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Darryl J. Ozimek MATH 791 Professor Bennett December 15, 2003 Effectively Teaching Vectors in an Introductory Physics Course 1. Introduction To effectively teach a concept (physical or mathematical) to an introductory course requires several crucial features the instructor must possess. First, the instructor must have mastery of the concept being discussed. Mastery includes content knowledge and pedagogical content knowledge. Also, the instructor should have some knowledge of how understanding a concept develops. Physics educators are confronted with discussing vectors within the first few class meetings (usually the first or second) of algebra-based and calculus-based introductory physics courses. The knowledge of vectors provides an essential foundation for beginning physics students. The introductory physics course(s) is the first experience many students have with vectors and research shows (Knight, 1995) that only approximately one-third of the students entering an introductory physics course have enough knowledge of vectors to begin study of Newtonian mechanics. Therefore, it is very important for physics instructors to effectively teach vectors with as much motivation for the student as possible. An example of motivational instruction may be providing examples and homework based on real-world student experiences. Motivational instruction may enhance student learning as well. Instructors must know their audience in order to provide effective instruction. They must know what prior knowledge the students come to class with each and every lecture, recitation, and/or laboratory. There must be a balance to the amount of time spent on quantitative and qualitative instruction to maximize student learning. Problem solving is a key aspect of 1

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Darryl J. Ozimek MATH 791

Professor Bennett December 15, 2003

Effectively Teaching Vectors in an Introductory Physics Course

1. Introduction

To effectively teach a concept (physical or mathematical) to an introductory course

requires several crucial features the instructor must possess. First, the instructor must have

mastery of the concept being discussed. Mastery includes content knowledge and pedagogical

content knowledge. Also, the instructor should have some knowledge of how understanding a

concept develops. Physics educators are confronted with discussing vectors within the first few

class meetings (usually the first or second) of algebra-based and calculus-based introductory

physics courses. The knowledge of vectors provides an essential foundation for beginning

physics students. The introductory physics course(s) is the first experience many students have

with vectors and research shows (Knight, 1995) that only approximately one-third of the students

entering an introductory physics course have enough knowledge of vectors to begin study of

Newtonian mechanics. Therefore, it is very important for physics instructors to effectively teach

vectors with as much motivation for the student as possible. An example of motivational

instruction may be providing examples and homework based on real-world student experiences.

Motivational instruction may enhance student learning as well.

Instructors must know their audience in order to provide effective instruction. They must

know what prior knowledge the students come to class with each and every lecture, recitation,

and/or laboratory. There must be a balance to the amount of time spent on quantitative and

qualitative instruction to maximize student learning. Problem solving is a key aspect of

1

effectively teaching any physical or mathematical concept; therefore the instructor must be

knowledgeable of many problem solving techniques and which ways are most effective

(minimum time spentquick solutions). The instructor must have a clearly defined set of

objectives for each and every lecture. A good attitude towards education and physics [or

mathematics] are also important characteristics for effective instruction. Each and every day the

instructor must be organized and prepared, for they are the facilitators of knowledge when they

walk into a classroom.

The paper below describes one way to effectively teach vectors to an introductory

physics class based on the authors’ views of the topic. Similar and different views from Arons

and Knight are combined with the author’s views. The theory described in this paper to

effectively teach vectors is also based on ideas of how understanding develops as discussed by

diSessa, Dubinsky, Piaget, Skemp, and Van Hiele.

2. Review of Literature

2.1 Review of Knight

All introductory physics course instructors must think of what prior knowledge their

students have when they come to class each day. During the first few days of class the topic of

vectors is usually discussed. All instructors should ask themselves the following question: How

much do my students know when they come to class? “Surveys (Knight, 1995) have found that

only about one-third of students in a typical introductory physics class are knowledgeable

enough about vectors to begin the study of Newtonian mechanics. Another one-third have

partial knowledge of vectors (e.g. a student who can add vectors graphically but not use vector

components), while the final one-third have essentially no useful knowledge of vectors.”

(Knight, 2004).

2

Students are easily confused with changes in notation. Most textbooks represent vectors

with boldface notation (e.g. F). But lately, more textbooks are beginning to use an explicit

vector notation (e.g. F ). “Students pay little attention to the boldface type.” (Knight, 2004).

According to Knight (2004), another notation difficulty is that most texts don't distinguish

between F as the magnitude of a force vector (positive values only) and F as the component of a

force vector (signed quantity) in one-dimension.

A difficulty with explicit vector notation is that some students will conclude that everything associated with a vector needs an arrow. You'll find that arrows over vector components (scalars) are not uncommon. This reflects an uncertain knowledge as to what constitutes a vector. (Knight, 2004).

“Textbooks differ as to whether the initial chapter on vectors includes the dot product and

cross product.” (Knight, 2004). An introductory physics course is the first experience many

students have with vectors and even have difficulty when discussed for the first time. The dot

and cross products are not needed until later in the course (typically, but dependent on the

instructor), so not discussing the dot and cross product during the first few days of class may

provide enhanced student learning as well as far less frustration. Once the students become more

familiar with vectors throughout the study of kinematics and into forces, then the dot and cross

products may be introduced.

As simple as the rules are, students need extensive practice to become familiar and

comfortable with vectors. Two class days are desirable, … Regardless of time, it is far

preferable to spend class time with students practicing vector problems rather than listening to a

lecture about vectors. (Knight, 2004).

According to Knight (2004), the first examples should focus on basic vector problems

and on the graphical method of adding and subtracting vectors without the use of a coordinate

system. Good class examples are:

3

bennett
Students pay little attention to the boldface type.”
bennett
This may be caused by the fact that lecturers often can't write boldface on the blackboard and students can't write in boldface in their notes. Good notation should be something the students see all the time and can use themselves.
bennett
As simple as the rules are, students need extensive practice to become familiar and comfortable with vectors. Two class days are desirable,
bennett
Is this realistic when we have only 45-60 class days available for all the material? Extensive practice will help with every topic, the trick is deciding how best to budget scarce class time. Is there evidence extra time spent on vectors will pay off in overall learning for the semester?

B

A

BA

BA

BA

+

2 c.

b.

a.

vectors theDraw

C

D

DC + vector theDraw

1-2.1 Figure

Textbook figures tend to draw the vectors in the ‘right-places,’ as in the question on the left, so

students need to face some less conventional situations. (Knight, 2004).

Knight (2004) suggests that coordinate systems and vector components are then

introduced.

It is better to introduce vector components without reference to unit vectors. Once students are comfortable with the decomposition of a vector into components parallel to the axes, then the unit vector becomes a convenient way to express this. (Knight, 2004).

He then comments: Physicists are rather cavalier in the choice of the angle to call θ, leading to

θ= cos FFx in some cases but maybe θ−= sin FFx in others. “Unlike math books, which

insist on defining θ as an angle measured from the positive x-axis, we tend to label and use

angles based on their convenience in the problem. You'll want to insist that students use a figure

to identify the angle they are using in their calculations.” (Knight, 2004).

According to Knight (2004), instructors should start with a few examples of finding the

components of a vector located at the origin but pointing into different quadrants. Then pose a

question such as Figure 2.1-2. The question in Figure 2.1-2 returns to the issue of whether the

4

bennett
Textbook figures tend to draw the vectors in the ‘right-places,’ as in the question on the left, so students need to face some less conventional situations. (Knight, 2004).
bennett
These are good problems and it is also very important that students learn to deal with pictures that aren't all drawn in the same (right) orientation.

location of the vector influences the properties of the vector. Many students will have initial

difficulties with a vector not drawn at the origin.

A

2-2.1 Figure

y

x

.A

Find the x- and y-components of vector 6 units

30°

Confusion of where a vector is with the direction it points is a big source of the difficulties students have throughout the study of motion and force. It's important to attack this problem early with numerous examples. (Knight, 2004).

Next, draw the same vector in the second quadrant. Because the vector is drawn at a point where

x is negative, many students will want to give Ax a negative value. (Knight, 2004).

Knight (2004) suggests the following: Once students can find components reliably, they

can try vector arithmetic problems. For example, give them vectors E and F and ask them to

find quantities such as 3E - 2F . Next, practice going the reverse direction with questions such

as “Vector B = 3 ˆ - 4 ˆ . Describe this vector as a magnitude and a direction.” ι j

Textbooks often define the direction of a vector as θ = tan-1 (Bx / By), but this gives a

negative angle if one component is negative and an angle in the wrong quadrant if both

components are negative. Students are confused by this, I recommend first selecting an angle

between 0° and 90° to specify the directionwhich is what physicists usually do in

practicethen using tan-1 (|By|/|Bx|) or tan-1 (|Bx|/|By|). Conclude by asking students to find the

components of vectors parallel and perpendicular to a tilted line. Even students familiar with

5

bennett
θ = tan-1 (Bx / By),
bennett
Computer languages now usually define an atan2(x,y) function that is slowly moving onto calculators that addresses this difficulty. And what is wrong with a negative angle?

vectors find this difficult, but it's clearly a prerequisite to working successfully with forces on

inclined planes. The figure below [Figure 2.1-3] is a good example. (Knight, 2004).

B

3-2.1 Figure

4 units

What are the components of vector parallel to and perpendicular to the surface? B

20°

2.2 Review of Arons

“Most presentations of the concept of a vector start with the representation of

displacements in two dimensions and develop the process of addition of such quantities in an

intuitive way. This is, without question, the most reasonable and effective starting point, and

students have relatively little difficulty with the ideas in the early stages.” (Arons, 1997). Many

textbooks suggest to obtain the negative of a vector, one reverses the direction of the original

arrow. In other words, often used by textbooks (i.e. Glencoe Physics: Principles and Problems,

1999), one may simply multiply by the scalar factor of [-1] to obtain the negative of a vector.

Arons (1997) believes that the reason is obvious to us, but it turns out to be far from obvious to

many students. The students typically hesitate to ask for any reason that may enhance their

understanding of vector subtraction and simply memorize the assertion of multiplying by the

scalar factor of [-1] without any understanding.

A more effective way of introducing the operation of subtraction is to adopt the systematic procedure of mathematics and ask what must be added to a given vector to obtain a zero vector. (Arons, 1997).

“This serves to define subtraction by tying it to the original starting point, namely addition, and

gives the student logical continuity rather than abrupt change and unsupported assertion.”

(Arons, 1997).

6

bennett
This serves to define subtraction by tying it to the original starting point, namely addition, and gives the student logical continuity rather than abrupt change and unsupported assertion.”
bennett
I am dubious about this approach. Students in physics have intuitive notions of addition and subtraction and taking a negative by multiplying by -1 is almost certainly closer to their intuitive picture of taking a negative than is the more formal approach of asking what must be added to get to 0. This reads more like what a theorist thinks than like actual student thinking, and I would like to see clear evidence this approach works better in actual instruction before accepting it.

According to Arons (1997): “Another property of vectors, frequently taken for granted in

instruction without being made explicit, is that of ‘movability.’ Many students tenaciously hold

an initial view that vectors are ‘attached to point.’ One can see how this notion gets planted:

Displacements, the first vectors encountered, begin at a fixed position, and a sequence of

displacements proceeds from point to point with the arrows head to tail, each tail rooted at the

initial fixed position; velocity vectors appear to be attached to particles; concentrated forces act

on object at a point.”

The fact that vectors are not attached to points requires explicit discussion if it is to be understood and used in attacking problems. Many students would benefit from more exercises and drill in graphical handling of vector arithmetic than are usually available in textbooks. (Arons, 1997).

Simply defining a vector as a quantity with a magnitude and a direction is not a complete

definition. The commutative laws of addition and subtraction must be included. For example, a

non-vector quantity with a magnitude and a direction is finite angular displacements (because

they do not commute on addition). Finite angular displacement is a concept most students may

not be knowledgeable about, but may be illustrated by a simple demonstration/activity. Have the

students rotate their textbook (or a piece of paper) through two successive 90° displacements

about two different axes and show the textbook/paper winds up in two entirely different final

orientations if the order of rotations is reversed. The final part of the definition of a vector

resides in behavior with respect to transformation under rotation of coordinate axes. (Arons,

1997).

I find that an understanding of these distinctions, and of the need of extension of the basic definition beyond the requirement of commutation in addition and subtraction, comes far more easily to students in more advanced courses if they have the advantage of having been gradually exposed, in introductory courses, to the simpler ideas outlined above, instead of suddenly encountering all of them de novo at the advanced level. (Arons, 1997).

According to Arons (1997): “The concept of orthogonal (or Cartesian) components of

vectors seems so simple and transparent to teachers, and manipulations, when the Cartesian axes

7

are given in a problem, are so easily memorized by students, that many significant student

difficulties in this area go unnoticed. Interviews with students, however, reveal very significant

gaps in understanding.” (Arons, 1997).

Consider the two diagrams shown in Figure 2.2-1 (Arons, 1997). “If one draws diagram

(a) and asks the student to ‘show graphically how large an effect the vector represented by the

arrow (perhaps a force or a velocity) has along the direction indicated by the line,’ many students

find themselves at a loss and are unable to answer the question. If one draws diagram (b) and

asks the same question, still more students are unable to answer. (In the latter case the difficulty

has been enhanced by the fact that the line does not pass through the tail of the arrow...).”

(Arons, 1997).

What is the magnitude of the “effect” of the vector in the direction indicated by the line?

1-2.2 Figure (a) (b)

Students are unable to solve the problem indicated in Figure 2.2-1 because nothing triggers the

student. There is no mentioning of the word “component”, no Cartesian axes is provided in a

familiar orientation, and no angle is labeled with a symbol that is familiar to the student. “In

other words, students exhibiting this difficulty have not formed an understanding of the

concept.” (Arons, 1997).

8

Practice problems are needed for many students beyond what is provided by most

textbooks in the case of graphical addition and subtraction of vectors in interpreting components

and in adding and subtracting vectors arithmetically by the use of rectangular components.

(Arons, 1997).

3. Précis of Thoughts on How Understanding Develops

The theory presented in this paper is a particular example of how a student understands a

concept [vector]. There are many other theories that may work as well or even better depending

on the particular situation (e.g. teacher style, student abilities/disabilities, teacher-student

relationship, environment, etc.). In order to effectively teach our students, instructors must be

aware of where their students are when they come to class each and every lecture. In effectively

teaching any context, diSessa sums it up nicely in the following quote:

We need to start where students are in terms of activities as well as where they are in terms of concepts. (diSessa, 1994).

According to Knight (1995), only about one-third of the students entering a typical introductory

physics class are able to understand enough about vectors to begin Newtonian mechanics.

Therefore, it is very important for instructors to begin at the very basics when introducing

vectors to students. It is equally important for the instructor to make learning as fun as possible,

for this may be the first experience many students have had with physics. For the above to

happen, instructors must have mastery of the content, pedagogical content knowledge, and some

knowledge of how understanding develops.

From Piaget's psychological viewpoint, new mathematical constructions proceed by

reflective abstraction (Dubinsky, 1991; Beth & Piaget, 1966). “According to Piaget, the first

part of reflective abstraction consists of drawing properties from mental or physical actions at a

particular level of thought.” (Beth & Piaget, 1966).

9

“Reflective abstraction is a concept introduced by Piaget to describe the construction of

logico-mathematical structures by an individual during the course of cognitive development.

Two important observations that Piaget made are first that reflective abstraction has no absolute

beginning but is present at the very earliest ages in the coordination of sensori-motor structures

(Beth & Piaget, 1966) and second, that it continues on up through higher mathematics to the

extent that the entire history of the development of mathematics from antiquity to the present day

may be considered as an example of the process of reflective abstraction.” (Piaget, 1985).

Reflective abstraction begins at an early age and continues throughout life. People (students)

utilize reflective abstraction no matter where their current level of understanding happens to be.

Students, people in general, are continuously constructing physical structures of experiences in

their everyday life. Therefore, reflective abstraction plays a crucial role in the understanding of

a physical [or mathematical] concept. Students continue to utilize reflective abstraction in

situations that increase in complexity as well as difficulty in order to understand a physical [or

mathematical] concept.

To develop new physical [or mathematical] understanding, the subject proceeds through

three major kinds of abstraction. According to Piaget, these are empirical, pseudo-empirical,

and reflective abstraction. Empirical abstraction is the knowledge gained from an experience

with an external object in which the subject performs (or imagines) actions on the object.

Pseudo-empirical abstraction is the intermediate step between empirical and reflective

abstraction. Pseudo-empirical abstraction occurs after the actions have taken place. In the

intermediate step the subject engages with the external object and teases out properties of the

actions introduced into the object. In the final step of developing new physical [or mathematical]

understanding, the subject internally undergoes reflective abstraction. During the final step, the

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subject reflects all actions on the object and develops a schema (conceptual structure) of the

knowledge gained.

“More generally, Piaget considered that it is reflective abstraction in its most advanced form that leads to the kind of mathematical thinking by which form or process is separated from content and that processes themselves are converted, in the mind of the mathematician, to objects of content.” (Dubinsky, 1991; Piaget, 1972).

I wish to note that reflective abstraction occurs internally. Therefore to change one's knowledge

and build new understanding of a portion of a physical [or mathematical] concept, the subject

proceeds through a complete cycle of empirical, pseudo-empirical, and reflective abstraction. In

order to begin the process of developing new understanding of the entire physical [or

mathematical] concept, reflective abstraction is used to assimilate various schemas to perform

new actions [empirical abstraction] on the external object.

Through time, the subject will discover, or be presented with, advanced topics. When I

speak of an advanced topic, I mean one in which the subject has no existing schema based on

any empirical or pseudo-empirical abstraction of the object (or concept). Dubinsky lists various

kinds of constructions that occur during reflective abstraction (heavily based on the work by

Piaget) for when the subject encounters a new topic. The first is interiorization and is referred to

as “translating a succession of material actions into a system of interiorized operations” (Beth &

Piaget, 1966). The subject constructs an internal process (interiorization) as a way of making

sense of the topic encountered when using symbols, communicating by language, and drawing

diagrams when posed with an advanced topic. The subject may use a coordination of two or

more processes to construct a new one. Also, the subject may use encapsulation (conversion) of

a dynamic process into a static object. The subject may learn to apply an existing schema to a

wider collection of phenomena in which Dubinsky would say that the schema has been

generalized. Finally, Dubinsky adds a fifth construction process to Piaget's first four. The fifth

11

process is internal in which the subject reverses the original process to construct a new process.

(Dubinsky, 1991).

During the reflective abstraction process, the subject will internally build a schema or

several schemas. “A subject's tendency to invoke a schema in order to understand, deal with,

organize, or make sense out of a perceived problem situation is her or his knowledge of an

individual concept in mathematics.” (Dubinsky, 1991, p. 102). Research (Skemp, 1987, p. 26)

has shown schematically learnt material was not only better learnt, but better retained.

Once the student has assimilated the content with the appropriate schemas through

reflective abstraction, he or she will be at a certain “Van Hiele level” for vectors. Van Hiele

(1986) classified five levels of thinking:

First level: the visual level Second level: the descriptive level Third level: the theoretical level; with logical relations, geometry generated according to Euclid Fourth level: formal logic, a study of the laws of logic Fifth level: the nature of logical laws When studying vectors, I believe the students proceed through developing schemas by reflective

abstraction and therefore move through different Van Hiele levels. According to my theory,

which will be elaborated on in the next section, the Van Hiele levels for vectors are:

First level: the geometric level Second level: the descriptive level Third level: the algebraic (commutative and associative laws, multiplication of a vector by a

scalar) and vector component level Fourth level: the multiplication of vectors by vectors and vector transformation level

A result of undergoing reflective abstraction, developing schemas, and being at a Van

Hiele level for vectors, is the students have proceeded through a genetic decomposition. A

genetic decomposition of a concept according to Dubinsky (1991) is a description of the

mathematics involved and how a subject might make construction(s) that would lead to an

understanding of it. I will extend Dubinsky’s definition of genetic decomposition to mean a

12

description of the physics involved as well as the mathematics involved in understanding a

physical/mathematical concept.

In summary, the construction of various physical concepts may be described through

each of the four kinds of reflective abstraction: interiorization, coordination, encapsulation, and

generalization. The student uses previous schemas or a combination of several previous

schemas to proceed through reflective abstraction and, ultimately, understand a physical concept.

Once the student has assimilated the schemas through reflective abstraction, he or she will

proceed through various Van Hiele levels for vectors. Throughout the entire process, a genetic

decomposition of vectors will be assimilated of student understanding that may be very

beneficial for the instructor to effectively teach a particular topic, in my case vectors.

4. Development of Effectively Teaching Vectors Based on How Understanding Develops

To effectively teach vectors to students, the instructor must posses a degree of mastery of

the content. The instructor must also know what his or her students’ prior knowledge happens to

be at the time of instruction. To understand a physical [or mathematical] concept, the subject

[student] must be able to recognize patterns, organize knowledge (chunking), recall previous

knowledge rapidly, and have strong metacognitive skills (Bransford, 2000). Also, to understand

a physical [or mathematical] concept, the student’s must have the ability to successfully solve

problems within several contexts (e.g. physical and abstract situations). Understanding a concept

means being able to understand in low and high levels of abstraction as well as concrete

contexts. Students should also actively monitor their learning. Actively monitoring ones own

learning results from practice/drill problems with the concept being discussed during class as

well as outside of class. Comparing and contrasting various problems in several contexts is also

achievable when one understands and therefore a necessity when the instructor chooses problems

13

for his or her class. In short, the instructor must posses content knowledge, pedagogical content

knowledge, and knowledge of how understanding develops to effectively teach a concept.

Schoenfeld (1985) implies student’s need: resources, heuristics, control, and belief. In

order for a student to understand a physical [or mathematical] concept, the student must have

resources that consist of knowledge learned from previous experiences (in educational situations

as well as non-educational situations). Heuristics is the ability to apply the resources learnt to

logically solve the given problem(s). Control is the ability to evaluate one’s own thought

process. If the student is not making progress through the assigned (or unassigned) material, the

student must evaluate the situation and look at a different resource or heuristic. Once the student

understands a concept, they do not choose paths of solving problems that are more time

consuming. Beliefs allow the subject to use all of the useful instruction and disregard any

information that does not lead to a quick solution. When the students have the necessary

resources, heuristics, control, and beliefs, they are able to begin understanding a physical [or

mathematical] concept.

A crucial aspect of effectively teaching vectors to an introductory physics class suggested

by Arons and Knight is to provide numerous practice problems. As indicated by both Arons and

Knight, textbooks do not provide enough practice for students to understand vectors to begin

study of Newtonian mechanics. Textbooks also draw the vectors in the ‘correct places,’ which

may cause difficulties for the students when they begin discussing forces. Therefore, instructors

should provide many abstract problems that focus on the concept being discussed as well as

context-rich problems that relate to the student’s everyday life. These problems as suggested by

Arons and Knight should also consist of situations where the vectors are in less conventional

places (i.e. right-hand side question of Figure 2.1-1).

14

One tool that instructors may utilize to effectively teach a concept is to provide a genetic

decomposition (Dubinsky, 1991) of the concept being addressed. A genetic decomposition

benefits the students by enabling them to visualize how the definition of the concept is

constructed as well as how all parts of the concept (object) relate to one another. The genetic

decomposition also provides the instructor with the objectives he or she will want to address

during the course of instruction. An example of a genetic decomposition of vectors is provided

on page 22.

To effectively teach any concept to students, the instructor should begin with a complete

definition of the concept being discussed as suggested by Arons (1997). Instructors should tell

the students that throughout the course of instruction, the definition will become more evident

even though they may not understand all parts of the definition at the current time of instruction.

Arons (1997) brings up an interesting point; he believes that an understanding beyond the typical

definition of a vector (magnitude, direction, and also the commutative and associative laws

which are usually not explicitly stated by instructors) comes easier to students in advanced

courses if they have been gradually exposed to what completely defines a vector (magnitude,

direction, commutative and associative laws, vector transformation upon rotation of the

coordinate axes) in introductory courses. I feel that the above comment from Arons is one that

all introductory physics instructors should carry out. There may be far greater retention of what

defines a vector if the students are gradually exposed to the definition throughout the

introductory course(s).

Instructors should be very conscious to what notation they are using throughout the entire

course. Knight (2004) observed that textbooks differ in the representation of vectors. Also, he

found that students are easily confused with changes in notation. Therefore as Knight suggests,

15

bennett
To effectively teach any concept to students, the instructor should begin with a complete definition of the concept being discussed as suggested by Arons (1997).
bennett
My experience in mathematics also makes me suspicious of this claim. Students usually aren't ready to understand a complete definition, and it is very far from clear that giving the full definiton early helps them develop a correct concept image.

instructors should be consistent with how they represent a vector. The most convenient and

proper representation of a vector should consist of a letter with an arrow above the letter (i.e.

F ). When discussing the magnitude of a vector, the instructor should write the vector as it

appears with the ‘absolute value signs’ (i.e. F ). Students will then explicitly see the symbol

they are now using is indeed a vector quantity (noted by the letter with the arrow above the

letter), but only the magnitude of the vector is currently being used in the explanation or practice

problem. Changes in notation or sloppiness by the instructor (or textbook) do not aid in student

understanding or retention. Staying consistent with the notation you are using will be beneficial

to student understanding and retention throughout the introductory course(s) as well as retention

in advanced courses.

Understanding a physical [or mathematical] concept begins with reflective abstraction

(Dubinsky, 1991; Beth & Piaget, 1966). All students are continuously constructing new physical

structures of their everyday experiences beginning at a very early age (Beth & Piaget, 1966).

The instructor must be aware of the current state of mind of his or her students in accordance

with their everyday experiences. Instruction should be focused on real-world applications of

physical concepts as well as abstract contexts. Therefore for students to understand a physical

[or mathematical] concept, the instructor should be aware of how reflective abstraction occurs in

the students’ mind.

In order for a student to begin to understand vectors, the student proceeds through three

types of abstraction. First the student undergoes empirical abstraction on an object from his or

her previous experiences with the object (for many students, the previous experience they may

have with vectors may be from what the instructor discusses when defining a vector and vector

notation). The student performs or imagines actions on the object. In the context of beginning to

16

understand the graphical addition vectors, the student will perform or imagine actions (i.e. tip-to-

tail method of adding vectors graphically) on the object (the vectors). The first typical

experience the students have with vectors is to begin by adding displacements in two dimensions

and develop the process of addition as suggested by Arons (1997). When the students

graphically add vectors for the first time (interiorization), they should be given practice problems

without a coordinate system as suggested by Knight (2004). All too often textbooks draw the

vectors in the ‘correct positions’ and the students need to face less conventional ways of

graphically adding vectors. A benefit from carrying out the above process of instruction is when

students encounter force problems, they will have experience with adding vectors that are not

originating from exact positions.

Next the student proceeds through pseudo-empirical abstraction. Once the actions have

taken place (placing the tail of the second vector to the tip of the first vector), the student teases

out the properties of the actions placed on the object. The properties of the actions when first

experiencing the graphical addition of vectors may be the student or instructor asking the

following question: What happens when I reverse the order of the vectors being added?

Finally the student develops new physical [or mathematical] understanding of the

graphical addition of vectors through reflective abstraction. At this point, the student reflects on

all actions introduced to the object(s) and the properties he or she posed on the object(s). Also

when new understanding of graphically adding vectors occurs in the students’ mind, they are

able to understand what Arons refers to as ‘movability.’ Examples provided by the instructor

should include problems in which the vectors do not appear as attached points and may be solved

in a manner in which they are not attached (see Figure 2.1-1).

17

The above process of empirical abstraction, pseudo-empirical abstraction, and reflective

abstraction is then repeated and coordinated for all additional topics discussed (graphical

addition with coordinate systems, decomposition of vectors into components parallel to axes,

algebraic addition, unit vectors, multiplication of a vector by a scalar, graphical & algebraic

subtraction of vectors). Throughout the process, the student begins to develop schemas

(conceptual structures) of the concepts being discussed and generalizes the concepts (applies

existing schemas to a wider collection of phenomenai.e. decomposition of vectors). Also

throughout the process of developing new understanding of vectors, the level of student thinking

may be expressed by the corresponding Van Hiele level for vectors. The end result is a coherent

conceptual structure (large schema) of vectors (definition, notation, graphical & algebraic

addition, components, unit vectors, graphical & algebraic subtraction of vectors) and the students

are now at the third Van Hiele level for vectors. As time progresses throughout the introductory

course, multiplication of a vector by a vector and transformation of vectors upon rotation of

coordinate axes may be elaborated upon to complete the definition of a vector (see Genetic

Decomposition of Vectorspage 22).

I agree with Knight that once problems based on the graphical addition of vectors is

introduced, coordinate systems should be the next topic of discussion. However, I do not agree

that vector components should be introduced at the same time as coordinate systems. Only once

the students have had successful practice with graphically adding vectors with and without a

coordinate system should vector components be discussed.

When introducing vector components to the student for the first time, instructors should

follow Knights suggestion: Start with a few examples of finding the components of a vector

located at the origin but pointing to different quadrants. Then pose a question such as Figure

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2.1-2 followed by a similar question with the vector in the second quadrant so that students are

able to generalize (Dubinsky, 1991) whether the location of the vector influences the properties

of the vector (also see Figure 2.2-1). Students will benefit by experiencing problems similar to

Figure 2.2-1 because there are no key terms/words that may trigger the students to find the

component of the vector. The students will also begin to see the importance of correctly drawing

a figure and labels (i.e. angles) with the correct mathematics (unlike traditional mathematics

which define θ as an angle measured from the positive x-axis) as Knight suggests.

I also agree with Knight that once students can find components successfully, algebraic

addition of vectors should be the next topic of discussion (examples are included in section 2.1

Review of Knight). At the same time of algebraic addition, multiplication of a vector by a scalar

may be introduced to provide additional practice for the students. Up to this point the students

have extensive practice with graphically and algebraically adding vectors with and without a

coordinate system and are able to easily find components of vectors with and without a

coordinate system. Then the unit vector may be introduced as a convenient way to express

decomposition of vectors parallel to the axes.

Textbooks often define the direction of a vector in a conventional manner [i.e. θ = tan-1

(Bx / By)], but in some cases provided by the instructor (or textbook), the students may need to

adjust the definition the textbook gives for the direction of a vector. At this time the instructor

may select an angle between 0° and 90° to specify the direction and ask students to find the

components of a vector parallel and perpendicular to a tilted line as suggested by Knight (see

Figure 2.1-3). Then examples or practice problems may follow with less conventional angles

(i.e. angles > 90°).

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When discussing subtraction of vectors algebraically, I concur with Arons suggestion: A

more effective way of introducing the operation of subtraction is to adopt the systematic

procedure of mathematics and ask what must be added to a given vector to obtain a zero vector.

When introducing subtraction of vectors to students in the manner mentioned by Arons, the

students will have less difficulty with the concept of algebraic subtraction of vectors as opposed

to simply multiplying the vector by the scalar factor of [-1]. As Arons comments, carrying out

the above step for defining subtraction ties into the original starting point (addition) and provides

logical continuity.

Textbooks also normally include sections on the dot (scalar) and cross (vector) products

when vectors are first introduced. Students will not need this knowledge for some time (time

depends on the instructor and course) and should not be completely discussed (only as part of the

definition). Once the students have had time to experience geometric and algebraic applications

of vectors through several contexts (kinematics and forces for example), then the dot and cross

products should be discussed in detail. As the students become more familiar with vector

notation and graphical/algebraic problems involving vectors, students may benefit by teachers

postponing instruction of multiplication by vectors until needed.

Once the student’s move through the genetic decomposition of vectors, they appear to be

at certain Van Hiele levels of thinking for vectors. The first level is the geometric level. The

students are at this level when they are able to solve graphical addition problems of vectors. At

the next level, the descriptive level, the students are able to define a vector, represent a vector

with correct notation, and graphically add vectors. The students then continue to develop

schemas through reflective abstraction and proceed toward the third Van Hiele level for vectors,

the algebraic and vector component level (of addition). At this time, subtraction of vectors is

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introduced. The students remain at the third Van Hiele level for vectors of addition (one large

schema for addition), but need to move to a lower level when discussing subtraction. The

student’s use portions of the schema they have developed through reflective abstraction for

addition of vectors to develop a large schema for subtraction of vectors (graphical and algebraic

subtraction of vectors). Once the two large schemas are developed for addition and subtraction

of vectors (graphically and algebraically) the students may continue to construct the final

definition of a vector (multiplication of a vector by a vector and vector transformation) through a

similar process and ultimately fully understand a vector (fourth Van Hiele level for vectors).

5. Application of Effectively Teaching Vectors

Discussion of any questions based on why I chose the method below to effectively teach

vectors to an introductory physics class is described in the previous section. To effectively teach

vectors to an introductory physics class, the following steps (Genetic Decomposition – page 22)

should be implemented in chronological order according to the numbers indicated in the boxes.

6. References

Arons, Arnold B. (1997). Teaching Introductory Physics. John Wiley & Sons, Inc. New York, NY.

Bransford, John, Ann Brown, and Rodney Cocking, (ed.) (2000) How People Learn, National Academy

Press, Washington, DC.

diSessa, Andrea (1994), "Comments on Ed Dubinsky's Chapter," in Mathematical Thinking and Problem

Solving, Schoenfeld, Alan (ed.), pp.248-256, Lawrence Erlbaum Associates, Hillside, NJ.

Dubinsky, Ed (1991), "Reflective Abstraction In Advanced Mathematical Thinking," in Advanced

Mathematical Thinking, Tall, David (ed.), pp.95-123, Kluwer Academic Publishers, Boston, MA.

Knight, Randall D. (2004). Five Easy Lessons: Strategies for Successful Physics Teaching, Addison

Wesley, San Francisco, CA.

Schoenfeld, Alan (1985), Mathematical Problem Solving, Academic Press, New York, NY.

Skemp, Richard (1987), The Psychology of Learning Mathematics (Expanded American Edition),

Lawrence Erlbaum Associates, Hillside, NJ.

Van Hiele, Pierre (1986), Structure and Insight: A Theory of Mathematics Education, Academic Press,

New York, NY.

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bennett
the final definition of a vector (multiplication of a vector by a vector and vector transformation) through a similar process and ultimately fully understand a vector (fourth Van Hiele level for vectors).
bennett
I don't think this is "the final definition." In advanced physics as well as mathematics, we often encounter function spaces acting as vector spaces and if using a full definition has any value, it is to prepare students to expand their notion of vector to situations where geometric notions are of limited use like this.
bennett
6. References
bennett
You should look at some of the other articles in Tall's volume on Advanced Mathematical Thinking (which is where I got Dubinsky's article from). In particular, Vinner's article on The Role of Definitions and Harel and Kaput's article on Conceptual Entities and their Symbols address many of the ideas about the role of definitions and notation in developing conceptual understanding that come up in your paper.

Numbers Symbols Unknowns

VariablesAddition/Subtraction/ Multiplication/Division

Formula

PythagoreanTheorem

Trigonometric Functions for Right Triangles

(i.e. ) AA

4. Notation: Scalar (i.e. )A

22. Multiplication of a Vector by a VectorDot (scalar) Product

Cross (vector) Product

23. Transformation of Vectorsupon rotation of coordinate axes

6, 9, 12, 16, 20. Movability

15, 19. Algebraic Addition

17. Multiplication of a Vector by a Scalar

14. Associative Law

21. Graphical & Algebraic Subtraction what must be added to a vector to obtain a zero vector

13. Commutative Law

18. Unit Vector

10. Resolution Decomposition of vectors intocomponents parallel to the axes

8, 11. Graphical Additionwith Coordinate System

5. Graphical Addition without Coordinate System

3. Notation: Vector (i.e. )

Magnitude

2. Definition of a Scalar: Magnitude with Appropriate Unit

1. Definition of a Vector: Magnitude with Appropriate Unit & Direction

Commutative & Associative Laws Transformation of Vectors

7. CoordinateSystems

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