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Revisiting Students' Reflection in Mathematics Learning: Defining, Facilitating, Analyzing, and Future Directions

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http://www.hrpub.org Universal Journal of Educational Research 9(1): 154-160, 2021
DOI: 10.13189/ujer.2021.090117
Revisiting Students' Reflection in Mathematics Learning:
Defining, Facilitating, Analyzing, and Future Directions
Shin-Yi Lee
Department of Education, University of Taipei, Taiw an
Received November 15, 2020; Revised December 24, 2020; Accepted January 28, 2021
Cite This Paper in the following Citation Styles
(a): [1] Shin-Yi Lee, "Revisiting Students' Reflection in Mathematics Learning: Defining, Facilitating, Analyzing, and
Future Directions," Universal Journal of Educational Research, Vol. 9, No. 1, pp. 154 - 160, 2021. DOI:
10.13189/ujer.2021.090117.
(b): Shin-Yi Lee (2021). Revisiting Students' Reflection in Mathematics Learning: Defining, Facilitating, Analyzing, and
Future Directions. Universal Journal of Educational Research, 9(1), 154 - 160. DOI: 10.13189/ujer.2021.090117.
Copyright©2021 by authors, all rights reserved. Authors agree that this article remains permanently open access under
the terms of the Creative Commons Attribution License 4.0 International License
Abstract Students rarely reflect on mathematics.
Reflection requires higher order cognition that has proven
difficult to teach. Much of the literature in mathematics
learning has highlighted the significance of reflection
without, however, offering much enlightenment on the
current research status of students' reflection within
mathematical situations. In this study, a wide range of
academic literature associated with studentsreflection
within mathematical situations is reviewed. Three
important aspects then emerge from the literature review:
how to operationally define, individually facilitate, and
analyze studentsreflection within mathematical situations.
Based on the three emerging important aspects, it would
be worthwhile for future studies to develop an operational
definition of reflection in mathematics learning, to research
how students’ reflection relates to their cognition and
metacognition within mathematical situations, to invent
individual reflective strategies in mathematics learning and
to investigate the effect of such strategies on their
mathematical performance, to devise techniques to analyze
students’ reflective behaviors within mathematical
situations, and to explore the extent to which students’
reflection could be promoted in mathematics learning. It is
hoped that this article will develop ideas for future practice
and research on studentsreflection within mathematical
situations about students’ reflection and the role it plays in
mathematics learning.
Keywords Communication, Mathematics Education,
Problem Solving, Reflection
1. Introduction
Reflection plays an essential role in students’
mathematics learning [1-4]. Carpenter & Lehrer [5]
identified reflection as one of “five forms of mental
activity from which mathematical understanding emerges”
(p. 22). Furthermore, studies have indicated that
understanding can be enhanced by reflection [6,7,8].
Kilpatrick [9] stated that reflection instructional
approaches were based on the belief that students learn
not only by doing but also by thinking about what they
and others do. Despite its importance in mathematics
learning, little is known about students reflection within
mathematical situations [10]. Even less is known about
the current research status of studentsreflection within
mathematical situations. Three key aspects emerge from a
comprehensive literature review on the topic. These
aspects were how to operationally define, individually
facilitate, and analyze students reflection within
mathematical situations. With respect to the aspects, the
present study is to discuss ideas of reflection and the
relationship between reflection and metacognition in
mathematics learning, to present activities facilitating
students’ reflection in mathematics classrooms, and to
consider ways to analyze students’ reflective behaviors
within mathematical situations.
2. Ideas of Reflection in Mathematics
Learning
In mathematics learning, various ideas of reflection
Universal Journal of Educational Research 9(1): 154-160, 2021 155
have been used based on Dewey [11], Piaget [12], Polya
[13], Schön [14] and so on. Dewey [11] indicated that
reflection is “the kind of thinking that consists of turning a
subject over in the mind and giving it serious and
consecutive consideration” (p. 3). Through reflection,
previous thoughts are reviewed, experience and learning
are connected, and deep learning can take place [11].
Piaget [12] used the notion of reflective abstraction to
describe an individual’s construction of
logico-mathematical structures by his or her personal
reflections. Reflective abstraction is critically important to
the theory of constructivism [15].
In mathematical problem solving, Polya [13] used the
term of “look back” to represent the ideas of reflection
[16,17]. He claimed that students can consolidate their
knowledge and develop their problem solving ability by
“looking back at the completed solution, by reconsidering
and reexamining the result and the path that led to it” (p.
14).
In his seminal work on reflection, Schön [14] created
the terms “reflection-in-action” and “reflection-on-action”.
The former enables students to reflect during their
learning activity and serves to reshape what they are doing
while they are doing it. The latter refers to thinking back
on what has been done to develop questions or ideas about
past activities. Schön [18] proposed reflection as a means
to integrate theory with practice.
While various ideas of reflection have been used in
literature on mathematics learning, the concept of
reflection is not clear [19-23]. To facilitate theoretical and
empirical work on students’ reflection in mathematics
education, more studies are needed to develop an
operational definition of reflection in mathematics
learning.
Further literature indicates that reflection is a
precondition to promote students’ metacognitive
development [20,24]. When students reflect, they are
encouraged to develop their cognitive and metacognitive
skills [16,17,25], both of which contribute greatly to their
mathematical performance [26,27,28]. Little, however,
was known about how reflection relates to cognition or to
metacognition [20]. Studies should be undertaken to
investigate how students’ reflection relates to their
cognition or metacognition in mathematics learning.
3.Activities to Facilitate Students’
Reflection
Literature has indicated that students rarely reflect on
mathematics [10,13,29,30,31,32,33,34]. Their
“consciousness of reflection in mathematical thinking is
low” and they have limited strategies to reflect on their
mathematical activities [10]. They often “adopt what
initially seems like a reasonable approach to a problem
and then proceed to follow it without further evaluation of
their decision as long as they are able to keep working”
[30] (p. 10). “Doing so, they miss an important and
instructive phase of the work” [13]. Indeed, to maximize
their learning opportunities, students should be
encouraged to reflect on their learning progress and
outcome during and after their learning processes. As
Dewey [11] maintained, reflection must be promoted in
students’ learning activities.
Reflection requires higher order cognition that has
proven difficult to teach [10]. Instruction in other
techniques requiring students to reflect may be fruitful.
Literature has suggested that communication and problem
solving encourage students to reflect [7,35-40]. This
section will review how these reflective strategies were
implemented in mathematics classrooms and their effects
on students’ mathematics learning.
3.1. Communication
Communication means “participating in social
interaction, sharing thoughts with others and listening to
others share their ideas” [41] (p. 5). Researchers have
described the relationship between communication and
reflection [42,43,44]. NCTM [44] claimed “reflection and
communication are interwined processes in mathematical
learning” (p. 61). Carpenter & Lehrer [43] described the
relationship between communication and reflection and
stated that “the ability to communicate or articulate one’s
ideas is an important goal of education, and it also is a
benchmark of understanding. … Articulation requires
reflection, and, in fact, articulation can be thought of as a
public form of reflection” (p. 22). From sharing their
solution methods, students are encouraged to think more
deeply about their own ideas in order to explain, clarify or
justify them [41]. Hiebert et al. [41] concluded that
“students who reflect on what they do and communicate
with others about it are in the best position to build useful
connections in mathematics” (p. 6).
Some research has investigated how to implement
reflection and communication in mathematics classrooms.
This section attempts to present different means which
have been used to evoke students’ reflection through
communication and how they influenced students’
mathematics learning.
Sfard [45] used the term commognition as a
combination of communication and cognition and argued
that communication and cognition are two sides of the
same coin. Thinking, when it occurs in a linguistic form,
cannot be separated from language[45] (p. 100). Within
the commognitive framework, language is viewed as the
main means of intellectual progress, and the acquisition of
numerical knowledge is conceptualized as using
numerical discourse in an objectified way. Sfard [45]
claimed that the perspective of commognition is useful in
understanding numerical thinking and abstracting process.
Brink [35] let first-grade students make an arithmetic
156 Revisiting Students' Reflection in Mathematics Learning: Defining, Facilitating, Analyzing, and Future Directions
book about sums for next year’s first-grade students. The
study found that writing the book made students reflect on
their own arithmetical knowledge. In the study conducted
by Cobb, Boufi, McClain & Whitenack [36], reflective
discourse in which what the students and teacher do
subsequently becomes an object of discussion in the class
was implemented in a first-grade mathematics classroom.
Students’ evolving arithmetical conceptions and strategies
were investigated in the study. The findings showed that
although reflective discourse did not inevitably lead to
each child’s reflecting, some students did demonstrate
mathematical development as classroom discourse
progressed.
Powell & Ramnauth [7] used multiple-entry logs to
prompt students’ reflection on mathematics text. In
multiple-entry logs, students wrote down a mathematics
text, their reflection on the text, and sometime later their
reflection on previous text or reflection on the same sheet
of paper. The mathematics text could be from a textbook,
a problem, a discussion, a computer screen, or from any
other learning materials. An instructor discussed with
students their multiple-entry logs and wrote down
responses to their reflection. Powell & Ramnauth [7]
indicated that multiple-entry logs served as vehicles that
prompted “learners to reflect on and to deepen their
understandings of mathematics” (p. 13).
Lafortune, Daniel, Schleifer & Pallascio [37] studied
philosophical dialogue in an elementary school
mathematics classroom and found that students used
significantly more higher-order thinking skills than
lower-order thinking skills at the end of the study. In their
study, students led the dialogue to question peers’ ideas
on philosophico-mathematical and meta-mathematical
questions. The teacher acted as the mediator of the
dialogue to help the students justify their opinions and
clarify their ideas in the classroom.
3.2. Problem Solving
With Dewey’s and Schön’s work as the foundation for
most models and theories of reflection, literature shows
that reflection can be precipitated by a perceived problem
[38]. In mathematics classrooms, reflection can be
promoted by solving mathematical problems either
cooperatively or individually.
Cooperative problem solving has been largely
suggested to promote students’ reflection while solving a
mathematical problem [39,40,46-50]. In cooperative
mathematical problem solving, reflection is stimulated not
only by the problem but also by the communication
among the students. Since cooperative problem solving
allows students to consciously think about what is being
done in the solution process through problem solving and
communicating, it should facilitate the building of
relationships between ideas and increase their
mathematical understanding. As suggested by Johnson &
Johnson [51], cooperative learning developed students’
higher quality cognitive strategies.
Artzt & Armour-Thomas [46] studied the problem
solving behaviors of seventh-grade students of varying
ability solving a mathematical problem in small groups.
They found that “Within the groups, students returned
several times to such problem-solving episodes as reading,
understanding, exploring, analyzing, planning,
implementing, and verifying” (p. 137).
Wheatley [40] implemented problem-centered learning
in primary mathematics classrooms in which students
were asked to solve mathematical problems in pairs, to
develop good arguments to support their solutions, and to
present their solutions to the class. In problem-centered
learning, teachers’ role was not to demonstrate how to
solve problems in details but to facilitate their students’
construction of mathematical understanding. The results
of this study indicated that “elementary students engaged
in problem-centered learning develop greater
mathematical competence than students taught using the
conventional explain-practice method even on
standardized tests which feature computation and standard
types of problems” (p. 531-532).
When solving problems individually, students may
reflect on their learning experience or on their solution
process. Literature indicates that students rarely reflect on
their solution process while solving problems individually,
especially when they get an answer of a problem
[13,16,17,29,30,32,33,34]. Venezky & Bregar [52]
indicated that “perhaps what grade schoolers need most
for improving problem solving abilities is not new and
more sophisticated instruction in mathematical concepts,
but simply more practice in evaluating solutions and
solution plans” (p. 126). “Even fairly good students, when
they have obtained the solution of the problem and write
down neatly the argument, shut their books and look for
something else” [13] (p. 14). More studies need to be
conducted to facilitate students’ reflection either during or
after their individual problem solving process and to
investigate the effect of such reflective strategies on their
mathematical performance. What follows reviews the
limited literature that focuses on feasible ways to promote
students’ reflection in individual problem solving and
their effects on students’ mathematical performance.
Polya [13] identifies five questions to prompt students
to look back at the completed solution: “Can you check
the result? Can you check the argument? Can you derive
the result differently? Can you see it at a glance? Can you
use the result, or the method, for some other problem?” (p.
xvii). Researchers and educators also have identified
approaches that can facilitate students’ reflection after
individually solving mathematical problems and most of
them have been centered on Polya’s ideas
[16,17,29,32,53]. As Silver [54] (cited in Kilpatrick [9])
noted, “many of Polya’s heuristic suggestions
aredesigned to get the problem solver to reflect on his
Universal Journal of Educational Research 9(1): 154-160, 2021 157
or her progress in problem solving and to assess the
effectiveness of the procedures being used” [9](p. 10).
Based on Polya’s ideas of “look back”, what follows
present ways to promote reflection in individual problem
solving and their effects on students’ mathematical
performance.
Checking the result “can be applied not only to the final
result but also to intermediate results” [13] (p. 60). When
students obtained an answer to a mathematical problem,
the answer should be examined not only for mathematical
accuracy and completeness but also for reasonableness
and practicality [13,17]. Jacobbe [32] investigated the
effectiveness of checking the reasonableness of the
answer in helping students overcome translation
difficulties in algebraic representations. In the study, the
students were asked to check their problem solving results
by substituting specific values for the problem variables.
The findings of the study indicated that the students
improved their problem solving performance in algebraic
representations.
Multiple solutions contribute greatly to promote
students’ reflection [13,17,29,41]. Through multiple
solutions, students are encouraged to solve problems in
different ways, to reflect on what they have done during
the solution processes, and to build connections among
the different solutions. These connections significantly
influence students’ conceptual understanding [55]. Krulik
& Rudnick [17] claimed that multiple solutions
encouraged students to reflect and helped them think
about what they have done.
There has been research exploring different means to
facilitate students’ reflection, among which writing books,
reflective discourse, multiple-entry logs, philosophical
dialogue, cooperative problem solving, and Polya’s look
back strategies have been discussed in the above section.
Little, however, has been done to develop individual
reflective strategies in mathematics learning and to
investigate the effect of such strategies on their
mathematical performance. The literature on analysis of
students’ reflective behaviors will be presented next.
4.Analysis of Students’ Reflective
Behaviors
Investigators usually determine existence or level of
students’ use of reflection based on the context of their
verbal activities and actions. Tape recording and verbatim
transcriptions of interviews, dialogue, discussion or
classroom discourse are usually used to analyze students’
reflective behaviors [36,37,46,49]. In mathematical
problem solving, think-aloud techniques were often used
to analyze students’ reflective behaviors while solving
mathematical problems. In this section, several coding
schemes using think-aloud techniques to analyze students’
reflective behaviors while solving mathematical problems
will be considered. Then, utilization of questionnaires will
be discussed.
4.1. Coding Schemes
In mathematical problem solving, Polya [13] used the
term of “look back” to represent the ideas of reflection
[16,17]. Schoen & Oehmke [56] developed a coding
scheme for analyzing students’ ability to look back when
a problem is solved. The scheme was developed to
analyze individual think-aloud protocols and “consisted of
student moves after a tentative solution was reached” (p.
221). The results indicated that the students rarely looked
back during their think-aloud interviews. What follows is
a brief description of the scheme.
Many students stopped as soon as they had an answer
and were given a score of 0 for step 4 [looking back].
Briefly, a score of 1 was assigned if some uncertainty was
expressed but no systematic check was made, a score of 2
was assigned if a check was attempted but was either
incorrect or incomplete, and a score of 3 was assigned if a
valid check of the computation, conditions, or
reasonableness of the solution was carried out. (p. 221)
With the definition of “examination of what was done
or learned previously” of “look back,” Lee [57] developed
a linguistic approach to analyze students’ “look back”
strategies when they solved problems in multiple solution
methods. A coding scheme was developed to identify key
utterances which themselves could suggest “examination
of what was done or learned previously” during the course
of mathematical problem solving. For instance, a student
said, I miscalculated right here. The word
miscalculated” was regarded as an indication of “look
back” because it suggested that the student went back to
check a prior calculation. It should be noted that
indications of “look back” need not be limited to those
identified by the study because of the variety of words or
phrases subjects might use to express their thoughts.
Furthermore, Lee [57] categorized all of the indications
of look back identified in the students’ “think aloud
protocols into four categories based on their function in
the students problem solving in multiple solution
methods. The four categories were recall, review,
check, and compare.” What follows presented the four
categories.
An indication of look backwas categorized as recall
if the student tried to look back at past experience that he
or she learned before solving the problem. An indication
of look backwas categorized as reviewif the student
tried to look back at a portion of a previous solution
process he or she just made during the course of problem
solving. An indication of look backwas categorized as
checkif the student tried to verify the outcome of a part
of a solution process. Finally, an indication of look back
was categorized as compareif the student tried to
compare a latter solution approach or result with a former
one.
158 Revisiting Students' Reflection in Mathematics Learning: Defining, Facilitating, Analyzing, and Future Directions
The results indicated that the students who looked back
more frequently tended to perform better in multiple
solution methods in the problems. Moreover, the students
tended to review and to compare more frequently than to
recall or to check on the problems during their problem
solving processes in multiple solution methods.
4.2. Questionnaires
While most studies coded transcriptions of students’
verbal activities and actions to analyze their reflective
behaviors, several studies used questionnaires to assess
students’ use of reflection in mathematics learning. For
example, to determine the extent to which the students
reported the use of reflection when receiving mathematics
instruction promoting reflection, Zhang, Zhao & Yang [10]
developed a questionnaire with four scales of frequency of
reflection: reflect when asked, reflect occasionally, reflect
voluntarily, and reflect very often. The results indicated
that although more students reflected after the instruction,
most students still didn’t reflect. For those who didn’t
reflect, their reasons for not reflecting were lack of
motivation and lack of reflective strategies.
5.Future Directions
Reflection is usually neglected by students when
learning mathematics. Little is known about how it should
be defined operationally, facilitated individually, and
analyzed in mathematics learning. The primary purpose of
this paper has been to draw attention to these three
important aspects of students reflection within
mathematical situations. What follows are the directions
for future mathematics education practice and research
with respect to the above three aspects. It would be
worthwhile for future studies to develop an operational
definition of reflection in mathematics learning, to
research how students’ reflection relates to their cognition
and metacognition within mathematical situations, to
invent individual reflective strategies in mathematics
learning and to investigate the effect of such strategies on
their mathematical performance, to devise techniques to
analyze students’ reflective behaviors within
mathematical situations, and to explore the extent to
which students’ reflection could be promoted in
mathematics learning. It is hoped that this article will
develop ideas about students’ reflection and the role it
plays in mathematics learning.
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