Post on 22-Mar-2018
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
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
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
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
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
10
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
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
18
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°).
19
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
20
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.
21
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
22