A word not often associated with mathematics is “obvious”. We usually ignore the obvious. That which is obvious is most often in the background, but mathematics tends to raise the obvious, almost in relief, and pulls it to the front of the stage. Mathematics looks at how those obvious things are hooked together, and casts these relations in symbolic representations. So, when we talk about abstraction in the mathematical sense, why do so many think of it as difficult? In part this is because in mathematics we’re often asked to attend to things, which in normal everyday life we don’t attend to, because they’re obvious.
The Algebra Project’s Five-Step Curricular Process is a pedagogical approach that aims to bring students from an intuitive understanding of the world around them to a deeper understanding of the abstract concepts and procedures of the mathematics they are learning in school. After students have moved through the first step of the process – a shared concrete experience – they are asked, in the second step, to make a picture or model of that experience. This step provides affordances to student learning at a number of different levels. The process of drawing pictures or making models gives students the chance to bring their imagination and creativity to the task as an artistic endeavor itself. For example, when the shared event is a trip, either on the subway, a bus or even a walking tour, students often paint a mural capturing the things they find most interesting about the trip. This step in the curricular process, for most if not all students, does not feel like they are doing mathematics. At this stage students are not yet dealing with anything that they recognize as mathematics but are simply representing their experience in a way that seems intuitive to them. “It’s art!” But this step in the process is actually the first step in a process of abstraction that will lead to what students clearly recognize as “real”, or more familiar, mathematics.
When students construct a picture or a model of the event, that model is a representation of the experience. It emphasizes those aspects that the students viewed as important or necessary and ignores other aspects of the experience. That process of attending to certain features of the event and ignoring others is the first act of abstraction in mathematizing the common shared experience. Even if only intuitively, some, and at times, all of the critical features of the shared experience are being communicated through the pictures and models which students create.
If we consider how human beings process information, large portions of our brains are dedicated to processing language on the one hand and visual information on the other. This second step of creating a picture or model is paramount to creating a scenario in which the natural ways a learner processes information is developed and leads into a deeper understanding of mathematics.
If his reading of the philosopher and logician Willard Van Orman Quine’s regimentation and structuring of ordinary discourse is most prevalent in the last two steps of the Five-Step Curricular Process, which we call (4) “Feature Talk” and (5) “Symbolic Representation”, then the first three steps, (1) A Shared Concrete Experience, and (2) A Picture/Model, and (3) “People Talk”, can be traced back to Algebra Project founder and Civil Rights Movement veteran Bob Moses’ apprenticeship in community organizing with Mrs. Ella J. Baker. A hidden figure of sorts to many outside of the 1960s civil rights movement, Ella Baker was deeply involved in the community organizing wing of the Civil Rights Movement. Bob was mentored by Ella in community organizing and practiced that tradition in, he referred to as, “the Mississippi Theater” of the Civil Rights Movement.
Bob came to understand that a primary tool of community organizing was the meeting, where the people most affected by an issue would come together in order to find and exercise their own voice on whatever issue local people may be grappling with. In this sense, Bob also saw the classroom as a meeting place, and consequently, it was seen as a place for teachers and students to organize themselves as a community around their common interests. In order to build a common interest around mathematics and to make it their own both teachers and students share a common experience that they will jointly mathematize and begin a process of abstraction that starts with a pictorial representation or model of the event.
For the past few paragraphs we’ve been talking about the cognitive side of the process. The other side is the socio-emotional side. Mathematics is a human activity, and the classroom is a place – a meeting tool – where teachers and students develop common interests that coincide as an interest in mathematics. They do this when they work in small groups to negotiate ideas between each other. The construction of a mural or model by a team or the whole class is a ‘low stakes’ task where students begin to work out methods of collaboration and negotiation with a set of ideas that they wholly own.
This process undergoes further elaboration when students move to Step Three of the Five-Step Curricular Process and then must create, collaborate, and negotiate their ideas about their shared experience verbally in what we refer to as “People Talk.”