Vargas,
X; Oktac, A; Trigueros, M (2006). Using
Apos theory to analyze the learning of the concept of vector space
[Version electrónica]. 3th Young European Society for Research in
Mathematics Education YERME 3. Jyväskyla,
Finland. Available here
USING APOS THEORY TO ANALYZE THE
LEARNING OF THE CONCEPT OF VECTOR SPACE
Xaab Nop Vargas, Cinvestav-IPN, México. xvargas@cinvestav.mx
Asuman Oktaç, Cinvestav-IPN, México. oktac@cinvestav.mx
María Trigueros, ITAM, México. trigue@itam.mx
Abstract. We report about a research study that aims at
describing student understanding of vector spaces using APOS
framework. In this paper we focus on three interview questions that
were asked to undergraduate students with the purpose of identifying
possible difficulties and errors in solving problems related to
vector space structure and observing the adequacy of the relations
that students might form between the vector space and the field.
1 BACKGROUND
Vector space theory, being abstract in nature and having an
epistemological status different from most mathematical topics taught
at the undergraduate level, is a major source of difficulty for
beginning linear algebra students ([1], [2]).
The identification of the nature of these difficulties and their
association with the way with which students construct the concept of
vector space is of great importance on the way to the development and
implementation of good instructional strategies. APOS
(Action-Process-Object-Schema) Theory provides a research tool that
has been successfully used in other areas of mathematics such as
abstract algebra and calculus, for similar purposes.
Previously we have reported on a possible genetic decomposition of
the concept of vector space and analyzed activities that students can
perform to make the necessary mental constructions ([3]). According
to this theoretical analysis, the construction of the vector space
schema for an individual requires the coordination of four schemas:
axiom, binary operation, function and set. As a result of working on
specific examples, different properties of the structure are
interiorized and later encapsulated to form an object that can be
called ‘vector space’.
2 METHODOLOGY
In order to test the viability of our genetic decomposition, and also
to study in depth the nature of student understanding and to identify
sources of difficulties in students with respect to the construction
of the vector space concept, we designed an empirical research study
which followed the steps that we describe here:
I. Observation of an introductory Linear Algebra course during
one semester at an undergraduate institution in Mexico, in which
engineering majors were being taught using the pedagogical component
of APOS framework. This pedagogy is based on the ACE (Computer
Activities – Class Discussion – Exercises)
cycle and the CAS Maple was used to handle the computer constructions
(for a detailed description of this cycle, see [4]). The textbook
chosen was the one written by the RUMEC (Research in Undergraduate
Mathematics Education Community) ([5]).
II. Interviews with 6 students form the same course about the
notions of vector space and subspace. Two of these students were
identified as belonging to the top level (we denote them as A1 and
A2) according to their grades on the first exam, two of them as
performing at the medium level (M1 and M2), and the remaining two
were classified as being poor (B1 and B2), but otherwise they were
chosen randomly within these three subgroups.
The interview consisted of 17 questions about the concepts of vector
space and subspace. Here we present three of these questions
(numbered 8, 9 and 12 in the instrument), together with our a
priori analysis of them and related student performance.
Question 8. Is a vector space over? (With the usual
operations) Q represents the rational numbers.
A priori analysis of Question 8: This question is about a case
in which the field is a subset of the set that we want the student to
check whether it forms a vector space over the given field. The fact
that Q is a subset of R might cause difficulties for students. Some
of them might think that for this reason it is not necessary to
verify the axioms that satisfy a vector space. We are interested in
observing if the students can see the role that the field plays in
this case, but in particular we want see if they confuse the elements
of the two sets. We are equally interested in identifying strategies
in dealing with this kind of problems and seeing whether somebody who
answers ‘yes’ has it clear what properties the vector space
should satisfy.
Question 9. Is a vector space over? (With the usual
operations)
A priori analysis of Question 9: Like the question 8, here we
would like to analyze the role that the field of scalars plays when
students think about a vector space. In this case the set to be
verified whether it forms a vector space is a subset of the field. We
are interested in observing similar phenomena as in the question 8.
Note. With these two questions we hope to have a general panorama
about students’ construction related to the vector space and the
field’s role, where there is an inclusion relation between the two
sets.
Related to these questions, we also asked:
Question 12. Let be a vector space over and. Is a vector
space over ?
A priory analysis of Question 12: In this question we asked if
any subset of a vector space over a field also forms a vector space
over the same field. We are interested in observing about student
difficulties related to the proprieties that a subset of any vector
space might have. Some of them might think that a subset of a vector
space is also a vector space, or give a counterexample to show that
this is not necessarily the case.
3 STUDENT PERFORMANCE
In this section we summarize students’ performance related to the
three interview questions.
Student A1 says, for the question 8, that the set of real numbers it
isn’t a vector space over, because is a vector space only over. In
his answer we observe that he didn’t check the axioms of a vector
space; we think that this is because the field is a subset of and
this might cause difficulties. He can’t differentiate the elements
of andhe says “for example. A rational, I don’t know, this is
a rational and you add to it another retrieval, you get a real, then
it is satisfied”. When the interviewer asks him what is
satisfied, he adds “the sum”. In this case he’s
operating over the elements of, but checking that the result belongs
to R.
In the case of Question 9 he says “yes, yes, yes…, we suppose,
this is a real.., a real isn’t it? And this is another real, I
don’t know what results, but this sum is not rational. Then not,
because the closure of sum isn’t satisfied”. We can see that he’s
operating with both sets, and he can’t make a difference as to
which should set satisfied the vector space axioms.
He answers correctly the question 12 and he gives the following
example: He says that V is a vector space over K, and he adds:
“if we have U as it isn’t a vector space because ”
A2 says that R is a vector space over Q, argumenting “because Q is
a subset of R”. When the interviewer asks her what happens with the
axioms she says that all are satisfied by the same reason. When she
checks the closure of the sum she operates with the elements of Q. We
think that this student confuses the vector space axioms with those
of the field.
About the question 9, she starts by saying that it isn’t possible,
because R isn’t a subset of Q. Searching for an example, she thinks
about some real number and she takes and adds zero to it. She says
that the closure of the sum isn’t satisfied because the result
doesn’t belong to Q. Clearly she confuses the elements of the
vector space and the elements of the field.
In question 12 she answers correctly, giving the following example:
“when they give you a line in, the whole line passing through the
origin, there you have a vector space, but if you take a segment of
this line in which you don’t pass trough the origin, this segment
belongs to whole line…, but this segment doesn’t pass trough the
origin, it isn’t a vector space”
For Question 8, M1 writes: “then, the operations you do with the
real numbers, result in real numbers, and all rationals are real.
Then the operations you perform on rational numbers result in real
numbers, then R is a vector space over Q.”
For Question 9, he writes: “Yes, because all the elements of Q
belong to R, so if all the axioms for vector space are checked, with
elements of Q, are done on real numbers and result in real numbers”.
For Question 12, he writes: “Yes, because all elements of U
belong to K, and the operations to check the properties of the vector
space are done on elements of K”.
M2 explains “because, if I take any two real numbers, their sum
does not necessarily turns out to be a rational number” when
answering Question 8. When the interviewer questions him if there is
an axiom that is not satisfied, he says “the sum” and he
writes “the sum’s closure is not satisfied”.
In the case of Question 9 he is confused between the elements of the
sets. He says “yes, that is what I said, isn’t it?, in the
other direction it is possible, all rational are real, if I take
two…, rational when I sum the result is a real number, and if I
multiply that with another rational the result is real always as
well. Q is a subset of R. Then, yes, all the properties are
satisfied”. When the interviewer asks him about the other
axioms he is not sure if zero is an element of Q.
About Question 12 he says “… yes because it is a subset, then
it inherits all the properties V has, isn’t it?” When the
interviewer asks him about an example he writes “”
When solving Question 8 B1 says “the rational numbers are
container in the real numbers”, and he writes, “then the
operations done in real numbers…and the rationals are all of them
real numbers, then, the operations that you do on rational numbers
will be performed with real numbers, so R is a vector space on Q”
When the interviewer asks him to explain, he says “well, here
you have the rational numbers, they are contained in the real
numbers, then the operations you do with …rational numbers result
only in real numbers…no, the other way round, the operations with
real numbers can result in rational numbers, then it is a vector
space on Q” The interviewer asks “Is that the only reason?”
and he replies “yes, it is”. For Question 9 he writes
“Yes because the elements of Q are contained in R, then if you
do all the operations that define a vector space with the elements of
Q, you are using real numbers and the results with be real numbers”,
later he says “as Q is contained in R, all the operations
performed on Q will result in numbers in R, then it is a vector
space” The interviewer asks “The fact that one set is
contained in other is enough?” and he replies “Yes, Q is a
vector space”
For Question 12 B1 says that U is a vector space over K, because U is
a subset of V.
B2 can’t answer Question 8. He says “I can’t remember an
example” and he adds that he can’t state formally the axioms.
For Question 9, he is not sure, but he writes “No, because not
all the elements of Q are elements of R”
B2’s first reaction to Question 12 is, “first we need to check
if U is a subset of V, then if U is a subset of V; we need to check
that it is a subset”, when the interviewer tells him that U is
a subset of V, B2 answers “yes, then, yes, can I give you an
example?”, he gives as a subset of, then he says that this is
always satisfied.
4 CONCLUSIONS
We observe that Questions 8 and 9 were particularly difficult for all
students who took part in the study. In terms of APOS theory we can
say that these students have not encapsulated yet the concepts of
vector space and field, they are not able to clearly differentiate
their roles in the definition of vector space. their schema for those
concepts. When the field is a subset of a set which is probably a
vector space (question 8) or conversely when the set which they need
to prove is a vector space is a subset of the field, students are
confused. As was observed in the results of the previous section,
students show a tendency to rely on the property of being subset to
conclude it is a vector space, ignoring the axioms.
Another difficulty that is revealed by these two questions
corresponds to the use of the elements of the sets, vector space and
the field. We observe that there students show confusions between the
elements of those sets, particularly when the students want to check
the axiom of the closure of the sum. In this case, they take one
element of the field and an element of the vector space instead of
two elements of the vector space. In those situations, the students
start analyzing which is the field and do not verify the validity of
the axioms of the vector space.
About question 12, we observe that the majority of students think
that if is a vector space over and, then is a vector space over. This
indicates that students’ schema for vector space and subset is in
an early stage of development. The following table shows how students
answered to the above mentioned questions.
|
student
|
Question
|
Answer
|
|
A1
|
12
|
Correct
|
|
A2
|
12
|
Correct
|
|
M1, M2, B1, B2
|
12
|
Incorrect
|
|
A1, A2, M1, M2, B1,
B2
|
9
|
Incorrect
|
|
A1, A2, M1, M2, B1,
B2
|
8
|
Incorrect
|
We hope that a detailed analysis of the observations made during the
lessons of the course, and the students’ answers to the other
questions in the interview will shed more light into students’
sources of difficulty and will also allow us to make some didactical
suggestions to improve the design of the course and to propose
complementary activities for the students.
REFERENCES
[1] Dorier, J-L. A general outline of the genesis of vector space
theory. 1995. Historia Mathematica, 22(3), 227-261.
[2] Dorier, J-L. Meta level in the teaching of
unifying and generalizing concepts in mathematics. Educational
Studies in Mathematics, 1995. 29(2),
175-197.
[3] Trigueros, M. and Oktaç, A. La Théorie APOS
et l'Enseignement de l'Algèbre Linéaire. 2005.
Annales de Didactique et de Sciences
Cognitives, vol. 10, 157-176.
[4] Asiala, M., Brown, A., DeVries, D., Dubinsky, E., Mathews, D. et
Thomas, K. A Framework for Research and Curriculum Development in
Undergraduate Mathematics Education. 2006. Research in Collegiate
Mathematics Education II, CBMS Issues in Mathematics Education,
6, 1-32.
[5] Weller, K., Montgomery, A., Clark, J., Cottrill, J., Trigueros,
M., Arnon, I., Dubinsky, E. Learning Linear Algebra with ISETL. 2002.
available at:
This research study is funded by Project CONACYT
2002-C01-41726S.
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