the!effect!of!geometric!representation!of!vector

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

The Effect of Graphical Addition of Vectors 2

Table of Contents

ABSTRACT ................................................................................................................................................ 3

RATIONALE .............................................................................................................................................. 3

LITERATURE REVIEW ........................................................................................................................... 5

METHOD .................................................................................................................................................... 8

SUBJECTS: .................................................................................................................................................................... 8

PROCEDURE FOR TREATMENT ................................................................................................................................ 9

QUALITATIVE DATA SPECIFICS ............................................................................................................................ 13

DATA ANALYSIS ................................................................................................................................... 16

VCQ PRE-‐TEST ....................................................................................................................................................... 16

EXPOSING STUDENTS TO GRAPHICAL ADDITION OF VECTORS ...................................................................... 21

TREATMENT: AN ALTERNATIVE APPROACH TO TEACHING AND ADDING FORCES ................................... 23

THE NET FORCE CONCEPT .................................................................................................................................... 30

MATHEMATICAL METHOD -‐ THE “VECTOR COMPONENTS” CONCEPT ......................................................... 43

COMPARISON OF SOLUTION METHOD ................................................................................................................ 44

FCI PRE-‐TEST AND VCQ/FCI POST TEST ........................................................................................................ 47

FCI RESULTS ........................................................................................................................................................... 48

STUDENT SURVEYS ................................................................................................................................................. 50

CONCLUSION ......................................................................................................................................... 53

WORKS CITED ....................................................................................................................................... 56

APPENDIX .............................................................................................................................................. 57


Abstract

While vectors play a significant role in understanding basic concepts in

physics, most high school and college students demonstrate only rudimentary

working knowledge of vectors even after taking a yearlong Physics course. In

developing the force model, careful consideration to the vector concept and its

operations is necessary, yet studies have shown that most instructors and textbooks

hastily introduce students to abstract orthogonal decomposition of vectors when

solving force problems. The purpose of this study was to evaluate how students’

understanding of vectors and forces were impacted by introducing them to

graphical addition of vectors before algebraic manipulation of vector components.

The research did not find any substantial positive correlation between graphical

methods of vector addition and improved understanding of forces and vectors.

However, most students, particularly students who struggle with math, benefited

from the geometrical approach and, consequently, improved their ability to solve

dynamics questions. Likewise, students who have learned how to add vectors

graphically increased their problem-‐solving skills and added another useful “tool” to

their repertoire.

Rationale One of the key tenets of modeling is to develop in students the ability to

represent the physical world in multiple ways (Wells, Hestenes, & Swackhamer,

1995). Students are encouraged to explain the model in various graphical and

mathematical representations, thus deepening their own understanding of physics.


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).

Literature Review

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


materials, force vectors in two-‐dimensions are quickly introduced, thus prompting

teachers to introduce students to mathematical algorithms for solving problems.

Most instructors first introduce the idea of a vector geometrically but hastily lead

students into breaking vectors into orthogonal (perpendicular) components to solve

problems algebraically (Nguyen & Meltzer, 2003;Flores & Kanim, 2004; Megowan,

2005). Dr. David Hestenes characterizes this practice of breaking vectors into

components as the “vector coordinate virus.” Students start to believe that

components are more important than the vectors themselves (Hestenes, 1992).

Despite teachers’ efforts, research has shown that more than 50% of students

cannot carry out two-‐dimensional vector addition, even after a full semester of

Physics (Nguyen & Meltzer, 2003).

One potential explanation for students’ difficulties with reasoning about

vectors may rest in the fact that teachers devote very little time to geometrical

representation and manipulation of vectors. Arons, in his book Teaching

Introductory Physics, suggests that “many students would benefit from more

exercise and drill in graphical handling of vector arithmetic than usually available in

textbooks” (Arons, 1997). Flores and Kanim also state that after instruction is

modified to include more emphasis on graphical vector manipulation, students’

ability to add vectors in two-‐dimensions improves greatly (Flores & Kanim, 2004).

Graphical addition of vectors reinforces the idea of “free vector” in space. The “free

vector” concept is what Poynter designates as the highest stage in student

conceptual understanding of vectors. By “free vector,” she means a vector that can

be translated (moved freely) as long as magnitude and direction do not change


(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.

Method

Subjects: 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


70% Brazilian and 30% from other nationalities, mostly Asians. I taught three

different Physics classes; two regular Physics courses of 19 and 10 students each

(29 students total; a mix of sophomore and juniors in each class), and one

International Baccalaureate (IB) course of 10 students (9 seniors and 1 junior). It’s

important to note that students in the second year IB class (equivalent to an honors

class) are seemingly stronger than others, and they all have had a year of regular

Physics with me. I did not teach these second year students by the modeling

curriculum, and they did not receive the treatment for vectors described below.

Procedure for Treatment 1. Permission:

All participating students, along with their parents or guardians, signed an

assent/consent form acknowledging their participation in the study. If permission

was not received from both a student and a parent or guardian, that student was not

included in the study. The names of individuals participating in the study were kept

confidential.

2. Pre-‐assessment of students’ abilities:

During the first two weeks of instruction, I examined students’ basic knowledge

of two different concepts. Students took two pre-‐tests: the Force Concept Inventory

(FCI) and the Vector Concept Quiz (VCQ). The FCI assessed their basic

understanding of force concepts, and the VCQ assessed the students’ ability to

interpret and carry basic operations with vectors.

3. Unit 1:


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).

4. Treatment:

I applied the treatment from Modeling Physics units 2-‐5 (“Constant Velocity”

through “Unbalanced Forces”). The goal of this research was to not rush students

into breaking two-‐dimensional vectors into orthogonal components for an algebraic

solution of vector problems. I judiciously emphasized the geometrical

representation and graphical solution of vector addition during the treatment.

I based the treatment mostly on three simple modifications to the modeling

cycle: 1) the introduction of vector concepts after the constant velocity unit, 2) the

use of computer-‐aided simulations to visually represent vector-‐addition problems,

and 3) the emphasis on geometrical addition of vectors prior to algebraic

manipulation of vector components, especially when dealing with vectors at an

angle (not collinear); which is common in units 4 and 5.

Unit 2 – I introduced the vector concept of velocity at the end of unit 2. Students

worked through the entire unit without modifications, completing the end of the

unit assessment just as presented in the modeling materials. After completing unit 2,

I introduced the idea of velocity vectors by developing a “relative velocity” activity


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


methods of solving for the resultant vector. I encouraged students to revisit

previously given worksheets and rework the problems algebraically, but few

proceeded to do so. I also introduced inclined-‐plane problems at the end of unit 5.

Treatment ended at the end of Unbalanced Forces Particle Model (UFPM), and

when students took the unit 5 assessment they had the choice to solve problems

either graphically or algebraically. I collected most of the worksheets and all of the

assessments for units 4 and 5 for analysis of students’ solving procedures. I asked

students to record screencasts of vector addition problems and various PhET

(Wieman, Adams, Loeblein & Perkins, 2010) simulations involving forces and

vectors during the development of units 4 and 5.

For the remaining units (6-‐9), students could choose their method for solving

vector quantities problems. At the end of the course I administered a student survey

to probe into their feelings and/or reactions to graphical addition of vectors.

5. Assessment:

Students were assessed in different ways. They took the pre and post tests of the

FCI and the VCQ. I applied both post-‐tests in the last week of school. I collected most

of the worksheets in units 4 and 5 and all of the end-‐of-‐unit assessments. I asked

students to produce somewhere between three to five screencasts, no longer than

five minutes each, on various topics throughout treatment. At the end of the course I

asked students to solve two very similar inclined plane balanced-‐force (equilibrium)

problems; the first problem students had to solve by geometric addition of vectors,

and the second problem by algebraic addition of vector components (see appendix). The

purpose was to compare students’ solving ability using different methods. Also I


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

Qualitative Data Specifics 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:

• Jing (http://www.techsmith.com/jing.html) by Techsmith

• Screenr (http://www.screenr.com) web based

• Screencast-‐o-‐matic (http://www.screencast-‐o-‐matic.com) web based

When “screencasting” the solution to a specific problem, I asked students to

scan (or take a picture) of the completed work and go over it in detail, step-‐by-‐step,

in less than five minutes. Students had to show me how much they understood of

the problem and the basic physics concepts.

When screencasting PhET simulations on forces or vectors, I asked students

to “explore” before recording. The main goal with simulations was for students to

demonstrate learned concepts through the ease and power of Internet simulations.

Students had the freedom to create and script whatever they wanted as long as it

gave me insight into their thinking and knowledge of physics.

All screencasts were assigned after I had introduced and discussed the

concept in class, except for one (vector addition). All screencasts were also

supposed to be done at home, where everyone had computers and a quiet place to

record. I gave students one week to complete and submit screencasts, and I could

not make the assessment worth more than 4% of their overall class grade, because


the work was done and completed outside of the classroom (this is part of school

policy).

I assigned a total of seven screencast activities to my students. The table

below lists all the specific questions and simulations assigned to them:

Unit Screencasts Collected

2 • One question on one of the worksheets from unit 2 (practice)

4 • Screencast of problem 4 of stage three worksheet • Screencast of PhET “Motion and Forces: Basics”

(http://phet.colorado.edu/en/simulation/forces-‐and-‐motion-‐basics)

5

• Screencast of PhET “Motion and Forces” (http://phet.colorado.edu/en/simulation/forces-‐and-‐motion)

• Screencast of PhET “Ramp: Forces and Motion” (http://phet.colorado.edu/en/simulation/ramp-‐forces-‐and-‐motion)

• Screencast of PhET “Vector Addition” (http://phet.colorado.edu/en/simulation/vector-‐addition)

• Screencast of question 17 on unit test.

Table 2 -‐ Screencasts Assigned

The first screencasts were mainly assigned with the objective of

troubleshooting the system and making sure students were comfortable with the

process. In the beginning, I had to help some students fix “computer issues” related

to recording and submitting screencasts.

Getting students to submit screencasts was always a struggle. I had to give

them constant reminders, but I still failed to collect work from several of my low

performing students. I was not surprised to notice that submission rates for high

achieving students were much better. As the year and units progressed, the

submission rate declined with an increasing number of students claiming “technical

difficulties” for failing to submit work. I collected less than 20% of the screencasts

assigned, and most of these were from responsible students who knew the content


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

Data Analysis

VCQ Pre-‐Test

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.


Looking at the breakdown for each question (graph 1) and the percentage of

correct responses, it became apparent that students initially had some basic

knowledge of words like “magnitude” and “direction” as they relate to vectors. This

can be observed by the percentage of correct responses to the first four questions in

the pre-‐test, all with averages higher than 50%.

0

20

40

60

80

100

1 2 3 4 5 6 7

% Correct

Question Number

VCQ Pre-‐Test

Figure 3 -‐ Question Breakdown in VCQ

Figure 2 -‐ VCQ Pre-‐Test Scores


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


“cuts” through the middle of the two. Nguyen & Meltzer found similar results for

question 3 in their study and concluded that this question was perhaps flawed and

provided no significant insight into students’ vector thinking.

When analyzing question 7, I noticed that a few students answered it

correctly but didn’t provide an explanation, which makes it difficult to understand

how much of their response was pure guess. The most common misconception was

to think of vectors as scalar; thus two vectors of equal size added together give you

the same magnitude for the resultant vector despite the different angles between

them. Students used words like “same length,” “equal size,” and “same magnitude”

when referring to vectors in figures A and B to justify their answers. Another

notable misconception was to believe resultant from A was larger because of a

larger angle between the vectors.

The VCQ pre-‐test led me to conclude that students starting Physics have

some kind of correct intuition about vector properties like magnitude, direction, and

collinear vector addition (maybe because they can somehow get away with treating

vectors as scalars). But students have little to no idea about adding vectors at an

angle, which is one of the main challenges teachers face in teaching dynamics and is

the focus of this study.

Exposing Students to Graphical Addition of Vectors

In Modeling Physics curriculum, the ideas of quantities like velocity,

displacement and forces as being vectors are not thoroughly developed or

addressed. In unit 2, we construct the constant velocity particle model and use “+”

and “-‐” signs to depict the vector nature of velocity. The arithmetic is simple and


straightforward, but I still wanted my students to graphically entertain the idea of

velocity as vectors. After a brief introduction to vectors, primarily discussing how

an arrow could represent a vector’s magnitude and direction (by its length and the

direction it points), I left students to work on a velocity addition worksheet (see

appendix B) on their own. The worksheet given to the students was divided into two

parts: the first introducing students to the addition of collinear vectors, and the

second the addition of orthogonal vectors. More than 80% of the students were able

to correctly answer questions about velocity addition in one-‐dimension. This

reinforces students’ aptitude to add vectors in one-‐dimension during early stages of

instruction.

Although students had no trouble properly answering questions involving

addition of one-‐dimensional vectors, their drawn vector-‐diagrams on graph paper

had serious shortcomings. It was very difficult for students to scale up vectors

properly or consistently on a piece of paper. Some students drew lines, not arrows,

to represent velocity, and the ones who attempted to include arrows would draw

“arrow heads” after scaling vectors, hence overextending the length of the vector.

Subtracting vectors was not easy for students either. They didn’t think vectors could

“run” on top of each other or probably didn’t like the aesthetic aspect of it. This led

some students to abandon the head-‐to-‐tail method of vector addition.

The second part of the worksheet addressed for the first time in the course

perpendicular velocity addition. The question asked students to calculate the

ground speed of a plane flying with side winds at 90° to the direction of flight. It’s

important to note that some of my students have seen similar questions in their


Math class. Several students didn’t answer the question and claimed they didn’t

understand the problem, though some of them attempted to draw a vector diagram

for the question. Most students lacked a proper, well-‐defined method and seemed to

be confused with the direction that vectors (arrows) should point. Other students

(53%) correctly used Pythagorean’s theorem (i.e. they didn’t stick to graphical

methods but resorted to math) to solve the question although most of their

diagrams lacked an appropriate system to represent the addition of vectors. It was

clear that students were unfamiliar with graphical vector diagrams and operations.

Before starting unit 4, I worked in class on graphical addition of vectors so students’

transition into treatment procedures would be smoother. I didn’t want students

resorting to math or the Pythagorean theorem for their solutions.

Treatment: An Alternative Approach to Teaching and Adding Forces Forces are an abstract and difficult concept to teach, even to the experienced

modeling teacher. The treatment included a three-‐stage scaffold activity to help

students move from the abstract idea of forces to a more kinesthetic and concrete

visualization of the concept. For this activity we set up a ring stand in the middle of

the room, and then we put a ring with ropes attached to it so students could pull and

“feel” the forces (see diagram on next page). The goal was to keep the ring in static

equilibrium so that it wouldn’t touch the stand and then discuss how changes

conveyed to one rope would affect the other forces if the ring were to remain in

equilibrium. Before allowing students to play with the set up, I asked them to

discuss some question in small groups and whiteboard their prediction. We tried to

keep the emphasis on the effects of changes in forces. Because forces are vectors,


changes could be interpreted as in magnitude and direction. After discussing

questions for each stage, a quick “play” time with the apparatus followed, and I gave

the students a worksheet where they were strictly instructed to use only graphical

addition of vectors to solve force problems. At the end of stage 3, students were also

given a force table to quantify some of the activities in stages 1, 2 and 3.

Stage Concepts Socratic Dialogue Questions

1 2 Collinear Forces

1. In order to maintain static equilibrium in the ring, what needs to happen when one of the students pulls harder on the rope? 2. What needs to happen if the same student changes direction, but not strength of pull?

2 4 Orthogonal Forces

1. In order to maintain static equilibrium in the ring, what needs to happen if only one force changes magnitude? 2. How do the forces depend on each other?

3 3 Forces at an Angle

1. What modification do we need to make to maintain equilibrium if we only have 3 students pulling on the ropes? 2. When in equilibrium, who is pulling the hardest? How do you know? 3. Would any arrangement allow all 3 students to pull with the same force? 4. If one student increases the pulling force, how does it affect the other 2 forces?

Table 4 -‐ Description of Force Activity 1

Figure 6-‐ Diagram of Activity Set-‐Up


The worksheets following both the first two activities revealed a couple of

interesting bits about students’ first attempts in adding force vectors geometrically.

In the stage 2 worksheet, question 1 asks:

For each problem, draw a force diagram and add the force vectors graphically.

1. In an experiment a student found that a wooden block, of mass 5 kg, needed a force of 4 N to make it slide over a desk at constant speed. Find the normal force and the frictional force on the block.

At this point in the course I had formally introduced scaling, graphical

representation of vectors and head-‐to-‐tail method of vector addition. I had also

instructed students on the use of rulers and protractors. This is the graphical

representation of the vector addition that I was hoping students would come up

with for the question above:

5 kg 4 N

Fearth on block

Ftable on block

Fstudent on block Ftable on block

Force Diagram Graphical Addition

Figure 7 -‐ Question in Stage 2 Worksheet


While most of the students correctly depicted the force diagram as I had

expected, none of them added the vectors the way I had imagined. Perhaps the

question was too easy to compel a graphical approach because 54% of the students

correctly answered the question by only drawing the force diagram. Another 27%

added the horizontal vectors separately from vertical ones, and 18% added by

overlapping the vectors on top of each other (see table 4). All my students responded

correctly to the question, and this led into a whiteboard discussion about multiple

ways to represent vector addition correctly. I used this question to stress the

commutative property of vectors, which became more obvious and applicable to

students during stage 3.

Percentage of Students Method of Solution 55% Force diagram only

27%

18%

Table 5 -‐ Method of Solution

Stage 3 of this scaffolding activity really addresses the issue that this study is

trying to examine: what is the effect of graphical addition of vectors on students’

understanding and ability to solve force problems? How can we avoid jumping into


trigonometry and complicated math algorithms when confronted with problems of

forces at an angle? Rulers, graphing paper, and protractors replaced students’ lack

of knowledge in trigonometry and vector operations.

The balanced force particle model is a great way to introduce graphical

addition of vectors. If all the forces are balanced, the sum of all force vectors are

zero, and graphically the vectors “close” the shape that is being drawn. We started

working with only three balanced forces (the geometrical representation always

forming a triangle) and then moved onto four balanced forces problems. At the end

of the introduction to forces at an angle in stage 3, I gave students another

worksheet and asked them to explain the solution to the last question using

screencasts. I encouraged the students to think about the questions that were

discussed during the ring activity and discuss some of them when answering the

question. Question 4 for worksheet 3 asks:

For each problem, draw a force diagram and add the force vectors graphically. 4. The diagram shows a box of mass 1.5 kg resting on a floor with a force 10 N acting on it.

Calculate the normal force and the force of friction.

Listening to screencasts and examining worksheets from students, I came to

the conclusion that most students approached the solution in the following way:

1.5 kg

10 N

37o

Figure 8 -‐ Question in Stage 3 Worksheet


In all force diagrams, the weight force was drawn at about the same length as the normal force.

Students then would draw a known quantity in a known direction-‐-‐in this example, the gravitational force pointing straight downward. (Appropriately scaled)

Followed by another known vector, in this case the 10 N force at a 37 degrees angle with the horizontal.

Students would close the trapezoid shape knowing that the last 2 vectors had a 90 degrees angle between them. They would measure the length of the unknown vectors and use the scale to come up with the final answer. (Fn = 21 N, Ff = 8 N)

Table 6 -‐ Development of the Solution

Despite my insistence on the use of arrows, some students still used lines to

represent vectors. Students who missed the question either failed to have a proper

force diagram, couldn’t apply a consistent scale, or misused the protractor. Of the

majority who correctly answered the question, I have included some interesting

comments made by students in screencasts:

15N

15N

10 N

15N

10 N

?

?


STUDENT A (with reference to solving by adding vectors graphically): That is

an easy way because you don’t have to do anything mathematically.

STUDENT B: The larger the scale is, the more accurate the answer will be.

STUDENT B: Doubling the 10N force doubles the force of friction and increases

the normal force by 6N.

I was puzzled by this last comment from the student until I noticed that the

source of the frictional force was only dependent on the 10 N force, but only part of

the normal force was dependent on the same 10 N force (more precisely, 6N out of

21N). The student correctly grasped how changes in one force vector had an effect

on other vectors in a visual way. If I were to ask a question like this in my previous

years of teaching, I would have seen students fumble with the calculator because

they thought a mathematical algorithm would tell them what the answer was

supposed to be. This student truly understood what he was doing and was able to

express it clearly.

Also at the end of unit 4, when going over the unit assessment, I was able to

observe what was becoming evident in classroom work from some of my students.

One student clearly used trigonometry and geometrical addition of vectors to

answer questions. He used both methods as a way to “check” his answer and make

sure he didn’t make a mistake when drawing forces on graph paper. An increasing

number of students started using a mathematical model of vector components

before the end of unit 4 (all of this without me mentioning anything about trig or

vector components). Students began to ask me about this different way of solving


the problem, and I would confirm that there were alternative ways of arriving to an

answer. I noticed several students inquiring others about this different method that

apparently didn’t require graphing. It was exciting to see students become

interested in an alternative way of solving problems without any prompting from

me.

The Net Force Concept

Building the “Unbalanced Force Particle Model” in unit 5 was a big challenge.

For the initial couple of weeks into this new unit, students went back to only dealing

with problems in one-‐dimension, like the classic elevator questions. One-‐

dimensional problems are not made easy by the arduous task of adding vectors

graphically, and students quickly took notice of that. Even when dealing with

multiple forces at perpendicular angles, they quickly assimilated that summing

horizontal and vertical forces independently was enough for quickly solving the

problem. Because time was short and I did not think this process was detrimental to

them, I didn’t push too hard for graphing methods.

Finally when they were once again faced with vectors at an angle, I insisted

that students use the graphical addition of vectors to come up with a solution. Using

unit 4 as the basis, I encouraged students to develop the graphical solution in a

manner similar to the previous unit. I recorded interviews with a couple of students

after the completion of unit 5 hoping to understand a little bit of their thinking when

solving unbalanced forces problem. I decided to give them a problem that could

easily be solved by using PhET’s vector addition simulation, making it simpler for


students to manipulate vectors so they could concentrate on the physics concepts

rather than graphing skills. The question asked:

A 12 kg box is pulled by a 85 N force at an angle of 45° above the horizontal. The frictional force is 30 Newtons.

a. Draw a quantitative force diagram!! b. Find the acceleration of the box in the horizontal direction.

I helped the students determine a suitable scale (10 N = 1 grid box) and aided

them in properly graphing the 85 N force at an angle (again, I wanted students to

focus on the physics rather than some small “technical” issues). The first student I

interviewed was Cindy, a sophomore who calls herself a non-‐math student and who

finished the course with a “C.” Her pre-‐FCI score was 5 and her post-‐FCI score was

15, which is below the Newtonian threshold. The Hake gain was just below the class

average of 0.46. After reading the problem and getting herself acquainted with the

software, she started putting together a force diagram for the problem. She

managed to identify the correct number of forces (this is an important fact as we

will see later) and managed to draw them to scale except for the normal force.

Below is a picture of her initial set up.

45°


CINDY: And this (pointing to the normal force) is less than this (pointing to the

gravitational force).

TEACHER: Why? Why do you say less?

CINDY: Because it’s being pulled up . . . so the box exerts less force on earth . . .

because this is being pulled up.

TEACHER: Do you know how much that should be up?

CINDY: Ya. I should know by the calculations, but I don’t know.

TEACHER: Okay, fair enough . . . does the problem say what it is?

CINDY: Ehhh . . . no.

TEACHER: Okay. It doesn’t say.


CINDY: Oh but . . . I can . . . uhh . . . wait . . . it’s 85 like up and then down . . . no,

but then this is another thing . . . wait . . . I know how to do this (she is

looking at the question at this moment).

TEACHER: Can you go back to your drawing there?

CINDY: Okay

TEACHER: Okay, you gotta have . . . this is the . . . what you call this . . . what you

have on the screen right there?

CINDY: Ummmm, force diagram?

TEACHER: This is the force diagram right? Could you rearrange those vectors

and add them together using the head to tail method?

CINDY: (long pause and then she starts to move some vectors on the screen. She

makes a mistake on the magnitude of the normal force, which she

doesn’t realize at first) There is something missing here (pointing to

the gap between friction and normal force). Aaahhhhh.

TEACHER: Let me ask you another question . . . let me ask you another question.

Can you show me with your mouse which one is the normal force? (At

this point I’m attempting to bring her attention back to the mistake on

the normal force.)

CINDY: This one (correctly points to the normal force).

TEACHER: Is this vector, is the head of this vector at the tail of the next vector?

CINDY: Umm, is the head of the vector? (She sounds confused.)


We proceed to clarify the fact (in the next minute or so) that the normal force

is not correctly set up as can be observed in the picture above. After she

understands what the mistake is, she pulls the normal force vector downwards and

continues.

TEACHER: Now do you think this is the correct length for this vector?

CINDY: Wait this is seven (she is looking up the vector magnitude).

TEACHER: Should it be seven? (pause) That is my question. Should it be less or

more, what do you think?

CINDY: It should be less because (she mumbles something I don’t understand). I

will make less, and I will put it in here.

TEACHER: So what is it now?

CINDY: That is six . . . ooohhhh that is what we calculated . . . aahhh, so . . . oh

what is this thing in here (pointing to the gap again).


TEACHER: I ask you, what you think that thing is? What kind of problem is this

that you’re solving, is this a, ahh . . . it’s balanced forces or unbalanced

forces kind of problem?

CINDY: Unbalanced force.

TEACHER: What you think it means “ unbalanced force?”

CINDY: Like it’s moving.

TEACHER: It’s moving?

CINDY: Um hum, oh oh no no, it’s accelerating.

TEACHER: It’s accelerating.

CINDY: Umm ya.

TEACHER: So, if it’s accelerating that must be what? (referring back to the gap)

CINDY: Ahhh, there must be like it plus (it is hard to understand what she is

saying but then she proceeds) . . . in the forces so this is, so this is what

causes the acceleration?

TEACHER: Okay . . . we have a name for this; we call it the net force.

The interview continues for another few minutes trying get her to answer

part b of the question (find the acceleration). She struggles with the math even after

my repeated attempts to walk her through the problem step-‐by-‐step. Finally, after a

lot of guiding, she manages to correctly divide the net force by the object’s mass and

give a correct answer to the problem. It is obvious that the minimal math comprised

in the problem is enough to stump her and prevent her from quickly arriving at an


answer. On the other hand, I was stunned by how much conceptual understanding

she was able to verbalize by using the graphical addition of vectors.

Cindy’s first reaction was to be “uncomfortable” that vectors wouldn’t add up

to “zero,” therefore their geometrical shapes would not “close” anymore. Still she

managed to explain the concept of the net force in her own words when she pointed

to the gap and said, “this is what causes the acceleration.” She also knew that the

normal force had to be smaller than the gravitational force, a difficult concept for

some students to grasp.

In the beginning of the interview I purposely asked her how many forces

were acting on the box, to which she answered “four” (correct). At the end of the

interview I gave her three different representations of the graphical addition of

vectors (see below) for the same problem she had just answered (one was

incorrect) and asked a few more questions.


TEACHER: Now and one final question, is that third one, the one on the right

(referring to the diagram with a gap on the right) is that diagram okay

or not? What is the difference between that diagram and the middle

one?

CINDY: It’s incomplete.

TEACHER: Is incomplete? Why?

CINDY: Becauuusee, it’s missing.

TEACHER: What is it missing?

CINDY: The net force.

TEACHER: Okay, one final question, how many forces should you have on your

force diagram?

CINDY: Five.

TEACHER: Five?

CINDY: Five, you know what? One, two, three, four, five (she counts all the

forces on the middle diagram, which is like the one she drew).

TEACHER: Okay.

Cindy’s unmistakable change of mind when it came to the number of forces

that belonged in free-‐body diagram was something I repeatedly observed among my

lower preforming students. After completing the graphical addition of vectors, some

students were conflicted about the number of forces acting on the object in the first

place. I probably helped reinforce this idea by always asking my students to draw

the net force on their diagrams. The visual aspect of the net force is a positive thing,


giving students a sense that there is an overall residual force accelerating the object

in a specific direction, but several students could be led to believe that the net force

is like another force that should be included in the force diagram.

In hindsight, I would have pushed students towards the third diagram above

as a strategy for solving unbalanced forces problems. I believe it’s good to reinforce

to the students that you only draw on the graph real forces and the “gap” exists

because there is a net overall pull in a certain direction that gives the object

acceleration. Had I done things this way, I believe I would have struggled a little less

with my students down the road about drawing net forces on their force diagram.

The second student I interviewed was Paul, a bright junior who always

insisted on finding out what the “right” answer was for each assigned problem. Paul

clearly favored the mathematical solution over graphical addition, claiming it gave

him a more precise and accurate answer in less time. He finished the course with an

“A”. His pre-‐FCI score was 8, and his post-‐FCI score was 29, well above the

Newtonian threshold. Hake gain was 0.95, which is much better than the class

average of 0.46.

I gave Paul the same question that I gave Cindy and helped him set up the

scale and the first diagonal vector. Paul was a little confused when he initially tried

to set up the free-‐body diagram. He paused for long periods and used a piece of

paper to jot down notes, which I was not able to see. The interview starts at the

moment he is trying to sort out part “a” of the question (drawing a quantitative force

diagram).


PAUL: Let me just get a paper to draw the diagram.

TEACHER: Okay, no problem.

PAUL: (Long pause – sounds like he is trying to calculate something. A little

mumbling and another long pause) The frictional force . . . oh . . . the

force to the right in this case would be thirty newtons . . . so if the force

are right . . . 30 newtons (another long pause) one second (another

long pause than I hear him mumbling numbers).

TEACHER: (At this point I interrupt and try to help him with the question.) How

many forces do you have in your force diagram?

PAUL: Three forces.

TEACHER: Can you name them?


PAUL: Force of friction to the right . . . or the force of the ground on the box . . .

aahhh, the force of the earth on the box, the force of gravity

downwards . . . 120 newtons . . . aannd the force in which the box is

pulled in the diagonal to the left at 45 degrees which is 85 newtons.

TEACHER: Now does the ground exert only one force at the box?

PAUL: Ah, the ground exerts . . . ahhh, two forces the frictional force and the

upwards force.

TEACHER: Okay.

PAUL: Uhhh . . . (long pause) ahhhh.

TEACHER: (At this point I decided it was best to show him the force diagram

and move on to the question.) Uhmmm, take a look at that file that I’m

sending you.

PAUL: Wait one second (he downloads the file) saving the file.

TEACHER: Okay.

PAUL: (After looking at the diagram that I drew for him) That makes life easy.

TEACHER: Do you agree with that force diagram?

PAUL: Ya I do.

TEACHER: So can you go back there and try and . . . and see what happens?

PAUL: I will just draw the force diagram that will be easier. (He starts to

reconstruct the force diagram that I showed him.)

PAUL: (at the end, in reference to the normal force) And the upper force on the

(I can’t tell what the word is here) I have no idea what it is. (He goes on

and gives a brief description of each force – talks about how the normal


force should be less than gravity – and how that is the only force he

doesn’t know yet.)

TEACHER: Okay, is there a net force on this problem?

PAUL: Ummmm, net force on this problem? Prooobably, yeah there is.

TEACHER: How do you know?

PAUL: Because there is a force… uh “pera ai” (Portuguese, meaning wait)…

wait one-‐second . . . let me take a look at that. (long pause) Uhhhh, how

do I know that? It does have a net force because the box ahhh . . . (long

pause again) that is a good question.

TEACHER: Are the forces balanced or unbalanced? What do you think?

PAUL: The forces are . . .the force are unbalanced.

TEACHER: If the forces are unbalanced, describe the motion of the object.

PAUL: The object is accelerating.

TEACHER: Accelerating, right? Accelerates in what direction?

PAUL: It is accelerating to the left.

TEACHER: Okay, so is there a net force? Yes or no?

PAUL: Ya, there is a net force.

TEACHER: And the net force is what direction, do you think?

PAUL: The net force is going to be . . . ah the net force is going to be to the left . .

. but . . .

Paul proceeded to quickly graph the force-‐vector addition for the problem,

and, unlike Cindy, he breezed through the math to get the correct answer for the

acceleration of the box. I gave him a final question showing three different


representations of the force diagram and asked him to describe which one was

correct, just like I had done with Cindy. He easily identified the diagram to the right

as the correct one and the middle one as another correct representation by showing

the net force.

I was perplexed by how difficult it was for him to initially set up the force

diagram and to identify the kind of problem he was solving. After conquering this

initial “bump on the road” it was not difficult for him to complete the question, and

he felt all around more comfortable solving it. Paul’s difficulty in setting up the

problem and, most importantly, identifying the kind of problem he was solving, is

revealing in showing how important the initial stages are in problem solving.

Several students fall short of understanding some initial basic ideas, thus making

problem solving nearly impossible.

Comparing Paul and Cindy’s interviews, Paul had a harder time setting up the

problem, but he could just zip through the math, even without some basic

conceptual understanding of the problem. Cindy made quicker progress by using

graphical representation, although she was hindered by her limited mathematical

skills. I’m compelled to hypothesize that graphical addition of vectors really helps

students with low math ability but does little to help students with a strong math

background. Exposing students to another representation of vector addition

improves the likelihood of successful problem solving.


Mathematical Method -‐ The “Vector Components” Concept I have anecdotal evidence that several of my students were already using trig

to solve problems involving vector addition, even before I introduced vectors

components to them. A few students had recognized that there were alternative

ways of solving problems, and they were eager to avoid all the time-‐consuming

strategy of adding vectors graphically. I introduced the idea of vector components

much the same way I introduce other concepts in class. I directed them to play with

PhET simulations of vectors and discover some of their properties on their own. I

asked students to produce screencasts of the PhET simulation on vectors to see how

much they already understood of the concept. One student said,

“This vector (referring to the resultant vector) is the sum of the broken down

vectors on the y-‐axis (y components) and the sum of the broken down vectors

on the x-‐axis (x components).”

It’s incredible how much all the graphical work done with vectors helped

students assimilate properties of vectors without me having to tell them about it.

The student above demonstrated how easily he understood the idea that vectors

could be broken down into components and that addition of vectors could be done

by adding orthogonal components independently. All I had to do to help them was to

introduce sine and cosine equations to find the orthogonal components. I avoided

delving into conceptual understanding of trigonometry and strictly gave them an

algorithm to find the vector components. Students who struggled with remembering


the trig equations, or even applying the algorithm, I encouraged to use graph paper

and protractors to find vector components using the graphical method. One

interesting side effect of teaching geometrical addition of vector was students’

newfound interest in trigonometry and math. They were pleased to learn a shorter,

alternative way to solve physics problems.

After concluding unit 5, and therefore the treatment, students had the choice

of solving problems by either graphical or algebraic methods. I noticed students’

preferences were towards the algebraic method because it represented the “path of

least resistance” to them. With the exception of a couple of students, who stuck to

graphical methods to the end of the course, most students’ use of graphical addition

dwindled progressively with the course.

Comparison of Solution Method Finally during the last week of the course, I was interested in comparing

students’ ability to use graphical and algebraic methods of solution when solving an

inclined-‐plane question. At this point, most students were using trig equations when

faced with forces at an angle, and just a few were sticking with geometrical addition.

I decided to give my students two inclined-‐plane questions: the first one to be

solved by adding vectors graphically, and the second one by using trigonometry.

They were both similar “balanced forces model” types of questions, and students

had as much time as they wished to work on each question. I was very clear that

they had to stick to the asked solution method even if they didn’t know or

remember how to solve the problem in that method. I chose an inclined plane

question because it represents one of the most challenging types of problems in


dynamics for high school students, and the understanding of vector components and

their addition is at the heart of it.

Graphical Solution Algebraic Solution 1/ The diagram below shows an inclined plane (ramp). Friction prevents the mass from sliding down the inclined plane. a) Calculate the size of the frictional force acting on the mass: b) Calculate the size of the normal reaction force between the inclined plane and the mass:

2/ A mass is held in equilibrium on a frictionless inclined plane as shown in the diagram. a) Calculate the tension force, T, in the string: b) Calculate the normal reaction force between the inclined plane and the 3.4 kg mass:

Table 7 -‐ Questions using 2 solution Methods

I was hoping for a higher percentage of correct responses in both questions,

but this was the end of the school year, and I don’t think students had a lot of

motivation to give their best in these questions. The number of student who

correctly answered the question by geometrical addition of vectors (36%, or 10

students) were more than twice the number of students who could answer the

question correctly by using vector components (14%, or 4 students) (see Table on

next page). I was flabbergasted; I didn’t expect such a difference, especially

considering most students were heavily leaning towards mathematical models by

the end of the course.

35°

m = 2.5 kg m = 3.4 kg

46°


It is important to note that all the students who answered question 2

correctly also answered question 1 correctly. This fact confirmed something I had

suspected all along: students who can answer the question algebraically are very

likely to also answer it graphically; but students who can answer the question

graphically are not always able to answer it algebraically. Graphical addition

method enhances students’ ability to understand and solve a physics problem

without making it more difficult for them to solve it mathematically.

One student missed the algebraic solution because his calculator was set to

“radians” rather than “degrees,” a common mistake that we have all seen, but this is

a nonissue when solving the problem graphically. Some of the students who missed

the question by using the graphical method had the correct drawing of the forces,

but they misused the scale to get a final answer. A quick, on-‐the-‐spot, survey (hands

up in the air) of students’ feelings right after this activity confirmed what I had

suspected all along: the majority of students felt and believed they were more likely

0

5

10

15

20

25

Graphical Algebraic # of Correct Responses

Solution Method

Inclined Plane Question

Correct

Incorrect

Figure 10 -‐ Graph Comparing both Methods of Solution


to have gotten the question right by using graphical addition than by using math

alone.

FCI Pre-‐Test and VCQ/FCI Post Test At the end of the course, I also gave the students the Vector Concept Quiz

post-‐test. Comparing both the histograms below for the pre and post test, it is easy

to see students’ better performance in the post-‐test (MEAN = 4.40, SD = 2.12) as

compared to the performance in the pre-‐test (MEAN = 3.16, SD = 1.84).

I was interested in the effect of treatment on students’ vector knowledge as

measured by the VCQ. Based on non-‐directional paired samples, t-‐test at alpha =

0.05, I reject the null hypothesis that the population mean of the pretest takers is

equal to the population mean of the posttest takers, t(24) = 3.72, p< 0.001. The

differences in the mean cannot be attributed solely to statistical chance, therefore I

conclude there was a difference in students’ vector concept understanding as

measured by the VCQ, between pre and post treatment scores.

The number of students who correctly answered all questions in the quiz

jumped from two in the pre-‐test to seven in the post-‐test.

Figure 11 -‐ Comparison of VCQ Pre and Post Tests


It is also interesting to look at some of the individual questions in the quiz,

especially questions 5, 6 and 7. Below is the breakdown of correct response for each

question in the post-‐test.

There was almost no difference at all between pre and post-‐tests for the first

3 questions but a significant one for questions involving addition of vectors

(questions 4-‐7). Looking specifically at question 5 and 6 (the two question involving

addition of vectors at an angle) the percentage of correct responses more than

doubles for them.

FCI Results I gave the students the FCI pre-‐test in the first week of school, and a post-‐test

during the last week of school. The histogram below comparing the pre-‐test to the

post-‐test shows students improvement in scores. There was on average a 10-‐point

gain between the pre and post-‐tests. Twelve students got 60% or more on the FCI

post-‐test demonstrating satisfactory Newtonian thinking.

0 10 20 30 40 50 60 70 80 90 100

1 2 3 4 5 6 7

% of Correct Response

Question Number

VCQ Question Breakdown

Pre-‐Test

Post-‐Test

Figure 12 -‐ VCQ Pre and Post Test Question Breakdown


The scatter plot below displays students’ posttest scores vs. pretests scores.

All students managed to improve their scores, with some improving dramatically.

When looking at the normalized gains, also as known as Hake gain, the class had an

average gain of 0.46 (MEAN = 0.46, SD = 0.27). Gains in the range 0.7 < (<g>) > 0.3

are characterized as medium gains and my students average falls within this

category.

Figure 13 -‐ Pre and Post FCI Test Scores

Figure 14 -‐ FCI Post Vs. Pre Scatter Plot


Student Surveys At the end of the course, I administered a five-‐question survey of my students

asking them to reflect on all the work we did with vectors and forces throughout the

year. One of the questions asked the students to rate how helpful they thought the

treatment was on broadening their understanding of forces concepts; 61% of my

students rated the treatment as being helpful or very helpful, while 17% weren’t

sure, and 22% thought it didn’t help or made no difference. Students’ comments

reflect some of the study’s findings. Following are some quotes from students who

found the treatment very helpful:

“Because if we forgot how to do it through the math we had a "plan b."

“Because math is more abstract, but drawings clear out all the doubts. to (sic)

understand physics, we have to be able to visualize the problem, and drawings

help.”

“It helped me visualize what I would later do mathematically. For me, math

way is easier, but I know that for many students, solving graphically was more

appealing.”

“If we weren't sure about the mathematical solution then we could just add

them and see.”


“I like it because i (sic) can then visualize the problem and understand how the

forces are working, instead of doing algebraically and not understand how the

angles and sides related.”

Like the students above, a lot of students referenced the visualization aspect

of the graphical method as being helpful. Another question in the survey asked

students to choose their preferred method of solution. The majority of them (87%)

said they preferred the mathematical method to the geometrical one. Here are some

explanations from students:

“I already know trigonometry, and also I prefer to use math as it is faster and

more precise.”

“Even though [the] graphical method is easier to understand, I always prefer to

use the fast way.”

“Because [the mathematical method] gives more exact answers and in less

time.”

“Although geometrical addition is very helpful, I [would] rather solve problems

by using mathematical manipulation, as it is faster and still is theoretically the

same as the graphical addition.”


This was not surprising either; the predominant reason for their preferences

appears to be the ability to solve questions faster and have more “accurate”

answers. Finally I asked students if I should continue to teach graphical addition of

vectors next year and why. Only three students didn’t think it was worthwhile to

spend time adding vectors graphically, while 90% thought I should continue

implementing the treatment in years to come. Here are some of the students’

responses to this question:

“Because some people may find [the] graphical method easier than

mathematically, so its always beneficial to teach both mehods [sic], and then let

the student choose which method to apply in his studies.”

“Because [knowing how to add vectors graphically] gives multiple ways of

answering the question”

“It is better to know how to answer one problem five different ways than five

problems one way.”

“Because the method of graphically adding vectors is a very simple and

comprehensible method that the students can use. Also, it is another option that

the students can use to solve dynamics problems, in case they don't understand

the Mathematical Manipulation of Force Vectors.”


The last two quotes characterize the major findings of this study in a nutshell.

Modeling Physics curriculum trumpets the idea of multiple representation of the

physical world, and geometrical addition of vectors empowers students, specially

those lacking math skills, to look beyond mathematical algorithms and models when

trying to make sense of the world around them. Most, nearly all, of my students

were very receptive of treatment methods and could see value in them. Some

students thought we spent too much time on graphical techniques, and I can

understand their perspective, but at the end they understood they were better off

knowing this unusual method.

Conclusion

After looking closely at student artifacts, field notes, screencasts and

interviews, I cannot emphatically say that graphical methods of vector addition

improves students’ understanding of forces and vectors. Perhaps students’

conceptual gains were due to the Modeling method of instruction; perhaps it was

due to some other unknown factor—there is no way of telling based on the data

collected. However, the data show reasonable evidence of students’ improved

problem solving skills prompted by the geometrical approach to vector addition.

Additionally, introducing another way to solve a physics problem made occasionally

demanding physics problems more “student-‐friendly” by “ripping” out the obscure

math calculations. Teachers always appreciate when they can focus on physics

concepts without the math distraction, and graphing aids in that. As some students

pointed out, the decreased emphasis on the math helped them visualize and solve


the problem more effectively. Although the treatment may not have significantly

contributed to improved scores, if nothing else, it added another instrument to

students’ problem solving “toolbox.”

On the issue of qualitative data collection, I had high hopes that screencasts

would be an effective tool for gathering information on students’ thought process.

While I was still able to utilize screencast to review data, I was disappointed when

many students, especially low performing ones, did not create a screencast. Given

the opportunity to conduct the study again, I would definitely stick with screencasts,

as I believe they are a great pedagogical tool, however, I would have used them in

different ways.

The study made me reflect on ideas to improve students’ screencasts

participation and quality. To improve participation, students need to see value in

making and/or producing a screencast. “Value” is a tricky word to define when it

involves students’ motivation. I could simply assign grades (the universal student

motivator) to screencasts, but I want students to see and understand the cognitive

benefit that comes with verbalizing their thinking. Maybe one way to do that is to

allow students to work together in recording a screencast or maybe even allowing

them to be formally assessed through screencasts. In order to improve quality, I

think I need to give students more specific instructions before recording (i.e.

“discuss your observations of the net force,” or “explore the multiple ways in which

forces can be added in this problem”). I believe this would prevent students from

being vague when discussing their work in screencasts. Students cannot avoid

difficult questions or concepts when you ask them directly. Lastly, I firmly believe in


formative assessment as a way to support instructional design, and screencasts can

be an effective way of evaluating students’ learning.


Works Cited Knight, R.D. (2004). Five easy lessons : strategies for successful physics teaching. San

Francisco, CA: Addison Wesley. Flores, S., & Kanim, S. (2004). Student use of vectors in introductory mechanics.

American Journal of Physics, 72(4), 460-468.

Nguyen, N., & Meltzer, D. (2003). Initial understanding of vector concepts among students in introductory physics. American Journal of Physics, 71(6), 630-638.

Poynter, A., & Tall, D. (2005a). Relating Theories to Practice in the Teaching of Mathematics. (in preparation).

Watson, A., Spyrou, P., & Tall, D. (2002). The relationship between physical embodiment and mathematical symbolism: the concept of vector. Mediterranean Journal of Mathematics, 1(2), 73-97.

Knight, R.D. (1995). The vector knowledge of beginning physics students. Physics Teacher, 33, 74-78.

Arons, A.B. (1997). Teaching Introductory Physics. New York, NY: John Wiley & Sons, 107-111.

Hestenes, D. (1992). Mathematical viruses. Clifford Algebras and their Applications in Mathematical Physics, 3-16.

Perkins, K., Adams, W., Dubson, M., Finkelstein, N., Reid, Sam., & Wieman, C. (2006) PhET: Interactive simulation for teaching and learning physics. The Physics Teacher, 44, 18-23.

Megowan, C. (2005). “Vectors – at the Confluence of Physics and Mathematics.” Arizona State University.

Wells, M., Hestenes, D., & Swackhamer, G. (1995). A modeling method for high school physics instruction. American Journal of Physics, 63, 606-‐619.

C. E. Wieman, W. K. Adams, P. Loeblein & K. K. Perkins (2010). Teaching physics using PhET simulations. The Physics Teacher, 48, 225-‐227.


Appendix

VCQ


Unit 4 Ring/Ropes and Force Table Activity

Materials:

• Large Ring • Rope • Force Table • Hanging Masses • Spring scales

Instruction: This activity will be developed in 3 stages. For each stage, students participate in Socratic dialogue in small groups followed by whiteboarding session. After large group discussion, student are encourage to perform the kinesthetic activity and then they will be given force tables to quantify the idea of forces acting in equilibrium. Each stage ends with a worksheet for practice. Stage 1 – Collinear Forces. In stage 1 and 2 students will be asked to pull on the rope attached to the ring in such way so the ring doesn’t touch the ring stand in the center. This system is described to be in static equilibrium. Question for small group discourse:

1. How do the students need to be aligned so the ring doesn’t touch the stand?

2. Compare the magnitude of the forces being applied by the students on the rope

3. How should student 2 respond to student 1 change in force (increase/ decrease) so the system remains in static equilibrium?

Stage 2 -‐ 4 Orthogonal Forces. In stage 2, 4 students will be asked to pull on the ropes attached to the ring at right angles to each other. The ring is not supposed to touch the stand in the center. Question for small group discourse:

1. What happens if student 1 pulls differently (softer/harder) on the rope? Does everyone need to change how hard they pull on the rope? Why?

2. If I place spring scales between the ropes and the ring, compare the readings on the scale for the 4 students pulling on the ring: (qualitative comparison)

Stage 3 – 3 Forces at an Angle (other than 90). Now we remove 1 student from the set up above (stage 2). Before students observe what will happen or how the arrangement should look like so it stays in equilibrium, ask them to predict what will happen. After prediction, ask the 3 students to hold the ring in equilibrium. Question for student discourse:


1. How is this arrangement different than when we had 4 students pulling on the ropes?

2. Who is pulling the hardest? How do you know? How could you tell for sure?

3. Is/are there any arrangement that would allow them to pull on the rope with the same force?

4. Describe the effects of student 3 pulling harder on the rope. Does it change how student 1 and 2 needs to pull on the rope so it stays in equilibrium?

Every stage classroom activity is followed by force table activity in groups to quantify the qualitative analysis. Students should whiteboard conclusions following every stage and complete the worksheets.


5 kg

5 kg

Stage 1 Worksheet


1. A box rests on the floor as show in the diagram below. Calculate the force of the floor on the box:

2. A ball is suspended by cables as shown in the picture below. Calculate the Tension on each cable:

8 kg


Stage 2 Worksheet


1. In an experiment a student found that a wooden block of mass 5 kg needed a force of 4 N to make it slide over a desk at constant speed. Find the normal force and the frictional force on the block. 2. An aircraft flies horizontally at a constant speed and has the forces acting on it as it flies. Please draw a free body diagram and label possible agents (causes) of the forces. Compare the magnitude of the forces qualitatively.

5 kg 4 N


Stage 3 Worksheet


1. The object hung from the cables has a weight of 20 N. What is the tension on each cable?

2. The cable at left (T1) exerts a 30 N force. Find the weight of the ball and the tension force of the other cable?

30ο T2

T1

530 37º


3. A man pulls a 50 kg box at constant speed across the floor. He applies a 200 N force at an angle of 30°. What is the value of the frictional force and the normal force?

4. The diagram shows a box of mass 1.5 kg resting on a floor with a force 10 N acting on it. Calculate the normal force and the force of friction.

1.5 kg

10 N

37o

the!effect!of!geometric!representation!of!vector

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