Standards-Based Interventions in Elementary
Mathematics
Marika Ginsburg-Block, Department of Educational Psychology,
University of Minnesota
America's national education goals acknowledge the importance of
preparing youth to achieve in challenging subject areas such as mathematics
and science (i.e. Goal 5). In fact, Goal 5 calls for U.S. students to lead
the world in mathematics by the year 2000-next year. The recent release of
the Third International Mathematics and Science Study (TIMSS) sheds light on
the achievement of America's students relative to their same-grade peers
around the world. Compared to 4th-grade students in 26 countries, American
4th-graders were among the top performers in mathematics achievement,
scoring above the international average. Our 8th-graders fared much worse;
they ranked 20th out of the 41 participating nations, performing below the
international average (Peak, 1996). However, the most recent report from the
National Educational Goals Panel indicates that our students are moving
toward our national education goals (1998). Between 1990 and 1996, the
performance of Grade 4 and Grade 8 students in mathematics improved in 28
states.
In contrast to these positive trends, national statistics indicate that
students, particularly those living in low-income urban areas, are still not
performing adequately in mathematics. In fact, over 67% of students in
grades four and eight living in low-income urban areas fail to show basic
levels of mathematics achievement (Children's Defense Fund, 1994). Many
factors contribute to this alarming trend of academic failure in American
urban education, including the political and social context of our society
and the ever-diminishing expectations that we have for our urban students.
Researchers from the National Center on Education Statistics report that
urban schools have the greatest percentages of students living in poverty.
In 1990, 30% of urban school children were living in poverty. Urban schools
also have to accommodate larger enrollments, more behavior problems (e.g.
absenteeism, weapons possession, and student pregnancy), and greater student
mobility, while receiving fewer resources than suburban or rural schools.
These findings indicate a growing need in urban education to identify and
investigate methods that promote positive outcomes for diverse elementary
learners. Multiple approaches are needed to address the complex challenge of
increasing student achievement. Educators focus on innovations in curriculum
and instructional strategies in the effort to improve educational outcomes.
In mathematics education, the National Council of Teachers of Mathematics
(NCTM) developed Curriculum and Evaluation Standards for School Mathematics
(NCTM, 1989), which have led mathematics teachers to adapt innovative
curricula and teaching methods. The NCTM Standards call for a more
conceptual elementary mathematics curriculum, making use of strategies that
promote active learning. Major themes found throughout the Standards include
the use of concrete materials, problem solving strategies, and interaction
with peers to promote conceptual learning. Recent revisions have been made
to the NCTM Standards (now called Principles and Standards), however, the
message remains constant- that all students deserve a mathematics education
of high quality (NCTM, 1998).
The goals of the Standards provide a comprehensive vision of optimal
student outcomes that math educators and educational psychologists agree
upon. Through them, students are expected to achieve the following goals: to
value mathematics, to feel confident about their ability to do mathematics,
to become problem solvers, and to develop the ability to communicate and
reason mathematically (NCTM, 1989).
Innovative Programs in Mathematics Education
The efficacy of mathematics strategies and programs based on one or
more of the principles of the NCTM Standards is documented in empirical
literature. At the elementary level, some of the most prominent work in
this area includes Project IMPACT (Campbell, 1996), Hiebert & Wearne's
(1992) research on conceptually-based instruction, and the research of
Cobb and his colleagues (Cobb, Wood, Yackel, Nicholls, Wheatley,
Trigatti & Perlwitz, 1991) on problem-centered approaches to
mathematics. The Algebra Project (Moses, Kamii, Swap, & Howard, 1989),
the Cognition and Technology Group at Vanderbilt's Jasper Series (1993),
and the QUASAR PROJECT (Silver & Stein, 1996) represent innovative
efforts to reform math education at the middle school level. Through
teacher enhancement and support, Project IMPACT promotes use of
constructivist approaches to the mathematics curriculum in predominantly
minority elementary schools. The major tenants of this program are (a) a
policy of understanding in which each student receives equal
expectations and support in learning mathematics, and (b) a
constructivist approach that focuses on the active construction of
knowledge through problem solving and collaboration with peers.
Evaluation data revealed that students in IMPACT schools scored
significantly higher on achievement tests, particularly on areas of
greater mathematical abstraction, compared to students enrolled in
comparable-site schools (Campbell, 1996). Hiebert & Wearne (1992)
studied the effects of conceptually based instruction on the performance
of suburban/rural first graders on problems involving place value and
two-digit addition and subtraction. These investigators developed an
instructional approach that helps students make connections through
several principles including the use of (a) external representations
(physical, pictorial, verbal, symbolic) as tools, (b) practice in order
for students to become familiar with these representations, (c)
representations to solve problems, and (d) class discussions to focus on
how to use, compare and contrast different representations. Students who
learned in this way performed significantly better than controls on
place value and two-digit addition and subtraction as well as using the
tens and ones structure of the number system. Hiebert and Wearne
attribute these differences to both content and pedagogical differences
between groups. Cobb et al. (1991) examined the effects of a
standards-based elementary problem solving curriculum on student
outcomes. They evaluated the impact of a Grade 2 problem-solving
curriculum for addition and subtraction based on the NCTM Standards.
More specifically, these researchers describe their approach as
encompassing a Vygotskian use of social interactions through small group
learning, and an emphasis on effort and persistence as measures of
success, stemming from the literature on motivation. Cobb et al. found
that students who participated in this method of instruction
out-performed controls on state-administered mathematics assessments and
curriculum-based assessments. These students also reported valuing
collaboration more than controls and did not identify with competition
as much as controls. In my own work on Project P.L.U.S. (Partners in
Learning in Urban Schools), I developed and evaluated two instructional
methods for enhancing the mathematics achievement, academic motivation,
and self-concept of low-achieving urban elementary students. The two
methods, Problem Solving, and Peer Collaboration were also based on the
NCTM Standards. Problem Solving required students to share problem
solving methods in a small group, identify multiple strategies and
solutions to problems, and make use of manipulatives, while Peer
Collaboration consisted of a structured peer tutoring format and a
student-managed group reward contingency. We conducted an empirical
study to evaluate the effectiveness of these methods for 3rd and 4th
grade students at an urban elementary school located in a northeastern
city (Ginsburg-Block & Fantuzzo, 1998). One hundred and four low
achieving, low-income, predominantly minority students were assigned
randomly to 1 of 4 groups: control, problem solving, peer collaboration,
and problem solving + peer collaboration. Over a 7-week period, these
students met twice weekly for 30-minute mathematics sessions. The
findings indicated that students who participated in either Problem
Solving or Peer Collaboration methods significantly out-performed their
peers in mathematics word problems and computation, academic motivation,
and perceived social competence. Student reports of academic competence
were linked to their participation in Problem Solving, reflecting the
importance of opportunities for small group interaction and support in
mathematics learning. These results are exciting for several reasons, 1)
we were able to bring together peer tutoring with an innovative
mathematics curriculum, 2) the procedures were effective in promoting
not only student achievement, but also motivation and self-concept,
reflecting a holistic perspective of student accomplishment, 3) the
study was conducted in a rigorous manner and 4) the procedures were
specifically designed for low-achieving urban elementary students.
Implications for Research and Practice
In reviewing the literature on the most innovative programs in
elementary math education, it is sometimes difficult to draw conclusions
that point to specific recommendations addressing the diverse needs of
elementary students in urban settings. One reason for this difficulty is
that few empirical studies have been conducted with diverse and at risk
populations at the elementary level. Of the projects reviewed here, only
Project IMPACT and Project P.L.U.S. were conducted with urban elementary
schools. This finding is consistent with a common criticism of the NCTM
Standards and its supporting documentation that it makes limited
reference to diverse populations of learners (Mercer, Harris & Miller,
1993). A second reason is the paucity of scientific research in this
area. The constraints of designing and conducting educational research,
often do not permit the evaluation of intervention components for their
specific contributions to student outcomes. For example, of the research
described previously, only Project P.L.U.S. allowed for the evaluation
of specific methods, linking them to specific student outcomes. Several
additional criticisms lead to these suggestions for future research.
First, schools need a more comprehensive assessment of the goals of the
NCTM Standards. Future studies need to more fully evaluate the
effectiveness of programs in achieving several additional student
outcomes. These outcomes should include measures of (a) mathematical
communication, (b) mathematical reasoning and (c) student attitudes and
beliefs about mathematics. The NCTM Standards call for students to learn
to communicate and reason mathematically, the fourth and fifth goals of
the standards, respectively (NCTM, 1989). Mathematical communication
requires learning the language of mathematics, including the signs,
symbols and terms. Mathematical reasoning requires students to learn to
make conjectures, gather evidence and build arguments to support their
ideas. The Standards suggest that students will learn to communicate and
reason mathematically by engaging in problem situations in which they
have an opportunity to read, write and discuss their ideas. The second
goal of the Standards, "becoming confident in one's own ability",
addresses the need to enhance student attitudes and beliefs about
mathematics. In his research on problem solving, Schoenfeld (1985) has
found that students' beliefs about mathematics may decrease their
ability to solve novel problems. For example, if students believe that
all mathematics problems should be solved in five minutes, they are less
likely to persist in solving problems that may realistically require
more time. An understanding of student perceptions about mathematics is
important for determining the effectiveness of mathematics programs.
Mathematics programs need to send clear messages to students about the
values and expectations they hold for student mathematics learning, and
assess the extent to which these messages are accepted and understood by
students. Second, research should be conducted in which the duration of
the mathematics intervention is varied. Do students need an entire
mathematics curriculum based on innovative strategies or supplementary
periods of innovative instruction? The studies presented in this article
varied from evaluating an instructional unit to a year-long curriculum.
An understanding of the optimal length of the intervention would result
in reducing the cost for educators (i.e. minimizing the added resources
and class time required to sustain the intervention) while maximizing
the benefits to students. Third, an important feature of several of the
mathematics projects that were detailed earlier is that they were
designed, implemented and found to be effective with a predominantly
minority, urban, academically "at-risk" elementary school population.
Teachers need a thorough examination of why these intervention programs
are effective with this student population. Although the learning style
literature points to specific features of these programs, such as their
cooperative approach to learning (Widaman & Kagan, 1987) and use of
varied task presentation formats (Allen & Boykin, 1992) to explain their
effectiveness with African American students, further study is warranted
to confirm these hypotheses. Despite the shortcomings of the literature
and the need for more thorough research, there is still much that we can
do to improve the quality of elementary mathematics education for urban
students. Some practical implications from the literature include: 1)
assessing the extent to which urban school systems support curriculum,
textbooks, assessment procedures and professional development
opportunities that are consistent with the literature, recommendations
and goals of the Standards, 2) providing teachers with the training and
support they need to fully implement the Standards in their classrooms,
3) providing equal expectations and support to all learners in
mathematics, 4) incorporating strategies that focus on the active
construction of knowledge, and 5) facilitating validated approaches to
elementary mathematics learning, such as problem solving and peer
collaboration. Based on my work on Project P.L.U.S., here is an example
of a classroom sequence that incorporates problem solving and peer
collaboration strategies.
Reciprocal Peer Problem Solving
| 5 min |
Pairs of students work together on a warm-up exercise.
Each pair shares their strategies with the larger group. |
| 15 min |
Pairs work on problem cards, alternating between
"teacher" and "student" roles. Problems require students to develop
multiple strategies and solutions and make use of manipulatives
(e.g. base-ten blocks, counters, containers, etc.). They are
selected to reflect a ratio of 8 known to 2 unknown problems, based
on student performance at pretesting. |
| 10 min |
Students take a 10-item quiz individually (again,
based on their level of performance). Scores are combined for each
pair and used to determine student success. Success is achieved when
pairs meet their self-determined goal. Three "successes" result in a
reward that is previously selected by each student pair (e.g.
helper). |
Conclusion
Replicable validated programs, such as Project IMPACT and Project
P.L.U.S., that demonstrate effectiveness with target populations, should
guide mathematics reform. Furthermore, teachers should be able to
implement these programs with reasonable effort and resources that are
readily available in school settings (Mercer, Harris & Miller, 1993).
The literature indicates that, although teachers are aware of the
Standards and curriculum developers are incorporating the Standards into
their textbooks, reform efforts are not necessarily reaching our
classrooms (Peak, 1996). In order for these reform efforts to be
successful, politicians, administrators, educational researchers,
teachers, parents, business persons and others must come together to
support educational reform, attending to both fiscal and cultural
policies that will provide leverage for such change to occur.
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