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students who make use of vectors to solve problems involving forces and ... assessments for units 4 and 5 for analysis of students' solving procedures.
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The Effect of Geometric Representation of Vector Operations on the Development of the Force Concept Principal Investigator: Colleen Megowan Co-‐Investigator: Kley Feitosa Action Research required for the Master of Natural Science degree with concentration in Physics July 2013
Vectors play a significant role in geometrically representing and developing several physics concepts. If students are to achieve success in physics, strong basic working knowledge of vectors is essential. However, research has shown that more than half of the students enrolling in college Physics do not possess such knowledge, including students who had been exposed to vectors in their high school Physics and Math classes (Knight, 1995; Nguyen & Meltzer, 2003). This is a dismal number. How can teachers improve on developing students’ understanding of vectors? To answer this question we will have to look at how the concept of vectors is usually introduced in the high school classroom. It is typical in the modeling curriculum to avoid the word “vector.” Students initially develop concepts of displacement, velocity and acceleration in one-‐ dimension, thus an explicit use of “+” or “-‐” signs preceding the numerical quantity is more than enough to describe direction. Teachers usually introduce geometrical representation of vectors, i.e. arrows having a magnitude (length) and direction, with motion maps and free-‐body diagrams, but little geometric manipulation is done, if any at all. When teachers introduce forces, students go from concrete physical models of forces, based on their own experiences, to abstract mathematical models of adding horizontal and vertical vector components. In the process of developing the idea and model of forces, there seems to be a rapid and precocious jump to algebraic manipulation of vector quantities. Students are quickly introduced to orthogonal vector components when forces “at an angle” (in two-‐dimensions) are present. Almost no time is given to developing geometrical manipulation of vectors,
so it’s no surprise students rarely use vector ideas to solve mechanics problems at the end of their Physics course. The purpose of the study is to analyze students’ development of the “free” vector concept by emphasizing geometric vector operations. I encouraged students to draw scaled vectors on graph paper and perform simple operations (i.e. adding and subtracting) geometrically. After exhausting geometric addition of vectors, I introduced students to algebraic manipulations of vectors. I used screencasts, think-‐ aloud interviews and student artifacts, like worksheets and tests, as the primary source for evaluating students’ progress towards a deeper understanding of vectors and forces. Conceptual gains were measured with the pre and post-‐treatment scores of the Force Concept Inventory (FCI) and the Vector Concept Quiz (VCQ).
There are large numbers of students that go through introductory Physics courses without significantly learning vector concepts. Vectors are used extensively in physics, yet very little attention is given to how and to what extent students learn about them. Studies have found that about half of students who have completed an introductory Physics course have no useful knowledge of vectors (Knight, 1995; Nguyen & Meltzer, 2003). Students’ lack of qualitative understanding of vectors becomes apparent when they begin to study forces. After traditional instruction, the number of students who make use of vectors to solve problems involving forces and acceleration is minimal (Flores & Kanim, 2004). Additionally, in the modeling
(Poynter & Tall, 2005a; Watson, Spyrou & Tall, 2002 ). Adding vectors graphically also requires students to translate vectors on paper, which in turn helps them stay away from the notion that vectors are “attached to points” (i.e. forces in a free-‐body diagrams) and cannot be moved (Arons, 1997). Anna Poynter has spent several years studying students’ vector concepts and their implications for Physics and Math classrooms. She developed a framework in which students’ cognitive development of vector concept experiences five distinct stages. In the “embodied world” (the physical world), the highest stage is the idea of the two-‐dimensional “free vector.” An effective way to promote students to this highest stage is by focusing on the effect of the action rather than the action itself. She points out that a student whose focus is on the effect of a vector translation is usually more successful in understanding vectors in different contexts (i.e. as a displacement or a force) and understanding the commutative property of vector addition (Poynter & Tall, 2005a). We have seen how emphasis on graphical operation of vectors should promote better understanding of vectors, however, Knight and Arons suggest yet another way to strengthen students’ concept of vectors by using computer-‐aided instruction (Knight, 1995; Arons, 1997). With the help of computers, students can go one step further and easily examine the effects of changing or translating vectors. Computer software allows students to make quick changes to vectors (i.e. size and direction), modify their arrangement, and immediately observe the effect of the changes—something that would be difficult and time consuming on paper. The use of simulations allows students to easily control the visual representations (vectors)
and immediately help them establish “cause-‐and-‐effect relationships” (Perkins, Adams, Dubson, Finfelstein, Reid & Wieman, 2006). After students’ vector concepts have been developed carefully by geometrical operations and representation, they will be ready for the introduction of orthogonal components. Students should be led to verbally articulate vector representation at different angles. Beside only breaking down vectors into horizontal and vertical components, they should also be asked to break vectors into parallel and perpendicular components of inclined surfaces (Knight, 2004). As outlined above, students have little understanding of vectors even after a whole year of physics instruction. The common practice of introducing orthogonal components of two-‐dimensional vectors and quickly moving towards algebraic solution of problems doesn’t seem to be an effective way to develop the “free vector” concept. Students should benefit from geometrically adding and representing vectors. Additionally, the use of computer simulations should help students concentrate on the effect of adding vectors rather than the action itself. Modifying instruction in the modeling cycle to address these shortcomings is the purpose of this study.
Investigator: I work at Escola Maria Imaculada (Chapel School), an American private school located in São Paulo, Brazil. The school has a student body of approximately 700 students from K-‐12 and serves mostly affluent families who seek bilingual education for their children. The student breakdown is in the vicinity of
In regular Physics, I started the year by going over unit 1 of the modeling cycle. During this time, I gave students the pre-‐tests, and we got acquainted with modeling techniques and the use of graphical software (logger pro). During the first week I also introduced students to software applications for recording “screencasts”; i.e. Jing (http://www.techsmith.com/jing.html) by Techsmith, Screenr (http://www.screenr.com; web based) and screencast-‐o-‐matic (http://www.screencast-‐o-‐matic.com; web based).
with students. I gave them an extra worksheet with parallel and perpendicular velocity addition questions. I used the worksheet to introduce students to scaling and graphical addition of vectors. Students were not allowed to solve for the resultant vector algebraically, even if they already knew how to deploy this method. After everyone completed the assignment, I collected the worksheet from students for analysis. Unit 3 – I didn’t make any modifications to unit 3 (“Constant Acceleration”). Units 4 and 5 – The students’ first encounter with vectors in two-‐dimensions happens in unit 4. I introduced students to the idea of forces as vectors by using a scaffolding activity that involved force tables (^) (see appendix). The activity was structured in three stages with each stage followed by a worksheet that included modified problems from the modeling worksheets and some additional problems. In stage 1, students worked on the idea of two collinear forces acting in equilibrium. In stage 2, students developed the concept of four orthogonal forces acting in equilibrium. In stage 3, they developed the concept of three or more forces acting at an angle in equilibrium. I followed each activity by giving worksheets where students had to first scale force vectors on graph paper and then add them graphically with the aid of rulers and protractors. This prompted students to translate vectors as “free” moveable vectors when working from free body diagrams to graphical representation of vector addition. All worksheets for unit 4 and 5 in the modeling cycle were modified to not include inclined-‐plane questions or any question that prompted students to use trig in its solution (^) (see appendix). At the completion of unit 5, I introduced students to the ideas of orthogonal components of vectors and algebraic
invited a total of five students in regular Physics to think-‐aloud recorded interview sessions after the treatment. No more than one think-‐aloud interview per student was recorded. Modeling Cycle Units Treatment Duration Assessment and Collected Artifacts Unit 1 Scientific Thinking No Treatment 1 -‐2 weeks Pre-‐Test of FCI and VCQ Unit 2 Uniform Velocity Introduce Velocity Vector / Start Scaling and Graphical Addition of Vectors. 2 -‐3 weeks Velocity Vector Worksheet Unit 3 Uniform Acceleration No Treatment 3 weeks No Assessment Unit 4 Balanced Forces 3 Stages Scaffolding Force Activity / Graphical Addition of Vectors 3 -‐4 weeks All worksheets and Assessments. Phet Simulations Screencasts Unit 5 Unbalanced Forces Graphical Addition of Vectors and Introduction to Orthogonal Components of Vectors 3 -‐4 weeks All Worksheets and Assessments. Phet Simulations Screencasts Units 6-‐ 9 No Treatment 16 weeks Post-‐Test of FCI and VCQ, Student’s Interviews and Survey Table 1 -‐ Summary of Treatment
The number of students participating in the treatment was very small (twenty-‐six in all). The low number of participants in the study unlocked a new possibility for collection of data by allowing me to look more closely into students’ work and thought processes. I chose “screencasts” as one of the tools for collecting qualitative data, because it offered me the ability to listen in on students’ thinking when answering problems or “playing” with simulations. I had seen screencasts in
use by other physics teachers in the blogosphere, and I had limitedly explored its use in my own classroom the school year before the study. Screencasts record students’ computer screen and audio input for a maximum of five minutes, which is the case for most of the free available software. In my study, I encouraged students to use one of the following three free softwares:
very well. I do not have the answer as to why I only managed to collect a dismal number of screencasts. Perhaps the stakes were too low (minimal points), perhaps students never felt confident in their work and thought they had nothing to gain from submitting this type of assessment. I’ll never know, so this led me to depend more on other student artifacts, (i.e. worksheets, tests and field notes) as sources of qualitative data. Figure 1 -‐ Frequency of Screencast Submission
I gave the Vector Concept Quiz (VCQ) pre-‐test (^) (see appendix) to all my students in the first week of school. I used the pre-‐test to measure students’ basic initial ability to understand and add vectors. The VCQ was developed by Nguyen & Meltzer from
Iowa State University and was used on thousands of students during the academic year of 2000-‐2001. The test is composed of seven questions, all presented in graphical form, and covers concepts like properties and addition of vectors. The table below shows the breakdown of questions and the concepts it evaluates: Question Number Concepts Evaluated 1 & 2 Properties of vectors. Listing vectors of equal magnitude (question 1) and direction (question 2). 3 Finding the resultant vector direction. Orthogonal (perpendicular) vector addition. 4 Finding and drawing the resultant vector of two collinear (parallel) vectors. The vectors are in opposing direction. 5 & 6 Finding and drawing the resultant vector of two vectors at an angle (neither collinear nor orthogonal). 7 Comparison of the magnitude of resultant vector by adding two equal vectors at two different angles. Table 3 -‐ Concepts Evaluated in VCQ (Nguyen & Meltzer, 2003) Each question was assigned one point (max pts. = 7) and students only received the mark if they answered the entire question correctly. Before the treatment (MEAN SCORE = 3.16, SD = 1.84) a high number of students answered question 3 correctly. As I will discuss further in the paper, I believe there was a disproportionate number of correct answers to question 3, and this may indicate a problem with the question itself. The histogram of the pre-‐test below confirms students’ incomplete understanding of vectors entering my Physics course.
Questions 5, 6 and 7, all of which address addition of vectors at an angle, are of special interest to this study. Questions 5 and 6 (see below) overlaid the vectors on top of a grid encouraging students to draw solutions, with question 6 asking for an explanation of reasoning. In the pre-‐test, only 14% of students answered questions 5 and 6 correctly. Looking more closely at question 6 —because it asks students to provide an explanation to their answer—of the students who attempted to solve the question, most suggested the idea that vector B should point in a direction that makes vector R “cut” right in the middle of them. Students cultivate this idea that the resultant Figure 4 -‐ Questions 5 & 6 in VCQ (Nguyen & Meltzer, 2003)
vector is one that separates the other two vectors right in the middle, no matter the size of the vectors being added. Below are some excerpts of students’ explanations to question 6: STUDENT A: R must be in the middle of A and B , so B is horizontal. STUDENT B: Since the resultant is almost 90° towards A , B should be almost 180 ° towards A. STUDENT C: The angle from A to R has to be equal to from R to B. I found little evidence of a specific method of adding vectors in both questions 5 and 6. Students mostly tried to draw a vector B about the same length as A in a direction that would make R appear in the middle of them. This line of thinking may be the reason why so many students (90%) answered question 3 correctly (“D”) in the pre-‐test. Question 3 doesn’t ask for an explanation, and students could have gotten away with thinking that the direction of the resultant vector is simply a vector that Figure 5 -‐ Question 3 in the VCQ
