Geometry, an important branch of Mathematics, has a place in education for the development of critical thinking and problem solving, furthermore, that geometrical shapes are parts of our lives as they appear almost everywhere, geometry is utilized in science and art as well. This paper defines geometry teaching and puts forth why it has been given an important place in teaching mathematics. The major issue the paper deals with is to facilitate teaching Geometry through employing same useful preaches.

ResearchGate Logo

Discover the world's research

  • 20+ million members
  • 135+ million publications
  • 700k+ research projects

Join for free

Journal of Education and Training

ISSN 2330-9709

2018, Vol. 5, No. 1

www.macrothink.org/jet

Perspectives on the Teaching of Geometry: Teaching

and Learning Methods

Hamdi Serin (Corresponding Author)

Ishik University, Iraq

E-mail: Hamdi.serin@ishik.edu.iq

Received: November 7, 2017 Accepted: December 28, 201 7 Published: January xx, 2018

doi:10.5296/jet.v5i1.xxxx URL: http://dx.doi.org/10.5296/jet.v5i1.xxxx

Abstract

Geometry, an important branch of Mathematics, has a place in education for the development

of critical thinking and problem solving, furthermore, that geometrical shapes are parts of our

lives as they appear almost everywhere, geometry is utilized in science and art as well. This

paper defines geometry teaching and puts forth why it has been given an important place in

teaching mathematics. The major issue the paper deals with is to facilitate teaching Geometry

through employing same useful preaches.

Keywords: Geometry teaching, Approaches, Learner

Defining Geometry Teaching

Geometry, as one of the most important branches of Mathematics, has a very significant place

in education. Most of the items that we mostly see and use in our environment are composed

of geometrical shapes and objects. Utilizing these objects and shapes efficiently depends on

understanding the relations among them. We also make use of geometrical thoughts in

solving problems (like painting, lining-wall etc.), in defining the space and running our

profession as well. Geometrical shapes and objects are a part of our jobs and works. Making

effective use of these objects depends on defining them and understanding the relation

between the object and its duty (Altun, 2004:217).

The subjects in geometry are the ones that firstly draw attention of the people. The

requirement to divide a piece of surface properly gave birth to geometry which is the

information of measurement of objects and shapes and expression by the numbers. That's

why this course has direct place in people's daily lives (Fidan, 1986).

Geometry is area of study of mathematics dealing with shapes and space. This area of study

has an important role in developing students' critical thinking and problem solving skills

(Pesen, 2006). Students start to understand and express the world around them by means of

geometry and they analyze and solve the problems. They can also express from the

perspective of the shapes to understand the abstract symbols better. Within this context, they

Journal of Education and Training

ISSN 2330-9709

2018, Vol. 5, No. 1

www.macrothink.org/jet

can understand the shapes around them and can set up connection between daily life and

mathematics.

The first inspiration sources of the mathematics phenomenon are the nature and the life. It is

more required and easier to relate its geometrical side of this phenomenon. What people have

done on behalf of geometry is to see the existing and undeniable truths in the nature and to

take these relations to the new truths and new relations by discovering the relations among

them (Develi and Orbay, 2003). People make decisions in their works and jobs by depending

on their information regarding geometric shapes and objects. Carpenters measure the angles

for house building. Engineers decide on which angles will shape slope of a highway road.

Gardeners plan the geographical formations and positions on which flowers are grown (MEB,

1999:1-3).

The following items can be among some reasons why geometry is given place in mathematics

teaching at schools (Baykul, 2005:363).

1) Critical thinking and problem solving occupy an important role amongst mathematical

studies at school. Geometry studies provide significant contribution to the skills of

critical thinking and problem solving.

2) Geometry subjects give assistance in teaching other topics of the mathematics. For

instance, geometry is utilized to gain the concepts regarding fraction and decimal

numbers; rectangles, squares, areas and circles are mainly used to teach the techniques of

the operations.

3) Geometry is one of the most important parts of the mathematics which is used in daily

life. For example, the shapes of the rooms, buildings and shapes used for ornaments are

geometric shapes

4) Geometry is a device which is used a lot in science and art as well. As an illustration, it

can be said that architects and engineers use geometric shapes a lot; geometrical

characteristics are used quite much in the physics and chemistry.

5) Geometry helps students gain much more awareness about the world in which they live

and appreciate its value. For example, the shapes of crystals and the orbits of the space

objects are geometric.

6) Geometry is a tool that will help students have fun and even make them love

mathematics. For example, they can have enjoyable games with geometrical shapes

through cutting, pasting, rotating, parallel displacement and symmetry. It is required that

a person who will be in charge of teaching and training of students must have

comprehensive knowledge of the subject and must know the growth and development of

human closely.

Geometry is one of the primary courses which are difficult to learn and comprehend for

students. It is a fact that the success level in geometry is low. As a result of this, mathematics

and geometry is a nightmare for most of the students (Akın and Cancan, 2007) because

mathematics is a system on its own. One of the reasons lying beneath this failure is that

Journal of Education and Training

ISSN 2330-9709

2018, Vol. 5, No. 1

www.macrothink.org/jet

geometrical thinking skills of the students are lower than expected. Thus, different teaching

methods must be applied to be able to improve these skills and to make the teaching much

more efficient. Within this context, geometry needs a strong pedagogical approach besides

deep knowledge to be able to provide an enjoyable and intellectual atmosphere for students.

The role of teacher is to guide students to have a better and comfortable thinking rather than

to force students to think in his/her own limits because in today's pedagogical view, knowing

much or having deep knowledge of any subject is not of high importance; the way how

teachers present or guide to get the information occupies more significance. Hence, according

to new teaching approaches, it is required for teachers to try to understand the codes and

perceptions of any students rather than expecting students to understand what is hidden in

teacher's mind.

Geometry Teaching and Learning Methods

Geometry is basically divided into two categories as conceptual part and graphical part.

Teaching these two categories require different approaches. The conceptual parts must be

transformed into perception by visualization; that is to say by the graphical parts. As Duval

mentioned that "their synergy is cognitively necessary for proficiency in geometry" (Duval,

1998). According to him, there are several approaches to the graphical parts, especially

discussing, teaching and interpreting the diagrams: immediate perpetual approach referring

to the interpretation of the diagrams; operative approach that is used to determine the

sub-configurations for problem solving; discursive approach referring to the description of

the problems given.

Fischbein (1993) also treats formal and content parts, in other words figural and conceptual,

as two sides of a coin. In this sense, the teachers must focus on the both figural side and

conceptual part with a specific care since the teachers will describe the geometrical objects

and their relations to each other while figural part will refer to these abstract objects. As

Berthelot stated, in traditional geometry teaching, the theoretical properties, [figural parts] are

assimilated into graphics (Berthelot and Salin, 1998). In this method, students are expected to

read any diagram or graphic only by looking and transform the data given by the diagram in

their minds; this used to be nightmares of the students.

In modern teaching, geometry is supported by the technology besides making use of some

parts of the traditional methods. For example, invariance was put forward by the

Mathematician Felix Klein in 1872 who described geometry "as the study of the properties of

a configuration that are invariant under a set of transformations ….. that [can be illustrated in]

all angle theorems like Thales' theorem and triangles [for example the sum of all internal

angles equals to 180◦]" (as cited in Jones, 2002). For the invariance, the use of d ynamic

software like GeoGebra will be quite useful since students will be able to see any immediate

change and the relations of the angles as whole (Jones, 2002).

Symmetry is another key term for teaching geometry. The conceptual part is illustrated the

visualization process as aforesaid. Defining the equalities between the angles, such as

symmetry in triangles, students must be given a perception about symmetry which can be

Journal of Education and Training

ISSN 2330-9709

2018, Vol. 5, No. 1

www.macrothink.org/jet

illustrated by using Interactive Whiteboard. These boards are very useful to draw such items

as it gives chances to save, to highlight, to use ruler and to zoom in and out (Jones, 2002).

Geometry has been seen as a subdomain of Mathematics throughout the history and even

today in primary schools, geometry is given under Mathematics. As Jones uttered that it "is a

wonderful area of mathematics to teach [and learn] …. It is open to many different

approaches …. [Which] has a long history intimately connected with the development of

mathematics" (Jones, 2002). In this sense, teaching and learning geometry somehow goes

parallel with mathematics training and education.

Most studies have been conducted on the learning styles since 1940 and many learning styles

have been developed. Guild, in his conversation with Brandt, has mentioned that there are

three different learning styles.

The first one is the individual awareness. This one is in fact the view of all learning styles

theories but some educators like Gregorc focuses on this comparing to the other ways.

The second one is curriculum design and implementation of it to the educational processes.

When it is known that the individuals learn in different ways, multi-directional educational

methods can be used. Kolb, McCarthy, Butler et al. are some of the researchers adopting this

idea.

The last approach is diagnostic approach. Individuals' learning styles that can be counted as

key elements are diagnosed and then they are matched with the materials and the education to

be prepared for individual differences. Rita Dunn, Kenneth Dunn and Marie Carbo can be

shown among those who adopt this approach (Brandt, 1990).

There are two different perspectives about how we get the information. The initial one is how

we perceive the information and the latter is how we process the knowledge we perceive.

Each of us comprehends the truths differently and place them in our mind with separate styles.

We are aware of the truths by feeling or observing or thinking or by practicing etc.

(McCarthy, 1987; Morris and McCarthy, 1990). 4MAT learning styles have been taken

account because it is focused on the process and perception of the information on the basis of

McCarthy's learning style. There are four types of learner in 4 MAT learning styles.

McCarthy names tem as first type learners (semiotic learners), second type learners (analytic

learners), third type learners (learners via common sense) and fourth type learners (dynamic

learners). Basic characteristics of the individuals having these learning styles have been

described below (McCarthy, 1982; McCarthy, 1987; McCarthy, 1990).

First type learners (semiotic learners) perceive the knowledge/information through the

concrete experience and process it via reflecting observatory method. They identify their

living and experiences with themselves. They learn the thoughts by listening and sharing.

They are the thinkers who trust in their own experiences. They are very successful in

analyzing direct experiences from different views. They give importance to comprehend the

lowdown of they learn. They need individual caring. They seek for the answer for "Why?"

Journal of Education and Training

ISSN 2330-9709

2018, Vol. 5, No. 1

www.macrothink.org/jet

Second type learners (analytic learners) perceive the knowledge/information through the

abstract experience and process it via reflecting observatory method. They set up theories by

combining observation with existed data. They need to know what specialist know. They

learn via experiences and views by evaluating the accuracy of the information they face.

They give importance to systematic thinking. They like details. They can abolish the

problems through logic, reasoning and analysis. They reanalyze the phenomena when the

conditions turn the things into an incomprehensible manner. They like conventional classes;

the schools are ideal for such students. They seek for the answer for "What?"

Third type learners (learners via common sense) perceive the knowledge/information via

abstract conceptualization method and process it via active experience. They focus on the

result. They unify the theory and practice. They learn the theories by testing. They learn well

through manual techniques. They are perfect in problem solving. They do not like being

given the answers; they need the method to solve the problems on their own. They pay

attention to the strategical thinking. They are the students who are directed to the skills. They

make experiments and they forward an idea on these experiments. They want to know of the

formulas. They seek for the answer for "How?"

Fourth type learners (dynamic learners) perceive the knowledge/information through the

concrete experience and process it via active experience. They unify experience and practice.

They learn via trial and error method. They have confidence in self-discovery. They are

excited about new things; they like researching. They reach at the correct results in the

situations in which logical conditions do not exist. They solve the problems by their intuitions.

They like taking risk. They want to know what they can do with the objects and the formulas.

They are impatient sometimes. School is a tedious place for such individuals. The school s

cannot satisfy the needs of those individuals as they demand to be persuaded for their interest

through different methods. They seek for the answer for "What if …?"

It is very important to evaluate individuals' learning styles for the process of learnin g and

teaching (Hein and Budny, 2000). The information which is obtained by the determination of

students' learning styles is helpful to decide on the method to be employed (Akkoyunlu,

1995). As Babadoğan (2000) stated that if the learning styles of the in dividuals are

determined, then it will be much easier to understand that how the persons learn and what

type of teaching method will be applied. Hence, the teacher can formulate appropriate

teaching environments both for himself/herself and for the students. There are many

researches that matching teaching and learning styles affect the students' success (Scales,

2000). It is expected that if the teachers' pay required attention to the teaching strategies that

will co-operate with the learning style of the students, students will be more successful.

Peker and Dede (2005) have analyzed the relationship between learning styles of

mathematics teaching candidates and their attitudes towards geometry in their study. The

results have shown that the candidates have different learning styles; they have mentioned

that even if their learning styles do not affect their attitudes against geometry, learning styles

are the factors which must be taken into consideration from secondary school to university,

even to the in-service training.

Journal of Education and Training

ISSN 2330-9709

2018, Vol. 5, No. 1

www.macrothink.org/jet

Trying to materialize the geometry subjects requiring abstract thinking skill by using the tools

like ruler, pen, paper and etc. may not be enough to internalize and visualize the relevant

concepts and the rules. Hence, it is necessary to provide learning the geometry subjects in a

much more detailed way, to follow the new tendencies in the education technologies

regarding the visualization of the abstract concepts and to make use of them appropriately.

References

Akın, Y., & Cancan, M. (2007). Matematik Öğretiminde Problem Çözümüne Yönelik

Öğrenci Görüşleri Analizi. Kazım Karabekir Eğitim Fakültesi Dergisi, 16, 374-390 .

Akkoyunlu, B. (1995). Bilgi Teknolojilerinin Okullarda Kullanımı ve Öğretmenlerin Rolü.

Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 11 , 105-109.

Altun, M. (2004). Matematik Öğretimi. İstanbul: Alfa Yayıncılık.

Babadoğan, C. (2000). Öğretim Stili Odaklı Ders Tasarımı Geliştirme. Milli Eğitim Dergisi,

147, 61-63.

Baykul, Y. (2005). İlköğretimde Matematik Öğretimi. An kara: Pegema Yayıncılık

Berthelot, R., & Salin, M. H. (1998). The role of pupils' spatial knowledge in the elementary

teaching of geometry. In C. Mammana & V. Villani (Eds.), Perspectives on the Teaching

of Geometry for The 21st Century (pp. 71–77). Dordrecht, The Netherlands: Kluwer.

Brandt, R. (1990). On Learning Styles: A Conversation with Pat Guild. Educational

Leadership, 48(2), 10-13.

Develi, H. M., & Orbay, K. (2003). İlköğretimde Niçin ve Nasıl Bir Geometri Öğretimi, Milli

Eğitim Dergisi , 157, 115-122.

Duval, R. (1998). Geometry from a cognitive point of view. In C. Mammana & V. Villani

(Eds.), Perspectives on the Teaching of Geometry for The 21st Century (37–52).

Dordrecht, The Netherlands: Kluwer.

Fidan, N. (1986). Okulda Öğrenme Ve Öğretme. Ankara: Kadıoğlu Matbaası

Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics,

24(2), 139 –162.

Hein, T. L., & Budny, D. D. (2000). Styles and Types in Science and Engineering Education.

Paper Presented at International Conference on Engineering and Computer Education ,

Sao Paulo, Brazil.

Jones, K. (2002). Issues in the Teaching and Learning of Geometry. In L. Haggarty (Eds),

Aspects of Teaching Secondary Mathematics: perspectives on practice. (Chapter 8 pp.

121-139) London: Routledge Falmer.

McCarthy, B. (1982). Improving Staff Development through CBAM and 4MAT. Educational

Leadership, 40(1), 20-25.

Journal of Education and Training

ISSN 2330-9709

2018, Vol. 5, No. 1

www.macrothink.org/jet

McCarthy, B. (1987). The 4MAT System: Teaching to Learning Styles with Right/Left Mode

Techniques. Barrington: Excel, Inc.

McCarthy, B. (1990). Using the 4MAT System to Bring Learning Styles to Schools.

Educational Leadership, 48(2), 31-37.

MEB. (1999). İlköğretim Okulu Matematik Dersi Öğretim Programı, Ankara: Kocaoluk

Yayıncılık.

Morgan, H. (2010). Teaching with the Interactive Whiteboard: An Engaging Way To Provide

Instruction. Focus on Elementary, 22(3), 3-7.

Peker, M., & Dede, Y. (2005). Matematik öğretmeni adaylarının öğrenme stilleri ve geometri

tutumlari arasindaki ilişki. Sakarya Üniversitesi Eğitim Fakültesi Dergisi, 10, 61 – 69.

Pesen, C. (2006). Matematik Öğretimi (3. Baskı). Ankara: Pegem yayıncılık.

Scales, A. Y. (2000). The Effect of Learning Style, Major, and Gender on Learning

Computer- aided Drawing in an Introductory Engineering/Technical Graphics Course.

Unpublished PhD thesis, North Carolina State University.

Copyright Disclaimer

Copyright reserved by the author(s).

This article is an open-access article distributed under the terms and conditions of the

Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

... Geometry is an important and essential branch of mathematics, which studies shapes and space, in two-dimensional (2D) and three-dimensional (3D), along with their characteristics (Aktaş & Aktaş, 2012;Paulina, 2007, in Fabiyi, 2017Serin, 2018). Understanding of geometry concepts is essential and it should be developed effectively in learning mathematics because it allows students to analyze and interpret the world they live in as well as equip them with tools they can apply in other areas of mathematics (Özerem, 2012). ...

... Traditional learning methods made students learn passively (Li, 2016) since they have too much time listening to facts and watching teachers introduced geometry concepts and theorems on the board and in front of the classroom without any students' contribution in formulating the knowledge (Abdelfatah, 2011). Whereas, geometry needs a strong pedagogical approach besides deep knowledge to be able to provide an enjoyable atmosphere for students (Serin, 2018). Serin (2018) also added that the role of a teacher is to guide students to have better and comfortable thinking rather than to force students to think in his/her limits. ...

... Whereas, geometry needs a strong pedagogical approach besides deep knowledge to be able to provide an enjoyable atmosphere for students (Serin, 2018). Serin (2018) also added that the role of a teacher is to guide students to have better and comfortable thinking rather than to force students to think in his/her limits. Therefore, it is necessary to find a learning source that can be used to present the concepts understandably, so that it can help students have a better understanding of geometry concepts and encourage active involvement in their learning process. ...

  • Risma Nurul Auliya
  • Munasiah Munasiah

The study investigates the impact of using AR in learning 3D geometry on students' attitudes and conceptual understanding. The research was carried out using a quasi-experimental design, in particular by using a conceptual understanding test and attitude towards geometry questionnaire, in a public secondary school in Rangkasbitung, Banten, Indonesia. Two groups of students were used for data collection; the experimental (31 students) and the control groups (30 students). The data analyzed by independent sample t-test, using a significant level of 0.05. The results of this study were: (1) students' conceptual understanding in mathematics in AR group was better than the control group, (2) students' conceptual understanding in mathematics for both groups was in the medium category, and (3) students' attitude towards geometry in AR group was better than the control group. Therefore, the appropriate method that is used in learning geometry can affect students' conceptual understanding and attitude towards geometry. Additionally, the learning environment also needs to be teachers' attention, because it can affect students' involvement in the learning process.

... On the other hand, skills trained in abstract contexts can be applied in different practical contexts. This aim of geometry teaching was confirmed by many authors [23][24][25]. In addition, the authors of [25] also stressed the role of manipulation and the creation of geometrical images in developing the so-called spatial intuitive skills. ...

... contexts. This aim of geometry teaching was confirmed by many authors [23][24][25]. In addition, the authors of [25] also stressed the role of manipulation and the creation of geometrical images in developing the so-called spatial intuitive skills. ...

  • Maria Kmetova Maria Kmetova
  • Zuzana Nagyová Lehock

Creating a mental image of our spatial environment is a key process for further abstract geometric thinking. Building a mental representation can be understood as a part of the process of visualisation. From the wide concept of visualisation, in this article, we will focus on the part where the mental representation of spatial relations, mental objects and mental constructions are created, and their manifestations as a 3D physical object and its plane representations arise. Our main goal is to follow the transition between 2D and 3D representations of physical objects and also to observe how and when such a transition happens in students' thinking. For that purpose, we also use Tangram, because manipulation with the Tangram pieces in space and filling out planar figures by them indicates the transition between 3D and 2D. Our research, using an action research methodology, was conducted on the students of three 5th grade primary school classes as a part of a larger long-term project. We pointed out a relationship between spatial abilities and the perception of 2D–3D relationships in students' mind.

... Transformation content is one of the most significant resources for students to learn [1,2]. Transformation geometry is essential since numerous transformation ideas are utilized in daily life [3,4] and offers a valuable visual source for recognizing arithmetic, algebraic and statistical concepts [5]. In addition, Edwards [6] confirms that the transformation of geometry offers students sufficient potential to enhance their ability to visualize space and think geometrically. ...

... For example, the researchers presented the material to the eighth-grade students using puppets with geometric shapes (Yilmazer & Keklikci, 2015) or using visualization to illustrate functions (Makonye, 2014). Likewise, another learning geometry approach is materializing the topics (Serin, 2018;Wierzchon & Klopotek, 2018). Various efforts to learn geometry show that visualization is the standard method (Baiduri et al., 2020;Bråting & Pejlare, 2008;Claudia et al., 2015;Presmeg, 2006). ...

  • Hanna Arini Parhusip Hanna Arini Parhusip
  • Hindriyanto Dwi Purnomo
  • Didit Budi Nugroho
  • Istiarsi Saptuti Sri Kawuryan

Geometry is one of the particular problems for students. Therefore, several methods have been developed to attract students to learn geometry. For undergraduate students, learning geometry through surface visualization is introduced. One topic is studying parametric curves called the hypocycloid curve. This paper presents the generalization of the hypocycloid curve. The curve is known in calculus and usually is not studied further. Therefore, the research's novelty is introducing the spherical coordinate to the equation to obtain new surfaces. Initially, two parameters are indicating the radius of 2 circles governing the curves in the hypocycloid equations. The generalization idea here means that the physical meaning of parameters is not considered allowing any real numbers, including negative values. Hence, many new curves are observed infinitely. After implementing the spherical coordinates to the equations and varying the parameters, various surfaces had been obtained. Additionally, the differential operator was also implemented to have several other new curves and surfaces. The obtained surfaces are useful for learning by creating ornaments. Some examples of ornaments are presented in this paper.

... Türkiye ve Almanya'da uygulanan matematik dersi öğretim programı incelendiğinde ise kazanımların önemli bir bölümünün geometri ile ilgili olduğu görülmektedir (MEB, 2018a; Ministerium für Schule und Bildung des Landes Nordrhein-Westfalen [MSB NRW], 2019). Matematikte önemli bir yere sahip olan geometri öğrenme alanı, birçok alanda eleştirel düşünme ve problem çözme becerilerin gelişimi açısından önemli bir araç olarak görülmekte (National Council of Teachers of Mathematics [NCTM], 2000;Serin, 2018) ve çevrenin daha gerçekçi biçimde tanınmasını, değerlendirmesini ve analiz etmesini kolaylaştırmaktadır. Örneğin, geometri öğrenme alanı kapsamında yer alan "geometrik cisimler ve şekiller", "uzamsal ilişkiler", "geometrik örüntüler", "geometride temel kavramlar" alt öğrenme alanları ile öğrencilerin şekil modelleri oluşturmaları beklenmekte ve geometrik cisimleri günlük hayattan verilen örneklerle sınıflandırmaları istenmektedir (MEB, 2018a). ...

Bu çalışmada, Türkiye ve Almanya'da uygulanan 5. ve 6. sınıf matematik dersi öğretim programları geometri öğrenme alanı bağlamında karşılaştırılmış ve alt öğrenme alanları, kazanımlar, uygulama önerilerine yönelik benzerlik ve farklılıklar ortaya konmuştur. Nitel araştırma yaklaşımının benimsendiği bu çalışma, bir ülkeler arası karşılaştırmalı eğitim çalışmasıdır ve doküman incelemesi ile toplanan veriler betimsel analiz ile çözümlenmiştir. Yapılan analizler sonucunda, Türkiye'de uygulanan programın alt öğrenme alanlarının nicel olarak daha fazla olduğu ve konuların sınıf düzeylerine göre bölünerek dağıtıldığı görülmüştür. Almanya programında ise alt öğrenme alanlarının daha sade ve genel ifade edildiği sonucuna ulaşılmıştır. Diğer taraftan, Türkiye programındaki kazanımların daha ayrıntılı sunulduğu ve çeşitli etkinlik örneklerine yer verildiği belirlenirken; Almanya programındaki kazanımların daha genel ve birleştirilmiş bir şekilde tek bir kazanım altında ele alındığı, etkinliklere ise öğretmenlerin karar verdiği ya da çekirdek öğretim programına dayalı olarak hazırlanan örnek okul programında yer verildiği görülmüştür. Türkiye programında öğrencilerden beklenen işlem becerisinin Almanya programına göre daha fazla olduğu gözlemlenmiştir. Her iki ülkenin ortak uygulama önerisi olarak konuların somut modeller ve gerçek yaşam durumları ile ilişkilendirilerek sunulduğu belirlenmiştir. Ayrıca, Türkiye ile diğer ülkelerin matematik dersi öğretim programlarını karşılaştıran çalışmaların en çok programın ögelerine yoğunlaştığı görülmüştür. ABSTRACT In this study, a comparison of the 5th and 6th grade school mathematics curriculum of Turkey and Germany in the context of geometry learning domain has been made, similarities and differences are revealed about the sub-learning domains, learning outcomes, implementation suggestions. This study, in which a qualitative research approach was adopted, is a cross-national comparative education study in terms of its subject and data collected by document analysis were analyzed by descriptive analysis. As a result of the comparisons made in the context of the geometry learning domain, it was seen that the Turkey program has more sub-learning domains, more detail, divided into two grade levels. In the Germany program, it was concluded that sub-learning domains were expressed more simply and generally. The outcomes in the Turkey program presented in more detail including examples of various activities; the outcomes in the Germany program were handled in a more general and unified manner under a single outcome, were decided the activities by teachers. Moreover, it was observed that in terms of processing skill expected from students, the Turkey program is more comprehensive than the Germany program. It was seen that the subjects were presented in relation to concrete models and real life situations as an implementation suggestion in the both countries. In addition, it was observed that the studies compared in terms of mathematics curriculum of Turkey and the other countries most focused on the elements of the program.

... Namun, beberapa penelitian menyebutkan bahwa tingkat pemikiran geometris siswa tingkat pendidikan dasar dan menengah masih berada di bawah tingkat yang diharapkan (Alex & Mammen, 2012;Serin, 2018). Kesulitan siswa pada materi geometri karena konsep dasar yang harus dikuasai lebih abstrak daripada bidang pembelajaran matematika lainnya (Özdemir, 2017;Adelabu, et al., 2019). ...

Geometry is an essential subject for students. However, geometry, especially the cone, remains a challenging material for students. Besides, we need the proper media and context for learning mathematics on cone material. This research aims to develop a learning trajectory that utilizes the Megono Gunungan tradition as a context in the learning process using the Indonesian Realistic Mathematics Education (PMRI) approach. The research method used is Design Research, which consists of three phases of the study: preliminary design, experimental design, and retrospective study. This study's outcome is designing a learning trajectory on cone material using the Megono Gunungan tradition. This learning trajectory comprises four activities, including observing the interactive video of the Megono Gunungan tradition to discover cone elements, finding the concept surface area of a cone using origami paper, and grasping the cone concept volume with magic seeds, and solving contextual problems related to the cone. Studies have shown that using the Megono Gunungan context could help students improve their understanding of the cone concept. Furthermore, this study hoped could be an inspiration for exploring another local wisdom that can be a context in learning mathematics.

... Geometri merupakan salah satu cabang matematika yang penting dalam kurikulum matematika di berbagai dunia (Serin, 2018). Pada konteks kurikulum matematika di Indonesia, hal tersebut dapat dibuktikan dari distribusi penyebaran kompetensi dasar (KD) mata pelajaran matematika untuk satuan pendidikan Sekolah Dasar (SD) pada materi geometri mendapatkan porsi yang cukup besar (35%) dibandingkan dengan materi statistika dasar (10%). ...

Penelitian ini bertujuan untuk mengeksplorasi kesulitan-kesulitan siswa sekolah menengah pertama dalam menyelesaikan permasalahan luas permukaan limas. Penelitian ini merupakan penelitian kualitatif dengan desain exploratory case study. Proses eksplorasi pada penelitian ini dilakukan pada partisipan yang berjumlah 13 siswa kelas VIII pada salah satu SMP Swasta di Kabupaten Indramayu. Pengumpulan data dalam penelitian ini menggunakan tes, wawancara, dokumentasi, dan catatan lapangan. Sedangkan teknik analisis data yang digunakan mengadopsi modifikasi analisis data dari Miles dan Huberman yang terdiri dari pengumpulan data, pengodean (coding), reduksi data (data reduction), penyajian data (data display), dan penarikan kesimpulan (verification). Berdasarkan temuan penelitian, didapat bahwa kesulitan belajar yang dialami siswa pada saat menyelesaikan soal materi luas permukaan limas yaitu: (1) kesulitan dalam mengidentifikasi masalah yang terdapat pada soal; (2) kesulitan dalam mengoneksikan konsep luas permukaan dengan konsep materi prasyarat; (3) kesulitan dalam menentukan strategi penyelesaian soal tersebut; (4) kesulitan dalam menggunakan operasi yang melibatkan perkalian bilangan pecahan atau bentuk akar; (5) kesulitan dalam menggunakan formula konsep luas permukaan limas. Selain itu hasil temuan ini memberikan implikasi kepada guru untuk dapat mempersiapkan desain bahan ajar atau pembelajaran untuk mengatasi kesulitan tersebut.

... Οπωσδήποτε τα Μαθηματικά είναι μάθημα που απαιτεί την ενεργοποίηση διαφόρων δεξιοτήτων που σχετίζονται με την Γλώσσα (Λεκτική), την Μνήμη (Εργασίας και την Μακροπρόθεσμη), την Οπτικοχωρική Αντίληψη, τις Λογικές και Μεταγνωστικές Δεξιότητες. Όπως αναφέρει ο Serin (2018): Κάποιοι μαθητές, αντιλαμβάνονται τη γνώση, την πληροφορία μέσω της συγκεκριμένης εμπειρίας και την επεξεργάζονται μέσω της αντανακλαστικής μεθόδου παρατήρησης. Είναι οι στοχαστές που εμπιστεύονται τις εμπειρίες τους. ...

  • Γιάννης Νικολόπουλος

Η εισήγηση έχει σαν βάση την βιβλιογραφική ανασκόπηση και την εμπειρία της μακρόχρονης διδασκαλίας, επίσης υποστηρίζεται έμμεσα από έρευνα στα πλαίσια διατριβής που εστιάζεται στον Όγκο των Στερεών και αξιοποιεί τα επίπεδα Van Hiele σε μαθητές του Γυμνασίου. Το ερώτημα που απασχολεί είναι πώς και πόσο μπορούμε να αναπτύξουμε τα Χαρισματικά παιδιά; Αλλά ας δούμε πώς ορίζουν οι 'ειδικοί' σήμερα την Χαρισματικότητα; Υπάρχουν πολλοί ορισμοί που έχουν 'ξεπεράσει' ή αλλιώς δεν αρκούνται στον Δείκτη Νοημοσύνης (Δ.Ν.) με ακραίο παράδειγμα τον Gardner που προτείνει τα οκτώ είδη Νοημοσύνης. Εμείς στηριζόμαστε σε ορισμό που διατύπωσαν οι Renzulli και Reis (1994), όπου η Χαρισματικότητα συνίσταται σε συμπεριφορές βασικών ανθρώπινων γνωρισμάτων: 1. Την Νοητική Ικανότητα, με τον Δ.Ν. άνω του Μέσου Όρου. 2. Του υψηλού βαθμού προσήλωσης στο έργο (πιθανώς λόγω παρελθόντων γνώσεων και επίσης ενδιαφέροντος) και 3. Του υψηλού βαθμού δημιουργικότητας. Για να εκπαιδεύσουμε Χαρισματικούς Μαθητές πρέπει καταρχάς να επιμορφώσουμε κατάλληλα εκπαιδευτικούς και να δούμε τις ενότητες στις οποίες θα επενδύσουμε για να βελτιώσουμε τους εν λόγω μαθητές. Να σημειώσουμε ότι εκτός από τα Γλωσσικά πλεονεκτήματα που διαθέτουν οι χαρισματικοί, έχουν επίσης σπουδαίες επιδόσεις στα Μαθηματικά.Θα στηριχθούμε στην θεωρία των Επιπέδων Van Hiele σε συνδυασμό με Ζώνη Επικείμενης Ανάπτυξης (ZEA) του Vygotsky. Επίσης θα σταθούμε στον τρόπο που η σημερινή κοινωνία αντιμετωπίζει τα παιδιά που τα ονομάζουμε Χαρισματικά και τι κάνουν οι αρμόδιοι φορείς στην κατεύθυνση της μόρφωσης.

  • Keith Jones Keith Jones

This chapter analyses a range of key issues in the teaching and learning of geometry. These include the nature of geometry, why geometry is important in the curriculum at school level and beyond, what geometry can be included at the school level, the aims of teaching geometry, and how geometry can be best taught and learnt. Also addressed are the uses of information and communication technology in geometry education. The chapter concludes that the twenty-first century is one where spatial thinking and visualisation are vital areas for education.

  • Bernice McCarthy

The seven-step Concerns-Based Adoption Model (CBAM) outlines the stages people move through when adopting an innovation. The 4Mat System applies learning style research and research on brain dominance to teaching practices. When combined, the two systems form a comprehensive model for staff development. (PGD)

  • Bernice McCarthy

4MAT is an eight-step instructional cycle that capitalizes on individual learning styles and brain dominance processing preferences. The four major learners (imaginative, analytic, common sense, and dynamic) can use 4MAT to engage their whole brain. Learners use their most comfortable style while being challenged to function in less comfortable modes. Includes 13 references. (MLH)

  • Efraim Fischbein

The main thesis of the present paper is that geometry deals with mental entities (the so-called geometrical figures) which possess simultaneously conceptual and figural characters. A geometrical sphere, for instance, is an abstract ideal, formally determinable entity, like every genuine concept. At the same time, it possesses figural properties, first of all a certain shape. The ideality, the absolute perfection of a geometrical sphere cannot be found in reality. In this symbiosis between concept and figure, as it is revealed in geometrical entities, it is the image component which stimulates new directions of thought, but there are the logical, conceptual constraints which control the formal rigour of the process. We have called the geometrical figuresfigural concepts because of their double nature. The paper analyzes the internal tensions which may appear in figural concepts because of this double nature, development aspects and didactical implications.

Matematik Öğretiminde Problem Çözümüne Yönelik Öğrenci Görüşleri Analizi

  • Y Akın
  • M Cancan

Akın, Y., & Cancan, M. (2007). Matematik Öğretiminde Problem Çözümüne Yönelik Öğrenci Görüşleri Analizi. Kazım Karabekir Eğitim Fakültesi Dergisi, 16, 374-390.

Öğretim Stili Odaklı Ders Tasarımı Geliştirme. Milli Eğitim Dergisi

  • C Babadoğan

Babadoğan, C. (2000). Öğretim Stili Odaklı Ders Tasarımı Geliştirme. Milli Eğitim Dergisi, 147, 61-63.