Corresponding Author : Friend Pandapotan Nainggolan
E- mail : [email protected]
INTRODUCTION
The importance of algebra in mathematics
has been expressed by many researchers. Jupri & Drijvers (2014) state that algebra is very important for
achievement in other mathematical domains such as analytic geometry, calculus,
and statistics. Therefore, initial algebra learning, which includes students'
first steps in this domain, is certainly an important phase in algebra learning
(van Amerom, 2002). Research on algebra is often done
through the development of algebraic thinking, where this thinking is needed to
analyze deeper mathematical structures (Kiziltoprak
& Kose, 2017). Similarly, research on early
algebra is often done through the development of early algebraic thinking.
The education curriculum in Indonesia has
accommodated early algebra learning at the junior high school (SMP) level.
Based on 2017 revised 2013 Curriculum, the basic material for initial algebra
is found at the level of 7th-grade junior high school students in
semester 1 which includes algebraic forms and linear equations and inequalities
of one variable. The subject matter of algebraic forms includes: coefficients,
variables, constants, and terms in algebraic form ;
arithmetic operations in algebraic form; and simplification of algebraic forms.
The subject matter of linear equations and inequalities of one variable
includes: statements; open sentence; and solving one-variable linear equations
and one-variable linear inequalities.
In Indonesia, students often experience
difficulties in understanding early algebra, including difficulties related to
the application of arithmetic operations (Jupri &
Drijvers, 2014), students' difficulties in completing
algebraic expressions (Muchoko et al., 2019), and difficulties in solving algebraic
expressions. equation (Jupri, Drijvers,
& Heuvel-panhuizen, 2014). Difficulties related
to the application of arithmetic operations, for example students' errors in
using the nature of arithmetic operations and interpreting the equal sign (Jupri & Drijvers, 2014).
Judging from the characteristics, there is
a difference between arithmetic and algebra. Arithmetic is very close to
calculation, while algebra focuses on relations (Carpenter et al., 2005). For example, in arithmetic, the equals
sign often refers to calculating and writing numerical answers. For example,
when faced with ..., students tend to get a specific answer 5 and do not view
it as or (Jupri & Drijvers,
2014), whereas in algebra, the equal sign indicates a relation or has the
meaning "algebraically equivalent to" (Herscovics & Linchevski, 1994). For example, when writing becomes ( Jupri & Drijvers,
2014).
On the other hand, arithmetic and early
algebra are closely related, especially in the nature of number operations.
However, often students do not understand how the basic nature of number
operations is applied in calculations, so as a result, students do not realize
that arithmetic and algebra are based on the same basic ideas (Carpenter et al., 2005). For example, in arithmetic, number
sentences will be very easy to complete if students subtract 56 from 56 first
instead of doing calculations sequentially from left to right.
For some students, learning algebra is a
natural progression that builds on their understanding of arithmetic (Harbour, Karp, & Lingo, 2017), however, for many
students, learning algebra is completely different from the experience of
learning arithmetic, and they find themselves experiencing a number of
difficulties. in the transition (from arithmetic to algebra) (Cai & Moyer, 2008). The shortcomings resulting from the way
of learning arithmetic during the transition from arithmetic to algebra have an
influence on the development of algebraic thinking (Kiziltoprak
& Kose, 2017). Therefore, the transition period
of students from (thinking) arithmetic to (thinking) algebra becomes a vital
phase in determining student success in mastering algebra well.
Relational thinking is said to be a bridge
between arithmetic thinking and algebraic thinking (Kiziltoprak
& Kose, 2017).
Relational thinking involves the meaning
of the equals sign , the use of the basic properties of operations and number
relationships (Carpenter et al., 2005), and making strategic decisions (Harbour et al., 2017; NCTM, 2017). Strategies that can be
used include "takeaway" or taking 3 out of 100 or taking 41 out of
39. Another strategy, by viewing subtraction as a distance, is by counting
backwards, for example counting from 3 to 100 or from 39 to 41. Students who
choose the strategy " take away� on subtraction and choose a �distance�
strategy to be believed to demonstrate relational thinking (NCTM, 2017).
(Kindrat & Osana, 2018) states that the center of algebraic
reasoning is relational thinking, which involves coordinating quantities in
mathematical expressions, often without calculation, using flexible reasoning
about quantities and converting mathematical expressions into equivalents. For
example, sentence numbers (Kindrat & Osana, 2018). The equal sign allows students to ignore
the 1986 number on each side and only focuses on equations that allow students
to reduce the computational level (Kindrat & Osana, 2018).
The hallmark of relational thinking is not
determined by computation, but rather by mathematical structure and
generalization (Carpenter et al., 2005). To illustrate, when faced with a problem
such as 25 + 17 = 22 + , students who understand the
equals sign meaning "equal to" can solve it either computationally
(i.e., 25 + 17-22) or by relational reasoning, which would require checks the
relationship between the sums on both sides of the equation (ie, "25 is 3 more than 22, so the answer must be 20
because 3 is greater than 17 to balance it."). In this context, relational
thinking involves examining the equation, paying attention to the structural
relationship between 25 on the left side of the equation and 22 on the right
side, and compensating for differences by adjusting the number 17 (Kindrat & Osana, 2018).
Previous research has shown that students
experience serious misconceptions about the meaning of the equal sign (�=�) (Baiduri, 2015) Jupri & Drijvers, 2014). Students perceive the equals sign as an
operator to perform calculations, state results, or as a signal to write down
the next answer (Harbour et al., 2017; Jones &
Pratt, 2012 ; Kiziltoprak
& Kose; 2017). Other research on relational
thinking related to the use of number sentences, either true or false or open
number sentences, has brought much success in developing relational thinking (Banerjee, 2011) (Carpenter et al., 2005). A number of studies have also succeeded
in designing relational thinking-based teaching, including through pre-service
training for teachers (Fisher et al., 2019), as well as several projects related to
the development of the teaching profession in terms of This has also shown
positive implications for the development of students' relational thinking (M. L. Blanton & Kaput, 2005).
A number of researchers have also
categorized relational thinking based on research findings to see students'
level of relational thinking (Kiziltoprak & Kose, 2017; M. Stephens & Wang, 2008). One of them is Kiziltoprak & Kose (2017)
which groups students based on the operation process carried out into three
themes, namely the operation process based on relational thinking, consisting
of sub-themes using the basic properties of arithmetic operations and using
relations between numbers; the process of introducing operations to relational
thinking, consisting of a sub-theme of explaining relations after finding the
unknown and prerelational thinking; and
results-oriented operating processes.
In solving a math problem, the curriculum
encourages students to apply a variety of strategies, but more than just
applying strategies, relational thinking involves making strategic decisions
(NCTM, 2017), students integrate relational thinking strategies with computational
skills so as to prepare students to succeed in advanced mathematics. (Harbour et al, 2017). In relation to problem solving
strategies, (M. Blanton et al., 2019) through their strategic approach, divides
relational thinking into three categories: structural strategies
(compensation), computational strategies, and operational strategies.
Therefore, relational thinking is closely related to problem solving
strategies. mathematics and on this basis this research analyzes mathematical
problem solving strategies and selects a preliminary algebraic topic.
(Baiduri, 2015) has researched relational thinking which
focuses on understanding the equal sign and analyzing students' strategies in
solving equivalence equations through written test analysis. This study
analyzes relational thinking with a wider scope, namely the aspect of using the
relationship between numbers, and applying the basic properties of number
operations. This study also analyzes students' strategies in solving initial
algebraic problems that are more varied (not only solving equivalent
equations), which include several indicators with the topic of algebraic forms
and one-variable linear equations. This research focuses on the analysis of
relational thinking which includes three indicators (interpreting the equal
sign as equality, using the relationship between numbers, and applying the
basic properties of number operations) and analysis of strategies in the form
of ways that students use when solving problems related to shapes. algebra and
linear equations of one variable.
METHOD
This study aims to analyze students' relational thinking and early algebra
problem solving strategies so that with qualitative methods it is hoped that
complete and in-depth analysis results can be obtained. Judging from the
characteristics of qualitative research according to Creswell (2017), this
study uses natural conditions, namely students as direct data sources, and
researchers are key instruments. The data collected tends to be in the form of
words, namely from the results of written tests and student interviews about
the number sentenced and initial algebra, in other words descriptive. This
research also places more emphasis on the students' process of solving any
given problem rather than the final result and tends to analyze the data
inductively.
Data collection was carried out in two stages through two different
methods. The first stage, to obtain initial data about students' relational
thinking, was carried out a written test technique. This written test is given
to students in the form of number sentence questions and initial algebra
questions . So that researchers can see the emergence of
indicators, in each number sentence a column is given to write down the reasons
(justification) for the answers given. The second stage, to validate the
results of the written test, conducted an interview technique. The interview
technique is conducted online (in the network), namely by telephone. This is
because when the interview has entered the COVID-19 pandemic period and the
related school applies distance learning (online) so it is not possible for
researchers to conduct interviews directly (face to face) at school . So that researchers can see the
emergence of indicators, in each number sentence a column is given to write
down the reasons (justification) for the answers given.
RESULTS AND
DISCUSSION
A.
Student's relational thinking in
completing number sentences
Based on the results of the written test and relational thinking
interview for each student/subject, the following are the results of the
research found:
1. Subject S1
In interpreting the equals sign , S1 has understood it as equality or balance. Problem
38+45 = ... + 47 is answered correctly by first adding the number on the left
side of the equals sign (38 and 45), then subtracting it from the number on the
right side (47) as shown in Figure 4.1. This shows that the subject thinks that
the sum of the numbers on the left side must be the same as the sum of the
numbers on the right.
Figure 1. Answer 2.1 by S1
As for the second indicator, using the
relationship between numbers or compensation, it has not been seen in the S1
subject's answer. All questions related to this indicator are solved by means
of computation or ordinary calculations. For example, true or false number
sentence 45+28 = 47+26
Figure 2. Answers 1.5 and 1.6 by S1
From the answers above, subject S1
compares the results of operations on each side of the equal sign to determine
whether the results are the same or not. That is, for these questions, the S1
subject has only just arrived at the understanding of the meaning of the equal
sign as equality, but has not shown relational thinking in the second
indicator.
For the third indicator, using the basic
nature of number operations, subject 1 already has knowledge of the basic
properties of number operations (commutative, associative, and distributive).
Subject 1 understands the commutative nature as the nature of the exchange of
numbers (based on his learning experience in elementary school) and uses it in
solving questions about commutative indicators (questions 1 to problem 3). The
subject believes that the truth value of these statements can be determined
without having to do calculations. That is, subject 1 only performs
calculations as a step to verify answers, not as the main step in completing
number sentences 1.1-1.3 , as shown in Figure 3.
Figure 3 Answers 1.1-1.3 by S1
The written answer above shows that
subject 1 understands that the commutative property applies to the operations
of addition and multiplication of numbers but does not apply to operations of
subtraction of numbers.
Likewise for the distributive nature, the
subject did not complete the number sentence 2.5 to completion (see Figure
4.4). Answer 2.5 shows that subject 1 assumes two unknown numbers as , which means that the subject thinks that the two
numbers must be the same. Even though the dots given mean answers that might be
different and it seems that S1 has not thought of this. This can be seen after
being given instructions in the form of a general formula for the distributive
property ( ) , the subject can solve it by giving one
of the correct answers, namely 7 and 7.
Figure 4.
Answer 2.5 by S1
Thus, the subject of S1 has shown relational
thinking in several indicators, namely interpreting the equal sign and applying
the nature of number operations. The subject is able to interpret the equals
sign as a sign of equality even though he has not been able to express it
through words.
2. Subject S2
For the first indicator, interpreting the
equal sign as equality, the written answer from S2 has shown this understanding
as equality or balance, namely the equality of the results of number operations
on each side of the equals sign. However, when interviewed, S2 only mentioned
the meaning of the equal sign as �result� and had difficulty finding other
meanings in words. In addition, there is a unique answer written by S2, he adds
the numbers on the left side and gets the result 83, then writes 47 + ? = 83. This method is referred to by subject 2 as a
�logic way�, not in a direct way 83 � 47. The final result or answer given by
the subject is correct, but there are errors in writing, 47 +
? = 83 = 36, that 83 should not be equal to 36. The equal sign in this
sentence shows the meaning of the equal sign as a symbol to write the final
answer or a place to write the answer. In this finding, the final answer is
still correct (36), it's just that there is an incorrect use of the equal sign
in the completion process (see Figure 5).
Figure 5 Answer 2.1 by S2
Indicators using the relationship between
numbers or compensation have not been seen in subject 2. Like subject 1, S2
tends to do calculations for all number sentences related to this indicator.
For example in problem 2.3 (see Figure 6). From this answer, implicitly the
subject has applied commutative and associative properties in solving problems.
However, they have not been able to find a simpler way ,
namely by looking at the relationship between the numbers 856 and 857. The
incorrect use of the equal sign is also found in this answer.
Figure 6
Answer 2.3 by S2
In applying the basic nature of number
operations, subject 2 has been able to apply commutative and associative
properties in solving problems, but the subject does not mention these
properties as "commutative" and "associative" properties
because they forget the terms. S2 correctly answered question 1.1 with a
logical reason, "because it has the same number and sign". From the
results of the interview, it means that the numbers are the same, as are the
signs (positive/negative). S2 also added, "If the numbers are the same but
the sign is different, the result will be different". Likewise for problem
1.2, the subject does not do the calculation because he believes that the result of the first operation
of each side will be different, namely between 35-8 and 8-35. S2 includes a logical reason in the
written answer, because if the smaller number is subtracted from the larger
number, it will result in a negative. This implicitly states that the commutative
property does not apply to addition.
The uniqueness of S2 is seen in solving
problems related to associative properties. Problem 1.4 in the form of a true
or false number sentence can only be solved by S2 by performing calculations
according to the order of operations and admitting that they do not know any
other way. However, in question 2.4 which is an open number sentence, the
subject solves it by applying the associative nature (see Figure 7) even
without a good knowledge base, in the sense that the subject does not know that
the method he does is based on the associative nature.
Figure 7 Answer 2.4 by S2
Based on the results of the analysis of each relational thinking
indicator, the master's subject has understood the meaning of the equal sign as
equality, but on the other hand, there are still many writings that indicate
the equal sign as a symbol to write the results or answers that come next.
3. S3 Subject
From the snippet, S3 interprets the equal
sign as "result", meaning a sign that shows the result. In addition,
S3 also interprets it as a "sign of equality", meaning that the value
or result of the operation of both sides of the equals sign is the same, as in
the problem 24+46 = 46+24. That is, S3 calls 24+46 = 46+24 as an equation (should
be equality). Thus, the researcher understands the meaning of this subject 3
that the equal sign indicates the result or indicates an equality.
In general, S3 subjects solve number
sentence problems by calculating . Likewise, the
arguments or reasons for the answers given are dominant in the calculation
reasons, such as the equality of the results of the operation; and some logic,
such as in the number sentence 24 + 49 = 49 + 24 S3 provides the argument
"because if the addition is reversed the same , the
number is still the same" (meaning that if the placement of the numbers is
reversed, the number remains the same). Subjects also did not appear to use
number linkages or compensation in solving questions related to this indicator.
Regarding the application of the basic
nature of number operations, from the results of interviews with question 1.1,
the subject believed that the value of the statement 24 + 49 = 49 + 24 was
correct because it was only reversed by the placement of the numbers, as well
as for multiplication. However, the subject is still unfamiliar with the term
"commutative". Likewise with associative properties, S3 can only
solve problems 102 + 591 � 591 = ... with sequential calculations (Figure 4.8).
The subject said he did not know the basic properties of number operations. The
subject also doubted whether in elementary school had learned about this topic.
Figure 8 Answer 2.4 by S3
An interesting finding from the S3 subject
is that when given an interrogative approach, in solving questions about
indicators of distributive nature, the subject is able to find alternative
answers (besides 7 and 7) for question 2.5, namely 6 and 8. The subject argues
because the number is 14.
4. Subject S4
When the subject S4 was asked by the
researcher what the meaning of the equals sign meant, he answered
"comparison" or "result". The purpose of comparison
according to subject 4, namely when compared the results between the right and
left sides of the equal sign must be the same. From this answer, even though
the word choice is not quite right, S4 has shown a fairly good understanding of
the meaning of the equal sign as equality.
Further analysis related to answer 1.5 by
S4 will be discussed in the discussion section (analysis per indicator).
Likewise for problem 2.3, the subject can see the relationship between the
numbers 856 and 857. However, S4 has not been able to apply compensation for
questions with subtraction operations. Subjects were only able to perform
ordinary calculations for the other two questions.
One of the problems regarding the application of
associative properties is solved by S4 in two ways ,
namely by applying associative properties and sequential calculations (Figure
9). For the application of distributive properties, subject S4 is able to find
the right solution by calculating and confirming the correctness of the answer
by replacing (substituting) the empty part with the solution obtained.
Figure 9
Answer 2.4 by S4
Thus, S4 has shown the ability of relational thinking
in interpreting the equal sign as equality and is also able to use the
relationship between numbers or compensation (for addition), but the subject
has not shown a good understanding of the properties of number operations even
though he has been able to apply commutative and associative properties. .
5. Subject 5
Seeing this written answer from S5, the
researcher gets the first impression as a result of work that tends to be
"original" (see Figure 4.10 left). However, after being dug through
interviews, several unique findings were obtained from this S5. The subject is
able to mention some basic properties of number operations such as commutative
and associative in his argument, but has not shown the level of understanding.
For example, for a number sentence with commutative indicators, the subject is
able to mention the commutative reason which means exchange. However, when
asked to explain further what he meant , the subject
failed to provide an explanation and seemed still confused. That is, S5 has
knowledge of the basic nature of number operations but understands them.
In the open number sentence 38+45 = ... +
47, the written answer is wrong, which is 38. The argument presented by S5
against this answer is incorrect and tends to be made up ..
When explored through interviews, the reason is "using the distribution
formula, associative". But again, the S5 has not succeeded in explaining
to the researchers what it means. On the other hand, in interpreting the equals
sign , S5 already understands and interprets it as
equality. However, the second indicator of relational thinking, compensation,
has not been seen in S5.
Figure 10 Answers 1.1, 1.2, and 2.3 by S5
Some of
the argument columns in the number sentence looked empty, when confirmed, S5
had many questions that he forgot how to get the answer or solution he wrote.
The unique findings of S5 are found in the solution of question 2.5, with
indicators of distributive properties. S5 got the right answer, namely 10 and
4. When asked the reason, it was because 10 was the tens and 4 was the unit.
Maybe this answer is influenced by previous learning experiences related to
place value on numbers in elementary school.
6. Subject 6
From the results of the interview with S6,
it was obtained information that S6 interpreted the equal sign as an equality
of results or balance. For number sentences related to indicators using
interrelationships between numbers or compensation, S6 is able to solve all
questions with correct answers by means of ordinary calculations or
calculations. The subject cannot resolve by means or other possible arguments
to avoid computation. For example, for question 1.5, the following are the
results of an interview with S6.
Likewise, for number sentences related to
indicators using the basic nature of number operations, all problems are solved
by dominantly result- equivalent-oriented calculations (Figure 4.11) and some
logic. Thus, in general, S6 has shown a good ability to interpret the equals sign , but other indicators have not been seen. The subject
of S6 is still very dominant in doing calculations in solving problems.
Figure 11 Answers 1.5-1.6 by S6
From the results of data analysis on the
results of written tests and interviews of all subjects that have been
described, in general the relational thinking of each subject as seen from each
indicator can be summarized as shown in Table 4.1. The ST code indicates that
the relational thinking indicator has been shown, the BT code indicates that
the relational thinking indicator has not yet emerged, while the MT code
indicates an introduction to relational thinking, namely subjects who (i) are able to interpret the equal sign as equality but
there is still a tendency to use the equal sign as a symbol to write the next
answer, (ii) able to use number relations or apply the basic nature of number
operations but not based on proper knowledge or subject capable of avoiding or
simplifying computations by using logical reasoning.
Table 4.1 Relational Thinking Subject per
Indicator
|
INDICATOR
|
NUMBER QUESTION
|
RELATIONAL
THINKING
|
|
S1
|
S2
|
S3
|
S4
|
S5
|
S6
|
|
1. Meaning sign "="
as equality
|
2.1
|
ST
|
MT
|
MT
|
ST
|
MT
|
ST
|
|
2. Using linkage between numbers or compensation
|
1.5
|
BT
|
BT
|
BT
|
ST
|
BT
|
BT
|
|
1.6
|
BT
|
BT
|
BT
|
BT
|
BT
|
BT
|
|
2.2
|
BT
|
BT
|
BT
|
BT
|
BT
|
BT
|
|
2.3
|
BT
|
BT
|
BT
|
ST
|
BT
|
BT
|
|
3. Using nature operation number
|
1.1 commutative
summation
|
ST
|
MT
|
MT
|
BT
|
BT
|
MT
|
|
1.2 no commutative
subtraction
|
ST
|
MT
|
BT
|
BT
|
BT
|
MT
|
|
1.3 commutative and
associative
multiplication
|
ST
|
MT
|
BT
|
BT
|
BT
|
BT
|
|
1.4 associative
summation
|
BT
|
MT
|
BT
|
BT
|
BT
|
BT
|
|
2.4 associative
summation
|
BT
|
MT
|
BT
|
MT
|
BT
|
BT
|
|
2.5 distributive
|
BT
|
MT
|
MT
|
BT
|
BT
|
BT
|
|
2.6 distributive
|
BT
|
BT
|
BT
|
BT
|
BT
|
BT
|
B.
Students' strategies in solving initial
algebra problems
The following will describe the analysis
of student responses in solving initial algebra problems, both from the results
of written tests and interview results.
1.
Subject S1
In writing, S1 was able to solve the first
four questions of the six algebra questions given. The first problem with
indicators using the properties of operations (commutative, associative, and
distributive) in algebraic form has not been able to be solved correctly on all
related problems. Problem 1.1 is answered correctly and accompanied by the
right argument, namely that 2 = 2 is true because the commutative property of multiplication applies. As for the
completion of sections 1.2 and 1.3, several errors were found in understanding
algebraic expressions, such as + 5 = 5 , (3 + ) = 3 . It can be concluded that the strategy used by S1 in the first question
is to apply the properties of operations to algebraic form. Unfortunately, S1
does not yet have a good understanding of some algebraic expressions as shown
in Figure 4.12.
Figure 12 Answers to the Algebra Test no.1
by S1
The last two questions about solving problems related to linear
equations of one variable were not answered in writing by S1.
2.
Subject S2
In solving the first problem about
algebraic forms, S2 tends to focus on the given operation. In the first part,
the subject did not understand the meaning of " " and " " . The second part was said by S2,
"The method is the same, both are multiplied, the result will be the
same". The subject thought that brackets in algebraic form always meant
multiplication. In this problem, the brackets only show the grouping. Likewise
for the third part, according to S2 because the left side has multiplication,
the right side is addition, the subject concludes "the method is different , the results are different". operations that
apply to him.
Picture 13 Answer Algebra Test number 1 by Subject 2
As for the second question, from the written answer the subject
answered "different" but not accompanied by a clear reason. When
confirmed through the interview, the subject sounded confused, he tried to
rework so that he got 3 + 7 3 + 3 = 15
3 + 3 , then 3 + 7 = 15. At
this point, the researcher tried to give direction whether this equation is the
same with equation (i) and he answered the same. Finally , the subject concludes that in both equations has the same value . Thus ,
S2 quite understands the concept of balance in algebraic equations, but
procedurally it seems that he has not mastered how to solve equations well.
This is supported by written data for questions about finding variable values.
The correct answer from S2 is only one of three parts and this correct answer
is obtained from an inaccurate process (Figure 14).
Figure 14 Answers Algebra 4.a by Subject 2
Figure 4.14 shows that the final answer
written by S2 is correct, but there is a wrong process, namely when S2 tries to
simplify the algebraic form by subtracting 3 on both sides. This is supported
by the next answer to the problem of finding the value of from equation 2 3 = 15 as shown in Figure 4.15. The subject tries to find
a solution to the equation by trying to replace the variable with a number
(substitution).
Figure 15 Algebra 4b by Subject 2
The last two problems were not answered by
S2 on the grounds that there was not enough time. However, from the results of
the interview, the subject was able to understand the meaning of questions 5
and 6 and was able to make an algebraic model for the length of the rectangle
at number 5 with a little direction from the researcher. For the last question,
the subject can also give verbal and relatively spontaneous answers by using
the "guess and check" strategy.
DISCUSSION
1. Student's relational thinking in
completing number sentences
Based on the findings that have been described in the
previous section, the relational thinking analysis of the subject seen from
each indicator is as follows.
a.
Define the equals sign (=)
When dug deeper into each subject by
referring to the results of the written test (especially question 2.1), all
these words tend to refer to the meaning of the equal sign as equality or
balance. There is no subject who writes the answer 83 in the number sentence
38+45 = ... +47 as the meaning of the equal sign as arithmetic-specific and
non-relational as stated by (Pang & Kim, 2018). The word "result" uttered by
the subject turned out to be not completely meaningful as stated by (Warren, 2006), namely as a sign to find results or as a
symbol before writing an answer. Because the subject also understands the equal
sign by showing the same result between the results of the operation to the
right of the equal sign and the result of the operation to the left of the
equal sign. This is also supported by the correct answer in the sentence number
2.1. On the other hand, there are findings in the written answers of several
subjects that indicate the meaning of the equal sign as a symbol used before
writing the answer or writing the next answer, for example answers 2.1-2.3 by
S2 as shown in Figure 4.21.
Figure 21 Answers 2.1-2.3 by S2
The answer above shows that the subject of S2 has implicitly understood
the meaning of the equal sign as equality, but there is still a tendency to use
the equal sign as a symbol to write the next answer. Another example is answer
1.2 of S3 (Figure 4.22).
Figure 4.22 Answer 1.2 by S3
In general, the appearance of this
indicator tends to be uniform in all subjects. This finding may be partly due
to the maturity factor of the subject who is already in grade 7 of junior high
school, while the majority of similar studies on this indicator were conducted
on elementary school age , grade four to grade 6. For
example, research by Falkner et al. (1999); Kieran (1981); and (Molina & Ambrose, 2006) who showed that the majority of 6th grade
elementary school students had difficulty finding answers to non-canonical
expressions such as 8+4=...+5.
b.
Using the relationship between numbers or compensation
In general, the research data indicate
that almost all subjects have not been able to use the relationship between
numbers or make compensation in completing number sentences
From the written data, all subjects used
the concept of the similarity of results or the concept of the balance of the
equal sign . However, from the interview data, it was
found that in addition to the usual computational methods on the written test,
S4 subjects were able to use the relationship between numbers or compensate as
another way of solving problems 1.5 and 2.3. For example, for a number sentence
1.5, 45+28=47+26, the subject explains, "The first one is 45 plus 28, the
second is 47 plus 26. The 28 is, the 2 are 45-in, so 47 adds 26 , that's the
same." However, the researcher tried to visualize the statement of subject
4 as follows:
This way of thinking of subject 4
implicitly uses the relationship between the numbers 28 and 26, i.e. 28 is 2
more than 26, which is then used to change the similarity of numbers so that
the same form is obtained with the number to the right of the equals sign.
According to (Carpenter et al., 2005), a student who considers an expression or
equation as a whole may notice that one number is more or less than another
number.
Some subjects believe that there is no
other way to solve the problem, this is in line with the findings of Kiziltoprak & Kose (2017),
namely that there are students who think they will not be able to find the
numbers to be written in the related boxes ( dots) without doing calculations .
The reason for this situation is considered to be the fact that students learn
arithmetic on a result-oriented basis and that they focus on calculations
rather than on the relationship between numbers and operations (Kiziltoprak & Kose, 2017).
c.
Apply nature base number operation
Students' learning experiences also seem to have a great influence
on this indicator, such as the subject of S1 whose appearance of the relational
thinking indicator is most visible in this indicator, he has existing knowledge
about the basic properties of number operations which he obtained in elementary
school.
In general, the findings of the written test and interview data on
these third indicators can be categorized into four categories. First,
relational thinking, which is the subject of applying the basic nature of number
operations with sufficient knowledge about the basic nature of number
operations. Second, the subject applies the nature of number operations without
sufficient knowledge or the subject is able to avoid doing calculations based
on logical reasoning. Third, the subject knows the nature of number operations
but has not been able to apply it in solving problems. Fourth, the subject does
not know so that he is unable to apply it in solving the problem. This
categorization briefly can be seen in table 4.3.
Table 4.3
Categorization of Indicators Applying Basic Properties of Number
Operations
|
Knowledge
- Application
|
Know
|
Not Know
|
|
Capable apply
|
I
|
II
|
|
Not able to apply
|
III
|
IV
|
In relation to the data in Table 4.1, category I (knows and is
able to apply) is included in relational thinking (ST), category II (able to
apply, but does not know) and category III (knows but cannot apply) is included
in the introduction to relational thinking (MT), while category IV (don't know
and don't apply) is included in the unseen relational thinking. The basis for
this grouping is the various theoretical frameworks that discuss relational
thinking that leads to the selection of strategies and how these strategies are
executed (Baiduri, 2015) (Harbour et al., 2016), but with the right foundation of conceptual knowledge. rather
than just descriptive knowledge or procedural thinking (�Equality Relation and Structural Properties,� 2010). This is because the goal of relational thinking is an
understanding of why a method or strategy may or may not be implemented or
applied and this requires an understanding of the nature of number operations (Carpenter et al., 2005).
Therefore, categories II and III cannot be categorized into relational
thinking, but because they can be a good foundation for the development of
relational thinking, in this study, they are classified as an introduction to
relational thinking.
From Table 4.1, for the three indicators, it can be seen from 42
items that only 3 items are indicated as relational thinking (ST), 11 items are
included in the category of introduction to relational thinking (MT), and the
remaining 28 items have not shown relational thinking. The three items of
relational thinking that appear are limited to the application of commutative
and associative properties, while for number sentences with the application of
distributive properties, all subjects tend to do calculations. Open number
sentences 14 � 28 = . . .� 28 + . . . � 28 in this study tends to produce only
answers 7 and 7 and students generally cannot recognize the form of the
question as ( + ) � = ( � ) + ( � ) so
they cannot find other answers. This is
in line with Kiziltoprak's research (2017) which
found that the biggest difficulty of students was in applying the distributive
nature.� have also conducted research on
relational thinking that focuses on the distributive nature.
This has been revealed by Kiziltoprak
& Kose (2017), namely students are not familiar
with mathematical expressions such as number sentences that involve operations
on both sides of the equation due to the limited number of student books that
provide such number sentences. In fact, completing number sentences can improve
relational thinking (Carpenter et al., 2005).
Furthermore, (Banerjee, 2011) �concluded that arithmetic with number
sentences is very useful for bringing students to algebra.
Based on previous studies, the dominant factor influencing the
development of students' relational thinking is the teacher's learning design (Carpenter et al., 2005).
(Carpenter et al., 2005)
used the term scaffolding to refer to the way teachers develop students'
relational thinking framework, while Kiziltoprak
& Kose (2017) mention the term teaching
interrogative approach to refer to questions that lead students to develop
their relational thinking. Thus, due to the importance of teacher-made teaching
designs, many studies recommend training for teachers to implement relational
thinking-based teaching. This can be in the form of web-based professional
development programs and in-service training (Kiziltoprak
& Kose, 2017), or pre-service training for
teachers to develop relational thinking-based teaching (Fisher et al., 2019).
Several projects related to the development of the teaching
profession in this regard have also been carried out (M. L. Blanton & Kaput, 2005).
Students taught by participating teachers showed significantly better
understanding of the equals sign and used relational thinking during interviews
than students taught by non-participating teachers (Jacobs et al., 2007).
2.
Strategy Solution
Problem Algebra Beginning
a.
Initial
algebra problems related to the application of the properties of operations in
algebraic form (questions 1 and 3)
In determining the truth value of 2 = 2
, all subjects were able to determine that the expression was true by just
looking at its shape, but only a few of them were based on the right reasons.
Among them there are also those who still misunderstand simple algebraic forms
such as saying that 2 times is denoted by squared ( S5 and S6 ). Only S1 shows a good understanding of the application of
this commutative property. For the application of the associative property, 3 + (
+ 5)
= (3 + ) + 5, S2 and
S3 answered correctly that the statement was true, while the other
subjects stated that the statement was false. For example, S2 states that
"this statement is true because the method is the same ,
both are multiplied". That is, S2 understands that brackets "()"
in algebraic form always mean multiplication and he ignores the existing
addition sign so that he concludes so. It seems that the associative nature of
this algebraic operation is still "foreign" for all subjects.
As for the application of the distributive property in algebraic
operations, 3(4 � 5 ) + 7 = 3.4 � 5 + 7 , the majority
of subjects have actually known this property by calling it
"rainbow times", but not all subjects recognize that this statement
involves the application of those properties. Only S3 answered correctly
accompanied by the right process even though it still included procedural
knowledge, in the sense that S3 did not really understand this as part of the
properties of operations on algebraic forms, but only carried out a series of
procedures based on previous learning experiences.
Uniquely, although they both involve the application of
operational properties in algebraic form, many errors are found in questions in
the form of true and false statements, while for algebra questions in the form
of determining variable values from equations, some subjects tend to be able to
solve equations 4a and 4b. For example, the subject of S5, the answers to
questions 1a-1c showed a lack of understanding, but in question 4 he was able to
solve it as shown in Figure 23. Through the interview, it was found that S5
solved this algebraic question procedurally because he only did a series of
procedures without understanding the concept, in this case the meaning of the
equals sign and the application of the properties of algebraic operations.
Figure 23 Answers to Algebra 4a-4c by S5
b.
Initial
algebra questions related to the meaning of the equals sign in algebraic form
(questions 2 and 4)
Solve and compare seems to be the dominant strategy chosen by
students in solving problem number 2. The subject solves the equation one by
one, namely looking for the value of in equation 3 + 7
= 15 and the value of in equation
3 + 7 � 3 = 15 � 3 , then compare the values. None of the subjects was
able to see the relationship between these two equations as equivalent. As for
solving problem number 3, only S6 is able to say mathematical sentences that
represent the problem as 5 + 4 = +
7 but verbally (from interviews). Meanwhile, S1 and S3 wrote it in
one sentence as 7+2=5+4 (S1) and 5+4=9=7+2=9 (S3) while S4 and S5 wrote it in
two different sentences (see table 4.2 indicator column third). All these
sentences do not yet indicate there is a problem to be solved. The meaning of
the equal sign which means the symbol for writing the answer is still visible
on the subject of S2 who wrote it as 2=7+2=9.
c.
Problems
related to solving problems related to linear equations of one variable
(problems 5 and 6)
Almost all subjects had difficulty in solving the fifth and sixth
questions in the form of word problems with linear equations of one variable so
that not many subjects reached the stage of determining and implementing
strategies. However, both of them made a mistake in choosing the formula,
namely length times width (the formula for the area of a rectangle) as the
formula for the perimeter of a rectangle. In understanding the problem, S3 also
made a mistake, namely understanding that 5 cm is the width of a rectangle
(Figure 4.24). The S4 has departed from the correct understanding and is able
to understand the relationship between two elements, namely the length and
width of a rectangle. Unfortunately, there is a wrong process in executing the
strategy, namely an error in applying the formula. This error leads to an
incorrect final answer.
Likewise for the sixth question, only S3 and S4 were able to
independently understand the relationship between the weight of the object on
each arm of the scale and were able to solve this problem using a guess and
check strategy. Thus, the ability of S3 and S4 to view the problem from a
structural perspective has been seen in solving this last problem. As for the
other subjects, admitted difficulties in understanding the problems given so
that they could not carry out the next process, namely choosing and implementing
strategies. Thus, judging from the problem-solving strategy, it does not appear
that there are subjects who solve this problem with strategies that involve
formal algebra such as modeling equations and solving them to obtain solutions.
Figure 24
Answers to Algebra number 5 by S3
Based on the categorization of problem solving strategies by Hejn�, Jirotkov� & Kratochvilov� (2006), the majority of strategies used by
students in solving problems about early algebra tend to be included in
procedural strategies, because the subject does not involve mathematical
structures such as linkages or relationships between elements or application of
the nature of the operation.
If it is associated with the results of the analysis of students'
relational thinking, it appears that there are some linkages with early algebra
problem solving strategies. The strategy used, namely solve and compare, has a
tendency to be similar when the subject completes the number sentence by
looking at the equality of results. The S3 and S5 subjects who still have a
tendency to interpret the equal sign as a symbol to write answers, have not
been able to understand the problem given in question 2.
At the completion of the initial algebra test, questions number 5
and 6 which implicitly understand the problem require an understanding of the
relationship of numbers. For example, in understanding the statement that the
length of the rectangle is 5 cm more than the width, this shows the
relationship between numbers. In relation to this indicator, S4 is able to
understand the problem in question number 5 (but incorrectly applies the
formula) also understands and solves problem number 6. Other subjects who have
not been able to use the relationship between numbers in completing number
sentences tend to have difficulty or misunderstand (eg.
S3 on question 5) questions 5 and 6 on the initial algebra test.
When viewed from the strategy carried out, these three categories
are included in the category of procedural strategies (Hejn�,
Jirotkov� & Kratochvilov�,
2006; Kieran, 2007) because they do not involve an understanding of the
mathematical structure. Only the subject of S1 shows relational thinking
(structural strategy) about commutative properties and this strategy is also
seen in algebra question 1.a, namely by understanding the commutative nature of
algebraic forms. Thus, it is suspected that an understanding of the properties
of operations on whole numbers seems to help students understand the properties
of operations in algebraic forms, because basically arithmetic and algebra come
from the same basic idea (Carpenter et al., 2005).
This understanding will encourage students to apply structural strategies in
solving algebraic problems. This is in line with the results of previous
studies that encourage relational thinking is very important because it gives
meaning to arithmetic, leads to a conceptual understanding of numbers and the
nature of numbers, and improves students' thinking about mathematical
generalizations (Jacobs et al, 2007; Molina, Castro, & Mason, 2006;
Stephens, 2007).
CONCLUSION
Based on the
results of research conducted on six grade VII junior high school students in
Bandung, students' relational thinking when completing number sentences only
applies to several subjects for each relational thinking indicator, namely the
first 3 indicator items: each item is seen in S1, S4, and S6; The second 2
indicator items are visible on the S4; and the third 3 indicator items are seen
in S1. In interpreting the equal sign , all subjects
were able to interpret it as equality, but in some subjects there was still a
tendency to write the equal sign as a symbol to write the next result or
answer. In using number relationships and applying the basic nature of number
operations, the majority of subjects do computations dominantly and/or apply
procedural knowledge-based operations. Similarly, in solving the initial
algebraic problem about algebraic forms and PLSV, it was found that the
dominant tendency of students to consider algebraic expressions from a
procedural perspective rather than a structural perspective (based on
mathematical concepts or structures). Thus, if students are given an initial
algebra problem on the topic of algebraic forms and PLSV, then students tend to
solve the problem from a procedural perspective.
REFERENCES
Baiduri, B. (2015). Mathematics education students�
understanding of equal sign and equivalent equation. Asian Social Science,
11(25), 15�24. Google Scholar
Banerjee, R. (2011). Is Arithmetic Useful for the Teaching
and Learning of Algebra? Contemporary Education Dialogue, 8(2),
137�159. Google Scholar
Blanton, M. L., & Kaput, J. J. (2005). Characterizing a
classroom practice that promotes algebraic reasoning. Journal for Research
in Mathematics Education, 36(5), 412�446. Google Scholar
Blanton, M., Stroud, R., Stephens, A., Gardiner, A. M.,
Stylianou, D. A., Knuth, E., Isler-Baykal, I., & Strachota, S. (2019). Does
early algebra matter? The effectiveness of an early algebra intervention in
grades 3 to 5. American Educational Research Journal, 56(5),
1930�1972. Google Scholar
Cai, J., & Moyer, J. (2008). Developing algebraic thinking
in earlier grades: Some insights from international comparative studies. Algebra
and Algebraic Thinking in School Mathematics, 70, 169�182. Google Scholar
Carpenter, T. P., Levi, L., Franke, M. L., & Zeringue, J.
K. (2005). Algebra in elementary school: Developing relational thinking. Zentralblatt
F�r Didaktik Der Mathematik, 37(1), 53�59. Google Scholar
Equality Relation and Structural Properties. (2010). Proceedings
of CERME, 6. Google Scholar
Fisher, M. H., Thomas, J., Jong, C., Schack, E. O., &
Dueber, D. (2019). Comparing preservice teachers� professional noticing skills
in elementary mathematics classrooms. School Science and Mathematics, 119(3),
142�149. Google Scholar
Harbour, K. E., Karp, K. S., & Lingo, A. S. (2016).
Inquiry to action: Diagnosing and addressing students� relational thinking
about the equal sign. Teaching Exceptional Children, 49(2),
126�133. Google Scholar
Herscovics, N., & Linchevski, L. (1994). A cognitive gap
between arithmetic and algebra. Educational Studies in Mathematics, 27(1),
59�78. Google Scholar
Kindrat, A. N., & Osana, H. P. (2018). The relationship
between mental computation and relational thinking in the seventh grade. Fields
Mathematics Education Journal, 3(1), 1�22. Google Scholar
Molina, M., & Ambrose, R. C. (2006). Fostering relational
thinking while negotiating the meaning of the equals sign. Teaching Children
Mathematics, 13(2), 111�117. Google Scholar
Muchoko, C., Jupri, A., & Prabawanto, S. (2019).
Algebraic visualization difficulties of students in junior high school. Journal
of Physics: Conference Series, 1157(3), 32108. Google Scholar
Pang, J., & Kim, J. (2018). Characteristics of Korean
students� early algebraic thinking: A generalized arithmetic perspective. In Teaching
and learning algebraic thinking with 5-to 12-year-olds (pp. 141�165).
Springer. Google Scholar
van Amerom, B. A. (2002). Reinvention of early algebra:
Developmental research on the transition from arithmetic to algebra. Google Scholar
Warren, E. (2006). Comparative mathematical language in the
elementary school: A longitudinal study. Educational Studies in Mathematics,
62(2), 169�189. Google Scholar
