Measuring student knowledge and skills

Examples from Competency Class 2
You have driven two thirds of the distance in your car. You started with a full fuel tank and
your tank is now one quarter full. Do you have a problem?
Mary lives two kilometres from school, Martin five. How far do Mary and Martin live from
each other?
Figure 3.
Examples from Competency Class 2
You have driven two thirds of the distance in your car. You started with a full fuel tank and
your tank is now one quarter full. Do you have a problem?
Mary lives two kilometres from school, Martin five. How far do Mary and Martin live from
each other?

---

Measuring Student Knowledge and Skills
46
OECD 1999
assessment of students’ responses to such items are very difficult. However, as this class forms a crucial
part of mathematical literacy, as defined in OECD/PISA, an effort has been made to include it in the
assessment, even though with only limited coverage.
An example problem from Class 3 is given in Figure 4.
“Mathematisation”
Mathematisation, as it is used in OECD/PISA, refers to the organisation of perceived reality through
the use of mathematical ideas and concepts. It is the organising activity according to which acquired
knowledge and skills are used to discover unknown regularities, relationships and structures (Treffers
and Goffree, 1985). This process is sometimes called horizontal mathematisation (Treffers, 1986). It requires
activities such as:
– identifying the specific mathematics in a general context;
– schematising;
– formulating and visualising a problem;
– discovering relationships and regularities; and
– recognising similarities between different problems (de Lange, 1987).
As soon as the problem has been transformed into a mathematical problem, it can be resolved with
mathematical tools. That is, mathematical tools can be applied to manipulate and refine the mathemat-
ically modelled real-world problem. This process is referred to as vertical mathematisation and can be rec-
ognised in the following activities:
– representing a relationship by means of a formula;
– proving regularities;
– refining and adjusting models;
– combining and integrating models; and
– generalising.
Some fish were introduced to a waterway. The graph shows a model of the growth in the combined weight
of fish in the waterway.
Suppose a fisherman plans to wait a number of years and then start catching fish from the waterway. How
many years should the fisherman wait if he or she wishes to maximise the number of fish he or she can
catch annually from that year on? Provide an argument to support your answer.
kg
100 000
20 000
40 000
60 000
80 000
0
9
1
2
3
4
5
6
7
8
Years
Figure 4.