A study of undergraduates’ understanding of vector - decomposition of forces on inclined planes

Students often struggle to decompose a vector, especially when calculating the force components of an object on an inclined plane. In this study, we designed an online vector survey in the context of physics and implemented it in a primarily undergraduate university. The study focuses on the students’ learning difﬁculties and teaching strategies associated with vector decomposition on inclined planes. The analysis of the students’ responses indicates that translating a vector to the position with geometric convenience can help them identify angles and calculate vector components on inclined planes more accurately; when students are provided with the deﬁnition of the angle in the vector component formula, their performance improves.


I. INTRODUCTION
Applying the knowledge of vectors to kinematics and dynamics is an essential skill for solving physics problems at the undergraduate level.Vectors first appear in the calculations of physical quantities such as displacement, velocity, and acceleration and then in the analysis of free body diagrams.The learning di culties that undergraduates encounter in vectors are an obstacle in their successful pursuit of physics courses.
Decomposition is one of the first lessons that is taught about vectors.It is an essential step in the calculations of two-dimensional motions and the analysis of free body diagrams in applications of Newton's laws.The intensive usage of vector decomposition and its close correlation to undergraduates' success in introductory level, calculusbased physics courses, make understanding it well an absolute necessity.
Various e↵orts have been made to address students' learning di culties in vector decomposition, particularly in simple cases.Knight's study in 1995 included two decomposition questions on algebraic and geometric aspects in Vector Knowledge Test and the results indicated that less than half of undergraduates had su cient vector knowledge before taking a calculus-based physics course [1].Barniol and Zavela asked students who had finished introductory physics courses three decomposition questions in graphical format in the 20 question Test of Understanding of Vectors.Based on the responses, they developed a taxonomy of the most frequent errors [2].Our recent investigation revealed that several learning di culties are due to a lack of understanding of the connections between the algebraic and geometric aspects of vectors [3].
With the framework of physics, vector-related applications become more challenging.Aguirre's investigation of vector kinematics in projectile motion showed that high school students had numerous incorrect preconceptions with components of a velocity vector and half of undergraduates who had completed a course in mechanics maintain their preconceptions and kept their intuitive beliefs [4,5].In Mechanics Baseline Test, Hestenes and Wells included kinetic vectors and superposition of force vectors.The lowest scores of test questions came from the ones requiring vector properties [6].Modifications to physics lecture instruction had been made and implemented based on curricula such as Tutorials in Introductory Physics and Physics by Inquiry, but the improvement of students' performance in questions about forces and Newton's second law was not more than moderately successful [7][8][9].The investigations of Flores et al. show that students' correct response rate of vector questions in the context of forces and dynamics is lower than 50% on average after implementing the modified lecture of Tutorials in Introductory Physics.
Previous studies and our teaching observations show that the majority of students perform worse on force decomposition when an object is placed on an inclined plane than when it is on a flat surface.The study of Mikula and Heckler showed that 50% of students could calculate the component of the weight parallel to the surface of the inclined plane correctly and only around 10% of them could correctly calculate the component of the weight perpendicular to the surface of the inclined plane, which is far below the requisite accuracy needed for such fundamental skills [10].
In extension of previous research, we focus on students' learning di culties behind two major common errors (Sin/Cos error and MultiDivid error) quantitatively.In order to address common errors and a lack of understanding of the connections between the algebraic and geometric aspects of vectors in the context of physics, we also investigate the e↵ectiveness of two methods that we can easily apply to improve students' performance in vector decomposition on inclined planes by using (1) translation of vector arrows (2) restatement of the angle definition in the vector component formula.

II. STUDY DESIGN
In three consecutive semesters in the academic year of 2020 and the spring semester of 2021, a survey focusing on vector decomposition on inclined planes was designed and implemented in an undergraduate-based university in the United States.The students taking University Physics at the calculus level were invited to take the survey and 69 students completed and submitted the survey voluntarily (32 students took version A and 37 did version B) with the motivation of improving their performance in classroom participation.
The survey was designed for online implementation, which is unavoidable during the pandemic period.We chose the format of multiple-choice questions so that it would be convenient for participants to submit their responses.The number of participants was su cient to give statistically significant results as shown in Section III.
The survey consists of four multiple-choice questions with two versions [11].All four questions are related to identifying and calculating the Y component (the component perpendicular to the surface of the inclined plane) of the weight of an object on an inclined plane shown in Figure 1.In order to calculate the components of the weight of an object on an inclined plane correctly, students need to master graphical techniques of resolving vectors into components, including realizing the significance of di↵erent projection axes (XY coordinates), formulating right-angle triangles, identifying angles and then using trigonometric ratios correctly to compute the components.These skill elements are included and divided to be tested separately.
The di↵erence between the two versions is designed to address the proposed method (1).The weight of an object is marked at the center of mass in the diagrams in Version A while in Version B, it is marked at the Positions with Geometric Convenience (PGC), which is the base of the object in this case.It is not our intention to obey the typical rules of free body diagrams where forces are attached at the center of mass.The translation of vectors to PGC is applied when students analyzing the diagrams to calculate components.It is an optional tool for students' convenience.Students can take advantage of the fact that vectors keep their directions and magnitudes in translation.Translation of vectors is also inevitable in graphical methods of vector addition and subtraction.As shown in Table I, the multiple choices of each question are designed and categorized according to the students' common responses in previous studies [2,10,12].Sin/Cos error is that the student switches sine and cosine functions when trying to get a component.Angle error happens when a wrong angle is used in decomposing a vector in the geometric aspect.MultiDivid error is the category of choosing magnitude to divide the trigonometric function to get a component, e.g.F/sin(✓) instead of F •sin(✓).Component error is often caused by the lack of understanding of the concept of components of a vector and the inability to identify components in a graph.Sign error occurs when the answer has the wrong sign.The use of Complementary Trigonometric Functions (CTF) e.g.F • cos(90 ✓) instead of F • sin(✓), is not an error.Some choices fall into more than one category.
The above-listed common errors are in two tiers.Sin/Cos error and MultiDivid error are in the upper tier and they can be caused by Angle error, Component error or other learning di culties.In order to identify them, we utilized intermediate steps of calculating components for Questions 1 and 2 as shown in Figure 1.Question 1 was designed to identify the reason for making Sin/Cos error.Sin/Cos error can come from applying a wrong trigonometric function or having an Angle error.In Question 1, students are asked to select either ↵ or to be equal to ✓. Question 2 is intended for MultiDivid error.Students who have di culty in applying the trigonometric function for opposite/adjacent sides in a right triangle and those who cannot identify components (choice 2a) can end up with the same error.Here, students choose either diagram a or b for the Y component of the weight.
Questions 3 and 4 have the same topics but with different choices.They are designed as pairs to test the improvement of implementing the proposed method (2).The study of Mikula and Heckler indicates that a notable proportion of students appear to be confused about when to use sine and cosine functions, instead, they simply use a memorized algorithm using a given angle [10].In teaching, we also notice that students often question the reason for associating sine angle (the angle of inclination) for the X component of gravity when the X-axis is aligned with the incline whereas the formula of the X component is magnitude times cosine angle.The learning di culty is possibly correlated to the inadequate understanding of the definition of the angle in the component formula.
In Questions 3 and 4, we intend to test whether and how much the restatement of the definition of the angle can help students to find the relevant components.The di↵erences between the two questions are that the definition of the angle in the component formula is given before Question 4 and CTF are available in multiple-choice options of Question 4. The hint of the angle definition helps students to identify the angle in the graph and build a connection between component formula and the geometric aspect of vectors.

III. DISCUSSION
In this investigation, our analysis is based on all the submissions, 32 students in Version A and 37 students in Version B. The average score of overall performance is 47%.(It is the same as the average di culty index as shown in Table II, when all 4 questions are counted equally.)The students who took survey Version A and the ones who took the survey Version B received grades of 44% and 51% respectively.The validity of the survey is assured by adapting questions in previous studies and common examples in introductory physics courses.The survey was also reviewed by 5 experienced Physics instructors in di↵erent institutes, and it was edited based on their feedback.In Table II, we present three types of statistical parameters which commonly applied to evaluate the reliability and discriminatory power of a test [13].Most evaluation data for the survey questions falls in the desired ranges, but not the di culty index P in Question 3 and 4. The di culty index is calculated by taking the ratio of the number of correct responses on a question to the total number of participants.The higher the di culty index is, the easier the question is.The lower than desired di culty index of Question 3 is the reason behind this investigation.As shown in the table, P B in Question 3 increases to 0.19 when the proposed method ( 1) is implemented and P A in Question 4 increases to 0.22 when the proposed method (2) is implemented.With both proposed methods, the di culty index reaches 0.35, falling in the desired range.
The discriminatory index D measures discriminatory power, the extent to how well a single test item can distinguish the students with and without robust knowledge.If many items have high D values in a test, the test is useful in separating strong and weak students.The point-biserial coe cient r pbs is a reliability index, which measures the consistency of a single test item to in the whole test.If an item has a high r pbs value, the students with high total scores are more likely to answer item correctly.

A. Learning di culties behind common errors
The accuracy of the first three questions is 80%, 65%, and 16% respectively.Accuracy drops dramatically from Questions 1 and 2 to Question 3.This means that most students can identify the angle ✓ and the Y component of weight successfully, nevertheless, they don't end up with a correct answer for the Y component in Question 3. The following is an analysis of the reasons behind the above large drop in correct response rate.Figure 2c shows Sin/Cos error and Sign error are the two main errors in students' responses.MultiDivid error is minor.In Question 3, 48% of the total participants made Sin/Cos error (3a and e).While going through individual responses, we found that 79% of these students correctly answered Question 1 of identifying angles correctly.It means that they had reached the stage of trying to calculate the adjacent sides of a triangle, where a cosine function should be applied.This result suggests that approximately 4/5 of the learning di culties behind students' Sin/Cos error are caused by either not being able to select a correct trigonometric function for the calculation of adjacent sides or failing in reasoning processes.
For MultiDivid error (3c, d, g and h), the survey re-sults show that the major learning di culty is also about trigonometry.33% of the students who had MultiDivid errors made mistakes in Question 2. The students who answered both Question 1 and Questions 2 correctly didn't make MultiDivid errors.In other words, nearly 1/3 of the learning di culties behind MultiDivid error on inclined plane questions are attributable to not being able to identify the Y component in graphs and the remainder are potentially caused by a lack of knowledge of trigonometry.
As shown in Figure 2a and 2b, Angle error in Question 1 and Component error in Question 2 are minor with comparably high correct response rates, but students' performance on vectors is expected to be improved if more practice in triangle geometry and the emphasis on the concepts of vector components are given in class.

B. Comparison of results with/without applying the proposed methods
The di↵erence between the two versions is the placement of the vectors at the center of mass or PGC.In Question 2, identifying components is irrelevant to the placement of the weight vector, so the comparison between two versions should not show di↵erences.Consistent with our prediction, P A and P B of Question 2 are nearly the same in Figure 2b.The data also indicates that the students who participated in both versions have a similar level of understanding of vectors.
The proportion of correct responses to Questions 1, 3 and 4 are higher in Version B. The advantage of translating a vector to PGC comes from the fact that students can identify the angles between the vector and its components with greater ease.87% of students can identify the angles next to the weight vector in the diagram when the vector arrow is marked at the bottom of the object.In comparison, 72% answer Question 1 correctly when the weight arrow is marked on the center of mass of the object in Version A. The proportion of correct responses to Question 3 and 4 in Version B is nearly 50% higher than the ones in Version A. The t-Test of the survey Questions 1, 3 and 4 in Version A and B indicates the improvement statistically significant (p = 0.0321).This result shows that the proposed method (1) e↵ectively helps students to decompose a force on an inclined plane.
Not only does it improve students' performance of force decomposition in comparably complex graphical settings, but translating a vector to PGC also helps them to have a better understanding of vector translation in space in general.Nguyen's investigation of vector addition, magnitude and direction in graphical form indicated that students seem to lack the understanding that the magnitude and direction of a vector are strictly preserved by parallel transport [14].We believe that the method (1) can help students' performance of the calculation of vector addition in the geometric aspect.
The results in Question 4 show that the students' performance improved when the definition of the angle in the component formula is given.The correct response increases from 16% in Question 3 to 29% in Question 4. We conducted a t-Test to compare students' performance in Questions 3 and 4. The result showed a statistically significant improvement (p=0.0190).A more substantial improvement is expected in teaching practice, by the considerations of our survey design and implementation.First, Question 3 and 4 have the same topic and are next to each other in the survey; the hint of the angle definition before Question 4 may not catch students' attention to rethink and redo the question.The other factor is that students may not be confident with using CTF without much exposure to CTF representations.
With a good understanding of the definition of the angle in the component formula, students' skills at vector decomposition were enhanced with the addition of alternative CTF options which allowed them to avoid Sin/Cos and sign errors.As going through individual responses, we find that 77% of the students who chose CTF options answered the questions correctly.Sin/Cos error (4d) was greatly reduced to 6% in Question 4 in Version A and 11% in Version B as shown in Figure 2d.Applying the vector component formula directly also avoided the mistake of omitting signs, which is another major common error as shown in our Question 3.

IV. CONCLUSION
This study shows that the students' learning di culties behind Sin/Cos and MultiDivid errors in the calculation of force decomposition on inclined planes are trigonometry (major), geometry and concept of components (comparably minor).The results of our investigation also prove that the proposed two methods are effective to improve students' performance, (1) translate vectors to the position with geometric convenience; (2) restate the definition of the angle in vector component formula.The suggested teaching strategies include reviewing vector knowledge along with related trigonometric knowledge before teaching vectors in physics, informing students that they can translate a vector with the full preservation of its magnitude and direction and they can translate forces to the positions with geometric convenience in force decomposition, emphasizing the angle definition and labeling the angle in graphs when introducing vector component formula.
It's imperative to research further on vector related teaching and learning because of the importance of its applications to physics and in the face of students' inadequate understandings of vectors.In the next stage, we plan to include the investigation of Sign error and implement the surveys in the format of open-ended questions so that students' reasoning in the calculation of vectors can be observed and interpreted.

FIG. 1 :
FIG.1: Free body diagrams in the survey[11].The upper left two diagrams are for Q1 in Version A and B; the two diagrams at the bottom left are for multiple-choice A and B of Q2 in Version A; the right one is for Q3 and Q4 in Version A. For the diagrams of Q2, Q3 and Q4 in Version B, the vector of weight is marked at the base of the object.

FIG. 2 :
FIG. 2: Students' responses to each multiple-choice option, where the correct option is capitalized.The responses in Version A are in black bars and the ones in Version B are in gray bars.

TABLE II :
[13]average di culty index of each question in survey Version A and B (P ), the ones for Version A (PA) and for Version B (PB) are listed respectively.Item discriminatory index (D) and point-biserial coe cient (r pbs ) for each question are included.The table also includes the average statistical parameters of all test questions in this study and the desired values (Des.Val.)takenfrom reference[13].