Metonymies and Object Formation
Vector Space Theory
Introduction
Work on metonymy mainly focuses on them as literary devices, rather than cognitive constructs,
that are used to encode information. Presmeg (1998;1997) and Lakoff and Johnson (2000)
however 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
they both 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 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 concept. They may aid in making
sense of symbolism apparent in advanced abstract concepts, furthermore they may support a
process of meaningful knowledge formation (Presmeg, 1998; 1997).
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. Moreover, mathematical
symbols are ones that are put for various mathematical entities such as number families. The
symbol “x” for example can represent natural numbers 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 mathematical attributes of the linear algebra concept linear
independence. The attributes of the mathematical object recognized from the image however are
subject to the individual’s interpretation of the image. The geometric image may in fact be both a
metaphor and a metonymy. On the one hand, an individual may consider the image as having
similarities with various aspects of linear independence. On the other hand, after the initial
consideration of the image as a metaphor, the same individual may begin considering the image
only as a symbol that is 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 “a house.” In
mathematics, an illustration of a circle taken to represent the class of all circles can 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.
Studies on metonymy have mainly been at the pre-secondary level documenting the role and
influence of metonymy on reasoning and object construction. There have not been very many
studies documenting the role and use of metonymies at the college especially at the advanced
level topics. Linear algebra is one of the advanced mathematics courses with high degree of
abstraction and symbolism, which require learners to be able to work with abstract
representations. Our worked showed frequent use of metonymic reasoning to make sense of
these 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, as a result
a new mathematical object.
Various theories such as APOS (Oktac, Maria….) and THE CK¢ THEORY1 BY BALACHEFF
by Marraci???????? are used in interpreting and understanding conception of linear algebra. We
use the framework of metonomical reasoning in our interpretation of student work in our study
of cognitive processes involved in the reasoning and formation of linear algebra concepts…..
We concur with Presmeg (1997) that it is when metonymies change their contextual signification
to the internal relations that meaningful reasoning may become possible.????
Method
The data discussed in the paper came from our work with two groups of students enrolled in
the three sections of a matrix algebra course at a Southwest University in USA—one traditional
and the other 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 was enrolled in a module section. We use an alphanumeric name “SA21,” to refer to
him throughout the paper. He is an Hispanic-American student majoring in mathematics with a
minor in secondary education. He was interviewed toward the end of April, 2009. Interviews
began with a set of pre-determined questions on basic vector space concepts such as linear
independence, span and spanning set. These questions were structured based on the learning
difficulties reported in the literature (XXXX, 2009; Sierpinska, 2000). During the interviews,
interviewer asked new questions in an attempt to extract more information and further tap
into students’ thought processes. A qualitative approach, namely the constant comparison
method (Glaser, 1992), is used to analyze SA21’s responses.
Get Figure at minute 16.32
Below, we will discuss SA21s metonymic use in his reasoning and how they shape his knowledge.
Existing Metonymies
Linear independence
New Object
Matrix ----set
Identity……>independent vectors
Solution
xi …………..> vector
known values assigned to vector corresponding to xi
linear combination
Values for linear combination
Figure 1. Metonymies displayed in reasoning.
Results and Discussion
Existing Metonymies
Responses provided here came from student responses to one question “Define the linear
independence of a set of vectors.” Interview began student sharing his definition of linear independence,
and continued with student responding to new questions. What readers will be observing in the results
section as they read the excerpts is SA21’s attempt to understand how two seemingly dissimilar (to the
student) aspects of the same concepts may be related. SA21 shared two main ideas for the linear
independence/dependence of a set of vectors. One was his notion of linear combination. With this
notion, SA21 was able to identify linearly dependent sets provided that he could obtain a linear
combination resulting in one of the vectors of the set. The second idea he held throughout the
interview, which appeared to be independent of the first idea, is the identity form of a matrix. He
appeared to have represented vectors with a matrix and searched for an identity form as a result of
Gauss-Jordan elimination process. He would identify linearly independent sets if matrix is reduced to an
identity matrix, and linearly dependent if it is not. He was also able to write parametric representations
of solutions to a vector equation. He however was not able to explain how the parametric
representation of solutions would result in linear combinations for vectors of sets. He was prompted
continuously by the interviewer to discuss his understanding of this connection. While attempting to
explain the potential relation between the two ideas, toward the end of the interview, he began to
apply his metonymies, and came up with a new notion of how they may be related. We believe SA21
was, at the start of the interview, unaware of any connections between the two ideas, but toward the
end began to consider potentiality due to interviewer’s persistent prompt for him to think about them,
leading to the formation of new mathematical object. SA21 regularly used metonymies to reason
throughout the interview. In this paper we will discuss the metonymies he used and how he reasoned
using them.
Before discussing SA21’s responses, let’s provide one of the examples student gave during the interview.
After sharing his notion of a linear independence, he was asked to give an example. He gave the set of
vectors seen in figure 1 below and proceeded to explaining how he identified the set as a linearly
1
1
 2
 3
 4
5 
dependent set: {u   , v   , w   } He considered a matrix whose columns were formed by
the vectors of the set. After applied Gauss-Jordan process, he circled the identity form the first two
columns formed writing both parametric and vector representation of solutions. Next, after asked to
provide a linear combination specifically using his solutions????, using the identity form and a relation
observed among the columns of the matrix (third column being the result of the sum of 2 times the first
and the second columns), he wrote a linear combination of the first two vectors u and v resulting in the
third vector, w as seen in figure 1 upper right corner. Some of the responses provided below are
revealing SA21’s attempt to explain how one may obtain a linear combination among vectors of a set
originating from parametric or vector solution forms. For clarity, let’s obtain a linear combination among
u, v and w using SA21’s parametric representation seen in figure 1. Given the parametric representation
of solution in figure 1 bottom right corner, one solution is (-2, -1, 1). That is, this solution satisfies the
vector equation x1u+x2v+x3w=0 for the vector u, v and w given in figure 1. Thus, -2u-v+w=0. Solving the
equation for w gives a linear combination, w=2u+v. Readers are asked to pay attention especially to the
meanings SA21 attributes to symbols xi and ai throughout the interview. One notices that we in our
work above considered xis as symbols for the unknowns of the vector equation, and a solution as the
values for the unknowns. This allowed us to use the values of xi and the equation to write the vector w
as a linear combination of the vectors u and v.
Figure 2. View from student work during the interview.
SA21’s interview indicates a frequent use of metonomies in his reasoning, further these metonomies
appear to dominate his knowledge of linear independence. Student SA21 was asked to share his
definition and understanding of linear independence. SA21’s initial response indicates that he may have
been using the term “linear independence” to stand for linear combination ideas.
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. (Student wrote
down: A set of vectors: {u1, u2,…, un}; a1u1+a2u2+…+anun; a1,a2,…,an are real numbers and identifies
a1u1+a2u2+…+anun as the key component).
Even though later in the interview he states that his initial description was for linear combination, not
for linear independence, his responses throughout the interview however indicates that the metonomic
use displayed in his initial description of linear independence is a more dominant factor in his
reasoning.
SA21 was regularly prompted to provide elaboration. While elaborating, SA21 integrates other
metonomies in his reasoning. His metonomic use of “matrix” standing for “set” for example fits well
with his overall notion of linear combination ideas. He consistently regards matrices as representing
vectors of sets, and attempts to find linear combinations among the columns of matrices. This can be
observed in the excerpt below. When given a set SA21 goes straight to a matrix whose columns are the
vectors of the set.
SA21: So, then here, so I would… to determine independence or dependence I know…so I just build my coefficient
matrix… (student comes up with five vectors u1=[2;3], u2=[1;1], u3=[3;4], u4=[5;6], and u5=[9;10]) [matrix->set,
columns->vectors]
Once he runs a process called Gauss-Jordan Elumination (reference????), he points to the final matrix product and
states that “it is linearly dependent.” When he is asked what he means by “it.” He states:
SA21: Uh… for the set? I would s… I don’t know if I would say for the set or for the matrix… [matrix->set]
SA21 continues to consider matrices as sets throughout the interview. As seen in the excerpt below, while searching
for linear combinations among columns of matrices, he integrates another metonomic use of “identity form”
representing “lndependent vectors.” He focuses on identity form among columns of matrices to eliminate some of
the columns (mainly the earlier columns) as independent vectors and tries to form linear combinations for the
remaining columns.
SA21: Yeah, I just wanna focus on these 2 (student points at the first 2 columns in the matrix)… 'cause I'm pretty
sure once I get the identity here there's not gonna be much I can do here…(student points at the last 3 columns in
the matrix) [matrix->set, columns->vectors, identity->independence]
SA21 …….., I was to… we have identity here (student points at the first 2 columns of the matrix [1,0;0,1]),
but this is not (student points at the last column of the matrix [2;1]) and this means that this is dependent
on this (student states that the last column of the matrix is dependent on the first 2 columns of the
matrix)… so I like to write what we have, so I'll write x1… I like to use x's… equals minus x2… x3… x2 equals
minus x3… x3 is our independent variable (student writes x1=-2x3, x2=-x3, x3=x3) So then, from here I can
just see that we have a dependent… linearly dependent set… (ai are different than xis??? As what they
represent???)
The second excerpt above introduces another metonomy, “xis” set fort as “vectors.” This appears to be
the most dominant metonomy in his knowledge. SA21 infact seems to attribute symbols fixed meanings
and reason with these meanings throughout the interview. Initially, SA21 considers ai as symbols
representing known values in a linear combination but later reserves them for unknowns and chooses
the symbols xi as standing for known values in a linear combination. He fails to connect xi and ai
throughout the interview. This may be attributed to his metonomic use of each symbol for different
meanings. Later on in the interviews xi begins to stand for two different objects; as vectors and as
known values for each vector xi represents. This does not appear to cause any conflict in his knowledge.
In the excerpt below, SA21 formed a matrix using a set of vectors and labeled each column as x1, x2, x3
similar to
. It is clear from his responses below that student is using x1, x2, x3 to
represent three vectors of the set.
73.SA21: Now, if I was to write, like how I did that last one [the one with three vectors using columns of
matrix????] so you have… I think I have… five vec… so I have x1, x2… [marked each column on a coefficient
matrix with x1, x2, x3] now this is where I would probably get a little tricky… [variables->vectors???check the
video]
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… [variables->vectors]
SA21: 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 (considering x3 as a vector) or our third vector
(pointing at [-2,,-1,1]) will be dependent that's kinda like how I think about it. (student rewrites the vector
[x1;x2;x3] equal to x3[-2;-1;1]).
When SA21 was prompted for further explanation on potential connection between a vector equation
and xi values he obtained, he appeared to begin thinking about roles of the symbols ai and xi but still not
sure if they are the same. One can observe clearly on his responses below his metonomic usage of the
symbols and how each appears to have distinct objects associated with them; ai unknowns vs. xi known
values.
33.SA21: Which is this… we know, and these are unknowns (student points at statement
a1u1+a2u2+…+anun=0, particularly to a1u1, and refers to a1 as the unknown part ). So I want to say
that…could it… I think that a1 and this [pointing to x1= x2= x3=] should be the same, is that what you are
trying to say? [ai vs. xi???]
It is clear with his statement “is that what you are trying to say?” that up to that point in the interview,
he was not thinking about symbols xi and ais as representing the same concept. After this point on the
interview though, SA21 begins to try to consider potentials of ai and xi relating. That is when he appears
to attempt to form another metonymy using his existing metonymies. As SA21 was repeatedly asked
whether he considers potential association between vector equations and linear combinations he was
providing, he began questioning whether ais and xis are related. This appeared to have encouraged him
to interpret them using his existing metonymies leading to the formation of a new metonymy. His
existing metonymic reasoning appeared to have let to the formation of solution standing for linear
combinations. Specifically, solutions of the form x1=….x3=x3 appears to represent coefficient values for
linear combinations.
The use of xis as vectors and coefficient values for vectors fits well with his reasoning of solutions
representing linear combinations. That is, SA21 considers xis representing coifficient values for
corresponding vectors. He is not able to see any connection between his metonomic use of solution as
linear combination and solutions as values for the unknowns ais of his equation. This is appeareant in hs
attempt to explain how his linear combinations he states and vector equations are related.
48.SA21: Yeah, I think this would work just for simple, but I think this is the key for maybe a larger set, so I'm
thinking, if… if we were to have a bigger set, I would actually have more and more, and then from here it
would maybe tell me if I could write one, 'cause I'm thinking… I'm thinking so… if this is a… since this is our
independent vector [appear to point x3(-2, -1, 1)], these depend on it [pointing to first two columns of a
matrix], so I just have to substitute this [appear to point x1= x2= x3=..] into here [pointing to a matrix] and I
could get… I could probably write one [meaning linear combination] into this equation
[a1u1+a2u2+….+anun=0]…
77.SA21: I was wondering if I was to form another matrix but I was like… whoo why… because say, say these
are not values I guess now, these are other vectors (student points at the vectors [-1;-1;1;0;0], [-1;-3;0;1;0],
and [-1;-7;0;01]) and these are unknowns (student points at the variables x3, x4, and x5) and I was wondering
if I was to form a… the 5 by 3 what would it give me…
78.,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… [variables->vectors]
79.SA21: I think well I… I wanna say these two or why… I would say these two because I put them in this order,
so I would say that these 2 would be independent (student points at vectors u1=[2;3] and u2=[1;1]) and these
independent vectors(student points at vectors u3, u4, and u5) these depend on these (student points at
vectors u1 and u2)… so then I think this is telling me that… I wonder if I was to put a x3 and then say some
value… [identity->independent]
80.SA21: Oh, no… what do I want to say?... so I know this is my unknown [pointing to x1 in
(x1,x2,x3..)=x1(2,..)+x2(2,…)+….] for my very first vector… this is the second one (pointing to x2 in
(x1,x2,x3..)=x1(2,..)+x2(2,…)+….] this is for 3rd and 4th… so I wonder if I was to put… I think maybe it’s telling
me that if I was to look at it this way, I have… do I have a minus 1? (student rotates notepad 90 degrees
counterclockwise)…so I am thinking If I want to express the third vector [pointing to x3 in x3(-1,-2, 1,0)]…This is
my third vector [pointing to u3 in a set], my x3.
81.SA21: So I wanna say that suppose I wanna try this [third vector??? Check] as a combination of this… it’s
telling me that if I was to have the… if I pick any value for x3 suppose I want 2, can I write in this here?
86.SA21: so I want x3 to be equal just some 2. and its telling me that I can write a linear combination of this
third vector which is I’m gonna say… what was it? … [3;4]…
Along with his idea of xis provide coefficients for linear combinations as seen in the above excerpt,
he now interpret xi s in (x1, x2, …)=x3(-1,-1,1,0) as referring to his unknowns and values in (-1,1,1,0) as their values, still however considering xis representing vectors. Consider the excerpt
below. SA21 is considering the values in (-2,-2,2,0) as the coefficients to consider a linear
combination for the third vector where x3 stands for the third vector and each entry value in (-2,2,2,0) associated with corresponding vector identified by the symbols in (x1,x2, x3,..). That is , he
now identifies the first vector with x1 component in (x1,x2, x3,..), and consider the first component
value, -2, in (x1,x2, x3,..)=(-2,-2,2,0) as the value of unknown and also the coefficient to be used for
the first vector.
83.SA21: that if I wanna write this [pointing to u3=[3,4????] ] as a combination of all these… then I can
substitute these 2 into here [pointing to x3(-1,-1,1,0) so it’s gonna give me -2, -2, 2, 0, 0 so since this [pointing
to -2 in (-2,-2,2,0)] belongs to my, my first unknown, the… I wanna say that I can express u1 as a -2u1 and
since we are adding them, it’s telling me that it can uh… I should have wrote… a minus since then I have a
minus 2u2 would equal my u3, my third vector, I think that’s kinda like…
SA21: that’s going to give me -4, -6… [student writes -2[2;3]+2[1;1]=[-4;-6]]
85.SA21: So, this is… actually -2, -3.. ok, hold on… this should be a -2… but what I think is I didn’t multiply the
third vector by the 2. So I should have a 2… (student tries to explain is statement -2[2;3]-2[1;1]=[-4;-6]+[-2;-2])
86.SA21: So, that should be a 3 times 4 which would give you 6 and then an 8 it doesn’t look like I’m getting
what I want… (student multiplies 2[3;4]=[6;8]) [linear combination]
87.SA21: If I add these collectively [[-4;-6]+[-2;-2]], probably… if I were to add them I’d get a minus 6 and
minus 8, and I have the opposite. [linear combination]
88. SA21: I’m focusing on… since I want to express these [check the video]… I know somehow this [x3(-1,2,1,0)]] has to just tell me that I can express this third vector [meaning u3 in a set] somehow with a
combination with these numbers (student points at the statement x3[-1;-2;1;0] ) [linear combination] x3>vector.
Formation of New Metonymy
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 ….or I would say these two [pointing to u1 and u2 in a set] because I put them in this order. So I
would say that these two would be independent and these independent vectors these dependent on
these [pointing to the last three vectors in a set] so and I think this is telling me that I wonder I was to
 x1 
  2
x 
  1
 2
 
put of x sub 3 like think some value?...I know this [pointing to x1 and -2 in  x3   x3  1  ] is my
 
 
 x4 
0
 x5 
 0 
unknown for my very first vector this is the second one, this is the third and fourth. 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
 x1 
  2
x 
  1
 2
 
the third vector [circling x3 in  x3   x3  1  ]….I wanna say this [pointing to u3 in a set] is my third
 
 
 x4 
0
 x5 
 0 
vector my x sub 3 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 a set of vectors]…then I can substitute this,
 x1 
  2
x 
  1
 2
 
two, into here [pointing to  x3   x3  1  ] so it is going to give me…..since this belongs to my first
 
 
 x4 
0
 x5 
 0 
unknown [pointing to -2 in [-2,-1,1,0,0] ] I wanna say that I can express u sub one as a minus two u one
…since we are adding them it is telling me that …will equal my u sub 3 my third vector? Kind of like…
I: Can we verify it?
Carrying out the computation of 2u1+v…
SA21: I didn’t multiply the third vector with two…this should be a two…getting a six and an eight….If I
was to add them…It does not look like I am getting what I want [meaning the third vector]…I just know
that I messed up somewhere…
I: What are you focusing on, on that page?
SA21: I am focusing on since I wan 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  ]
 
 
 x4 
0
 x5 
 0 
somehow with a combination with these numbers [pointing to [-2,-1,1,0,0]] ….I don’t know I just kind of
left them [meaning x4 and x5] alone… Since these are independent I mean these are dependent vectors
 x1 
  2
  2
  2
x 
  1
  1
  1
 2
 
 
 
[pointing to  x3   x3  1   x 4  1   x5  1  ] that dependent on somehow these [pointing to u1
 
 
 
 
 x4 
0
0
0
 x5 
 0 
 0 
 0 
 2
1
 3
5 
9
and u2 in a set {u1   , u 2   , u 3   , u 4   , u 5   } ] two. I don’t really need to worry
 3
1
 4
6 
10
  2
 2
  1
  1
 
 
about these [pointing to x4  1   x5  1  ]. I just want to express one of them but…
 
 
0
0
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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 make the case
that metonymic use may be as common, if not more often, at the advanced college mathematics
courses. SA21 interview responses painted a portrait of how a student may be functioning with
metonomies and how his/her reasoning may be influenced by them. Furthermore, we observe among
this student’s responses that metonomic reasoning may have shaped the students knowledge of the
concepts, and resulted in a new metonymic knowledge.
This infact may have implications for learning and teaching of advanced mathematics courses especially
courses such as linear algebra that heavily use formal language; symbolic approach, which may be
encouraging the formation of metonymies with fixed meanings as was the case with xi representing only
vectors or known values in our student’s knowledge.
Toward the end of the interview, SA21’s responses clearly showed that Student was reasoning mainly
with his metonomies and integrating them into his ideas of linear combination. In fact his metonymies
appear to have lead to the formation of a knowledge of solutions of the form x1=, x2=, x3=x3, standing
for coefficient values for linear combinations….
Our case further supports the earlier studies that in fact metonymies appear to be cognitive constructs
with meanings associated to (Presmeg,?????), not just literally devices to aid with recalling (reference…).
They need to be taking with utmost importance and considered their role when one’s knowledge of
advanced mathematics concepts.