Adelson, J., Carroll, S., Casa, T., Gavin, M., Sheffield, L. & Spinelli, A.
Journal of Advanced Academics
Prufrock Press
18(4)
This study is a research grant project designed to create an enriched and accelerated curriculum for mathematically talented elementary students.
Summary
To date, there has been very little researchbased mathematics curriculum
for talented elementary students. Yet the gifted education and mathematics
literature suggest support for curriculum that is both enriched and
accelerated with a focus on developing conceptual understanding and
mathematical thinking. Project M3: Mentoring Mathematical Minds is a
5year Javits research grant project designed to create curriculum units
with these essential elements for talented elementary students. These
units combine exemplary teaching practices of gifted education with
the content and process standards promoted by the National Council
of Teachers of Mathematics. The content at each level is at least one
to two grade levels above the regular curriculum and includes number
and operations, algebra, geometry and measurement, and data analysis
and probability. The focus of the pedagogy encourages students to
act as practicing professionals by emphasizing verbal and written communication.
Research was conducted on the implementation of 12 units
in 11 different schools, 9 in Connecticut and 2 in Kentucky. The sample
consisted of approximately 200 mathematically talented students
entering third grade, most of whom remained in the project through
fifth grade. Students in this study demonstrated a significant increase in
understanding across all mathematical concepts in each unit from preto
posttesting. Thus, Project M3 materials may help fill a curriculum void
by providing appropriate accelerated and enriched units to meet the
needs of mathematically talented elementary students.
More than 25 years ago, the National Council of Teachers of
Mathematics ([NCTM], 1980) in their Agenda for Action stated,
“The student most neglected, in terms of realizing full potential,
is the gifted student of mathematics. Outstanding mathematical
ability is a precious societal resource, sorely needed to maintain
leadership in a technological world” (p. 18). Unfortunately, the
results of the Trends in International Mathematics and Science
Study ([TIMSS], 2000, 2004) showed that U.S. students continue
to fall far below their international peers on the mathematics
assessment. In fact, the gap increased from 4th to 12th
grade, by which time only two countries had students performing
significantly lower than the United States (TIMSS, 2000).
The most talented students in the United States also compared
unfavorably with their peers. While 40% of eighthgrade students
in Singapore and 38% of eighth graders in Taiwan scored
at the most advanced level on the 2003 TIMSS mathematics
assessment, only 7% of U.S. eight graders scored at this level
(TIMSS, 2004). Clearly, U.S. students, including the top ones,
are not measuring up internationally.
On the national level, results from the National Assessment
of Educational Progress (NAEP) indicated that although student
performance increased in mathematics, a large percentage
of students still were not performing at an acceptable level (Perie,
Grigg, & Dion, 2005). In fact, 70% of U.S. eighthgrade students
cannot solve a word problem involving more than one operation.
Moreover, there was a frightening shortage of students performing
at the highest level. Only 5% of fourthgrade students and
6% of eighthgrade students performed at the “advanced” level.
It is at this level that eighthgrade students are expected to use
abstract thinking, which is a cornerstone of highlevel mathematics.
Whether we look at international or national measures,
the U.S. system clearly is failing. How can we change this situation
to help talented math students, especially those of diverse
backgrounds, learn more mathematics and achieve at higher
levels?
Curriculum for
Mathematically Talented Students
One of the first steps in addressing the needs of these students
is to provide effective, highlevel curriculum. However, to
date there is a paucity of researchbased mathematics curriculum
for mathematically talented elementary students. Nevertheless,
the mathematics and gifted education literature suggests that
there may be support for curriculum that focuses on both mathematical
content and processes, combines acceleration and
enrichment practices, addresses the range and diversity of students’
mathematical talents through differentiation, and encourages
students to process mathematics in ways similar to those of
practicing professionals.
Addressing Mathematical Content and Processes
In the latest reform movement, the National Council of
Teachers of Mathematics (2000) has not only outlined what students
should learn (i.e., the number, algebra, geometry, measurement,
and data analysis and probability content standards) but
also how they should learn mathematical content. The process
standards encourage students to problem solve, communicate,
reason, make connections, and use different representations as
they engage with mathematics. Some elementary mathematics
curricula based on the NCTM (2000) standards, including
Math Trailblazers, Everyday Mathematics, and Investigations in Number, Data,
and Space, have students employ these mathematical
processes as they study these content areas. In addition, these
curricula are conceptbased and focus on significant mathematical
ideas. Research on the implementation of these curricula
indicates that students using these curricula do as well as other
students on traditional measures of mathematics achievement,
even on measures of computational skill. Furthermore, on formal
and informal assessments of conceptual understanding and ability
to solve problems, students using the reformbased curricula
generally do better than other students (Carroll & Isaacs, 2003;
Carter et al., 2003; Mokros, 2003; Putnam, 2003). Thus, research
has shown that curriculum developed using the NCTM (2000)
content and process standards is effective. However, these curricula
were designed for the general student population and not
specifically for talented students.
As with all students, the curriculum used with mathematically
talented students should be based on the NCTM (2000)
content and process standards, but they also should “explore topics
in more depth, draw more generalizations, and create new
problems and solutions related to each topic” (Sheffield, 1994, p.
21). In addition, the focus of curriculum for students with mathematical
talent should be problem solving (NCTM, 1980, 2000;
Sheffield, 1994; Wheatley, 1983). Problem solving is interrelated
with the other mathematical processes, which include communication,
connections, reasoning, and representations.
A Combination of Acceleration and Enrichment
Research studies on the different programming models of
acceleration and enrichment in the area of elementary mathematics
are limited and reveal mixed results. Robinson, Shore,
and Enersen (2007) stated that acceleration enables students to
cover content efficiently. However, they cautioned that acceleration
alone does not attend to the development of the highlevel
mathematical thinking characteristic of talented students.
Stanley, Lupkowski, and Assouline (1990) viewed acceleration
as a good fit for only a small percentage of students. On the
other hand, it is not an uncommon practice for programs that
focus on enrichment to have students work on a “puzzle of the
week” or “fun” mathematics activities, which are enjoyable but
may not deepen student mathematical understanding. Sowell
(1993) reviewed five studies that focused on the use of enrichment.
In a study focused on elementary students, fourth graders
outperformed the control groups on cognitive measures and also
improved in attitudes towards mathematics. However, fifth and
sixth graders were not significantly different from the control
group in their achievement or attitudes towards math.
Sheffield (1999) pointed out that “services for our most promising students
should look not only at changing the rate or the number of mathematical
offerings but also at changing the depth or complexities of the mathematical
investigations” (p. 45). Using both acceleration and enrichment as a programming
model at the elementary level is promising, although only limited research has
investigated this dual strategy. In one study, when exposed to a highlevel
curriculum that focused on developing mathematical reasoning, talented students
in grades 2–7 made significant achievement gains and were satisfied with the
curriculum (Robinson & Stanley, 1989). In another study, Moore and Wood (1988)
found that students in grades 3–7 learned mathematics more quickly using both
acceleration and enrichment than they would have if they were using the regular
math curriculum. Finally, Miller and Mills (1995) found that students of varying
highability levels in second through sixth grade
made large achievement gains when placed in a program using
both acceleration and enrichment. Thus, the answer to the most
appropriate programming for talented elementary mathematics
students may be a combination of acceleration and enrichment.
Mathematically Talented Students
and Their Need for Differentiation
Mathematically talented students approach, perceive, and
understand mathematics differently than other students. For
instance, they are able to skip steps in the logical thought process
when solving mathematical problems, can flexibly use
problemsolving strategies, and have a “mathematical cast of
mind” (Krutetskii, 1968/1976, p. 302). In defining mathematical
promise, the Task Force on Mathematically Promising Students
identified it “as a function of ability, motivation, belief, and experience
or opportunity.” They also stated that this definition recognized
that students who are mathematically talented “have a
large range of abilities and a continuum of needs that should be
met” (Sheffield, 1999, p. 310).
Due to these characteristics, the curriculum must be differentiated
for these students; that is, the content, process, and
products used with these students consistently must be modified
in response to their learning readiness and interests (Tomlinson,
1995). There are very few studies to date that study the effects of
differentiation on achievement of talented elementary students.
In one study with upper elementary students, Tieso (2003) found
that using an enhanced or differentiated mathematics unit with
aboveaverage students from all socioeconomic backgrounds
resulted in significant achievement gains compared to using a
unit from the regular mathematics textbook.
Students Processing Mathematics Like Professionals
As Pelletier and Shore (2003) and Sriraman (2004) have
found from their studies, mathematically talented students think
about mathematics in ways similar to the ways that experts or
professional mathematicians operate. Two renowned mathematicians,
Jacques Hadamard (1954) and George Polya (1954),
believed that the sole difference between the work of a professional
mathematician and the work of a student is in the degree
of sophistication they possess. Thus, both are capable of being
creative and analytical in solving problems and in posing new
problems at their respective levels.
In fact, encouraging students to act and work like professionals
is an approach that has been in place in the field of gifted
and talented education for quite a while. One of the hallmarks
of the Enrichment Triad Model is the placement of students
into the role of the “practicing professional” to pursue problems
of particular interest to them (Renzulli, 1977). More recently
Tomlinson et al. (2002) identified a special curriculum called the
“Curriculum of Practice,” whose intent is to provide opportunities
for talented students to use the skills and methodologies
of a discipline by having them function as a practicing professional
in the discipline. Experts recommend using this curricular
approach to help students construct and apply knowledge in a
particular discipline and thus gain a deeper understanding of the
subject; however, research on its effect on mathematically talented
elementary students is needed.
Development of the Project M3
Curriculum Units
In 1995, the NCTM Task Force on the Mathematically
Promising urged that, “new curricula standards, programs, and
materials, should be developed to encourage and challenge the
development of promising mathematical students, regardless
of gender, ethnicity, or socioeconomic background” (Sheffield,
Bennet, Berriozabál, DeArmond, & Wertheimer, 1995, p. 8). In
response, a collaborative team of experienced mathematicians,
mathematics educators, and leaders in the field of gifted and talented
education developed Project M3: Mentoring Mathematical
Minds curriculum units under the auspices of a U.S. Department
of Education Javits Program research grant. Following the recommendations
set forth in the literature, the units engage students
in both advanced and enriched content as they process
the mathematics like practicing mathematicians. Additionally,
the lessons are differentiated to meet the range of needs of talented
students. A general description of the units and how they
address the literature recommendations is provided next; this is
followed by a more thorough description of one unit to provide
a more concrete example of how these recommendations were
implemented.
Addressing the Literature Recommendations
Project M3 has developed a total of 12 units, with 4 units
at each of three levels primarily aimed at students in grades 3,
4, and 5. The content of individual units at each level is based
on one of the NCTM (2000) content standards, including: (a)
number and operations, (b) geometry or measurement, (c) data
analysis or probability, and (d) algebra. Table 1 summarizes the
primary content presented in each unit (for a more elaborate
explanation of unit concepts and lessons, see Adelson & Gavin,
2006; Casa & Gavin, 2006; Casa, Spinelli, & Gavin, 2006; and
Gavin, Casa, & Adelson, 2006).
The content in the units is accelerated by at least one to two
grade levels. The units also are enriched with interesting and highlevel
mathematical investigations. One way this occurs is with an
emphasis on the NCTM (2000) process standards, particularly
problem solving, realworld connections, and communication.
Communication provides a unique avenue for enrichment. The
Project M3 units include Chapin, O’Connor, and Anderson’s
(2003) talk moves (e.g., agree/disagree and why, adding on) to
help teachers facilitate verbal discussions and focus on significant
mathematics. Students also write about the mathematics in
two indepth and highlevel writtenresponse questions in each
lesson. The verbal and written communication helps engage students’
thought processes that resemble those of practicing mathematicians.
“Part of learning mathematics is learning to speak
like a mathematician” (Pimm, 1987, p. 76). Project M3 developed
the Student Mathematician’s Journal to support written communication.
In addition to worksheets used with the investigations,
the journals include two “Think Deeply” questions for each lesson
to engage students in writing about significant mathematical
concepts.
In accordance with prior research (Tieso, 2003; Tomlinson,
1995), Project M3 units provide differentiated instruction so students can work at their own level of understanding. Two unique
features, “Hint Cards” and “Think Beyond Cards,” are available
for each lesson and offer students support or further challenge
when necessary. Hint Cards are for students who have had little
prior experience with certain concepts and who may need a little
help in getting started or moving along. Think Beyond Cards
are for those students who have a firm grasp of the concepts
presented in the lesson and are ready for further challenge. They
ask students to expand their knowledge by using deeper, more
complex reasoning.
Unit Example
Thirdgrade students studying Unraveling the Mystery of
the MoLi Stone: Place Value and Numeration (Gavin, Chapin,
Dailey, & Sheffield, 2006) took on the role of mathematicians
at an archeological dig as they tried to decipher the numerical
markings on a stone. To do this, they explored the essential
concepts of place value: mainly patterns, groupings, and symbols.
In the lessons, they investigated differences between place
values, various bases, and other numeration systems (including
the Egyptian and Chinese systems). One enrichment activity
required students to apply their understandings of all of these
concepts to create their own numeration systems. Acceleration
occurred when students studied bases other than base10, as
these concepts typically are taught at much higher grade levels.
For example, one Think Deeply question asked students to
consider the similarities and differences between the base3 and
base10 numeration systems. To help differentiate instruction, a
Hint Card guided students to examine the values of the different
places in a number. Similarly, at the other end of the spectrum,
a Think Beyond Card had students ponder, “Does having place
values in the base10 system help us add and subtract more easily
or quickly? Why or why not?”
Research Design
There were several components to the Project M3 research
study that examined the impact of the units on student achievement.
Study participants included students using the Project M3
units (intervention group) and those using the regular curriculum
(comparison group). This paper addresses the efficacy of the
intervention with respect to student understanding of the concepts
in the units. Data collection from both the intervention
and comparison groups on the Iowa Tests of Basic Skills and
openended questions based on the released NAEP and TIMSS
items is still in progress. Results on these components of the
research study are forthcoming. The purpose of the study being
presented was to determine if the Project M3 units had a positive
impact on the intervention group’s understanding of mathematical
concepts presented in the units. The following question was
addressed: Was there an increase in mathematical understanding
for mathematically talented students after exposure to the
Project M3 curriculum units?
Sample
The sample consisted of approximately 200 mathematically
talented students entering third grade, most of whom remained
in the project through fifth grade. In order to be inclusive and
encourage the inclusion of typically underrepresented groups
in gifted programs, Project M3 implemented multiple methods
of student identification to include minorities, second language
learners, and female students as project participants. These measures
included a teacher rating scale, teacher feedback on class
performance and prior achievement, and a nonverbal ability
test.
Student participants were from nine schools in Connecticut1 and two schools
in Kentucky. There was an almost equal breakdown by gender (50% females in
grades 3 and 5 and 49% in grade 4). As indicated in Table 2, more than 40% of
students
were eligible for meal subsidies, and the sample was composed
of students from diverse racial and ethnic groups.
Professional Development
Teachers participated in a 2week summer training to increase
their mathematical content knowledge and to implement teaching
strategies developed to promote enrichment learning and
mathematical communication. Teachers also attended four to six
professional development sessions throughout the academic year
prior to teaching each unit. These sessions included training on
how to score pre and posttests using the rubrics, as well as time
to score their class sets with the supervision of the professional
development team. A professional development team member
visited each school every week the Project M3 units were taught to
ensure fidelity of treatment and offer individualized assistance.
Methodology
Instrumentation. Teachers in the 11 schools implemented the curriculum units
(see Table 1) with Project M3 students beginning in grade 3 and progressing
through grade 5, and they administered pre and posttests for each curriculum
unit. Professional mathematics educators, in conjunction with the curriculum
writers, created the tests and rubrics for each unit. They developed
test questions to determine students’ understanding of the major
mathematical ideas in each unit. Most questions were openresponse
items that asked students to justify their answers (e.g.,
Which digit in the problem 56 + 42 should be replaced by a 7 to
get the largest sum, and why?). The Project M3 staff designed the
unit rubrics to identify students’ various levels of understanding
(e.g., 2 points for replacing the 4 tens with the 7 and 1 point
for replacing the 5 tens with the 7). They made efforts to make
clear the distinction between points to be awarded by providing
sample responses and identifying misconceptions.
Scoring of the Pre and Posttests. The Project M3 staff used the
student responses on the pretests to identify approximately five
samples for each question that ranged in levels of responses; they
then came to a consensus about how to score them according
to the rubric. They used these samples during the professional
development sessions prior to the teaching of each unit to train
teachers on how to score the tests.
In addition to scoring their class pretests, teachers also scored
the posttests using the rubrics soon after completing each unit.
Project M3 staff doublescored all pre and posttests. If the first
and second set of scores on any subcomponent of any question
did not match, another staff member triple scored it. Expert
scorers discussed any other discrepancies further until a consensus
was reached, thus insuring interrater agreement.
Research Results
The researchers conducted paired t tests on the total scores
for each unit pre and posttest. Table 3 summarizes student
achievement gains, including the pretest and posttest mean
scores with their respective standard deviations as well as effect
sizes (Cohen’s d calculated with pooled standard deviations) for
pairs of data for all units. For each of the 12 Project M3 units,
similar results were achieved. The total scores for each unit indicate
statistically significant gains from pretest mean to posttest
mean at the p < .01 level of statistical significance. In addition,
the effect sizes were all large (Cohen, 1965) and ranged from
1.55 to 3.49.
Students in Project M3 began each unit with a mean pretest
score ranging from 7 to 36% of the total score possible, as noted
in Table 3. Although talented students typically might score
higher than this on an assessment, the researchers designed the
curriculum, testing, and scoring to be very rigorous to challenge
students and to avoid a ceiling effect. At the end of each unit,
students earned 48 to 77% of the total score, showing remarkable
improvement, with mean percent total gains from 30 to
55%. The almost entirely openended unit tests and their rubrics
required a great deal from students in explaining their answers
using precise and accurate mathematics and mathematics vocabulary,
and students made great strides in this process. Moreover,
94 to 100% of students, regardless of school or SES, made gains
from pretest to posttest for each unit.
Discussion, Limitations,
and Future Directions
The data indicate that the use of the Project M3 units by students identified
with mathematical talent produces significant gains in the understanding of the
mathematical concepts outlined in the curricular units. This considerable
advancement in student understanding of unit concepts occurred over a relatively
short period of time (each unit took approximately 6 weeks to implement).
According to Cohen (1965), an effect size equal to or greater than .80 is
considered to be large, and, as noted, the effect sizes for the Project M3 units
ranged from 1.55 to 3.49. It appears that the design of the curriculum units, in
combination with the professional development offerings, contributed to these
findings. However, a limitation of this study is that it is not possible to
isolate how much of the growth could be attributed to individual components of
the units and/or the professional
development support that was offered. Future investigations
might explore these areas further.
As already noted, students who participated in Project M3
represented a wide variety of racial and ethnic backgrounds. Also,
almost half were eligible for meal subsidies. There was a purposeful
selection of schools to ensure that the curriculum could meet
the needs of those students who had already exhibited mathematical
talent and those who had mathematical promise but
may not have had the opportunity to demonstrate their ability.
Future studies might investigate the differences between gender,
racial/ethnic groups, and lower and higher socioeconomic status
groups.
Other possible investigations emerge from these findings
that have meaning for both researchers and teachers. What
are the longterm advantages of students being exposed to the
Project M3 curriculum units? That is, will participation in Project
M3 impact their deep understanding of mathematics in middle
school, high school, and beyond? Will these students select majors
in mathematics and go on to become leaders in the field?
Another area of research interest is an examination of the
role of verbal discussion in written communication. The researchers
believe that the model of verbal discourse that was used in
Project M3 had an impact on students’ understanding and ability
to communicate that understanding in writing. A research study
to explore this further could have farreaching implications on
the teaching and learning of mathematics.
Researchers also need to study grouping options. In most
classrooms, students in this study were grouped by ability as a
top group in a particular grade level and taught by one of the
gradelevel teachers. In two situations, a teacher of the gifted
and talented, rather than a gradelevel teacher, taught a class
of students. The curriculum should be tested in other settings,
such as cluster groups, pullout groups, and selfcontained gifted
classrooms.
In conclusion, the review of literature indicates there is very
little researchbased curriculum that is specifically designed for
or is appropriate to meet the needs of mathematically talented
elementary students. This study suggests that curriculum based
on the recommendations set forth in the mathematics education
and gifted and talented literature can help students learn
advanced mathematics. Thus, Project M3 units may help fill this
curricular void and provide an accelerated and enriched program
to meet the needs of talented elementary students.
Author Note
The work reported herein was supported under the Jacob
K. Javits Gifted and Talented Students Education Act, Grant
Award Number S206A02006106 as administered by The Office
of Elementary and Secondary Education. However, the findings
and opinions expressed in this report do not necessarily reflect
the position or policies of the U.S. Department of Education.
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