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¢ THEORY
1
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.

Existing Metonymies
Linear independence
New Object
Matrix --- set
Identity……> linear independent Solution x i
…………..> vector
known values assigned to vector corresponding to xi linear combination Values for linear combination
Figure 2. Metonymies displayed in reasoning.
Results and Discussion
Responses 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 added questions apart from the interview questions prepared in advanced. What readers will be observing in this section as they read excerpts is SA21’s attempting 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 among the vectors of a set resulting in one. The second idea he held throughout the interview, which appeared to be independent of the first idea, focuses on the identity form of a matrix. Whenever a set with vectors given where a linear combination is not seen right away he appears to proceed directly (skipping corresponding vector equations) to representing vectors with a matrix and searching for an identity form via Gauss-Jordan elimination process. As a result, he identifies accurately linearly independent sets depending on whether rref
(reference textbook??????) of matrices are in identity form or not. Specifically, he was also able to write both parametric and vector representations of solutions to a vector equation using rref forms.
Throughout the interview SA21 attempted 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. 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 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. SA21 regularly used his metonymies to reason. In this paper we will discuss his metonymies and how he reasoned using them.
Before discussing SA21’s responses, let’s provide one of the examples student gave in order to provide a context for the excerpts included. After sharing his notion of a linear independence, SA21 was asked to give an example. He gave the set { u

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1
1
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, v



2
3


, w



4
5


} and proceeded to explaining how he identified the set as a linearly dependent set: He considered a matrix whose columns were formed by the vectors of the set. After applying the Gauss-Jordan process, he obtained the rref of the matrix,


1
0
0
1
2
1

 , circling the identity form as seen in figure 1 below. Using the rref form he identifies the set as linearly dependent indicating that the last column of the matrix is depended on the first two, proceeding to writing both parametric and vector representation of solutions (see figure 1). Next, after asked to consider how his solution forms imply linear dependence of the set of vectors, he, switching to his first notion, writes a linear combination 2u+v=w directly from the numerical entries of the vectors of the set , ignoring his solution forms, as seen in figure 1 underlined upper right corner. Many of the excerpts 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.
In order to provide a context for the excerpts included in the paper, let’s obtain linear combinations among u, v and w using SA21’s parametric representation seen in figure 1. Given the parametric representation of solutions, one solution would be (-2, -1, 1) where the independent variable x
3
=1. That is, this solution satisfies the vector equation x
1 u+x
2 v+x
3 w=0 for the vectors u, v and w. Thus, -2u-v+w=0.
Solving the equation for w, one would obtain the vector equation w=2u+v, indicating that w is a linear combination of the vectors u and v. Furthermore, considering x
3
=2, one would obtain -4u-2v+2w=0 leading to one of the vectors written as a linear combination of the other two, for instance u=1/2v-1/2w, indicating that the vector u is a linear combination of the vectors v and w.
For the remainder of the paper, we will make a case for SA21’s use of his existing metonymies while responding to interview questions and portray a picture of his endeavor to form a new metonymy (a mathematical object) using the existing ones via excerpts from the interview.
Figure 2. View from student work during the interview.

Existing Metonymies
Linear Independence Linear combination
SA21’s interview indicates a frequent use of metonymies in his reasoning, further these metonymies appear to dominate his knowledge of linear independence . As depicted in figure 2 above, overarching metonymy that SA21 reasons with appears to be the linear combination ideas. Student SA21 was asked to share his definition and understanding of linear independence. SA21’s initial response indicates that he was considering 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 [ pointing to a
1 u
1
+a
2 u
2
+…+a n u n
; a1,a2,…,an are real numbers]
Even though later in the interview he stated that his initial description was for linear combination, not for linear independence, his responses throughout the interview however indicates that his metonymic use displayed in his initial description of linear independence has been a more dominant factor in his reasoning.
Matrix Set
SA21 was regularly prompted to provide elaboration. While explaining, SA21 integrates more metonymies in his reasoning. His metonymic 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, never mentioning their association to vector equations, and looks for linear combinations among the columns of matrices. This can be observed in the excerpt below. For a set of vectors he came up with SA21 goes straight to a matrix whose columns are the vectors of this set.
SA21 : So, then here, so I would… to determine independence or dependence I know…so I just build my
coefficient matrix…
Next, he points to the rref of his matrix 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 … I don’t know if I would say for the set or for the matrix…

Identity Form Linear independence
SA21 continues to consider matrices as sets throughout the interview. As seen in the excerpt below, while searching for linear combinations among the columns of a matrix, he integrates another metonymy, the use of “identity form” representing “ linear independence.” He focuses on identity form among the columns of matrices to identify “independent vectors.” Furthermore, he uses these vectors to come up with linear combinations for the remainder of the vectors in the set as observed in the two excerpts below.
SA21: Yeah, I just wanna focus on these 2 [pointing to the first 2 columns of a 2x5 matrix]… 'cause I'm pretty sure once I get the identity here there's not gonna be much I can do here [meaning the last three columns cannot form an identity].
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 entry (2,1)] 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 x's… equals minus x2… x3… x2 equals minus x3… x3 is our independent variable [see the parametric representation in figure 1 bottom corner] So then, from here I can just see that we have a dependent… linearly dependent set… x i vector
The second excerpt above introduces another metonymy, “xis” set forth for “vectors.” This appears to be the most influential metonymy. SA21 in fact seems to attribute symbols with unchanging meanings and reason with these meanings throughout the interview. Initially, SA21 considers “a i
“ as symbols representing known values in a linear combination but later reserves them for unknowns and chooses the symbol “x i
“ as standing for known values. 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 put of x sub 3 like think some value?...I know this [pointing to x
1
and -2 in






 x x
2 x
3 x
4 x
5
1







 x
3
 







1
0
0
2
1







] is my
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 the third vector [circling x
3
in






 x
1 x
2 x
3 x
4 x
5







 x
3
 







1
0
0
2
1







]….I wanna say this [pointing to u3 in a set] is my third 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, two, into here [pointing to






 x x x
1
2
3 x
4 x
5







 x
3
 





 0

1
0
2

1






] so it is going to give me…..since this belongs to my first
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






 x x x x x
5
1
2
3
4







 x
3

1
0
0
2

1






 






] telling me that I can express this third vector [pointing to x3 in






 x x
2 x
3 x
4 x
5
1







 x
3


1
0
0
2
1














] 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 x
4 and x
5
] alone… Since these are independent I mean these are dependent vectors
[pointing to






 x x x x x
5
1
2
3
4







 x
3
 


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



1
0
0
2
1







 x
4

1
0
0
2

1






 


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


 x
5

1
0
0
2

1






 






] that dependent on somehow these [pointing to u
1 and u
2
in a set { u
1



2
3


, u
2 about these [pointing to
 x
4

1
0
0
2

1






 








1
1


, u
3




3
4


, u
4



5
6

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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.