A new approach to the assessment of children's mathematical competence

The report is a translation of part of a Swedish paper entitled “On Students' Achievement in Mathematics after Finishing Comprehensive School”. The intention of the investigation was to

- diagnose the retention of some basic skills in some topics in algebra and geometry,

- reveal difficult steps in the learning processes in these topics.

Starting with a complicated question, e.g. the equation \(\frac{{3x - 2}}{2} = \frac{x}{3}\) a sequence consisting of 5–15 problems were constructed. Each new problem followed by the preceding one by taking away one or two details. \(\begin{gathered} 3(3x - 2) = 2x \\ {\text{ 9}}x - 6 = 2x \\ {\text{ 7}}x - 6 = 0 \\ {\text{ 7}}x = 6 \\ \end{gathered} \) is an example of a sequence belonging to the equation above. From about 10 complicated problems (“top-items”) and their sequences, in all 130 items, 10 sub-tests were put together in such a way that the pupils who took the test were not aware of the sequences but found no connection between the problems. Many surprising results were found, e.g., that the students scored higher on 14/(x+2)=2 than on 4/x=3, that the difficulty in finding the area of a triangle depended on the position of the triangle and that the problems “Simplify a/a ², a²/a, a/a” were of quite unequal difficulty. A discussion about the students' thinking in patterns and mechanically learning ends the report.