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Computers can be used to enhance a learner's knowledge of mathematics, focusing on what can be done above and beyond with pencil and paper alone (Pea, 1986). Using computers as cognitive tools to assist learners in learning powerful mathematics that they could have approached without the technology should be a key goal for research and development-not only learning the same mathematics better, stronger, faster, but also learning fundamentally different mathematics in the process (Jonassen and Reeves, 1996; Pea, 1986).
In the words of Papert(1980, 145), "The computer allows, or obliges, the child to externalize intuitive expectations. When the intuition is translated into a program it becomes more obtrusive and more accessible to reflection" and can thus be used as material "for the work of remodelling intuitive knowledge."
Technology facilitates the observance of pattern and relationships, to create a virtual environment for exploration and conjecturing, to create simulations, to provide an effective means for using mathematical tools and operations, ... to implement some algorithms or procedures, ... to access or organize data, to support a conjecture or general statement with experimental evidence, to check paper-and-pencil calculations, to facilitate the teaching of programming fundamentals, and to highlight the limitations of technology. In addition, technology allows for the elimination or reduction in emphasis of some topics or skills such as complicated long division done by paper and pencil. Technology also suggests new content such as computer graphics, dynamical systems, and fractals.
Technology affects what learners learn and how learning is accomplished. Educators need to understand and be able to use technology in an ever-growing number of ways consistent with how people use it outside the classroom. (Robinson, Robinson, and Maceli, 2000:123)
Using computers can change teaching. The resulting change in perspective, which for some educators took as long as a year to achieve was characterised by:
a lessening of control and greater use of guided discovery learning that made use of discussion and group work;
a willingness to learn along with the learners;
a desire to plan lessons involving the computer where its role is a tool for learning;
an ability to make mathematics and its implications, and not the computer, the focus of concern.
Two important influences in bringing about the shift were the constant availability of the computer in the classroom, and the encouragement, motivation and assistance provided by the support staff.
For primary school learners computer environments are satisfactorily dynamic and interactive to help learners coordinate spatial and numeric concepts, and to link screen images to real three-dimensional space to develop understandings of perspective, proportion and angle. However, in these studies educators played a significant role in encouraging educator-learner and learner-learner discussion, and this encouraged learners to use higher-order thinking skills and metacognition. The computer provided immediate graphical feedback on the reflective modifications children made to their thinking about aspects of the problem tasks and how to solve them. This enabled the children to build and test ideas for themselves and, in so doing, make sense of mathematical ideas and problem-solving strategies.
DYNAMIC GEOMETRIC SOFTWARE
Dynamic geometry software, such as Cabri Geometry and The Geometer's Sketchpad, has the potential to return geometry to greater prominence in school mathematics curricula. A crucial factor when working in a dynamic geometry environment is that geometric constructions based on visual appearance alone-so called by-eye methods-are not drag-resistant, that is they do not retain the required geometric properties when individual components are dragged. Hence, dynamic geometry software forces learners to think very carefully about the properties of the figure they wish to draw. Visual constructions provide essential scaffolding for Cabri constructions at the start of an investigation. The use of Cabri enhances the use and understanding of appropriate geometric language, particularly when learners actively discussed their constructions with each other (Cashman, 1997; Vincent and McCrae, 1999a). However, design of worksheets and educator intervention and whole class discussion may be essential for the software usage to be effective (Stewart, 1999). In one study, the learners using dynamic geometry software improved in the van Hiele geometry levels (Vincent, 1998). Studies on angles have been particularly successful (Brown, 1997; Dix, 1999; Redden and Clark, 1997).
Using a socio cultural perspective on learning, the proposed roles for technology should be recognised by curriculum developers and researchers and educators. The lowest level is that of technology as master. Here the complexity of usage limits learner activity to a few operations over which they have competence and the learner, without sufficient mathematical understanding, blindly accepts the output produced. In the next level, technology as servant, the user is in control, applying the technology as a reliable fast mechanical aid to complete mathematical tasks whose output is largely regarded as authoritative, and not questioned. In the third level, technology as partner, the technology is seen as a companion with which to explore, and not just a tool for doing something. There is also an awareness that an outcome needs to be judged against criteria other than the technology-produced response, with a consequent recognition of the need to balance the authorities of mathematics and technology.
Recently, several authors have described technology, pedagogy, and content knowledge (TPACK) as a type of teachers' knowledge needed for teachers to understand how to use technology effectively to teach (Mishra & Koehler, 2008; Niess, 2005, 2006; AACTE Committee on Innovation and Technology, 2008). Koehler and Mishra (2005) claimed that TPACK is the integration of teachers' knowledge of content, pedagogy, and technology (Figure 1).
Figure 1. Components of technological pedagogical content knowledge.
Given the changing nature of technology, it is important that teachers develop a model of teaching and learning that goes beyond the specifics of a technology tool so that they are able to make informed decisions about appropriate uses of technology in mathematics. A key feature in our approach to preparing teachers to teach mathematics with technology is to integrally develop teachers' TPACK. Teachers need to understand that critical instructional decisions they make are grounded in their understandings of each domain (technology, pedagogy, and content) and influenced by their beliefs and conceptions. We hypothesize that by integrally developing teachers' understanding of mathematics, pedagogy, and technology with a focus on learner thinking, we will help teachers develop a more complete picture of what is needed when teaching mathematics with technology and, in turn, be prepared to make informed decisions about appropriate uses of technology.
Technology can amplify learners' abilities to solve problems or reorganize the way learners think about problems and their solutions. Technology tools can be used to generate large lists of pseudorandom numbers quickly, and to generate graphical representations or compute least squares regression lines efficiently.
Through dynamic features of dragging, the linking of multiple representations, and overlaying measures on graphs, technology tools can be used in ways that extend what teachers may be able to do without technology to help learners reorganize and change their statistical conceptions. For example, overlaying statistical measures such as a mean on a graphical representation can help change the way teachers and learners conceptualize these measures in relation to a bivariate distribution, particularly since the statistical measures update as data is changed by the user dragging points in the graph. This visualization is not possible without technology and can provide learners with a way of reorganizing their conceptions of bivariate distributions.
Teachers need to know how to capitalize on the power of technology to create lessons that assist learners in developing understandings of mathematics. An instructional model that engages prospective teachers in solving mathematics tasks using technology tools and encourages them to reflect on those experiences from the perspective of a teacher provides an integral learning experience that is similar to what they will encounter when placed in a classroom.
Van Hiele Math Cognitive Development Scale
Learners are expected to recognize shapes and comprehend their properties. Learners' developmental stage is the key that one should be aware of. There are theories developed based on studies of Piaget's and Pierre and Dina Van Hiele's to explain and help us on understanding of development of geometrical thinking. In Piaget's work, there are two major themes related to geometrical thinking. Firstly, development of geometric ideas follows a definite order. Topological relations develop first, followed by Euclidean relations. It develops over time by integrating and synthesizing these relations to their existing schemas. Secondly, mental representation of space develops through progressive organization of the learner's motor and internalized actions. Ideas about space evolve as learners interact with their environments. These two themes are supported by research (Clements and Battista, 1992).
Van Hiele Theory proposes that learners move through different levels of geometrical thinking. These levels are as follows:
Level 0 (Pre recognition): Since learners do not comprehend visual characteristics of shapes, they are unable to identify many common shapes.
Level 1 (Visual): They can only recognize shapes as whole images.
Level 2 (Descriptive/Analytic): Learners by observing, measuring, drawing, and model making can recognize and characterize shapes by their properties.
Level 3 (Abstract/Relational): Learners can distinguish a shape based on certain properties which it has.
Level 4 (Axiomatic): Learners can establish theorems with an axiomatic structure. According to Van Hiele Theory, geometric thinking levels of learners in elementary and middle school are at most level 3. Thinking at level 4 is necessary for high school geometry. According to the Van Hiele theory, the levels are progressive that learners move from one level of thinking to the next. Curriculum developers and teachers should take these levels into consideration by enriching learning environment to help learners to progress to a next level (Burger and Shaughnessy, 1986).
Math Cognitive Development Scale by Dave Moursund
Dave Moursund developed a six-level Piagetian-type scale for school mathematics. It is an amalgamation and extension of ideas of Piaget and the van Hieles. The first three levels are particularly relevant to elementary school learners.
Level 1. Piagetian and Math sensorimotor.
Infants use sensory and motor capabilities to explore and gain increasing understanding of their environments. Research on very young infants suggests some innate ability to deal with small quantities such as 1, 2, and 3. As infants gain crawling or walking mobility, they can display innate spatial sense. For example, they can move to a target along a path requiring moving around obstacles, and can find their way back to a parent after having taken a turn into a room where they can no longer see the parent.
Level 2. Piagetian and Math preoperational.
Children begin to use symbols, such as speech. They respond to objects and events according to how they appear to be. The children are making rapid progress in receptive and generative oral language. They accommodate to the language environments they spend a lot of time in. Children learn some folk math and begin to develop an understanding of number line. They learn number words and to name the number of objects in a collection and how to count them, with the answer being the last number used in this counting process. A majority of children discover or learn "counting on" and counting on from the larger quantity as a way to speed up counting of two or more sets of objects. Children gain increasing proficiency in such counting activities. In terms of nature and nurture in mathematical development, both are of considerable importance during this stage.
Level 3.Piagetian and Math concrete operations.
Children begin to think logically. In this stage, which is characterized by 7 types of conservation: number, length, liquid, mass, weight, area, volume, intelligence is demonstrated through logical and systematic manipulation of symbols related to concrete objects. Operational thinking develops (mental actions that are reversible).While concrete objects are an important aspect of learning during this stage, children also begin to learn from words, language, and pictures/video, learning about objects that are not concretely available to them. For the average child, the time span of concrete operations is approximately the time span of elementary school (grades 1-5 or 1-6). During this time, learning math is somewhat linked to having previously developed some knowledge of math words (such as counting numbers) and concepts. However, the level of abstraction in the written and oral math language quickly surpasses a learner's previous math experience. That is, math learning tends to proceed in an environment in which the new content materials and ideas are not strongly rooted in verbal, concrete, mental images and understanding of somewhat similar ideas that have already been acquired. There is a substantial difference between developing general ideas and understanding of conservation of number, length, liquid, mass, weight, area, and volume, and learning the mathematics that corresponds to this. These tend to be relatively deep and abstract topics, although they can be taught in very concrete manners.
Level 4.Piagetian and Math formal operations. Van Hiele level 2: informal deduction.
Thought begins to be systematic and abstract. In this stage, intelligence is demonstrated through the logical use of symbols related to abstract concepts, problem solving, and gaining and using higher-order knowledge and skills. Math maturity supports the understanding of and proficiency in math at the level of a high school math curriculum. Beginnings of understanding of math-type arguments and proof. Piagetian and Math formal operations includes being able to recognize math aspects of problem situations in both math and non-math disciplines, convert these aspects into math problems (math modelling), and solve the resulting math problems if they are within the range of the math that one has studied. Such transfer of learning is a core aspect of Level 4.
Level 5. Abstract mathematical operations. Van Hiele level 3: deduction.
Mathematical content proficiency and maturity at the level of contemporary math texts used at the senior undergraduate level in strong programs, or first year graduate level in less strong programs.
Good ability to learn math through some combination of reading required texts and other math literature, listening to lectures, participating in class discussions, studying on your own, studying in groups, and so on. Solve relatively high level math problems posed by others. Pose and solve problems at the level of one's math reading skills and knowledge. Follow the logic and arguments in mathematical proofs. Fill in details of proofs when steps are left out in textbooks and other representations of such proofs.
Level 6. Mathematician. Van Hiele level 4: rigor.
A very high level of mathematical proficiency and maturity. This includes speed, accuracy, and understanding in reading the research literature, writing research literature, and in oral communication (speak, listen) of research-level mathematics. Pose and solve original math problems at the level of contemporary research frontiers.
Many secondary learners are on van Hiele visual or analysis levels. In order for a learner to cope with the demands of an axiomatic system as required in secondary school, however, s/he needs to be on the van Hiele ordering level. Learners who have not received sufficient experience on the visual and analysis levels resort to memorisation to cope with the demands of formal school geometry. It is in the primary school that the learners require experiences on the visual and analysis levels in preparation for activity on the van Hiele ordering level.
Learners are surrounded by spatial settings and the ability to perceive spatial relations is regarded as important for everyday interaction in space. Smit (1998) stresses the importance of these skills:
Without spatial sense it would be difficult to exist in this world - we would not be able to communicate about position, relationships between objects, giving and receiving directions or imagine changes taking place regarding the changes in position and size of shapes.
Furthermore, some research has suggested a link between spatial sense and general performance in mathematics itself. For example, Presmeg (1992) stresses the importance of visual imagery in general reasoning skills in mathematics and Guay and McDaniel (1977) suggest that high mathematics achievers at elementary school have greater spatial ability than low achievers and that there is a relationship between mathematical and spatial thinking for pupils with high as well as low spatial ability.
Guay and McDaniel (1977: 211) define "low-level spatial abilities" as those requiring the visualisation of two-dimensional configurations, but no mental transformations of these visual images. "High-level spatial abilities" are characterised as requiring the visualisation of three-dimensional configurations, and the mental manipulation of these images.
There are numerous assessment reports revealing that learners fail to learn basic geometric concepts especially geometric problem solving (Kouba et al., 1988; Stigler, Lee and Stevensen, 1990; the International Study Center, 1999). The current elementary and middle school geometry curriculum do facilitate opportunities for learners to use their basic intuitions and simple concepts to progress to higher levels of geometric thought. Learners going through such experience in elementary school do not have the necessary geometric intuition and background for a formal deductive geometry course in high schools (Hoffer, 1981; Shaughnessy & Burger, 1985). Deficiencies on conceptual and procedural understanding of learners cause problems for the later study of important ideas such as vectors, coordinates, transformations, and trigonometry (Fey et al. 1984).
CONCRETE TO ABSTRACT
According to Piaget and Inhelder (1967 :43), action is of paramount importance in the development of geometric conceptualizations. The child "can only 'abstract'. . . the idea of a straight line from the action of following by hand or eye without changing direction, and the idea of an angle from two intersecting movements" .
Indeed, physical actions on concrete objects are necessary. But learners must internalize such physical actions and abstract the corresponding geometric notions. Logo can facilitate this process, thus promoting a transition from concrete experiences with geometric ideas to abstract reasoning. For example, by first having children form paths by walking, then using Logo; children can learn to think of the turtle's actions as ones that they themselves could perform. They seem to project themselves into the place of the turtle. In so doing, they are performing a mental action--an internalized version of their own physical movements.
Dynamic geometry software allows pupils to explore and learn geometrical facts through experimentation and observation. Pupils can construct figures on the screen and then explore them dynamically. When an independent point or line is dragged with the mouse, all dependent constructions remain intact. They can be used to understand what stays the same and what changes under different conditions. They can motivate pupils to explain and prove. Dynamic geometry software can be used in a variety of ways in Key Stage 3:
â€¢ exploring and learning about the properties of shapes;
â€¢ studying geometric relationships and learning geometrical facts;
â€¢ transforming shapes;
â€¢ working with dynamic images to make and test hypotheses about properties of shapes;
â€¢ making and exploring geometric constructions;
â€¢ constructing and exploring loci.
A Pre-test, Post-test model was used to measure the effect of the intervention programme on the geometric performance. The programme was implemented at a South African urban primary school with the sample consisting of 40 English-speaking learners from a Grade 7 class.
The instruments used for the research consisted of a test that was administered as pre-test and post-test. The test consisted of:
1. Matching 2 dimensional shapes in different orientation.
2. Identifying nets of cubes, tetrahedra and octahedron
The pre-test was administered at the commencement of the research project. This was then followed by an intervention programme. The learners were now exposed to POLY and National Library of Virtual Manipulatives where they were able to manipulate 3 dimensional shapes.
The post-test was administered at the conclusion of the contact session.
1. Shape Matching Questions
In this example, you are asked to look at two groups of simple, flat objects and find pairs that are exactly the same size and shape. Each group has about 25 small drawings of these 2-dimensional objects. The objects in the first group are labeled with numbers and are in numerical order. The objects in the second group are labeled with letters and are in random order. Each drawing in the first group is exactly the same as a drawing in the second group. The objects in the second group have been moved and some have been rotated.
ANALYSIS OF RESULTS
Learners from a grade 7 class were divided into a control and experimental group. There were 20 learners in each group. Learners in the control group were taught with traditional method. The traditional instruction method in this study was lessons given by a teacher, use of textbooks and other materials, and a clear explanation of procedural knowledge and conceptual knowledge of 2 dimensional and 3 dimensional shapes to learners. The teacher demonstrated 2 and 3 dimensional shapes using the chalkboard and the textbook. The learners did not have any tasks that made use of representations on computers (see Table 1).
Table1 Computer based Instruction vs. Traditional Instruction
Computer Based Instruction
(n = 20)
(n = 20)
The instruction included lessons on 2 and 3 dimensional concepts.
The instruction included lessons on 2 and 3 dimensional concepts.
Forms of instruction
The learners worked in pairs on a computer.
The learners worked in groups without using any computers.
Learners received instruction using a computer-assisted instructional software.
Learners did not use any computer-assisted instructional software.
The learners in the experimental group were taught with the free software "Poly." This programme enabled the learners to manipulate the nets of solid shapes. They were able to view various orientations of the nets of the solids. Thus the results revealed that the technology could be attributed to the improved results. This is clearly seen in table 2, graph1 and graph 2.
Table 2 Analysis of Responses of Experimental and Control Pre and Post Test
Pre Test - More than 50% Correct
Post Test - More than 50% Correct
Pre Test - More than 50% Correct
1. Shape matching
2. Nets of cubes
Graph 1 : Experimental Group - Pre and Post Test
Graph 2 : Control Group - Pre and Post Test
Table 3 gives the statistical results that were computed. Once again the benefits of using technology can be seen.
Table 3 : Statistical Analysis of Experimental and Control Group
Technologies such as CDs, mobile phones, digital cameras, and personal digital assistants are common accessories in the digital home and workplace. The World Wide Web is increasingly becoming part of most schools and classrooms. Online education has a number of benefits over traditional computer-based technologies. Clearly, greater access is provided to those learners studying at a distance or unable to mainstream into regular classrooms, as well as those learners who wish to learn at their own pace (Santoro, 1995). This type of flexible teaching can enable learners to assume greater responsibility for their own learning (Schwier and Misanchuk, 1993; Winn, 1997). Collaboration is further enhanced because online technologies are also effective in allowing computers with different platforms and browsers to work together in a learning environment, in contrast to other computer technologies that are platform specific (Hosie and Schibeci, 2001). Changes in the way teaching and learning are conceptualized have paralleled changes in technology.