Metonomies in mathematical reasoning : Vector Space Theory
Introduction
Some of the work on metonymy and metaphors in mathematics focus on them as
literary devices that are used to encode information rather than as cognitive
constructs. In contrast, Presmeg (1998;1997) views metaphor and metonymy as
cognitive structures, which is similar to the views of Lakoff and Johnson (2000).
The act of using one object to stand for another is considered as functioning with
metaphors or/and metonymy. The difference between the two constructs
however is that metaphor links one domain of experience with another domain,
and gives meaning from similarities while metonymy uses one element or
relevant attribute of a class to stand for another element. Presmeg (1998) argues
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 help learners in making sense of the construct accompanied by
ambiguities inherent in its representations. Metaphors and metonymies are
conceptual entities that may influence one’s reasoning of an abstract
mathematics concept (Presmeg, 1998; 1997). The influence of the conceptual
entities can, for the most part, be advantages for learners in their reasoning, and
furthermore they may support a process of meaningful knowledge formation. At
times however, they may manifest themselves in learner’s reasoning causing
incomplete or inaccurate understanding. As Walkerdine stated “To reflect on the
internal relations alone we have to ignore the metaphoric content of a statement
which might distract from the focus on the logical relations innate in the
statement, namely by directing attention to the practice to which the statement
refers” (Walkerdine, 1982, p. 138: cited in Presmeg, 1997). We would like to
further add that in order to reflect on the internal relations we also have to ignore
the metonymic aspect of statements otherwise solely depending on the
metonymic role of statements may lead to the rote application of metonymies,
which may lack meaningful reasoning. We concur with Presmeg (1997) that it is
when metonymies change their contextual signification to the internal relations
that meaningful reasoning may become possible.
Many of us witness students using shortcuts and/or procedures and
processes with very little understanding of the relations present in them. This is
especially the case in mathematics courses. Majority of our students can
accurately respond to items but may not be able to explain their reasoning. They
may easily apply a procedure accurately to one situation but may not able to
apply the same procedure to a slightly different situation or recognize situations
where the procedure can be applicable. For instance, many calculus students can
evaluate the area of a region accurately using definite integrals that is derived
from the partitioning of regions (Riemann sum approach) but cannot apply the
reasoning inherent in the Riemann sum approach to the volume of a solid. This
behavior may be the result of learners’ focus on the metonymic aspect of definite
integrals put for area but not on the logical relations inherent on the statement.
Through a matrix algebra student’s reasoning with metonymies and metaphors,
we will build a case that the sole use of the two conceptual constructs in the
absence of focus on the relations inherent in them may account for the student’s
incomplete knowledge of linear independence.
Metaphor
A metaphor can be defined as an implicit analogy (Presmeg, 1998). Presmeg
describes a metaphor having both ground and tension. According to her,
similarities between concepts constitute the ground and differences constitute
the tension. For instance, for the mathematical statement, “A is an open set,” the
tension of a metaphor may be the physical idea of openness (an open space view)
without a boundary and the mathematical idea of an open set with a boundary.
Consider a set of all points that satisfy the inequality, x2+y2<1. Here, the open set
is bounded by the unit circle. Considering that similarities between concepts
(source and target) are mainly determined by students based on past
experiences, rather than being given to them (XXXX, 2007), students, in this case,
may apply the no boundary characteristic of the source concept, and come to a
conclusion that the particular set is not open since it has a boundary.
Metonymy
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.
Partitioning of a Region
Riemann Sum
Definite Integral
Area of a Region
Signified
Signifier1 Signifier 2 Signifier 3
Signified 2 Signified 3
Fig. 1. Example of the chaining process. Display of the Model is adopted from Presmeg (1997).
The theory of chaining of signifiers (a semiotic model), which defines “the sign as
a combination of a ‘signified’ together with ‘signifier ‘ ” (Presmeg, 1997, p274),
can be considered involving metonymic structures (Presmeg, 1997). According to
Presmeg (1997), in a semiotic model, all signifiers are metonymic since they are
considered in place of signified. Furthermore, as a result of a reification process,
each signifier in turn is considered as a new signified object. We go further to add
that the chaining and reification process of signifier to signified gives way to
meaning construction among objects. An example for the theory might be
definite integrals from calculus. Definite integral may be put for, by learners, the
area of a region. Here, area is considered as the signified and definite integral as
the signifier. Learners may use definite integrals without regard to the underlying
relation between the integral and the area of regions. To construct meaning
however, one may need to carry out a reification process by considering Riemann
sums as signifiers for definite integrals (signified). One may further enrich his/her
understanding if the partitioning of regions is considered as metonymies
(signifier) for Riemann sums (signified) and so on (see fig. 1). We think that one’s
understanding becomes richer with each new step on the chaining process. We
will demonstrate the particular behavior through our matrix algebra student’s
interview responses (see Findings and Discussion section).
Conceptual
understanding is the rich connections between multiple aspects of a concepts…..
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—one traditional and the other implementing an interactive webmodule 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 a generic name “Mario,” to refer to him
throughout the paper. He is a Hispanic mathematics student with a 4.0 GPA. 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 (See Fig. 2 for sample interview tasks).
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 students’ thought processes. A qualitative
approach, namely the constant comparison method (Glaser, 1992), is used to
analyze Mario’s responses.
1. Define the linear independence of a set of vectors.
2. Given a linearly independent set {u1, u2, u3, u4} in Rn. Prove/Disprove that the set is {u1, u2+5u1,
u3, u4} is linearly independent. [i.e. if it is linearly independent, prove it is; if it is not linearly
independent, explain why it is not linearly independent].
3. Given the set {u1, u2, u3, u4} where the vectors u1, u2, u3 are on the same plane and u4 is not.
Determine if the set {u1, u2, u3, u4} is linearly independent. Explain your answer.
Fig. 2. Sample Interview Tasks.
Question:
How the solutions to a vector equation relate to linear combination among vectors.
Metonomies displayed in reasoning:
Linear independence
Matrix ----set
Identity……>independent vectors
xi …………..> vector
known values assigned to vector xi represent.
solution…….> coefficient values for vectors
linear combination
Results and Discussion
SA21’s interview indicates a frequent use of metonomies in his reasoning. Student SA21 was asked to
share his definition and understanding of linear independence. Mario’s initial response indicates that he
may have been using the term “linear independence” to set 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 description was for linear combination, not for
linear independence as seen in the excerpt below, his responses throughout the interview however
supports his initial description of linear independence standing for linear combination.
Mario was regularly prompted to provide elaboration for his responses. In his elaboration, Mario
integrates other metonomic usage into his initial idea of linear combination. His metonomic use of
“matrix” standing for “set” 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.
15. 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]
71. 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]
While searching for linear combinations among columns he also integrate another metonomy of
“identity form” representing “lndependent vectors.” He focuses on identity form among columns of
matrices to eliminate some vectors as independent ones and tries to form linear combinations for the
remaining columns (representing vectors).
55.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]
61. SA21: Was… I got a… I have a… how I’m gonna say this.. so I have a…I can't express any of the…
these other vectors as identity… (student points at the last 3 columns of the row reduced
echelon matrix)
The most visible metanomic usage appears to be on his use of symbols “xis” as “vectors.” Mario infact
seems to atribute symbols entailing fixed meaning and reason with these meanings throughout the
interview. Initially, he indicates ais are known values in a linear combination but later reserves them as
unknowns and chooses the symbols xis as known values in a linear combinations, furthermore xis begin
to stand for both vectors and coeffiecient values for each vector interchangeable causing no conflict.
19.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???)
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]).
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???]
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]
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]
The use xis as vectors and coefficient values for vectors fits well with his reasoning of solutions
representing linear combinations. That is, Mario considers xis representing coifficient values for each
vector they represent. 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]…
85.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]…
87.SA21: that if I wanna write this as a combination of all these… then I can substitute these 2 into
here, so it’s gonna give me -2, -2, 2, 0, 0 so since this [poinying 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…
88.SA21: that’s going to give me -4, -6… [student writes -2[2;3]+2[1;1]=[-4;-6]]
Linear combination
1. SA21: Okay, so… when I (inaudible…) your definition. So I try to write it as close as I can, so… so
I think of linear independence so… I think we have a set of vectors , so I’ll just write… and then I
remember, so… 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… (5:04) times au and we’ll just keep on going…
2. 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).
Matrix->set
16. 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]
55.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]
62. SA21: Was… I got a… I have a… how I’m gonna say this.. so I have a…I can't express any of the…
these other vectors as identity… (student points at the last 3 columns of the row reduced
echelon matrix)
72. 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]
Xis->vectors
19.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???)
25.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]).
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???]
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]
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]
Solution->coefficients for vectors
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]…
85.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]…
87.SA21: that if I wanna write this as a combination of all these… then I can substitute these 2 into
here, so it’s gonna give me -2, -2, 2, 0, 0 so since this [poinying 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…
88.SA21: that’s going to give me -4, -6… [student writes -2[2;3]+2[1;1]=[-4;-6]]