Measuring student knowledge and skills

Examples from Competency Class 3
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.
Examples from Competency Class 3

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Mathematical Literacy
47
OECD 1999
Thus the process of mathematisation occurs in two different phases: horizontal mathematisation,
which is the process of translating the real world into the mathematical world, and vertical mathematisa-
tion, that is, working on a problem within the mathematical world and using mathematical tools in order
to solve the problem. Reflecting on the solution with respect to the original problem is an essential step
in the process of mathematisation that seldom receives adequate attention.
One can argue that mathematisation occurs in all competency classes because, in any contextualised
problem, one needs to identify the relevant mathematics. However, in OECD/PISA, the kind of mathema-
tisation that is required in Competency Class 3 is of particular importance. It is that form of mathematisa-
tion that goes beyond the mere recognition of well-known problems.
The differing complexity of mathematisation is reflected in the two examples below. Both are meant
for students of 13 to 15 years of age and both draw upon similar mathematical concepts. The first requires
simple mathematisation and the second requires more complex mathematisation.
The item in Figure 5 is an example of a Competency Class 2 item because it requires relatively low
levels of mathematisation.
The mathematisation required in the example in Figure 6 is from Competency Class 3 as it requires
the student to recognise the relevant mathematics and to develop and then communicate a mathemati-
cal argument.