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Didactics of mathematics I
Ro~ena Bla~kov
BRNO 2013
Content
Preface & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & . 3
Didactics of mathematics and its role in a system of sciences & & & & & & & & . 5
How we understand the notion didactics of mathematics .. 5
Didactics of mathematics particularities ....... 6
The relationship between mathematics and didactics of mathematics ... 7
The relationship between general didactics and didactics of mathematics 7
The aim of didactics of mathematics . 8
1.2.1 Didactics of mathematics focused on a curriculum content ... 8
1.2.2 Didactics of mathematics focused on pupils cognitive processes .. 9
Curriculum documents 11
Framework of educational programme . 12
Aims of the educational area Mathematics and its application .. 12
Educational area Mathematics and its application .. 13
Educational programme Basic school .... 15
Curriculum of basic school mathematics 1979 .. 16
Mathematics curriculum of the nineyear basic school (ZD) 1973 17
Didactical principles 18
Teaching methods 22
Communication 25
Creation of basic notions . 29
Notions and their properties .... 29
Notions introduction in mathematics . 30
Notions creation process... 37
History of teaching mathematics 40
Bibliography and recommended literature .. 44
PREFACE
Dear students,
the introduced text is dealing with preliminary chapters of mathematics (math) didactics which are the starting point of thinking about the possibility of an effective math teaching implementation. Its meant for students of a followup master study programme for teaching general educational subjects and math.
The whole text is divided into 7 chapters. Each chapter has its own aim which should be fulfilled after its study, and you should be able to answer concept questions which appear at the end of the chapter. Math didactics study requires an active and independent approach, deals with special and didactical literature, and with math textbooks. There is still being solved the problem, how to use theoretical knowledge in a particular pedagogical job with pupils, and there exists the ceaseless confrontation of their successfulness. The presented approaches are general, but teachers work with each class and with each individual pupil is unrepeatable, unique, and requires teachers a creative approach. We assume math knowledge as a necessary theoretical basis of school math, acquaintance with methods of work at math, pedagogicalpsychological relations of learning and teaching, and mainly a good relationship with children.
The first chapter reflects the relation of math didactics to individual disciplines it is closely connected with, i.e. the relation to math and general didactics.
The second chapter introduces valid curriculum documents which were issued by the Ministry of Education, Youth and Sports, and which deal with teaching math. For the comparison, there are also presented examples of two documents from the past which show us the approach to math curriculum in this time.
The third and fourth parts pay attention to didactical principles and teaching methods you already know from your pedagogy and general didactics study. The aim of these chapters is the implementation of methods and principles in teaching math.
The fifth chapter focuses on various kinds of communication in teaching math. Apart from a common communication teacherpupil, it is necessary in math to acquire other types of it, mainly symbolic and pictorially illustrative communications.
The essential part of the text is the sixth chapter which deals with forming math notions. The most important thing is a teachers approach to forming each math notion and its properties so as to get right ideas, which could be enriched in an educational process and which could form an information system.
In the last chapter there is a brief history of math teaching in the Czech lands. Monitoring both the math teaching development at our schools and changes of a math content
as a school subject, it is very informative for teachers. The text provides illustrational examples and demonstrations which enable to compare them with your own approaches to math and its teaching.
Study of math didactics is a permanent educational process and it looks for more effective ways of math teaching at a basic school. I wish you success and a lot of nice experiences at your study.
Author
DIDACTICS OF MATHEMATICS AND ITS ROLE IN A SYSTEM OF SCIENCES
Aims: While studying this chapter
youll understand didactics of math tasks
youll be able to characterize the role of didactics of math in a system of sciences
youll be able to understand the importance of math theoretical ground and its transformation into the system of primary school math curriculum
The guide of study
In the first chapter there are presented some approaches for defining didactics of math and the relationship of math with other science disciplines it has a close connection with. The relationship between didactics of math and scientific math is an essential theoretical ground of school math curriculum.
What the term didactics of math means
It is not possible to use more precise opinion than the one which J. A. Komensk presented in his book Analytical Didactics (Komensk, 1947): Didactics means the ability to teach well. Teaching is an activity which enables to transfer somebodys knowledge to somebody elses one so as to master it.
The definition of the term didactics of math occurs in various publications mostly with the aim to show its role among scientific disciplines which are math, pedagogy, and general didactics. Let us mention some of them.
Dictionary of school math (1981) under the headword didactics of math says: Didactics of math it is a boundary scientific discipline between math and pedagogy which deals with various matters of school math at all types of schools, i.e. its content and methods how to teach and how to learn math.
P. KvtoH (1982) understands didactics of math in this way: Didactics of math is a scientific discipline which examines relations of teaching math in accordance with tasks determined by society.
B. Novk(2003) says: Didactics of math is usually considered a special didactics (subject, possibly branch didactics) in a sense of educational theory in math. It is a science with its own structure, logic and the way of thinking. We can distinguish four dimensions in it: content, pedagogical, psychological, and constructive.
Didactics of math is a scientific discipline which solves special math teaching tasks at individual levels and types of schools. It defines aims and a content of a math curriculum, it recommends appropriate teaching methods and procedures, organizational forms of teaching, it takes into account psychological relations of learning, and provides teaching technology. At present didactics of math is dealing with the role of a pupil and a teacher in an educational process, it studies processes which are going in pupils and teachers minds during teaching math, solving problem tasks, and using math in practise.
In the past there also appeared titles such as Theory of math teaching, Math methodology, Math pedagogy. For example J. Mikul
k (1982) says: The aim of Math pedagogy is complex examining of a math education system, all its elements, interrelations, and relations to didactic and nondidactic environment of the system.
Considering the relationship between math and didactics presents M. Hejn (1990):
Term teaching math consists of two words. The first word expresses the subject content, the second one an activity which is carried out by a teacher. Both math and teaching have their own structure, logic, and the way of thinking. Between the both fields there is a big difference. Math works with idealized objects, axiomatically accurately with full argumentation.
Teaching is dealing with people and each effort to axiomatise the structure of math methodology leads inevitably to reality violating. In math methodology similarly as in any other real scientific discipline there exist phenomena, objects, situations, and examples which are typical and crystalline, but there are also foggy, marginal, and unclear ones. It is caused not by the lack of our knowledge, but by the heart of the matter.
Task for you
Think about your perception of Didactics of math and express your expectation of this subject.
Think about these utterances: learn something by oneself, teach somebody, teach somebody something.
1.1.1. Didactics of mathematics particularities
Didactics of math and math as a school subject have their own distinct particularities which set them apart from other branch didactics and school subjects.
The main particularities are:
Big abstractness of math. Math terms were created on the basis of real life situations abstractness (anybody can never see a straight line, plane or number, but everybody has their images in a brain). Ideas are created thank to our intuition, and much later its possible to create a system based on deductive approaches.
Math is a subject in which knowledge and understanding of higher level elements depends on understanding and knowledge of lower level elements.
In some cases motivation of math curriculum is problematic because its difficult to find a real example in a practical life (e.g. for multiplying two negative numbers), or practical usage is very rare (e.g. modification of fractional expression).
Teaching math cannot depend only on formularization of relationships, rules, and formulas which students should remember.
Such approaches as: I will say it to them (it means a teacher to pupils) or I will show it to them are not effective enough in a teaching process. Pieces of knowledge are not transferable. A pupil should acquire math knowledge by his/her own concrete and thinking activities.
Didactics of math cannot teach students everything which is necessary for them to be able to teach math and transfer their knowledge. It cannot be its aim not only regarding the scope of math at all types of schools. But teaching math could be of a greater value because it provides methods of work and an ability of perception, thus teachers can try to lead their pupils on the way of knowledge. Didactics can recommend certain procedures which were proved in a practical life, but it should enable them to have enough space for their own creative work. Didactics of math should avoid two extremes: approaches which have their basis only in math, assume the beauty of its rationality and its results. But they assume teaching a pupil who wants to learn math and who is interested in solving problems and thinking (theory highlighting, too big expertise), or approaches which deal with detailed and very practical instructions which could be influenced let us say by the only experience without the help of math notions creation at childrens minds, sometimes with mistakes (practical orientation, overusing of methodology).
The relationship between mathematics and didactics of mathematics
Math as a scientific discipline gathered during its historical development a huge amount of knowledge and it develops all the time. The pieces of knowledge are arranged into logical wholes. Individual parts of math are created in a deductive manner by a system of axioms. Individual notions are defined accurately and links among wellestablished notions are looked for. When generalizing we look for more general notions, not only new theories have been arising, but also a language by which the theories could be described, and so on..
The aim of math didactics is to determine which parts of math theory will be taught at basic schools, how to present the pieces of knowledge to be understandable and adequate to pupils age and abilities. We are looking for the way how to transfer pieces of knowledge to pupils, in which order and form while respecting scientific correctness of appropriate curriculum. Chosen topics should belong to foundations of current math, they should form a selfcontained system we could continue with in both further studies and practical life. It is necessary to carry out curriculum didactical transformation, i.e. the choice of math pieces of knowledge as a scientific discipline and their processing into the system of math curriculum at basic schools. It is necessary to create a system which ensures the development of knowledge, abilities, manners, values, and pupils personal qualities. In the currently valid Framework educational program these demands are formulated as key competences.
The relationship between general didactics and didactics of mathematics
Didactics is a theory of teaching (in Greek didakstein teach). General didactics is focused on general questions of teaching, partly on an educational content, partly on a process which characterizes teachers and pupils activities and in which pupils acquire the content. The aim of general didactics is general solving of aims, content, methods, and organizational forms of teaching.
Didactics of math solves special questions of math teaching at individual levels and types of schools. It determines the content of a math curriculum, recommends suitable methods and teaching procedures, takes into account psychological relations of learning, and ensures teaching technology. Didactics of math fulfils a lot of various tasks the most important of which are the transformation of a scientific discipline into the system of a school math, the process of communication during a teaching process, and the development of pupils key competences.
1.2 The aim of didactics of mathematics
It is not possible to prefer one of the following requirements, but it is necessary to carry out and at the same time everything which is stated further. Reasons for such approach are proved by longtime experience. If a teacher has very good expertise, but he has a lack of emotional intelligence and pedagogical abilities, it is as problematic as having big enthusiasm and a good relationship with children, but inadequate professional knowledge.
1.2.1 Didactics of mathematics focused on a curriculum content
Math as a scientific discipline contains a lot of pieces of knowledge and only a small part of them belongs to the math curriculum content at basic schools. Scientific math pieces of knowledge cannot be presented in their abstract and theoretical form or in an axiomatic system in the way they are created in math. It is necessary to carry out so called didactical transformation of a theoretical math ground into math curriculum so as the curriculum would be adequate to pupils of a certain age, it would be presented in an understandable language, and there would be used math apparatus the pupils have at their disposal. The curriculum should not be in contrast with math correctness. Things a pupil learns at a lower basic school level should be learnt in the way the pupil does not have to learn certain pieces of knowledge differently in the future (so called relearning). E. g. explanation it is not possible to divide by zero we can account for in a certain way in the 2nd 3rd year at the basic school, in a different way in the 7th year at the basic school, and in another way at the grammar school or university when using the term limit, but at all cases mathematically correctly.
Math teachers should be completely clear about math notions and relations; they should realize what they know about the notions from their special preparation, what is mentioned in a math curriculum at a certain type of school, and how the notions and relations among them work. They should not see a gap between their theoretical preparation and math curriculum. It is possible to give an example using the topic Functions and inverse functions which are the basic notions in math analysis just in the first year at university, and the topic Square number and square root in the 8th year at the basic school. When teaching square root, students use it well, they give an example EMBED Equation.3 = 8, but they cannot answer pupils question why EMBED Equation.3 = (8), when (8) . (8) = 64 in the satisfying way.
1.2.2 Didactics of mathematics focused on a pupils cognitive process
The main criterion for a math teacher successful job is his approach to children. A student who is interested in working with children at a basic school, mainly at its second level, is highly appreciated. If somebody would like to become a teacher, he usually does his best to fulfil the idea. For successful math teaching it is necessary to follow how a pupil perceives what is submitted to him, how he can deal with abstract mathematical notions, which procedures are optimal for pupils, whether the pupil can see the same in a cognitive process as his teacher can see in it. Every child is a distinctive individuality, has his own math model, which is necessary to reveal and develop. At the same time it is necessary to respect the reality that creating math pieces of knowledge is not transferable (only information is transferable). When working with children a receptive teacher notices pupils train of thoughts and uses them appropriately, possibly he guides them delicately. The teacher also focuses on pupils with specific learning needs in math, either on gifted pupils, or pupils with specific learning disorders. Communication mastering with pupils in math lessons requires a lot of knowledge and mainly much thinking. A teacher should respect the students personality, ensure positive atmosphere in a teaching process and enough experience during new knowledge cognition, and he should support the desire for education. The aim is not only pupils math education, but also shaping pupils personality and creation of his emotional relationship to math and work.
1.2.3 Didactics of mathematics focused on methods of work
A math teacher uses in his work both math work methods (analysis, synthesis, math induction, deduction, generalization, abstraction etc.), and of course work teaching methods, including all accessible tools of modern informational and communicational technologies. The choice of methods should be adequate to the topics.
Among teachers from practice and students there prevails opinion that transmission approach to math teaching, when a teacher demonstrates required approaches and pupils repeat them, is the most optimal and reliable. Knowledge transfer seems to be the best for them. The effort to persuade them about possibilities of other approaches meets with scepticism. But when they learn about some other procedures and approaches, e.g. constructivist ones, they recognize their priority. They realize pupils mainly remember experiences and pieces of knowledge created by their own discoveries. There exists certain gap between theoretical mastering of teaching methods and their implementation into a teaching process.
Principles of constructivist approaches to math teaching are formulated e.g. in Hej:n, KuYina: Dt akola a matematika (Hejn, KuYina, 2009, s. 194, Stehlkov 2004, s. 13). The speech is about these principles (it is given in a short way):
I Math is understood as a specific human activity, not only its result.
II The essential part of math activity is looking for coherences, solving tasks and problems, creating conceptions, claims generalizing, their verifying and giving logical arguments.
III. Construction of pieces of knowledge, they are not transferable, they rise in a mind of a learning man.
Knowledge creation is based on the learners experience.
The ground of math education is creation of thoughtprovoking environment for creativity.
The construction of knowledge development is supported by social interaction in a classroom.
It is important to use various types of presentations and pieces of knowledge structuring.
Communication has a very important role.
It is necessary to assess the educational process at least from three aspects: math understanding, math teaching mastering, application of math.
Cognition based on information recalling leads to a formal cognition.
Tasks for you
Reflect your attitude to math, remember if your attitude was influenced by
somebody, either in a positive or in a negative way.
Familiarize yourself in details with principles of constructivism.
Study the publication: Hejn, M., KuYina, F.: Dt, akola a matematika.
Think about possibilities of using transmissible and constructivist approaches in teaching math.
Exercises
Solved exercise
The problem shows the determination of the sum of an infinite geometric series when using a topic dealing with infinite geometric series and their sum, and the transformation of the topic into the math curriculum for basic schools.
Determine the sum of the series EMBED Equation.3
using the relation for the sum of infinite series
tools of a basic school pupil.
sn = EMBED Equation.3
sn= EMBED Equation.3 EMBED Equation.3
We will mark the sum of the series with a symbol x: EMBED Equation.3 = x  . 3
We will multiply the equality by three and after adjustments we will get:
EMBED Equation.3 EMBED Equation.3 = 3x
EMBED Equation.3 x = 3x
x = EMBED Equation.3
Concept questions
Use the procedures in task 1.3.1 and calculate:
1. Determine the sum of an infinite geometric series EMBED Equation.3
2. Write these rational numbers with the help of a fraction with the integer in a numerator and denominator: a) 0, EMBED Equation.3 b) 0, EMBED Equation.3 c) 0,5 EMBED Equation.3 d) 0,45 EMBED Equation.3
Solution: 1. sn= 2
a) EMBED Equation.3 EMBED Equation.3 b) EMBED Equation.3 c) EMBED Equation.3 d) EMBED Equation.3
Summary
Math as a school subject is a very important part of education and a cultural overview of a man. For math teachers at the second level of a basic school it is necessary to get theoretical preparation, which is a prerequisite for their successful pedagogical activity. Professional preparation should contain special mathematical, didactical, and pedagogicalpsychological parts. These parts should be interconnected and students should not see any sharp distinction among them.
CURRICULUM DOCUMENTS
Aim
After studying this chapter you should be able to:
understand changes which were carried out within a school system in the connection with the implementation of the framework educational program
understand how math can contribute to the development of pupils core competences
study in details the educational content of the educational area Math and its application
Under the term curriculum in the Pedagogical dictionary there are given three meanings:
1. Educational program, project, plan.
Course of study and its content.
The content of all pupils experience they get at school and during activities which are linked with school, its planning and assessment. (Pedagogical dictionary, 1998, s.118).
Curriculum documents are created at two levels state and school ones. The state level is represented by the National program of education and by framework educational programs. The school level is represented by school educational programs.
2.1 Framework educational program
Framework educational programs are based on a new educational strategy which points out the development of pupils core competences, their interconnection with the educational content and the implementation of gained knowledge and abilities in a practical life. They are based on the concept of longlife learning, they express the expected level of education for all graduates of individual educational periods.
The framework educational program emphasizes the development of core competences which is the summary of knowledge, skills, abilities, attitudes, and values important for a personal development and selffulfilment of each member of a society. The framework educational program for basic education includes these core competences:
Learning competence, problems solving competence, communicative competence, social and personal competence, civil competence, working competence.
Individual subjects are presented in socalled educational areas. Educational area Math and its application is introduced by the characteristics of the educational area, aims, and the educational content for the first and second levels of a basic school. In the framework educational program there are allocated altogether for the educational area Math and its application 16 lessons of math for the 6th 9th year of a basic school. The headmaster can increase the number of lessons from the available time allocation. The framework educational program also contains socalled crosscurricular topics and speaks about the education of pupils with specific educational needs.
2.2 Aims of the educational area Mathematics and its application
Education in the given educational area tends to forming and developing core competences by leading a pupil to:
usage of math pieces of knowledge and skills in practical activities estimating, measuring, and comparing of sizes and distances, orientation
development of pupils memory by numerical calculations and assuming necessary mathematical formulas and algorithms.
development of combinatorial and logical thinking, critical reasoning, and understandable and factual arguments by solving mathematical problems.
development of abstract and exact thinking by acquiring and using basic math terms and relations to get to know their typical features, and on the basis of these qualities to be able to determine and categorize the math terms.
creation of a math tools supply (arithmetic operations, algorithms, task solving methods) and effective usage of acquired mathematical apparatus.
perception of a real world complexity and its understanding: developing experience with math modelling (a math approach to real situations), the evaluation of a math model and the scope of its usage; awareness of the reality which is more complicated than its math model, the appropriate model can be suitable for various situations and one situation can be expressed by various models.
analysis of the problem and of the plan of its solving, results estimation, choice of the correct procedure for problem solving and the correct result evaluation with regard to the task or problem conditions
precise and brief expression with the help of the math language and symbols, carrying out analyses and recordings during tasks solving, and the improvement of a graphic skills
cooperation development during problem and applied tasks solving which express situations of everyday life, then using acquired solution in a practical life; the realization of math possibilities and of the fact that we can come to a result by various ways.
development of own abilities and possibilities while solving tasks, permanent selfcontrol at each step of tasks solving, systematicness, persistence, and accuracy development, acquiring ability to express a hypothesis on the basis of personal experience or experiment and their verification or rebuttal by contrary examples.
2.3 Educational area Mathematics and its application
The content of the educational branch of knowledge Math and its application is divided into four thematic areas:
Number and variable
Dependences, relations, work with data
Geometry in a plane and space
Nonstandard application tasks and problems
In each topic there are formulated expected outputs which are obligatory for school educational programs creation, and the subject content is briefly defined there. An example of the topic Dependencies, Relations, and Data Processing:
DEPENDENCIES, RELATIONS, RELATIONS AND DATA PROCESSING
Expected outputs
A pupil
looks for, evaluates and processes data
compares data sets
determines relations of a direct and inverse proportion
expresses functional relation by a chart, equation, graph
uses functional relations and transfers simple real situations into math
Curriculum
dependencies and data examples of dependencies from a practical life and their features, drawings, schemes, diagrams, graphs, charts, frequency of a symbol, arithmetic mean.
mathematical functions rectangular coordinate system, direct proportion, inverse proportion, linear function.
Math Standards are being worked out in which indicators for individual points from FEP are processed in details, and there are stated illustrative tasks as well. E.g. For the expected output A pupil determines relations of a direct and inverse proportion there are stated indicators:
1. A pupil creates a chart, graph, and equation for a direct and inverse proportion which are based on the task text.
2. A pupil determines a direct and inverse direction according to the task text, chart, graph and equation.
3. A pupil uses direct and inverse proportion for tasks solving.
Illustrative task:
Milan earns some extra money in an advertising agency by rewriting data from questionnaires into PC. The number of processed questionnaires (d) is directly proportional to the number of minutes (m) spent working at PC. Milan measured he can rewrite 8 questionnaires in 20 minutes.
Fill into the chart the time Milan needs to complete the given number of questionnaires.
Fill into the chart the number of questionnaires Milan rewrites within the given time.
Number of minutes (m)2030Numer of questionnaires (d)6 820Copied from Math Standards for upper primary school, available online at HYPERLINK "http://www.MSMT.cz" www.MSMT.cz
According to Framework educational program for basic education each school works out its own School educational program.
In the past there was formed math curriculum. We introduce examples of two types of the curriculum to compare with a new approach which was applied in FEP (Framework educational program).
2.4. Educational program Basic school (1996)
Course of study (syllabus): 6th 9th grade of a basic school (Z): 4 lessons of math
Math curriculum: curriculum is divided according to individual years and topics. At the end of each topic there is stated what a pupil should know and additional topics.
Specific goals
Math together with teaching Czech is the basis of a basic school education. Math provides pupils with knowledge and skills which are necessary for a practical life and they are prerequisites for successful activities in most branches of the professional preparation and of various curriculum specializations at high schools. It develops pupils intellectual abilities, their memory, imagination, abstract thinking, and a logical judgement. At the same time it contributes to forming certain personality features, such as persistence, diligence, and criticalness.
Knowledge and abilities gained in math are the prerequisite for the cognition of scientific branches, economics, and technology and PC usage.
The aim of math teaching is to enable pupils to learn:
to carry out mathematical operations with natural numbers, decimal numbers and fractions both by heart and in writing; if solving more difficult tasks use a pocket calculator rationally;
to solve tasks from practice and use numerical operations including percentage and simple interest calculation;
to carry out estimations of solutions and to assess their validity, to carry out necessary rounding;
to read and use simple statistics tables and diagrams;
to use a variable, to understand its meaning, solve equations and inequalities and use them when solving tasks;
to record and graphically demonstrate the dependencies of quantitative phenomena in nature and in society, and work with some particular functions during practical life tasks solving
to solve metric geometry tasks, to calculate circumferences and areas of plane figures,
surfaces and volumes of solids, use basic relations among plane figures (identity,
similarity);
to have a good sense of direction in a plane and space, to use a coordinate system,
understand the relation between numbers and points as the basis of PC projects
representation;
to prove simple mathematical propositions and to deduce logical conclusions from
given assumptions.
Example of the 9th grade topic: Basics of financial math
interest Subject content
principal  interest calculation, determination of the
interest period number of interest period days
interest interval  simple interest
interest rate  compound interest
simple interest  solving mathematics word problems
compound interest from practical life
What a pupil should know
 to calculate an interest from the given principal for a certain period with given interest
rate
 to determine the required principal
 to perform simple and compound interest
 to calculate the interest rate from the interest
Examples of extending topics
solving real problems from parents practice
foreign exchange, foreign currency, currency conversion
solving compound interest tasks
2.5 Curriculum of the basic school from 1979
Teaching plan: th 5th 8th grade of the basic school 5 math lessons per week in each year.
Math curriculum: curriculum is divided according to years and topics including the number of lessons for each topic. At the end of each year there are mentioned revision and summary of the topics.
Characteristics of the math conception for the second level of the basic school
We present the chosen part of the whole text:
Pupils of the 5th 8th grade learn decimals, whole numbers, rational numbers, as well as numerical performance with these numbers, they familiarize with some irrational numbers (, square and cube roots, values of goniometric functions); they learn how to solve practical tasks, express real situations in a mathematical way with the help of equations and inequations&
They will acquire the term reflection and mainly the term function (direct proportion, linear function, square function, inverse proportion, goniometric function )
they will acquire the size of an angle, an area and a volume of geometric figures, they will learn how to use formulas for calculations
Example of one topic the 6th grade
Percentage (15 lessons)
Decimal numbers revision. Percentage. Simple word problems for percentage practising.
Pupils will revise operations with decimal numbers. They learn the term percentage as one hundredth of the number. They will realize the significance of percentages for comparing quantitative aspects of natural and social phenomena, mainly phenomena of the economic life, industry, and agriculture development. If solving basic word problems with percentages (it means for the base and the number of percentages calculation the formula EMBED Equation.3 is used
(
 percentage part, p the number of percentages, z base). The variables domain is the set of all positive decimal numbers.
In each period of changes the proportion between mathematical theory and its application is solved. It is rather difficult to ensure the balance between these two approaches because of the number of math lessons at the basic school and because of the increasing number of math topics which are being implemented into school math as the result of the math development as a science.
2.6. Math curriculum of nineyear basic school from 1973
Teaching plan: 5 math lessons per week in the 6th  9th grade, 1 drawing lesson per week in the 9th grade. In the teaching plan there is mentioned optional subject Math exercises with two lessons weekly.
Math curriculum: The subject content is divided into individual years and topics for each of them including the number of lessons for each topic. At the end of each year there are planned 46 revision lessons, in the 9th grade there are 15 revision lessons.
The introduction to math curriculum is elaborated in details and we choose from it:
Pupils math thinking is being developed mainly by acquiring math knowledge in a logical system and on the basis of their interrelations, by data relations analysis, in word problems and when forming and solving equations, by the analysis and substantiation while solving constructional tasks, and by using acquired knowledge in practice. Pupils learn how to prove theorems adequately and how to express themselves logically precisely. A teacher systematically focuses on the language level of pupils oral and written formulations and he/she is a model for them because of using an apt, accurate, and correct language.
There are also introduced detailed instructions for teaching arithmetic, algebra, and geometry, as well as didactic principles (life and school linkage, activeness, permanence, adequacy, clearness, individual approach).
Example of one topic the 9th grade:
Similarity (40 lessons, 5 of them aimed at topography)
Similarity of geometric figures, similarity of triangles. Theorems of triangles similarity (without proves), rightangle triangles similarity. Practical usage of similarity.
Topographical mapping: plane table, observation station, engineering zoning.
Sine, cosine, and tangent of an acute angle. Tables of sin , cos , tg for solving rightangle triangles in simple cases.
Task for you:
Study in details individual key competences and say how teaching math contributes to their development.
Study Framework educational program for basic education, mainly the part which is devoted to the educational area Math and its applications.
Familiarize yourself with the school educational program at school where you teach.
Tasks for thinking
What is according to you more convenient for a teacher curriculum for individual years, topics, including the number of lessons, or the framework of topics, and great latitude for creating school educational programs?
It is necessary to specify certain outputs and standards for appropriate educational levels.
Summary
Math as a school subject has a very important position both among all educational subjects at a basic school and in curriculum documents. Except mother tongue, foreign language, and informatics it has its own independent educational area in the Framework educational program. It is important for teachers to know both all documents relating to math education, and expected outputs and standards which enable pupils to continue successfully their other types of study and use math pieces of knowledge in everyday life.
3 Didactical principles
Aim:
To realize whether and how didactical principles as the most general rules can contribute to reaching better results in math teaching.
To follow whether and how they can contribute to the pupils personality development.
We can distinguish these three groups of didactical principles from the point of view of math teaching:
Principles resulting from educational aims and pupils competences development
Principles dealing with the content of math teaching
Principles which influence the process of learning and teaching math by the subject content
We can incorporate these principles into individual groups:
Ad a) principle of scientism
principle of purposefulness
principle of upbringing impact of education
principle of school and life connection
principle of theory and practice connection
Ad b) principle of adequacy
principle of systematic nature
principle of successiveness
principle of clarity
Ad c) principle of awareness
principle of activity
principle of permanency
principle of individual approach to pupils
principle of feedback
Principle of scientism
In teaching math we respect that math content as a school subject is based on scientific math, and chosen pieces of knowledge which form curriculum belong to foundations of current math and make up a compact system.
The principle of scientism cannot be broken even at the introductory phase of forming math visions. Didactical simplification must not mean distortion or deformation of a notion and pieces of knowledge cannot be implemented in the way which would lead to relearning in the future. Following the principle of scientism a teacher should think about the subject content according to the scheme:
To be aware of the knowledge on the given term from math theory, how the terms are defined and in which system they are presented.
To be aware of the way the terms are expresses in school math, which of their qualities are mentioned, what and how is rationalized.
What the system of curriculum is, which pieces of knowledge precede and follow the
given notion, how a pupil will use the subject content for his further studies or
practical life.
Respecting the principle of scientism doesnt mean that it is necessary to use a deductive approach for forming notions. Pupils are not made to define notions, but notions are formed within correct definitions.
Principle of purposefulness
One of the teachers main goals is to make clear basic aims of math teaching. The aim of the educational area Math and its application is
explicitly formulated in the Framework educational program. The aims of each lesson should be formulated in the way to be concrete, achievable, and checkable. Too much generally determined aims or a formal approach to their formulations are useless.
Principle of upbringing impact of education
Math as a school subject significantly contributes to the complete pupils personality development by its content and methods of work. It leads pupils to creativity, persistence, systematicness, selfactivity, decisiveness, sense of accuracy, responsibility, cultural values understanding, ability of communication with other people, cooperation, curiosity, and to the need of permanent education.
Principle of school and life connection
Math teaching noticeably contributes to this principle realization because math is the part of all social life spheres. There doesnt exist any branch of human activity without math knowledge. The motivation of individual school math topics is usually based on their application in a practical life.
Principle of theory and practice connection
Math develops according to the society needs and at the same time math influences society development. At school math it is necessary to consider the proportion of essential theoretical knowledge of individual topics and the proportion of its application. Respecting this principle it is important to avoid two extreme approaches on one hand it is practical orientation overestimating and underestimating the role of math theory, on the other hand it is focusing on theoretical knowledge without the necessary connectivity with a practical usage.
Principle of adequacy
Math teaching should be in balance with the level of pupils development mainly with regard to his logical thinking and the development of his ability to generalize and carry out abstractions. It is also necessary to have an adequate content and methods of work in math teaching.
Principle of systematic nature
Math curriculum is organized systematically in a logical succession which is necessary to respect. New subject content should followup the previous one, we should proceed from easier topics to more complicated ones, from a concrete to an abstract matter. The elements of higher level subject content require very good knowledge of lower level subject content elements. Math calls for regular, constant, systematic, and more often work in lower loads. Pupils like teachers systematic work, they hate randomness and chaos.
Principle of clarity
J. A. Komensk expressed the principle in the most apposite way: Everything should be demonstrated to all our senses as much as possible. It means visible things should be demonstrated to our sight, audible things to our ear, and tangible ones to our sense of touch. If something could be perceived by all senses it could be demonstrated to more senses.
Math ideas are created on the basis of manipulative activities with concrete objects, later with symbols so as to get demanded abstracts. In math teaching, it is bad if there is both too much clearness (if a pupil doesnt need it any more), and if there is a lack of it. The opinion can facilitate good pupils a piece of knowledge understanding, weaker pupils could even comprehend the piece of knowledge. The opinion in math teaching fulfils motivating role it should prepare a pupil to comprehend actively in a psychological way and it should be a source of a pupils interest. It has also a didactical role  it should arouse and regulate forming of right images on the basis of a sensory perception and it should facilitate subject content comprehension.
Principle of awareness
A pupil should master the subject content thanks to conscious comprehension so as to understand the content and the sense of it for his further studies and everyday life. Understanding relationships among math notions, the ability to give reasons for procedures, and applying topics in new situations enable to prevent teaching math from formalism. We show students not only HOW to act in solving math tasks, but we also show them WHY we act in this way.
Principle of activity
Quality pieces of knowledge could be formed only on the basis of active pupils approaches. The teaching process respect an independent pupils activity in all its parts according to the model: a teacher should manage, a pupil should work. The pupils activity is visible both in cognition processes, and in a moral sphere, when a pupil must be made to a strenuous activity which is necessary in math.
Principle of permanency
In math teaching it is necessary to guarantee the durability of acquired pieces of knowledge and skills which could be used by a pupil in the following topics and practice. A teacher can use an elaborated revision system and topics in new situations. At the same time it is necessary to take into account certain psychological relations of remembering and forgetting.
Principle of individual approach to pupils
Respecting pupils individual oddities is an expression of teachers pedagogical mastery. Pupils differ by their individual qualities, level of their education, approaches to learning and work, pace of work, features of character, perception, memory etc. Each pupil is a personality who should be respected. Pupils with specific educational needs rate special attention. The speech is about handicapped and health disadvantaged pupils, about socially disadvantaged ones, and about talented and extraordinary talented pupils.
The conditions for educating these pupils are given in the Framework educational programme for basic schools (RVP ZV, 2006, p. 99).
Principle of feedback
A math teacher should permanently know whether pupils understand the subject content, how they are able to use it in applications, and they should adjust their teaching, methods and forms of work to this ability. In math it is impossible to teach new topics if the previous ones are not acquired properly. Pupils should be aware of their procedures and thoughts correctness, as well as their possible mistakes. It assumes mutual communication among a teacher and pupils.
Concept questions:
Give examples of not respecting the principle of scientism at school math.
Give examples of topics in which you can put into effect the principle of clearness:
By a factual pupils activity.
By using ICT technologies, interactive boards etc.
How can you carry out the principle of individual approach to pupils with specific learning handicaps, mainly with dyscalculia?
How will you develop math talented pupils?
Acquaint with the assignment of Math Olympiad for the 6th 9th year of a basic school (eg. on websites of the Union of Czech Mathematicians and Physicists). Solve the tasks.
6. Acquaint with Kangaroo competition (KLOKAN) and with the tasks for the 6th 9th
year of a basic school.
4 Teaching Methods
Aim
acquaint with teaching methods and their classification from the pedagogical point of view
modification of teaching methods for math teaching
Curriculum guide
This chapter provides brief outline of teaching methods both from the pedagogical conception, and from the point of view of their usage in individual phases of a teaching process.
From the pedagogical point of view teaching methods could be divided according to J. MaHk and V. `vec (MaHk, `vec, 2003, p. 49) in the following way:
4.1 Summary of teaching methods
Classical teaching methods
Verbal methods (narration, explanation, lecture, working with a text, dialogue)
Illustrativedemonstrational methods (demonstration and observation, work with an image, instruction)
Proficiencypractical methods (imitation, manipulation, trying, experimentation, production methods)
Activating methods
Discussion methods
Heuristic methods, problems solving
Situation methods
Staging methods
Didactical games
Complex teaching methods
Frontal teaching
Group and cooperative teaching
Partnership teaching
Individual and individualized teaching, independent pupils work
Critical thinking
Brainstorming
Project teaching
Teaching via drama
Open learning
Teaching via TV
Computer assisted teaching
Elearning
Using ICT
4.2. Teaching methods with regard to teaching process phases
According to individual teaching process phases we can distinguish the following teaching methods:
Motivational methods
introductory motivation
continuous motivation
Expositional methods
methods of knowledge direct reporting (lecture, narration, description, explanation, instruction)
methods of mediate transmission through illustration (observation, demonstration, manipular activities, working methods, game)
heuristic methods (dialogic methods, problem methods)
independent work methods
methods of unintended learning
Fixation methods
methods of knowledge revision
methods of skills practice
Diagnostic and classification methods
methods of didactic diagnostics
methods of evaluation
methods of classification
Teaching math should ensure pupils would acquire certain amount of knowledge and skills, information, facts, and at the same time they should be able to think about something, make decisions, and develop key competences. That is why it is necessary in math teaching to stop using methods which provide students with complete knowledge which would be only reproduced by pupils. The choice of teaching methods should contribute to creating situations which would make pupils form their knowledge on their own on the basis of their activities and thought processes
4.3. Transmission and constructivist approaches
Transmission approaches in a teaching process are based on passing complete knowledge. This way of teaching is according to a lot of teachers the most effective and least timeconsuming way of cognition. A teacher is dominant, everything is explained to pupils, and it is expected from them to remember the topics.
It is typical for constructivist approaches to form knowledge actively in a pupils minds. The pupil is dominant, discovers (either on his own or with the teachers help) new information and pupils are expected to use the knowledge and to remember it in this way. This approach uses methods when pupils work independently, make experiments, make models, and carry out manipular activities. They test acquired knowledge and use it for solving various tasks of a different difficulty level. It is well known experience that what a pupil acquires by his thought activities or through his experiences, it is easier to remember.
These two approaches shouldnt be viewed as contrasting ones. In math teaching these two approaches should supplement each other suitably because pupils should have particular facts in reserve, which could be obtained without constructivist approaches, but with understanding. We can give an example from a practical life solving motion tasks. At the beginning a lot of pupils appreciated the approach which provides them with instructions how to proceed in tasks solving. After solving several analogical tasks, they started thinking about problems; they familiarized themselves with the problems, and were able to solve even more demanding tasks.
The task for you
Study the publication: MaHk, J., `vec, V.: Vukov metody (Teaching methods)
Give examples of a didactical game which would provide pupils with the environment for math knowledge discovering.
Think about your own experience of using transmission and constructivist approaches in math teaching.
5. Communication in Math
Aim
To realize how math contributes to the communication competences development.
To be able to present one situation via various communicative tools (text, picture, symbolic language).
To be able to realize the importance of the pictorial illustrative communication when solving word problems.
In math teaching we can face these basic types of communication:
Communication in the area of reading a math text
Verbal communication
Verbal symbolic communication
Graphic communication
Graphic symbolic communication
Pictorial symbolic communication
Pictorial illustrative communication
Communication in the field of a math text reading
Reading a math text, word problems, and application tasks and a text transcript into math language mainly in word problems and application tasks  it is for many pupils a hard nut to crack. Pupils are troubled with reading the whole text, understanding the text, and acquiring the length of a text. They are not able to understand the question of the task in connection with reading the task assignment, they often answer another question which was not mentioned in the text and may not relate to the task solving. Some pupils have problems to understand used expressions in the task text. Other problems occur when reading a symbolic notation, i.e. numbers, mainly multi figures or decimals, reading numerical expressions, powers, roots, and expressions with a variable.
Verbal communication
The precondition for a pupil to be able to express himself in math properly is to understand math notions, terms, and relations. It is necessary to have a clear idea of each notion in terms of its correct definition, although pupils do not have to learn such definitions. If speaking about the verbal communication, both a teacher and a pupil should concentrate on essential phenomena, on facts which are fundamental for the given notion or topic, it is necessary to limit less important properties and to characterize the notion precisely. It is a big artistry to express an idea by our own words and to keep the meaning of the notion.
If developing the verbal communication, we should perceive whether
pupils have enough opportunity for a verbal communication,
pupils understand teachers verbalization
pupils understand teachers questions,
pupils are not refused if their verbalization is not correct or formulated well,
pupils can see and perceive what their teacher assumes,
pupils know what their vocabulary is and how they can understand notions which are used.
Verbal symbolic communication
Appropriate verbal interpretation of math symbols is connected with understanding of individual symbols meanings. Pupils should acquire a verbal expression of writing numerals (symbols 0, 1, 2, , 9), writing numbers with the help of these numerals, symbols which express equality (=), inequality (<, >), symbols of math operations, brackets, then writing powers, roots, symbols of sets etc. Many pupils have problems with correct and understandable reading math symbols, keeping the order of running operations, using symbols for a correct calculation.
Graphic communication
The development of the culture of a graphic expression is the most important tool of the graphic communication because pupils ability to record an idea in a written way proves their good math level. Nearly all math notations, e. g. writing numerals, numbers, notations of written operations algorithms, brief recording of tasks assignments, procedures of their solving and answers, all of them are for a lot of pupils very demanding. Also a written expression modification, which is the prerequisite for a correct calculation, is very difficult for some pupils. The reason of making some mistakes could be careless recording. There exists a well known rule that pupils exercise book reflects teachers board. It is necessary to take into account that pupils neat written expression doesnt mean understanding and acquiring math topics. Very often it happens that pupils copy from a board a perfect teachers notation, their writing in exercise books is perfect as well, but pupils do not understand what they are writing at all. Nowadays pupils use for writing more and more PC. Education with the help of PC but without using textbooks and exercise books is tested at some schools. In such case it is necessary to understand all recordings which occur on the PC screen.
Graphic symbolic communication
The same problems which occur in the graphic communication can be seen in the graphic symbolic communication. The relation numeral, number  number notation is the expression of a notion understanding and its graphic processing with the help of a symbol. Notation of all other symbols requires mainly understanding such operation or relations which are expressed by the symbol. It is noteworthy that for a lot of pupils symbolic math notation is more legible and understandable than recording in a text, so it is often helpful to them if working with a math text.
Pictorial symbolic communication
Depiction of a math situation via an image, e. g. a symbolic depiction of a word problem or a construction problem with the help of a simple schematic image, enables and facilitates both weaker and skilled pupils solving the task. It is important to use a correct symbolic expression and to express a real situation in the task.
Another example of an easier illustration of relations among numerical data is e. g. diagrams used in statistics. The symbolic illustration of numbers in diagrams is much more legible than numbers notation e.g. in charts.
Pictorial illustrative communication
In pictorial illustrative communication pupils use images to express math notions and relations. It is possible to show pupils, with the help of images, the assignment of word problems and an outline of their solving etc. When illustrating geometric figures an image can often facilitate its solving. Using images for properties rationalization or for proofs of geometric theorems (e.g. theorems about triangles sides and angles properties, Pythagorean theorem, Euclids theorem) explain the particular topic very illustratively. Special attention is necessary for graphic illustration of solid geometry tasks in a plane, e. g. in a free parallel projection. Graphic task solving requires except a well acquired task theory also a certain level of a depth perception. In this context it is possible to remind the requirement on a teachers written and graphic presentation on a board.
We monitor which communication processes are under way when deducing a new notion, e. g. when deducing a term fraction.
Verbal communication: a pupil says: half an apple, a quarter of a cake,
Pictorial illustrative communication: a pupil can see (or draw) images of concrete objects.
Pictorial symbolic communication: a pupil draws a circle and divides it according to an assignment, he colours in an appropriate part of the whole.
Symbolic communication: a pupil writes down an appropriate fraction EMBED Equation.3 , EMBED Equation.3 .
We will let a pupil work independently as much as possible. This approach helps to form assumptions after a lot of similar situations, so the pupil moves from understanding a fraction as a part of the whole to understanding the fraction as a number (rational number) and gets the required abstraction.
Solved problem
We illustrate various kinds of communication by the example of a word problem:
Marek and Filip saved altogether 1000,  crowns, Marek saved by 250,  crowns more than Filip. How much money did each of them save?
Graphic illustration: Filip altogether 1000
Marek
250
Symbolic communication: numeric expression: (1000 250) : 2 arithmetic solution
equation: x + (x + 250) = 1000 algebraic solution
Solution: Filip has 375 crowns, Marek has 625 crowns.
Concept exercises
In each task give a graphic illustration (graphic symbolic communication), arithmetic solution, and solution by an equation (symbolic communication).
A barrel with water weighs 64 kg. When on the first day 28% of water was poured off and on the second day one third of the remaining water was poured off, the barrel with the rest of water weighted 34 kg. What is the weight of the empty barrel and how much water was in it at the beginning?
Fresh mushrooms contain 90% of water, desiccated mushrooms contain only 12% of water. Calculate how many kilograms of fresh mushrooms we have to pick up to get 3 kg of desiccated mushrooms.
Prove Pythagorean theorem (in several ways) by using an image.
Solution:
Barrel 14 kg, water 50 kg.
26,4 kg.
Other types of communication would be worth examining, e. g. nonverbal communication, communication by an act etc., which are also very important for pupils math success.
Except the above mentioned types of communication during a lesson, it is necessary to think about the communication teacherpupil, about pedagogical communication, which significantly influences teachers and pupils work efficiency. Polite language, tactful and sensitive behaviour, striving for understanding, respect of somebody else opinions, and the ability to work with a question, they are the expression of a teachers high professional standard. Each teachers word, the way and tone it was pronounced, has in a specific situation its importance and educational impact.
FORMING OF BASIC NOTIONS IN MATHEMATICS
Aim:
% To realize the need of math notions knowledge and possibilities of its didactic transformation into basic school math curriculum
% To understand the difference between inductive and deductive approaches to math teaching at a basic school.
Study guide
Forming of all math notions in math teaching at a basic school assumes that a math teacher has a clear idea of each math notion, is able to define it precisely, and knows its other properties. Thanks to his expertise the teacher is able to form proper pupils ideas of notions in accordance with right definitions, but he doesnt require definitions knowledge from pupils.
6.1 Notions and their properties
First of all lets mention how we understand the meaning of the word notion. According to the Normative Czech Dictionary (1998, p. 286):
Notion is:
A general idea (of people, objects, phenomena, actions) the content of which is defined by the important properties summary.
The idea or opinion, view, to create a right idea about something, to comprehend something, to understand something,
We will understand notion as one of scientific cognition forms which reflects in thinking essential properties of examined objects and relations.
Each notion has a certain content and scope.
Notion content it is the summary of all features which are typical for an appropriate notion.
Notion extent it is a set of all objects which properties are given by the content.
If the notion content is extended, its scope will be narrowed and vice versa.
Example: The content of the parallelogram notion: it is a quadrilateral which opposite pair of sides is parallel. The scope of this concept is formed by all parallels. If we extend the content of this notion, e.g. we join the identity of the adjacent sides, only a square and a rhombus belong to the scope of the notion.
According to the type we can distinguish individual and general notions. An individual notion is formed only by one object, e.g. null set, Euclidean space. A general notion contains more than one object, e.g. triangle, circle etc.
Next we distinguish concrete and abstract notions. Concrete notions reflect concrete objects, e.g. a cube, a block, a sphere. Abstract notions originate as objects of thinking, e.g. a straight line, a set, a number etc.
Pupils start getting to know new notions step by step, and the more they are familiar with their content and scope, the more they understand them. Each notion we work with must be explained to pupils properly, we should also use drawings, models, and real illustrations to get an appropriate amount of ideas of a given notion.
We create notions in a certain system. To be clear we classify notions. The notions classification must fulfil all attributes of sets decomposition into classes:
% It is necessary to carry out the classification according to the same feature
% The classification must be utmost and complete it must include all elements of an
appropriate set (the scope of a notion).
% The classification must be carried out in the way which ensures having all the classes
disjunctive each element of the classified set is included just into one class.
Example: Classification of numeric fields:
Complex numbers
Imaginary numbers Real numbers
Irrational numbers Rational numbers
Negative numbers Nonnegative numbers
Zero Positive numbers
Task for you:
Mention the classification of a mutual position of two straight lines in a space.
Classify triangles a) according to sides b) according to interior angels.
Classify quadrilaterals.
6.2 Introduction of notions in Math
If forming any theory there must be fulfilled the requirement of a precise definition of all notions which are used in the theory. This process could not be realized if some notions were not considered as basic ones. E.g. to define the notion straight line it would be necessary to define e.g. the notion line, this notion would require the definition of other notions, and this
process would never end. Thats why basic notions are used in math first, their meanings are given precisely by axioms, and from basic notions there are presented those ones which are derived with the help of definitions.
6.2.1 Basic notions
Primary concepts are such ones the given theory is based on, e.g. in geometry there are following notions: point, straight line, and plane. The meaning of primary notions is introduced with the help of axioms.
Axioms are theorems whose truth criterion is the practice. Axiomatic system must be complete, unquestionable; any axiom cannot be derived from other ones.
Euklides was the first human who formulated the axiomatic geometry and five axioms. This system was improved by D. Hilbert. Axiomatic theory of arithmetic was presented after finding the set theory.
Math definition
Other derived notions are implemented with the help of definitions. Math without definitions could exist but it would not be well arranged because each notion would have to be defined all the time.
Math definition is comprehended as a grammatical sentence by which the meaning of a math notion is precisely defined (Drbek, 1985, p. 41).
According to School Math Dictionary a definition is an implementation of a new notion with the help of notions known previously. More precisely: a definition is the equivalence on one side of which there is a new notion, and on the other side there are only notions known earlier. (School Math Dictionary, 1981, p. 28).
From the set of various definitions we will mention nominal and constructive definitions which occur at school math most often.
The definition, by which the name of the defined notion is implemented, is called nominal definition. E.g. a quadrilateral which has two opposite pairs of sides collinear is called parallelogram.
The definition, which implements the way of a new notion forming, is called constructive definition. E.g. there is given a point S and a nonnegative real number r. The circle is a set of points in a plane which have a distance r from the point S.
It is clear that some notions could be defined by various ways. E.g. the notion triangle could be defined in the following ways:
D 1. Three different points A, B, and C are determined, but they are not collinear. Triangle ABC is an intersection (common part) of half planes ABC, ACB, and BCA.
D 2. There are given three different points A, B, and C, which are not collinear. The triangle ABC is a set of all points X in a plane which belong to the line segment AY and at the same time the point Y belongs to the line segment BC.
D 3. There are given three different points A, B, and C which are not collinear. The triangle ABC is a closed polyline ABC which is unified with its inner area.
At the basic school it is necessary to work with definitions of notions very sensitively. It is more important for a pupil to understand the notions, to be able to present their examples, to notice relations with relative notions, and then to be able to verbalize and understand the notion definition. A lot of notions are formed intuitively and later at the secondary school we start with definitions of notions. Pupils would answer difficultly the question what e.g. a straight line, line segment, number etc. is, but they have already formed ideas of the notions.
Incorrect definitions
If defining notions, we face a lot of drawbacks some of which we present:
Excess definition it contains more properties of the defined notion than necessary.
Broad definition it contains less characters than necessary to define a notion. The set of objects, which belong to a notion defined in this way, is more comprehensive than the set of objects which belong to a precise definition.
Narrow definition it contains more characters than necessary to define a notion. The set of objects which belong to a concept defined in this way is narrower than the set of objects which belong to a precise definition.
Circle definition the first notion is defined with the help of the second notion, and speedily the second notion is defined with the help of the first notion.
Tautology definition a notion is defined by itself, although in another formulation.
Concept questions
Find mistakes in the given definitions and define the notions correctly.
Circle is a set of points which have the same distance from the given fixed point.
Parallelogram is a geometric figure which opposite sides are parallel.
Parallelogram is a quadrilateral which has opposite couples of sides parallel and identical.
Two geometric figures are similar if they are alike.
A number is divisible by two if it is an even number. Even number is a number which is divisible by two.
Two straight lines are perpendicular to each other if they form a right angle. Right angle is the one which is formed by perpendiculars.
The length of a line segment is a distance of two points. The distance of two points is the length of a line segment.
Define the notions: rectangle, square, area of a rectangle, body volume, decimal number, and fraction.
Solution
1. Extended definition, it can also include a sphere.
A point S is given and a real number r > 0. A circle k(S, r) is the set of all points X in a plane, which have distance r from the point S.
2. Extended definition. There exist also other geometric figures which opposite sides are
parallel, e. g. regular hexagon.
3. Excess definition.
Parallelogram is a quadrilateral which has both pairs of opposite sides parallel.
Tautology definition.
Two geometric figures are alike if and only if there exists such a real number k > 0 that for each two pairs of points (X,Y), (X, Y) there applies XY  = Xݴ .
The 5th, 6th, and 7th definitions are circle definitions.
5. Natural number is divisible by two if there is at the place of unites some of the
numerals 0, 2, 4, 6, 8. Natural numbers which are divisible by two are called even
numbers.
6. The first sentence is the correct one. Right angle is an angle which is identical with its
side angle.
7. The length of an abscissa is a positive real number which shows the abscissa multiple
of the unit abscissa. The distance of two points is the abscissa length.
Math theorem
Notions properties are expressed by math theorems.
Example:
If the natural number n is divisible by three, then the sum of its digits is divisible by three.
If in a triangle there holds that a2 + b2 = c2, then this triangle is rectangular with legs a, b and with hypotenuse c.
Math theorem is a true proposition with a true math content. Most of math theorems are in a form of the general quantification of implication forms with one or more variables. For one variable it is possible to formulate the math theorem with the help of symbols
( EMBED Equation.3 x EMBED Equation.3 D)[A(x) EMBED Equation.3 B(x)
where D is the domain of the propositional forms, A(x) is called an assumption, B(x) is a proposition and we say that the theorem is in a conditional form. Each theorem should have a clearly stated assumption. In theorems formulated in textbooks it happens very often that the assumption is not stated. E. g. the theorem The sum of triangle interior angels is a straight angle should be better expressed in the conditional form: If the geometrical figure is a triangle, then the sum of its interior angels is a straight angle.
From the previous theorem we will form the inverse theorem by substituting the assumption for the proposition.
( EMBED Equation.3 x EMBED Equation.3 D)[B(x) EMBED Equation.3 A(x)]
If we form negations of propositional forms A(x) a B(x), we will get the changed theorem
( EMBED Equation.3 x EMBED Equation.3 D)[B(x) EMBED Equation.3 A(x)]
Task for you
Find in math textbooks for basic schools several math theorems.
6.2.4. Proofs of math theorems
In math we use basic types of proofs: direct proof, indirect proof, proof by contradiction, and proof by mathematical induction. These proofs have the biggest significance for teachers work. We do not usually prove math theorems at basic schools, but we should verify each theorem, i.e. we should find the way how to substantiate an assertion. There are pupils who are pleased if the teacher shows them the proof of a math theorem.
Direct proof
Direct proof of the theorem A(x) EMBED Equation.3 B(x) lies in the idea that the assumption is true and we will form a chain of implications which followup one after another.
A(x) is true
A(x) EMBED Equation.3 A1(x), A1(x) EMBED Equation.3 A2(x) . An(x) EMBED Equation.3 B(x)
Example:
If the last binary number of the natural number n is divisible by four, then the natural number n is divisible by four.
We will express the natural number n in an extended notation:
n = an . 10n + an1 . 10n1 + + a3 . 103 + a2 . 102 + a1 . 101 + a0, then
n = 100 . (an . 10n 2 + an1 . 10n3 + + a3 . 101 + a2 ) + a1 . 101 + a0
the first part of the mathematical formula is divisible by four (100 = 4 . 25) if there is (a1 . 101 + a0), which is the last binary number of the number n, then the number n is divisible by four.
Indirect proof
The essence of the indirect proof is the fact that instead of the theorem A(x) EMBED Equation.3 B(x) we will prove the changed theorem B(x) EMBED Equation.3 A(x).
Example: If the number n2 is divisible by three, then the number n is divisible by three.
From the given theorem we will form the changed one: If the number n is not divisible by three, then the number n2 is not divisible by three.
If the number n is not divisible by three, then the number n is in the form = 3k + 1 or n = 3k + 2.
Then n2 = 9k2 + 6k + 1= 3k(3k + 2) + 1 or n2 = 9k2 + 6k + 4 = 3k(3k + 2) + 4. Thus none of these numbers is divisible by three and the original theorem is true.
Proof by contradiction
Proof by contradiction is based on the fact that there cannot be true a math theorem and at the same time its negation. We suppose that the theorem A(x) EMBED Equation.3 B(x) is not true, but its negation (A(x) EMBED Equation.3 B(x)) is true.
Example: Prove that EMBED Equation.3 is not a rational number.
We suppose that the theorem is not true, i.e. its negation is true: EMBED Equation.3 is a rational number. So EMBED Equation.3 = EMBED Equation.3 , where p, q are integers, q EMBED Equation.3 a p,q are relatively prime. After modifications we will get: 5 = EMBED Equation.3 , 5 q2 = p2 so p2 is a multiple of number 5 and also p is a multiple of number 5. We can write p = 5k , then 5q2 = 25p2, q2 = 5p2, so q is a multiple of number 5. Numbers p and q share the same divisor, number 5, which is a contradiction with the assumption of the theorem. So EMBED Equation.3 is not a rational number.
Proof by mathematical induction
The basis of the math induction proof is one of Peans axioms of arithmetic of natural numbers.
We will prove that the theorem is true for the first element.
We suppose that the theorem is valid for a certain kand we will prove that the theorem is true for k + 1.
Example: Prove that 1 + 3 + + (2n 1) = n2 is true.
6.5.1 V(1): 1 = 12
6.5.2 V(k) EMBED Equation.3 V(k + 1)
1 + 3 + + (2k 1) + (2k + 1) = k2 + 2k + 1 = (k + 1)2
Concept questions
What math preparation should a teacher of the second level of basic school get from the point of view of forming notions?
How will you explain to a pupil in the 8th year of basic school that EMBED Equation.3 is not a rational number?
How will you explain to a pupil in the 6th year of basic school that we cant divide by a zero?
Introduce a geometrical interpretation of the theorem on the odd numbers sum:
1 + 3 + + (2n 1) = n2
5. Define an axis of a line segment, express the theorem on each point properties of a
line segment axis, and prove the theorem.
6. Define an axis of an angle; express the theorem on properties of each point of an angle
axis, and prove the theorem.
Pupils familiarize with math notions step by step and these notions are more understandable to them if they know better their content and scope. For pupils it is impossible at a certain level to acquire the whole content and scope of an idea. A teacher doesnt require from pupils notions definitions, but in the sense of the right notion meaning he forms the notion gradually.
At the end of the chapter let us give examples from older textbooks to show how authors introduced basic notions, i.e. a point, line, and plane.
1. Textbook Josef Vin: Geometry for the fourth year of secondary schools, published in
Prague in 1927:
Basic spatial notions
Tiny things like a grain of sand, particles of chalk or graphite, a vertex of a cube, lead us to a point image. A taut thread, a cube or a ruler edge, awake our image of a straight line. A sheet of paper, a table top, a cube side, is an imperfect image of a plane. A point, a line, and a plane are basic geodetic figures.
By a point motion we will create a line (a straight line or a curved line).
By a straight or curved line motion we will create a surface (a plane or a curved surface).
By a surface motion a solid figure is created.
He continues:
Geometric procedure and method
Items by which we illustrate points, straight lines, planes, and other geometrical figures are imperfect images of these figures: perfect geometrical figures in fact do not exist and they are only our images.
Pieces of knowledge which we find at simple basic figures and which we get from our experience and points of view, we express by rules (axioms).
We join basic geometrical figures into the new and more complicated ones. For each new geometrical figure we must say from which figures and how it was created. It is done by enumerations (definitions).
Examining these new figures we get new pieces of knowledge and properties which we express by propositions (theorems). In each theorem we can find an assumption and affirmation.
2. The textbook Jan Vojtch: Geometry for the fourth grade of secondary schools, published in Prague in 1921:
We choose a point as the first basic notion and an element of all geometrical figures. Next basic notions are a straight line as the simplest line and a plane as the simplest surface.
The image of a straight line is represented by a taut thread and a ray of light; the idea of a plane is given by a sheet of paper, a board, and a calm water surface. A straight line and a plane (as lines and planes generally) are formed from points. Basic theorems which emerged from observations are:
 Through two different points there can be always led a straight line and it is only one.
 Through three points which are not collinear there can always be put a plane and it is only
one.
 If a plane contains two points of a straight line, it contains the whole straight line.
3. Textbook Eduard ech: Geometrie pro 1. tYdu stYednch akol (Geometry for the first grade of secondary schools), published in Prague in 1946:
Drawing of straight lines
An imaginary line has no thickness. Lines are straight and curved.
The expression straight line represents the whole unlimited straight line.
The expression line segment (abscissa) represents a straight line which is limited on both sides.
A point has no size; it has only a certain location.
Task for you
Recall the axiomatic system of forming basic geometric notions.
Look up in present math textbooks how the basic geometric ideas like point, straight line, and plane are introduced.
Look up in present math textbooks how the notions half line, half plane, line segment, and angle are presented.
6.3. Notions forming process
For math teachers it is important to monitor how individual notions are created in pupils minds through their own cognition and without any help of other people. As an example we can use forming a notion of natural number, when a threefour year child starts to understand quantity and gradually, independently creates ideas of natural numbers up to five. The fact that an adult teaches a child the series of numbers (to count up to ten) does not influence it at all. The cognitive process consists of several phases; the basic ones are motivation, experience, and cognition. Detailed information about this process is provided e.g. by Hejn (1990, p. 23)
The theory of the notions forming process can be found mainly in pieces of work by J. Piaget (1966), L. S. Vygotsk (1970), and other psychologists. Hejn (1990, p. 28) presents results:
Notions forming process is
The result of a particular mans activity and his communication with other people. By this activity a man acquires complete, historically created meanings
It is an organic part of the whole mans mind development.
It does not form notions in our minds separately, but in a structured way in complicated semantic nets.
We study notions forming process as the process divided into four phases (Hejn, 1990, p. 28):
Syncretic phase from many experiences there is singled out such group of them which are associated with a future notion. In the group there have been differentiated neither images, nor activities, nor vocabulary. E.g. manipulation with rounded objects (a round shape).
Phase of concrete objects images a notion is gradually predifferentiated, but it stays connected with concrete and real phenomena. The manipulation with the notion is dealing with objects. E.g. a sphere is being separated from a circle (a ball, a small wheel).
Phase of intuitive abstract images a notion is becoming an element of arising idealized and abstract images. Manual operations are gradually replaced by mental ones. E.g. circle and its drawing.
Structure phase a notion is becoming an element of an axiomatic theory.
Drawbacks of notion forming process
Words or symbols are joined with a wrong image
The image of some notions is created imperfectly, e.g. the notion angle is understood by pupils only within angle arms drawn on a piece of paper or within a small arc, by which it is marked, the notion triangle they understand only as a polyline without inner points, they dont distinguish such notions as number, numeral, circle, etc. Pupils are getting familiarized with procedures how to carry out something, but they miss information why to do it in this way. E.g. if dividing fractions proceed as follows: you will invert the second fraction and multiply the first fraction by the second one. The operation will be deduced without any understanding.
There is preferred verbalism; memorizing propositions without any understanding and formalism when pupils consider formally acquired pieces of knowledge as correct ones.
Words and symbols dont have assigned any image
Pupils use a lot of notions without understanding their meaning. They dont know what a decimal or fraction is, they dont understand the notion function, etc. They work with notions formally, but they miss a correct image of a notion.
Notions lack language expression
Pupils often argue: I know it but I cant express it. A teacher answers: You only suppose it, if you knew it, you would be able to express it. But it is necessary to realize that a pupil, who thinks and finds a piece of knowledge by his own mental activity, does not have to be able to verbalize it immediately; he needs some time to make an effort and to express his idea verbally. It is more valuable to form correct images of notions, to verbalize them gradually, and define them according to the pupils age.
How can we contribute to the correction of these drawbacks?
Since the first year at basic school we create all notions on the basis of pupils concrete activities, so first at an object level and later at an abstract intuitive level. We use as many demonstrations of the notion we would like to form as possible.
We will free a teaching process of only verbalism and formalism which slow down the cognitive pupils development and lead to a lack of interest in math. E.g. if we only present the relation for a pyramid volume calculation without any derivation, it is a formal approach.
We carry out an early diagnosis of notions understanding by discussing about them with pupils and by monitoring pupils thinking. In written assignments we monitor the whole procedure, not only the result.
We will teach pupils to work with a mistake.
Although the above mentioned procedures are timeconsuming and require a lot of teachers proficiency to involve the whole class into the educational process, we will appreciate their cognitive value.
Exercises
Work with a mistake. How many mistakes will you find?
5a + 4b = 9ab 4a2b + 4ab2 = 8a2b2 a2 + a3 = a5
7a 4a = 3 6xy2 2xy2 = 4xy2 a2. a3 = a6
3a + 5a = 8a2 8a2 : 2a2 = 4a2 EMBED Equation.3
3a . 5a = 15a2 2x3y2 . 5 x3y2 = 10x6y4
EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3
Find a mistake in solving the task:
a = EMBED Equation.3 we will multiply both parts of the equation by four
4a = 6b
14a 10a = 21b 15b we will notate the difference; e.g. in this way
15b 10 a = 21b 14 a
5(3b 2a) = 7(3b 2a)
5 = 7
Summary
One of the most important factors in teaching math is to assure that pupils will understand the subject content, they will have clear ideas about each notion, will be able to formulate properties of notions and give reasons for them. A teacher has to have perfect knowledge of each notion and its qualities to be able to ensure everything mentioned above, and he should be able to prove expressed theorems. Although the proofs of expressed theorems are not carried out at basic schools, nearly at each class we can find a pupil who would like to know, why a certain theorem is true, and he is able to understand the proof.
7 HISTORY OF TEACHING MATHEMATICS
History of teaching math is not separable from the Czech math development.
7.1. After establishing Prague University by Golden Sicilian Bull, issued by Charles IV on 7th April 1348, the Czech lands had a university. The university consisted of four faculties: Faculty of Liberal Arts, Faculty of Medicine, Faculty of Law, and Faculty of Theology.
There were taught grammar, rhetoric, dialectic, and quadrivium: arithmetic, geometry, astronomy, and music.
At Charles University there taught important teachers, e.g. Jan KYiaean zPrachatic (1368 1439), Jan `indel (1373 asi 1455) an astronomer, Old Town Square Astronomical Clock (Staromstsk orloj), Tadea Hjek zHjku (1525 1600), Tycho de Brahe (1546 1601), Johan Kepler (1571 1630), Joost Brgi (1552 1632).
The first textbook of counting:
1530 OndYej Klatovsk from Klatovy:
Nowe kni~ky wo poc~tech na Cifry a na lyny, przytom niektere velmi u~yte
n regule a exempla mintze rozlyc~n podle biehu kupetzkeho kratze a u~yte
znie sebrana.
1567 JiY Brnnsk:
Kn~ka, vn~ obsahuj se za
tkov umn aritmetickho tj. po
tom na cifry neb na liny pro pacholata a lidi kupeck.
7.2. In the second half of the 16th and at the beginning of the 17th century crafts start to develop, so it was necessary to determine distances and land acreage precisely, to develop trade, and thus there were higher demands for math knowledge. Upper grades of basic compulsory education (maeanka) are being established and there are taught reading and numbers notation, addition, subtraction, doubling, halving, multiplication, division, fractions, rule of three, division in a given proportion, measurements recalculating etc. The level of students knowledge was poor, they usually learnt by heart. The book by imon Podolsk from Podol (born in 1562) is focused on practical surveying tasks and mainly on earth measures, and it contributed to introducing uniform measures in the Czech lands.
We have to mention J. A. Komensk personality (15921670) who formulated a lot of requirements dealing with education and school in general. Mainly his principles which facilitated the world cognition in an educational process have been valid till now (principle of purposefulness, successiveness, systematicness, consciousness, clearness, activeness, emotionality, adequacy, permanence).
7.3. In the 17th and 18th centuries a big development of math begins in the world, mainly in the field of functions and infinitesimal calculus. In our country after the Battle of White Mountain there was certain stagnancy. Elementary schools were in the hands of municipalities and church. The Institute of Engineering Education was established in 1707 and there was paid a big attention to math.
7.4. In the second half of the 18th century a big revival of Czech math starts and its development is connected with names such as Josef Stepling (17161778), who educated a lot of students, and with his financial help Klementinum observatory was established. To his former students belonged, e.g. Jan Tesnek (17281788), Stanislav Vydra (17411804) who taught elementary math at Prague University (book Po
tkov aritmetiky). His former student was also Bernard Bolzano (17811848).
In the 60th of the 18th century math cultivation in Czech language begins. Among a lot of mathematicians we can mention Vojtcha Sedl
ka, Rudolfa Skuherskho, Josefa Studni
ku.
In 1869 the Union of Czech Mathematicians and Physicists was established which was a very important event. Its predecessor was the Association for free lectures in mathematics and physics which was founded in 1862.
7.5. At the end of the 18th and 19th centuries the Czech lands were characterized by these features: developed industry, business, the Enlightenment period, the idea of a progress, the end of ignorance, education was required. These requirements contributed to reforms which were introduced by the enlightened ruler Marie Terezie. Forming the uniform education system and its democratization were very important for our education development.
Let us mention the most important changes:
1774 elementary education reform
 established normal schools and courses for teachers
 principal threeclass schools (at least at one city of the region)
 trivial school (in small towns or at vicarages) where Czech was a teaching language
 education should have followedup to the previous one and it should have been broadened
 sixyear school attendance was recommended
1775 grammar school reform
 to be accepted to the grammar school it was necessary to be ten years old and to have
appropriate knowledge tested by an entrance exam
1869 general education act
 state had a decisive role in education
 eightyear compulsory school attendance
 fouryear education for teachers
 introduced new subjects including math
1877 Czech primary schools
 the aim of teaching math is to be able to solve skilfully oral and written practical numerical tasks
Czech upper grade schools (maeanky)
the aim is to carry out elementary numerical operations confidently and deftly, to solve calculation of townsmen living and simple trading accounting
(Teaching math aim: Pupils, learn to carry out confidently and swiftly elementary numerical operations and operations with unusual numbers, using usual advantages and reductions, and learn solving calculation of townsmen living. Finally, practise simple trading accountancy.)
7.6. 20th century
1915  Czech primary schools
 practical numerical tasks from life accountancy, saving, measures and weights, currency,
calculation of lengths, areas, volumes, and assessments. Learn four basic operations with
whole numbers (i.e. natural numbers), decimal numbers, and fractions which often occur.
1932 upper grade schools
 swift solving of numerical tasks according to the needs of business and public life, math
thinking habit, calculating with common numbers.
1948 the first school act transition to unified school
 the 1st fiveyear level primary school
 the second fouryear level secondary general education school
 the third level grammar schools and vocational schools
1953  54  the second school act
 eightyear school attendance (basic education)
 elevenyear secondary school (3 years of secondary education)
 parts of math are arithmetic, algebra, geometry, trigonometry.
1960 nineyear basic schools
 the first fiveyear level
 the second fouryear level
 gradual change of elevenyear secondary schools into an independent nineyear basic school
(ZD) and Secondary general education school (SVV).
 Math curriculum content:
 Arithmetic: Four basic math operations with whole numbers, decimals, fractions,
properties of operations, their usage in examples from practical life. Math thinking
development.
 Algebra: counting with common numbers.
 Geometry plane geometry, solid geometry practical tasks solving.
1968 fouryear grammar schools act
1976 gradual testing of a new math teaching conception
Integration of setlogical conception of teaching math since the first grade of the basic school.
First fouryear level
Second fouryear level
5th8th grade algebra enhancement, mainly the following notions: mapping, functions, equations, inequations.
1983 integration of sets conception into all classes of the basic and secondary school
The aim of math teaching at the 1st level of basic school is ensuring children to learn basic pieces of knowledge on natural numbers and geometric figures in space with the help of illustrative sets teaching and widely developed activities. Children should learn applying their pieces of knowledge and skills in real situations and using them as a specific tool of thinking. (Math operations with natural numbers, equations, inequations, geometric figures.)
Teaching math in the 5th 8th grades follows the 1st 4th grades and it should enable pupils to acquire math methods as an efficient tool for solving practical situations, and at the same time it should provide essential knowledge for subsequent studies at secondary schools. (Math operations with rational numbers, equations, systems of equations, quadratic equations, inequations, modification of mathematical expressions, mapping, functions, geometrical mapping etc.)
1986 math curriculum modification in 1983 (simplification)
 Curriculum subject content divided into grades and topics, number of lessons
1990 changes in the school system
 State, private, church schools
1996 obligatory nineyear school attendance
 Educational programs: Basic school majority of schools
Primary school some schools 1st level
Citizens school (Ob
ansk) some schools 2nd level
National school (Nrodn) few schools 1st level
Curriculum of the Educational program Basic school  subject content is divided into grades and topics without number of lessons.
7.7. 21st century
2004 Framework Educational Programs
Since the 1st September 2007 education in the 1st and 6th grade according to School Educational Program
A pupil and the development of his key competences are becoming the centre of the educational process (competence for learning, solving problems, and communicative, personal, social, civil, and work competences).
Educational area Math and its application
Four topic areas (math for the 2nd level of basic school):
Number and variable
Dependencies, relations, work with data
Plane and space geometry
Nonstandard application tasks and problems.
Schools work out their own School educational programs in which they divide the subject content into grades so as to ensure fulfilling of expected results and developing of pupils key competences.
Math standards are elaborated in more details.
LITARATURA
[1] BALADA, F.: Zdjin elementrn matematiky. Praha: Sttn pedagogick nakladatelstv, 1959.
[2] BLA}KOV, R., MATOU`KOV, K., VAGUROV, M.: Texty kdidaktice matematiky pro studium u
itelstv 1. stupn zkladn akoly, 1.
st. Brno: Masarykova univerzita, 1987.
[3] BROCKMAYEROV FENCLOV, J., APEK, V., KOTSEK, J.: Oborov didaktiky jako samostatn vdeck disciplny. In: Pedagogika ro
. XLX, 2000, s. 23 37.
[4] ECH, E.: Geometrie pro1. tYdu stYednch akol. Praha: J MF, 1946
[5] DIV`EK, J. a kol.: Didaktika matematiky. Praha: Sttn pedagogick nakladatelstv, 1989.
[6] DRBEK, J. a kol.: Zklady elementrn matematiky. Praha: Sttn pedagogick nakladatelstv, 1985.
[7] GAVORA, P. a kol.: Pedagogick komunikcia na zkladnej kole. Bratislava: Veda, 1988.
[8] HEJN, M. a kol. Teria vyu
ovania matematiky. Bratislava: SPN, 1990.
[9] HEJN, M., KUXINA, F.: Dt, akola a matematika. Praha: Portl 2009.
[10] HEJN, M., NOVOTN, J., STEHLKOV, N. (eds.): Dvacet pt kapitol zdidaktiky matematiky, 1. a 2. dl. Praha: PdF UK, 2004.
[11] KOMENSK, J. A.: Didaktika analytick. Praha: 1947.
[12] Kolektiv: Slovnk spisovn
eatiny.Praha: Academia 1998.
[13] Kolektiv: Slovnk akolsk matematiky. Praha: SPN, 1981.
[14] KVTOG, P.: Kapitoly zdidaktiky matematiky. Ostrava: PdF, 1982.
[15] LEBLOCHOV, J.: Historie vyu
ovn matematice v
eskch zemch. Diplomov prce. Brno: Pedagogick fakulta MU, 2005.
[16] MAGK, J., `VEC, V.: Vukov metody. Brno: Paido, 2003.
[17] MARE`, J., KXIVOHLAV, J.: Komunikace ve akole. Brno: CDVU 1995.
[18] MIKULK, J.: Didaktika matematiky. Praha: SPN, 1982.
[19] NOVK, B.: Vybran kapitoly zdidaktiky matematiky 1. Olomouc: PdF, 2003.
[20] NOVK, B.: Vybran kapitoly zdidaktiky matematiky 2. Olomouc: PdF, 2005.
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[22] PIGET, J.: Psychologie inteligence. Praha: SPN, 1996.
[23] PRnCHA, J., WALTEROV, E., MARE`, J.: Pedagogick slovnk. Praha: Portl 1998.
[24] SPAGNOLO, F., I}MR, J.: Komunikcia vmatematike. Brno: PYFMU , 1993.
[25] STEHLKOV, N.: Nkter komunika
n jevy vhodinch matematiky. In: Sbornk pYspvko Dva dny sdidaktikou matematiky 2003. Praha: PedF UK, 2003.
[26] `IMONK, O.:vod do akoln didaktiky. Brno: MSD, 2005.140 s. ISBN 8086633330.
[27] VIN`, J.: Geometrie pro
tvrtou tYdu stYednch akol. Praha: 1927.
[28] VYGOTSKIJ, L. S.: Myalen a Ye
. Praha, SPN, 1970.
[29] VOJTCH, J.: Geometrie pro IV. tYdu stYednch akol. Praha: 1921
[30] Rmcov vzdlvac program pro zkladn vzdlvn. Dostupn: HYPERLINK "http://www.vuppraha.cz" www.vuppraha.cz
[31] U
ebn osnovy zkladn devtilet akoly. Matematika. Praha: SPN, 1973.
[32] U
ebn osnovy zkladn akoly, matematika 5. 8. ro
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[33] U
ebn osnovy pro 1. 9. ro
nk Vzdlvac program Zkladn akola. Matematika. Praha: Fortuna 1996.
Working with polynomials at school
Ro~ena Bla~kov
Exponents: Properties and applications
Multiplication xm . xn = xm+n
Division xm : xn = xm n
Power (xm)n = xm.n
Power of Product (xy)n = xn.yn
Power of Quotient EMBED Equation.3
Zero exponent x0 = 1
Negative exponent xn= EMBED Equation.3
Exponent and Equations:
If two powers are equal and the bases are like, then the exponents are egual.
If am = an, then m = n. 63 = 6x means x = 3.
If two powers are equal and the exponent are egual, then the bases are like bases.
If an = bn, then a = b. x3 = y3 means x = y.
Using Symbols
Monomial 2x, 3a, xy
Binomial 3x + 2y, 5a 3b
Trinomial 3x + 2y 5z, a + 2ab + b
etc.
Using Exponents: Product of Binomials
x.(x + 4) = x2 + 4x
(x + 2).(x + 4) = x (x + 4) + 2(x + 4) = x2 + 4x + 2x + 8 = x2 + 6x + 8
Square of a Binomial
(x + 3)2 = (x+3)(x+3) = x(x+3) + 3(x+3) = x2 + 3x +3x + 9 = x2 + 6x + 9
Thing of diagram to help you multiply binomials mentally:
B2
A2
(A + B)(A + B) = A2 + 2AB + B2
AB
AB
The phrase below may help you2
First Outer Inner Last
Exercise
(5xy). (4x2y3) 52. 53  5 . 52
EMBED Equation.3 = ? for x =1, y = 1 EMBED Equation.3 = ? for x = 2, y = 3
(2a3b0)3  EMBED Equation.3
2 ( y  3)( y + 6) 3(y + 2)2 EMBED Equation.3
(a + b)2 = a2 + 2ab + ? 102x + 1 = 103x  8
(x 2y)2 = x2  ? + 4y2 (x 3)2+ (x + 3)2
(3u + 2v)2 = ? + 12uv + ? x2 y2 (x y)2
EMBED Equation.3 EMBED Equation.3
Problem  solving: analyzing algebraically
1.
Statement Use algebra to analyse the steps
Step 1Thing of a numbern is the number chosenStep 2Add 5 to your numbern + 5Step 3Multiply your answer in Step 2 by 2 2(n + 5) = 2n + 10Step 4Add your original number to your answer in Step 3 2n + 10 + n = 3n + 10Step 5Subtract 4 from your anter in Step 43n + 10 4 = 3n + 6Step 6Divide by 3(3n + 6): 3 = n + 2Step 7Subtract 2 from your answer in Step 6n + 2 2 = n
What answer did you get? Do you get the same answer each time?
Pick any whole number.
Add 4 to your number.
Then multiply by 3.
Then subtract your original number.
Then divide by 2.
Then subtract 6.
Pick any whole number.
Double your number.
Then add 6.
Then add your original number.
Then divide by 3.
Then subtract your originl number.
Pick any whole number.
Subtract 5.
Then double your anter.
Then add one more than your original number.
Then divide 3
Then add 3.
Try the following example on a frond. It will make you apper to be a mind reader.
Give these instruction Your friend completes Think: Algebra
To a frond these calculations
Step 1 Pick a number. 12 x
Step 2 Add 10 to it. 22 x + 10
Step 3 Then double the result. 44 2x + 20
Step 4 Then add one more than
the number. 57 3x + 21
Step 5 Then divide by 3. 19 x + 7
Ask your friend for the final result obtained. 19  7 = 12
FUNCTIONS AND GRAPHS
Ro~ena Bla~kov
Definition 1. Let X be a set of real numbers. A function f with domain X is a rule that assignis to each number x EMBED Equation.3 X exactly one real number f(x).
The number f(x) assignet to x by f is called the image of x under f , ort he value og f at x.
We read f(x) as f of x .
The set of all image sof elements of X is caled the range of f.
Definition 2. The graph of a function f is the set of all points (x, f(x)) in the coordinate plane, with x in the domain of f.
Definition 3. A constant function is defined by a formula of the form
f(x) = c c a constant.
Example:
Sketch the graph of the function: a) f(x) = 2,5 b) f(x) =  3,5
EMBED Equation.3
Definition 4. A linear function is defined by a formula of the form
f(x) = ax + b
with a, b constants and a EMBED Equation.3 0.
The graf of a linear function is always an oblique line.
Example:
Sketch the graph of the function:
f(x) =3x b) f(x) = EMBED Equation.3 x c) f(x) =  3x d) f(x) =  EMBED Equation.3 x
Sketch the graph of the function:
f(x) =2x + 3 b) f(x) =2x  3 c) f(x) =  2x + 3 d) f(x) = 2x  3
Sketch the graph of the function: f(x)= EMBED Equation.3
Definition 5. A guadratic function is defined by a formula of the form
f(x) =ax2 + bx + c
with a, b, c constants and a EMBED Equation.3 0.
The graf of a quadratic function is always a parabola.
Example
Sketch the graph of the function:
f(x) = x2 b) f(x) = 3x2 c) f(x) = EMBED Equation.3 x2 d) f(x) =  x2
Sketch the graph of the function:
f(x) = x2 + 1 b) f(x) = x2 2 c) f(x) = (x + 1) 2 d) f(x) = (x 2) 2
Sketch the graph of the function:
f(x) = (x 3)2 + 2
Sketch the graph of the function:
f(x) =  (x 1)2 + 1
Definition 6. A polynomial function is defined by a formula of the form
f(x) = an xn + an1 xn1 + & + a1 x + a0,
with n nonnegative integer and a0, a1, & an constants.
Example:
Sketch the graph of the function: f(x) = x3
Definition 7. A rational function is the ratio of two polynomial functions:
f(x) = EMBED Equation.3
with a, b, c constants and c EMBED Equation.3 0 and ad  bc EMBED Equation.3 0.
Example:
Sketch the graph of the function: f(x) = EMBED Equation.3
Definition 8. A root function is defined by a formula of the form
f(x) = EMBED Equation.3
with n a positive integer.
Example:
Sketch the graph of the function: f(x)= EMBED Equation.3
Exercises:
State in words the rule defined by each of the following functions:
fx) = x 3 b) f(x) = EMBED Equation.3 c) f(x) = EMBED Equation.3 + 2
Give a formula for each of the following rules:
The image of x under f is 1 more than twice x.
f(x) is the absolute value of 1 less than 3 times x.
Find the domain of each function:
f(x) = x b) f(x) = 5 c) f(x) = EMBED Equation.3 d) f(x) = EMBED Equation.3
Sketch the graph of each function:
f(x) = 3x 2 b) f(x) = EMBED Equation.3 c) f(x) = EMBED Equation.3 d) f(x) = 1 x, if x<0
e) f(x) = EMBED Equation.3 f) f(x) = EMBED Equation.3 g) f(x) = 1  %x%.
PAGE \* MERGEFORMAT52
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