Derrick, C. L. (2005). Improving
student retention through articulation and reflection. Instructional
Technology Monographs 2 (1). Retrieved <insert date>,
Testing and accountability are part of the educational structure
today. From elementary school to graduate level instruction, tests
are used as a convenient way to assess knowledge for a large group
of students. Elementary students must participate in state testing,
such as the Georgia Criterion Reference Competency Test (Georgia
Department of Education, 2004). High school students are faced
with the pressure of performing well on the Scholastic Aptitude
Test in order to be granted admission into many colleges and universities.
Many graduate programs require the Graduate Record Exam as part
of the application process. Whether we agree with it or not, testing
is an assessment tool that will probably remain a part of the
educational system. Students must be able to retain and apply
their knowledge in testing situations and in the real world. Research
indicates students are not able to transfer their knowledge between
the real world and the classroom (Chaney-Cullen & Duffy, 1999).
In order to accomplish this, students need to be given the opportunity
to explore concepts, develop an understanding of fundamental mathematical
processes and algorithms and use the combination of the two to
solve authentic problems. Memorizing the algorithm without understanding
may not be helpful when faced with an unscripted problem. Likewise,
a lack of procedural knowledge may hinder ones ability to focus
on higher level problem solving. I think that a blend of instructional
theories, such as constructivism, cognitive apprenticeship, project-based
learning, and the instructivist approach will facilitate learning
that transcends the retention and testing gap.
My problem is based upon the need to improve student retention
and application by incorporating articulation and reflection in
the mathematics classroom in an attempt to answer the following
Will articulation and reflection improve retention and application?
Does the quality of journal responses contribute to student success?
The study will focus on sixth grade math students. By having
students write and reflect about mathematical topics and procedures,
retention and application will evolve naturally. The purpose of
this study is to determine if articulation and reflection increase
retention and application.
Piagetâs perspective focuses on how logical reasoning is
developed and the role school plays in the process. Questions
and examples that lead to further investigation, activities that
challenge students, and finding out the ways students are thinking
are important for cognitive development (as cited in Green &
Gredler, 2002). Vygotskyâs focus on cognitive development
revolves around conceptual thinking, categorical perception, logical
memory, and voluntary attention. There is a correlation between
the written language and the transfer to other situations. Reciprocal
teaching is a good example of a constructivist teaching method
(Green & Gredler, 2002). This ties into the articulation and
reflection strategies used in the cognitive apprenticeship model.
One must develop an understanding of the meaning of articulation
and reflection and how it fits into the educational setting. Then,
determine if this is an effective teaching method.
My study is designed to evaluate the implementation of writing
and reflecting in the math classroom. A brief overview of cognitive
apprenticeship will be presented, followed by a more in-depth
review of articulation and reflection. As I embarked on the review
of literature, I found limited research on articulation and reflection.
Collins, Brown, and Newman were often cited. Most of the available
literature was written from an expository perspective on the effects
of writing in the mathematics classroom. I located a few studies
in Dissertation Abstracts International that looked promising,
but were not readily available due to cost and time constraints.
Therefore, I will summarize my findings based upon the literature
I was able to review at the time.
The cognitive apprenticeship instructional model is based on
the premise that learning should be authentic and have meaning.
In reviewing the literature, several strategies emerged, such
as modeling, coaching and scaffolding, articulation and reflection,
and exploration (Brill, Kim, and Galloway, 2001). Modeling may
take the form of a demonstration or verbalizing the process. Coaching
and scaffolding refer to the role of the facilitator and exploration
addresses the process of finding knowledge independently. Articulation
and reflection focus on the conveyance of ideas through verbalizing
or the writing process and the opportunity to revisit these ideas.
Articulation and Reflection
It is important to explore the meaning of articulation and reflection.
Collins, Brown, and Newman, (1987) state ãArticulation
includes any method of getting students to articulate their knowledge,
reasoning, or problem-solving processes in a domainä (p.
18-19). In addition, they also state, ãReflection involves
enabling students to compare their problem-solving process with
that of an expert other, other students, and ultimately, an internal
cognitive model of expertiseä (p.18-19).
Jurdak and Zein (1998) conducted a study entitled, The Effect
of Journal Writing onAchievement in Attitudes toward Mathematics.
The study looked at middle school students (11-13 years old) in
Beirut . The experimental group participated in journal writing
while the control group did not participate in the journal writing.
The following criteria were taken into account: mathematical communication,
conceptual understanding, procedural knowledge, problem solving,
and achievement in mean test scores, and the attitudes toward
mathematics. The Mathematics Evaluation Test was used, along with
a Likert questionnaire and student evaluation essay to measure
attitudes. Jurdak and Zein (1998) found that journal writing studentsâ
mean score was higher for conceptual understanding, procedural
knowledge, and mathematical computation. There was little difference
between the two groups in problem solving, attitudes, and school
achievement. The researchers were surprised that school achievement
test scores were not impacted by the journal writing process (Jurdak
& Zein, 1998).
Another study of ninth grade Algebra I students confirms the
need to incorporate writing in the math curriculum. This study
looked at relationship between writing and metacognition. The
studentsâ writing indicated that this framework of metacognition
exists. Writing allowed students to share their thought processes
and reasoning with others. Writing could support the development
of skills needed for problem-solving (Pugalee, 2001). Pugalee
indicates a lack of research to substantiate writing activities
in mathematics and the hopes that his study will encourage more
research in this area.
In an essay entitled, Advanced Math? Write? Brandenburg indicates
that through the incorporation of writing, students retained what
they learned ( Brandenburg , 2002). She advises teachers to consider
the time needed to grade journals and to develop a rubric to facilitate
assessment. Brandenburg (2002) was very open to various writing
ranging from definitions to self-evaluation and felt the process
helped students ãdeepen their understanding and retentionä
(p. 67-68). Havens (1989) indicated that her students developed
better conceptual understanding and application instead of relying
on memorization. Borasi and Rose (1989) focused on incorporating
journal writing in a college math class. They concluded the following
benefits: writing helps develop a better understanding and articulation
and reflection improved learning and problem solving. They confirmed
one of my rationales for undertaking this study; students will
internalize meaning if they use writing to explain concepts and
Albert and Antos (2000) wrote about a journal writing project
in which fifth graders wrote about how they used math in the real
world. Students shared and reflected on their writings, wrote
problems and worked in groups to solve them. They contend that
the writing process allowed the students to bridge the gap between
the classroom and meaningful situations. Similar comments regarding
the understanding of math concepts increasing through writing
and sharing and providing an avenue to make comparisons were echoed
(p. 527). In addition, writing helped students develop confidence
and ownership for their learning. (p. 528). Shepard (1993) shares
the opinion that the writing process facilitates the development
of meaningful conceptual learning.
Another article, written by Johanning (2000) reiterates similar
sentiments regarding writing in the mathematics class. Writing
provides students a chance to develop and clarify what they have
learned. This includes learning from oneâs mistakes. Students
were involved in a process of writing their solution, discussion
in groups, and revision. The studentsâ writing showed the
reasons for approaching a mathematical problem. The process of
sharing with others gave students a large knowledge base to apply
Language is compared to the acquisition of knowledge and concepts,
both are evolving based upon new experiences. This supports the
logical connection between the articulation and reflection to
increase the retention and amount of knowledge acquired within
the time limits of the educational setting. Students need to have
declarative and procedural knowledge for math computations in
order to move on to the more complex problems. Students become
focused on the computational skill, instead of the problem-solving
aspect. A common method of instruction for diverse students is
to teach the concepts, operations, and the problem-solving strategies
using a direct approach (Bottge, 2001).
One aspect of this study is to look at inert knowledge and ritual
knowledge, which is part of the math classroom, and how to help
students retain and apply this information. Inert knowledge is
something that is not called upon very often, used primarily to
pass a test and then not utilized. Ritual knowledge consists of
the steps needed or the routine to follow in order to arrive at
a response. Knowledge does not have to be one or the other. Educators
need to make ritual knowledge lessons more meaningful for students
and inert knowledge lessons need to include ways for students
to be more active in their learning process (Perkins, 1999). Brown
and et al. share the opinion students may be able to pass a test
but not be able to apply the concept in the real world. Students
would benefit from journal writing because it provides the opportunity
to connect the concept and algorithm with the real world. Teachers
benefit because they can prompt further reflection and follow
the thought processes of students and guide when necessary.
Based on their perceptions of their intelligence, children may
see problem solving as an opportunity for learning or an activity
that promotes failure (McGuinness, 1993). Writing and reflection
might provide students with an opportunity to be active participants
in a risk free environment and change their negative perceptions
about themselves. Through articulation students share thought
processes, make comparisons, and may discover new ideas. The teacher
can facilitate the transfer of knowledge by helping students make
other connections (McGuinness, 1993).
D.C. Phillips depicted the roles of the learners as active, social,
and creative. The idea being that active learning is acquired
through discussion and investigation. The social learner relies
on discussion and communication with others and the creative learner
may look at knowledge as a process of being created. Active engagement
of the learner may foster retention and application of knowledge
(as cited in Perkins, 1999). Overall, the literature indicates
the process of articulation and reflection is a worthwhile strategy
to implement in mathematics instruction.
Again, the minimal amount of information and research on articulation
and reflection in mathematics opens the door for further research
in this area. In general, writing and reflection are considered
to be worthwhile endeavors, but formal research is difficult to
My study will attempt to find the connection between articulation
and reflection and the ability to retain and apply basic knowledge
and heuristics. The assessment instrument for this study consists
of a fraction computation assessment. This does not mean that
conceptual learning and higher level problem solving are not taking
place throughout the learning process. This study represents a
narrow aspect of the learning process, retention and basic application.
Additional or more in-depth research would be needed to fully
evaluate conceptual learning and problem-solving abilities through
the use of articulation and reflection.
The purpose of this study is to determine whether articulation
and reflection improve retention and application in the mathematics
The study took place in a suburban middle school located in the
southeastern United States. Most of the students were from middle
to upper middle socioeconomic backgrounds. In order to efficiently
manage the collection of data, sixth grade students in one class
were invited to participate in this research. Eight students elected
to participate in the study.
A fraction computation test consisting of twenty problems was
developed to evaluate basic skills. The same test was used as
a pretest, posttest, and surprise test. A rubric was created to
assist in the evaluation of studentsâ journal writing. The
journal evaluation instrument was designed using a component point
system to decrease subjectivity. All responses were evaluated
using the same rubric during the same time period. The criteria
for the fraction computation entries were: 1.) Explain in your
own words the mathematical procedure you used to solve this problem.
2.) Write an example problem and include the solution. 3.) Explain
in detail how or why you might use this skill in the real world.
4.) Write down any questions or ideas you have that you would
like to share.
In order to increase validity and reliability, the fraction computation
assessment was designed to represent all four computations in
equal amounts. In addition, the same assessment was used every
time to eliminate format as a variable. Objectivity was minimized
by using a point system to evaluate the journal responses and
responses were evaluated during the same time period.
Students were given a pretest prior to the teaching of fraction
computation in their class. After the completion of the computational
section of the fraction unit, students took the same test as a
posttest. They were told in advance to prepare for the test. Approximately
one month later students were given the same test, but were not
told about the test in advance.
Students were encouraged to compile their journal writings in
a composition notebook or folder. All journal responses were assigned
after the pretest but prior to the first posttest. Some class
time was provided for journal writing and students were instructed
to complete some entries for homework. Students were encouraged
to write in a timely manner. Teacher assigned journal writings
were required to explain the procedures and application of addition
and subtraction, multiplication, and division of fractions.
Results and Discussion
The quantitative test data was collected using a written test
format. The pretest provided an initial starting point or baseline.
Confounding factors such as instructional strategies should be
taken into account. All the participants had the same instructor.
Students were encouraged to write and share their journal responses
for fraction computation. Time was allotted during class to write
and articulate their responses. In addition, students were encouraged
to complete their writings or discussions at home.
The data indicated that 100% of the participants increased their
score on the first posttest; which ranged from 25-95 percentage
points. Approximately 30 days later the second posttest was administered.
In order to use this as a tool to evaluate retention, students
were not told in advance. All three assessments were exactly the
Comparison of the posttest and second posttest assessments indicated
split results. Half of the participants showed an increase in
their second posttest score, while the remaining 50% showed a
decrease. Interestingly, both groups were relatively close in
their range of percentage points. The range of decrease was 10-25
percentage points, while the increase was 5-25 percentage points.
A score of 85% or higher on the first posttest was demonstrated
by 80% of the participants who submitted some journal responses.
The participants who did not submit any journal responses scored
75% or below on the first posttest. The second surprise posttest
represented a decrease in 50% of the scores and an increase in
50% of the scores. Of the two highest journal response scores,
one showed a decrease of 20 points and the other an increase of
10 points. Two participants who did not earn any journal points
scored 85% on the second posttest.
Based upon the assessment data comparing the pretest and first
posttest scores, the articulation and reflection of fraction computation
appeared to be beneficial to all participants. Even though all
participants did not submit journal entries, they may have benefited
by the sharing of the journal entries during class time. Students
were encouraged to share even if they did not have their written
journal response with them.
It is difficult to isolate articulation and reflection as the
sole reason for an increase in achievement on the first posttest.
Certainly, other factors such as motivation, study habits, and
instructional strategies have an impact on learning. The data
from the pretest to the first posttest supported the implementation
of articulation and reflection as an effective avenue for learners
to construct meaning and validated articulation and reflection
as a component of best practices for educators.
The retention of fraction computation was explored even further
by looking at the results of data from the second posttest score
and the quality of journal responses. While the data indicated
the same number of participants increased as decreased, it should
be noted that greatest decrease occurred for a participant who
did not earn journal points and the greatest increase occurred
for a participant who also did not earn any journal points. Overall,
75% of the students, scored an 80% or higher on the second posttest.
Again, other factors that are difficult to assess may have played
a role in this. Therefore, the data does not substantiate whether
the quality of journal writing had a direct impact on retention
as noted by the second posttest scores.
Some of the inconsistencies discovered when comparing the journal
responses to achievement gains warrant the need for further studies
in the area of articulation and reflection in all classrooms.
This study does not provide data to support that articulation
and reflection is the best way or 100% effective in developing
long term retention. Maybe different results would be derived
if another study included more participants and or a longer period
of time. Perhaps others will continue to research this aspect
of education in the quest for more effective classroom strategies
to meet the needs of all learners.
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