams wrote a letter to state schools chief Delaine Eastin warning her that California schools might lose federal funds because the state school board, to Eastin’s dismay, voted in favor of math standards that — horrors — mandate that third-graders memorize multiplication tables and fourth-graders master long division.

Williams was appalled at this rejection of new-new math. He derided the board for buying into a “wistful or nostalgic approach” that “has chronically and dismally failed.” He apparently failed to notice that California’s commitment to trendy math — write about math, but you don’t have to be right about math — put California fourth-graders so far behind in the National Assessment of Educational Progress math test that they scored behind every NAEP-taking state but Mississippi and Louisiana.

Another reason not to trust federal educrats with a math test is the debacle California educrats created in their 1994 California Learning Assessment System (CLAS) tests. Before state lawmakers put CLAS out of its misery, the the state department of ed directed scorers to give students with wrong answers, but nice essays, higher scores than students who gave correct answers, but didn’t embellish them with happy-face prose.

This there-is-no-wrong answer philosophy comes straight from the “deep, balanced mathematical learning” playbook, dear to Williams and other basics-hostile faddists.

House Education and Workforce Committee Chairman Bill Goodling, R-Pa, fears that national tests will create national curricula. If he’s right, and federal faddists dictate what goes in the tests, national tests ineluctably would dumb-down curricula nationwide. Local districts would be faced with the choice of teaching math-lite or living with low-test scores because their students aren’t adept at writing about how happy they are about math.

Reading is different. You would hope that federal swells could figure out a way to test reading ability without mucking that up too much. But good reading tests already exist. Some schools use them. California is about to launch its own tests and shouldn’t need a federal tests.

A national reading test, therefore, may be a waste of money that otherwise could be spent on needed teacher training for those teachers who were never schooled in sound phonics instruction.

“Why would you waste $100 million to tell half the kids they don’t read well?” Goodlng asked during a recent interview.

It wouldn’t be a waste if you knew the new test, unlike others, would be a good measure of student literacy. You’d do it if you trusted this test would prompt schools to teach failing students the basics. You would do it if you trusted D.C. educrats to recognize a strong curriculum. But do you?

You can read Debra J. Saunders on The Gate at www.sfgate.com.

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URL: http://sfgate.com/cgi-bin/article.cgi?file=/chronicle/archive/1998/02/15/ED68913.DTL


©2005 San Francisco Chronicle

Experts Attack Math Teaching Programs

Thursday, November 18, 1999

Experts Attack Math Teaching Programs

Education: Top mathematicians and scientists urge U.S. to withdraw endorsement of methods that leave out basic skills. Federal official says change is unlikely.

By RICHARD LEE COLVIN, Times Education Writer

Nearly 200 top mathematicians and scientists, including four Nobel laureates, are urging U.S. Secretary of Education Richard W. Riley to withdraw the government’s endorsement of math programs that experiment with nontraditional teaching methods.

The strongly worded letter expresses outrage that some of the 10 widely used programs leave out such basic skills as multiplying two-digit numbers and dividing fractions.

“These curricula are among the worst in existence,” said David Klein, a Cal State Northridge math professor who was one of the letter’s authors. “To recommend these books as exemplary and promising would be a joke if it weren’t so damaging.”

Those signing the letter fear that a government endorsement of the programs will be a powerful force pushing teachers and school districts to use “dumbed down” instructional materials and methods. Several said they see the letter, which is to be publicized widely today, as providing a countervailing argument.

Klein was joined by math professors and physicists from UC Berkeley, Stanford University, Harvard University, the University of Chicago and elsewhere. The signers also include two winners of the Field Medal, which is the top honor in the field of mathematics, and Nobel laureates in physics Steven Chu (1997), Sheldon Lee Glashow (1997), Leon M. Lederman (1988), and Steven Weinberg (1979).

“I’m hoping to provide ammunition for teachers who are under pressure to adopt some of these programs,” said Richard Askey, who holds an endowed chair in math at the University of Wisconsin.

More broadly, those signing the highly unusual letter want the federal government to refrain from taking sides in the continuing national education debate that some have dubbed the “math wars.”

Linda P. Rosen, Riley’s top math advisor, said the endorsements are not likely to be withdrawn. She said Congress directed the department to create the panel of experts that made the recommendations and that the intent was to help school districts make informed choices when purchasing math programs.

But, she said, such decisions remain “absolutely locally based” and that school districts must take local opinions into account.

Steven Leinwand, a member of the federal panel that judged the books, defended the selection process.

“Every one of the programs designated as exemplary had real, clean data that showed test scores going up,” said Leinwand, a consultant to the Connecticut Department of Education.

But he acknowledged a difference of opinion among mathematicians as to what constitutes good mathematics. “These programs do not teach kids to do five-digit by three-digit long division problems,” he said. “Instead, they teach all kids, not just a few kids, when and why people need to divide.”

Rosen and others said the letter represents an escalation in the back-and-forth rhetorical struggle over how to promote mathematical understanding without sacrificing the ability to compute accurately.

As a result of the controversy, the nation’s leading mathematics education group and the leading proponent of nontraditional methods, the National Council of Teachers of Mathematics, has sought input from professional mathematicians. Some fear that this letter will hamper those efforts.

“I have an uncomfortable sense that everyone is talking past one another,” Rosen said. “What’s missing in the whole darn thing are the students . . . and that’s very unfortunate and just devastating to all of us who care deeply about young people.”

Hung-Hsi Wu, a UC Berkeley math professor and a co-author of the letter, acknowledged that not all mathematicians agree with it. But, he said, he wrote it out of a sense of “social obligation” to improve math instruction.

The use of nontraditional instructional materials by schools has sparked protests in communities across the nation. Usually, those who complain are parents who are concerned that their children are failing to learn fundamental skills, that solving algebraic equations is being de-emphasized and that math class has been downgraded to math appreciation class, leaving high school graduates unprepared for college-level courses.

In California, at least, the traditionalists have gained the upper hand. The state adopted standards for math classes that stress memorization of multiplication tables and only limited use of calculators, as well as an understanding of concepts such as place value.

As a result, the state rejected, or did not consider, all of the math programs recommended by the federal government except for a part of one, so school districts are prevented from using state textbook funds to buy them.

Still, the materials on the federal recommended list remain in widespread use across the state and have been the focus of protests by parents in Palo Alto, Escondido, Torrance, Simi Valley, Los Angeles and many other cities around the state.

Similar battles have occurred nationally.

In the upper-middle-class Atlanta suburb of Fayette County, Ga., for example, parents protested the use of a program called Everyday Math. That program recommends the use of calculators beginning in kindergarten as a device to help children count, and teaches children an ancient Egyptian method of two-digit multiplication as well as the one more commonly used in the United States.

As a result of the parents’ protest, the school district began paying more attention to basic skills and added an after-school math tutoring program for high school students.

“The children haven’t learned the basic facts. They move on to advanced math before they’ve laid a strong foundation and they’re in a muddle,” said Amy Riley, a leader of the parent protest there.

Rick Blake, a spokesman for the Everyday Learning Co., defended the program, saying the company has overwhelming evidence that students do well on computation as well as on more advanced topics.

“We’ve got kids doing algebra by the fifth grade,” he said. “This is not fuzzy math. It’s hard.”

In Plano, Texas, the parents of more than 600 middle-school students demanded alternatives to Connected Math, a program that the expert panel called “exemplary.” The school district has refused and the parents have filed a federal lawsuit.

The text of the letter sent to the U.S. Department of Education can be found at http://www.mathematicallycorrect.com/

The text of the U.S. Department of Education report on math programs can be found at

http://www.enc.org/ed/exemplary/

Search the archives of the Los Angeles Times for similar stories about:  United States – Education, Mathematics, Education Reform.

Education Panel Lays Out Truce In Math Wars

Education Panel Lays Out Truce In Math Wars

Effort to Fix ‘Broken’ System Sets Targets for Each Grade, Avoids Taking Sides on Method

Wall Street Journal  By JOHN HECHINGER  March 5, 2008; Page D1

A presidential panel, warning that a “broken” system of mathematics education threatens U.S. pre-eminence, says it has found the fix: A laserlike focus on the essentials.

The National Mathematics Advisory Panel, appointed by President Bush in 2006, is expected to urge the nation’s teachers to promote “quick and effortless” recall of arithmetic facts in early grades, mastery of fractions in middle school, and rigorous algebra courses in high school or even earlier. Targeting such key elements of math would mark a sharp departure from the diverse priorities that now govern teaching of the subject in U.S. public schools.

The panel took up its work amid widespread alarm at the sorry state of math achievement in America. In the most recent testing by the Program for International Student Assessment, released late last year, U.S. 15-year-olds achieved sub-par results among developed nations in math literacy and problem-solving, behind such countries as Finland, South Korea and the Netherlands.

“Without substantial and sustained changes to the educational system, the United States will relinquish its leadership in the twenty-first century,” reads a draft of the final report, due to be released next week by the Department of Education.

MATH ESSENTIALS

 

The National Mathematics Advisory Panel is expected to call for the following “critical foundations” or benchmarks for U.S. school children.

Fluency with whole numbers:

  1. By the end of grade three, students should be proficient with the addition and subtraction of whole numbers.
  2. By the end of grade five, students should be proficient with multiplication and division of whole numbers.

Fluency with fractions:

  1. By the end of grade four, students should be able to identify and represent fractions and decimals, and compare them on a number line or with other common representations of fractions and decimals.
  2. By the end of grade five, students should be proficient with comparing fractions and decimals and common percents, and with the addition and subtraction of fractions and decimals.
  3. By the end of grade six, students should be proficient with multiplication and division of fractions and decimals.
  4. By the end of grade six, students should be proficient with all operations involving positive and negative integers.
  5. By the end of grade seven, students should be proficient with all operations involving positive and negative fractions.
  6. By the end of grade seven, students should be able to solve problems involving percent, ratio and rate and extend this work to proportionality.

Geometry and measurement:

  1. By the end of grade five, students should be able to solve problems involving perimeter and area of triangles and all quadrilaterals having at least one pair of parallel sides (i.e. trapezoids).
  2. By the end of grade six, students should be able to analyze the properties of two dimensional shapes and solve problems involving perimeter and area, and analyze the properties of three-dimensional shapes and solve problems involving surface area and volume.
  3. By the end of grade seven, students should be familiar with the relationship between similar triangles and the concept of the slope of a line.

Source: Draft of National Mathematics Advisory Panel final report

Unlike most countries that outperform the U.S., America leaves education decisions largely to state and local governments and has no national curriculum. School boards and state education departments across the country are likely to pore over the math panel’s findings and adjust their teaching to make sure it aligns with the nation’s best thinking on math instruction. The federal government could also use the report to launch a national program in math instruction, as the government did for literacy after findings from a similar advisory panel on reading in 2000.

The math panel’s draft report comes amid the so-called math wars raging in the nation’s public classrooms. For two decades, advocates of what has come to be known as “reform math” have promoted conceptual understanding over drilling in, say, multiplication and division. For example, to solve a basic division problem, 150 divided by 50, students might cross off groups of circles to “discover” that the answer was three. Some parents and mathematicians have complained about “fuzzy math,” and public school systems have encountered a growing backlash.

The advisory panel’s 19 members include eminent mathematicians and educators representing both sides of the math wars. The draft of the final report declines to take sides, saying the group agreed only on the content that students must master, not the best way to teach it.

The group said it could find no “high-quality” research backing either traditional or reform math instruction. The draft report calls a rigid adherence to either method “misguided” and says understanding, which is the priority of reform teachers, and computation skills, emphasized by traditionalists, are “mutually supported.”

Larry Faulkner, the panel’s chairman and president of the Houston Endowment, a philanthropic foundation, said in an interview that the group had “internal battles” but decided “it’s time to cool the passions along that divide.” The panel held 12 meetings around the country, reviewed 16,000 research publications and public-policy reports and heard testimony from 110 individuals.

The advisory group also doesn’t take a position on calculator use in early grades, a contentious issue among educators and parents. The draft says the panel reviewed 11 studies that found “limited to no impact of calculators on calculation skills, problem-solving or conceptual development.” But the panel, noting that almost all the studies were more than 20 years old and otherwise limited, recommended more research on whether calculators undermine “fluency in computation.”

Still, the draft report says calculators shouldn’t be used on tests used to assess computation skills. Some states allow disabled children to use calculators on tests of arithmetic.

The draft report urges educators to focus on “critical” topics, as is common in higher-performing countries. The panel’s draft report says students should be proficient with the addition and subtraction of whole numbers by the end of third grade and with multiplication and division by the end of fifth. In terms of geometry, children by the end of sixth grade should be able to solve problems involving perimeter, area and volume.

Students should begin working with fractions in fourth grade and, by the end of seventh, be able to solve problems involving percent, ratio and rate. “Difficulty with fractions [including decimals and percents] is pervasive and is a major obstacle to further progress in mathematics, including algebra,” the draft report says.

These benchmarks mirror closely a September 2006 report by the National Council of Teachers of Mathematics, which many viewed as a turning point in the math wars because it recognized the importance of teaching the basics after the group for years had placed more emphasis on conceptual understanding.

Francis Fennell, president of the math teachers group and a panel member, said the group’s specific recommendations could help parents determine whether their kids are on the right track.

The draft report recommends a revamp of the National Assessment of Educational Progress, a widely followed test administered by the Education Department, to emphasize material needed for the mastery of algebra, especially fractions. The draft calls for similar changes to the state tests children must take under the federal No Child Left Behind Law.

The document urges publishers to shorten elementary and middle-school math textbooks that currently can run on for 700 to 1,000 pages and cover a dizzying array of topics. Publishers say textbooks often must cover a patchwork of state standards.

 

 

Welcome to Prof. David Klein’s Home Page

Welcome to Prof. David Klein’s Home Page

California State University Northridge

Department of Mathematics

 

 
 

Office: Faculty Office Building Room 127Telephone: (818) 677-7792

Email: david.klein@csun.edu

Fax: (818) 677-3634

Office Hours  Mon, Wed, 11:30-12:30 & by Appointment

Classes (Spring 2005)
Math 250 Third Semester Calculus
TTh 4 to 5:15 p.m., JR 212
Math 651B Math of Relativity & Gravitation
TTh 5:30 to 6:45 p.m., JR 215

 

B.S. Physics 1975, UCSB
B.A. Mathematics 1975, UCSB
Ph.d. Applied Mathematics,
Cornell University, 1981
Professional InterestsMathematical Physics, Statistical Mechanics, General Relativity, Mathematics Education

Research Articles:  Mathematics, Mathematical Physics, and Physics research articles, preprints and reprints


K-12 Mathematics Education

The State of State Math Standards 2005, by David Klein
with Bastiaan J. Braams, Thomas Parker, William Quirk, Wilfried Schmid, and W. Stephen Wilson
Technical assistance from Ralph A. Raimi and Lawrence Braden
Analysis by Justin Torres
Foreword by Chester E. Finn, Jr.

  • A Case Study: An Email Debate on the quality of education research in mainstream education journals between Dr. Pendred Noyce, head of the Noyce Foundation, and David Klein focusing on the article, “The impact of two standards-based mathematics curricula on student achievement in Massachusetts” by Riordan, J. E., & Noyce, P. E. Journal for Research in Mathematics Education, 32(4), 368-398 (2001).
  • C-SPAN Panel Discussion/Debate on mathematics education. This link includes a complete transcript of a panel discussion and debate hosted by the American Enterprise Institute on March 4, 2002 and nationally broadcast by C-SPAN:
    Does Two Plus Two Still Equal Four?What Should Our Children Know About Math?

    Panelists:

    Mike McKeown, Brown University
    Gail Burrill, Michigan State University and former NCTM President
    David Klein, California State University, Northridge  (For Klein’s Presentation click here)
    Lee V. Stiff, President of the National Council of Teachers of Mathematics
    Tom Loveless, Brookings Institution

    Moderator: Lynne V. Cheney, AEI

    (“Real Player”; click here to get free download)

  • California State Adopted Middle School Math programs for 2001, official Content Review Panel Reports
  • The Freedom to Agree, by David Klein, California Political Review, Vol. 9, No. 4,  pp 15-16, 31, July/August 1998
  • Open Letter to CSU Chancellor Charles Reed in support of the California Mathematics Standards for K-12.  Endorsed by more than 100 college and university mathematicians, and by Jaime Escalante.  January 1998

Other Education Related Links

Mathematically Correct, a nationally influential parents’ organization to improve mathematics instruction in K-12

New York City Honest and Open Logical Debate on mathematics education reform, or NYC HOLD, was “formed in order to address mathematics education in the New York City schools and to provide an organization for parents, educators, mathematicians and other concerned citizens to work to improve the quality of mathematics education”

Core Knowledge Foundation, founded in 1986 by E.D.Hirsch Jr.
What Elementary Teachers Need to Know, College Course Outlines for Teacher Preparation, from the Core Knowledge Foundation.  Syllabai are available for Math, Science, History, Geography, Music, Art.

National Council on Teacher Quality
American Board for Certification of Teacher Excellence
These two organizations provide high quality teacher certification programs.
Save Our Children from Mediocre Math, a parents’ group in Thousand Oaks, California
TeachMath.net, created by math teacher E. Toby Earl

Parents Concerned with Penfield’s Math Programs, a parents’ group in Penfield, New York
Bas Braams’ education page
Bill Quirk’s home page
Ralph Raimi’s home page
Hung Hsi-Wu’s home page

CSUN and Political Issues

CSUN United for Peace and Justice, a coalition of CSUN faculty, students, and university community members

Mechanisms of Western Domination: A Short History of Iraq and Kuwait

Correcting math’s blackboard bunglers

Nov 5, 1998


Correcting math’s blackboard bunglers


(Accompanying Ramirez cartton shows a man at a desk, nameplate on desk says “public education” and the man is reading “Why Johnny can’t read” – upside down.)

They say you can’t fight city hall – but a group of California parents calling itself Mathematically Correct has taken on the statehouse itself and won the right to restore a rigorous math curriculum to public education.

They also say that as California goes, so goes the nation.

The erstwhile Golden State has indeed been the locus of a nationwide contagion of educational “reforms;’ from Whole Language to anti-American propaganda masquerading as history to science-as-radical-ecologism. Since California is the largest public school market, half the country uses textbooks originally pitched to that market.

The Whole Math fad, better known to its traditionalist enemies as fuzzy math, has since 1989 become notorious for its verbalizing and visualizing approach to mathematics, its use of concrete “manipulatives” instead of abstract number concepts, its demotion of the teacher from classroom leader to “co-discoverer” and its reliance on calculators even in the lower grades.

Hallmarks of “fuzziness” also include a fixation on “everyday, real- world” applications, on “method” or “process” rather than domain-specific content, on “higher-order thinking” rather than memorization of facts, and on team-work rather than individual achievement –

The idea of math without numbers sounded dazzlingly innovative to many educators; it was in conveying any sort of mathematical proficiency to students that the miraculous new method broke down.

Parents watched in horror as their children whipped out calculators to determine 10 percent of 470. Fuzzy algebra texts droned on trendily for a hundred pages before getting down to a single equation. State test scores nosedived. One half to two thirds of freshmen entering the Cal State University system needed at least one year of remedial math, despite being among the top third of graduating seniors.

Among those horrified parents were the founders of Mathematically Correct (MC), “dedicated to the proposition that 2+2=4.”

Cofounder Martha Schwartz recalls the night she and her husband “discovered we were not alone!” She was a college geology professor “reacting to the damage done to good kids and the suffering of the best teachers,” her husband Rick, a high school chemistry teacher whose opposition to “fuzzy science” had earned him “un-veiled threats” from district superiors.

At an American Chemical Society meeting in San Diego that the Schwartzes attended in October 1995, Michael McKeown, a molecular biologist at the Salk Institute, closed his remarks with a critique of whole math.

‘After a minute or so of this, Rick and I were almost jumping up and down in our chairs. Within a week or so we had formed Mathematically Correct!’

The fledgling force began by contacting anyone and everyone they had heard of who dissented from the new order in math education. Email and the Internet were crucial to this molecular process. All the individuals contacted had hitherto believed their school districts’ solemn protestations that they were “the only ones who had a problem with the math reform?’

Paul Clopton of UC. San Diego, instrumental in setting up the group’s website, points out that “when parents get together, the bureaucrats’ first defense fades away. As each new parent told their story, we were constantly re- energized”

At first, efforts focused on convincing local districts – Petaluma, Novato, Escondi do, San Diego, Torrance and others – to get rid of existing whole math programs. Larry Gipson, a design engineer consultant and cofounder of MC, led a (successful) fight in Escondido because “I didn’t want my kids experimented on. . . . They were telling the kids to invent their own math out of thin air.”

Mr Gipson formed Parents for Math Choice and lobbied his school board. for just tat choice between traditional and whole approaches. Today 70 percent of district parents opt for traditional, and parental per-mission is required before any experimental program is implemented.

Larry Gipson jokingly refers to himself as “the token conservative” of Mathematically Correct, and indeed, contrary to alarums sounded by the fad- ridden National Council of ‘Teachers of Mathematics (NCTM), most MC members are politically liberal even if academically traditional. Martha Schwartz, during her own local fight, found herself annoyed most by having to point out “that as a secular Jewish geology teacher and registered Democrat, I was not, as charged, a Christian fundamentalist conservative – but that those were all legal things to be!’

Mrs. Schwartz continues, “I’m always outraged when people claim females and minorities can’t learn math or science like ‘regular people! “Lest that sound like a distortion of her opponents’ real views, listen to the former head of the NCTM, Jack Price, in an April 1996 debate with Michael McKeown on San Diego radio;

“What we have now is nostalgia math. It’s the mathematics we’ve always had, good for the most part for high-socioeconomic-status white males!’

Despite some successes at the local level, MC decided to aim for an overhaul of the whole state math framework. Revised every seven years, it is this which dictate’ content and methodology prospective textbooks. The group pushed for tough new K- 12 content standards and for MC to be represented on the appointed Academic Standards Commission. They I hammered legislators with data non grata about whole math’s dismal showing on all manner of tests, wrote open letters and critiques, and repeatedly gave testimony.

The standards adopted by California at the end of 1997 are a realization of Mathematically Correct’s belief that, in Mr. McKeown’s words, “Mastery of the basic the key prerequisite for effective problem solving and one of most effective ways to build understanding!’

The new standards have been denounced by the superintendent of public instruction, who has criss-crossed the state urging teachers to simply ignore them. But an independent review by the Fordham Foundation of America’s state math standards recently rated California’s No.1 and even compared them favorably to Japan’s.

Can the group now declare victory and go home? MC member Leslie Schwarze reflects philosophically that “the battle’s been going on since the 1700s -whether children are innately good or need to be civilized through parenting.

“This baffle will never be over.”

Marian Kester Coombs is a contributing writer for The Washington Times, specializing in education topics.


Reprinted by permission

Content Review of CPM Mathematics

Content Review of CPM Mathematics

Wayne Bishop
Department of Mathematics and Computer Sciences
California State University, Los Angeles


NOTE: CPM withdrew its application to California so this report is not based on its formal submission but, instead, on the document that CPM supplies as part of the Teacher’s Version entitled, “Correlation of CPM Mathematics 1, 2nd ed. (Algebra 1, v. 6.0) and the California Mathematics Standards”, hereafter, “Correlation”. Although Professor Bishop was a member of both the 1999 and 2001 state adoption cycle Content Review Panels, any official role as a CRP member ended with the conclusion of the 2001 cycle so this report is that of an experienced private citizen, not an official CRP review. Nonetheless, the criteria used herein were developed from the state criteria that Professor Bishop used for the official reviews of the 2001 adoption cycle. He is, however, more than happy to testify informally, by legal deposition, or in person, as to the quality and consistency of this report in comparison with those which he formally helped to prepare.


Overall Summary

With regard to mathematics content, this program does not sufficiently address the content standards and applicable evaluation criteria to be recommended for adoption.

In summary, most of the program is below the specified standards level and there is too much of an assumption that work will be done in teams. Although the publisher claims that all standards are met, several are clearly not met and several more identified herein as met are, in fact, not adequately met. Finally, there is a systemic misconception as to what is meant by logical argument in mathematics. If a statement looks to be true, students are told to put it into their “Tool Kit”, then to be available in all settings of study and assessment thereafter.

Evaluation of Content Criteria

Criterion

Met Criterion

1.  The content supports teaching the mathematics standards at each grade level (as detailed, discussed, and prioritized in Chapters 2 and 3 of the framework).
NO

2.  Mathematical terms are defined and used appropriately, precisely, and accurately.
NO

3.  Concepts and procedures are explained and are accompanied by examples to reinforce the lessons.
NO

4.  Opportunities for both mental and written calculations are provided.
NO

5.  Many types of problems are provided: those that help develop a concept, those that provide practice in learning a skill, those that apply previously learned concepts and skills to new situations, those that are mathematically interesting and challenging, and those that require proofs.
NO

6.  Ample practice is provided with both routine calculations and more involved multi-step procedures in order to foster the automatic use of these procedures and to foster the development of mathematical understanding, which is described in Chapters 1 and 4.
NO

7.  Applications of mathematics are given when appropriate, both within mathematics and to problems arising from daily life. Applications must not dictate the scope and sequence of the mathematics program and the use of brand names and logos should be avoided. When the mathematics is understood, one can teach students how to apply it.
NO

8.  Selected solved examples and strategies for solving various classes of problems are provided.
NO

9.  Materials must be written for individual study as well as for classroom instruction and for practice outside the classroom.
NO

10.  Mathematical discussions are brought to closure. Discussion of a mathematical concept, once initiated, should be completed.
NO

11.  All formulas and theorems appropriate for the grade level should be proved, and reasons should be given when an important proof is not proved.
NO

12.  Topics cover broad levels of difficulty. Materials must address mathematical content from the standards well beyond a minimal level of competence.
NO

13.  Attention and emphasis differ across the standards in accordance with (1) the emphasis given to standards in Chapter 3; and (2) the inherent complexity and difficulty of a given standard.
NO

14.  Optional activities, advanced problems, discretionary activities, enrichment activities, and supplemental activities or examples are clearly identified and are easily accessible to teachers and students alike.
NO

15.  A substantial majority of the material relates directly to the mathematics standards for each grade level, although standards from earlier grades may be reinforced. The foundation for the mastery of later standards should be built at each grade level.
NO

16.  An overwhelming majority of the submission is devoted directly to mathematics. Extraneous topics that are not tied to meeting or exceeding the standards, or to the goals of the framework, are kept to a minimum; and extraneous material is not in conflict with the standards. Any non-mathematical content must be clearly relevant to mathematics. Mathematical content can include applications, worked problems, problem sets, and line drawings that represent and clarify the process of abstraction.
NO

17.  Factually accurate material is provided.
YES

18.  Materials drawn from other subject-matter areas are scholarly and accurate in relation to that other subject-matter area. For example, if a mathematics program includes an example related to science, the scientific references must be scholarly and accurate.
YES

19.  Regular opportunities are provided for students to demonstrate mathematical reasoning. Such demonstrations may take a variety of forms, but they should always focus on logical reasoning, such as showing steps in calculations or giving oral and written explanations of how to solve a particular problem.
NO

20.  Homework assignments are provided beyond grade three (they are optional prior to grade three).
NO


Notes on Individual Criteria

1.  The idea of the shortfall here is explained in more detail in the Additional Comments at the end of the review. From the Correlation, a sufficient percentage of the CA Algebra Standards are addressed in some form as to ostensibly meet Criterion 1 but there is much less present than CPM indicates. In regard to some standards, more than three-fourths of the indicated citations are stretched beyond the limit of what the standards writers clearly had in mind.

2.  CPM-1 is deliberately constructivist in regard to such things. It is unfortunate as well because, if a student has not accurately built his Tool Kit, there will be severe difficulties since there is no glossary or even clearly stated terms.

4.  Some might disagree with this assessment because too much written work is often required. The problem is that it is often a misdirected effort that does not provide sufficient mental and written calculation of a genuinely algebraic nature – for example, a Guess and Check table of values in word problems. More opportunity for standard “by-type” approaches is needed and more use and confirmation of standard skills such as arithmetic or rational function..

7.  See #4 above. The program delights in major “daily life” problems except that the forest is lost for its trees. The entire Unit 7 “Big Race” is an example, and the introductory section of Unit 12 entitled “Problems Solving with Distance, Rate, and Time.” It is almost beyond belief that students could then never have seen d = rt but it is true. The authors hold such an anti-“by type” bias that it happened; e.g., no such items are mentioned in the Assessment Handbook for either Team or Individual tests.

8.  See #7 above. Another example is the absence of I = Prt; there is not so much as a mention of the terms. Similarly with the ideas of direct and inverse proportionality that the Framework deliberately discusses in Chapter 3. The self-proclaimed goal of the program is simply not met and, ironically, somewhat by design.

9.  The program has such a pedagogical bias toward group work that it is not clear what, if anything, is expected of students outside of the classroom environment and includes such little direct instruction that it would be extremely difficult for a student who had to miss class to fill in the gaps. There is an accompanying Parent Guide but it is not clear that all parents would have a copy and, beyond that, it really is not much help.

11. Properties of exponents are shortchanged. The Pythagorean Theorem is just given (9: 12) when algebraic proofs are easily available, the quadratic formula is used without proof for a couple of chapters before a proof is given that only a leap of insight would call a proof.

12.  Nothing close.

13.  There is almost no emphasis given to standards of any kind, let alone the ideas of Chapter 3 of the Framework. For example, the second subheading is “Basic Skills for Algebra 1” and includes Standards 4.0-7.0, 9.0, and 15.0. A glance at the list in Criterion 1 shows that these basic skills are inadequately developed in CPM-1.

14.  There is a great deal of irrelevancy, especially in Volume 1, but these are not supplemental.

15.  The entire Volume 1, so the first half the course, would be better left in the closet. Almost anything mathematical is Grade 7, if not below, yet the time requirements are huge. For example, Unit 3:1 is a silly “Algebra Walk”, literally, a human graphing exercise that is at the Grade 5 standard, AF 1.4 and 1.5, yet would take an entire class period to get organized, go outside, an conduct the exercise.

19.  Regular opportunity is present but, reiterating Criterion 1, Standard 24.0 is not met. The entire course confuses heuristics and inductive reasoning, one form of mathematical reasoning, with logical argumentation.

20.  It is not clear, even from the Teacher’s Version that says “Homework begins here,” what is to be homework and what is to be done as work in class as a team.

Standard by Standard Evaluation

Standard

Evaluation

Standard 1.0 Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable:1.1 Students use properties of numbers to demonstrate whether assertions are true or false. Met.
Standard 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents. Not met, no fractional exponents and properties of exponential expressions with the same base are not confirmed until Unit 10: 40 and 43.
Standard 3.0 Students solve equations and inequalities involving absolute values. Very weakly met, only the simplest of absolute value equations.
Standard 4.0 Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x – 5) + 4(x – 2) = 12. Met, but “cups and tiles” all the way through Volume 1!
Standard 5.0 Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step. Met.
Standard 6.0 Students graph a linear equation and compute the x- and y-intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4). Weakly met in 12: 99 ff but weakly assessed and not in the Two-Year Final at all.
Standard 7.0 Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula. Met.
Standard 8.0 Students understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point. Weakly met. 12: 110, 120 meet the perpendicular  specification but they are not assessed or used regularly enough to be confirmed.
Standard 9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets. Met.
Standard 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. Met, but inadequate. No division of polynomials except simplification.
Standard 11.0 Students apply basic factoring techniques to second- and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials. Met, but inadequate. Essentially no perfect square trinomials but 13: 79 hints at it.
Standard 12.0 Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms. Met, but too many are already in factored form and the skills are barely assessed.
Standard 13.0 Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques. Met, but too many are already in factored form and the skills are barely assessed.
Standard 14.0 Students solve a quadratic equation by factoring or completing the square. Inadequate. There are quite a few by factoring so “or” is satisfied. (See #19).
Standard 15.0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems. Not met. There is too much of a program bias against “by type.”
Standard 16.0 Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions. Inadequate.
Standard 17.0 Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression. Inadequate.
Standard 18.0 Students determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion. Inadequate.
Standard 19.0 Students know the quadratic formula and are familiar with its proof by completing the square. Not met. The standard is to “know”, while it is always available in the student’s Tool Kit and the proof is weak since completing the square is weak and not assessed.
Standard 20.0 Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations. Weakly met and not assessed, the last topic of the course.
Standard 21.0 Students graph quadratic functions and know that their roots are the x-intercepts. Met, but weakly, more systematic methods are needed.
Standard 22.0 Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points. Weakly met
Standard 23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity. Very weakly met. 13: 77 purports to address it but most won’t see the relation.
Standard 24.0 Students use and know simple aspects of a logical argument:24.1 Students explain the difference between inductive and deductive reasoning and identify and provide examples of each.

24.2 Students identify the hypothesis and conclusion in logical deduction.

24.3 Students use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion.

Not met. The entire course confuses heuristics with logical argumentation.
Standard 25.0 Students use properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements:25.1 Students use properties of numbers to construct simple, valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions.

25.2 Students judge the validity of an argument according to whether the properties of the real number system and the order of operations have been applied correctly at each step.

25.3 Given a specific algebraic statement involving linear, quadratic, or absolute value expressions or equations or inequalities, students determine whether the statement is true sometimes, always, or never.

Weakly met.


Comments Regarding Assessment Philosophy

Clear indication of the fact that the CPM program is not serious about students meeting the CA Mathematics Content Standards for Algebra follows from looking critically at their own words about, and examples of, assessment. For example, from the CPM Assessment Handbook, Math 1, “A good first step is trying group tests. [P. 3]” “We recommend for this course that you don’t give the usual type of in-process quiz that evaluates students’ mastery of skills while they’re still in the early stages of learning them. [P. 9]”

Why the de-emphasis on early skill mastery? It is philosophical, “When we test the students mastery of skills too early, the students focus is diverted from understanding where and how the skill is to be used.”

As shown below, CPM has carried this philosophy, that does have some element of validity, beyond logic to a point that lack of mastery of skills – ever – still allows students to get good grades in the course without having developed skill mastery. For example, the model final included in the CPM Outline for a Two-Year Program has no mention of parallel or perpendicular lines, no items that require an equation given two points or given one point and the slope, one simple item that implies factoring (solve x2 + 6x + 8 = 0) but no simplification of rational expressions, no mention of completing the square or quadratic formula, etc.

A strong student working at the level of the CA Grade 7 standards, i.e., a good pre-algebra background, would do very well on this final with no formal course, let alone two years of “algebra”! The proposed Team Final does address some of these in a minimal way but the two addition of rational expressions items are very easy and all but one of the expressions in the two quotient items are already in factored form, etc. Again, there is no mention, or use of, slopes in regard to parallel or perpendicular lines, completing the square, the quadratic formula, etc. So to argue, as CPM does, that the CA standards are met, just a bit delayed until mastery has had time to sink in, is nonsense. They are not assessed because they have not been mastered by sufficiently many students.

To put this in perspective, the reviewer’s daughter is in a school that uses one of the California approved texts at the appropriate grade level. Three-quarters of the way through her sixth grade program she was presented the first-year final of the CPM two-year program and had no trouble setting up the percentage word problems (“43 is what percent of 125”, etc.) in mathematical equation form out-loud as she read them. When she was shown the title, “Algebra – Part 1, Final Exam”, she laughed and then excused herself, “I’m sorry, but it’s kind of funny.”  Excluding the CPM specialized “Diamond Problems” that supposedly lead to eventual factoring (of which there are none, even on the Part I Team Final), she would have been able to do nearly every item correctly, and will be able to do all of them by the middle of seventh grade per the California Standards for that grade.

Indicative is the outrageous presentation of the only real word problem, “Solve the problem and write an equation. You may do this in either order. If you do not need a guess and check chart to solve the problem use it to define your variables.” That is, use algebra to “solve” it or don’t. A table of trials is perfectly OK half-way through algebra, in fact its inclusion is mandated even for those students perfectly capable of solving the problem entirely algebraically.

Looking more deeply at the assessment philosophy, “the emphasis should be on the mathematical thinking evident in the work and on what the student knows, not on what the student does not know.[P. 9]” Several pages of the Assessment Handbook are devoted to scoring holistically. “Holistic scoring means just writing the score by the problem 0, 1, 2, 3, or 4 and not making corrections on the students’ papers. [P. 9, Bold is original.]”  Still, the language, including that of the portfolios and the journals is sufficiently imprecise as to allow the possibility of clear, objective, individual student evaluation, “Assessment includes testing basic knowledge and skills, but it encompasses much more.”

Comments Regarding Research Support

The reality of this program is that the standards are not met and genuine assessment would quickly confirm that fact. The evidence and testing, both globally for all CPM students and locally as a teacher tries to assess a student’s knowledge, are entirely inadequate and the conclusions of studies as described in the Teacher’s Version is not nearly as conclusive as the writers imply.

For example, the first paragraph of the page entitled “Research Summary, Comparison of CPM and Traditional Students” is in regard to use of the CSU/UC Mathematics Diagnostic Testing Program (MDTP), data from eight schools that purport to verify that students in seven of the eight learned more in CPM-1 than in their traditional counterparts. Ignoring the fact that “traditional” is not defined and the schools are not named so it is impossible to see exactly what CPM-1 was being compared against, this study did not use the MDTP. It used only 20 of the available 50 MDTP Elementary Algebra items. Two word problems specifically designed for CPM evaluation were also included in the test. Key components of a traditional Algebra I course which are largely or completely absent from CPM -1 were omitted. The following tables indicate the breakdown of the original MDTP items into its subscales as well the distribution of MDTP questions used in this study:

MDTP Subscale

# Items
in MDTP

# Items
in CPM Test

Linear & Quadratic Equations

13

5

Arithmetic

6

5

Geometry

7

5

Graphs

6

5

Rational Expressions

5

0

Exponents and Square Roots

6

0

Polynomials

7

0

Further down the Research Summary page is “CPM End-of-Year Assessment”. The very nature of the description is indicative of the CPM approach, “we gave two questions to …” Two questions, even with “presentations of complete solutions”, is not the kind of algebra assessment most people envision when they read the subtitle.

Another measure of CPM “success” is the state SAT-9 scores, in which a page of data purports to prove that CPM schools are more successful than “their peers who use other curriculum materials,” but there is so much missing as to make the data almost meaningless. As a start, the SAT-9 is not algebra! Of course, there are some exercises, ratio and proportion problems for example, that lend themselves to nice algebraic representation, but it is not algebra at the level of then President Clinton’s assessment, “Algebra is algebra!”

A comparison of the California Algebra Standards Test would be useful data, but even that (which CPM chose not to publish) would be comparing CPM-1 students against a far less homogeneous group, some using an even more aggressively “reform” curriculum. Beyond that, some schools use CPM for regular classes and a more traditional program for more advanced ones. That could be taken as evidence, probably supportable, that CPM is preferable to “general math” but hardly an argument for using it in place of a traditional college preparatory curriculum as its name would imply.

Finally, CPM cannot be trusted to give us an honest picture. The “MDTP” study that they continue to use is a clear indicator of that fact with its 20 of the 50 MDTP items. Since the public lacks the names of the CPM schools in the CPM summary sheet, it is impossible to do a quick comparison of SES factors, for example, to see if most of the CPM schools in these counties might have had a head start even before any choice of mathematics curriculum.

Comments Regarding Assessments in CPM

A much clearer vision of how far short CPM falls on more traditional end-of-course assessments is contained within their own Teacher’s Version and, most explicitly of all, within their “Outline for a Two-Year Program”, the guide to teachers for setting up the same program but over a two-year, less-demanding schedule referred to above. This document supplements the Teacher’s Version guides for constructing unit tests and tells what the designers really have in mind for verified competence. It is far off the algebra standards of California or, beyond that, of any other set of standards for algebra.

Here is one way that the assessment materials are designed to appear to be sufficiently demanding of standards-level competence when they are not. After Unit 3, the Assessment Handbook itself does not have model exams (they are in the Two-Year Program guide), but it does have item banks, by unit, along with instructions for constructing the tests themselves. Indicative in these instructions is “If you give an individual test …” That is, even the act of having an individual, bottom-line assessment can not be taken for granted in a CPM environment.  Going onward with the quote, “it would be best to make this a very short test.”

Most indicative of all, however, is the test bank itself. The instructions recommend, “no more than one question of any type,” which would be reasonable advice if the items in each set were, in fact, of the same type. That is obviously not the case, the sets are constructed so that it is possible to avoid confirmation of the ideas involved at the level that a cursory look could imply.

Standards Representation in CPM Test Items

The test-bank items are not numbered (so as to make it easy to omit the item entirely) but some representative examples are the following from the indicated unit with item number counting from the first item in that unit looking at every reference given for the particular standard in the Correlation for the CA Standards for Algebra I .

Standard 8  Students understand the concepts of parallel and perpendicular lines and how these slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point.

Unit 7: 49, and 73 These words are not mentioned in these references, nor in the assessment-bank, Team or Individual. Ref. 7:83 looks at parallel for slope of 2/3 but not in general.

Unit 11: 101 This item does have students note that the slope of one specific line is -5/3 when the original line was 3/5 and that the lines are perpendicular, but without verification other than they look like it. Students are to “Record your observation in Your tool kit.” These words are not mentioned in the Study Team Questions (Team) but in #2 of the Individual Test (Individual), parallel is to be recognized by slope, without graphing. No student constructed equations are expected except when given two points.

Unit 12: 110, 122 Both of these meet the standard for perpendicular slope. Neither parallel nor perpendicular is mentioned in either the Team or Individual test banks.

Unit 13: 28, 61, 102 Items 28 and 102 meet the standard and 61 is borderline.

Standard 11.  Students apply basic factoring techniques to second- and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials.

Units 0-8: diamond problems 0: 3 ff, 2: 53, 78, 4:52, 6:70, 82, 96 All of these references from Volume 1 are so far off the standard as to be open to a charge of lying. For example, the last, 6: 96 consists of 11 exercises of multiplying a monomial or binomial times a binomial, not the reverse. Granted, converting expressions in factored form into un-factored form is helpful but it is does not begin to address, let alone meet, the standard.

Unit 8: 2-4, 10, 12 These are only preliminaries to factoring; e.g., 4 is only recognizing if an expression is written in any kind of factored form, 10 and 12 are tile-pushing with the factored forms already given. 8: 18 finally is an actual factoring but only with algebra tiles, 19 and 20 tie factoring into the earlier “generic rectangle” extension thereof, 31 is factoring out monomial common factors. 8: 50-51 does treat the difference of two squares and 8: 52 is ten mixed factoring problems, including one of them. 8: 57-63 meet the standards along with 70-71 and 77-80. The Unit 8 Team #1 has six expressions to be factored, including one difference of two squares and #3 uses factoring to solve quadratic equations in one variable, none of which is a difference of two squares, exactly as in the Individual, four quadratic equations to be solved, three that still need to be factored, two that are trinomial, and none that are the difference of two squares. In the Two-Year Program, that includes models of actual tests, not “select from the following,” the Unit 8 Individual Test includes six factoring items, five with assistance and one stand alone. There is one difference of two squares item, but not “by type” but only by coincidence. The test includes no quadratic equations to be solved.

Unit 9: 90 does FOIL factoring (not in the CA Standards) and this is the only reference in the Correlation but, in fact, they missed some, 9: 21, 42, etc. However, the Team Questions only include two factoring exercises and one quadratic equation to be solved. The Individual test includes a choice of three or two factoring items, one of which includes a difference of two squares, and block of five equations to solve that includes one quadratic. By contrast, the Two-Year Unit 9 Mid-Unit Individual Test (there is no unit test because of the up-coming First Semester Test, contains no factoring items nor quadratic equations to be solved. The First Semester Individual Final has two, x2 – 7x + 12 and x2 + 5x. There are no difference of squares items and no perfect square trinomials.

Unit 10: 1-3 ff, 17 Recurring exercises do confirm the ideas. The quadratic formula is simply given in 10: 86 so it is not clear whether or not factoring will continue in solving quadratic equations. In regard to assessment, several Team and Individual items have radical expressions or decimal approximations indicating that they are not to be factored since completing the square is not introduced until three units later. Somewhat surprisingly, given the end-of-course exams, the Two-Year Unit 10 Individual Test does have several factoring problems including the advice to look for difference of two squares and a simplification of a rational expression that requires factoring both numerator and denominator (See CA Standard 11).

Unit 11-13 practice in homework 13:79 hints at perfect square trinomials but this standard is not met under almost any level of generosity. The Two-Year Part 2 end-of-course Individual Final Exam is the most indicative, here. Instead of demonstrating that students finally have mastered these ideas, there is exactly one factoring exercise, #6b) Solve: x2 + 6x + 8 = 0. There are no simplification of rational expressions, let alone multiplication or division of them.


Standard 12.  Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to lowest terms.

There are some, but see the last line of the Standard 11 remarks above. Since the assessment specifications allow for picking and choosing – they’re deliberately not numbered – it is impossible to say to what extent the program expects individual student competence.


Standard 13.   Students add, subtract multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques.

There are some, but see the last line of the Standard 11 remarks above. Since the assessment specifications allow for picking and choosing – they’re deliberately not numbered – it is impossible to say to what extent the program expects individual student competence. Indicative of how far short the program is of the intended standard, there is one example given in the 1999 Mathematics Framework to demonstrate the intent of this standard: Solve for x and give a reason for each step: 2 / (3x + 1) + 2 = 2/3. There is no equation of this level of difficulty to be solved in the two volumes.


Standard 14.  Students solve a quadratic equation by factoring or completing the square.

The same remarks apply; completing the square is an afterthought at the very end of the book, Unit 13: 67, and only with the CPM insistence on an overuse of so-called “algebra tiles” belying the problem with an odd or fractional middle term, and students are simply not expected to use it. In fact, the disclaimer at the beginning of the Unit 13 Individual Test admits as much, “We really do not expect many students to begin to master the topics in Unit 13. So, we provide a little extra assistance so we can still test them on these topics,” followed by inclusion of the quadratic formula with no items that require completing the square, not even with an even middle term and a pile of algebra tile. The Part 2 Individual Final is the most indicative, of course, neither is ever needed. The one quadratic equation is already in standard form and factors easily.


Standard 15.  Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.

From the Correlation comment one would expect this standard to be well met, “Word problems and investigations are at the core of the CPM program. Students regularly solve word problems without categorizing them “by type.” It is the second sentence that belies the first. Nearly all of the Unit 4-9 exercises are below the level of the algebra standards, many at the 6th grade standards, just far wordier and often with a great deal of irrelevant fluff. A good example is Unit 7, the entire “Big Race” premise. It is a complicated d = rt exercise without saying so but therefore not on standard at all, but a year or two off. Even among the last exercises that the Correlation indicates, Unit 13: 1, 12-14, 52, 62, 72, 78, 101, only 62, 78, and 101 meet the intended standard and nothing of the kind appears on the Two-Year Final. The only genuine word problem leads immediately to a pair of simultaneous linear equations. That qualifies but only as a small part of the intent of the standard. More ordinary problems are described in words and that is to be commended, but that is not an acceptable excuse for avoiding the others.

Other standards are far wide of the mark as well, and by design. CA Standard 24 is claimed to be met in the Correlation by lots of “Explain your answer…” and there is much of that in CPM-1. Most of them are not close to what the standard says and means. The books can be opened almost anywhere to see examples but using one that they chose:

Unit 10: 33 The item consist of three parts, solving a quadratic equation in standard form by factoring, graphing the corresponding function with these points as zeros and finally the supposed logical argumentation, “How are (a) and (b) related?” This not an unreasonable exercise, even good. The problem is that the informal argumentation of the exercise has nothing to do with the listed standards:

Standard 24.  Students use and know simple aspects of logical argument:

24.1 Students explain the difference between inductive and deductive reasoning and identify and provide examples of each.
24.2 Students identify the hypothesis and conclusion in logical deduction.
24.3 Students use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion.

Or another example on the same theme, 10: 96. Again, the exercise is reasonable; it is asking students to expound on the connection between the discriminant term of the quadratic formula and factorability. In fact, it would meet 24.1 if were it a bit more direct, something like, “Argue that if b2 – 4ac is a perfect square, then …” instead of “Explain what the values of sqrt(b2 – 4ac) tells you about factorability of the polynomial?” Few students would recognize that this new tool constitutes proof that the polynomial can or cannot be factored; they will not distinguish it from an earlier “logical argument” 10: 74 prior to the introduction of the quadratic formula, “Is the expression x2 + 6x + 2 factorable? Explain your answer.” Even assuming knowledge of uniqueness of factorization (which should get acknowledged in a formal setting), lacking the rational roots theorem, the ability to complete the square that (unconscionably!) is not introduced until three chapters later, or the quadratic formula, it is not at all clear what a “good” explanation would be! “I tried everything I could think of!” perhaps?

 

 

Final Observations Regarding The Curriculum Materials

In spite of being too far off the algebra standards to warrant state approval but on the positive side, the books themselves have, or rather, Volume 2 has some distinct advantages over more traditional texts. The regular, mixed review of exercises is excellent. So is requiring students to present sensible explanations in support of their conclusions. If the authors of the Teacher Version introduction materials were to abandon their strong allegiance to what’s often known as “Authentic Assessment,” i.e., team tests, journals, portfolios, and individual observation and get back to lower case authentic assessment that algebra teachers everywhere have traditionally done, the program would be greatly enhanced. Yes, there should be more “by type” word problems and they should be part of the regular mixed review exercises, more formal logical argument, actual use of completion of the square in a way that students are expected to be able to use it, etc., but that is not the worst problem. The worst problem is pedagogical.

Volume 2, in the hands of a competent traditional algebra teacher, could be reasonably effective; short of the standards, but effective. If all of it, through Unit 13, is covered and if individual students do all of the exercises, they will master enough of the skills to be considered competent at the level of algebra 1, capable of going forward successfully in mathematics that depends on these concepts and skills. The problem is, and it is convicted as fatal by the included Individual Tests in the Two-Year Program guide, there is no assurance – nor any genuine effort to confirm – that this is what will happen. A little group work from time-to-time is fine, having stronger students help weaker ones master the material likewise, but that is not the described philosophy; in fact, it is completely the reverse. Similar is the work with algebra tiles. A little at the beginning? Sure, why not. Plastic toys instead of mental manipulation far into the course is very different. But, although serious, the worst parts are not the leisure introduction to factoring, that does have supportable logic, or too much graphing calculators, or being too far off the standards. They are two; one is pretending that everyone is mastering algebra while letting strong students carry weak or lazy ones, and the other is the heavy-handed, non-algebra, time-wasting in Volume 1.

The latter problem is associated with the first pedagogically and philosophically although the supporters would phrase it differently, of course. From the Introduction and Overview, “Telling has little to do with promoting learning – students must construct their own understanding.” In a sense, this is true since we do not learn by “direct download” but neither do we learn much mathematics by activity-based insight. Being told wrongly by a convincing fellow student, a situation common in un-led team settings, is far worse than being told what is correct by a competent teacher. Yet, “The daily activities in this course will require much more work in study teams and much less introduction and explanation of ideas by the teacher.” Much of Volume 1 actually detracts from developing algebraic competence. Almost all of the mathematical content is at the level of the Grade 7 standards or below, e.g., the equations to be solved all are, but the activities are still very time consuming and sometimes frustrating. The worst of all, however, is not teaching the power of algebra itself. Unit 4: 123 is TOOL KIT CHECK UP and it is mandated that it contain Guess and Check tables, and “cups and tiles to model solving equations.” This is not algebra and it is not college preparatory math, no matter what it calls itself. Eventually, Volume 2 starts teaching some algebra but it is too little and too late.

Mathematically Correct presents The Pythagorean Theorem

Mathematically Correct presents

The Pythagorean Theorem

G. D. Chakerian and Kurt Kreith


At a recent school meeting, a group of Davis parents and teachers used the Pythagorean theorem to illustrate the difference between a constructivist vs. traditional approach to teaching. Their goal was to provide other parents with a basis for responding to a recent decision by the Davis Board of Education. For in Fall, 1996 Davis junior high schools will offer a choice between two different courses in Algebra 1, one emphasizing constructivist pedagogy and the other relying on a more traditional deductive approach.

As set forth in the currently used text Themes, Tools and Concepts, one constructivist approach to the Pythagorean theorem is based on the use of geoboards. A traditional approach appropriate to Algebra 1, one found in many algebra texts, is based on the dissection of a square.

 

   The former calls on students to use rubber bands to build a right
   triangle on a geoboard, use rubber bands to enclose the squares
   defined by the triangle's legs and hypotenuse, and then look for
   patterns in the areas of the squares so generated.  

   The latter asks the student to visualize two different dissections
   of a square of size (a+b) x (a+b).  Using the usual notation of a,
   b, and c for the legs and hypotenuse of a right triangle, one such
   dissection corresponds to a^2 + b^2 + 2ab and the other corresponds
   to c^2 + 2ab .  Equating these two expressions yields the usual
   symbolic representation of the Pythagorean theorem: a^2 + b^2 = c^2.
   
   [^2 notation indicates squared terms]

Laudable as the use of experimentation as a prelude to mathematics may be, there are serious dangers hidden in this constructivist approach to the Pythagorean theorem. For instance, while it is easy to construct right triangles on a geoboard by orienting the legs of the triangle along the horizontal and vertical axes, the example given in Themes, Tools and Concepts suggests that the student should use more general orientations. The question that then arises is, “how is the student to know whether a triangle with such general orientation is, or is not, a right triangle?” (It is very easy to construct geoboard triangles with one angle imperceptibly close, but not quite equal, to a right angle.)

The only mathematical solution is to use the Pythagorean theorem itself (or more precisely, its converse) to confirm that such a triangle is in fact a right triangle. However, this is precisely the knowledge that the student is being urged to construct! Thus, implicit in this particular constructivist approach to the Pythagorean theorem is the notion that the student should build his or her own knowledge by “eyeballing” right angles.

Training a generation of carpenters to rely on “eyeballing” right angles would be a national disaster (none of us would let such a carpenter touch our house). Yet, in the name of constructivism, we seem to be encouraging a generation of children to erect this pillar of mathematical knowledge on just such a basis.

Another problem arising in this constructivist approach to the Pythagorean theorem is that of calculating the areas of the squares built on the sides and hypotenuse of a geoboard triangle. The usual formula “Area = Side x Side” requires that we first determine the lengths of the sides of these squares. However, unless these squares are aligned with the geoboard’s vertical and horizontal axes, finding the lengths of their sides also requires the Pythagorean theorem!

An alternative way of finding the areas of “tilted squares” is to use an advanced mathematical result called Pick’s theorem. While children can be taught to use Pick’s theorem at an early age, any semblance of a mathematical understanding of this tool is well beyond the traditional high school curriculum. It may also be that students will be taught to calculate areas by counting unit squares, and pieces thereof.

Another possibility is to enclose such a “tilted square” within a larger square whose sides are parallel to the axes of the geoboard. While this provides an appropriate way of calculating areas, note that it corresponds to the dissection “(a+b)^2 = c^2 + 2ab” arising in the traditional proof of the Pythagorean theorem cited above. That is, this particular approach to implementing the “discovery process” takes the student half way to actually understanding the Pythagorean theorem. However, on ideological grounds, it stops short of conveying the gift of understanding.

Some may argue that it doesn’t really matter which method is used to teach the Pythagorean theorem – i.e., that both methods lead to the same result. However, this is not true. Experiments with the geoboard correspond to a cumbersome verification of the Pythagorean theorem in rather special circumstances (the geoboard’s discrete structure is well suited to experimentation, but it fails to represent the more general structure of the Euclidean plane). The traditional dissection approach corresponds to a proof of the theorem, providing an answer to the question “why.” One approach sets the stage for discoveries that lead to conjectures; the other emphasizes properties of area that lead to the understanding of an important truth.

Many parents believe that the most important end of education in any field is to raise the question “why,” to know when an answer might or might not exist, and to demand an answer when it can be given. Having children “discover” a hodge-podge of mathematical properties, without providing answers to which properties are true and why, is to deny them a real mathematical education.

While manipulatives can be powerful tools for leading students through a discovery process that reinforces mathematics, the haphazardly planned use of manipulatives can be destructive. An essential adjunct to “hands-on” mathematics is an effort to organize ideas and develop the capacity for mathematical thought and reason. Experiments performed under the tutelage of unskilled guides can lead students into a chaotic jungle, one in which their minds become entangled in an underbrush of mismatched concepts to which they, their parents, and their future teachers will be hard pressed to bring order.