DATA ANALYSIS
5.6 TANGRAMS USED TO ATTAIN VAN HIELE LEVEL 3
This question was probed using the following question: "Can we call a square a rectangle?" Ninety percent of the learners answered this question in the affirmative. The one learner who answered differently did have a logical explanation. He stated that a square could not be a rectangle because a rectangle must have two long sides and two short sides, whereas in a square all the sides are of the same length. This brings to fore the debate about definitions between inclusive or exclusive and partitional and hierarchical definitions. According to De Villiers (1994), hierarchical is when the given definitions
provide for the inclusion of special cases, ego a parallelogram is defined so as to include squares, rhombi and rectangles, whereas partitional definitions, is when learners prefer to define a parallelogram as a quadrilateral with both pairs of opposite sides parallel, but not all angles or sides equal.
However, in a constructivist approach, learners are allowed to formulate their own definitions irrespective of whether they are partitional or hierarchical. Moreover, one can see that hierarchical definitions are more economical than partitional, and furthermore partitional, besides being longer will have to include additional properties to ensure the exclusion of special cases. An advantage of hierarchical definitions is that all theorems proved for that concept automatically apply to its special cases, for example: if we prove that the diagonals of a parallelogram bisect each other, we can immediately conclude that it is true for rectangles, rhombi and squares. If we classified them partitionally, we would have to prove separately in each case, for parallelograms, rhombi and squares (De Villiers, 1996b).
A second question was posed whether a rectangle was a square, and 90% of the learners answered "no" while one learner stated ",yes". On further questioning and enquiry the one learner, stated that two squares made one rectangle; therefore a rectangle has squares, and hence we could call a rectangle a square. Itwas clear from the answer that learners do not look at concepts holistically, but rather in parts. The following excerpt from an interview conducted represents justification of above:
RESEARCHER: Can we call a rectangle asquare~
LEARNER: Yes. Ifwe divide the rectangle exactly in the centre you get two squares, therefore the rectangle ismade up of squares and it can be called a square.
Although not necessarily correct, the learner actually believed that the longer side of the rectangle was twice the length of the shorter side.
Although only two questions probing the attainment of van Hiele level 3 was asked, the researcher regards this as being adequate proof that learners have attained van Hiele level 3, as this study concentrated only on squares and rectangles. The final interview question was to elicit from learners whether they have truly attained van Hiele level 3. The learners were required to complete following question: "a square is a rectangle " (complete).
This ultimately summarised the learners' ability to attain van Hiele level 3. This statement was correctly answered by 90% of the learners, further providing proof that they have attained the required level. This indicated that many learners have the ability of attaining level 3 if they are provided ample opportunities to explore and discover properties by using manipulatives. Yet again, there was evidence that these grade eight learners were able to grapple with their manipulatives to arrive at this complex deduction. The following are excerpts from interviews conducted:
LEARNER]: A square is a rectangle with all sides equal.
LEARNER2: A square is a rectangle with all sides equal in length.
LEARNER3: A square is a rectangle with all sides equal in length and diagonals equal in length.
LEARNER4: A square is a rectangle having no long and short sides but all equal sides.
LEARNER5: A square is a rectangle with smaller sides but equal sides as well.
LEARNER6: A square is a rectangle having all sides equal.
LEARNER7: A square is a rectangle having all sides equal in length.
LEARNER 8: Asquare is a rectangle because it has the same properties with one addition that the square has four equal sides.
LEARNER 9: Asquare is a rectangle but with all sides equal.
The questions relating to the definition of the rectangle and the square were not precisely answered as most learners gave all the properties of the two figures instead of giving a clear and concise description of both figures. The following are excerpts from interviews conducted:
RESEARCHER: Can you define a squarefor me?
LEARNER 3:
LEARNER 6:
LEARNER 2:
LEARNER 1: A square has four equal sides andfour 90· angles. When the diagonals cut each other 90· angles are formed around the centre.
When the diagonal cuts the square two equal triangles are formed.
A square is a shape that has four 90° angles and all sides are equal.
A square is a shape that has 4equal sides and4corner angles are 90· .
A square has all sides equal and each angle equal to 90°.
A square has four equal sides and four equal angles.
A squareisa diagram/object which has 4,90° angles and 4 equal sides.
LEARNER 7: A square must have all the sides equal and each angle must be LEARNER 4:
LEARNERS:
equal to90° .
LEARNER 8: A square has four equal sides andfour 90° angles.
LEARNER 9: A square has four equal sides andfour angles of 90° each.
LEARNER 10: A squareisafour-sidedfigure with equal sides and equal angles.
The following are excerpts from the researcher's interview relating to the definition of a rectangle:
RESEARCHER: Can you define a rectangle?
LEARNER 1: A rectangle is a shape with opposite sides that are equal.
All angles are equal to 90°. A rectangle is not a square.
LEARNER2: A rectangle is a shape that has four 90° angles and opposite sides equal.
LEARNER3: A rectangle is a stretched square with opposite sides the same length but all4sides don't have the same length, but each angle must be 90°.
LEARNER4: A rectangle has opposite sides equal and each angle equal to 90°.
LEARNER5: A rectangle is a figure with4equal 90°angles and parallel sides are equal.
LEARNER6: A rectangle is an object, which has two equal long sides and two equal short sides and all 90° angles.
LEARNER7: A rectangle has all the angles equal to 90° and2short and2long equal sides.
LEARNER8: A rectangle has2long equal sides and2short equal sides with each angle equal to 90° .
LEARNER9: A rectangle is a quadrilateral shape with opposite sides equal and each angle 90° .
LEARNER 10: A rectangle is a four-Sided figure with the parallel sides equal and each angle measuring 90°.
These learners' definitions were irrelevant to this research, and therefore it did not necessitate the intervention of the researcher. However, it must be acknowledged that the activity could have provided a useful platform for the establishment of concise definitions.