Effective Practices in Mathematics
As the call for critical literacy has fueled interest in reading and writing across academic disciplines,
so has a movement for “quantitative literacy” influenced the ways in which the developmental
mathematics curriculum is structured and delivered. A set of standards conveyed by the American
Mathematical Association of Two Year Colleges (AMATYC, 2006)
recommends that two-year college mathematics programs focus
on eight standards of intellectual development:
• Problem Solving
• Modeling
• Reasoning
• Connecting with other disciplines
• Communicating
• Using technology
• Developing mathematical power
• Linking multiple representations
In addition, the organization also establishes standards of recommended pedagogy, including:
• Teaching with technology: modeling the use of appropriate technology in teaching
mathematics
• Active and interactive learning: fostering interactive learning through student writing,
reading, speaking, and collaborative activities so that students can learn to work effectively
in groups and communicate about mathematics both orally and in writing
• Making connections: actively involving students in meaningful mathematical problems that
build upon their experiences, focus on broad mathematical themes, and build connections
with branches of mathematics and between mathematics and other disciplines
• Using multiple strategies: interactive lecturing, presentations, guided discovery, teaching
through questioning, and collaborative learning
• Experiencing mathematics: learning activities including projects and apprenticeships that
promote independent thinking and require sustained effort
Further reports from this organization recognize the importance of student engagement in learning
activities, and recommend the use of group work, case studies, and projects (U.S. Department of
Education, 2005). In general, the movement to a more “learner-centered” environment constitutes
the most substantial reform of mathematics education over the past few decades.
Another issue with implications for success in mathematics is the recency of prior preparatory
course completion. In a study of five community colleges in Virginia, Waycaster (2001a) reinforces
the need for students in foundation-level courses to enroll immediately after succeeding in the
previous level math course, citing an almost 15 percent difference in performance when contrasting
student groups (9). In addition, the study cites significant differences in student success when
students completed the recommended preparation, reinforcing both prerequisite enforcement and
careful curriculum sequencing.
Among the practices currently informing the direction of developmental mathematics education in
community colleges, the following initiatives are of note:
Addressing Environmental Factors. In their review of literature concerning environmental factors
relating to student achievement in mathematics, Higbee and Thomas (1999) identified a number
of affective considerations that impacted performance. These included students’ attitudes, selfconcept,
and confidence in mathematics, as well as math anxiety, test anxiety, low motivation, and
misplaced sense of locus of control. These same researchers also examined cognitive factors such
as preferred learning style and critical thinking skills. Based on this body of research, educators
are beginning to explore various techniques to address the barriers and mismatches identified,
including increased use of collaborative learning and verbalization of the problem-solving process.
Author Sheila Tobias (Overcoming Math Anxiety) concurs that the predominant causes of math
anxiety derive from environmental factors created by teachers, leading to destructive student
self-beliefs. These obstacles include timed tests, overemphasis on “one right method/one right
answer,” humiliation at the blackboard, classroom atmospheres of competition, and the absence
of discussion in typical math classrooms (Armington, 2003). Her suggestions for relieving math
anxiety and re-envisioning math instruction to respond to the more prevalent verbal learning style
of many developmental math students continue to influence the way developmental mathematics
instruction is delivered in today’s classroom.
Small Group Instruction. In a study of preparatory algebra
students at a large urban university, DePree (1998) demonstrated
that those taking course sections taught in a small group
instructional format had higher confidence in their mathematical
ability and were more likely to complete the course than
those in comparison courses with traditional instructor-led
teaching. This was particularly true of students from traditionally
underrepresented groups (Hispanic, Native American, and female
students). Among those completing the courses, there was no
significant difference in overall course grades.
Problem-Based Learning (PBL). Based on a constructivist approach,
this instructional strategy emphasizes the learning and application
of mathematical concepts in connection with student exploration of
a complex problem, usually deriving from a “real world” situation. Problems are posed in such a
way that students need to gain new knowledge in order to solve the problem, and most problems
have multiple correct solutions. Problem-based learning involves students gathering information,
identifying possible solutions, evaluating the various alternatives, choosing a solution, interpreting
results, and defending conclusions. Since complex problems are often solved collaboratively,
this method also promotes teamwork, shared responsibility, and skill development for peer-topeer
mathematical communication. Proponents feel that PBL leads to deeper understanding of
mathematical concepts and avoids learning by imitation that may occur in traditional algorithmic
approaches. Studies have shown that students who learn through a problem-based approach exhibit
higher achievement on both standardized tests and on project tests dealing with realistic situations
than do students taught in traditional content-based learning environments (Boaler, 1998).
Contextual Learning. Cognitive science teaches that students retain information longer and can
apply it more effectively if it is learned in context. With respect to developmental mathematics,
an approach gaining favor is the teaching of mathematics “across the curriculum:” the notion
that applied mathematics delivered in conjunction with business, technical, or other professional
preparatory coursework enhances student motivation and acquisition of mathematical skills. This
may also take the form of curricular enhancements in traditional developmental math courses, in
which standard math concepts are enhanced with problems, examples, or applications from other
fields. A stronger emphasis on reading/math integration (i.e., analyzing word problems, building
mathematical vocabulary, and teaching reading skills as they relate to learning from a math
textbook) has also been suggested as a means to leverage interdisciplinary skills and help students
see connections between vital components of a developmental curriculum (Haehl, 2003).
Use of Manipulatives. In a study of middle school students, Moyer and Jones (2001)
conclude that the use of manipulatives to illustrate mathematical concepts may promote more
autonomous thinking, curiosity, and understanding among math students. The study asserts
that “communicating the value of representations and the importance of being able to move
flexibly among different representational systems, including manipulatives, visual images, and
abstract symbols, helps students develop a deeper understanding
of mathematics” (30). The study suggests that the practice
diversifies instructional delivery and may provide students with
additional points of access when contrasted with traditional
lecture models.
Use of Technology. A great deal of literature in recent years
has addressed the use of technology in developmental math
instruction. This includes technology primarily used by teachers
(e.g., presentation technology), students (e.g., calculators), or both
(e.g., computer-assisted instruction, or CAI). A seven-year study
in five Virginia colleges examined developmental math classes of 10
instructors whose primary instruction was either lecture with lab or
individualized computer-aided instruction to determine how student
outcomes from these courses compared to those of traditional lecture courses. Results from
this study indicated that student pass
rate was independent of the manner of instruction used
(Waycaster, 2001b).
An extensive review of recent studies examining computer-assisted instruction found mixed results
at a variety of colleges, each implementing slightly different forms of computer-assisted instruction
(U.S. Department of Education, 2005). These included self-paced or lab-based instruction with
products such as Academic Systems (internet-delivered curriculum combining lecture, practice
and self-administered tests), ALEKS (a nonlinear, nontraditional internet-based course), or PLATO
(a popular computer-based program for K-adult learners). Instructor-created distance learning
courses were also examined, as were courses using computer algebra systems (CAS; programs
that manipulate mathematical expressions in both symbolic and numeric forms). The authors of this
extensive review find studies crediting CAI and CAS with higher, lower, or no difference in pass rate, no
difference or higher rates of persistence to higher level math, and no difference in final grades compared
to developmental math sections taught in traditional instructor-led formats. They ultimately conclude,
however, that offering a variety of instructional formats may allow students more options for choosing
a modality that best suits their particular learning styles. They also reiterate the views of Boylan and
AMATYC that, for technology to be effective, it should be used as a supplement to, rather than a
replacement for, regular classroom instruction (U.S. Department of Education, 2005.)
Further examples and recommendations for effective practices in
mathematics can be found in
Effective Practices for Developmental Mathematics, Vols. 1 and 2, 2002 and 2003, published under NADE
SPIN (National Association of Developmental Education – Special Professional Interest Network,
Thomas Armington, editor).
Effective Practices in English as a Second Language (ESL)
Any discussion of effective practices for ESL must first recognize the inherent diversity of student
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