Developing geometry for primary school learners

Computers can be used to heighten a scholar ‘s cognition of mathematics, concentrating on what can be done above and beyond with pencil and paper entirely ( Pea, 1986 ) . Using computing machines as cognitive tools to help scholars in larning powerful mathematics that they could hold approached without the engineering should be a cardinal end for research and development-not merely larning the same mathematics better, stronger, faster, but besides larning basically different mathematics in the procedure ( Jonassen and Reeves, 1996 ; Pea, 1986 ) .

In the words of Papert ( 1980, 145 ) , “ The computing machine allows, or obliges, the kid to project intuitive outlooks. When the intuition is translated into a plan it becomes more noticeable and more accessible to reflection ” and can therefore be used as stuff “ for the work of remodelling intuitive cognition. ”

Technology facilitates the observation of form and relationships, to make a practical environment for geographic expedition and conjecturing, to make simulations, to supply an effectual agencies for utilizing mathematical tools and operations, … to implement some algorithms or processs, … to entree or form informations, to back up a speculation or general statement with experimental grounds, to look into paper-and-pencil computations, to ease the instruction of programming basicss, and to foreground the restrictions of engineering. In add-on, engineering allows for the riddance or decrease in accent of some subjects or accomplishments such as complicated long division done by paper and pencil. Technology besides suggests new content such as computing machine artworks, dynamical systems, and fractals.

Technology affects what learners learn and how acquisition is accomplished. Educators need to understand and be able to utilize engineering in an ever-growing figure of ways consistent with how people use it outside the schoolroom. ( Robinson, Robinson, and Maceli, 2000:123 )

Using computing machines can alter learning. The ensuing alteration in position, which for some pedagogues took every bit long as a twelvemonth to accomplish was characterised by:

a decrease of control and greater usage of guided find acquisition that made usage of treatment and group work ;

a willingness to larn along with the scholars ;

a desire to be after lessons affecting the computing machine where its function is a tool for acquisition ;

an ability to do mathematics and its deductions, and non the computing machine, the focal point of concern.

Two of import influences in conveying about the displacement were the changeless handiness of the computing machine in the schoolroom, and the encouragement, motive and aid provided by the support staff.

For primary school scholars computing machine environments are satisfactorily dynamic and synergistic to assist scholars organize spacial and numeral constructs, and to associate screen images to existent 3-dimensional infinite to develop apprehensions of position, proportion and angle. However, in these surveies educators played a important function in promoting educator-learner and learner-learner treatment, and this encouraged scholars to utilize higher-order thought accomplishments and metacognition. The computing machine provided immediate graphical feedback on the brooding alterations kids made to their thought about facets of the job undertakings and how to work out them. This enabled the kids to construct and prove thoughts for themselves and, in so making, make sense of mathematical thoughts and problem-solving schemes.


Dynamic geometry package, such as Cabri Geometry and The Geometer ‘s Sketchpad, has the possible to return geometry to greater prominence in school mathematics course of study. A important factor when working in a dynamic geometry environment is that geometric buildings based on ocular visual aspect alone-so called by-eye methods-are non drag-resistant, that is they do non retain the needed geometric belongingss when single constituents are dragged. Hence, dynamic geometry package forces scholars to believe really carefully about the belongingss of the figure they wish to pull. Ocular buildings provide indispensable staging for Cabri buildings at the start of an probe. The usage of Cabri enhances the usage and apprehension of appropriate geometric linguistic communication, peculiarly when scholars actively discussed their buildings with each other ( Cashman, 1997 ; Vincent and McCrae, 1999a ) . However, design of worksheets and pedagogue intercession and whole category treatment may be indispensable for the package use to be effectual ( Stewart, 1999 ) . In one survey, the scholars utilizing dynamic geometry package improved in the new wave Hiele geometry degrees ( Vincent, 1998 ) . Studies on angles have been peculiarly successful ( Brown, 1997 ; Dix, 1999 ; Redden and Clark, 1997 ) .

Using a socio cultural position on acquisition, the proposed functions for engineering should be recognised by course of study developers and research workers and pedagogues. The lowest degree is that of engineering as maestro. Here the complexness of usage bounds learner activity to a few operations over which they have competency and the scholar, without sufficient mathematical apprehension, blindly accepts the end product produced. In the following degree, engineering as retainer, the user is in control, using the engineering as a dependable fast mechanical assistance to finish mathematical undertakings whose end product is mostly regarded as important, and non questioned. In the 3rd degree, engineering as spouse, the engineering is seen as a comrade with which to research, and non merely a tool for making something. There is besides an consciousness that an result needs to be judged against standards other than the technology-produced response, with a attendant acknowledgment of the demand to equilibrate the governments of mathematics and engineering.

Steering Model

Recently, several writers have described engineering, teaching method, and content cognition ( TPACK ) as a type of instructors ‘ cognition needed for instructors to understand how to utilize engineering efficaciously to learn ( Mishra & A ; Koehler, 2008 ; Niess, 2005, 2006 ; AACTE Committee on Innovation and Technology, 2008 ) . Koehler and Mishra ( 2005 ) claimed that TPACK is the integrating of instructors ‘ cognition of content, teaching method, and engineering ( Figure 1 ) .

Figure 1. Components of technological pedagogical content cognition.

Figure 1

Given the altering nature of engineering, it is of import that instructors develop a theoretical account of learning and larning that goes beyond the particulars of a engineering tool so that they are able to do informed determinations about appropriate utilizations of engineering in mathematics. A cardinal characteristic in our attack to fixing instructors to learn mathematics with engineering is to integrally develop instructors ‘ TPACK. Teachers need to understand that critical instructional determinations they make are grounded in their apprehensions of each sphere ( engineering, teaching method, and content ) and influenced by their beliefs and constructs. We hypothesize that by integrally developing instructors ‘ apprehension of mathematics, teaching method, and engineering with a focal point on scholar thought, we will assist instructors develop a more complete image of what is needed when learning mathematics with engineering and, in bend, be prepared to do informed determinations about appropriate utilizations of engineering.

Technology can magnify scholars ‘ abilities to work out jobs or reorganise the manner scholars think about jobs and their solutions. Technology tools can be used to bring forth big lists of pseudorandom Numberss rapidly, and to bring forth graphical representations or calculate least squares arrested development lines expeditiously.

Through dynamic characteristics of dragging, the linking of multiple representations, and covering steps on graphs, engineering tools can be used in ways that extend what instructors may be able to make without engineering to assist scholars reorganise and alter their statistical constructs. For illustration, covering statistical steps such as a mean on a graphical representation can assist alter the manner instructors and scholars conceptualize these steps in relation to a bivariate distribution, peculiarly since the statistical steps update as informations is changed by the user dragging points in the graph. This visual image is non possible without engineering and can supply scholars with a manner of reorganising their constructs of bivariate distributions.

Teachers need to cognize how to capitalise on the power of engineering to make lessons that aid scholars in developing apprehensions of mathematics. An instructional theoretical account that engages prospective instructors in work outing mathematics undertakings utilizing engineering tools and encourages them to reflect on those experiences from the position of a instructor provides an built-in acquisition experience that is similar to what they will meet when placed in a schoolroom.

Van Hiele Math Cognitive Development Scale

Learners are expected to acknowledge forms and comprehend their belongingss. Learners ‘ developmental phase is the key that one should be cognizant of. There are theories developed based on surveies of Piaget ‘s and Pierre and Dina Van Hiele ‘s to explicate and assist us on apprehension of development of geometrical thought. In Piaget ‘s work, there are two major subjects related to geometrical thought. First, development of geometric thoughts follows a definite order. Topological dealingss develop foremost, followed by Euclidian dealingss. It develops over clip by incorporating and synthesising these dealingss to their bing scheme. Second, mental representation of infinite develops through progressive organisation of the scholar ‘s motor and internalized actions. Ideas about infinite evolve as scholars interact with their environments. These two subjects are supported by research ( Clements and Battista, 1992 ) .

Van Hiele Theory proposes that scholars move through different degrees of geometrical thought. These degrees are as follows:

Degree 0 ( Pre acknowledgment ) : Since scholars do non grok ocular features of forms, they are unable to place many common forms.

Degree 1 ( Visual ) : They can merely acknowledge forms as whole images.

Degree 2 ( Descriptive/Analytic ) : Learners by detecting, mensurating, pulling, and theoretical account devising can acknowledge and qualify forms by their belongingss.

Degree 3 ( Abstract/Relational ) : Learners can separate a form based on certain belongingss which it has.

Degree 4 ( Axiomatic ) : Learners can set up theorems with an self-evident construction. Harmonizing to Van Hiele Theory, geometric thought degrees of scholars in simple and in-between school are at most flat 3. Thinking at degree 4 is necessary for high school geometry. Harmonizing to the Van Hiele theory, the degrees are progressive that scholars move from one degree of believing to the following. Course of study developers and instructors should take these degrees into consideration by enriching larning environment to assist scholars to come on to a following degree ( Burger and Shaughnessy, 1986 ) .

Math Cognitive Development Scale by Dave Moursund

Dave Moursund developed a six-level Piagetian-type graduated table for school mathematics. It is an merger and extension of thoughts of Piaget and the new wave Hieles. The first three degrees are peculiarly relevant to simple school scholars.

Degree 1. Piagetian and Math sensorimotor.

Babies use centripetal and motor capablenesss to research and derive increasing understanding of their environments. Research on really immature babies suggests some unconditioned ability to cover with little measures such as 1, 2, and 3. As babies gain creeping or walking mobility, they can expose unconditioned spacial sense. For illustration, they can travel to a mark along a way necessitating traveling around obstructions, and can happen their manner back to a parent after holding taken a bend into a room where they can no longer see the parent.

Degree 2. Piagetian and Math preoperational.

Children Begin to utilize symbols, such as address. They respond to objects and events harmonizing to how they appear to be. The kids are doing rapid advancement in receptive and productive unwritten linguistic communication. They accommodate to the linguistic communication environments they spend a batch of clip in. Children learn some common people math and get down to develop an apprehension of figure line. They learn figure words and to call the figure of objects in a aggregation and how to number them, with the reply being the last figure used in this numeration procedure. A bulk of kids discover or larn “ numbering on ” and numbering on from the larger measure as a manner to rush up numbering of two or more sets of objects. Children addition increasing proficiency in such numeration activities. In footings of nature and raising in mathematical development, both are of considerable importance during this phase.

Degree 3.Piagetian and Math concrete operations.

Children begin to believe logically. In this phase, which is characterized by 7 types of preservation: figure, length, liquid, mass, weight, country, volume, intelligence is demonstrated through logical and systematic use of symbols related to concrete objects. Operational thought develops ( mental actions that are reversible ) .While concrete objects are an of import facet of larning during this phase, kids besides begin to larn from words, linguistic communication, and pictures/video, larning about objects that are non concretely available to them. For the mean kid, the clip span of concrete operations is about the clip span of simple school ( grades 1-5 or 1-6 ) . During this clip, larning math is slightly linked to holding antecedently developed some cognition of math words ( such as numbering Numberss ) and constructs. However, the degree of abstraction in the written and unwritten math linguistic communication rapidly surpasses a scholar ‘s old math experience. That is, math larning tends to continue in an environment in which the new content stuffs and thoughts are non strongly rooted in verbal, concrete, mental images and apprehension of somewhat similar thoughts that have already been acquired. There is a significant difference between developing general thoughts and apprehension of preservation of figure, length, liquid, mass, weight, country, and volume, and larning the mathematics that corresponds to this. These tend to be comparatively deep and abstract subjects, although they can be taught in really concrete manners.

Degree 4.Piagetian and Math formal operations. Van Hiele degree 2: informal tax write-off.

Thought begins to be systematic and abstract. In this phase, intelligence is demonstrated through the logical usage of symbols related to abstract constructs, job resolution, and gaining and utilizing higher-order cognition and accomplishments. Math adulthood supports the apprehension of and proficiency in math at the degree of a high school math course of study. Beginnings of apprehension of math-type statements and cogent evidence. Piagetian and Math formal operations includes being able to acknowledge math facets of job state of affairss in both math and non-math subjects, convert these facets into math jobs ( math modeling ) , and work out the ensuing math jobs if they are within the scope of the math that 1 has studied. Such transportation of acquisition is a nucleus facet of Level 4.

Degree 5. Abstract mathematical operations. Van Hiele degree 3: tax write-off.

Mathematical content proficiency and adulthood at the degree of modern-day math texts used at the senior undergraduate degree in strong plans, or first twelvemonth alumnus degree in less strong plans.

Good ability to larn math through some combination of reading required texts and other math literature, listening to talks, take parting in category treatments, analyzing on your ain, analyzing in groups, and so on. Solve comparatively high degree math jobs posed by others. Pose and work out jobs at the degree of one ‘s math reading accomplishments and cognition. Follow the logic and statements in mathematical cogent evidence. Fill in inside informations of cogent evidence when stairss are left out in text editions and other representations of such cogent evidences.

Degree 6. Mathematician. Van Hiele degree 4: asperity.

A really high degree of mathematical proficiency and adulthood. This includes velocity, truth, and understanding in reading the research literature, composing research literature, and in unwritten communicating ( speak, listen ) of research-level mathematics. Pose and work out original math jobs at the degree of modern-day research frontiers.

Many secondary scholars are on new waves Hiele ocular or analysis degrees. In order for a scholar to get by with the demands of an self-evident system as required in secondary school, nevertheless, s/he demands to be on the new wave Hiele telling degree. Learners who have non received sufficient experience on the ocular and analysis degrees resort to memorization to get by with the demands of formal school geometry. It is in the primary school that the scholars require experiences on the ocular and analysis degrees in readying for activity on the new wave Hiele telling degree.

Learners are surrounded by spacial scenes and the ability to comprehend spacial dealingss is regarded as of import for mundane interaction in infinite. Smit ( 1998 ) stresses the importance of these accomplishments:

Without spacial sense it would be hard to be in this universe – we would non be able to pass on about place, relationships between objects, giving and having waies or imagine alterations taking topographic point sing the alterations in place and size of forms.

Furthermore, some research has suggested a nexus between spacial sense and general public presentation in mathematics itself. For illustration, Presmeg ( 1992 ) stresses the importance of ocular imagination in general logical thinking accomplishments in mathematics and Guay and McDaniel ( 1977 ) suggest that high mathematics winners at simple school have greater spacial ability than low winners and that there is a relationship between mathematical and spacial thought for students with high every bit good as low spacial ability.

Guay and McDaniel ( 1977: 211 ) define “ low-level spacial abilities ” as those necessitating the visual image of planar constellations, but no mental transmutations of these ocular images. “ High-level spacial abilities ” are characterised as necessitating the visual image of 3-dimensional constellations, and the mental use of these images.

There are legion assessment studies uncovering that scholars fail to larn basic geometric constructs particularly geometric job resolution ( Kouba et al. , 1988 ; Stigler, Lee and Stevensen, 1990 ; the International Study Center, 1999 ) . The current simple and in-between school geometry course of study do ease chances for scholars to utilize their basic intuitions and simple constructs to come on to higher degrees of geometric idea. Learners traveling through such experience in simple school do non hold the necessary geometric intuition and background for a formal deductive geometry class in high schools ( Hoffer, 1981 ; Shaughnessy & A ; Burger, 1985 ) . Lacks on conceptual and procedural apprehension of scholars cause jobs for the ulterior survey of of import thoughts such as vectors, co-ordinates, transmutations, and trigonometry ( Fey et al. 1984 ) .


Harmonizing to Piaget and Inhelder ( 1967:43 ) , action is of paramount importance in the development of geometric conceptualisations. The kid “ can merely ‘abstract ‘ . . . the thought of a consecutive line from the action of following by manus or oculus without altering way, and the thought of an angle from two crossing motions ” .

Indeed, physical actions on concrete objects are necessary. But scholars must internalise such physical actions and abstract the corresponding geometric impressions. Logo can ease this procedure, therefore advancing a passage from concrete experiences with geometric thoughts to abstract concluding. For illustration, by first holding kids form waies by walking, so utilizing Logo ; kids can larn to believe of the polo-neck ‘s actions as 1s that they themselves could execute. They seem to project themselves into the topographic point of the polo-neck. In so making, they are executing a mental action — an internalized version of their ain physical motions.

Dynamic geometry package allows students to research and larn geometrical facts through experimentation and observation. Students can build figures on the screen and so research them dynamically. When an independent point or line is dragged with the mouse, all dependent buildings remain integral. They can be used to understand what stays the same and what alterations under different conditions. They can actuate students to explicate and turn out. Dynamic geometry package can be used in a assortment of ways in Key Stage 3:

aˆ? researching and larning about the belongingss of forms ;

aˆ? analyzing geometric relationships and larning geometrical facts ;

aˆ? transforming forms ;

aˆ? working with dynamic images to do and prove hypotheses about belongingss of forms ;

aˆ? devising and researching geometric buildings ;

aˆ? constructing and researching venue.


The Sample

A Pre-test, Post-test theoretical account was used to mensurate the consequence of the intercession programme on the geometric public presentation. The programme was implemented at a South African urban primary school with the sample consisting of 40 English-speaking scholars from a Grade 7 category.

The Instruments

The instruments used for the research consisted of a trial that was administered as pre-test and post-test. The trial consisted of:

1. Matching 2 dimensional forms in different orientation.

2. Identifying cyberspaces of regular hexahedrons, tetrahedra and octahedron

The pre-test was administered at the beginning of the research undertaking. This was so followed by an intercession programme. The scholars were now exposed to POLY and National Library of Virtual Manipulatives where they were able to pull strings 3 dimensional forms.

The post-test was administered at the decision of the contact session.

1. Shape Duplicate Questions

In this illustration, you are asked to look at two groups of simple, level objects and happen braces that are precisely the same size and form. Each group has about 25 little drawings of these two-dimensional objects. The objects in the first group are labeled with Numberss and are in numerical order. The objects in the 2nd group are labeled with letters and are in random order. Each pulling in the first group is precisely the same as a drawing in the 2nd group. The objects in the 2nd group have been moved and some have been rotated.


Learners from a class 7 category were divided into a control and experimental group. There were 20 scholars in each group. Learners in the control group were taught with traditional method. The traditional direction method in this survey was lessons given by a instructor, usage of text editions and other stuffs, and a clear account of procedural cognition and conceptual cognition of 2 dimensional and 3 dimensional forms to scholars. The instructor demonstrated 2 and 3 dimensional forms utilizing the blackboard and the text edition. The scholars did non hold any undertakings that made usage of representations on computing machines ( see Table 1 ) .

Table1 Computer based Instruction V. Traditional Direction


Computer Based Instruction

( n = 20 )

Traditional Direction

( n = 20 )

Instructional content

The direction included lessons on 2 and 3 dimensional constructs.

The direction included lessons on 2 and 3 dimensional constructs.

Forms of direction

The scholars worked in braces on a computing machine.

The scholars worked in groups without utilizing any computing machines.

Learning environments

Learners received direction utilizing a computer-assisted instructional package.

Learners did non utilize any computer-assisted instructional package.

The scholars in the experimental group were taught with the free package “ Poly. ” This programme enabled the scholars to pull strings the cyberspaces of solid forms. They were able to see assorted orientations of the cyberspaces of the solids. Therefore the consequences revealed that the engineering could be attributed to the improved consequences. This is clearly seen in table 2, graph1 and graph 2.

Table 2 Analysis of Responses of Experimental and Control Pre and Post Test

Experimental Group

Control Group


Pre Test – More than 50 % Correct

Post Test – More than 50 % Correct

Pre Test – More than 50 % Correct

1. Shape fiting




2. Internets of regular hexahedrons




3.Netsof tetrahedra




4.Netsof octahedra




Graph 1: Experimental Group – Pre and Post Test

Graph 2: Control Group – Pre and Post Test

Table 3 gives the statistical consequences that were computed. Once once more the benefits of utilizing engineering can be seen.

Table 3: Statistical Analysis of Experimental and Control Group



Pre Test

Post Test

Pre Test









Std dev.









Technologies such as Cadmiums, nomadic phones, digital cameras, and personal digital helpers are common accoutrements in the digital place and workplace. The World Wide Web is progressively going portion of most schools and schoolrooms. Online instruction has a figure of benefits over traditional computer-based engineerings. Clearly, greater entree is provided to those scholars analyzing at a distance or unable to mainstream into regular schoolrooms, every bit good as those scholars who wish to larn at their ain gait ( Santoro, 1995 ) . This type of flexible instruction can enable scholars to presume greater duty for their ain acquisition ( Schwier and Misanchuk, 1993 ; Winn, 1997 ) . Collaboration is farther enhanced because online engineerings are besides effectual in leting computing machines with different platforms and browsers to work together in a acquisition environment, in contrast to other computing machine engineerings that are platform specific ( Hosie and Schibeci, 2001 ) . Changes in the manner instruction and acquisition are conceptualized have paralleled alterations in engineering.