The researcher used an action research and task-based interview process with ten 8th grade students to demonstrate that the use of manipulatives, namely tangrams, was effective in improving students' understanding of the properties of a square and a rectangle. The students who were interviewed were able to understand the properties of the square and rectangle well, resulting in significantly better pre-test scores. If the work of others has been used, this is duly acknowledged in the text.
LITERATURE REVIEW
CRITIQUE OF VAN HIELE THEORY
Depending on the level of the problem, some students use several levels at the same time. Treffers (1987) points out that the van Hiele theory was proposed at a time when geometry was not part of the primary school curriculum in the Netherlands. The conclusion was that the results of the experimental group, which interacted with the manipulators, were higher than those of the control group.
A CRITIQUE OF MANIPULATIVE USE
Manipulative materials should be used regularly in a total mathematics program in a manner consistent with the goals of the program. However, since they do not see the sizes of the three angles as interdependent, they are often misled by the appearance of the angles. For example, students progress beyond an empirical generalization that the sum of the angles of a triangle is a right angle to a belief, based on logical reasoning, that it must necessarily be so.
COMPARISON OF VAN HIELE THEORY TO PIAGETIAN THEORY
Moreover, they suggest that van Hiele and Piaget do not support the idea that clear explanations are what define good teaching. They claim that Piaget would suggest that it depends on activity, while van Hiele would suggest that it depends on language. We therefore note that van Hiele (1986), in a similar vein to Piaget, was interested in general characteristics, and his model reflects this.
THE SOLO TAXONOMY
Biggs and Coli (1991) consolidated the evolving structure of the SOLO taxonomy, consisting of developmental modes and levels within each mode. Biggs and Coli is' (1982) SOLO Taxonomy focuses on student responses, as opposed to levels of thinking or stages of development. Biggs and Collis (1982) wanted the SOLO taxonomy to be of particular interest to the educator.
COMPARISON OF VAN HIELE TO SOLO TAXONOMY THEORY
On the other hand, van Hiele did not assume that these "intermediate" levels existed at all. When evaluating van Hiele's model as a theory, we can conclude that it has advantages that distinguish it from the SOLO taxonomy and Piaget's theory. This can be attributed to the fact that van Hiele were both teachers and their theory grew out of this context.
THEORETICAL AND CONCEPTUAL FRAMEWORK
CONSTRUCTIVISM
Because from the constructivist perspective the student is responsible for his own learning process, the role of the teacher in the learning process must be adapted. The role of the teacher in a constructivist setting is to help students develop new insights and connect them to previously learned knowledge. One must provide multiple representations of reality and also represent the natural complexity of the real world;
RADICAL AND SOCIAL CONSTRUCTMSM
According to Schifter (1993), in the construction of knowledge, students construct understandings when they try to make sense out of their experiences. Historically, schools have been organized around recitation teaching, that is, the teacher conveys knowledge to be remembered by the students, who in turn recite the information back to the teacher (Hausfather, 1996). Zone of Proximal Development (ZPD) is the difference between an individual's current level of development and his or her potential level of development (Vygotsky, 1978).
According to Rogoff (1998), children develop most when they participate in activities slightly beyond their competence with the help of adults and other capable children.
COMPARISON BETWEEN VYGOTSIAN THEORY AND PIAGETIAN THEORY
Van Hiele's geometry learning theory provided this study with a useful conceptual framework for interpreting and analyzing students' levels of understanding. A student may recognize a square, for example, but will not be able to list any properties of a square. At this stage of students' interaction with manipulatives, namely tangrams, the student engages in research activities and tries to find common relationships that can lead to a hypothesis.
He may even be able to compare or rank shapes based on overall appearance. Using the tangram, the student may be able to fit two squares into a rectangle or two isosceles triangles into a square. However, the student is unable to find relevant relationships between different figures, for example, the relationship between a rectangle and a square.
However, it would be presumptuous to think that the learners in this study will be able to reach this stage, as they are not expected to construct evidence at this level. A child at level n will answer most questions at that level, but may not be able to answer questions at level +1. In the study, the researcher made the learners familiar with the different pieces of the tangram sets by making different shapes and patterns.
The learners in the research had to work through a series of activities in the interview schedule.
RESEARCH DESIGN AND METHODOLOGY 4.1 INTRODUCTION
- ACTION RESEARCH
- METHODOLOGICAL APPROACH
- RESEARCH DESIGN
- SAMPLING
- CONTEXTUAL FACTORS - THE SCHOOL AND THE LEARNERS
The researcher thinks that there is a need for some kind of change, or improvement in the teaching and learning of geometry, especially the properties of square and rectangle. According to Cohen, Manion, (2000) the purpose of the research should help establish the methodology and design of the research study. Thus, the purpose of this study is to determine whether the use of manipulatives was effective in improving students' understanding of square and rectangle features in grade eight at a secondary school in KwaZulu-Natal, South Africa.
This study focused on the student and on understanding and interpreting the student's understanding, reasoning, and techniques used regarding the properties of the square and rectangle. According to Pillay (2004), research design refers to the logical structure of the research so that unambiguous conclusions can be drawn. In addition to the research methodology and instrumentation, the research also depends on the quality of the chosen sample.
Most of the students at this school come from middle to upper middle income families. This school offers mathematics as one of the compulsory subjects in the curriculum from eighth to twelfth grade. Thus, the selected students were selected using a purposive sampling strategy based on the needs of the research.
Participants had little or no knowledge of the properties of a square and a rectangle.
DATA ANALYSIS
- OVERVIEW
- PROPERTIES OF THE SQUARE USING TANGRAMS
- TANGRAMS TO DEDUCE PROPERTIES OF DIAGONALS - SQUARE
- TANGRAMS USED - PROPERTIES OF RECTANGLE
- TANGRAMS USED - PROPERTIES OF RECTANGLE DIAGONALS
- TANGRAMS USED TO ATTAIN VAN HIELE LEVEL 3
They also found that angles and sides are still the same regardless of rotation. Once again, 100% of the students correctly identified that the two lines were the same length. 100% of students were able to state that all angles are equal to 900, although some stated that all angles are equal, meaning that every angle is 900.
100% of the students were able to state that the diagonals cut the area of the square. However, surprisingly only 40% of students were able to state that opposite sides were parallel. The following observations enabled the researcher to answer the second critical question about the diagonals of the square.
Eighty percent of the learners could say that the diagonals of a square bisect the area of the square, while one learner said that the diagonal increases the area. Some of the learners superimposed all the angles (refer to figure 1.4) and found that they were all equal. 60% of the learners could say that "the diagonals bisected each other", not necessarily using the same vocabulary.
Questioner: Do you think the diagonals of the rectangle will have the same properties as the square.
CONCLUSIONS AND RECOMMENDATIONS
- DEFINITIONS
- RECOMMENDATIONS
- LIMITATIONS
- CONCLUSIONS
The researcher observed that redundancy, or a lack of economy was evident in definitions and characteristics, which suggests that teachers must be aware of the adequacy, sufficiency and equivalence of some characteristics and definitions as pointed out by De Villiers (2003). This study proves that learners, who knew very little about squares, rectangles and certainly nothing about diagonals from their pre-test, were able to grasp and list these properties with ease. The researcher therefore suggests that workshops and training be made available to teachers on how and when to use concrete manipulatives.
The researcher was able to obtain valuable information about diagonals of square and rectangle that would otherwise have been long and laborious. The results of this study support the initiation of other studies to replicate these findings, not only in geometry, but also in other areas of mathematics. This study sample consisted of only ten learners; therefore, the results could not be generalized, but it is useful to determine that manipulations can help make teaching simpler and make learning much easier.
Even a qualitative analysis is compromised because the researcher, the author of this study, was the participant in this study who analyzed the classroom experiences. In conclusion, the researcher affirms that to achieve satisfactory results in mathematics, geometry and to promote active participation in mathematics, the authorities need to make some crucial changes in the way mathematics, especially geometry, is taught at the secondary school level. The results of this study confirm my initial speculation that geometry can be presented in a form that is much easier to understand if concrete manipulatives are used, as this provides a visual vehicle that subtly conveys the concepts to students.
The results of this study should help educators determine an effective way to approach mathematics instruction using manipulatives.
What can you say about the length of the lines connecting the opposite vertices of the square? What does this tell us about the diagonals of the square and its area? RESEARCHER: What does this tell us about the diagonals of the square and its area?
I took this angle and placed it on each angle of the square and they matched exactly.
