Teaching the concept of velocity in mathematics classes
Regina Dorothea Möller
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Regina Dorothea Möller. Teaching the concept of velocity in mathematics classes. Konrad
Krainer; Naďa Vondrová. CERME 9 - Ninth Congress of the European Society for Research
in Mathematics Education, Feb 2015, Prague, Czech Republic. pp.1853-1858, Proceedings of
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Teaching the concept of velocity
in mathematics classes
Regina Dorothea Möller
University of Erfurt, Erfurt, Germany, [email protected]
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 has shown 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.
Keywords: Concept of velocity, historical development,
teacher education.
tios such as a half, one quarter and three quarters and
the second refers to the decimals within the context
of magnitudes. The third is the appearance of tables
in application problems. In these problems prices of
products are given – like 1 kg of apples cost 75c – and
the question is how much is to pay for 2 kg / 3kg / 5kg.
These three kinds of anticipations occur in math classes because of the application principle. Students see
these kinds of numbers and these kinds of questions
in their daily life and math classes respond to this
phenomenon by introducing these numbers and these
tables without giving a rigid mathematical reasoning.
INTRODUCTION
The causes to investigate the historical development
of the concept of velocity are mathematical problems
which lately can be found in German fourth grade
classes.
Several pictures are given which show people or machines such as cars or trains each with information
about the respective velocity: the train covers 150 km
in an hour. The assigned task is to fill out tables in
which the students write the distances for different
time spans: 1h, 2h, and so forth. Above the assigned
tasks one can read the title velocities. One can observe
at once that like in other cases in which an introduction into the field is in the focus of interest – here a
start with a series of different observable motions and
a leading to the question how they can be quantified –
there is an emphasis on the computational aspect
under which the ideas that have led to the possible
to computations, have disappeared (Doorman & van
Maanen, 2009). Instead of an approach pinpointing
the quantities in question there is a given table to be
filled out by the students.
This kind of problem can be considered as an anticipation. There are at least three other kinds of anticipation tasks on the elementary level: one refers to the ra-
CERME9 (2015) – TWG12
What kind of anticipation is done when students solve
this kind of velocity problem? Since the students are
to fill out tables, which are methodically a representation of functions one could argue that functions
have arrived in elementary math classes since the
inspection of Felix Klein (1905) to make functions a
subject matter in math classes. It also could be understood as an example of an (anti-) didactical inversion
(Freudenthal, 1983, pp. 305ff.). In any case it is obvious
that this is not following the historical development
and it is not showing an elementary approach which
is possible at this stage of mathematics classes.
Since this kind velocity problem is not really an application task, since only tables need to be filled out,
one could argue that they give a mathematical way to
compute velocities which the students can observe in
their daily lives because our modern world presents
this phenomenon. Since the formula is given later at
secondary I level one chooses tables to compute.
Having in mind the way magnitudes are introduced
in math classes on the elementary level (finding representatives, studies of comparisons with chosen
measurement objects and afterwards with agreed
upon measurements objects, leaning the standardized
measurement unity and at last solving of application
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Teaching the concept of velocity in mathematics classes (Regina Dorothea Möller)
problems), the velocity problems in focus do not show
any such procedure although velocity is the first composed magnitude that the pupils encounter.
Since this is a mathematical concept one would expect
a series of steps that lead to a definition. A possible
approach would be the didactical triangle of Bruner
(1960) in which he argues for an approach that encompasses the enactive, iconic and symbolic level.
Another didactical theory of leaning concepts is given by Vollrath (1984) who outlines in general what
kind of different steps lead to an understanding of
mathematical concepts.
tance 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 calculus also considering
the idea of (planetary) movements.
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
Another concern is the fact that the concept of velocity idea of his far-reaching mathematical understanding,
can be looked upon as a real mathematical modelling we have no clear idea what he understood by velocity.
procedure (e.g., Blum & Leiß, 2005). Observable move- Assured work is passed on us of Aristotle (384–322
ments can be measured in two dimensions: length BC), who investigated the phenomenon of motion
and time span. It could be arranged as a project for qualitatively and verbally (Aristotle, Physics, 1829).
students in which the definition of velocity is the end He distinguished three types of motion: motion in
product of their investigative endeavour.
undisturbed order, such as the celestial spheres, the
“earthy” motion such as the concept of the rise and
This point also leads to the question why a fundamen- fall, and the violent motion of bodies that needs an
tal phenomenon like the concept of velocity lacks any impulse (cf. Hund, 1996, p. 29). Although his remarks
historical approach in textbooks. The students could touched the phenomenon of velocity, his conceptions
measure for example the free fall of objects getting an proved wrong later on: “Aristotle came close to the
idea of how scientists in the middle ages approached concept of velocity when in the sixth book, the words
problems of velocity.
‘faster’ (longer distance in the same time, same route
in shorter time) and the ‘same speed’ are explained.”
The paper focuses on the historical development (Hund, 1996, p. 30)
of the concept of velocity with the idea that Newton
might not have thought primarily of it as a function Galilei (1564–1642) succeeded in a better understandsince he was still following Galilei’s proportional ing of the concept of velocity, as he did not rely on his
theory. The development starts with some inherent direct perception: “The means of scientific evidence
philosophical aspects since in times of Newton the was invented by Galilei and used for the first time. It
subject matter still belonged to the so-called natural is one of the most significant achievements, which
philosophy.
boasts our intellectual history [...]. Galilei showed
that one cannot always refer to intuitive conclusions
PHILOSOPHICAL AND HISTORICAL ASPECTS
based on immediate observation because they sometimes lead to the wrong track” (Einstein, 1950, p. 17).
The concept of velocity is one with a long tradition,
similarly like the history of calculus (e.g., Doorman As Weisheipl (1985) points out, Galilei struggled with
& van Maanen, 2009). Embedded in the concept of the Aristotelian concept of nature (pp. 8ff.). Aristotle
motion already Greek mathematicians, especially considered nature as an active principle. “Nature is
Aristotle (384–322 BC), had ideas about velocity which a source not only of activity but also of rest” (p. 22)
he combined with his observations of the spheres and which has an impact on the understanding of moof the free fall of objects. Before the next step was done tion (p. 49ff.) He also still pondered over the idea of
by Galilei (1564–1642), Nicole of Oresme (1360) used Parmenides “all change is illusion” and the one of
graphic representation of changing qualities. Later Heraclitus “everything is flux”. Galilei can be underGalilei used experiments to argue for the statement stood as being at the brink of Aristotelian sight of nathat there is a quadratic dependency between the dis- ture and the one later proposed in Newton’s Principia
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Teaching the concept of velocity in mathematics classes (Regina Dorothea Möller)
and Descartes’ Principia. Newton formulated the
principle of inertia like this: “Everybody preserves in
its state of rest or of uniform motion in a straight line,
unless it is compelled to change that state by forces
impressed upon it (Weisheipl, 1985, p. 69).
Another author, Palmerino (2004), argues for a new
sight of the reception of Galilei’s theory since like any
other theory it was not at once in the way accepted
in which textbooks nowadays present for example
the free fall and projectile motion (Palmerino, 2004,
p. 140). During the last decades it became apparent
that the ideas that Galilei, Descartes and Newton
presented at their times were different from each
other. Also the European continent was not aware
of Galilei’s and Newton’s theories and later Leibniz
focused very much on his functional approach using
variables. That is, the way the theory of motion and
velocity is presented nowadays is much more formed
by Leibniz than by Galilei and Newton although it was
very much their ideas that came through eventually.
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, Dahan-Dalmedico, 1994, p.
228ff.)”. The difficulties mathematicians had at that
time are 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, pp. 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
1855
Teaching the concept of velocity in mathematics classes (Regina Dorothea Möller)
(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, pp. 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.
sions 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), there are a diversity of ideas
which can be reflected.
DIDACTIC CONSEQUENCES
Figure 2: Extended didactical triangle
Figure 3: Aspects of history
This example shows again, that the mathematical
subject matter is not always the ‘first philosophy’ in
mathematics education (e.g., Ernest, p. 213). 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
appear 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.
However, Freudenthal (1977, vol. 1, p. 188) argues that
one can interpret measure indications as function
symbols; 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.
In the light of the use of language in math classes,
Vygotsky (1964) elaborated on the relationship between everyday experiences and logical reasoning.
Everyday experiences may interfere with scientific
reasoning which in the case of the concept of velocity
occurred historically for quite some time because
of the lack to understand the invisible forces. It is
therefore necessary for teacher students to learn to
discriminate between the appearance or motions and
its scientific reasoning.
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 conclu-
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.
1856
Teaching the concept of velocity in mathematics classes (Regina Dorothea Möller)
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 (Doorman & 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.
They recognize that
――
the feature of the motion is the continuous change
of position of subjects and 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
To arrive at an overall conception of the concept of velocity in mathematics education, a strategy is needed
which leads from an intuitive to a structured taxonomy (Vollrath, 1984, p. 14). Vollrath has given a general
theory to teach mathematical concepts. The following
steps outline the procedure scheme for the specific
concept of velocity which teachers could have in mind
before teaching the respective classes (Vollrath, 1984,
pp. 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)
――
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)
1857
Teaching the concept of velocity in mathematics classes (Regina Dorothea Möller)
They can
Lange, L. (1886). Die geschichtliche Entwicklung des
Bewegungsbegriffs und ihr voraussichtliches Endergebnis,
――
bring the definition v = s / t into relation to everyday life contexts.
Schluss. In W. Wundt Town (Ed.), Philosophische Studien, 3,
643-691.
Mauthner, F. (1997, Neuauflage). Wörterbuch der Philosophie,
――
calculate uniform velocity.
Band 3. Vienna, Austria: Böhlau Verlag.
Newton, I. (2014). Mathematische Grundlagen der
――
appreciate that the concept of velocity is a fundamental physical phenomenon that can be described quantitatively by mathematical means.
Naturphilosophie, Ed. Ed Dellian. Sankt Augustin: Academic.
Palmerino, C. R., & Thijssen, J. M. M. H. (Eds.). (2004). The
Reception of the Galilean Science of Motion in
Seventeenth-Century Europe (Boston Studies in the
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.
Philosophy of Science, 239), Dordrecht, The Netherlands:
Kluwer Academic Publishers.
Pfeiffer, J., & Dahan-Dalmedico, A. (1994). Wege und Irrwege –
Eine Geschichte der Mathematik. Basel, Switzerland:
Birkhäuser.
Piaget, J., (1996). Einführung in die genetische
Erkenntnistheorie, 6. Frankfurt am Main, Germany:
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. It is essential for their teaching to
have a grip on solid knowledge and cultural heritage
in the subject matter they are to teach.
Suhrkamp.
Vollrath, H.-J. (1968). Die Geschichtlichkeit der Mathematik als
didaktisches Problem, Neue Sammlung, pp. 108–112.
Vollrath, H.-J. (1978). Rettet die Ideen! Der MathematischNaturwissenschaftliche Unterricht, 31, 449–455.
Vollrath, H.-J. (1980). Sachrechnen. Frankfurt am Main,
Germany: Klett.
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