1. 12th
grade pupils’ understandings of the
distinction between function and relation.
Andonis Zagorianakos, Panagiotis Spyrou
Department of Mathematics, University of Athens, Greece
PME34, Belo Horizonte, Brazil

2. The aim of the study is to explore the 12th
grade pupils’ awareness
of the distinction between the concepts of function and arbitrary relation.
The main research question investigates
the awareness of the “discrimination between
the concept of function and relation”
(Sierpinska 1992, U(f)-12).
A secondary research question linked to the main question is concerned with
the “discrimination between the dependent and the independent variables”
by the 12th
grade pupils
( Sierpinska 1992, U(f)-5 )

3. The awareness of
the “discrimination between the concept of function and relation”
(Sierpinska 1992, U(f)-12).
The “discrimination between
the dependent and the independent variables”
( Sierpinska 1992, U(f)-5 )
Exploring the 12th
grade pupils’ awareness
of the distinction between the concepts of function and arbitrary relation.
Exploring the 12th
grade pupils’ awareness
of the distinction between the concepts of function and arbitrary relation.
The aim of the study
M
ain
Secondary

4.  …conceived the concept of the
rectangular coordinate system
 …investigated natural laws that describe
one quantity as dependent on another.
Nicole Oresme
Dirichlet
“if a variable y is so related to a variable x that
whenever a numerical value is assigned to x
there is a rule according to which a unique value of y
is determined, then y is said to be a function of the
independent variable x”
14th
century
1837
Bernoulli, Euler, Cauchy

5. The notion of function sprouted out of the relational ‘‘universe’’,
as a means to describe and measure natural phenomena.
The function definition states the necessity of the dependent
coordinate being uniquely determined and not always the inverse.
The Dirichlet definition expressed precisely, for the first time,
the insight of a mediated measure in the concept of function:
To estimate the dependent variable y, and to achieve it
although there is no immediate access to y, is to measure it through x.
Therefore the independent variable is the mediating variable which gives access to
the dependent variable, resulting in the priority of the latter.

6. Having a long experience of teaching at the Greek secondary education
environment, the researcher was aware of the fact that the teaching of
function during the first years of school:
 is oriented towards a common perception of function
 there is a dominant use of one-to-one functions
 there is a focus on the relational aspect of biunique functions

7. research questions
[1] Can the pupils recognize the difference between the concepts of
function and of an arbitrary relation?
(1st, 2nd
and 3rd question of the questionnaire)
[2] Can the pupils distinguish the order of the variables x and y,
and the asymmetry that they have?
(3rd, 4th question of the questionnaire)

8. x y
-1 0
0 1
1 2
2 3
3 4
x y
-3 2
-2 2
-1 2
0 2
1 2
x y
5 3
-3 2
5 1
0 1
-3 6
The Questionnaire given to the pupils:
❶ Which of the following relations are function relations? Make the necessary
corrections to the rest of them, in order to transform them into functions.
❷ Give two examples of arbitrary relations that are not functions
(one described graphically and one analytically (with an algebraic formula)).
❸ What happens to the graphical representations of the following functions if the line on
the graph is rotated by 90°? Are they still functions? Give a short justification in your
answer.
❹ In which situation(s) does a 90° turn of a function’s graphical representation
represents a function? Which general rule would you use?
Phase 1

9. Phase 2 of the research took place after one week and included 10 semi-structured
interviews, which were selected according to the variety of pupils’ responses to the
research questions.
The results from the questionnaires and the interviews confirmed the problematic
areas anticipated at the outset of the research:
resultsresults
It is evident that the pupils experienced difficulties in answering all four questions.
Most of the correct answers were based on stereotypical examples.
The research revealed the extended use of mnemonic rules, which were detached
from the condition that they were supposed to serve.
In the interviews it is more obvious how the VL-test substitutes for the direct
application of the definition.
The questionnaires were distributed to 19 pupils of a typical secondary school and a
particular class was chosen because mathematics is a main course in their curriculum.

10. DiscussionDiscussion
Beyond the evidence that the pupils have serious problems recognizing the difference
between the concepts of function and of an arbitrary relation, the research has also
revealed the extended use of mnemonic rules, which were detached from the condition
that they were supposed to serve.
The necessity of such mnemonic rules emerges from a teaching of function that is
detached from any pragmatic reference and presents it as a symbolic convention.
As an effect the claims of the definition appear as disconnected components,
thus needing mnemonic rules in order to be retained.

11. We suggest that a possible answer for the teaching of the concept is the association
with certain activities, where the independent variable is pursued and used as a means
for the estimation of the target, namely the dependent variable.
The concept of function as action would unify the components of the definition
and establish them as necessities.

12. “Concerning definitions: Introduction of the concept of function
set–theoretically as a particular kind of relation is little justified both from
didactical and epistemological points of view. Informal definitions resembling
that of Dirichlet are sufficient at the secondary level. On higher levels, when
the notion of relation is studied for its own sake, Peano’s definition may be
discussed and the students should be brought to discriminate between the
roles and meanings of the concepts of relation and function in mathematics.”
It is in this sense that we perceive Sierpinska’s didactical conclusion:

13. Thank you
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