My research will describe an alternative sixth grade curriculum and pedagogy that intends to address two major issues in math education: pervasive negative attitudes and beliefs about mathematics and low performance levels on assessments involving unfamiliar problems and higher order thinking. This qualitative case study will follow a curriculum based on giving students minimally defined problems that will serve as a catalyst for student-posed problems, which in turn will shape a student-formed content curriculum. Students will begin by exploring these minimally defined problems from a constructivist viewpoint. The lessons will continue with students creating and exploring their own variations, generalizations, and extensions.
Here are some of the characteristics of minimally defined problems: they are purposely vague; they have multiple solutions methods; they have multiple solutions; they require students to determine and make assumptions. Student-posed problems are mathematically rich problems posed by students instead of the teacher. Variations, extensions, and generalizations are types of student-posed problems. Variations involve minimal changes to a problem such as changing a number. Generalizations involve changing the problem to address a set of variations. Extensions are a bit more nebulous, but for now will be defined as new problems inspired by the original. Students will then cycle between creating and exploring subsequent student-posed problems and the corresponding conceptual, skills-based, and procedural tools necessary to solve these problems. The resulting student-formed content curriculum is the sum of the concepts and skills that students learn in order to make progress on their own problems.
How little information and scaffolding can we give students where they can still progress mathematically? What if students started with nothing but the following?
Could a classroom structure be developed so that deep mathematics still happened? How would you measure success in this structure? One major aspect of my research will be a reflection on the meta-curriculum—the necessary structure to ensure that students are successful in creating, exploring, and solving mathematical problems—needed for student success.
My second research question will focus on student beliefs and attitudes about mathematics. How will this environment affect students’ productive dispositions, or “habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy” (Kilpatrick et al., 2002, p. 116)? How will beliefs (such as the length of time students think it should take a good math student to solve a problem) and attitudes differ from other structures and change over time? Will students be more engaged? Will students have a greater sense of relevance and ownership?
My third and final research question concerns mathematical content. What mathematics will students explore if they are given open-ended problems that are not intended to lead to a particular concept or skill? I will study the mathematics content that students develop, and compare this to an age-appropriate traditional curriculum.