Metonymies and Object Formation
Vector Space Theory
Abstract: Paper discusses the use of existing metonymies in reasoning with advanced mathematics tasks,
and the role of these metonymies in the formation of a new object (metonymy).
Introduction
Various theories such as APOS (Parraguez and Oktac, 2010), Balacheff’s theory of conceptions and
Fischbein’s theory of tacit models (Maracci, 2003) are used in interpreting and understanding the
cognition of linear algebra concepts. We use the framework of metonymy as cognitive construct
(Presmeg, 1998; 1997) in our interpretation of a linear algebra student’s interview responses. Studies on
metonymy have mainly been at the pre-secondary level documenting the role and influence of metonymy
in reasoning and object construction. There have however not been very many studies documenting its
role and use at the college level especially at the advanced level topics such as linear algebra and analysis.
Linear algebra is one of the advanced mathematics courses with high degree of abstraction and
symbolism, which require learners to be able to cope with abstract representations. Our work identified
frequent use of metonymic reasoning while working with abstract representations. In this paper, we
discuss a linear algebra student’s reasoning with metonymies and his attempt to use these metonymies to
construct another metonymy, thus a new mathematical object.
Framework
Work on metonymies mainly focuses on them as literary devices, rather than cognitive constructs, that are
used to encode information. Presmeg (1998; 1997) and Lakoff &Johnson (2000) on the other hand view
metaphor and metonymy as cognitive structures. The act of using one object to stand for another is
considered as functioning with metaphors or/and metonymies. Even though both constructs are used to
represent objects and concepts, they do differ in that a metaphor links one domain of experience with
another domain, and gives meaning from similarities (Authors, 2007) while a metonymy uses one
element or relevant attribute of a class to stand for some part of the same class via association. Presmeg
(1998) states that “metaphors, metonymies, and the imagery and symbolism which accompany them are
essential components in the representation of mathematical constructs for an individual.” She further
supports her argument by explaining that these entities may help learners in making sense of the construct
accompanied by ambiguities inherent in its representations. In fact, metaphors and metonymies are
conceptual entities that may influence one’s reasoning of an abstract mathematics task. They may aid in
making sense of symbolism apparent in advanced abstract topics, furthermore, support a process of
meaningful knowledge formation.
Presmeg (1998;1997) considers two types of metonymies. One of which, namely metonymy proper, is
defined as “a figure by which one word is put for another on the account of some actual relation between
the things signified” (Webster). An example of this kind is “We studied Gauss.” Here, the word “Gauss”
is used to indicate Gauss’ work (Gauss
Gauss’ work). Moreover, mathematical symbols can be put for
various mathematical entities such as number families. The symbol “x” for example can represent real
numbers (x
real number) even though the symbol x and the numbers are two unrelated objects.
Another example of a metonymy proper may be a geometric image of a plane which may represent the
mathematical attributes of vector spaces. The attributes of the mathematical object recognized from the
image however are subject to the individual’s interpretation of it. The geometric image may in fact be
both a metaphor and a metonymy (Authors, 2010). An individual may first consider the image as having
similarities with various aspects of vector spaces, and after the initial consideration of the image as a
metaphor, the same individual may begin considering the image as a symbol that is solely put for the
concept itself.
Second type of metonymy is considered as figure of speech. In this type, a part is used to represent the
whole or vice versa (Presmeg, 1998). An example of this kind may come from the sentence, “I’ve got a
roof over my head.” Here, the part “roof ” stands for the whole “house” (roof
house). An illustration
of a circle taken to represent the class of all circles can also be considered as the metonymy of this kind.
Presmeg (1997) however argues that this example may go beyond the figure of speech type to metonymy
proper for the signifier may not be an element of the class represented. In other words, because the
elements of classes are mental constructs, and an act of interpretation by an individual is involved in
setting up the metonymy, the individual may use the illustration to consider a class of circles that are not
closely related to the illustration. Hence, the illustration may become an example of a metonymy proper.
Method
The data discussed in this paper came from our work with two groups of students enrolled in three
sections of a matrix algebra course at a Southwest University in USA—one traditional and the other two
implementing an interactive web-module that provided the geometric representations of abstract linear
algebra concepts. Students volunteered for a set of interviews conducted during spring 2009. The student
whose interview responses discussed in this paper is from a module section. We use an alphanumeric
name “SA21,” to refer to him throughout the paper. He is an Hispanic-American majoring in mathematics
with a secondary education minor. He was interviewed toward the end of April, 2009. Interview began
with a set of pre-determined questions on basic vector space concepts such as linear independence, span
and spanning set, and included new questions whenever necessary. Pre-set questions were structured
based on the learning difficulties reported in the literature (Authors, 2010; Sierpinska, 2000). A
qualitative approach, namely the constant comparison method (Glaser, 1992), is used to analyze the
responses.
Existing Metonymies
Linear independence
New Metonymy
Matrix
Identity
xi
set
linear independence
Solution
vector
Linear combination form
Values for linear combination
Figure 1. Metonymies displayed in SA21’s reasoning.
Results and Discussion
Data provided here came from SA21’s interview responses to a question “Define the linear independence
of a set of vectors.” Interview began student sharing his definition of linear independence, and continued
with more relevant questions. What readers may observe in this section is SA21’s attempts to understand
how two seemingly dissimilar (to the student) aspects of the same concept are related.
SA21 shared two main notions with metonymies embedded for the linear independence/dependence of a
set of vectors. One was his notion of linear combination. With this idea, SA21 was able to accurately
identify linearly dependent sets provided that he could obtain a linear combination among the vectors of a
set resulting in another vector of the set. The second idea he held throughout the interview focused mainly
on the identity form of a matrix. Whenever a set with vectors given, where a linear combination is not
accessible easily, SA21 appeared to proceed directly (skipping vector equations) to representing vectors
with a matrix and searching for an identity form via Gauss-Jordan elimination process, which is a part of
one of the approaches included in the textbook (Johnson, Reiss and Arnold, 2001) and covered in class.
Using the two notions, SA21 was able to accurately identify linearly independent/dependent sets without
feeling any need to make connections between the solutions of vector equations and linear combinations.
Throughout the interview SA21 was prompted continuously by the interviewer to discuss his parametric
representation of solutions and its connection to the linear combinations he provided. During his attempts,
toward the end of the interview, he began to apply his existing metonymies, and came up with a new
notion of how the two entities may be related. We believe that SA21 was, at the start of the interview,
unaware of any connections between the two objects, but toward the end he began to consider the
potentiality.
Before proceeding with SA21’s responses, in order to provide a context for the responses included in this
section, let’s discuss one of the examples student gave. After sharing his notion of a linear independence,
1
1
 2
 3
 4
5 
SA21 was asked to give an example. He gave the set {u   , v   , w   } and continued to
explain how he identified this set as a linearly dependent one: first, he considered a matrix whose
columns were formed by the vectors of the set. After applying Gauss-Jordan elimination process, he
1 0 2
 , circling the identity
0 1 1 
obtained the row reduced echelon form (rref) of the matrix, which is 
form as seen in figure 2, I. Furthermore, using the rref form he identified the set as linearly dependent
reasoning that the last column of the matrix is depended on the first two. He proceeded to write both
parametric and vector representation of solutions (see figure 2, IV, III respectively). Next, when asked to
explain how his solutions may imply the linear dependence of the set, he wrote the linear combination
2u+v=w directly using the numerical entries of the vectors of the set ignoring his solution representations
(see figure 2, II). In fact, many of the excerpts provided in the results section are revealing SA21’s
attempts to explain how one may obtain a linear combination among vectors of a set using parametric or
vector solution forms.
At this point, let’s share our perspective of how a linear combination can come from a solution. Consider
the parametric representation of solutions seen in figure 2, IV. One solution would be (-2, -1, 1) with
variable x3 assigned value 1. That is, this particular solution satisfies the vector equation x1u+x2v+x3w=0
for the vectors u, v and w (This connection appeared to have been missing in SA21’s knowledge during
the interview). That is, -2u-v+w=0. Solving the equation for w, one would obtain the vector equation
w=2u+v, offering a linear combination of the vectors u and v for w. Considering x3=2 as another
example, one would obtain the equation -4u-2v+2w=0 leading to one of the vectors (say u) written as a
linear combination of the vectors v and w; that is u=-1/2v+1/2w. For the remainder of the paper, we will
discuss SA21’s use of his existing metonymies while responding to interview questions and portray a
picture of his effort to form a new metonymy (a mathematical object).
Figure 2. View from SA21’s work from his interview.
Existing Metonymies
Linear Independence
Linear combination
SA21’s interview displays a frequent use of metonymies in his reasoning. Moreover these metonymies
appear to dominate his knowledge of linear independence. As depicted in figure 1 above, overarching
metonymy that SA21 reasons with appears to be the linear combination form. When student SA21 was
asked to share his definition and his understanding of linear independence, SA21’s initial response
indicated that he was considering the term “linear independence” to stand for “linear combination” ideas
as seen in the following excerpts (some of the phrases are made bold by the authors):
SA21: Okay ….I think of linear independence so… I think we have a set of vectors, so I’ll just write… like you have u1,
u2, so we can go all the way to however many we want. Then I… so, I know that they are independent if, suppose
we have, so we have a1 which is like some real number… times an and we’ll just keep on going…
SA21: So I think that’s kind of close to what you wanted. Since this is the key component [pointing to
a1u1+a2u2+…+anun; a1, a2,…,an are real numbers].
Even though later in the interview SA21 stated that his initial description was for linear combination and
not for linear independence, his responses throughout the interview however appeared to sustain the view
that the metonymic use of “linear independence” for “linear combination” has been a more dominant
factor in his reasoning.
Matrix
Set
SA21 was regularly prompted to elaborate his responses. While elaborating, he integrated other
metonymies as part of his reasoning. His metonymic use of “matrix” standing for “set” for example fits
well with his overall notion of linear combination ideas. He consistently regarded matrices as representing
vectors of sets, (lacking the mention of their association to vector equations) and looked for linear
combinations among the columns. This can be observed in the excerpt below:
SA21: So, then here, so I would… to determine independence or dependence I know…so I just build my coefficient
matrix…
SA21 goes straight to a matrix whose columns are the vectors of his set. Next he points to the rref of this
matrix and states that “it is linearly dependent.” When asked what he means by “it.” He says:
SA21: Uh… for the set? I would … I don’t know if I would say for the set or for the matrix…..
Identity Form
Linear independence
SA21 continued to consider matrices as sets throughout the interview. While searching for linear
combinations among the columns of a matrix, he introduced another metonymy, the use of “identity form”
for “linear independence.” In the conversation below, for example, when SA21 is asked to explain how
he identified the linear dependence of a set without considering solution forms he reasons with his
metonymy, “identity form.” Here, he focuses on the identity form among the columns of matrices to
identify “linear independence.” Furthermore, he uses these vectors to come up with linear combinations.
I: …you stopped you did not write it [meaning a solution set]. You directly said this [pointing to a set of vectors] is
linearly dependent, and reasoning for that was?
SA21. …I cant express these other vectors [pointing to the last three columns of a 2x5 matrix] as identity …What I
would want is I want identity that is the key, for a 3 by 3…we want something like this [meaning an identity form]
to me that [meaning identity form] says that that [pointing to vectors of an identity form in a matrix] is linear
independent…
He next gave, after prompted to provide a solution set, the following response still reasoning with his
metonymy of identity form.
SA21 ... we have identity here [pointing to the first 2 columns of a 2x3 matrix], but this is not [points at the last
column with values (2, 1), see figure 2, I] and this means that this is dependent on this [meaning that the last
column of the matrix is dependent on the first two columns] so I like to write what we have, so I'll write x1, I like to
use xs, equals minus x sub 2, x sub 3, x sub 2 equals minus x sub 3 and then x3 is our independent vector [see the
parametric representation in figure 2, IV] So then, from here [pointing to the parametric form seen in figure 2, IV] I
can just see that we have a dependent… linearly dependent set…
xi
vector
The excerpt above further reveals another metonymy, “xis” set forth for “vectors.” This appeared to be
the most influential metonymy in SA21’s reasoning. He, in fact, seemed to attribute symbols with fixed
meanings and reason with these meanings throughout the interview. Initially in the interview, SA21
considered “ai “ as symbols representing known values in a linear combination but later reserved them for
unknowns and chose the symbol “xi“ to stand for “vectors.” His shift to the symbol xi to represent vectors
is apparent in the phrases “I like to use xs” and “x3 is our independent vector.” His persistence to use the
fixed meaning attributed to xis can also be observed in the excerpts below. SA21 forms a matrix using a
set of vectors and labels each column as x1, x2, x3, x4 and x5 respectively as seen in
.
It is clear from these responses that student is using x1, x2, x3, x4 and x5 to represent the five vectors of a
set.
SA21: Now, if I was to write, like how I did that last one [pointing to a set with three vectors] so you have… I have
five vectors. So I have x1, x2 [marking each column on a coefficient matrix with x1, x2, x3, x4 and x5]…
SA21: I’m thinking… so I’m already saying that I think I’m saying that my x1 and x2 are independent vectors… and
that x sub…Well, the way I can think about. It is. I know we would rewrite this as x1, x2, this is gonna equal some
x3… and this is gonna be -2, -1, and 1… so then I just see that x3 or our third vector will be dependent that's kinda
like how I think about it.
Formation of New Metonymy
After prompted for further explanation on the potential connection between a vector equation and a
parametric representation of solutions, toward the end of the interview, SA21 began comparing the roles
of the two symbols ai and xi. One can distinctly observe, on his responses below, his metonymic use of
the symbols and how each symbol continues to hold a distinct meaning.
SA21: Which is this… we know, and these are unknowns [pointing a1 in a1u1+a2u2+…+anun=0]. So I want to say
that… I think that a1 and this [pointing to a parametric solution in figure 2, IV] should be the same, is that what
you are trying to say?
It is evident with the phrase, “is that what you are trying to say?” that up to that point in the interview,
SA21 was not considering the symbols ai and xi holding the same meaning. After this point however he
began to consider the potentiality of them relating.
As seen in the excerpt below, while attempting to connect the two symbols, he uses his existing
metonymies. He begins with reiterating his metonymy of xis representing vectors. Next, he uses the
metonymies of “columns” for “vectors,” and “identity” for “linear independence.” That is, SA21
considers the first two vectors of a set (which forms the first two columns of a matrix) as linearly
independent vectors reasoning with his metonymy of “identity form,” and concludes the linear
dependence of the last three vectors of the set. He next, for the first time in the interview, begins
considering xis as unknowns, at the same time, reserving them as representing vectors. xis now embrace
two meanings. The two meanings xis hold furthermore appear to imply that xis may be representing
coefficient values for linear combinations.
SA21: So I am really saying that I think of saying that my x sub 1 and x sub 2 are independent vectors, and that
 2
 3
1
1
 3
 4
5 
6 
9
10
….or I would say these two [pointing to u1 and u2 in the set {u1   , u 2   , u 3   , u 4   , u 5   } ]
because I put them in this order [implying they would lead to identity form]. So I would say that these two would
be independent and these independent vectors these [pointing to the last three vectors in the set] dependent on
these [pointing to the first two vectors in the set] so and I think this is telling me that I wonder I was to put of x sub
 x1 
  2
x 
  1
  ] is my unknown for my very first
3 like think some value?...I know this [pointing to x1 and -2 in  2 
 x3   x3  1 
 
 
 x4 
0
 x5 
 0 
vector this is the second one, this is the third and fourth [pointing to x2, x3 and x4]. So I wonder if I was to put. I
think may be telling me that if I was to look at it this way…..so I am thinking if I want to express the third vector
 x1 
  2
x 
  1
2

 ]….I wanna say this [pointing to u3 in the set] is my third vector my x sub 3
[circling x3 in  
 x3   x3  1 
 


 x4 
 0 


 0 

 x5 

because I gave it this…so I wanna say that suppose I wanna write this as a combination of this it is telling me that if
I was to have that. If I pick any value for x sub three, suppose I want two so I want x sub 3 equal just some two. It is
telling me that I can write a linear combination of this third vector…as a combination of all of these [pointing to
 x1 
  2
x 
  1
2

 ] so it is going to
the vectors in the set]…then I can substitute this, two, into here [pointing to  
 x3   x3  1 
 


 x4 
 0 
 x5 
 0 
give me…..since this [pointing to -2 in (-2,-1,1,0,0)] belongs to my first unknown. I wanna say that I can express u
sub one as a minus two u one [writes -2u1] …since we are adding them [meaning -2u1 and –u2] it is telling me that
[-2u1–u2] will equal my u sub 3 my third vector...
In the excerpt below, SA21 is using the values in (-2,-1,1,0,0) as the coefficients to form a linear
combination for the third vector represented by the symbol x3 and he is considering each entry value in (2,-1,1,0,0) associated with one of the symbols x1,x2, x3, and x4 respectively. That is, he now identifies the
first vector with x1 and considers the first component value, -2, as the coefficient value for the first vector
and so on.
I: What are you focusing on, on that page?
SA21: I am focusing on since I want to express these I know somehow this has to. This [pointing to
 x1 
  2
x 
  1
 2
 
 x3   x3  1  ] telling me that I can express this third vector [pointing to x3 in
 
 
 x4 
0
 x5 
 0 
 x1 
  2
x 
  1
 2
 
 x3   x3  1  ] somehow
 
 
 x4 
0
 x5 
 0 
with a combination with these numbers [pointing to (-2,-1,1,0,0)] ….
From this point on in the interview, SA21 consistently considered “solutions” (represented by parametric
representations or vector forms) as values for “coefficients of vectors forming linear combinations.” This
new notion appeared to have developed into a new metonymy (and a mathematical object) for SA21.
Conclusion
There are many studies documenting metonymic reasoning at the pre-college level, but very little at the
college level, especially in advanced mathematics topics. This paper outlined one case to discuss the
metonymic use at the college level, specifically with linear algebra topics. SA21’s interview responses
provided a portrait of how a mathematics student may be functioning with metonymies. We observed
among the particular student’s interview responses that metonymic reasoning may have led to the
formation of a new metonymic knowledge.
Our findings in fact may have implications for the learning and teaching of advanced mathematics
courses especially courses such as linear algebra that heavily uses formal symbolic language, which may
encourage the formation of metonymies with fixed meanings as was the case with xis representing only
vectors or known values in SA21’s knowledge.
The case we discussed in this paper further supports the earlier studies in that metonymies appear to be
cognitive constructs with meanings associated to (Presmeg, 1998; 1997), not just literally devices to aid
with recalling. They need to be taken with utmost importance and paid close attention to their role in
one’s knowledge and reasoning of advanced mathematics concepts.
Finally, this paper reported findings of one linear algebra student’s metonymic reasoning. They by no
means can be taken as generalization to all linear algebra learners. Future research, utilizing the work
reported here, is in need with expanded population.
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Authors (2007; 2010)…