The purpose

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

Perspective (HEP)

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.

Keywords: Calculus,

Interplay between Physics and Mathematics,

1

INTRODUCTION

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

perspective.

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

research.

2

INTERPLAY BETWEEN MATHEMATICS AND PHYSICS

2.1

Theoretical

Context

Maths is

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” (4).

Actually,

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).

From the

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.

Maybe the

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).

This

“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.

2.2

Educational Context

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).

3

Possible

matches: reflections and directions

3.1

A pathway from pre-calculus to

calculus

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

is possible.

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

intuition.

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.

3.2

Bridging

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.

At primary

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.

From the

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.

Thus,

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.

Then, at

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).

3.3

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

Perspective (HEP)

·

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

computation).

4

FINAL

REMARKS

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.

Furthermore

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.

The

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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