Counter:

RELATIONSHIPS AND FUNCTIONS

Marie Kubínová & Na?a Stehlíková

The concept of relationship (function) and the connected concepts certainly belong to the primary school curriculum and general education. This fact will probably not be doubted. However, what we have to think about is the transformation of new views about the use of functions in solutions to problems in science (borderline disciplines, mathematical modelling, etc.) and society (economic actions, data elaboration, interpretation of information from various sources, etc.), along with the didactic interpretation of functions in school educational programmes and textbooks.

The approach we would like to promote in this text is the genetic approach enriched by new views:

The development of the concept of function in the history of mathematics is observed and connections to real life are introduced;
The approach is used in the teaching of mathematics beginning with the first grades of primary school – function is presented as a relationship or regularity there;
The concept of function is built gradually by work with a large number of isolated models of function (Hejný, 2003);
Function is defined as ‘matching’ or ‘prescription’;
The ability to read properties of relationships or functions from their graphs or diagrams and to construct the graph of function (or diagram) without the use of formal mathematical analysis (including limit, continuity, derivatives, etc.) and with the knowledge of the properties of functions which can be found with the help of calculators or mathematical software is emphasised. Pupils need a strong conceptual understanding of function as well as procedural fluency.

The starting point of our approach in this text is that the definition of the concept of function is the final, not the beginning phase of the concept creation process. To ‘know the concept’ is something different to ‘knowing the definition of the concept’. At the primary school, it is sufficient if most concepts from the area of functional thinking are known intuitively. On the other hand, we want to prepare in a propaedeutic manner some secondary mathematics concepts such as function, graph of function, extremes of function, sequence, infinitely small, infinitely big, limit, continuity, etc.

If we consider the ontogenesis of the concept of function as its shortened phylogenesis in the history of mathematics, it follows that the period of building it in the pupil’s knowledge structure must be very long. That is why we suggest preparing the concept of function very slowly with marked connections to other subjects to strengthen the importance and use of functions in modelling different situations and phenomena which appear in pupils’ world or which they will meet in their future life.

The tasks in this Unit are mainly aimed at the development of the following pupils’ abilities:

to observe the world around them and describe it in mathematical terms (to the extent given by the level of their mathematics) – that is gather, record and interpret data;
to use different tools to describe real-life and mathematical phenomena (tables, graphs, diagrams, schemes, pictures, algebraic symbols, verbal descriptions – first using their own vocabulary, later also the mathematical vocabulary);
to understand that each of these representations describes how the value of one variable is determined by the value of another (in other words, that they are different ways of describing the same relationship);
to fluently move among the different representations and realise their advantages and disadvantages;
to ‘discover’ (construct) mathematical concepts (e.g., relationship, extreme), their properties (e.g., periodic, continuous, monotonous, discrete) and relations between them (e.g., direct proportion as a special case of linear function, relationship between a table and a graph of a certain function), procedures (e.g., how to get information about the shape of a function, how to use the graph of direct proportion for drawing a graph of linear function).

During the work on the IIATM project, we elaborated and trialled different kinds of problems. To make their presentation here more transparent, we divided them into four Subunits (the division of problems into Subunits is not clear-cut as the Subunits overlap):

Natural Observations which include problems in which pupils are asked to observe natural phenomena (such as temperature, measures of their bodies) and to table and plot the data;
Planning a Trip is a topic field in which the data are given to pupils and they should interpret them;
Real-Life Problems concern actual graphs of functions of some real-life phenomena;
Revision of Functions includes some suggestions how to diagnose pupils’ knowledge of functions.

The division of the Unit into the four Subunits can also be interpreted in terms of the paths

from reality (pupils’ experience) ® to mathematics and

from topic oriented language ® to mathematical language.

Subunit 1 is the closest to other subjects such as biology, physics and geography. Its topics are to be dealt with in other subjects and not only just in mathematics. Mathematical models originating in this Subunit contribute to pupils’ understanding of the world around them. That is, the data, tables and graphs elaborated here bring new information for pupils. We use reality but not real-life problems – how a child grows or how days and nights change are not problems to be solved but phenomena to be observed. In Subunit 2, functions (graphs) get into the foreground. Pupils learn to interpret graphs. In Subunit 3, real-life contexts are taken in order to elicit basic properties of functions and graphs of functions. In Subunit 4, mathematics itself and functions in particular come to the fore without any (apparent) connection to real life.

TOP

Charles University Prague, The Faculty of Education
Department of Mathematics and Mathematics Education