Algebra for Second Graders
The School at Columbia University Sets a Trend
By Jaimal Yogis
It looks like a typical second grade classroom. Seven-year olds sprawl out across the carpet playing with multicolored tiles. They assemble patterns that look like flowers, pyramids, rocket ships and houses with chimneys.
But this is neither art class nor playtime. These second-graders are learning algebra.
“I’m trying to make a ‘number times two’ pattern that looks like a flower,” says Charlotte, 7, explaining the “explicit” functional rule of a pattern that looks like a sunflower made of green triangles and yellow diamonds.
By constructing “growing” patterns with tiles and then trying to predict what will happen to these patterns as they increase, the youngsters are diving head first into algebra concepts before many of them can pronounce the word.
Here at The School at Columbia University, a new private elementary school next to the Columbia campus, second graders are beginning the second year of an experimental early algebra program. The results of the first year show that after five months of teaching basic algebra through patterns, second graders at The School, and their counterparts at a similar lab school at the University of Toronto, had a much better grasp of multiplication and algebraic reasoning than students who began with a more advanced mathematical base. The Columbia children were compared to a control group of second graders at an elite Toronto private school.
“The overall story is that more and more early algebraic reasoning is being thought of as very important to later mathematical understanding,” said Joan Moss, a researcher at the Institute of Child Study at the University of Toronto, who initiated the project. “We discovered that by helping students merge their knowledge of visual patterns with what they know about addition and multiplication, we are helping them understand mathematical relations well beyond their grade level.”
The pre-algebra project is The School’s first major research endeavor and the results are already eliciting national and international attention. “We’re receiving invitations to conferences all around the world,” said Moss.
The School opened last year with students who are half children of Columbia faculty and half from the surrounding community – from Harlem down to Columbus Circle. Because of The School’s university location and diverse student body, administrators believe studies here will have a broad impact on public education nation wide, much as projects from the laboratory schools at UCLA and the University of Chicago have had in the past.
America is poised for change in algebra teaching. The nation’s eighth graders placed a grim 28th compared to those in other industrialized nations in a 2003 international math test. The project at The School is a small piece of a an early algebra movement that has been in motion for more than a decade, but has yet to trickle down to most public schools. Researchers at The School and the University of Toronto think they can change that.
The National Council of Teachers of Mathematics published new standards in 2000 that suggested students as young as kindergarten should begin “algebraic reasoning.” But researchers at The School said that the council does not explain how elementary teachers, many of whom are not prepared to teach algebra, should implement these standards.
“One of the most talked about subjects at national math conferences right now is why American students have such problems transitioning to algebra in high school from hard arithmetic,” said Arana Shapiro, one of the second grade teachers at The School. “We’re trying to develop a curriculum that can smooth that transition out.”
The School’s curriculum is not just playing with blocks. It’s about developing conceptual ability in mathematics. The children use a computer program called Knowledge Forum, which allows them to talk with each other about math problems on their own laptops. Last year, the students even chatted on the database with second graders in Toronto about certain patterns. The teachers also interview the students privately, video tape the interviews, then show them to the class to facilitate discussion about different ways to arrive at the same answer.
“This has reaffirmed my whole belief in kids talking about ideas,” said Patti Macdonald, one of the second-grade teachers. “We’re always asking kids to do things, make things. This shows how important it is to let kids take activity and translate it into ideas.”
In one of the private interviews from last year, Macdonald asked Rohit, 7, about a pattern of blocks that increased by three in each position. “Is this a pattern?” Macdonald asked the boy, who sat restlessly in a red sweatshirt, peering down at a series of red squares and green triangles.
“Well it’s kind of a repeating pattern,” Rohit said, scratching his black hair. “It adds a three every time. It adds three, six, then adds another three.”
The questions then became more difficult. Macdonald asked Rohit to find the pattern’s algebraic rule – only not in those words – then to predict the 7th, 8th and 10th positions of the pattern. Rohit paused; then got it. “Number times three,” he said confidently, “number” being the exponent of the equation “n x 3”. Rohit was then able to get the 7th position instantly by multiplying seven times three: 21.
It’s not only impressive that Rohit was able to get the equation, which many middle school students would not be able to do. Rohit was using multiplication at a time when the students had not even learned what multiplication was.
“That was the amazing thing about this,” Macdonald said. “The children had a better conceptual understanding of multiplication than the control group, who formally learned it. It might have been difficult to say, ‘what I’m doing is multiplication,’ but they could understand the concept and explain it.”
Rohit’s interview and others were then showed to the class. The students seemed to “get” the ideas approach. When Macdonald asked them why math interviews might be helpful, the then second grader Jocelyn said: “I think it’s a good idea because everybody has a different idea. You can take half of that idea and half that idea and put it together to make a whole idea.”
“I think we should try to see if there are different ways to figure it out,” Zarina responded. “Some people use strategies...if you take two answers and put them together it might not turn out the right answer...then you have to take them apart again and figure out another person’s strategy. It’s kind of connecting the sides.”
Parents of second and third graders at The School said they were happy that their children are enjoying math so much and unanimously said that they trusted The School was not using their children as “lab rats.” But some are still a bit wary of the new conceptual approach to math, even though it is balanced with a hefty dose of traditional arithmetic teaching.
“I’m not comfortable with it because I didn’t grow up with it,” said Rudy Arnold, whose daughter Sydney was in the second grade last year. “It’s not tangible. I still like the instant gratification of 2 x 2, 3 x 2.”
Arnold, who envisions her 6 and 8-year olds at Harvard or Yale, said her children absolutely love math now. But she added: “I just hope my daughter knows her multiplication tables by the end of the third grade.”
It is skepticism like Arnold’s that has kept many American schools from heading toward early algebraic learning, unlike those in countries with top math scores like the Netherlands, Singapore, Japan and China. The conceptual approach does not always lend itself to standardized test results, which have grown in popularity in recent years.
But America’s international test scores are not even close to those of the top math countries, which tend to teach algebra much earlier than schools in the United States. “The math teaching in educationally successful countries is much richer,” said Kaput, Chancellor Professor Mathematics at Dartmouth College and one of the original advocates of early algebra. “The contrast is very stark.”
It’s that stark contrast that catapulted Kaput and a group of math professors around the country to lobby for a new approach to algebra education in the late 1980’s. Using the successful models of countries like the Netherlands, Kaput said the group influenced the math teachers’ council to change its standards in 1990 and suggest early algebraic reasoning in all schools. With more research and evidence in the late 90’s, the national council made these standards even stronger in 2000.
The changes have had an impact. Since 1990, algebra has been taught more widely at high schools and middle schools; and national math scores for fourth and eighth graders have increased every year. Last year, high school seniors math SAT scores climbed to their highest level in 35 years, which many experts attributed to algebra being taught in the eighth grade, rather than later in high school.
Cathy Seely, President of the NCTM, said that the Council’s new standards are more integrated than people realize. “Unfortunately it’s hard to document how many schools, or teachers within those schools, might be teaching patterning and algebraic ideas at the elementary grades,” said Seely. “My impression is that it’s pretty common in the US and Canada.”
But Kaput said that the integration process has a long road ahead. Many elementary teachers are still not prepared to teach algebra, he said. And according to the 2003 ACT college entrance exams, only about 40 percent of U.S. high-school seniors were qualified to take college-level algebra.
It’s too early to tell if the council, researchers like Kaput and new studies like the one at The School will fundamentally change American math teaching. But the US will have to do something if it wants to keep up with international standards. Seely said that Japan, Singapore and China are all shifting their curriculum to incorporate algebra even earlier.
But whether or not second graders across the country start hitting the exponential functions, students at The School are going to continue. Macdonald and Shapiro have beefed up their algebra curriculum for this year and expect the results to be even more convincing. They will soon present their findings to the American Educational Research Association.
Successful as the project was, the teachers say it has a long way to go. “We are really amazed by what the kids are capable of,” Shapiro said. “We’ve also learned from this that we have a lot to learn about teaching.”