"Mathematics Education: Clear or Fuzzy?" was the title of panel discussion at NYU's Steinhardt School of Education on April 29. The audience of about 70 comprised clears, fuzzies and undecideds, including mathematicians from the Courant Institute at NYU, professors and administrators from the Steinhardt School, and others from city colleges and public schools. The event was one in a series of education policy breakfasts hosted by NYU.

The clear vs. fuzzy debate - the uninhibited call it the math wars - has been going on for a number of years. On this panel, unlike the outside world, the clears outnumbered the fuzzies two to one. But as usual, the two sides spoke different languages. The speakers were R. James Milgram, professor of mathematics at Stanford, a leading math warrior and co-author of the current California standards and framework in mathematics; Ricki Goldman, professor of information systems at the New Jersey Institute of Technology and a researcher in math education for minorities; and Daniel L. Goroff, professor of the practice of mathematics at Harvard and associate director of the Derek Bok Center for Teaching and Learning. Each panelist was equipped with an Apple laptop to show slides.

James Milgram began by noting that in the late eighties and nineties, the lack of home-grown talent in Silicon Valley was so severe that the technology industries recruited thousands of foreign engineers and programmers. Yet, in 1998, 80 percent of those admitted to the California state university system failed the mathematics entrance exam, a test pitched to a sixth-grade level. Milgram and his colleagues based the new math standards for California public schools on methods used abroad, especially in Japan and Singapore. "The critics called it 'drill and kill,'" said Milgram, but nonetheless, since the standards were implemented, math scores in statewide tests of kindergarten through fifth grade rose significantly - without additional teacher training. (By contrast, the constructivist curricula have required extensive teacher training.)

"We 'back-mapped' for algebra by eighth grade, because if they don't get algebra by eighth grade, they can't proceed," said Milgram. "The problem-solving component was improved over the old California and NCTM standards. Now, in the K-through-6 program every third lesson is problem-solving."

Math performance in Los Angeles is improving, Milgram said, showing graphs of student test scores. In San Diego, where the city used money from foundations to buy their own books and have used Connected Math, TERC and other constructivist curricula, scores are declining.

In the districts making progress, he cautioned, "It's not a complete improvement." California still does not teach math at the level achieved by many foreign countries. There is resistance: "Education schools don't want real math content." Addressing the problem, Milgram is currently directing a grant from the Department of Education in a project where prospective teachers are taught by mathematicians. "I see a glimmer of hope," he concluded.

Ricki Goldman advocates constructivist approaches to science and math education. She has established a laboratory, according to the event program, "dedicated to understanding the role [in what is not specified] of learning, new media and human interaction" and is concerned with education for "groups underrepresented in the scientific and technical professions."

"Children see math everywhere," she said. "Kids examine their own math ideas. There is a culture of math." The Everyday Math curriculum recently adopted by New York City public schools "doesn't go far enough in constructivism."

She showed a video of a little black girl adding 1/2 and 3/4. The girl didn't know the algorithm for adding fractions, but she drew two circles and divided them, then added two quadrants to three quadrants and got the right answer. She clearly knew what she was doing, but it took her a couple of minutes.

Goldman said that when she was growing up in Winnipeg, she was surrounded by mathematics: "I walked along the hypotenuse of a right triangle to get to school." She continued, "The process [of teaching math] is messy ... We must empower children ... The fun is in figuring it out ... Children are like architects ..."

She showed more videos, one of white boy incomprehensibly explaining what she translated as "recursion. ... He's thinking metaphysically about his work." In another video, a boy explained breathlessly how a laser strikes a piece of metal to play a compact disk. She showed another slide: "Does there need to be a MATH WAR?"

"We do need standards, but we also need to celebrate differences," she said. A slide reading "1 = 1/1 = 1.0 = 100%" illustrated what she called "the need for wholeness and infinity."

Another click: "E-VALUE-ation." "Evaluation should be a motion picture of a child's growing understanding," she explained. More slides: "The Ah-ha ... The big picture"; "Core mathematics? No. Core ideas? Yes."

Finally, she asked rhetorically, "Do we discard the work of educators? I think not," presumably referring those who developed the constructivist curricula. "When we expect the same answer from everybody, we focus on teaching, not learning," she asserted, concluding, "Dancing is more than the steps. No one dances the same. ... Math is that dance. How do we create a culture of dancing?"

An audience member asked, "What about problems that aren't easy, like 1/2 + 1/7?" alluding to a well-known critique of the TERC program. Goldman answered, "We're talking about different ages here. I like your question. Could we be designing technologies to help kids see this?"

Daniel Goroff of Harvard began with a problem in conditional probability: "Donald Jones is either a librarian or a salesman. His personality can be described as retiring. What are the odds that he is a librarian?" The audience overwhelmingly voted that Jones was a salesman, and indeed they were right. Goroff elaborated: "Male salesmen outnumber male librarians by about 100 to 1. P = (A???????????????????????????????????? ?????)????????????????????????B?A?, but people often confuse the two." He went on with slides to state Bayes' law "P(S?I? = P(S); ??I??? = P(L) where I = information."

It was unclear whether everyone was still with him, but he made it easier: "If 80 percent of CEOs had pets as children, what does that tell us? Does it mean it's a good thing? What if 100 percent of people who don't make it in business had pets as children? "So who needs conditional probability? Brains don't naturally construct these ideas. ... Probability and statistics are a new emphasis in the curriculum, even in Everyday Math. Teachers need to know this." While Goroff did not show videos, he recommended that we look at the U.S. and Japanese classroom videos from the Third International Math and Science Survey, which demonstrate the difference between American and Japanese teaching methods a crucial difference.

In our country, he said, the classroom script is:

- The teacher reviews the homework.
- The teacher works problems on the board.
- The students do the problem at their desks.

In Japan:

- The teacher presents a "rich problem" or hatsuno.
- The students work the problem individually.
- The students work the problem in groups.
- The teacher walks around looking at the students' work.
- Finally, the teacher reviews the students' various answers at the board.

He stressed that when the Japanese teacher walked around the classroom, it was not to help the students, but to find examples of all the students' approaches to the problem, both right and wrong. The teacher would then choose students to go to the board and work the problem in various ways. The teacher never helps an individual student in class, which would embarrass both parties.

"The emphasis is on development of social capital, on holding groups responsible," said Goroff. Another difference in Japan is that teachers from adjacent grades prepare lessons together, discussing the students' anticipated misconceptions: "The possible states of students' minds, that's what teachers have to know and engineers don't."

"The Japanese think it's crazy that American teachers don't work in groups," he added. At Harvard, they are using some Japanese methods, creating lesson plans that anticipate students' difficulties.

The question period began. A black teacher asked about cultural differences in student assessment.

Milgram replied, "In Japan, there are not a lot of tests in the early grades, but in high school the pressure of exams takes over with a vengeance."

Goldman mentioned cultural differences in the U.S. and strategies for dealing with diversity: "Look at the art and science of pedagogy."

From the audience, Fred Greenleaf of the Courant Institute objected, "You can't teach what you don't know."

Goldman responded, "But if you believe learning is a process, the teachers work together to develop knowledge."

Greenleaf insisted, "But you can't teach it if you don't have sound training in math. Many K-through-6 teachers come out of school with little knowledge."

Goldman replied, "But testing makes people afraid of knowledge."

Greenleaf's colleague Sylvain Cappell asked, "Regarding the curriculum change in New York: The city uses Everyday Math as a model. Where will we be in three or four years?"

Goroff replied, "Everyday Math, of all the reforms, has the potential for more math substance than most. But it's extremely demanding of teachers. So it depends on teacher development. It's possible to teach from any textbook. But I don't know how to solve it." Goldman added, "I'm more concerned with the politics of inequity. ... But it looks like a lot of fun."

Milgram responded: "I was responsible for the decision not to accept Everyday Math in California. At first I did, but then I realized we couldn't. We can't use it in New York, either. The teachers' manual is a mess! In four years, the situation is going to be a disaster."

A teacher asked from the floor, "Where is the responsibility? How do we put things in place?"

Milgram said, "It took thirty years for Japan to get where it is. For us, it won't happen overnight. We need teacher content knowledge. We have to concentrate on future teachers. It's going to take time." Goldman said, "I see a lot of hope in education research. We have to reexamine schools of education."

Another listener commented, "I'm a mathematical physicist at Rutgers. There's a chasm between math and math education. Pretty soon one falls into this pit. Mostly the two sides don't contradict each other. In New Jersey, there's strong collaboration between mathematicians and teachers. We need to stop the war!"

The audience applauded.

A woman who said she was from New Visions for Public Schools said, "Having mathematicians teach elementary school teachers is not so good. The gap is language."

On that note, the proceedings were closed and the participants thanked. As the audience milled around, Goldman was talking excitedly to an audience member. "You know, math is a dance!" she said. "I even wore my dancing skirt today!"

For this reporter, Sylvain Cappell had the last word: "All this talk of peace! Clausewitz says, 'The aggressor is always in favor of peace. He wants to occupy the enemy's territory unopposed.'"

Jessica Raimi is an interested onlooker.

NYU breakfast - May 13, 2003

(Return to the NYC HOLD main page.)