Access and Equity in Mathematics Education
Roland Pourdavood
Cleveland State University
r.pourdavood@csuohio.edu
Abstract
This qualitative research investigates secondary mathematics students’
dispositions in two introduction to calculus classrooms taught by the same
teacher. The research questions are: What is the relationship between
mathematics classroom discourse and students’ mathematical dispositions? And,
how may students’ mathematical dispositions be strengthened and sustained as
they take more participating role in an inquiry-based classroom? Data sources
include survey questionnaires, transcripts of audiotapes from interviews with the
participating students, and transcripts of audiotapes from two classrooms
discourses, researcher’s field notes, and students’ written solutions to various
non-routine problems. The findings of the study suggest that the classroom
teacher’s teaching strategies are mainly responsible for transformation of
students’ attitudes and beliefs. Furthermore, sustaining students’ mathematical
dispositions requires competent teachers who have good understanding of
mathematics contents and pedagogy relative to how students learn mathematics.
Access and Equity in Mathematics Education
The call for mass access and equity in mathematics education cannot go
unheeded. Failure to educate all students mathematically restricts job
opportunities in a technological society and thus has economic and social
implications. Conversations about access and equity issues, however, are
obscured by complexity. That is, a complex tangle of educational shortfalls,
personal fears, established attitudes, strongly held beliefs and entrenched
practices, serves to shroud issues and to stifle conversations. The professional
literature, on the other hand, serves to illuminate issues and to invite
conversations.
Failures and fears of mathematics students are widely discussed in professional
literature, as are beliefs, attitudes and practices of mathematics teachers.
Particularly, both mathematical illiteracy and mathematical phobia in America are
documented (Moses and Cobb, 2001). The roots of mathematical illiteracy and
mathematical phobia are traced to conventional ways that mathematics is taught;
that is, with myriad details and procedures without connections to concepts or to
general structures (Kabasakalian, 2007). Most high school mathematics teachers
perceive mathematics as a rigid and fixed body of knowledge. They strongly
believe that teachers are responsible for transmitting that body of knowledge to
their students (Staples 2007). As such, conventional instruction clings to a
predominantly behaviorist theory of teaching and learning.
As an alternative to conventional instruction, the professional literature suggests
problem solving and communication within a context of inquiry-based instruction.
Because the nature of mathematics is neither rigid nor fixed, inquiry-based
instruction focuses on the idea that mathematics should be taught in ways that
invite students to investigate mathematical problems and that encourage students
to use mathematical discourse. That is, to make conjectures, to talk, to question,
and to agree or disagree about problems in order to discover important
mathematical concepts (Stein, 2007). “A major product of discourse is the
transformation of the participant.” (Manouchehri & St. John, 2006). Discourse not
only promotes the development of shared understandings and new insights but
also helps to build meaning and permanence for ideas and to make them public.
Classroom discourse contributes to deeper analyses of mathematics on the part
of teachers as well as students. The increasing number of schools choosing to
adopt inquiry-based mathematics programs underscores the prominence of
communication in mathematics and the importance of language use in
mathematics classrooms (Davis, 2008).
The professional literature also underscores the power and the benefits of
inquiry-based mathematics instruction (NCTM, 2000; Truxaw and DeFranco,
2007; Steen, 2007). The power of inquiry-based mathematics instruction resides
in collaborative and student centered environments where students reason and
construct their understanding as part of a learning community (Staples, 2007).
Students benefit directly from participating in inquiry-based classes by being able
to influence how ideas are developed and meanings are made in the classroom
(Gee, 2003). The benefits accrued from interactivity in the classroom help
students to develop competence: “A student’s ability to influence or manipulate
an environment can affect his or her commitment to learning.” (Choppin, 2007).
Students who participate in inquiry-based mathematics classrooms not only
structure understanding as part of a learning community but also tend to develop
positive identities with respect to mathematics; their beliefs about mathematics
more accurately reflect the nature of the discipline than do those of their
traditionally schooled peers. As such, there is consensus among researchers: “To
become proficient in mathematics, students need to participate in mathematics
discussions and conversations in classrooms.” (Kotsopoulos, 2007).
The quality of classroom discourse depends on the ability of students to process
language in order to build on the ideas of others. The vehicle that promotes
mathematical thinking is the ability to process language, an ability that many
middle school and high school students still need to learn and develop. Formal
mathematical terminology is an indispensable component of mathematical
communication. Noting the importance of communication in the mathematical
process, students need to know the meaning of mathematics vocabulary words,
whether written or spoken, in order to better understand and communicate
mathematical ideas (Gay, 2008). Similarly, for students to be able to make and
state mathematical observations on their own, they need to take ownership of the
thinking that must be done and they need to break away from the belief fostered
by much of the schooling process that authority resides only in books and
teachers. Based on the premise that explicitly referencing and building on the
ideas of others is a feature of academic and professional discourse, teachers can
help students overcome their resistance to classroom discourse by encouraging
students to verbalize what they mean and to reiterate what their peers have said.
Participating in a mathematical community through discourse is an important step
for mathematical learning and conceptual understanding; therefore,
communication is necessary for pondering, refining, discussing, and amending
mathematical ideas. To better meet the demands of college and work place, K-12
students need to practice verbal expressions of mathematical concepts. Despite
the need to practice verbal expressions of mathematical concepts and despite the
power and benefits of inquiry-based instruction, teachers and students resist.
Teachers do not foster inquiry-based learning because helping students to rethink
themselves as learners requires too much of the teacher’s time and energy. For
example, one of the most prominent images of inquiry-based instruction is that of
the teacher as facilitator. Because the level and effectiveness of classroom
discourse depends heavily on the facilitation skills of the teacher, the professional
development of the teacher significantly contributes to the student’s mathematical
understanding “Teachers do not conduct instructional conversations because
they do not know how.” (Truxaw, et al., 2008) Students may not engage in
instructional conversations because they, too, may not know how. Furthermore,
students who are not receptive to inquiry-based mathematics instruction usually
fail to view mistakes as opportunities for learning and fail to view learning as a
collaborative process for all.
Endeavoring to illuminate issues and to invite conversations, this study examines
important relationships and then describes how the relationships change over
time. More precisely, this study examines relationships between dispositions of
high school mathematics students and classroom discourse. Furthermore, this
study examines how the dispositions of high school mathematics students
change over time as students take more participatory roles in classroom
discourse. As such, this study contributes to the body of research not only by
exploring beliefs, attitudes, and practices of high school mathematics students
but also by describing problem solving and communication as necessary
components for promoting access and equity in mathematics education.
Design of the Study
The focus of this study is to connect theory and practice more intimately by
concentrating on classroom discourse in two Introduction to Calculus classes
taught by the same teacher. As such, this study investigates social interaction
among the teacher and students and how social interaction influences the
mathematical dispositions of students. The research question for this study,
therefore, has two parts: How are the mathematical dispositions of students
related to mathematical discourse in the classroom and how are the
mathematical dispositions of students strengthened and sustained as students
take more participatory roles in an inquiry-based mathematics classroom? To
investigate mathematical dispositions of students in the Introduction to Calculus
classes, this study uses survey questionnaires and interview questions that were
developed by the researcher. Data sources include results of survey
questionnaires, transcripts of audio tape recordings from one-on-one interviews
with participating students, transcripts of audio tape recordings from classroom
observations, written responses to various non-routine problems, and the field
notes of the researcher.
Methodology
This qualitative and descriptive study is grounded in constructivist inquiry (Guba
& Lincoln, 1994; Lincoln & Guba, 1985); as such, the research is context specific,
namely the two Introduction to Calculus classes taught by the same teacher. The
researcher does not intend to generalize the findings to all settings. Rather, he
intends to share the ideas and experiences of the principals in the hope that
readers may identify with the research context and apply the findings to their own
particular settings. Furthermore, this study uses social constructivist theory to
explain and interpret the mathematical dispositions of students as they solve nonroutine problems and communicate their solutions. Social constructivist theory
contends that an interactively open learner can be understood mostly through the
learner’s interaction with the environment; that is, learners cannot be studied in
isolation. Social constructivist theory recognizes the importance of individual
identity, individual autonomy, and the notion of context. Accordingly, individuals
build learning and knowing within the social and cultural milieu. The notions of
experience and social interaction play pivotal roles in constructivist theory (Cobb,
1994).
This study investigates the change processes for attitudes, beliefs and practices,
especially related to classroom discourse as a medium for meaning making and
understanding.
The recursive relationship between student voice and classroom discourse is,
therefore, pivotal for understanding and explaining the mathematical dispositions
of students. Classroom discourse that is guided by student voice provides open,
respectful, and democratic environments for teaching and learning mathematics.
Because learning is an unending process, inquiry-based mathematics classrooms
are built upon foundations of student empowerment and encourage both
individual sense making and negotiated meaning. Teachers and students work
with diverse groups toward a common vision of mathematics learning consistent
with nurturing student mathematical literacy and empowerment. As such, the
mathematical dispositions of students are intertwined with their capabilities to
explore, conjecture, reason, and communicate their mathematical thinking.
Therefore, the mathematical dispositions of students cannot be studied in
isolation apart from the context of the learning environment.
Context of the Study
The demographics of the school community, and the histories of the students all
help to explain the context for the learning environment in this study. The public
high school is located in a first ring suburb of a large Midwestern city in the United
States. Typically, about 40% of the graduating seniors complete a calculus
course in high school. The school mathematics program offers two different
advanced level calculus courses and one honors level calculus course. One of
the advanced level courses focuses on deductive treatments of differential
calculus, integral calculus, sequences and series; enrollment is 40 ± 5 students
per year; the average annual score is 4.8 ± 0.2 on a 5 point scale. The other
advanced level course focuses on deductive treatments of functions, limits,
differential calculus, and integral calculus; enrollment is 60 ± 5 students per year;
the average annual score is 3.2 ± 0.3 on a 5 point scale. The honors level course
focuses on inductive treatments of functions, limits, differential calculus, and
integral calculus; enrollment is 60 ± 5 students per year. Regarding access and
equity, enrollment in the honors level course represents the diversity of the school
community more closely than does the enrollment in the advanced level Calculus
courses. The title of the honors level course is Introduction to Calculus.
The student histories help to explain the context for this study. All of the students
in the Introduction to Calculus classes complete full year courses in Algebra I,
Geometry, Algebra II and Pre-Calculus in grades 8 through 11. Without
exception, every student in the Introduction to Calculus classes is a drop-out from
a level of mathematics classes leading to an advanced level calculus course.
Motivated by low grades to voluntarily withdraw from the advanced level, some
students in the Introduction to Calculus classes feel that they were unjustly
weeded-out. On the other hand, many students in the Introduction to Calculus
classes assume that they are less intelligent than students in advanced level
calculus classes. In either case, the attitudes and beliefs of students in the
Introduction to Calculus classes are confounded by feelings of anger, failure,
deficiency, guilt and frustration that persist well into the school year. The
mathematical dispositions of the students profoundly impact the culture of the
classroom.
A Search for Meaning
The classroom teacher deliberately plans lessons that engage students in
problem solving and communication. The lessons are best described in terms of
four pedagogical strategies described in the Hafferd-Ackles, Fuson, & Sherin
(2004) study. As such, Strategy 1, Establishing Expectations, dominates the
classroom discourse for roughly the first six weeks of the school year; the
classroom teacher communicates his expectations regarding how to talk and
write about mathematics. Strategy 2, Mathematics Language, dominates the
classroom discourse roughly from week seven through week twelve; the
classroom teacher encourages students to shift from using informal language
riddled with pronouns to using more formal language clarified with standard
mathematical vocabulary. Strategy 3, Mathematics Community, dominates the
classroom discourse roughly from week 13 through the end of the first semester;
students formalize their language and become more confident in their abilities to
solve problems and communicate mathematically. Strategy 3 and Strategy 4,
Establishing Formal Discourse, are prominent throughout the second semester;
students exhibit a sense of mathematical empowerment and communicate their
ideas using the more formal mathematical language that is established by the
classroom community. The four pedagogical strategies not only guide
mathematical discourse in the Introduction to Calculus classroom but also enable
students to construct the bridges that connect problem solving with their
mathematical dispositions.
1. Expectations: Students listen and sequester tentative responses; teacher leads
and dominates.
2. Language: Students reflect and contribute truncated responses; teacher
prompts and pursues.
3. Community: Students speak and attempt formal discourse; teacher observes
and facilitates.
4. Discourse: Students analyze and dominate formal discourse; teacher monitors
and assists.
Discussion
Understanding student perceptions of mathematics is important to mathematics
educators because it provides a better understanding of the ways students
experience learning difficulties. To better understand student perceptions of
mathematics, this study examines four pedagogical strategies that engage
students in mathematical discourse as part of an inquiry-based mathematics
classroom. First, this study finds that the discursive teaching strategies used by
the classroom teacher are mainly responsible for transforming attitudes and
beliefs of participating students. Strategy 1 and Strategy 2 explain a process by
which the teacher plays a dominant role during the discursive process; Strategy 3
and Strategy 4 explain a process by which students can acquire and develop the
dominant role. Over time, without overly emphasizing the place and voice of the
teacher, the teacher can minimally engage in the discourse by opting for more of
a student led discourse and by acting solely as a guardian and facilitator of the
process. Second, this study finds that mathematical dispositions are related to
problem solving and classroom discourse as media for constructing meaning and
understanding. In this study, the classroom teacher deliberately connects word
problems with mathematical dispositions. Third, this study finds that mathematical
dispositions change over time when students engage in problem solving
situations. At the onset, all of the participating students are more disposed to
conventional instruction, believing that they cannot learn effectively unless the
teacher tells them when their answers are right or wrong, even though most of
them agree, simultaneously, that figuring out answers to problems by themselves
is perhaps more beneficial. The responses of the participating students are fairly
typical in the United States, where most students primarily experience
conventional instruction in which the teacher demonstrates mathematical
procedures and students practice the procedures. Fourth, this study finds that
mathematical dispositions depend significantly on prior experiences. Each
student brings to class a range of prior experiences. The experiences significantly
affect the receptivity of the student to novel teaching strategies and the
endeavors of the teacher to engage the student. In this study, when asked about
their mathematical experiences in middle school and high school mathematics
classes, the participating students focus on procedures and correct answers
rather than on experimentation and discourse. Fifth, this study identifies several
challenges facing school leaders who support inquiry-based mathematics
instruction. One challenge is epistemological differences among teachers and
students. Effectively implementing inquiry-based mathematics instruction requires
a teacher to possess a deep understanding of mathematics and curricular
materials as well as a deep understanding of how students think, reason, and
develop proficiency. The staying power of conventional instruction is another
challenge. “The need for meaningful classroom discourse is now universally
accepted among education researchers.”(Springer & Dick, 2006) Its
implementation, however, falls short of the goals outlined by the NCTM
Standards. Conventional instruction still dominates the educational landscape,
especially in secondary school mathematics classes. In most conventional
mathematics classes, univocal discourse prevails. The classroom teacher
deliberately uses four instructional strategies to create learning opportunities that
engage all of his students. The learning opportunities promote problem solving
and discourse that, over time, turns into collaborative communication.
Collaborative communication transforms and sustains the mathematical
dispositions of the students in the Introduction to Calculus classes.
References
Choppin, J. M. (2007). Teacher-orchestrated classroom arguments. Mathematics
Teacher, 10(4), 306-310.
Cobb, P. (1994). Where is the mind? Constructivist and sociocultural
perspectives on mathematical development. Educational Researcher, 23,
13-20.
Davis, J. D. (2008). Connecting students’ informal language to more formal
definitions. Mathematics Teacher, 101 (6), 446-450.
Gay, A. S. (2008). Helping teachers connect vocabulary and conceptual
understanding. Mathematics Teacher, 102 (3), 218-223.
Guba, E. G. & Lincoln, Y. S. (1994). Comparing paradigm in qualitative research.
In N. K. Denzin & Y. S. Lincoln (Eds.), Handbook of qualitative research
(pp.105-117). Thousand Oaks, CA: Sage Publications.
Hufferd-Ackles, K. Fuson, K. C. & Sherin, M. G. (2004). Describing levels and
components of a math-talk learning community. Journal for Research in
Mathematics Education, 35, 81-116.
Kabasakalian, R. (2007). Language and thought in mathematics staff
development: A problem probing protocol. Teachers College Record, 109
(4), 1-21.
Kotsopoulos, D. (2007). Mathematics discourse: It’s like hearing a foreign
language. Mathematics Teacher, 101 (4), p. 301-305.
Lincoln, Y. S. & Guba, E. G. (1985). Naturalistic inquiry. Beverly Hills, CA: Sage
Publications.
Manouchehri, A. & St. John, D. (2006). From classroom discussions to group
discourse. Mathematics Teacher, 99, 544-551.
Moses, R. P. & Cobb, C. E. (2001). Radical Equations: Math Literacy and Civil
Rights. Boston: Beacon Press.
National Council of Teachers of Mathematics. (2000). Principles and standards
for school mathematics. Reston, VA: Author.
Springer, G. T. & Dick, T. (September, 2006). Making the right (discourse)
moves: Facilitating discussions in the mathematics classroom.
Mathematics Teacher 100 (2), 105-109.
Staples, M. (2007). Supporting whole-class collaborative inquiry in a secondary
mathematics classroom. Cognition and Instruction, 25 (2), 161-217
Steen, L. A. (November, 2007). How mathematics counts. Educational
Leadership, 65 (3),8-14.
Stein, C. C. (2007). Let’s talk. Promoting mathematical discourse in the
classroom. Mathematics Teacher, 101 (4), 285-289.
Truxaw, M. P. & DeFranco, T. C. (2007). Lessons from Mr. Larson. An inductive
model of teaching for orchestrating discourse. Mathematics Teacher, 101
(4), 268-272.
Truxaw, M. P., Gorgievski, N., & DeFranco, T. C. (2008). Measuring K-8
teachers’ perceptions of discourse use in their mathematics classes.
School Science and Mathematics, 108 (2), 58-68.