Revitalizing Algebra: Unstructuring Tasks
REvitalizing ALgebra (REAL) is a Partnership between San Francisco State University (SFSU) and five local Northern California school districts. The goal is to improve the performance of all students (particularly minority students) in algebra, both in K-12 schools and in college. We did two years of work with each of two cohorts. The first year included an intensive two-semester weekly class and a three-week summer session. Each cohort of 27 participants consisted of approximately equal numbers of secondary algebra teachers, graduate student instructors for remedial algebra at SFSU, and math majors aspiring to be teachers. In addition to participating in the REAL class, each undergraduate math major worked ten hours per week in an algebra class of one of the participating secondary teachers. In this report we will focus on the secondary teachers. During the second year, secondary teacher leaders worked with teams of teachers in their own departments at their schools using common daily preparation time (paid for by the REAL project) to help each other examine the day-to-day effectiveness of their teaching. They also began to wrestle with more challenging pedagogical issues as a basis for permanent growth in their department teaching culture.
Prior to working with teachers, we sat in on classes in their schools. Generally, we visited classes of students in the lowest track. During those initial classroom visits, most teachers were asking questions that required short computations and little or no reasoning. On further inspection the problems, even those originally designed to be open to multiple solution methods, had been augmented with "scaffolding" which served to reduce the work to a series of small steps that required little thinking. For example, directions were added to tell students to make a chart and look for a pattern. Sometimes the directions even gave the column headings for the chart. In other cases explorations were limited. For instance, in a problem that originally asked students to come up with many kinds of function output patterns, directions channeled students to only the linear patterns and then broke down the process of finding the patterns to a step-by-step algorithm.
How much scaffolding or structure is effective? By restricting the choices and creativity of students, in principle a teacher could gain
- more certainty of the mathematics being used and less complexity of managing different groups working in different ways;
- more control over any resulting whole class interaction and greater ease of grading student work;
- more student confidence as they succeed at tasks their teacher thinks they can accomplish;
- more student confidence as they feel certain which "direction" their thinking should go.
On the other hand, by providing less structure, one might gain
- student creativity, flexibility, active problem solving, and the sense that mathematical struggle is an essential part of math and not something to be ashamed of;
- student motivation to engage in mathematical discourse;
- student exploration, and pride of ownership of their solutions;
- student belief that math is more than a small number of computations to be done quickly and a large number of problems whose solution methods must be memorized; and
- student engagement and interest in the mathematics!
Our concern was that most of the low-track algebra classrooms, which were mostly serving minority students, displayed none of the gains (1) - (4) of more rigid structure and lacked all of the benefits (5) - (9) of less structure. We saw students who were unsure of themselves and unable to take any risks. The goal was to be right, not to learn through multiple attempts to solve the problem. Students would hide their work from each other and give up when they didn't find answers quickly. Classes were boring and students were not engaged.
