Using Vector and Conventional Approach in Calculating Distance in a Three Dimensional Objects: An Experimental Study

Many of high school students find difficulty in studying three dimensional geometry. They find difficulty in understanding three dimension objects such as cube, cuboid, pyramid, etc. Most students fail to make correct perception about the distance to calculate, when the problems are dealing with lines or planes. Only students with good spatial capability can solve such problems. This research focused on making new approaching method to understand and then calculate the distance in a three dimensional objects. The approach is vector approach, including develop connection vectors, normal vector of a plane, and calculating distance from a point to certain line or plane using vector theorems. The Approach was tested by experimental study to a class of XII grade of high school students in SMAN 1 Parongpong, Indonesia, consists of 30 students, and conducted in six meeting during July until August 2017. A similar class treated with the conventional spatial approach as the control class. SPSS 20 was used to analyze the research data. Based on the research data analysis, the conclusion of the study was that the students treated with vector approach achieve higher performance improvement compare with the students treated with conventional approach. By using vector theorems to solve three dimensional problems, we turn the theory into action.


Introduction 1.Background Of The Study
In the High School level Mathematics curriculum in Indonesia, the Three Dimension is one of the subjects that must be taught to students.The problems related to the subject also always appear both in the National Examination and in the entrance test for Higher Education.There, the students learn how to calculate the distance and angle between the elements of geometry contained in various types of three dimensional objects, including cubes, rectangular prisms, and pyramid.The geometrical elements in such problems are points, lines and planes.
To solve the distance and angle problems, all of the high school mathematics textbooks in Indonesia use the spatial approach, and the calculation use conventional formulas, including Pythagoras formula, triangle area formula, trigonometric formulas, and geometric congruency formula.The approach can easily found in Sukino (2012) and Kemendikbud Indonesia (Matematika Kelas XII Kurikulum 2013Revisi 2018, n.d.)The spatial approach consist of several steps.It begin with simplifying the plane into a line, then simplify the problem of three dimensions (space) into a two-dimensional problem (plane), and end with the use of conventional formulas above.Until now, there is no book that discusses the use of vector and matrix approaches to analyze and solve the problem of calculating distance and angle.
With the spatial approaches and conventional formulas, researchers found many complaints from students regarding the problem of calculating these distances and angles.Some complaints that often appear include: 1. Students are not sure of their own geometrical interpretation when simplifying threedimensional problems into two-dimensional problems.2. It is quite difficult to imagine the position of the line or plane intended by practice questions or exam questions, especially in imagining the location of the line of projection of a line on a particular plane.
3. The time to work on a three-dimensional question is more than 5 minutes, while the average time to do the National Examination questions should be 3 minutes.
These difficulties and complaints led to a decline in students' interest in the Three Dimensions chapter in particular, and geometry in general.
Thus, a new approach is needed to be taught to students who have difficulty using the spatial approach when calculating distance in a three dimensional solid.

Research Question
Based on the background of the study, the questions to be answered through this research are (1) Is there any significant improvement in problem solving ability in students who are given a vector approach?(2) Are there any significant differences in problem solving ability improvement of students who are given a vector approach to students who are given a spatial/conventional approach?

Methodology
This chapter gives an overview of research variables, research design, and research instruments.

Research Variables
The variables in this study are (1) Dependent Variable : problem solving abilities of the students in three dimensions geometry, especially the distance calculating problems.(2) Independent Variables : the use of vector approach and (vs.) conventional spatial approach.

Research Design
This research is an experimental quantitative study of 12 th grade of high school students in Parongpong Public High School, in particular Science Class 1 as the experiment group (EG) and Science Class 2 as the control group (CG).Each class consists of 30 students, and the initial abilities of the two classes are not significantly different.The Figure 2.1 show the flowchart of the study.

Research Instrument
The research was conducted in six weeks, from July 2018 until August 2018 , at Public High School I Parongpong, West Bandung, West Java Province.
Starting with the pretest to students both in the experimental class and in the control class.Pretest consists of five questions, including the problem of distance between points, distance from point to line, and distance from point to flat plane.
In subsequent meetings the experimental class receives the three-dimensional learning using a vector approach, while the control class uses a conventional spatial approach.Especially in the experimental class matrix and vector theorems are also taught, because The study ended with a post test that was identical to the pre test.Both pre-test and post-test have an essay form, and are assessed based on the assessment rubric created by the researcher.
The test results of the control group and experimental group were processed using SPSS 20 to obtain normalized gain for each group.The gain will be differentiated according to its categories based on Hake (2002), i.e.
(1) low category if the gain is less than 30%, (2) medium category if the gain is between 30% to 70%, (3) high category if the gain is more than 70%.

Vector Approach vs Spatial Approach
This chapter gives brief overview on the aproaches used in the research.
According to Rahadi (2018), the vector approach to calculate distance in a three dimensional objects consists of 1. Formation of vectors in the three dimension objects.2. Formation of normal vectors in the planes located inside cubes or prisms.3. Operation of adding and subtracting vectors 4. Dot products 5. Cross product 6.Orthogonal projection of a vector to another vector 7. Phytagoras.
The problem of distance in space in three dimensions can be classified as follows (1) Distance between two points (2) Distance from point to line (3) Distance from point to flat plane (4) The distance between two parallel lines (5) The distance between two parallel planes (6) Distance of lines to parallel planes (7) The distance between two lines that are not intersecting but not parallel.
As explained in Indonesia (2018), the spatial approach to calculate distance in three dimensional objects consist of Phytagoras formula, geometry congruency principle, triangle area formula, and simplifying the problems into two dimensional objects.The same steps can also be found in Sukino (2012).
In order to make clear the difference and similarity, the two approaches in this study will be used to solve the problems in example 3.1 and 3.2.Example 3.1.Given the cube ABCD.EFGH with the edge length of .The point P is the midpoint of edge DH as ilustrated by Figure 3.1.Calculate the distance from point P to the plane ACH.
the test.Researchers also found that there were some students who still did not understand what to do to solve the three dimensional problems given in the test.
Meanwhile, the students in the experimental group who are given vector approach achieve higher improvement in their ability to solve three dimensional distance problems.Their average score increase from 10.80 in the pre-test to 47.97 in the post-test.It also shown that the experimental group reaches gain 41%, which is categorized as medium category based on Hake.
Since the normalized gain was normally distributed and the data is homogenous, then the independent sample t-test was done to answer the second question in 1.2.The result of the t-test shown in the Table 4.2.
As shown in the Table 4.2, the significance of the test is 0.000 , less than .It means the null hypotesis is rejected, in another word the two groups have significantly different gain, with 95% confidence interval of the difference.
Furthermore, considering the experimental group has higher gain than the control group based on the Table 4.1, it can be concluded that the vector approach producing higher problem solving ability improvement in three dimensional problems rather than the conventional/spatial approach among the high school students.

Conclusions and Recommendation
Based on the results and the data analysis, the conclusions of the research are (1) The students who were given a vector approach experienced an improvement in Dimension Three problem solving abilities, in the medium improvement category.
(2) The students who were given a vector approach reach significant higher improvement in Dimension Three problem solving abilities rather than the students who were given a conventional spatial approach.
In order to improve the quality of the approach, the research recommendations are (1) It is necessary to establish matrix and vectors theorems before the vector approach is taught to students, because several students already forgot what they have learned about matrix and vector in the 10 th grade.(2) Interactive learning tools and geometry learning media are needed to be created to make it easier for students to master this vector approach.
Figure 2.1 Flowchart Of The Study

Table 4 .
2. T-Test for Equality of Means between the EG and the CG