Download enacting proof in middle school mathematics

ENACTING PROOF IN MIDDLE SCHOOL MATHEMATICS
Kristen N. Bieda
Michigan State University, East Lansing, Michigan, USA
Although justifying and proving have been gaining in importance in school
mathematics in all grade levels and across all content areas over the past
decade, little research has examined how tasks designed to develop students’
competencies in justifying and proving are enacted in the classroom. This paper
presents findings from a study of the implementation of proof-rich tasks in
classrooms utilizing the Connected Mathematics Project (CMP) curriculum.
Findings from this work suggest that middle school students’ classroom
experiences with justification and proof are insufficient for developing desired
conceptions of mathematical proof. Implications of this work for curriculum
development and teacher education will be discussed.
INTRODUCTION
Proof in school mathematics within the United States since the decline of the
“New Math” movement in the late 1960’s has typically been instantiated as a
topic in a secondary-level geometry course. However, unlike other topics in
mathematics education, little attention has been paid to how the process of
proving is taught. Students’ experiences with proving at the middle school level
is becoming increasingly important, as the research indicates that students
struggle with producing mathematical proof within and beyond the high school
geometry course (Chazan, 1993, Harel & Sowder, 1998, Hoyles, 1997, Weber,
2003). Given that tasks exist in middle school curricula, particularly the
Connected Mathematics Project (CMP), that provide opportunities to engage
students in justifying and proving their reasoning (G.J. Stylianides, 2007), it has
been unknown, to this point, the process of enacting proof-related tasks. The aim
of the work presented here is to shed light upon the process of learning to prove
in middle school classrooms from the perspective of understanding a pedagogy
of proving as it currently emerges in practice —specifically, to answer the
question: How do curricular opportunities intended to engage middle school
students in justifying and proving become enacted?
CONCEPTUAL FRAMEWORK
As the mathematics education community has understood the potential and
importance of developing students’ competencies in proof and more research is
starting to examine the pedagogy of proving, there is a need for a widely held
definition of proof that takes into account the process of proving as it evolves in
classrooms. The following framework of proof, which guided the work
presented here, aligns conceptions of proof from the discipline with the demands
of school mathematics:
Proof is a mathematical argument, that is, a connected sequence of assertions for or
against a mathematical claim, with the following characteristics: (i) it uses
statements accepted by the classroom community (set of accepted statements) that
are true and available without further justification; (ii) it employs forms of
reasoning (modes of argumentation) that are valid and known to, or within the
conceptual reach of, the classroom community; and; (iii) it is communicated with
forms of expression (modes of argument representation) that are appropriate and
known to, or within the conceptual reach of, the classroom community. (A.J.
Stylianides, 2007, p. 6)
One critical feature of the definition is the prominence to the role of the
classroom community. As proof emerges as a product of discussions about what
is required to justify a conjecture for all cases and how various forms of an
argument influence a proof’s potential for justification, students learn that proof
goes beyond convincing one’s self and convincing a friend, but it must convince
the broader classroom community.
With Stylianides’ (2007) proof framework in mind, the design of this research
draws from a framework of the implementation of a mathematical instructional
task (Stein et al., 1996). The process of enacting mathematical tasks begins with
the curricular task either developed by a teacher or written in a textbook (task as
written), evolves as the teacher conceptualizes the task to be implemented (task
as intended), the task is introduced and taken up by the students (task as
implemented), and student learning results (task as enacted). Along each stage
of the model, there are several factors that influence the success of the task
enactment. For example, the teacher’s content knowledge, both pedagogical and
mathematical, plays a significant role in how the teacher decides to use the task
from its intended form to its implemented form. As this study seeks to
understand the pedagogy of enacting proof-related tasks, the focus of the
research is on how the task as intended is implemented and becomes enacted in
the classroom.
METHODOLOGY
Data was collected through observing lesson implementations in seven middleschool teachers’ classrooms. Three 6th grade teachers, two 7th grade teachers,
and two 8th grade teachers and their students participated in the study. Each
lesson observed was identified to contain a task that engaged students in
generating conjectures prior to the observation. All of the classrooms in the
study utilized the Connected Mathematics Project (CMP) curriculum, a
curriculum where 40% of the tasks are proof-related (G.J. Stylianides, 2007). At
least six lessons were observed per teacher; there were a total of 49 observations
across all seven classrooms.
During the lesson observation, the researcher took fieldnotes along with
audiotaping to document the most salient classroom events and discussions
related to students’ conjecturing and justifying. The fieldnotes were analyzed to
distinguish elements of the implementation as proving events. A proving event
consisted of an initial response provided by a student to a task eliciting a
conjecture, with the teacher’s and students’ feedback to the student’s initial
conjecture. Elements of the fieldnotes were coded using the A.J. Stylianides’
(2007) proof framework at the turn level; thus, individual student and teacher
moves related to each part of the proving process were coded as separate
instances. The product of this analysis provided information as to what parts of
establishing proof were enacted in the proving events, as well as how frequent
each of these activities was in the proving process for each classroom. Once the
parts of the proving events were determined, the data were utterances from the
teacher and the students were coded using a constant comparative method of
coding (Glaser & Strauss, 1967), focusing on the teacher’s and students’ moves
during proving events.
RESULTS
Analysis of the fieldnote corpus revealed that while proof-related tasks were
generally implemented as written (71% of the time), roughly half of the
opportunities to prove, students’ generalizations provided in response to the
tasks, were never supported with justifications (see Table 1).
Grade
Level
Number of Problems
as Implemented
Number of
Opportunities to
Prove1
Number of Opportunities
to Prove Never Supported
with Justifications
6th
31
65
24
7th
13
28
17
8th
8
16
9
TOTAL
52
109
50
Table 1: Opportunities to prove generated from problems implemented
Table 1 shows that there was little difference in the rate at which students in 7th
or 8th grades produced justifications compared to 6th graders, suggesting that
students have difficulties with justification and proof throughout middle school
and teachers fail to address the needs of students as they learn to justify their
reasoning. Data analysis revealed several trajectories that proof-related tasks
followed during implementation (see Figure 1). Although each task ultimately
resulted in either being enacted or not enacted, depending upon whether
justifications were provided to generalizations in response to proof-related tasks,
there were three outcomes in terms of the teacher’s and students’ responses to
the generalization.
1
As each problem was implemented, more than one opportunity to prove could be generated.
As such, the totals reflect situations where more than one opportunity to prove emerged
during the implementation.
Student(s) Make
Generalizations (i.e.
claims, conjectures)
Student(s) Justify Generalizations
(Proof-Related Problem Enacted)
Student
to
Student
Feedback
Teacher
Feedback
No
Feedback
Given
Student(s) Do Not Justify
Generalizations
(Proof-Related Problem Not Enacted)
Teacher
Sanctions
Conjecture
without
Justification
Given
Teacher
Request
for Student
to Student
Feedback
No
Feedback
Given
Figure 1: Trajectory of Proof-Related Tasks
Of the 50 proof-related problems not enacted, where students did not provide a
justification for their generalizations, the most prevalent response was for no
feedback to be given to the students’ generalizations (42% or 21 out of 50). In
essence, these emergent opportunities to prove were ignored by the classroom
community. The following excerpt from Mr. Zeff2’s 8th grade class illustrates
this type of response, or lack thereof.
In this excerpt, students work on an investigation from CMP based on the classic
“wheat and chessboard” problem, using a context of a king providing a reward
to a peasant for saving the life of the king’s daughter. The king allows the
peasant to stipulate her reward, and the clever peasant suggests that the king
place 1 ruba, comparable to a penny, on the first square of a chessboard,
followed by 2 rubas on the second square, and 4 rubas on the third square,
continuing in the same manner of doubling the previous amount until all of the
squares had been covered. An alternative plan presented, Plan 2, reduces the size
of the chessboard to 16 squares, but the king has to triple the amount of rubas he
places on each subsequent square. In the exchange below, Mr. Zeff leads a
discussion about the two plans:
Mr. Zeff: Ok, what are the similarities between both plans?
Brian: One gets multiplied by 2 and the other is multiplied by 3.
Damon: Yeah, Plan 2 is always higher.
Mr. Zeff: Let’s have a little caution on that one, just from what we see so far.
Lakesha, will you share your equation?
2
All names presented here are gender-appropriate pseudonyms.
Lakesha: I got
.
Mr. Zeff: Usually I start the equation with r equals. You can do it anyway you want.
[writes
on the overhead projector]
Mr. Zeff: What are we doing over and over again?
Brian: Multiplying by 3.
Mr. Zeff: Right, and just by trial and error you see that it is always one box before
it. What is another reason...why do you subtract one from it?
Brian: You triple every single square but one.
Mr. Zeff: Which one?
Brian: The first square.
Mr. Zeff: Ok, so we didn’t triple the first time so it’s always one less.
Mr. Zeff: Which plan is better?
Damon: Plan 1
Mr. Zeff: Which is better for the King?
Brian: Plan 2.
Mr. Zeff: [puts up graph of exponential equation on overhead projector] OK, now
let’s look at your graphs. You should have a very, very straight line at the bottom.
Throughout the excerpt, we see few requests from Mr. Zeff for students to
provide justification for their generalizations. As a result, he implicitly sanctions
their generalizations without insisting upon justification. Further, Mr. Zeff’s
decision to have students continue sharing answers, without justification,
reinforces a classroom culture where justifying and proving are not essential
components of everyday mathematical practice.
Although space limitations do not allow for a full presentation of results and
examples from the data, there was evidence of missed instructional opportunities
during the proving events across the middle school classrooms. Across all grade
levels, there were only 28 opportunities to prove supported by justifications
consisting of general arguments. When examining how teachers addressed
students’ justifications as they were made public, nearly 30% of the
justifications, primarily based on general arguments, never received feedback.
Further, when a teacher sanctioned a justification with a positive appraisal, it
was just as likely that it was a justification based on non-proof arguments as it
was a justification based on general arguments.
CONCLUSIONS
The results in this study reveal that middle school students produced few general
arguments in response to proof-related tasks, a finding not surprising
considering the literature on students’ difficulties with proving. Stein et al.
(1996) defined several factors that contribute to the decline of high-level
cognitive activities such as generating proof, including time constraints and lack
of accountability. As the excerpt presented illustrated, the relative lack of
sufficient critical feedback once a proof-related task was enacted (e.g., pushing
students to move beyond examples-based justifications) contributed to relatively
few justifications based on general arguments. A confluence of factors,
including time constraints, lack of accountability, and lack of curricular support,
worked against the development of a community in the classroom where one’s
reasoning, once public, was expected to be scrutinized. During the study, I will
present additional data to support these conclusions.
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