On 4 April 2001, Boston College will release the Benchmarking results of the Third International Mathematics and Science Study-Repeat (TIMSS-R) in Washington, D.C., at the National Press Club. The press conference is scheduled to begin at 10:00 a.m. Eastern Time. The main announcement will take place from 10:00-10:30 a.m., followed by general remarks and a Q&A session.
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The press conference speakers, in order of appearance, are as follows:
The TIMSS 1999 Benchmarking Study includes 13 states and 14 districts or consortia of districts from all across the Untied States. The TIMSS 1999 assessments were administered to representative samples of eighth-grade students in these districts and states during the spring of 1999 and following the same guidelines as those established for the 38 countries participating in TIMSS-R.
For a list of the 27 TIMSS-R Benchmarking participants, see http://www.timss.org/timss1999b/participants.html
While educators and policymakers debate whether 8th graders can be readied to learn algebra, Sigrid B. Frawley sits in front of her kindergartners with a magic bag.
She puts three tokens in the bag and pulls five out. Then she asks the students: "What's the rule?"
"Add two," is the answer.
Next, she puts four tokens in and pulls six out. "What's the rule?" she asks.
The 5-year-olds don't know it, but they're talking about an algebraic equation: x+2=y.
Students at Walter Stillman Elementary School in this suburb of New York City, in fact, get doses of algebra starting in kindergarten and lasting throughout their careers here. By the time they reach middle school, one-third of them will be ready for algebra in the 7th grade. Almost all take the course in the 8th grade.
"The answer to, 'How do you get them there?' is: You give it to them early," said Principal William B. Greene.
American middle schoolers are increasingly being moved into algebra courses. In what was once a rite of passage in high school, now middle schoolers--some as early as 7th grade--are now expected to learn about the mathematics of variables and quadratic equations. In California, for example, all students are expected to learn algebra in the 8th grade, according to the state's standards.
In a 1998 survey, 95 percent of high school graduates had taken algebra, a 14 percent increase from eight years earlier, according to the Council of Chief State School Officers.
To get students ready for that leap, the consensus among mathematicians and educators is that students need to be introduced gradually to algebraic concepts throughout the elementary school years. The only debate is how to teach it.
"The student of algebra need not begin with a formal course in the subject," the National Research Council concluded in a January report on teaching mathematics. "From the earliest grades of elementary school, students can be acquiring the rudiments of algebra."
"Elementary school really is the critical place for fixing America's algebra problem," said James Kaput, a professor of mathematics at the University of Massachusetts Dartmouth. "We're only slowly coming to terms with that."
At Stillman and at Tenafly's other three elementary schools, teachers follow the Everyday Mathematics curriculum and textbook series developed at the University of Chicago.
In addition to her magic bag, Ms. Frawley integrates equations with variables into her arithmetic lessons. She'll write the equation 3+__=5, and ask her students to fill in the blank.
Second graders are encouraged to draw graphs of equations similar to the ones created from the objects Ms. Frawley draws from her bag. If the rule is to add 2 to every number, the students are asked to find a place for everything on the x and y axis.
By 5th grade, students are completing word problems with introductory algebraic reasoning. For example, they're told that mules travel six miles per hour. They're asked to calculate how far the mules will have traveled after one, two, and six hours. Then, they're asked to figure out how many hours the animals took to walk 30 miles and 48 miles.
If they saw the equation as 6x=y, as they would in an algebra class, they might not understand it. But with the concrete examples in front of them, about 85 percent of the students can solve the equations, said Mireille Bany, one of the school's 5th grade teachers.
"We've done it so many times that they know how to do it," she said.
Such preparations are important, math experts say, because middle schoolers often have a difficult transition into algebra.
After years of solving problems using basic arithmetic skills, they struggle with the mathematical reasoning and manipulations required to succeed in algebra.
Students who learn arithmetic come to think of the equals sign as a function, much the same way the plus sign instructs them to add, according to Thomas P. Carpenter, the director of the National Center for Improving Student Learning and Achievement in Mathematics at the University of Wisconsin-Madison.
The equals sign, however, "expresses a relationship rather than a command to do something," he said. If students don't understand that, they can't solve such beginning algebraic equations as 3x+5=15.
"Fifteen isn't the answer," Mr. Carpenter said, "but that's how they are thinking about it in arithmetic"...
While there's wide agreement that algebra should be introduced in the early grades, differences remain over how to teach it. California's math standards-which follow the traditional method of teaching the subject-emphasize mastering addition, subtraction, multiplication, and division in the early grades. Algebraic concepts are introduced as ways to apply the algorithms.
"You should be doing certain kinds of variable equations all the way along," said Wayne W. Bishop, a professor of mathematics at California State University-Los Angeles and one of the authors of the California standards. "You can't just assume you're going to dump everybody into 8th grade algebra if you haven't done some preparation with elementary linear equations."
Teachers should mix simple word problems into arithmetic as early as 1st grade, Mr. Bishop said. They should even use "x" as a variable as part of the teaching, he said.
By contrast, the National Council of Teachers of Mathematics' standards downplay the algorithms and treat algebraic principles more fully. In addition, teachers are encouraged to use real-life examples to illustrate the mathematics. One example in the standards has students filling in the blanks of a chart projecting the cost of bunches of balloons. The chart says that one balloon costs 20 cents, two cost 40 cents, three cost 60 cents, and four cost 80 cents. Students are expected to complete the series that extends to a bunch of seven balloons. Using variables such as "x" is "too abstract," contends Lee V. Stiff, the NCTM's president.
"When it comes to doing algebraic manipulations, [students] can see how they did it with numbers, and then they can do it with variables," said Mr. Stiff, a professor of mathematics education at North Carolina State University in Raleigh.
But even though young students are able to learn simple algebra, the ability of many elementary teachers to help them learn it is open to question.
"You have elementary school teachers who do not know what algebra is about, so they're not in the position to think about how the arithmetic they're teaching will mesh with algebra later," said Roger Howe, a professor of mathematics at Yale University and a member of the panel that wrote the National Research Council report.
Elementary school teachers generally don't excel in mathematics in high school and often aren't expected to take high-level courses in college. They often come to their first jobs afraid that they are unable to teach the subject, said Ms. Frawley, the kindergarten teacher here in Tenafly.
Mr. Carpenter's center at the University of Wisconsin is working with teachers in Los Angeles, Phoenix, and San Diego as part of a larger project to get teachers thinking about how they can introduce variables into their grade school math curricula.
Good teachers need to "build algebra eyes and ears," said Mr. Kaput of the University of Massachusetts, so that they can take advantage of any opportunity to teach about the subject. Mr. Kaput and his colleagues are working with 350 teachers in Fall River, Mass., to help them engage young students in algebraic thinking.
"What we do is teach the teachers how to take a problem and build a series of problems off of it," Mr. Kaput said. "It helps build [students'] computational skill and builds on the deeper understanding that we're after."
Here in Tenafly, new teachers attend a one-week seminar on Everyday Mathematics before they enter the classroom. They then meet with mentors once a month to learn more about the goals of the program.
The key to making the experience work, Ms. Frawley said, is to avoid the term "algebra" and focus on the real-life applications of the mathematics of the subject. "To say to them, 'You're teaching algebra in a kindergarten class,' that would throw them," she said of the new teachers. "They think of that as a high school class."
But teachers have been able to pick up on the program and have fully integrated algebraic thinking into the early grades in a matter of three years, Ms. Frawley said.
Sometimes, they're teaching the subject without even thinking about it. One morning early this month, Terry Moore, a 3rd grade teacher at Stillman Elementary School, is filling out the March calendar on a whiteboard. He writes the numbers symbolizing the first seven days of the month in their boxes. He calls on a student to tell him the number of every Wednesday of the month. The simple answer is to add seven to Wednesday the 7th. As Principal Greene watches, he points out that the exercise is a simple algebraic equation: x+7=y.
Later, Mr. Moore is asked if he thought of the exercise as an algebra lesson. "Only since you mentioned it," he said. "It wasn't set up to do that, but you could make a function of it."
Each day she comes to the sprawling campus on the outskirts of this quintessential leafy college town, Anne Connell has a mission, one that could affect the educational destinies of thousands of students. "I am the one who selects the 'Question of the Day' on the [Educational Testing Service] Web site," said Connell, her cheery eyes turning somewhat diabolical as she speaks. "When I'm working on that SAT question, I try to make sure it's just right, something that will challenge."
Those simple little letters--SAT--grind an awful lot of fear through the souls of prospective college students. And it is Connell and her cohorts in these low-slung buildings at the Educational Testing Service in the rolling central New Jersey countryside who help determine those teenagers' fate. They are the creators of the SAT, or SAT I (formerly the Scholastic Aptitude Test), the 138-question test most colleges still require for admission. Nearly 2.4 million SAT and 2.2 million PSAT (Preliminary Scholastic Aptitude Test, the "practice" for the SAT) tests were taken in 1999-2000...
Recently, University of California President Richard C. Atkinson suggested that his university and others should de-emphasize the use of the SAT in college admissions. But as the SAT comes under renewed fire from administrators, academics, parents and students, the people who make up the test go about their jobs with calm and serious purpose, confident of the test's staying power.
After all, the recent California uproar is not the first time the SAT as a universal tester has been called into question. "It comes in waves," said Gretchen W. Rigol, vice president of higher education services at the College Board, which employs the Educational Testing Service to make the SAT and administers the test. "I'll tell you why it comes in waves. Until every discernible group in this country gets the same scores--Asian Americans, Latinos, males, females, the handicapped--people are going to say, 'Why is that?' One of the answers they like to come up with is that the instrument is biased. But we do our best to make that not be the case...
Those who create the SAT at the testing service are primarily middle-aged and middle-class, most with an educational background of some sort--teaching or administrative...
"We have children, too," said Robin O'Callaghan, who creates questions, among other duties, in her job as director of math skills for the SAT. "We know how important the SAT can be in their lives."
Like any team members, they have their jargon. For instance, they would call the "Question of the Day" a misnomer. "They are items, not questions," said Chancey Jones, the soft-spoken, grandfatherly executive director of the test division, who has been a math specialist at the teaching service for 35 years after an initial teaching career. "We are testing aptitude, not making them recall facts. So it's hard to call them questions"...
The nonprofit testing service, which employs 2,100 people full-time, also makes up the tests for such standardized exams as TOEFL (Test of English as a Foreign Language) and GRE (Graduate Record Examinations) and the SAT II (often known as "achievement tests," those college-entrance exams that measure skills in specific academic areas). But since so many students from so many backgrounds take the SAT, a vast number of items is needed--up to 1,500 for more than 20 versions of the test in some years--and it is a major focus of the testing service.
Creating items in their final form for the test is a collaborative process, much like, say, writing a screenplay. Rarely does an item come through unedited, even from the mind of seasoned writers. A typical item can go through as many as six to eight reviews. Committees both inside the ETS and outside the campus look over every question--the committees outside comprising academics and sometimes laypeople and students around the country...
It often takes a question 18 months to wend its way from that first thought to an actual SAT test. In addition to the regular reviews, each question gets two different fairness reviews. The first is straightforward. "Basically, it is this: Will this item offend any particular group?" said Sydell Carlton, chair of the ETS Fairness Steering Committee...
Then there is differential item functioning. If this piece of jargon sounds like computerese, well, it is. Each item gets a dry run before it takes an official place on the test. The SAT has seven sections, but only six of them count toward a score. One of the sections--the identity of which is never revealed--is used solely for refining the test, and its results are analyzed by computer for ethnic, racial and gender correlations. If one is particularly out of whack on an item, the item is eliminated...
The items--60 in math and 78 in verbal in each test--go from easy to hard within each section... Though there are 47 people at ETS who create items as part of their jobs, about a third of the original items are contributed by freelancers. Jones said it gives the test a greater geographic diversity and lets the staff work more with reviewing and coordinating items into whole tests.
The SAT was developed in 1926 but didn't come into widespread use as a college-entrance standard until after World War II, when masses of students started applying to colleges and universities. And it has changed remarkably little over the years. Even the fairness reviews are well-entrenched, having been instituted in the late 1960s...
There are some kinds of items, though, that despite some statistical biases, still make it into the test. Female students, said O'Callaghan, tend to do better on straightforward math questions dealing only with computation. Males tend to do better with practical ones, using everyday language.
"I don't know whether it's that boys figure out batting averages or go to the garage more with their parents, but it seems to correlate that way," she said. Since it would be too hard to get away from using practical language, those questions will stay, with the hope that females will get better at them as time goes on...
...Growing numbers of schools such as Dickinson say an SAT-optional policy helps lure strong students who otherwise might shy away because of lower SAT scores.
Not that everyone is signing up. The SAT has long provided admissions officers with what they consider a valuable yardstick to help put a student's high school grades in national perspective. Among those who have no intention of making the scores optional are Temple University, the University of Pennsylvania, and Pennsylvania State University.
But now that the head of the nation's largest state system of higher education has weighed in against the SAT, critics of the test hope more colleges will follow the lead of mavericks such as Dickinson...
In late February, the debate escalated when Richard C. Atkinson, president of the 170,000-student University of California system, gave a speech to hundreds of college presidents in which he proposed that California drop the SAT...He concluded that "America's overemphasis on the SAT is compromising our educational system."
Less focus on the SAT would let educators concentrate on improving the curriculum and resource needs of the nation's elementary and secondary schools, he argued.
Atkinson's proposal to drop the SAT I - and replace it with a test linked to curricula - still is subject to approval by the university's Board of Regents. But it has already sent shock waves through the
"He adds a huge amount of clout," said Bob Schaeffer of FairTest, a critic of the SAT. Last year, of the 6.9 million score reports that students sent to colleges, 7.6 percent went to the California system.
"A decision by California to abandon the SAT could be enormous," said Brian O'Reilly, executive director of the SAT program for the College Board, which oversees the SAT. "There's a real concern about the spillover effect it could have on other schools"...
In response to Atkinson's speech, College Board president Gaston Caperton said: "The key to students' success and opportunity is not to scapegoat the SAT I, but to...confront the tough issues that the standards-based school reform movement has been addressing: radically improving curricula, teacher training, and accountability"...
...After a rough decade of losses caused by a heavy investment in computer-based exams, ETS last year--for the first time in its history--hired a businessman, not an educator, to run the company. And looking to seize a large chunk of the pre-college testing market, it launched a for-profit subsidiary, ETS K-12 Works. ETS president Kurt Landgraf, former CEO of DuPont Pharmaceuticals, hopes to double ETS's overall revenues within five years, to more than $1 billion a year. "The future for testing is in K-12," says Landgraf. "It's the biggest initiative we have." His golden ticket may be ETS's new "e-rater," a nifty tool that can grade essay questions in under a second, using advanced artificial-intelligence technology. ETS claims the scores the e-rater spits out match those given by human graders 97% of the time. That's as accurate as a second human reader.
The company has a ready market in states looking for high-quality test designers. Today just three companies (conveniently, the three biggest school-textbook publishers) develop nearly all K-12 tests, and there is a severe shortage of psychometricians--specialists trained in educational measurement and test design...
While ETS is mining the whole K-12 market, the College Board has its eye on middle schools. This spring the company will unveil new math and English curriculums and tests designed to be like AP courses for seventh- and eighth-graders. College Board president Gaston Caperton says middle schools "are crying out" for such programs. Researchers at the College Board have also developed an SAT for eighth-graders, complete with developmentally appropriate math and verbal reasoning sections, to get kids thinking about college even sooner than they already do.
Not to be left out of the testing boom, the $400 million test-prep industry is also expanding. One might have expected John Katzman, founder and CEO of The Princeton Review, one of the two leading SAT-prep companies, to be at least a little concerned by University of California president Richard Atkinson's push to abolish the SAT. In fact, Katzman is ecstatic, calling the SAT "a vestige from another era" that "should be discarded at the first possible moment." It's a position he can afford to take, as his company, which is in the process of going public, recently launched homeroom.com, a potentially profitable interactive tool meant to help kids prepare for their state exams.
So here's the key question: When historians look back on this moment in American education, will they see a) the beginning of the end of the SAT; b) a national frenzy over school testing in general; or c) the dawn of the testing industry's greatest boom? Try d) all of the above.
Cooperative Learning in Undergraduate Mathematics: Issues That Matter and Strategies That Work (The Mathematical Association of America, 2001), edited by Elizabeth C. Rogers, Barbara E. Reynolds, Neil A. Davidson, and Anthony D. Thomas. $31.50; MAA members $23.95.
Mathematics is one of those subjects that is learned alone. Lectures are the dominant mode of teaching. Homework is done at one's desk. Group projects are few and far between. When it comes to math, you may be sitting in a classroom full of other people, but that's where the sense of community usually begins and ends.
For centuries, that has been the case. By fits and starts, however, communal learning has been creeping into mathematics, and now, as a result of a new book from the Mathematical Association of America, "group think" may take center stage in the undergraduate classroom once and for all. Cooperative Learning in Undergraduate Mathematics, the latest volume in a series dedicated to the teaching of undergraduate math, presents the collective experiences of 17 authors who have used cooperative learning in their classrooms--and who say that it works.
Cooperative learning encourages students to work, study, and learn in small groups, usually two to five people. That's a much more conducive setting than a lecture hall, the contributors argue, for the development of critical--and creative--reasoning skills. In small groups, students feel more at ease asking questions and trading ideas, which they might hesitate to do in a more formal setting. And by working together, they can tackle problems that might be beyond the abilities of any one of them individually. Some students are good at basic computation; others have advanced computer skills. "In a well-functioning cooperative-learning group," the book points out, "students learn to recognize and draw on each other's skills."
The real boon of cooperative learning is that it provides social support in the often intimidating mathematics classroom. The strategy, the editors say, is to view learning as a social activity, and to approach math as an interesting topic for inquiry and discussion. One of the tenets of cooperative learning, after all, is that students learn by "talking, listening, explaining, and thinking with others. The very act of explaining an idea or concept causes students to reach for a deeper understanding of that idea."
Of course, grasping the pedagogical merits of cooperative learning is only part of the battle; creating a classroom climate receptive to group learning is also crucial. The editors do their best to deliver on that front as well, providing suggestions that range from the relentlessly practical (how to physically arrange a classroom or use the chalkboard to promote cooperative learning) to the philosophical (how to resolve conflict within or among groups, or facilitate peer tutoring among students).
The contributors to this volume are the first to confess that sometimes their experiments in group learning have fallen short of the goal. But more often than not, their efforts have produced results that outstripped their expectations. That's why they wrote for this volume in the first place: to provide readers with a primer on what works, and why. And the contributing authors believe in practicing what they preach. From start to finish, Cooperative Learning was a joint effort. Each chapter was written--and rewritten--by small groups of authors, and then underwent a critique by the group as a whole.
The Eisenhower National Clearinghouse (ENC) has as its mission the identification of effective curriculum resources, the creation of high-quality professional development materials, and the dissemination of useful information and products to improve K-12 mathematics and science teaching and learning.
ENC produces a free magazine, ENC Focus, which is available on the web (http://www.enc.org/focus/) and in hard copy. The theme of the current issue is "Teaching in the Standards-Based Classroom" (http://www.enc.org/focus/standards/):
"Virtually every national standards document, every state framework, and every local set of standards calls for fundamental changes in what and how teachers teach. The challenge for teachers is to implement the vision for mathematics and science classrooms called for in the standards. This issue describes that vision and suggests ways to use the standards mandated in your school to improve your practice--to help you teach in your standards-based classroom."
COMET is sponsored in part by a grant from the California Mathematics Project.
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