3rd International e-Conference on Optimization, Education and Data Mining in Science,
Engineering and Risk Management 2013/2014 (OEDM SERM 2013/2014)
Mathematics - Introduction to the Curricular Process (Part 2)
Jan Singer
University of South Bohemia, Czech Republic
singerj@seznam.cz
Abstract
When teaching mathematics at universities, the curricular process consists of the following three steps:
1. Teaching text, 2. Mind of the student, 3. Application by the student. The first step consists of
mathematical parts (for example arithmetic, algebra, geometry, differential calculus, integral calculus).
In the present work, the second step (for example logics, deduction, mathematical operations, tables,
graphs) of the teaching texts are discussed. The third step (for example physics, dosimetry, chemistry,
biology, financing) will be analyzed in next works.
Keywords: Curricular process, mathematics, teaching text, the mind of the student, the application by
the student, arithmetic, algebra, geometry, differential calculus, integral calculus
I. INTRODUCTION
In teaching mathematics at universities, particularly at other than technological faculties, the
form of the curricular process is rather brief, for example compared to physics, chemistry,
dosimetry, radiation protection, etc. There is neither laboratory teaching nor development as is
the case in natural sciences. Mathematics is a discipline dealing with abstract entities and
searching for laws between them. It prevalently employs three steps of the curricular process
in usual sequence 2,3.
teaching text → mind of the student → application by the student
These three steps can characterize the efficacy and quantity of the curricular process.
Pedagogues and students enter the process here. The three steps are detailed in the examples
below.
1. Teaching text
→ quantity
Four basic fields of interest → structure → basic mathematics
→ space
→ change
2. Mind of the students
Basic thinking of a student in mathematics is a logic way of thinking - mathematical logics
aimed at concepts such as the demonstration, theory, axiom, model and also completeness,
doubtlessness, decisiveness. Further parts are the deduction, calculations (mathematical
operations), for example with the use of tables, graphs, etc.
3. The application by the students to the above mentioned branches can be
exemplified as follows
1
arithmetic → │
© Publishing House Curriculum. ISBN 978-80-87894-01-9
2
3
logics→│physics
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3rd International e-Conference on Optimization, Education and Data Mining in Science,
Engineering and Risk Management 2013/2014 (OEDM SERM 2013/2014)
algebra→│
deduction→ │dosimetry
geometry→ │mathematical operations→│chemistry
differential calculus →│
tables→│biology
integral calculus→│
graphs │finances
In the next chapter, a second part of the curricular process is considered, particularly problems
of logic and mathematical operations. The first part of the curricular process was published in
a previous work (Singer 2014).
II. LOGIC
1. History of logic (Wikipedia 2014)
Aristoteles is recognized as a founder of logic. However, modern logic is characterized by
putting increased emphasize on experience acquired from the surrounding world. Its
important representatives are Gottfried Wilhelm Leibniz, Bernard Bolzano, George Boole,
Gottlob Frege, George Cantor and Bertrand Russel.
In the work “Wissenschaftslehre” (Science History, 1837), Bernard Bolzano brought a
concept of the sentence itself, which is abstract with respect to particular realizations. This is a
written sentence (a track on paper), a spoken sentence (oscillations of acoustic medium) and
thought sentence (nerve impulses).
Gottlob Frege compiled an obviously first text dealing with modern semantics entitled “Über
Sinn und Bedeutung” (About Sense and Meaning, 1892). According to Frege, every word
(sentence) has its sense as well as meaning (see theory A). The expression either directly
defines its meaning or defines the meaning through the mediation of its sense (so called
theory B).
Some topics of logic are closer to philosophy, some others to mathematics. A definition of
“Mathematical logic” is thus sometimes employed.
2. Mathematical logic (Sochor 2001, Enderton 2001, Mendelson 2001)
Mathematical logic is a scientific discipline dealing with problems occurring at a boundary
between logic and mathematics. It is focused on the investigation, formalization and
mathematical approaches particularly in those fields of logic, based on which mathematics is
established. The whole mathematics is based on its logic and also on the theory of sets.
Mastering these branches is necessary to provide exact ways of mathematical expressions,
formulation of mathematical theorems and construction of respective proofs. The German
mathematician George Cantor is recognized as a founder of the set theory.
Mathematical logic provides a firm axiomatic framework for the whole mathematics, and its
maximum exactness makes all the mathematical results indisputable. The answer to most
mathematical questions is the most famous result of the whole logic: the Gödel
Incompleteness Theorem.
The set theory is frequently referred to as “the world of mathematics”. Any other
mathematical discipline can be considered as a part of the set theory.
3. Principal disciplines
Contemporary mathematical logic can be divided into three extensive disciplines, which are
tightly associated one with another.
The theory of proofs deals with the construction and study of different formal deductive
© Publishing House Curriculum. ISBN 978-80-87894-01-9
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3rd International e-Conference on Optimization, Education and Data Mining in Science,
Engineering and Risk Management 2013/2014 (OEDM SERM 2013/2014)
systems serving as a basis for the formal proof concept. It employs a solely finite method,
most frequently applied to finite sequences of characters or words.
The theory of models is focused on studying the general concept of the mathematical structure
and validity of a certain declaration within this structure. It is particularly interested in
concepts such as the homomorphism of structures, possibilities of their definition and
axiomatic use, saturation and elementary immersion. It quite commonly takes advantage of
the set theory infinite methods, and results, which can be achieved in the theory of models, are
frequently dependent on accepting or refusing a certain additional set axiom (axiom of choice,
generalized hypothesis of continuum).
The theory of arithmetic studies formal arithmetic systems such as Robinson and Peann
arithmetic and structures of sets of natural numbers definable in them. It is tightly related to
theoretical informatics, particularly to the theory of recursion and theory of complexity. It is
also involved in possibilities of the “arithmetization of logic”, i.e. expression of certain logic
concepts, procedures and declarations in the language of natural numbers and in consequences
of this “arithmetization” (they include for example the famous Gödel theorems of
incompleteness.
III.MATHEMATICAL OPERATIONS
1. Definition of mathematical operation (Wikipedia 2014, Daníčková, Houdek 2004)
The operation in mathematics, logic and information is a procedure, which takes given inputs
(also termed arguments, input values or operands) and produces one or several values (also
termed output values, results or outputs). The most frequently encountered operations are the
unary and binary ones. For a unary operation, only one input value is required (it can be
exemplified by trigonometric functions), whereas two input values are necessary for a binary
operation (as for example in summing, subtracting, multiplying or dividing).
However, operations do not concern numbers only. For example, in logic, the values truth and
untruth can be combined by the operations conjunction and disjunction, vectors in linear
algebra can be added and subtracted, sets in the set theory can be summed and subjected to
conjunction, and functions in the theory of categories can be assembled.
Operations need not be necessarily defined for all the values imaginable. For example the
procedure of division is not defined for real numbers, if the second argument equals 0. The
arguments, which is the operation defined for, form a range of definition, and the values,
which may be produced by the operation, form a range of values.
2. Applications to mathematics
The substructure of an algebraic (for example subgroup etc.) is formed just by those subsets,
which are closed for all the operation. For example, a subset of a group forms a subgroup just
if all the group operation are closed to the given subset. In the verification, nullary operation
must not been omitted (the set of all integers larger than 10 could be otherwise erroneously
considered as a submonoid of the monoid (Z, +, 0) of all the integers).
3. Examples of operations
Arithmetic operations
Basic operations performed with numbers, so called arithmetic operations (counting
operations), include the addition, subtraction, multiplication, division and raising to a
power.
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3rd International e-Conference on Optimization, Education and Data Mining in Science,
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Logic operations
Operations on statements, as e.g. the unary logic operation negation or the binary
operations conjunction, disjunction, implication, equivalence, XOR, NAND a NOR.
Linear algebra
In the vector space, the operations opposite vector (unary), and addition or subtraction of
vectors and scalar and vector multiplication (binary) are defined.
Operations on functions
On sets of representations, the unary operation is an inverse representation and the binary
operations are sums of representations.
Table 1 Examples of operations
On sets
Logic
Arithmetic
In linear
algebra
Geometric
With functions
summing, conjunction, difference, complement, symmetrical difference
negation, conjunction, disjunction, implication, equivalence
summing, subtracting, multiplying, dividing, raising to a power, root
extraction
multiplication of scalars and matrices, sum and difference for vectors
and matrices
(with vectors only) – vector, tensor, external and mixed products
(with matrices only) – transposition
(congruent) – identity, shift, rotation, central and axial symmetry
(homothetic) - homothety
differentiation, integration
REFERENCES
[1] (Singer 2014) Singer J. Mathematics – Introduction to the Curriculum Process, 3rd International eConference OEDM – SERM 2013, December 1., 2013 – March 31., 2014, in Press.
[2] (Sochor 2001) Sochor A., Klasická matematická logika (Classical mathematics and logics – in Czech),
Karolinum, textbooks, Praha 2001
[3] (Enderton 2001) Enderton M. R., A Mathematical Introduction to Logic, Academic Press, San Diego, 2001
[4] (Mendelson 2001) Mendelson E., Introduction to Mathematical Logic, CRC Press, Boca Raton, 2001
[5] (wikipedia 2014) cs.wikipedia. org, 2014
[6] (Daníčková, Houdek 2004) Daníčková S., Houdek F., O povaze královny věd aneb Matematika (About the
nature of the queen of science, mathematics – in Czech), Akademický bulletin, Akademie věd ČR, květen 2004
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