Communication in Mathematics Lessons
Wolfram Eid, Otto-von-Guericke-University Magdeburg
wolfram.eid@ovgu.de
Abstract
Communication is one of the famoust people’s cultural techniques. It exists in several kinds. And oral outcomes could be created in different ways and kinds, too.
There are outcomes with descriptive character, but also other with inzitative or inventive character. The both last are directed for initializing comments and reactions
by the participants of communication. The first does not do so.
But these various aspects are not found in mathematics lessons in generality. Pupils’ oral outcomes are often rarely and reduced of single words or fragments of
sentences. Communication with a descriptive character has a majority all over the
complete processes of communication in mathematics teaching.
That is the reason to think about possibilties for intensivation processes of communication in mathematics teaching directed espacially to processes with inzitative or
inventive character. One important part of mathematics lessons is the process of
mathematical modelling. It is a very complex process and pupils should be able for
communication with other pupils if they like to solve the complete process successful. In the report the descriptive, inzitative and inventive aspects of communication
shall be described with the help of examples. Also potencies for communication
shall be shown for the process of mathematical modelling on the one hand. Possibilities for teaching mathematical modelling in lessons of mathematics with the aim
of initialing processes of communication between pupils shall are described on the
other hand. The potencies shall are focused on special kinds of working as heuristic-experimental working, object orientad modelling and using analogies espacially
for geometrical contents and backgrounds.
The Character of Communication
Mathematics and its applications are using mathematical knowledge for modelling
in mathematical areas at most for mathematics teaching in schools. Wishes and
aims have been documented in several kinds already in the last years but the existing situation is not satisfying enough in generality. One reason of this unfortunate
situation might be the connection between modelling and language. Modelling
makes it necessary to write down spoken expressions of ideas and thinking processes for instance. Oral communication could be realized for instance by
formulating and explaining rules
giving reasons for ... (a solution, a step for solving a problem, ... )
describing and annotating ways of solution
assessing of Statements
reviewing a process of solving a problem or task and summarizing knowledge.
But the existing situation has to be described as similar bad as for the aspect of
modelling. Some important problems are the following e.g.:
Teachers call upon pupils not enough to speak loudly.
Productive speaking (especially speaking in coherent sentences) is not stimulated enough.
Different teaching methods are rarely as stimulant to speak.
Book-tasks, which could have a benefical effect on intellectual and creative work
are used only sometimes (standard-tasks are used very often).
Teachers often do not give enough time for discussions of different ways or
methods to get a solution of a task or of wrong solutions in their lessons.
Tasks to describe mathematical facts with own words (and reverse) by pupils
are very rarely.
Communication media (natural language, written assembly language, drawings,
pattern) are used only spontaneously.
Pupils like to speak in lessons with a minimum of words. So their sentences are
often not correct in the natural language and without aesthetics. Problems in
notification are so programed.
That is the reason to think about connections between language and modelling and
to find possibilities for stimulating pupils' communication in the class-room. To do
so it is necessary to make clear the subject of language. A lot of definitions are
existing about this subject. Three aspects could be focussed out of all of them:
Language is a system of oral or nonoral signs (the lexis) with some rules of
connections between them (the grammar) and semantic contents (linguistic aspect).
Language could mean also to use the instrument "language" to tell something
(psychological aspect).
Language also specified alienated activity of people to change information (social aspect).
For teaching, especially mathematical teaching the second aspect is the most important one. It is determined by the system of the used language (natural language,
mathematical language, symbols, etc.) on the one hand and by the orientation of
the aim of the specified communication on the other hand.
Four in their character different categories of methods for communication could be
distinguished; components written in italic have an espacially significance for mathematics teaching:
descriptive character
inventive character
giving a talk reporting
summarizing explaining exposing disproving
informing
describing
giving reasons
arguing
generalizing
defining
discovering
proving
drawing conclusions
claiming
-
inzitative character
wishing
ordering
demanding inatructing
supposing asking
contactive character
asking
greating
wishing
congratulating
But observations in lessons of mathematics have been shown descriptive communication is the mostly used kind of communication and it seems language or communication becomes poor by reducing to single words or parts of sentences at
most. (An analysis of the communication of pupils in 20 random selected lessons
of mathematics has shown that near the half of all oral outcoms by pupils have a
descriptive character and that near 80 % are only single words or fragments of
sentences.)
That is the reason for thinking about kinds of teaching stimulating non-descriptive
components of language and to find out possibilities of mathematical modelling for
it. A view about the complete process of mathematical modelling (compare figure
1) shows different phases in the process defined by different dominated components of communication. (In addition to the given dominating components other
components have also an influence to the communication, but even a smaller one,
and transitions are possible, of course).
phase of mathematical
application
task
applications
(problem)
phase of heuristical
moments
algorithms
theoremes
phase to reach exactness
supposition
terms
process of mathematical modelling inwards mathematics (/1/)
main component(s) of communication
giving the problem
descriptive components
finding a way for solving the problem inzitative components
the developing-algorithm-phase
inzitative/inventive components
the phase of reviewing
descriptive components
phase
1
2
3
4
Initiating Communication by Heuristic Elements
In addition it is thinkable to stimulate communication between pupils with a spezified character by selecting one of the described phases above. For instance selecting the phase of heuristical moments and creating corresponding tasks for the
teaching of mathematics it should be possible to stimulate communication with an
inzitative character. A simple kind to do so could be an intuitive-heuristic working
for solving a problem instead of solving with the help of algorithms. The difference
between both kinds shall be shown with the help of a task about similar triangles
below.
Given the areas (in square inches) of two similar triangles (F1 = 6,4; F2 = 2,4)
and the side a1 = 4 inches. Calculate the altitude h1 in this triangle and the
corresponding quantities in the other triangle!
Is the construction of both triangles possible by the given dates? /2/
solving about algorithmitical working as usually
2F
a
h
8
h1  1 , F1  F2 '  k 2  F2  k 
, a2  1 , h2  1
a1
3
k
k
solving about intuitive-heuristic working
I. F1 = 6,4 sqin; a1 = 4 in 
F1 : F2 = 8:3 =
II.
III.
a2
h2
a1 : a2
h1 : h2
h1 = 3,2 in
F1 8 12,8 sqin
 
F2 3
a2  h2
2
2,4
2
4:3
2 in < a2 < 3 in
2,5 ▼
a2
1
4,8
4
4,8
F2 = 2,4 sqin
a 2  h 2 = 4,8 sqin
3
1,6
4:3
2

a2, h2  
(1,6 in < h1 < 2,4 in)
2,4
▲ 2,45
h2
1,92 ▲
2
▼ 1,96
al : a2
1,6
1,67
1,632...
hi : h2
1,67
1,6
1,632...
6
,
More strictly directed to the developing of faculties
in heuristical thinking and car7
6
rying out inzitative dominated processes of communication by pupils is using heuristical strategies for working in mathematics lessons for instance the strategy of
using analogies. Some ideas about that have been written down in /3/.
Another way could be directed by the kind of working in sciences (for instance
physics, chemistry) that are using the experiment to get new discoveries. One typical example to do so is the following given sequence of tasks:
task 1: Reconstruct the following experiment with a calculator and give reasons for
the results to see!
Input
Display
17,3 °
17.3
SIN
2.9737-01
INV
2.9737-01
SIN
17. 3
task 2: What do you think about the result in the next example?
Input
Display
154°
154
SIN
4.3837-01
INV
4.3837-01
SIN
.......
Test your supposition with the calculator and complete: sin 154° = sin ……….!
task 3: Give other pairs of angles using the same sequence of steps by a calculator!
task 4: Give a general supposition about a connection between data of the sinus
function in the I. and II. quadrant.
Last but not least teaching mathematics could be directed very
strongly by the kind of working in
sciences using hypothesises on
the way to new discoveries. The
principle idea and a practical example of that shall be shown on
the following two pages. /5/
What's the angle between a cube's diagonal and its ground-area?
1. The pupils give a hypothesis after
they have read the task and thought
about it.
„ The angel is the same in
all kinds of cubes."
2. The pupils try to find some ideas to
overcome the problem. Calculations in triangles are suggested
about the made
drawings.
d
a

stimulating communication;
Other pupils are going to think about
the given hypothesis, make some
drawings and so on.
communication with discussions;
Pupils like to give reasons for their
ideas and ask the others about the
correctness of their ideas, too.
stimulating communication;
Some pupils could have the idea to
use the Pythagoras' theorem. So
other pupils give their thoughts on the
inputs of the task, for example Are
there enough data in the task to overcome the problem in this way?
3. The pupils give statements for experimental tests.
„Cubes with edges of 4 cm, 5 ft.,
25 in. ... have the same angles."
stimulating communication;
4. The pupils carry out the experiments.
describing communication;
Pupils describe their procedures for
the experiments.
establishing communication;
Pupils establish the truth of the statements.
5. Pupils get and test a solution.
The tested statements are true.
The hypothesis could be so, too.
6. Generalization of the solution by
means of variables. Pupils also
think and speak about the used
strategy.
(new actions are triggered off)
communication
with inventive
character:
The pupils give general reasons for
the truth of the hypothesis:
tan  
a
2a 2

1
2
2
Demonstration-orientated versus Object-orientated Teaching
The phase to reach exactness of the process of modelling is dominated by communication with inzitative or inventive character. If the teacher tries to dominate
these components he should think about the kind of modelling. A view to lessons
in mathematics shows a situation as follows in generality: An algorithm or a calculus has been developed or given by the teacher and after that it is used to solve
series of several and different kinds of tasks (method-orientated teaching). The
calculus of curve-tracing is a typical example for it. Another possibility of modelling
could be the examination of one object with several methods, a more objectorientated modelling (compare the following two figures).
Characteristics of mathematical object-orientated modelling shown at he example of
geometrical objects (compare also /4/)
description with
developing new methods of mathematical
the aim
the help of
modelling
getting to know new
well-known
and unknown (geometrical) objects
methods
using new methods of
(single objects or
ma-thematical modelclasses of obling to get to know
jects)
more about new objects
symmetrical
examining
transforming
and completing
zooming
filling up
working with
coordinates
(exhaustionsmethod)
The idea of this kind of modelling shall become clear by the example of modelling
a formula for the area of triangles. The triangle is a geometrical object and for that
we could use the different kinds of examining.
For the examination of the object "triangle" four important different kinds are existing:
modelling about completing triangles to other geometrical figures
modelling about transforming triangles in other geometrical figures
modelling about counting up standard squares for rectangled triangles (a
special kind of the method of exhaustions)
modelling about infinitesimal working and thinking.
Using the last called kind above the principal way of thinking and teaching the
mathematical content is shown in the following figure.
P3
P3
P2
P3
P3
P1
P2
first step:
(1)
A= AP1 + 2AD1
(2)
A= AP1 + 2AP2 + 4AD2
(3)
A= AP1 + 2AP2 + 4AP3 +8AD3
(4) A= AP1 + 2AP2 + 4AP3 +8AP4 +16AD4
second step:
side
P1
P2
½g
¼g
P3
1
/8 g
P4
1
/16 g
altitude
third step:
½h
1
¼h
/8 h
1
/16 h
(1) A= ½ g ·½ h + 2·AD1 = ¼ g·h + 2·AD1
(2) A= ¼ g·h + 2·¼ g·¼ h + 4·AD2 = 3/8 g·h + 4· AD2
(3) A= 3/8 g·h + 4· 1/8 g· 1/8 h + 8·AD3 = 7/16 g·h + 8·A D3
(4) A= 7/16 g·h + 8· 1/16 g· 1/16 h + 16· A D4 = 15/32 g·h + 16·A D4
∶
A= ½ g ·h
After finding the result above it could be possible to take
the cutted parallelograms together as shown beside.
The parallelograms are each a part of the given triangle.
Their areas are a part of the complete area of the triangle
and they all together complete the area of the triangle as
better as growing up the number of parallelograms. With it the triangle could be
formed to a parallelogram with the same altitude as the given triangle and the half
of the ground side. The born idea is to transform the given figure in another figure
(with the same area) for that pupils the value of the area could calculate (perheps
easy) already. So the pupils could examine the triangle with two methods at least
- the method of exhaustion and the method of transforming. And there a analogy
to later contents of mathematics is possible: calculating the volume of y pyramid
(compare /4/).
Further Aspects
In addition to the described ideas initiating many and diverse communication by
the process of mathematical modelling is to remember that this aim could be realized also on other ways without an explizit modelling character. Viewing the natural
language grammatical aspects could determine the process of teaching. Some aspects have been written down below.
grammatical aspect
chosen textstructure
kind of relation between
the given data and
those that have to be
found
linguistic levels that
have been used
But viewing mathematics teaching so called "open tasks" could initiate many and
diverse communication, too. The potency of communication for this kind of tasks
will become clear by solving
the given example.
What is the value
of A und B?
References
/1/ HEUGL, Helmut (1998). Computeralgebrasysteme - das gelobte Land des
Mathematikunterrichts? Klagenfurter Beiträge zur Didaktik der Mathematik, In
Mathematische Bildung und neue Technologien (reports, 8. Intern. Symp. Didaktik
der Mathematik, Klagenfurt 1998), 127-146. Teubner, 1999. HB: Bb5047.
/2/ PIETZSCH, G. a.O.(1986): Mathematics, Vol. 8, Berlin 1986: People and
Knowledge Publishing House
/3/ EID, Wolfram (2007): "Geometrical analogies in mathematics lessons"; Teaching Mathematics and its Applications 10.1093(2007) teamat/hrl022
/4/ EID, Wolfram (1997): "Object-orientated modelling in geometry"; lnt. J. Math.
Educ. Sci. Techn. 28(1997)4, S. 473 – 479
/5/ EID, Wolfram (1997): "Verbalizirag ideas in mathematics teaching"; Int. J.
Math. Educ. Sci. Techn. 28(1997)2, S. 161 - 183