Expert answer:Reading Reflection Framing the structural role of
Solved by verified expert:Directions: Answer each of the following questions based on your reading “Framing the structural role of mathematics in physics lectures: A case study of electromagnetism” by Karam. (attached)1. This paper differs from the ones we have read previously in that it focuses more on instruction than on student understanding. What do you think the value of this research perspective is, if any?2. Which of the categories (summarized in Table II) do you use the most in your instruction? Why?3. This case study is of a university lecture environment. A high school environment is often much different. How are you preparing your students for this transition? (Note: There is a movement to reduce traditional lecture in the university setting, but it has not reached all physics courses yet.)
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PHYSICAL REVIEW SPECIAL TOPICS – PHYSICS EDUCATION RESEARCH 10, 010119 (2014)
Framing the structural role of mathematics in physics lectures:
A case study on electromagnetism
Ricardo Karam
Faculty of Education, University of Hamburg, 20146 Hamburg, Germany
(Received 27 September 2013; published 28 May 2014)
Physics education research has shown that students tend to struggle when trying to use mathematics in a
meaningful way in physics (e.g., mathematizing a physical situation or making sense of equations).
Concerning the possible reasons for these difficulties, little attention has been paid to the way mathematics
is treated in physics instruction. Starting from an overall distinction between a technical approach, which
involves an instrumental (tool-like) use of mathematics, and a structural one, focused on reasoning about
the physical world mathematically, the goal of this study is to characterize the development of the latter in
didactic contexts. For this purpose, a case study was conducted on the electromagnetism course given by a
distinguished physics professor. The analysis of selected teaching episodes with the software Videograph
led to the identification of a set of categories that describe different strategies used by the professor to
emphasize the structural role of mathematics in his lectures. As a consequence of this research, an analytic
tool to enable future comparative studies between didactic approaches regarding the way mathematics is
treated in physics teaching is provided.
DOI: 10.1103/PhysRevSTPER.10.010119
PACS numbers: 01.40.Fk, 01.40.gb
I. INTRODUCTION
Many physics instructors complain that their students do
not know enough mathematics. However, it is quite clear
that the domain of basic mathematics skills does not
guarantee success in physics, since “using mathematics
in physics is much more complex than the straightforward
application of rules and calculation” [1]. Despite the
deep interrelations between physics and mathematics—
confirmed by both historical and epistemological studies
[2–7]—it is not uncommon in physics education to regard
mathematics as a mere tool to quantify physical entities and
express relations between them. This instrumental view of
the role of mathematics leads to an artificial separation
between the mathematical and the conceptual aspects of
physical theories, which becomes evident when educators
sustain that physics instruction should concentrate on the
latter to the detriment of the former.
Previous research into the role of mathematics in physics
education has focused mainly on the learning perspective
by analyzing students’ reasoning in thinking-out-loud
problem-solving sections and interviews [8–12]. Some
research outcomes highlight difficulties faced by students
in transferring mathematical knowledge to physical contexts [13], assigning meaning to physics equations [14],
using mathematical structures to model physical situations
[15], and other tasks associated with a deep understanding
Published by the American Physical Society under the terms of
the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and
the published article’s title, journal citation, and DOI.
1554-9178=14=10(1)=010119(23)
of the interplay between physics and mathematics. But how
are these difficulties related to instruction? If one observes,
for instance, that students often focus on the instrumental
role of mathematics when they solve problems (e.g., plug
and chug), it is likely that this attitude is related to certain
features of the instruction they had. Therefore, focusing on
the role of mathematics in physics from the teaching
perspective—a facet rather overlooked in current physics
education research—should enhance our understanding of
the origins of students’ difficulties and provide possible
solutions to overcome them. This assumption determines
the guidelines of this work.
Taking the multifaceted character of the role(s) of
mathematics in physics into account, an overall distinction
between the technical dimension—associated with an
instrumental (tool-like) use of mathematics—and the structural dimension—related to the use of mathematics as a
reasoning instrument to think about the physical world
(Sec. II) is proposed. The central aim of this research is to
characterize the development of the structural dimension in
physics teaching approaches. For this purpose, a case study
was conducted on the electromagnetism course given by a
distinguished professor in the undergraduate introductory
level (Sec. III). The analysis of these lectures led to the
identification of a set of categories to describe the professor’s didactic discourse. These categories are described,
justified, and exemplified in Sec. IV and then applied in
the coding of two teaching episodes extracted from the
lectures (Sec. V).
The intended contributions of this work are twofold.
First, I characterize the didactic choices made by the
professor to teach his students how to frame the physical
world mathematically and illustrate the complexity of this
010119-1
Published by the American Physical Society
RICARDO KARAM
PHYS. REV. ST PHYS. EDUC. RES 10, 010119 (2014)
process with several examples extracted from his electromagnetism lectures. Second, an analytic tool to enable
future comparative studies between lectures (or lessons)
regarding the way mathematics is treated in physics
teaching is provided.
stresses that “the challenge is to seriously consider the
design and use of mathematics as an important subject for
Physics Education Research.” Sherin ([8], p. 482) argues
in the same direction by stating that “we do students a
disservice by treating conceptual understanding as separate
from the use of mathematical notations.”
Physics education research literature provides various
means to categorize students’ different ways of using
mathematics to solve physics problems, which seem to
be broadly classifiable according to this general technical
or structural distinction. Among them, Sherin’s symbolic
forms [8] certainly belong to the structural dimension, since
they are connected with a deep understanding of how to
represent conceptual schemata—which are very basic
mathematical relations between physical quantities—in
terms of equations. The epistemic games proposed by
Tuminaro and Redish [9] could possibly be classified into
the ones focusing merely on technical aspects (plug and
chug and transliteration to mathematics) and the ones
demanding a deeper or structural understanding (mapping
meaning to mathematics and mapping mathematics to
meaning). Similarly, the classification of Walsh et al.
[10] of students’ problem-solving approaches into a scientific (structural) and other less structured approaches like
plug and chug and memory based (both technical) seem to
fit in such an overall distinction. The categorization of Bing
and Redish [11] of students’ epistemological framing could
also be separated into technical (calculating and invoking
authority) and structural (physical mapping and mathematical consistency) approaches.
It is not the intention, of course, to imply that all the
subtleties contained in each of these categories can be
incorporated by such a broad distinction, but rather to focus
on differentiating a superficial from a deep understanding
of physics regarding the use of mathematics. After consulting several physics graduate students and faculty
members, Chasteen et al. [23] identified this focus on
such deep understanding (mathematical sophistication [23],
p. 924) as one of the major learning goals of physics
courses and an essential trait of what it means to “think like
a physicist.” Furthermore, the nontriviality of the ability
to use mathematical structures to think about the physical
world is explained by Redish [24] with the argument that
math in physics is “semantically different” from simply
doing math [25]. According to him, physicists make
different use of constants and variables, put great value
on dimensional analysis, and often blend conceptual
physics with mathematical symbols when interpreting
equations. (See also Ref. [31] for a comprehensive list
of differences between math in physics and math in math.).
With the purpose of presenting a synthetic view of this
technical-structural distinction, Table I displays different
epistemological ways to use mathematics in physics, as
well as different views on this interplay, which are
associated with each dimension.
II. TECHNICAL AND STRUCTURAL DIMENSIONS
The intermediate portion of mathematical science,
which consists of calculation and transformation of
symbolic expressions, is most essential to physical
science, but it is in reality pure mathematics. Everything connected with the original question may be
dismissed from the mind during these operations, and
the mathematician to whom they are referred may be
doubtful whether his results are to be applied to solid
geometry, to hydrostatics or to electricity. But as we are
engaged in the study of Natural Philosophy we shall
endeavour to put our calculations into such a form that
every step may be capable of some physical interpretation, and thus we shall exercise powers far more
useful than those of mere calculation—the application
of principles and the interpretation of results (Ref. [16],
p. 672, our emphasis).
This quotation—extracted from Maxwell’s inaugural
lecture given in 1860 at King’s College—implies that,
although mathematical calculations play an essential role in
physics (natural philosophy at that time), physicists should
constantly pursue physical interpretations of each step
when performing them. This message illustrates an overall
distinction between two ways of using mathematics in
physics, namely, a technical and a structural one. Following
Maxwell’s quotation, the technical dimension is associated
with the mathematical calculations without any connection
with physical phenomena, whereas the structural one is
related to the use of mathematics to reason about the
physical world.
When translating this dichotomy (it is more likely a
duality) into physics teaching and learning practices, it is
possible to distinguish between the development of technical and structural skills [17–19], which implies different
kinds of questions to be asked to students, different teaching
approaches, assessments, materials, and so forth. In fact, the
domain of technical skills has already been proven to be
insufficient (although probably necessary) for success in
physics courses [20]. The excessive focus on this instrumental dimension in instruction has had a major impact on
students’ lack of interest in physics [21] and has motivated
physics education researchers to advocate in favor of a focus
on conceptual physics. Nevertheless, the intrinsic mathematical nature of the physical sciences [4] reveals that
understanding in physics is strongly connected with the
ability to think about the world with mathematical structures
(structural dimension). In this sense, Hestenes ([22], p. 104)
010119-2
FRAMING THE STRUCTURAL ROLE OF …
TABLE I.
PHYS. REV. ST PHYS. EDUC. RES 10, 010119 (2014)
Technical-structural distinction concerning the role of mathematics in physics.
Technical (instrumental, procedural)
Structural (relational, organizational)
Blindly use an equation to solve quantitative problems (plug and
chug)
Focus on mechanic or algorithmic manipulations
Use arguments of authority; rote memorization of equations and
rules
Fragmented knowledge: memorize different equations for each
specific case (e.g., free fall and vertical throw)
Derive an equation from physical principles using logical
reasoning
Focus on physical interpretations or consequences
Justify the use of specific mathematical structures to model
physical phenomena
Structured knowledge: connect apparently different physical
assumptions through logic (e.g., Snell’s law and Fermat’s
least-time principle)
Recognize profound analogies and common mathematical
structures behind different physical phenomena (e.g.,
central force field)
Mathematics seen as reasoning instrument
Mathematics seen as essential to define physical concepts
and structure physical thought
Identify superficial similarities between equations (e.g., d ¼ 12 gt2
and K ¼ 12 mv2 both seen as “half a constant times a squared
variable”)
Mathematics seen as calculation tool
Mathematics seen as “just another” language used to represent
and communicate
As physics education researchers, we are all engaged in
finding ways to enable our students to understand this
difficult science deeply, which involves their ability to map
meaning to mathematics and vice versa [9], blend conceptual and formal mathematical reasoning [12], adopt
scientific approaches [10], think like physicists [23], or, in
other words, recognize the structural role of mathematics in
physical thought and apply sophisticated strategies consciously to problem solving. However, the truth is that we
do not have clear guidelines on how to teach students to do
that effectively. One reasonable possibility for finding such
guidelines is to look for specific characteristics in the
didactic discourses of excellent lecturers “in action.” Based
on this hypothesis, a case study was conducted on the
electromagnetism lectures given by a distinguished
professor of physics majors at the introductory level.
Thus, the central research question of this study can be
formulated as follows:
What are the main features of the didactic discourse of
an experienced physics professor who focuses on the
structural role of mathematics in his electromagnetism
lectures?
In the search for an instrument to characterize the
lectures, an analytic tool was developed, which is likely
fruitful for the conduction of future comparative studies,
although this hypothesis is not investigated here. The
methodology used to analyze these lectures and the reasons
for the selection of this particular case study are described
in the next section.
III. METHODOLOGY
The methodological design of the research was
outlined according to the goal of investigating the role
of mathematical reasoning in physics lectures. Given the
innovative character of this approach and the absence of
categories for such analysis in the literature, conducting a
case study [32] seemed the most appropriate alternative.
According to Gerring ([33], p. 40) “case studies tend to be
more useful when the subject being researched has not yet
been explored in a systematic manner or when it is
considered from a new perspective.” Moreover, it is
acknowledged that case studies can penetrate situations
in ways that are not always susceptible to numerical
analysis [34].
It is important to justify the decision of studying the
lectures of this professor in particular. In fact, several
reasons influenced this choice. The professor is quite well
known in the Physics Department of the University of
São Paulo for having a great ability to explain things clearly
and for encouraging his students to reason about the
physical meaning underneath the mathematical formalism.
Furthermore, he has been teaching introductory courses
for more than 30 years. Another motivation is related to
the students’ approval of his lectures. At the end of each
semester, students were asked to rate the quality of the
lectures according to the following criteria: (1) professor’s
interaction with the class, (2) preparation of the lectures,
and (3) quality of explanations, using a 4-point Likert scale
items (very good, 100%; good, 75%; regular, 50%; poor,
25%). It was possible to have access to students’ evaluations of his courses since 2005, which makes a total of 317
students (the average number of students per class being
50). The approval of his lectures is attested by expressive
numbers—99.13% for (1), 99.53% for (2), and 98.82% for
(3)—which makes him one of the best-evaluated professors
in the department.
Thus, the starting research hypothesis was that the
lectures of this particular professor would provide several
examples of the emphasis on the structural dimension. If
this were correct, it would be possible to better characterize
the focus on this dimension in didactic situations. In order
to confirm the validity of this hypothesis, 10 lectures of the
professor on special relativity were recorded. This pilot
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RICARDO KARAM
PHYS. REV. ST PHYS. EDUC. RES 10, 010119 (2014)
study revealed the excellent quality of his explanations and
left no further doubt concerning the appropriateness of a
deep investigation of his lectures.
Moreover, both the analysis of the electromagnetism
course and the interviews provide an a posteriori validation
for this choice. In one the interviews, for instance, the
professor clearly explains why he does not emphasize the
technical dimension in his lectures. In his words, “You
cannot learn the technique at once; practice is really
important. I feel that it is not possible to learn this in
the lecture. […] Thus, what I prefer to do in my lectures is
to explore the multidimensionality of the physical knowledge and exercise a bit each of its dimensions. In reality, my
hope is that they become aware of this broad picture.”
The research data of the main study consist of the
recordings of 40 lectures (total of approximately 60 hours
of video) from an introductory course [35] on electromagnetism given for physics majors in 2009 [36]. The
camera was directed toward the professor and special
attention was given to the moments when mathematical
reasoning took part in the exposition. The textbook used in
the course is a compendium of lecture notes on electromagnetism elaborated by the professor and other colleagues through the years. The 40 chapters of the textbook
are related to the (ideal) program of the 40 lectures to be
given in the course.
The compendium of lecture notes has quite a different
style, compared with traditional introductory level physics
textbooks (e.g., Refs. [37–40]). The main features that
distinguish them are as follows: (1) intense presence of
epistemological discussions (e.g., What is physics? What is
the relationship between physics and mathematics? How
does electromagnetic theory enable us to reason about the
world?), as well as dictionary definitions of essential terms
(e.g., law, principle, explanation), (2) few examples (typical
problems) are presented and discussed in depth, each step
being carefully justified in their resolution, (3) a small
number of problems and conceptual questions are proposed
at the end of each chapter. Students are encouraged to use
this material as support for individual study, but it is not
required for the attendance of the lectures.
The evaluation of the course was carried out by means
of four individual exams throughout the semester. The
final average score M is calculated as follows [41]:
M ¼ 0.3× (sum of the two highest grades among the
first three exams) þ0.4× (grade on the fourth exam).
Among the 81 students enrolled at the beginning of the
period analyzed, 11 left it after the first exam, and 59 of
the remaining 70 were approved. Even though a systematic comparative study was not conducted, 84% of
approval can be considered a successful result considering the history of students’ performance in the first
course on electromagnetism.
A preliminary analysis of the course syllabus pointed out
moments in which the intertwined relationship between
mathematics and physics would be addressed. Central
concepts like charge, density, flux, and electric current
would be introduced and properly mathematized.
Mathematical operations such as derivatives, multiple
integrals, vector operators (gradient, divergence, and curl),
among others, would be physically interpreted in the
context of charges and fields. Each one of Maxwell’s
equations would be address …
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