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math effecive practices

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 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


• 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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