EDUCARE:
International Journal for Educational Studies, 8(2) February 2016

JADITH TAGLE, RENE R. BELECINA & JOSE M. OCAMPO, JR.

Developing Algebraic Thinking Skills among
Grade Three Pupils through Pictorial Models

ABSTRACT: During the primary grades, young children work with patterns. At an early age, children have a
natural love for Mathematics, and their curiosity is a strong motivator as they try to describe and extend patterns of
shapes, colors, sounds, and eventually letters and numbers. At a young age, children can begin to make generalizations
about patterns that seem to be the same or different. This kind of categorizing and generalizing is an important
developmental step on the journey toward Algebraic thinking. Algebra instruction has traditionally been postponed
until adolescence, because of the assumptions about psychological development and developmental readiness.
Concrete operational children tend to be capable of mental operations as long as they relate to real objects, events, and
situations. As they mature, they are able to work with more abstract concepts without the aid of concrete objects. The
study attempted to determine the effect of using pictorial models on the Algebraic thinking skills of grade three pupils.
The one-group pre-test – post-test experimental research design was used in this study. Twenty-eight grade three pupils
participated in the study. To determine the effect of using pictorial models, an Algebraic thinking skills test was given
to the pupils before and after using pictorial models. Results showed that the use of pictorial models significantly
improved the Algebraic thinking skills of the pupils. Interviews from the pupils revealed that pictorial models helped
them to solve problems easier. The findings suggest that Algebraic thinking can be taught even at the early age.
KEY WORDS: Algebraic Thinking; Young Children; Patterns; Letters and Numbers; Pictorial Model.

About the Authors: Jadith Tagle, M.A.T. is a Teacher at the Faculty of De La Salle School-Greenhills, Philippines. Rene R.
Belecina, Ph.D. is a Full Professor at the CGESTER PNU (College of Graduate Studies and Teacher Education Research, Philippine
Normal University). Jose M. Ocampo, Jr., Ph.D. is a Full Professor at the Faculty of Education Sciences PNU in the Philippines. The
authors can be contacted via their e-mails at: jadithtagle88@gmail.com, rrbelecina@yahoo.com, and juno_6970@yahoo.com

How to cite this article? Tagle, Jadith, Rene R. Belecina & Jose M. Ocampo, Jr. (2016). “Developing Algebraic Thinking Skills among
Grade Three Pupils through Pictorial Models” in EDUCARE: International Journal for Educational Studies, Vol.8(2) February, pp.147-158.
Bandung, Indonesia: Minda Masagi Press and UMP Purwokerto, ISSN 1979-7877.

Chronicle of the article: Accepted (November 30, 2015); Revised (January 20, 2016); and Published (February 28, 2016).

Algebra instruction has traditionally been
postponed until adolescence, because of the
assumptions about psychological development
and developmental readiness. Concrete
operational children, according to J. Piaget
(1952 and 1994) and others, tend to be capable
of mental operations as long as they relate to
real objects, events, and situations. As they
mature, they are able to work with more
abstract concepts without the aid of concrete
objects (Piaget, 1952 and 1994; Blair, 2003;
Simatwa, 2010; Bautista & Francisco, 2011;
and Vorpal, 2012).

INTRODUCTION
During the primary grades, young children

work with patterns. At an early age, children
have a natural love for Mathematics, and their
curiosity is a strong motivator as they try to
describe and extend patterns of shapes, colors,
sounds, and eventually letters and numbers.
At a young age, children can begin to make
generalizations about patterns that seem to be
the same or different. This kind of categorizing
and generalizing is an important developmental
step on the journey toward algebraic thinking
(Seeley, 2004; Warren, 2007; and Stump, 2011).

JADITH TAGLE, RENE R. BELECINA & JOSE M. OCAMPO, JR.,
Developing Algebraic Thinking Skills

Some researches provide evidence that
young students, aged 9-10 years, can make
use of Algebraic ideas and representations
typically absent from the early Mathematics
curriculum, and thought to be beyond
students’ reach (Bednarz & Janvier, 1996;
Blanton & Kaput, 2004; and Badger &
Velatini, 2010). One of these researches comes
from a 30-month longitudinal classroom
study of four classrooms in a public school in
Massachusetts, with students from grades two
to four. The data help clarify the conditions
under which young students can integrate
algebraic concepts and representations into
their thinking (Carraher et al., 2001).

An increasing number of Mathematics
educators, policy makers, and researchers
believe that Algebra should become part
of the elementary education curriculum.
The NCTM (National Council of Teachers
of Mathematics), in 2000, and a special
commission of the RAND (Research And
Development) Corporation, in 2003, have
welcomed the integration of Algebra into
the early Mathematics curricula. These
endorsements, however, do not diminish
the need for research. On the contrary, they
highlight the need for a solid research base
for guiding the Mathematics education
community along this new venture (NCTM,
2000; and RAND Corporation, 2003).

Within the past decade, discussions
and opinions pertaining to school Algebra
have changed dramatically. Professional
organizations, policy makers, Mathematicians,
Mathematics educators, administrators, and
teachers, who once considered Algebra as a
course just for university-bound students, now
espouse the notion of Algebra for all students.
Overarching policy statements from these
groups and individuals indicate a widening of
the range of topics that constitute Algebraic
thinking to encompass now more than just
the structural aspects of Algebra (Ferrucci,
2004). This broader description of thinking
Algebraically has led to the introduction of
Algebraic ideas into the curriculum at much
earlier grade levels.

It is now widely understood that preparing
elementary students for the increasingly
complex Mathematics of this century requires

an approach different from the traditional
methods of teaching Arithmetic in the
early grades, specifically an approach that
cultivates habits of mind that attend to the
deeper, underlying structure of Mathematics
(Booker et al., 2004; Blanton & Kaput, 2008;
and Brizuela & Schliemann, 2003). Hence, it
explains the purpose of this study on pictorial
models in developing Algebraic thinking of
primary pupils was made with the hope that
it could help improve the basic Mathematics
foundation of our learners.

Students and teachers need to appreciate
that there can be a number of ways to visualize
a problem, as well as number of ways to solve
a problem non-visually. Some students might
benefit from visualization more than others.
Sometimes, students resist using visual models,
when a solution is readily apparent to them.
Mathematics teachers should always be open
to various approaches that can be introduced
to the pupils for them to develop higher-order
thinking skills.

Young children today need a different
kind of Mathematics from what their parents
learned. This study was intended to introduce
the use of pictorial models that can be of great
help in helping the primary pupils formulate
and manipulate Algebraic expressions, and
understand relationship between quantities in
the problem and leading them to a strategy in
solving it.

This study is indeed important to the
Mathematics teachers and the learners as
well, because it established a basis on what
must be given more focus on the elementary
curriculum, and how the Arithmetic thinking
can be developed into Algebraic thinking,
thus promoting a deeper understanding of the
Mathematical concepts.

CONCEPTUAL FRAMEWORK
This study was based on the following

theories: Jerome S. Bruner’s Constructivism
and Development Learning and Visual
Learning Theory, in 1960. Jerome S. Bruner’s
theoretical framework is based on the notion
that learners construct new ideas or concepts
based upon existing knowledge. Learning is an
active process. Facets of the process include
selection and transformation of information,

EDUCARE:
International Journal for Educational Studies, 8(2) February 2016

decision making, generating
hypotheses, and making meaning
from information and experiences
(Bruner, 1960).

Jerome S. Bruner (1960)
postulated three stages of intellectual
development. The first stage,
he termed “Enactive” transpires,
when a person learns about the
world through actions on physical
objects and the outcomes of these
actions. He calls second stage “Iconic”,
when learning takes place through the use of
models and pictures. The final stage is termed
“Symbolic”, which occurs when the learner
develops the capacity to think in abstract
terms. Based on this three-stage notion about
the development of the intellect, Jerome S.
Bruner recommended using a combination
of concrete, pictorial, and symbolic activities
believing that this will lead to more effective
learning (Bruner, 1960).

Jerome S. Bruner (1960) inspired an
approach in teaching Mathematics, which is
called the CPA (Concrete-Pictorial-Abstract)
approach. Concrete components include
manipulatives, measuring tools, or other
objects that the students can handle during
the lesson. Pictorial representations include
drawings, diagrams, charts, or graphs that are
drawn by the students or are provided for the
students to read and interpret. Abstract refers
to symbolic representations, such as numbers
or letters that the student writes or interprets
to demonstrate understanding of a task
(Bruner, 1960).

Research has shown that visual learning
theory is especially appropriate for the
attainment of Mathematics skills for a wide
range of learners. Understanding abstract
Math concepts is reliant on the ability to “see”
how they work; and children naturally use
visual models to solve Mathematical problems.
They are often able to visualize a problem
as a set of images. By creating models, they
interact with Mathematical concepts, process
information, observe changes, reflect on their
experiences, modify their thinking, and draw
conclusions (Warren & Cooper, 2005; and
Murphy, 2006).

One example of a pictorial model is

a structure comprised of rectangles and
numerical values that represent all the
information and relationships presented in
a given problem. The rectangles replace the
unknown represented by letters in equations.
The rectangle, known as a unit, becomes the
“generator” of the model about which other
relations are constructed. The model method
can be used to solve an Arithmetic problem,
where pupils work with known values to solve
the unknown. As pupils progress, the model
method is used to solve Algebraic problems
involving unknowns, part-whole concept, and
proportional reasoning.

In view of the theoretical bases of this
study, the conceptual paradigm is shown in the
figure 1.

Algebraic thinking skills are organized into
two general categories: (1) problem solving
skills; and (2) representation and reasoning
skills. Young children are capable of making
generalizations and constructing ways of
representing them (Kaput, 2004; and Windsor,
2007). These generalizations make powerful
Mathematical ideas accessible to students to
solve problems and to deepen understanding.
Generalization and formalization involve
the articulation and representation of
unifying ideas that make explicit important
Mathematical relationships. Thus, these forms
of thinking build directly on conceptions of
understanding as constructing relationships
and reflecting on and articulating those
relationships (Carpenter & Lehrer, 1999).

Problem solving is knowing what to do,
when one doesn’t know what to do. Students
who have a tool kit of problem solving
strategies, e.g. guess and check, make a list,
work backwards, use a model, solve a simpler
problem, etc., are better able to get started
on a problem, attack the problem, and figure

Pictorial
Models

Algebraic Thinking Skills:
 Representation and

Reasoning
 Problem Solving

Figure 1:
Conceptual Paradigm

JADITH TAGLE, RENE R. BELECINA & JOSE M. OCAMPO, JR.,
Developing Algebraic Thinking Skills

out what to do. Giving students opportunities
to explore Math problems, by using multiple
approaches or devising Math problems that
have multiple solutions, allows students to not
only develop good problem solving skills, but
also to experience the utility of Mathematics.
Children initially solve problems by modeling
the problem situations using physical
materials. By reflecting on the modeling
strategies, children abstract these strategies so
that they no longer need the actual materials
to solve the problem (Yeap, 1997; and
Carpenter & Lehrer, 1999).

The ability to use and make connections
among multiple representations of
Mathematical information gives us quantitative
communication tools. Mathematical
relationships can be displayed in many forms:
visually, i.e. diagrams, pictures, or graphs;
numerically, i.e. tables and lists; symbolically;
and verbally. Often a good Mathematical
explanation includes several of these
representations, because each one contributes
to the understanding of the ideas presented.
The ability to create, interpret, and translate
among representations gives students powerful
tools for Mathematical thinking (Carpenter &
Levi, 2000; Carpenter, Franke & Levi, 2003;
and Cai & Knuth, 2005).

Shelley Kriegler (2008) pointed out that
the ability to think and reason is fundamental
in Mathematics. Understanding of the core
ideas can influence one’s success in solving
word problems, the strategies they use in
their solution processes, and the justifications
they provide for the solutions. Inductive
reasoning involves examining particular cases,
identifying patterns and relationships among
those cases, and extending the patterns and
relationships. Deductive reasoning involves
drawing conclusions by examining a problem’s
structure (Kriegler, 2008).

The development of Algebraic thinking
does not emphasize manipulating symbols,
but rather encourage children to make explicit
ideas and to construct ways to represent
those ideas for thinking about them and for
communicating them. In this context, E.
Yackel (1997) posited that non numerical
reasoning about quantities is foundational
to Algebraic reasoning. Algebraic reasoning

in the elementary level can be accomplished
through activities that encourage children
to move beyond numerical reasoning to
more general reasoning about relationships,
quantities, and ways of notating and
symbolizing (Yackel, 1997).

In the teaching of Mathematics, words,
numbers, and pictures should come together
to clearly demonstrate what is taking place
(Murphy, 2006). Using pictorial models, pupils
can help make sense of complex data. Through
the children’s image-making, they have the
capacity to internalize and make connections
to other areas of learning.

The main purpose of this study was to
determine the effect of the use of pictorial
models in developing Algebraic thinking
among primary pupils. Specifically, this study
sought answers to the following questions: (1)
What are the pupils’ Algebraic thinking skills
before and after using pictorial models in terms
of the following: representation and reasoning,
and problem solving?; (2) Is there a significant
difference between the pre-test and post-test
mean scores of the participants?; and (3) What
are pupils’ experiences on the use of pictorial
models?

METHODS
Research Design. This study utilized the

experimental method of research, specifically
the one-group pre-test – post-test design. The
experimental research design is appropriate to
this study, because it is the only design that can
truly test a hypothesis concerning cause-and-
effect relationship (Sevilla et al., 1992).

Participants of the Study. There were 28
grade three pupils at the La Salle Green Hills
in the Philippines, who participated in this
study. These pupils were heterogeneously
grouped according to their mental ability: nine
pupils belong to the Above Average group,
13 from the Average group, and six from the
Below Average group. The classification of the
pupils was based from the result of the SAT
(School Ability Test) given by the Guidance
Office, when these pupils were in grade two.

Grade three pupils were chosen as the
participants of the study, since they belong
to the primary level and they are being
trained to analyze word problems using
pictorial models. LSGH (La Salle Green

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International Journal for Educational Studies, 8(2) February 2016

Hills) Mathematics Area, in the Grade School
Department, started introducing the use of
pictorial models in solving word problems to
grades one to seven pupils in the beginning of
school year 2011-2012.

Though the use of pictorial method is not
fully implemented yet, the participants have an
idea on how to apply the method. However, there
is still a need to explore other means of applying
this method, and how useful the method is in
developing the Algebraic thinking of the pupils.

RESEARCH INSTRUMENTS
The following research instruments were

used to gather the needed data and information
relevant to the study.

Algebraic Thinking Skills Test. The test
was developed to measure the following
Algebraic thinking skills adapted from Shelley
Kriegler (2008), namely: problem solving
skills, representation skills, and reasoning skills
(Kriegler, 2008). The test was divided into two
parts, which are as follows:

Part I (Multiple Choice). This part consists of
20 multiple-choice items. These items aim to
test the pupils’ ability to generate, represent,
and justify generalizations about fundamental
properties of Arithmetic.

Part II (Problem Solving). The second part
consists of five open-ended word problems,
which are Algebraic in nature. This part tested
how the pupils apply pictorial models in
arriving at their answers (Kriegler, 2008).

In developing the test instrument, a table of
specifications was prepared. This instrument
underwent content validation through the help
of three teacher-experts: a high school Algebra
teacher, an elementary Algebra teacher, and
a grade school Mathematics coordinator. The
solution and the answers to problems were
checked according to the rubric provided.

Scoring Rubric. The scoring rubric was used
to determine how the participants performed
in problem solving. A rubric is a set of criteria
used to determine scoring for an assignment,
performance, or product. Analytical rubric is
an example of a scoring designed to assess
students’ work based on specified criteria and
different degrees of quality of the assignment.

In problem solving, the pupils were given
scores according to the following criteria:

accuracy, understanding, communication, and
Mathematical reasoning and strategies. Again,
the three teacher-experts helped in preparing
this scoring rubric.

Interview Guide. This was used to identify
how the pupils answered the test questions,
particularly the problem solving part. Hence,
this was helpful in verifying answers of the
participants. Informal interviews were also
conducted during the teaching-learning
process, wherein the pupils explained how the
pictorial models helped them arrived at their
answers.

Lesson Plans/Work Plans. The work plans
served as guide in conducting the lesson. A
seven-week semi-detailed plan was prepared
before the experiment took place. Topics
covered were fractions and decimals. The
learning activities were categorized into two
parts: (1) the preliminary activities, which
are composed of drills, mental problem, and
review; and (2) the developmental activities,
which include motivation, presentation of
the lesson, comparison and abstraction,
generalization, and application. The
preliminary activities present mental problem
that can be solved using the pictorial models.
Each lesson focused mainly on developing
algebraic thinking of the pupils.

Data Gathering Procedure. To gather data
relevant to the study, permission to administer
the tests was sought from the grade school
principal, assistant principal, student activities
coordinator, and Mathematics coordinator.
With the approval of the request, the list
of materials needed for the study were
immediately prepared.

The process of data gathering involved three
stages: Pre-Experimental Stage, Experimental
Stage, and Post-Experimental Stage.

Stage 1: Pre-Experimental Stage. A pre-test
was administered to the pupils a week before
the actual implementation of the study. This
was done to identify how the pupils think prior
to the discussion of the topics.

Stage 2: Experimental Stage. The lessons on
fractions and decimals were taught for seven
weeks, following the work plans prepared. The
pupils was exposed to drills that involve finding
the value of unknown and problem solving
that involve addition and subtraction of whole

JADITH TAGLE, RENE R. BELECINA & JOSE M. OCAMPO, JR.,
Developing Algebraic Thinking Skills

numbers, fractions and decimals, using the
pictorial method.

The lessons on fractions and decimals were
introduced using the CPA (Concrete-Pictorial-
Abstract) approach from Jerome S. Bruner
(1960). The use of manipulations like food/
bacon shared with the class, which the pupils
represented through drawings and blocks
greatly helped the pupils in understanding the
concepts. During this phase, interviews were
conducted as follow-ups to the pupils’ answers
during the discussion.

Stage 3: Post-Experimental Phase. The post-
test was given after seven weeks of exposure
to the pictorial models to find out if there are
improvements in the Algebraic thinking of the
respondents. Afterwards, the pupils were asked
to summarize how the use of pictorial models
helped them in their Mathematics class.

Data Analysis Procedure. The following
statistical tools were used in this study. Firstly,
Mean and Standard Deviation. These were
used to describe the pupils’ scores in the
Algebraic thinking skills test before and after
using the pictorial models.

Secondly, t-test for dependent samples.
This was utilized to determine if a significant
difference between the pre-test and post-test
mean scores of the pupils.

RESULTS AND DISCUSSION
Table 1 presents the Mean and Standard

Deviation of the pupils’ scores in part I of
the test, which measures their representation
and reasoning skills. The scores in the
reasoning and representation skills show a
marked improvement. This result may be

attributed to their conceptual understanding
of the Mathematical concepts. Conceptual
understanding refers to their integrated and
functional grasp of Mathematical ideas.

It can be deduced that the pupils developed
a deep understanding of the concepts learned,
and they were able to manipulate numbers and
represent value of the unknown after using
pictorial models.

M. Burns (2004) said that teachers in the
lower grades routinely focus on teaching
procedures, rather than on conceptual
understanding. Pupils are able to perform
tasks, but they do not understand why they
work (Burns, 2004). Some researchers say that
the difficulty to transfer from rote Arithmetic
operations of quantities to Algebraic thinking
is due to the lack of conceptual understanding
(cf Desforges, Hughes & Mitchell, 2000; Burns,
2004; Kaput, Carracher & Blanton eds., 2008;
and Clark, 2009). For the primary pupils to
understand the concepts of fractions and
decimals better, pictorial models should be used.

As J. Kaput, D. Carracher & M. Blanton
eds. (2008) suggest, the primary goal of
early Algebra is that pupils learn to see and
to express generalization in Mathematics.
Algebraic thinking cannot be developed
without the pupils’ understanding of the
numerical relationships, operations, and
properties (Kaput, Carracher & Blanton
eds., 2008:60). The NCTM (National
Council of Teachers of Mathematics), in
2000, emphasized that “Algebra is more than
the manipulation of symbols”; and that its
study should start from the earliest years of
schooling (NCTM, 2000).

Table 1:
Results of the Pre-Test and Post-Test of the Participants

in Terms of Representation and Reasoning (Multiple-Choice Type)

Mean Standard Deviation
Pre-Test 7.11 2.53
Post-Test 14.29 2.54

Table 2:
Results of the Pre-Test and Post-Test of the Participants

in Terms of Problem Solving (Open-Ended Type)

Mean Standard Deviation
Pre-Test 3.26 1.81
Post-Test 6.23 2.51

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International Journal for Educational Studies, 8(2) February 2016

Table 2 presents the pupils’
performance in problem solving.
Though there is an increase in the
scores of the pupils, there is still a
need to focus more in developing their
problem solving skills. Mathematical
problem solving has been instrumental
in achieving a variety of goals within
the Mathematics curriculum. A
classroom environment that values
and promotes problem solving can
facilitate Algebraic thinking.

In using pictorial method, a block
may represent one-fourth and four
blocks may represent one whole. In
this regard, the pupils will be able to
understand the relation between 1/4
and 1 whole. Pupils will also learn to
represent values using blocks, as two
blocks as to 2/4 and three blocks are
equal to 3/4.

Illustrations, in figures 2 to 5, show
some excerpts from the pupils’ work
on how they applied the pictorial
model in answering word problems
that are Algebraic in nature.

Figure 2 shows that the pupil used
a circle to represent a set of apples.
Each half was labeled as four. Two
halves equal eight. The picture shows
that the concept of half in relation to
a whole must be fully understood by
the pupils. The pupils had to utilize
their knowledge of fractions to draw
the model properly. In teaching
fractions, the CPA (Concrete-
Pictorial-Abstract) approach is helpful
in developing the pupils’ conceptual
understanding (cf Bruner, 1960; Walle, 2004;
and Diezzman & McCosker, 2011).

The answer proved that the pupil was able
to show visual representation of the concrete
objects. Thus, this helped him/her visualized
Mathematical operations during problem solving.

Problem A:
Mark gave 4 apples to Lorna. If what he shared represents 1/2 of his apples,

how many apples did he have at first?

Figure 2:
Pupil’s Solution to Problem A

Problem B:
Marko and Anna have P 24.00 together. If Marko has twice

as much as Anna’s, how much does Anna have?

Figure 3:
Pupil’s Solution to Problem B

Table 3:
Results of the t-Test for the Comparison of the Pupils’ Pre-Test and Post-Test Mean Scores

Mean
Standard
Deviation

Mean Difference t-value
Critical
Value

Interpretation

Pre-Test 7.11 2.54
7.18 16.22 2.77 Significant

Post-Test 14.29 2.53

Using the pictorial model, the pupil used
two blocks to represent Marko’s money. Two
blocks represent Marko’s since he has twice
as much as Anna’s. The pupils should be able
to understand the concept of sum, in relation
to its parts/addends to be able to solve this

JADITH TAGLE, RENE R. BELECINA & JOSE M. OCAMPO, JR.,
Developing Algebraic Thinking Skills

problem. As suggested by Carolyn
Kieran (2004), pictorial equation
solving can help students to focus
on both representing and solving
the problem rather than on merely
solving it (cf Lee et al., 2004; Kieran,
2004; and Ronda, 2004). See figure 3.

The solution in figure 4 shows that
the pupil drew 3 blocks for Jeremy’s
number of stickers, since Janna has
only one-third of Jeremy’s. In this
kind of problem, if the pupil doesn’t
fully understand the concept of
fraction in relation to a whole, wrong
representation could be made, as three
units for Janna instead of the other
way around. This only proves that
teaching Algebraic thinking should
start from developing the conceptual
understanding of the pupils.

It can be seen from figure 5 that
the pupil subtracted the difference
from the total (12.40 – 3.40) to get
the remaining parts. Since there were
2 blocks left, he equally divided 9.00
into two equal parts, hence each
number has 4.5. The first number is
3.4 more than the second number,
which is 4.5. In this solution, three
operations were used: addition,
subtraction, and division. Though the
pupil was able to arrive at the correct
answer, the way he represented the
numbers must be corrected. That is a
bigger number should represent a bigger block.

Comparison of the Pupils’ Pre-Test and Post-
Test Mean Scores. Table 3 shows that there is
a significant difference between the pre-test
and post-test of the participants. This implies
that the use of pictorial models had a positive
effect on the development of the pupils’
Mathematical thinking. This result is consistent
with positive statements on the effectiveness of
pictorial models in Mathematics learning.

It seems that the use of pictorial models
provides the pupil opportunities to deepen
their understanding of Mathematical concepts,
apply their knowledge of the four basic
operations, and model the problem situation
through representation. As cited by Stuart
Murphy (2006), visual learning strategies can

Problem C:
Janna and Jeremy have 36 stickers together. Janna’s stickers’ are 1/3 of

Jeremy’s. How many stickers does Janna have?

Figure 4:
Pupil’s Solution to Problem C

Problem D:
The sum of two numbers is 12.40. The first number is 3.40 more than

the other number. What are the two numbers?

Figure 5:
Pupil’s Solution to Problem D

make a profound difference in a student’s
depth of understanding about Mathematics
(Murphy, 2006).

It is a powerful teaching tool for kids who are
natural visual/spatial learners, for children who
are English language learners, and for students
of all learning modalities (cf Katz, 2006;
Murphy, 2006; and Richarson, Sherman &
Yard, 2009). By using visual learning strategies
in the teaching of Mathematics, teachers can
increase the learning potential of children.

Supporting the result of this study is found
in a case study made by Swee Fong Ng (2004).
Swee Fong Ng and others said that through the
model method (pictorial models), pupils with
no knowledge of formal Algebra are provided
with a tool to construct pictorial equations to

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International Journal for Educational Studies, 8(2) February 2016

solve increasingly challenging word problems,
involving simple part-whole relationships as
far as those that require proportional reasoning
(cf Fong Ng, 2004; Ernest, 2006; and Krulik &
Posamentier, 2009).

Pupils’ Experiences on the Use of Pictorial
Models. Pupils’ experiences on the use of
pictorial models were obtained through formal
interviews. Based on the conducted oral
interviews, most of the pupils said that they use
the pictorial model only “sometimes”. That is
when it is applicable to the problem presented.
The pupils were one in saying that not only have
the pictorial models helped them in improving
their Mathematical thinking skills, but they have
also developed their Algebraic thinking skills
unconsciously (interview with Respondents A,
B, C, and D, 19/12/2011).

Generally, the pupils find the use of pictures,
“blocks” or rectangles helpful in understanding
concepts and word problems deeply. Some of
the pupils’ comments were as follows:

“Pictorial model helps me compute better” (interview
with Respondent A, 19/12/2011).

“It makes the word problem easier” (interview with
Respondent B, 19/12/2011).

“It helps me analyze word problems well” (interview
with Respondent C, 19/12/2011).

“It is easy to understand numbers through drawings,
that is through the use of rectangles” (interview with
Respondent D, 19/12/2011).

The comments of the pupils only proved
that the use of pictorial models aided them in
solving word problems. These clearly show that
through the use of pictorial models, pupils were
able to represent numbers using bars or blocks
and learned how to model problem situations.
Similar findings were found in the study of N.
De Guzman (2009) and others, where the grade
five pupils perceived the block model as a useful
tool in solving word problems (cf Charlesworth
& Radeloff, 1978; Carruthers & Worthington,
2003; and Guzman, 2009).

Since equations were introduced
through pictures, pupils were able to use
this to represent quantitative relationships.
This relates with J. Cai (2005) and othres’
contention that “pictorial equations” do
not only provide a tool for students to solve

Mathematical problems, but they also provide
a means for developing pupils’ Algebraic ideas
(cf Lew, 2004; Cai, 2005; and Jackson, 2009).

Based on the data gathered, the following
are the findings of this study. Firstly, there is
a significant difference between the pre-test
and post-test mean scores of the pupils on
representation and reasoning skill and problem
solving skill. Secondly, the pupils were able to
answer word problems, which are Algebraic
in nature through the use of pictorial models.
Thirdly, the pupils perceived pictorial models as
helpful tools in analyzing word problems as the
models make them understand word problems
better. Finally, fourthly, teaching Algebraic
thinking can start at an early age (Maletsky &
Sobel, 1988; Nebres, 2006; Sousa, 2007; Lee &
Lee, 2009; and Ptylak, 2010).

CONCLUSION
Based on the findings of the study, the

following conclusions were drawn. The use
of pictorial models has a positive effect in
developing Algebraic thinking of primary
pupils. Primary pupils can engage in powerful
Mathematics structures if given appropriate
learning activities. One of these activities is
engagement in problem solving through the
use pictorial models to deepen understanding
of word problems. In the pictorial models,
the use of blocks as units representing the
unknowns provides a link to more abstract
ideas, like letters representing the unknowns.

From the findings and conclusions, the
following recommendations are given:

The use of pictorial models should be
introduced to the pupils, when solving word
problems. Pictorial representations, drawings,
and models may lead children to understand the
symbols which seem abstract to them initially.

Mathematics teachers should allow pupils
to think out of the box or to find creative
ways in solving word problems. Mathematics
teachers should be provided with frequent
opportunities for high quality training.
Teachers should be exposed to a wide variety
of approaches, such as the use of pictorial
models, which they can introduce to the pupils
to develop their higher-order thinking skills.

Further studies should be done also in
the public schools to determine whether

JADITH TAGLE, RENE R. BELECINA & JOSE M. OCAMPO, JR.,
Developing Algebraic Thinking Skills

this approach has a positive effect on the
performance of the pupils in Mathematics.
Future researchers are encouraged to conduct
similar studies that may help the pupils develop
their Algebraic thinking skills.1

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Pupils at the La Salle Green Hills in the Philippines
(Source: http://www.interaksyon.com, 20/5/2015)

The use of pictorial models has a positive effect in developing Algebraic thinking of primary pupils. Primary pupils can
engage in powerful Mathematics structures if given appropriate learning activities. One of these activities is engagement
in problem solving through the use pictorial models to deepen understanding of word problems. In the pictorial models,
the use of blocks as units representing the unknowns provides a link to more abstract ideas like letters representing the
unknowns.