INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIII, Issue VII, July 2024
www.ijltemas.in Page 170
Handling Difficult Topics in Linear Algebra Through Pedagogical
Approaches
Dr. Robert Kati
Department of Curriculum and Pedagogy, Kibabii University, Kenya
DOI: https://doi.org/10.51583/IJLTEMAS.2024.130720
Received: 15 July 2024; Revised: 06 August 2024; Accepted: 10 August 2024; Published: 20 August 2024
Abstract: Linear algebra is a fundamental branch of mathematics that plays a crucial role with a wide range of applications to the
natural sciences, to engineering, to computer sciences, to management and social sciences, and more various fields. However, for
many students, certain topics within linear algebra including include eigenvalues and eigenvectors, determinants, and abstract
vector spaces. can be challenging to grasp. This paper explores nine (9) pedagogical approaches to effectively teach and learn
these difficult topics, aiming to enhance students' understanding and retention of linear algebra concepts.
Key words: Linear algebra, difficult topics and pedagogical approaches
I. Introduction
Linear algebra is concerned with vector spaces, vectors, linear functions, the system of linear equations, and matrices. These
concepts are a prerequisite for sister topics such as geometry and functional analysis. Linear algebra is one of the most central
topics of mathematics. While some students may find the basics of linear algebra manageable, more advanced topics can prove
challenging. Difficult topics often include eigenvalues and eigenvectors, determinants, and abstract vector spaces. Teaching linear
algebra without providing concrete examples of concepts can lead to students simply memorizing definitions and rules. Numerous
studies have demonstrated that incorporating technology into instruction among other pedagogical approaches, is an effective way
to make concepts more tangible and comprehensible. This paper examines various pedagogical strategies that can help instructors
and students navigate these challenging areas of linear algebra effectively.
Visualization Techniques
One of the difficulties in linear algebra arises from its abstract nature. According to Carlson, Johnson, Lay, and Porter (1993),
teaching without concretizing the concepts of linear algebra drives students to memorize the definitions and the techniques. Harel
(2000) stated that the geometric visualization of the concepts of linear algebra can support students about meaningful learning but
if this is done extensively, this can prevent students from making generalizations to multidimensional spaces. Visualizing
concepts can greatly aid comprehension. Graphs, diagrams, pictures and geometrical shape or models are a tool for visualization
of the abstract concept in mathematics. By means of these, human reason sets up a relation between physical or external world
and the abstract concepts (Konyalioglu, 2003). For instance, when teaching eigenvalues and eigenvectors, using graphical
representations to demonstrate how matrices transform vectors can make these abstract concepts more tangible. Interactive
software and 3D visualization tools can enhance the learning experience. According to Konyalioglu et al, (2011), utilizing a
visualization approach in teaching linear algebra can significantly benefit students who struggle with excessive abstraction.
Presenting concepts visually in this manner can be highly effective. These illustrative examples have the potential to enhance
their engagement and performance. Teaching linear algebra can be challenging, but employing visualization, especially through
vector geometry, can make it more accessible and engaging for students.
In teaching linear algebra, visualization techniques can make abstract concepts more tangible. For instance, using graphing
software or 3D models to represent vectors and transformations helps students grasp vector spaces and linear transformations.
Visual aids like matrices displayed as grids can clarify operations like matrix multiplication. Eigenvectors and eigenvalues can be
illustrated through dynamic animations showing how vectors stretch or shrink. However, potential problems include students
becoming overly reliant on visual aids, which may hinder their ability to understand concepts abstractly. Additionally, those with
limited spatial reasoning skills might struggle, and complex visualizations can sometimes oversimplify or misrepresent intricate
ideas.
Real-Life Applications
Numerous studies highlight the need to motivate students to study mathematics by presenting problems related to real life and
concrete problems, (Yilmaz, & Mierlus,2020, Yilmaz et al, 2020). Showcasing real-life applications of linear algebra can motivate
students and help them understand the practical significance of challenging topics. For example, using linear transformations to
explain image compression or Markov chains to model real-world processes can make abstract concepts more relatable. Other
real life applications of linear algebra include: optimization, encoding data as vectors in a vector space, greedy algorithims, linear
models among many others. In his paper on, “Some Applications of Linear Algebra and Geometry in Real Life”, Bonanzinga,
(2022) provides some real-world motivated examples illustrating the power of linear algebra tools as the product of matrices,
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIII, Issue VII, July 2024
www.ijltemas.in Page 171
determinants, eigenvalues and eigenvectors. A key application of linear algebra in real life is in projecting a three-dimensional
view into a two-dimensional plane, handled by linear maps.
Using real-life applications in teaching linear algebra can enhance understanding by connecting abstract concepts to practical
situations. For instance, demonstrating how linear algebra is used in computer graphics can illustrate transformations and vector
spaces. Explaining its role in data science, such as in principal component analysis (PCA) for data reduction, makes eigenvalues
and eigenvectors more relatable. In engineering, showing applications in systems of linear equations for circuit analysis can
provide practical context. However, potential problems include the risk of students focusing too much on specific applications
rather than grasping underlying principles. Additionally, real-life examples might sometimes be too complex, leading to confusion
rather than clarity.
Scaffolding
The term “scaffolding” is often used to characterize the various contributions made by agents and artifacts to an individual’s
learning. Once the learner is proficient in executing some target skill, the litany goes, this scaffolding can be removed. Nachowitz
(2018) described instructional scaffolding as requiring teachers to “model their writing processes using think-aloud protocols,
followed by collaborative practice, feedback with guided instruction, and individual student practice until mastery is achieved” (p.
11). Citing real problems as an example can lead to meaningful learning because student can aloud think about respected
mathematical problem in scaffolding process then relates to self-mind and self-around until be able response to problem. In this
moment, meaningful learning will occur and student can motivate to problem solving and it's interesting that student will not has
fear of difficult problem and can solve via teacher or instructor (Amiripour et al, 2012).
Breaking down complex topics into smaller, more manageable parts can facilitate learning. Start with simpler concepts, and
gradually introduce more challenging ones. For example, when introducing vector spaces by defining them as, “A vector space V
over a field F is a set V with an addition operation + and scalar multiplication operation · by elements of F that satisfy given
axioms”, it is important for the concept of a field F be well recapitulated to learners before they internalize what a vector space is
additionally when teaching abstract vector spaces, begin with concrete examples in familiar vector spaces like IR² or IR³ before
moving on to more abstract spaces.
Scaffolding in teaching linear algebra involves breaking down complex topics into manageable steps, gradually increasing
difficulty as students build understanding. For example, starting with basic vector operations before progressing to vector spaces
and transformations helps students develop foundational knowledge. Introducing matrices with simple addition and multiplication
before tackling more complex topics like eigenvalues and eigenvectors ensures comprehension at each stage. However, potential
problems include the risk of oversimplifying concepts, leading to a superficial understanding. Additionally, students might
become dependent on the scaffolding process, struggling to apply their knowledge independently. Balancing support and
independence is crucial to avoid these pitfalls.
Active Learning
Integrating active learning into mathematics classrooms necessitates a departure from the conventional lecture-style teaching
approach, in favor of one that fosters constructive student interactions. For instance, altering the norms governing mathematical
communication can encourage students to engage in activities such as explaining concepts to their peers, making conjectures, or
providing justifications. Active learning also encompasses the promotion of beneficial intra-student activities, encompassing
aspects like students' mathematical reasoning, written reflections, and individual task work. Regardless of whether the tasks
emphasize procedures, applications, or concepts, each presents valuable opportunities for active learning (Boyce, & O’Halloran,
2020).
Active learning can be encouraged through problem-solving sessions, group discussions, and hands-on activities. For instance,
when teaching determinants, students could be given matrices to calculate determinants themselves. This hands-on approach
fosters a deeper understanding of the underlying principles.
Active learning in teaching linear algebra engages students through interactive and participatory methods. For example, group
activities where students solve linear systems collaboratively enhance problem-solving skills. Using clicker questions during
lectures to test understanding of vector spaces and transformations keeps students engaged. Incorporating hands-on projects, like
coding applications of matrix operations, allows practical application of concepts. However, potential problems include the
challenge of adequately covering the syllabus due to time constraints of active learning activities. Additionally, some students
might feel uncomfortable participating in group work or interactive sessions, potentially hindering their learning experience.
Balancing active methods with traditional instruction is essential.
Technology Integration
Leverage technology to enhance learning experiences. Online platforms, mathematical software, and linear algebra-specific apps
can provide students with interactive tools to explore and experiment with difficult concepts. Interactive software and graphing
calculators can provide students with dynamic visual representations of matrices, vectors, and transformations. This not only
helps demystify abstract concepts but also encourages active exploration, enhancing understanding. Integrating technology into
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIII, Issue VII, July 2024
www.ijltemas.in Page 172
the teaching and learning of linear algebra is pivotal for fostering a more engaging and effective educational experience. Linear
algebra, known for its abstraction and complexity, can become more accessible and comprehensible with the judicious use of
technology. Online platforms and collaborative tools facilitate group problem-solving and peer-to-peer learning. Students can
work together on linear algebra problems, fostering a sense of community and offering diverse perspectives on solving
mathematical challenges. Incorporating technology in linear algebra education requires teacher training and ongoing professional
development to effectively leverage these resources. When used thoughtfully, technology can transform the learning experience,
making linear algebra more engaging, accessible, and applicable in today's digital age (Jimoyiannis,2010).
Technology integration in teaching linear algebra can greatly enhance comprehension and engagement. Utilizing graphing
software allows students to visualize vector spaces and transformations dynamically. Interactive apps can help demonstrate
concepts like matrix multiplication and eigenvalues. Online platforms provide simulations and virtual labs, offering hands-on
experience with abstract ideas. However, potential problems include the digital divide, where some students may lack access to
necessary technology. Additionally, over-reliance on technology can lead to a superficial understanding if students focus more on
using tools than grasping underlying principles. Technical issues and the learning curve associated with new software can also
impede progress.
II. Conceptual Understanding Over Memorization
Emphasize conceptual understanding over rote memorization. conceptual understanding has to do with comprehension of
mathematical concepts, operations, and relations while rote memorization has to do with the use of repetition to keep information
in the brain. Teachers need to encourage students to grasp the underlying principles and theories rather than relying solely on
formulas. This approach enables them to apply their knowledge to a wider range of problems and scenarios. A significant
indicator of conceptual understanding is being able to represent mathematical situations in different ways and knowing how
different representations can be useful for different purposes. According to Al-Mutawah, el al (2019), conceptual understanding
reflects a student's ability to reason and comprehend mathematical concepts, operations and relations which will be helpful in
solving non-routine problems.
Focusing on conceptual understanding over memorization in teaching linear algebra helps students grasp the "why" behind
concepts. For example, instead of just memorizing matrix operations, students can explore why certain transformations work
through visual proofs and applications. Discussing the geometric interpretation of vectors and spaces promotes deeper
comprehension of their properties and relationships. Encouraging students to derive formulas and solve real-world problems
fosters critical thinking. However, potential problems include the possibility that some students might struggle without concrete
steps to follow, and the depth of conceptual discussions may take more time, potentially slowing curriculum coverage. Balancing
conceptual insight with procedural fluency is crucial.
Multiple Representations
Present concepts using diverse representations, such as matrices, equations, and geometric interpretations. For example, when
teaching matrix multiplication, demonstrate how it relates to composition of linear transformations and systems of equations.
Encouraging students to solve mathematical problems by employing diverse representations, definitions, theorems, and properties
is a fundamental pedagogical approach. This method not only aids in problem-solving but also enables students to grasp the
interconnections among various mathematical concepts and fosters the creation of novel mathematical knowledge. By exploring
different perspectives and approaches to problems, students develop a deeper understanding of the underlying principles. They
learn to discern patterns, identify relationships, and appreciate the versatility of mathematical concepts. This process cultivates
critical thinking skills and mathematical fluency. Moreover, encouraging multiple approaches to problem-solving promotes
creativity in mathematics. It empowers students to think outside the box, formulate conjectures, and explore uncharted
mathematical territory. This active engagement in mathematical exploration can lead to the discovery of new theorems and
insights (Ervynck, 1991; Levav-Waynberg & Leikin, 2012; Silver, 1997).
Using multiple representations in teaching linear algebra enhances understanding by presenting concepts in various formats. For
example, vectors can be represented graphically, numerically, and algebraically, helping students see their properties from
different perspectives. Matrix operations can be shown through symbolic manipulation, visual grids, and real-world applications
like transformations. Linear systems can be solved using algebraic methods, graphing, and matrix techniques, reinforcing
connections between approaches. However, potential problems include overwhelming students with too many representations at
once, which can lead to confusion. Additionally, some students might struggle to transition between different formats, requiring
careful integration and clear explanations to ensure cohesive understanding.
Peer Teaching and Collaborative Learning
There is need to encourage peer teaching and collaborative learning. Students explaining concepts to their peers can solidify their
own understanding while helping others. In a linear algebra classroom, the peer tutoring strategy can be viewed as a scenario in
which students assist their peers in understanding linear algebra concepts, facilitated by a teacher who serves as a guide and
facilitator. Various research studies have documented that peer tutoring in college mathematics significantly influences the
intellectual and ethical growth of students. In this context, the language employed among peers fosters collaboration and
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIII, Issue VII, July 2024
www.ijltemas.in Page 173
teamwork, enabling all participants to freely share their ideas, grasp diverse concepts, shoulder responsibility, and demonstrate
resourcefulness (Abdurrahman et al, 2020). Group activities can foster a supportive learning environment where students can ask
questions and seek clarification. Incorporate collaborative learning activities that motivate students to learn linear algebra and
promote an environment for them to develop social and communication skills. According to Andam et al, (2016), the traditional
way of teaching and learning where the teacher decides on what goes in the classroom has a limited space in the 21 st century
mathematics classrooms, and that the cooperative learning approach must be encouraged by all since it promotes greater students'
participation in the teaching and learning process and environment.
Peer teaching and collaborative learning in linear algebra involve students working together to understand concepts. For instance,
students can explain vector operations or matrix manipulations to one another, reinforcing their knowledge. Group projects, such
as solving linear systems or applying eigenvalues in real-world scenarios, promote teamwork and deeper comprehension. Peer
tutoring sessions allow advanced students to help peers struggling with complex topics. However, potential problems include
unequal participation, where some students might dominate while others remain passive. Miscommunication or incorrect
explanations among peers can also lead to misunderstandings. Careful monitoring and structured activities are essential to
mitigate these issues and ensure effective collaboration.
III. Assessment and Feedback
Regular assessments and constructive feedback should be provided to gauge students' comprehension and track their progress in
linear algebra. Feedback should be tailored to address common misconceptions and difficulties encountered by students.
According to Stacey, & Wiliam, (2012), assessment should be regarded as an intrinsic component of teaching and learning, rather
than as the final outcome of the educational process. In this role, assessment offers a valuable chance for both teachers and
students to pinpoint areas of comprehension and areas of confusion. Armed with this insight, students and educators can expand
on their comprehension and actively work to convert misconceptions into meaningful learning experiences. Therefore, the time
allocated to assessment becomes an essential contributor to the overarching objective of enhancing the mathematics education of
every student. Mathematics assessments can serve as a valuable tool for enhancing the work of students and teachers alike. It is
essential for students to develop the skills to monitor and evaluate their own progress in mathematics. When students are actively
encouraged to assess their own learning, they gain a heightened awareness of their knowledge, learning methods, and the
resources they utilize when tackling mathematical problems. This conscious understanding of available resources and the capacity
for self-monitoring and self-regulation are pivotal aspects of self-assessment, which successful learners employ to foster a sense
of ownership over their learning and encourage independent thinking (National Research Council, 1993).
Assessment and feedback in teaching linear algebra are crucial for monitoring progress and guiding learning. For example,
frequent quizzes on vector operations and matrix manipulations provide immediate insight into students' understanding.
Assignments requiring application of linear transformations and eigenvalues offer opportunities for detailed feedback. Peer
assessment in group projects encourages collaborative learning and self-reflection. However, potential problems include the time-
consuming nature of providing individualized feedback and the possibility of students feeling overwhelmed by frequent
assessments. Balancing formative assessments with constructive feedback is essential to avoid discouraging students while
ensuring they receive the guidance needed to master complex linear algebra concepts.
IV. Conclusion
Handling difficult topics in linear algebra through pedagogical approaches requires creativity and adaptability on the part of
instructors. By using visualization techniques, real-life applications, scaffolding, active learning, technology integration,
emphasizing conceptual understanding, multiple representations, peer teaching, and effective assessment, educators can make
linear algebra more accessible and engaging for students. With these strategies, students are more likely to overcome challenges
and develop a deeper and lasting understanding of this fundamental mathematical field.
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