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 students’ reflection

within mathematical situations is reviewed. Three

important aspects then emerge from the literature review:

how to operationally define, individually facilitate, and

analyze students’ reflection 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 students’ reflection 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 students’ reflection 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

are…designed 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 student’s 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 back” was 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 back” was categorized as “review” if 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 back” was categorized as

“check” if the student tried to verify the outcome of a part

of a solution process. Finally, an indication of “look back”

was categorized as “compare” if 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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