TEACHING THE CONCEPT OF VELOCITY IN MATHEMATICS CLASSES
Regina Dorothea Möller
University Erfurt, F. R. Germany
regina.moeller@uni-erfurt.de
The genesis of the concept of velocity refers to a centuries-old search within the
context of motion. This led to Newton's definition which is still taught in school.
The historical development shows that both mathematics and physics classes
have their respective characteristic manner using this term. However the
mathematical potential for teaching this concept is by far not exhausted.
INTRODUCTION
Motion is a characteristic of our world. Mankind and animals move naturally within
a range of long or short distances. Plants move too, however since they stay at one
place we do not call the stretching into the ground and into the air a motion. The way
animals move depends on their body structure: some fly, run or swim. Humans can
do all this too, some using technical equipment.
In our time, we use technical achievements which enable us to travel distances in
different velocities: bicycles, cars, ships, trains, airplanes and even rockets. With the
use of technical equipment our perception of velocity and their imprints on human
bodies has changed. In the last century a train with a velocity of 60 km per hour was
considered fast. Today this is a normal speed on a highway.
The concept of velocity is one with a long tradition. Embedded in the concept of
motion already Greek mathematicians, especially Aristotle (384-322 BC), had ideas
about velocity which he combined with his observations of the free fall. Before the
next step was done by Galilei (1564-1642), Nicole of Oresme (c. 1360) used graphic
representation of changing qualities. Later Galilei used experiments to argue for the
statement that there is a quadratic relationship between the distance travelled and the
falling time. Even later Newton (1642-1727) defined velocity using the concept of
force that initiates motion. Leibniz (1646-1716) developed the differential and
integral computation also considering the idea of (planetary) movements.
Speed problems can be found on every level of math classes. On the elementary
level, they are looked upon as physical applications and are solved arithmetically. On
the secondary level they are used to show functional perspectives and later
differentiation is used to compute instant velocity. However, although speed is an
integral part of our modern society, a “red line” of development of the concept of
velocity is not apparent in math (and physics) classes throughout the levels. One
observes an emphasis on the computational aspect under which the ideas that have
led to the possible to computations, have disappeared (e.g. Doorman & van Maanen,
2008).
First, a summary of the development of the concept of velocity is given starting with
some inherent philosophical aspects. This review results in a recommendation for
teaching the concept of velocity using Vollrath’s conceptual framework (1984). The
formulated steps include some historical ideas concerning the development to enable
the teacher students during their studies and teachers to reflect on their
understanding before planning to teach this concept in class.
PHILOSOPHICAL AND HISTORICAL ASPECTS
In the following sections, two aspects are discussed: Firstly, the formal definition of
the concept of velocity was preceded by a struggle for a clarification of the concept
of motion. Secondly, the concept of velocity embodies a circular reasoning.
Although a lot of the work of Archimedes (287-212 BC) concerning mechanics is
transmitted and gives an idea of his far-reaching mathematical understanding, we
have no clear idea what he understood by velocity. Assured work is passed on us of
Aristotle (384-322 BC), who investigated the phenomenon of motion qualitatively
and verbally (Aristotle, Physics, 1829). He distinguished three types of motion:
motion in undisturbed order, such as the celestial spheres, the “earthy” motion such
as the concept of the rise and fall, and the violent motion of bodies that needs an
impulse (cf. Hund, 1996, p.29). Although his remarks touched the phenomenon of
velocity, his conceptions proved wrong later on: "Aristotle came close to the concept
of velocity when in the sixth book, the words ‘faster’ (longer distance in the same
time, same route in shorter time) and the ‘same speed’ are explained (Hund, 1996,
p.30)”.
Galilei (1564-1642) succeeded in a better understanding of the concept of velocity,
as he did not rely on his direct perception: "The means of scientific evidence was
invented by Galilei and used for the first time. It is one of the most significant
achievements, which boasts our intellectual history [...]. Galilei showed that one
cannot always refer to intuitive conclusions based on immediate observation because
they sometimes lead to the wrong track" (Einstein, 1950, p.17).
Only Newton (1643-1726) introduced the concepts of absolute time and absolute
space and opened up the exact relationship between force and motion: The power
does not get the motion upright (Aristotle), but it causes its change (acceleration).
While Aristotle argued by inspection, Newton made an abstraction, as he looked
upon length and time as not necessarily bounded materially.
In today's linguistic usage, we understand motion as a change in position in the
(Euclidean) space over a certain period of time. Lengths and time periods are
conditions for the quantification of such motions. On this basis, the (average)
velocity is defined as the quotient of the distance travelled over the time required.
A circular argument is obvious on closer inspection: time depends on a movement
and vice versa, because time is measured using motion (Mauthner, 1997, vol 3, p.
438.). In the hourglass sand runs through, in an analogous clock the pointer moves,
and for the period of a year, we follow the cycle of the earth around the sun.
Likewise, the idea of space is connected to motion, for only through movement we
perceive the space. Mauthner said: "His [The people, note of the author] language
makes it impossible for him to understand the metaphorical tautology of the
preposition ‘in’. Only rigorous reflection will enable him, to at least understanding
the metaphorical of the preposition (in time). In space means something like ‘in the
space of the room’, for the time as much as ‘in the space of time’ (ibid, p. 443)”.
Even Piaget refers to this circular argument: "Speed is defined as a relationship
between space and time - but time can be measured solely on the basis of a constant
velocity" (Piaget, 1996, p.69). For him the concepts of space, time and speed are
mutually dependent.
MATHEMATICAL ASPECTS OF THE VELOCITY CONCEPT
Towards a functional terminology of velocity Oresme (ca. 1320-1382) sought the
help of experiments to assign a rate of change to certain intensities and concluded:
“All things are measurable with the exception of numbers …. (Pfeiffer, DahanDalmedico, 1994, p. 228ff.)”. The difficulties mathematicians had at that time is well
summarized there.
The reasoning of Aristotle and Galilei were based on their observations of linear
motions. However, both had also planetary motions in mind. For motions on a
curved path you need two different aspects: the direction and the magnitude of a
velocity vector. It is this distinction which led, in modern terms, to a vectorial
description and thus to a further clarification of the concept of velocity, which is thus
a generalization of the concept of velocity on a straight line. Bodies on a straight line
have the same speed, in the same direction and the same quantity. Since then, the
following statement is true: the change in force and velocity are vectors with the
same direction (Einstein, 1950, p. 38). We observe an idea of permanence, because
all statements that apply to velocities along curved paths must also apply to linear
trajectories.
The cause of this observation is given by an idealized thought experiment, which
confirmed the theory (cf. Einstein, S.25ff.); this is yet another idea that came to an
effect only at the time of Galilei. Since then, the mathematical language has been
used in physics to reason for not only qualitative but also quantitative conclusions.
As soon as you engage in quantitative calculations, one deals with quantities
(“Größen”). With respect to the concept of velocity you have the dimension (the
quotient of distance and time) and the measured value (an element of real numbers),
which is a composite physical quantity.
Griesel analysed the subject matter of quantities on the primary level (length, weight,
time periods) (1973, vol. 2, p.55ff.) as a technical background for the didactics of
quantities. This presentation does not fit the quantity of velocity (and is not
mentioned there) because it requires a description as an element of a vector space,
which can be higher than one-dimensional.
An analysis of the mathematics of physical quantities can be found in Whitney
(1968). He systematically described simple and compound quantities with means of
linear algebra. This mathematical framework allows to consider a treatment of the
velocity both in vectorial and elementary, one-dimensional form. However this
mathematical approach has not been made fruitful for school reality, and in general it
can be stated that the "quantities are not taught systematically" (Kienle, 1994, p. 42),
especially not in vocational schools which has been criticized for some time (cf.
Sträßer, 1996).
However, Freudenthal (1973, vol.1, p.188) argues that one can interpret dimensional
designations as a function sign; an idea that was not previously addressed in class.
Another functional aspect occurs in two ways with the concept of velocity: The
distance-time function leads with considering of the difference quotients to the
average speed and the transition to differential quotient to instantaneous velocities
that are themselves again functions, namely, the velocity-time functions.
It has taken over 2000 years for the concept of velocity to be defined consistently out
of the concept of motion - in today's usage an act of mathematical modelling. It is
therefore a prime example of a mathematical and interdisciplinary concept
development - with both mathematical and physical - mechanical - representations
throughout history. The intuitive conclusions drawn by Aristotle led to difficulties
and proved much later untenable. Only an idealized thought experiment led
eventually to a verifiable physical theory. The concept of velocity is an example of a
mathematical modelling of a qualitative knowledge with more potentials as there are
quantitative statements and other insights. The knowledge of such phenomena, the
resulting misconceptions and the trodden paths of knowledge are essential
components of mathematical education and exemplify scientific processes.
The genesis also shows a potential for didactical perspectives of mathematics
education. Despite the scarce representation of this topic in mathematics class (in
many curricula of the German provinces the concept of velocity is only mentioned
only one time on the secondary level; in Bildungsstandards, 2003, 2012) there are a
diversity of ideas which can be reflected.
DIDACTIC CONSEQUENCES
This example shows again, that the mathematical subject matter is not the ‘first
philosophy’ in mathematics education (cf. Ernest, 2013). Although of great
importance for the development of mathematical and physical theory, it plays only
the role of a physical application in mathematics classrooms. Hidden under
algorithmic manipulations, the ideas that lead to the definition of concept of velocity
do not show to pupils in class. Since this topic also disappears during teacher
training, there is an extra afford needed to pull the history into the light.
This afford can be done in three ways: there is a chronology of historic development
which can be shown on a line. There is the genesis which makes apparent who under
which presumptions found a concept or an algorithm. And last but not least the
narratives that are carries along to influence at least the speed in which new ideas are
accepted (Dörfler, 2014).
The genesis of the concept of velocity is influenced by interdisciplinarity for
centuries. Therefore, it is wrong to neglect it in mathematics education by pointing to
the physics education or to avoid it in the classroom at all. In mathematics teaching,
it owns several important functions:
Basic everyday experiences relate to the phenomena of velocity and can be used on
several grade levels by taking up Vollrath's idea and take seriously measurement
processes as a basis of experience (Vollrath, 1980). This can be addressed in
propaedeutic form at the primary level and in a quantitative manner at secondary
levels.
Under the current relations of applications of mathematics education, the concept of
velocity is a prime example of mathematical modelling of everyday phenomena of
motions which can be quantified normatively, allowing measurements and
calculations. According to the current understanding these are necessarily factual and
methodological skills.
As a first composite quantity the concept of velocity can be addressed in lower
secondary education at many occasions. Before getting to the usual algorithmization
(van Maanen, 2009) teaching this concept allows for teaching the fundamental idea
of measuring. This is a valuable and vital contribution to the implementation of the
rules in the educational standards and curricula and guiding principles would be met.
The implementation of the items listed would be an important part in promoting
mathematics education.
To arrive at an overall conception of the concept of velocity in mathematics
education, according to Vollrath a strategy is required (Vollrath, 1984, p.14), which
leads from an intuitive to a structured taxonomy. The following steps outline the
procedure scheme which could have in mind before teaching the respective classes
(Vollrath, 1984, S.202ff). It would mean to get a clear view on the subject matter and
teach perhaps more than what is given by the textbooks:
1. The concept as a phenomenon (intuitive understanding of the concept)
They recognize that
• the feature of the motion is the continuous change of position of the objects.
• for describing motion two quantities, length and time intervals, are required.
• with a constant distance the one object is faster that needs less time.
Conversely, at constant time, the object is faster that has travelled a longer
distance.
2. The second term and his wealth aspect (constructive understanding of the
term)
They recognize that
• rest and motion of a body are always give relative to a reference system.
• with a rectilinear and uniform motion in equal intervals of time, equal
distances are covered.
• for a linear and non-uniform motion the same time intervals can be covered in
different journeys.
• for a linear and uniform motion, velocity can be described as the ratio v = s / t.
3. The concept is a carrier of properties (substantive understanding of the term)
They recognize that
• for mathematical calculations of the body and the mass, they are idealized to a
point.
• the ratio s / t possess a constant (average) value relative to the total distance
and total time.
• at constant velocity s to t proportional.
• the quantity velocity possesses the dimension "distance per time" with the
units m / s or km / h.
4. The concept as a tool for problem solving (problem-oriented understanding of
the term)
They can
• bring the definition v = s / t into relation to everyday life contexts.
• calculate uniform velocity.
• appreciate that the concept of velocity is a fundamental physical phenomenon
that can be described quantitatively by mathematical means.
The discussion of the velocity concept in the context of the stage scheme shows the
potential that this concept already has in mathematics education. From this more
elaborated standpoint the potential for the interdisciplinary character can be
developed further and more in detail.
The four steps in the concept development to enlighten teacher students in their
knowledge of this topic also gives a good background to approach the topic in an
ethical manner (Ernest, 2013). It is essential for their teaching to have a grip on solid
knowledge and cultural heritage in the subject matter they are to teach.
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