TRACES OF EUROPE - ERASMUS KA2 NORWEGIAN MEETING

1. The program
 What is ’Developing learning?’
 The Principles of the Zankov
Education System
- Excamples of tasks from first
to fourth grade
 What is emphasized?
 Conditions for Learning and
Development

2. What is ’Developing learning’?
The goal of elementary school education according
to the Zankov aducation system is the general
development of each child, i.e. the development of
fundamental elements of a child’s psyche, mind,
will and emotions.
 - develope joy of learning
- vill to optain goals
- ability to cooperate
- develope endurance
 The program is meant for all subjects

3. It’s a teaching program for all pupils
All pupils shall feel well to work with the
mathematics.
- weak pupils
- strong pupils
- pupils who have less confidence in
themselves or are reserved

4. The language is important
 «Without language, no thought»
(Vygotskij)
 The purpose:
- give the pupils a language they can use to
explain their ideas and thoughts
- to make it easier for them to communicate with
each other and the teacher.
The pupils learn the mathematical language from
the first grade: Addition, subtraction, sum,
difference, the value of the sum etc.

5. Excamples of mathematical language:
sum
5 + 3 = 8 the value of the sum
term term

6. Principles of the Zankov
Education System
 1 Teaching at an optimal level of difficulty
 2 Emphasiszing theoretical knowledge
 3 Proceeding at a rapid pace
 4 Developing students’awarness of the
learning process
 5 The purposeful, systematic development
of each student

7. The principle of teaching at an optimal level of difficulty
 1 L.Z.:”If the teaching material and methods of its
learning are such that there is no obstacle for
students to overcome, then the development of
children is sluggish and weak”
Difficulties that are possible to overcome are
placed in the zone of their proximal development.
The teacher has to teach at the optimal level of
difficulty to stimulate the intensive development of
children.
 The teacher or a more competent pupil gives help.

8. 1.grade: What cloth is missing?

9. 1.grade: How many threesomes in each figure? (6 and
3) How many squares? (3)

10. 1.grade: Find the colours, the pattern
and the numbers in the missing house

11. 3.grade: What is alike and unlike between the
equations? Do they have the same solution?
How do we solve them?
31 + (x + 16) = 84
40 + (x + 16) = 93
49 + (x + 16) = 102
58 + (x + 16) = 111

12. 4.grade - What is alike and unlike between the
expressions? Do they have the same value?

13. 2 The principle of emphasizing theoretical
knowledge
 The students are led toward eplaining specific
dependencies and making conclusions. The
children like to discover connections and make
conclusions on the basis of their own observations.
 Children learn the concepts of equality and
inequality , they compare figures on pictures,
numbers and figures

14. 1.grade: What is alike and unlike between the
expressions?
2 + 3 3 + 2
4 + 2 2 + 4
2 + 3 3 + 3 4 + 3 5 + 3
What do you see/discover?
0 + 5 = 5 6 + 0 = 6
8 + 0 = 8 0 + 7 = 7

15. 2.grade: Analyse and place the signs (>=< ) without
making any calculation.
Study the expressions below, what do you see?

16. 4.grade: What is alike and unlike between the
expressions? Do they have the same value?
372 – (45 – 29)
372 – 45 – 29
372 – (45 + 29)

17. 3 The principle of proceeding at a rapid pace
in the study of the curricular material
 The rapid pace of study suits children’s needs.
They are more interested in learning something
new rather than repeating many times the material
previously studied.
 Learning new knowledge alternates with repeating
the previously learned.
 The previous learned will be present in a different
way to make the learning stuff more various for the
children.

18. 1.grade: The tasks are presented in different ways.
Put in the right signs >=<

19. 2.grade: Put in the right signs, < = >

20. 3.grade: What kind of lines are not a radius? Why?
Two and two discuss, use the concept: circle center,
circle line, straight and curved line
(Various tasks in the same lesson)

21. Draw a line and find two threesomes.
Find 4 solutions.

22. The solutins:

23. 4 The principles of developing the students’
awarness of the learning process
 The awarness of the learning process
taking place within them, the awarness of
what they have known before and what new
is being learned during the process of
studing a certain subject.
 How is the knowledge to be learned linked
together?
- What do I need to know to solve this
problem? What do I know?
 The children reflect on their own learning
and thinking

24. 2.grade: True or not true?
a) Are those equals true?
81 – 54 : 9 ∙ 4 + 2 = 14
81 – 54 : 9 ∙ 4 + 2 = 18
b) Use parenteses to change the order on the
operations so that the equals become true.
81 – 54 : 9 ∙ 4 + 2
1 23 4
= 81 – 6 ∙ 4 + 2 = 81 – 24 + 2
= 57 + 2 = 59

25. Solution:
( 81 – 54 ) : 9 • 4 + 2 = 14
27 : 9 • 4 + 2
(81 – 54 ) : 9 • (4 + 2 ) = 18
27 : 9 • 6
The pupils have to calculate a lot before
they can write the answer!

26. 5 The principle of purposeful and systematic
work on the development of each student
 If there are no pathological deviations, all
children should be able to progress in their
development. The process of evelopment
takes place either slowly or unevenly,
depending on individual potentials,
characteristics of nervous systems,
personal experiences etc. Weak and strong
students should learn together. They help
and challange each other.

27. All the pupils should have learned those basic
skills during the 1., 2. and 3. grade:
 1.grade: Some combinations of
addition. (up to ten)
Write the numbers in a correct way.
 2.grade: Addition (more than 10)
Multiplication up to 5 ∙ 5.
 3.grade: Multiplication up to 10 ∙ 10.
After this, there is no limit what to learn!

28. What is emphasized?
 Counting and comparing
 Observe pictures – what picture does not fit
in with the others?
 Analyse and observe what’s alike and
unlike between figures, patterns and the
relations between numbers and
expressions
 To discuss, explain and justify
 Cooperate with each other in the class –
argue and help one another
 Tasks with several solutions
 To learn different strategies of calculating

29. Excamples on task with several solutions.
Which picture does not fit in with the others? Why?
Observation av numbers, colours,other characteristics

30. What picture does not fit in with the
others? Several solutions

31. 3.grade: One answer, but many ways to get there!
32 + 19 = 32 + 18 + 1
32 + 19 = 32 + 8 + 10 + 1
32 + 19 = 19 + 1 + 31
32 + 19 = 32 + 8 + 11
32 + 19 = 19 + 11 + 21
32 + 19 = 2 + 9 + 30 + 10

32. Subtraction - mental arithmetic
horizontal setup
 43 – 16 = 43 – 13 – 3
 100 – 78 = 100 – 70 – 8
 45 – 19 = 45 – 20 + 1
 45 – 19 = 45 – 15 - 4
 312 – 174 = 112 – 60 - 2

33. 2.grade: Different strategies.
Find the value of the sum 36 + 27, vertical
setups. How did Ida,Tor and Leon think?

34. 3.grade: Excample multiplication:

35. Conditions for Learning and Development
 Vygotskij: ”Children’s thinking begins with
astonishment.”
 A child is astonished and emotionally involved only
if he or she feels comfortable, when the
relationships between the teacher and the students
are sincere, friendly, and warm. Consequently, the
atmoshpere of trust and care supports the
development of the selfrespect and self-
acceptance of children, strong will and belief in
own abilities. In such conditions even the weakest
student, receving support of peers and a teacher
participates in the joint creative work.

36. Conditions continue
 Summarizing, everybody in the classroom is
encouraged to follow the following rules:
- Listen and hear
- Look and see
- Think and ponder
- Enjoy and experience
To provide this learning environment and engage
young pupils into this learning process is not an easy
task for any teacher. It requires great efforts and
commitment from a teacher. But teaching results will
be rewarding for a teacher as well as for students.

37. The last words:

38. Read more:
www.matematikklandet.no
Also books for kindergarden

39. Reference: