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Файл:Integration of Natural Science and Maths in Scientific Thought and Education. The materials of Russian - German Seminar in Moscow - Cologne, 2014
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The other example is the radian system of angular measurement, first being
used in the 9th grade physics while studying angular velocity and then properly
discussed in the 10th grade maths.
What’s the best way to integrate maths and physics studies?
The next important question is how to organize the students’ activities in an
integrated approach. Should it be the separate integrated lessons or the whole
integrated course? At the same time, students are to have good knowledge and skills
both in maths and science and be able to apply them to the interdisciplinary
assignments. So, what is the best way to form a complete picture and not to lose
sight of significant details?
«A mom pervasive problem is that integration means different
things to different educators»
According the first model five different types of science and mathematics
integration can be distinguished: discipline specific, content, process methodological,
and thematic integration.
1) Discipline Specific Integration. This approach to integration involves an activity
that includes two or more different branches of mathematics or science. For
example, discipline specific integration might include activities involving algebra
and geometry in mathematics and activities infusing biology, chemistry, and
physics in science.
2) Content Specific Integration. Content specific integration involves choosing an
existing curriculum objective from mathematics and one from science. An activity
is planned which will involve instruction in each of these objectives. It is content
specific because it conforms to the previously developed curriculum, infusing the
objectives from each discipline. In this type of integration, the challenge to the
teacher is to weave together the existing programs in science and mathematics
with objectives from two separate and distinct curricular For example, suppose
that the content objective for mathematics is measurement and the science content
objective is the study of dinosaurs. Then, using masking tape on the gym floor,
they create life-size dinosaurs.
3) Process Integration. Another approach to integrating curriculum in mathematics
and science is through the use of real-life activities in the classroom. By
conducting experiments, collecting data, analyzing the data, and reporting results,
students experience the processes of science and perform the needed mathematics.
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4) Methodological Integration. The Standards of the NCTM state that students
should be able to “apply mathematical thinking and modeling to solve problems
that arise in other disciplines” (NCTM, 1989, p. 84).
5) Thematic Integration. The thematic approach begins with a theme which then
becomes the medium with which all the disciplines interact. McDonald and
Czemiak (1994) describe how the theme “Sharks” can be used to design an
integrated curriculum unit. In another example, the theme could be oil spills: in
mathematics you would be working withvolume, surface area, and cost of
cleanup; in science, you would be working with density and environmental
aspects of oil spills. However, the thematic unit goes beyond integration of
mathematics and science by including all other disciplines typically found in
elementary and middle schools. For example, this unit would include an
investigation of the economic and social implications of oil spills. In thematic
integration, while science and mathematics integration is important, integration of
individual disciplines is subsumed under the integration implied by the
investigation of the thematic topic.
The other model is called “The Correlated Science and Mathematics
Model”. Integrating science and mathematics traditionally means linking the two
disciplines in some manner (Davidson, Miller & Metheny, 1995). Science integrates
mathematics by using mathematics either as a tool to work science problems (e.g.,
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solving genetics or rate problems) or to actually teach a science concept (e.g., using
ratios in the equation to teach photosynthesis). Similarly, mathematics integrates
science by using science applications to explain or practice mathematics concepts or
to employ the science to reinforce students’ interest in mathematics or to enable the
students to recognize the broad utility of mathematics. The need to infuse
mathematics and science more completely was recognized by West and Tooke
(2001) and termed Correlated Science and Mathematics (CSM). The CSM PD
model was developed in 2006 and has been continually revised and refined. The
CSM model is unique in that it integrates science and mathematics in a more
comprehensive manner than other integration models. Each discipline is taught with
seven fundamental goals: (a) teaching for conceptual understanding, (b) using each
discipline’s proper language, (c) using standards-based learning objectives, (d)
identifying the natural links between the disciplines, (e) identifying language that is
confusing to students, (f) identifying the parallel ideas between the disciplines when
possible, and (g) using 5E inquiry format in science and mathematics when
appropriate (see Figure). The CSM approach to PD is “centered in the critical
activities of the profession-that is, in and about the practices ofteaching and
learning” (Ball & Cohen, 1999, p. 13). A new continuum is proposed that spans
from pure mathematics to pure science with the midpoint now representing a
correlated lesson including each of the seven goals of CSM.
This model has a modified version, called ‘Balance Model’, where the balance
replaces the continuum.
As previously stated, Lonning and DeFranco, Huntley and Roebuck and
Warden all used a continuum in their models regarding content. In the balance model,
however, balance replaces the continuum. As the model is an attempt to design a
long-term curriculum, the balance between science and mathematics needs to be
preserved. For instance, if a curriculum is designed for one year, the integrated
curriculum should not favour science over mathematics. Balance should be achieved
by giving an equal share of time to both disciplines in the process. The balance model
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was preferred, as it represents an equal integration of the content of both disciplines
in this study (Fig. 2).
LOGARITHMS AND ASTRONOMY
Arseniia Burkova
Moscow State Pedagogical University
The history of sciences is bound up with each other. When we present pupils some
material on, mathematics we don’t usually tell them the reasons of its appearance and
don’t explain what for people have created this or that mathematical concept.
However if we look deep in the history we’ll find out that discoveries in different
scientific fields wouldn’t have been made but for mathematical instruments, and
mathematical discoveries have been performed for their usage for society, economic,
scientific needs. This is the way logarithms were discovered.
In the beginning of the 17 century trigonometric functions, multiplication and
abridged multiplication formula were known. They were widely used in financial
sphere, insurance and astronomy. But multiplication and division need a lot of time
for the implementation, and time that needed for astronomical calculations was
unutterable. Convenient instruments that could turn multiplication into addition were
necessary. It was John Napier who invented logarithms to do it.
The way we understand logarithms nowadays is different from what has been
proposed by Napier. First ideas of logarithms appeared from matching members of
geometrical progression and arithmetical made from serial numbers of members of
geometrical progression. The idea was proposed by Archimedes: he noted that the
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result of multiplying of two members of geometrical progression is equal to the
member of this progression which’s number is equal to the sum of serial numbers of
these members minus one. So we can correlate multiplication in geometrical
progression and addition in arithmetical; involution in geometrical progression and
multiplication in arithmetical.
But for practical usage the slowly growing progression was necessary. The difference
between its members should be small in a way to cover almost all the numbers. After
realization of al the calculations the results could have been used in practice. But in
this way logarithm would have been determined for a discreet series of numbers only.
To reach the generality the function should have been proposed.
Full sinus considered to be 10 000 000. As decimals weren’t in use yet, logarithms
were numbers consisted of eight symbols and less. The logarithm of full sinus was
taken as zero. Logarithms of other sinuses were positive number increasing with the
angle decrease.
Napier invented the way to present logarithm as a function. There are two lines: the
length of the first in equal to the full sinus; the second is infinite. One dot is moving
along the first line still slowed; the second dot is moving along the second line
evenly. The start their movement simultaneously and from same position. The
velocity of the first dot is proportional to the distance left. The let the distance
covered by the first dot be the logarithm of the distance covered by the first.
This way in the language of up to date mathematics Napier proposed the kinematic
model of logarithm that can be decrypted in differential form:
Using this idea Napier made a table of logarithms, that could ease different
calculations. Years after this Pierre-Simon Laplace said, that logarithms had extended
mathematician’s life.
In 1594 one friend of Napier informed Tycho Brahe, a famous astronomer, about new
convenient way of calculations. But it was his disciple who had the chance to use it.
It was Johannes Kepler, young astronomer, who had a lot of ideas about the universe
organization and planet movement. He is famous for his three laws, but they wouldn’t
appear but for the logarithms.
Tycho Brahe had a huge observational data about planet and stars movement, but this
data was to be analyzed. All casual ways of analyses such as trigonometry weren’t
useful and convenient: it could demand few human lives. Kepler constructed an
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equation which described the movement of a body along the elliptical orbit in the
two-body problem: M = E - e sin(E), where M is the meananomaly, E is the
eccentricanomaly, and is the eccentricity. As we can notice, the eccentricanomaly is
present twice in the equation: by itself and under the sight of sinus. The equation
can’t be solved by usual methods, and the method of selection and approximation is
too lingering. The astronomical data was necessary for sailors, but they were waiting
for the results for 22 years, during which Kepler was calculating the parameters of
stars and planets movements. Kepler finished his work only after he had learned
Napier’s tables and presented the results as Rudolphine Tables. These Tables were
used by seafarers during next hundred years.
Years after this Pierre-Simon Laplace said, that logarithms had extended
mathematician’s life.
Literature:
1. Jakob Uspensky. A Short History of Logarithms// LKI, 1923 – 88p
2. O. Gingerich. Johannes Kepler and the Rudolphine Tables // The Great Copernicus Chase
and other adventures in astronomical history. — Cambridge, 1992
BRACHISTOCHRONE PROBLEM
Florian Lange,
University of Cologne
What does the name Brachistochrone mean? It is composed of the two greek
words brachistos (the shortest) and chronos (time). The name describes the problem
quite well, since the question is to find the fastest path between two points A and B in
a vertical plane. Let us suppose, that a ball starts at point A with velocity zero and
slides down to point B under the inuence of gravity. We want to construct a path
between A and B in such a way, that the ball reaches B in the shortest amount of
time. To simplify this problem we neglect friction and suppose that our ball has no
spatial extension. Such an idealized body with mass, but without spatial extention is
called a point mass.
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The problem was posed by the swiss mathematician Johann Bernoulli in the
end of the 17th century. He solved the problem himself, however he did not publish
his solution first. He wanted to give other mathematicians the chance to find the
solution and he particularly intended to challenge his own brother Jakob Bernoulli.
Jakob and Johann always competed with each other and both of them wanted to be
more successful. This competition led on the one hand to new scienti_c results but on
the other hand to some childish arguments. Finally both of them solved the problem.
Johann's solution was quite clever and intuitive, while Jakob's solution was more
general. Here I want to present the idea of Johann Bernoulli's solution, as it is
physically motivated and can be understood with knowledge acquired at school.
Since we assume that the motion of the point mass is frictionless, the only
force acting on it is the gravitational force Fg = m· g, with m the mass of the point
mass and g the gravitational acceleration, which is close to the surface of the earth
roughly 9,81m/s. In general the gravitational force, which point mass m1 at positions
r1 acts on point mass m2 at positions r2 is given by
where G denotes the gravitational constant. In our case one mass is equal to the mass
of the earth ME and one radius is equal to the radius of the earth RE. Because of the
act that RE is much larger than the altitude of point A, we obtain with RE – r ≈ RE:
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In order to find a solution of the Brachistrochrone problem we use Snell's law
of optics. Let us suppose, that a ray of light passes from one medium into another
one, like from air into water. Snell's law says that
where α
is the angle of incident, α2 is the angle of refraction and v1, v2 are the
1
propagation velocities of light in medium one and two.
Snell's law is a direct consequence of Fermat's principle, saying that a ray of
light always follows the path of shortest time. The idea of Johann Bernoulli was to
consider the point mass as a ray of light, which passes through different media. First
we divide the vertical plane in different layers and suppose, that on the one hand the
velocity of the point mass is constant in each layer and on the other hand Snell's law
is fulfilled at each interface. By using Snell's law it is guaranteed that the constructed
path is the fastest connection between A and B. In the second step we increase the
number of layers and make them thinner and thinner until the path between A and B
becomes continuous. Then α is the angle of climb and v is the velocity at each point
of the curve.
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To determine the velocity we can make use of the basic principle of energy
conservation. The sum of kinetic energy and potential energy is at each time constant.
At the starting point the velocity is given by v = 0 and the y-coordinate is y(0)
= 0, so we get E
= 0. We can rearrange the equation above and obtain for the
tot
velocity
Next we want to find another expression for sin α. By definition the slope at a
point (x,y) is equal to the derivative of y(x) with respect to x. We can make use of the
Pythagorean Theorem and obtain
Finally we arrive at
Here c is a constant and r is defined as
. Such an equation, depending
on y(x) and on the derivative of y(x) is called a differential equation. We will not
explicitly solve this equation, but rather state the solution. The x- and y-coordinate of
the curve which solves the last equation can be written as a function of a parameter θ
in the following way:
Here θ lies between 0 and θ0 ϵ (0; 2π) and θ0 and r are constants, depending on
the coordinates of point B. A curve which is parameterized in such way is called a
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cycloid, as it can be constructed with the help of a circle. The cycloid is the locus of a
point on the rim of a circle, which rolls along a straight line. In our case this point
initially lies at the origin and the circle rolls along the x axis. When the circle has
turned through an angle θ, its centre has covered a distance rθ, which is equal to the
length of the corresponding segment of the circle.
The cycloid is not only the solution of the Brachistochrone problem, it is also a
Tautochrone. This means that a point mass always needs the same amount of time to
reach the lowest point of the curve, independent of the starting point.
The Brachistochrone problem is one of the most famous problems in classical
physics, providing the possibility for an interesting experiment at school. Probably it
is not doable to present this solution of the Brachistochrone problem during a normal
lesson, however one can maybe do it with interested pupils within the scope of a
project group.
INTEGRATION OF MATHEMATICS AND BIOLOGY IN EDUCATION (AT
SCHOOL)
Tatiana Ozharovskaia,
Margarita Rodionova
Moscow State Pedagogical University
Abstract
The article describes one of the methods which can be used to integrate mathematics
and biology at school. Authors suggest a list of laboratory works held in Russian
schools which claim using of mathematical methods and some calculations or other
mathematical skills. These integrative labs are made for students to develop practical
skills of both subjects at the same lesson. Also this list is written for teachers to
consider the necessity of integrative lessons for students’ education and their future
development.
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