of this paper is to investigate the interplay between physics and mathematics
in educational context. Although mathematics has always been closely related to
physics, they are different in objects, aims, methods and tools. Nevertheless,
mathematics provides physics with the language it needs. However, this language
has to be presented through its historical, cultural and epistemological
development to be properly understood. Our attempt focuses on the rising tide
of the mathematizing of physical problems with the aim to comprehend better the
troubles it generates for students and to propose some solutions to be
introduced in the theory of didactics. More precisely, the question of the
rearrangement of physics from pre-calculus to calculus will be approached
through historical and epistemological analyses. Under this approach, learning is conceived as a permanent activity
of building and re-structuring, in an elliptical way, the key mathematical
concepts and functions needed to analyze physical models. Our research will
involve the following three dimensions:
1) Historical and Epistemological
2) Cognitive and Learning Perspective (CLP)
3) Didactic Perspective (DP)
The main goal of this type of analysis is
to discuss the role and the study progression of mathematical concepts involved
in the learning of physics, and to explore a possible didactic itinerary for a
better integrated course of mathematics and physics. Specifically, our purpose
is to study the transition from primary to secondary school and from this to
the university in the mathematics field, with the ambition to re-design a
curriculum in which the steps from basic to advanced mathematics would be
coherent and smooth.
Interplay between Physics and Mathematics,
The chief question for teachers of
Mathematics and Physics is to determine what the nature of the relation between
these two disciplines is. This question has received different answers depending
on the frameworks of knowledge (from philosophico-epistemological to scientific
perspectives) in which it was asked (1). Moreover in each framework the
answers have underwent changes several times throughout history. As time
progressed, following the lesson of history, the mathematics seems to play increasingly
a bigger role in the development of modern science. In the current situation Math
is shown as vital organ in the body of Physics, without which it can not even
be born. (2).
Our attention goes beyond the pure
conceptual relationship between the two disciplines, aiming rather at
pinpointing how their interdependency can meaningfully bring effective changes
into the methods, objects and curriculum of the Physics’ teaching and learning.
Therefore, the picture of a linear and organic process of learning should be revised
by taking into account the connection between Mathematics and Physics in a merged
This article is structured as follows. In section 2, we give an
overview of the theoretical and educational context about the interplay between
Mathematics and Physics. Section 3 offers didactical matches of theoretical
ideas exposed in section 2 focusing particularly on the pathway from
pre-calculus to calculus and on the three dimensions to be taken into
consideration. Section 4 is devoted to outline final remarks for future
INTERPLAY BETWEEN MATHEMATICS AND PHYSICS
unavoidably everywhere. Whatever you do, wherever you go, mathematics is there.
How mathematics really works and controls our world is fascinating. It seems
that natural laws have chosen to be expressed in the language of mathematics.
Galileo Galilei (1623) in his book “The Assayer” (3) states that “The
book of nature is written by the hand of God in the language of
mathematics.” The same statement is made by Paul Dirac (1939) three
centuries later, “God chose to make the world according to very beautiful
mathematics imposes its rules to the structure of the universe. Physics is one
of the branches of natural sciences where mathematics has been directly and
significantly applied as Richard Feynman (1965) stated: “The strange thing
about physics is that for the fundamental laws we still need mathematics. The
deeper we penetrate nature, the more the math “disease” persists. It gets more
abstruse and more and more difficult as we go on. Why? I have not the slightest
idea” (5). So there is always an air of mystery about the relationship
between mathematics and physics, a mystery Eugene Wigner (1959) alluded to
saying that “The miracle of the appropriateness of the language of mathematics
for the formulation of the laws of physics is a wonderful gift which we neither
understand nor deserve” (6).
beginning Physics as Natural Philosophy has been deeply connected with
Mathematics and this mutual influence has played an essential function in both
their developments. Nevertheless the onset of mathematization in physics is commonly
dated back to the times of Galileo, Newton, Descartes and Huygens. However,
from the time we can call “The birth of modern science” until today there still
was a long way until the mathematical structures used in physics were fully
developed and reached their modern undoubted relevance and it is always an
admirable mystery how the math is so appropriate to the objects of reality. It
seems that nowadays mathematics continues to impose the rules to the universe.
problematic question is not why Maths works but why we do need the language of
mathematics to explain physics and why we do not use another language. This
question is hard to answer but maybe it is because when we evolve in a new
range of parameters, when for instance we deal with the world of very small
things (quantum mechanics) or with the world of heavy things (black holes) it
is simply inescapable to admit that the understanding of physical world
precisely requires a grip on math that cannot be achieved without considerable
mental effort. In some ways the impact of this discovery on mathematics has
been quite remarkable. New
areas of mathematics have been brought in and have been shown to be influenced
by ideas from physics. Sir Michael Francis Atiyah stated in an interview
(2004): “There is a kind of quantum mathematics evolving and physics shows what
you need to do and how to develop it; in this way we could see it as a natural
evolution from discrete to continuous and in that sense it may seem perfectly
logical and will gradually make a better and better picture” (7).
“unbelievable effectiveness” of mathematics in physics should
influence the teaching of physics and its methodology. It seems that the first
and most important task of a teacher is to make students conscious of this
mystery from the very beginning of their curriculum.
In physics education, mathematics is usually
seen as a mere tool to describe and calculate, whereas in mathematics
education, physics is typically viewed as a possible field for the application
of mathematical concepts. Furthermore physics and mathematics are taught
separately both in secondary school and at the university.
This compartmentalization is likely to
generate many theoretical misunderstandings in the process of learning certain
important concepts. For instance, students showing adequate computational
skills, still lack the ability to apply these skills in a meaningful way.
Moreover, it often happens that some mathematical concepts have to be applied
in physics courses even before they have been taught or mastered by learners.
In a few words, students display a largely nominal rather than substantial
comprehension of physical and mathematical ideas. It means that students focus
their attention more on basic comprehension and memorizations of concepts and
formulas, than on critical thinking.
What might possibly be done to avoid this
lack of harmony in the pedagogical curriculum? Although many studies (8) have
been devoted to the problem of teaching together mathematical and physical
contents, the problem of how to use math in teaching physics actually remains
unsolved. One of the key concerns of physics teachers, at whichever level they
teach, is their students’ familiarity with mathematics. It is rather common
that students do not know enough maths to be able to succeed in physics. The
absence of some basic mathematical abilities, as it has already been
suf?ciently demonstrated, is a significant reason for students’ failure in
physics courses. However, it is equally accepted that the mastery of these
skills does not guarantee success in physics (9; 10; 11; 12).
matches: reflections and directions
A pathway from pre-calculus to
As it is well known (13), calculus is a
difficult subject for teachers and learners. The study of (the?) mathematical
concept of limit, which is at the basis of calculus, often appears problematic
for a secondary school student if taught through using an abstract tool such as
the notion of inequalities. For that reason, it would be useful for the student
to combine this abstract approach with a more intuitive one, because human mind
learns mostly intuitively. Generally speaking, it seems especially fruitful to
use an intuitive approach in the treatment of any subject where this approach
Newton and Leibnitz made the mathematical
formalism progress by introducing calculus to present nice significant results
in mechanics. After a century, Heisenberg and Schrödinger used that calculus
separately to express their ideas. The discovery of quantum physics was due to
a new mathematical formalism called “new calculus” by Heisenberg himself (14).
So physics and mathematics have complementary roles in scientific discovery and
it is not always possible to discriminate them within this process. It is
difficult, in modern physics, to ever have empirical, purely physical
principles, completely free of specifically mathematical conceptual expression.
Actually, the physics principles have to be
operative by themselves in the field of gravity and electromagnetics even when
working with finite intervals. Computer programs usually use finite-interval
arithmetic to resolve problems for which a symbolic solution cannot be found by
using calculus tools. This way, the pre-calculus description of physics has
gained some contemporary validation. Thus, it would be fundamental to develop
the calculus introducing the variation and change problems and reinforce the
Nowadays, numerical methods are introduced
by digital technologies in accordance with the use of calculus. It is,
therefore, highly significant to present the subject to students in the
cultural context where it originates because it provides strong motivation and
gives greater coherence to the approached material.
a didactic itinerary
The task of
displaying the role of the language of mathematics is vital for the development
of mathematical proficiency and physics understanding. The insights of the
importance of mathematics in physics education strongly depend on the age of
the students. More precisely, the transition from
primary to secondary school and from this to the university in the mathematics
field should be investigated with the aim to re-design a curriculum in which
the steps from basic to advanced mathematics would be coherent and smooth.
school, the concepts can be explored in a non-technical, even non-mathematical
approach, where, practically, only arithmetic is used. It is only in this first
discovery of the wonders of physics by children that the application of
mathematical formulae is not required.
end of secondary school to the first two years of upper schools, the introduction
of algebra is traditionally broken up into two subparts: literal calculation (i.e.
expressions including letters and numbers) and equations. Pupils are used to solving
similar exercises using algorithms mechanically. This formal training leads
students to acquire the techniques of solving problems without any use of scientific
reasoning. However the role of algebra in the school could be a very powerful
tool for connecting different scientific domains.
teaching to solve the equation doesn’t mean to expose
students to a long period of solving more and more complex equations. Students
lose the motivation because they do not understand why they have to study
mathematics. It is important to give them a lot of examples related to the
field of geometry, physics and scientific domains in general that allow them to
build meanings and to understanding the availability of mathematical concepts.
the second stage, at a higher level, math upgrades and students have to master
basic knowledge of geometry, algebra and trigonometry. Finally, at the
conclusion of the secondary school, and especially in advanced courses,
calculus is required; calculus referring here almost exclusively to the differentiation
and integration procedures in one variable used in the study of rate of change. At this point the students are
able to revisit physics concepts (e.g. energy and work done by a constant or
variable force) using the language of calculus in order to expand the depth of
understanding about the mysterious and engaging connexion between mathematical
and physical domains of the reality (15).
Three Dimensional Analysis involved
The main goal of our analysis is to discuss the role and the study (the?
of the?) progression of mathematical concepts involved in the learning of
physics, and to explore a possible didactic itinerary for a better integrated
course of mathematics and physics. Under this approach, learning is conceived
as a permanent activity of building and re-structuring, in an helicoid way, the
key mathematical concepts and functions needed to analyze physical models.
The following three dimensions should be
taken into consideration:
1) Historical and Epistemological
To give historical and
epistemological case studies that show how physics problems motivate the
creation of mathematical concepts but also how “pure” mathematics are used to
derive conclusions about the “real” world.
2) Cognitive and Learning Perspective (CLP)
To help students to avoid any
possible misconceptions by starting from geometric intuition.
To investigate the spontaneous
image of concepts a student has at different ages and to follow the evolution
of these images.
To guide learners to
formalization and systematization of the results.
To examine the semantics of
words associated with physical variables.
To refine and reinforce the
physical arguments with the progression of math tools.
3) Didactic Perspective (DP)
To prepare classroom or
homework sets to facilitate and increase the skills of students in exploring,
analyzing and transferring knowledge in real context.
To use computers to help
building and visualizing concepts (numerical analysis, sequence, approximated
Success in calculus depends to a large extent on knowledge
of the mathematics before calculus: algebra, analytical geometry, functions,
and trigonometry (16). We
intend not only to identify weaknesses that students might display in these
areas but to make final-year high school students able to justify the use of
the differential in physics, to assign meaning to the differential expressions
and to perceive how they are supposed to use calculus in physics.
In this way we expect to identify students’
conceptions and the persistence of these conceptions and to focus the attention
on difficulties students can have in mathematizing physical situation by
objectively gathered data.
we expect to evaluate the minimal mathematical toolkit necessary to understand
physics and likely to remove any possible barrier between what teachers teach
and what students learn. The
exact amount of mathematics essential to understand (ing?) some determinate
aspect of physics depends upon the degree of analytical depth with which the
student is approaching the subject.
previous activities would model a suitable and creative environment in which
mathematical tools become more internalized throughout the whole path towards
the university. The realization of the project would create a kind of bridge
enabling the students to progress from one stage to the next all the way from
the elementary to the advanced treatment of scientific topics.
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