Available online at www.sciencedirect.com
ScienceDirect Procedia - Social and Behavioral Sciences 209 (2015) 519 – 525
International conference “Education, Reflection, Development”, ERD 2015, 3-4 July 2015, Cluj-Napoca, Romania
A tool for solving geometric problems using mediated mathematical discourse (for teachers and pupils) Nava Kivkovicha* a
Doctoral School “Education, Reflection, Development”, Babeș-Bolyai University Cluj-Napoca, 7 Sindicatelor Street, Cluj-Napoca, 400015, Romania
Abstract The research deals with teaching strategies in geometry. A strategic mediating tool to resolve geometric problems was developed. Underpinning theories are: Piaget's theory (1960), Van Hiele (1959), Vygotsky's socio-cultural communication approach and Feuerstein's theory of mediated learning (1998). The research is based a combination of teaching strategies focused on significant mediation between the material and heterogeneous groups of learners. The innovation is in providing a holistic response to the needs of pupils and geometry teachers. Mixed methods research will examine the influence of this tool on high school pupils and their attitudes to mathematics in general and geometry in particular © 2015 The Authors. Published by Elsevier Ltd. ThisLtd. is an open access article under the CC BY-NC-ND license © 2015 The Authors. Published by Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Scientific Committee of ERD 2015. Peer-review under responsibility of the Scientific Committee of ERD 2015 Keywords: teaching strategies; geometry; mediation; mathematics; pupils' attitudes
*
Corresponding author: Tel: +972-52-2906258 Email address:
[email protected]
1877-0428 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Scientific Committee of ERD 2015 doi:10.1016/j.sbspro.2015.11.282
520
Nava Kivkovich / Procedia - Social and Behavioral Sciences 209 (2015) 519 – 525
1. Introduction – Rationale Formal geometry is the most problematic subject in middle school and high school math studies. The difficulty stems from the need to understand the language of mathematics in the field of geometry while integrating it with prior knowledge, parallel to the level of the student's mental development. To help students overcome the difficulties in learning geometry a program was developed based on a tool combining use of teaching strategies, through mediation, such as proper use of mathematical and geometric language in particular, including visual marking and identifying memory supports to prior knowledge and using it in the process of finding a solution. School's teaching-learning-evaluation processes are focused on turning private knowledge into public knowledge for the students. Through "innovative pedagogy", it will be possible to ensure the required achievements, as well as the quality of the teaching-learning and assessment processes. It can be said that quality (good) teaching is teaching that creates motivation and desire to learn. The students are active in the learning process, generate meaning from it, and reach the set goals (Vidislavski, Peled and Pevsner, 2010). Characteristics of quality teaching, learning and evaluation processes as detailed in the Director General's Circulars in Israel (2012) emphasize perceptions of learning as a personal, conscious and intelligent process that takes place in a social context within reciprocal relations between significant adults and the peer group. The significance of focusing teaching-learning-assessment processes in particular and emphasis on the individuality of each pupil while adapting teaching to their differences with regard to their aims, needs, areas of interest, preferences, previous experiences and giving them the opportunity to experience a wide range of processes to construct knowledge in a range of frameworks and pathways, while removing barriers between the worlds outside and within schools. 1.
Theoretical Foundation and Related Literature
Learning geometry as a subject has resulted in many theories that sometimes supported or opposed one another. This article will review three theories on the subject of learning geometry and discuss pupils' abilities: Piaget's theory, Van Hiele's theory and Vygotsky's socio-cultural communication approach. It will then review the subject of teaching strategies concentrating on the Mediated Teaching approach and teacher-pupil discourse. According to Piaget's model, the development of thinking and awareness is part of biological development. The structure of thinking is hierarchical, with concrete thinking at the bottom and formal thinking at the top. According to Piaget (1969), transition from stage to stage is a result of situations of imbalance that constitute a stimulus to reach the next stage on the developmental ladder. According to Piaget's theory (Piaget, Inhelder & Szeminska, 1960), there are three stages, to which he refers as periods: the first is the pre-operational stage (ages 4-7), the second is the concrete operative stage (ages 7-12) and the thirst if the formal operative stage (ages 12 and over), which is the stage at which Euclidean qualities are acquired by causality of logic. That is to say that thinking at this stage is no longer dependent on the concrete, and deductive, conjectural thinking has developed. That is to say that development, according to Piaget, is a result of evolution from stage to stage. The transition mechanisms are created as a result of maturation, interactions with the social environment and self-regulation or balance. The accepted central theory dealing with the development of geometric thinking is the Van Hiele couple's theory (1959). Like Piaget, their theory deals with different stages of mental development in geometry learners, according to which the acquisition of skills is basic to learning geometry, but there is a need to pupils to reach a required level of mental development in order to understand geometry (Sarfaty & Patkin, 2011). In contrast to Piaget's theory, the Van Hiele theory is based on the assumption that progress from one level to the next is dependent on teaching more than age or biological maturity, and even that teaching methods can influence progress or lack thereof in different ways. According to Hershkovitz (1990) and Patkin (1994), the Van Hiele theory, follows in the footsteps of the two key aims of geometry learning: geometry, as a body of knowledge that helps pupils to arrange their geometrical environment, is expressed at the first and second levels whereas geometry as a deductive system is expressed and the fourth and fifth levels. Mental development can be arranged hierarchically in five stages according to stages of development: recognition, analysis, ordering, deduction, and rigor. Van-Hiele's theory can be summarized by the following two hypotheses: The first is the highest purpose of learning geometry is to present a deductive system and the second is
Nava Kivkovich / Procedia - Social and Behavioral Sciences 209 (2015) 519 – 525
521
that without mastering geometry as "information space", there is no possibility of understanding geometry as a "deductive system". In contract, Vygotsky (1978) emphasized socio-cultural processes as the root of cognitive internal changes. Vygotsky transferred the focus of interest to the developing discourse between learners and teachers - learning is a process that takes place in an interpersonal space between learners and significant others, in areas in which learners want to develop their abilities and knowledge. A developing child has the ability to learn and this is put into practice through contact with an environment mediated by an adult, who serves in an active role and is aware of this role. The role of adults as teachers is to identify this need and create for learners a zone of proximal development as a result of which Vygotsky defined a developmental situation as a situation in which a learner is already capable of acting without help to perform a task (Zellermayer & Kozulin, 2004). According to Vygotsky (1978), the development process starts with interactions between children and adults. This interaction is internalized over time and becomes an internal cognitive tool. A learner's participation in dialogues with adults is what creates high level learning. That is to say that a pupil's progress from spontaneous concepts (which children develop naturally) and more formal, scientific concepts, whose roots are in culture and school teaching. Consequently, he argued that good learning is the only thing that advances real development in children. In reality it creates the following zone of proximal development. Learning does not occur all the time, it takes place in a context that is culturally authentic for learners and with regard to the quality of learning. The development of scientific concepts occurs when reciprocal links are created between new scientific and daily concepts of children. As Mathematics is considered to be an exact science in which precise concepts are used, it is important to use them as intended by their definitions and thus, correct and appropriate mediation by teachers is necessary in order to help children move from spontaneous to scientific concepts. In contrast to traditional teaching, this mediation, requires a different way of teaching and learning and focuses on encouraging pupils to think and converse about mathematical problems and concepts, in order to develop independent, investigative and understanding learners rather than those who focus on repetition and memorization. 2.
Teaching Strategies In order to test these approaches and methods of developing pupils, we will examine those methods and approaches that have produced better results in developing pupils and making them think, deal with and resolve problems. Teaching strategies are defined as the means used by teachers to achieve the aims of a lesson (Schroeder et al., 2007). In studies about learning strategies, researchers generally agree about the contribution of learning strategies to the quality of learning. The more supportive and transferrable a strategy is, the more effective it is in teaching and learning (Nissim, Barak & Ben-Zvi, 2012). Therefore, a mediated learning approach on which the tool to solving geometrical problems will be based, will now be presented. As previously mentioned, a number of approaches blossomed at the beginning of the 20 th century, such as Vygotsky's (1978), whose approach was found to be relatively limited in terms of the perception of learners' potential to change dramatically with regard to the development of tools in different area of functioning and interpretation of changes that individuals undergo. Professor Feuerstein's approach (1998) chose to focus on qualities of mediated teaching interactions of adults and children. According to Feuerstein (1998), people, especially children, have the ability to think. This ability has a decisive influence on people's ability to choose, make decisions rationally and organize data in order of preference. People are able to acquire not only knowledge or skills, but new cognitive constructs. Feuerstein (1998) maintained that cognitive modifiability is an option but not everyone realizes it, as in order to do an investment of effort and means is required. This ability is acquired through mediation, meaning through mediated learning in which people interact with their environment through direct exposure to stimuli. According to Feuerstein's theory of mediated learning (1998) the concept of mediation is defined as a teaching behavior based on adults' intentions to transfer information or any type of message to children, focus their attention, clarify connections, include them in emotional experiences and awaken them to develop values, tendencies to action, world views and the like. This is learning that transfers not only information or skills, but mainly provides ways of observation, terms of reference and ways of looking for the connections between them There are three definitive criteria necessary for mediated interactions to take place. The first is intentionality and reciprocity. According to Tal (2004), it is a conscious attempt by mediators or learners, to focus the attention of
522
Nava Kivkovich / Procedia - Social and Behavioral Sciences 209 (2015) 519 – 525
another party in an interaction, to awaken their reciprocity and relationship with the partner. Egozi and Feuerstein (1987) added that the principle of intentionality and reciprocity can be expressed through verbal or non-verbal behavior or a combination of the two. Intentionality and reciprocity are regarded as a definitive condition, but are not enough for mediated learning. Two additional criteria must be used: transcendence and meaning. Transcendence, according to Egozi and Feuerstein (1987) is intrinsically the nature of sound mediated learning that encompasses just a small number of stimuli that reach children, such that most situations are a direct confrontation between people and stimuli. In order for intentionality and reciprocity to exist, and that it be considered a mediated activity between adult and child, it must train recipients of mediation to deal with stimuli in an unmediated way. Through transcendence, mediators produce a continuous expansion to recipients' systems of needs, beyond basic existing ones. Through the principle of transcendence, children learn to deepen and expand their knowledge and means of investigation and to better organize available knowledge that is beyond learning. That refers to meta-cognitive foundations that lead learners to high levels of thinking and processes of reaching conclusions. Moreover, this principle can appear in interactions between mediators and learners (Tal, 2004). Another criterion is mediation of verbal and emotional meaning. Klein (1991) defined mediation of meaning to include reference to verbal and non-verbal-emotional meaning, thoughts, feelings, phenomena, things, animals, etc. Egozi and Feuerstein (1987) defined verbal meaning broadly to include purposes of interaction, comparisons between phenomena and connections to past and present processes. Similarly, Doll (1999) went even further and emphasized that most learning activity's purpose is to encourage thinking and personal expression and as such, it must concern significant involvement of learners in learning that is meaningful to them. These criteria are integrated into mediated learning and create a unique quality in interactions between children and mediators that promote flexibility, emotion and willingness to understand what is taking place and to generalize beyond a single phenomenon. Therefore, mediators must use all available tools to mediate as completely as possible. Thus, how can teachers be seen as mediators in current learning processes? Darling-Hammond (2005 in Levy-Feldman, 2010) compared teachers to orchestra conductors, who to those watching from the side appears to be waving their arms in the air, but their ability to identify, discern, arrange, encourage, communicate and time is that which produces the final result. Teachers, with their expertise and knowledge, are able to lead a group of learners from one level of understanding to another. Therefore their teaching requires creativity, involvement and flexible understanding, over and beyond learning information, about their learners - how they learn, what motivates each one of them, their previous experiences, perceptions etc. In addition, teachers who take a dialogic stand with regard to their pupils present themselves as neither knowing nor not knowing everything all the time. This stance invites active understanding, that is to say, that pupils must process knowledge and deal with questions and teacher must serve as role models, which does not mean that they should appear as an empty vessel, the opposite is true, teachers must have a deep understanding of lessons' contents, but they must also show that they are able to learn from their pupils, as if they have removed the uniform of their profession (Peled & Blum-Kulke, 1996). A further emphasis in dialogic discourse is that questions are based on active listening and enable negotiation about knowledge that gradually comes into being. This knowledge is perceived as something that should be created with the help of consideration, doubts and examination. Teachers must exercise responsible judgment in properly managing this process (Epstein-Yanay, 2006). The relationship between participants is reciprocally mediated, with emotional and cognitive empathy as well as flexibility in exchange of viewpoints (Ron 1994 in Peled & Blum-Kulke, 1996). Like Egozi and Feuerstein (1987), Wolf (2004) similarly defined strategies as a basis for quality interaction that must guide teachers in every communicative situation in class. According to Wolf, honest and focused listening, devoting time to response, and allowing learners to express themselves in a variety of ways. Wolf (2004) added that there are six general strategies upon which teachers can rely to promote cognitive aims: mirroring, reference to content, thinking, informing and pointed questions. Teachers must ask questions whose answers are not predictable. The final strategy is language - developing language abilities. By using this strategy, teachers simultaneously promote communication aims and develop language skills in their pupils. 3.
Author's Contribution to Existing Theory and Practice in the Educational Field
Nava Kivkovich / Procedia - Social and Behavioral Sciences 209 (2015) 519 – 525
523
To help pupils overcome learning difficulties in geometry, a program was devised that combines use of teaching strategies. The importance of geometry is manifested in the fact that it is one of the core subjects in elementary and junior high school curricula. The subject has a deductive structure, including unique ways of thinking and investigation. Geometrical shapes in general and polygons in particular, is one of the central subjects that accompany pupils from elementary to junior high school. The subject is learned according to the curriculum in the fifth grade, and expanded upon from seventh grade onwards. A teaching program that comprises 8-10 lessons including the teaching of the geometry of polygons with an emphasis on strategic tools to solve problems in geometry, This program integrates focused teaching strategies acquired by pupils through classroom teaching that give students tools to cope with solving other problems. Using this strategy is appropriate for every pupil, does not require repetition or oral memorization by pupils, which have not proven to be tools for solving problems in geometry. Using these strategies enable pupils to develop skills to solve problems at any level. Teaching these strategies will be implemented through mediated teaching in class by teachers in parallel with teaching the contents of the curriculum. Teaching that advances learning with comprehension is characterized by leading pupils to high level thinking. Learning rich in thought is also appropriate for weak pupils, encouraging pupils to think and to discuss problems and mathematical concepts develops independent, investigative and understanding learners, in contrast to those who deal with repetition and memorization. This type of discourse is known as mathematical discourse. In the literature that was reviewed it appears that the program is based on Van Hiele's (1959) theory that discusses acquisition of skills as a basis of learning geometry through understanding the need for pupils to reach a level of mental development required for comprehension. Additionally, the program assumes, as Sarfaty and Patkin (2011) claimed, based on Van Hiele, that progress from one level to another is more dependent on teaching than on age or biological maturity. Another theory that guided the program was the socio-cultural communication approach and in particular Vygotsky (1978). The program is based on the fact that pupils are capable not only of acquiring knowledge or skills, but also new cognitive constructs. This ability is instilled in pupils using mediated learning in which they interact directly with their environment through non-mediated exposure to stimuli so that not only are knowledge and skills transferred to pupils, but also means of observation, ways of relating with and ways of looking for connections between them. Furthermore, the program is based on Feuerstein's (1998) approach whereby three criteria are necessary for mediated interaction to take place are integrated into the program and create interaction between children and mediating teachers. Beyond that, this connection has a unique quality that promotes flexibility, sensitivity and willingness of children to understand what is occurring and to generalize beyond a single phenomenon. The learning strategies that will be utilized in the framework of the recommended program to learn formal geometry, are based on the characteristics of processes that enable quality teaching-learning-evaluation processes. This a result of positive and correct experiences of mediated discourse processes, about discussed by Wolf (2004), by using mediation tools that include communication strategies in order that teachers will produce total and quality mediated learning. As reviewed, it is supremely important to control and manage teachers' strategies when planning and during lessons by performing situational assessments at every stage. Mathematical discourse requires a process of mediating the causality of mathematical phenomena between teacher and student (Regev & Shimoni, 2000). Therefore, the discourse is guided by presenting a problem and finding possible ways of solution. 4. Author's Contribution to the Topic Different studies (Hershkovitz, 1992; Kramersky, 1996; Pelach-Borowitz, 2004; Shalev, 2002) have dealt with the link between using teaching strategies and pupils' achievements. Practice strategies based on computers, using drawings in geometry and Van Hiele's levels of thinking model (Idris, 2009; Kutluca, 2013; Patkin, 1994) to solve problems in geometry as well as different strategies used by mathematics teachers in teaching computational geometry, trigonometry (Aydoğdu, 2014). There are those that dealt with different factors and thinks between them and the ability to solve problems in geometry, such as: motivation, feelings, drawing skills (Bailey, 2014). Others examined the connection between work memory ability of children and achievements in mathematics (Holmes &
524
Nava Kivkovich / Procedia - Social and Behavioral Sciences 209 (2015) 519 – 525
Adams, 2006). Other studies that were reviewed examined strategies focused on algebra and numbers as well as teachers' perceptions of using learning strategies but no studies were found that dealt with the construction of models of strategies in teaching geometry and their use in high schools. The program developed by the author is based on theories such as: Piaget's (1960) theory, Van-Hiele's (1959) theory, Vygotsky's (1978) socio-cultural communication approach and others. In addition the tool is based on an integration of teaching strategies focused on meaningful mediation between material and groups of heterogeneous learners for teachers and pupils to generate complete and quality mediated learning. The innovation in integrating this mediated strategic tool is that it provides a holistic response to pupils', as well as teachers', needs in teaching geometry using one common tool. This tool provides a response to teachers' as well as pupils difficulties and through it both sides are given a solution in the learning process as well as a solution to problems in geometry. No studies dealing with this subject were found. As such, the researcher chose to conduct research whose aim is to examine the influence of a mediated strategic tool in geometry and its use on pupils' achievements in solving problems in geometry and their views with regard to mathematics and geometry in particular. The chosen research population consisted of high school students aged 15-16. The research will be carried out using mixed methods and will include active research including questionnaires, interviews, and exams. After the data is collected, a summary and conclusions that can assist teachers in teaching the subject of geometry to pupils will be presented. 5. Conclusions It is possible to conclude that there are strategies that assist and advance pupils and it is a teacher's responsibility to understand and implement them in lessons in order to carry out a mediated discourse that will advance pupils to more effective learning processes. As such, the tool that teachers can use to produce complete and quality mediated learning, is dialogic discourse. Mathematics includes verbal and non-verbal aspects. Mathematical discourse revolves around different subjects. Teachers who teach geometry must develop great skill in the formal part of the subjects and integrate them into verbal explanations that have specific wording in geometry, as well as non-verbal aspects. This obligates teachers to have high discourse capabilities, be suitably qualified, in order that they can fulfill the role of mediators between contents and learner. Using innovative pedagogy it will be possible to guarantee reaching required goals, as well as the quality of teaching-learning-evaluation processes. Another conclusion is that advancing learning processes and their mediation by teachers requires control and strategic management by teachers at the same time as lesson planning, situational assessment at every stage emphasizing listening, knowledge and identification through dialogue, how every pupil thinks, what the pupil's current level is and what difficulties they have encountered, and be capable of responding to each pupil. The process finding a solution jointly by teachers and pupils, with correct use of geometric language through visual marking and finding memory supports of previous knowledge and using them through asking questions, will advance pupils' authentic mathematical activity, make them encounter problems, many different methods and solutions, advance their creativity and may even provide them with a bridge from concrete to abstract thinking in geometry, in dealing with more abstract and broader situations in their lives. References Aydoğdu Z. M. (2014). A research on geometry problem solving strategies used by elementary mathematics teacher candidates. Journal of Educational and Instructional Studies in the World, Volume: 4 Issue: 1 Article: 07 Bailey, M. Taasoobshirazi, G. Carr, M. (2014). A multivariate model of achievement in geometry. Educational Psychology. Vol. 26, No. 3, June 2006, pp. 339–36 Doll, W. E. (1999). A post-modern perspective on curriculum. Hebrew Version: Tel Aviv Poalim Publications (In Hebrew Egozi, M. and Feuerstein, R. (1987). The mediated learning theory and its role in teacher-education. Dapim, 6, pp. 16 – 34 (Hebrew) Epstein-Yanay, M.(2006). Teacher education – Is it a possible mission? Organizational Analysis, 10, 70-80. [Hebrew] Feuerstein, R. (1998). Man as a changing being: on mediated learning theory. Tel Aviv: Ministry of Defense Publications (In Hebrew). Hershkovitz, R. (1990). Cognitive aspects in teaching and learning geometry. Alon Lemorei Hamatematika, 9-10, pp. 24 – 38, 20 – 27 (In Hebrew) Hershkovitz S. (1992). The contribution of schemes for solving two-echelon word problems in math. Thesis for Ph.D., Haifa University, Israel Holmes, J. and Adams, J. W. (2006). Working memory and children's mathematical skills: implications for mathematical development and mathematics curricula, Educational Psychology., 26 (3). pp. 339-366
Nava Kivkovich / Procedia - Social and Behavioral Sciences 209 (2015) 519 – 525
525
Idris, N. (2009). The impact of using geometer sketchpad on Malaysian achievement and Van Heile geometric thinking. Journal of Mathematics Education, Vol. 2 no. 2 pp. 94-107. Klein, P. (1991). Moral assessment and parental intervention in infancy and early childhood: New evidence. In: R. Feuerstein, P. Klein & A. Tannenbaum (Eds.), Mediated Learning Experience: Theoretical, Psychosocial and Social Implications, London: Freund Publishing House. Kramersky, B. (1996). The development of mathematical language in different learning environments. Thesis for Ph.D., Bar Ilan University, Ramat Gan, Israel. Kutluca, T. (2013). The effect of geometry instruction with dynamic geometry software; GeoGebra on Van Hiele geometry understanding levels of students. Global Educational Journal of Science and Technology, Vol. 1 (1) pp. 1 – 10. Levy-Feldman, I. (2010). Who are you the "accomplished" teacher? Hachinuch Usvivato. Vol. 32 pp. 87 – 104 (In Hebrew) Ministry of Education (2012). Director General Circular 9(A), retrieved from: http://cms.education.gov.il/EducationCMS/Applications/Mankal/Templates/HoraotKevaFreeContent.aspx?NRMODE=Published&NRNOD EGUID=%7bA3BE1402-899B-461C-8631E1E722B0A35%7d&NRORIGINALURL=%2fEducationCMS%2fApplications%2fMankal%2fEtsMedorim%2f3%2f31%2fHoraotKeva%2fK-2012-9-1-3-1-43%2ehtm&NRCACHEHINT=NoModifyGuest#3 Nisim, Y. Barak, M. and Ben-Zvi, D. (2012). Role Perception and Teaching Strategies of Teachers who Integrate Advanced Technologies in their Lessons. Dapim, Vol. 54 pp. 193 – 218 Patkin, D. (1994). Training teachers for assessing geometric thinking level. Dapim, 19, pp. 37 – 47 (In Hebrew) Patkin, D. (1994). Influence of using computers on levels of geometric thinking. ALEH- Mathematics Teaching Journal, 15, pp. 29 – 36 (In Hebrew). Pelach-Borowitz, D. (2004). Identifying strategies and good teaching skills in the mathematics learning unit in the sixth grade. MA Thesis, TelAviv University, Israel. Peled, N. Blum-Kulka, S. (1996). Dialogic in class discourse. Helkat Lason, 24 pp. 60 – 61. Tel Aviv: Leveinsky college of Educatio Piaget, J., Inhelder, B. & Szeminska, A. (1960). The child's conception of geometry. New York: Harper Torchbooks Piaget, J. (1969). The child's conception of number. Norton: New York. Regev, H. Shimoni, S. (2000). Speaking Mathematics – What for? Why? How? ALEH- Mathematics Teaching Journal, 25, pp. 77 – 89 (In Hebrew). Sarfaty, Y. Patkin, D. (2011). The effect of solid geometry activities of pre-service elementary school mathematics teachers on concepts understanding and mastery of geometric thinking levels. Hachinuch Usvivato, 33 pp. 177 – 193 (In Hebrew) Schroeder, C. M., Scott, T. P., Tolson, H., Huang, T-Y. & Lee, Y-H. (2007). A meta-analysis of national research: Effects of teaching strategies on student achievement in science in the United States. Journal of Research in Science Teaching, 44(10), 1436-1460. Shalev, P. (2002). Teachers' perceptions of the use by students with learning disabilities of learning strategies. MA Thesis, Tel-Aviv University, Israel. Tal, K. (2004). Literacy fostering mediation as an example of applying development-congruent teaching. Dapim, 37, pp. 10 – 41 (In Hebrew) Van Hiele, P.M.,La Pensee de I'Enfant et La Geometrie, Bulletin de I'Association des Professeurs de Mathematiques de I'Enseignement Public 198, 199-205 (1959) Vidislavski, M., Peled, B. and Pevsner, O. (2010). Adjusting schools to the 21st century. Eureka, Tel Aviv University, No.30 pp 1-6 (In Hebrew). Retrieved from: http://www.matar.ac.il/eureka/newspaper30/docs/05.pdf (27.12.2013) Vygotsky, L. S. (1978). Mind in society: the development of higher psychological processes. Cambridge, MA: Harvard University Press. Wolf, D. (2004). Quality Interaction in class. In: Dialogue in teaching and learning. two options. Jerusalem: Ministry of Education – Elementary School Division (In Hebrew). Zellermayer, M. Kozulin, A. (2004). Learning in social context: development of high psychological processes. Tel-Aviv: Hakibbutz Hameuchad. pp. 113 – 128 (In Hebrew)
