For more than thirty years, Paul Goldenberg has worked on the front lines of the “math wars,” watching as educators, researchers, and communities battle over the relative merits of competing approaches to mathematics education: Traditional “drill and practice” instruction versus “reform math”; back-to-basics versus “Standards-based” methods. Having witnessed the latest round of skirmishes in the conflict, Goldenberg is more convinced than ever that educators need to look beyond these polarizing positions to create mathematics materials that are excellent and effective for a large and diverse student body. “Learning mathematics shouldn’t be a trade-off: either you memorize your multiplication tables or you learn to think conceptually,” says Goldenberg. “I want my kid to do both. I think most parents do.”
To prove the point, Goldenberg and his EDC colleagues are developing an elementary school curriculum that draws on the best of traditional and reform methods. The curriculum is a collaboration of EDC, the National Science Foundation (the driving force behind reform mathematics), and Harcourt School Publishers, a leading publisher of basal textbooks. “We can develop mathematics materials that reflect national standards and embody the National Science Foundation’s goals of applying best research-based practice to mathematics education, but that also appeal to traditional communities and schools,” says Goldenberg.
For inspiration, Goldenberg has looked back 40 years to a classic mathematics curriculum called Math Workshop. “When you first skim through an old version of Math Workshop, you see lots of calculation. It looks like a basal-type curriculum,” he says. “But when you read it closely you realize that the exercises guide students very thoroughly through the ideas behind the calculations. This curriculum has always shown me that there don’t have to be ‘math wars,’” says Goldenberg
The new Math Workshop will feature a heavy dose of skill-building calculations, while emphasizing understanding. Students get plenty of opportunity to practice skills, but the skill-building is embedded in investigations of new ideas and solving of problems. “Nothing should be arbitrary in mathematics,” says Goldenberg. “Kids should always understand why things work.” The curriculum introduces core mathematical concepts beginning in kindergarten, and makes those concepts concrete with number games and physical representations.
For example, across the early elementary grades Math Workshop students explore intersecting lines as they work to understand what multiplication is and how it differs from addition. MW uses the same image in varying multiplication contexts such as identifying the intersections of streets and avenues in a town as well as combining shirts and pants to make outfits (Figure A).
This model offers an illustration of the differences between addition and multiplication. While addition describes the total number of items in the original sets (pants + shirts, 2+3=5), multiplication involves the pairing of each item of one set with each of the other (each pair of pants is paired with each shirt, (2x3=6), (Figure B). Further, children can use this representation as they consider the relationship between 2x3 and 3x2 (commutativity) since it is possible to see the six dots as two rows of three or three columns of two, without changing the total number of dots.
Making mathematics accessible to all kinds of learners is an explicit goal of the project, according to Goldenberg. “You don’t want to handicap kids in math who are struggling with reading,” he says. As a result, the student worksheets feature very little written instruction. This is designed to reach visual learners and students who have difficulty reading English, and to build mathematical ways of working. “In mathematics, information is conveyed differently from text-reading—it is not left to right, top to bottom,” says Goldenberg. “Our worksheets give kids practice in looking for patterns and applying them. They need to use reasoning and deduction skills all the time. This is a mathematical way of thinking and it even supports test-taking skills.”
Although much of the new Math Workshop was inspired by the original curriculum, there are significant differences between the two. For example, certain topics that are considered critical in today’s math classes, like data analysis, were not covered in the original materials. The new teacher’s guides are also written differently. The original Math Workshop did not include lesson plans. Instead it provided activities and other materials for teachers to draw on as they developed their own lesson plans. “Today we have to provide lesson plans—teachers are rarely given the time to create their own—but we are trying to make them respectful and flexible,” says Goldenberg.
Of course, the biggest changes to hit mathematics education since the original publication of Math Workshop are the Standards movement and the No Child Left Behind act, with its strong emphasis on state frameworks and testing. To ensure it meets all state-level demands, the new Math Workshop incorporates state framework guidelines from Texas, California, New York, Illinois, Florida, North Carolina, and Massachusetts, as well as the guidelines proposed by the National Council of Teachers of Mathematics. “We take this obligation very seriously,” says Goldenberg, “because the kids, the teachers, and the school all get punished if the kids don’t do well on these tests. We have to protect our kids from the consequences of these tests.”
Teachers also rely increasingly on curriculum materials to ensure they are meeting state standards. “The curriculum is like a safety net for them,” says Goldenberg. “They have to know that the materials are covering all the bases. But they also need room to use their own judgment. It’s a hard balance to strike, but we are committed to getting it right.”
In pursuit of that balance, the curriculum is being extensively pilot- and field-tested. A team of seven researchers conducts classroom observations and teacher interviews, as well as collecting teachers’ written comments about each lesson; they also track and synthesize information about kids’ understanding. All of this information is used to guide revisions.
Researcher Lynn Goldsmith shared some anecdotal reports from the field-tests conducted so far: “We repeatedly hear that teachers are surprised to see their kids thinking as deeply as they are. This curriculum offers activities for kids that lets them engage with mathematical ideas and skills in powerful ways. Teachers feel they’re seeing greater mathematical understanding among their students than they’ve seen in years past. The kids also really enjoy it. We’ve had teachers tell us that for the first time in their careers, kids are choosing math games during ‘free choice’ times and saying that math is their favorite part of the day.” Teachers are also reporting that they, themselves, are learning a lot. “Many say they’re seeing more connections among mathematical topics that they’ve taught quite separately in the past.”
In the next phase of the research, the team will investigate how the curriculum affects teachers’ mathematical understanding. “Some of our teachers are now in their third year with the curriculum, so we can track their learning,” says Goldsmith. The curriculum features a “learn-by-doing” model of professional development, designed to allow teachers to gain a more profound understanding of fundamental mathematics through the natural course of their daily work. “The curriculum is designed to be accessible to districts that don’t have a budget for massive in-service training before using the curriculum,” says Goldenberg. “So on the surface it looks like any other curriculum, but it provides a lot of implicit support for teachers.”
Inspiring teachers to understand and embrace mathematics education has been Paul Goldenberg’s passion for more than 30 years. “A lot of people think that the problem with elementary school teachers is that they don’t know math. I don’t think that’s the biggest problem. A curriculum appropriate for kindergarten must be largely about reading and writing numbers. Teachers may be excited about their kids’ development, but not about the math itself: there’s not much there, typically, to feed creative adult minds. Making the math intellectually stimulating to the teachers, providing some surprises for them to discover while keeping the content appropriate for the children, makes the teaching more interesting, less routine. Teachers’ interest matters to the students. We want our materials to open up that potential for teachers as well as students.”
Originally published on January 1, 2005