Transition Process of Procedural to Conceptual Understanding in Solving Mathematical Problems

This article aims to describe the transition process from procedural understanding to conceptual understanding in solving mathematical problems. Subjects in this study were three students from 20 fifth grade students of SDN 01 Sumberberas Banyuwangi selected based on the results of the students’ answers. The transition process from procedural to conceptually based on three aspects: (1) identify problems in the use of an algorithm, (2) the process algorithm, (3) connect multiple concepts to transform into another shape through the symbolic/picture representations. The results showed that the majority of students (18 students out of 20 students) only meets two (2) aspects of 10 students (50%) can identify the algorithms and the use of algorithms, 8 students (40%) able to use algorithms and connect with other forms. While other students (two students from 20 students) of 10% that meet only three aspects of the transitions. Thus, understanding the procedural has an important role in developing a conceptual understanding. Because the component/aspect of procedural understanding exist on components/aspects of conceptual understanding. Thus, the association acquired several components/aspects that support the process of transition from procedural understanding to a conceptual understanding. Keyworsd: procedural understanding, conceptual understanding, mathematical problems


Background
Mathematical is also a form of science that are arranged in a systematic, logical and structured.Each student must learn mathematics with understanding (Marchionda, 2006;Isleyen & Isik, 2003).Variety of mathematical concepts can be understood only after the students have acquired skills in using a procedural concept which then led to a better understanding (Rittle- Johnson et al., 2001).The skills to use the procedure is necessary so that students do not encounter obstacles and learning objectives in class right on target, especially in learning mathematics (Rahaimah & Noraini, 2013;Mousley, 2004).Some studies show that success in solving mathematical problems is supported by ideas that allow a deep understanding (Stylianou, 2002;Parkinson & Redmond, 2002).In solving the problem, students need the support of a strategy that will govern the interpretation and manipulation of information in analyzing and interpreting for the selection procedures and making the right decisions with the ability to think (Johnson, 2010;Lager, 2006;Geary, 2004).At a deep level of understanding of the students began to be able to make the connection between mathematical ideas and make a generalization of the concept armed with the basic capabilities already understands (Potter & Kustra, 2012).
There was a significant relationship between procedural and conceptual understanding of mathematics (Hutkemri, 2010;Hiebert & Lefevre, 1986).procedural and conceptual understanding is also important in school mathematics, especially in problem solving (Krulik & Rudnick, 1996;NWREL, 2000).Thus, procedural understanding can assist conceptual understanding in mathematics (Lim, 2002).
Procedural understanding is the ability to solve problems by using a procedure that is efficient, accurate, true to add, subtract, multiply and divide (Askew, 2012, p. 55;Kilpatrick et al., 2001, p. 18).While, conceptual understanding is the ability to classify objects that make up the concept, applying the concept in the algorithm, presents the concept in various forms of mathematical representation, and linking the various concepts (Effandi & Norliza, 2009;Romero & Mari, 2006;Chappell & Killpatrick, 2003).In addition, a conceptual understanding of knowledge that involves a thorough understanding of the basic concepts in mathematics an algorithm (Marchionda, 2006;Angel, 2007).Some aspects of your procedural understanding: ability to reason through a situation, ability to use the algorithm in a problem situation, ability to perform noncomputational skills such as rounding and ordering, ability to verify the truth of the procedure, and select and implement the appropriate procedures (National Assesment of Educational Progress, 2003).Kilpatrick et al. (2001, p. 18) suggests three aspects of the conceptual understanding that comprehension of mathematical concepts, operations, and relations.Comprehension of mathematical concepts can be done by identifying and using appropriate concepts to solve problems.Operations can be known through the use of shape and determine the results of operations that fit correctly.Relations can be done through a representation (picture or symbol numeric).Thus, the process of transition from procedural understanding to conceptual understanding of using the three aspects, which can identify problems in the use of an algorithm, the process an algorithm, connecting some of the concepts to transform into another shape through the representation of a symbol or picture.These three aspects are believed to represent the students 'understanding of the ability of a procedural and heading ability of the students' understanding conceptually.

Objectives of the Study
Procedural understanding has important aspects of the conceptual understanding of learning mathematics, the purpose of this study is to describe the process of transition from procedural understanding to conceptual understanding of of solving mathematical problems.

The Participants
Subjects in this study were three students from 20 fifth grade students of SDN 01 Sumberberas Banyuwangi selected based on the results of the students' answers.The transition process from procedural to conceptually based on three aspects: (1) identify problems in the use of an algorithm, (2) the process algorithm, (3) connect multiple concepts to transform into another shape through the symbolic/picture representations.Based on these criteria, selected 1 (one) student each of the criteria be used.It is considered be representative of aspects the transition procedural to conceptual understanding.

Instruments
The instrument used in this study is all about tests and structured interviews.Test questions about the description given in the form that is tailored to a predefined aspect.While structured interviews necessary to dig or search for more in-depth understanding of the transition process prosesural to students' conceptual understanding.So in this way will be able to help and obtain complete information in analyzing the results.
Instrument test items as follows.
Adi berada 13 meter di kiri tiang bendera.Ati berada 6 meter di kanan Adi.Sedangkan Popy berada 9 meter di kiri Ati.Jika posisi tiang bendera dianggap titik nol, berapa meterkah jarak Popy dari tiang bendera?Berikan penjelasan dari jawaban anda!Translate in English: Adi is located 13 meters to the left of the flagpole.Ati is 6 meters on the right Adi.While Popy is 9 meters to the left Ati.If the flag pole position is considered the zero point, how many meters distance Popy from the flagpole?Please provide an explanation of your answers!Table 1.Components of trancitions the procedural-conceptual understanding

Komponen Transisi
Indikator/Sub-Component Identify problems in the use of an algorithm Finding a logical basis for using an algorithm or specific operations.
Finding a logical basis for applying an algorithm to solve the problems The process algorithm Verify the use of algorithms/computing Selecting the use of algorithms/computational precise and correct Connect multiple concepts to transform into another shape through the symbolic/picture representations Shows the relationship/linkages with some of the concepts in the process an algorithm to solve the problem Shows interpretation through other forms of representation to solve the problem correctly

Procedure
Problems test was given to all participants (20 students) who voluntarily agreed to participate in solving a given problem.Before conducting the test, students were given instructions/rules read by researchers, are: (1) students are given 30 minutes to solve the problem; (2) students are not allowed to collect the answer before the appointed time finish; and (3) students are not allowed to ask or see the results of his friend work.After completing the test, researchers examined the results and conduct a structured interview to students who meet the specified three aspects.The interviews were conducted one day after the test is performed.

The Data Analysis
Data is obtained from answers to students who meet the three aspects of the transition, are identify problems in the use of an algorithm, the process algorithm, and connect multiple concepts to transform into another shape through the symbolic/picture representations.Data were analyzed quantitatively and qualitatively.Quantitative data to show the percentage of each aspect of the transition.While, qualitative data to describe the process of transition based on the specified aspect.The instructions coding used in qualitative data are shown in Table 2 below.

Results Analysis Data
The results of the data obtained were analyzed quantitatively and qualitatively.Analysis of quantitative data obtained to show the percentage of each aspect of the transition.While the analysis of qualitative data to describe the process of transition based on the specified aspect.

Discussion
The results showed that there were 10 students (50%) can identify problems and use of algorithms with indicators of the transition is to find a logical basis for the use of an algorithm (certain operations) and apply the an algorithm in solving the problem, select the use of algorithms/computational precise and correct; 8 students (40%) able to use algorithms and connect it to any other form (representation) with indicators of the transition is to choose the use of an algorithm/computation is accurate and correct, shows the relationship/linkages several concepts to the algorithm in the process of solving problems, and show the interpretation through the representations to other forms in solving the problem in correctly.Moreover, very few students can identify the problem, the use of an algorithm, and connect multiple concepts into another form (representation) is as much as 2 students (10%) with indicators of the transition is to find a logical basis for using the algorithm (specific operation), select use an algorithm/computation is accurate and correct, and shows the interpretation through the other forms of representation to solve the problem in correctly.
The findings of this study showed that most of the students understanding of the procedural and little understanding conceptually students.The learning activities are the most important thing is to build understanding (procedural and conceptual) in the classroom (Mousley, 2004).This is because the understanding of mathematical concepts is an important part in the process of learning mathematics (Hasnida & Zakaria, 2011).According to Potter and Kustra (2012), at the level of deep understanding of the students began to be able to make the connection between mathematical ideas.The linkage between mathematical ideas is one component in the conceptual understanding.
According to Wisconsin (2007), with a deep conceptual understanding of students who can give meaning to mathematics through the ability to apply their knowledge.Developing procedural knowledge have a positive effect on conceptual understanding (Rittle- Johnson et al., 2001;Johann et al., 2005).Furthermore Mary and Heather (2006) argue for successfully resolve the problem, students must develop a better understanding of the procedural and conceptual.With a conceptual understanding, students see the relationship between concepts and procedures that can provide arguments to explain some facts.In addition, the conceptual understanding help students avoid many critical errors in problem solving.More specifically stated NCTM (2000,20) that it when students gain conceptual understanding the specific mathematical topics, the students can see connections between concepts and procedures that can give reasons to explain some facts that are a consequence of other facts.
Other findings from the discussion explained that if students learn a procedure without understanding will require extensive training so that steps procedure can be performed easily and correctly.Thus, the use of the procedure can strengthen and develop conceptual understanding.Components transition from procedural understanding to conceptual understanding able to help the teacher comprehend the thinking of students in depth and know the relationship is strong enough to complement each other.So with some procedural transmission components such conceptual to identify problems in the use of an algorithm, the use of an algorithm, and connect with other forms (of representation) able to facilitate teachers in the learning process that refers to a conceptual understanding, although the level of students' understanding of different.

Conclusions
From the discussion described, it could be concluded that it procedural understanding has an important role in developing a conceptual understanding.This is due to the component/aspect of procedural understanding exist on components/aspects of conceptual understanding.Thus, from this association gained some components/aspects that support the process of transition from procedural understanding to conceptual understanding.Components/aspects of the transition in question are (1) identify problems in the use of algorithms with the indicator are find a logical basis for the use of an algorithm (certain operations) and apply an algorithm to solve problems; (2) process an algorithm with the indicator are verifies the use of algorithms/computational and choose the use of an algorithm/computation is accurate and correct; (3) connecting some of the concepts to another (representation) with the indicator are shows the relationship/linkages several concepts to the algorithm in the process of problem solving and interpretation through the representation to show another form in correctly.

Figure
Figure 1.S1 h has not provid f the procedure e flag pole ?.A negative value words, the use o cedure used b ion and use o Such as, Adi a he final result Thus, S1 is said Figure 2. The results of the u iew given S2 th at is written gi S2 has other e ge) of understa

Figure
Figure 5.The eference to a s as negative v he operation to is 16 meters (to

Table 2 .
Coding of aspects transition the procedural-conceptual understanding