Content uploaded by Toto Nusantara
Author content
All content in this area was uploaded by Toto Nusantara on Apr 21, 2016
Content may be subject to copyright.
Vol. 11(8), pp. 732-742, 23 April, 2016
DOI: 10.5897/ERR2016.2719
Article Number: 1B733A858138
ISSN 1990-3839
Copyright © 2016
Author(s) retain the copyright of this article
http://www.academicjournals.org/ERR
Educational Research and Reviews
Full Length Research Paper
Local conjecturing process in the solving of pattern
generalization problem
Sutarto1, Toto Nusantara2, Subanji2 and Sisworo2
1Institut Keguruan dan Ilmu Pendidikan (IKIP) Mataram, Indonesia.
2Department of Mathematics, Universitas Negeri Malang, Indonesia.
Received 25 February, 2016; Accepted 14 April, 2016
This aim of this study is to describe the process of local conjecturing in generalizing patterns based on
Action, Process, Object, Schema (APOS) theory. The subjects were 16 grade 8 students from a junior
high school. Data collection used Pattern Generalization Problem (PGP) and interviews. In the first
stage, students completed PGP; in the second stage, work-based interviews were conducted by the
researchers to understand the process of conjecturing. These interviews were video tapped. The
results show that the local conjecturing process can be grouped into two, namely local conjecturing
based on proximity and local conjecturing by contrast. The local conjecturing process occurs at the
stage of action in which subjects build a conjecture by observing and counting the number of squares
separately. At the stage of process, the object and scheme were perfectly performed.
Key words: local conjecturing, problem solving, and pattern generalization.
INTRODUCTION
Pattern generalization is an important aspect in the
activities of school mathematics (Dindyal, 2007; Vogel,
2003; Zazkis and Liljedahl, 2002). In line with this idea,
Küchemann (2010) states that generalizations should be
the core of the activities of mathematics at school.
Generalization is certain kind of conjecture, which is
obtained from particular to general reasoning
(Yerushalmy, 1993). Further, a generalization based on
inductive reasoning is called a conjecture (Ramussen
and Miceli, 2008).
A conjecture is a logical statement, but whose truth has
not been confirmed (Cañadas and Castro, 2005; Ontario
Ministry of Education, 2005; Mason et al., 2010; Reid,
2002). The process of producing conjecture is called the
conjecturing process. Conjecturing process is the mental
activity used in building a conjecture based on one’s
knowledge (Sutarto et al., 2015). Mental activity in
building conjecture is a process that occurs in the mind
that can be seen through the behaviour of students in
problem solving.
Associated with the conjecturing in the school
mathematics is a process of building new knowledge for
the students. This is in accordance with the statement by
Lee and Sriraman (2010) that conjecturing in
mathematical problem solving in learning process is to
construct new knowledge for the students according to
the existing knowledge that students already have.
A significant contribution to the conjecturing process
*Corresponding autor. E-mail: sutarto_zadt@ymail.com.
Authors agree that this article remain permanently open access under the terms of the Creative Commons Attribution
License 4.0 International License
has been given by several researchers (Cañadas, 2002;
Cañadas et al., 2007; Polya, 1967; Reid, 2002). Polya
(1967) suggests a four-step process of inductive
reasoning in solving problems, namely (1) observation of
particular cases, (2) conjecture formulation based on
previous particular cases, (3) generalization, and (4)
conjecture verification with new particular cases. Reid
(2002), in the process of inductive reasoning in the
context of empirical induction of a finite number of
discrete case, uses the following stages of (1)
observation of a pattern, (2) the conjecturing (with doubt)
that this pattern applies generally, (3) the testing of the
conjecture, (4) the generalization of the conjecture, and
(5) using generalizations to prove. Cañadas (2002)
suggests seven steps to describe the process of
inductive reasoning, namely (1) observing cases, (2)
organizing cases, (3) searching for and predicting
patterns, (4) formulating a conjecture, (5) validating the
conjecture, (6) generalizing the conjecture, and (7)
justifying the generalization. Furthermore, Cañadas et al.
(2007) use seven stages that describe the process of
inductive reasoning of Cañadas (2002) as one type of
conjecturing process of empirical induction of a finite
number of discrete cases. These studies have not
revealed the conjecturing process done by students in
pattern generalization problems.
Associated with the process of conjecturing, Cañadas
et al. (2007) argue that one of the familiar conjecturing
processes in mathematical problem solving is
conjecturing of empirical induction of a finite number of
discrete cases. This type of conjecturing is frequently
found in problems involving pattern of numbers. In
solving problems involving numbers to a consistent
pattern, the seventh stage of conjecturing process does
not always happen as there are many factors that
influence such type of task or characteristics of the
students involved (Cañadas and Castro, 2005).
Sutarto et al. (2015) explained that subject classification
based on observing the case in conjecturing process: 1)
observing and counting the number of the square at the
down-left side up-right, the primary, the left and right
without discerning the Black and White boxes (3
subjects), 2) observing and counting the number of Black
and White boxes separately (5 subject) , 3) Observing
and counting the number of the box without discerning
the Black and the white boxes (8 subject).
Based on that explanation, conjecturing processes of
students in the solving of pattern generalization problems
are grouped into two: local conjecturing and global
conjecturing. The local conjecturing process is the activity
in building a conjecture by observing the problems
separately, while the global process of conjecturing is a
activity in building a conjecture by observing the problem
as a whole. The local conjecturing process is rarely per-
formed by students when solving pattern generalization
problems. Therefore, in this study we will describe the
process of local conjecturing in pattern generalization
Sutarto et al. 733
problems.
The process of local conjecturing in problem solving of
pattern generalization is analysed by using the APOS
theory, because it is a theory that can be used as an
analytical tool to describe the development of a scheme
by a person on a mathematical topic as totality of
knowledge related (consciously or unconsciously) to the
topic (Dubinsky, 2001).
This theory is based on the hypothesis that a person’s
knowledge of mathematics will be a tendency to cope
with a situation of mathematical problem by building
actions, processes, and objects and arrange them in a
scheme to understand the situation and solve the
problem (Dubinsky and McDonald, 2001). Asiala et al.
(1996) considered that APOS theory begins with
manipulating previously constructed mental or physical
objects to form actions; action is then interiorized to form
processes which are then encapsulated to form object.
Object can be encapsulated back to the processes from
which they were formed; Finally, action processes and
objects can be organized in schemas. This theory is
called the APOS (actions, processes, objects, and
scheme).
Indicators of local conjecturing process
Researchers describe the process of local conjecturing
using the theory of Cañadas et al. (2007) on the seven
stages of the process of conjecturing of empirical
induction of a finite number of discrete cases. The
explanation on the seven stages of the conjecturing
process is as follows: (1) Observing cases is the initial
activity carried out on particular cases of the problem
posed; (2) Organizing case is an activity that involves the
use of strategy facilitating the work in certain cases.
The most common strategy used in organizing cases is
registering or sorting the data; (3) Searching for and
predicting patterns is one’s activity when observing
repetitive and regular situations, one naturally imagines
that the pattern may apply to the next cases of unknown;
(4) Formulating a conjecture is making a statement about
all possible cases, based on empirical facts, but with an
element of doubt or in other words conjecture is a
statement that has not been validated; (5) Validating the
conjecture is the activity performed to justify the
conjecture generated based on specific cases but not in
general; (6) Generalizing the conjecture is an activity on
changing confidence related to the generated conjecture,
that the conjecture is valid in general; (7) Justifying
generalization is the activity performed to justify
generalizations. Justifying the generalization involves
giving reasons that explain conjecture with the intention
of convincing others that the resulting conjecture is
correct.
Based on the explanation of the seven stages and
indicator of conjecturing processes adapted from Sutarto
734 Educ. Res. Rev.
Table 1. Indicators of local conjecturing process
Stages of conjecturing process
Indicator
Observing the case
Initial activities against specific cases of problems presented as:
- Observing and counting the number of square section on lower left, upper
right, main, left and right without distinguishing a black square and a white
square
- Observing and counting the number of horizontal and vertical square-section
without distinguishing a black square and a white square
- Observing and counting the number of black squares and white squares
separately
Organizing the case
Activities that involve the use of strategies that facilitate work in certain cases
such as:
- Writing down the number sequence
- Making a list or table to associate number 1 with the picture number 1,
number 2 with picture number 2, number 3 with picture number 3, and so on
- Writing down symbol that indicates a similar pattern as the bottom line, circle,
or other
Searching for and predicting the
pattern
Activities of observing certain objects either organized or disorganized and
thinking about the next object that is not yet known such as:
- Calculating the difference between a square to the first, second, and third
image and thinking about the next object
- Calculating the difference between a black square and a white square to the
first, second, and third image and thinking about the next object
Formulating the conjecture
Making a statement about all possible cases, based on empirical facts, but
have not been validated as:
- Stating pattern or formula of image n applicable
- Declaring the n formula of black square 2n + 1, white square 2n + 2 and the
n formula for black and white squares (2n + 1) + (2n + 2)
- Stating the general formula or the number of squares of n image = 4n + 3
Validating the conjecture
Activities carried out to establish the truth of the conjecture produced by
certain new cases but not in general, such as:
- Validating the particular case to establish the truth of the conjecture
generated, for example, for the first, second, third image, ...
- To sketch the next object that represents the next reachable pattern to
establish the truth of the conjecture generated, for example, for the fourth, fifth,
sixth image, ...
Generalizing the conjecture
Changes related to the confidence of the generated conjecture, that the
conjecture is valid in general such as:
- Stating pattern or formula of then image
- Believing the formula of n black square as 2n + 1, white square as 2n + 2
and the formula for n black and white squares (2n + 1) + (2n + 2)
- Believing the general formula or the number of squares for n = 4n + 3 applied
in general.
Justifying the generalization
Giving the reasons that explain the generalization with the intention of
convincing others that the resulting generalization is correct, such as:
- Justifying generalizations based on specific cases.
et al. (2015), we obtained indicators for local conjecturing
process in problem solving of pattern generalization as
presented in Table 1.
Pattern generalization
Many mathematicians claim that mathematics is referred
Sutarto et al. 735
Figure 1.The Pattern Generalization Problem (PGP)
to as the science of patterns (Resnik, 2005; Tikekar,
2009). Learning about pattern is important and needs to
be taught early. NCTM (2000) recommends that students
participate in patterning activity from a young age, in the
hope that they will be able to: (1) make generalizations
about geometric and numerical patterns, (2) provide
justification to their conjecture, (3) state the rules of the
patterns and functions through verbal forms, tables, and
graphs. Based on some of these opinions, it can be
concluded that patterns are an important matter to be
taught from an early age to train children to reason.
Pattern generalization is the activity to make a general
rule pattern based on specific examples. Specific
examples can be graphic, numeric, verbal and algebraic
pattern (Janvier, 1987). For the aim of this study, the
information provided through specific cases stated in the
linear-shaped graphic pattern, because the graphic
pattern allows students to observe in different ways.
Wertheimer (1923) states Gestlat law of proximity,
similarity and closure. The law of proximity states that
when individuals perceive an assortment of objects they
perceive objects that are close to each other as forming a
group. The law of similarity states that elements within an
assortment of objects are perceptually grouped together
if they are similar to each other. The law of closure states
that individuals perceive objects such as shapes, letters,
pictures, etc., as being whole when they are not
complete.
In particular, patterns are seen by some researchers as
path to transition to algebra because they are a
fundamental step to build the generalization that is the
essence of mathematics (Zazkis and Lijedahl, 2002). In
generalizing patterns, it is not enough to declare a
general rule and order patterns verbally but must also
state the general rule of pattern with symbol.
The aim of the study
The aim of this study is to describe local conjecturing
process in the solving of pattern generalization problem
based on APOS theory.
METHODOLOGY
Subjects
Researchers asked 42 students at junior high school (SMPN 3
Malang) to complete pattern generalization problem. After
experiencing saturation data in the subject, 16 students produced a
formula or general rule symbolically.
Instrument
There are two types of instruments used, main and auxiliary
instruments. The main instrument is the researchers themselves
who act as planners, data collectors, data analysts, interpreters,
and reporters of research results. The auxiliary instrument used in
this study is a Pattern Generalization Problem (PGP) and interviews.
The problem given aims to obtain a description of the process of
conjecturing of the students, while the interview used was
unstructured interview. The PGP is presented in Figure 1.
Procedure
In the first stage, students completed PGP. In the second stage, the
researchers conducted work-based interviews to understand the
process of conjecturing and then the researchers recorded them by
using a handy cam.
Data analysis
This study is a qualitative research with descriptive exploratory
approach. At the data analysis stage, the activities conducted by
researchers were (1) transcribing the data obtained from interviews,
(2) data reduction, including explaining, choosing principal matters,
focusing on important things, removing the unnecessary ones, and
organizing raw data obtained from the field (3) encoding the data
from PGP answer sheet and interviews refer based on indicators of
local conjecturing process are presented in Table 1, (4) describing
the local conjecturing process in the solving of pattern
generalization problem based on APOS theory, and (5) conclusion.
RESULTS AND DISCUSSION
Based on the results of the analysis of the PGP answer
sheets and interviews, we obtained data on the local
conjecturing process undertaken by students in solving
Figure 1.The Pattern Generalization Problem (PGP)
736 Educ. Res. Rev.
Table 2.The results of the conjecturing process
presented by students when solving pattern
generalization problem.
Conjecturing process
Local
Global
Proximity
Contrast
3
3
10
18.75 %
18.75 %
62.5 %
pattern generalization problem. The local conjecturing
process in pattern generalization problem solving can be
grouped into two general categories: (1) local
conjecturing based on proximity and (2) local conjecturing
by contrast. Of the 16 students who did conjecturing
process and produced a formula or symbolic general rule,
10 students did global conjecturing process, and 6
students did local conjecturing process. Six students
were grouped into two, 3 students (subject S1, subject S2,
dan subject S3) did the proximity conjecturing, and 3
students (subject S4, subject S5, dan subject S6) by
contrast. The results of the process of conjecturing are
presented in Table 2.
After experiencing saturation in the process of data
collection, there were 6 students that did the local
conjecturing. Of the 6 students, we chose one subject
that did local conjecturing process based on proximity,
that is, the subject S3, and one subject who did local
conjecturing process by contrast, that is, the subject S6.
Local conjecturing based on proximity
At the stage of action, subject S3 realized that the images
(first, second, and third image) formed a pattern. S3
observed cases by observing and counting the number of
square section at the lower left, upper right, main, left and
right without distinguishing black square and white
square of the first, second, and third image. Here are
excerpts of the interview and the work of S3.
I : What did you think when reading this issue?
S31 : The pattern, off course.
I : What kind of patterns?
S32 : The pattern of addition.
It is always 4, namely from right, top, left and bottom, to
the first, second, third, fourth, and fifth and onwards
(pointing to the results of the work). Based on the number
of squares on the first, second, and third images, subject
S3 organized the case by writing a symbol indicating a
similar pattern by circling the first, second, and third
images. This was confirmed by the transcript of the
interview S33 and the work of S3 in Figure 2.
S3 3: To be able to see a pattern, I saw the addition at
Figure 2.The work of S3.
each end of the square that I circled.
At the process stage, S3 internalized the action by
finding and predicting patterns. The activities were
carried out by calculating square difference of the first,
second, and third image, that was 4 squares, as well as
thinking about the fourth, fifth, sixth image and so on.
This was confirmed by the transcript of the interview S32
and the work of S3 (Figure 3).
At the object stage, S3 encapsulated process to
formulate a conjecture by connecting between the first,
second, and third image to the difference and the number
of the main part of the image given. For example, for the
first image, it was written ; the student wrote one
for the first image, wrote four because there were four
different images, wrote 3 for the number of main section
was 3. For the second image, S3 wrote , and
the third image, S3 wrote . Furthermore, S3
formulated the general formula for n image, that was
. This was confirmed by the results of the work
of S3 (Figure 4) and interview transcript S34. S3 validated
conjecture by calculating conformity of the formula with
the number of square of the fourth image, namely
. The formula obtained was correct as the
number of square at the third image plus 4 equal to 19.
This is evidenced by the following interview transcript of
S35 and the work of S3.
S3 4: So for the first image I created a formula; for
example, the first image was multiplied by the addition,
and I got 4, then right, up, left, down was one, and I
added all, and the result was 4. Once multiplied, then
added with the main figure.
S35: I’m sure, I tried to count in the fourth image
, the result is the same as for the third image,
3 plus 4 is 19.
At the scheme stage, S3 believed the general formula
produced was correct, based on the results of
the validation. By believing that formula, S3 was
generalizing. Next on justify generalization stage, S3
pinpointed the specific examples as done in validating the
conjecture in order to convince others that the conjecture
was generated correctly. This can be shown from the
following interview excerpts.
I: Ok. How did you explain to others that the resulting
Sutarto et al. 737
Figure 3. The work of S3.
Figure 4. The work of S3.
Figure 5. S3 thinking structure.
formula is true (pointing to the work of S3).
S36: I'm going to show you an example.
I: Examples like what?
S37: Suppose the first image is true, the
second image is true, the third image
is true, then is true and so on.
Based on these data, the structure of thinking of S3 is
based on the stages of the conjecturing process and
APOS theory. S3 thinking structure is presented in Figure
5 and code description of thinking structure Table 3.
Local conjecturing by contrast
At the action stage, subject S6 had been aware that the
b
a
c
e
f
g
h
i
j
l
k
m
d
738 Educ. Res. Rev.
Table 3. code description of S3 thinking structure.
a :
The problem posed that is to find out general
formula to decide the number of square on
image n
k :
Believing that the n formula is
b :
Observing and counting the number of squares
on the first, second, and third image separately
l :
Justifying the n formula
c :
Counting the number of squares on the bottom
left, upper right, main, left, and right without
differentiating the black and white squares on
the first, second, and third image
m :
Finish
d:
Making a list to order the pattern
Order of activity
e:
Counting the difference of the first, second, and
third image and thinking about the next object
Validating, example from i to h, back to i;
from j to c, back to j, and so forth
f:
Stating the difference of the numbers
sequence, that is 4
Action
g:
Connecting the first, second, and third image by
adding the main part
Process
h:
Example, the first image
The second image
The third image
Object
i:
The general formula of the n image is
Scheme
j:
Validating using the fourth image
Initial and end of activation
first, second, and third image formed a pattern. S6
observed and counted the number of black and white
squares of the first, second, and third image separately to
find a common formula of the number of square-n. Here
are excerpts from the interview with S6.
S61: Counting this, one by one, how many black squares,
how many white squares (pointing at the image)
I: What do you mean?
S62: There are 3 black squares on the first image, 4 white
squares. There are five black squares and 6 white
squares. There are four black squares and 5 white
squares on the fourth image. Each image has 2 different
squares. And so on.
Based on the number of black and white squares of the
first, second, and third image, S6 organized case by
ordering number sequences. This is demonstrated by the
work of S6 in Figure 6.
At the process stage, the subject S6 internalized the
action by finding and predicting patterns. The activities
were carried out by calculating the difference between
the second and the first image, between the third and the
second image respectively for black and white squares.
The difference of black and white square was 2. S6 then
thought the difference for the fourth image, the fifth
image, and so on. The interview excerpt with S62 confirmed
this.
At the object stage, S6 performed encapsulation process
to formulate a conjecture by looking at the relationship
between the first image with the number of black squares
plus white squares of the first image. S6 also looked at
the relationship between the second image with the
number of black squares plus white squares of the
second image, and so on. By looking at the relationship,
S6 formulated a conjecture to determine the number of
black squares on image n = 2n + 1, to the white square
on image n = 2n + 2. Furthermore, S6 validated conjecture
by looking at the suitability on the number of squares on
the first, second, and third image, and counted the
number of squares on the fourth and fifth image. Here is
the interview excerpt with S6 and the work of S6 in Figure
7.
S63: Er .... No, no, I mean it. This must have something to
do with this (pointing to the work). We are told to count
pattern of the first image, and n. From the first image,
meaning we must count the next pattern, the hundredth
image, or maybe thousandth image for example so we
could easily do that. We also have to count the
relationship with this (pointing to the work). It makes me
conclude this. , . , .
Thus, the second image is , this
continues. ,
I : Then how?
Sutarto et al. 739
Figure 6.The work of S6
Figure 7.The work of S6.
S64: Try to find the fourth image.
I : How can you be sure with this answer?
S65: Because I have counted the fourth image, the fifth
image, and I matched with the first, second, and third
image, and the result is correct.
At the scheme stage, S6 justifies generalization with the
aim of convincing others that the conjecture produced is
correct by way of explaining how to get the formula and
calculate the number of squares as done in validating the
conjecture. This is shown in the following interview
excerpt.
I : How do you explain that the formula you produced is
correct?
S6 6: I explain how I come to and count the number of
black and white squares (pointing at the work)
Based on these data, the structure of thinking of S6 can
be described based on the stages of the conjecturing
process and APOS theory. S6 thinking structure is
presented Figure 8 and code description of S6 thinking
structure Table 4.
Local process conjecturing scheme based on APOS
Theory
This section describes the scheme of local conjecturing
process based on the APOS theory. In the action stage,
S3 and S6 observed cases and organized a separate
case. The activity then became the foundation in building
conjecture. S3 observed and counted the number of
square from bottom left, top right, main, right and left
separately. These activities were in accordance with the
Gestlat law in the observation that is the Law of
Proximity, in which a person tends to perceive elements
adjacent to each other as a specific form (Wertheimer,
1923). S3 observed cases by observing and counting the
number of black and white squares separately. These
activities were in accordance with the Gestlat law in the
observation that is the low of similarity, whereby one
tends to perceive the same stimulus as a whole
(Wertheimer, 1923).
At the stage of process, subject internalized the action
to find and predict patterns by looking at the difference
between the number of squares of the first and second
image, the second and third image, and thought that the
next image had the same pattern. At the object stage, S3
formulated conjecture by doing encapsulated process to
formulate conjecture by connecting between the first,
second, and third image to the difference and the number
of the main part of the image given. From this activity, S3
formulated a conjecture to calculate the
number of images to n square. S6 did the encapsulation
process to formulate a conjecture by looking at the
740 Educ. Res. Rev.
Figure 8. S6 thinking structure.
Table 4. Code description of S6 thinking structure.
a :
The problem posed that is to find out general
formula to decide the number of square on
image n
k :
Believing that the general formula for black
squares and white squares
b :
Observing the number of squares on the first,
second, and third image separately
l :
Justifying the n formula
c :
Counting the number of black and white
squares separately
m :
Finish
d :
Wiring down the number sequence of black
squares 3, 5, 7, and white squares 4, 6, 8
Order of activity
e:
Counting the difference of the first, second, and
third image
Validating, example from g to c, back to g;
from l to i, back to l, and so forth
f:
Stating the difference of the black squares 2
and the white squares 2, and thinking about the
next object
Action
g:
Connecting the first, second, and third image by
adding the number of black and white squares
Process
h:
Example:
The first image black squares
The first image white squares
Object
i:
The general formula of black squares image
and white squares image
Scheme
j :
Validating using the fifth image that is
+
Initial and end of activation
relationship of the number of black square plus white
square of the first image. S6 also looked at the relationship
between the number black square plus white square of
the second image and so on. The activity of S6 to
formulate a conjecture to determine the number of black
squares on n image of , and the number of white
squares on n image of . At the process and the
object stage, this was done perfectly by the subject to
produce a conjecture.
At the scheme stage, S3 and S6 generalized conjecture
as to believe that the resulting conjecture was correct. In
justifying generalizations, with the aim of convincing
others that the resulting conjecture was true, S3 and S6
used specific examples obtained at the stage of action
and object. In justifying generalizations, S3 and S6
performed it in their own way. This is in accordance
with the statement of Caraher et al. (2008) that students
do not just simply use the notation or symbols but also
present and give a mathematical reason, make
conclusions and generalizations on their own way. The
scheme stage was also done perfectly.
Based on the description above, it can be described
that the scheme of local conjecturing process by students
in the solving of pattern generalization problem has been
based on APOS theory. Local conjecturing process
occurs at the stage of action, in which the subject built
conjecture by observing and counting the number of
squares separately and internalizing the action to the
b
a
c
e
f
g
h
i
j
l
k
m
d
Sutarto et al. 741
Figure 9. Scheme of local conjecturing process.
process stage, encapsulating the process to becoming an
object. Then at the stage of scheme, all phases of APOS
have been done perfectly. Here is the process of local
conjecturing process scheme (Figure 9).
Conclusion
Local conjecturing process in the solving of pattern
generalization problem is composed of local conjecturing
process based on proximity and local conjecturing by
contrast. Local conjecturing process based on proximity
happens at the action stage, in which the subject builds a
conjecture by observing and counting the number of
square separately based on without proximity
differentiating a black square and a white square; and at
the process stage, object and scheme is done perfectly.
Local conjecturing process by contrast occurs at the
action stage, in which the subject builds a conjecture by
observing and counting the number of squares separately
between black squares and white squares, and at the
process stage, object and scheme is done perfectly.
Certainly, the results of the current study mean much
and have implications for the development of science.
For the solving of graphic patterns problem, teachers
need to consider aspects of observing the problem
separately based on proximity and contrast in the
process of building a conjecture.
Limitation of the study
The results of this study are limited to the data collected
from the eighth grade students and this study has not
described the process of global conjecturing in pattern
generalization problem.
Conflict of Interests
The authors have not declared any conflicts of interest.
REFERENCES
Asiala M, Brown A, DeVries DJ, Dubinsky E, Mathews D, Thomas K
(1996). Framework for research and curriculum development in
undergraduate mathematics education. Res. Coll. Math. Educ. 6:1-
32.
Cañadas MC (2002). Razonamiento inductivo puesto de manifiesto por
alumnos de secundaria. Granada: Universidad de Granada.
Cañadas MC, Castro E (2005). A proposal of categorisation for
analysing inductive reasoning. In M. Bosch (Ed.), Proceedings of the
CERME 4 International Conference. Sant Feliu de Guíxols, Spain.
pp. 401-408
Cañadas MC, Deulofeu J, Figueiras L, Reid DA, Yevdokimov O (2007).
The conjecturing process: Perspectives in theory and implications in
Observing cases
Organizing cases
Action
Process
Finding and
predicting patterns
Formulating
conjecture
Validating
Conjecture
Justifying
Generalisasi
Generalizing
conjecture
Validating
Conjecture
Justifying
Generalisasi
Object
Scheme
742 Educ. Res. Rev.
practice. J. Teach. Learn. 5(1):55-72.
Caraher DW, Martinez MV, Schielmann AD (2008). Early algebra and
mathematical generalization. ZDM Math. Educ. 40:3-22.
Dindyal J (2007). High school students’ use of patterns and
generalisations. In: J. Watson and K. Beswick (Eds), Proceedings of
the 30th annual conference of the Mathematics Education Research
Group of Australasia. Hobart, Tasmania, Australia: MERGA Inc. pp.
236-245.
Dubinsky E (2001). Using a theory of learning in college. TaLUM 12:10-
15.
Dubinsky E, McDonald M (2011). APOS: A constructivist theory of
learning in undergraduate mathematics education research. In D.
Holton et al. (Eds), The teaching and learning of mathematics at
university level: An ICMI Study. Kluwer Academic Publisher. pp. 273-
280.
Janvier C (1987). Translation processes in mathematics education. In
C. Janvier (Ed.), Problems of representation in the teaching and
learning of mathematics. Hillsdale, New Jersey:LEA. pp. 27-32.
Küchemann D (2010). Using patterns generically to see structure.
Pedagogies: Int. J. 5(3):233-250.
Lee KH, Sriraman B (2010). Conjecturing via reconceived classical
analogy. Educ. Stud. Math. 76(2):123-140.
Mason J, Burton L, Stacey K (2010). Thingking mathematically second
edition, England: Pearson Education Limited.
Nasional Council of Teacher of Mathematics (2000). Principles and
standards for school mathematics. Reston, VA: NCTM.
Ontario Ministry of Education (2005). The Ontario Curriculum: Grades 9
and 10 mathematics. Toronto, ON: Queen’s Printer for Ontario.
Pólya G (1967). Le découverte des mathématiques. París: DUNOD.
Reid D (2002). Conjectures and refutations in grade 5 matematics. J.
Res. Math. Educ. 33(1):5-29.
Resnik MD (2005). Mathematics as a science of mathematics. Oxford:
University Press.
Sutarto Toto N, Subanji S (2015). Indicator of conjecturing process in a
problem solving of the pattern generalization. Proceding International
Conference on Educational Research and Development (ICERD),
UNESA Surabaya. pp. 32-45.
Tikekar VG (2009). Deceptive patterns in mathematics. Int. J. Math.
Sci. Educ. 2(1):13-21.
Wertheimer M (1923). Untersuchungen zur Lehre von der Gestalt: II
(Investigations in Gestalt Theory: II). Psychol. Forschung 4:301-350.
Yerushalmy M (1993). Generalization, induction, and conjecturing: a
theoretical perspective. In Schwartz, J.L., Yerushalmy, M., and
Wilson, B. The geometric supposer: What is it a case of?. Hillsdale,
NJ Lawrence Erlbaum Associates.
Zazkis R, Liljedahl P (2002). Generalization of patterns: the tension
between algebraic thinking and algebraic notation. Educ. Stud. Math.
49:379-402.
