THEORETICAL OVERVIEW - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/25936/11... · THEORETICAL OVERVIEW 2.1 Conception of Reflection-A Dynamic Prospect ... (As cited by Erikson,

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THEORETICAL OVERVIEW

2.1 Conception of Reflection-A Dynamic Prospect

2.2 Functional Architects of Reflection

2.3 Multi-pronged Approaches towards Reflective Discourses

2.4 Celebrating the Radiance of Reflective Learning Practices

2.5 Fueling Mathematics Learning through Reflection

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Theoretical Overview

School of Pedagogical Sciences, Mahatma Gandhi University

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Introduction

Every rational being has the natural tendency to inquire and

experience; and then through reflection, find patterns and webs so as to

integrate meaning and seek additional opportunities to propitiate the

human propensity for learning. Educational practices thus need to provide

avenues that help individuals recapture, sustain and liberate the natural self

organizing learning tendencies inherent in all human beings. And it is in

this context that reflection should assume an undeniable place in the

curriculum of promoting excellence among our learners. Empowerment

through reflective practices has thus emerged as vital dimension in

scientific and everyday domains of human life. Thus an exploration of

reflective practices is inevitable in accrediting one’s own experiences and

in putting the heart back into learning. The present chapter thus aims at

developing a rich resource for building meaningful and lifelong learning

through coherent frameworks and scholarly researched practices.

2.1 Conception of Reflection –A Dynamic Prospect

Thinking about reflection and reflective practices has evolved over

many decades. Numerous educators and philosophers have contributed to

this body of knowledge. John Dewey is frequently recognized as the

modern day originator of the concept of reflection, although he drew on

the ideas of earlier educators, such as Aristotle, Plato and Confucius. To

these educators, reflection served as the pivot to transform lives through

extracting meaning from one’s own inner experiences. This is evident

through the words of Confucius uttered almost 25 centuries ago that,

“Learning cannot occur without silent reflection”.

Dewey Thought of reflection as a form of problem solving that

chained several ideas together by linking each idea with its predecessor in



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order to resolve an issue. According to him, reflection involves not simply

a sequencing of ideas, but a consequence - a consecutive ordering in such

a way that each idea determines the next as its proper outcome; while each

outcome in turn leans back on or refers to its predecessors. The successive

portions of a reflective thought grow out of one another and support one

another; they do not come and go in a medley. Each phase is a step from

something to something-technically speaking it is a term of thought

(Dewey, 1933).

Several attempts have been made to provide a clear succinct

definition of reflection. As a result an academic purview of reflection and

reflective thought has been evolved. The educational theories of Dewey

(1933), Schon (1983), Kolb (1984) and Boud et al. (1985) provide the

fundamental principles while Gibb (1988) ,Cowan (1998), Brockbank and

Mc Gill (1998) ,Moon (1999a, 1999b, 2004), Mezirow (2000), Kember et

al. (2000), Fade (2005) have extended and enhanced our understanding of

reflection. Given below are the changing view and enduring perspectives

on reflection (Table 2.1).

Table 2.1. Understanding the Domain of Reflection

Year Educationist Definition

1933 John Dewey

‘An active persistent and careful consideration of any belief or supposed form of knowledge in the light of the grounds that supports it and the further conclusions to which it tends’. He viewed reflection as an integration of attitude and the skill of judgment in methods of inquiry and attempts to resolve problems through rational thought process

1985 Boud et al. Reflection is an important human activity in which people recapture their experience, think about it, mull it over and evaluate it.



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1985

1995

2000

Boud et al.

and Slock

Hausen

Reflection is a generic term for those intellectual and affective activities in which individuals engage to explore their experiences in order to lead to a new understanding and appreciation

1991 Mezirow Reflection is the process of critically assessing the content, process or premise(s) of our efforts to interpret and give meaning to an experience.

1993

Reid

(As cited by Allyn & Turnock, 1993.)

Described as an active process rather than passive thinking. She states, ‘Reflection is a process of viewing an experience or practice in order to describe, analyse, evaluate and so inform learning about practice.’

1996 Morrison

Reflection as ‘a conceptual and methodological portmanteau’. According to Morrison, the manner in which reflection is commonly used has shuttled between the process of learning and the representation of that learning

1996 Loughram The deliberate and purposeful act of thinking which centers on ways of responding to problem situation in teaching and learning.

1996

Daudelin

(As cited by Erikson, 1996.)

Reflection is the process of stepping back from an experience to ponder, carefully and persistently, its meaning to the self through the development of inferences: learning is the creation of meaning from past or current events that serve as a guide for future behaviors.

1998 Pierson

Regarded reflection as both complex and elusive concept. He suggests that reflection is a purposeful inter subjective process that requires the employment of both calculative and contemplative thinking

2002 Cambill -Jones and Campbill John

Describe reflection as an, “inner dialogue with oneself whereby a person calls forth experiences, beliefs and perceptions”.



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2002 Rusko, Roskos and Vukelich

Explains reflection by adding that the dialogue should both inform and transform knowledge and action

2002 Bergsgaard and Ellis

Reflection may be take the form of self observation to gain insight about the reasons for one’s own thinking and one’s own actions and their consequences

2004 Moon

Reflection is thinking for an extended period by linking recent experience to earlier ones in order to promote a more complex and interrelated mental schema which involves looking for commonalities, differences and interrelations beyond their superficial elements.

2005 Fade Reflection involves describing, analyzing and evaluating our thought

2009 Mertler Reflection is defined as the action of critically exploring what you are doing, why you decided to do it, and what its effects have been

The dynamic connotations within the reflective matrix indicate that:

Reflection is the efficacious judgment of one’s thought process

As within the transformative paradigm knowledge is processed

from crude form into meaningful and creative associations.

There is a possibility to apply apt discourses.

It hinges on calculative and contemplative thinking.

It is the basis for learning representations

It is an action plan for future learning and slot for integrating

previous experiences.

It capacitates visualized actions.

It cultivates cognitive resonance aspects of problem solving behavior



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The primary focus on reflection thus lies on processing issues and in

the application of theory to practice. Three operational forms of reflection

are in vogue in the reflective writing at various educational levels. They

have been delineated here.

2.2 Functional Architects of Reflection

Reflectivity in its functional form can be identified as comprising of

three prominent constructs. They are reflective thinking, reflective learning,

and reflective practices. Reflective thinking is the heart of any intellectual

process and stimuli for reflective judgment. Reflective thinking builds an

awareness of problematic zones, convinces one of possibilities to solve them

and provides the grounds for supporting evidences. As a result of such mental

engagements, meaningful knowledge gets constructed and the individual is

placed on the road of acquiring entrepreneurial skills of decision making,

problem solving, leadership quality, communicative skill and the like.

Reflection helps students to develop higher order thinking skills and

decision making capabilities by prompting them to:

Relate new knowledge to their prior understanding

Think in both abstract and concrete terms

Apply specific strategy to novel tasks

Understand their own thinking and learning strategies

Activate efficient decision managing capacity in their learning

episodes

Exhibit professionalism in their practices

Reflective thinking develop the expertise and insights for the

intelligent organization of syntactic stages of knowledge pyramid through

connecting, sharing and structuring that accumulate into wisdom .



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The outcome of reflective thinking fabricates reflective learning which is

placing a student in a state of doubt, perplexity or mental difficulty so as to

prompt him to search, hunt and inquire extensively in order to resolve the doubt

and dispose the perplexity. It requires an exploration of past experience,

attention to feelings and reexamination of information that would facilitate the

emergence of appropriate solutions and better understanding of concepts.

As individuals reflect on their learning they gain important information

about how they perceive the efficacy of their planning, experimenting, data

gathering assessment and self modification. These experiences provide

opportunities to practice the habit of continuous growth through reflection,

called as the reflective practice spiral. It is in essence a professionalized form

of reflective learning. As Bright (1996) puts it reflective practice “is the

process which underlies all forms of high professional competence”. From

the organizational perspective reflective practice is a powerful norm that is

required for continuous improvement of teaching and learning practices that

result in high levels of student achievement. Reflective practice is the means

by which learning, renewal and growth continue throughout the development

of education. In educational literature John Dewey and Schon are two of the

most cited contributors to foundational concepts of reflective practice.

Reflective practice is a way of thinking that fosters personal learning,

behavioral change and improved performance. Through systematic inquiry

and analysis, it is a way for individuals to create meaningful and enduring

change by changing themselves. Reflective practice requires an environment

of support. It requires an organizational climate that encourages trust and

openness of communication, critical dialogue, risk taking and collaboration.

It requires people to be willing to analyse their own behavior and explore

thoughts, feelings and actions. Reflective practice as a human resource

strategy enhances individual learning, creating change and shaping learning



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organization. It provides information needed for people to effect positive

changes in performance. It respect the right of individuals to exercise self

direction and even more, enhances their ability to exercise control over their

own learning and their own actions.

Educators need to stress that reflection is an important part of any

learning and this may mean that there is a need for greater specificity and

precision in the learning outcomes of modules and courses. Because reflection

is part of the thought process, there is a potential danger that academicians and

educators assume that all students are automatically reflective learners. But

literature reviews reveal that reflection does not occur by change; it has to be

fostered through specific exercises, techniques and tools.

2.3 Multi-pronged Approaches towards Reflective Discourses

Sustaining a large vision and creating a coherent educational experience

for students requires constant orchestration of the various teaching learning

strategies, student tools and various theories introduced into the school.

Teaching learning strategies determine the approach for achieving the

learning objectives and are included in the pre-instructional activities;

information presentation, learner activities and testing follow through.

Reflective practice adopts very different set of strategies incorporating key

principles drawn from constructivism; experiential learning and situated

cognition (Osterman & Kottkamp, 2004). Reflective learning strategies can

have noticeable effect on learner engagement, metacognitive stances of

learning habits and performance levels. It provides opportunities for learners

to design; construct and manage their own learning episode which enables

them to become active life long learners.

Frameworks that assist learners in unleashing understanding and in

eliciting self explanations have been widely researched and a few

prominent facilitative modes are presented here.



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a). Van Manen’s Type III Mode of Reflection (2002)

In the model the networking of reflection focused on three types namely

recollective reflection, active or interactive reflection, and anticipatory

reflection. Here the learners are encouraged to mine the past experiences to

evolve deeper and new insights and to indulge in discussions during learning

process which culminates in the framing of action plans with a vision of its

future consequences. The model is depicted in Figure 2.1

Figure 2.1. Van Manen’s Type III Mode of Reflection

b). Zeichner and Liston’s Dimensional Mode of Reflection (1996)

This model gives five different dimensional foci of reflective practice

as rapid reaction, repair, review, research and retheorise and research.

This is a different approach to uncover the personal theories of learners

and makes them explicit. Figure 2.2 illustrates this model.



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Figure 2.2. Zeichner and Liston’s Dimensional Mode of Reflection

In this model learning starts with immediate response to a situation.

The learners then assess the situation by reframing it and the learner

formulates an action plan with new insights through framing and

reframing the learning.

c). Seidel & Blythe’s Four ‘ward’ Reflection (1996)

Steve Siedel suggests that a useful framework for approaching

reflective practice is one which involves the learner in looking backward,

looking inward, looking outward and looking forward. Here the learners

are encouraged to consider the personal experiences and actions with an

inquiry mind and allowed to work around occurring events in the light of

personal purposes and beliefs and find possible solutions. Sometimes the

learners envision the future events and reflect on it to prepare plans of

action to workout.

d). Scanlon and Chernomas (1997)

They described a model of reflective learning based upon the work of

Atkins and Murphy (1993). This model suggests three stage of reflection

as in Figure 2.5.



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Figure 2.3. Scanlon and Chernomas’ Model of Reflection

Awareness, the cornerstone of reflection initiates the process of

reflection when an individual is confronted with a discomfort or lack of

information in a problematic situation. Curiosity or excitement regarding

the need to learn more also triggers the first stage of this model. The

second stage of reflection entails critical analysis of the concept, situation,

event or need for knowledge, taking into account one’s current knowledge

and the application of new information. The final stage ‘new perspective’

emerges as subsequent to applying new information, indicates that the

learner has gained insight into understanding the concept, situation or

event.

e). Jack Mezirow’s Operational Level of Reflection (1998) provides a

comprehensive, logical and most importantly workable frame work of

reflective thinking in no less than seven levels, which is a fine sub division

of levels to produce an operational assessment scheme. The investigator

developed the evaluation tool of reflective thinking in par with this

framework. The different levels are depicted in Figure 2.4

Figure 2.4. Mezirow’s Operational Levels of Reflection



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Mezirow separates reflective actions from non-reflective action. He

identified three types of non-reflective actions namely, Habitual action,

introspection and thoughtful action. There were two levels of reflective

action of which the lower or less critical level was subdivided into content

and process reflection. Mezirow labels the more critical form of reflection

as premise reflection. Detailed explanations are provided in the

methodology chapter.

f). Jay and Johnson’s Accelerated Levels of Reflection (2002) have

outlined a systematic classification of reflective thought profiling three

dimensions as in Figure 2.5

Figure 2.5. Jay and Johnson’s Accelerated Levels of Reflection

This framework sketches three different levels. The initial stage

involves describing a situation and the second stage points to the weighing

up of a situation from different perspectives. The final stage namely

critical reflection involves learners looking at all the possible perspectives

of a problematic situation and at all the players involved: facilitator,

learner, learning environment, institution and the society.

g). Evertson’s Explorative Levels of Reflection (2006) presents a

conceptual frame work representing a continuum of multiple levels as

follows.

Evertson’s Exploratory Levels of Reflection

Surface Reflection

• Strategies, Methods , Approaches



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Pedagogic Reflection

• Educational goal, Underlying theory, Linking theory and Practice

Critical Reflection

• Moral and ethical implications, Consequences.

Self Reflection

• Personalistic, Reflection within, Dialogic

This framework implies a functioning understanding of the

constructed nature of knowledge and a metacognitive stance. It gives a

critical orientation of different phases of reflection.

Some more suggested teaching learning strategies promoting reflective

practices and their significant aspects are described in Table 2.2.

Table 2.2. Promotion of Reflection through Prominent Strategies

Sl. No. Strategy Name Sketches of strategy highlights

1

Mapping:- • Concept

mapping • Thinking maps • Mind maps and

the like

• Visual transformational language/tool of learning, planning and instructor

• Visual patterns of thinking, communicating and synthesizing thoughts

• Meta language for learning • Student centered, collaborative approach • Explicit processing of information

2

Personal narratives or inventories: • Reflective

journaling • Portfolios • Critical incident

analysis • Left hand column• Questioning • Personal

inventories

• Purposeful collection of student work • Artifacts of self reflection and self

evaluation of learning • Metacognitive growth of the learner • Vividly remembered event • Unplanned and unanticipated • A sense of student’s daily thoughts,

perceptions and experience in the classroom

• Uncovering assumption • Identification of problematic event



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• Prepares a dialogue with left hand column and right hand column

• Improve ability to communicate • Powerful role play activity • Explore underlying beliefs • Diagnostic and counseling

3

Direct observations:

• Video taping • Audio taping • Structured

observation • Role playing

and simulations

• Unfiltered and true record of events • Thoughtful and sustained dialogue

possible • Gathering information of individual

behaviour • Developing effective group process skills• Observable dimensions of classroom

practice • Judgmental information • Structure hypothetical situation and

reflect on it

4 • Problem based

learning • Problem solving

• Learning is driven by challenging problems

• Collaborative work • Both curriculum and process • Accessing prior knowledge and

reflecting • Problem solving proficiency • Naturalistic learning from experience

5 • Let me learn

• Interactive learning model • Depicts the brain as a triune system • Systematic enquiry into one’s own

practice

6

Personal developmental planning or profiling

• Self appraisal system • Organised collection of evidence and

reflection on the experience

7 Peer and self assessment

• Assess learner’s own work or others • Constructively critical of the learning

process



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Even though a series of approaches and strategies have been explored

for activating internal processes in learning it has been observed that Problem

Based learning, Journaling and Thinking maps provide rich resources for

engineering thought and in establishing narrative patterns of learning. The

detailed version of the three select reflective learning strategies namely

Problem Based learning, Reflective Journaling and Thinking maps have been

synthesized with an aim towards bringing a balance in teacher preparation.

2.4 Celebrating the Radiance of Reflective Learning Practices

The purpose of reflective practices is to get learners into the habit of

thinking about their experiences. The select strategies namely, Reflective

journaling, Problem based learning and Thinking maps capacitate the

learners to start internal and external voices of reflection making a

commitment to modifications, plans and experimentations.

2.4.1 Toning up reflection through Problem based learning

PBL – Emergence and Meaning

PBL is an approach to learning that has grown breadth and depth

across the world since 1970’s. It was evolved from innovative health

sciences curricula introduced and has spread across the globe and across

most disciplines in North America. The great teacher Socrates presented

students with problems that through questioning enabled him to help them

explore their assumptions, their values and the inadequacies of their

preferred solution.

Problem Based Learning is a term used within educators for a range of

pedagogic approach that encourage students to learn through the structured

exploration of a research problem. It describes techniques that make students

take an active, task oriented and self directed approach to their own learning

and provide students with insights into the research process.



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It is a way of constructing knowledge using problems as the stimulus

and focus for student activity and is based on the idea that individuals

fashion their understanding largely through what they experience (Ronis,

2008). It starts with problems rather than with explosion of disciplinary

knowledge helping students identify problems, pose their questions,

research answers, report results and create a stake in their own learning.

PBL moves learners towards the acquisition of knowledge and skills

through a staged sequence of problems presented in context, together with

associated learning materials and support from teachers. The emphasis of

PBL is on learning processes of enquiry which proceed by asking what

needs to be known to address and improve a particular situation.

The assumption behind PBL is that learning through problem

situation is much more effective than memory based learning for creating a

usable body of knowledge. In PBL the learners were asked to explore a

problem situation. Through such an exploration students are expected to

examine the gaps in their own knowledge and skills in order to decide

what information they need to acquire depending on the nature and scope

of problems which were presented as stimulus.

The multiplicity of learning approaches, active participations in the

construction of new knowledge , sufficient time for metacognitive

activities like knowledge assimilation and reflection, scope for

socialization and risk taking and making connection emotionally,

physically and cognitively to the content; all delineates PBL process in

equipping the learners with higher thinking abilities .The philosophical

underpinnings to the PBL process that explain why one would chose to

make a shift to PBL is the relevance of learning that occurs in the

classrooms. Real world problem scenarios and the selection of learning

needs by learners themselves contribute to holding the student’s interest in



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developing a deeper understanding of the content and increasing the

retention of new information. PBL ends up orienting students towards

meaning making over fact-collecting through contextualised problem sets

and situations. Students are encouraged to take responsibility for their group

and organize and direct the learning process with support from a facilitator.

It can be used to enhance content knowledge and foster the development of

communication, problem solving and self directed learning skill.

PBL could be defined in many ways .Barrows, one of the major

contributors to the field, defined PBL as ‘The learning that results from the

process of working towards the understanding of a resolution of a

problem. The problem is encountered first in the learning process’

(Barrows and Tamblyn; 1980)

An inquiry based PBL approach to learning on the other hand, is an

approach based upon a teaching philosophy in which the learning process

itself is the goal, one in which individual’s quest for knowledge and search

for solutions are the learning objectives [Ronis, 2008].

A working definition of PBL may be that it is and inquiry process

that resolve questions, curiosities, doubts and uncertainties about complex

phenomena in life.Lambros (2004) defined PBL as a teaching method

based on the principle of using problems as the starting point for the

acquisition of new knowledge. The principal idea behind PBL is that the

starting point for learning should be a problem, a query or a puzzle that the

learner wishes to solve (Boud, 1985).

The following definition of PBL draws together the points about it in

the most comprehensive manner. PBL is both a curriculum and a process.

The curriculum consists of carefully selected and designed problems that

demand from the learner acquisition of critical knowledge, problem



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solving proficiency, self directed learning habits and team participation

skills. The process replicates the commonly used systematic approach to

resolving problems or meeting challenges that are encountered in life and

career. Official descriptors generally describe it as an instructional strategy

in which students confront contextualized ill structured problems and strive

to find meaningful solutions collaboratively and reflectively. PBL is a

student centered instructional strategy in which students collaboratively

solve problems and reflect on their experiences. The above mentioned

definitions throw light into the following characteristics of PBL.

Learning is driven by challenging open-ended problems.

Students work in small collaborative groups.

Multiplicity of learning approaches.

Teachers take on the role as facilitators of learning and

students, the active problem solvers.

Sufficient provision for metacognitive actions

Accessing prior knowledge and reflecting.

Both curriculum and process.

Systematic problem solving proficiency.

Naturalistic learning from experiences.

Choosing effective PBL problems for the learners should acknowledge

that they enjoy being challenged and appreciate the opportunity to work

things out for themselves. They are developmentally ready not just to access

new information but also to determine its relevance and to apply it. Selected

PBL problems should allow ample opportunities for students to demonstrate

their growing and more mature abilities to conceptualize the problem by

brainstorming, possible linkages to course content, experiment with new

approaches in the light of reflection process and reassessing the experience in



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the light of its outcomes. Evidence indicate that peer support, active decision

making and planning help reduce the stress levels that these students often

feel and can tackle PBL problems that are longer in duration, richer in content

and more complex in problem solving challenges.

Figure 2.6. Attribute Web of Problem Scenario

There were two main discussions about facilitating PBL groups.

Concerned the power to control learning activities.

When and how to make interventions.

Through out the PBL process the teachers’ role is very active as the

facilitator and guide and sometimes as ‘expert resource’ in order to provide

some of the needed information. For the most part, the teacher is monitoring

the process and progress of the students, helping them to explore the intended

learning objectives, and reassuring them or redirecting them as needed. It is

important to understand that the teacher is any thing but absent from the



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dynamics and effectiveness of this learning approach. The teacher is pivotal

to the opportunities available to the students in the PBL methodology and

should know how a student is learning more, retaining more, and becoming

able to apply more and be a better collaborator or negotiator as a result of

engaging the PBL methodology.

Constructs of PBL

PBL as one of those pedagogic phenomenon that is ,both a learning

activity and an assessment , is a student centered learning strategy in

which learners confront contextualized, ill structured problems similar to

real world situation which entail deep learning and responsibility for their

own learning. It is essential that learners determine their own learning

needs, or learning issues, based on the problem they encounter. Effective

facilitation will prompt students into the intended learning issues and

needs. PBL is also renowned for shaping learners in collaborating team

work and allows the development of critical thinking, reflective thinking,

content knowledge and problems solving skills.

In PBL the learner may encounter the following ingredients.

Figure 2.7. PBL Ingredients



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PBL process in action

Mathematics rests on the premise that learning is an active process in

which learning mathematics is something pupils do, not something that is

done to them. PBL process engage learners actively in self directed

learning processes with a concerted group effort. This exploratory method

of learning is so effective for mathematics education as it captures pupils’

interest and forge significant learning links, amplify deeper understanding

and expatiate retention of new information. The characteristic, approaches

and principles underlying PBL can be compiled in to certain guidelines as

a framework for designing and implementing PBL process in mathematics

classes. The guidelines compiled as operational components of PBL

process are enlisted as:

The problem is encountered first in the learning process before

any preparation or formal study has necessarily occurred.

The problem is presented to students in the same way, that it

would present itself in reality

Students work with the problem in a manner which permits their

ability to reason and apply knowledge to be challenged and

evaluated, in a manner that is appropriate to the current level of

learning.

Needed areas of learning are identified as the problem is

explored and used as a guide to individualized study.

The skills and knowledge acquired by this study are applied

back to the problems to evaluate the effectiveness of learning

and to reinforce learning.

The learning that has occurred in work with the problem and in

individualized study is summarized and integrated into the



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students existing knowledge and skills. (Barrows, 1986; Barrows

and Tamblyn, 1980).

There are many ways in which PBL can be implemented to

synthesize mathematics and make it more creative, relevant and global. It

offers how the learners frame problem heuristics; how they participate in

the learning process; how they discern the resources for needed

information; how the teachers can make use of the community as a

learning resource and explains how such activities can be initiated and

integrated into classroom learning. A generalized version of all these

pathways is depicted in Figure 2.8

Figure 2.8. Semantic Map of PBL Process

Problem Scenario • Whole class presentation • Different format • Ill structured in design • Challenging to the level of

learning/thinking

Communicate Learning • Reflections on learning • Peer review • Tutor review • Individual contributions

Presenting Solutions • Whole group discussion • Discussion in isolation • Analysis of data • Produce artifacts

PBL Tutorial – Small Group • Collect data • Sharing information • Peer teaching • Working together

Exploring resources • Independent research • Facilitation • New list of learning issues

New hypothesis

Identify learning issues • Brainstorm ideas • Small group discussion • Need to learn • What they ‘ Know ’ • What they ‘ don’t know ’ • Action plan for working

on problem



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In this study the investigator made use of kaAMS model PBL

strategy to develop the select reflective learning design. The various phase

of PBL strategy are detailed in Chapter IV.kaAMS model of PBL

incorporates various components to prompt student’s reflective thinking as

follows:

Provide questions designed to prompt students to identify and

clarify overall and subordinate problems

Provide many opportunities to engage students in gathering

information to look for possible causes and solutions

Provide activity sheets and ideas to help students evaluate the

evidence they gather

Provide questions that prompt students to consider alternatives

and implications of their ideas.

Provide questions and activities that prompt students to draw

conclusions from the evidence they gathered and pace solutions

Provide opportunities for students to choose and implement the

best alternative.

Encourage students to monitor and re-evaluate their finding

throughout the entire unit.

Rationale behind PBL

PBL fosters self-directed learning, independent and interdependent

research, inquiry, hypotheses generation, discernment of credible sources

and resources, new information acquisitions, hypotheses testing, hypotheses

revision and confident decision making. Other benefits include increased

communication skills, integration of technology and instruction, higher

retention of learned content, integrating and synthesizing new content,



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applying new content to appropriate situations and accurately self assessing

to determine knowledge gaps. These are the characteristics and attributes

that serve all learners in both formal and informal learning situations. PBL

is introduced and continued for many reasons including:

Acquiring subject matter knowledge

Motivating students to learn.

Helping student retention

Developing student’s thinking skills

Facilitating students learning how to learn

Linking theory and practice

Encouraging students to integrate knowledge

Having a sense of belongingness and friendship

Help to realise and develop learner identity

Help to develop independence in enquiry and the ability to

contest and debate

Help to ensure that learning is attuned to the world of work

Factors basic to the success of PBL

The emphasis of PBL approach is on learning process of enquiry

which proceeds by asking what needs to be known to address and improve

a particular situation. This is quite different from some of the garbled

versions of discovery learning which imply that students are supposed to

invent knowledge which is already known. The knowledge which students

use needs to be identified and applied in the context of the presented

situation. Critical reflection is central to effective action and an enquiry

oriented approach is a useful means of promoting this vital skill.



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The following are some of the most important factors in the success

of PBL.

Commitment to the philosophy of PBL - this will enable the

facilitator to overcome difficulties.

Understanding of the philosophy of PBL – facilitator can act

according to the changed roles of the learner and himself.

A realistic acceptance of the role change.

The ability to model process skills-such as reflective thinking,

critical thinking, problem solving, learning to learn.

Frequent opportunities for students to gain feed back.

An acceptance that students will need time to change their

learning process and practice new skills.

Pragmatic and realistic approach.

For the successful implementation of PBL all our strengths and skills

as teachers will be required. The behavior and belief of teachers should be

changed when complex difficulties arise. Teachers need the ability to

explore options and generate creative solutions in cooperative contexts.

Commitment, determination and team work along with self knowledge and

understanding of the learning process are essential for the successful,

promising and exciting implementation of PBL.

2.4.2 Out fielding reflective learning through Journaling.

Reflective journaling

Dewey (1963) noted that, “Thinking is the accurate and deliberate

institution of connections between what is done and its consequences.” P

(151). For Dewey it is reflection on activities that make them meaningful

experiences. Dewey mentioned that one way to organize these reflections



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is by writing about them after they have occurred. With an attempt to

make the connections, journals provide such opportunities and the writing

and subsequent sharing of reflections can contribute significantly to

developing a culture of inquiry. Journaling makes invisible thoughts

visible. It provides a means of describing practice and identifying and

clarifying thoughts, beliefs perspectives, challenges and hopes for practice.

It is a way to put our thoughts linearly or sequentially on paper and offers

a private place for honest accounting and review.

Journal writing has been recognized by educators as an effective

strategy and a formal tool to promote reflective thinking and learning.

Writing about what we do and what we think and what we have learnt

disciplines us to become more thoughtful, reflective and analytic. Our

writing needs to demonstrate active and reflective engagement in the

experiences and ideas we encounter. Reflective learning journals enable us to

create a record of reflective thought and meaning we are making while

engaging in learning process. If the journaling is done conscientiously, the

reflection in each learning journal will be unique as each learner has his own

response to any one experience. Regularly writing up our learning in a

journal has the potential to provide us with a systematic approach to our

development as a critical, reflective and constructive learner.

When learning is to be incorporated into everyday practice, a

personal sense of the diverse experiences is needed. Reflective journaling

could be a better support in this respect as it assists reflective process.

Journals can provide an opportunity to make explicit thinking on a range

of issues of personal significance through reflective process.

Reflective Journals are the corner stones of reflective practice work.

Reflection on practice or learning by keeping a Journal is a fundamentally



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dynamic rewarding process (Benner,1984; Beveidge, 1997 and Schon,

1983). Journals are a collection of expressions of thinking and explorations.

It reflects the personal reflective background of the learner in the context of

the experience and understanding. Journaling is an interactive process

which needs commitment, energy and open mindedness from the learner.

Getting started

Keeping reflective learning journals may feel threatening to some

and they might be inhibited before they start. Therefore learners need to

be introduced to the idea with proper care and guidance. Techniques for

getting started (Knowles, 1993; Gibb, 1988; Garry & Cowan, 1986)

include the following tips.

Prior to introducing the journal, create the right atmosphere to

demonstrate their purpose and value. The best way to do this is

by discussing one’s own personal experiences of reflection and

how they have progressed in their learning, teaching or research.

Use a series of exercises in reflective writing to introduce the idea

After students have kept journals for a year get them to write

guidance notes for those students who will be introduced to

journals in the following year.

The reflective writing in a learning journal may be considered as a

cyclic process which is really an aid to learning often in the form of a raw

material of learning. We could almost say that the reflective Journals are the

learner’s personal curriculum in a written form. The investigator used a five

stage cyclic process of Reflective Journaling in this study which is described

in Chapter IV in detail. Within the purview of this cycle of reflective writing

the process of writing in journals is mapped in Figure 2.9.



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Figure 2.9. A Map of Reflective Writing in Journals

Shaping the journal to suit you

Journals can be created in different shapes, sizes and forms. The design

and structure of the Journal is significant rather than the recording mode.

There are three types of format that have been used for journal writing.

Structured Journals follows a specific or structured format. It imposes

certain form of constraints in the manner in which it is written. This aids to

obtain reflections in a uniform format or range of format which makes the

comparison objective and scientific. Students, for their part are able to follow



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a template, which serves to provide guidance to students on approaching and

developing Journals. (John, 1994).

Unstructured Journals allows the learners to produce their own format

and design the writing accordingly. This unstructured nature of journals

makes it difficult to compare with other formats used by learners resulting

in subjective ascertaining of reflection and learning.

Dialogue Journal is used methodologically to train student expression and

reflection, and was explored by Garmon (1998) and Peyton (1993). Peyton’s

model resembled a mentor/mentee relationship requiring a consistent one-to-

one interface and a dialogue as a vehicle for communication. Station et al.

(1988) defined dialogue journals as a method to encourage the exchange and

development of ideas between two or more writers.

Brookfield (1995) suggested that students write about whatever

seems important, but also be provided with a set of question as initial

prompts. The questions focus on what the students learned about

themselves and their emotional moments, significant or surprising events

and the area of satisfaction or dissatisfaction. In the present study as the

learners fall into adolescent group, the investigator made use of a semi

structured journaling format in which the facilitator provided journaling

prompts through out the programme. This made the investigator to assess

the reflective writing of learners objectively and scientifically.

Factors which shape reflective writing.

There are many forms of expressing reflection, namely in speech, in

film, in graphic portrayal, music and in paper. In expressing or representing

learners’ reflective thoughts on paper we should shape and model the

content of our reflections according to many factors. According to Moon,

2004 these might include:



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The reason why you are writing reflectively.

Whether others are going to see what you have written and who

they are.

Your emotional state at the time of writing and your emotional

reactions to what you are writing.

Related to the above, how safe you feel about the material and

anyone seeing it.

What you know about reflective writing and how able you are

to engage in it.

Quality of reflective writing

Reflective Journal is more an organic tool, to support the process of

reflection rather than to be seen as a created product. Reflective writing

need be pitched according to the purpose of writing it. The challenge is at

least to go beyond descriptive writing which is very superficial and to go

into a deeper level to serve its purpose. A comparison between these two

may yield a better picture of reflective writing which was adopted by the

investigator in the present study. It is described in Table2.3.

Table 2.3. Comparison between Reflective Writing and Descriptive Writing

Reflective writing Descriptive writing Involves process of learning &

thinking in a ‘non-tidy’ order Is the product of thinking which

is tidily ordered Intended for the purpose of learning Intended for the representation of

learning Writing style is subjective Wring style is objective May be structured, unstructured or

dialogue Totally structured in an approved

pattern Conclusion may be further areas of

reflection Ends in conclusion

Subject matter may be undefined and personal

Subject matter is clearly defined



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From this we could conclude that reflective writing holds the possibility

of deepening our self understanding and a new perspective to our lives.

The benefits of regular reflective writing

Reflective writing enables documentation of experiences, thoughts,

questions, ideas and conclusions that sign point our learning journey.

Planning and engaging in action and then thinking about how to improve

the action involves reflection and evaluation .Learning comes through

reflection, evaluation, decision, action and again reflection.

Literature [Selfe et al. (1986), Langer, (2002)] suggests that reflective

learning journals facilitate critical reflection; particularly it assists learners in

conceptualizing and reconceptualizing abstractions through reconfigurating,

reframing and relating it to practice. Besides, the learners developed

better problem solving strategies through journaling which needs critical

reflection.

Knowing how we learn makes us a more effective learner. Thinking

about our learning and writing things down help to clarify our thoughts and

emotions in this regard. There is evidence that the use of learning journal

facilitate metacognitive abilities which leads to self inquiry, self managing

learning process and self esteem [Holly (1989), Norton, Ownes, and Clark

(2004)]. Indeed the literature offers evidence that learners regardless of the

course of topic improve their reflective learning capacity by keeping journals.

Reflective writing helps us to focus and participate in the

development of an effective independent and active learner. It helps us to

discover the strategies and processes that work well for us and to evaluate

our own performance as an independent learner which will effect in

recognizing the responsibility for becoming life long learners. Studies

which corroborate this argument are given below.



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Abbas & Gilmer (1997) explored the use of learning Journals as an

interaction between student and instructor, designed to stimulate active

learning .Taggart & Wilson (1998) expanded this concept by suggesting

strategies to enhance learner’s reflective capabilities while writing

Journals. Thorpe (2004) suggested that reflective learning journals are

recognized as a significant tool in promoting active learning among

nursing students. While writing in a journal we not only record what

happened or what was observed, but also develop new understanding, and

insights and thereby new hypothesis and new perspectives of the

phenomena. Reflective journaling empower the above mentioned orbits of

a learner which is schematically represented as in Figure.2.10

Figure 2.10. The Orbit of Reflective Journals



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Moon (2004) suggested that there are few accounts in the literature

which clearly indicates the manner in which learning journals have been

used to facilitate learning. The analysis of the existing literature on

learning journals summarizes common themes about their use as learning

tools like:

Act as a facilitator of learner’s moral, personal, psychological,

emotional and cognitive development.

Journals are the transition from theory to practice or praxis in

learning.

Used to facilitate critical reflections

Reflective writing helps to remember something later, it is a

record to look back on (Holly, 1989)

The act of putting pen to paper engages our brain

Writing a journal allows us to look at ourselves, our feelings

and our actions in a different way

Journal writing allows us to clear our minds

Making journaling part of our routine means that we do actually

take time out to reflect on what might be happening in our

practice and in our lives generally (Rainer, 2004)

Reflective writing is a way of learning and helps to evaluate our own

learning performance as a learner remembers that reflective writing is a

life skill, not just an academic requirement. By engaging in Reflective

journaling learners are taking an active role in learning and recognizing

our personal responsibility for life long learning. No doubt, regular

Reflective journaling is extolled by those learners who ought to become

life long learners.



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2.4.3 Generating ambience of reflectivity through Thinking Maps

As Educators our greatest challenge and opportunity is to engage and

conduct research in order to develop a strong foundation and constant

orchestration of the various teaching learning strategies and student tools

which enables the stake holders to frame a sustainable vision and to create

coherent educational experiences. One such current research has been

about how the brain learns, linking its application to the best practices

drawn from research on teaching and learning. Knowing more about how

the brain works, how it processes information has profound implication on

how we teach and how we participate in effective instruction and learning.

So it would be more beneficial to couple the theoretical research about the

brain with the educational research about the practice. The power of visual

representation of an idea or process is widely accepted. This allows the

development of a holistic understanding that words alone can not convey;

because the graphical form allows representation of parts and whole in a

way that is not available in the sequential structure of text. These visual

representational techniques are major instructional formats available to

teachers.

Visual tools are nonlinguistic symbol systems used by learners,

teachers and leaders for graphically linking mental and emotional

associations to create and communicate rich patterns of thinking [Hyerle,

2009]. They are a natural bridge between brain and mind and high

intellectual performance supported all learners in transforming static

information into active knowledge, thus offering a complimentary

representational system. In the Ausubelian view the growth of knowledge in

an individual was characterized by the gradual development of more

complex and more differentiated structure organized in a hierarchical

pattern. Instantiating the process of knowledge growth that Ausebel formed



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is subsumption-a powerful and coherent structure that is a semantic

relationship within the hierarchy.

There are number of different map structures that can help students to

represent and organize what they know and can find out as well as the

thinking and learning process behind it. These map structures have been

used to refer to a variety of strategies designed to portray, graphically and

visually, a relationship of concepts or ideas. Some of the visual

transformational tools are:

Clustering Concept mapping

Think links Word webbing

Semantic mapping Mind maps

Graphic organizers Thinking maps

Structured overview Idea branches

Plot maps

According to Fisher (1995), learning a concept is a gradual process in

which the learner builds up successive approximation. Finer and finer

distinctions are made, and the learner increases understanding by

developing and widening the network of related ideas. The use of thinking

maps allows the learner to create an understanding of the world by making

connections, by creating links, by exploring and testing links-a basic

process of all creative thinking. Thus creating a Thinking map is one way

of representing and communicating one’s understanding of concepts.

On a global level Thinking maps can be described as a synthesis of

three types of visual tools viz. brain storming webs, graphic organizers and

thinking process tools such as concept mapping. According to Hyerle though

these visual tools offered useful ways of accessing knowledge, they had some

weaknesses that could not be overlooked. Early mind-mapping technique



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lack the consistent structure and deeper level of complexity required for

today’s classrooms. The familiar graphic organizers were static backline

masters focused on isolated content tasks. These tools are task specific

organizers because they usually focus on a specific content task rather than

the learner. Synthesizing the best qualities of these types of visual tools; the

creative dynamisms of brain storming webs, the organizing structure of

graphic organizers, and the deep cognitive processing found in concept maps

that Thinking maps have evolved as a powerful language for learning.

Thinking maps-defined

Thinking maps developed by Dr David Hyerle (2004) are described

as a transformational language for learning, planning and instruction.

They are the explicit visual representations of thinking process that foster

and encourage life long learning. Thinking maps enable all learners to

communicate what and how they are thinking. Through this language all

learners convey, negotiate and evolve meanings with others and within

themselves through visual patterns of thinking. Thinking maps are really a

meta language for learning-an interrelated set of thinking patterns for

communicating and synthesizing our thinking.

Thinking maps-How it works

Thinking maps are eight visual-verbal learning tools, each based on a

fundamental thinking skill defined and animated by maps and introduced

as a common visual language for thinking and learning across whole

learning communities(Hyerle 2004). These visual tools are used together

as a set of tools for showing relationships and as a common language for

meaningful learning.

The consistency and flexibility of each of the Thinking maps

promotes:



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Student centered and cooperative learning;

Concept development, reflective thinking and creativity;

Clarity of communication;

Continuous cognitive development.

Thinking maps can help learners become independent, reflective, life

long problem-solvers and learners. They are common visual language that

facilitates easy transfer and continuous development of thinking processes

across disciplines, grade levels and schools. Thinking maps help students

do just what the brain is built to do; to think in patterns of content in order

to solve problems.

Thinking maps as a language are eight cognitive skills’ each

represented and activated by graphic primitives. It identifies the coherence

and interdependency of the eight fundamental cognitive skills that ground

thinking and learning .The eight cognitive skills are defining in context,

describing attributes, comparing and contrasting, classification, part-whole

spatial reasoning, sequencing, cause and effect reasoning and reasoning by

analogy. Each of the graphic primitives that visually define and animate

each cognitive process is closely attuned to and reflects the cognitive

pattern. Each of the cognitive process is influenced, animated and

transformed by the cultural frames that surround these behaviors. This

means that everyone may understand and utilize the cognitive process of

categorization, but the categories carry a different language, content,

process for development and forms within and across cultures. A ‘Meta

cognitive’ frame may be drawn around any of the maps at any time as a

‘meta-tool’ for identifying and sharing one’s frame of reference including

personal histories, culture, belief systems for the information found

The primitives and their specificities are explained in Table 2.4



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Table 2.4. Graphic Primitives and their Specificities

Primitives Name of Map Specificities Expanded

Maps

Circle Map • Used for seeking context • Effective brain storming tool • Frame of reference

Bubble

map

• Process of describing attributes

• Identify traits • Identify properties

Double Bubble

Map

• Comparing two things • Contrasting two things • Prioritizing information

Tree Map

• Inductive and deductive classification

• Categorize • Sort • Organize

Brace Map

• Identifying part-whole relationships

• Analyse • Break into parts • Support spatial reasoning

Flow Map

• Sequencing • Ordering • Showing time lines, cycles,

actions, steps, directions • Relations between sub stages

Multi Flow Map

• Seek cause-effect • Shows inter relationships of

feedback effects

Bridge Map

• Creating and interpreting analogies

• Develops analogical reasoning • Metamorphic concepts of

deeper content learning



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These graphic primitives are the basic element of thinking maps.

Each line, circle or square of primitives represents a relationship or a

boundary and provide consistent visual vocabulary for creating maps and

thinking on a high order about fundamental acts of cognition. Each of

these Primitives is described and the expanded maps are depicted in the

following section.

Circle map

Circle map is used as a visual tool for putting things in context. The

‘thing’ is represented in the center circle and the contextual details are

shown in the outside circle. The circles show how we create mental

boundaries when we try to define something. The basic map can be expanded

by drawing a frame of reference around the two circles [Figure 2.11]. The

visual frame and the circles provide reflective tools for learning.

Figure 2.11. Circle Map for Thinking –making

Bubble map

Bubble map [Figure 2.12] is used to represent the process of

abstracting from and projecting qualities onto things, and to identify

sensory, logical and emotional qualifications we are making. The circle in

• Frame of reference

• Context • Thing



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the middle is the thing being qualified; the lines extending outward each

represent the abstraction or projection process, and the outside circle are

the abstracted place holders for adjectives and phrases.

Figure 2.12. Bubble Map for Qualification

Double bubble map

Double bubble map is the expanded form of double map for the

process of comparing and contrasting the qualities of two things [Figure

2.13].The middle circles are the perceived common qualities of the two

things being compared while the outside circles describe the unique

qualities of the two things respectively.

Figure 2.13. Double Bubble Map for Comparing and Contrasting



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Tree map

The underlying thinking process for a tree map is classification and

organization [Figure 2.14]. The map represents a top down pattern or

sorting information, with the general term on the top and specific groups

and specimens below.

Figure 2.14. Tree Map for Classification.

Brace map

The brace map helps identify whole and part relationships: the whole

thing is written on the left side above the line and the braces represent the

physical joints between the parts. The lines are the place holders for ‘major

parts’ followed by the sup parts. [Figure 2.15].It is used for something

concrete that can be broken into components or subparts.

Figure 2.15. Brace Map for Structure Analysis



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Flow map

Flow map operates with events having sequential stages and sub

stages. It is a useful starting point for thinking about different kinds of

operations and can be prioritize them using these maps [Figure 2. 16].

Figure 2.16. Flow map for Operation Analysis.

Multi-Flow map

Multi-Flow maps are one of the most powerful maps. It is an

expanded version of flow maps investigating cause-effect patterns of

thinking. The event is written in the middle [Figure2.17].

Figure 2.17. Multi-Flow Map for Cause-Effect

Bridge map

A Bridge map is used to illustrate analogies and metaphors. The line

of the bridge represents the ‘relating factors’ that is transferred across

relationships. The relating factor is the common relationship that a student

finds to exist between two or more pairs of things [Figure2.18].



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Figure 2.18. Bridge Map for Seeing Analogies.

The learners energize thinking and represent their cognitive

processes using these graphic primitives. They are used together, linked

together and visually scaffold to create other products of learning such as a

piece of writing. Learners and teachers shape and reform the static content

knowledge by transforming it into maps. They could very well organize

their thinking by making use of these maps. Also if the maps are

expanded and integrate with words, symbols and numbers learners face

boundless nature of their learning.

Two major themes of thinking maps as a transformational language

are ‘construction of knowledge’ as a framework for learning and

‘communities of learning’ expressing the communal quality of the

education experience. This is described in Figure 2.19.

Figure 2.19. In the Landscape of Thinking Maps



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Thinking maps-learner’s perspective

Thinking maps, a network of visual information gave students a

method and structure that emphasized the relationship of pieces of

information to each other. Because the patterns are embedded across the

concept, recalling one bit of information on the map essentially fires the

retrieval of the entire map. Visual tools like Thinking maps that pattern

information, support what the brain does naturally to process information

by reflecting both its highly structured as well as its adaptive capacities to

embed patterns within patterns.

Thinking maps as transformational language could be used as tools

for developing students’ executive functioning and metacognitive habits of

mind. By executive functioning we mean planning an approach to a task,

organizing a sequence of actions or series of data points, holding action

sequences in working memory until they are executed inhibiting actions

that are irrelevant to the task at hand, deciding what to attend to and what

to do, monitoring and evaluating behaviour and adjusting behavior and

emotions in response to perceived success or failure (Denckla, 1998;

Singer & Bashier, 1999). Thinking maps help students to visualize

patterns that go beyond the word or sentence. They offer student with

varied learning abilities and learning styles a means for organizing their

thinking and understanding of the world and hence facilitate learners to

overcome executive function problems.

Thinking maps a visual pattern of learning process fosters the

development of metacognitive habits of mind which effects in self

regulated learning. Thinking maps changes the way of talking between

teachers and learner; learners each other; and learners to themselves .Using

thinking maps the cognitive processes like think, classify sequence,

analogy and brainstorm could be taught to learners directly. They are



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asked to consider what cognitive routes they will take before they set out

on their journey towards the internal working of the mind. That

consideration takes place through an explicit reflective and constructive

dialogue between teachers and learners which helps in shaping a culture of

reflective thinking and metacognition.

Thinking maps – Hallmarks

Thinking maps, a visual, metacognitive tool for learning has five

critical attributes; the awareness of which leads to better development of

the maps. These attributes are portrayed in Figure2.20.

Figure 2.20. Hallmarks of Thinking maps

Reflective: - Thinking maps unveil what, how and why one is thinking in

patterns. Learners can reflect upon the patterns of content and the process

of thinking which leads to metacognition. In addition when the learners

draw a frame of reference around a map it represents their metacognitive

frame. Teachers can also reflect on and informally assess the learning and

thinking processes of the learners.



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Consistent: - As thinking maps are grounded on eight primitives, each

map has a unique but consistent form that visually reflects the cognitive

skill being defined.

Flexible: - The graphs primitives and the thinking skill for each map leads

to the flexibility in form and the infinite number of ways the map can grow

and be configured.

Developmental: - Because of the consistent graphic primitives and

flexible use of it, any learner may develop a map and expand it to show his

thinking. Every learner can use the primitives as he wishes and thus can

produce his own configurations of the content.

Integrative:-The key dimensions of integration-thinking process and

content knowledge are integrated together. In thinking maps when the

learner develops maps using primitives they are integrating the content

knowledge to their thinking processes.

Mathematical knowledge is either declarative or procedural. Using

the different graphic patterns of thinking maps both declarative form

and procedural form can be depicted which helps in scaffolding students

learning by translating abstract thinking processes into explicit and

tangible visual representations. Students and teachers indicated that

thinking maps fostered students’ ability to articulate how they were

thinking or to reflect with a metacognitive stance in order to assess.

Impact of Thinking maps

Learners, teachers or even administrators not only receive

information, but they also need to interpret, rethink and relate it to their

own schemas of understanding. As there is information flow to, from, and



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among students and teachers, this strategy is a useful procedure for

investigating, visualizing and organizing information.

Thinking maps serves many purposes. It allows us to explore our

understanding of key concepts and helps us to make meaningful patterns

of our knowledge and understanding by linking ideas using appropriate

primitives to develop maps. It is also an aid in planning process, since it

shows how we categorize, link and organize ideas and is a visual means

for communication and evaluation. Thinking maps stimulate active

thinking through the cognitive skills of analysis, categorization, synthesis

and reflection of key elements of what we know or have done and thus it

makes this visually oriented strategy applicable within the context of any

topic in the school curriculum. These mapping strategies allow teachers

and teacher educators to cover topics in greater depth, where meanings can

be identified and created in an organized and ongoing way as the maps

provide a framework that can be viewed, elaborated on, adapted and

developed overtime.

Thinking maps are highly useful to learners, teachers and administrators

as an instructional and evaluation tool as they move towards an understanding

of shared meaning, the creation of new knowledge and facilitate the move

from representational meaning to richer conceptual meaning. Thinking map is

a useful artifactual tool for probing teachers’ understanding- whether it is to

explore an understanding of a limited aspect of the topic, to see whether they

are able to relate distinct topics, to find out whether they appreciate those

concepts and to check whether students understand the reasons for a

lesson.



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Figure 2. 21. Nurturing Effect of Thinking Maps

Thinking maps, a flexible consistent common visual founded on the

nexus of patterns of thinking and non linguistic representations support

both the process and product of mathematical knowledge. Using the

different graphic patterns of Thinking maps both declarative forms and

procedural forms of mathematical knowledge can be depicted which helps

in scaffolding student’s learning by translating abstract thinking process

into explicit and tangible visual representations. Students and teachers

indicated that Thinking maps fostered student’s ability to articulate how

they were thinking or to reflect with a metacognitive stance in order to

assess. An analysis of their own maps and reflections would provide

further information about their understanding as well as a powerful



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learning experience and allowed a light torch in probing their capacity for

reflection and critical thinking there by make them empowered over their

learning.

2.5 Fueling Mathematics Learning through Reflection

Mathematics is one of humanity’s great achievements of sophistication

and beauty that epitomizes the power of deductive reasoning. Mathematicians

seek out patterns, formulate new conjectures and establish truth by

rigorous deduction from appropriately chosen axioms and definitions.

2.5.1 Dimensional focus of mathematics proficiency

Mathematics, with its system of psychological tools and

mathematical thinking dynamic provides the vehicle for the formulation,

organization and articulation of human thought. Today, mathematics is

used throughout the world as an essential tool in many fields, including

natural science, engineering, medicine and the social sciences.

Psychological tools of mathematics refer to symbolic devices and schemes

that have been developed through socio-cultural needs to facilitate mental

activity dealing with patterns and relationships. The structuring of these

tools has slowly evolved over periods of time through collective, generalized

purposes of the transitioning needs of the transforming cultures.

Mathematical psychological tools range from simple forms of

symbolization such as numbers and signs in arithmetic to the complex

notations and symbolizations that appear in calculus and mathematical

physics such as differential equations, integral functions or Laplace

transforms.

Kinard (2000) defines rigorous mathematical thinking as the

synthesis and utilization of mental operations to:

Derive insights about patterns and relationships.



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Apply culturally derived devices and schemes to further

elaborate these insights for their organization, correlation,

orchestration and abstract representation to form emerging

conceptualizations and understandings.

Transform and generalize these emerging conceptualizations

and understandings.

Transform and generalize these emerging conceptualizations

and understandings into coherent, logically-bound ideas and

networks of ideas.

Engineer the use of these ideas to facilitate problem solving and

the derivations of other novel insights in various contexts and

fields of human activity; and,

Perform critical examination, analysis, introspection and

ongoing monitoring of the structures, operations, and processes

of rigorous mathematical thinking for its radical self

understanding and its own intrinsic integrity.

Mathematical thinking is a dynamic that structures a logical framework

and an organizing propensity for numerous socio-cultural endeavors through

its discovery, definition and orchestration of those qualitative and quantitative

aspects of objects and events in nature and human activity. It is through

mathematical thinking that the human mind can attempt to discover and

characterize underlying order in the face of chaos; structure in the midst of

fragmentation, isolation and incoherency; and, dynamic change in the context

of steady behaviors and constancy. Mathematical thinking structures and

creatively manipulates growing systems of thought.

As the learner acquires and utilizes these mathematical psychological

tools to generate, transform, represent, manipulate and apply insights



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derived from patterns and relationships, mathematical thinking is unfolding.

Here the learner is rigorously mediated to utilize his/her day to day

perceptions and spontaneous concepts to construct mathematical concepts

and procedures that builds a ‘functional mathematical system’, which is

the set of component processes that must be coordinated for an individual

student to engage in optimal developmentally appropriate mathematical

thinking and problem solving (Andrews, Saklofski, Janzen,2001).

During the twentieth century the meaning of successful mathematics

learning underwent several shifts in response to changes in both society

and schooling. The research in cognitive psychology and mathematics

education have led us to adopt a composite comprehensive view of

successful mathematics learning National research council (2001) have

chosen mathematics proficiency to capture completely all aspects of

expertise, competence, knowledge and facility in learning mathematics

successfully.

As a goal of instruction, mathematical proficiency provides a better

way to think about mathematics learning. Mathematics proficiency implies

expertise in handling mathematical ideas. Learners with mathematical

proficiency understand basic concepts, are fluent in performing basic

operations, exercise a repertoire of strategic knowledge, reason clearly and

flexibly and maintain a positive outlook towards mathematics. As NRC

(2001) suggested mathematics proficiency have five interwoven and

interdependent strands namely conceptual understanding, procedural fluency,

strategic competence, adaptive reasoning and productive disposition which

provides a framework for discussing the knowledge, skills, abilities and

beliefs that constitute mathematical proficiency (Adding it up: Helping

children Learn mathematics, a report published by the National Research

Council, 2001).



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Figure 2.22. Five Strands of Mathematics Proficiency.

Source:NRC 2001

Conceptual understanding refers to an integrated and functional grasp

of mathematical ideas which enables the learners to understand why a

mathematical idea is important and the kinds of context in which it is useful.

They can organize their knowledge into a coherent whole, which enables

them to learn new ideas by connecting those ideas to what they already

know. A significant indicator of conceptual understanding is being able to

represent mathematical situations in different ways and to see how the

various representations connect with each other. Conceptual understanding

is a wise investment that pays off for students in many ways.

Procedural fluency refers to knowledge of procedures, knowledge of

when and how to use them appropriately and skill in performing them

flexibly, accurately and efficiently. By studying algorithms as general



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‘procedures’ students can gain insight into the fact that mathematics is

well structured and that a carefully developed procedure can be a powerful

tools for completing routine tasks. Both accuracy and efficiency can be

improved with practice which can help learners to maintain fluency,

flexibility and facility without sufficient procedural fluency learners could

not deepen their understanding of mathematical ideas or solving

mathematical problems.

Strategize competence refers to the ability to formulate mathematical

problems, represent them and solve them. They should know a variety of

solution strategies as well as which strategies might be useful for solving a

specific problem. To become proficient problem solvers, students learn

how to form mental representations of problems, detect mathematical

relationships and devise novel solution methods when needed. Flexibility

of approach is the major cognitive requirement for solving non-routine

problems.

Adaptive reasoning refers to the capacity to think logically about the

relationships among concepts and situations which stems from careful

consideration of alternatives, and includes knowledge of how to justify the

conclusions. In mathematics adaptive reasoning is the glue that holds

everything together, the lodestar that guides learning.

Productive disposition means the tendency to see sense in mathematics,

to perceive it as both useful and worthwhile, to believe that steady effort in

learning mathematics pays off, and to see oneself as an effective learner and

doer of mathematics. If learners are to develop the other strands of

mathematics proficiency they must believe that mathematics is

understandable not arbitrary; that with diligent effort, it can be learned and

used; and that they are capable of figuring it out. Developing a productive



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disposition requires frequent opportunities to make sense of mathematics, to

recognize the benefits of perseverance and to experience the rewards of sense

making in mathematics.

Hence our view of mathematical proficiency goes beyond being able

to understand, compute, solve and reason but includes a disposition toward

mathematics that is personal. Mathematically proficient learner believe

that mathematics should make sense that they can figure it out, that they

can solve mathematical problems by working hard on them and that

becoming mathematically proficient is worth and effort. Moreover they

possess and use these strands of mathematical proficiency in an integrated

manner; so that each reinforced the other.-

Children today are growing up in a world permeated by mathematics.

A knowledge system which capacitates the learners to visualize the world

through a mathematical lens is the dare need of the nation. Moreover it is

necessary for the citizen to use accurate, logical, precise and rigorous

thinking now that even before. Learners have to develop not only their

powers of mathematical thinking but also a positive attitude and a capacity

to take responsibility for their own progress. A reflective mind set to

mathematics learning provides its stake holders the whole momentum to

architect such a system of knowledge and wisdom to manage one’s own

future.

2.5.2 Towards a synthesized framework of reflection`

To activate the internal process of Mathematics learning by way of

reflective practice, a practice based approach to the learning situation is to

be designed and implemented. The investigator selected a two stage

approach to reflective activities in mathematics classroom as Moon (2004)

suggested. The two stages are:



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Presenting reflection

Facilitating deeper reflection

The phase presenting reflection helps the learners to get started in

reflective tasks. Approaches, suggestions and ideas are depicted in

Figure 2.23.

Figure 2.23. Presenting Reflection – A Lay Out

The second phase facilitating deeper reflection provides the learners

with the circumstances which make them aware of what is involved in

deeper reflection. The suggestions, approaches and ideas that can

facilitate deeper reflection are summarized in the Figure 2.24.



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Figure 2.24. Facilitating Deeper Reflection – A Cognitive Vision

The first phase involves discussion and exercises and the provision of

examples that introduce the idea of reflection and ensure that students

come to a reasonable understanding of what is required in reflective

writing. The task for students is to learn to be able to manage a basic form

of reflective writing. Then the second stage is introduced with more

activities which focus on deepening the process of reflection. There is no

suggestion of a direct relationship between the stages and the levels in the

framework. Different learners will achieve different levels sometimes in

accordance with different learning events.

There is no one best approach to the presentation or to the deepening

of reflection. At both stages, the investigator suggests the use of multiple



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approaches, providing different ideas and activities around reflection

rather than just verbal instruction.

Remember that reflective practice is a life skill, not just an academic

requirement. By engaging in reflective learning we are taking an active

role in learning and recognizing our personal responsibility for our own

life long learning. Where reflection prospers, it is seen by many students

as a major significant feature of their development in all spheres that it can

promote self authentication, existential self realization, empowerment and

transformation.

….. …..

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THEORETICAL OVERVIEW - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/25936/11... · THEORETICAL OVERVIEW 2.1 Conception of Reflection-A Dynamic Prospect ... (As cited by Erikson,