THOUGHT AND ACTION IN CONTEXT: AN EMERGING PERSPECTIVE OF
TEACHER PREPARATION
|
Sarah
B. Berenson |
Laurie
O. Cavey |
|
North
Carolina State University |
North
Carolina State University |
|
berenson@unity.ncsu.edu |
locavey@unity.ncsu.edu |
This
teaching experiment examined one high school preservice teacher’s thoughts and
actions when given a task of planning a lesson to teach rate of change to
algebra 1 students. Pre and post interviews were used to probe the preservice
teacher’s knowledge of school mathematics and her ideas of pedagogy. Results indicated that her knowledge of
ratio and proportion was incomplete and sometimes incorrect, even though she
was a successful mathematics student at the university. Despite incomplete understanding,
she planned to involve her students in collecting, analyzing, and interpreting
data with rich choices of problem types and strategies. This protocol holds
pedagogical promise for teacher educators’ as a powerful learning and
assessment tool for prospective teachers.
Introduction
The preparation of teachers has been marked by
extraordinary changes over the past two decades, generating considerable
interest and study by mathematics educators. Schulman (1986) described the
knowledge base of teaching that grounded a number of investigations (For
example, Ball, 1990; Cooney, 1994; Simon, 1995). Some of these findings suggest
that knowledge of mathematics, particularly for elementary teachers, may be an
inhibiting factor in their professional development. Some studies of preservice
teachers’ pedagogical and pedagogical content knowledge focused on the
knowledge of children’s thinking (See Tirosh, 2000). Others examined tasks,
activities, and representations used by preservice teachers to teach school
mathematics (Blanton, 1998). More recently, Ma (1999) extended the notion of
mathematical understanding to include knowledge of subject matter along with
the explanations and approaches to teaching mathematical ideas to elementary
school children. It is from this
comprehensive perspective that this study asks the question: How do prospective
high school teachers envision teaching ratio and proportion in general, and
rate of change in particular to Algebra 1 students? Over the past 25 years, a
number of studies have provided insights into the components of students’
proportional thinking (For example, Noelting, 1980; Lachance & Confrey,
1996). Few studies have examined how prospective secondary mathematics teachers
think about teaching ratio and proportion.
Schoenfeld (1999) noted that we are considerably distant from possessing
a theoretical perspective for education that unifies how we think and act. He
questions:
Is it possible to build robust theories of how we
think and act in the world – theories that provide rigorous and detailed
characterizations of “how the mind works,” in context? (p.5)
The inability to link cognition and context
theoretically creates some obstacles for researchers. For this study, we
recognized the lack of a unifying theory, but viewed our research questions
with the contextual binocular vision of Ma (1999). One lens focused on the
preservice teachers’ knowledge of ratio, proportion, and rate of change in
school mathematics. The other lens provided insight into their planned approaches
to teaching ratio, proportion, and rate of change. It is with these two lenses
that we viewed prospective teachers’ thoughts and actions within the context of
lesson planning. This qualitative research study derives its tradition from
that of a teaching experiment. Individually, ten preservice teachers engaged in
activities that were designed to access their thinking about teaching
mathematics. The context of this investigation was purposefully naturalistic in
terms of the developed protocol, with interviews before and after the lesson
planning activity.
Preservice
Teachers. Due to limitation of
space, data from one preservice teacher is reported and analyzed here. Planning
to teach high school mathematics, Chris, age 20, transferred from a community college
to the university. At the time of study, she was taking her first mathematics
education methods course with 40 hours of school internship and had completed 6
rigorous mathematics courses beginning with the engineering calculus sequence. Her 3.8 GPA indicated successful college
experiences as a student.
Protocol. The research protocol contained the following
components: a) pre-plan interview (10 minutes); b) lesson planning activity
(30-45 minutes), and c) post-plan interview (30 minutes). In the pre-plan
interview, the prospective teacher was asked to recall his or her personal
experiences and the connections of ratio and proportion to other school math
topics. They were asked for a definition of “rate of change” and if needed,
given examples of rate of change for clarification. Then the following activity
was posed: Plan a lesson to introduce
rate of change to a class of Algebra 1 students. Connect the lesson to ratio
and proportion. A number of instructional materials and a methods textbook
were supplied in a quiet corner to simulate the conditions of a teacher’s
planning activity. As much time as needed was given to complete the activity
before the preservice teacher explained her/his plan for the lesson.
Plan of
Analysis. All transcripts, videotapes,
artifacts, and field notes were reviewed to select a subject with a strong
academic record and a comprehensive lesson plan to gather as much information
as possible about prospective teachers’ thoughts and actions. Chris’s
transcripts were initially coded to determine major categories of data. From
these categories a second sort of the transcripts coded specific actions and
thoughts described within each category. These thoughts and actions were
matched to the initial categories and then reviewed to develop initial
conjectures about high school teacher preparation.
Two lenses of thought and action were used to analyze
the data of the lesson planning activity and the supporting mathematical
knowledge of the preservice teacher. The lesson plan analysis identified three
major categories and subcategories in the initial sort of the data. These
categories were activities
(students’, teacher’s), tools (students’,
teacher’s) and problems (contexts,
types, strategies). Chris’s knowledge of school mathematics that emerged from
the pre- and post-plan interviews, as well as, the planned lesson supported all
of the major categories of the lesson planning activity. The aspects of Chris’s
school mathematical knowledge that were examined were rate, ratio, fraction,
proportion, and slope.
Table 1 contains Chris’s 1) Lesson Plan Activities
and Tools, 2) Problems Posed in the Lesson Plan, and 3) School Subject Matter
Knowledge. Chris’s selected student activities for her plan to engage and
involve her students in data collection. The post interview revealed that this
activity was suggested to Chris from the array of instructional materials
available in the lesson planning corner. She portrayed her role of teacher as
leading and guiding throughout the process, while her students had access to a
number of tools such as tape measures, stopwatches, and charts. Graph tools for
both teacher and students were added by Chris as she gained new mathematical
understanding of school mathematics during the post-plan interview. The
problems that Chris selected for her lesson indicated a flexibility of thought
in posing problems that initially drew upon the students’ data. From these rate
problems, Chris then planned to pose rate comparison problems and then to find missing
x and y values of different ratios. The number of different types of problems
was enhanced with four different solution strategies found within her plan:
equivalent fractions, cross
multiplication, finding unit of rate, and within strategies. The two problem contexts selected were
distance/time and item/cost and would be familiar contexts for algebra 1
students.
Lesson Plan Activities and
Tools
|
Teacher Activities |
Teacher Tools |
Student Activities |
Student Tools |
|
·
Gives
explanations/ directions ·
Asks
questions ·
Gives
examples ·
Poses
problems |
·
Graphs! |
·
Collect
data ·
Answer
questions ·
Calculate
answers ·
Graphs
rates! |
·
Tape
measure ·
Stopwatch ·
Graphs! ·
Table/chart |
Problems in Lesson Plan
|
Problem Types |
Problem Strategies |
Problem Contexts |
|
·
Missing
value (x) ·
Missing
value (y) ·
Finding
rate (y/1) ·
Comparing
rates |
·
Set
up 2 equivalent fractions ·
Cross
multiplication ·
Finding
unit of rate ·
Within |
·
Distance/time ·
Item/cost |
School Subject Matter Knowledge
|
Definition of Ratio |
Examples of RoC |
Connections |
Unresolved |
|
·
Ratio
is a fraction ·
Rate
is a ratio ·
Ratio
is a comparison |
·
Acceleration ·
Speed ·
Displacement |
·
Remembers
rate of change from physics ·
Remembers
how to write a ratio ·
Discovers
slope is a ratio! |
·
Meaning
of proportion |
Note: “!” Indicates idea emerging in interview
Chris’s successful academic record in college
mathematics supported the planning of a lesson to involve her students
mathematically. The involvement was inextricably linked to learning school
subject matter as the plan moved from collecting data to analyzing the data to
solving problems of rate of change posed by the teacher. However, Chris’s
knowledge of school mathematics was not fully developed in several areas. Her
rendering of a ratio was in reality a fractional representation and therefore,
she considered all ratios to be fractions. In addition, Chris had never
considered slope as a ratio until the post interview where she assimilated the
new information and introduced graphing tools into her lesson plan. While
Chris’s knowledge of slope seemed to be enhanced, the researchers were not able
to resolve Chris’s understanding of proportion.
As we examined Chris’s representations of teaching,
depth of mathematical knowledge, and connections to school mathematics we saw
pieces of a puzzle that were emerging to create a dimensional and connected
picture. Beginning her study of teaching, we noted that Chris had many
strengths. Some of the pieces were
assembled, but not all were connected. Her memories of school mathematics were
incomplete and initial applications were drawn from more recent experiences in
college physics.
We conjecture that with modification, the lesson
planning task has potential as a powerful learning and assessment tool for
teacher educators. The preservice teachers all responded positively to the
individual experience that was adapted to their personal beliefs, philosophy,
and knowledge of teaching secondary mathematics. Thinking aloud with verbal and
written communication was an important tool to give voice to and validate the
preservice teachers’ ideas. The interview dialogues were non-judgmental and
provided information on an as-needed basis to the preservice teachers. Probing
questions on the interviewer’s part, assisted these future teachers in thinking
beyond numerical answers to deepen, and in some cases change, their
understanding of school mathematics. The lesson planning corner provided
pertinent school subject matter information needed to plan the lesson and the
instructional materials suggested possible lesson planning activities for
students and teachers. As teacher educators, we can assess our prospective high
school mathematics teachers on an individual basis, understanding more clearly
how their thoughts about school mathematics determine their actions in the
classroom.
Ball D.L. (1990). Prospective elementary and secondary
teachers’ understanding of division. Journal
for Research in Mathematics Education. 21(2), 132-144.
Blanton, M.L. (1998). Prospective Unpublished doctoral
dissertation, North Carolina State University, Raleigh.
Cooney, T. (1994) Teacher education as an exercise in
adaptation. D. Aichele & A. Cosford, (Eds.), Professional development for Teachers of Mathematics: 1994 Yearbook.
Reston, VA: National Council of Teachers of Mathematics.
Lachance, A., & Confrey, J. (1996).
Mapping the journey of students’ explorations of decimal notation through ratio
and proportion. Paper presented at the annual meeting of the American
Educational Research Association, New York.
Ma, L. (1999). Knowing
and teaching elementary mathematics. Mahwah, NJ: Lawrence Erlbaum.
Noelting, G. (1980). The devfeloment of proportional
reasoning and the ratio concept. Part I: Differentiation of stages. Educational Studies in Mathematics, 11,217-253.
Schoenfeld, A.H. (1999). Looking toward the 21st
century: Challenges of educational theory and practice. Educational Researcher, 28(7), 4-14.
Simon, M. (1995). Reconstructing mathematics pedagogy
from a constructivist perspective. Journal
for Research in Mathematics Education. 26,114-145.
Tirosh, D. (2000). Enhancing prospective teachers’ knowledge of children’s conceptions: the case of division of fractions. Journal for Research in Mathematics Education. 31(4), 5-25.
Table 1. Thoughts and Actions for Teaching Rate of Change to Algebra 1 Students
Lesson Plan Activities and Tools
|
Teacher Activities |
Teacher Tools |
Student Activities |
Student Tools |
|
· Gives explanations/ directions · Asks questions · Gives examples · Poses problems |
· Graphs! |
· Collect data · Answer questions · Calculate answers · Graphs rates! |
· Tape measure · Stopwatch · Graphs! · Table/chart |
Problems in Lesson Plan
|
Problem Types |
Problem Strategies |
Problem Contexts |
|
· Missing value (x) · Missing value (y) · Finding rate (y/1) · Comparing rates |
· Set up 2 equivalent fractions · Cross multiplication · Finding unit of rate · Within |
· Distance/time · Item/cost |
School Subject Matter Knowledge
|
Definition of Ratio |
Examples of RoC |
Connections |
Unresolved |
|
· Ratio is a fraction · Rate is a ratio · Ratio is a comparison |
· Acceleration · Speed · Displacement |
· Remembers rate of change from physics · Remembers how to write a ratio · Discovers slope is a ratio! |
· Meaning of proportion |
Note: “!” Indicates idea emerging in interview