Proof is a core idea in the discipline of mathematics, but an understanding of what proof is, how proving is carried out, and what a completed proof signifies has been shown to be lacking at all levels of education, both in the United States and elsewhere.  Many national reports have called for increased and more coherent attention to proof across the K-12 curriculum (Ball & Bass, 2003; Carpenter, Franke, & Levi, 2003; NCTM, 2000; RAND Mathematics Study Panel, 2002), and "constructing viable arguments" is one of the eight mathematical practices identified as standards in the Common Core State Standards for Mathematics (2010). However, in practice little movement has been made toward consistently incorporating ideas about proving into the elementary grades.  With little attention to proving, how one might come up with a proof, or what the significance of a proof is, elementary school mathematics is not preparing students for their encounters with proof in later years. 

From our own classroom-based research, we have found that representations of the operations (physical objects, pictures, diagrams, or story contexts) are accessible and generative tools for young students' proofs of general claims about the behavior of the operations. For example, consider a claim that is often implicit in students' calculation work: Given an addition expression, if 1 is subtracted from one addend and added to the other, the sum stays the same. To prove this claim, students may hold two sticks of cubes to represent the two addends and show that moving 1 cube from one stick to the other does not change the total number of cubes. They might say, "It doesn't matter how many cubes are in either of the sticks, the total will always stay the same."

Through examining and analyzing students' arguments, we have established the following criteria for use of representations for proving general claims about the behavior of the operations:

1.     The meaning of the operation(s) involved is represented in diagrams, manipulatives, or story contexts.

2.     The representation can accommodate a class of instances (for example, all whole numbers).

3.     The conclusion of the claim follows from the structure of the representation.

In order to successfully implement national calls for developing ideas about proof early in the grades, teachers must themselves understand proving as a mathematical process, learn how young students can engage in proving, and figure out how to integrate an emphasis on proof and proving into their core curriculum.  We have found that elementary teachers can learn to use a small set of instructional routines to systematically engage students to develop the habits of noticing, articulating, representing, and justifying general claims about the operations in the context of core grade-level content.  Further, we hypothesize that, through using a teaching model based on these routines, teachers will increase their knowledge about proof and about students' learning about proof.

The routines (e.g., Number of the Day, Is This Number Sentence True?) provide a starting point.  However, it is not simply the mechanical use of the routine that leads to student learning; the routines must be embedded in a teaching model that provides support for teachers in effective use of these routines to support the learning of proof.  We have identified several phases of a class's investigation of a generalization leading to proof. The teacher supports students as they:

1.     Notice a generalization about the behavior of an operation or operations.

2.     Articulate a claim about the operation.

3.     Explore specific cases of the claim with a variety of representations.

4.     Use their representations to prove their claim.

5.     Compare different operations with respect to this claim.

Each of these phases calls upon different skills, knowledge, and competencies on the part of the teacher. For example, to help students notice a generalization, teachers must identify a generalization that is fruitful for students to pursue at their grade level; they must formulate questions and problems that require students to think about connections between problems; and they must select and sequence examples to draw students' attention to the mathematical relationship that underlies the general claim.

In this work session, we will present a series of video clips, each representing one phase of the model. We will also share our own list of skills, knowledge, and competencies teachers must call upon to effectively enact each phase. We will elicit from participants what teacher knowledge and skills they see in each video clip and ask them to comment on our developing list. Specifically we will ask:

á       Are the items on our list of competencies essential for teachers to enact this phase of the model?

á       What is missing from the list?

á       Of the items on the list, what should receive priority to address in a professional development program?

Timetable:

15 minutes: We lay out our perspective on proof at the elementary grades. We present a general claim to participants and have them create a representation-based proof.

10 minutes: We present the five-phase teaching model for investigating and proving a general claim.  We share our list of competencies for each phase and pose the central questions for the session.

60 minutes: We present the five video clips (each between 2 and 7 minutes long). Following each clip, participants consider what the teacher did in the clip and what she needed to know to be able to do it. Participants review and respond to our list of competencies.

5 minutes: Closing comments.

This session addresses instructional interventions in that participants examine the actions of teachers in classroom video clips. It also addresses professional learning as participants consider what knowledge, skills, and competencies teachers call on to enact these moves and implications for professional development.