Mathematics in physics: analysis of students' difficulties - ESERA

Mathematics in Physics: Analysis of students' difficulties
Olaf Uhden¹ and Gesche Pospiech¹
¹Chair of Didactics of Physics, Technische Universität Dresden
Abstract: The deep interrelation between physics and mathematics is an important fact to
understand the success of physical science. Therefore it also influences classroom teaching
which leads to additional problems due to mathematical difficulties and knowledge transfer.
The teaching and learning of physical concepts seems to be hindered or even neglected. But
could not mathematics be used supportively for a physical understanding? In order to
approach this question it is important to carefully analyse the difficulties students experience
while dealing with mathematics in a physical context. With this aim we observed 15 to 16-
year old students from different schools of higher education. They worked at an interactive
whiteboard on special physical-mathematical tasks which were designed in order to address
the translation process between math and physics. Findings suggest that students focus on a
technical use of mathematics. Especially understanding the connection between formulas and
physical meaning seems to be problematic for most students. Therefore the conceptual
meaning of mathematical descriptions in physics should be stressed in classroom teaching.
Keywords: Mathematics in Physics; Mathematization; Technical and structural skills;
Mathematical Modeling; Knowledge Transfer
INTRODUCTION
“Our previous experience justifies the belief that nature is the realization of the
most basic mathematical thoughts. […] Experience remains certainly the sole
criterion of usefulness of a mathematical structure for physics. But the truly
creative principle is in mathematics.” (Einstein, 1956, p. 116, our translation)
The quotation by Einstein sheds light on the deep interrelation between mathematics and
physics. It is not the technical toolbox mathematics provides but rather the creative aspect
producing new insights into physical structures that hints at a deep conceptual relation
between both sciences. Under the perspective of discussions about the importance of “Nature
of Science” for science teaching (Lederman, 1992), the conceptual entanglement between
mathematics and physics should also have consequences for the teaching and learning of
physics. The way theoretical physicists use mathematics, i.e. for modeling and investigating
physical behavior, is also an important aspect of the nature of physics.
The importance of taking care of mathematical problems in physics education is also
supported by research findings related to knowledge transfer. Generally it can be stated that
transfer does not succeed to the desired extent, since knowledge and understanding is
acquired context-specific (Brown, 1989). That implies that also the transfer of mathematical
knowledge into physics domain can not be expected to happen automatically. On the contrary,
it can be suspected that transfer form mathematics to physics is an especially difficult case as
there are large differences in the use of and approach to mathematics by physicists. The use of
mathematical language in physics differs in many ways from its use in mathematics (Redish,
2005).
On the other hand, the problems and difficulties arising with the use of mathematics within
physics education must not be ignored. The objection that the application of formulas and
calculations would hinder an understanding of physical concepts, lead to routine and
“senseless” computing activities and brings additional mathematical difficulties with it, has to

e taken seriously. However, one can assume that these problems stem in large part on the
way in which mathematics is involved in physics teaching. Already Richard Skemp, a pioneer
in mathematics education research, pointed out the difference between instrumental and
relational understanding (Skemp, 1976). By focusing on instrumental skills like rote
manipulations and learning rules, it is not possible to achieve a deep (relational)
understanding of mathematical concepts. But especially for knowledge transfer a relational
understanding is indispensable. Therefore, it is particularly important in physics classes to
stress this kind of understanding of the relationship between physical behavior and
mathematics.
THEORETICAL FRAMEWORK
A deep analysis of the role of mathematics in physics leads to the distinction into a technical
and a structural role. Whereas the technical role describes the tool-like use of mathematics
(e.g. calculating numbers), the structural role represents the close connection between physics
and mathematics and their structural entanglement (e.g. the possibility to predict new physics
from mathematical equations). In classroom situations the technical skills seem to be
dominant, although the structural skills – e.g. mathematizing and interpreting – allow a
meaningful use of mathematics. Moreover, a focus on the structural role of mathematics
would be more in line with the nature of physics than the current instrumental use of
mathematics (for a deep investigation of this issue and the derivation of a corresponding
didactical model, see Uhden, Karam, Pietrocola & Pospiech, 2011).
A shift in the teaching practice towards the structural skills, and thereby towards the
translation between physics and mathematics, requires the consideration of relevant didactic
concepts. With the translation to and from mathematics the so called “Grundvorstellungen”
(vom Hofe, 1992) – a theoretical concept used in mathematics education research to describe
the translation between ideas or meaning and mathematical operations – emerge as they
mediate between the mathematical and physical world. The term “Grundvorstellungen” might
be best translated as “basic ideas” or “mental models”. It refers to mathematical objects (i.e.
mathematical operations, symbols etc.) and the ideas and conceptions connected to them. For
example, a basic idea of “addition” can be understood as a sum of contributions, i.e. the
operation of adding two amounts to get the whole amount as a result. On the other hand, one
can think of “addition” as changing an initial state to a final state – as it is the case for
describing changes in temperature. Obviously there is not only one correct basic idea of a
mathematical operation. Rather it is important to activate the basic idea corresponding to the
specific problem one deals with. Therefore one needs to have more than one basic idea
available for activation. Enabling a flexible use requires experience and knowledge based on
comprehension or, in other words, a relational understanding.
In the physical context these “Grundvorstellungen” have to be connected to an understanding
of the physical concepts and be activated in accordance to the physical situation. Sherin
(2001) investigated the meanings college students assign to formulas and classified them as so
called “symbolic forms”. A symbolic form is an entity in a formula which is associated with a
corresponding physical behavior, e.g. a fraction stroke with a variable in the denominator
means that the resulting behavior is inversely proportional to this variable. But as it is the case
with the “Grundvorstellungen”, there also exist more symbolic forms for one formal structure.
Sherin claims for ongoing research in the same direction with students at a younger age (i.e.
high school and secondary school) and for teaching strategies which strengthen the use of
symbolic forms. Also an in-depth analysis of its interplay with students' strategies and
difficulties is still owing.

RESEARCH DESIGN
The delineated argumentation leads to the main research question which needs to be answered
by a qualitative study: What kind of difficulties do students have with mathematical reasoning
in physics?
For animating the students to use structural skills and for investigating the resulting
difficulties students experience thereby, we developed special mathematical physics tasks in
order to address different aspects of translating between physics and mathematics. By
working with these tasks the students are challenged to actively carry out the translation and
to establish the connection between mathematics and physics not on a calculational but on a
meaningful basis. More specifically the tasks contain amongst other aspects
the creation of a formula on the basis of physical reasoning
the interpretation of the special cases of a formula
to draw conclusions based on a formula for physical behavior
explaining the meaning of a formula
The theoretical assumptions about the nature of the connection between physics and
mathematics, as exposed in the model by Uhden et al. (2011), guided the construction of the
tasks. The degree of difficulty was validated by the observations made in the pilot study
which also gave hints for improvements and additional ideas. The physical topic is mechanics
due to the fact that this is the topic with the highest degree of mathematics involved in
secondary school.
In total 30 students from different schools of higher education in Germany (“Gymnasium”)
were invited to work on the tasks. The students are from class level 9 and 10 – i.e. 15 to 16-
years old – and their school grades indicate a selection of mainly good students, but with
more satisfactory grades than very good ones.
As the students worked in pairs at an interactive whiteboard we were able to conduct the
observation without a video camera because the whiteboard provides the possibility to record
the speech and writings simultaneously. The students were requested to discuss the problems
with each other and get a joint solution in order to animate them to speak aloud and express
their ideas and thoughts. This makes it possible to follow their lines of reasoning and thought
processes to some degree. The recorded speech was transcribed and placed in relation to the
writings. The transcripts were then evaluated according to the framework of qualitative
content analysis (Mayring, 2008).
RESULTS
The observed difficulties were categorized along 4 main categories (which were each divided
into subcategories):
Mathematical understanding
Superficial translation
Connection between formula and meaning
Interference with intuitive notions
The category “Mathematical understanding” refers to pure mathematical difficulties where no
explicit connection to physics can be observed. That can pertain to problems with
mathematical reasoning or to the already mentioned basic ideas. Nevertheless, these kind of
difficulties have influences on the physical argumentation: If an increase in the denominator

leads to an increase of the whole fraction, the interpreted physical behavior will also be
wrong.
The category “Superficial translation” is related to a strategic aspect. The established
subcategories indicate that the connection between physics and mathematics is guided by
superficial criteria. For example, for many students the association of formulas and units
seems to be based on rote learning and the formula v=s/t is used once velocity is mentioned,
no matter what physical circumstances are present.
The “Connection between formula and meaning” is an essential category that is in close
correspondence to the mentioned symbolic forms (Sherin, 2001). The classified difficulties
demonstrate a severe lack of understanding of equations and its related physical meaning. For
example, the role of physical magnitudes as being a function, parameter, variable or constant
seems to be confused or disregarded by some students. That led to discussing the influence of
a function on a parameter with the result of confusion.
The category “Interference with intuitive notions” includes that students reject an actually
successful connection between physical and mathematical argumentation due to implications
that are in contradiction to their intuitive notions. If, for instance, the physical description is
not understood as an idealization, the (idealized) mathematical results are considered as
unrealistic, leading to a shortened interpretation of the results.
IMPLICATIONS
The outlined difficulties are a plea for a change from the technical to the structural
mathematical skills in physics teaching and learning. Especially the observed superficial
translation seems to be cultivated by a focus on a technical use of mathematics. And also the
mathematical understanding can be assisted by teaching structural skills without delegating
this task to mathematics education.
Furthermore, the categorized difficulties with the translation between formula and physical
meaning offer the possibility to be addressed explicitly in the classroom in consistence with
the symbolic forms. That could help building the foundations for a meaningful use of
mathematics in physics teaching. Tasks that address the connection between physics and
mathematics, as well as a general qualitatively oriented use of mathematical structures, are
further possibilities to stress and support the understanding of structural skills.
Acknowledgements
Thanks to the European Social Fund (ESF) and to the Free State of Saxony for the financial
funding of this PhD-project.
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