For
the last 15 years there has been controversy about the teaching of mathematics
in the
One
area of emphasis in the NCTM standards is the role of reflection on and
communication of math ideas by students. In
particular, the standards for Grades 6-8 require that students be able to
“monitor and reflect on the process of mathematical problem solving.”
(National Council of Teachers of Mathematics [NCTM], 1989, Problem Solving
section, ¶ 1). The standards call for teachers to “regularly ask students to
formulate interesting problems based on a wide variety of situations, both
within and outside mathematics.” (NCTM,
1989, Problem Solving section, ¶ 9). In addition, teachers should use both oral
and written communication in mathematics so that students can “think through
problems; formulate explanations; try out new vocabulary or notation; experiment
with forms of argumentation; justify conjectures; critique justifications; and
reflect on their own understanding and on the ideas of others.” (NCTM, 1989,
Communication Section, ¶ 20).
Writing-to-learn
is a practice in education which has grown along with the rise in constructivist
practice. Among other things it
encourages learners to reflect on and communicate their learning.
There is quite a bit of anecdotal literature on writing-to-learn as
applied to mathematics. The primary
purposes of writing-to-learn in mathematics, as reported in the literature, are
to explore students’ attitudes toward math and math learning, and to allow
students to reflect on their understanding, log their learning process, and
solve specific mathematical problems (Connolly, 1989) (Countryman, 1992).
Schools in
Methodology
and Method
8th
grade math teachers across the district were invited to participate, either as
part of the writing group or the control group.
Roughly half of the teachers, spread across the four middle schools in
the district and one K-8 choice school, volunteered to use writing-to-learn in
their classes. 618 students will be in the writing group, and 617 students in
the control group. Scores for ESL and special
needs students will be included in the study, and will be analyzed separately as
well. Differences in pre- and
post-test scores between the study and control groups will be analyzed using
students t-test. Any differences
between the study and control groups revealed by the pre-tests will be accounted
for by an analysis of covariance.
The structure of
the autobiography, the formal math paper, and the math journal will be
prescribed and will be consistent across all classrooms using them.
Teachers will be encouraged to use information from the students’ math
writing to adjust their teaching to better meet students’ needs, but how this
is done will be left to the discretion of individual teachers.
Teachers in the writing-to-learn group will participate in a one-week
training workshop where they will be introduced to writing-to-learn in math,
will write their own math autobiographies and keep a math journal as they work
through some advanced mathematics problems.
Teachers will write a reflective essay about their experience of the
training as well. Thereafter, the
teachers will participate in quarterly workshops of three hours each, to further
explore the techniques of writing-to-learn, to ask questions, express concerns,
and clarify understanding, and to share with each other their experiences with
the journals and their responses to them.
A thorough literature review for this study would include a review of
literature about constructivist theory and mathematics teaching.
This would be included for two reasons.
First, the practice of writing-to-learn is situated within the
constructivist theory of learning, in that is seen as a tool which helps
students construct knowledge. Teachers
who report using writing-to-learn in their math classrooms generally approach
math from a constructivist point of view, or are led to a more constructivist
view through the use of writing to learn (Countryman, 1992) (Fennema et al.,
1996). Secondly, one limitation of the current study is that the schools
participating in the study use the knowledge-transmission model of learning
rather than the constructivist model. Therefore,
while using writing-to-learn, a strategy situated in constructivist theory, the
teachers’ general instructional practices are situated in transmissionist
theory. This disparity may influence how writing-to-learn is implemented in the
study. For the purposes of this
paper I will only mention enough about constructivist theory and mathematics
teaching to provide a minimal background for the reader of the paper.
The literature review would also include a review of studies relating
teachers’ knowledge of student thinking to the teachers’ instructional
practices. Two such studies have
been included here, but further exploration of this area would be beneficial.
The review would also include studies of the effectiveness of
writing-to-learn across disciplines. Since
there are few studies of the effectiveness of writing-to-learn in mathematics, I
would include some studies across disciplines as validation of the technique.
Since its development in the 1970’s, ideas about writing-to-learn have
evolved, and studies have been done to measure its effectiveness.
These studies have shown mixed results, and ideas about which kinds of
writing most greatly impact learning have been refined (Tynjälä, 2001). For
the purposes of this paper, I will only include a brief historical sketch to
create context.
A thorough review would also include some studies of the role of
metacognition in mathematics learning and the relationship between metacognition
and writing-to-learn. While anyone
familiar with the field of writing-to-learn would be aware of the relationship
between metacognition, learning, and writing-to-learn, this would not
necessarily be true of mathematics teachers, and an understanding of this
relationship is important for understanding one of the strengths and purposes of
writing-to-learn.
I
would also include a section on the writing techniques teachers report using in
the implementation of writing-to-learn in mathematics.
Two books which explore this seem to be pivotal works in that the first
is cited by research studies on writing-to-learn in math, and the second is
cited in anecdotal accounts by teachers of using writing-to-learn in math
classrooms (Connolly, 1989) (Countryman, 1992). While these reports include
teachers’ beliefs that the techniques improve students’ attitudes and math
learning, the results have not been verified through research.
Since the techniques that the current study uses will be drawn from those
recommended by these two authors and reported anecdotally in the teaching
literature, I include them in the literature review.
Finally, the literature review would include studies of the effectiveness
of writing-to-learn in mathematics, of which there are few.
I found only one study that looked at the relationship between
writing-to-learn used as a technique in mathematics teaching and students’
scores on standardized tests such as the MTBS-M, possibly in part because such
tests themselves are somewhat controversial as a measure of real learning and
reasoning ability. That study was
not in a peer-reviewed journal and had many problems, so was not included in
this review.
The
traditional view of learning, one that has dominated education in this country,
is the knowledge transmission framework, which sees learning as the transmission
of knowledge from teacher or text to student, and the reproduction of that
knowledge by the student. Current
thinking supports a constructivist or social constructivist theory of learning,
which describes learning as a cognitive activity, the active process of
knowledge construction, within a social context (Tynjälä, 2001).
Constructivism
as applied to the teaching of mathematics has been explored in the literature
for quite a few years (Narode, 1987), and is the basis of the NCTM’s Curriculum
and Evaluation Standards for School Mathematics.
Simon (1995) notes the problem that since the traditional view of
learning mathematics has been so pervasive, researchers have had limited access
to teachers with well-developed constructivist perspectives.
This has led to a disconnect between research on learning, which is
focused on constructivism, and research on teaching, which has focused primarily
on traditional instruction. Because of this, there are few fully developed
models of constructivist mathematics teaching which have been systematically
evaluated for effectiveness.
In
a 4-year longitudinal study, Fennema et al. (1996) attempted to determine if
there is a relationship between teachers’ growth in understanding their
students’ thinking, their own classroom instruction and subsequent student
learning. Over the 4-year period, data were collected from 21 1st-, 2nd- and
3rd-grade teachers and their classes. The
researchers gathered baseline data about teachers and their students and for the
next three years led a teacher-development program that focused on children’s
thinking, supported teachers in applying their new understanding in the
classroom, and studied teachers through observation, interviews,
paper-and-pencil instruments, and informal interactions. They assessed the
mathematics learning of the children before the teacher development program and
throughout the following three years.
The
researchers created a research-based model of children’s mathematical
thinking. The children’s thinking
was illuminated for the teachers using videotapes of children solving
mathematical problems, which the teachers analyzed.
After the teachers had abstracted their own understanding of children’s
thinking, the researchers shared their model with the teachers.
Teachers were encouraged to assess the validity of the model in their own
classrooms, eliciting information about students’ thinking through
conversation which the teachers recorded. Formal
and informal data about teachers’ instruction and beliefs were collected
through audiotape transcriptions of classroom observations, interviews, CGI
Belief Scale scores and field notes of informal interactions.
Building on previous research, the researchers defined levels of
constructivist-based teaching, and categorized each teacher’s movement, over
time, through the four levels. To measure student learning, a test for each
grade level was developed. Estimates
of internal consistency of the tests, estimated using Cronbach’s alpha, were
.84, .81 and .77 for the 1st-, 2nd-, and 3rd-grade tests respectively.
Gains for each teacher over the period of the study represented changes
in each teacher’s class means from Year 0.
The
researchers found that scores on the concepts and problem-solving test showed
consistent patterns of improvement for every teacher at each grade level, with
most of the improvement occurring within the first year, improving from between
1.7 to .95 standard deviations for 1st-grade teachers, an average of
.81 standard deviations for 2nd-grade teachers, and increasing more
slowly in 3rd grade classes, with an average increase of half a
standard deviation per year. Over
the duration of the study, 90% of the teachers had become more cognitively
guided in their teaching, believing that children’s thinking was the
appropriate place to start in their mathematics instruction, with their role
being to guide the development of the students’ understanding.
This
was a thorough study, with a strong research base and detailed description of
procedures and analyses. The researchers give a thorough explanation of their
methods, present results in tabular and graphic form, and give detailed
explanations of the levels of cognitive teaching. The teachers’ development
was undoubtedly influenced by the teacher development program, so the results
would not necessarily be generalizable to a population of teachers whose
development was not guided in this way. I
would rate it a 4.8. It is relevant to the current study in that it illuminates
the complex relationships between children’s thinking, the teacher’s use of
that thinking in the instruction process, and student performance on measures of
concept and problem-solving. According
to the authors, the study “provides strong evidence that knowledge of
children’s thinking is a powerful tool that enables teachers to transform this
knowledge and use it to change instruction.” (Fennema et al., 1996, p. 432).
They also conclude that, in conjunction with findings of other studies, this
study “provides a convincing argument that one major way to improve
mathematics instruction and learning is to help teachers understand the
mathematical thought process of their students.” (Fennema et al., 1996, p.
432).
Miller
(1992) studied the benefits to teachers of impromptu writing prompts in algebra
class. Their research questions
were: what can teachers learn about
their students understanding of mathematics by reading their written responses
to writing prompts? and, are their instructional practices influenced as a
result of these responses? Three
teachers from a large urban high school agreed to participate in the study.
Two were Algebra I teachers whose students were mostly in grades 9 and
10, and one was an Algebra II teacher whose students were mostly in grades 11
and 12. There were 28 and 32
students in the Algebra I classes, and 25 students in the Algebra II class.
A broad range of mathematical ability was represented in all classes.
The study was interpretive, with information gleaned from students’
responses to timed, in-class impromptu writing assignments, teachers’ writings
about what they were learning from students’ writings, and field notes.
The
prompts were varied, some asking students to produce a clear expression of their
understanding of a mathematical concept, skill or generalization, and some
asking for an expression of feelings about how the class was going.
Prompts were grouped into categories.
Writing prompts were carefully structured and were used four out of every
five classroom days. Teachers spent
some time weekly reviewing the students’ writings and provided the
investigators with their own writings, reflecting on their impressions gained
from reading students’ writing. The
duration of the study was one semester. Student
entries were read and analyzed and data summaries were constructed which
addressed the question: what can teachers learn about students’ understandings
of mathematics from their responses to prompts?
During their discussions, the researchers posed a second question:
did teachers’ instructional practices change as a result of what they
learned?
The
teachers found that the student writings revealed their misunderstandings in
surprising ways. The students were
unable to articulate concepts that the teachers assumed the students understood.
The teachers’ assumptions were based on the students’ ability to
answer correctly the mathematical problems associated with the concepts.
At the end of the project, the researchers found that instructional
practices of teachers had been changed in at least five ways:
reteaching immediately, delaying an exam because of a revealed lack of
understanding, designing a review based on information from student writings,
initiating private discussions with individuals who held misconceptions, and
using writing prompts during a lesson to ascertain understanding.
In addition, teachers noted the need to be very explicit and provide
examples when everyday language was used in mathematical context.
Both teachers and students reported that writing about mathematics at the
beginning of class helped to shift their focus from what had previously been
going on to the content at hand. The
researchers also noted an improvement in the attitudes of the students and
teachers toward each other as they used this means of communication.
Students expressed gratitude at having their thoughts listened to.
At the end of the semester, both students and teachers said they wanted
to continue the writing.
The
researchers noted that the benefits of this technique are limited by the
students’ ability to express themselves in writing, which some students
struggled with. Students were not
directed to write in ways that were grammatically correct and researchers found
some entries to be confusing enough so that it was impossible to interpret what
the student was saying. At one point
researchers interpreted a students’ answer to indicate a misunderstanding, but
when the teacher talked with the student, she found that the student had
misrepresented in writing what she meant. Thus
the researchers recommended that teachers be sensitive to the limitations of
written responses as windows to students’ thinking. This study was small, with
a limited sample, but was well-organized. The
report includes only a brief literature review in the introduction, and previous
studies are not referred to throughout the paper.
Comments of teachers and students are given throughout, but I would like
to have seen more. The results were
not categorized or tabulated, and this would have been useful. The results were
not generalized inappropriately. I
would rate this study a 3.
Writing-
to-learn.
In
the 1970’s, Writing Across the Curriculum programs, later called
writing-to-learn, were developed based on an understanding that writing could be
a tool for learning and thinking as well as a means of recording and expressing
(Tynjälä, 2001). Since then, the
practice of writing across the curriculum (also called writing-to-learn) has
spread through all disciplines, from elementary schools to universities.
In
her history of writing-to-learn and the development of ideas about the writing
process, Tynjälä makes the distinction between ‘knowledge telling’ writing
tasks and ‘knowledge transforming’ writing tasks. Knowledge telling tasks
fit more comfortably within the transmission framework, and knowledge
transforming writing tasks fit within the constructivist framework (Tynjälä,
2001). Within the constructivist
framework, writing is a cognitive act, and the writing process a thinking
process. Through writing, students
can reflect on and revise ideas, and explore and clarify their own thinking.
Connolly
claims that writing-to-learn in mathematics is “most basically about
developing students’ conceptual understanding of these subjects by developing
their capacity of use the languages of these fields fluently.” (Connolly,
1989, p. 4). In addition, he claims that informal classroom writing can
“retain natural curiosity, promote confidence in reason’s ability to
construct order by trial and error, even in problematic circumstances, and
overcome the anxiety that occurs when educations stresses answers, not options,
and product, not process.” (Connolly, 1989, p. 6).
Much of the
current literature about using writing-to-learn in mathematics classes is
anecdotal in nature, and appears in practical teaching journals which are not
peer-reviewed. However, in one of
the earliest works on writing-to-learn math, Connolly (Connolly, 1989)
identified the following writing techniques to be used in learning math: freewriting,
to eliminate distractions, focused freewriting, a way to initiate exploration of
a term, issue, question or problem, attitudinal writing, to discover attitudes
that affect aptitudes for learning, reflective, probative writing, metacognitive
process writing, which allows the learner to observe how one reads, takes an
exam, or solves problems, writing that explains errors, that questions,
summarizes, or defines, the creation of new problems, learning logs, and writing
to solve specific problems.
Countryman
(1992) outlines a comprehensive approach to writing-to-learn mathematics.
She includes such activities as freewriting, learning logs, math
autobiographies, writing about math problems, writing formal papers, and writing
test questions. The freewrites may
or may not be about mathematics, but her experience is that usually students
choose to write about math-related topics. Learning
logs are a record of the students’ experiences of the material, their own
work, their progress, and the class in general.
Students may be given prompts, or be asked to write main ideas of writing
assignments, definitions of new terms, and descriptions of new methods.
They may be asked to compare and contrast different procedures.
They may be asked to think through and create a verbal expression of
their problem-solving strategy before they solve problems mathematically.
In formal papers, they may be asked to report on mathematical ideas or
concepts.
There
are few studies that evaluate student performance using writing-to-learn.
One of the reasons for that may be that relatively few math classrooms
have embraced the constructivist approach, which is the context in which the use
of writing-to-learn math makes the most sense.
In addition, assessments of learning mathematics often test computation
skills and a student’s rote memorization of a procedure, rather than testing
conceptual understanding or real-world problem-solving ability.
In most studies that test math performance following the use of
writing-to-learn strategies, tests were created by the researchers.
One such study is the Fennema study, mentioned earlier, which shows a
positive relationship between writing-to-learn and math achievement (Fennema et
al., 1996).
Lesnak
(1989) studied the use of writing-to-learn in college remedial algebra classes,
using increased academic achievement as the principal measure of success.
He taught four basic algebra classes of 26 students each.
Two of the classes were taught using traditional techniques, and two used
the same techniques plus writing-to-learn strategies, which are provided in
chart form in the article. The
students were placed in the classes randomly. More than 90% had taken at least
one year of algebra at the secondary level with little or no success.
The rest were taking algebra for the first time.
The mean averages of final scores for the study and control group were
compared. Using “a statistical
test of the difference between two means” (not specified beyond this), Lesnak
found a difference of 3.2% in final scores, with the writing-to-learn group
scoring higher, a difference that is significant at a level of 4.6%.
More important to Lesnak were the value that students placed on the
activity in written evaluations at the end of the year.
Lesnak refers to these as his ‘qualitative’ results.
This
study was not published in a peer-reviewed journal but in a book (and also as an
ERIC document) that is cited by researchers now doing research on
writing-to-learn in math. It is not
in APA format, and is sloppily reported. There
is no literature review, and there are no charts showing results.
As a quantitative study it seems to be sound in its methods, but there is
really not enough information given to assess that.
Lesnak’s ‘qualitative’ results are impressive, but the study was
not conducted as a qualitative study, and therefore the study lacks credibility
and reliability. There has been no
attempt to categorize student responses, and only one means of information
collection was used, the end of year final writing assignment.
I would rate the study a 1.5, and include it here only because of the
paucity of good studies.
In
their study, Rudnitsky, Etheredge, Freeman and Gilbert (1995) attempted to
answer the question “Does learning problem structure through a
writing-to-learn approach enhance children’s problem solving?”
A secondary question was whether a treatment that teaches structure in a
meaningful way through students’ writing will lead to better retention of
knowledge. The participants were 401
3rd- and 4th-grade students from 21 classrooms representing six schools.
Classrooms were randomly assigned to one of two treatment conditions, the
writing treatment or the solving treatment, which involved a focus on solving
word-problems. Teachers of four 3rd-
and 4th-grade classes that were unable to participate in the study
agreed to act as a control group, and administer the pretest, posttest, and
retention test to their students. These
classes used the traditional mathematics curriculum which include solving word
problems, but without a focus on them. Following
training workshops, the teachers in both treatments were provided with the
instructional sequence, plans, and materials for each day of instruction, and
the procedures were carefully modeled. A
pretest was administered to students at the beginning of the study.
This was followed by the 16 days of instruction, during which researchers
were not in the classrooms. At the
conclusion of instruction, the posttest was given, and ten weeks later, the
retention test was administered.
The
pretest, posttest and retention test were equivalent forms of the same test.
Questions were developed and piloted in two 4th-grade classes not
included in the study. The tests
consisted of six word problems each, and one point was given for each correct
answer. If the students performed
correct operations but made computational errors, credit was given for a correct
answer. The data were analyzed using
a mixed factorial analysis of covariance. The
pretest score was used as a covariate to adjust posttest and retention test
scores. Sex, grade, and treatment
condition were the between-group factors. Time of test, post or retention, were
the within-groups factors and were treated as repeated measures.
Students rather than classes served as the unit of analysis.
The researchers found that the mean problem-solving score on the posttest
for the writing treatment (3.53) was significantly higher than for the solving
treatment (3.23) or the control group (2.28), and that the mean score for the
solving treatment was significantly higher than for the control group.
Males in the writing treatment (3.81) did significantly better than males
in the solving treatment (2.89) or the control group (2.36), and females in the
writing treatment (3.47) and the solving treatment (3.39) performed
significantly better than females in the control group (2.11), though there was
no significance difference between the treatment groups for females.
At the time of the retention test 10 weeks later, both treatment groups
performed significantly better than the control group (2.23), and the writing
treatment (3.74) performed significantly better than the solving treatment
(2.96).
The
authors conclude that both their hypotheses, that the writing-to-learn approach
would result in better arithmetic word-problem solving than the other approaches
used, and that the writing treatment would produce more enduring learning, were
supported by the data. The
author’s belief that this writing treatment is particularly powerful was
strengthened by the fact that the students using the writing-to-learn approach
did no actual problem-solving until the 9th day of the study, and
that, at that time, they worked each other’s problems rather than problems
constructed to look like the test problems, as the solving treatment did. This
is a well-designed and conducted study. The
authors have included tables of results, and provide a strong theoretical
foundation through their literature review.
Statistics used were appropriate to the design of the study and the
questions being researched. One
limitation of the study was the short time period over which it was conducted.
In addition, only a narrow range of problem types was addressed.
The researchers conclude that the study would have been stronger had the
children been given more opportunities to solve problems in their writing, and
if the more time had been spent training teachers.
The authors give no information about themselves besides their
institutional affiliation. Because of these limitations, I would rate the study
a 4.
For the last 20 years there has been controversy about the teaching of
math in the
In today’s political climate with its emphasis on standardized testing,
it is difficult to use authentic measures of assessment, such as students’
ability to think about problems in meaningful ways, to support instructional
practices. Ideally, real learning by
students will also be reflected in less meaningful assessments, such as
standardized tests, although there is little published evidence that this is so.
The current study will attempt to see if there is a relationship between
the use of writing-to-learn in mathematics and both achievement, as measured on
the MTBS-M, a standardized test, and attitude toward math, across a large
population of students. If I were to
do this study, I would include in my literature review more studies on the
efficacy of writing-to-learn, not limited to math, and on the relationship
between metacognition, one of the major functions of writing-to-learn, and math
learning. Because the students’
math journals will themselves be rich sources of information, I would probably
follow this study with a heuristic study, finding themes in the math journals
and categorizing entries according to theme to gain a richer understanding of
how the journals functioned for students. Additional
questions would be: What themes emerge from student math journals over the
course of the year? How can this information inform teacher practice?
What kinds of writing best encourage the sort of thinking that leads to
learning? In addition, the
literature suggests that student learning improves both through the process of
writing itself and from improved instruction geared more closely to their needs
that results when teachers read their writing and change their practices.
It would be fruitful to explore these relationships in greater depth.
References
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Battista,
M. (1999) The mathematical miseducation of
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strategies to support sense-making and discussion in mathematics classrooms: An
exploratory study. Journal for Research in
Mathematics Education, 29,
275-305.
Connolly,
P. (1989). Writing and the ecology of learning. In P. Connolly & T. Vilardi,
Eds. Writing to learn mathematics and science (pp. 1-14). New York:
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Countryman,
J. (1992). Writing to learn mathematics: Strategies that work.
Fennema,
E., Carpenter, T.P., Franke, M.L., Levi, L., Jacobs, V.R., & Empson, S.B.
(1996). A longitudinal study of learning to use children’s thinking in
mathematics instruction. Journal
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Miller,
L.D. (1992). Teacher benefits from using impromptu writing prompts in algebra
classes. Journal for Research in Mathematics Education, 23, 329-340.
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R. (1987). Metacognition in math and science education. (ERIC Document
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Mathematics. Retrieved
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Tynjälä,
P., Mason, L., & Lonka, K. (2001). Writing
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