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 two interview questions that
were asked to undergraduate students with the purpose of identifying
possible difficulties with the vector space structure and observing
the adequacy of the relations that students might form between
different elements of the genetic decomposition of the vector space
concept.
2006 Oktac,
A; Trigueros, M; Vargas, X (2006). Understanding of Vector Spaces- a
viewpoint from APOS Theory [Versión electrónica]. Proceedings of
the 3rd
International Conference on the Teaching of Mathematics at the
Undergraduate Level. Turkish Mathematical Society, Istambul, Turkey, available here
Understanding of Vector Spaces- a
viewpoint from APOS Theory
Asuman Oktaç, Cinvestav-IPN, Mexico. oktac@cinvestav.mx
María Trigueros, ITAM, Mexico. trigue@itam.mx
Xaab Nop Vargas, Cinvestav-IPN, Mexico. xvargas@cinvestav.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 two interview questions that
were asked to undergraduate students with the purpose of identifying
possible difficulties with the vector space structure and observing
the adequacy of the relations that students might form between
different elements of the genetic decomposition of the vector space
concept.
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 (Researh 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 two of these questions (numbered
6 and 7 in the instrument), together with our a priori
analysis of them and related student performance.
Question 6. Is it possible to have a vector space containing
only one element?
A priori analysis of Question 6: This question is about an
‘extreme case’ and might cause difficulties for students. Some of
them might think that in order to perform binary operations one needs
at least two elements, or perhaps three, in the case of checking
associativity. We are interested in observing whether the axioms of
‘inclusion of the zero vector in the vector space’ and ‘closure
with respect to addition and scalar multiplication’ are
interiorized and present in the students’ discourse when responding
to this question. 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 unique element
should satisfy.
Related to this question, we also asked:
Question 7. Is it possible to have a vector space containing
only two elements?
A priori analysis of Question 7: Apart from observing what we
mentioned above in the case of Question 6, with this question we
would like to analyze the role that the field of scalars plays when
students think about a vector space. In traditional courses, the
field is almost always taken to be the set of real numbers with the
usual operations and hardly other possibilities are mentioned. A
student who restricts himself/herself to the field of real numbers
might answer that this is not possible, as all the vectors in the
vector space should also have their (real-) multiples contained in
the same space.
3 Student performance
In this section we summarize students’ performance related to the
two interview questions.
Student A1 says that the set consisting of the zero vector would be
the only possibility for Question 6.
In the case of Question 7 he gives the example {(0,0) , (1,1)} and
afterwards writes U = {(x,x) Z2
│ k Z2}indicating
that it is a vector space over Z2.
A2 thinks that a vector space with only one element cannot exist,
giving the following argument: If that element were different from
zero, adding it to itself would yield a different element, and hence
it would not satisfy the closure axiom. On the other hand, if that
element were zero, then it would not have a multiplicative inverse.
We think that this student confuses the vector space axioms with
those of the field.
About a vector space containing two elements, she starts by saying
that it is possible. Searching for an example, she thinks about a
line in R2. After the interviewer reminds her that a line
would have infinitely many elements, she cannot find any example and
concludes that it is not possible.
For Question 6, M1 writes: “yes, {} because it is its own
inverse and all the operations will give the same element”.
For Question 7, he writes: “No, because if there are 2 elements,
we would need an additive inverse. There has to be a zero, an element
to do operations, and its inverse. And with 2 elements there isn’t”.
M2 gives {} as an example for Question 6. When the interviewer
questions him about the possibility of verifying the axioms with one
element, he gives a satisfactory answer.
In the case of Question 7 he says it is possible, but cannot think of
an example.
B1’s first reaction to Question 6 is, “Depends what element it
is”. He says that we have to check various axioms and see if
the element satisfies them. He then says that it is not possible,
because “by definition a vector space has to have at least two
elements”. He writes {v1} and explains that there
has to be an element from the field, and another from the vector
space. He adds: “So, only one element doesn’t allow us, how do
you call it? .. verify the axioms, because we don’t have a field,
nor… we would only have one element so it is not possible to check
if it is a vector space or not”. When the interviewer tells him
“and if I told you over any field K?”, B1 repeats: “if
you tell me one element over a field K?” He then says that in
that case “it can be done… if you have the scalars”.
However, thinking about it more, he concludes that it is not possible
either, because we would need two elements v1 and v2
to be able to check, for example, the commutativity axiom.
Similarly, he concludes that to check associativity he would need a
third element v3; because if he cannot check the property
it is not satisfied and hence there cannot be a vector space with
only two elements.
B2 immediately proposes {} as a vector space for Question 6,
indicating that it is closed under addition. When the interviewer
repeats the question as to whether it is a vector space, he doubts
himself and comments that the subject of vector spaces was
particularly difficult for him.
Before starting to work on Question 7, he asks whether he can define
the field himself. The interviewer responds affirmatively to this
question and B2, after thinking a little bit and writing different
sets, decides that V = {0,1} would be an example. When the
interviewer asks what the field is, he says he doesn’t remember,
and then writes K = R. After this he cannot justify his response nor
think of another example, but he comments that he is pretty sure
there is one.
In order to have elements for comparison, we applied a questionnaire
that contained the same two questions to a group of master’s
students in Mathematics Education, who are taking a course on the
teaching of Linear Algebra. Out of 10 students who are registered for
the course, 8 responded the questionnaire; the other two were absent
that day.
All of these students gave the vector space containing only the zero
element as an answer for Question 6. For Question 7, there were three
types of answers: 4 students said it was possible, but they gave
erroneous examples. Two students said it was not possible, arguing
that the axiom of closure of addition would not be satisfied. The
remaining two tried to obtain a vector space structure with two
elements, but did not succeed.
4 Conclusions
We observe that Question 7 was particularly difficult for all but one
student who took part in the study. We explain this fact in terms of
APOS theory as follows: Students have not encapsulated the concept of
binary operation and have not formed a rich schema of this concept.
In spite of explicit instruction about structures other than the real
numbers with the usual operations, it is particularly difficult for
them to apply these ideas in the context of a new structure that is a
vector space. In particular, the interpretations of students about
the axioms stay within the field of real numbers and correspond to
the visual image that they have about operating with real numbers. As
a result, coming up with an example of a structure that would satisfy
the vector space axioms becomes impossible.
Question 6 was answered correctly by 4 out of 6 students that we
interviewed (although one showed doubts about it). The difficulties
we observed can be explained in similar terms corresponding to our
analysis of Question 7.
Another difficulty that is revealed by these two questions
corresponds to the use of variables in mathematics. Thinking that
each letter in an expression has to represent a different value has
its roots in earlier experiences of the students. It has been shown
that even university students can work with variables only in very
simple and straightforward problems. Students can consider variables
as general numbers in simple expressions, but when they are used for
generalizations of properties, interpreting their meaning and
operating with them becomes an obstacle for most students. The
Algebra they have learnt at school is based on blind manipulation and
they have probably not had enough opportunities to interiorize the
different roles variables can play in different problems and within
the same problem (For a detailed account of difficulties with
variables, see [6]).
We were expecting to see that in the case of students who were
following a Linear Algebra course that made use of the ACE cycle, the
use of finite fields and computer activities would enrich students’
mathematical world and would allow them to consider different
structures in solving problems. In fact this happened in the case of
at least one student and in some other students we observe that there
is something that makes them think about the possibility of a vector
space with only two elements. We hope that detailed analysis of the
course and the students’ answers’ to the other questions in the
interview will shed light into this difficulty and will also allow us
to make didactical suggestions to improve the design of the course
and propose other activities for the students.
5 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:
[6]
Trigueros, M. and Ursini, S. Starting college students' difficulties
in working with different uses of variable. 2003. Research in
Collegiate Mathematics Education. CBMS Issues in Mathematics
Education (Vol. 5, pp. 1-29). Providence, RI. American
Mathematical Society.
This research study is funded by Project CONACYT
2002-C01-41726S.
Paper 345 (oral presentation); Main Theme: Educational Research; Secondary Theme: Specific Courses.
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