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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
.pdf
see a little bit astronomy during the geography lesson. And this is not the limit. Some
attentive students can find the connection between sunlight and growth of some
plants and some natural objects` disposition. It is well-known that sunflower follows
the sun during the day, an ant-hill needs the warmth of the sun so it collects it by the
shallow surface, and so on. All these objects may serve as indicators of intensity of
sunlight. The motion of the sun is oriented according to the cardinal directions so
biological objects may give a hint in orienting. Now it is the collaborative work with
biology.
Such an integration of science is also possible during the laboratory work. For
example an experimentalist may take a bottle with narrow throat and add some
vinegar or lemon acid diluted with water in it. A balloon with baking soda inside
should be put on the throat of the bottle and flipped over it. so the content of the
balloon mixes with the liquid inside the bottle. The chemical reaction starts and the
gas appears because of it fills the balloon. If the balloon is released it will fly by the
help of the gas leaving it. This simple chemical reaction of soda and vinegar may
become not only a good example of chemical transformation of substance but also be
used as a model of one physical phenomenon- reactive motion. Reactive motion we
can see used in space engineering, aircraft engineering. Some living organisms use
this way of motion like lobsters, ink fish, octopuses. Consequently within the
framework of one lesson a teacher may show the connection between such different
branches of science as chemistry, physics and biology. It is rather important to
remember that we study nature not from the separate points of view but at large. And
this should be shown to pupils.
PISA, COMPETENCES AND MATHEMATICS
Felix Beschorner,
University of Cologne
In this paper I shall summarize the presentation “Pisa, Competences and
Mathematics” which was given within the German-Russian exchange program of the
University of Cologne and the Moscow State Pedagogical University in Cologne in
November 2014. In the beginning I will briefly recapitulate a crucial experience for
the whole German school system which took place in the year 2000 – the “PISA
shock”. In the following I will give an overview over reforms in the area of teaching
mathematics in Germany of the last years which were provoked by the “PISA shock”.
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I will focus on the effect of these reforms in terms of creating new exercises in
mathematics.
The Programme for International Student Assessment (PISA) is a worldwide
study by the OECD which was first performed in 2000 and then repeated every three
years. At that time all 28 member states of the OECD – especially Germany –
participated in this survey. Even the Russian Federation as a Non-OECD country
participated in the first study.
In the year 2000, the German results created a big wave of excitement within
Germany, which was called the PISA shock. The most shocking part was that
Germany’s mean performance in all literacy scales was significantly below the
OECD average (see Table 1). In addition to that, the gap between the scores of the
5% lowest and the 5% highest achieving students was larger than in any of the other
participating countries (see Table 1). These results dominated the headlines and
political discussions for a long time. In 2008, the German Times wrote that “The
"PISA shock" undermined German self-confidence. Germans had always tacitly
assumed that they led the world in education, and yet now an international
organization had shown that, in the 21st century, their education system was well
below average“. (5)
So how did Germany’s education system react on this shock? Obviously
Germany looked beyond borders and asked for the differences between the German
school system and the other ones, which performed better in this study. Many
proposals were made and a lot of reforms were initiated. The one I will focus on is
the implementation of “Educational Standards” in mathematics and thus the
introduction of the concept of “competences”. In order to understand you have to
know, that Germany is a federal republic and that the field of educational decisions is
up to the particular state. The “Educational Standards” was a further attempt to raise
the general level of all the states to a similar one. The creation of concepts of
“competences” was an attempt to change the concept of teaching from a “deficitoriented” one to a “strength-oriented” version. “Deficit-oriented” stands for a kind of
teaching, where the focus lies on knowledge gaps. In contrast “strength- or
competence-oriented” means a kind of teaching where the focus lies on skills making
students stronger. It represents a change in perspective.
The Curriculum – The Concept of Competences
What does competence mean? In fact there is no real definition of the concept of
competences. Prof. Rainer Lersch from the University of Marburg once said: „Wer
nichts weiß, ist nicht kompetent, aber wer mit seinem Wissen nichts anfangen kann,
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Table 1: Results from PISA 2000 (4)
auch nicht“ (3). If you try to translate this quotation you will get something like this:
Someone who doesn‘t know anything is not competent, neither is someone who is
unable to apply their (his/her) knowledge.
It is easier to describe this concept by means of the math curriculum of NRW Nordrhein-Westfalen (the state Cologne is in).
In Table 2 you can see the competences of the math curriculum for Gymnasium
of the state NRW. On the right side you can see the content-oriented competences.
These are describing the classical content you have to deal with in school which were
not modified by the reforms recording to the results of PISA 2000. The processoriented competences - like “modelling” and “problem solving” on the left side -
were recently introduced. These competences describe the knowledge of students by
means of the question: which general manners are relevant for the typical work as a
mathematician (1)? This Table 2 gives just an overview of the competences you have
to provide. In the following Table 3.1 and Table 3.2 are giving an intense impression
of exact expectations in the end of form 8. You can find such expectancy for any
form in the curriculum.
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Table 3.1: Expectations for mathematics in end of form 7/8 out of the curriculum for mathematics in
Gymnasium of the state NRW (6)
Table 2: curriculum for mathematics for Gymnasium of the state NRW (6)
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Table 3.2: Expectations for mathematics in end of form 7/8 out of the curriculum for mathematics in
Gymnasium of the state NRW (6))
Strategies to Create New Exercises
Going hand in hand with these new requirements for education – especially
concerning the process-oriented competences – there was a growing demand for a
change in the “culture of exercises”. Traditional exercises just asked for a simple
solution and that was the only thing that counted. In these new exercises one has to
include the pursuit of process-oriented competence. Since then one has to consider
these demands in the marking of an approach. To create such new exercises I will
present six strategies elaborated by the ISB - Staatsinstitut für Schulqualität und
Bildungsforschung in Munich (2). If you read books of some other authors (1) you
will find much more of these strategies.
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1. Create “open exercises”
Traditional exercise:
b is the base, and h is the height of a parallelogram.
b=8 cm and h=3,5 cm are given.
Draw the parallelogram and calculate the area
“Open“ exercise:
Search for a parallelogram with an area of 24 cm². You must name at least two
different answers for b and h.
Additional “opened“ exercise:
Give as many combinations as possible for b and h to create a parallelogram with an
area of 24 cm². If you are sure that there are no other combinations, explain why you
think so.
In this case you just need to invert the direction of the exercise. Then the claim of the
task is much more problem solving than before. You have to explore the
mathematical connections between the variables. Furthermore, the conceptual
formulation motivates students to think further to communicate, discuss and argue.
2. Leave out basic information
Determine the content of the barrel you can see on
the picture. Work with your partner.
Present your result and argumentation
In this exercise no data is provided. So the student has to estimate some key data
beforehand. What are the values we need? Is it possible to estimate them? Which
approaches are the best to get these data?
The students have to get these pieces of information out of the picture. Then they
must present their approaches and judge other ones.
The one big thing about that task is that there is no correct solution. It is all just about
the way you solve the exercise.
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3. Ask for the solving strategy knowingly
Solve the following equations in the most efficient way. Give arguments why your
approach is the most efficient one.
a) 3x² - 4x + 4 = 0 b) (x+8)² = 121
c) 30x² = 6x d) 11x² - 22 = 0
Generalize your thoughts
In traditional exercises the students just had to solve the equations. Through this kind
of conceptual formulation the calculation is not foregrounded anymore. Obviously
process-oriented competences are needed to handle this task.
4. Ask the students to create their own exercises
In this kind of task students have to show that they have totally understood the topic.
The more complicated the exercise is, the better the student have understood the
topic.
5. Ask the students to think out of the box
The sum of two natural numbers add up to 195. Which numbers are we looking for, if
the one number is four times bigger than the other one?
… if the one number is twelve times bigger than the other one?
… if the one number is n-times bigger than the other one?
- Specify for which n the exercise has a solution.
- In order to make the exercise solvable for more n‘s, we are looking for additional
three-digit numbers other than 195. Name some suitable numbers and explain how
to find them
- Which numbers other than 195 should be considered in order to make the exercise
solvable for as few n‘s as possible? State the number of possible solutions for n.
In this exercise the focus lies on the competence “problem solving”. At first you have
to find some special cases, but then you have to generalize more and more. Therefore
the students have to review different approaches. Even this conceptual formulation
motivates to discuss in groups.
6. Use adequate „operators“
- Calculate the area of a triangle, which base is 5 cm and height is 4 cm.
- Justify the fact that the area of a triangle does not change, if you slide the edge C
upon a parallel line to AB.
- Describe how you can change a given triangle, in order to double the area.
- Specify the formula used to calculate the area of a triangle
- Give your opinion on the following assumption: „If you double the base and the
height of a triangle at the same time, you will quadruplicate the area“
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These so called “operators” were introduced by the “educational standards”. In
school the operators are used as a kind of vocabulary the students have to learn, so
that they directly know what to do. In this example you can see, that all these
exercises deal with the same topic. Just by using these operators you can shift the
focus on particular competences.
References:
1. Bruder, Leuders & Büchter, 2008: „Mathematikunterricht entwickeln. Bausteine für
kompetenzorientiertes Unterrichten“, Cornelsen,
2. ISB, 2015: “Kompetenzorientierung im Unterricht”. Online-Document:
http://www.isb.bayern.de/download/8385/kompetenzorientierung_im_unterricht_1.ppt.
Access: 16.01.15
3. Lersch, 2010: „Wer nichts weiß, ist nicht kompetent... Aber wer mit seinem Wissen nichts
anfangen kann, auch nicht!“ - Bildungsstandards, Kerncurricula und Kompetenzorientierter
Unterricht. In: Bildung bewegt. AfL Hessen, Nr.9/Juni 2010, p.: 4-7
4. MPHD, 2015: “PISA 2000: Overview of the Study. Design, Method and Result”. Online-
Document: https://www.mpib-berlin.mpg.de/Pisa/PISA-2000_Overview.pdf Access:
16.01.15
5. The German Times, 2015: “The PISA story”. Online-Document: http://www.german-
times.com/index.php?option=com_content&task =view&id=3205&Itemid=12 Access:
16.01.15
6. QUA-LiS NRW, 2015: “Kernlehrplan Mathematik für das Gymnasium – Sekundarstufe I
(G8) in Nordrhein-Westfalen”. Online-Document:
http://www.schulentwicklung.nrw.de/lehrplaene/upload/lehrplaene_download/gymnasium_g
8/gym8_mathematik.pdf. Access: 16.01.15
INTEGRATION OF MATHEMATICS AND NATURAL SCIENCES IN
EDUCATION
Svetlana Biriukova
Moscow State Pedagogical University
Integration of mathematics and physics studies in school
In this work we don’t discuss the question why do we use math to describe
nature. As it is accepted in theoretical physics, we postulate that for some reasons
mathematical description of natural phenomena is highly effective.
The correlation between physics and mathematics
The correlation between subjects of physics and mathematics reflects the
interconnection between the corresponding sciences. These correlations can be
contingently divided into three groups:
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1) Physics sets tasks and provides mathematical methods required to address
mats
physics
vector and operations on vectors
examples of vector quantities ( ) and
operations on them
Coordinate system
flat and spatial Cartesian coordinate system
The radian system of angular measurement, the
relationship between degrees and radians
problem solving, contributing to the formation
of mathematical language
linear function and its graph
equations of coordinates
( ) and velocities
( ),
motion graphs
quadratic function, quadratic equation
equations of coordinates
(
),
equation of the Trajectory
( )
Trigonometric functions
problem solving
them. These methods serve subsequently as a basis for the development of
the mathematical theory.
2) Applying a developed mathematical theory to the analysis of physical
phenomena, that leads to appearing of a new physical theory and disclosure
of new problems.
3) Physical theory in its development is based on the mathematical apparatus,
which is being improved while using in physics.
The implementation of interdisciplinary correlations between physics and
mathematics in the 10th grade (Mechanics)
Why do we need to integrate mathematics and physics studies in school?
There are several reasons for integration mathematics and physics studies in
school. First of all the formation of modern scientific world-view becomes possible
only owing to interdisciplinary. We also need to overcome the fragmentation of
students’ knowledge to teach them to apply their attainments from one field to
another. There are many teachers’ claims that students are often unable to perform
39
mathematical operations in the context of a physics problem, despite successful
performances in their math classes.
In a physics or an engineering course, problems are often presented in real
world contexts, using words, figures, and tables to organize and communicate the
situation to be solved. Students are expected to take these situations and to create
mathematical equations from which they can perform procedures. Students also need
to dissect equations and to describe relationships between multiple variables. By
contrast, in a mathematics course students are often given bare equations and asked to
perform routine procedures on them (Meel, 1998; Park & Travers, 1996; Clement,
Lochhead, & Monk, 1981).
As a result, students, while solving tasks in physics, may operate with symbols
and formulas paying no attention on their physical meaning.
This leads to another issue in applying mathematics to science courses, which
is the connection of mathematical symbols to physical meaning. There is a difference
between the interpretation of certain symbols in a mathematics classroom and the
interpretation of those same symbols in a science classroom (Torigoe & Gladding,
2007; Gainsburg, 2006; Glazebrook, 2001). The following example illustrates this
difference.
The main difficulties of integration physics and maths in school are following:
1) Physical concepts using in math class aren’t always timely formed in the
course of physics or vice versa.
2) There are some physical concepts, which are not introduced in the course of
mathematics at all.
3) There is a difference between the physical and mathematical interpretation
of certain symbols
4) The inconsistency of terminology and symbols
Let us examine the first point in more detail. In some cases, new mathematical
concepts are introduced first in physics classes and then in maths ones. For example,
the concepts of argument dx and increment of the function df are introduced in the
maths curriculum in the 10th grade, and in physics – already in the 9th grade while
studying Instantaneous Velocity. In this place of the physics course these concepts are
still vaguely expressed, moreover, Time is a scalar, and Displacement is a vector
quantity, whereas in the 10th grade math curriculum the concept of incremental is
introduced only for scalar quantities. We also don’t discuss zero (null) vector.
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