communication jarmilas presentation

1. Communication
in the classroom
Jarmila Novotná
Charles University in Prague
Faculty of Education
e-mail: jarmila.novotna@pedf.cuni.cz

2. Programme
 Language and communication
 Posing questions
 Didactical situation of formulation
 Games focusing on communication and
terminology
 Other activities

3. Communication and language used in
the classroom
In the past: main focus of the teacher’s
language
Nowadays:
• More focus on the communication between
the teacher and students and also among
students
• Attention shifts from the language of texts
to the language of discourse
Important elements of a constructvist form of
teaching: discussion with the teacher and
discussion among students

4. Posing questions
„Good“ question
• Requires more than mere referring to known
facts.
• Students may learn something when they
answer it and the teacher can learn something
about his/her students from their answers.
• There exist several answers that could be
accepted.

5. Creation of “good” questions
Starting from the end
• a) Define the topic
• b) Create a closed question and find the
answer to it
• c) Transform the formulation and create a
“good” question

6. Creation of “good” questions
Phase a)
Rounding
Phase b)
12 seconds
Phase c)
My coach told me that I covered 100 m
approximately in 12 seconds. What
time could his stop-watch show?

7. Creation of “good” questions
Phase a)
Phase b)
Phase c)
Rounding
12 seconds
My coach told me that I covered 100 m
approximately in 12 seconds. What
time could his stop-watch show?
Area
6 cm2
How many triangles with the area
6 cm2 could you draw?

8. Creation of “good” questions
Phase a)
Phase b)
Phase c)
Rounding
12 seconds
My coach told me that I covered 100 m
approximately in 12 seconds. What
time could his stop-watch show?
Area
6 cm2
How many triangles with the area
6 cm2 could you draw?
Fractions
3½
The product of two numbers equals
3½. Which numbers might it be?

9. Creation of “good” questions
Phase a)
Phase b)
Phase c)
Rounding
12 seconds
My coach told me that I covered 100 m
approximately in 12 seconds. What
time could his stop-watch show?
Area
6 cm2
How many triangles with the area
6 cm2 could you draw?
Fractions
3½
The product of two numbers equals
3½. Which numbers might it be?
Money
35 EUR
The customer paid 35 EUR in the shop.
What did he buy and what was the
price of the products?

10. Creation of “good” questions
Phase a)
Phase b)
Phase c)
Rounding
12 seconds
My coach told me that I covered 100 m
approximately in 12 seconds. What
time could his stop-watch show?
Area
6 cm2
How many triangles with the area
6 cm2 could you draw?
Fractions
3½
The product of two numbers equals
3½. Which numbers might it be?
Money
35 EUR
The customer paid 35 EUR in the shop.
What did he buy and what was the
price of the products?
Diagram
x
xxx
xxx
xxxxx
xxxxx
12345
What might this diagram represent?

11. Creation of “good” questions
Adapt a commonly used question
• a) Define the topic
• b) Choose a common question
• c) Convert it into a “good” question

12. Creation of “good” questions
Phase a)
Geometry
Phase b)
What is a
square?
Phase c)
What do you know about a square?

13. Creation of “good” questions
Phase a)
Phase b)
Phase c)
Geometry
What is a
square?
What do you know about a square?
Subtraction
731 – 256 =
Replace the digits so that the
difference of the two numbers is
between 100 and 200.

14. Creation of “good” questions
Phase a)
Phase b)
Phase c)
Geometry
What is a
square?
What do you know about a square?
Subtraction
731 – 256 =
Replace the digits so that the
difference of the two numbers is
between 100 and 200.

15. Example of “good” questions from
outside mathematics
Biology: How do crayfish walk?
Misconception: Backwards!
Method: careful observation of crayfish in a pellucid aquarium
Crayfish walk backwards when in danger, when they fear
something; but when at ease and when they look for food,
they walk forwards. And because crayfish when observed by
people in nature feels endangered, the myth that crayfish walk
(only) backwards was born.
Careful observation of crayfish in a pellucid aquarium can
uproot this misconception passed on in literature and folk
tales.

16. Example of “good” questions from
outside mathematics
Interdisciplinary situation
A “good” question in interdisciplinary situations is such a
question which combines both the above mentioned
approaches: mathematical modelling as well as observation of
real situation typical for natural sciences. The following
example illustrates this approach.

17. Example of “good” questions from
outside mathematics
Interdisciplinary situation
Closed question: A driver wanted to cover 200 km on a
motorway. Most of the way he was driving at the allowed
speed limit 130 km/h. However, he spent half an hour in a
tailback driving only 80 km/h. What was his average speed?
(The answer is: 115,5 km/h.)
“Good” question: You have an important meeting in a town
which is 200 km away. You can use a motorway where the
speed limit is 130 km/h. The meeting starts at 10 o’clock.
What time should you leave home to be at the meeting in
time?

18. Use of “good” questions in the
classroom
1. Posing a “good” question
• Not only posing the question, but also
verification that everybody understands.
• Students should have the opportunity to ask
the teacher e.g. what it means to answer the
question.
• Students’ task: to find the answer; the
teacher does not do it for them.

19. Use of “good” questions in the
classroom
1. Posing a “good” question - example
Two fifth (2/5) of students of the school visit daily the
school library. How many students can be in the school
and how many from them visit daily the library?
Recording on the table/distributing sheets with the
assignment …
Collective reading of the problem
Explanation of one or more students’ question (in their
language)
The teacher’s help does not contain any hint for the
solving process.

20. Use of “good” questions in the
classroom
2. Students are finding answers to the posed
question
•
Recommended form: group work
•
Too many students do not know how to start 
interrupt the work and discuss together
•
If it does not help  pose a simplified question
•
The teacher observes what students do but does not
intervene in their work
•
It is not necessary to wait till all groups find the
answer

21. Use of “good” questions in the
classroom
2. Students are finding answers to the posed
question – example
Two fifth (2/5) of students of the school visit daily the
school library. How many students can be in the school
and how many from them visit daily the library?
Simplification, e.g. other fractions (½, ¼, …)
Help, e.g. the use of buttons, a picture, …
Question for fast solvers, e.g. What happens when 3/5
instead of 2/5 of pupils visit the library?

22. Use of “good” questions in the
classroom
3. Whole class discussion
Groups present their solutions and explain why they
choose their solving procedure.
Each group summarizes the procedure, the teacher
records all answers of all groups on the blackboard
Recommendation: pose further questions similar to the
original one so that they can see that their procedure
is/is not applicable more generally

23. Use of “good” questions in the
classroom
3. Whole class discussion – example
Two fifth (2/5) of students of the school visit daily the school
library. How many students can be in the school and how many
from them visit daily the library?
Examples of answers:
• In the school, any number of students can be.
• In the school, there are 100 students, 40 of them visit the
library.
• The number of students in the school is a multiple of 5, e.g. 5,
10, 15, 20, … . 2, 4, 6, 8, … are visiting the library
Difference in the level of generality of proposed solutions, in its
level of real life perspective, …

24. Use of “good” questions in the
classroom
4. Summary
• If everything works as it should, some students
should be able to do the summary without the
teacher.
• The fact that a group presents the correct answer
does not mean that they understand everything.
Therefore it is useful to summarize important places
and explain them.
• Also here it is useful to assign students further
questions similar to the original one in order to show
them that their procedure is applicable more
generally.

25. Use of “good” questions in the
classroom
4. Summary – example
Two fifth (2/5) of students of the school visit daily the school
library. How many students can be in the school and how many
from them visit daily the library?
Similar question might be e.g. :
• In a survey it was found that ¾ of population like Beatles. How
many people answered the question and how many of them like
Beatles?

26. Use of “good” questions in the
classroom
Use of “good” questions in the classroom
Different course of the lesson
Other expectations of the teacher
Fast reaction to students’ proposals and
recognizing of correct answers
Managing the classroom discussion that
allows finding doubtful, incomplete etc.
answers

28. Role of the teacher
•
•
•
•
When starting the activity – instructions,
motivation
During the activity, the teacher does not offer
hints, students work on their own as much as
possible
Final discussions: The teacher should be a
moderator of the debate; assures that the
debate rests in the objective, factual domain
He/she does not decide about the correctness
but helps students learning to explain, use and
accept legitimate arguments and to defend
their statements

29. Role of the teacher
•
•
Final discussions, the teacher should be a
moderator of the debate; assures that the debate
rests in the objective, factual domain
He/she does not decide about the correctness but
helps students learning to explain, use and accept
legitimate arguments and to defend their
statements
The teacher may delegate these activities to a
student or a group of students

30. Examples of activities
• General, adaptable to any content
• Focusing on concrete thematic units

31. Is it true?
It is not possible to construct a triangle with the
lengths of sides 4, 5 and 9.
YES
NO
Do not know
Diagonals in each trapezium divides in two.
YES
NO
Do not know
A right-angled triangle cannot be equilateral.
YES
NO
Do not know
The area of a triangle depends only on its height.
YES
NO
Do not know
All face of a cuboid are always rectangular.
YES
NO
Do not know
The number 675 is divisible by 3.
YES
NO
Do not know
7 is a divisor of 504.
YES
NO
Do not know
The smallest prime is 1.
YES
NO
Do not know

32. Sprint
Every
square
has
just three sides of the same length.
At least one
rectangle
does not
have
just one right angle.
trapezium
No
perpendicular diagonals.
parallelogram
just two parallel sides.
kite
at least two sides of the same length.
perimeter inversely proportional to its
area.

33. Graph
Classroom activities (motivation)
What does this graph represent?
What do the axes represent? Which units may be used?

34. Graph – Possible follow-up activities
The house where Peter lives has the altitude exactly 300 m. In
the morning, Peter set out for a short excursion .
What does the axis x represent?
What do the data on the axis y represent? In which units?

36. Examples of games focusing on
communication and terminology





They may be used in any part of the lesson.
They may serve as a tool for motivation,
practicing, revision etc. but also for
“discovering”.
They can be easily adopted to the age, level
and learning styles of individual students.
All types of communication can be applied.
They can comply with various learning styles
of students and teachers.

37. Examples of games focusing on
communication and terminology

How many things can you think of
that...?
In groups, students try to think of and note down as
many things as they can that fit a given definition.
After two or three minutes, pool all the ideas on the
board, or have a competition to see which group can
think of the most items. The definitions might be:
... are rectangular,
... are round?
... you can divide into seven pieces?
... are smaller than zero?

38. Examples of games focusing on
communication and terminology

Blackboard Bingo
Write on the board 10 to 15 Maths items (e.g. numbers,
a simple equation, a triangle, ... ). Tell the students
to choose any five of them and write them down.
Then read out the items, one by one in any order, but
in a different way than they are written on the board
(e.g. instead of 4, you might say 16/4, instead of
2x + 1 = 0, you might say -1/2, and so on). If the
students have got the items you call out, they cross
them off. By shouting „Bingo“, they will tell you they
have crossed off all their five items.

39. Examples of games focusing on
communication and terminology

What’s the explanation?
This activity practices the properties of mathematical
objects. Describe to the students or show them on a
picture a property of a mathematical object. They
ask Yes/No questions in order to find out what is the
true explanation for the presented property.
Here are some examples of the situations:
 It holds for any positive numbers a, b, c, d: ½(a + b)  (ab).
 9 999 999 999 is not a square of a natural number.
 In the triangle ABC with the angles α, β, γ it holds:
cotg α cotg β + cotg β cotg γ + cotg γ cotg α = 1.
 It holds for the area S of a rhombus ABCD with the angle α
at the vertex A: S = a2 sin α.

40. Examples of games focusing on
communication and terminology

Noughts and Crosses
This game is played as the typical Noughts and Crosses game, i.e.
two teams, one has got noughts, the other crosses. The winner
is the team who first get the full line of their symbol (i.e. a line
of crosses or noughts).
The teacher prepares a square number of tasks/questions (e.g. 9,
the topic depends on what he/she wants to practise) and
draws a grid on the blackboard as the following one:
Each number corresponds to one task/question.
The teams take turns in picking up the numbers
of tasks/questions. If they answer the question
right, the teacher makes a cross/nought instead
of the number of the question/task. In this example,
the winner must get three noughts/crosses in the
line or on the diagonal.
1
2
3
4
5
6
7
8
9

41. Examples of games focusing on
communication and terminology

Noughts and Crosses
This game is played as the typical Noughts and Crosses game, i.e.
two teams, one has got noughts, the other crosses. The winner
is the team who first get the full line of their symbol (i.e. a line
of crosses or noughts).
The teacher prepares a square number of tasks/questions (e.g. 9,
2
3
the topic depends on what he/she wants 1 practise) and
to
draws a grid on the blackboard as the following one:
10
11
9
Each number corresponds to one task/question.
The teams take turns in picking up the numbers
8
of tasks/questions. If they answer the question 16
right, the teacher makes a cross/nought instead
of the number of the question/task. In this example,
15
the winner must get three noughts/crosses in the
line or on the diagonal.
7
171 18 2
19 3
24
20 12 4
4
5
6
23 22 21
7
14
6
8
9
13
5

42. Examples of games focusing on
communication and terminology

Unusual view
This activity could be designed as a lead-in when
starting the chapter of a 3D space. Draw familiar
objects from an unusual point of view. Ask the
students to identify it. Encourage different opinions.
The objects might be:

43. Examples of games focusing on
communication and terminology

Correcting mistakes
This activity could be used to identify and correct
mistakes and to encourage monitoring by students of
their own mistakes.
Write up a few problems on the blackboard that have
deliberate mistakes in them. If you wish, tell the
students in advance how many mistakes there are in
each sentence. With their help, correct them.

44. Some examples of possible problems:
67.98 - (63.7 - 9.72) x (-2) - 1 = 67.98 - (63.7 + 9.72) - 1 =
= 67.98 + 63.7 - 9.72 - 1 = 131.68 - 9.72 - 1 = 122.96 - 1= 121.96
Mark the invisible lines of this prism:
-42/25 - 5/25 : 1/12 - 6/12 = - 47/25 : (-5)/12 = -564/300

45. Examples of games focusing on
communication and terminology

Jumbled numbers
Choose a few numbers and write them up on the
blackboard.
E.g.
2
0
1
8
Then ask students to:
(an example)
Use the four digits to make a number which is more
than ... .
Use the four digits to make a number which is less
than ... .
Use the four digits to make a different number which
is less than ... .

46. Examples of games focusing on
communication and terminology

Brainstorm round a concept
Take a concept the class has recently learnt (e.g.
fractions), and ask the students to suggest all the
words and ideas they associate with it. Write each
suggestion on the blackboard with a line joining it to
the original word (you create something like a mind
map). This activity can be done as individual or pair
work. It can be used, for example, as an introduction
to the next topic, or for comparison, or it can just
help the teacher to realise how the students perceive
and understand the concept.

48. Geometrical puzzles


In the proposed activity the learners are asked to work in
pairs, one of them providing the other with a sequence of
instructions for the drawing of a geometrical figure. Both
learners are then asked to describe the figure and to define it.
The activity seems to correspond with a good way of dealing
with mathematical notions, introducing them through a nice
mixture of theoretical and practical activities.

49. Geometrical puzzles
The basic rules are:
 Learners work in twos.
 A member of each pair is given a piece of paper with the name
of a (plane or solid) geometrical figure that must be kept
secret from the partner until the end of the activity.
 The first learner provides the other with a sequence of
instructions how to draw the figure.
 Only unitary instructions that correspond with a single
graphical activity of the partner are allowed. For example, the
instruction “Draw a segment” is allowed, the instruction “Draw
the axis of the segment AB” is not allowed, because it requires
the determination of the middle point M of the segment AB
first, and the perpendicular in M to AB after.
 Each instruction given/received is written on a sheet of paper
by both learners.

50. Geometrical puzzles
The basic rules are:
 If necessary, an instruction can be repeated, but neither
modified nor explained.
 For the drawing pupils make use of a sheet of squared paper
and a pen (no pencil, ruler, compasses, etc.). No deletions are
allowed.
 The in-progress drawing cannot be shown.
 When the sequence of instructions has come to an end, the
final drawing is shown and compared with the name of the
given geometrical figure.
 Both learners are asked to give the name, the description and
finally the definition of the geometrical figure.
 Discussion in the whole class, rooted in the final drawings and
the given directions, concludes the activity.

51. Geometrical puzzles - aims
General
 The development of awareness and critical attitudes towards
the use of language and its interpretation.
 The awareness of how important it is to use specific and
unambiguous language.
 The increase of the learners’ capacity to understand and to
elaborate oral instructions.
 The stimulation of “critical” listening to instructions.
 The improvement of the ability of reading, understanding,
respecting and applying the rules of the didactical activity.
 The acquisition of the notion of simple instructions.
 The ability to respect the pace of the classmates.
 The capacity to state the reason for the choices made and used
during the activity.

52. Geometrical puzzles - aims
Mathematical









The improved usage of mathematical language.
The reinforcement of the knowledge of geometrical language.
The improvement of drawing skills.
The consolidation of geometrical knowledge.
The ability to visualize three-dimensional objects from their twodimensional representations and to represent solid figures on the
plane.
The capacity of describing basic plane and solid geometrical figures by
pointing out the properties necessary and sufficient to define them.
The development of the ability to find a proper balance between the
description and the definition of a plane or solid geometrical figure.
The awareness of the relevance of the definition in geometry.
The capacity to compare and evaluate different kinds of input from
the debate in the context of the correct construction of the concept of
geometrical figures.

53. Geometrical puzzles
Material
 Sheet of paper, pencil/pen, form for recording instructions
Assignments and questions for students
Hints for students
 Instructors
 Drawers

54. Geometrical puzzles – Instruction
sheet

55. Geometrical puzzles – Examples

56. Geometrical puzzles – Examples

57. Geometrical puzzles

60. Didactical situations


Didactical situation= an actor (e.g. the teacher) organizes a
plan of action with the intention of modifying or causing the
creation of some knowledge in another actor (a student)
A-didactical situation: partially liberated from direct
teacher’s interventions, enabling students to develop their
knowledge individually
Devolution
Situation
 of action
 of formulation
 of validation
Institutionalisation
A-didactical
situation
Didactical
situation

61. Group work
Important to choose
• instructions
• possibilities for using the activity
• organizational forms
…so that communication is developed