Rheta Rubenstein:
Cultivating students better than weeding

"Because of the changes in technology, because of the changes in our world, because we want more students to be successful, because we want them to value and appreciate mathematics and because we want them to be thinkers, not machines -- for all those reasons we wanted a stronger curriculum and one with which all students, not just a selected few, could be successful."

At the college level, she said, the slogan was "Make mathematics a pump, not a filter." "We wanted to move more students forward rather than keep more of them out of advanced mathematics," she said.

Before she wrote Integrated Mathematics, Rubenstein co-authored A Core Curriculum: Making Mathematics Count for Everyone. Initiated by the NCTM, A Core Curriculum was one of several addenda books written to address some of the most important aspects of the standards," she said. "The notion of the core curriculum is to have one set of objectives for all students. Even if we modify instruction to teach ideas concretely as well as abstractly, we can have a quality curriculum that all students work toward learning," she said. That was the concept, Rubenstein said, behind Integrated Mathematics.

"It was an ambitious objective -- to bring a wider range of mathematics to a wider range of students," Rubenstein said. "I think of it as a two-dimensional stretch. We're strengthening the curriculum and we're trying to recreate it in a way that is accessible to a wider range of students than had been studying it in the past."

"When Houghton Mifflin asked me and Tim Craine and others to write Integrated Mathematics, we worked from NCTM's original recommendations and the syllabi in the core curriculum addenda book," she said. "In the end, of course, some things changed, but the textbook series remained centered on the NCTM standards."

In trying to make math accessible for all students, Rubenstein and her coauthors incorporated the following features into their three-part Integrated Mathematics series, now published by McDougall Littell:

• Applications-oriented materials. "We wanted the students to see the usefulness of mathematics right off the bat. So applications, rather than coming at the end of sequences of instruction, are used to initiate lessons," she said.
  
  
• The use of tasks, discourse and environment. The National Council of Teachers of Mathematics professional standards for teaching mathematics talk about tasks, discourse and environment as three ways to achieve the goal of making mathematics accessible to everyone, said Rubenstein: "By incorporating tasks, we wanted students to do very challenging, interesting mathematics from which important concepts could be highlighted. Discourse means students talk to each other as well as with the teacher. They communicate with mathematical symbols, with stories, with diagrams, with tables of data and with materials. The idea of "environment," means we create a supportive classroom climate where students realize we're working together to learn mathematics."
  
  
• Active engagement. The books use Explorations for students to work as teams to develop mathematical ideas. "We have 'talk it over questions' where students process new ideas," she said. "We know that students learn best when they have a chance to digest new ideas and make them theirs. Such processing needs to occur intermittently throughout instruction, not just on homework."
  
  
• Projects. Students are invited to study in-depth math applications related to each unit. This often includes their creating a product to illustrate their learning. For example, Rubenstein said, in a unit on sequences, students track changes in costs due to inflation.
  
  
• Self-assessment questions. These questions have students ask themselves things like what are three methods you know for solving this question? Which one do you prefer? said Rubenstein: "This way they are thinking about how they're learning and which patterns are more successful for them."
  
  
• Visual learning strategies. Based on the knowledge of co-author Stuart Murphy, a visual learning specialist, the book incorporated color and other visual devices. "He helped us learn how to make the book more three-dimensional so we could make it more accessible to a wide range of students, not all of whom are strong readers," she said. "We wanted to give them other ways to get into the mathematics besides just text."
  
  
• Content integration. Geometry, algebra, trigonometry, and other strands, are connected where appropriate. For example, Rubenstein said, through a carpentry example, the book connects slope, similar triangles, proportions and tangent ratios -- algebra, geometry and trigonometry -- to look at the steepness of a roof. "The point is that when you're doing mathematics in life it doesn't come to you in capsules that say 'this is geometry and this is algebra'," she said. "You have to recognize what mathematics to apply where." Also, she said, students see the unifying idea, in this case ratio, that underlies and connects the various topics.
  
  
• Multicultural applications. "We tried to make the book multicultural to help youngsters see that people across time and around the world have contributed to mathematics," she said. "We hope that current learners will also be contributors in the future. I'd like to break down the notion that math has always existed in one way and help students see that it's created by human beings like them and they, too, can continue to enrich the legacy. Also, we wanted students who come from other countries to see that we honor and value their culture's contributions to our world."

"When the materials finally came out, teachers I had worked with before said 'Now I know what you're talking about!'" Rubenstein said. "They couldn't picture before how anyone could weave the subjects together. We worked hard to keep the development in sequence from lesson to lesson, chapter to chapter, and year to year. Teachers had trouble imagining how they were going to do the cooperative learning, the explorations, the use of technology, the use of materials, so we tried to mold staff development into the books too. We tried to write very clear directions and questions for the students so the teachers could manage everything rather easily."

In 1995 Rubenstein also co-authored Facilitating Change in School Mathematics, a monograph containing tools to help leaders, curriculum coordinators, principals, or lead teachers work with colleagues in their district. "It helps them think about why we need to change our curriculum and how we can do that in ways that are equitable and effective for teachers and their students," she said. Previously, she co-authored two books in a series produced by the University of Chicago School Mathematics Project and directed by Dr. Zalman Usiskin, Functions, Statistics and Trigonometry, and Advanced Algebra. Although their development pre-dated the standards, the University of Chicago materials were the first to give prominence to the idea of applications and technology, said Rubenstein. Functions, Statistics and Trigonometry was also a highly integrated book, she said: "It included topics not normally taught together. The topic of functions is a part of algebra, statistics is data analysis, and trigonometry is a bridge between algebra and calculus, but because some key concepts like transformations link those topics, we could teach them together in one course."

Between 1984 and 1987, she co-authored with Tim Craine Geometry Developed with Algebra, which was informally published by the Detroit Center for Professional Growth and Development. This grew out of a course that she and Crain had developed together and taught at Renaissance High School in Detroit. "It was an effort to integrate algebra into a high school geometry course," Rubenstein said. She and Craine also did a series of workshops with teachers in Detroit and its suburbs to help them teach from these materials, which were also popular and highly accessible to students, said Rubenstein. "We tried to maintain and extend their algebra skills throughout the year while they were studying geometry," she said. "We also tried to do the more concrete geometry early in the year with logic and proof later in the year so they were proving things that were already familiar."

"A lot of the things I learned from writing Geometry Developed with Algebra and FST were folded into Integrated Mathematics," she said. "Integrated Mathematics really represents a lot of my history."

Rubenstein said she really learned to love math at 13, when she read The Magic House of Numbers by Irving Adler. "My cousin Norman, who was two weeks older, made it into school a semester ahead of me," she said. "When he was in the eighth grade, I saw him with The Magic House of Numbers. It showed all these interesting things about math. At the time, what we were doing at my school was very boring -- long multiplications, long division -- all these repetitive exercises. I was not enjoying math, but when I saw The Magic House of Numbers, I was really excited. I thought, well, if Norman can do this, certainly I can! So that was one of the first things that interested me. I also had very good math teacher in the ninth grade, and we used early versions of the school mathematics study group (new math) materials. In the 10th grade, in speech class, we had to present a topic we had researched. I chose topology and in doing the research discovered in the library many wonderful mathematics books that were not textbooks. They also had a very positive effect."

Rubenstein, currently a mathematics professor at Schoolcraft College in Livonia, Michigan, said she tries to use Howard Gardner's theory of multiple intelligences in her teaching. The idea is that people are intelligent in a variety of ways: linguistically, kinesthetically, musically, visually, interpersonally and intrapersonally as well as mathematically. "So I try to incorporate multiple approaches in my teaching," she said. For example, to support students with linguistic intelligence, she explains etymologies of math words to help them link the meaning to their roots. "I use diagrams for visual learning, groups for interpersonal learning, and reflective writing for intrapersonal learning," she said. For musical learners, she has created songs and raps. "For kinesthetic learners I use materials and graph aerobics where students actually stand up and use their body to shape the graphs," she said. "I try to do these things to help the students who may be learning in different ways to grasp the ideas." She has given several presentations on this topic at professional conferences.

Prior to teaching at Schoolcraft Community College, Rubenstein taught math education to pre- and inservice teachers at the University of Windsor. She also taught for 20 years at the high school and junior high school level in Detroit, Michigan.

Rubenstein has worked with multiple co-authors on many of her projects, and says she really enjoys working with a team: "I get a great deal of enjoyment from that process. I love to share ideas with others." How do they make it work? In Integrated Mathematics, she said, the senior authors did all of the outlining and sequencing for the three books, and then each senior author and other authors were responsible for writing certain chapters. "With textbooks I've worked on there's been an editorial team that's given the work one voice at the end," she said. "I've been very fortunate to work with excellent editors." When working with multiple coauthors, she said, it helps to have a leader. "The main thing is to have a shared philosophy or sense of what you value as an approach," she said. "I see collaboration as a teaching-learning process. You're sharing your thinking with others."

Rubenstein says she can write anytime, but works best at a computer: "When computers first came out, I thought I could never write at a computer. But now I find the computer helps me think. I can draft and revise with it easily." When does she feel she's completed a project? "When I have the hardcover book in my hand," she said. "Then to celebrate, I go ballroom dancing with my husband." She also enjoys traveling with her husband, Howard. Their favorite spots: France and Thailand.

— reported by Kim Pawlak, 1998