Theoretical Foundation of the Developmental Mathematics Department
Serves as the guiding principle for the design and implementation of instruction and services
As with most programs, our Developmental Mathematics (DM) instruction is based on a blend of
several learning theories that support mathematical skill development, conceptual understanding,
critical thinking, and improved study strategies. All strategies are implemented with the longterm goal of a smooth transition to credit level mathematics. The learning theory that most
holistically describes our philosophy would be Constructivism, rooted in the work developed by
Piaget (1950) which suggests that knowledge and meaning are constructed (generated) through
experiences. Bruffee’s work (1993) extends this theory through the position that “knowledge is
‘constructed through negotiation with others’ in communities of knowledgeable peers.” His work
with collaborative learning provides a strong basis for our program. He states (p.9),
“Collaborative learning is not a universal educational cure-all, and it does not make obsolete such
time-tested pedagogical activities as lecture, drill, and recitation.” This is particularly important
in mathematics classrooms where a variety of activities are blended. Bruffee continues (p. 207),
“Collaborative learning does not exclude lecturing. It only changes the social context and the
authority structure in which a lecture is delivered.”
According to Richard E. Mayer (2004), “…active venues such as group discussions, hands-on
activities, and interactive games are classified as constructivist teaching.” Our DM pedagogy
encourages active learning. Our belief is that professors do not teach mathematics but rather
facilitate the learning through experiences that support the construction of knowledge. As
practiced by our program, instruction is primarily guided discovery, rather than pure discovery.
While we believe in the instructional effectiveness of discovery learning, we have found that it is
too time consuming for full coverage of a standardized curriculum. Due to the tremendous time
required to allow a student to “discover” patterns or a process, we believe the coverage of the
curriculum suffers. Thus, the “compromise” is guided discovery. Mayer, p. 18 states: “Yet a
dispassionate review of the relevant research literature shows that discovery-based practice is not
as effective as guided discovery.” Our department believes discovery learning is effective, but we
do not believe it is as efficient as guided discovery.
An example may help clarify our instructional process. For the topic of simplifying exponential
expressions, an all too often common approach would be to place the rules on the board and ask
the students to use them to simplify the expressions. We do not do that. We have a “drag it out”
approach to teaching this concept until the students can discover the patterns. The use of this
guided discovery strategy is exemplified by the following instructional method.
Simplify: . The steps would be:
Presentation of this information is followed by an instructor-led discussion of the pattern that
emerges. The student, with guidance, would notice that 2 and 3 are related to the 5 by addition.
Thus, they discover the rule . While this is a simple example of guided discovery as practiced by
our department, it captures the essence of our philosophy. Students explore the mathematical
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situation and formulate the general rules from this experience. We encourage students to discover
concepts but, due to time constraints of a set curriculum, we provide guidance and direction to
decrease the time needed for that discovery.
The mathematical content and the andragogy are interrelated. What is covered, or can be covered,
depends, in large part, on the instructional techniques being used. We encourage an active
learning environment with activities designed to encourage student collaboration. To use active
learning effectively, students must be involved in creating their own learning experiences; this is
accomplished by having students discuss, while the instructor monitors and guides the direction
of the discussion, as appropriate.
While our current practice is collaborative learning, faculty are encouraged to take that to a level
of cooperative learning, where each student is responsible for participation. There must be
accountability within group work evidenced by ratings, responsibilities, and other instructorestablished requirements. Formative evaluation occurs as the facilitator (instructor) questions the
students, using those responses to adapt the teaching. Open-ended questioning, observations, and
conversations direct the instructor through the class session. According to Wood (1995, p.204),
“…teaching is by definition an interactive collective activity in which it is the intention of the one
teaching to influence the development of his or her students’ thinking.”
We practice the role of instructor as facilitator as described by Cobb (1995). Others (Gamoran, et
al. 1998, Rhodes and Bellamy 1999) have described a similar role for instruction. While the
lecturer stands at the front of the classroom and presents a structured curriculum, the facilitator
serves as a guide for students to draw conclusions based on patterns. Facilitators effectively use a
questioning technique rather than a telling approach. They wait for a student response and then
direct the student to an accurate mathematical statement using correct mathematical terminology.
Language is emphasized as a tool for deepening the understanding.
The instructor-as-facilitator philosophy minimizes the mathematics anxiety experienced by many
students. We believe this anxiety is primarily a product of classical conditioning. Students have
had negative experiences in their past and, as a result, have low expectations for their college
experience. Past failures contribute to a cycle of low student expectations and low performance.
Beyond the mathematics, many times these students require advice and instruction in study skills,
test-taking, note-taking, and confidence building. We are aware that an environment of mutual
respect and trust contribute to a healthy learning environment; therefore, support services include,
but are not limited to: a Math Lab, Study Sills Seminars, Counseling services, College Success
courses, and group tutoring.
We do, in reality, teach using a textbook written more for the behaviorist. This philosophy tends
to break topics into measurable skills to be learned, tested, and built upon in subsequent topics.
Textbooks have objectives, introduction of topic, examples, self-checks, and exercises written
with graduated levels of difficulty. All culminate in a chapter review and test. Although this
could be construed as limiting the ability to teach for true understanding and to reach the
constructivist level, if judiciously applied, the text will serve as an organizing and standardizing
mechanism for the curriculum. Our goal is to teach for relational understanding rather than just
instrumental understanding as defined by Skemp (1993). Students will work less on
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memorization and more toward conceptual understanding that will aid in transition to the next
level. This may relate to the transition to the final, formal stages of development discussed by
Piaget. The focus on student understanding rather than manipulation is supported by the
standards (NCTM, 1991, AMATYC 2004) established by several organizations in the field.
The use of technology to increase student learning is supported by Brownstein (2001, p. 246),
“The ability to express ideas for the teacher and learner now includes, in addition to text, sound,
graphics, and motion. Where paper is static, multi-media is alive.” The technology used by our
professors includes graphing calculators, spreadsheets, PowerPoint, videos, mathematics
software, and interactive programs. While we offer use of the current technology to support
learning, we are sensitive to the fact that computer and calculator expenses may be a discretionary
expense rather than a need for many households. Thus, we are ever mindful of the required
technology component. We work under the realization that effective use of technology is
dependent on availability and the learning style of the student. We recognize that one size does
not fit all; therefore, we offer computerized settings, hybrid online coursework, as well as selfpaced and traditional classroom instruction.
The DM department uses two standardized measurements for students. At the beginning of the
mathematics sequence, a college-wide assessment is used for student placement. At the end of
each course, a departmental final exam is used as an evaluation tool to collect summative data.
This allows for meaningful data analysis among sections, increases consistency across all class
sections, and provides the student an opportunity to “pull it all together” in a demonstration of
mastery before advancing to the next level.
Developmental Mathematics coursework is presented with a systematic and ongoing review of
the curriculum, teaching methodology, assessment, and evaluation. This is done with a focus on
serving the needs of students. To ensure that all faculty provide active learning in the classroom,
the department offers teaching technique workshops bi-annually.
References
Bauersfeld, H., Cobb, B., (1995). The Emergence of Mathematical Meaning: Interaction in
Classroom Cultures, Lawrence Erlbaum Associates: Hillsdale, NJ.
Brownstein, B. (2001). “Collaboration: The Foundation of Learning in the Future,” Education,
122(2), 240-247.
Bruffee, K. A. (1993). Collaborative learning: Higher education, interdependence, and the
authority of knowledge. Baltimore: Johns Hopkins University Press, 3.
Cobb, P., Bauersfeld, H., (1995). The Emergence of Mathematical Meaning: Interaction in
Classroom Cultures. Lawrence Erlbaum Associates: Hillsdale, NJ.
Gamoran, A., Secada, W.G., Marrett, C.A., (1998). The organizational context of teaching and
learning: changing theoretical perspectives, in Hallinan, M.T. (Eds), Handbook of Sociology of
Education.
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Mayer, R. (2004). “Should there be a three-strikes rule against pure discovery learning? The case
for guided methods of instruction.” American Psychologist, 59 (1), 14-19.
National Council of Teachers of Mathematics. (1989). Curriculum and Evaluation Standards for
School Mathematics. Reston, VA: The Council.
National Council of Teachers of Mathematics. (1991). Professional Standards for Teaching
Mathematics. Reston, VA: The Council.
National Council of Teachers of Mathematics. (2000). Principles and Standards for School
Mathematics. Reston, VA: The Council.
American Mathematical Association of Two-Year Colleges (AMATYC). (1995). Cohen, D.
(Ed.). Crossroads in Mathematics: Standards for Introductory College Mathematics Before
Calculus. Memphis, TN: American Mathematical Association of Two-Year Colleges.
American Mathematical Association of Two-Year Colleges (AMATYC). (2006). Blair, R. (Ed.).
Beyond Crossroads: Implementing Mathematics Standards in the First Two Years of College.
Memphis, TN: American Mathematical Association of Two-Year Colleges.
Piaget, Jean. (1950). The Psychology of Intelligence. New York: Routledge.
Rhodes, L. K., Bellamy, G. T. (1999). “Choices and Consequences in the Renewal of Teacher
Education.” Journal of Teacher Education, 50 (1), 17.
Skemp, Richard R. (1976) “Relational Understanding and Instrumental Understanding.”
Mathematics Teaching, (77), 2-26.
Wood, T. (1995). An Emerging Practice of Teaching, in Cobb, P., & Bauersfeld, H. (Eds), The
Emergence of Mathematical Meaning: Interaction in Classroom Cultures. Lawrence Erlbaum
Associates: Hillsdale, NJ.
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