6.3.1 Van Hiele levels according to Questionnaire A and Questionnaire B
The van Hiele levels of geometric thinking are regarded as a good descriptor of current performance in geometry and a reasonably good descriptor of future performance (Usiskin, 1982). This section will look at the van Hiele levels of the learners as revealed by the analysis.
The data analysis of this study showed that the majority of the grade 10 and grade 11 learners who participated in this study were operating at the visual, analysis and informal deduction levels (refer to Table 5.4, Table 5.13 and Table 5.15).
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Van Hiele Level Percentage of Learners Operating at a Particular Level Questionnaire A Questionnaire B
Visual Level 16% 34%
Analysis Level 52% 27%
Informal Deduction Level 31% 38%
Formal Deduction Level 1% 1%
Rigour 0% 0%
The table shows that the majority of the learners were operating at the first three van Hiele levels, with only 1% of the learners operating at the formal deduction level.
The data analysis also revealed that for learners to reach a higher van Hiele level they must first pass through the lower (preceding) levels, hence the criteria used to place a learner at a higher van Hiele level require that the learner first meet the requirements for placement at the lower levels. From Table 6.1 above, it can be deduced that of the 84 % of learners who reached the analysis level in Questionnaire A, 32% have proceeded to the informal deduction level and 1% managed to reach the formal deduction level. Similarly for Questionnaire B, of the 66% of learners who reached the analysis level, 39% have proceeded to the informal deduction level and 1% has reached the formal deduction level.
It was disturbing to see that many grade 10 and 11 learners were still operating at the visual level after spending years in primary school. These learners have not moved beyond the recognition of shapes and mentioning of properties and showed no evidence of knowing how the properties are connected. These findings concur with the findings of the studies by Siyepu (2005), and Atebe & Schafer (2011), whose studies indicated that the majority of the learners were found to be operating at the pre-recognition level and that a very small number of the students operated at the van Hiele level 2. This is consistent with research done by Mateya (2008) who carried out a study on grade 12 students and found that out of 50 students who participated in the study, 19 (38%) were at the pre-cognition level, 11 (22%) were at van Hiele level 1, 13 (26%) at the van Hiele level 2 and 4 (8%) were at the van Hiele level 3.
Similarly Senk (1989) and Usiskin (1982), indicated that many secondary school learners are on the visual or analysis levels of the van Hiele levels.
The findings that most of the learners were operating at the visual and analysis level mean that most of the learners involved in this study were operating at levels of geometry thinking which are lower than the mathematics curriculum requirements. Usiskin (1982) mentioned that the van Hiele level 3 is a “guidepost” for the learner to be able to master the art of proof in geometry. According to Usiskin (1982), many learners were found to be unfamiliar with basic terminology or geometric ideas even after many years of geometry lessons.
The study also revealed that the ability of the learners to work with problems at the van Hiele levels 3 and 4 is still a challenge as only a few were able to show competence at these levels.
This was evident from the learners’ responses to questions which involved proofs or needed
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learners to carry out more than two steps. The results showed that the majority of learners found it difficult to use deductive reasoning to prove that angles were equal, that triangles were congruent and that a shape was a parallelogram in question 6 for Questionnaire B. To be able to cope with the demands of the axiomatic system and proofs as required by the
curriculum, learners need to be on the ordering level of the van Hiele levels. To be at the ordering level learners must have passed through the analysis and visual levels.
The analysis of the results also revealed that sometimes learners have some degree of
competence but not much, for example, a learner was able to show that the shape in question 4 in Questionnaire B was a parallelogram but failed to prove that the shape in question 5 was a square. Yet both questions required thinking at the same van Hiele level. This shows that reasoning at a van Hiele level is not static but constantly developing; hence a learner can show some competence at a level but still struggle with other aspects at the same level. It must be noted that the van Hiele levels are not there to make judgements but are there to help diagnose barriers to progression that teachers need to address to their learners in order to improve. Some of the reasons why Van Hiele levels are important are discussed in the next section.
6.3.2 Differences in van Hiele levels according to Questionnaire A and Questionnaire B Table 6.1 above, showed that there were differences in the van Hiele levels of the FET learners according to Questionnaire A and Questionnaire B, even though some of the differences were minor. Sixteen percent of the learners were at the visual level for Questionnaire A while 34% of the learners were at the visual level for questionnaire B.
Questionnaire A consisted of multiple choice items, which means that when learners look at the given options, it is easier for them to compare and contrast. This also helps them to remember the correct option. However sometimes learners could have simply guessed the answer as there is a 20% chance of getting it right for those items with 5 options and a 25%
chance of getting it right for those items with 4 options. This was different with
Questionnaire B where learners were expected to work out their own answers and could leave questions unanswered if they didn’t know the answers.
Fifty two percent of the learners were at the analysis level for Questionnaire A and 27 % were at the analysis level for Questionnaire B. As was explained in the visual level, those learners who managed to guess correct answers in Questionnaire A were placed at the analysis level in the first instrument, hence making the number of learners operating at the analysis level greater than that at the analysis level for Questionnaire B. It became more difficult to guess the correct answer at the informal deduction level as the choices given required learners to think and reason at the informal deduction level. Thirty one percent of the learners were at the informal deduction level for Questionnaire A as compared to 38% of the learners for Questionnaire B.
The differential results obtained from the two instruments are not surprising considering that the issue has been raised previously by other researchers. Smith and de Villiers (1989) found that the results of the level placements for the 1465 grade 9 and grade 10 learners were
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different according to the two test instruments, one of which was a modification of the Usiskin instrument (Usiskin, 1982), which was also true for this study. In this study, Questionnaire A was a modification of the same instrument. The authors attributed the differences in placement to the mathematical process involved, for example, some learners did better when asked to draw shapes than when they were asked to identify shapes where orientation and shape varied.
6.3.3 Comparison between grade 11 and grade 10 learners according to the van Hiele levels
Table 5.13 shows that 47% of the grade 10 learners were at the visual level as compared to 21% of the grade 11 learners, while 39% of grade 10 learners were at the analysis level as compared to 15% of the grade 11 learners. Eleven percent was at the informal deduction for grade 10 while 64% of the grades 11 were at the informal deduction level. Only 3% of the grade 10 learners were at the formal deduction level whereas no grade 11 learner was at the formal deduction level. Overall it shows that the grade 11 learners performed better than the grade 10 learners in terms of the van Hiele levels. This might have been due to the exposure and experience of geometric concepts that they had compared to the grade 10 learners who have spent a fewer number of years in school and hence have had less exposure to geometric concepts.
6.4 Emergence of Pertinent Issues