Lack of set theory-relevant prerequisite knowledge
HAMIDE DOGAN-DUNLAP
University of Texas at El Paso, Department of Mathematical Sciences
El Paso, TX, 79968-0514, USA. Phone: (915)-747 6769
Email: hdogan@utep.edu
Many of our students struggle with college mathematics topics due to
a lack of mastery of prerequisite knowledge. Set theory language is
one such prerequisite for linear algebra courses. Many students’
mistakes on linear algebra questions reveal a lack of mastery of set
theory knowledge. The paper reports the findings of a qualitative
analysis of a group of linear algebra students’ mistakes on a set of
linear algebra questions. Paper also details an in-time intervention (a
pedagogical approach) to enhance students’ understanding of linear
algebra concepts through advancing their set theory knowledge.
Mathematics teachers can consider similar approaches to address their
students’ mistakes.
Keywords: Set theory; Prerequisite knowledge; Linear Algebra
1. Introduction
As college mathematics teachers, we all witness our students’ learning difficulties with
abstract mathematics concepts. Research indicates that formalism and the lack of set
theory knowledge are among the reasons for students’ struggles with and mistakes on
linear algebra concepts [1-3]. Dorier and Sierpinska [3] also add the nature and historical
background of linear algebra among the reasons. Formalism entails a wide range, from
the use of notations and symbols to the structures used to represent ideas. One such
representation tool is the language of set theory. Linear algebra makes use of set theory
language quite often. For instance, the vector space concepts such as subspaces and
spanning sets are often introduced through set theory-based representations. It is no doubt
that students need to have the knowledge of set theory as prerequisite for successful
learning and understanding of linear algebra concepts. Dorier, Robert, Robinet and
Rogalski in their paper [1] give various examples of a group of students’ work as
testimony to the necessity and the importance of set theory knowledge in responding
accurately to linear algebra questions. Referring to students’ incorrect responses, they
state: ‘The other incorrect responses show both the lack of appropriation of the notions in
question and the more or less inadequate mastery of set theory language’ (p 90). Brenton
and Edwards [4] look at their algebra students’ work and come to the conclusion that:
‘…Our investigations have convinced us that the most important
reason for the failure of good students to come to grips with the
concept of the quotient (factor) group is that they do not
understand what the elements of a quotient group are’ (p 32).
2. Student Mistakes on Linear Algebra Questions
A qualitative examination of the students’ responses on linear algebra exams from two
linear algebra sections (Fall 1999, N=29; and Spring 2002, N=45) offered in two different
institutions were used in the investigation. Students’ examinations were collected as part
of a longitudinal study investigating the effect of visual representations on students’
cognitive process in learning basic abstract linear algebra concepts. For further
information on the study and the methodology, see Dogan-Dunlap [5]. Both sections
were dominated by mathematics, science, and engineering students. Fall section had a
[INSERT FIGURE 1]
mixed population of US and international students, and the spring section comprised of
dominantly Hispanic-American students. The investigation revealed that the lack of
mastery of set theory knowledge may explain many of the mistakes occurred on students’
responses. The following three categories were documented as likely reasons for the
occurrence of these mistakes:
(C1). Inability to recognize appropriate criteria to determine membership.
(C2). Inability to distinguish between the general membership
description and the description of a single member.
(C3).
Inability to recognize various representations of the same set,
and particularly inability to describe other representations
using set theory language.
Figure 1 shows three linear algebra students’ incorrect responses on the exam
questions [5]. The three responses on figure 1 also typify the mistakes made on solutions
provided for each of the three questions. The student with the first response seems to be
confused between a vector and a set. This is considered as the category (C2) type
mistake. He considers (1, 2, 3) as a set rather than a single member of the set. Even
though the student provides the correct answer, his inaccurate reasoning may eventually
result in incorrect answers for similar questions with vectors in the form of (2y, -x+y,
x+y) for x, y≠0 and x≠y. Interestingly, the student was able to calculate the dimension of
the span of {a, b, c}. This could be credited to the fact that linear algebra provides many
procedures, and one can apply these procedures with no to very little understanding, and
arrive at correct answer. This particular student’s answer for the dimension question also
came from his application of the row reduced echelon form of a matrix whose columns
are formed by the three vectors, a procedure provided and worked on in class.
Another student’s incorrect response for question 2 appears to reveal her
inadequate mastery of set theory knowledge too. Her response indicates a reasonable
knowledge and an understanding of parametric representation yet her answer is not
correct. One explanation for this might be that the student may not have recognized the
vectors of the kernel as the vectors of R3. This however looks unlikely since she includes
the x, y, and z components of the members of the kernel through her parametric
representation. Even though the student arrives at an accurate parametric representation,
she is not able to transfer her first representation to the second form (set theory-based
representation). This is considered as the category (C3) type mistake.
The response for question three reflects a student’s lack of understanding of the
criteria that determine membership for a subset S of a set M 2, 2 (a set of all 2x2 matrices).
He uses any 2x2 matrix from M 2, 2 as the member of the subset S even though the set
theory-based algebraic representation of the subset S in the question specifically states the
members of S as those 2x2 symmetric matrices. This is considered as the category (C1)
type mistake. The student displays a reasonable understanding of a necessary condition
for a subset to be a subspace, and apply the condition, ‘closeness under addition’, but
uses inaccurate vectors and reasoning in doing so. His selection of the kinds of matrices
implies that he may lack an understanding of the description of the set S. He is not able to
extract the primary characteristics of the members of the set S from its set theory-based
description. It should also be noted that he states that the addition of corresponding
entries of the matrices results in a real number, which he says is necessary for the matrix
to be in M 2, 2 . He seems to be aware of the real number condition yet he shows a lack of
understanding that this condition is necessary but not sufficient to determine membership
for the subset S. There were some responses for the particular question where two
specific matrices with numerical entries (for the most part, they were 2x2 symmetric
matrices) used to show the closeness of the subset S under addition and scalar
multiplication. These responses are considered as the (C2) type mistakes since students
treated specific members (matrices with numerical entries) as representing all the
members of the subset.
It is undoubtedly clear that linear algebra primarily makes use of set theory
language, yet many linear algebra students lack the mastery of set theory knowledge as
prerequisite. Reinforcing set theory language as a prerequisite for linear algebra courses
may resolve many of the mistakes reported here. But in the mean time, what one can do
to help those students already in our classes displaying a lack of set theory knowledge
and mistakes similar to the ones reported?
One solution may be an in-time intervention to address the categories (C1)-(C3).
This may be done through carefully designed, student-centred, guided activities. The
author however is not claiming that all the learning difficulties linear algebra students are
experiencing can be addressed by a single approach. The claim is that a pedagogical
approach addressing above categories may minimize students’ mistakes originating from
an insufficient mastery of set theory knowledge by increasing the knowledge and
awareness of the relevant aspects of set theory language, and as a result decrease
frustration level. The rest of the paper details one such approach that is developed to
address linear algebra students’ mistakes due to the three categories listed above.
3. The Pedagogical Approach
The promise of the approach is to have students connect their real life experiences with
institutional (club) membership requirements to formal set theory language. A set of
questions in the form of a worksheet are structured around ‘club’ memberships and its
criteria. See figure 4 for the questions. While introducing the activity, students are told to
consider the ‘club’ as any organization such as fitness centers and golf clubs that requires
membership.
[INSERT FIGURE 2]
The purpose of the questions is to maximize discussions among students, and
create learning opportunities for students to become increasingly aware of the primary
aspects, relevant to the three categories, of set theory language. For instance, the
statement ‘list of evidence for membership’ is added to increase opportunities for
students to discuss issues relevant to the category (C1); the worksheet statement ‘reasons
for your decision on whether your description indicates all members or not’ is included to
encourage students to think about, and discuss how one can distinguish specific
membership representations from general membership representations. This is addressing
the issues relevant to the category (C2). The first row with the three club options on the
worksheet is added to have students compare and contrast various set theory-based
representations of clubs. Differing representations for three different sets (clubs) are
provided for students at the start of the activity to address issues relevant to the category
(C3). See figure 2 for the representations of three clubs.
In addition, a set of index cards for membership descriptions with varying
representational forms is provided for students to further think about membership criteria.
[INSERT FIGURE 3]
See figure 3 for a set of sample index cards. It is worth noting that some students are
given membership descriptions that are vectors from Rn, but not members of any of the
clubs provided in order to provide opportunities for students to engage in discourse, and
as a result, become aware of the fact that not every vector in Rn is a member of a club
even if the members of the club are vectors in Rn.
This activity can be completed in groups of three to four through group and class
discussions during a 50 minute class meeting. It is important that students not only
complete the worksheet but also fully participate in discussions in order to receive
support and feedback from their group members and classmates.
The particular approach is designed specifically for linear algebra students but it
can be modified for any mathematics courses that require the knowledge of set theory as
prerequisite. One can use the same structure and organization of the worksheet but
replace the sets of vector spaces (clubs) and vector space related membership descriptions
with more appropriate ones.
For further clarification, a group of linear algebra students’ work is described in
the remainder of the paper. A group of students enrolled in a first year linear algebra
section in Fall 2003 at a southwest four year university completed the activity during a 50
minute class time. There were 45 students, dominantly Hispanic, majoring in
engineering, computer science, science, and mathematics.
4. Student Work
 a  3b 
Student AL was given the algebraic expression, w   4b  , where a, b are real numbers,


2a  5b
as his membership description (See figure 4). To determine that the index card has a
1  3
membership description for the first club, V=Span{ 0, 4 }, AL needed to use the
  
2 5
definition of span as well as to recognize the vectors of R3. His response for the question
‘list of evidence for membership’ indicates that he knew the membership criteria for the
first club. He was able to show that his membership description is a linear combination of
the vectors used to describe the members of the first club. The three check marks under
the ‘Convinced’ column indicate that he was involved in a discourse on whether the club
he chose was the club he was a member of. His worksheet does not reveal whether the
discussion led him to discover primary membership criteria for the first club, class
discussions however revealed that in fact he became aware of them as a result of
dialogues with his group members. The reasons AL stated for his decision under the
‘Yes’ column implicitly indicate that during the activity he might have been encouraged
to think about the distinction between general membership descriptions and membership
descriptions for single members. His response ‘since a and b are real numbers…and the
subset W…’ and the instructor’s in-class observations point that the particular student
considered a new subset called W with members defined by his membership description,
and he made informal arguments in support of the new set W being the same as Club 1.
[INSERT FIGURE 4]
There were a noticeable number of students with membership descriptions that
are vectors from R3, and not in any of the sets given. These students initially declared
Club 1 as the club they were a member of. Their earlier arguments indicate that the
students at first considered vectors being in R3 as sufficient criteria for Club 1
membership. During their attempt to convince their group members however they were
challenged to come to realization that having a vector in R3 is a necessary but not
sufficient criteria for them to be a member of the first club.
Student BN was given the index card 3 in figure 3 as her membership
description. To determine that her index card has a membership description for Club 2,
student BN needed to use the criteria Ax=0, and be aware of R4 aspect of members of the
club. It is clear from the student’s work included on the worksheet under the ‘list of
evidence for membership’ column that the student was aware of both criteria for
membership (see figure 5). She included R4 criteria as the necessary condition, and the
matrix equation as the necessary and sufficient condition for Club 2 membership. BN’s
explanation for her decision that her membership description is a general description for
all members of the club is an informal argument which does not contain enough
information for an accurate interpretation of her knowledge. It is not clear for the
investigator what she means by ‘only one possible form of solution’.
[INSERT FIGURE 5]
During group and class discussions, both students were exposed to representations
they may not have been exposed to during their worksheet activity. For instance, if one of
AL’s group members had BN’s membership description then AL was asked by the
member to discuss the accuracy of his decision. This gave AL a chance to study the
representation and as a result increase his awareness of these forms of representations
addressing the category (C3) type difficulties.
Students not only needed to make decision on the appropriate criteria to be used
but also needed to decide whether their descriptions could be applied to other members
(whether it is general or specific). For instance, student BN needed to realize that the
description,
  2z 
 z  t  ,

w
 z 


 t 
where z, t are any real numbers, is the description for all the
vectors that are solutions for the matrix equation included in the description of Club 2.
On the other hand, a student with index card 1 in figure 3 needed to be aware that the
description is for a single member. That is, it does not describe anyone else but his/her
membership for Club 1.
Both group and class discussions revealed various issues students were struggling
with. For instance, many students including AL and BN were not able to provide formal
supportive arguments (proofs) for their decision on whether their membership description
indicates all members of a club or not. This led to further discourse and an additional
lesson on proving the equality of two sets with different representational forms.
Since the activity was in the context of a real life setting, it was easier for students
to communicate abstract concepts. After the activity, many students began to consider
abstract linear algebra concepts as less of a symbol manipulation. The instructor of the
particular linear algebra section believes that having students revisit basic set theory
concepts through a single activity resulted in a more meaningful learning of vector space
concepts.
5. Conclusion
This paper provided the results of a qualitative analysis of a group of first year linear
algebra students’ responses on examination questions, and recommended a pedagogical
approach to address some of the set theory related mistakes the particular group of
students revealed on their responses. The work of another group of linear algebra students
who completed the activities of the pedagogical approach is discussed to provide the
cognitive and social processes the group may have gone through. The arguments made
about these processes are strictly based on the classroom observations and the students’
work as displayed on their worksheets. A further investigation of the effect of the
approach on students’ cognitive and social processes in addressing set theory knowledge,
and its implications for the learning of linear algebra concepts is in order.
In courses where knowledge of set theory is a prerequisite and crucial for the
effective mastery of subject matter, activities similar to the one discussed in this paper
can be used to help students revisit, refresh, and perhaps advance their understanding of
set theory knowledge, and as a result, give students a chance to gain deeper
understanding of the content matter to be learned.
References
[1] DORIER, J., ROBERT, A., ROBINET, J., and ROGALSKI, M., 2000, The Obstacle of
Formalism in Linear Algebra. On the Teaching of Linear Algebra (Kluwer
Academic Publishers) Mathematics Education Library, 23, 85-124.
[2] DORIER, J., and ROBERT, A., 2000, On a Research Programme Concerning the
Teaching and Learning of Linear Algebra in the First Year of a French Science
University. International Journal of Mathematics Education in Science &
Technology, 31, Issue 1, 27-35.
[3] DORIER, J. and SIERPINSKA, A., 2001, Research into the Teaching and Learning of
Linear Algebra. In Derek Holton (Ed.) The Teaching and Learning of
Mathematics of University level. Kluwer Academic Publishers, DorDrecht, 255273.
[4] BRENTON, L., and EDWARDS, G. T., 2003, Sets of Sets: A Cognitive Obstacle. The
College Mathematics Journal, 34, n1, 31-38.
[5] DOGAN-DUNLAP, H., 2003, Visual Instruction of Abstract Concepts for Non-major
Students. The International Journal of Engineering Education (IJEE), 20. n4,
671-676.
Figures
Figure 1. Students’ responses on linear algebra questions.
Figure 2. Descriptions of clubs used in an activity in a first year linear algebra
course.
Figure 3. Membership descriptions on index cards used in an activity in a first year
linear algebra course.
Figure 4. Student AL’s worksheet from a first year linear algebra course.
Figure 5. Student BN’s worksheet from a first year linear algebra course.