In our introduction to this issue of Mosaic, we referred to Paulo Freire’s description of literacy as “reading the word and world.” That same phrase-with its dual emphasis on the concrete and the abstract—can be used to characterize EDC’s definition of mathematical literacy. For nearly four decades, EDC has been developing and promoting an approach to mathematics education that emphasizes mathematics as a system for understanding the world. Consider, for example, this introductory paragraph from Connected Geometry, EDC’s new high school curriculum:
Geometry is the attempt to understand space, shape, and dimension. Parts of “geometry”-earth-measuring-grew out of the age of explorers to map where they had been, and of landowners to determine their holdings. Other parts were invented by artists, who wished to portray convincingly what they saw with their eyes of saw in their minds, and by inventors and engineers who wished to make devices that would fit together and work. Geometrical ideas have also come from the needs of architects and builders whose work needs to be both strong and beautiful, and from surveyors, planners, and workers who must be able to assure that tunnels of railroad tracks built from both ends will actually meet in the middle.
Teaching students to see and use mathematics in this way can be a formidable challenge. In our curricula and staff development programs, we aim to build understanding of basic mathematical skills within the broader context of what we refer to as the “big ideas” and “habits of mind” that get at the essence of mathematical thinking [see “The Big Idea Behind Regrouping”].
Usually, when we think of what it is that makes up the content of mathematics, we think of learning things like the times tables or subtracting by borrowing. But each of these is just the tip of the iceberg. Underlying each is a much larger, deeper, and more important mathematical idea. For example, let’s focus on the concept of borrowing in subtraction. To solve the equation 23 minus 15, you would have to “borrow,” that is, cross out the 2 in 23, make it a 1, then put the 1 in front of the 3 to get 13, from which you would subtract 5 to get 8. A deeper way of looking at what is happening here is that you are breaking the number 23 into two separate numbers: 13 and 10. When you borrowed the 10, you really converted that 10 into 10 ones, and added them to the 3 to get 13 ones. So you have 13 ones and one 10. Then you can subtract the 8 ones from the 13 ones. The “big idea” here is that all numbers are built up from ones and tens and that it is possible to take any number apart accordingly. So, as long as we teach only the tip of the iceberg, the mechanics of borrowing, students will just be memorizing rules. If they forget the rule, they’re sunk. When they learn the “big idea,” however, they understand something fundamental about the way numbers work that they can apply in lots of other contexts.
—Barbara Scott Nelson, director of EDC’s Center for the Development of Teaching, as quoted in a roundtable discussion on
The Learner Online’s Guide to Math & Science Reform
This fall, we launched one of our most ambitious mathematics education projects: developing training materials for thousands of tutors participating in America Counts, the mathematics counterpart to America Reads. As part of the high-profile project—a collaboration between the Department of Education and the National Science Foundation—college students participating in the federal work-study program will be trained to tutor mathematics students in grades K-9.
Part of the challenge in developing a training program and materials for college students is confronting the traditional, limited role of the “math tutor.” Math tutors tend to be associated with the most narrow definition of mathematics; the tutor is someone who shows up to help struggling students complete their homework or prepare for a test.
Project co-directors Mark Driscoll and Betty Bjork say that one of the goals of America Counts is to train the college students to be mentors, rather than simply tutors. As such, the training will focus on the importance of relationships—relationships among mathematical ideas and relationships between themselves and the students. “Mentoring suggests an enriching and ongoing relationship between the younger and older student,” Driscoll comments. “We’re developing concept-rich approaches to mathematics, and we’re inviting the kids to talk with the mentors about the hurdles they face in their math studies.”
As those relationships develop, the mentors will look for opportunities to deepen the younger students’ understanding of mathematics. “Kids come to tutoring with an immediate need for getting their homework done for tomorrow,” Bjork acknowledges. “We want to allow time for that work but also to develop some strategies for the mentors to enrich the experience, to move beyond the immediate need in order to do something that complements what the student is doing in class and to provide some alternative ways to look at a problem. More of the same fails with a lot of kids. You need to provide some different avenues and different handles on the problem to get kids engaged and to help them experience some success. Developing a sense that they can do it goes a long way for children.”
To build conceptual thinking about mathematics—along with the needed skills—mentor materials are grouped around important mathematical concepts, including number sense, algebraic thinking, geometry and measurement, probability and statistics, and word problems. Materials are also grouped according to age so that a mentor can choose strategies for approaching statistics and probability with a kindergarten student or with a ninth grader. The materials and training are designed to enable the mentors to assess students’ current abilities, enhance their conceptual understanding, and address any needs for remediation. Each unit includes a pre-teaching assessment tool, a checklist of mathematics concepts addressed in the unit, and approximately 8-10 mathematical investigations.
The training takes place in two full-day sessions. The initial meeting features active engagement with mentoring materials in several of the content areas; simulations of mentoring sessions; the use of case discussions to provide insight into common mentoring dilemmas; and preparation for the logistical aspects of the mentoring process. The second day of training, occurring several months after the first, introduces the materials for other content strands and help mentors deal with challenges that have arisen in their first months of mentoring.
Eventually, the training materials will include videotapes of mentoring sessions that will serve as mentoring guides-in particular, modeling the use of diagnostic questioning to help mentors better understand student thinking. It is those kinds of dynamic interactions around mathematics that build relationships while also building the mathematical literacy of the participants-whether they are students, mentors, or experienced teachers.
Originally published on May 31, 2000