THE
INTERPLAY BETWEEN INSTRUCTION AND THE DEVELOPMENT OF MIDDLE SCHOOL STUDENTS’
ALGEBRAIC THINKING
|
Cynthia W. Langrall Illinois State University langrall@ilstu.edu |
John K. Lannin Illinois State University lannin@ilstu.edu |
Abstract: This study examined the interplay between
instruction and the development of middle school students’ algebraic
thinking. Interviews were conducted
with six students over a two-year period during their 7th- and 8th-grade
years. In grade 7, all students
received algebra instruction using units from the Connected Mathematics Project
(CMP) materials. In grade 8, three of
the students took a traditional algebra 1 course while the other three students
remained in the regular mathematics class following a typical pre-algebra curriculum
supplemented with two CMP units. Our
findings indicate that although the algebra 1 students obtained a level of
proficiency in algebraic symbol manipulation, their reasoning and sense-making
abilities were less pronounced than in grade 7 and generally not as robust as
those of the students who received the regular grade 8 mathematics instruction.
For more than a decade, the National
Council of Teachers of Mathematics (NCTM) has recommended a broad-based,
integrated curriculum for the middle grades that includes topics in
probability, statistics, measurement, geometry, and algebra (NCTM, 1989). In support of this recommendation, recently
published Standards-based curriculum
materials and even basal textbooks have included these topics as part of the
middle grades curriculum (Reys, Robinson, Sconiers, & Mark, 1999). More recently, the NCTM (2000) has called
for “significant amounts of algebra and geometry throughout grades 6, 7, and 8”
(p. 212) and has recommended that algebra and geometry instruction emphasize
the interconnections between these topics and among other mathematics
topics. Nevertheless, a trend adopted
by many school districts is to shift mathematics content from the high school
to the middle school in the form of a traditional algebra1 course in grade
8. According to Silver (1995) “mandated
algebra instruction in grade 8 can undermine those reform efforts directed
precisely at broadening and integrating the curriculum of middle grades” (p.
32). There is also the concern that the
content and teaching methods used in traditional algebra courses do not promote
conceptual understanding or the development of students’ algebraic reasoning
(Kaput, 1999).
The
purpose of this study was to investigate the interplay between instruction and
the development of middle school students’ algebraic thinking. The first phase of this two-year study
sought to describe the algebraic thinking of students whose mathematics
instruction in grade 7 included the study of algebra using materials from the
Connected Mathematics Project (CMP) curriculum. In a departure from traditional algebra instruction that focuses
on manipulating expressions and solving symbolic equations, the Standards-based CMP curriculum uses
contextual problem situations to develop students’ concepts of variable and
their understandings of the relationships among variables (Phillips,
1998). The second phase of the study
followed the development of the students’ algebraic thinking through grade 8. However, in grade 8, the students split into
two groups with one group receiving instruction that continued to incorporate
some of the CMP materials and the other group receiving instruction based
solely on a traditional Algebra 1 textbook.
The development of students’ thinking across grade levels and between
the regular 8th-grade mathematics and traditional algebra 1 students
was examined.
The theoretical perspective guiding our
work assumes that mathematics should be taught for understanding. We refer to the definition of Hiebert et al.
(1997) who indicated that understanding something involves seeing how it is
related or connected to other things that are known. Understanding is crucial to learning mathematics “because things
learned with understanding can be used flexibly, adapted to new situations, and
used to learn new things” (Hiebert et al., p. 10). According to Romberg and Kaput (1999), learning for understanding
“cannot be viewed as a mechanical performance or an activity that individuals
engage in solely by following predetermined rules” (p. 6). Yet it is this mechanistic view of teaching
and learning mathematics that characterizes traditional school mathematics
(Romberg & Kaput). Within this
theoretical perspective, we examined the development of students’ understandings
of algebra from two curricular orientations: one traditional and the other more
Standards based.
Participants
Grade 7 students from a Midwestern school
formed the population for this study.
Based on the results of a researcher-generated algebra problem-solving
assessment, six students were purposefully selected from this population to
represent contrasting levels of performance.
These six students participated in the study during their 7th-
and 8th-grade years of schooling.
The classroom teacher in this study
taught all mathematics classes for grades 7 and 8. He had 15 years of experience at the middle school level. His teaching style was traditional in the
sense that instruction was typically teacher directed during whole-class activities
but he frequently engaged students in small-group interactions that allowed
them to communicate and justify their thinking.
Year-One Instruction
In grade 7, approximately 9 weeks of
instruction focused on the study of algebra.
During this time, we observed the mathematics class on at least two
consecutive days each week to document the nature of algebra instruction. We also noted the classroom interactions of
the students, asked them to explain their reasoning as they worked on in-class
assignments, and obtained copies of their written work when it was collected by
the teacher. The CMP texts Variables and Patterns (Lappan, Fey,
Fitzgerald, Friel, & Phillips, 1998d) and Moving Straight Ahead (Lappan et al., 1998b) were the basis for
algebra instruction.
Year-Two
Instruction
In grade 8, three of the students were
selected for participation in a traditional Algebra 1 class based on their
overall 7th-grade mathematics performance and their performance on a
standardized algebra readiness test.
The textbook used in this class was Algebra
1: An Integrated Approach (Benson et al., 1991). The other three students remained in the “regular” mathematics
class. The curriculum for this class
incorporated the text Pre-Algebra: An
Accelerated Course (Dolciani, Sorgegnfrey, & Graham 1985), the CMP
texts Looking for Pythagoras (Lappan
et al., 1998a) and Thinking With
Mathematical Models (Lappan et al., 1998c), and was supplemented with
instructional units in geometry and statistics. Classroom observations similar to those in Year 1 were conducted
in Year 2.
Procedure
In Year 1, students were individually
interviewed on seven occasions to investigate the interplay between students’
algebraic thinking and instruction.
Using a semi-structured interview script, we posed problems similar to
the ones students were working on in class.
We also asked probing questions to assess levels of students’ conceptual
understanding, and attempted to identify any cognitive obstacles the students
were encountering. Interview topics
included linear and nonlinear relationships, tabular and graphic
representations, concept of slope, and solving linear equations. In Year 2, five interviews were
conducted. All of the students were
given the same interview items that represented a balanced distribution of
topics from the regular mathematics and algebra 1 curricula. As in Year 1, we asked probing questions to
assess students’ conceptual understanding and attempted to identify students’
misconceptions. Interview topics
included linear and non-linear relationships, interpretation of multiple
representations, concepts of slope and variable, generating expressions, and
writing equations. All interviews were
audio taped and later transcribed.
Data sources consisted of the transcribed
interviews, students’ written work, researcher field notes, and data displays
and summaries generated during the analysis.
These data were analyzed using a double coding strategy and data reduction
approach (Miles & Huberman, 1994) to discern key thinking patterns for each
student. Transcripts were analyzed to
characterize each student’s reasoning or solution strategy for each problem
presented in the interviews. The coding
rules used in this process were modified and refined throughout the
analysis. They pertained to students’
understandings of concepts (e.g., slope, variable), as well as their abilities
to describe patterns or functional relationships, generalize relationships
either verbally or with symbols, and use and interpret tabular, graphical, and
symbolic representations. Codes were
reconciled and organized in a Task Coding Matrix that allowed us to examine
within-case and across-case trends. Narrative
summaries were constructed for each student describing his or her reasoning,
solution strategies, and cognitive difficulties across all interview
problems. Also, cross-case summaries
were developed for each problem.
In Year
1, students examined the relationships among graphic, tabular, and symbolic
representations of linear situations.
In an attempt to uphold the philosophy of the CMP materials, the teacher
engaged the students in small-group problem-solving activities and encouraged
students’ explanation and justification of solution strategies. However, the teacher occasionally
supplemented the CMP lessons with worksheets designed to introduce and practice
procedures for symbolically solving linear equations. By the end of grade 7, three trends in students’ thinking were identified:
(1) all students could represent linear situations in symbolic, graphic, and
tabular forms; (2) students demonstrated an emerging understanding of the
concept of variable in the sense that they could use a variable when it was
defined for them but sometimes had difficulty identifying what a variable meant
in a problem context; and (3) half of the students (two who later participated
in regular 8th-grade instruction and one who took the traditional
algebra class) consistently referred to the problem context to make sense of a
problem situation while the other three students generally ignored the problem
context and focused on superficial patterns.
In Year
2, the traditional algebra 1 class focused on manipulating symbols in the
context of finding unknowns, using properties of generalized arithmetic, and
solving problems by representing situations symbolically. The typical lesson followed a mechanistic
approach (Romberg & Kaput, 1999) whereby the teacher demonstrated a
procedure, provided a few examples for students to work individually, and
assigned practice exercises. The
algebra instruction in the regular mathematics class continued to build on the
algebraic understandings introduced in grade 7. Students informally investigated the graphic, tabular, and
symbolic representations involving both linear and nonlinear
relationships. It should be noted,
however, that only one of the four designated grade 8 algebra units from the
CMP materials was used.
By the end of grade 8, differences in the
students’ understanding and reasoning emerged between the two instructional
groups. The algebra students were more
proficient at solving equations and simplifying expressions. Although all students could represent
situations in various forms when asked to do so, the students in the regular
mathematics class accessed a variety of representations to solve problems,
while the algebra 1 students focused exclusively on symbolic
representations. The regular
mathematics students referenced alternative representations to verify their
work, whereas the algebra 1 students rarely sought other representations to
confirm the validity of their work.
Although all students struggled to understand the different uses of
variables, students from the algebra group often referred to, and used,
variables as unknowns while ignoring the underlying meaning of their symbolic
form. In contrast, the regular
mathematics students maintained a dynamic view of variables and frequently
questioned what variables represented.
Justifying one’s thinking was another difference that emerged between
the two groups. The algebra 1 students
often stated “I just know this” rather than offering any justification of their
reasoning. For them, providing an
explanation did not seem to be an integral part of doing mathematics. The regular mathematics students justified
their reasoning when asked or provided an explanation as they talked their way
through solving the problem.
Our
results indicate that although students in the traditional algebra course
acquired some level of proficiency in algebraic symbol manipulation, their
reasoning and sense-making abilities were generally lacking. In many respects, the reasoning and
sense-making of the algebra 1 students was less pronounced than it had been in
grade 7 and was not as robust as that of the students who remained in the
“regular” grade 8 mathematics class.
The findings of this study underscore the concern that a traditional
algebra 1 course in grade 8 may promote a mechanistic view of algebra and fail
to develop concepts and understandings that would allow students to reason
algebraically in new situations.
References
Benson, J., Dodge, S., Dodge,
W., Hamberg, C., Milauskas, G., & Rukin, R. (1991). Algebra
1: An integrated approach.
Evanston, IL: McDougal, Littell.
Dolciani, M. P., Sorgegnfrey,
R. H., & Graham, J. A. (1985). Pre-algebra: An accelerated course. Boston, MA: Houghton Mifflin.
Hiebert, J., Carpenter, T.
P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., Olivier, A., &
Human, P. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.
Kaput, J. J. (1999). Teaching
and learning a new algebra. In E.
Fennema & T. A. Romberg (Eds.), Mathematics
classrooms that promote understanding (pp. 133-155). Mahwah, NJ: Earlbaum.
Lappan, G., Fey, J. T.,
Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998a). Looking
for Pythagoras. Menlo Park, CA:
Dale Seymour.
Lappan, G., Fey, J. T.,
Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998b).
Moving straight ahead: Linear relationships. Menlo Park, CA: Dale Seymour.
Lappan, G., Fey, J. T.,
Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998c).
Thinking with mathematical models:
Representing relationships.
Menlo Park, CA: Dale Seymour.
Lappan, G., Fey, J. T.,
Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998d).
Variables and patterns:
Introducting algebra. Menlo
Park, CA: Dale Seymour.
Miles, M. B. & Huberman,
A. M. (1994). Qualitative data analysis: An expanded sourcebook (3rd
ed.). Newbury Park: Sage.
National Council of Teacher
of Mathematics. (1989). Curriculum and Evaluation Standards for
School Mathematics. Reston, VA:
NCTM.
National Council of Teacher
of Mathematics. (2000). Principles and Standards for School
Mathematics. Reston, VA: NCTM.
Phillips, E. (1998),
Developing a coherent and focused K-12 algebra curriculum. In National Research Council (Ed.), The nature and role of algebra in the K-14
curriculum: Proceedings of a national symposium, (pp. 27-29). Washington, D.C.: National Academy Press.
Reys, B., Robinson, E.,
Sconiers, S., & Mark, J. (1999) Mathematics curricula based on rigorous
national standards: What, why, and how?
Phi Delta Kappan, 80, 454-456.
Romberg, T.J. & Kaput,
J.J. (1999). Mathematics worth
teaching, mathematics worth understanding.
In E. Fennema & T.A. Romberg (Eds.), Mathematics classrooms that promote understanding, (pp.3-18).
Mahwah, NJ: Earlbaum.
Silver, E. A. (1995). Rethinking “algebra for all.” Educational
Leadership, 52(6), 30-33.