Parents fight back

I came across this parent group site in Penfield. The parents are concerned about fuzzy math in their schools. This list of observations neatly encapsulates what's wrong with the fuzzy math plague:

Penfield parents have noticed several problems relative to the new math programs. Some of the most common problems and observations are noted below. Skip to the bottom of the list to see some of the more recent observations noted since parents have turned up the heat on the district:

Students are receiving high grades on their report cards, but they don't seem to exhibit appropriate math skills for their grade level.

Children are no longer being taught basic math facts. Students lack skills in division, multiplication, fractions etc...

Calculators are too widely used and kids are losing their ability to perform simple arithmetic without their calculators.

There is too much group work. Students no longer receive direct instruction. They may work for a long period of time on one question and never really learn the correct method to solve the problem.

There is no emphasis on finding the correct answer.

There are no reference materials. There is no textbook.

There is not enough practice so children do not retain the concepts.

Problems are very abstract and often frustrate the students, sometimes to tears.

Parents are concerned about the future of their children should they need to leave the district and relocate to a district using traditional math techniques. Their children will not know the math algorithms.

Many parents have placed their children in tutoring to compensate for the lack of instruction.

Several teachers have mentioned that they are unhappy with the programs, but they don't speak out for fear of retribution.

High school students who have experienced both math techniques have spoken out against the "reformed math".

Many universities have indicated that students from traditional math programs perform better in college math courses.

Children with learning disabilities, particularly those with reading/writing difficulties, are hurt by these programs.

Students are asked to solve problems using tools/algorithms that have not been taught.

The schools have taken away the ability for students to work independently. Parents are forced to teach math lessons at home.

Children no longer feel confident or successful. They hate math now.

There are big concerns about the way that standardized test scores are being interpreted by the school district administrators.

Complaints to teachers and counselors are not being addressed.

Core-Plus received a poor showing in the Michigan State University study by Hill and Parker.

The math materials don't match the NYS guidelines for the various grade level expectations.

High achievers are being held back.

Academic Intervention Services (AIS) are not effective.

The district has no plans in place to repair the damage done to students over the past few years.

They are experimenting with our kids again. This is another "Whole Language" fiasco, only now it's math.

There are inconsistencies in the implementation of the programs between schools and even within schools.

Students report that some science teachers are providing math instruction during/after science class due to missing math prerequisites.

I love this entry under more recent observations:

Traditional math worksheets coming home for homework just prior to standardized testing. Teaching to the test and cramming.

Looks like the fuzzies don't have full confidence in their crap programs. They want to have it both ways: Supplement with traditional math and attribute any success to their snake oil programs to keep the faith.

Social construct

Educationists in Australia are on a "social construction" kick. How many "multiple and/or conflicting interpretations" can there be for 2+2=4 or for the fact that an educationist would fall on his face if he jumped out of a window? Teachers are to be penalized for presenting factual knowledge:

TEACHERS should present knowledge as a "social construct" open to interpretation rather than undisputed facts, even in maths and the sciences, says a NSW guide to quality teaching.

The Quality Teaching framework, developed on behalf of the NSW Education Department, rewards teachers for presenting "problematic knowledge" in their lessons.

Under a coding system developed to assess teachers, lessons that present knowledge "only as fact and not open to question" score the lowest.

The highest score is given when "knowledge is seen as socially constructed, with multiple and/or conflicting interpretations presented and ... a judgment is made about the appropriateness of aninterpretation in a given context".

"Knowledge is treated as problematic when it involves an understanding of knowledge not as a fixed body of information but rather as being socially constructed, and hence subject to political, social and cultural influences and implications," the guide says. "Knowledge is not treated as problematic when it is presented only as fact, a body of truth to be acquired by students, or is treated as static and open to only one interpretation."

The guide specifically relates the idea of contested knowledge to the teaching of science, saying if it is difficult to see how a subject is problematic, look at its history.

Deconstruction is in full bloom.

Decreased attention

An excellent dissection of NCTM charlatanry.

The NCTM Standards have been developed by progressive "math educators", not by people with genuine knowledge of mathematics. For eighty years progressive educationists have rejected the idea of remembering any domain-specific knowledge. They say knowledge is changing too fast, and the facts of today will be obsolete tomorrow. Calculators and computers are the latest "proof" of this claim. The NCTM wields them as a double-edged sword, justifying the trashing of traditional math and offering the benefit of exciting "tools" for bypassing the difficulties of traditional "paper-and-pencil" math.

Progressive educationists believe important factual knowledge is already known intuitively or will be picked up naturally as a byproduct of real-world experiences. They claim that real-world experts rely on "higher-order skills" and "just-in-time" factual knowledge supplied by computers and reference materials. They say real-world experts never trust their own long-term memory.

Michigan tries "history"

Michigan produces history "standards" (called "content expectations") that sound more like a joke.

Indeed, the draft "expectations" say nothing about America before 1890, leaving the nation's foundation years, its crucial philosophical groundings and the Civil War to elementary and middle schools. In the post-1890 studies, no mention is to be found of Theodore Roosevelt, Franklin Roosevelt or Ronald Reagan. Same for Henry Ford, Rosa Parks, Andrew Carnegie, Douglas MacArthur and Earl Warren -- and Hitler, Stalin and Tojo. Similarly absent: the development of mass production and the rise of industrial unions, D-Day, defeat of fascist Germany and imperial Japan, the Korean War and the toppling of Soviet communism. In a flash of good sense, curriculum writers rejected the lead consultant's attempt to ban use of "America" and "American" as "ethnocentric," potentially offensive to the rest of the hemisphere.

Instead of history, there is an ideological agenda:

The standards do direct study of, for example, the environmental movement, the American Indian protests at Wounded Knee, Rosie the Riveter, the World War II internment of Japanese-Americans, acid rain, the automobile's contribution to global warming, consequences in the Persian Gulf of U.S. energy policy and alternatives to President Truman's use of the atomic bomb.

Mumbo-jumbo vs. quality education

As more and more schools adopt ed school mumbo-jumbo (authentic assessment, countless intelligences and learning styles, constructivism, etc.), it is reassuring that quality schools crop up now and then.

Teacher tests

In Why American Students Do Not Learn to Read Very Well: The Unintended Consequences of Title II and Teacher Testing Sandra Stotsky takes a look at teacher tests and finds that prospective teachers are not expected to have knowledge of research-based reading instruction:

This paper provides an analysis of the descriptions of the subject tests assessing reading instructional knowledge that prospective elementary teachers in this country take for licensure: those offered by Educational Testing Service, a variety of those provided by National Evaluation Systems, and the one offered by American Board for Certification of Teacher Excellence. I examined these descriptions to determine whether the tests appear to address three major components of a research-based approach to reading pedagogy (instruction in phonemic awareness, phonics, and vocabulary knowledge), the weights attached to knowledge of these three components, and the quality of the sample questions they provide.

I guess science is not needed if reading unfolds naturally, as whole language advocates believe. The trouble is that learning how to read is not natural like language acquisiton.

Pod people

Over at edspresso a prospective teacher has discovered that the pod people have invaded math education:

Math education is in a shambles, starting from the so-called standards put out by the National Council of Mathematics (NCTM) in 1989 and revised in 2000. These standards were then copied by many states that thought they were great. State boards of education paid no mind to the shrieks of horror from mathematicians, simply not believing that the resulting standards took the math out of mathematics in the name of fun, and whose approach for eliminating the achievement gap eliminated the mastery of any math knowledge that matters. The well-intentioned but ill-conceived standards have actually widened the gaps between the rich and the poor by motivating those who can do so to hire tutors for their children, to enroll them in learning centers like Sylvan and Kumon, or to put them in private schools.

Fuzzy math was ostensibly invented to make math "accessible" to what's termed "minorities". But as the writer points out, taking math out of math only widens the achievement gap.

Ed school "teacher"

Ever wondered what a teacher is? A school of education has the answer. Discovered by Chanman

Helpful jargon

England is abuzz with something called "personalised education" as shown in this BBC account. It looks similar to what's known here as differentiated instruction. As is usually the case in edland, these innovations are couched in inpenetrable jargon. One can never be sure of what one is dealing with:

There is a very tricky question which is bothering many people involved in schools today, namely: "What is personalised education?"

The question is important because "personalisation" is the current buzzword in the Department for Education and in schools.

Last October, the Prime Minister said the government's school reforms would lead to "personalised lessons" for pupils. The then Education Secretary, Ruth Kelly (my, how fast they change!) characterised the reforms as being about "personalisation and choice".

So, everyone is talking about it. It will eventually affect every child in a school. All teachers will have to learn how to teach it. But what does it mean?

Ask professional educators and you might get an answer like this: "Personalised learning is about learner-managed and co-constructed learning -- the shift from dependency to independence and interdependency -- and invitational learning and assessment."

I took this from a website dedicated to personalised education. If you can make sense of it, you are a much better person than me.

It also talked about the "re-integration of learning, life and community", making use of "catalogue and natural versions of curriculum and assessment" and "de-coupling of age-stage progressions".

That clears it up.

Up is down

One of the panel members of Bush's new National Mathematics Advisory Panel enters another dimension:

A very short story by Vern Williams

One night I walked into the 4 3/8 dimension and actually believed the following:

We should write about math but never do math.
Correcting students' papers using red ink is a threat to children's self esteem and that red pens should be banned from all public schools.
Howard Gardner was right about his multiple intelligence theory (I think that he claims about nine at the moment) and that schools should value bodily-kinesthetic ability and the intelligence of self as much as mathematical and linguistic ability.
The war on intellectual excellence is a great thing. It will make us all equal.
Teachers Unions are actually concerned about students.
Advanced courses and gifted programs should be banned because they are elitist and unfair. Since everyone is gifted in their own way (see Howard Gardner), why have special gifted programs?
There are no bored students in US public schools.
We can teach thinking even when there is no content to think about.
We should treat members of politically protected minority groups as victims.
We should never view our students as individuals but as members of racial and ethnic groups.
We should buy into the latest educational fad even if it's based on political correctness and has nothing to do with learning or common sense.
There is no money wasted on administration, specialists, and useless programs. In fact, we should have more of each.
I should join the NCTM.
I should join the NEA.
I should feel guilty because I teach smart kids.
I should feel really guilty because I enjoy teaching smart kids.

I finally woke up in a cold sweat from this nightmare and asked myself does anyone actually believe those things? The answer is a resounding yes. Unfortunately the people who believe them are running our school systems and colleges of education.

I do my very best to shield my students from the effects of educational fads, political correctness and anti-intellectualism that we experience every day in public schools.

Small schools malaise

Somehow I have the feeling that academics isn't high up on the agenda of this new small school, despite its lip service. This is a pattern with a lot of these new small schools. Their motto should be: Anything but academics.

The Green School is a New Century High School opening in September 2006 in Williamsburg, Brooklyn, New York City.

Our mission is to foster community values in sustainability; specifically self, relationships, community and society, while preparing students for Regents exams, college and careers.

Our School themes: Sustainability • Real-World Learning • Student Voice • Careers that Make a Difference

Strong applicants for a teaching position will demonstrate the following abilities to: • Link subject area curricula to the school’s themes and incorporate discussion of current events, hands-on experiences in the community, and meaningful connection to students’ interests and lives. • Create project-based curricula and use performance-based assessment. • Use inquiry-based approach to teach interdisciplinary classes. • Teach an advisory class that includes community building, identity development, reflective writing, and interest exploration. • Work collaboratively with other teachers to design curricula and assessments, and to support each other in improving teaching practices. • Involve students in decision making about topics of classes and electives, and support students in independent projects. • Work to develop internship opportunities for students around the theme of sustainability. • Design and teach curricula to heterogeneous classes including English language learners, special needs students, and accelerated students, together. • Will participate in developing and implementing new ideas into the school or “wearing more than one hat” (this is key in small schools). • Help plan and participate in field trips of various duration. • Develop curricula that uses the resources of NYC and gets students to apply their learning in real-world settings.

Febrile in Seattle

The Seattle Public Schools system has adopted a set of definitions dripping with far-left wackiness that purport to identify various forms of "racism". "Racism" itself (without a modifying adjective) apparently cannot be perpetrated by groups which in the view of the Seattle Public Schools have "relatively little social power," identified as "Blacks, Latino/as, Native Americans, and Asians." This would seem to exempt a sizable chunk of the population (known in the PC vernacular as "people of color") from being capable of committing this hideous crime. One would have thought that "racism" (without the adjectives) is a state of mind. Apparently it is an activity:

Racism:
The systematic subordination of members of targeted racial groups who have relatively little social power in the United States (Blacks, Latino/as, Native Americans, and Asians), by the members of the agent racial group who have relatively more social power (Whites). The subordination is supported by the actions of individuals, cultural norms and values, and the institutional structures and practices of society.

The next entry is on "individual racism." Just when I thought that individuals of all colorations are capable of "racism" after all, it turns out that, besides "telling a racist joke, using a racial epithet," the miscreant must believe in the "inherent superiority of whites." So, unless members of "targeted racial groups" perversely believe in such superiority, they are off the hook once again:

Individual Racism:
The beliefs, attitudes, and actions of individuals that support or perpetuate racism. Individual racism can occur at both an unconscious and conscious level, and can be both active and passive. Examples include telling a racist joke, using a racial epithet, or believing in the inherent superiority of whites.

But perhaps matters are even more weirdly complex than that. Perhaps members of "targeted racial groups" cannot be individuals at all, for we learn under "cultural racism" that the emphasis on "individualism" is also "racist". Since "targeted racial groups" cannot be "racist," they thus cannot hold "racist" beliefs such as a belief in the individual. Following this dizzying Seattle Public Schools logic, "targeted racial groups" might not exist as individuals at all:

Cultural Racism:
Those aspects of society that overtly and covertly attribute value and normality to white people and Whiteness, and devalue, stereotype, and label people of color as “other”, different, less than, or render them invisible. Examples of these norms include defining white skin tones as nude or flesh colored, having a future time orientation, emphasizing individualism as opposed to a more collective ideology, defining one form of English as standard, and identifying only Whites as great writers or composers.

Another revelation is that planning ahead (what these educationists call "having a future time orientation") is apparently also "racist". Now, believing that planning ahead is a function of pigmentation strikes me as truly racist. But these educationists are probably too dim-witted to realize this.

It should be noted that these dogmas have been far-left fare for a long time. They are not recent creations of Seattle schools. The inspiration listed as a source is Teaching for Diversity and Social Justice, 1197 eds. Adams, Bell & Griffin. Still, it is astonishing that a tax-financed, public entity would adopt a lunatic fringe creed as its guiding policy. An incidental benefit is that it becomes a little clearer what the proponents of "social justice" have in mind.

Nutty judge

The Los Angeles Times reports that a California judge thinks it is unfair to ask high school graduates to know some 8th grade math and ninth- and 10th-grade English since some went to lousy schools:

A California judge struck down the state's controversial high school exit exam Friday, potentially clearing the way for thousands of seniors who have failed the test to graduate with their class next month.

Alameda County Superior Court Judge Robert B. Freedman issued a preliminary injunction against the mandatory testing requirement, ruling it places an unfair burden on poor and minority students who attend low-performing schools.

Even answering little more than half the questions on the test is an onerous requirement:

This year's 12th-graders were the first class to face the testing requirement, which includes a section of eighth-grade math and another of ninth- and 10th-grade English. Students are required to answer little more than half of the questions correctly and can take the test multiple times. Students with learning disabilities were exempted from the test.

The judge's insistence on giving ignorant students a diploma confuses symbolism and substance. What good is a diploma if it does not stand for something? And how much are students helped if educational failure is shoved under the rug?

Political litmus test for math teacher

A Chicago public school is seeking a math teacher who needs to have a "social justice background" (whatever that is) of all things to qualify. Political indoctrination at taxpayer expense.

Cluster/Area 04/25
School Name/Address Greater Lawndale / Little Village School for Social Justice
3021 S. Kostner
Chicago, IL 60623 (or GSR #37)
Grade or Subject Mathematics
Certificate Requirements (Type 09) 6-12 w/Mathematics Endorsement

Other Information: Progressive educators with social justice background. Must be willing to create alternative assessments and work collaboratively.
School uses IMP curriculum.

UPDATE: See Darren's analysis of social justice math in the Comment section.

Farther shores

In Silly Season for the School Scholars, Frederick Hess and Laura LoGerfo report on the doings at the annual meeting of the American Educational Research Association (AERA).

Will this organization drown in its own irrelevance?

Here is a sampling:

One scholar of multiculturalism showed how to do away with injustice and racism, while promoting compassion and wisdom, in “Resisting Resistance: Using Eco-Justice and Eco-Racism to Awaken Mindfulness, Compassion, and Wisdom in Preservice Teachers.”

Other work promised to promote proper multicultural teacher attitudes: as with “Teaching White Preservice Teachers: Pedagogical Responses to Color-Blind Ideology” and “Overcoming Odds: Preparing Bilingual Paraeducators to Teach for Social Justice.” Breakthrough research on this front included “Discovering Collage as a Method in Researching Multicultural Lives” and “Artistic Code-Switching in a Collaged Book on Border Identity and Spanglish.”

Among the panels tackling the pressing questions of “queer studies” (formerly “gay and lesbian studies”) were “Queering Schooling and (Un)Doing the Public Good: Rubbing Against the Grain for Schooling Sexualities,” “The Silence at School: An Ethnodrama for Educators About the School Experiences of Gay Boys,” and “Working Against Heterosexism and Homophobia Through Teacher Inquiry.” Unfortunately, this work may have felt a bit conventional to those researchers fortunate enough to catch the 2004 analysis of ableist oppression in homoerotic magazines: “Unzipping the Monster Dick: Deconstructing Ableist Representations in Two Homoerotic Magazines.”

Reading Moloch

The NYT reports how, driven by NCLB, math and reading is driving out other subjects like science and history.

SACRAMENTO — Thousands of schools across the nation are responding to the reading and math testing requirements laid out in No Child Left Behind, President Bush's signature education law, by reducing class time spent on other subjects and, for some low-proficiency students, eliminating it.

Schools from Vermont to California are increasing — in some cases tripling — the class time that low-proficiency students spend on reading and math, mainly because the federal law, signed in 2002, requires annual exams only in those subjects and punishes schools that fall short of rising benchmarks.

I find it curious that educationists regard "reading" and learning science and history as being mutually exclusive. "Reading" is not some abstract, isolated skill but a practical tool that can be applied to many fields. Couldn't you learn a lot of history and science by reading? Whatever happened to reading across the curriculum?

In The Knowledge Deficit , E.D. Hirsch argues that reading instruction should be less concerned with "strategies" and should focus more on domain knowledge:

From Publishers Weekly
The notion of learning how to learn is a shibboleth in America's schools, but it distorts reading instruction, contends this provocative manifesto. Education theorist Hirsch decries a dominant "Romantic" pedagogy that disparages factual knowledge and emphasizes reading comprehension "strategies"—summarizing, identifying themes, drawing inferences—that children can deploy on any text. Such formal skills, he argues, are easily acquired; what kids really need is a broad background knowledge of history, science and culture to help them assimilate new vocabulary and understand more advanced readings. "Process-oriented" methods that apply reading comprehension drills to "vapid" texts waste time and slow kids' progress, Hirsch contends, and should be replaced with a more traditional, "knowledge-oriented" academic approach with a rich factual content. Hirsch repeats the call for a standard curriculum based on a canon of general knowledge (he touts his own core knowledge sequence as a model) made in his bestselling Cultural Literacy. That work drew fire from multiculturalists who accused Hirsch of promoting dead-white-male worship, but here he grounds his case in the latest cognitive-science research (with a healthy dose of common sense). Fluently written and accessible to teachers and parents alike, the book presents a challenge to reigning educational orthodoxies. (Apr. 24)

Humor

Teacher Arrested

At New York's Kennedy airport today, an individual later discovered to be a public school teacher was arrested trying to board a flight while in possession of a ruler, a protractor, a set square, a slide rule, and a calculator.

At a morning press conference, the attorney general said he believes the man is a member of the notorious Al-gebra movement. He is being charged by the FBI with carrying weapons of math instruction.

"Al-gebra is a fearsome cult," a Justice Department spokesman said. "They desire average solutions by means and extremes, and sometimes go off on tangents in a search of absolute value. They use secret code names like 'x' and 'y' and refer to themselves as 'unknowns', but we have determined they belong to a common denominator of the axis of medieval with coordinates in every country.

As the Greek philanderer Isosceles used to say, "there are 3 sides to every triangle."

When asked to comment on the arrest, President Bush said, "If God had wanted us to have better weapons of math instruction, He would have given us more fingers and toes."

Small is radical left

The promise of the small-schools movement was to create manageable, cozy entities as an alternative to the mammoth high-school jungles. Now it turns out that small schools are a vehicle for the implementation of a radical left agenda.

See, for example, this mission statement from a new small school being set up in Chicago that goes by the inspiring name of UPLIFT:

MISSION: Our mission is to continue to adapt and align our curriculum so that it is relevant, student-centered and adheres to the highest national standards. We will transform service delivery so that the theme of social justice is embedded in all subject areas.

Perhaps none of this is surprising since the father of the movement is none other than an SDS (Weathermen) fugitive turned Distinguished Professor of Education at the University of Illinois, Chicago.

Examples of other small schools turned into radical indoctrination centers abound as chronicled by Sol Stern.

Kozol made him do it

Sol Stern discusses how a quasi-official pedagogy permeating pre-collegiate education promotes political indoctrination:

At least the higher education professoriate denies that it favors using the classroom as a political bully pulpit. By contrast, the K-12 public school establishment has adopted a quasi-official pedagogy that encourages the classroom teacher to shape students’ beliefs on controversial issues like race, gender, sexual preference, and American foreign policy.

The documentation on this is so extensive that Jay Bennish might have a pretty good Nurenberg defense: “my union and my professional teacher association made me do it.”

For example, the National Education Association, the larger of the two national teacher unions, supports “the movement toward self-determination by American Indians/Alaska natives” and believes these designated victim groups should control their own education. It believes that all schools should designate separate months to celebrate Black History, Hispanic Heritage, Native American Indian Heritage, Asian/Pacific Heritage, Women’s History, Lesbian and Gay History. This nearly takes up the entire school calendar, leaving scant time for American history – or Geography, the subject that Mr. Bennish was supposed to be teaching when he went off on Bush and Bush’s Amerikkka.

After 9/11, the NEA posted guidelines on how teachers should discuss with their students the terrorist attack on our homeland. It was filled with multicultural psychobabble and stressed the need for children to be tolerant and to respect all cultures – while hardly saying a word about the fact that the country was at war with a vicious enemy out to destroy our tolerant society. The document came so close to apologizing for the 9/11 attack that a public outcry ensued, and the union was forced to remove the teacher guidelines from its website.

NEA-affiliated teacher organizations, such as the National Council of the Social Studies and the National Council of Teachers of English, carry on the political struggle by training teachers to focus inordinate attention in the classroom on issues of “diversity.” The NCSS believes that academic history – which some of its leaders have disparaged as "pastology" – is elitist and irrelevant. The organization has successfully lobbied state education departments to require little or no history. Instead, it has filled the schools with a hodgepodge of "global studies," "cultural studies," and "peace studies" that present all cultures and civilizations as equal in value.

If NCSS had its way, American education’s entire system would reflect a race- and gender-centered pedagogy. The organization's official policy paper, "Curriculum Guidelines for Multicultural Education," is one of the scariest documents in American education today, going far beyond the demand that social studies curricula reflect the grievances of a rainbow coalition of ethnic and racial groups. In the tone of a commissar's lecture at a political reeducation camp, the NCSS exhorts teachers, administrators, and other school employees to think and act multiculturally during every moment of the school day, lest they become accomplices of American culture's invisible but omnipresent racism. Teachers are instructed to scrutinize every aspect of the school environment – from classroom teaching styles and the pictures on the walls to the foods served in the lunchroom and the songs sung in the school assemblies – to be sure they reflect "multicultural literacy."

At the heart of the NCSS paper lies a fundamentally racist assumption: "[T]he instructional strategies and learning styles most often favored in the nation's schools," the guidelines declare, "are inconsistent with the cognitive styles, cultural orientations, and cultural characteristics of some groups of students of color." These students flourish under "cooperative teaching techniques" rather than the "competitive learning activities" that work for white kids.

We are left with this Orwellian conclusion by the Social Studies group: "Schools should recognize that they cannot treat all students alike or they run the risk of denying equal educational opportunity to all persons."

Spelling helps reading

Among the countless educationist lunacies is the belittling of correct spelling. This
article from American Educator shows that correct spelling supports reading.

A different take on "constructivism"

A reader of this site called Allison left a comment on my piece on arrested development that I found fascinaning. She describes how Husserl of phenomenology fame views constructivism and authentic learning. I have to admit, though, that I had hitherto been unaware that educationists have ever heard of Husserl, let alone have sufficient knowledge of phenomenology to distort it to absurd lengths:

How sad that the constructivists in education misused the ideas from such wonderful philosophers as Husserl and Wittgenstein.

Husserl, in particular, was terrified that within a generation, all modern knowledge could be lost. From his perspective, we were standing on such a tower of shoulders of giants that we could fall due to some calamity (war, plague, etc.) and we couldn't even reconstruct the society we'd had before. So he set out to find the "Authentic description" for things--for concepts, ideas, words, traits, algorithms, experiences. He was trying to write down a body of knowledge as best as possible so that we wouldn't have to start over with a blank slate.

This idea of his led him to be one of the founders of phenomenology, a philosophy much maligned for many unfair reasons. Between Husserl and Heidegger, phenomenology came up with any explanation for learning called the Hermeneutic circle, which explains that constructivism is a necessary component for authentic learning.

But in the circle, all of the rote learning is a REQUIREMENT before the constructivist reaching BECOMES authentic.

Here's an example: At first, you don't know things like your times tables. You don't know 6 times 7 off the top of your head. You must inauthentically calculate it, say by adding 6 7s. You are unsure of your answer, maybe. (And you have yet to REALLY be convinced that adding 7 6s produces the same answer.) You have doubt still. Over time, though, you learn your times tables by rote (still inauthentically at first) because you are forced to. So now when asked, 6 times 7 is 42 AND 7 time s 6 is 42. You don't think about why; it's just the rule.

Eventually, you learn the tables so well that they become known to you, and you have no doubt that 6 times 7 is 42.

Now, you start working on another problem: 2 times 21. Now, this isn't in your times table. You have doubt; you are forced to try and discover something you DO know that helps you solve the problem. In doing so, you may learn something fascinating: that 2 times 21 is 2 times 3 times 7. This may be one of the first times that you've even NOTICED factors before. You finally, unsurely at first, guess that maybe 21 times 2 is 6 times 7, because 2 times 3 is 6.

But now, you're beginning to guess something FASCINATING: that factors are associative! This is still unsteady to you, so you fall back on the KNOWN, the rote: and you start examining other multiples: 3 times 14, for example. lo and behold, this is 3 times 2 times 7!

This is an example of the hermeneutic circle at work: every time you learn something inauthentically, it becomes the basis for a future authentic learning. All learning is predicated on prior learning--and ironically, predicated on "learning" in such a way that you even FORGOT that it was strange that you knew that fact, and yet, this time around, that fact you were convince of, leads you to an A-Ha! you never saw before.

And over time, you know these truths so deeply that you KNOW all numbers have prime factorizations; then at some later layer, you understand the beauty of diophantine equations because of what you've "always known" about prime factorizations, etc.

So the original constructivists, who were trying to get at authentic learning, which always involves moving into the unknown, understood that you must ALWAYS predicate that unknown on the known. (In fact, ask a phenomenologist what the bottom layer of that predication is, and he'll probably tell you something fascinating: the top!)

Tutoring

Misguided pretentiousness and false rigor

The Chicago Public Schools are pushing something called the Chicago Math & Science Initiative

I checked out what is supposed to be taught in sixth grade and found that one quarter is devoted to this:

In Investigating Climate and Weather students begin by conducting a series of inquiries to connect their own experience of climate and weather to the planetary systems that govern weather events and climate change. Next, they investigate the evidence and associated scientific debate surrounding climate change. From their evaluations of this evidence, students predict climate and weather changes for the area in which they live.

I can't help it but somehow I find the idea that sixth grade pupils in Chicago schools will conduct a "series of inquiries to connect their own experience of climate and weather to the planetary systems that govern weather events and climate change" utterly laughable. Most of these kids can't even spell atmosphere and wouldn't know troposphere from ionosphere. Try throwing "adiabatic" at them (a reversible thermodynamic process executed at constant entropy and occurring without gain or loss of heat). Next they will "investigate the evidence and associated scientific debate surrounding climate change." For sure!

Weather systems and factors contributing to climate are some of the most complex things imaginable requiring an advanced and sophisticated store of knowledge. The top brains in science can't even come up with computer models that can fully account for this complexity. But Chicago pupils who lack even the rudiments of science will somehow pore over scientific papers and make predictions. This is delusional to the point of being comical!

What these students need is a systematic, coherent and age-appropriate grounding in major topics of science (physics, chemistry, earth science, biology, etc.) with increasing sophistication as they advance through the grades.

Instead of realistic, specific content goals for each grade, the Illinois state board presents vague and highly pretentious "descriptors" focused entirely on process and "inquiry". These "descriptors" are essentially the same for babes and high-school seniors and everything in between. What "content" knowledge requirements exist are rather vague, skimpy and applied to broad grade ranges.

The following is an excerpt from the Illinois Learning Standards: Classroom Assessments and Performance Descriptors:

Here is a portion of the "descriptors" for FIRST AND SECOND GRADE!!!

Descriptors
11A - Students who meet the standard know and apply the concepts, principles, and processes of scientific inquiry.
1. Describe an observed science concept using appropriate senses, making applicable estimations and measurements, predicting steps or sequences, describing changes in terms of starting and ending conditions using words, diagrams or graphs.
2. Begin guided inquiry asking questions using prior knowledge and observations, inferring from observations to generate new questions, or developing strategies to investigate questions.
3. Conduct guided inquiry following appropriate procedural steps and safety precautions as directed by teacher.
4. Collect data for guided inquiry identifying and using instruments for gathering data, making estimates and measurements, recording observations, or reading data from data-collection instruments.
5. Record and store data assembling pictures to illustrate data, or organizing data on charts and pictographs, tables, journals or computers.
6. Analyze and display results recognizing and describing patterns, noting similarities and differences in patterns, or predicting trends.
7. Communicate individual and group results identifying similar data from others, generalizing data, drawing simple conclusions, or suggesting more questions to consider.
11B - Students who meet the standard know and apply the concepts, principles, and processes of technological design.

Revised whole language golf instruction

The creators of a golf instruction program that applies modern educational theories have been forced to update their program, reports Kerry Hempenstall in Balanced golf instruction.

Here are some elements of the revised program:

We are also conscious of the developing golfer's learning style. We advise the visual learners to focus their learning transactions on watching the golf on TV (with the sound off) at every opportunity. The auditory learners actually go to golf courses, but wear blindfolds - better to focus attention on the sound of the ball being struck. They also make use of brain-based golf education employing a looped audiotape of a ball being struck. When played during sleep, this procedure repatterns the golfing region of the brain for these fortunate students. The kinaesthetic learners must actually swing the club regularly, but their oneness with the game is dramatically enhanced when they cannot see the ball. The feel is the thing. Because our teachers are so skilled they are also able to use multiple methods, tailor-made to parallel each of the multiple golf intelligences our students may display.

You can read about the original WL golf instruction program here.

MI disorder

A teacher in Denver writing in the NYT education forum illustrates the absurd lengths to which Gardner's so-called multiple intelligences are taken by educationists:

While you are on this one, my American History class is heterogeneous.

We know that there are seven separate and distinct learning modalities.

Kindly tell us how you will differentiate your teaching so that the tactile, auditory, visual, and kinetic [sic] learners will come away with the same knowledge as the readers and writers.

I would like to see how this teacher would teach history in the purely tactile and kinesthetic mode. At some point there has to be exposure to words, either by reading or listening. It's also dubious that people can be pigeonholed into one mode to the exclusion of other modes.

I also find it ironic that on the one hand educationists rail against expository instruction and on the other hand they show concern for "auditory" learners.

See here for a link to Willingham's debunking of modality theory. In Willingham's priceless observation, students learn best when content is presented in the subject's best modality.

http://www.aft.org/pubs-reports/american_educator/summer2002/curriculum.pdf

Superficial ed reporting

Fuzzy math enthusiasts like to justify their enthusiasm by invariably pointing to supposedly towering test scores attributed to fuzzy math. This self-congratulation then gets reported uncritically in the media. See, for example, Everyday Math multiplies in schools:

But administrators at Clayton and Edwardsville stand by the program.

"Look at the scoreboard," Keenoy said, referring to Clayton's high test scores.

Edwardsville has seen its math scores jump since it started the program in 2001 by up to 20 percent in a grade level, some of the highest overall in the area. And the number of students taking higher math classes in high school has nearly tripled, said Lynda Andre, assistant superintendent for instruction.

What education reporters fail to do is ask a series of critical questions and do some investigating. For example, reporters could ask: What math questions are on these tests? Are these tests possibly geared to fuzzy math? Is there math instruction outside of fuzzy math, e.g. from parents or tutors?

The article does point out that there is supplementary math instruction.

Both districts have addressed some of the perceived shortfalls by requiring teachers to supplement the program with timed tests and exercises on basic math facts.

Could that and other unreported factors like parent involvement and tutoring contribute to the supposedly high tests scores attributed to fuzzy math?

Math puzzle

I found this math problem at Kitchen Table Math.

"769. An old brainteaser by Leonty Magnitsky: If a man drinks a barrel of water by himself in 14 days and the same barrel with his wife in 10 days, how many days would it take his wife to drink the barrel by herself?"

First of all, I find this easier to solve when the wording is changed to a barrel of vodka.

Here is my solution:

The sum of the fractional parts (man and wife) of drinking in one day is equal to the fractional part of both drinking in one day:

Let x = days wife takes to consume the whole barrel

1/14 + 1/x = 1/10

Multiply each side by the LCD to clear fractions:

10x + 140 = 14x
4x = 140
x = 35

Eduvacuity

The Fordham Institute has published a major review of state science standards. According to the study, a handful of states has managed to produce respectable standards. However, the vast majority of state standards is afflicted with numerous problems, most notably with the "constructivist" anti-knowledge plague:

Educational Constructivism
Constructivism is not new. It was evident in the first draft
(1992) of the National Science Education Standards, where
it took the form of a claimed postmodern philosophy of science. That, in turn, incorporates one kind of constructivism (“social” constructivism) about knowledge, including scientific knowledge.The adopted philosophy was an application to learning standards of the increasingly popular educational constructivism,whose main tenet is that learning happens only by an individual’s action, his or her making and doing things in the world, not as a result of any conveyance of knowledge (as in teaching).10 A revision of that early draft eliminated the praise of postmodernism but left in place the notion that a learner can do no more than to construct knowledge, which is therefore personal, from things and events in his or her sensed environment. It is supposed to follow from this that scientific knowledge cannot be transferred from one person—a teacher (or from a book)—to another. The learning expectations of standards should therefore focus much more on process, the “doing”of science by the student, and much less on its reputed facts.11

The California science framework has received top marks and can be read here.

"Whole language" appreciation

Loony education theories can have unintended consequences: They can be a rich source of hilarity. So it is with the psychotic guessing game promoted by whole languagists.

Here, a writer gives thanks to the whole panoply of whole language inventors and promoters:

Thank you Whole Language. Thank you for your many pearls of wisdom. Thank you for Context Clues. Thank you for Prior Knowledge. Thank you for the Initial Consonant. Thank you for Picture Clues. Thank you for Miscues.

But most of all, thank you for my wife. The other day she and I were riding along the highway and saw a sign for a town called Verona, so my wife read "Veronica". It's very simple, you see. First she applied Context Clues (she knew we were looking for a name). Then she applied the Initial Consonant ("V"). Then she applied Prior Knowledge (she already knew of a name "Veronica"). She put these Whole Language strategies together and ... success! At least, as much success as we can expect, I suppose.

Thank you William S. Gray for inventing "Look-Say" and the "Dick and Jane" series of basal readers. Thank you A. Sterl Artley for helping Mr. Gray and for your phonics-bashing diatribes of the 1950s and 1960s. Thanks to the National Education Association for giving Mr. Gray and his friends two years of free promotion in the NEA Journal in 1930 and 1931. Together you all had managed to essentially eradicate phonics from America's public schools by the 1950s and early 60s, when my wife went to school.

But more importantly, thank you for my wife. A while back she was reading a pamphlet about something that was described as "venerable". Now that's a word you don't see every day, so what did she do but cleverly pull out her Whole Language skills? Context Clues, you see, told her that she was looking for an adjective. Next was the Initial Consonant "V". Then out came the Prior Knowledge -- she simply thought of an adjective she already knew that was about the right length and started with "V". And voila ... success again ... she came up with "vulnerable". Perfect! Well, at least as perfect as things get in publik ejukayshun, right?

Thanks Kenneth Goodman for reviving the floundering Look-Say, adding a few New Age twists and renaming it Whole Language back in the early 80s. Just like the Whole Earth Catalog and Whole Grains and everything else that was Whole ... what else could it be but wonderful? Without you, Kenneth, the evils of phonics might have returned, and then where would we have been?

Thank you Dorothy Strickland for "Emerging Literacy" -- the idea that kids are naturally inclined to read if only we will surround them with literature. Thanks to all the other Whole Language textbook authors who cranked out textbook after textbook that either omitted phonics entirely or disparaged phonics openly. Thank you Teachers College, Columbia, for promoting Whole Language to teachers' colleges worldwide. Can you even imagine how effective you were in eradicating phonics instruction throughout the English-speaking world?

Thank you International Reading Association (IRA) and National Council of Teachers of English (NCTE). For decades you appointed people like William S. Gray and Kenneth Goodman to lead your entire organizations in the fight against phonics. Somehow you raised hundreds of thousands or perhaps millions of dollars to pay PR firms to get their opinions so heavily quoted in the press that the public is now completely confused in its ideas about what works and what doesn't work in reading instruction.

But once again thank you for my wife. A while back she was reading about some Congregational Church. And do you know, even with the Context Clues and the Prior Knowledge (about what names churches might have, presumably) and the Initial Consonant, she still managed to come up with "Congressional Church". Even though this was years ago, I remember it like it was yesterday.

Thank you Alfie Kohn and Dennis Baron and Mike Ford and Gerald Coles and Harvey Daniels and Gerald Bracey and Susan Ohanian and Stephen Krashen and Jim Trelease and all the other propagandists who lash out continuously against successful practice in general and phonics in particular. Through your tireless efforts, the public is continually misinformed. Without the public's perpetual state of confusion and misinformation, Whole Language would not have survived a single day. Thank you for keeping Look-Say and Whole Language and Balanced Literacy alive to create yet another generation of people who can read as well as my wife does.

Speaking of my wife, last night she was reading a brochure aloud about a museum with an "eclectic" collection, and what do you suppose she said? You guessed it (and so did she): "electric"! Maybe the absence of the Initial Consonant threw her off.

Thank you Marie Clay for inventing the phenomenally expensive Reading Recovery, a program installed in virtually every public school, it seems, and designed to treat the educational effects of Whole Language by applying yet more Whole Language. Thank you for giving my school district more stuff like this to spend my tax money on. How is it that I am not clever enough to imagine things like this?

Thank you Richard Allington, current (2005) president of the International Reading Association, for your campaign of misinformation against Direct Instruction (a successful phonics-based program). The cleverness of your propaganda puts the Soviets, the Chinese Communists, and all the other tyrants of the 20th century to shame. You know of course that Direct Instruction (DI) participated in a huge study (Project Follow Through) in which all the participants except DI failed, and in which DI succeeded brilliantly. And so you twist this around to say that by virtue of its association in this study with the constructivist-favored instructional styles that failed so miserably, we should all conclude that DI must necessarily also have been a failure. Your logic, so typical of that of the IRA, the NCTE, and the rest of the Constructivist Cabal, is irrefutable.

But once again thank you all for my wife. Hardly a day goes by when she does not demonstrate the success of Look-Say, or Whole Language, or Balanced Literacy or whatever you all call it now. Really, it's so amusing I really can't even quantify it. I never know what she'll read next ... and neither does she! Just imagine all her Miscues!

The sheer unpredictability of listening to her read is astounding ... and unpredictability is the essence of entertainment, right? I mean, she might read "deleterious" as "delicious" or perhaps "injurious" as "injustice" or "parabola" as "parachute" or maybe "quintessence" as "quintuplet", or "signify" as "signature". I could go on and on almost endlessly. The laughs just never stop here. And all thanks to you. All of you.

So thank you, Whole Language. Where would we be without you? The possibilities just boggle the mind.

-- Anonymous

[Note: This author normally signs his work, but in this case declines because he doesn't want his wife identified in this manner.]

The e-learning route

High-ability students who are languishing in a regular school setting have a chance to pay big bucks and move as fast as they can for credit. All this thanks to Stanford's EPGY program.

Flexible inefficiency

The National Education Association helpfully offers A Parent's Guide to Helping Your Child with Today's Math but then only goes halfway as this example illustrates:

A farmer sends his daughter and son out into the barnyard to count the number of chickens and pigs. When they return the son says that he counted 200 legs and the daughter says she counted 70 heads. How many pigs and chickens does the farmer have?

A student well versed in algebra might do the following to set up the problem: p = pigs, c = chickens. p + c = 70 (heads) 4p + 2c = 200 (pigs have 4 legs and chickens have 2 legs). These two equations may be used to solve the problem. Students might solve this problem by "guessing and checking," or drawing pictures. Some methods of solving problems might be considered more "efficient." That may be true, but the correct answer can be found using multiple methods. Children think about mathematics in different ways depending on their prior experiences at home and school. By allowing students to think flexibly about numbers, we encourage them to "own" the math forever, instead of "borrowing" until class is over. (Answer: 40 chickens and 30 pigs)

Constructivism is rearing its ugly head again. With this kind of help, students are not going to "own" the algebraic method of solving this problems. They might not even "own" the inefficient guess-and-check method. If you are being taught properly and practice sufficiently to the point of overlearning, then you are more likely to "own" math knowledge than when left groping in the dark.

"Learning specialists" alarmed by academic achievement

What's the problem with schools? You guessed it! Students know too much. At least that's the concern voiced by three "learning specialists" (an education consultant, a professor emeritus at an ed school and a professor at a law school) in a commentary published in EdWeek.

In what sounds like a three stooges routine, the three "learning specialists" are alarmed by what they see as a trend toward academic achievement:

Perhaps now is a good time to ask this question: What are schools supposed to do for our children? As learning specialists, we see an alarming trend: Our education system increasingly is focusing not on developing children’s aptitude for learning—their ability to absorb new information quickly and solve problems creatively—but on their academic achievements—their mastery of particular subjects and skills as proven by performance on standardized tests.

Silly me. I would have thought that academic achievement demonstrates at least an "ability to absorb new information quickly," the purported goal of these specialists.

The authors of the anti-achievement piece do manage to offer "sobering" examples of imperial decline due to memorization and academic achievement. The cause-and-effect scenario painted here does seem a bit fishy to me. I kind of doubt that China went into a tailspin because a few mandarins had the ability memorize Confucian philosophy. Would we as a society suddenly have to live in caves in the unthinkable event that some of our bureaucrats (say, at the board of ed) suddenly had the urge to memorize a few poems by Whitehead and Tennyson?

Snippet from the trio's commentary:

This is a serious concern for our kids and our society. History offers sobering examples of what can happen when standardized achievements are elevated over open-ended abilities.

During the 18th and 19th centuries, imperial China—once the most technologically advanced civilization in the world—fell into decline as power passed into the hands of a mandarin class of bureaucrats selected for their ability to memorize Confucian philosophy. More recently, Japanese authorities have begun dismantling an education system that long relied on a uniform national curriculum and after-hours classes at juku “cram schools.” The Japanese believe this approach has stifled creativity, innovation, and independent thinking, contributing to the stagnation of the Japanese economy.

We worry that America is heading down a similar path. If promoting our children’s achievements becomes our sole focus, both our children and our society will suffer.

The essence of the anti-academic achievement position as far as I can distill it is a false dichotomy between learning academic subject matter on the one hand and "creativity, innovation, and independent thinking" on the other. Educationists worship at at the altar of ignorance in the name of "creativity" but ignorance is not a prerequisite for "creativity".

"Progressive" education impedes progress

In Why Traditional Education Is More Progressive, E.D. Hirsch makes a strong and plausible case that the dominant educational creed defeats the purported aims of its advocates:

I would label myself a political liberal and an educational conservative, or perhaps more accurately, an educational pragmatist. Political liberals really ought to oppose progressive educational ideas because they have led to practical failure and greater social inequity. The only practical way to achieve liberalism’s aim of greater social justice is to pursue conservative educational policies.

In combating the dominant ed creed, propopents of quality education must have a developed sense of how the language employed by the dominant ed creed is manipulative, charged, disparaging and stacked against them. Some of the phrases that come to mind are "sage on the stage, guide on the side," "chalk and talk," "drill and kill" "teach the whole child," "teach the child, not the subject," "less is more," "up is down," "freedom is slavery" (well, I made up the last two or, rather, stole the last one from Orwell). There is nothing comparable for those who value expository instruction and domain knowledge.

[Fortunately, not all educationist phrases are catchy. There is also some pretty dreary stuff. I culled this from a constructivist site: "This article discusses several active-learning techniques that instructors can use to help students construct knowledge, such as think-pair-share, guided reciprocal peer questioning, jigsaw, and co-op co-op."]

Other words deployed by the dominant ed creed to disparage the notion of imparting knowledge are "lecture", "active" and "passive". "Lecture" has the negative connotation of droning on without regard for the audience's (in this case the pupils') level of understanding and its capacity to follow while the captive audience sits by "passively". No teacher worth anything would teach that way. (Adherents of the dominant ed creed forget that listening attentively to explicit instruction is also being active, but the followers of the creed claim to have a monopoly on "active").

But the disparagement of any explicit instruction by labeling it "lecture" is so strong that explicit instruction is proscribed in many places. That is detrimental to achieving quality education. For example, good math instruction should consist of modeling (interactive modeling if appropriate) followed by guided practice, independent practice and review. In other words, there must be talk as required by the circumstances. And how would one teach history and other subjects without talking?

I coined the impressive, albeit cumbersome, phrase "empathetic, interactive expository instruction" as a counterpoint to the dreaded "lecture". Alas, it can't compete with the catchiness of "talk and chalk" or "drill and kill". (Now I have to put my phrase in rhyme form).

I include "empathetic" in my phrase to stress the importance of discovering and being being sensitive to Vygosky's fabled zone of proximal development (a jazzed-up way of saying that our instruction must be geared to the pupils' ability and level of comprehension).

In his article, E.D. Hirsch takes apart some of the conceits of educationists:

Unfortunately, many of today’s American educators paint traditional education as the arch-enemy of "humane" modern education. Even everyday classroom language unfairly pits the two alternatives against one another. Here are some typical descriptions used by progressives to compare old and new methods:

Traditional vs. Modern
Merely verbal vs. Hands-on
Premature vs. Developmentally appropriate
Fragmented vs. Integrated
Boring vs. Interesting
Lockstep vs. Individualized

Parents presented with such choices for their children’s education would be unlikely to prefer traditional, merely verbal, premature, fragmented, boring, and lockstep instruction to instruction that is modern, hands-on, developmentally appropriate, integrated, interesting, and individualized. But of course this is a loaded and misleading contrast. Let’s look at those simple polarities one at a time.

You can read the entire article here.

Math-challenged child does CMP

Fuzzy math is being rammed down people's throat and reasonable requests to provide an alternative are being denied.

From the NYT's Penfield article:

By last spring, these parents had discovered one another and their common exasperation with constructivist math. Jim Munch's father, Bill, a software developer at Kodak, drew up a petition asking the Penfield schools to offer pupils the option of taking traditional math. Nearly 700 residents signed it. Last June, the Board of Education turned down the request.

Giving parents a choice between fuzzy and real math is the democratic thing to do and is also a good political strategy. It should satisfy everyone. It's an inoffensive offensive. But I doubt that zealous educationists in a position of power will go along (and don't as the example above shows). Being responsive to reasonable popular wishes is not their thing. I also suspect that many parents are not conversant with the real vs. fuzzy math issues and won't know what to do with choice.

Just yesterday I talked to a parent of a sixth grader I am tutoring in math who had no clue of fuzzy math. (I tutor disadvantaged kids after hours in addition to my regular classes.)

I was helping the kid do homework. Part of the homework required the young girl to cut a sheet of paper into strips to make various fractions. The parent was aghast and thought it was a time-waster. I had to explain the purpose of the exercise. It was all news to her.

The school the kid is in uses the fuzzy math series Connected Math. The homework assignment was quite demanding and way beyond this pupil's abilities. She had neither a conceptual understanding of the task nor the requisite tools (computational skills, procedural knowledge, math facts) to accomplish the task had she had a conceptual understanding of the problem.

This is a key problem with fuzzy math. It is quite pretentious on the one hand, and refuses to teach the necessary skills on the other. The result: the kid was hopelessly drowning and getting straight F's.

Now what was the task? It was a real-world problem.

A class was holding a fundraiser to raise $300.00 in ten days. The progress was shown in the form of thermometers showing progress in two-day increments. The thermometers were all 8 1/2 inches long and showed the money raised so far on the various days in red. The fraction strips were to be used to determine the amount of money raised so far on the various days and then to plot the progress in a coordinate plane. The pupil was to make the strips and mark fractions from 1/2 to 1/12 on the various strips, and then use the strips for measurement.

Making fractions strips of 1/2, 1/4 and 1/8 is of course easy. It's not so easy to come up with 1/3, 1/5, 1/7, 1/9, 1/12. You could do time-consuming trial-and-error folding. Or you could divide 8 1/2 by the denominator of the various fractions if you know how (the child didn't know).

Even if you can get the numbers they don't work well with an inch ruler. You could approximate. Suppose you (meaning the kid) could accomplish all that. Then what?

The goal of the assignment is to come up with dollar amounts derived from the thermometers and then to plot these amounts over time (fundraising progress).

How are you going to derive dollar amounts if all you have is an 8 1/2 in long thermometer (representing $300) and a red bar on the thermometer without any numbers, marks or gradations (the length of the red bar represents the money raised so far)?

You could measure the red bar in inches, form a ratio (the red bar to total thermometer length ratio), calculate the decimal, multiply the decimal by 300.

However, the assignment calls for measuring the red bar with the "fraction ruler". Then you would know what fraction of 8 1/2 the red bar represents. You can then multiply the fraction by 300 to get the dollar amount. All this without instructions in CMP and without computational skills and procedural knowledge.

This is too complicated and frustrating for a math-challenged child who needs to learn at her level and make steady progress.

No wonder the kid is drowning. What a tragedy.

Alternative to fuzzy math denied

An illuminating story appeared in the NYT on how the fuzzy math plague plays out in a Rochester, NY, suburb:

LAST spring, when he was only a sophomore, Jim Munch received a plaque honoring him as top scorer on the high school math team here. He went on to earn the highest mark possible, a 5, on an Advanced Placement exam in calculus. His ambition is to become a theoretical mathematician.

So Jim might have seemed the veritable symbol for the new math curriculum installed over the last seven years in this ambitious, educated suburb of Rochester. Since seventh grade, he had been taking the "constructivist" or "inquiry" program, so named because it emphasizes pupils' constructing their own knowledge through a process of reasoning.

Jim, however, placed the credit elsewhere. His parents, an engineer and an educator, covertly tutored him in traditional math. Several teachers, in the privacy of their own classrooms, contravened the official curriculum to teach the problem-solving formulas that constructivist math denigrates as mindless memorization.

"My whole experience in math the last few years has been a struggle against the program," Jim said recently. "Whatever I've achieved, I've achieved in spite of it. Kids do not do better learning math themselves. There's a reason we go to school, which is that there's someone smarter than us with something to teach us."

After citing example after example of math cripples, the article has this gem that shows the arrogance of educationists:

By last spring, these parents had discovered one another and their common exasperation with constructivist math. Jim Munch's father, Bill, a software developer at Kodak, drew up a petition asking the Penfield schools to offer pupils the option of taking traditional math. Nearly 700 residents signed it. Last June, the Board of Education turned down the request.

The superintendent haughtily dismisses parent concerns. Many of the parents have extensive math backgrounds:

Susan Gray, the superintendent, attributed the criticism of the math program to "helicopter parents" who are accustomed to being deeply involved in all aspects of their children's lives. "Because the pedagogy has changed, the parents who knew the old ways didn't know how to help their children," she said. "They didn't have the knowledge and skills to support their children at home. There's a security in memorization of math facts, and that security is gone now."

YET many of the dissident parents have extensive math backgrounds and thus the ability to criticize the curriculum. It is also true that most of them tolerated the constructivist program for its first several years, until bitter experience drove them into rebellion.

Fuzzy math watch

Homeschool Math has a helpful list of fuzzy math books to avoid. The site also quotes Mathematically Correct on how the fuzzies operate:

"Mathematics achievement in America is far below what we would like it to be. Recent "reform" efforts only aggravate the problem. As a result, our children have less and less exposure to rigorous, content-rich mathematics. The advocates of the new, fuzzy math have practiced their rhetoric well. They speak of higher-order thinking, conceptual understanding and solving problems, but they neglect the systematic mastery of the fundamental building blocks necessary for success in any of these areas. Their focus is on things like calculators, blocks, guesswork, and group activities and they shun things like algorithms and repeated practice. The new programs are shy on fundamentals and they also lack the mathematical depth and rigor that promotes greater achievement."

National standards

NCLB leaves it up to the states to set academic standards. Schools must also demonstrate progress if the states want fed money and if the schools want to avoid being placed on the list of the infamous. This creates a perverse incentive leading to a "race to the bottom."

One way out of this disastrous state of affairs is to create national standards. This is, of course, fraught with danger as past experiences show, e.g. the experience with the Gary B. Nash history "standards". Such a project would attract the usual suspects. But I think the risk has to be taken. The advocates of sound, rigorous standards must organize and mobilize and be in a position to counter the inevitable attempt to dilute the standards and to institute educational lunacy.

It is therefore gratifying to read that Diane Ravitch is calling precisely for such standards. [Diane Ravitch's article was first published in the NYT but the article will soon disappear into the paid archives.]

The release last month of test results by the National Assessment of Educational Progress, which is part of the Department of Education, vividly demonstrated why varying state standards and tests are inadequate. Almost all states report that, based on their own tests, incredibly large proportions of their students meet high standards. Yet the scores on the federal test (which was given to a representative sample of fourth and eighth graders) were far lower. Basically, the states have embraced low standards and grade inflation.

Ed psych kitsch

Here is a novel perspective on a popular ed psych text. It's Anita Woolfolk's text I used in grad school.

So why don’t I like it? Why do I belittle it as kitsch? Kitsch, after all, usually refers to something of tawdry design or appearance created to appeal to nondiscriminating tastes—for example, those portraits of Elvis or bullfighters painted in neon colours on black velvet. But the Woolfolk text can hardly be accused of tawdry design or appearance. Indeed, measured against its competitors, it is presumably the crème de la crème.

Miseducative kitsch. In name-calling the Woolfolk text “kitsch,” I have in mind a less obvious, but perhaps more definitive, meaning of the term suggested by writer and critic Robert Fulford. In a recent CBC radio interview, Fulford dismissed so-called “victim-based art” as kitsch, but not necessarily because of any cheap or garish aesthetic qualities. Rather, Fulford argued, this “art” is kitsch because it seeks, by design, to compel the viewer to experience certain predetermined responses to it—in this case, sorrow, sympathy, compassion, and, perhaps, guilt. Fulford went on to liken victim-based art to the kitschy Saturday Evening Post cover art of Norman Rockwell, the artist whose slice-of-Americanlife paintings are typically unambiguously and irresistibly “cute” and, hence, admit of no other viewer response. For Fulford, what makes both victim-based and Norman Rockwell’s “art” quintessentially kitsch is that both contrive to over-determine and, consequently, to limit the viewer’s range of intellectual and emotional response. Neither allows for any interpretive or responsive ambiguity; both attempt to coerce thought and feeling.

Multiple ignorances

First we get "multiple intelligences." Now the rage is "multiple perspectives." From INTASC and numerous other educationist outfits we get "dispositions" that extol the virtue of "multiple perspectives."

The teacher appreciates multiple perspectives and conveys to learners how knowledge is developed from the vantage point of the knower.

The teacher appreciates and values human diversity, shows respect for students’ varied talents and perspectives, and is committed to the pursuit of “individually configured excellence.”

What is it with these "multiple perspectives?" It seems to me that at the bottom of this excitement is this postmodern notion that evidence, scholarship, objectivity and truth are illusory. There are only perpectives -- influenced, no doubt, by some "power" considerations.

This is dangerous drivel. What would stand in the way to conferring to creationism and to Holocaust denial the status of just another, equally valid, "multiple perspective?"

It would be much more appropriate to talk about "multiple ignorances."

UPDATE: See Chris Correa for a thoughtful examination of dispositions.

Heterodoxy lives!

http://www.discoverthenetworks.org/guideDesc.asp?catid=152&type=issue

Willful indifference

In Why Reading Teachers Are Not Trained to Use a Research-Based Pedagogy: Is Institutional Reform Possible? Sandra Stotsky argues that enough is now known (and has been known for decades) about how to teach reading effectively. The obstacles to teaching reading effectively are ideological. There is also a need to pretend that more research is needed to keep folks in academia artificially busy. Otherwise they might have to drive a cab.

Abstract: Reading instruction is one of the very few areas where it is not the case that “more research is needed.” Educational policy makers already have the theory and the evidence supporting it to guide the implementation of effective reading programs from K-12. In fact, they have had the theory and the evidence for decades. The central problem they face in providing effective reading instruction and a sound reading curriculum stems not from an absence of a research base but from willful indifference to what the research has consistently shown and to a theory that has been repeatedly confirmed. Using Jeanne Chall’s The Academic Achievement Challenge as a point of departure, I suggest why our education schools, through their influence on teachers, administrators, textbook publishers, and state and national assessments of students and teachers, have come to be the major obstacle to closing the “gap” in student achievement.

11th graders write

What are the ethics of posting samples of atrocious writing by students on weblogs? Should there be ethical concerns if the names of the producers of the hideous writing samples are not disclosed and the samples are otherwise slightly redacted to make identification difficult?

One of the sharpest education writers in the edusphere has done just that and now has convinced herself that posting such samples is unethical and will take down her post.

Here is a tiny snippet from these samples from 11th graders:

"in [title]. will has to deal with alot of racism bieng that she is a nigga during slavery. she sees her father get killed and then have to go home and find out that her mother is tooken buy the british. then will y goes to her aunt besty house and captian ivers try to put her back into slavery."

My view is that any lingering ethical concerns are far outweighed by the public service rendered by exposing this appalling state of affairs.

Read the rest of the samples before they disappear into electronic heaven.

cult comp

http://www.ifeminists.net/introduction/editorials/2005/1019.html

Science attempt

Something is brewing in the dismal area of science education. Not only will the NCLB science requirement kick in belatedly at some point, the fabled "nation's report card" body is stirring and drafting a "framework" for science education.

But all is not well. The good folks at the Fordham Foundation have assembled a
crack team to take a closer look at what the NAEP drafters are up to. The signs are not encouraging. Apparently, the discovering drafters are smitten by the fashionable but absurd educationist creed according to which barely literate pupils can "discover" the vast knowledge contributed by giants of science and accumulated over thousands of years on their own. Ignis fatuus is their guiding light.

Our basic position is that every child in America should receive a rich and rigorous science education in the primary and secondary grades, one that provides a broad understanding of key scientific concepts and ways of thinking. We reject the trendy notion that children, unaided, can “discover” key scientific concepts. Most of science must be taught if it is to be learned.

In these ways we demur from the “consensus” represented by several professional organizations that have offered national guidelines for science education (the National Science Education Standards of the National Research Council and Benchmarks for Science Literacy of the American Association for the Advancement of Science [AAAS]). Unfortunately, the NAEP Science Assessment Steering Committee encouraged the Framework’s authors to rely upon those very documents as their guide stars. This was a mistake. Just as the National Council for the Teaching of Mathematics (NCTM) is a partisan in the “math wars,” so, too, do these organizations represent one pole of the debate over science education and instruction. To follow their guidance is to “take one side” in an important debate rather than to strive for balance.

Let's not allow the reinvent-the-wheel crowd to follow the will-o'-the-wisp unhindered.

The Carnival is here

Make sure to check out this week's Carnival at the
Education Wonks. Lot's of math-related stuff. See especially the Alice entry at Kitchen Table Math and a lot more.

Math disaster

A friend of mine teaches chemistry at a community college in South Carolina and is pulling out his remaining hair. It appears that in his introductory chemistry and remedial classes the vast majority of students (recent high school graduates) is unable to do simple calculations involving percentages. Forget about dimensional analysis.

Something must have gone awfully wrong with these students' math instruction in high school and probably even earlier in elementary school. I am on the case investigating.

I teach math to eighth graders and know that teaching math successfully need not be rocket science. Most of my students can now convert fractions to decimals and percents (and vice-versa) in their sleep. They can also solve the three different types of percent word problems (unkown rate, whole and part) in their sleep.

The following is an aspect of how I teach percentages after teaching the concept that a percentage is an equivalent fraction with a denominator of one hundred.

I have my students create a table with three columns. Each column is labeled fraction, decimal, percent. This visualizes the operations and helps reduce confusion. They can go from left to right to start with fractions and end up with percents, or from right to left to start with percents and end up with fractions.

Converting percents to fractions generally poses no problems when the percent is a whole number, e.g. 47% --> .47 --> 47/100. A special problem arises when the percent is a fraction like 5-1/4 %. This is where the students need to realize that an additional step is required. Many students want to enter 5.25 in the decimal column. The task of the teacher is to focus on this problem and to show that the students must first convert to 5.25%, then to the decimal .0525 and on to the fraction.

A good way to drive home this point is to contrast the conversion of 5-1/4 (without %) and 5-1/4% to decimals and writing 5-1/4% and 5.25% in the percent column. Only now can 5.25% leave the percent column and move to the decimal column to become .0525. Bingo!

I also have my students solve percent problems as proportions and equations. I drill them in identifying the rate (percent), part and whole in any percent problem. For equations I have them use the rb=a formula. r=rate (percent), b=base (whole) and a=amount (part). This is also a nice algebra exercise since they need to solve for a, b or r as the case may be.

I once again have them make a table with three columns and several rows. The columns are labeled r, b and a.

Then I have them look at various percent problems and ask them to identify what's known and unknown. If the percent is unknown, a question mark goes into the box of the r-column and what's known (whole and part) goes into the respective boxes of the b-column and a-column.

If a (the part) or b (the whole) is unknown, the procedure is repeated accordingly. This works amazingly well and the visualization once again helps to minimize confusion.

An added advantage is that the students learn how to solve for the various variables and how to manipulate the rb=a formula, e.g. r=a/b, b=a/r.

It is amazing, though, how many of these 8th graders find this manipulation of variables difficult. Adults might consider this manipulation child's play. But it really represents a huge jump to this age group to go from the concreteness of numbers to the abstractness of variables (letters).

This concludes the lesson on how to teach percents for understanding. Instructivist will give other lessons on how to teach math successfully to middle graders as he sees fit.

Quantifying instruction

What is good teaching? Can it be quantified as educational researchers are trying to do. I, for one, am glad I am nowhere near a coding sheet and instead rely on observation, common sense and intuition.

Good teaching is far more than directly observable and measurable behavior. Good teachers possess traits and qualities that either cannot be quantified or are hard to quantify. I would even go so far as to claim that trying to quantity these traits and qualities is a case of scientism, which I idiosyncratically define as an attempt to quantify qualities that are ill-suited for quantification.

How, for example, do you go about quantifying an inspiring, radiating personality? How do you quantify a sense of humor, wit, a capability for empathy, a developed sensitivity to the zone of proximal development? How do you quantify imagination, creativity and a capacity to teach for understanding? How do you distinguish numerically between genuine and phony sentiment and what numbers would you assign to a story-telling capability?

I think education would be much better served if we approached the question of what makes a good teacher from a humanistic-philosophical perspective. But then, of course, we would first have to have a notion of desirable educational goals because "good" does not exist in a vacuum.

Arrested development

Constructivism reigns supreme in ed schools. It is the unquestioned doctrine that guides all aspects of teacher preparation. It should therefore be of some interest to find out what it is all about, if there is an empirical core that can be discerned from the thick fog that envelops hapless teacher candidates. The prospects are not good. Von Glaserfeld calls constructivism "a vast and woolly area in contemporary psychology, epistemology, and education."

Trying to pin down constructivism -- to see if it can be defined in a meaningful way and whether there is any sense that can be separated from nonsense -- is like searching for the unicorn.

At this point of my search, the best I can do is conclude that constructivists display a case of arrested development. Constructivists are stuck in the Piagetian sensorimotor or, at best, pre-operational stage. This infantilism that manifests itself in constructivists goes a long way in explaining educationist hostility to knowledge and educationist anti-intellectualism. But this infantilism is golden compared to the denial of objective reality by radical constructivist gurus like von Glaserfeld.

Let me elaborate a bit.

Constructivists cite Piaget and Vygotsky as their progenitors. Piaget did groundbreaking work on how toddlers develop intellectually and came up with different stages (really a continuum divided into stages to get a better handle on this development). For example, a toddler might recognize at some point that an object that is moved behind another object and disappears from sight does not cease to exist (permanence). Or a toddler might discover on his own that an object falls when released or that bumping into a wall is not a good idea or that a flame is hot.

The constructivists' fallacy is to carry this notion of discovery and individual experience to absurd lengths and apply it to later years -- to adolescence and even adulthood. Learning then becomes solely a matter of discovery and individual experience from which one constructs one's own knowledge and meaning. But you cannot "construct" broader knowledge ex nihilo and keep reinventing the wheel endlessly. You need external input. You need to benefit from the knowledge accumulated over thousands of years. This is where constructivism breaks down. Constructivism presents itself as a theory of learning based solely on experience. But personal experience is limited. Broader learning also needs to tap into an existing body of knowledge that constructivists disparage.

This otherwise imcomprehensible educationist hostility to knowledge and especially imparting knowledge becomes clearer when one considers the views of leading radical constructivist gurus like von Glaserfeld who seem to come straight out of the loony bin:

Von Glaserfeld is one of the leading apostles of radical constructivism. Radical constructivism rejects the traditional philosophical position of realism and adopts a relativist position. The traditional view of realism sees knowledge as a representation of an absolute reality - a world "out there" prior to having been experienced. The radical constructivists sees knowledge as "something that is personally constructed by individuals, in an active way, as they try to give meaning to socially accepted and shared notions." As von Glaserfeld himself says "knowledge is the result of an individual subject's constructive activity, not a commodity that somehow resides outside the knower and can be conveyed or distilled by diligent perception or linguistic communication"

This explains why educationists don't believe in an external body of academic knowledge that should be communicated to students. It explains why teachers are not allowed to teach, i.e. give explicit instruction.

On the other hand, how constructivists can claim Vygotsky as one of their own still remains a mystery to me. His notions of the zone of proximal development and scaffolding are sensible and don't rule out explicit instruction (despised by constructivists). His emphasis on socio-cultural factors doesn't fit in with "constructing one's own knowledge" either.

I wish someone could explain all these mysteries to me.

UPDATE #1: In the meantime, reader Rob has helpfully directed me to a scholarly article called Does No One Read Vygotsky’s Words? Commentary on Glassman that exposes attempts to Deweyize Vygotsky through omissions, distortions and inventions.

From the abstract:

In the May 2001 issue of Educational Researcher, Michael Glassman proposed several commonalities in the thinking of John Dewey and Lev Vygotsky. However, in addition to general problems in the article (misstatements about scholars’ writings and a reliance on unsupported inferences), the discussion misconstrues major concepts and topics addressed by Vygotsky’s theory of cognitive development—psychological tools, the role of the cross-cultural study, the zone of proximal development, and the nature of conceptual thinking. In addition, Glassman attempted to force Vygotsky’s goals into a Deweyan framework. The result is a misportrayal of Vygotsky’s work.

UPDATE #2: I dug out my ed psych text (Anita Woolfolk) we used in grad school to see what it says about Piaget and constructivism. From it I learn that knowledge is "constructed by transforming, organizing, and reorganizing previous knowledge. Knowledge is not a mirror of the external world, even though experience influences thinking and thinking influences knowledge. Exploration and discovery are more important than teaching." The quote is Woolfolk speaking and giving a summary of Piaget's purported views under the heading "Assumptions about Learning and Knowledge."

Woolfolk goes on to helpfully explain how knowledge is constructed citing Moshman (1982). Knowledge construction is directed by internal processes like Piaget's organization, assimilation and accommodation. This means that new knowledge is "abstracted from old knowledge." It turns out that "[k]nowledge is not a mirror of reality, but rather an abstraction that grows and develops with cognitivie activity. Knowledge is not true or false; it just grows more internally consistent and organized with development."

Talk about being self-referential. Where is the external input?

I don't know what to make of this. I can understand that we might have to readjust our thinking when we learn new things that might conflict with or supplement our previous knowledge. But apparently there is no input of new knowledge from an external source. Saying that knowledge is constructed by transforming, organizing, and reorganizing previous knowledge is purely self-referential. Previous knowledge is simply remixed and stirred the way you might mix the ingredients of a cake. Nothing new is added. How relevant is all this to teaching reading and writing skills, math, science, history, geography, literature or languages?

Neither relevant nor helpful. It's all nonsense -- nonsense on stilts that has managed to become the dominant creed of the ed establishment. Constructivism is to education what creationism is to science. Both lack an empirical basis and rely on some unfathomable, ineffable, magical, supernatural thing said to "construct" something out of nothing.

How refreshing, then, to have someone like Prof. Plum rip the mask off this pretentious drivel:

What is Constructivism?

Constructivism is big word that makes education perfessers think they are intelligent.

Constructivism is an invention that makes education perfessers think they know something that everyone else doesn't.

Constructivism is a set of statements about learning that are quite simpleminded and generally false.

"Knowledge can't be transmitted from one person to another. 'Learners' have to construct knowledge." [This very statement shows that constructivists don't believe what they say. Isn't the statement an effort to transmit knowledge?]

"Therefore, teachers should not teach directly by telling or showing (e.g., how to solve math problems). Instead, they should guide students as STUDENTS figure out concepts (what granite is) and strategies (how to sound out words, how to solve math problems)." [Constructing knowledge means NOTHING more than comparing and contrasting, identifying sameness and difference, making inductions and deductions. This is all OLD news. There is NO reason why teachers can't teach in a direct and focused fashion. In fact, students "construct knowledge" (figure things out) better--faster and with fewer errors--when they ARE taught directly, rather than expected to "discover" knowledge--which makes no sense, anyway. If knowledge is constructed, what IS there to discover?]

"How each person constructs knowledge is unique. Therefore, teachers should not arrange instruction in sequences. Instead, students should select learning tasks. Don't worry. They will select what they are ready for." [Unique in the DETAILS but not in the general logical operations by which human beings learn. If each person is unique, I guess physicians should not take their blood pressure.]

"Drill (distributed practice) is bad. It is boring. It is not needed." [Baloney!]

"Tasks should be 'authentic.' Holistic. Teach the fundamentals of chemistry in the CONTEXT of chemistry experiments. Teach phonics skills in the context of reading." [This is the prescription for keeping kids ignorant and unskilled and for leaving them demoralized.]

"Since each student's learning is unique and INTERNAL, you cannot use quantitative and standardized methods of assessment. It should be qualitative--how students feel and think about what they are learning." [This makes no sense. Body temperature is also "internal," but you can measure it quantitatively and with a standard instrument. Likewise, you can easily count how many math problems kids do correctly. This is a cop-out to protect constructivists from data that would ruin them.]

And from this set of sophomoric beliefs, you get whole language, fuzziest math, inquiry science, literature without literacy, and history without moral and political lessons.

...

Constructivist "theory" is a mishmash of overlapping platitudes and absurdities--"empty words and poetic metaphors" (Aristotle, Metaphysics). Taken separately, constructivist "propositions" are merely simpleminded. Taken together, they are indistinguishable from the verbal behavior of a person suffering from chronic schizophrenia.

"Reality is a construction."
"Knowledge is a construction."
"Experience is a construction."
"Experience is constructed with constructs."
"Constructs are constructed out of experience."
"Reality is knowledge."
"Knowledge is reality."
"Experience is reality."
"There is no knowable reality external to the knowing subject (the constructor)."
"Individuals and groups construct meaning as they interact with environments."
"Therefore, no statement can be more than relatively true."
"A current body of knowledge ('reality') is a context that shapes the construction of knowledge."
"Therefore, environment, knowledge, experience, meaning and reality are the same thing."

What does progressive/constructivist education actually look like in practice. Here we have a smartly written account from someone who is experiencing it first hand:

In my lesson during the first block, we read a lot of the primary source documents out loud together, and then I led a discussion about them and we took notes together. This was roundly condemned as the wrong approach, and so the next period I had to let them read and interpret in groups. As usual, they wasted a lot of time and I don't really think they understood it. Not that I'm sure they had totally understood it in the first block, since they didn't have the proper historical background and I can't give them quizzes, but at least I got some reasonable answers out of the class discussion. In the second block, everyone got something different, depending on their group. It was so frustrating. I don't want to learn to teach like this because I don't think it's right. But I can't contradict the teacher. But I need the practice. It's an education grad school moral dilemma. The best kind.

This is just the last paragraph of a fairly long post. Read the whole thing.

UPDATE #3: Here are more sources on constructivism:

http://mathforum.org/mathed/constructivism.html

What is Constructivism?

"Students need to construct their own understanding of each mathematical concept, so that the primary role of teaching is not to lecture, explain, or otherwise attempt to 'transfer' mathematical knowledge, but to create situations for students that will foster their making the necessary mental constructions. A critical aspect of the approach is a decomposition of each mathematical concept into developmental steps following a Piagetian theory of knowledge based on observation of, and interviews with, students as they attempt to learn a concept." [Emphasis added].
- Calculus, Concepts, Computers, and Cooperative Learning (C4L)

These prescriptions would explain the surfeit of math cripples. I know from my own experience teaching math that students thrive when having things explained to them combined with guided practice and independent homework in the form of distributed practice and overlearning.