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.
The presenters describe a five-phase model for a lesson sequence on investigating and proving generalizations about the behavior of the operations. Viewing a video clip to illustrate each phase, participants consider what the teacher did to support students' learning and identify knowledge the teacher called on to make that move.
Session Type: Work Session