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8/3/2019 Jasneet Kaur, Mohd Mamur Ali, Ashu Threja: Understanding of Geometrical and Algebraic Concepts at Elementary Level

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Understanding of Geometricaland Algebraic Concepts at

Elementary Level



Jasneet KaurPh D. Scholar, Dept of Educational Studies, Faculty of

Education, Jamia Millia Islamia, New Delhi

Mohd. Mamur Ali

Guest Faculty & Ph. D. Scholar, Faculty of Education, Jamia

Millia Islamia, New Delhi

Ashu Threja

Assistant Professor, Department of Elementary Education,

Miranda House, Delhi University



A mathematical idea or procedure is understood if itsmental representation is part of a network of

representations (Hiebert & Carpenter,1992).

A piece of mathematics is understood if one can Explain mathematical concepts and facts in terms of

simpler concepts and facts.Easily make logical connections between different facts and

concepts.

Recognize the connection when you encounter something

new (inside or outside of mathematics) that's close to themathematics you understand.

Identify the principles in the given piece of mathematicsthat make everything work. (Alfeld, 2004).

Understanding is a context-dependent concept. It is clearthat understanding can vary in degree or completeness.



Some of the Indian researches are focussed on the

Students’ achievement in mathematics (Patel, 1984; Tiwari,1986; Nilima Kumari, 1984; Gakhar, 1981; Jabbal, 1981;Nalinidevi, 1976; Katiyar's 1979)

Achievement in algebra (Joshi, 1970; Chauhan, 1982)

Error committed by students in algebraic and geometricconcepts (Bhatia, 1998; ; Singh, 1986 ; Bannerji, 2000,Rooprai,2002).



Reviews related to Understanding of Geometrical Concepts

There are studies (Alam, 1986 ; Obad, 1989; Kaul,1990) that investigatedthe relationship of maturation of space concept and of concept imagery with achievement of geometrical concepts” findings of the study revealed that conceptual maturity of a child is positively related to hisstage of development at a particular age

Gupta, (1989) has found that the understanding of axioms in geometry as self evident truths occurs in the course of growth between ages of 5th and 7th and are understood in order different from the order it is given.

Alqabbas (2001) did a study with two basic purposes i) understanding

students learning strategies which they use to solve problems inmathematics ii) Investigating the effectiveness of the teachingprogramme in mathematics with additional emphasis on teachinglearning strategies. The most important finding is that explicit teachingof learning strategies while teaching mathematics at elementary levelhelps students in improving their performance.



Rooprai (2002) identified errors in solving geometrical

problems using questionnaire and achievement test.Findings

Different types and classifications of errors viz. conceptual,

comprehension; procedural and calculative.

Major causes of errors in geometry is attitude towardsgeometry, poor understandings of geometry problems and lackof practice.



Kaur (2010) studied Developmental Changes inConceptual Understanding: A Study of Concepts of

‘Triangle’ and ‘Circle’ in Classes V and VII



The focus

to compare the perceptual and lexical aspects,

to investigate the figural aspects,

overall understanding of these concepts and

social factors(Pedagogical Practices in mathematics,Support available at home regarding learning mathematics,Resources available at school for mathematics.)contributing to the understanding of the students.

The analysis

Theoretical framework of the Houdement and kuzniak’ sgeometrical paradigms (2003) ,

Fischbein theory of figural concepts (1993) and

the mathematical meaning of ‘Triangle’.



TOOLS AND TECHNIQUES:Study was conducted in two phases

Phase 1 included the following tools and techniques: In order to investigate what the word ‘Triangle’ and ‘circle’ evoke

in the mind of the pupils, Four activity were prepared (Vighi, 2005)

Drawing

BrainstormingDefining

Identification

Classroom Observations were done in order to explore thepedagogical practices follow by teachers.

Semi structured interview for teachers for further informationabout the pedagogical practices.

Information schedule for students to get demographic information



ACTIVITY SHEETS FOR STUDENTS

Activity One: DrawingStudents were asked to draw 4 different triangles and threedifferent circles, using a sheet of blank paper and they were freeto draw either using geometrical tools or free hand.

Following tasks were carried out:

Draw a triangle

Draw a different triangle

Draw another different triangleDraw another triangle different from all three.

The children were also asked to explain the choices they made in their drawing and to explain the ‘differences.



Activity Two: Brainstorming

In order to further investigate, Brainstorming activity was carriedout.

Each child was asked about the think of a word ‘triangle’and to draw everything that the word suggest and they see around.

Activity Three: Defining Children were asked to give written explanation of what they

mean by ‘triangle’ in their own words.



Activity Four: Identification Children were asked to look at different shapes and their

features, identify their main characteristics and decide whether those are triangle or non triangle.



As far as developments of these concepts are concerned,researcher found a huge gap between the personal andgeometrical meaning of the concepts among students of class

V and VII.

Some of the major findings related to the Development of theconcept Triangle are as following:

Most of the students gave explanation about the difference that was based upon perceptual aspect of geometrical Orientationi.e. vertex with changing directions (Threja,2010)



Another significant aspect is that the need for diversification

led a number of children to change the sides of the triangle.They thus obtained what they referred to as a “moved”,‘inclined’ triangle’, or a triangle with curved sides or even, moresurprising, a triangle where outer sides are designed. Thusmany children do not have the idea of side or rather they have,a very specific idea, certainly not Euclidean.



The most significant result however was that the triangle is

essentially the same, equilateral and with one side horizontal.It is “the” triangle; some children call it the “normal/simpletriangle”. Students have a perception about triangle whichshould have a horizontal base and a point. The plausibleexplanation is that this type of triangle is usually drawn by

their teacher or most of the time it is in their textbooks.

Looking at the changes from class V to class VII there is notmuch difference found in the drawing of triangles as most of the students drew acute angled triangle which is considered asprototype (Hershkowitz, Vinner and Bruckheimer, 1987) of thetriangle except very few students of class VII who drew rightangled triangle as different one.



Though there is an improvement in the answers of class VII studentsthan class V, analysis of the data shows that instruction provided in theclass may responsible for the quality of answers with regard to thecontrol exerted by the conceptual constraints on the figural

interpretation of geometrical entities as the attributes explained by themost of the students while defining ‘Triangle’ were not being used while identifying those figures (consistent with on field experiences by Ashu Threja 2010).

Analysis of the data also shows that class V and VII students still belongto Geometrical Paradigm 1 (Houdement and Kuzniak,2003), asresponses of most of the students were based more on intuition andexperience than on properties and definitions. Here to note that as perthe three paradigms of geometry, students should be able to usegeometrical instruments in paradigm 1, but most of the class V students were not able to handle the compass.

Another significant aspect that emerges which is consistent with whatDuval explains about three cognitive processes i.e. student could notdraw equilateral triangle though he knows what it is. It meansdevelopment of reasoning or visualization does not assure thedevelopment of construction.



On field experiences (Threja,2010) observe the following

difficulties in understanding geometrical concepts at themiddle grades:

Learners are not able to explore different shapes and cite thedifference between the varied shapes with respect to the

relationships between the corresponding elements of variedgeometrical shapes.

They often lack the knowledge of the language associated withthe spatial concepts

Spatial concepts and skills that were not well learned in theprimary years cause problems into middle school learning of geometry.



Reviews related to understanding algebra

Chauhan (1982) evaluated the students’ achievement in algebra of class X students in Delhi schools. Number correlates, arithmeticoperational sequence and numbers group property were the bestpredictors of achievement in mathematics. General algebraic

reasoning was identified as the only factor common to tests inalgebra.

Bannerji (2000) identified learning difficulties in elementary algebraic concepts namely literal numbers, base, exponent, constant,

variables, coefficients, and algebraic expressions and their causes.

She found a very high error rate in defining the above mentionedconcepts and stating the rules of addition and subtraction of liketerms and algebraic expressions, identification of constants,

variables, types of algebraic expressions, base and exponents in theexponential forms etc.



Ali (2008) studied the understanding of the Algebraic

concept ‘Linear Equations with one variable’ of thestudents of grade VII.

The major focus of the study

To investigate the students’ understanding of linearequations and the difficulties faced by them in solvingthe linear equations using In-depth interview with anachievement test

Analysis Coding system to identify different solution procedures



Findings of the study

The students have a procedural thinking of solving equations such as they have to subtract the number by shifting from one side of the equal sign to theother and unknown should be always on the left side of the equal sign.

Meaning of equal sign ( Threja, 2010)

The results also showed the students' inability to operate with or on theunknown.

The study also explored other difficulties such as a tendency to ignore the

minus sign preceding the number and problems in the acceptance thatunknown number may come on right side of the equal sign and number onthe right side. The results were also consistent with the findings of Herscovicsand Linchevski, 1994.



On field experiences (Threja,2010) observe the followingdifficulties in understanding Algebraic concepts at the middlegrades:

Learners find it difficult to internalize that the variables can beused as unique values or as quantities that vary. It’s observedthat the students think of the former and not the latter.

Learners find it difficult to write the statements given inordinary language to algebraic expressions and equations.Student teachers find difficulty to link between children’sability to process written and spoken language, with theirability to process symbols in mathematical context. They also

find difficulty in formulating their own contextual situations which are familiar to them and expressing as algebraicexpressions and equations.



Simplifying expressions and equations: Students are unsure of

what steps to do and when and how to apply mathematicalproperties. It’s observed that learners find it difficult topreserve equivalence as they continue to simplify. They do notfocus on whole of the expression rather follow a mechanicalleft to right approach in reaching a solution.

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